Improving Spinal Cord Stimulation

Model-Based Approaches toEvoked Response Telemetry

By

J A M E S L A I R D - WA H

Graduate School of Biomedical EngineeringFaculty of Engineering

U N I V E R S I T Y O F N E W S O U T H WA L E S

A dissertation submitted to the University of New

South Wales in accordance with the requirements of the

degree of D O C T O R O F P H I L O S O P H Y.

AU G U S T 2 0 1 5

to Estee

C O N T E N T S

Contents i

List of Figures iii

List of Tables vii

List of Abbreviations viii

Acknowledgements x

1 Introduction 1

1.1 Chronic Pain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Evoked Response Telemetry . . . . . . . . . . . . . . . . . . . . 6

1.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Document Structure . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Background 10

2.1 SCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Dorsal Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Evoked Responses . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Intraspinal Evoked Potentials . . . . . . . . . . . . . . . . . . . 24

2.5 Electrophysiological Models . . . . . . . . . . . . . . . . . . . . 47

2.6 Current Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Virtual Ground 56

3.1 Artefact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Saline Bath Testing . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4 Human Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

i

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Evoked SFAP Model 89

4.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5 ECAP Model 120

5.1 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6 Cord Movement 175

6.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2 Current Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.3 Recruitment Control . . . . . . . . . . . . . . . . . . . . . . . . 184

6.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.5 Biphasic Stimulation . . . . . . . . . . . . . . . . . . . . . . . . 196

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7 Conclusions 202

7.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.2 ECAP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.3 Improving SCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.4 Further Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Bibliography 219

A Publications 247

B MRG Model Code 250

ii

L I S T O F F I G U R E S

2.1 Melzack and Wall’s gate theory of pain. . . . . . . . . . . . . . . . 11

2.2 Dermatomes of the human body. . . . . . . . . . . . . . . . . . . . 13

2.3 Examples of epidural leads. . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Fluoroscopic image of an implanted octopolar SCS lead. . . . . . 16

2.5 Cross-section of cervical vertebra. . . . . . . . . . . . . . . . . . . . 18

2.6 Cross-section of spinal cord . . . . . . . . . . . . . . . . . . . . . . 19

2.7 The NICTA MCS2 multichannel ERT system . . . . . . . . . . . . 26

2.8 Patient vs. epidural ground recording modes. . . . . . . . . . . . . 28

2.9 Ascending and descending responses . . . . . . . . . . . . . . . . . 33

2.10 Growth of fast response as current is increased. . . . . . . . . . . 34

2.11 Evoked responses in an ovine spinal cord. . . . . . . . . . . . . . . 37

2.12 Propagation in both directions from a central stimulation site,

recorded in an ovine subject. . . . . . . . . . . . . . . . . . . . . . 38

2.13 Evoked responses in a human spinal cord. . . . . . . . . . . . . . 39

2.14 Waveforms with increasing current, showing fast and slow re-

sponses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.15 Relationship between stimulus and response amplitudes. . . . . 42

2.16 ECG signal observed on SCS electrodes . . . . . . . . . . . . . . . 44

2.17 Human patient noise spectrum . . . . . . . . . . . . . . . . . . . . 45

3.1 Mechanisms of current flow through the electrical double layer

at the interface between a metal electrode and an ionic solution. 59

3.2 Electrical model of the electrode-tissue interface. . . . . . . . . . 61

3.3 SCS stimulation configurations. . . . . . . . . . . . . . . . . . . . . 67

3.4 Current flow during one stimulation phase. . . . . . . . . . . . . . 68

3.5 Common-mode potentials with different stimulation methods. . 70

3.6 Two schemes for an actively driven return electrode. . . . . . . . 72

iii

3.7 MCS virtual ground implementation. . . . . . . . . . . . . . . . . 73

3.8 Saline bath used for artefact measurements. . . . . . . . . . . . . 75

3.9 Artefact waveforms in a saline bath. . . . . . . . . . . . . . . . . . 76

3.10 Artefact amplitudes in a saline bath. . . . . . . . . . . . . . . . . . 77

3.11 Ratio of artefact amplitude to stimulus voltage. . . . . . . . . . . 78

3.12 Improvement in artefact amplitude using virtual ground. . . . . 79

3.13 Human artefact waveforms by stimulation mode. . . . . . . . . . 81

3.14 Human artefact waveforms by pulse width. . . . . . . . . . . . . . 82

3.15 Human artefact levels with and without virtual ground, E8 refer-

ence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.16 Human artefact levels with and without virtual ground, E4 refer-

ence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1 Cross-section of the thoracic spine model . . . . . . . . . . . . . . 95

4.2 Cable model of a myelinated axon. . . . . . . . . . . . . . . . . . . 95

4.3 Parametrised fibre geometry. . . . . . . . . . . . . . . . . . . . . . 97

4.4 Fibre threshold as a function of solver timestep . . . . . . . . . . 104

4.5 Fibre waveform as a function of solver timestep . . . . . . . . . . 105

4.6 The MRG axon model . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.7 Change in threshold current for different fibre diameters. . . . . 108

4.8 Ratio of threshold currents for different fibre diameters. . . . . . 109

4.9 Waveform of 15 µm diameter fibres in each model. . . . . . . . . 110

4.10 Normalised peak-to-peak amplitude. . . . . . . . . . . . . . . . . . 110

4.11 Ratio of peak-to-peak amplitudes. . . . . . . . . . . . . . . . . . . . 111

4.12 Conduction velocity as a function of diameter. . . . . . . . . . . . 111

4.13 Convergence study results . . . . . . . . . . . . . . . . . . . . . . . 117

5.1 Effects of floating-point precision on calculation . . . . . . . . . . 126

5.2 GPU model fibre threshold as a function of solver timestep . . . 127

5.3 GPU modelled fibre waveform as a function of solver timestep . 128

5.4 Comparison of SFAP waveforms between GPU and reference

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5 Residual difference between GPU and reference models . . . . . 130

5.6 Survey area of Feirabend et al. . . . . . . . . . . . . . . . . . . . . 134

5.7 Interpolated fibre diameter CDF . . . . . . . . . . . . . . . . . . . 134

iv

5.8 Interpolated fibre diameter PDF . . . . . . . . . . . . . . . . . . . 135

5.9 Fibre diameter CDF sampled according to Feirabend . . . . . . . 136

5.10 Properties of fibres in each diameter cell . . . . . . . . . . . . . . 139

5.11 ECAP waveforms, monophasic stimulation. . . . . . . . . . . . . . 141

5.12 ECAP growth curve, monophasic stimulation. . . . . . . . . . . . 142

5.13 ECAP peak latencies, monophasic stimulation. . . . . . . . . . . . 143

5.14 ECAP conduction velocity, monophasic stimulation. . . . . . . . . 144

5.15 Recruitment growth, monophasic stimulation. . . . . . . . . . . . 145

5.16 Mean diameter, monophasic stimulation. . . . . . . . . . . . . . . 145

5.17 Spread of recruitment with current, monophasic stimulation. . . 147

5.18 Ventral-lateral spread ratio, monophasic stimulation. . . . . . . 148

5.19 ECAP waveforms, biphasic stimulation. . . . . . . . . . . . . . . . 150

5.20 ECAP growth curve, biphasic stimulation. . . . . . . . . . . . . . 152

5.21 Early and late N1 peaks. . . . . . . . . . . . . . . . . . . . . . . . . 153

5.22 ECAP peak latencies, biphasic stimulation. . . . . . . . . . . . . . 154

5.23 ECAP conduction velocity, biphasic stimulation. . . . . . . . . . . 155

5.24 Recruitment growth, biphasic stimulation. . . . . . . . . . . . . . 155

5.25 Mean diameter, biphasic stimulation. . . . . . . . . . . . . . . . . 156

5.26 Spread of recruitment with current, biphasic stimulation. . . . . 158

5.27 Number of recordings made at each stimulus current in the hu-

man patient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.28 Single recordings made at two currents, and averages of all record-

ings at both currents. . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.29 Template components. . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.30 Scaling factors for the linear template, and the line fitted to each

electrode’s scaling factor. . . . . . . . . . . . . . . . . . . . . . . . . 163

5.31 Scaling factors for the common template, and the constant factor

fitted to each electrode. . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.32 Recordings after template subtraction and smoothing. . . . . . . 165

5.33 N1 and P2 peak latency on each electrode. . . . . . . . . . . . . . 167

5.34 Propagation delays between peaks observed on successive elec-

trodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.35 N1-P2 amplitude on each electrode as a function of current, and

lines fitted to the data for each channel. . . . . . . . . . . . . . . 169

5.36 Model results overlaid on human measurements. . . . . . . . . . 170

v

5.37 Peak times for human and modelled ECAPs. Model current is

scaled by a factor of 1.4. . . . . . . . . . . . . . . . . . . . . . . . . 171

6.1 Peak-to-peak amplitude of ECAPs on electrode 6 at various cord-

electrode distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2 Fibre recruitment at various cord-electrode distances. . . . . . . 177

6.3 Conduction velocity at various cord-electrode distances. . . . . . 177

6.4 Mean recruited fibre diameter at various cord-electrode distances. 178

6.5 Normalised amplitude, recruitment and mean diameter. . . . . . 180

6.6 Recruitment as a function of stimulus current. . . . . . . . . . . . 182

6.7 Recruitment as a function of recording amplitude. . . . . . . . . . 183

6.8 Relationship between conduction velocity and recruitment over a

range of distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.9 Current-distance curves. . . . . . . . . . . . . . . . . . . . . . . . . 187

6.10 Slopes of current-distance curves. . . . . . . . . . . . . . . . . . . . 188

6.11 Voltage-distance curves. . . . . . . . . . . . . . . . . . . . . . . . . 189

6.12 Slopes of voltage-distance curves. . . . . . . . . . . . . . . . . . . . 190

6.13 Recruitment as a function of I−kV . . . . . . . . . . . . . . . . . . . 192

6.14 Comparison of the performance of constant-current, constant-

voltage, and I-V control. . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.15 RMS deviation from the mean recruitment across all distances

for a range of desired setpoints. . . . . . . . . . . . . . . . . . . . . 194

6.16 Effect of k on recruitment control. . . . . . . . . . . . . . . . . . . 195

6.17 Effect of k on recruitment control on several electrodes. . . . . . 196

6.18 Effect of k on recruitment control with biphasic stimulation. . . 197

6.19 Comparison of the performance of constant-current, constant-

voltage, and I-V control with biphasic stimulation. . . . . . . . . 198

6.20 I-V control feedback loop. . . . . . . . . . . . . . . . . . . . . . . . . 200

vi

L I S T O F TA B L E S

2.1 Typical parameter values used in SCS. . . . . . . . . . . . . . . . 14

3.1 Composition of phosphate-buffered saline (PBS) used in the saline

bath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1 Electrical conductivity of model compartments. . . . . . . . . . . 94

4.2 Geometric parameters of nerve fibres, as derived from overall

fibre diameter D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Equations governing membrane current density Jm at each node

of Ranvier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Equations governing ion channel activation and inactivation vari-

ables at each node of Ranvier . . . . . . . . . . . . . . . . . . . . . 99

4.5 Constants governing neural model behaviour. . . . . . . . . . . . 100

4.6 Diameters of the 10 fibres used for timestep validation. . . . . . 103

4.7 Timestep values used for timestep validation. . . . . . . . . . . . 103

4.8 Initial meshing parameters . . . . . . . . . . . . . . . . . . . . . . 114

4.9 Number of elements in refined meshes . . . . . . . . . . . . . . . . 115

4.10 Fibre parameters for validation . . . . . . . . . . . . . . . . . . . . 116

5.1 Computation time of reference and GPU models . . . . . . . . . . 131

6.1 List of symbols used in Chapter 6. . . . . . . . . . . . . . . . . . . 185

vii

L I S T O F A B B R E V I AT I O N S

AP Action Potential.

ASIC Application-Specific Integrated Circuit.

CDF Cumulative Density Function.

COMSOL COMSOL Multiphysics.

CRPS Complex Regional Pain Syndrome.

CSF Cerebro-Spinal Fluid.

DC Dorsal Column.

DLC Dorsolateral Column.

DT Discomfort Threshold.

ECAP Evoked Compound Action Potential.

ECG Electrocardiogram.

EMG Electromyogram.

ERT Evoked Response Telemetry.

FBSS Failed Back Surgery Syndrome.

FIR Finite Impulse Response.

GPU Graphics Processing Unit.

ICT Information and Communication Technologies.

viii

IPG Implantable Pulse Generator.

MCS2 Multi-Channel System, mark 2.

MRG McIntyre/Richardson/Grill model.

NICTA National ICT Australia.

ODE Ordinary Differential Equation.

PBS Phosphate-Buffered Saline.

PCHIP Piecewise Cubic Hermite Interpolating Polynomial.

PDF Probability Density Function.

PT Paraesthesia Threshold.

RMS Root Mean Square.

SCS Spinal Cord Stimulation.

SFAP Single Fibre Action Potential.

SNR Signal to Noise Ratio.

UT-SCS University of Twente SCS model.

VAS Visual Analogue Scale.

ix

A C K N O W L E D G E M E N T S

Five years ago, Dr. John Parker offered me the chance to uproot myself,

travel a thousand kilometers, and spend my foreseeable future immersed in a

subject I knew nothing about. In many ways, the road has been much longer

than the distance or the time would seem to indicate; filled with byways,

wrong turns, and seemingly insurmountable obstacles, but also with the

adventure, discovery, and sheer joy that we find only at new frontiers. It has

been a great pleasure to be able to work with such a skilled scientist, and

I remain eternally grateful to Dr. Parker for granting me this opportunity,

and for his guidance along the way.

I must also express my deepest thanks to the team at Saluda Medical,

who took me in as one of their own. It has been a remarkable experience

to see this group grow from the erstwhile small research unit that was

NICTA Implant Systems to a fully-fledged medical device company, with

all hands passionate about turning our research into improved quality of

life for patients. Saluda have been kind enough to give me access to their

extensive library of unpublished human trial datasets, without which my

work would not have been possible. I would like to extend particular thanks

to Peter Single and Mark Bickerstaff, who have shown me an exceptional

amount of support and kindness over the years.

Many thanks are also due to Prof. Nigel Lovell, who has provided much

guidance and encouragement from the academic side throughout my candida-

ture, as well as helping me navigate the unfamiliar waters of the University.

Finally, I thank my partner Estee: a person I never expected to meet, met

on my travels along this new frontier. Words cannot express my appreciation

for everything she has done for me throughout this journey.

x

The purpose of models is not to fit the data

but to sharpen the questions.

SAMUEL KARLIN

CH

AP

TE

R

1I N T R O D U C T I O N

Spinal cord stimulation (SCS) has been used as a treatment for chronic

neuropathic pain for over 40 years. An electrode array is placed in the epidu-

ral space of the spine, and stimulation pulses are delivered continuously

to the array by an implanted stimulator. Significant pain reduction can

be achieved in this way, even though the underlying mechanisms are not

yet clearly understood [58], and several different systems are likely to be

involved [168]. Unlike simpler therapies such as opioid medications, SCS

does not cause systemic side effects, but it does cause perceptual side effects

as a result of sensitivity to the patient’s posture [164]. Improvement of the

therapy since its introduction has been limited by the nature of the available

data: the nerve fibres targeted by SCS cannot be accessed for direct measure-

ment in humans due to their location deep within the spinal canal. Animal

models have been used to study the origins of neuropathic pain [228], [230],

[243] and the effects of SCS [53], [79], [124], [125], [147], [260], although the

use of animal models for such a complex perceptual phenomenon renders

translation of experimental results difficult [24], [154]. Computer models of

SCS have also been used to study the results of stimulation [50], [91]. These

models predict which nerves will be recruited by a given stimulus. As the

neural recruitment cannot be measured in human subjects, these models

cannot be validated directly.

1

A number of indirect methods have been used to probe the mechanisms

of SCS in human patients. Quantitative methods for measuring percep-

tual information are available, including the Visual Analog Scale (VAS) for

pain levels [105], the use of Von Frey filaments for mechanical hypersen-

sitivity [53], [148], and Quantitative Sensory Testing (QST) for measuring

perceptual changes during stimulation [194], [235]. These have been used to

determine the effectiveness of SCS in treating chronic pain [117], [210], and

permit some investigation into the physiological effects of stimulation [188].

Physical measurements are also possible. While direct probing of individ-

ual spinal neurons is impractical, the ensemble behaviour of the spinal cord

and peripheral nerve trunks can be measured by electrodes placed outside

them. This is the evoked compound action potential (ECAP). Recordings

of responses to stimulation have been made by stimulating in the cord

and recording from peripheral nerves, and vice versa [8], [82], [132], [192],

[209]. A small number of experiments have performed both stimulation

and recording using spinal electrodes, placed several spinal segments apart

due to technical limitations [208], [232]. Recent developments have enabled

both stimulation and recording functions to be performed on a single elec-

trode array, of the type used in SCS [176]. The use of this evoked response

telemetry (ERT) allows the ensemble response of the nerves of the spinal

cord to be measured during therapeutic SCS, without the implantation of

additional electrodes. This lowers the bar for making electrophysiological

measurements in SCS patients to something that could be performed on a

routine basis; it also offers a source of real-time data that can be used to

compensate for patients’ changing postures, reducing the side effect profile of

SCS [178]. A feedback control technique is currently in closed trials for this

purpose [183], which adjusts the stimulus current to maintain a constant

recording amplitude.

SCS is a new and challenging environment for evoked response measure-

ment. Stimulation and recording are difficult due to the distance between

the electrode array and the target tissue. This distance varies from pa-

tient to patient as well as from posture to posture [84], [89], and at several

millimetres is much greater than the distances involved in other evoked

response measurement targets such as nerve cuff recordings and cochlear

implants [190]. This increased distance increases the ratio between the

2

required stimulus intensity and the amplitude of the resulting evoked re-

sponses; in SCS, this ratio can exceed one million [177]. This is problematic

because of the stimulation artefact: signals appear in the recording which

are caused by the stimulus rather than of neural origin. The high stimulus-

response ratio in SCS can result in a high artefact-response ratio, obscuring

the desired neural responses partially or completely. The recently developed

techniques are able to mitigate the artefact to the extent that recording is

feasible in most patients, and signal processing applied to the recordings in

order to extract measurements during therapeutic SCS [180]. In order to

achieve clinical relevance, it must be possible to perform ERT recording and

meaningfully interpret the results in a wide range of patients. This requires

improvements to the chain from recording through signal processing and,

ultimately, interpretation.

Feedback control of neurostimulation requires recordings to be made,

measurements to be taken, and for some change to stimulation made as

a result. This work aims to use model-based approaches to improve, un-

derstand, and apply ERT to improve patient outcomes in SCS. To this end,

three questions are examined:

1. Do models of corrupting signals enable better ECAP measurement?

2. Does a model of SCS explain observed ECAP features?

3. Can a model-based approach improve clinical SCS?

This work begins by addressing a primary corruptor of ECAP record-

ings, the stimulation artefact. Using understanding gained from an existing

model of artefact in SCS ERT [204], a new stimulation scheme was designed

which reduces the artefact at its source. This enables higher-resolution mea-

surement of evoked potentials, improving subsequent electrophysiological

experiments.

A model of SCS ERT is developed which simulates ECAP waveforms as

a function of stimulation. This is based on prior work in SCS modelling, but

is the first SCS model to consider the evoked response of the spinal cord.

This allows it to be validated directly against measured data from a human

patient. This ECAP model explains the origins of the major features of the

evoked responses.

3

The new model is then used to explore the effects of cord movement

within the spinal canal. The performance of existing control methods are

measured: traditional open-loop stimulation, and the constant-amplitude

feedback method currently under trial. Insight from the model’s behaviour

is then used to derive a new feedback algorithm with improved performance.

1.1 Chronic Pain

Neuropathic pain refers to pain which appears in the absence of a noxious

stimulus or acute injury, instead originating within the nervous system. Neu-

ropathic pain can be caused by disease or traumatic damage to nerves, and

can persist for many years as a chronic condition [22]. Symptoms can include

spontaneous painful sensations, as well as pain from normally innocuous

stimuli such as gentle touching or temperature changes (allodynia). The

perception of normal nociceptive pain can also be intensified (hyperalgesia).

Chronic pain is treated using a variety of methods. First-line thera-

pies include over-the-counter pain medications, cognitive and behavioural

therapies, exercise, and massage [236]. When these fail, the second line of

treatment is provided by chemical means: opioid painkillers or targeted

neural blocks via injection [2]. These are much more difficult to manage,

and introduce the potential for systemic side effects; opioid painkillers intro-

duce the potential for addiction, and injection blocks must be renewed on a

regular basis.

The last line of treatment involves surgical interventions. Surgical re-

moval of nerves (neurotomy) is used for a range of pain types [28], [32],

[121]; this is an irreversible operation, and can lead to new neuropathic pain

through deafferentiation [74], [220]. Implantable drug pumps can be used to

deliver anaesthetics directly to the intrathecal (subarachnoid) space [114].

This can yield pain relief where systemic opioids do not, and reduces the

required doses and the potential for overdosing. Implanted pumps need to

be regularly refilled using a needle through the skin, and they are equipped

with catheters which cross the blood-brain barrier into the cerebrospinal

fluid. This provides a potential route for serious infection [67].

Spinal cord stimulation is currently used as a last-line treatment for

4

chronic pain. An electrode array is placed in the epidural space proximal

to the dorsal columns (DCs) of the spinal cord, which predominantly carry

sensory information from the periphery to the brain. The electrode array is

sited to try and stimulate the spinal nerves that innervate the painful area

of the patient’s body. Stimulation is then delivered at a constant amplitude

and pulse repetition rate. This results in paraesthesiae, sensations often

described as pricking or tingling, perceived on the body area in question.

SCS is preferable to anaesthetics and neurotomy in that it has no sys-

temic side effects, although it still has the costs and risks associated with

an active implantable device. As with other treatments, the degree of pain

reduction varies widely between patients [11]. SCS also exhibits a signifi-

cant sensitivity to patient posture: movement of the cord within the spinal

canal changes the distance between the stimulating electrodes and the dor-

sal columns, causing the neural recruitment obtained by a fixed stimulus

intensity to change. If the stimulus intensity is too weak for the distance,

the patient will receive no pain relief; too strong, and they may experience

painful sensations or muscle activation. The natural position of the cord is

different depending on the patient’s posture, and movements such as walk-

ing or coughing can cause it to move rapidly [179]. This constantly changing

geometric relationship makes consistent pain relief difficult with current

technology; patients must either constantly adjust their stimulator, settle

for insufficient stimulation, or even avoid certain postures entirely [84]. One

SCS device manufacturer uses an accelerometer in the implanted electronics

package to detect postural changes [202]; this information is used to ad-

just the stimulus intensity. However, the accelerometry data is not a direct

measure of cord-electrode distance.

Until recently, research on SCS has been hampered by a lack of detailed

information on the effects of stimulation; the primary source of data avail-

able to researchers is perceptual data reported by patients. In order to bring

a data-driven approach to device design, models of SCS have been developed

to try and predict the effects of stimulation on spinal nerves [50], [91], [122].

These models share a common structure: an electrical-geometric model of

the spine and its tissues, coupled with neural models of the nerve fibres in

the relevant parts of the cord. These models are based on bringing together

a great many pieces of prior research: electrophysiology, anatomy, and his-

5

tological examinations of the nerve fibre populations of the cord. They are

used to simulate models of various nerve fibres; their input is a stimulus

waveform, and their output a determination of which of the modelled fibres

are activated as a result. This is a limitation: their output - determining

precisely which nerve fibres are activated - cannot be measured in a human,

and so they cannot be validated directly against human data.

Successful SCS therapy can result in substantial improvements in pa-

tient quality of life, without systemic side effects. In order to elevate SCS

to a second-line therapy, the postural sensitivity must be reduced, and suc-

cess rates improved; further elucidation of the therapeutic mechanisms is

required to understand how to proceed.

1.2 Evoked Response Telemetry

In 2010, researchers at NICTA Implant Systems demonstrated the

recording of evoked neural responses on a single array during SCS [177].

Evoked response telemetry allows the electrical activity of nerves to be

observed immediately after each stimulus pulse. Each recording is an ECAP,

a combination of signals from the action potential travelling along each

activated nerve fibre.

ERT gives an immediate measure of the effects of a particular stimulus,

rather than through the more distant peripheral or perceptual pathways.

This provides a new data source for examining the physiology of chronic

pain and the mechanisms of SCS. ERT can also be used to compensate

for postural variation directly: in a technique presently being trialled, a

feedback control system is applied to adjust the stimulus continuously, with

the aim of keeping the recorded ECAP amplitude constant.

The distance between the spinal cord and the electrodes means that the

stimulus pulses that must be applied are on the order of several volts in

amplitude, while the recorded responses are typically on the order of tens

of microvolts [176]. This leads to the major challenge of ERT: the stimulus

artefact. The artefact is a signal caused purely by the stimulus pulse. The

stimulus drives physical systems out of equilibrium, most notably at the

electrochemical electrode-tissue interface and in the ERT amplifiers. This

6

results in a decaying voltage observed on the recording as these reequilibri-

ate. This is problematic because it is time-correlated with the stimulating

signal and changes with the stimulus intensity. The correlation means that

unlike external noise sources, it cannot be removed by averaging. Both the

ECAP and artefact change with current, making it difficult to definitively

separate the two signal components. This is particularly problematic for

feedback control.

Ideally, the neural recruitment of SCS would be fixed: exactly the same

nerve fibres would be stimulated on every pulse. ERT feedback techniques

currently being trialled use the ECAP amplitude as a proxy for recruitment.

However, the ECAP recorded for a given level of recruitment will vary with

the movement of the cord, so constant-amplitude stimulation will not deliver

constant recruitment. At present, no method is known for estimating the

actual recruitment.

1.3 Approach

This work aims to address the key issues impeding therapeutic SCS:

the stimulus artefact, the interpretation of recordings, and the control of

recruitment. Improving these will result in broader applicability of SCS

therapy to chronic pain patients.

A novel stimulation scheme was developed in order to address the stim-

ulation artefact at its source. The scheme, known as virtual ground, is

designed to hold parts of the stimulator-patient system in near-equilibrium

during the stimulus, reducing the amplitude of the resulting artefact. This

technique has facilitated the collection of high-resolution ECAPs in scores

of patients. Virtual ground is designed to be suitable for implementation

in implanted devices, and it is general enough to be used in non-SCS neu-

romodulation applications. The artefact suppression of virtual ground also

assists in experiments with feedback, which have been conducted by the

Implant Systems group. These have shown promise, and the technology is

being developed into a commercially available therapeutic device.

A model of SCS ERT was developed to shed light on how recordings relate

to neural behaviour. This is structured similarly to existing SCS models,

7

but was extended with additional mechanisms to simulate the recording

process. The model’s behaviour was compared with high-resolution ECAP

recordings, resulting in the first quantitatively validated model of SCS. This

novel model provides explanations for many of the salient features of ECAP

recordings in human patients, and provides insight into which neuronal

elements are activated during therapeutic stimulation.

This model was then used to examine the effects of postural changes on

SCS, by varying the cord-electrode distance in the model. This demonstrated

the extreme variations in recruitment seen with traditional open-loop re-

cruitment. Constant-amplitude feedback control was simulated, showing

improvements over open-loop control but overcompensating for changes. Ex-

amination of the model’s behaviour led to the derivation of a new control law

which takes into account the distance-related effects on recording amplitude,

which in simulations results in further improvement in recruitment control

over the constant-amplitude method.

1.4 Document Structure

Chapter 2 describes the background to the problem. Overviews of clin-

ical SCS and the dorsal columns of the spine are provided. The methods

used to record spinal ECAPs are described, including example recordings.

Existing literature on neural recording and SCS modelling are reviewed.

Chapter 3 documents the development of virtual ground, a new stimu-

lation technique. The mechanisms of artefact generation are discussed with

reference to an existing model of artefact in SCS, and the virtual ground

technique is derived. Test results are presented.

Chapter 4 describes a model of evoked potentials in SCS at the single

nerve fibre level. The structure of the model as a combination of volumetric

and neural models is discussed, as well as its implementation. This model is

then subject to validation of its volumetric and neural components.

Chapter 5 combines multiple single-fibre models to model the ensemble

behaviour of the dorsal columns under stimulation. Performance issues

relating to the large number of fibres involved are addressed. Resulting

ECAPs are compared with a dataset recorded in a human patient.

8

Chapter 6 considers the effects of varying spinal cord position on the

results of spinal cord stimulation. A novel method for controlling neural

recruitment in the face of cord movement is introduced and its performance

measured using the SCS model.

9

CH

AP

TE

R

2B A C K G R O U N D

This chapter provides a tour of the necessary background material for

this work. A brief introduction to the techniques and anatomy of SCS is

provided for the unfamiliar reader.

The recent development of SCS evoked response telemetry is then dis-

cussed. SCS ERT provides a new ability to record compound action potentials

evoked by therapeutic stimulation; this has been used to make scores of

recordings in patients to date, forming a large corpus of data. This corpus is

largely unpublished, so the major findings and common characteristics are

shown and discussed, along with some example recordings. The potential

therapeutic application of ERT is also discussed, covering the use of feedback

to compensate for postural variation.

The stimulus artefact remains one of the limiting factors on the quality

of ERT recordings. The origins of artefact and methods used to reduce it are

both considered.

Finally, the underlying mechanisms of SCS for chronic pain remain

poorly understood. Attempts have been made to understand these by using

models of the stimulation process and its effect on spinal nerve fibres; a

history of these efforts and their results is given.

10

2.1 SCS

The origins of SCS can be traced to the gate theory of Melzack and Wall

in 1965 [144]. The theory describes a gating mechanism in the substan-

tia gelatinosa of the spinal cord, responsible for the transmission of pain

sensations. Large-diameter sensory fibres were believed to close this gate,

while small-diameter fibres act to open it, resulting in pain. This is shown

in Figure 2.1.

Figure removed from public copy

due to copyright restrictions

Figure 2.1: Melzack and Wall’s gate theory of pain. L and S represent large

and small fibre inputs, respectively. SG represents cells in the substantia

gelatinosa, acting to inhibit central pain transmission cells T. Reproduced

from [144].

This theory led to experiments in closing the gate. Early experiments

were performed by Wall and Sweet [242], who stimulated nerve bundles

supplying the painful areas - electrical stimulation favours the recruitment

of large fibres. Using transdermal electrical nerve stimulation (TENS) and

implanted electrodes in eight pain patients, they achieved substantial pain

relief during stimulation. Shealy et al. [206], later in 1967, then used epidu-

ral spinal cord stimulation for the first time, again with excellent results;

epidural electrode arrays are now the standard method of spinal cord stim-

ulation, which has continued relatively unchanged over the intervening

years.

SCS is now used for complex regional pain syndrome (CRPS) [21] and

failed back surgery syndrome (FBSS) [18]. It is also being used for is-

chaemic pain conditions, including refractory angina pectoris [104], crit-

11

ical limb ischaemia [102], and peripheral arterial disease [231]. Further

research is exploring the potential uses of SCS in treating painful diabetic

neuropathy [136], [186], phantom limb pain [134], [237], cancer pain [66],

headache [258], and visceral pain [106]. Despite its popularity and effec-

tiveness in treating a range of conditions, our understanding of the exact

mechanisms by which SCS interacts with the nervous system is still very

limited.

SCS treatment involves the implantation of an electronics package re-

ferred to as an implantable pulse generator (IPG). The IPG typically contains

a power source, telemetry electronics, and one or more stimulus pulse gener-

ators. The IPG is then connected to one or more electrode leads, which are

placed in the patient’s epidural space.

The electrodes must be placed to stimulate nerves coming from the

patient’s painful region. The nerve roots at each spinal level innervate

a distinct region of the body, known as a dermatome; these are shown

schematically in Figure 2.2. The level of initial electrode insertion is decided

by the dermatome or dermatomes on which the patient feels the pain. The

lateral placement may be on the midline, or laterally offset ipsilateral to

the pain location. After the initial insertion, the electrode position must be

refined. This procedure involves delivering stimulus pulses to electrodes on

the lead, which causes paraesthesiae in the patient - sensations commonly

described as tingling, pricking or buzzing. The stimulus configuration and

lead position are revised in order to ensure that the area of paraesthesia

covers the painful area of the body. Surgeons may use multiple leads to

achieve the necessary coverage. Paraesthesia coverage appears to be a

necessary condition for pain relief [116], [157], [242], although it is not a

sufficient condition; some patients experience paraesthesia without pain

relief [163].

12

(a) Branching and exit

levels of the spinal

roots

(b) Dermatome distribution over skin

Figure 2.2: The spinal cord branches into roots at various levels, each of which innervates a particular region of the body

known as a dermatome. The dermatomes shown here are approximate; there is substantial overlap between dermatomes.

Images ©Janet Fong, 2009 (http://www.aic.cuhk.edu.hk/web8/Dermatomes.htm)

13

The stimulator is programmed with one or more stimulus programmes,

determined by the device manufacturer’s representative at the time of im-

plantation. Required stimulating currents vary widely between patients,

postures and lead placements [40]. The range of parameters typically em-

ployed in SCS is shown in Table 2.1. It is not clear whether these differences

are purely due to differences in physical characteristics, such as spinal ge-

ometry and lead placement within the spinal canal, or whether there are

neurological differences as well. Neurological differences may be due to dif-

ferent populations of nerve fibres in the cord, which naturally vary between

patients [64]. The types, diameters, and paths of fibres also vary within

spinal segments and along the length of the cord [54], [129], [161].

Parameter Minimum Maximum

Current 0.5 mA 30 mA

Pulse width 40 µs 800 µs

Pulse frequency 10 Hz 100 Hz

Table 2.1: Typical window of parameter values used in clinical SCS [193],

[205].

An individual patient will have a relatively small range of stimulus

intensities between the paraesthesia threshold (PT), where the stimulus first

becomes perceptible to the patient, and the discomfort threshold (DT), where

the stimulus becomes unbearable. The mean ratio of DT:PT is approximately

1.4 with traditional SCS systems [16], [103], depending on lead placement.

Overstimulation or electrode misplacement can result in uncomfortably

strong paraesthesia, or even motor effects such as twitching.

The onset of paraesthesiae is commonly associated with that of pain

relief, but in some cases therapeutic effects have been noted with subthresh-

old stimuli [23]. Recent research into high-frequency stimulation, with a

repetition rate of 500Hz or higher, has suggested that paraesthesia-free pain

relief may be feasible [55], [56], [239].

14

Figure removed from public copy

due to copyright restrictions

Figure 2.3: Examples of epidural cylindrical (left) and paddle (right) leads.

Manufactured by Boston Scientific.

There are two broad classes of SCS leads: cylindrical and paddle leads;

an example of each type is shown in Figure 2.3. Cylindrical leads have

multiple cylindrical electrodes, on a flexible polymer lead. These typically

have between 4 and 8 contacts. Their small diameter (1.2 mm typical) allows

them to be inserted using a Tuohy needle, rendering the procedure no more

invasive than an epidural anaesthetic. Paddle leads have flat contact sur-

faces, exposed on one surface of a strip. These may be in a line, or arranged

in multiple columns; a lead may have as few as 4 or as many as 16 elec-

trodes. Paddle electrodes have much larger cross-sections than cylindrical

electrodes. Traditionally, this meant they could only be placed with an inva-

sive laminectomy procedure, although a percutaneous introducer for paddle

leads has been invented [57]. Paddle leads improve power consumption over

cylindrical electrodes due to their insulated back sides [165]. Some have

multicolumn arrays of electrodes, which permit adjustment of the lateral

position of the stimulation without revising the electrode position [167].

15

Figure 2.4: Fluoroscopic image of an implanted octopolar SCS lead.

In clinical use, patients are often trialled on SCS before proceeding to

a permanent implant. In the trial, a percutaneous lead is inserted. This

is then connected to an external trial stimulator system, which is worn by

the patient for the duration of the trial. If successful treatment is achieved,

the trial lead is removed, and a new lead and implant are installed. A

fluoroscopic image of an implanted percutaneous lead is shown in Figure

2.4.

The epidural leads are a frequent cause of problems in SCS. The electrode

position is critical to therapeutic success; lateral displacement in particular

can cause unintended stimulation of lateral spinal structures [14]. Leads of-

16

ten migrate after their initial placement [20], [174], [216], though migration

effects can sometimes be addressed by reprogramming the stimulator [205].

Leads are also prone to breakage as a result of repeated flexure [85], [143].

A more fundamental problem is that SCS is sensitive to the patient’s

posture. The leads are located in the epidural space, near the surface of the

dura. The target tissue, however, is part of the white matter of the spinal

cord itself, which floats freely in the cerebrospinal fluid. As patients change

posture, the cord can move by several millimetres transversely [89] as well as

axially [111]. This change in distance results in changes to the sensitivity to

stimulation [40], [84], [172], which in turn result in posture-dependent side

effects. A program that provides good pain relief in one posture may result

in painful overstimulation in another, or instead a lack of stimulation. This

can prevent patients from receiving therapy whilst walking, and coughing

can become extremely painful.

One manufacturer has attempted to address this problem by introducing

an accelerometer into their implant [201]. This is used to attempt to discrim-

inate between different postures, and select a suitable stimulation program

accordingly. This method requires that the accelerometer be calibrated to

the natural postures of the patient, and is not a measure of cord position.

Trials showed that the accelerometer-driven stimulation control reduced

the number of manual stimulation adjustments made by patients during

the trial period by 40% [202]. More direct methods of position measurement

have been considered, using transducers placed on the epidural array. An

ultrasonic technique was investigated [61], but never commercialised. More

recently, an optical approach using near-infrared light has been tested in a

cadaver [257]. This work required the injection of saline into the intrathe-

cal space to obtain suitable optical behaviour. The results indicate that

dynamic motion induces movements that cannot be accurately modelled

from acceleration measurements at a remote implant site.

2.2 Dorsal Columns

Spinal cord stimulation is believed to target Aβ axons in the dorsal

columns of the spinal cord; these columns are also known as the posterior

17

Figure 2.5: Cross-section of a cervical vertebra showing the cord within

the spinal canal. The cord proper is composed of the grey and white matter,

which are surrounded by cerebrospinal fluid (CSF). The fluid is enclosed

by the meninges, including the dura mater; these in turn are surrounded

by fatty tissue in the epidural region of the vertebral canal. Nerves enter

and exit the cord through the dorsal and ventral roots, also within the CSF;

these form a single nerve bundle on each side, and transition out through

the vertebral foramen.

Image ©Wikimedia user debivort (http://commons.wikimedia.org/wiki/

File:Cervical_vertebra_english.png)

funiculi. A cross-section of a vertebra is shown for reference in Figure 2.5,

and of the cord in Figure 2.6. The DCs are composed of long myelinated

axons, running largely along the axis of the cord. They occupy a roughly

wedge-shaped region of the spinal cord, extending laterally to the dorsal

roots. This area is divided into the gracile and cuneate fasciculi. The gracile

fasciculus occupies a medial position, and carries fibres entering below the

T6 spinal level. The cuneate fasciculus appears laterally above T6, carrying

fibres entering at T6 and above. The nerve roots entering at each spinal

level innervate a particular dermatome.

18

Dorsal columns

Subdural cavityDura mater

Arachnoid mater

Pia mater

Subarachnoid cavity

Spinal ganglion

Spinal nerve

Posterior root

Anterior root

Figure 2.6: Cross-section of the thoracic spinal cord. The dorsal columns

are present at the top of the image, towards the patient’s back.

The dorsal columns carry information affecting touch and vibration,

proprioception, nociception and motor behaviour [256]. Most of the fibres in

the DC carry sensory information from peripheral receptors. These fibres

have their cell bodies in the dorsal root ganglia, outside the cord. Their

axons extend in two directions from the cell body: one extension runs out to

the periphery and innervates target tissue, while the other enters the dorsal

column through the dorsal roots and dorsal horn. On entering the columns,

these fibres bifurcate, with collaterals both ascending and descending the

cord [99], [211]. These collaterals themselves then branch as they travel

away from the point of entry; the branches synapse in the grey matter, and

are involved in sensory and reflex processing.

The afferents entering the cord have a range of diameters, depending on

their origins. After entering the cord, they then narrow as they move away

from the point of entry and collaterals branch off them [99]. Only 25% of

fibres entering the dorsal roots project all the way to the brain in cats [76].

The remaining fibres terminate in the grey matter within a few segments of

their entry to the cord; the length of travel depends largely on what kind of

receptor they correspond to. Mechanosensory afferents dominate the fibres

that ascend to the brain; proprioceptive fibres tend to terminate in the

cord [161].

The detailed parameters of the fibres of the dorsal columns are not par-

ticularly well known; their individual paths, diameters, branching-points

19

and electrical properties are not known. The diameter distribution of fi-

bres near the DC surface has been measured using automated histological

techniques [64], giving some indication of the fibre types present there, but

studies of fibre paths and termination points have been restricted to very

small numbers of fibres [10], [76], [99], [161] or the ensemble behaviour of

larger bundles [54], [215].

20

2.3 Evoked Responses

The term "evoked potentials" refers to any electrical signal which results

directly from the application of a stimulus. Stimuli may be delivered directly

to sense organs, such as using flashes of light, tones, or mechanical impulses.

Alternatively, electrical or magnetic stimulation can be used to directly acti-

vate neural elements; this can be performed electrically using transdermal

stimulation or implanted electrodes, or transcranially using magnetic stim-

ulation. Electrically-evoked potentials are of particular interest in the study

of neuromodulation; these can provide a direct measure of the effects of a

neural stimulus.

The stimulus triggers a neural response, which can also cascade through

other neural circuitry or trigger a muscle contraction. Nerves and muscles

both generate external electric fields when they are activated; this field,

the evoked potential, is then recorded. Depending on the stimulated neural

systems and the site of recording, an evoked potential may consist of neu-

rogenic signals, myogenic signals, or a mixture of the two. When recording

neurogenic responses from nerves, the response results from the behaviour

of many fibres within the nerve; these are referred to as evoked compound

action potentials (ECAPs).

Current usage of evoked potential recording can be divided into three

broad categories: investigative electrophysiology, control of neuromodulation,

and surgical monitoring.

2.3.1 Investigation

Evoked potentials are being used in an investigative capacity in neu-

romodulation research. There are a number of neuromodulation therapies

in use today whose mechanisms are not fully understood, and the direct

measurements possible with evoked potentials promise to shed more light on

these. Vagus nerve stimulation (VNS) is used as a therapy for certain epilep-

tic and depressive conditions [42], [238]. ECAP recordings during VNS can

be made [63] and some results suggest that cardiac activity can be indirectly

evoked and measured via this nerve [173]. Deep brain stimulation (DBS) is

used as a therapy for movement disorders such as Parkinson’s and essential

21

tremor. DBS involves stimulating particular grey matter structures using

an implanted electrode array. Recent work has shown that it is possible

to record evoked potentials from these arrays immediately after stimula-

tion [182]. Evoked potentials have particular value in applying model-based

approaches to understanding neural mechanisms: neural models which

include ECAP generation can be validated against recorded ECAPs [109].

2.3.2 Control

Evoked potentials can be used to configure and control neuromodulation

therapies. A prominent example is found in cochlear implants, which have

a great many parameters which must be tuned in order to achieve optimal

auditory function. In particular, the relationship between stimulus intensity

and perceived loudness must be determined for each stimulating location on

the implanted electrode array. This is normally established through a fitting

process, in which a clinician delivers stimuli to the patient and the patient

describes their perceptions. This is time-consuming, and particularly chal-

lenging in non-verbal patients such as infants. Consequently, ECAP-based

techniques have been developed which aim to determine the necessary pa-

rameters automatically [33]. These are effective, but not necessarily optimal;

recording is not presently possible under the same conditions as used for

therapeutic stimulation, and it is not known how to accurately extrapolate

the results from one into the other domain [139].

Evoked potentials may be useful for ongoing control and adjustment,

rather than merely for initial configuration. Cochlear implant patients who

previously had natural hearing appear to undergo a relearning process

after implantation, requiring adjustments to their fit parameters; it may be

possible to manage this automatically. Recording through an extracochlear

electrode to determine the evoked neural responses further along the audio

processing pathways [140] would allow automatic adaptation over time.

2.3.3 Monitoring

Evoked potentials can also be used in surgical monitoring during po-

tentially hazardous spinal surgeries. The development of operations which

22

require extremely high corrective forces, such as for scoliosis, has increased

the risk of unintentional application of pressure to spinal nerves and con-

sequent damage [86]. In this application, the integrity of the spinal nerve

pathways is of interest, rather than any specific neural behaviours. Stimuli

are constantly delivered to some part of the desired pathway and monitored

at another for signs of degradation, and corrective action taken if necessary.

This early warning, while the patient is still unconscious, results in the de-

tection and rectification of problems which could otherwise cause permanent

functional losses for the patient [3], [166].

The various monitoring techniques can be divided according to pathway:

sensory pathways predominantly run in the dorsal columns of the cord, while

motor pathways are found in the lateral and ventral cord regions. Sensory

pathways can be stimulated through electrical stimulation of peripheral

nerves and then recording in spinal or cortical locations, resulting in so-

matosensory evoked potentials (SSEPs). Motor pathways are more difficult

to activate; either carefully placed spinal electrodes or cortical stimulation

are necessary to evoke motor potentials. These are then further divided into

myogenic motor evoked potentials (MEPs) and neurogenic motor evoked

potentials (NMEPs), differing in whether the recording is from an activated

muscle or from motor nerves directly.

2.3.4 SCS

Evoked potentials such as the SSEP are used by some surgeons in de-

termining optimal SCS lead placement [192], and can indicate whether

neurological deficits exist that may prevent SCS from being successful [213].

The use of an objective measure of lead placement is particularly critical in

cervical lead placements, where patients are under general anaesthesia.

The ability to trigger muscular contraction (MEPs) in the targeted body

area using the SCS electrode has been correlated with suitable placement

for therapeutic stimulation [207]. SSEP and MEP monitoring can be used

for surgical monitoring as well as a placement aid [192].

These techniques involve peripheral nerves and muscles in the measure-

ment, and require the use of peripheral or scalp electrodes in addition to

the SCS lead being implanted. Stimulation and recording can be performed

23

simultaneously within the spine, providing a more focussed measurement.

2.4 Intraspinal Evoked Potentials

Of particular interest with regard to SCS are the spinal cord potentials

(SCPs), recorded using electrodes placed in the epidural space of the spinal

cord. These potentials are recorded from the sensory fibres of the dorsal

columns; their presence was first observed in animals in the 1930s [73]. Early

human experiments in this area involved applying stimuli to peripheral

nerves and recording the potentials intraspinally [209], a form of SSEP

recording. Later work then successfully performed both stimulation and

recording within the spinal cord [208], although with some distance between

the recording sites: stimulation was delivered in the cervical spinal cord,

and recordings made in the lumbosacral enlargement.

Dorsal column recording never became popular for intraoperative moni-

toring, as scalp recordings can be used as a less invasive route to recording

sensory evoked potentials. In spinal cord stimulation, however, there is al-

ready an electrode array implanted near the dorsal columns. Recent work

has examined the recording of spinal ECAPs in SCS patients, in which both

stimulation and recording are performed on the same implanted electrode

array [176]. This new technique permits the observation of dorsal column

ECAPs during therapeutic spinal cord stimulation or lead placement pro-

cedures, without the use of extra electrodes. Spinal ECAPs have now been

collected in a number of patients, and work is now proceeding to determine

whether this information can be used to improve SCS therapy. This work

was carried out at NICTA’s Implant Systems Unit. This unit has now tran-

sitioned into a company named Saluda Medical, formed to exploit spinal

ECAP recording for therapeutic purposes. As part of these efforts, ECAP

recording has been performed in a number of human and ovine subjects.

The resulting datasets, referred to here as the Saluda recording corpus, are

currently not published.

The recording of spinal ECAPs required the development of new record-

ing systems to address the particular challenges of SCS recording - in

particular, the requirement that the amplifiers are not overloaded by the

24

stimulation pulse. The data described in this document were collected with

an evoked-response collection system known as the MCS2; an overview of

the device’s capabilities is presented here.

The characteristics commonly seen in spinal ECAPs within the Saluda

corpus are also discussed. The recordings contain evoked and spontaneous

potentials of neurogenic and myogenic origin, as well as electrical noise and

stimulation-induced electrical artefact. These different contributions can be

difficult to separate, posing challenges for practical applications of ECAP

recordings.

2.4.1 ECAP Recording

A custom system for ECAP collection was developed at NICTA. This

system, the MCS2, is designed to be connected to one or more implanted elec-

trode arrays; it is comprised of stimulation, amplification, and digitisation

subsystems.

This system is distinguished from off-the-shelf neural recording systems

by its ability to record small signals immediately after the application of

large stimuli; this rejection of stimulus crosstalk, or artefact, is one of its

primary figures of merit.

25

1

2

3

Patient

1

etc

AmplifiersCurrent source

switches

24

24

+

-

+

-

+

-

+

-

2

3

1:24P

N

Vppp

Vnnn

1:24P

N

Vppp

Vnnn

1:24P

N

Vppp

Vnnn

1:24P

N

Vppp

Vnnn

1:24

1:24

Inverting input

mutiplexerGround switches

Figure 2.7: The NICTA MCS2 multichannel ERT system. The system is

connected to up to 24 electrodes implanted in the patient. One amplifier is

provided for each electrode; these share a common reference potential. Four

independent current sources are provided, each with positive and negative

drive capability as well as a grounding switch. Switches are arranged with a

bus structure so that each electrode may be connected to any of the current

sources, system ground, or the amplifier reference inputs. A set of resistors

is included to meet safety requirements, preventing the accumulation of

stray charge. Amplifiers are capacitively coupled for biasing purposes.

26

This system connects to up to 24 epidural electrodes, with an individual

amplifier for each electrode. The outputs of each amplifier are digitised; data

is acquired on all channels simultaneously. A block diagram of the system

is shown in Figure 2.7. These amplifiers are differential, with a common

reference input. The reference may be selected from any of the 24 connected

electrodes, or taken from an external connection to the system. This permits

the use of a distant indifferent electrode for performing monopolar recording,

where available. Each amplifier is equipped with a blanking circuit which

isolates the input during stimulation and for a short time before and after.

This prevents the high-gain amplifier stages from being driven into clipping;

this is critical to the system’s artefact rejection capability. Each amplifier

is AC coupled, and has a low-pass filter characteristic with a 15kHz corner

frequency.

Stimulation capability is provided in the form of four independent cur-

rent sources. Each is capable of driving either positive or negative pulses,

with a 4 µs time resolution. The current sources can each be connected to any

combination of electrodes using software-controlled switches. This arrange-

ment allows for very complex stimulation schemes, including interleaving of

stimuli to different locations.

The stimulation and recording circuitry is assembled into a single unit,

including a microprocessor to control signal routing and stimulation. A

second unit contains the data acquisition unit, as well as batteries to power

the system. The system has fibre optic control and data links, and is isolated

from its chassis; this ensures that no current can flow from other sources in

the patient field through the implanted electrodes.

Reference Potentials

Experimental SCS recordings can be made in two situations: patients

undergoing external trials (with percutaneous leads) at any time during

their trial periods, and implant patients during the operation when the

leads are exposed. Intraoperative experimentation is quite challenging;

patients are often under general anaesthesia, and time is short. Conversely,

external trial patients generally spend some days on trial, and are capable

of conversation and free movement. As a result, the bulk of human ECAP

27

recording performed to date has been with external trial patients, who are

fitted with one or more percutaneous cylindrical leads.

+

-

+

-

+

-

+

-

+

-

+

-

(a) Patient ground recording, using a separate

electrode as the amplifier reference input.

+

-

+

-

+

-

+

-

+

-

(b) Epidural ground recording,

using one of the array electrodes

as the amplifier reference input.

Figure 2.8: Comparison of patient and epidural ground recording modes.

a) Patient grounding: an implanted plate or needle electrode outside the

spinal canal serves as a reference potential. b) Epidural grounding; one of

the array electrodes is used as the reference potential.

A complicating factor is the necessity of a reference input to the ERT

amplifiers. The signals being sought are of microvolt amplitude; differential

amplifiers are used to maximise noise rejection. The reference (inverting)

input to these amplifiers must be connected to a low-impedance part of the

patient. Dermal electrodes exhibit high impedances, picking up too much

noise for this application. Ideally, a separate, large electrode implanted in

the patient would be used. This reference electrode is also referred to as an

indifferent electrode in the neural recording literature. This configuration is

shown in Figure 2.8(a). This electrode would be placed somewhere outside

the spinal canal, so that it would not be exposed to the neural response

voltages, and allowing the voltage at each electrode to be measured inde-

pendently. In practice, a separate reference electrode is often not practical;

implanting one can be too invasive under trial circumstances. As a result,

28

one of the electrodes on the implanted array must often be chosen as the

reference input, shown in Figure 2.8(b). This reduces the available recording

channel count, and also complicates the resulting signal; since the reference

electrode is exposed to some of the neural response voltage, this signal is

superimposed in inverted form on all of the recorded channels.

Ethics

Human trials referred to in this work were performed under one of the

following ethics approvals, and with informed consent from the patients.

Title: Automatic Control of Spinal Cord Stimula-

tion

Location: Royal North Shore Hospital, Sydney, Aus-

tralia

Board: NSW Health Northern Sydney Central Coast

Human Research Ethics Committee

Protocol: 1111-474M

NEAF: HREC/11/HAWKE/133

SSA: SSA/11/HAWKE/266

Principal Investigator: Michael Cousins AM DSc MD MBBS FFP-

MANZCA

Title: A Feasibility Study Evaluating the SCS Sys-

tem Incorporating Feedback Control Using

Evoked Compound Action Potential (ECAP)

to Aid in Accurate Positioning of a Paddle

Spinal Cord Stimulation (SCS) Lead

Location: Thomas Jefferson University, Philadelphia,

USA

Board: IRB #152

FWA: #2109

Protocol: Control # 13F.466

Principal Investigator: Ashwini Sharan MD

29

Animal recordings were made during trials under the following ethics

approval.

Title: Measurement of Evoked Spinal Cord Poten-

tials in Ovis aries: An Acute Study

Location: Royal North Shore Hospital, Sydney, Aus-

tralia

Board: Royal North Shore Hospital Animal Care

Ethics Committee

Protocol: 1101-002A

Principal Investigator: Michael Cousins AM DSc MD MBBS FFP-

MANZCA

30

2.4.2 Spinal ECAP Features

Recordings made from epidural electrodes contain components from a

number of different sources. Electrical noise from the amplifier system is

always present, as is noise from electromagnetic interference in the patient

field. Implanted electrodes also pick up potentials generated by spontaneous

neural activity in the spinal cord, as well as myogenic potentials from muscle

movements and cardiac activity.

The delivery of electrical stimulus pulses to spinal electrodes can then

contribute a further set of components. The stimulus artefact is caused by

the stimulus pulse, but without any neural involvement; the pulse causes

charge accumulation at electrical and electrochemical interfaces, producing

decaying waveforms as the system reequilibriates after stimulation.

Of most interest in SCS are the components due to the stimulation

of nerve fibres in the cord. These components appear only above certain

threshold currents; these thresholds vary depending on the individual, their

posture, the lead placement, the stimulating electrodes chosen, waveforms,

pulse widths, and so forth. These components also tend to increase in ampli-

tude as the stimulus intensity is increased above the threshold.

The neurally-evoked components are divided into a fast and a slow

response. The fast response appears immediately during or after stimulation

and lasts up to three milliseconds, while the slow response appears from

two to twenty milliseconds after stimulation and have longer durations than

the fast response.

Fast Response

The fast response consists of a triphasic waveform lasting typically less

than 3 milliseconds [176]. This is the waveform expected from an action

potential in an axon travelling past a recording electrode [184], and appears

with a successively increasing latency on electrodes more distant from the

site of stimulation. This appears to be due to the direct recruitment of dorsal

column fibres, and is of prime interest in the context of SCS.

The three peaks are referred to as P1, N1, and P2, in time order. These

are consistent with action potentials propagating along nerve fibres past

the recording electrodes. The first peak, P1, is caused by a region of pas-

31

sive depolarisation being driven along inside axons. The following peak,

N1, is caused by the active depolarisation of the fibre membrane near the

recording electrode. The final peak P2 then results from the membrane’s

repolarisation.

The P1 peak is not observed on recording electrodes near the stimula-

tion site, as it has already passed by the time the stimulus ends and the

amplifiers begin to record. However, it is visible on more distant electrodes,

which the action potential volley reaches later after stimulation.

32

Rostral

Caudal

1

2 9

3

4 10

5

6 11

12

7 13

14

8 15

16

(a) Ascending

Rostral

Caudal

1

2

3

4

16

15

14

13

12

11

10

9

(b) Descending

Figure 2.9: ECAPs propagating in a human spinal cord. Both recordings

are made using an MCS2 under hospital conditions (ie. with the patient in

an unshielded treatment room). 100 stimuli are delivered and the resulting

signals averaged to reduce noise. Electrode connection diagrams shown at

right.

a): responses ascending towards the head from the point of stimulation.

Recorded on a Medtronic Specify 5-6-5 paddle lead, which has three columns

of electrodes staggered laterally. Stimulation was delivered in bipolar mode

between electrodes 8 and 16, and the visible responses are ascending towards

the brain from the stimulation site.

b): responses descending towards the sacrum from the point of stimulation.

Recorded on a St. Jude Lamitrode S-8, with eight collinear electrodes. Stim-

ulation was delivered in bipolar mode between electrodes 1 and 2, and the

responses are descending from the stimulation site.

Some example responses are shown in Figure 2.9. These are from two

patients with different electrode configurations. In 2.9(a), some of the initial

P1 peaks appear to be truncated by the start of recording, as the amplifiers

come out of blanking. The responses appear to arrive at the same time on

electrodes which are at the same cord level but displaced laterally. In 2.9(b),

most of the P1 peaks are clearly visible, but the P2 peaks are less clear.

Some stimulus artefact is visible at the start of recording in the form of a

decaying spike, but this is difficult to distinguish from a truncated P1 peak.

(a) Recording amplitude (b) Electrode

layout

Figure 2.10: a): Amplitude of the fast response on several electrodes as

current is increased, in an unconscious patient. A line is fitted to the linear

portion of one channel’s response. Recorded on a St. Jude "Penta" paddle,

shown in b). Stimulation is delivered on the circled red electrodes, and

recordings are made from the numbered electrodes.

The fast responses appear on all electrodes at the same threshold stimu-

lus current. They increase in amplitude as the stimulus intensity increases,

although this relationship has a different slope on each electrode. At suffi-

ciently high currents, the response begins to saturate; this, too, occurs at

roughly the same current on each electrode. This behaviour is shown in

Figure 2.10. The maximum comfortable stimulus is usually not more than

twice the threshold at which ECAPs begin to be observed. The saturation

plateau is never observed below the discomfort level.

34

Sometimes, the discomfort level is reached without ECAPs being mea-

sured at all. The reason for this is not yet understood; possibilities include:

• the stimulating electrodes are not over the dorsal columns, and the

discomfort comes from stimulation of other spinal structures;

• the recording electrodes are in the wrong location, too far from the

dorsal columns;

• the ECAP is too small to measure, being masked by the stimulation

artefact or electrical noise;

• a neuropathic condition prevents conduction in the direction of the

recording electrodes; or

Achieving good pain coverage is already very sensitive to lead placement,

with misplacement of the stimulation electrodes reducing or eliminating

the therapeutic window between the paraesthesia threshold and discomfort

threshold. The introduction of ERT introduces an additional requirement

to optimise the ECAP recording quality. Techniques for doing so are still in

their infancy.

Placement technique and amplifier performance can be expected to im-

prove as SCS ERT matures. Of more fundamental interest are the possible

involvement of neuropathic effects, or of non-DC neural elements. Testing

these hypotheses will require substantial further study.

Some chronic pain patients have systemic neuropathies which affect dor-

sal column fibres directly, for example via demyelination. In these patients,

significant changes in the ECAP morphology can be expected - potentially

including absence of propagating signals. This would not necessarily affect

the therapeutic effect, if this is obtained by a short-distance pathway such

as the collaterals from the DCs to the grey matter. In these cases, SCS ERT

may even be a helpful diagnostic tool. Abnormal ECAP morphologies have

been observed in a small number patients with systemic neuropathies in

closed Saluda trials; more data is required.

It is also possible that elements other than the dorsal columns could be

activated. Assuming the lead is correctly placed over the dorsal columns,

other candidates for recruitment are the dorsal roots and the dorsolateral

35

columns (DLCs). Direct nerve root stimulation is used for treating pain,

although not necessarily very effective in the long term [247]. Stimulation

of the dorsal roots should result in an ECAP, as the low-threshold Aβ fibres

entering through the root then ascend and descend the cord. The DLCs, like

the DCs, consist predominantly of long ascending and descending fibres, with

a broadly similar distribution of fibre diameters [64]. As such, stimulation

of the DLCs would also be expected to give rise to an ECAP.

36

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3 I =2mA

Time (ms)

Am

plit

ude (V

)

Figure 2.11: Evoked responses recorded over several segments in an ovine spinal cord, and a radiograph of the position of

the leads within the spine. A significant modulation in the ECAP amplitude correlates with electrode position within the

spinal segment. Bipolar, biphasic stimulation is delivered on the two leftmost electrodes.

37

Figure 2.12: Propagation in both directions from a central stimulation site,

recorded in an ovine subject.

The ECAP amplitude recorded on each electrode generally decreases

with distance, but is also affected by the position of the electrode within

the spinal segment. This is demonstrated in Figure 2.11, where ECAPs

travelling over several spinal segments are recorded simultaneously. The

segment-related modulation is sufficiently large that the ECAP amplitude

does not fall off monotonically with distance. A set of recordings from both

sides of a stimulus are shown in Figure 2.12.

38

0.0 0.5 1.0 1.5 2.0 2.5

Time (ms)

−60

−40

−20

0

20

40

60

80

100

Volta

ge

(µV

)

E5

E6

E7

(a) Recording made in patient ground mode, using a needle electrode

as reference

0.0 0.5 1.0 1.5 2.0 2.5

Time (ms)

−60

−40

−20

0

20

40

60

80

100

Volta

ge

(µV

)

E5

E6

(b) Recording made in epidural ground mode, using E7 as reference

Figure 2.13: Evoked responses in a human spinal cord. A needle electrode

was inserted through the skin and used as a reference electrode in 2.13(a).

The same recording is shown in 2.13(b), using an electrode on the array as

reference. This shows how the waveforms can be subtly distorted by the

choice of reference electrode; note, for example, the change in P1-P2 peak

amplitude ratio.

39

In many human patients, only a single electrode array is implanted.

This reduces the number of electrodes available for recording: the stimulus

electrodes cannot be used for recording, and the proximity to the stimulation

site also increases the artefact on the remaining electrodes. This often

renders the signal on the electrodes next to the stimulus unusable. An

example recording is shown in Figure 2.13(a). A single 8-electrode array is

implanted; stimulation is delivered on electrodes 1, 2 and 3, while electrode

8 is faulty (open circuit). This leaves few electrodes on the array on which

recording can be performed. A needle is inserted through the patient’s

skin to act as a reference potential for the differential amplifiers. Decaying

stimulus artefacts appear to be present on the recordings, and the P1 peaks

are not clearly distinguished. If it is assumed that the artefact waveforms

on all channels are somewhat similar, then any measurement of the peak

amplitudes and latencies must necessarily be distorted by the artefact’s

presence.

The situation is further complicated by the lack of a dedicated reference

electrode in many scenarios, such as longer-term trials where a needle

electrode is not suitable. In these cases, one of the implanted electrodes must

be used as a reference. This decreases the number of channels available

for recording, but perhaps more importantly, it superimposes the evoked

potential waveform from the reference electrode on to all the other channels

in inverted form. This is not necessarily an obvious effect. An example

is shown in Figure 2.13(b). In this case, electrode 7 has been used as a

reference; its voltage is subtracted from the other channels’ recordings. This

results in subtle shifts in the parameters of those other waveforms; for

example, the N1 latency measured on electrode 6 shifts earlier by 160 µs,

and the apparent N1-P2 amplitude on E6 reduces by 10%.

The stimulus artefact is present under all stimulation conditions. The

artefact increases with the stimulus intensity, though not necessarily in

linear proportion. The artefact overlaps the fast responses in time; as a

result, it is not possible at present to unambiguously distinguish the two

components of a given signal. The combined effects of artefact and refer-

ence selection make precise measurements of ECAPs via peak and trough

locations unreliable.

40

Slow Response

Slow responses are not generally recorded in human patients, as they

tend to occur only at stimulation levels higher than patients’ comfort limits.

They are regularly observed in anaesthetised ovine subjects. Slow responses

do not show triphasic waveforms or propagation behaviour, indicating that

they do not originate in the long fibres of the dorsal columns.

0.0 0.5 1.0 1.5 2.0

Time (ms)

−1200

−1000

−800

−600

−400

−200

0

200

Voltage

(µV)

(a) Fast response

0 1 2 3 4 5 6 7 8

Time (ms)

−1200

−1000

−800

−600

−400

−200

0

200

Voltage

(µV)

(b) Fast and slow response

0.1 mA

0.2 mA

0.3 mA

0.4 mA

0.5 mA

0.6 mA

0.7 mA

0.9 mA

1.0 mA

1.1 mA

1.3 mA

1.6 mA

1.8 mA

2.1 mA

2.3 mA

2.6 mA

2.8 mA

3.1 mA

3.3 mA

3.5 mA

3.8 mA

4.0 mA

4.2 mA

4.4 mA

4.7 mA

4.9 mA

5.7 mA

6.5 mA

7.3 mA

8.0 mA

8.7 mA

Figure 2.14: Waveforms recorded on one electrode in sheep, showing the

appearance of and changes in form as the current is increased from zero.

Each trace represents a different stimulus current. A biphasic stimulus is

used, with a tripolar electrode configuration approximately 60mm from the

recording electrode.

41

0 1 2 3 4 5 6 7 8 9

Current (mA)

0

200

400

600

800

1000

1200Voltage

(µV

p-p)

(a) Fast response amplitude (0-2ms after stimulus ends)

0 1 2 3 4 5 6 7 8 9

Current (mA)

0

200

400

600

800

1000

1200

Voltage

(µV

p-p)

(b) Slow response amplitude (2-8ms after stimulus ends)

E8

E11

E14

E17

E20

Figure 2.15: Relationship between stimulating current and response ampli-

tude across several electrodes in a sheep. Both measurements contain some

amount of current-dependent artefact, even when there is no response. A

tripolar, biphasic stimulus was applied using electrodes 1 through 3. Elec-

trode spacing approximately 7mm.

42

Fast and slow responses on a single electrode are shown in Figure 2.14.

Each trace represents a different stimulus current. The artefact and fast

response are visible before the 1 millisecond mark, while the slow response

appears only at the highest currents. The peak-to-peak amplitudes of the

fast and slow responses are shown in Figure 2.15, showing the onset of slow

response at approximately 5.7mA stimulus current.

The slow response is of little concern in SCS; stimuli delivered during

therapeutic SCS are below the discomfort level, and so are of insufficient

strength to evoke it.

Noise

The electrical noise performance of ECAP recording is largely limited by

the electromagnetic environment in which recording takes place. The MCS2

has an intrinsic noise of 1 µV RMS from 300-5000Hz, the band in which

the neural response energy is concentrated. The circuit from the patient

to the MCS consists of largely unshielded cables and adapters and can be

a metre or more in length; this results in both electric and magnetic field

pick-up. Non-evoked potentials in the body also contribute noise. Cardiac

activity (ECG) is easily identified by its period and characteristic waveforms,

and spontaneous muscular activity (EMG) is suppressed in anaesthetised

patients. However, spontaneous neural activity cannot be suppressed or

otherwise measured independently of the electrical noise.

43

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Time (s)

−30

−20

−10

0

10

20

30

40

50

60

Voltage

(µV)

Figure 2.16: Long period recording from SCS electrodes, showing an ECG

signal. This recording was made during stimulation at 60Hz; it has been

low-pass filtered at 30Hz to suppress the periodic stimulation spikes and

responses. The patient’s heart rate was approximately 74 BPM.

An ECG waveform from a human patient is shown in Figure 2.16. This

signal was obtained by low-pass filtering a period during which stimulation

was being delivered at 60Hz. The QRS complex of the ECG waveform in

particular gives rise to large spikes in the measured data. This can be

addressed if necessary by detecting the heartbeat and excising recording

segments which are likely to be contaminated by P, QRS, or T waves.

44

Figure 2.17: Noise spectrum recorded in an unconscious patient. 16 epidu-

ral electrodes are present; E24 is a needle electrode inserted through the

patient’s skin for use as a reference potential.

A spectrum of noise recorded in an unconscious human patient is shown

in Figure 2.17. A substantial low-frequency component is present, due in

part to the presence of an ECG waveform. Regular peaks are also observed,

corresponding to the odd harmonics of 60Hz; this indicates noise picked up

from the mains network in the operating theatre. This is an unavoidable

source of noise in modern buildings.

2.4.3 Current Problems

This recent work has demonstrated the feasibility of recording evoked

responses as part of spinal cord stimulation, using the same implanted

electrodes. The next step is to determine how best to apply this information

to improve spinal cord stimulation.

Work is ongoing in this area, with the goal of reducing or eliminating

45

variations in stimulation due to postural change [178]. This is addressed

by measuring the ECAP amplitude after every therapeutic stimulus, and

using a feedback system to control the delivered stimulus intensity. Closed

trials of feedback therapy indicate that this constant-amplitude feedback

outperforms traditional open-loop stimulation [183].

Constant-amplitude feedback is likely not an optimal solution to this

problem. The sensitivity to posture is believed to come from the movement

of the spinal cord within the cerebrospinal fluid; when the cord moves away

from the stimulating electrode, the stimulus intensity required to achieve a

particular level of neural recruitment increases, and vice versa. However,

the voltage recorded for a given level of recruitment will also change as

the distance between cord and electrodes changes. Constant-amplitude

control then overcompensates: if the amplitude remains the same as the

cord distance is increased, then the neural recruitment must be increased

also. In order to avoid this overcompensation, the relationship between

stimulation, recording, recruitment and distance must be understood.

Measurement of the ECAP is a challenge for feedback control. The arte-

fact waveforms vary with the applied stimulus current, as do the evoked

responses. In recordings within the Saluda corpus, the amplitude of the

artefact is also dependent on factors including the stimulation method and

the electrode configuration in use; this can lead to circumstances under

which the ECAP amplitude cannot be measured at all. There are some pa-

tients in whom feedback cannot currently be established due to an inability

to measure ECAPs. The artefact must be mitigated if the ultimate goal of

feedback control is to be realised in all patients.

46

2.5 Electrophysiological Models

Understanding the effects of neuromodulation requires knowledge of

the stimulated neural tissue - its direct response to stimulation, and how

that response then affects the patient. In spaces such as the spinal cord, the

stimulated population could encompass some tens of thousands of individual

nerve fibres. While the behaviour of individual fibres can be measured [99],

[211], [241], it is impossible with today’s technology to record the individual

behaviour of so many fibres at once; the invasive nature of microelectrode

recording also precludes its use in SCS patients. In such circumstances,

we can turn to models of neural behaviour. A model which incorporates

the mechanisms believed to be involved can be used to interpret ensem-

ble measurements such as ECAP recordings. The establishment of such a

model entails making a set of assumptions about the physiological features

involved, and the validity of the model’s results is contingent on the validity

of those assumptions. This is complicated by the piecemeal nature of avail-

able measurements of physiological parameters, reflecting our incomplete

understanding of the nervous system.

Ideally, models would be directly validated against measurements on

all of their component parts, as well as against the higher-level behaviours

of the targeted tissues. Validation gives some measure of confidence in

the predictive and explanatory power of the model. Models of SCS have

been created in the past, although their validation has been limited by

the available data: perceptual information and peripheral recordings both

invoke a number of non-spinal mechanisms, complicating validation.

This section gives a brief history of neural modelling, with specific focus

on the myelinated nerve fibres which form the dorsal columns. This in-

cludes the axonal membrane measurements that explain the basics of action

potentials, and the models subsequently based on them that consider the

behaviour of such membranes in myelinated fibres. Many such models have

focussed on the behaviour of nerve fibres after stimulation, and in particular

on determining under what circumstances a fibre may be triggered. This out-

put has been used in SCS models to date. Some neural models consider the

extracellular action potentials generated by nerve activity; a few consider

both the internal reaction to stimulation and the ensuing evoked potentials.

47

Finally, the existing SCS models are discussed, and an approach to an SCS

model which simulates evoked response telemetry is laid out.

2.5.1 Membrane Models

Nerve cells form long, tubular processes known as axons, along which

they transmit information in the form of action potentials. This behaviour

results from a special property of the cell membrane: it is studded with

voltage-gated ion channels, each of which permits the flow of certain metal

ions under particular conditions. A concentration gradient of each ion is

maintained by pumps, also in the membrane; as a result, a potential is

present across the membrane at rest.

Models of these ion channels’ behaviour rely on measurements on excised

nerve fibres. These are typically bathed in electrolyte, and microelectrodes

are used to establish connections to each side of the membrane. Voltage- and

current-clamp conditions are imposed on the membrane and its resulting

behaviour observed. Early work was carried out by Hodgkin, Huxley and

Katz [87], who established a model of conduction in an unmyelinated nerve

fibre. This included models of sodium and potassium ion channels; these

were controlled by gating variables whose behaviour was dependent on the

transmembrane potential, and solved using numeric integration.

Later work made similar measurements in myelinated nerve fibres,

which are insulated along most of their length by Schwann cells which

wrap around them many times, referred to as the myelin sheath. This

sheath is interrupted at regular intervals, exposing the axonal membrane

underneath; these exposed regions are known as the nodes of Ranvier. The

giant axon of the squid (Xenopus laevis) was an early target due to its

very accessible size, and simulated action potentials were published in the

1960s [68]. Measurements made in human dorsal column nerve fibres are

now available [203]. All models include sodium and potassium channels, but

some include other channels; for example, a non-specific channel [68], or a

persistent sodium channel [31]. Observations indicate that there may be five

or more potassium channels in human peripheral myelinated axons [191].

Measurements of membrane parameters obtained using single-fibre in

vitro experiments are fraught with hazards. For example, the physical han-

48

dling of the fibres can cause demyelination [203] or more subtle damage [12],

changing fibre behaviour. There appear to be potassium ion channels which

are only exposed after retraction of myelin from the paranodal region [46].

2.5.2 Axon Stimulation Models

Explaining axon behaviour requires not only knowledge of the membrane,

but also of the interactions between the individual nodes of Ranvier as well

as their response to an externally imposed stimulating field.

An early model of a myelinated axon is due to McNeal [142] in 1976; this

model treats each node of Ranvier individually, solving a set of membrane

current equations at each node. The internal potential of each node is then

connected to that of its neighbours by an impedance representing the con-

ductive axoplasm within the axon. This cable model is then used to estimate

the threshold stimulus at which an action potential will be initiated; the

computational resources available at the time did not permit solving the full

membrane equations and modelling the travelling action potential.

Later models extended this approach. The basic cable model treats the

myelin layer as a perfect insulator. Blight argues for the use of a double

cable model, with both resistance and capacitance of myelin being modelled

using multiple segments for each internode [25]. More recent models have

included the influence of paranodal potassium channels, paranodal leakage

currents, and extracellular potassium accumulation [30], [219]. Some of

these mechanisms have been shown to reproduce both depolarising and

hyperpolarising afterpotentials as measured in humans [137].

None of these models include mechanisms for extracellular action poten-

tial recording; however, they form the basis for models that can.

2.5.3 Extracellular Potential Models

The electric potential set up by a nerve’s action current in a volume

conductor was considered as far back as 1947 [126]. This was extended to a

lumped axon representation by Plonsey [184], who demonstrated that the

expected extracellular potential waveform should be triphasic, using only

analytic methods.

49

Later models began to use numeric methods as available computational

power increased. The single-fibre action potential (SFAP), which is the

extracellular potential due to a single fibre’s activation, can be calculated

straightforwardly using a cable model. Simulation is used to determine the

current flow waveform through each node of Ranvier after the stimulus is

applied. An electric potential model of the fibre geometry is then used to

calculate the external potential waveform [72].

A number of methods have been developed to calculate ECAPs; commonly,

this is addressed by ensemble methods, rather than individually calculating

and summing SFAPs for each fibre under consideration. Stegeman et al. de-

scribe the CAP for a nerve bundle consisting of fibres of different diameters,

each with different propagation characteristics [218]. The fibre diameter

distribution is discretised, and a representative SFAP is calculated for each

diameter bin; the resulting SFAPs are then scaled according to the fibre

density in each bin, and summed to arrive at an ECAP. This assumes that

all the fibres are located along the same line in space, and that an action

potential is initiated simultaneously, in the same location, in every fibre.

This appears sufficient to model recorded ECAPs where the fibre diameter

is known approximately [217].

2.5.4 Evoked Potential Models

Evoked potential models combine elements of both stimulation and ex-

tracellular potential models, considering both the effect of a stimulating

pulse and the resulting fibre behaviour together.

The straightforward approach is quite possible with modern computa-

tional power: calculating the field imposed on a fibre by a stimulus pulse,

simulating that fibre’s subsequent behaviour, and calculating the field im-

posed on a recording electrode by the action currents in that fibre. This

approach has been used both for cochlear implants [47] and deep brain

stimulation [109]. Both these models use a finite-element volume conductor

model to determine the transimpedances between electrodes and neural

elements, and a three-stage approach to simulation: imposition of a stimulus

on the fibres, calculating their behaviour, and then determining the voltages

imposed on the recording electrodes. This is a flexible approach, allowing the

50

combination of any spatial electrical model with any set of neural models.

Some models seek to reduce the computational power required to model

ECAPs; when dealing with populations of thousands of fibres, the neural

simulation step can be very time-consuming. Miller et al. propose a prob-

abilistic model [150] in which parameters of different fibre diameters are

measured, and then used to estimate an ECAP for a given stimulus. These

parameters include latency and latency variability (jitter) as a function of

stimulus intensity, and threshold current. The threshold behaviour of each

diameter was found to dominate the results. Such a model could be used

with a fully simulated set of single-fibre behaviours to produce a lower-cost

ECAP simulator.

2.5.5 SCS Models

A number of models have been designed to try and predict the behaviour

of the spinal cord under stimulation. The first published SCS model is due to

Coburn and Sin [50], [51], who modelled the potential distribution as a result

of a stimulus current using finite-element techniques. A three-dimensional

model of the tissues is built as a series of compartments of different conduc-

tivities, and a constant current is driven between the stimulating electrodes.

The field is then calculated by solving Poisson’s equation over the finite-

element approximation, and may be examined in the dorsal columns, where

the target fibres lie. This allows the determination of whether a particular

nerve fibre will or will not fire after a given stimulus.

In 1991, Holsheimer and Struijk published the first paper [91] on what

would become the UT-SCS model, developed at the University of Twente.

This model combines two components: an electric field model, and a neural

model. The field model is similar to Coburn and Sin’s, although its exact

geometry and implementation differs. The geometry is modelled on a grid

system, and solved using a custom successive-overrelaxation solver. As of

2014, a new volume model appears to be under development [212]. This

new model uses a commercial package for finite element analysis [52], and

promises better accuracy.

In the early embodiments of the UT-SCS model, nerve fibres were not

modelled directly, but instead the activating function for different fibre

51

positions was estimated. The activating function describes the tendency of

a particular electric field to activate a nerve fibre. (For a myelinated fibre,

the activating function is the second difference of the field at its nodes of

Ranvier.)

After 1992, a discrete time-domain neural model was implemented [225].

This model, based on one described by McNeal [142], describes the currents

flowing within a myelinated nerve as a result of an externally imposed

electric field. This requires that the ion channels be modelled numerically.

Initially, this was implemented using a model of rabbit axonal membrane

given by Chiu [226]. This was superseded by another model [249], [251]

based on the human membrane model of Schwarz [203], and morphometric

model of Behse [17]. The volume model is used to calculate the field imposed

on a fibre by a given stimulus waveform; the neural model is then run in

the time domain, to determine whether it does or does not fire as a result of

the stimulus.

The distribution of nerve fibres, in terms of their diameters and paths

within the cord, is not a fixed feature of the model. A great deal of work was

carried out by UT researchers to try and establish these parameters. Initial

efforts measured stimulation thresholds [250] in patients, using a transverse

tripole to steer the stimulation point laterally across the spine. The resulting

thresholds were then compared to modelled results to estimate the fibre

diameters. A later effort measured these directly [64]; dorsal columns in

cadavers were sectioned, and fibre populations directly measured using

computer vision techniques.

The model was validated in 1993 by experiments on patients [226],

comparing the paraesthesia thresholds experienced by patients with the

stimulation amplitudes of the model. The paraesthesia threshold was as-

sumed to correspond to the minimum stimulus intensity at which any fibres

were recruited. Discrepancies between model and measurements were found,

and presumed to be a result of insufficient data on the exact parameters

of the cord and its fibres. Later work after changing to a new human fibre

model improved matters somewhat [249].

A major finding from the UT-SCS model is the conclusion that only the

very largest fibres of the cord are activated, ranging in diameter from 16.4

to approximately 11 microns in diameter. This follows from the assumption

52

that the most sensitive fibres in the cord are those responsible for the onset

of paraesthesiae. This conclusion, in combination with the morphometric

studies of [64], led the authors to conclude that therapeutic stimulation

in SCS involves very few fibres - from a few dozen for multi-dermatome

stimulation, to a single fibre for single-dermatome coverage [95]. This also

implies that only a very shallow layer on the surface of the dorsal columns

must be involved; up to 300 µm deep.

The UT-SCS model has been used to investigate a number of scenarios.

Curved dorsal root fibres were simulated, and found to have generally lower

thresholds than the axial dorsal column fibres [224]. The effect of the cord

position was analysed [88], finding large changes in sensitivity with position.

This supports the hypothesis that cord movement is responsible for the

posture-related side effects of SCS.

Much use is made of the UT-SCS model to determine optimal stimula-

tion techniques. Electrode design is revisited repeatedly; from early work

published in 1991 [92], followed by more advanced work in 1995 through

1997 [93], [94], [98], the model is used to determine in increasing detail the

best geometries for selecting desired fibres in the dorsal columns. A new elec-

trode geometry was introduced in this work in 1996: the tranverse tripole,

where three electrodes are distributed laterally across the cord [223]. By

delivering different currents to the two outer electrodes, returning through

the centre, the focus of action potential initiation can be roughly steered

laterally across the cord. Simulation shows that the tripole also reduces the

field in the lateral regions of the cord, favouring the stimulation of dorsal

column fibres compared to dorsal roots or the dorsolateral columns, which

may be implicated in side effects.

The predicted steering behaviour was verified in a clinical setting [90],

[252], in which paraesthesiae were moved across the body by changing the

relative stimulus currents. This improves clinical outcomes by reducing

the need for lead revision to achieve the necessary paraesthesia coverage.

The usage range was also increased, with a mean of 1.7 DT:PT ratio [252],

compared to a typical mean of 1.4. This was attributed to the spatial se-

lectivity of the transverse tripole. A later multicentre clinical study of 41

implant patients [167] did not find a statistically significant difference in

pain control outcomes or VAS scores; this was complicated by stimulator lim-

53

itations which prevented certain configurations from being used. Research

on tripolar stimulation using the UT-SCS model continues [198]–[200], and

SCS implant manufacturer Medtronic now offers tripolar stimulation func-

tionality on many of its stimulators.

Other SCS implant manufacturers maintain SCS models with similar

structure to the UT-SCS model. St. Jude Medical have published details

of a simple model [108], [152] considering only fibres of 12 µm diameter

and aimed at understanding spatial control of paraesthesia using paddle

electrodes. Boston Scientific also maintain a model [122]; unusually, this

takes account of fibres extending down to 5.7 µm in diameter. This particular

embodiment appears to indicate that the broad fibre distribution is neces-

sary to understand mechanisms such as sacral shift of paraesthesia with

changing pulse-width [123], which contradicts conclusions drawn from the

UT-SCS model which suggest that only the largest fibres are stimulated [95].

2.6 Current Problems

In order for ECAPs to be useful for therapeutic SCS, they must be inter-

preted. This requires that their origins be understood, and that the recording

process distorts them as little as possible.

The interpretation of ECAPs relies on knowledge of the involved anatomy

and neurophysiology, which can be embodied in a model. To date, SCS models

have been used to predict neural recruitment. Perceptual measurements are

used as a proxy for recruitment in patients, but this is a link that has not

been validated. The advent of spinal ECAP recording offers a more objective

measure of recruitment. Recorded data thus far suggest that some of the

conclusions drawn from prior models may be incorrect; for example, the UT-

SCS model’s indication that only a very small number of fibres are involved

in therapeutic SCS. If this is true, the evoked ECAPs should have sub-

microvolt amplitudes [222] and exhibit quantisation effects at a therapeutic

stimulation level. The recordings in the Saluda corpus often have much

higher amplitudes and no signs of quantisation have been observed. A new

model is built in this work, which uses established techniques for evoked-

potential calculation in addition to those mechanisms previously considered

54

in SCS. This will allow examination of hypotheses about what happens

during stimulation of the cord, and the expression of these behaviours in the

ECAP. Tested hypotheses would then allow rational design of therapeutic

components such as electrode arrays and stimulation patterns to optimise

therapies for specific goals in the future.

As part of the pathway to validating such a model, the ECAP recording

process must be improved. Recordings are distorted by various signals,

including the stimulation artefact and various sources of uncorrelated noise.

Of these, the artefact is most troublesome: it cannot be removed by averaging,

which adds uncertainty to the measurement of ECAP features. This is

problematic both for exploration of the electrophysiology - including model

validation - and for use in feedback control of stimulation. In the following

work, the artefact is addressed at its source: a new technique for stimulation

allows the artefact to be reduced without changing the neural response.

This enables improved recordings to be used for validation, and is equally

applicable to feedback control.

55

CH

AP

TE

R

3V I R T U A L G R O U N D

The stimulus artefact is the component of a recording which results from

stimulation, but is not neural in origin. Instead, it is a result of electrical and

electrochemical components of the system being driven out of equilibrium

during stimulation. If the artefact amplitude is much larger than the de-

sired neural response, then the neural response may be completely masked.

The amplitude and waveform of the artefact both change as the stimulus

intensity is varied; this makes it difficult to determine precisely the relative

contributions of artefact and neural waveform components. This uncertainty

places limits on how much information can be extracted in any stimulation

scenario. This chapter details the development, implementation and perfor-

mance of a new stimulation system which reduces artefact at the source,

which can be used to improve performance in implantable neuromodulation

systems.

The sources of artefact are first discussed, along with models of artefact

behaviour. Existing techniques for artefact reduction are then reviewed,

including stimulation methods and signal-processing techniques. The basis

of a new method is then introduced. This method is tested both in vitro and

in vivo, demonstrating its performance.

56

3.1 Artefact

Charge can be stored in electrical components which are connected to the

electrodes. For example, capacitive coupling is used in many neurostimulator

designs as a safety measure. Parasitic capacitances are also found in long

cables, amplifier input stages, and stimulators [135]. These can be minimised

through traditional electrical design techniques.

Evoked response amplifiers can create recording artefacts as a result of

being overdriven by the stimulation pulse. The stimulation pulse is larger

than the evoked potential in most forms of neuromodulation; in SCS, the

stimulus is typically several volts in amplitude, while the subsequent evoked

response is in the tens of microvolts. The results of this overdrive can result

in artefact in different ways. Some amplifier designs end up in internal

disequilibrium after being overdriven, and take some time to recover from

this state. This can result in a complete block of the input signal during

the recovery time [27]. Overdrive is addressed using established electrical

engineering techniques. Kent et al. use a multi-stage amplifier with clamp

circuitry before each successive gain stage [107]. The clamps limit the degree

of disequilibrium, resulting in fast recovery after the stimulus. Another

approach is to disconnect the amplifier during stimulation, using something

akin to a sample-and-hold circuit [69], [153]. This is the approach taken in

the Saluda MCS2.

Charge is also stored at the electrode-tissue interface. The implanted

electrode array makes connection to the surrounding tissue through the

interstitial fluid, which is an ionic solution. This results in the formation of

an electrical double layer at the metal-solution interface. This couples the

electrode and surrounding tissue capacitively. If a sufficiently high voltage

is imposed across the interface, then redox reactions will begin to occur;

these result in charged species being generated near the electrode surface.

Any capacitive or ionic charge deposited at this interface manifests as a

potential, seen as artefact when it discharges after stimulation [135].

The capacitance of the double layer is distributed across the surface of the

electrode, and this interacts with the distributed resistance of the solution

to form a distributed RC network. Redox reactions begin to occur at suffi-

ciently high potentials across the electrode-electrolyte interface (hundreds

57

of millivolts), providing a highly nonlinear set of additional behaviours.

The magnitudes of the distributed RC and redox mechanisms vary with

the surface area of the electrode. Larger electrodes exhibit higher capaci-

tance, and consequently develop a lower potential for the same stimulating

current; this increases the current threshold at which redox reactions begin.

Consequently, redox reactions and the distributed nature of the double layer

are often not considered in treatments of microstimulation artefact, where

electrodes are extremely small and tissue resistive components dominate

the voltage drop during stimulation [27], [159].

Artefact varies even between identical arrays in different patients. The

impedance of the tissue interface is highly dependent on the surface condi-

tion of the electrodes; for example, roughening the surface electrochemically

vastly increases the surface area [146]. This increases the capacitance, and

consequently reduces the artefact voltage per unit of stored charge. Hy-

drophobic contaminants such as skin oil can reduce the effective surface

area, to opposite effect. Surface coatings are not necessarily stable after

implantation. Fibrous capsules form around implanted electrode arrays; this

is known to change the tissue impedance substantially (up to 2x), although

this affects the resistive components of impedance much more than the

capacitance [158]. Liquid ingress after implantation can also create inter-

nal conductive paths between electrodes. The electrical impedance of the

surrounding tissue at a macroscopic scale also plays a role. Even with identi-

cally placed arrays, the natural variation between different patients’ spinal

geometries and postures results in different inter-electrode impedances [1],

[6].

3.1.1 Charge Storage

Charge is built up along the interface if a net current flows into or out

of the electrode via an external circuit. This is shown, along with a mesh

approximation of the interstitial fluid, in Figure 3.1(a). In the diagram,

this current is imposed by a stimulation pulse, but net currents can also

flow through recording electrodes if they are connected to nodes with finite

impedance and an external field is imposed by stimulation elsewhere.

58

+

I

Electrode Electrode

(a) A net current flows between two electrodes in an ionic solution.

Electrode

Current

(b) A current flows through an electrode exposed to an electric

field, parallel to its surface.

Figure 3.1: Mechanisms of charge accumulation in the electrical double

layer at the interface between a large metal electrode and an ionic solution.

Tissue impedance is represented by a resistor mesh, and the distributed

double-layer capacitance by a capacitor at each end of the electrodes. In a), a

stimulation current is driven between two electrodes (thick bars), resulting

in a net current out of the left electrode and into the right electrode. In b),

a single electrode is shown. A voltage is imposed on the tissue mesh along

the electrode axis. No net current flows through the electrode, but the field

gradient causes current to flow into the left end of the electrode and out of

the right end.

59

A number of mechanisms can give rise to net current flows on recording

electrodes. Impedances between the electrodes and ERT system potentials,

including stray capacitance and protective resistor networks, will admit

some current when a voltage is imposed on the electrodes during stimulation.

Amplifiers which have protection diodes at their inputs can also admit large

current flows through their input terminals under some conditions. Careful

design of the entire electrical system is required to minimise these effects.

Imposed field gradients also cause currents to flow in the electrodes,

seen in Figure 3.1(b). These currents flow (and so store charge) even if the

electrode is disconnected from all apparatus, and cannot be controlled by

attached circuitry.

3.1.2 Models

Artefact behaviour can be approximated by models. The simplest models

of electrode artefact simply treat the electrodes as RC networks [27], [159].

This approach has the advantage of analytic tractability, at the cost of ex-

cluding redox dynamics and the distributed impedance effects of physically

large electrodes. Mayer et al. [133] describe a circuit model of an electrode-

electrolyte interface, using a Warburg impedance to model the double layer

impedance in concert with diffusion effects around the interface. Warburg

impedances [245] describe the behaviour of distributed complex impedances;

these cannot be modelled by single poles and zeros in the manner of individ-

ual resistors and capacitors. Instead, they can be considered as fractional

poles [62], which have behaviour described by fractional derivatives. Unlike

simple capacitors and inductors, in which current and voltage are ±90◦

out of phase with one another, a fractional impedance has an arbitrary but

constant phase, leading to their naming as constant-phase elements.

When sufficiently high potentials are imposed across the double layer,

ionic species present in the interstitial fluid will begin to react at different po-

tentials [100]; this too is dependent on surface condition of the electrode [39].

Some models also take some of these effects into account [29].

60

First electrode Space Second electrode

Figure 3.2: Electrical model of the electrode-tissue interface, simplified

from [204]. This represents the tissue spreading resistance as a network

of resistors. The distributed double-layer capacitance is discretised using

a number of individual capacitors. Nonlinear elements may be placed in

parallel with the capacitors to model electrochemical reactions.

A model of artefact specific to SCS has been proposed by Scott and

Single [204]. This model includes constant-phase elements to describe the

distributed capacitance of the electrode surface, as well as nonlinear effects,

the solution spreading resistance, and amplifier frontend impedances; it

has been validated directly against SCS electrodes in saline. The model is

based around discretising a distributed RC model of the electrode surface

and tissue, as shown in Figure 3.2.

The various models of artefact then inform approaches to mitigating

artefact, either through signal processing after recording or by modifying

the stimulus itself.

3.1.3 Signal Processing Methods

Signal-processing approaches to artefact removal rely on assumptions

about the characteristics of the artefact and/or neural components of the

recorded signal.

The very simplest approaches assume that the artefact occurs in a differ-

ent location in time [171], or that the artefact waveform can be approximated

by fitting a simple analytic function to the recording [240]. These require

that the artefact be very clearly distinguished from the ECAP waveform.

More advanced approaches take their assumptions from models of the

artefact. The simplest such approach is based on RC models of artefact

production: that the artefact waveform is constant, and that its amplitude

varies linearly with the applied stimulus current. In this case, the artefact

waveform can be measured at a low stimulus intensity - lower than the

61

threshold at which a neural response is first evoked. This pure artefact

waveform is used as a template. To process an evoked potential waveform,

the template is scaled according to the delivered stimulus intensity, and sub-

tracted from the recording [44], [45]. The accuracy of this approach depends

on the linearity of the artefact with respect to current. This approximation

is no longer true once redox reactions begin to occur; for platinum electrodes

in saline, this begins at interface potentials well below 500mV, and well

within the range of stimuli used in SCS.

Independent component analysis (ICA) has also been explored as a way

of separating artefact and ECAP without prior knowledge of either [5]. This

technique applies statistical techniques to many independent recordings

made with various stimulus amplitudes, with the idea that the artefact

and response signals can be distinguished as they change differently with

current. ICA relies on representation of the input signal as a linear combina-

tion of several template waveforms. This is more flexible in representing an

artefact waveform that changes shape with current, but offers no guarantees

about distinction between artefact and neural response.

More information can be gleaned for signal processing purposes if the

stimulus can be deliberately modified or repeated. Recordings can be made

in pairs, with the polarity of the stimulus inverted for the second recording.

This aims to ensure that the artefact is equal but opposite in the second

recording; the two recordings are then summed to cancel the artefact. This

alternating polarity approach does not require that the artefact be linear,

but does require that the nonlinearities be identical when the stimulation

polarity is reversed; this may not be true if the stimulating electrodes are

not physically identical. Another issue is that the response of the neural

tissue to an inverted stimulus is likely to be quite different [151].

A similar approach is often used in cochlear evoked potential measure-

ment, known as forward masking [112]. This relies on the refractory prop-

erties of nerve fibres: after they are recruited, there is a refractory period

during which they cannot be recruited again. In this method, three record-

ings are made: MP, with a masker pulse followed after some delay by a probe

pulse; and then one recording each of the masker and probe pulses alone,

M and P. The masker pulse is intended to place the stimulated nerves into

a refractory state, so that in the MP recording, the probe pulse recruits no

62

nerves, and only artefact is recorded. The M and P recordings are added

and the MP recording subtracted; this is intended to cancel the artefacts

between the single- and double-pulse recordings. This particular approach

makes the assumptions that the M and P pulses, when delivered together,

will affect precisely the same set of nerve fibres. However, the first pulse

will not only trigger some set of fibres, but will also push higher-threshold

fibres out of equilibrium without triggering them. This can result in an

additional recruitment during the probe pulse. The only way around this

is for the masker pulse to be so large as to recruit the entire nerve bundle

being examined.

All signal processing approaches are also limited by the input signal they

are given. If the input is subject to saturation effects, then no amount of

signal processing can recover the original signal. The difficulty of separating

artefact and signal numerically also limits the ultimate performance of these

techniques, and points to a different course of action: reducing the artefact

at its source, during stimulation.

3.1.4 Stimulation Methods

Modifying the stimulus allows the adjustment of artefact at its source.

However, modifications can also change the resulting neural response, lim-

iting the available methods. A classic technique for artefact reduction is

the use of balanced biphasic pulses [160]. A rectangular pulse is delivered,

followed immediately by a pulse of the opposite polarity. These have the

same duration and amplitude. These tend to have opposite effects on charge

storage: what is charged by the first pulse is then discharged by the second.

This applies to the redox products at the electrode-tissue interface as well;

many of the charged species created during the first pulse are recombined

during the second. Redox products can be harmful to tissue in the longer

term [37], and so charge recovery in some form is mandatory for chronic

implanted neurostimulators. Biphasic stimuli are often used for this purpose

already, including cochlear implants [49] and in the Saluda MCS2.

An extension of this technique is to use more complex pulse waveforms

that are optimised to reduce the residual artefact. Zero-forcing equalisation

has been used in intracortical microstimulation [48], using linear techniques

63

to design a multistep stimulation pulse and decrease artefact duration. This

assumes that the artefact behaves linearly with current, and so cannot be

applied directly to SCS.

Another approach to artefact is to intervene after stimulation in order to

discharge the residual charges more quickly. By shorting the stimulating

electrodes after stimulation, the remanent charge can recombine [59]. Some

active approaches have been proposed [27], [36], in which a feedback ampli-

fier is used to drive the electrode potential back to its pre-stimulation value

immediately after stimulation. Actively driving electrodes after stimulation

poses some safety issues; any discharge amplifier must necessarily have

high drive capability, and measures must be put in place to ensure that no

unintended stimuli are delivered during the discharge period.

Charge accumulation can also be limited by the electrical configuration

of the stimulation and recording systems. High input impedances on the

recording electrodes will reduce the amount of charge that is allowed to flow

into electrodes; however, resistive or capacitive impedances to ground are

required in many systems for safety or practical reasons. Using a stimulator

which is galvanically isolated from the recording apparatus will also reduce

the net current which flows through the recording electrodes, although

parasitic coupling between the two systems still permits some flow [135].

This approach requires the use of totally separate power supplies for the

stimulation and recording units, making it extremely impractical for use in

implantable devices.

3.1.5 Problem

The stimulus artefact is an unavoidable consequence of stimulating an

electrode-tissue interface. In SCS, the artefact overlaps the desired neural

responses in such a way that these are difficult to separate, and this limits

the accuracy of ECAP measurements for use in therapies. Linear techniques

developed both for signal processing and stimulation control are of limited

utility in SCS, as the high stimulation voltages used lead to a significant

degree of non-linearity in the artefact behaviour.

Reduction of the artefact at its source remains a viable option. Balanced

biphasic stimuli are already in use, for reasons of both chronic safety and

64

artefact reduction. The use of an isolated stimulator is known to improve

artefact. However, isolation also greatly increases system complexity. At a

minimum, the power supplies must be duplicated or an isolated converter

added; further, in multichannel ECAP recording there must be a multiplex-

ing system to select the stimulating electrodes, as well as data links to

control stimulation in synchrony with the recording process - all requiring

isolation. This precludes implementation in an implant, where space is at

a premium; even in a bench system such as the MCS2, such an approach

would be expensive.

This does not mean that the benefits of isolation cannot be achieved by

other means. An examination of the effects of isolation suggests another

route to artefact reduction.

65

3.2 Analysis

During stimulation, an electric field is set up in the tissue by the stimu-

lating current. Every electrode is exposed to this field, and some current will

flow into or out of them through the attached system, resulting in charge

accumulation at the electrode-tissue interface. The current flows can also

be decreased by reducing the imposed voltage on the electrodes. This is

constrained by the requirement that the system deliver precisely specified

stimulation current; this defines the voltage differential between the stimu-

lating electrodes, but leaves their common-mode potential free.

The imposed field can be considered as a differential plus a common-

mode component: the differential component differs on all electrodes, while

the common-mode component is identical for all of them. For the purposes

of analysis we define the differential components as summing to zero. The

differential component, then, is defined by the stimulation current, and so

isolated stimulation must achieve its benefits by affecting the common-mode

component in some way that other methods do not.

3.2.1 Stimulation Methods

Two non-isolated stimulation arrangements are shown in Figure 3.3. In

3.3(a), current is alternately sourced to and sunk from the stimulating elec-

trode, while the return electrode is grounded. This method requires positive

and negative supplies, and is implemented in the MCS2. In 3.3(b), only one

current source is used and only one power supply rail; the alternation of

current is achieved by switching the stimulus and return electrodes between

ground and the current source.

66

1 2

(a) Source-and-sink current source stimu-

lation

1 2

(b) Source-only current source stimulation

Figure 3.3: Stimulator configurations used in SCS. Each arrangement is

able to deliver a biphasic stimulus across electrodes 1 and 2. In a), using

two current sources to source and sink current through the same node. In b),

using a single current source which can only source current, in conjunction

with switches.

67

S1

S2

R2

R1

(a) Current flow with an isolated current

source

R1

S2

R2

S1

(b) Current flow with a non-isolated cur-

rent source

Figure 3.4: Current flow during one stimulation phase. Recording elec-

trodes R1 and R2 are connected to measurement amplifiers with some input

impedance to ground. Stimulation is delivered between electrodes S1 and

S2. a), with a fully isolated stimulator: a current circulates due to the exter-

nal field gradient driven by the stimulation current, but the total current

through the recording electrodes is zero. b), with a stimulator galvanically

connected to the amplifier ground, shows how a net current flows into the

measurement electrodes in addition to the circulating current.

68

The difference in total charge storage can be observed by considering

the current flows during one phase of stimulation, represented in Figure 3.4.

During isolated stimulation as seen in 3.4(a), a differential component is

driven into electrode R1 and out of electrode R2 due to the differing voltages

each is exposed to during stimulation. These are equal and opposite; without

a galvanic connection to the stimulator, no net current can flow into the

recording electrodes. With non-isolated stimulation as in 3.4(b), the same

differential component will flow; but now there is also a net current compo-

nent circulating through the system ground. This additional contribution to

current flow can increase charge storage and artefact.

3.2.2 Common-Mode Potential

The common-mode field can be considered as the potential imposed by

stimulation at a very distant point in the tissue, where the differential

field is cancelled out. This potential, Vcm, is actively driven by non-isolated

stimulation methods. Figure 3.5 shows the common-mode potential during

a biphasic stimulus, with four in-line electrodes as in Figure 3.4.

69

S1

S2

Vcm

R1

R2

(a) Source-only stimulation

S1

S2

Vcm

R1

R2

(b) Source-and-sink stimula-

tion

S1

S2

Vcm

R1

R2

(c) Isolated stimulation

Figure 3.5: Stimulation, common-mode, and recording potentials during

a biphasic stimulus, delivered to the configurations in Figure 3.3. S1 and

S2 are the stimulation and return electrodes. R1 and R2 are two recording

electrodes, placed to the outside of S1 and S2 respectively as in Figure 3.4.

An identical stimulus current waveform is driven in each configuration; only

the method of delivery changes.

In a), current is sourced to S1 while S2 is grounded, and then vice versa in

the second phase. As in Figure 3.3(b). In b), current is sourced to and sunk

from S1, while S2 is grounded, as in Figure 3.3(a). In c), a fully-isolated

stimulator does not drive Vcm.

70

Figure 3.5(a) uses a current source that can only source current, as in

Figure 3.3(b). This results in a common-mode potential with a large DC

component, leading to a net charge accumulation on all recording electrodes.

The use of biphasic stimulation does not cause cancellation of this charge,

as its sign is constant.

Figure 3.5(b) uses a current source that can both source and sink current,

as in Figure 3.3(a). This still has a substantial common-mode potential, but

its DC component is zero. This results in some cancellation of the stored

charge.

Isolated stimulation does not drive Vcm at all; instead, the common-mode

potential will be defined by the bias potential of the ERT amplifiers. This

is shown in Figure 3.5(c). This scenario has the minimum possible charge

accumulation.

3.2.3 Return Drive

The effects of isolation can be duplicated in a non-isolated system by

deliberately controlling the common-mode potential to minimise the net

current. This is achieved by actively driving the return electrode, instead of

connecting it to a fixed potential.

71

-1

R1 R2S1 S2

(a) Unity inverting drive

R1 R2S1 S2

+

-

(b) Feedback drive

Figure 3.6: Two schemes for an actively driven return electrode S2. S1 is the

stimulating electrode; R1 and R2 are two recording electrodes. (a), a unity-

gain inverting amplifier drives the return electrode, such that the voltages

applied to them sum to zero. (b), a high-gain amplifier drives the return,

using feedback to control the potential at one of the recording electrodes to

zero.

In the case of bipolar stimulation, this can be simply achieved by driving

the return electrode with an inverted form of the voltage waveform on the

stimulating electrode, such that the stimulus and return electrode voltages

sum to zero. This arrangement is shown in Figure 3.6(a). An implicit as-

sumption is that the two stimulating electrodes have identical impedances;

in practice, this is not the case, and a common-mode potential will be present

depending on the degree of mismatch. This scheme is also difficult to gener-

alise to multipolar stimulation.

A more advanced approach is shown in Figure 3.6(b). In this approach,

known as virtual ground, a reference electrode is used to estimate the

common-mode potential Vcm, and a feedback amplifier used to drive this po-

tential to zero during the stimulation pulse. This works with any multipolar

configuration of stimulation and return electrodes.

When an indifferent electrode is implanted in the patient at some dis-

tance from the array, its potential gives a good estimate of Vcm. When an

electrode on the array is used, the accuracy of the estimate will depend on

the distance from the stimulating electrodes.

72

1

2

3

Patient

1

etc

AmplifiersCurrent source

switches

24

24

+

-

+

-

+

-

+

-

2

3

1:24P

N

Vppp

Vnnn

1:24P

N

Vppp

Vnnn

1:24P

N

Vppp

Vnnn

1:24P

N

Vppp

Vnnn

1:24

1:24

Inverting input

mutiplexerGround switches

1:24

Multiplexer

+

-

Amplifier/driver

Virtual Ground

Figure 3.7: The NICTA MCS2 multichannel ERT system, now augmented

with virtual ground. The electrode selected as the recording amplifier ref-

erence is also fed into the virtual ground drive amplifier; the output of

this driver is then connected to the return electrode(s) via a switch set and

the electrode bus. Note that little change to the architecture is required to

accommodate the virtual ground system.

73

The addition of virtual ground to the MCS2 architecture is shown in Fig-

ure 3.7. The system design constrains the differential recording amplifiers

and virtual ground to share a reference electrode. A multiplexer allows the

virtual ground amplifier to drive any combination of electrodes.

3.3 Saline Bath Testing

The performance of virtual ground is measured in a bath of physiological

saline. This has the advantage of using real SCS electrodes, while avoiding

any ambiguity between neural response and artefact.

3.3.1 Method

A bath of phosphate-buffered saline solution (PBS) is prepared as a

tissue analog, with contents according to Table 3.1; this solution is 1/10 the

concentration of isotonic saline. This is used as an artefact standard due to

the lower conductivity of the epidural fatty tissue into which the electrode

array is placed, compared to the isotonic interstitial fluid [204]. The bath

is composed of a cylindrical volume of liquid 28 cm in diameter and 15 cm

deep in an insulating container. Figure 3.8 shows the bath used. A St. Jude

Medical (St. Paul, MN, USA) octopolar lead is suspended in the centre of the

bath, oriented vertically. This lead was explanted from a patient after their

SCS trial and chemically sterilised before use.

Compound Concentration (mg/L)

NaCl 800

KCl 20

Na2HPO4 144

KH2PO4 24

Table 3.1: Composition of phosphate-buffered saline (PBS) used in the saline

bath.

74

Figure 3.8: Saline bath used for artefact measurements. The test elec-

trode enters the bath through the centre of the lid. The metal bowl is not

electrically connected.

Electrode 4 was used as the stimulation electrode, and electrode 5 as the

return. Electrode 8 provided the reference both to the amplifiers and to the

virtual ground system. The stimulation method of Figure 3.3(a) was used

when virtual ground was disabled, in which electrode 5 was grounded as the

return in both phases.

Balanced biphasic stimulus pulses were delivered with currents ranging

from 100 µA to 12 mA. The upper limit was determined by the compliance

voltage of the stimulation subsystem. The polarity was alternated on every

stimulus. The stimulus pulses were 120 µs long, with a 50 µs inter-phase

gap and repeating at 44 Hz. 500 recordings at each current and with each

polarity were made; this was repeated with and without virtual ground.

Each set of 500 identical recordings were averaged to produce a single

waveform for measurement.

The artefact amplitude is estimated by calculating its root mean square

(RMS) value over a measurement window of 1.5 milliseconds. This interval

is chosen to reflect the ECAP duration on a nearby electrode. The DC compo-

nent of the signal in the window is subtracted before the RMS calculation, to

75

avoid any contribution from DC offset; DC content has no bearing on ECAP

measurement.

3.3.2 Results

0.0 0.5 1.0 1.5

Time (ms)

−200

−100

0

100

Voltage

(µV)

(a) VG off, AF

0.0 0.5 1.0 1.5

Time (ms)

0

100

200

Voltage

(µV)

(b) VG off, KF

0.0 0.5 1.0 1.5

Time (ms)

−50

0

50

Voltage

(µV)

(c) VG on, AF

0.0 0.5 1.0 1.5

Time (ms)

−40

−20

0

20

40

Voltage

(µV)

(d) VG on, KF

E1

E2

E3

E6

E7

Figure 3.9: Example artefact waveforms recorded at 2.0 mA. Each plot is

an average of 500 recordings. Biphasic stimulation on electrodes 4 and 5.

KF: cathodic first on E4; AF: anodic first.

Some example artefact waveforms are shown in Figure 3.9, recorded

at a single current. The notations AF (anodic first) and KF (cathodic first)

describe the polarity of the biphasic stimulation pulse delivered to electrode

4: AF refers to an anodic pulse followed by a cathodic pulse, and KF vice

versa. Without virtual ground, the artefact waveforms all have the same

sign with respect to the reference electrode potential. When virtual ground

is enabled, the sign depends on which side of the stimulating bipole each

electrode was on. In both cases, the sign is inverted when the stimulus

polarity is inverted.

76

0 5 10

Current (mA)

0

50

100

Voltage

(µV

RMS)

(a) VG off, AF

0 5 10

Current (mA)

0

50

100

150

Voltage

(µV

RMS)

(b) VG off, KF

0 5 10

Current (mA)

0

10

20

30

40

Voltage

(µV

RMS)

(c) VG on, AF

0 5 10

Current (mA)

0

10

20

30

Voltage

(µV

RMS)

(d) VG on, KF

E1

E2

E3

E6

E7

Figure 3.10: RMS amplitude of the artefact waveforms on each electrode

after DC removal. Biphasic stimulation on electrodes 4 and 5. KF: cathodic

first on E4; AF: anodic first.

The change in the artefact with the applied stimulus current is shown

in Figure 3.10. Without virtual ground, the artefact amplitude increases

somewhat linearly with current; the behaviour differs between electrodes

as well as between polarities. When virtual ground is enabled, the artefact

amplitudes are reduced. Nonlinearities are visible on all electrodes.

77

0 5 10

Current (mA)

−105

−100dB

(RMS)

(a) VG off, AF

0 5 10

Current (mA)

−106

−104

−102

−100

dB

(RMS)

(b) VG off, KF

0 5 10

Current (mA)

−140

−130

−120

−110

dB

(RMS)

(c) VG on, AF

0 5 10

Current (mA)

−140

−130

−120

−110

dB

(RMS)

(d) VG on, KF

E1

E2

E3

E6

E7

Figure 3.11: Ratio of artefact amplitude to stimulus voltage. Biphasic

stimulation on electrodes 4 and 5. KF: cathodic first on E4; AF: anodic first.

Figure 3.11 shows the ratio between the stimulus voltage and the artefact

voltage. The stimulus voltage is measured at the stimulator, and so includes

the drop across the electrode-tissue interface. Without virtual ground, this

ratio is quite variable at low currents but relatively stable above 5mA. When

virtual ground is enabled, the electrodes differ in behaviour; there appears

to be an inverse correlation between the artefact amplitude of electrodes on

opposite sides of the stimulating bipole.

78

0 5 10

Current (mA)

10

20

30

40

dB

(RMS)

(a) AF

0 5 10

Current (mA)

10

20

30

40

dB

(RMS)

(b) KF

E1

E2

E3

E6

E7

Figure 3.12: Improvement in artefact amplitude obtained by using virtual

ground. KF: cathodic first on E4; AF: anodic first.

Figure 3.12 details the improvement in artefact obtained by using virtual

ground. The behaviours with each polarity are quite different, although the

inflection points are similar at approximately 7mA.

3.3.3 Conclusions

The use of virtual ground in a saline bath reduces RMS stimulus artefact

by 10-40dB. The reduction achieved depends on several factors, including the

geometric relationship of the stimulating electrodes, the recording electrode,

and the reference electrode.

Electrodes nearest the reference showed the least improvement. This is

likely due to the differential configuration of the amplifiers; these electrodes

are exposed to fields of the same polarity as the reference during stimulation

and hence have broadly similar charges, leading to some cancellation during

measurement without virtual ground. Electrode 1 showed similarly low

improvement. This may be due to its geometry: it is at the tip of the lead

and has a domed metal end, giving it extra surface area compared to the

other electrodes. This mismatch changes its distributed capacitance, so for

the same current flow during stimulation, its accumulated voltage will be

somewhat less than that of the other electrodes.

The artefact shows non-linearity with current, with different behaviours

across different electrodes. The stimulus-to-artefact ratio varies by as much

as 5dB, with the greatest variation at low stimulus currents. This indicates

79

that template subtraction is of limited utility: it depends on a constant

stimulus-to-artefact ratio, and the template must be measured at a low,

subthreshold stimulation current. The artefact also behaves differently

when the polarity is inverted, which limits the performance of alternating-

polarity averaging as an artefact reduction method.

3.4 Human Testing

A set of measurements were made in a human patient comparing stim-

ulation with and without virtual ground. These provide a more limited

picture of artefact performance, the protocol consisting of stimuli of four

pulse widths but only one current apiece.

3.4.1 Method

The patient was implanted with a cylindrical octopolar lead (model un-

known, Medtronic, MN, USA). Stimulation was delivered to the three elec-

trodes at the rostral end of the array: electrode 2 was connected to a current

source, while electrodes 1 and 3 were connected together and used as the

return. Biphasic stimulation was applied in both cathodic-first (KF) and

anodic-first (AF) modes, and with and without virtual ground. In KF stim-

ulation, the central electrode of the tripole is driven first with a negative

pulse, and then a positive one; AF stimulation is the reverse.

Electrodes 4 and 8 were used in separate trials as the combined reference

electrode and virtual ground input. This left four electrodes free for recording

in each configuration. Four different pulse-widths were used: 40, 80, 120

and 160 µs, with the stimulation current being adjusted for each such that

the total charge of the stimulus pulse was 160 nC. 50 recordings were made

under each stimulation condition and averaged. None of these stimuli gave

rise to a visible ECAP on any recording channel.

3.4.2 Results

Example averaged artefacts are shown in Figure 3.13. Note the asymme-

try in artefact between cathodic-first and anodic-first stimuli, highlighting

80

−200

−100

0

100

200

Voltage

(µV)

E4 E5

0 1 2 3

Time (ms)

−200

−100

0

100

200

Voltage

(µV)

E6

0 1 2 3

Time (ms)

E7

AF FG

KF FG

AF VG

KF VG

Figure 3.13: Artefact recorded with 40µs, 4mA stimuli. Averaged over 50

recordings in each configuration. Electrode 8 is used as the reference, and

the signals on electrodes 4 through 7 are shown. Stimuli were delivered on

E2. Return electrodes were E1 and E3, both passively fixed to ground (FG)

and actively driven by virtual ground (VG).

the nonlinearity of the artefact generation process. The artefact is largest on

electrode 4, next to the stimulating tripole. On the more distant electrodes,

virtual ground substantially reduces the artefact produced.

81

−200

−100

0

100

200

300

Voltage

(µV)

E4 E5

0 1 2 3

Time (ms)

−200

−100

0

100

200

300

Voltage

(µV)

E6

0 1 2 3

Time (ms)

E7

40 µs

80 µs

120 µs

160 µs

Figure 3.14: Artefact recorded with different pulse widths. Polarity AF,

without virtual ground.

The variation in the artefact with different pulse widths is shown in

Figure 3.14. The artefact observed after 40 µs stimuli differs notably from

those observed after longer pulses.

82

E4 E5 E6 E7010203040506070

Artifact(µV

RMS)

(a) VG off, AF

E4 E5 E6 E7010203040506070

Artifact(µV

RMS)

(b) VG off, KF

E4 E5 E6 E701020304050607080

Artifact(µV

RMS)

(c) VG on, AF

E4 E5 E6 E701020304050607080

Artifact(µV

RMS)

(d) VG on, KF

E4 E5 E6 E7−10−5051015202530

Improvement(dB

RMS)

(e) Improvement, AF

E4 E5 E6 E7−5

0

5

10

15

20

Improvement(dB

RMS)

(f) Improvement, KF

40 µs

80 µs

120 µs

160 µs

Figure 3.15: Human artefact amplitudes with distant reference electrode.

Measured with and without virtual ground. Electrode 8 is the amplifier

reference input, as well as the virtual ground driver input. Biphasic stimuli

on electrodes 1, 2 and 3. KF: cathodic first on E2; AF: anodic first. In (a) and

(b), the artefact amplitudes with the return electrodes 1 and 3 connected

to the ERT system’s ground. In (c) and (d), the artefact amplitudes with

the returns driven actively by the virtual ground system. In (e) and (f), the

ratio of the two amplitudes expressed in dB. A positive value indicates a

reduction in artefact when virtual ground is employed.

83

Measured artefact amplitudes are shown with and without virtual ground

in Figure 3.15, with electrode 8 used as the reference potential for the ampli-

fier inputs as well as for virtual ground. Electrode 4, nearest the stimulating

tripole, is subject to the greatest level of artefact. The observed artefact

also depends on the stimulation polarity, despite the use of a biphasic stim-

ulation pulse. This effect is more pronounced at the shortest pulse width.

When virtual ground is enabled, the artefact is reduced on most electrodes.

Electrode 4, nearest the stimulus, experiences a slight increase in artefact

instead.

84

E5 E6 E7 E80102030405060708090

Artifact(µV

RMS)

(a) VG off, AF

E5 E6 E7 E80102030405060708090

Artifact(µV

RMS)

(b) VG off, KF

E5 E6 E7 E8010203040506070

Artifact(µV

RMS)

(c) VG on, AF

E5 E6 E7 E8010203040506070

Artifact(µV

RMS)

(d) VG on, KF

E5 E6 E7 E80

1

2

3

4

5

Improvement(dB

RMS)

(e) Improvement, AF

E5 E6 E7 E80.0

0.5

1.0

1.5

2.0

2.5

3.0

Improvement(dB

RMS)

(f) Improvement, KF

40 µs

80 µs

120 µs

160 µs

Figure 3.16: Human artefact amplitudes with reference electrode near

stimulus. Measured with and without virtual ground. Electrode 4 is the

amplifier reference input, as well as the virtual ground driver input. Biphasic

stimuli on electrodes 1, 2 and 3. KF: cathodic first on E2; AF: anodic first.

In (a) and (b), the artefact amplitudes with the return electrodes 1 and 3

connected to the ERT system’s ground. In (c) and (d), the artefact amplitudes

with the returns driven actively by the virtual ground system. In (e) and (f),

the ratio of the two amplitudes expressed in dB. A positive value indicates a

reduction in artefact when virtual ground is employed.

85

The experiment was repeated using Electrode 4, next to the stimulat-

ing tripole, as a reference; the results are shown in Figure 3.16. Artefact

performance is worse with or without virtual ground. Some improvement is

obtained on all electrodes when using virtual ground.

3.4.3 Conclusions

Artefact reduction at subthreshold stimulus levels has been demon-

strated with virtual ground. Best results are obtained when the reference

electrode is far away from the stimulus, and when the stimulating pulse

width is short. Improvements obtained with the reference at the maximum

distance from the stimulating tripole ranged from 3dB to 25dB, depending

on pulse width and electrode; artefact on the electrode next to the stimulus

was worsened under some conditions. Gains with the reference placed next

to the tripole were poor.

The difference in improvement between the two stimulation polarities

is greater than 5dB in some cases. The generated artefact is asymmetric

between stimulus polarities; this is likely to depend on the relative surface

area and geometry of the stimulating and return electrodes, which in turn

depends on the microscopic surface condition of the electrodes. Tripolar

stimulation would contribute to this asymmetry as the return current is

split between two electrodes, so the potentials at the electrolytic interface

are reduced at the returns. This results in different threshold currents

for the onset of redox reactions in each stimulation phase. This is in con-

trast to bipolar stimulation, in which the stimulation and return electrodes

are nominally identical, leading to similar thresholds with either polarity.

This suggests that gains from alternating-polarity averaging using tripolar

stimulation may be less than those using bipolar stimulation.

The waveforms seen without virtual ground in Figure 3.14 indicate that

artefacts are similar for the longer pulse widths, while those at 40 µs pulse

width differ markedly. This suggests that some additional redox process

begins at currents between the 2 mA used for the 80 µs pulse width, and

4 mA used for 40 µs.

The collected data are limited by the experimental protocol in use, partic-

ularly in the use of a single charge. The small number of recordings of each

86

stimulus also place some of the virtual ground recordings near the noise

floor; this may cause a slight underestimate of the improvement obtained

on electrodes 6 and 7.

3.5 Conclusions

Virtual ground reduces the levels of stimulation artefact by controlling

the potential imposed on one of the non-stimulating electrodes. This per-

mits the currents flowing through the input impedances of the recording

electrodes to be reduced, minimising the charge stored at the electrode in-

terfaces. Experiments in a saline bath and in a human subject demonstrate

this improvement, with gains ranging from 5 dB to 40 dB in RMS amplitude.

The improvement obtained depends on the stimulus current and electrode

configuration, as well as the geometric relationships between the electrodes.

The implementation of virtual ground in the MCS2 has enabled collec-

tion of improved data in the Saluda corpus. Recordings are now routinely

made using virtual ground, resulting in lower baseline levels of artefact.

The resulting improved ECAP-to-artefact ratio expands the range of circum-

stances under which ECAPs can be observed and detected. This directly

affects the application of feedback for therapeutic applications; the difficulty

of separating artefact from ECAP signal components means that artefact

can limit the control of neural recruitment.

Virtual ground may be combined with signal-processing techniques for

artefact reduction, such as alternating-polarity averaging and template

subtraction. The artefact exhibits a greater degree of nonlinearity with the

applied stimulus current when virtual ground is enabled, indicating that the

proportional gains from these techniques will be lower when virtual ground

is used. Further work is required to determine whether these nonlinearities

can be characterised by subthreshold stimuli for the purposes of subtraction.

As implemented, virtual ground is not identical to isolated stimulation;

instead of forcing the net current through the recording electrodes to zero,

the voltage imposed on the reference electrode is nulled. This can be ex-

ploited to mitigate one of the problems with differential recording in the

MCS2: the electrode selected as the differential reference potential has a

87

different input impedance than the others due to its connection to multiple

amplifiers, and so accumulates a different charge - which is then observed

as a common component on all the recording channels. Virtual ground es-

sentially eliminates charge storage on the differential reference, removing

this common component.

This is effective but may not be optimal. The technique could be further

generalised by controlling the reference electrode’s potential to a value other

than zero, which alternates according to the stimulus polarity. This would

allow the imposed voltage on any electrode to be controlled indirectly, and

may allow the optimisation of artefact on that electrode. Such an approach

would be useful in situations such as feedback control, where one electrode’s

signal is used for ECAP detection. Further study is required to determine

the potential effectiveness of this concept.

Virtual ground requires no isolation or extra power supplies, making it

ideal for implanted use. It has been implemented on an application-specific

integrated circuit (ASIC) in combination with a multichannel stimulation

and recording system; this ASIC is being used in an implanted ERT system

for feedback control of SCS.

A patent has been applied for [145] covering the technique and its imple-

mentation. This work was performed in cooperation with Peter S. Single of

Saluda. The driven-return concept and initial implementations are entirely

the author’s; its final implementation and stabilisation in the MCS2 are due

to Single.

88

CH

AP

TE

R

4E V O K E D S F A P M O D E L

We wish to create a model of the spinal ECAP, as observed in SCS. This

must model the behaviour of all of the individual fibres activated by stimu-

lation. The number of Aβ fibres activated by SCS is unknown; model-based

estimates for this number range from less than ten [95] to approximately one

thousand [123]. The total number of fibres in the dorsal columns is on the

order of one hundred thousand [64], providing an upper bound. Modelling

the behaviour of fibres on this scale places limits on the possible complexity

of the model. Consequently, we seek the simplest possible model that will

reproduce the characteristics of spinal cord behaviour.

4.1 Approach

We begin by building a model of the evoked potential from a single fibre,

the evoked SFAP. The ECAP can then be derived by summation of the SFAPs

of the individual fibres under consideration, an approach taken by existing

ECAP models [47], [109], [150]. This model must take into account the

geometry of the spinal cord and surrounding tissues, as well as the electrodes

used for stimulation and recording. To this end, the approach of Coburn

and Sin [50], [51] is adopted, in which a 3-dimensional electrical model is

used to calculate the electric fields in the cord as a result of stimulation.

89

The field imposed on a nerve fibre passing through the cord can then be

calculated according to its position, and the fibre simulated in the time

domain with this stimulus imposed. Existing SCS models perform this step

merely in order to determine whether or not a fibre is recruited, and the

process terminates there.

We extend the existing approaches at this point: the fibre is simulated

during the stimulus, and then afterwards for as long as recording is de-

sired. The currents flowing through its nodes of Ranvier are stored, and

the electrical model applied again: this time to calculate the field observed

at the recording electrodes as a result of the neural activity, resulting in a

simulated SFAP waveform.

In this chapter, the design and implementation of the geometric and

neural model components are discussed. A basic spinal geometry is defined,

including a model of a commercial percutaneous electrode in the epidural

space. A simple neural model is presented, based on voltage-clamp mea-

surements made on human dorsal column fibres. Validations of both model

components are performed in order to detect any gross errors in imple-

mentation. The neural model is validated by comparison with an existing

published model of a nerve fibre, and the geometric model is subjected to a

mesh sensitivity analysis. This results in a model suitable for calculating

the SFAP for any nerve fibre in the dorsal columns.

4.2 Implementation

The simulation of an SFAP for a particular nerve fibre requires several

steps. Firstly, the parameters of the fibre must be established: its diameter

and its path through the dorsal columns. From these, the locations of the

nodes of Ranvier must be calculated, and the electrical parameters of the

neural model determined; these include the internodal conductance and the

surface area of each node.

Given the locations of the nodes, two transfer functions must be de-

termined: in the forward direction, the voltages imposed on the nodes by

currents applied to the stimulating electrodes; and in the return direction,

the voltages imposed on the recording electrodes by the action current

90

through each node.

Finally, the behaviour of the fibre itself can be simulated. A set of dif-

ferential equations models the behaviour of the neural membrane at each

node of Ranvier. This process calculates the imposed fields using the forward

transfer function, and the resulting recording voltages using the return

transfer function. This simulation is time-stepped for the desired recording

duration, resulting in the simulated SFAP waveform.

4.2.1 Transimpedance

The transfer functions through which the neural and geometric models

interact can be calculated using an approach similar to that used in previous

models of SCS [50], [92]. These models consider the spine and surrounding

tissues to be composed of resistive compartments. The transfer functions

between the electrodes and the nodes of Ranvier then take the form of

electrical transimpedances.

This can be formalised for a model of E electrodes and N nodes of Ranvier.

In the forward step, the voltages at each node of Ranvier Vn(t) are functions

of the electrode stimulus currents Se(t). As the tissue is assumed to be

purely resistive, the effects of the current at each electrode are linearly

superimposed:

Vn(t)=E�

i=1

Gn,eSe(t) (4.1)

The matrix G is the transimpedance matrix, which describes the tran-

simpedance between each electrode and each node of Ranvier, in ohms.

This may also be represented as a matrix equation, with V and S vectors:

V (t)=GS(t) (4.2)

This is a similar formulation to the use of z-parameters in network theory.

These imposed voltages Vn(t) are then fed into the nerve fibre model

M, which is used to calculate the nodal current flows, In(t). This is highly

nonlinear, and requires the voltages at all nodes of the fibre as input.

I(t)= M(t,V (t)) (4.3)

Finally, the geometric model is used again, this time to calculate the

recording electrode voltages Re(t) as a function of In(t). Again, this is a linear

91

transimpedance problem, and can be written using the same transimpedance

matrix G:

R(t)= I(t)G (4.4)

The transimpedance matrices are the same in both the forward and

return directions due to the ohmic nature of the tissue model. According to

Lorentz’s reciprocity theorem [127], the relationship between a current and

a resultant electric field remains the same if the points where the current is

sourced and measured are interchanged. Consequently, if applying a current

to an electrode results in a voltage at some node of Ranvier, then applying

the same current to that node of Ranvier would result in an identical voltage

appearing at the electrode, even though the voltages at other points in the

model will differ. This shows that the geometric model need only produce G

as output; this is a time-invariant matrix, and so can be calculated just once

for a given geometry.

The columns of the transimpedance matrix are straightforward to com-

pute using the geometric electrical model. A test current is driven through

one of the electrodes on the array, and the resulting fields at the nodes of

Ranvier are solved for. A test current of T amperes on electrode e will yield

some voltage Vn at node n. The transimpedance column is calculated as:

Gn,e =Vn

T(Ω) (4.5)

This can be repeated for each electrode in turn, yielding one column of the

matrix for each simulation.

This requires that some part of the model be constrained to act as a

ground for the driven current to return to, simulating a distant indifferent

electrode. In clinical practice, stimuli are generally applied in bipolar or

tripolar fashion, where a current is driven into one electrode and returned

on one or more other electrodes. This can be implemented using the tran-

simpedance matrix as defined above, by driving the return electrodes with

opposing currents. For example, if a current of 1mA is to be driven on the cen-

tral electrode in tripolar stimulation, then a total current of −1 mA should be

driven on the two return electrodes. Ideally, this should result in cancelling

the effect of currents through the distant ground, which is generally not

present in clinical work.

92

A more accurate approach is to calculate columns in the transimpedance

matrix that directly capture the effects of multipolar stimulation. The dis-

tant ground is removed from the model; instead, the return electrode(s) are

configured as the ground points, and the stimulating electrode driven in the

usual fashion. This results in a single transimpedance value for each node

of Ranvier that correctly captures the effect of multipolar stimulation. This

can also be performed to determine transimpedance columns for differential

recording, by grounding the electrode that will be used as the recording

reference.

4.2.2 Geometric Model

A three-dimensional model of the spinal cord and surrounding tissues is

created by taking a transverse cross-section of tissues representative of the

thoracic spinal cord at the T10 level. The bone structure and the outline of

the dura are traced from photographs of a sectioned female cadaver [4]. The

cord outline and the grey matter are recreated from a micrograph [162]. This

cross-section is then extruded along the axis of the cord for a total length of

30 cm, forming a set of 3-dimensional compartments. Each compartment is

assigned an electrical conductivity according to Table 4.1. The white matter

has an anisotropic conductivity due to the large number of axially oriented

fibres with their insulating myelin sheaths; the other compartments are

isotropic.

This approach is the same as that taken by previous SCS models, and

does not incorporate the changes in the spine along its axis. This simplifi-

cation is used in this model due to a lack of detailed geometric data, and

to computational limitations. The dimensions of the dural and epidural

spaces change along the length of the cord [89]; this model considers only

stimulation at a single level. Structures external to the spinal canal such as

the vertebrae also vary along the axis of the spine. For the purposes of this

model, the ligamentum flavum and spinous process are assumed to present

an axially uniform conductor.

An electrode array is also modelled, corresponding to an ANS (now St.

Jude Medical, St. Paul, MN, USA) Octrode percutaneous lead. This lead is

1.2 mm in diameter, featuring eight electrodes. Each electrode is 3 mm in

93

Tissue Conductivity (S ·m−1)

Cerebrospinal Fluid 1.7

Grey Matter 0.23

White Matter (transverse plane) 0.083

White Matter (axial) 0.60

Epidural Fat 0.04

Bone 0.04

Table 4.1: Electrical conductivity of tissue compartments within the model.

Sourced from [14].

length, spaced on 7 mm centres. The electrodes are numbered from E8 at

the base to E1 at the tip; E1 has a closed metal end. The body of the array

is treated as an insulator, and the electrodes are configured as isopotential

surfaces due to the high conductivity of platinum relative to the tissue

compartments.

The electrode array is placed in the epidural space, 0.5 mm above the

dura. The position of the cord within the dura is a configurable parameter,

in order to allow the variation of behaviour with cord position to be modelled.

The cord-electrode distance is configurable from 1mm through 5mm, limited

by the extent of the dural cross-section; this corresponds to measurements

made with magnetic resonance imaging [89]. A cross-section of the model,

including electrode, is shown in Figure 4.1.

COMSOL (version 4.1, [52]) is used to perform finite-element meshing

and to solve for fields. A tetrahedral mesh is used, with a mesh refinement

step during simulation to ensure that the mesh is suitably fine in the regions

of high field derivatives.

When monopolar simulations are performed, the four outer faces of the

model in the transverse plane are grounded. The rostral and caudal faces

are not grounded. When multipolar simulations are performed, the return

electrodes are grounded instead.

The COMSOL simulation step is performed once for each stimulating

and recording electrode configuration. The resulting fields throughout the

model are stored in the COMSOL native format for each simulation. This

allows the expensive finite-element solving steps to be carried out once

94

-15 -10 -5 0 5 10 15

x (mm)

-10

-5

0

5

10

15

y(m

m) Cortical bone

Epidural fat

CSF

White matter

Grey matter

Electrode array

Figure 4.1: Cross-section in the XY plane of the volumetric model, shown

with cord-electrode distance set to 4.5mm.

for each geometry and electrode configuration. Interpolation of the solver

results to determine the transimpedances at specific nodal locations is then

a low-cost procedure.

4.2.3 Neural Model

Each nerve fibre is modelled with a discrete cable model approach, as

described by McNeal [142].

Im Im ImCm Cm Cm

Ve,n−1 Ve,n Ve,n+1

Vi,n−1 Vi,n Vi,n+1Ra Ra

Figure 4.2: Cable model of a myelinated axon. The exposed membrane at the

nodes of Ranvier is represented by the Ve terminals, whilst the Ra branches

represent the conductance of the axoplasm inside the axonal membrane.

Cm is the membrane capacitance, and Im the total ionic current across the

membrane.

95

In this model, the internodal axon segments behave as resistors, and

the exposed membrane at each node has some capacitance. A model of the

active membrane describes the nonlinear current flow, represented by Im in

Figure 4.2.

The internodal conductance Ga depends on the diameter and length

of the internodes, whilst the membrane currents at the nodes of Ranvier

depend on the nodal surface area. There is a relationship between the

diameter and length of the internodal and nodal fibre segments. Each fibre

is parametrised by the diameter of its outer myelin sheath, D. From this,

the diameter of the node of Ranvier dn and of the internodal segment di are

calculated. The nodal length Ln is fixed at 1 µm, and the internodal length

L i is also calculated from the fibre diameter. The equations used, and their

sources, are listed in Table 4.2. A schematic representation of the geometry

is shown in Figure 4.3. The nodal and internodal segments are treated

as cylinders of resistive axoplasm, and the myelin sheath is considered a

perfect insulator.

Parameter Symbol Size (µm) Source

Fibre diameter D

Internodal diameter di 0.7D [70]

Nodal diameter dn 0.37di [196]

Nodal length Ln 1 [197]

Internodal length L i −3.7+98D−1.03D2 [155]

Table 4.2: Geometric parameters of nerve fibres, as derived from overall

fibre diameter D.

96

Figure 4.3: Parametrised fibre geometry.

For each nodal location, the transimpedance values between that node

and each electrode are interpolated from the COMSOL results. Applied stim-

uli are described as current waveforms through each stimulating electrode

set. These are multiplied with the transimpedance matrix to determine the

external voltage waveform at each node (Ve,n in Figure 4.2).

The current Im is due to ionic currents through voltage-gated ion chan-

nels. Schwarz et al. [203] published an empirical model of the ion channels

in human dorsal column axons. The gating variables and their governing

equations are inferred from fitting curves to voltage clamp measurements

performed on individual nodes of Ranvier. This model incorporates one

sodium and two potassium currents, as well as a leakage current. This

model is used to calculate the ionic current density Jm at each node inde-

pendently. The basic equations are reproduced in Table 4.3, written in terms

of E, the membrane potential at each node:

E =Vi,n −Ve,n

97

The derivative of the membrane voltage at each node is given by:

En =Jm A+Ga(Vi,n+1 −Vi,n)+Ga(Vi,n−1 −Vi,n)

C0A(4.6)

with nodal area

A =πLndn (4.7)

and internodal conductance

Ga =πd2

i

4L iρ i

(4.8)

A and Ga depend on the nodal and internodal geometry, respectively; all

other model terms are independent of the fibre’s geometry. The ends of the

fibre are subject to sealed-end boundary conditions.

Jm = JNa + JKf + JKs + Jleak (4.9)

JNa = m3hPNaEF2

RT

[Na]o − [Na]i exp(EF/RT)

1−exp(EF/RT)(4.10)

JKf = n4 gKf(E−EK ) (4.11)

JKs = sgKs(E−EK ) (4.12)

Jleak = gleak(E−Eleak) (4.13)

(4.14)

Table 4.3: Equations governing membrane current density Jm at each node

of Ranvier

Each type of ion channel is controlled by one or more gating variables

(m,h,n, s); each of these has different dynamics, but all are controlled by

the membrane voltage at the node. These dynamics are described in a set of

ordinary differential equations (ODEs), set out in Table 4.4. Constants used

in the calculation are listed in Table 4.5.

The problem of simulating a nerve fibre then requires solving these

equations at each node of Ranvier for the current waveforms through their

membranes, which in turn allows the SFAP at the recording electrodes to

be calculated. This is an initial value problem: the initial state of all the

variables must be specified in order to determine the subsequent evolution

of the system. For a fibre of N nodes, there are 5N variables; the membrane

98

Rate constant Form A (ms−1) B (mV) C (mV)

αm I 1.86 −18.4 10.3

βm II 0.0860 −22.7 9.16

αh II 0.0336 −111.0 11.0

βh III 2.30 −28.8 13.4

αn I 0.00798 −93.2 1.10

βn II 0.0142 −76.0 10.5

αs I 0.00122 −12.5 23.6

βs II 0.000739 −80.1 21.8

(a) Constants for rate equations

Form Equation

IA(E−B)

1−exp

�

B−E

C

�

IIA(B−E)

1−exp

�

E−B

C

�

IIIA

1+exp

�

B−E

C

�

(b) Forms of rate equations

Derivative Equation

m 2.2(T−293.15)/10[αm(1−m)+βmm]

h 2.9(T−293.15)/10[αh(1−h)+βhh]

n 3.0(T−293.15)/10[αn(1−n)+βnn]

s 3.0(T−293.15)/10[αs(1− s)+βss]

(c) State variable derivatives

Table 4.4: Equations governing ion channel activation and inactivation

variables at each node of Ranvier

99

Constant Value Unit Description Source

PNa 5×10−3 cm ·s−1 Maximum mem-

brane sodium per-

meability

[203]

[Na]o 154 mmol Extracellular

sodium concen-

tration

[203]

[Na]i 35.0 mmol Intracellular

sodium concen-

tration

[203]

EK −84.0 mV Potassium equi-

librium potential

[203]

Eleak −84.0 mV Leakage equilib-

rium potential

[203]

gKf 21.4×10−3 S ·cm−2 Maximum con-

ductance due to

fast potassium

current

[203]

gKs 42.8×10−3 S ·cm−2 Maximum con-

ductance due to

slow potassium

current

[203]

gleak 42.8×10−3 S ·cm−2 Maximum con-

ductance due to

leakage current

[203]

C0 2.0 µF ·cm−2 Membrane capac-

itance

[68]

ρ i 110 Ω ·cm Axoplasm resis-

tivity

[142]

T 310.15 K Model tempera-

ture

F 96514.0 C ·g−1 ·mol−1 Faraday’s con-

stant

R 8.3144 J ·K−1 ·mol−1 gas constant

Table 4.5: Constants governing neural model behaviour.

100

voltage E as well as the gating variables m,h,n, s must be independently

calculated for each node. The fibre is assumed to be at rest at the beginning

of the simulation. This requires that the five nodal variables be identical at

each node, and that the derivative of each variable is zero; these equilibrium

values are calculated and used to initialise all the nodes.

The system of ODEs describing the fibre are highly nonlinear, and do not

admit a closed-form solution. Numeric integration is used instead, in which

the solution is calculated for discrete time-steps by sampling the derivative

functions. Various numeric integration techniques are available; these have

different convergence characteristics and computational costs. Integration

techniques may be divided into explicit and implicit solvers. Explicit solvers

sample the derivative functions at various points and combine the results

to determine the solution at the next timestep. Implicit solvers go further,

estimating the Jacobian of the variables and solving a set of algebraic

equations at every timestep. Implicit solvers are much more expensive

computationally, but are able to deal with problems which mix very different

time scales (referred to as stiff problems). In this instance, since we are

interested in the precise waveforms of the fibre’s behaviour, the timestep

will be much shorter than the time constants of the nodal variables. These

variables have similar time constants (to within an order of magnitude).

This problem is nonstiff, and so an explicit integrator may be used. For this

problem, a fourth-order explicit Runge-Kutta numeric integrator is chosen,

with a fixed time step. The time step chosen must be validated to ensure

that the solver’s results are converged; an overly large timestep may result

in erroneous solutions. A reference model is established with the equations

of Table 4.4. The equations are implemented in C, using double-precision

(64-bit) floating-point arithmetic. An existing ODE solver from the GNU

Scientific Library (version 1.16, [71]) is used.

4.3 Validation

The implemented models must be validated to ensure that they are

correctly implementing the desired behaviours.

Both component models have parameters that trade off computational

101

cost against accuracy. In the geometric model, the mesh refinement step has

a parameter that determines the degree of refinement; a higher refinement

results in more tetrahedra in the mesh, increasing accuracy but taking

longer to compute. The neural model has an adjustable timestep size; too

long a timestep will result in poor convergence of the ODE solver, while

shorter timesteps again take longer to compute.

Both of these can be resolved through convergence analyses, where the

parameter in question is varied to determine when the model is sufficiently

converged. This requires the determination of some quantitative measure

of the model’s results. Given the intended use of this model in determining

the behaviour of fibres under stimulation, the threshold currents required

to stimulate a nerve fibre are used as the measure for the neural timestep

convergence analysis. This also gives a convenient measure for the finite ele-

ment mesh convergence analysis, since the output of the finite element model

is to be used with the fibre model. The sensitivity of a fibre to stimulation

is a function of the imposed field and its derivatives [185], [246], rendering

this a more appropriate measure than the absolute voltages measured at

points in the model.

The neural model is quite complex. As a safeguard against gross imple-

mentation errors, it can be validated by comparing its behaviour to that

of another model. Here, a model of a human motor fibre is used as a ref-

erence [137], [138], and comparisons made of the major properties of each

model’s results.

4.3.1 Timestep Convergence

The optimal timestep is determined by constructing 10 nerve fibres of

various diameters, and measuring their thresholds at timesteps ranging

from 200 ns to 20 ms.

102

Diameter (µm)

3.0

4.5

6.0

7.5

9.0

10.4

11.9

13.4

14.9

16.4

Table 4.6: Diameters of the 10 fibres used for timestep validation.

Timestep (µs)

0.2

0.5

1.0

2.0

5.0

10.0

20.0

Table 4.7: Timestep values used for timestep validation.

The diameters are listed in Table 4.6, and the timesteps in Table 4.7.

Fibres were placed at the surface of the dorsal columns, on the midline. A

midrange cord-electrode distance of 3.25 mm was used. A biphasic stimulus

was delivered on the first three electrodes of the array: an anodic pulse on

electrode 2 was followed by a cathodic pulse, while electrodes 1 and 3 acted

as the return. A pulse width of 120 µs and an interphase gap of 200 µs were

used. The stimulus began 100 µs after the start of simulation. The timesteps

used were chosen to ensure that the start and end times of each pulse fell on

an integer number of samples. This ensures that the total charge delivered

remains identical across timesteps, allowing the results to be compared

directly. Thresholds were calculated using a binary search technique to a

precision of better than 0.1%. Whether a fibre has fired is determined by

103

examining the membrane potential of a node 10 nodes away from the centre

of the stimulating electrode; if this potential exceeds −30 mV at any time

within a 5 ms long simulation, the fibre is deemed to have fired. (The resting

potential is approximately −84 mV at equilibrium.)

10−3 10−2

Timestep (milliseconds)

100

Threshold(normalised) 3.0 µm

4.5 µm

6.0 µm

7.5 µm

9.0 µm

10.4 µm

11.9 µm

13.4 µm

14.9 µm

16.4 µm

Figure 4.4: Fibre thresholds as a function of solver timestep, relative to

the shortest timestep. Tested for a range of fibre diameters; the traces are

identical for the finer timesteps. The solver diverges to infinity for timesteps

longer than 5 µs, and no threshold is determined.

104

0 1 2 3 4 5

Time (ms)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

Voltage

(nV)

(a) 3 µm fibre

0 1 2 3 4 5

Time (ms)

−60

−50

−40

−30

−20

−10

0

10

20

30

Voltage

(nV)

(b) 16.4 µm fibre

0.2 µs

0.5 µs

1 µs

2 µs

5 µs

Figure 4.5: Fibre waveforms for the largest and smallest fibres, simulated

using different timesteps. The traces are so similar across timestep values

as to be indistinguishable.

The resulting thresholds are shown in Figure 4.4, relative to the thresh-

old at the smallest timestep. Waveforms for the smallest and largest fibres

are shown in Figure 4.5 after a stimulus of 1.1 times threshold, showing

the SFAP simulated on electrode 8 at the different timesteps. No deviation

105

in either threshold or waveform is observed up to a timestep of 5 µs. Above

that value, the model becomes numerically unstable; infinite values begin

to appear and thresholds are no longer calculated. A timestep of 1 µs is se-

lected for subsequent work, yielding a high-resolution recording and leaving

a large safety margin.

4.3.2 External Model

Having implemented a neural model from scratch, there is a risk that

some error in the implementation may cause incorrect results to be calcu-

lated. In the absence of SFAP measurements directly from the spine, the

neural model cannot be directly validated. An alternative approach is to com-

pare the results obtained from the model to those of an existing myelinated

nerve model. While this does not guarantee that the model accurately repre-

sents a dorsal column sensory fibre, it does allow for gross implementation

errors to be ruled out.

A suitable model of a myelinated motor axon was published by McIntyre,

Richardson and Grill et al. [137] (MRG model). Source code implement-

ing this model in the NEURON [41] simulation environment is available

online [138]. The MRG model implements a single myelinated axon, and

contains parameter sets allowing simulation of a range of fibre diameters.

The MRG model uses a different axon model to that described in this work

(the SCS model). The conductivity and capacitance of the myelin around the

internodes is included, and longitudinal leakage currents under the myelin

are also considered. The physical structure of the MRG model is shown in

Figure 4.6(a), and the associated electrical network in Figure 4.6(b).

The membrane model used at the nodes also differs between the two fibre

models. The SCS fibre model uses an exact implementation of the membrane

dynamics described by Schwarz et al. [203], measured from human nodes

of Ranvier. In [137], the MRG authors describe bringing together parts of

the membrane dynamic equations and constants from a number of sources.

That model does not include a fast potassium channel, and adds a persistent

sodium channel. The geometric parameters of the fibre also differ from

the SCS model; parameters for a fixed set of fibre diameters are provided,

ranging from 5.7 µm to 16 µm.

106

Figure removed from public copy

due to copyright restrictions

(a) Physical structure of MRG axon

Figure removed from public copy

due to copyright restrictions

(b) Electrical network of MRG axon

Figure 4.6: The MRG axon model, reproduced from [137]. This model in-

cludes the effects of currents flowing in the periaxonal space between myelin

and axon, as well as directly through the myelin itself.

The MRG and SCS nerve models have different aims. The MRG model

was designed to determine the effects of fibre structure on post-stimulation

afterpotentials within the fibres, and as such is quite rich in physical detail.

The SCS model is designed as a high-speed model, intended only to capture

the action potentials ensuing from a single stimulus. As such, the results

from the models are not expected to be identical. Agreement to within an

order of magnitude and with trends in the correct directions indicates that

there are no gross errors in the implementation of the SCS model.

Method

A test harness was written in Python for the MRG model. This permits

controlling the extracellular potential and recording the membrane current

at each segment of the fibre. These mechanisms can then be used to simulate

extracellular stimulation and recording in the same fashion as in the SCS

fibre model, using a calculated set of transimpedances. The adapted MRG

model’s code is reproduced in Appendix B. For each experiment, identical

axons are configured in each model, 200 nodes of Ranvier long. The fibre

diameter and internodal spacing are set to the same values in both models.

Other diameter-dependent parameters are calculated within each model.

Each experiment is repeated for each fibre diameter supported by the MRG

model.

107

The stimulation and recording electrodes are modelled as points in an

infinite homogeneous conductor of resistivity ρ = 300Ω ·cm. The electrodes

are placed at 10 mm and 50 mm along the fibre, respectively; both are at a

distance of 3 mm away from it. Transimpedances are calculated between

the electrodes and the nodes of Ranvier for the SCS fibre model. Treating

both as points leads to the transimpedance for a given distance,

Z(d)=ρ

4πd

The more complex MRG model treats the internodal region as a set of

equal-sized segments; as the myelin sheath is assumed to conduct, this

also requires the transimpedances to the exterior of the paranodal and

myelinated segments of the fibre. These are calculated using the centre point

of the relevant segments. Simulation is carried out using a 1 µs timestep for

both models.

Results

6 8 10 12 14 16

Diameter (µm)

0.5

1.0

1.5

2.0

2.5

3.0

Relativecurrent

MRG

SCS

Figure 4.7: Change in threshold current for different fibre diameters. Nor-

malised relative to a 10 µm fibre in each model. Stimulus pulse width 40 µs.

Thresholds for each fibre diameter are calculated by repeatedly simu-

lating the application of a single 40 µs cathodic pulse, using a logarithmic

108

6 8 10 12 14 16

Diameter (µm)

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

ThresholdratioSCS:M

RG

Figure 4.8: Ratio of threshold currents (SCS relative to MRG) for different

fibre diameters. Stimulus pulse width 40 µs.

binary search to determine the threshold to within 1%. These are shown in

normalised form in Figure 4.7. Both models show an inverse relationship

between simulated fibre diameter and threshold. The ratio of the results

of the two models is shown in Figure 4.8. Results for fibres greater than

9 µm in diameter agree to better than 10%, while the SCS model threshold

is nearly 25% lower for a 5.7 µm fibre.

Evoked waveforms are simulated for each fibre diameter using a single

large stimulus pulse (30 mA, 40 µs). An example waveform is shown in

Figure 4.9. The SCS model produces waveforms with lower latency, lower

amplitude, and with a pronounced and separated P2 peak at approximately

850 µs. The P2 in particular contrasts with the very broad P2 of the MRG

model at around 900 µs.

109

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−0.15

−0.10

−0.05

0.00

0.05

0.10

Voltage

(µV)

MRG

SCS

Figure 4.9: Waveform of 15 µm diameter fibres in each model.

6 8 10 12 14 16

Diameter (µm)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Relativeam

plitude

MRG

SCS

Figure 4.10: Peak-to-peak amplitude, normalised relative to a 10 µm fibre

in each model.

The change in peak-to-peak amplitude with fibre diameter is shown in

Figure 4.10, and the ratio of the two models’ results in Figure 4.11. Both

increase with the fibre diameter. The SCS model amplitudes are always

lower than those of the MRG model; this difference increases at larger

diameters.

110

6 8 10 12 14 16

Diameter (µm)

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

AmplituderatioSCS:M

RG

Figure 4.11: Ratio of peak-to-peak amplitudes (SCS relative to MRG) for

different fibre diameters.

6 8 10 12 14 16

Diameter (µm)

20

30

40

50

60

70

80

90

100

Con

ductionvelocity

(m/s)

MRG

SCS

Figure 4.12: Conduction velocity as a function of diameter.

The conduction velocities for each diameter are shown in Figure 4.12.

Both increase roughly linearly with diameter; the SCS model is consistently

around 10 m ·s−1 faster than the MRG model.

Basic metrics vary similarly between the SCS and MRG models, agreeing

within a factor of 2. The specific differences in results are due to the quite

111

different equations implemented in each model. The addition of conductive

paths through the myelin sheath in the MRG model changes the fibre’s

internal behaviour, such as the propagation of action potentials from node to

node; meanwhile, the currents through the myelin also provide an additional

contribution to the SFAP. The differences in nodal membrane equations

also contribute. The depolarisation characteristics of the membrane are

dominated by the behaviour of the sodium current; the formulation of this

current differs between the models, leading to changes in threshold and con-

duction velocity. The repolarisation is controlled by the potassium current;

here, the SCS model has an additional channel type, which may give rise

to the pronounced P2 peak as seen in Figure 4.9. This is consistent with

measurements showing that sensory fibres (as in the SCS model) can have

twice the potassium conductance of motor fibres (MRG model) [175].

The results indicate that the model implementation is free of gross errors,

and conforms to behaviours expected of large myelinated nerve fibres. The

absolute values of thresholds and SFAP voltages cannot be validated directly.

However, the variation in these values with diameter appears consistent

with previous work, and so it can be expected that an ECAP calculated from

a sum of SFAPs will also be correct, subject to some scaling in stimulus

current and recording voltage.

Running on an Intel Core i7 920 clocked at 2.66GHz, he MRG model

required an average of 40.1 seconds to simulate a 200-node fibre for 3 ms,

while the SCS model required 13.7 seconds on average for the same task.

The MRG model is more than three times slower than the SCS model. This is

due in part to its implementation in the NEURON simulation environment;

this environment provides a great deal of run-time flexibility, at the expense

of raw speed. However, neither model is sufficiently fast to directly model

all the fibres involved in SCS: even if the simulation area is restricted to the

superficial 300 µm of the dorsal columns, approximately 15,000 fibres must

be simulated [64]; more than 60 hours of CPU time would be required to

simulate the result of a single stimulation pulse. Further work is required

to enable the simulation of such large populations.

112

4.3.3 Mesh Convergence

In order to solve for the electric field in the spine, the geometric model

must be meshed – approximating the geometry with a series of polyhedra,

which allows the continuous field equations to be solved on a finite number

of elements. In order for this approximation to yield correct answers, the

mesh has to be fine enough to capture the nonuniformities of the field. The

number of elements must also remain low enough for the problem to be

computationally tractable.

In the spinal cord model, the electric field around the active electrode

is quite nonuniform. The model extends far beyond the electrode region, in

order to capture long travelling responses; at any substantial distance from

the electrode, the field is generally smooth. This means that meshing the

model with a uniform element size would be very inefficient. Adaptive mesh

refinement techniques can be used to solve this problem. An initial mesh,

generated uniformly, is used to calculate an initial approximation to the

solution. This information is used to determine which mesh elements are

subject to the highest gradients, and so delivering the poorest approxima-

tions. Some fraction of the worst elements are then refined by splitting or

remeshing, improving the resolution in those critical areas. The problem is

then solved on the new, refined mesh.

Element and mesh parameters must be correctly selected for the solution

to be accurate. The use of adaptive refinement in COMSOL dictates the

use of tetrahedral elements. An element order must also be chosen; this

specifies the order of polynomial used for the approximating basis functions.

Either first-order (linear) or second-order (quadratic) elements may be suit-

able. First-order elements are represented by 4 nodes, while second-order

elements have 10; this increases the computational complexity greatly, but

allows better representation of curved surfaces (as piecewise polynomials).

The adaptive refinement fraction must also be specified to get a suitably

accurate mesh.

A convergence study was carried out to determine the optimal element

and meshing parameters. This involves generating meshes with a range of

parameters. Solutions from each mesh are used to simulate a group of nerve

fibres. Changes in the threshold and responses of these fibres are used to

113

determine the effects of parameter choices.

Two geometric configurations were used, with the cord in the extreme

dorsal and ventral permissions the model permits - a total movement of

4 mm. For each geometry, meshes were generated using both first- and

second-order elements. An initial mesh was generated using COMSOL’s

"Fine" preset, which has parameters shown in Table 4.8. Mesh refinement

was then applied using a single iteration of the "Rough global minimum"

selection algorithm, splitting chosen elements on their longest edge. The pa-

rameter for this algorithm, the element growth rate, determines the number

of elements that the refined mesh will contain. The growth rates used, and

the number of elements in the resulting meshes, are shown in Table 4.9. An

electric field solution was calculated from a single electrode for each mesh,

as described in Section 4.2.1.

Parameter Value

Maximum element size (mm) 25

Minimum element size (mm) 3

Maximum element growth rate 1.45

Resolution of curvature 0.5

Resolution of narrow regions 0.6

Table 4.8: Parameters used to generate the initial mesh of the spinal cord

model. These settings correspond to the COMSOL "Fine" parameter set.

114

RefinementLinearNear

LinearFar

QuadraticNear

QuadraticFar

0.1 96483 96550 96538 95998

0.4 153835 164748 155784 168244

0.8 239822 253414 239085 250817

1.1 310719 337558 318265 341550

1.5 363592 414700 370267 421781

1.8 466065 497696 463971 489759

2.2 557506 603869 555269 606835

2.5 648774 728851 657146 706380

Table 4.9: Number of elements in refined meshes. The exact number of

elements depends on the geometry (cord near or far from electrode), as well

as the selected element order (linear or quadratic).

The fibre group was generated to span the full parameter space that

is to be used for later simulation work. Fibres were placed in the dorsal

column region, running parallel to the cord between both ends of the model.

Diameter D, depth y, and lateral placement x are all swept, as shown in

Table 4.10. For each (D, y, x) tuple, five fibres were generated with different

offsets in the z-axis. These offsets differ by one-fifth of the internodal spacing,

which ensures that artefacts caused by the precise location of nodes of

Ranvier with respect to mesh features can be observed. This results in a

test population of 320 fibres.

115

Parameter Minimum Maximum Unit Steps Description

d 8 18 µm 4 Axon diameter

x 0 1 mm 4 Lateral displace-

ment from the

midline

y 0 1 mm 4 Depth beneath the

surface of the dorsal

columns

Φz 0 0.8 5 Axial displacement,

scaled by the intern-

odal length of that

fibre

Table 4.10: Parameters used to generate the nerve fibres used for validation.

320 total fibres were generated, using every combination of parameters.

The threshold for each fibre is determined under each mesh. A bipha-

sic, bipolar stimulus is used, with an 80 µs pulse width. The stimulus is

delivered on the first and second electrodes, and responses are measured

differentially between the same electrodes. Binary search is used to find the

fibre thresholds. An expected threshold for each fibre is calculated from one

of the meshes, and a search is carried out around this value for each mesh to

determine the exact value. This search uses a minimum step size of 0.3125%

of the expected value. The evoked response is also calculated for each fibre

and mesh. A stimulus of 1.2 times the expected threshold is delivered to

each fibre, and the resulting electrode voltage is stored for 1.5 ms after the

end of the stimulus.

The resulting thresholds and responses are used to determine the mesh

convergence as a function of its parameters. Results from the finest mesh for

each configuration are used as a reference to which the others are compared.

This results in a single reference threshold T0, and single reference response

waveform R0(t), for each (D, x, y,Φz) class. Each individual fibre under each

mesh is then compared to the reference values. The variation in threshold

for a fibre with threshold T is described by the threshold deviation

ET =

�

�

�

�

T

T0−1

�

�

�

�

116

and the deviation of a response R(t) is calculated from the RMS value of

the response difference as

ER =RMS(R−R0)

RMS(R0)

Results

0.0 0.5 1.0 1.5 2.0 2.5

Mesh refinement parameter

0.0

0.5

1.0

1.5

2.0

2.5

Devia

tion

(perc

en

t)

(a) Linear elements, large cord dis-

tance

0.0 0.5 1.0 1.5 2.0 2.5

Mesh refinement parameter

0.0

0.5

1.0

1.5

2.0

2.5

Devia

tion

(perc

en

t)

(b) Linear elements, small cord dis-

tance

0.0 0.5 1.0 1.5 2.0 2.5

Mesh refinement parameter

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Devia

tion

(perc

en

t)

(c) Quadratic elements, large cord

distance

0.0 0.5 1.0 1.5 2.0 2.5

Mesh refinement parameter

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Devia

tion

(perc

en

t)

(d) Quadratic elements, small cord

distance

Response

Threshold

Figure 4.13: Results of the convergence study. Mean deviation of threshold

and response values from those obtained with refinement parameter 2.5.

Each data point is the average deviation for all fibre types and placements.

Results of the convergence study are shown in Figure 4.13, showing

the deviations measured from the finest meshes for each scenario. The

deviations all decrease as the mesh is refined more; when the results stop

changing with the mesh, mesh convergence has been achieved, and the

sensitivity to further refinement is minimal.

The linear elements can be seen to converge more slowly than the

quadratic. In Figure 4.13(a), it is not clear that the results have in fact

117

converged, even looking at relatively fine meshes. The quadratic elements

show clearer convergence; less than 0.5% average deviation is obtained at a

refinement parameter of 1.2, whether the cord is close or far away.

Quadratic elements, with a refinement parameter of 1.2, are selected as

the default for further modelling work.

4.4 Conclusions

A model has been presented which determines the evoked response

expected from a single fibre by combining geometric and neural models.

This architecture is taken from previous SCS models, which consider fibre

recruitment thresholds as the model output [51], [96], [122]. The approach is

extended to calculate evoked action potentials by computing the time course

of membrane currents in the neural calculation, and using the geometric

model to determine the potential at the recording electrodes.

Validation of the solver and mesh discretisations show that both the

neural and geometric models are converged, with good safety margins. Com-

parison of the neural model results to those obtained from a published

neural model of motor fibres [137] indicate that no gross errors are present

in the neural model implementation.

This model balances cost and accuracy in order to support the simulation

of high fibre counts in SCS. The neural component of the model is based

on measurements of the membrane behaviour of human dorsal column fi-

bres, and uses a minimal model of fibre geometry with few parameters. The

model treats the nodes of Ranvier and internodal segments as cylindrical

conductors, and the myelin as a perfect insulator. Some more complex mod-

els indicate that conduction through and around the myelin plays a role in

neural activity, particularly in afterpotentials present for some time after

firing [25], [137], [219]. These can cause threshold changes with repeated

stimulation; changes of up to 20% are observed in [137] up to 100Hz stimula-

tion rate. As SCS is typically delivered using continuous stimulation, this is

likely to manifest as a steady-state difference in threshold; 20% is well below

inter-patient variation levels. Potassium accumulation under the myelin is

also implicated in conduction block with high-frequency stimulation (above

118

300Hz) [19], [34]; this is well above the range used in SCS therapy, and is

not presently of interest.

The structure of the SCS model allows expensive computations to be per-

formed up-front. The finite element solver need only be run once for a given

geometric and electrode configuration. Calculating the transimpedances for

the nodes of a particular fibre is then relatively fast, allowing large num-

bers of fibres to be created. Finally, the simulation of a fibre response to a

stimulus requires only the precalculated transimpedances and the neural

model; determining the response to different stimuli is extremely fast. This

approach is similar to that used in recent models of ECAPs in deep brain

stimulation [109] and cochlear implants [47].

While the model was designed for performance, there are many tens of

thousands of fibres in the dorsal columns, and measurements of characteris-

tics such as the ECAP growth curve require delivery of a range of stimulus

amplitudes. Even with the neural model implemented in C, computing the

response of 100,000 200-node fibres to a single stimulus would take 380

hours with the computer used for validation. More advanced techniques are

required to simulate a population of this scale.

119

CH

AP

TE

R

5E C A P M O D E L

The evoked compound action potential results from the simultaneous

response of many nerve fibres to a single stimulus. Each responding fibre

contributes to the ECAP waveform; when fibres do not interact significantly,

such as in the fibres of passage in the dorsal columns, the ECAP can be

obtained by summing the SFAP generated by each individual fibre after

stimulation.

This chapter describes an ECAP model using summation of SFAPs to

model the response of the dorsal columns. A fibre population is established

using histological measurements of the distribution of fibre properties in

human dorsal column sections [64]. According to this distribution, tens of

thousands of fibres are present in the dorsal columns. Simulating individual

fibres on this scale using the SFAP model described in the previous chapter

is impractical due to the speed of that mode. Two methods are introduced for

reducing the time required to simulate dorsal column ECAPs: the fibre model

speed is increased, and the total number of fibres to be simulated is reduced.

A speed increase is obtained by leveraging the high numeric performance of

modern graphics processing units (GPUs); this technique is implemented

and validated against the CPU-based model. The simulation count is reduced

by grouping similar fibres together for the purposes of simulation. This

involves dividing fibres up by their parameters, and simulating a single

120

representative fibre for each resulting parameter cell. The resulting cell

SFAPs are then scaled according to the number of fibres expected within

that cell. Where there are many similar fibres, as in the dorsal column

distribution, this approximation provides a significant speed increase. These

techniques are combined to allow simulating wide ranges of geometric and

stimulation parameters in a matter of days.

Simulated ECAPs are presented; their characteristics are compared with

those recorded in a human subject. The simulated ECAPs explain several

features of the human ECAPs, and suggests that therapeutic SCS operates

within a specific region of recruitment.

5.1 Computational Cost

Prior studies in SCS have focussed largely on the behaviour of individual

fibres, and have not attempted the simulation of tens of thousands of fibres.

The pioneering work by the UT-SCS team made the assumption that the

perceptual threshold of the patient is that of the very largest fibres of the

DCs [95]. Patients have a comfort threshold on the order of 1.4 times their

perceptual threshold [103], so therapeutic stimulation is delivered from 1 to

1.4 times the perceptual threshold. The threshold for a given nerve fibre is

roughly inversely proportional to diameter, leading to the conclusion that

fibres smaller than 16.4/1.4 or 11.7µm should not be recruited. This led the

UT-SCS team to omit small fibres from consideration. Since the smallest

fibres are by far the most numerous, this greatly reduces the number of

fibres to be simulated; only 1-2% of fibres are expected to be larger than

10.7µm in diameter [64]. Therapeutic stimulation was hypothesised to be

the result of as little as a single fibre [95]. This assumption and its resulting

conclusions thread their way through subsequent SCS modelling work. More

recent efforts even restrict themselves entirely to a single fibre diameter [97],

[108], [152]. However, work by Lee et al. [123] shows that a broader mix of

fibre diameters provides an explanation for the perceptual effects of varying

the stimulus pulse width.

Simulations of single nerve fibres in the SCS SFAP model show that the

SFAP amplitude of a single nerve fibre is in the hundreds of nanovolts at

121

most, compared with observed ECAPs of tens to hundreds of microvolts in

the therapeutic range. ECAP recordings and the results in [123] suggest

that the large-fibre assumption is best avoided; all types of fibres found in

the DCs must be simulated to determine their relevance to ECAPs, requiring

that another method be found to render the problem tractable.

5.1.1 GPU Acceleration

In recent years, a new computational technology has become available

for general use: GPU computing. This involves offloading calculations on

to the highly parallel hardware built into modern computer graphics cards

(graphics processing units, GPUs). GPUs differ from the central processing

units (CPUs) that power computers in that they are designed to efficiently

apply identical operations to many data units concurrently. This architecture

results in very poor performance for general computing tasks where small

amounts of data are processed at a time, but can vastly outperform CPUs

on highly parallelisable tasks. Solving the equations that govern nerve fibre

behaviour is a problem that would seem eminently suited to GPU-based

approaches: the same ODEs are being solved at every node on a fibre for

every timestep. Ideally, each node would be allocated to a separate processing

unit on a GPU, permitting their results to be calculated concurrently.

Constraints

For the purposes of this analysis, an NVIDIA GTX 570 (NVIDIA, Santa

Clara, USA) is used. This card, released at the end of 2010, contains 15

processing units known as Streaming Multiprocessors (SMs). Each SM

contains 32 Streaming Processors (SPs), each of which can perform up to

two floating-point operations per clock cycle. This results in a theoretical

peak computational performance of 1400 GFLOPS (billions of floating point

operations per second). By comparison, the Intel CPU on which the reference

model is run is theoretically capable of 43 double-precision GFLOPS at best,

some 33 times slower.

GPU processing is not without its drawbacks. A number of the archi-

tectural features of GPUs place stringent requirements on the structure

and execution behaviour of code in order to ensure performance, and the

122

architectures vary between manufacturers as well as between generations

of hardware.

GPUs are designed for graphics tasks, where numeric precision is not

critical. Consequently, they are designed primarily for dealing with single-

precision (32-bit) floating-point data. Some GPUs are incapable of dealing

with double-precision at all, while others can, but with a severe performance

penalty. Any GPU implementation of the nerve fibre model would need to

be able to use single-precision floating-point maths in order to achieve good

performance.

The NVIDIA GPU computing architecture describes parallel computa-

tion in terms of threads. A processing job ("block") is composed of multiple

threads which execute together; these are subdivided into smaller groups

("warps") of threads, which are fed with the same instruction stream. The

execution flow of different threads in a warp need not be identical, but any

divergent branches will be executed serially. For example, if only one the

threads in a warp enters a conditional sequence of code (branches), the re-

maining threads will remain idle until the conditional sequence is complete.

This constitutes a substantial penalty for divergent execution, potentially

causing each thread to execute in sequential order and remove any advan-

tage of parallelism. Ideally, all of the threads within a warp will always take

identical branches.

GPUs have a complex memory hierarchy. A small amount of memory is

present on the GPU chips themselves and can be accessed extremely fast,

but the main memory provided is off-chip and has access latencies in the

hundreds of cycles. Depending on the available resources, the GPU may be

able to continue with other processing while waiting for memory requests,

or it may stall while waiting for data. Main memory must be used sparingly

or not at all if peak performance is desired. There are also special require-

ments on memory access ordering. Under some circumstances, simultaneous

requests by multiple threads can be coalesced into a single bus transaction.

If the coalescing conditions are not satisfied, then the threads must issue

individual bus transactions in a serial fashion, multiplying the overall la-

tency in a similar fashion to execution divergence. The best performance

is obtained when individual threads access linearly contiguous values in

memory.

123

This suggests a natural mapping from the nerve fibre model to a GPU

implementation: using one thread per node of Ranvier. Since the compu-

tations are identical at each node, this permits the code to be structured

to minimise divergent branches. The electrical network of the nerve fibre

also requires that neighbouring nodes communicate, but no information is

shared between more distant nodes; these shared accesses can be mapped

to a single contiguous array, resulting in efficient memory access coalescing.

The NVIDIA architecture limits blocks to 1024 threads. This limits the

length of a simulated fibre to 1024 nodes of Ranvier.

Adaptation

In order to achieve good performance, it is necessary to modify the model

code to use single-precision floating point numbers. This requires replacing

the GSL library’s double-precision ODE solver with a direct implementation

of a fourth-order explicit Runge-Kutta method [118], as used in the reference

model. The remainder of the code remains unchanged. Compiler options for

fast maths are also enabled (NVIDIA CUDA’s –use_fast_math). This has

the effect of reducing accuracy in the edge cases of special functions such as

exp, but improving the computation speed.

Reducing the numeric precision also leads to potential numeric insta-

bility problems. The model equations must be slightly modified in order to

avoid issues when dividing by extremely small numbers. This is addressed

by calculating an approximation of the desired function in regions where

instability is likely; outside this region, the original function is calculated.

The problematic expression is of the form:

efun(z)=z

ez −1(5.1)

This expression appears in two areas in the model. The gating equations

for αm, αn and αs are computed in the form:

αx(E)= Aexpmod(B−E,C) (5.2)

where A,B,C are the constants associated with that particular equation,

and the expmod function is defined as:

expmod(x, y)= yefun(x/y) (5.3)

124

The second usage is in the sodium conductance term in the membrane

current equations:

gNa = PNaEF2

RT

[Na]o − [Na]i exp(EF/RT)

1−exp(EF/RT)(5.4)

In the reference model, this is calculated using the following, assuming E is

in millivolts:

z = 10−3 EF

RT(5.5)

gNa = 10−3PNaF([Na]iefun(−z)− [Na]oefun(z)) (5.6)

When |z| is very small, both the numerator and denominator become

small. These should cancel; this expression approaches 1 as z approaches

0. However, floating-point numbers are quantised, and the numerator and

denominator do not cancel well. This is shown in Figure 5.1. When calculated

with 64-bit floating-point arithmetic, no problem is visible in this numeric

range. However, when using 32-bit floating point, the quantisation error

results in large aberrations from the desired value, and below |z| = 1 ·10−8,

the result is infinite (ez −1= 0).

125

−0.0004−0.0002 0.0000 0.0002 0.0004

z

0.9994

0.9996

0.9998

1.0000

1.0002

1.0004

1.0006

efun(z)

(a) 64-bit floating point

−0.0004−0.0002 0.0000 0.0002 0.0004

z

0.990

0.995

1.000

1.005

1.010

efun(z)

(b) 32-bit floating point

Figure 5.1: The function efun(z)= zexp(z)−1

, calculated using both 64-bit and

32-bit floating-point arithmetic. The 32-bit calculation becomes increasingly

unstable near z = 0.

126

This expression can be replaced with some terms of its Taylor series over

the unstable region:

efun(z)=

1− z2

|z| < 10−3

zez−1

|z|≥ 10−3(5.7)

The accuracy of the GPU model, including this substitution, can then be

assessed by comparing its behaviour to that of the reference model.

Performance

The resulting model’s behaviour is then compared to the reference model

in order to determine the accuracy of the adapted code. The timestep anal-

ysis of subsection 4.3.1 is repeated, in which the thresholds of 10 fibres

are calculated. Figure 5.2 shows these thresholds, normalised against the

threshold calculated using the reference model.

10−3 10−2

Timestep (milliseconds)

0.990

0.995

1.000

1.005

1.010

Threshold(normalised) 3.0 µm

4.5 µm

6.0 µm

7.5 µm

9.0 µm

10.4 µm

11.9 µm

13.4 µm

14.9 µm

16.4 µm

Figure 5.2: Fibre thresholds calculated with the GPU model for various val-

ues of solver timestep. Normalised relative to the reference model’s thresh-

olds.

127

0 1 2 3 4 5

Time (ms)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0Voltage

(nV)

(a) 3 µm fibre

0 1 2 3 4 5

Time (ms)

−60

−50

−40

−30

−20

−10

0

10

20

30

Voltage

(nV)

(b) 16.4 µm fibre

0.2 µs

0.5 µs

1 µs

2 µs

Figure 5.3: Fibre waveforms for the largest and smallest fibres, simulated

using different timesteps in the GPU model. The traces are so similar across

timestep values as to be indistinguishable.

128

0 1 2 3 4 5

Time (ms)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

Voltage

(nV)

(a) 3 µm fibre

0 1 2 3 4 5

Time (ms)

−60

−50

−40

−30

−20

−10

0

10

20

30

Voltage

(nV)

(b) 16.4 µm fibre

Reference

GPU

Figure 5.4: Comparison of the waveforms of the largest and smallest fibres

computed with the reference and GPU models. The traces are so similar

between the models as to be indistinguishable.

129

0 1 2 3 4 5

Time (ms)

−0.00025

−0.00020

−0.00015

−0.00010

−0.00005

0.00000

0.00005

0.00010

0.00015

Residual

error(nV)

3.0 µm

4.5 µm

6.0 µm

7.5 µm

9.0 µm

10.4 µm

11.9 µm

13.4 µm

14.9 µm

16.4 µm

Figure 5.5: Residual difference between GPU and reference models for a

range of diameters. The reference model SFAPs are subtracted from the

GPU model SFAPs to obtain the residual.

130

SFAP waveforms simulated with the GPU model are shown in Figure 5.3.

These are compared with the reference model’s waveforms in Figure 5.4, and

the residual error of the floating-point approximation is shown in Figure 5.5.

The speed of the model is assessed by computing the response of 1000

fibres to a single stimulus. The test fibre set consists of 10 fibres with diam-

eters logarithmically distributed between 3 and 16.4 µm, each duplicated

100 times. Each fibre is 240mm long. The logarithmic distribution roughly

approximates the distribution of fibres in the dorsal columns, where smaller

fibres dominate the population [64]. These fibres are each simulated for 5

milliseconds using both the reference and GPU models, with a timestep of 1

microsecond. The reference model is run on an Intel Core i7 920 clocked at

2.66GHz, and running Linux 3.19.3. The machine has 48GB of RAM and

performs no other work during calculation. Eight parallel worker threads

perform the computations for the reference model, in order to take maximal

advantage of the multiple cores of this CPU. The GPU model is run in the

same machine, using a single GTX 570. NVIDIA driver version 346.59 is

in use, and the GPU code is compiled with CUDA compiler version 7.0.27.

The simulation process is repeated three times with each model, and the

resulting run times are tabulated in Table 5.1.

Run Reference (s) GPU (s)

1 5436.2 38.9

2 5472.7 38.9

3 5412.5 38.9

Table 5.1: Computation time of reference and GPU models, in seconds,

for 3 runs. Each run involves computing the SFAPs of 1000 fibres over 5

milliseconds, on each of 5 recording electrodes.

Conclusions

The reference model has been successfully ported to an NVIDIA GPU

architecture. The resulting model’s behaviour results in thresholds, shown

in Figure 5.2, that match the reference to better than 0.01%, the resolution

of the binary search which was used to determine them. The resulting

waveforms also appear visually identical; the residual difference between

131

the two models’ SFAP waveforms is more than 105 times smaller (peak to

peak) than the waveforms themselves. This indicates that the conversion

from double-precision to single-precision floating-point arithmetic does not

affect the simulation results. The GPU model fails to converge for timesteps

greater than 2 microseconds; this differs from the reference model’s stability

at the 5 microsecond timestep. The 1-microsecond timestep still suffices for

stable computation with a substantial safety margin.

The computation speed is increased greatly by using the GPU architec-

ture: for the 1000-fibre test, the GPU performed 140 times faster than the

reference model. This improvement can be used to increase the number of

fibres simulated. The computational cost is determined largely by the total

number of nodes of Ranvier and duration of the simulation; approximately

300,000 node-milliseconds can be simulated per minute using a single GPU.

Measurement of the superficial 300 µm of the dorsal columns indicate

that on the order of 18,000 fibres may be found in this region alone, and

that small fibres are more prevalent than a logarithmic distribution would

indicate. This suggests that on the order of 1 hour would be required to

simulate a 5-millisecond response to a depth of 1 millimetre. This is tractable

for small numbers of responses, but performing detailed current sweeps

under various geometric configurations requires many response calculations.

A further reduction in computation time is required to explore wide domains.

5.1.2 Parametric Grid

The tractability of the problem can be improved by making an approxi-

mation, using the fact that fibres which are similar in position and diameter

will have similar thresholds and SFAPs. Simulations of even the largest

fibres, when situated on the very surface of the cord, have SFAPs with

amplitudes less than 0.1µV. Such fibres are also very rare, numbering in

the tens according to histological measurements [64]; smaller fibres, with

smaller SFAP amplitudes, dominate the population of the dorsal columns.

This means that the incremental contribution of each fibre to a visible (tens

of microvolts) ECAP is negligible. The transimpedance also varies smoothly

within the dorsal columns, meaning that fibres which are nearby in the

physical space experience very similar fields.

132

This suggests an approach that groups similar fibres together. This can

be done by dividing the parameter space into a regular grid of cells. One

fibre is simulated for the parameter values at the centre of each cell; its

SFAP can then be scaled in amplitude according to the number of fibres

whose parameters fall inside that cell. This approach lends itself well to

situations such as SCS where a continuous fibre distribution is present;

this can be integrated over each cell to determine the appropriate scaling

factor. Similar approaches have been used previously to simulate compound

action potentials, both in the periphery [217], [253] and for the auditory

nerve [150].

Each fibre is parametrised by three values: the lateral position, depth,

and diameter. The lateral position is specified as the fibre’s distance from

the midline of the cord; the depth is measured from the surface of the dorsal

columns. Fibres are assumed to run along the axis of the cord, with constant

diameter and position.

A scaling factor must then be calculated for each grid cell to determine

the effective number of fibres within it. The best information available

on fibre distribution is given in [64], which has detailed information on

the relative prevalence of fibres of different diameters. Some variation in

the distribution of fibres laterally is also given, but without much detail;

variation with depth is not described at all. As a result of previous SCS

modelling efforts, it is believed that fibres are only activated to a depth of

0.3mm in SCS; as a result, [64] examines only fibres extending to that

depth. As a result of the available data, the model described here treats the

fibres as uniformly distributed in space both laterally and below the surface

of the dorsal columns. A maximum depth of 1mm is chosen as a compromise

between computational cost and accuracy.

5.1.3 Distribution

The diameter distribution is extracted from [64], which covers an area

of the dorsal columns as shown in Figure 5.6. The distribution there is pre-

sented in terms of the axon cross-section in µm2. The figures corresponding

to the male subject are extracted and used to derive a cumulative distribu-

tion function CDF(a), which calculates the density of fibres smaller than a

133

Figure removed from public copy

due to copyright restrictions

Figure 5.6: Area of the cord surveyed by Feirabend et al., reproduced from

[64]. The dorsal columns (DC) and dorsolateral columns (DLC) were sepa-

rately surveyed to a depth of 300 µm below their dorsal surface.

given cross-sectional area a; this is in units of fibres per µm2. The CDF is

used to calculate the density of fibres in each grid cell. The source data are

shown in Figure 5.7, along with the CDF obtained by cubic interpolation.

0 50 100 150 200

Fiber cross-section (µm2)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

Density

(µm

−2)

Measured

Interpolated

Figure 5.7: Cumulative distribution of fibres by cross-sectional area. Source

data from [64] are interpolated using PCHIP (Piecewise Cubic Hermite

Interpolating Polynomial) cubic interpolation.

134

0 50 100 150 200

Fiber cross-section (µm2)

10−8

10−7

10−6

10−5

10−4

10−3

Density

(µm

−4)

Figure 5.8: Fibre distribution probability distribution function (PDF) calcu-

lated from the interpolated CDF.

135

Figure removed from public copy

due to copyright restrictions

Figure 5.9: The source data reproduced from [64], with the interpolated

fibre diameter CDF overlaid in blue. The DC0-300 data are obtained from

histological study of dorsal column fibres to a depth of 300 µm below the

surface of the cord. The male subject’s data are used for the ECAP model.

The DLC0-300 data are measured from the dorsolateral columns and are not

used here.

a) the overall distribution of the counted fibres by fibre cross-sectional area.

b), c), d) detailed breakdown of the counted fibres of larger cross-sectional

areas.

Common polynomial interpolation algorithms can introduce non-monotonic

artefacts, as they optimise for a smooth interpolated result. Here, the PCHIP

(Piecewise Cubic Hermite Interpolating Polynomial) algorithm is used for

interpolation, as it is designed to preserve monotonicity. This is visible in

the calculated PDF in Figure 5.8. Densities are calculated from the source

data based on a 2700×300 µm survey area. The resulting CDF is used to

reconstruct the original figures from which the data were measured, and

overlaid for comparison; this is shown in Figure 5.9.

136

5.1.4 Scaling

For a grid cell extending from diameter D i to D i+1, lateral displacement

x j to x j+1, and depth yk to yk+1, the effective number of fibres inside it is

calculated as:

Ni jk = (CDF(D i+1)−CDF(D i))(x j+1 − x j)(yk+1 − yk) (5.8)

A representative SFAP is calculated for a fibre placed at the centre of

each grid cell, which is to say with

D = (D i +D i+1)/2 (5.9)

x = (x j + x j+1)/2 (5.10)

y= (yk + yk+1)/2 (5.11)

(5.12)

The resulting SFAPi jk are then scaled and summed to derive the ECAP:

ECAP=�

i, j,k

Ni jkSF APi jk (5.13)

5.2 Results

The ECAP model was used to calculate the response of the dorsal columns

to stimulation at various currents. Therapeutic stimulation typically uses

biphasic stimulation, which can cause fibres to be recruited at two different

times. This can complicate the resulting ECAPs. Results are first presented

here for monophasic stimulation, which avoids these issues; biphasic results

follow. The biphasic results are then compared with recordings taken from a

human patient under biphasic stimulation.

Simulations were carried out on a population of axial fibres parametrised

by their lateral position, depth within the dorsal column, and fibre diameter.

Fibre positions were considered uniformly distributed in the dorsal columns

to a depth of 1 mm from the dorsal surface and extending 2.5 mm laterally

on each side of the midline. Fibre diameters from 2.9 µm to 17 µm were

considered. The model extends for 300 mm along the cord axis. This area

contains approximately 61,000 fibres according to the distribution function.

137

The 3-dimensional parameter grid was subdivided into cells. The lateral

position and depth axes were divided into twelve and nineteen equal-sized

intervals, respectively, while the diameter axis was divided into twenty-

nine logarithmically sized intervals. A representative fibre was modelled for

each cell, with parameters according to the centre of that cell. Each cell was

assigned a scaling factor, which is the number of fibres expected to fall within

its bounds according to the fibre distribution. Scaling factors were calculated

using the diameter distribution established in Section 5.1.3 and the scaling

method of Section 5.1.4. This resulted in a set of 3306 representative fibres.

Each fibre’s first node of Ranvier was aligned with one end of the model, so

the nodes of different diameters do not align in the central region where the

electrode array is placed, to avoid any interference issues.

The geometric model was then used to calculate the transimpedance

values between the electrodes and the nodes of Ranvier of the simulated

fibres for a cord position in the centre of the model’s range, with a distance of

3.2 mm between the electrode array’s ventral surface and the dorsal column

surface; this is in the middle of the model’s movement range.

Properties of the fibre distribution are shown in Figure 5.10. The peak-

to-peak amplitude of a given fibre’s SFAP varies with the fibre diameter,

while the number of fibres decreases with increasing diameter despite the

increasing cell width. The amplitude of the fibre nearest the electrode (on

the surface at the midline) is shown in Figure 5.10(a). The number of fibres

in cells of different diameters is shown in Figure 5.10(b); the diameter

distribution is constant throughout the dorsal columns. The maximum

expected contribution of all fibres of a diameter is shown in Figure 5.10(c),

suggesting that fibres from 6 to 10 µm in diameter would be expected to

dominate the response if complete recruitment were achieved.

138

2 4 6 8 10 12 14 16 18

Diameter (µm)

0

10

20

30

40

50

60

Amplitude(nV)

(a) Amplitude of individual SFAPs

2 4 6 8 10 12 14 16 18

Diameter (µm)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Fibers

(b) Number of fibres

2 4 6 8 10 12 14 16 18

Diameter (µm)

0

10

20

30

40

50

ScaledAmplitude(µV)

(c) Scaled contribution

Figure 5.10: Properties of fibres in each diameter cell. a) N1-P2 amplitude

of the SFAP for the fibre of each diameter nearest to the electrode array. b)

Number of fibres of each diameter across all depths and lateral displace-

ments. c) Product of amplitude with number of fibres, indicating a maximum

contribution to be expected from each diameter class.

139

5.2.1 Monophasic Stimulation

SFAPs were calculated for each representative fibre at a range of currents.

A tripolar stimulus was applied with a cathodic polarity on electrode 2 and

returning through 1 and 3. A pulse width of 120 µs was used. Currents

ranging from 0 to 30 mA in steps of 50 µA were used. Each representative

SFAP was multiplied by the scaling factor of its associated parameter cell,

and the scaled SFAPs summed to obtain the ECAP for each current.

The waveforms simulated at a range of currents are shown in Figure

5.11.

140

−100

−50

0

50

Voltage

(µV)

2.0 mA 6.0 mA

−100

−50

0

50

Voltage

(µV)

10.0 mA 14.0 mA

−100

−50

0

50

Voltage

(µV)

18.0 mA 22.0 mA

0.5 1.0 1.5 2.0 2.5

Time (ms)

−100

−50

0

50

Voltage

(µV)

26.0 mA

0.5 1.0 1.5 2.0 2.5

Time (ms)

30.0 mA

E4

E5

E6

E7

E8

Figure 5.11: ECAP waveforms on each electrode at a range of currents with

monophasic stimulation.

The model produces triphasic waveforms similar to SFAPs generated

with the single-fibre model. Growth of the ECAP with current was measured

using N1 and P2 peaks. For the purposes of analysis, the global minimum

of the waveform was taken as the N1 peak, while the maximum of the

waveform from N1 to the end of recording was taken as P2.

141

0 5 10 15 20 25 30

Current (mA)

0

50

100

150

200

250

300

N1-P2Voltage

(µV)

(a) Full sweep

1 2 3 4 5 6 7 8 9 10

Current (mA)

0

20

40

60

80

100

120

N1-P2Voltage

(µV)

(b) Detail

E4

E5

E6

E7

E8

Figure 5.12: Growth of N1-to-P2 peak-to-peak amplitude on each electrode

as current is increased. Monophasic stimulation. Detailed view shows the

region up to several times threshold.

The N1-P2 amplitude growth is shown in Figure 5.12. No response

is observed at currents below 3 mA. Above this current, the slope of the

curve increases for some time before stabilising, forming a foot and then a

relatively linear region. At currents above 15 mA, the growth rate reduces

142

as the available fibres are saturated.

0 5 10 15 20 25 30

Current (mA)

0.2

0.4

0.6

0.8

1.0

1.2

N1Latency

(ms)

(a) N1 (full)

0 5 10 15 20 25 30

Current (mA)

0.5

1.0

1.5

2.0

2.5

P2Latency

(ms)

(b) P2 (full)

1 2 3 4 5 6

Current (mA)

0.6

0.8

1.0

1.2

N1Latency

(ms)

(c) N1 (detail)

1 2 3 4 5 6 7 8 9 10

Current (mA)

1.0

1.2

1.4

1.6

1.8P2Latency

(ms)

(d) P2 (detail)

E4

E5

E6

E7

E8

Figure 5.13: Latencies of N1 and P2 peaks simulated on each electrode

with monophasic stimulation. Shaded area shows stimulus period, ending

at 0.22 ms. Detailed views show the region up to several times threshold.

The change in latency of the peaks is shown in Figure 5.13. A rapid

increase in latency is observed at low currents, in the region corresponding

to the foot of the growth curve. The N1 latency begins to decrease slightly in

the saturation region, while the P2 latency continues to increase throughout

the measured current range.

143

0 5 10 15 20 25 30

Current (mA)

45

50

55

60

65

70

75

N1velocity

(m/s)

Figure 5.14: Conduction velocity calculated from the N1 latency between

electrodes 6 and 8, a distance of 14 mm. Monophasic stimulation.

A group conduction velocity value can be estimated from the differences

in the N1 peak latency across electrodes, shown in Figure 5.14. The velocity

falls rapidly in the foot region of the growth curve, changing relatively little

at higher currents.

144

0 5 10 15 20 25 30

Current (mA)

0

10000

20000

30000

40000

50000

60000

Fibers

Figure 5.15: Number of fibres recruited at each stimulus current with

monophasic stimulation.

0 5 10 15 20 25 30

Current (mA)

4

6

8

10

12

14

16

MeanDiameter

(µm)

Figure 5.16: Mean diameter of recruited fibres with monophasic stimulation.

145

The total population of recruited fibres are calculated using the cell

scaling factors: each representative fibre which was recruited on a given

stimulus contributes according to the number of fibres its cell represents.

The number of recruited fibres as a function of current is shown in Fig-

ure 5.15, and the mean diameter in Figure 5.16. The recruitment curve

shows a foot, a linear region, and a slow saturation as current is increased.

The mean diameter of recruited fibres drops rapidly in the foot region, and

slowly thereafter.

146

2 4 10 30

Current (mA)

0.0

0.2

0.4

0.6

0.8

1.0

Max

imum

Depth

(mm)

(a) Maximum Ventral Spread

2 4 10 30

Current (mA)

0.0

0.5

1.0

1.5

2.0

2.5

Max

imum

Lateral

Pos

(mm)

(b) Maximum Lateral Spread

2 4 10 30

Current (mA)

0

1

2

3

4

5

RecruitmentArea(m

m2)

(c) Recruited Area

3.0 µm

3.2 µm

3.4 µm

3.6 µm

3.8 µm

4.1 µm

4.3 µm

4.6 µm

4.9 µm

5.2 µm

5.5 µm

5.9 µm

6.2 µm

6.6 µm

7.0 µm

7.5 µm

7.9 µm

8.4 µm

9.0 µm

9.5 µm

10.1 µm

10.8 µm

11.4 µm

12.2 µm

12.9 µm

13.7 µm

14.6 µm

15.5 µm

Figure 5.17: Spread of recruitment with current for each fibre diameter cell

in the model. Monophasic stimulation. a) Depth of the deepest recruited

cell of each diameter. b) Position of the most lateral recruited cell of each

diameter. c) Total area of recruited cells of each diameter.

147

2 4 6 8 10 12 14 16

Diameter (µm)

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Ratio

Ventral

Lateral

Figure 5.18: Ratio of the currents at which fibres at the most ventral and

most lateral extents of the model region are first recruited, compared to the

thresholds for the fibres at the surface midline cell (closest to the electrodes).

Monophasic stimulation.

148

Detailed recruitment changes with current are shown in Figure 5.17.

Recruitment proceeds from largest fibres to smallest for a given position,

and from medial fibres on the surface to more lateral and ventral fibres. As

the current is increased, the lateral extent of recruitment increases rapidly,

while the ventral extent (depth) increases more slowly. Figure 5.18 shows

the relationship between the ventral and lateral extent of recruitment: the

most lateral fibres begin to be recruited at around 1.6 times the threshold

of those nearest the electrode, while the deepest fibres do not begin to be

activated until more than 2.2 times the nearest fibres’ threshold.

5.2.2 Biphasic Stimulation

The simulation process was repeated using biphasic stimuli over the

same current range. Stimulation polarity was anode-first on electrode 2

returning through electrodes 1 and 3, with a stimulus pulse-width of 120 µs

and an inter-phase gap of 200 µs. Example waveforms simulated with bipha-

sic stimulation at a range of currents are shown in Figure 5.19.

149

−100

−50

0

50

Voltage

(µV)

2.0 mA 6.0 mA

−100

−50

0

50

Voltage

(µV)

10.0 mA 14.0 mA

−100

−50

0

50

Voltage

(µV)

18.0 mA 22.0 mA

1.0 1.5 2.0 2.5

Time (ms)

−100

−50

0

50

Voltage

(µV)

26.0 mA

1.0 1.5 2.0 2.5

Time (ms)

30.0 mA

E4

E5

E6

E7

E8

Figure 5.19: ECAP waveforms on each electrode at a range of currents with

biphasic stimulation.

At low currents, the model produces waveforms similar to those obtained

with monophasic stimulation. The waveform changes as the current in-

creases. At first, the latency of the ECAP peaks appears to increase with

current; then, at higher currents, a second set of peaks begins to appear at

an earlier time. As the current is increased further, the early peaks grow,

while the later peaks diminish. The early peaks also appear to undergo

150

latency increases with increasing current.

The shift from a later to an earlier time can be explained by the second-

cathode effect, where the first phase of biphasic stimulation causes recruit-

ment near the return electrodes. This results in an action potential volley

that starts earlier in time as well as nearer to the recording electrodes,

adding an earlier ECAP to the signal. This also reduces the ECAP evoked

by the second phase of stimulation, as any fibres stimulated in the first

phase will still be in their refractory period; this causes the shrinkage and

eventual disappearance of the original, later ECAP. This causes discontinu-

ous jumps in the estimated amplitudes and latencies as the second-cathode

effect becomes dominant. The N1 value on the nearest electrode is no longer

measured once it moves past the beginning of the recording and into the

stimulus phase.

151

0 5 10 15 20 25 30

Current (mA)

0

50

100

150

200

N1-P2Voltage

(µV)

(a) Full sweep

1 2 3 4 5 6 7 8 9 10

Current (mA)

0

20

40

60

80

100

120

N1-P2Voltage

(µV)

(b) Detail

E4

E5

E6

E7

E8

Figure 5.20: Growth of N1-to-P2 peak-to-peak amplitude on each electrode

as current is increased. Biphasic stimulation. Detailed view shows the region

up to several times threshold.

The N1-P2 amplitude growth is shown in Figure 5.20. At currents above

10 mA, the peak amplitude reduces, forming a trough at 15mA and then

increasing again. This is due to the second-cathode effect interacting with

the peak-detection measurement of amplitude; the trough represents the

152

point where the early N1 peak is equal to the late N1 peak. On the electrodes

nearest the stimulus, the N1 peak begins to overlap the stimulus at high

currents, and peak amplitude measurement becomes impossible.

0 5 10 15 20 25 30

Currents (mA)

−120

−100

−80

−60

−40

−20

0

20

40

Voltage

(µV)

1.20 ms

0.77 ms

Figure 5.21: Voltages on electrode 6 sampled at two fixed times, correspond-

ing approximately to the early and late N1 peaks.

The voltages at single points in time corresponding roughly to the early

and late peaks are shown in Figure 5.21. The late voltage initially grows

downward, but swings positive as the P2 wave of the early response grows

and the late response diminishes. The early voltage starts upward, corre-

sponding to the P1 lobe of the late response, before growing downward with

the early response.

153

0 5 10 15 20 25 30

Current (mA)

0.6

0.8

1.0

1.2

1.4

1.6N1Latency

(ms)

(a) N1 (full)

0 5 10 15 20 25 30

Current (mA)

0.5

1.0

1.5

2.0

2.5

3.0

P2Latency

(ms)

(b) P2 (full)

1 2 3 4 5 6

Current (mA)

0.8

1.0

1.2

1.4

1.6

N1Latency

(ms)

(c) N1 (detail)

1 2 3 4 5 6 7 8 9 10

Current (mA)

1.2

1.4

1.6

1.8

2.0

2.2

P2Latency

(ms)

(d) P2 (detail)

E4

E5

E6

E7

E8

Figure 5.22: Latencies of N1 and P2 peaks simulated on each electrode.

Biphasic stimulation. Shaded area shows stimulus period, ending at 0.54 ms.

Detailed views show the region up to several times threshold.

The change in latency of the peaks is shown in Figure 5.22. A rapid

increase in latency is observed at low currents, in the region corresponding to

the foot of the growth curve. A discontinuity is seen around the current where

the second-cathode trough appears in the growth curve; this reflects the peak

detector’s shift from the late to the early component of the ECAP. The N1

latency stops changing substantially with current above the second-cathode

threshold, while the P2 latency continues to increase over the measured

current range.

The group conduction velocity is shown in Figure 5.23. A discontinuity

is visible where the N1 peak detection shift occurs; this occurs at a different

current on each electrode.

154

2 4 6 8 10 12

Current (mA)

45

50

55

60

65

70

75

N1velocity

(m/s)

Figure 5.23: Conduction velocity calculated from the N1 latency between

electrodes 6 and 8, a distance of 14 mm. Biphasic stimulation.

0 5 10 15 20 25 30

Current (mA)

0

10000

20000

30000

40000

50000

60000

Fibers

Figure 5.24: Number of fibres recruited at each stimulus current. Biphasic

stimulation.

155

0 5 10 15 20 25 30

Current (mA)

4

6

8

10

12

14

16

MeanDiameter

(µm)

Figure 5.25: Mean diameter of recruited fibres. Biphasic stimulation.

156

The number of recruited fibres as a function of current is shown in

Figure 5.24, and the mean diameter in Figure 5.25. Both are similar to the

results obtained with monophasic stimulation.

157

2 4 10 30

Current (mA)

0.0

0.2

0.4

0.6

0.8

1.0

Max

imum

Depth

(mm)

(a) Maximum Ventral Spread

2 4 10 30

Current (mA)

0.0

0.5

1.0

1.5

2.0

2.5

Max

imum

Lateral

Pos

(mm)

(b) Maximum Lateral Spread

2 4 10 30

Current (mA)

0

1

2

3

4

5

RecruitmentArea(m

m2)

(c) Recruited Area

3.0 µm

3.2 µm

3.4 µm

3.6 µm

3.8 µm

4.1 µm

4.3 µm

4.6 µm

4.9 µm

5.2 µm

5.5 µm

5.9 µm

6.2 µm

6.6 µm

7.0 µm

7.5 µm

7.9 µm

8.4 µm

9.0 µm

9.5 µm

10.1 µm

10.8 µm

11.4 µm

12.2 µm

12.9 µm

13.7 µm

14.6 µm

15.5 µm

Figure 5.26: Spread of recruitment with current for each fibre diameter cell

in the model. Biphasic stimulation. a) Depth of the deepest recruited cell of

each diameter. b) Position of the most lateral recruited cell of each diameter.

c) Total area of recruited cells of each diameter.

158

Detailed recruitment changes with current are shown in Figure 5.26,

and are similar to those obtained with monophasic stimulation.

5.2.3 Human Data

A set of recordings made in a human SCS patient are used for comparison.

The patient was implanted with a percutaneous electrode with similar

geometry to that used in the model (Octrode, St. Jude Medical, St. Paul, MN,

USA). Tripolar stimuli were delivered on electrodes 1 through 3, which were

at the rostral end of the array. Biphasic stimuli were used, with a pulse

width of 120 µs and interphase gap of 200 µs. Stimulation was repeated at

60Hz. Recordings were made on electrodes 5 through 8. The patient had a

perceptual threshold of approximately 4 mA, and recordings were made up

to the patient’s discomfort threshold.

These recordings were made intraoperatively; a needle electrode was

inserted under the patient’s skin to provide a reference potential for the

MCS2 amplifiers. Virtual ground was enabled to reduce the stimulation

artefact. Stimulation was manually controlled, ramping up to the patient’s

discomfort threshold over several minutes. This resulted in 2957 individual

recordings at 51 individual currents, ranging from 625 µA to 5.625 mA in

steps of 100 µA. The manual ramping process led to different numbers of

recordings being made at each current; the counts are shown in Figure 5.27.

159

0 1 2 3 4 5 6

Recordings

0

50

100

150

200

250

300

350

400

450

Figure 5.27: Number of recordings made at each stimulus current in the

human patient.

160

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−20

0

20

40

60

80

100

Voltage

(µV)

(a) 1.5mA, single

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−20

0

20

40

60

80

100

120

Voltage

(µV)

(b) 1.5mA, averaged

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

0

100

200

300

400

500

600

700

Voltage

(µV)

(c) 5.5mA, single

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−100

0

100

200

300

400

500

600

700

Voltage

(µV)

(d) 5.5mA, averaged

E5

E6

E7

E8

Figure 5.28: Single recordings made at two currents, and averages of all

recordings at both currents.

161

The recordings for each current are averaged to reduce noise; some indi-

vidual recordings and their corresponding averages are shown in Figure 5.28.

A travelling neural response is visible at 5.5mA, but not at 1.5mA. The arte-

fact on electrode 5, nearest to the stimulating tripole, shows a substantial

artefact; a response is visible, superimposed. Due to the small number of

averages in many cases, a great deal of variation due to noise remains from

current to current.

Artefact Estimation

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Time (ms)

−4

−2

0

2

4

6

8

Voltage

(normalised)

E5

E6

E7

E8

Common

Figure 5.29: Components of the templates used for artefact subtraction.

An individual component is fitted to the artefact on each electrode, and a

common component is fitted to all of them.

The artefact is then estimated and removed using a template method.

Two templates are used; each is applied in such a way as to minimise poten-

tial cancellation of neural responses. The first template has a component

for each electrode, while the second uses one component for all electrodes.

These are shown in Figure 5.29.

162

1 2 3 4 5

Current (mA)

−20

0

20

40

60

80

100

Voltage

(µV

RMS)

E5

E6

E7

E8

Figure 5.30: Scaling factors for the linear template, and the line fitted to

each electrode’s scaling factor.

For the per-electrode template, each electrode’s template waveform is

scaled as a linear function of the applied stimulus current, with different

slopes. This template is generated by averaging all of the recordings below

1.5mA, where the artefact behaves linearly. The fitted values are shown in

Figure 5.30.

163

1 2 3 4 5

Current (mA)

−10

−5

0

5

10

15

Voltage

(µV

RMS)

E5

E6

E7

E8

Figure 5.31: Scaling factors for the common template, and the constant

factor fitted to each electrode.

The second template is identical on all channels (the constant template);

this is used at a fixed amplitude, which differs on each electrode. This

addresses a second signal component which becomes evident when the linear

template has been subtracted; this appears above approximately 1.7mA, has

roughly constant amplitude, and is identical on all channels. The identical

waveform and near-identical amplitude suggest that this is related to the

reference electrode. This template is obtained by averaging responses from

1.725mA to 2.925mA after subtraction of the linear template. The fitting

values are shown in Figure 5.31.

164

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−50

0

50

100

150

200

250

300

Voltage

(µV)

(a) 3.0mA, averaged

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−50

0

50

100

150

200

250

300

350

400

Voltage

(µV)

(b) 3.5mA, averaged

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−10

−5

0

5

10

15

20

Voltage

(µV)

(c) 3.0mA, subtracted

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−10

−5

0

5

10

15

20

25

30

35

Voltage

(µV)

(d) 3.5mA, subtracted

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−6

−4

−2

0

2

4

6

8

Voltage

(µV)

(e) 3.0mA, smoothed

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Time (ms)

−10

−5

0

5

10

15

20

Voltage

(µV)

(f) 3.5mA, smoothed

E5

E6

E7

E8

Figure 5.32: Recordings made at two currents, subjected to averaging,

template subtraction, and then smoothing. A propagating response is visible

at 3.5mA but not at 3.0mA.

165

After subtraction, recordings are smoothed by convolution with a nor-

malised Hamming window of length 9. This acts as a low-pass filter, al-

lowing the estimation of the peak positions in the presence of noise. The

recordings after averaging, artefact subtraction and smoothing are shown

in Figure 5.32.

Measurements

The positions and amplitudes of N1 and P2 peaks were estimated from

the smoothed waveforms. Propagating responses can be discerned above

3.3mA; P2 peaks are clear only above 4.3mA. This restricts the measurement

of ECAP amplitude to currents above this level.

166

3.5 4.0 4.5 5.0 5.5 6.0

Current (mA)

200

300

400

500

600

700

800

900

N1latency

(µs)

(a) N1 latency

4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

Current (mA)

800

900

1000

1100

1200

1300

1400

1500

1600

P2latency

(µs)

(b) P2 latency

E5

E6

E7

E8

Figure 5.33: N1 and P2 peak latency on each electrode.

The measured peak latencies are shown in Figure 5.33. The latencies

vary somewhat at low currents, where small amplitudes leave their mea-

surement subject to artefact and residual noise. At higher currents, they

appear to increase slightly with current.

167

3.5 4.0 4.5 5.0 5.5 6.0

Current (mA)

60

80

100

120

140

160

180

200

N1delay

(µs)

(a) N1 delay

4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

Current (mA)

50

100

150

200

250

300

350

P2delay

(µs)

(b) P2 delay

E5-E6

E6-E7

E7-E8

Figure 5.34: Propagation delays between peaks observed on successive

electrodes.

Propagation delays between peaks on successive electrodes are shown in

Figure 5.34. The group conduction velocity estimated from the N1 peaks is

approximately 45 m ·s−1 between E5 and E6, 74 m ·s−1 between E6 and E7,

and 55 m ·s−1 between E7 and E8.

168

4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

Current (mA)

0

20

40

60

80

100

Voltage

(µV)

E5

E6

E7

E8

Figure 5.35: N1-P2 amplitude on each electrode as a function of current,

and lines fitted to the data for each channel.

The growth of the response with current is shown in Figure 5.35. The

amplitude increases linearly throughout the measurement region; linear

fits to each electrode’s response yield an x-intercept of 4.1-4.2mA.

Discussion

The human recording shows a linear increase in ECAP amplitude above

4.1mA; the peak latency increases slightly with current in this region. This

corresponds to the linear region of the model behaviour, where the diameter

distribution of recruited fibres remains relatively constant.

ECAPs are visible with stimuli as low as 3.1mA with an amplitude on

the order of 5 µV peak to peak, although they are too small to be accurately

measured. Estimates of peak times suggest that the latency continues to

vary with current. This supports the model result showing a foot region

in the amplitude growth curve where the current-voltage relationship is

sublinear, but the peaks cannot be estimated with sufficient accuracy to

determine whether a change in the latencies occurs here.

169

4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.60

20

40

60

80

100

Human E5

Human E6

Human E7

Human E8

Model E5

Model E6

Model E7

Model E8

Figure 5.36: Peak-to-peak voltages from the model overlaid on measured

results from humans. Model current is scaled by 1.4 to obtain an x-intercept

of 4.1mA for the linear region. Model amplitudes are scaled by 3.9, 2.9, 2.3

and 1.5 on electrodes 5 through 8, respectively.

The model and human results can be roughly compared by scaling the

model current to achieve a similar x-intercept for the linear region. The

model results have an x-intercept of approximately 3 mA, and so are scaled

by a factor of 1.4. After scaling, recruitment in the model consists of approxi-

mately 400 fibres when stimulated near the perceptual threshold at 4.1 mA,

and 3000 fibres near the discomfort threshold at 5.7 mA.

The measured ECAP amplitudes drop off much more rapidly with dis-

tance than those in the model; the model amplitudes must be scaled to

achieve a similar amplitude, as shown in Figure 5.36. Model results are

between 1.5 and 4 times lower in amplitude than those recorded in the

human patient.

The peak latency values also differ. The model predicts evenly spaced N1

and P2 peak values across the electrodes with a propagation delay of 120 µs,

and a delay of approximately 425 µs between N1 and P2 on all electrodes.

The propagation delays are not constant in the recorded data, and differ

between N1 and P2 peaks. The N1-P2 delay on electrodes 4 and 5 is similar

to the modelled value, but increases to 525 µs on electrode 6 and 700 µs on

170

electrode 7. N1 propagation delay varies from 100 µs to 150 µs, with E4-E5

being relatively slow and E5-E6 relatively fast. In contrast to this, the P2

delays from E4 through to E7 are very similar at approximately 150 µs per

electrode. Broadly, the human N1 peaks are slightly earlier than the model

predicts, but with a similar delay between electrodes; the P2 peaks are close

to predicted values at the nearest electrodes, but arrive later with increasing

distance.

3.5 4.0 4.5 5.0 5.5

300

400

500

600

700

800

(a) N1 latency

4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6600

800

1000

1200

1400

1600

(b) P2 latency

Human E5

Human E6

Human E7

Human E8

Model E5

Model E6

Model E7

Model E8

Figure 5.37: Peak times for human and modelled ECAPs. Model current is

scaled by a factor of 1.4.

The human and modelled latency results are compared in Figure 5.37.

The N1 peaks measured from the model show unexpectedly low values on

electrodes 5 and 6 around 4mA stimulus current; this may be due to artefact

171

contamination, as they do not appear on more distant electrodes.

5.3 Conclusions

A model has been presented which simulates spinal ECAPs based on ex-

pected spinal geometry and nerve fibre populations. The ECAPs are treated

as a summation of SFAPs originating from the individual involved fibres.

The resulting ECAPs exhibit basic characteristics similar to those collected

from human patients. Detailed comparison with recordings from a single

patient show good agreement on basic measurements, and suggest explana-

tions for observed behaviours.

In order to handle the large number of fibres present within the dorsal

columns, two techniques were used. The SFAP model was ported from a

standard CPU to run on a GPU, which provides a high degree of parallelism.

This is an increasingly popular technique in modern scientific computing.

The port provided a speedup of 140 times on the system in use, allowing a

corresponding increase in the number of considered fibres. This performance

increase did not suffice for calculating behaviour as the stimulus current is

changed; hundreds of different stimuli must be simulated, increasing the

workload again. An approximation technique was used which groups fibres

with similar behaviours together, allowing a reduced number of represen-

tative fibres to be simulated. This technique has previously been used for

calculation of ECAPs in other nerve groups [150], [217], [253]. This was used

to reduce the number of fibres simulated by a factor of 18, allowing sweeps

of 600 steps to be calculated within 2 days.

5.3.1 Results

The model ECAPs exhibit basic features seen in human ECAP recordings.

The ensemble waveform is triphasic, propagating away from the stimulation

site and diminishing in amplitude. The recruitment curve exhibits three

distinct regions of behaviour: a foot above the stimulation threshold, in

which the range of recruited diameters rapidly increases; a roughly linear

region, in which the diameter range changes little but the total number of

172

fibres increases greatly; and a saturation region at high currents, where the

larger fibres in the modelled region are completely recruited.

Behaviour corresponding to the linear region is commonly observed in

human patients during therapy; saturation behaviour can also be observed

in unconscious patients. The foot region is not generally visible, likely due

to its low amplitude.

Detailed examination of a set of human recordings shows amplitude

growth consistent with a foot and linear region. Peak latencies and propaga-

tion delays are comparable between model results and recorded data. The

recording amplitudes correspond to within an order of magnitude, but show

a rapid diminution with propagation distance which is not reproduced by the

model. The overall amplitude difference could be explained by differences in

the number of fibres activated, error in the SFAP amplitude expected from

a single fibre, or an incorrect cord-electrode distance; the available data do

not permit these causes to be distinguished.

5.3.2 Further Work

The model assumes that the dorsal column fibres are axially oriented,

constant in diameter, and run the full length of the modelled region. All

three of these assumptions are known to be incorrect; fibres change in di-

ameter and position as they move away from their entry point to the dorsal

columns, and few fibres continue from their points of entry to the brain [65],

[99], [214]. This provides a potential explanation for the large reduction in

amplitude with distance observed in human recordings; fibres that reduce

in diameter, terminate, or move deeper into the dorsal columns will con-

tribute less to an ECAP. Further work is required to determine the effects of

these individual classes of variation on the ECAP as a whole. Parametrising

different fibre paths, termination points and diameter changes increases

the dimensionality of the fibre parameter space; this will require a differ-

ent structural approach in order to avoid an exponential increase in the

necessary computation time for mapping this space.

Artefact and noise limit the resolution of waveform details in the record-

ings, particularly at low amplitudes. This precludes examination of the

precise behaviour in the foot region of the operating curve. This could be

173

improved upon by recording a series specifically for this purpose, taking a

large number of recordings at each current to permit noise reduction through

averaging.

5.3.3 Application

This is the first model to explain the origin of SCS ECAPs. The model

results indicate that therapeutic SCS operates primarily in the linear region

of the recruitment curve, in which a broad range of fibre diameters are

recruited. This is supported by measurements of amplitude, conduction

velocity and latency values in an example human patient, which correspond

with the activation of large numbers of relatively small fibres in the model.

These results conflict with those obtained with most previous SCS models,

which drew the conclusion that a narrow range of diameters is recruited [95].

This resulted from an assumption that the perceptual threshold is equal

to the threshold of the most sensitive fibre in the dorsal columns (largest

and nearest the electrodes). When aligned with the human data, the ECAP

model predicts a recruitment of several hundred fibres at the perceptual

threshold, with diameters ranging from the maximum of 16.7 µm down to

7.5 µm. The activation of a range of diameters is not unexpected; one prior

SCS model has shown that this mechanism is necessary to explain the

caudal shift of the perceived stimulation field as the stimulus pulse width is

increased [123]. This is particularly relevant to the design of new clinical

techniques, including electrode and stimulus arrangements.

The ECAP model has been shown to reproduce the expected characteris-

tics of ECAPs, and explains the origins of some of their features. Next, the

model can be used to shed light on one of the major side effects of SCS: its

postural sensitivity. It has long been recognised that the position of the cord

within the spinal canal changes with posture [89], and this is believed to

change its sensitivity to stimulation, giving rise to posture-related over- and

understimulation. An ECAP-based therapy is now under development which

aims to reduce these effects using feedback control to maintain a constant

ECAP amplitude. Using the model, it is possible to examine this technique

and determine its potential performance before it is deployed in humans.

174

CH

AP

TE

R

6C O R D M O V E M E N T

One of the primary goals of this work is to shed light on the effects of

cord movement during SCS, which at present causes posture-dependent

side effects due to changes in recruitment [172], [195]. ECAP recording

provides the opportunity to measure cord behaviour in real-time, and trials

of feedback control of the stimulus intensity are currently ongoing. In these

trials, the stimulus is controlled to maintain a constant ECAP amplitude;

this is expected to maintain relatively constant recruitment.

There are two distance-dependent transfer functions involved in ECAP

recording. The first is in stimulation: at a greater distance, a higher current

is needed to stimulate the same nerve fibres. The second is in recording:

at a greater distance, the same neural recruitment results in a smaller

ECAP. The constant-amplitude feedback approach takes no account of the

recording transfer function, so that recruitment may actually increase as

the cord distance increases. An approach which considers both transfer

functions could improve the performance of feedback control. Such a method

is necessarily limited by the impossibility of directly measuring neural

recruitment in humans; it must be possible to fit any parameters to the

patient without that information.

In this chapter, the ECAP model is used to explore the behaviour of

the cord at different positions within the spinal canal. The performance of

175

constant-current and constant-amplitude stimulation regimes are demon-

strated. Options for more advanced methods are discussed, and a new control

method is introduced which results in improved performance.

6.1 Simulations

ECAP simulations were performed for each of 10 different cord positions,

varying the cord-electrode distance from 1.7 mm to 5.2 mm. Monophasic

stimuli were used to avoid confounding measurements with the second-

cathode effect.

0 5 10 15 20 25 30

Current (mA)

0

50

100

150

200

250

300

350

N1-P2Voltage

(µV)

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.1: Peak-to-peak amplitude of ECAPs on electrode 6 at various

cord-electrode distances.

The peak-to-peak amplitude on one electrode for various cord positions

is shown in Figure 6.1. As the cord is moved closer to the electrode array,

the threshold is lowered and the growth slope and saturation amplitude

increase. The basic form of the curve appears to be relatively unchanged.

176

0 5 10 15 20 25 30

Current (mA)

0

10000

20000

30000

40000

50000

60000

70000Fibers

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.2: Fibre recruitment at various cord-electrode distances.

0 5 10 15 20 25 30

Current (mA)

45

50

55

60

65

70

75

80

85

90

N1velocity

(m/s)

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.3: Conduction velocity at various cord-electrode distances.

177

0 5 10 15 20 25 30

Current (mA)

4

6

8

10

12

14

16

MeanDiameter

(µm)

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.4: Mean recruited fibre diameter at various cord-electrode dis-

tances.

178

The recruitment curve in Figure 6.2, estimated group conduction ve-

locity in Figure 6.3, and mean recruited diameter in Figure 6.4 all show

similar effects, namely lower thresholds and higher rates of change at closer

distances.

179

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Current (relative)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Voltage

(relative)

(a) N1-P2 Voltage

0 1 2 3 4 5

Current (relative)

0

10000

20000

30000

40000

50000

60000

70000

Fibers

(b) Fibres Recruited

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Current (relative)

4

6

8

10

12

14

16

MeanDiameter

(µm)

(c) Mean Diameter

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.5: Amplitude, recruitment and mean diameter normalised for each

distance relative to N = 5000 fibres. Each curve is normalised in current

by a factor determined from the recruitment curve. The amplitude curve is

further normalised in voltage by its value at N = 5000.

180

The changes with distance are more than a simple scaling of fibre thresh-

olds. To demonstrate this, the growth, recruitment, and mean diameter

curves are scaled and overlaid in Figure 6.5. The current required to recruit

5000 fibres is used to normalise the current axis, while the voltage curve is

further scaled according to the measured voltage at the 5000-fibre current.

The 5000-fibre point is chosen as a suitable operating point in the linear

region of the curve.

The amplitude curves in Figure 6.5(a) differ on both sides of the nor-

malisation point, particularly at higher currents. The recruitment curves in

Figure 6.5(b) are quite similar across the whole range examined; the mean

diameters are less similar. The mean fibre diameter and conduction velocity

drop more rapidly with increasing current at closer distances. This suggests

that distance has a different effect on different fibre diameters, with smaller

fibres being more affected than larger ones.

181

6.2 Current Methods

In order to compare stimulation methods, a target recruitment of 5000

fibres is chosen; this is within the linear recruitment region, as commonly

observed in therapeutic stimulation.

6.2.1 Constant Current

0 2 4 6 8 10 12 14

Current (mA)

0

5000

10000

15000

20000

25000

30000

35000

40000

Fibers

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.6: Recruitment as a function of current, showing the use of a fixed

stimulus amplitude (vertical line). The set current is chosen to recruit 5000

fibres at the medial cord position.

Recruitment after the application of a fixed stimulus current is shown in

Figure 6.6. A setpoint current is selected as that required to recruit 5000

fibres at the medial cord position; the intercept of the vertical setpoint line

with the recruitment curves shows the number of fibres recruited at each

distance using that stimulus. The number of fibres recruited using constant

current ranges from zero to nearly 35000 across the examined cord positions.

This change highlights the degree to which traditional SCS is sensitive to

cord position.

182

6.2.2 Constant Amplitude

0 10 20 30 40 50 60 70 80

Voltage (µV)

0

2000

4000

6000

8000

10000Fibers

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.7: Recruitment as a function of recording amplitude, showing the

use of a constant recording amplitude (vertical line). The set amplitude is

chosen to recruit 5000 fibres at the medial cord position.

The use of constant-amplitude feedback is shown in Figure 6.7. The

setpoint voltage is again chosen for 5000 fibre recruitment at the medial

distance, and the amplitude of the ECAP on electrode 6 is used for feedback.

This electrode is chosen in line with clinical experience from the Saluda

corpus; while the ECAP amplitude is higher near the stimulation (here on

electrodes 1 through 3), the artefact is also higher, and recording is often

not possible on the one or two electrodes nearest the stimulus.

The variation is much reduced, with recruitment now varying from

approximately 3000 to 6500 fibres across the distances examined. This

improvement correlates with early data from closed human trials, indicating

an improvement in postural sensitivity is obtained with constant-amplitude

feedback over constant-current. The variation is reversed with regard to

constant-current stimulation: now the recruitment is increased at larger

cord distances.

183

6.3 Recruitment Control

The signal amplitude is not the only source of information available from

SCS ECAP recording.

Measurements of spectral content of the waveform, either through trans-

forms or through a set of filters, could allow the responses from different

distances to be discerned. The output of such a detector could then be fitted

to the patient’s perceptual response at different distances. Such an approach

would need to be based on analysis of a large corpus of patient data, and the

computational requirements of signal transforms are not presently compati-

ble with low-power implants.

45 50 55 60 65 70 75 80 85 90

N1 velocity (m/s)

0

10000

20000

30000

40000

50000

60000

70000

Fibers

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.8: Relationship between conduction velocity and recruitment over

a range of distances.

A computationally cheaper approach would be to use times obtained from

peak detection in some fashion. Figure 6.8 shows the conduction velocity

estimated from N1 peaks at a range of distances. The estimated conduction

velocity does not vary consistently with either distance or recruitment, as

well as highlighting the sensitivity of peak measurements to noise.

One piece of information always available during SCS: the stimulus

current. If the relationship between stimulation and recording transfer

functions with distance is understood, then a control method could use some

184

combination of the applied current and ensuing evoked potential to control

the stimulus.

These transfer functions can be approximated analytically. A number of

symbols are introduced in the following section; these are listed in Table 6.1

for convenience.

N Number of recruited fibres

I Stimulus current

x Cord-electrode distance

A(x) Stimulation transfer function

n Stimulation transfer function parameter

m Recording transfer function parameter

Si Amplitude of a single-fibre response (SFAP)

V Amplitude of a multi-fibre response (ECAP)

T0 Normalised threshold current of all fibres

Ti(x) Threshold current of a single fibre

I0(x) I-axis intercept of linear portion of growth curve

M(x) Slope of linear portion of growth curve

R Ratio of stimulus current I to intercept current I0

xi Cord position in posture i

Mi Growth curve slope in posture i

I i Growth curve intercept in posture i

Vi Growth curve voltage at some multiple R of I i

Table 6.1: List of symbols used in this section.

6.3.1 Stimulation

We wish to derive an expression for the relationship between the stim-

ulating current I and the number of recruited fibres N - the stimulation

transfer function. This function will depend on the distance x between the

target tissue and the stimulating electrodes. N will be equal to the number of

fibres with thresholds lower than the stimulation current; these thresholds

Ti will also vary with distance:

N =�

i

[I ≥ Ti(x)] (6.1)

185

The change in Ti with x can be approximated by a simple analytic model.

If we consider a single myelinated nerve fibre exposed to a point current

source at distance x and with internodal length L, the voltage at the ith

node of Ranvier is of the form:

I�

x2 + (iL)2(6.2)

assuming that the 0th node is at the point on the fibre nearest the electrode.

The propensity of the fibre to be activated by a given stimulus is approxi-

mated by a function known as the activating function [189]. This represents

the net depolarisation current being applied to each node of Ranvier on the

fibre, and has a threshold behaviour; if the depolarisation is sufficient at

any node, the fibre will fire. For a myelinated fibre, the activating function

is given by the second difference of the field along the fibre. This has a

maximum at the node nearest the electrode, with value

I

�

2�

x2−

1�

x2 +L2−

1�

x2 + (−L)2

�

(6.3)

Thus the threshold will vary with distance as:

Ti(x)∝

�

1

x−

1�

x2 +L2

�−1

(6.4)

This is not a particularly tractable expression. The internodal spacing in

the dorsal columns is generally less than the cord-electrode distance; in the

region L < x, the fourth and higher derivatives of Ti are quite small, and

the behaviour approximates:

Ti(x)∝ xn (6.5)

with n ∈ [1,3].

Returning to the ensemble behaviour, the total number of fibres recruited

by a given stimulus depends on the fibre thresholds Ti. In the linear region

of recruitment, N increases linearly with I, so the Ti can be assumed to be

uniformly distributed. Under this assumption, the total number of fibres

recruited above threshold varies as:

N ∝ Ix−n −T0 (6.6)

where T0 is a normalised threshold corresponding to the threshold of the

most sensitive fibres at x = 1.

186

1 2 3 5

Distance x (mm)

1

3

8

20

CurrentI(m

A)

Figure 6.9: Current-distance curves calculated from the ECAP model results.

Each line shows the relationship between current and distance for a fixed

recruitment ranging from N = 1000 (bottom) to N = 20000 (top) in steps of

1000, on log-log axes.

This relationship is examined using the ECAP model results. For a given

recruitment value N, the corresponding stimulus current for each distance

x is calculated. This is shown in Figure 6.9 for recruitment values up to

N = 20000, within the linear portion of the recruitment curve. These lines

are not exactly straight; recruiting fibres at larger distances requires dispro-

portionately higher current, indicating a shift in the power law. Treating it

as constant and using linear regression over log-log axes results in n ≈ 1.64

at low recruitment, dropping to n ≈ 1.55 at higher recruitment levels.

187

0 5000 10000 15000 20000

Recruitment N

1.54

1.56

1.58

1.60

1.62

1.64

1.66

Slope

Figure 6.10: Slopes of lines fitted to Figure 6.9. The slope is an estimate of

the value of n in Equation 6.6.

6.3.2 Recording

In a similar fashion to the stimulation transfer function, the recorded

voltage V varies in a distance-dependent manner with the neural recruit-

ment. The propagation of an action potential along a myelinated fibre is

too complex for meaningful analytical treatment; instead, simulations can

be used, as performed by Struijk [222], who simulated a collection of myeli-

nated nerve fibres with a point recording electrode and found that the SFAP

amplitude Si of a single fibre at distance x followed a law

Si(x)∝ x−m (6.7)

where m = 1 close to the fibre, and m = 3 in the far field. m = 1 is expected

where x � L, and the nearest node’s current dominates the recording; m = 3

in the far field results from the tripolar nature of the travelling action

potential.

The ECAP voltage V results from the summation of SFAPs, and so de-

pends on the spatial and diametric distribution of recruited fibres. Different

diameter fibres have different SFAP amplitudes, but their proportions are

fairly constant over the linear portion of the growth curve. If we also assume

that the spatial distribution of recruited fibres varies less than x, we can

188

approximate the ECAP amplitude as:

V ∝ Nx−m (6.8)

1 2 3 5

Distance x (mm)

7

20

70

200Voltage

V(µV)

Figure 6.11: Voltage-distance curves calculated from the ECAP model

results, using electrode 6. Each line shows the relationship between ECAP

amplitude and distance for a fixed recruitment ranging from N = 1000

(bottom curve) to N = 20000 (top curve) in steps of 1000, on log-log axes.

189

0 5000 10000 15000 20000

Recruitment N

−0.80

−0.75

−0.70

−0.65

−0.60

−0.55

Slope

Figure 6.12: Slope of lines fitted to Figure 6.11. The slope is an estimate of

the value −m in Equation 6.8.

The recording transfer function in the ECAP model is examined using

voltage-distance curves at constant recruitments, shown in Figure 6.11.

Slopes are shown in Figure 6.12, indicating values for m on the order of 0.9

at the beginning of recruitment, dropping to 0.6-0.65 over the linear region.

The decreasing value of m corresponds to the broadening of the range of

recruited fibre diameters. The diameter-dependent conduction velocity of

myelinated fibres results in a dispersion of the action potential volley; this

results in a longer region of the nerve trunk that is contributing to the

ECAP via membrane currents. The triphasic field radiated by each fibre

then cancels somewhat, resulting in a field that falls off more rapidly with

distance.

6.3.3 Recruitment Control

These findings can be combined to compensate for the changes in cord

movement and ensure constant recruitment. It is assumed that the transfer

function parameters, m and n, can be determined; these are related to the

electrode geometry and configuration, but not expected to change during

therapy. The cord position x is not known.

190

Using controlled constant-current stimulation means that the stimula-

tion current I and ECAP voltage V can both be measured on each stimulus.

Substituting (6.8) into (6.6) gives:

N ∝ I

�

AN

V

�−n/m

−T0 (6.9)

where A is the constant of proportionality of equation 6.8. In order to main-

tain constant N, it follows that the expression

Im/nV (6.10)

must be constant. For the purpose of recruitment control, this is most easily

expressed as:

y= IkV (6.11)

where

k =m

n(6.12)

and k > 0.

This captures the transfer function behaviour in a single parameter, k.

The stimulus current can then be adjusted using a feedback algorithm to

maintain the relationship between I and V , with the desired value of y

being driven to a chosen set-point. A higher value of y will result in higher

recruitment.

6.4 Performance

The performance of the new feedback method (the "I-V" method) is

examined by measuring the setpoint-recruitment curve for various distances.

For this comparison, m and n were estimated to be 0.6 and 1.6, respectively;

this yields a value of k = 0.37.

191

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ik · V

0

2000

4000

6000

8000

10000

Fibers

5.2 mm

4.8 mm

4.4 mm

4.0 mm

3.6 mm

3.3 mm

2.9 mm

2.5 mm

2.1 mm

1.7 mm

Figure 6.13: Recruitment as a function of I−kV , showing the effect of a

constant I-V control (vertical line). k = 0.37.

The relationship between I−kV value and recruitment is shown in Fig-

ure 6.13. The setpoint shown in the figure is chosen for 5000 fibre recruit-

ment at a cord-electrode distance of 3.2mm, in the middle of the range.

Across the full range of cord positions, the recruitment remains within a

narrow range.

192

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Cord-electrode distance (mm)

−100

0

100

200

300

400

500

600

700

Deviation

(%)

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Cord-electrode distance (mm)

−40

−20

0

20

40

60

Deviation

(%)

I

V

I + V

Figure 6.14: Comparison of the performance of constant-current, constant-

voltage, and I-V control. The setpoint for each is chosen to achieve recruit-

ment of 5000 fibres at 3.2mm distance. Deviation is measured from the

desired 5000 fibres.

The behaviour of I-V control is compared with the existing techniques

of constant-current and constant-voltage control in Figure 6.14. The ex-

tremely poor performance of constant-current stimulation is visible in the

full view; the detail view shows the improved performance of I-V control over

constant-amplitude control. Constant-amplitude control results in variations

193

of greater than −30/+50% from the setpoint, while I-V control maintains

constant recruitment to better than ±5%.

0 2000 4000 6000 8000 100001200014000

Target (fibers)

0

5

10

15

20

25

30Deviation

(RMS%)

V

I-V

Figure 6.15: RMS deviation from the mean recruitment across all distances

for a range of desired setpoints.

The performance of voltage and I-V control across a range of desired

recruitment values is shown in Figure 6.15. For each desired recruitment, a

suitable setpoint was determined for each algorithm at the cord distance of

3.2mm; the recruitment was calculated for each cord position with that set-

point, and the RMS deviation from the mean recorded. I-V control results in

less variation in recruitment across a wide range of setpoints. Performance

is reduced at very low recruitment levels. This is due to the rapid change

in m as shown in Figure 6.12; the ideal value of k increases throughout the

foot region, up to a maximum of 0.6 at the ultimate threshold.

194

0.0 0.2 0.4 0.6 0.8 1.0

k

0

5

10

15

20

25

30

35

Meandeviation

(RMS%)

Figure 6.16: Effect of different values of k on performance of recruitment

control. Electrode 6 is used for feedback. For each value of k, the RMS devia-

tion across all positions was calculated for a range of desired recruitment

values, as in Figure 6.15. The average of these deviations forms the mean

deviation for each value of k.

The I-V control method requires that a suitable value for k to be chosen.

A sensitivity analysis was conducted, varying k between 0 and 1. For each

value, an RMS deviation measurement was made as in Figure 6.15. The

overall error for each value of k was estimated by averaging the RMS devia-

tion for all considered recruitment levels, shown in Figure 6.16. With k = 0,

the current term is removed from the feedback equation, and I-V control is

equivalent to constant-amplitude control. Figure 6.16 shows that I-V control

outperforms constant-amplitude control on electrode 6 for any value of k

between 0 and 0.8.

195

0.0 0.2 0.4 0.6 0.8 1.0

k

0

5

10

15

20

25

30

35

40

45

Meandeviation

(RMS%)

E4

E5

E6

E7

E8

Figure 6.17: Effect of different values of k on performance of recruitment

control on different electrodes. For each value of k, the RMS deviation across

all positions was calculated for a range of desired recruitment values, as in

Figure 6.15. The average of these deviations forms the mean deviation for

each value of k.

The recording transfer function depends on the recording electrode in use.

Geometric factors may differ between electrodes, and the dispersion of the

action potential volley increases as it travels away from the stimulation site.

Increasing dispersion decreases m, so the correct value of k can be expected

to be lower with increasing distance from the stimulus. This is seen in

Figure 6.17. If recording on electrode 4, closest to the stimulus, the optimal

k is approximately 0.45, but any value between 0 and 1 outperforms constant-

amplitude feedback. For electrode 8, the optimal value is approximately 0.27,

with values between 0 and 0.55 outperforming constant-amplitude.

6.5 Biphasic Stimulation

The preceding results were presented using monophasic stimulation,

in order to clearly demonstrate the principles without consideration of

the second cathode effect and its complication of results. The techniques

described are equally applicable when biphasic stimulation is used.

196

0.0 0.2 0.4 0.6 0.8 1.0

k

0

10

20

30

40

50

Meandeviation

(RMS%)

E4

E5

E6

E7

E8

Figure 6.18: Effect of different values of k on performance of recruitment

control, with biphasic stimulation and on different electrodes. For each value

of k, the RMS deviation across all positions was calculated for a range of

desired recruitment values. The average of these deviations forms the mean

deviation for each value of k.

The sensitivity analysis for k is repeated with biphasic stimulation in

Figure 6.18. The best results obtained are similar to those in the monopha-

sic case, although the optimal value for k is somewhat lower. I-V control

outperforms constant-amplitude control for k values between 0 and 0.65.

197

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Cord-electrode distance (mm)

−40

−20

0

20

40

60

Deviation

(%)

I

V

I + V

Figure 6.19: Comparison of the performance of constant-current, constant-

voltage, and I-V control with biphasic stimulation. The setpoint for each is

chosen to achieve recruitment of 5000 fibres at 3.2mm distance. k = 0.35 for

I-V control.

Performance of I, V, and I-V control are compared with biphasic stimula-

tion in Figure 6.19. A k-value of 0.35 was used for I-V control. Voltage mode

control results in a variation of greater than −30/+50% across the examined

range of distances, while I-V control maintains recruitment within less than

+10/−0% of the initial setpoint.

6.6 Conclusions

The ECAP model demonstrates a substantial variation in response to

stimulation current with variation in the cord position. This finding is consis-

tent with results from previous models [88] and clinical measurements [84],

[96], and is believed to be responsible for postural sensitivity and related side

effects in SCS [172]. Extremely large changes in neural recruitment are ob-

served in the ECAP model when the cord is moved during constant-current

stimulation.

Constancy of neural recruitment with cord movement is seen to improve

by controlling the stimulus amplitude to achieve a constant ECAP amplitude;

198

human trials of this technique are ongoing. Constant-amplitude control is

shown to overcompensate for cord movement: recruitment increases as the

cord moves away from the electrode, as the recording amplitude for a given

recruitment level decreases.

A new method, I-V control, is developed here to account for this effect;

this further reduces the fluctuation of recruitment with cord movement.

This method uses a single parameter, k, to capture the relationship of the

stimulation and recording transfer functions.

Analytic modelling of spinal cord stimulation suggests that the current

required to recruit a given nerve fibre varies with the cord-electrode dis-

tance according to a power law. Simulation results show that the ensemble

behaviour of a modelled dorsal cord is consistent with a power law with

constant parameter within the linear region of the operating curve.

Prior studies of recording in a single fibre indicate that a distance-

dependent power law determines the recording amplitude [222] when using

a point electrode in a homogeneous medium, with distinct near- and far-field

effects. Simulation results with the ECAP model show that a similar power

law applies, including a dependency on the diameter distribution of the

recruited fibres.

Simulations performed with the ECAP model indicate that the new I-V

control method outperforms constant-amplitude control across a wide range

of setpoints and distances with a fixed parameter k, reducing variation

across the distance range from greater than 20% RMS to less than 5%

RMS. The improvement obtained depends on the degree of variation of the

recording transfer function, given by m. If the recording function varies

not at all (m = 0), then I-V control is equivalent to constant-amplitude

stimulation; the higher the value of m, the more constant-amplitude control

will overcompensate changes in distance, and the more improvement I-V

control can deliver.

6.6.1 Fitting

The I-V control method requires a single parameter, k, which will depend

on the patient’s spinal geometry as well as the stimulation configuration. k

can be determined by determining the patient’s most comfortable stimulus

199

level in a range of postures, and fitting k to the resulting stimulus/ECAP

values.

Further work is required to determine whether it is possible to deter-

mine k directly from ECAP recordings, without reference to the patient’s

percept. The recording transfer function between recruitment, distance, and

ECAP amplitude is complex, particularly as it depends on the dispersion

characteristics of the recruited fibre population. Amplitude measurements

alone cannot distinguish changes in dispersion from changes in recruitment.

6.6.2 Implementation

Figure 6.20: Feedback loop to maintain constant recruitment using I-V

control. Incoming measurements of ECAPs are multiplied with an exponen-

tiated version of the stimulus current used to generate them. An error signal

is generated and fed into a controller, which determines the next stimulus

intensity.

I-V control can be implemented as a feedback loop in the manner shown

in Figure 6.20. The setpoint is a unitless value determined by the patient’s

desired level of stimulation. A discrete-time controller G(z) is used to control

the stimulus current based on the error signal; this may take the form of a

simple gain or a more complex system such as a PID controller.

This particular scheme has the potential to improve loop response speed

in addition to providing better control accuracy. In constant-amplitude feed-

back, the patient’s transfer function (from stimulus to ECAP amplitude)

varies with posture; this limits the maximum controller gain that can be

applied while keeping the loop stable, and in doing so limits the potential

bandwidth. If I-V control is implemented by scaling the ECAP amplitude

at the input to G(z), then when the loop is tracking correctly the scaling

200

compensates for the changing transfer function of the patient. This means

that the gain of G(z) can be maximised without compromising stability,

increasing the speed with which the loop can respond.

This technique has yet to be tested in human subjects. The implemen-

tation in an ERT-capable implant requires only software changes. The I-V

control law, however, introduces a nonlinear element into the feedback loop.

This will require extensive validation before it can be deployed.

201

CH

AP

TE

R

7C O N C L U S I O N S

The recent development of evoked response telemetry in SCS provides

a new source of objective information on the effects of stimulation. This

technology is still in its infancy; the neural origins of SCS ECAPs have

not previously been documented. ECAP-based feedback is currently being

trialled to reduce the posture-related side effects in SCS therapy, and the

techniques presently being tested to control stimulation are necessarily

simple. This work set out to pursue better measurement, understanding,

and application of evoked potentials using model-based approaches, seeking

to answer three core questions:

1. Do models of corrupting signals enable better ECAP measurement?

2. Does a model of SCS explain observed ECAP features?

3. Can a model-based approach improve clinical SCS?

7.1 Measurement

The stimulus artefact was selected for investigation as the most promi-

nent and well-modelled corrupting signal present in SCS ERT. Examina-

tion of an electrical model of SCS artefact [204] led to a novel stimulation

technique now known as virtual ground. With virtual ground, the return

202

electrodes are actively driven to reduce the voltage swing applied to the

recording electrodes, without a change in the stimulating current waveform

delivered to the tissue. Substantial improvement in artefact power was

demonstrated both in vitro and in vivo; the improved signal to noise ratio

broadens the range of conditions under which ECAPs can be observed.

Galvanically isolating the stimulation system from the recording sys-

tem is a common technique for minimising artefact [135]. This reduces

charge storage on the recording electrodes, although parasitic capacitance

between the stimulator and patient limits the achievable reduction [141].

Galvanic isolation is also impractical in an implanted device; independent

or transformer-isolated supplies are bulky and reduce efficiency. Virtual

ground addresses both issues. Actively controlling the potential imposed on

the recording electrodes effectively bootstraps the patient-stimulator capaci-

tance, without requiring a second power source or changing the stimulation

waveform.

Most recent research on the stimulus artefact has explored new signal

processing techniques for removing artefact after recording. Existing work

on prevention during stimulation largely involves modifying the stimulus

waveform, which affects the neural response [35], [48]. One new development

is a system which actively discharges each electrode independently after

stimulation [27], [36], [156], designed for microelectrode array recording

and presently requires an artefact-free reference electrode. Virtual ground

appears to be the only published technique which controls the artefact

during stimulation without changing the stimulus current waveform.

7.1.1 Sample Size and Measurement

Virtual ground performance was measured in a single human patient.

While anecdotal reports from Saluda trials indicate that virtual ground is of

benefit in a wide range of patients, quantitative performance measurements

have not been made. The definition of an artefact measurement is a compli-

cated question. In the present work, the signal power is used as a measure

of artefact; this can only be measured accurately in the absence of neural

responses, limiting measurements to the sub-threshold stimulation region.

Determination of the threshold is nontrivial; as demonstrated in the human

203

responses considered in Section 5.2.3, responses can be present at stimulus

levels well below those at which they are easily visible.

Different artefact components are sometimes seen with opposing polarity,

leading to cancellation. A practical measure of artefact also needs to take

into account the intended use of the recorded signal. For the purposes of

feedback control, only components which may interfere with ECAP detec-

tion are relevant. This depends on the ECAP detection method in use, and

further requires that measurements be made above the recruitment thresh-

old. Application of closely spaced stimuli can allow differential control of

neural response and artefact [35], [112], which would enable suprathreshold

artefact separation.

An electrical network model could also be used to separate the contri-

butions of the individual involved electrodes, based on the SCS artefact

model established in [204]. This would allow measurements of artefact to

be separated into multiple parameters without cancellation. This may be

complicated by postural change, as the interelectrode impedance network is

known to change with posture [1].

7.1.2 Reference Selection

In its present form, virtual ground requires that a single reference elec-

trode be selected as input to the virtual ground driver. The performance of

virtual ground has been shown to be strongly dependent on the choice of

reference. At present, the optimal reference electrode must be determined

and selected by the user. This is not ideal for clinical use, where the ad-

ditional step takes precious time and introduces room for user error. An

automated method to determine the correct reference electrode is likely to

be of great utility. This is dependent on the development of a measure of

clinically relevant artefact.

7.1.3 Controlled Variable

As presented in this work, virtual ground controls the common-mode

potential applied to the stimulating electrodes in such a way that the ref-

erence electrode potential is nulled. Greater control could be exercised by

204

driving the reference potential to some nonzero value, which may differ

between stimulation phases. Cancelling artefact by adjusting the voltages

imposed on recording electrodes has previously been demonstrated using

other approaches [48], [135]. Explicit control over the common-mode would

provide a pathway to optimise the elimination of artefact on specific elec-

trodes through a cancellation process. Further investigation is required to

determine the feasibility of this concept.

7.2 ECAP Model

A new model of SCS was created with the aim of reproducing ECAP fea-

tures. This model uses a similar structure to previous SCS models, consisting

of the UT-SCS model [92] and its descendants. The ECAP model combines a

volumetric electrical model of the spinal tissues and electrode array with

models of individual nerve fibres within the dorsal columns. Modelling the

ECAP requires two major changes from previous models. Firstly, the SFAP

from each nerve fibre after stimulation must be calculated, and the SFAPs

summed to derive the ECAP. Where previous models stop simulation of

a fibre when they determine whether or not it has fired, the ECAP model

simulates each fibre for the full time period of recording, and calculates the

SFAP from the membrane currents flowing in the fibre.

The second change relates to the computational tractability of the model.

Previous models have examined small numbers of fibres due to untested

assumptions about the fibre diameters involved in SCS, and also need only

simulate each for a short time to determine whether they were recruited.

In order for the ECAP model to simulate the full populations of the dorsal

columns for multiple milliseconds, it was necessary to introduce a para-

metric grid approximation in which one simulated fibre stands in for many

similar actual fibres. This is not dissimilar to prior approaches in mod-

elling compound action potentials in the periphery [217], [253] and auditory

nerve [150], allowing arbitrary fibre populations to be simulated with re-

duced computational effort.

This is the first SCS model to simulate ECAP waveforms. The model

predicts three distinct regions of behaviour as the stimulation current is

205

increased: a foot, linear region, and saturation. In the foot region, the range

of recruited diameters increases rapidly, from large to small fibres; in the

linear region, the diameter range varies little, but the spatial extent of

recruitment continues to increase.

Examination of a set of human recordings showed behaviour consistent

with operation in the model’s linear region, with some responses discernible

at lower currents that strongly suggest the presence of a foot. Linear growth

is a phenomenon observed in both humans and sheep; the Saluda corpus

indicates that therapeutic levels of stimulation commonly fall within the

linear growth region.

An increase in peak latencies with increasing current is observed in the

model, just as it is in human and ovine dorsal column recordings [180], [181].

This can be explained by the increasing recruitment of smaller, slower fibres;

the mean conduction velocity is reduced and the observed peak latency shifts

later. This supports the hypothesis that therapeutic SCS recruits a wide

range of fibre diameters.

The ECAP amplitudes observed in patients range from tens to hundreds

of microvolts; in the model, thousands of activated fibres are required to

reach these amplitudes. If the perceptual threshold of stimulation is as-

sumed to correspond to the x-intercept of the growth curve, then the model

predicts substantial recruitment at 1.4 times threshold: at the midrange

cord position, fibres as small as 6 µm in diameter are recruited, and fibres

13 µm and larger are recruited at the model’s deepest extent of 1mm below

the cord surface, with just over 5000 fibres activated in total.

This result stands in contrast to conclusions drawn from previous work

in SCS modelling. Past studies of SCS modelling have made the assumption

that the perceptual threshold for SCS is identical to the threshold at which

fibres are first recruited, which corresponds to activation of the largest and

most sensitive fibres in the cord [91]. A commonly cited statistic states

that the discomfort threshold is, on average, 1.4 times the paraesthesia

threshold [103], [233]; roughly speaking, this assumption means that fibres

down to 1/1.4 the diameter of the largest fibres should be stimulated at the

discomfort threshold, a range from approximately 16.4 to 11µm. The result

of this assumption is the conclusion that only very few fibres are stimulated

in therapeutic SCS [95] - as few as a single fibre. This conclusion has been

206

used in most later SCS models, which then restrict their considerations to

large diameters, or even a single diameter only [108], [131]. These large

fibres are rare, numbering a few hundred at most; the ECAP model predicts

that the activation of this population alone would result in extremely small

ECAPs, with much faster conduction velocities than are observed in practice.

The assumption has also been contradicted by results of one other SCS

model: Lee demonstrated that a model incorporating a range of diameters

can explain the phenomenon of caudal shift, in which longer stimulation

pulse widths lead to a spread of the paraesthesia region caudally on the

patient’s body [123].

The large-fibre assumption has particular bearing on the understanding

of side effects due to overstimulation, which include unpleasant sensations

and undesired motor effects. The dorsolateral columns (DLCs), located later-

ally outside the dorsal roots, consist of long axial myelinated fibres and are

understood to carry pain and temperature information. The DLCs include

many large fibres similar to those in the dorsal columns [64], but they are

not likely to be recruited under the large-fibre assumption [95]. Their in-

volvement may now have to be reassessed. Simulation of DLC fibres within

the ECAP model would provide some indication of the likely onset of recruit-

ment, and it may be possible to distinguish DLC responses in recordings

using laterally placed electrode arrays.

A second notable corollary of the large-fibre assumption is that neural

recruitment is limited to a layer at most 400 µm beneath the dorsal col-

umn surface [95]. In the ECAP model, larger fibres are stimulated at the

full model depth of 1mm around the end of the foot region; this suggests

that recruitment may extend much deeper than previously thought. The

dorsal columns are organised by modality, with cutaneous mechanosensory

afferents near the surface, and deep proprioceptive afferents in a deeper

layer [161]. A third set of fibres has been observed in between these two

consisting of post-synaptic fibres [10], most of which were found to ascend

to the brain. There are indications that these fibres may carry nocicep-

tive signals [43], [255]. This suggests a potential role for postsynaptic and

proprioceptive fibres in either the therapeutic or side-effect components of

SCS [115].

207

7.2.1 Unexplained Features

While the human data comparison shows good agreement on many

parameters, some aspects of human results are not fully explained. In

particular, the recorded amplitude changes more rapidly with propagation

distance than the model predicts. Dorsal column fibres are not uniform linear

cylinders, as used in the model; they follow complex paths near their points

of entry to the cord, changing in diameter and in many cases terminating

near their points of entry [99]. This is supported by recordings made in

unconscious patients at high currents in which responses can be observed

over multiple spinal segments. The amplitudes vary not only with distance

but with the location within the spinal segments, even increasing at some

points. Dorsal root fibres also pass into the cord through the CSF in the

vicinity of the dorsal columns. While these have been hypothesised to be

responsible for some of the side effects seen with overstimulation [95], no

conclusive evidence is yet available. Dorsal root activation should be visible

in ECAPs, but cannot yet be distinguished from dorsal columns.

This work used a fibre diameter distribution taken from a particular

individual, and a neural membrane model from a nonhuman species. It is

known that fibre populations vary significantly between individuals [64],

and ion channel behaviours can also change in individuals with neuropathic

conditions [83], [119], [221], [229], [230]. The relationship between the de-

tailed morphometry of the ECAP, the fibre distribution, and the ion channel

model requires further study. Techniques already exist for estimating diam-

eter distributions in peripheral nerves [13], [81], [248]. Similarly, some work

has been done on indirect measures of ion channel behaviour [38]. A better

understanding of these relationships in dorsal column ECAPs would have

benefits for diagnosis, long-term tracking, and for individualised treatment.

This will require an approach which considers the fibre population, ion

channels, and geometric factors simultaneously; this is a high-dimensional

problem. It is unlikely that these dimensions can all be measured simply

from ECAP current sweeps; more complex stimulus waveform sequences

will be needed to distinguish their effects. A sensitivity analysis using the

existing model would give initial direction to the effort, indicating which

parameters might be measured from existing datasets.

208

Arbitrary fibre paths, diameter changes, and branches can all be mod-

elled using the SCS model developed in this work. The higher degree of

parametrisation required to specify these for each fibre will bring new chal-

lenges. Firstly, the computational power required will increase, as the high

variation between individual fibres’ parameters is likely to preclude the use

of the parametric grid approximation. Given the continuously falling cost of

GPUs and their ability to work in parallel, this is not a significant hurdle to

further work.

The second and more substantial challenge is limited knowledge of fibre

paths and parameters. Most investigations of the dorsal columns to date

have looked at the paths or properties of a few individual fibres [10], [65],

[75], [99], [101], [161], [214]. Of more direct utility in modelling are tracto-

graphic surveys using magnetic resonance diffusion tensor imaging [129]

to determine the directions of axons and collaterals within volumetric cells.

These measurements suggest that fibre paths could be selected using a

statistical method which conforms to these measurements, reducing the

dimensionality of the problem. The ultimate in fidelity would be attained by

direct histological measurement of each fibre’s properties. Much attention is

presently being devoted to the measurement of fibre paths within the brain,

including staining techniques and software that reconstructs fibre paths

from serial slices of neural tissue. This technology is now at the point where

entire small animal brains can be processed [149]. The relative simplicity

of the dorsal columns compared to the brain suggest that this may soon

become a practicable approach.

7.2.2 Inter-Patient Comparison

The model results have been compared in detail with only a single pa-

tient in this work. Evidence from the Saluda corpus indicates that there is

considerable variation between patients in some characteristics. The per-

ceptual and discomfort thresholds of stimulation and the associated ECAP

amplitude vary by more than an order of magnitude between patients [176].

The ECAP amplitude is seen to increase roughly linearly with current above

some threshold in all patients; the visibility threshold can be above or be-

low the perceptual threshold. Conduction velocities are also similar in all

209

patients, suggesting that similar ensemble populations are being recruited.

Perception and discomfort thresholds are known to vary greatly with

electrode position [233] and the stimulated contact combination [15]. Model

results indicate that the recording amplitude is also a function of geometry.

It is possible to assess the position of the cord in the spinal canal using

magnetic resonance imaging (MRI) [89]; this would be of assistance in

further validation of the model.

7.3 Improving SCS

One of the primary drawbacks of SCS is its sensitivity to the patient’s

posture [60], [195]; this was selected as an area for further investigation. The

question of cord movement was examined using the validated ECAP model.

Varying the cord-electrode distance resulted in changes in the sensitivity

to stimulation, as well as to the amplitude of the resulting ECAPs. Greater

than sixfold variation in sensitivity was observed with a cord movement over

a range of 4.5mm; simulation demonstrated that this variation translates

into widely differing levels of recruitment when open-loop, constant-current

stimulation was applied. The magnitude of this change is in line with ECAP

recordings from patients. ECAP amplitudes have been observed to jump by

as much as ten times during motions such as coughing [179].

The development of ECAP recording was motivated by a desire to address

this variation using direct closed-loop control. An ECAP-based therapy is

currently under development which incorporates evoked response telemetry

capability into an implant; this is used to control the stimulus amplitude

to maintain the ECAP amplitude at a fixed level. The model was used to

examine the expected performance of constant-amplitude feedback, and

showed that this technique does result in improved constancy of recruitment

as the cord-electrode distance changes. However, it also demonstrated that

constant-amplitude feedback does not yield constant recruitment: instead, it

overcompensates, resulting in greater recruitment when the cord is further

from the electrode. This was shown to be due to the changing relationship

between recruitment and recording amplitude with distance.

Analysis of the stimulation and recording transfer functions with respect

210

to distance in the model suggests that these can be modelled as power laws,

which have previously been predicted both for stimulation [185] and record-

ing [222]. Knowledge of the relationship between these transfer functions

allows a feedback algorithm to split the difference: instead of controlling

current or voltage alone, a function of both variables can be controlled,

cancelling distance-related effects. This new technique, referred to as I-V

control, was demonstrated on the model. With a single parameter capturing

the model behaviour, I-V control improved constancy of recruitment as much

as tenfold over constant-amplitude control; this improvement was achieved

over a wide range of target recruitment setpoints.

I-V control now needs to be tested in human patients. Implementation

of the algorithm requires only software changes to the feedback implants

currently being prepared for trials. It is possible that better loop perfor-

mance can ultimately be obtained with I-V control than with constant-V

control. Under constant-V control, the loop gain depends on the patient’s

posture, and has been observed to vary by a factor of 10 in Saluda trials. This

then requires that the controller be configured very conservatively to ensure

stability. With I-V control, the loop can be configured such that the loop gain

remains almost constant with posture, allowing a more aggressive controller

to be used. The introduction of the nonlinear I-V control law will, however,

require extensive validation to ensure unconditional stability before it can

be deployed for patient use. The fibrous capsule which grows around the elec-

trode array after implantation is known to change the electrode impedance;

this should have no effect on a constant-current stimulus, but this remains

to be confirmed. Preparation for preclinical trials is ongoing.

Measurement of cord position using MRI would be helpful in validat-

ing this technique, as well as for developing automated fitting techniques.

Performing scans with implanted leads and stimulators requires that the

entire implant system be designed and validated for MRI compatibility; no

qualified ERT-capable SCS implant is currently available.

211

7.4 Further Questions

The presented work demonstrates the great utility of modelling in im-

proving SCS, with several model-derived techniques in various stages of

progress from simulation to implantation. There are a number of other

fronts within SCS on which these new approaches have yet to be brought to

bear. These include improvements to the rational design of therapeutic sys-

tems, as well as further advances in understanding of the electrophysiology

of SCS.

7.4.1 Artefact

The stimulus artefact remains a barrier to ECAP recording. This is

a particular problem for electrophysiological investigations; noise can be

dealt with by averaging, but artefact is highly correlated with the desired

responses. Virtual ground provides much improved performance compared

to traditional stimulation techniques, but investigation of low-amplitude

signals such as those in the foot region is still limited. Low-amplitude signals

are of particular interest in determining precisely which neuronal elements

are activated at the perceptual threshold; this is a step to determining

whether these are the elements responsible for the therapeutic benefits of

SCS, leading in turn to much more targeted systems of stimulation and

monitoring.

Gains are most likely to be made through model-based signal processing

techniques. At present, estimation and subtraction techniques are limited

by the nonlinear nature of the artefact, and the overlap in time of the neural

responses with the fastest and most problematic components of the artefact

waveforms. Virtual ground reduces the sources of nonlinearity by reducing

the electrode potentials at the recording electrodes during stimulation,

which can prevent electrochemical reactions from occurring. This should

allow the use of the body of existing techniques which assume a linear

artefact behaviour, such as independent component analysis [5]. Information

from ECAP modelling will be of use in discriminating between artefact and

responses at suprathreshold stimulation levels.

212

7.4.2 Electrode Design

Current electrode arrays for SCS are designed largely from a clinical

user’s perspective, particularly to provide good control over perceptual cov-

erage. Some efforts have been made with previous SCS models to validate

control over coverage [108], [130], or to optimise for selectivity [131]; how-

ever, these models may have made erroneous assumptions about the fibre

populations involved.

The introduction of a model validated against ECAP recordings opens

the door to an optimisation-based approach to electrode design. The new use

of ECAP recording in SCS also introduces additional goals: arrays now need

to provide high recording amplitudes, low noise, and low artefact. These

goals may conflict. A greater number of electrodes, as used for improving

control over coverage, provides also a greater number of potential sources of

artefact [204]. Larger electrodes provide several advantages: the interface

impedance is reduced, reducing power consumption and Johnson noise. The

increased area also provides a greater double-layer capacitance, which re-

duces the voltage associated with a given stored charge, as well as increasing

the current levels at which redox reactions begin; these can be expected to

reduce artefact. However, a larger electrode will also be exposed to a greater

voltage gradient across its surface during stimulation, which may contribute

to artefact. The recording amplitude is also affected by the axial extent of the

electrode; a longer electrode covers more of the tripolar field induced by the

travelling action potential, resulting in partial cancellation and reducing the

recording amplitude in a diameter-dependent fashion. This favours axially

short electrodes [9].

In its current form, the SCS ECAP model can be used to explore the

tradeoff space between power consumption, control over stimulation cov-

erage, and recording amplitude. Further work on artefact models will be

required to be able to include relevant properties in the optimisation process.

7.4.3 Neural Fidelity

The present work does not consider the effects on ECAPs of complex

stimulus waveforms, which may provide a way to probe the neural popu-

lation of the cord in more detail. Exposure to stimulation fields can alter

213

the threshold behaviour of neurons for some time thereafter, on scales from

microseconds to many minutes [26], [110]; this can be measured in the

cord using sequences of multiple pulses [78]. The stimulation pulse width

also has a diameter-dependent effect on the stimulation threshold [80]. The

neural model used in the present work is deliberately simple, but more

complex models have been developed which seek to explain post-stimulation

variations in behaviour using additional mechanisms [137]. A detailed ex-

ploration of the power efficiency and fibre selectivity of different stimulation

paradigms would benefit from a maximally accurate fibre model.

7.4.4 Response Localisation

Proper placement of the electrode array is known to be critical to the

ultimate benefit obtained with SCS; this is a time consuming process, which

must be performed using trial stimulation with perceptual feedback from

the patient, using fluoroscopic guidance, or a combination of the two. Lead

migration after implantation necessitates reprogramming or causes loss of

therapy in upwards of 20% of patients [7], [143], [216]. ECAP measurement

during lead placement may be able to guide lead placement more directly.

Peripheral recordings of evoked responses (SSEPs) and muscle responses

(EMG) stimulated by SCS leads have previously been used to predict out-

comes [213] and determine optimal lead placement [192] when using leads

implanted in the cervical region, where the use of head fixation means that

most patients are placed under general anaesthesia during the procedure.

Recording evoked responses on multi-electrode arrays provides a great

deal of information about the location of stimulated fibres, particularly

when using paddle leads with multiple electrodes placed laterally across

the cord. Early measurement results suggest that the lateral position of the

array with respect to the cord midline can be determined, and the responses

of dorsal roots distinguished from those of the dorsal columns [170]. The

modulation in ECAP amplitude seen as the action potential volley traverses

spinal segments and dorsal roots may provide a source of axial location

to complement the observed transverse information, and modelling efforts

should determine how these responses can be observed and interpreted. With

both axial and lateral location known, the use of peripheral measurements

214

and radiographic imaging during placement can be reduced or avoided.

7.4.5 Simple Models

Some applications would also benefit from a model that is substantially

simpler than that presented here. For example, a model that could be fit-

ted to a particular patient’s responses could be useful for implant fitting,

diagnostics, and long-term monitoring - but the nonlinear optimisation tech-

niques necessary to fit a high-dimensional model require many thousands

of simulations, and the stability of such a fit would be difficult to determine

with the complexity of the present model.

With the application of a single fixed stimulus with varying current, as in

the results presented here, the resulting behaviour is monotonic in several

ways. Fibres are recruited from largest to smallest, shallowest to deepest,

and from nearest the electrode to furthest away. Fibres that fire at one

current will always fire at higher currents; the only substantial difference in

these experiments is whether they fire on the first or the second stimulation

pulse of a biphasic stimulus (the second-cathode effect), a transition that

also happens at a particular current for each fibre. The SFAP of a given fibre

depends almost entirely on whether, when and where it was triggered.

This suggests an alternate processing of the model, by calculating thresh-

old currents and SFAPs for each fibre under each potential firing circum-

stance (such as each of the two pulses in a biphasic stimulus). The ECAP

evoked by a given stimulus current would then be determined simply by

summing the SFAPs of all fibres with equal or lower threshold. The cal-

culation of a growth curve with data in this form is essentially free, and

has the same resolution as the threshold determinations; time-consuming

and largely redundant simulations of all fibres with each current step are

avoided. For fibres of simple geometry such as all-axial fibre bundles, this

could then be abstracted further by fitting an equation to the threshold func-

tion within the fibre parameter space. This form of model would allow rapid

iteration with different fibre distributions, both in spatial and diametric

parameters.

This can be taken further by breaking the SFAP generation down to

its functional components. The SFAP originates from the passage of a cur-

215

rent through the nodes of Ranvier of each nerve fibre; if all fibres use the

same membrane model, the membrane current waveforms are very similar,

varying with nodal diameter. Given the vector of transimpedances from

these nodes to a recording electrode, the SFAP can be calculated using a

convolutional process which takes the internodal propagation time into

account. This then permits access to some of the more powerful tools of

linear algebra. Such formulations have previously been used for simulating

peripheral nerve recording [227], [253], and for then estimating the conduc-

tion velocity distribution in a fibre bundle from recordings [254]. A similar

approach has been used to model ECAPs of the auditory nerve after cochlear

stimulation [150].

7.4.6 Advanced Control

As with constant-amplitude feedback, I-V control makes the assumption

that perceptual intensity of stimulation varies with overall neural recruit-

ment. It is likely that intensity — and particularly the onset of discomfort

— is affected particularly by fibres carrying particular kinds of informa-

tion. At present, the populations responsible for therapeutic benefit and

for discomfort are not precisely known, but such information could allow a

more advanced controller to maintain or avoid recruitment in a particular

subpopulation using information from changes in the ECAP morphology.

These control methods also assume that only the cord-electrode distance

changes as a result of postural change. In practice, the cord moves axially

and laterally as well as dorso-ventrally. Detecting and compensating for

these movements is not possible with a single-input, single-output controller.

Axial movement can be determined by changes in the ECAP amplitude

and the stimulation sensitivity across several electrodes. Both the threshold

and the ECAP amplitude vary periodically within each spinal segment [77],

[177]. The stimulation position can then be adjusted axially to compensate,

for example by stimulating on two adjacent electrodes and adjusting the

current ratio between them. This technique is known as current steering,

presently used in manual programming techniques [131].

Lateral movement can only be detected when using a paddle electrode,

where recording electrodes at different lateral locations can be used to deter-

216

mine movement relative to the midline of the cord, including translation and

rotation [169]. This can then be used to change the lateral position of the

stimulus; again, varying the current ratio between two or more electrodes

allows the effective stimulus position to be varied.

Two major classes of problem need to be solved to achieve this. The

first is to develop algorithms to estimate the axial or lateral position of

the electrode array, which will require further mapping experiments, as

well as a way to fit the system to individual patients. The second class

is to find a way to compensate for electrode movement without affecting

the patient’s perceived stimulation. Lead placement and programming to

achieve therapeutic coverage currently require intensive manual effort from

expert clinical engineers, and it is not yet clear whether an automated

adjustment approach can be sufficiently simple to run in closed loop.

7.4.7 Neuropathies

There are numerous hypotheses about long-term changes to neural

behaviour being involved in chronic pain [120], [187], and about the effects

of SCS on this class of behaviours [244]. Some of these changes may be

observable using ECAP recording; for example, post-injury sprouting of fibre

terminals in the dorsal horn has been observed in animal models [113],

[128], [259], which might be expected to affect post-synaptic responses. This

is a prime target for further modelling work; histological investigations of

these changes have necessarily been confined to animals, and animal models

of pain are fundamentally unverifiable. Relevant information may well be

present in existing ECAP recordings, but we must know where to look.

If pain states are observable, then SCS systems could be used not only

to treat but to diagnose and monitor neuropathic conditions. This is particu-

larly important given that a great deal of chronic pain is caused by spinal

surgery; in one study, 30% of patients successfully undergoing lumbosacral

spinal surgery subsequently developed problems with pain [234]. Implanting

an SCS system for monitoring purposes at the time of surgery could permit

the early detection and treatment of problems.

217

Closing Remarks

This work presents the first model of SCS which has been able to be di-

rectly validated against human data. This model serves to explain several of

the salient features of human ECAP recordings, and yields new information

on the fibre populations targeted by this therapy. This model also permits

the first quantitative assessment of the strengths and weaknesses of voltage

feedback control of SCS. This provides the basis for a novel control technique

which promises to further improve control performance.

This work also demonstrates the practical application of models to im-

proving ECAP measurement. A novel stimulation technique reduces the

stimulation artefact at its source; this has now been implemented in SCS im-

plants, and can be expected to benefit other applications of evoked response

telemetry, such as in fitting cochlear implants.

SCS has been in use since the 1970s. The core technology remains

unchanged today, and understanding of its mechanisms of action remain

limited. The development of evoked potential recording opens the door

to a vast new stream of information; we must bend our efforts to deeper

understanding if we are to utilise it to its full potential.

218

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BM R G M O D E L C O D E

The external model used in the validation process was changed some-

what from its published form for this purpose, and so the new version is

reproduced here in full. The SCS model itself is too large for publication in

this format.

The original McIntyre-Richardson-Grill model code, sourced from [138],

is factored into three files:

• MRGglobals.hoc, setting default parameters and calculating diameter-

dependent ones,

• MRGaxon.hoc, implementing the electrical network model, and

• AXNODE.mod, implementing the membrane conductance model.

The hoc files have been split in two from the original to facilitate wrap-

ping in a test harness. This harness is in mrg.py, and depends on the

Python interface to NEURON. Code has been added to permit adjustment

of the membrane parameters from within Python, as well as to change the

form of the slow potassium current to match that used in the SCS fibre

model. Neither mechanism is used during validation. Finally, interfaces

in mrg_validate.py are used to establish both models with standard pa-

rameters. Default parameters to validation_model() result in unmodified

MRG and SCS fibre models.

250

Listing B.1: MRGglobals.hoc

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−2/02

Cameron C. McIntyre

SIMULATION OF PNS MYELINATED AXON

This model i s described in de ta i l in :

McIntyre CC, Richardson AG, and Gri l l WM. Modeling the e x c i t a b i l i t y o f

mammalian nerve f i b e r s : in f luence of a f t e r p o t e n t i a l s on the recovery

cy c l e . Journal o f Neurophysiology 87:995−1006, 2002.

This model can not be used with NEURON v5 .1 as errors in the

e x t r a c e l l u l a r mechanism of v5 .1 e x i s t re la ted to xc . The or ig inal

stimulations were run on v4 . 3 . 1 . NEURON v5 .2 has correc t ed the

l imi ta t ions in v5 .1 and can be used to run th i s model .

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/

proc model_globals ( ) {c e l s ius =37v_ in i t=−80 //mV//

dt =0.005 //ms//

tstop=10//Int race l lu lar stimuluation parameters//

istim=2delay=1pw=0.1

//topo log i ca l parameters//

axonnodes=21//morphological parameters//

fiberD =10.0 //choose from 5.7 , 7 .3 , 8 .7 , 10.0 , 11.5 , 12.8 , 14.0 , 15.0 ,

�→ 16.0

paralength1=3nodelength =1.0space_p1 =0.002space_p2 =0.004space_i =0.004

// e l e c t r i c a l parameters//

rhoa=0.7e6 //Ohm−um//

mycm=0.1 //uF/cm2/lamella membrane//

mygm=0.001 //S/cm2/lamella membrane//

g_pas_mysa=0.001g_pas_f lut =0.0001g_pas_stin =0.0001}

proc dependent_var ( ) {paranodes1 =(axonnodes−1)*2paranodes2 =(axonnodes−1)*2axoninter =(axonnodes−1)*6axontotal=axonnodes + paranodes1 + paranodes2 + axoninter

i f ( fiberD ==5.7) { g=0.605 axonD=3.4 nodeD=1.9 paraD1=1.9 paraD2=3.4�→ deltax=500 paralength2=35 nl =80}

i f ( fiberD ==7.3) { g=0.630 axonD=4.6 nodeD=2.4 paraD1=2.4 paraD2=4.6�→ deltax=750 paralength2=38 nl =100}

i f ( fiberD ==8.7) { g=0.661 axonD=5.8 nodeD=2.8 paraD1=2.8 paraD2=5.8�→ deltax=1000 paralength2=40 nl =110}

i f ( fiberD ==10.0) { g=0.690 axonD=6.9 nodeD=3.3 paraD1=3.3 paraD2=6.9�→ deltax=1150 paralength2=46 nl =120}

i f ( fiberD ==11.5) { g=0.700 axonD=8.1 nodeD=3.7 paraD1=3.7 paraD2=8.1�→ deltax=1250 paralength2=50 nl =130}

251

i f ( f iberD ==12.8) { g=0.719 axonD=9.2 nodeD=4.2 paraD1=4.2 paraD2=9.2�→ deltax=1350 paralength2=54 nl =135}

i f ( fiberD ==14.0) { g=0.739 axonD=10.4 nodeD=4.7 paraD1=4.7 paraD2=10.4�→ deltax=1400 paralength2=56 nl =140}

i f ( fiberD ==15.0) { g=0.767 axonD=11.5 nodeD=5.0 paraD1=5.0 paraD2=11.5�→ deltax=1450 paralength2=58 nl =145}

i f ( fiberD ==16.0) { g=0.791 axonD=12.7 nodeD=5.5 paraD1=5.5 paraD2=12.7�→ deltax=1500 paralength2=60 nl =150}

Rpn0=( rhoa * .01 ) / ( PI * ( ( ( ( nodeD / 2 ) +space_p1 ) ^2) −((nodeD / 2 ) ^2) ) )Rpn1=( rhoa * .01 ) / ( PI * ( ( ( ( paraD1 / 2 ) +space_p1 ) ^2) −((paraD1 / 2 ) ^2) ) )Rpn2=( rhoa * .01 ) / ( PI * ( ( ( ( paraD2 / 2 ) +space_p2 ) ^2) −((paraD2 / 2 ) ^2) ) )Rpx=( rhoa * .01 ) / ( PI * ( ( ( ( axonD / 2 ) +space_i ) ^2) −((axonD / 2 ) ^2) ) )interlength =( deltax−nodelength −(2* paralength1 ) −(2*paralength2 ) ) /6}

Listing B.2: MRGaxon.hoc

/*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−2/02

Cameron C. McIntyre

SIMULATION OF PNS MYELINATED AXON

This model i s described in de ta i l in :

McIntyre CC, Richardson AG, and Gri l l WM. Modeling the e x c i t a b i l i t y o f

mammalian nerve f i b e r s : in f luence of a f t e r p o t e n t i a l s on the recovery

cy c l e . Journal o f Neurophysiology 87:995−1006, 2002.

This model can not be used with NEURON v5 .1 as errors in the

e x t r a c e l l u l a r mechanism of v5 .1 e x i s t re la ted to xc . The or ig inal

stimulations were run on v4 . 3 . 1 . NEURON v5 .2 has correc t ed the

l imi ta t ions in v5 .1 and can be used to run th i s model .

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/

objectvar stim

create node [ axonnodes ] , MYSA[ paranodes1 ] , FLUT[ paranodes2 ] , STIN[ axoninter ]access node [ 0 ] //APD

proc i n i t i a l i z e ( ) {for i =0 ,axonnodes−1 {

node [ i ] {nseg=1diam=nodeDL=nodelengthRa=rhoa /10000cm=2insert axnodeinsert extrace l lu lar xraxial=Rpn0 xg=1e10 xc=0}

}for i =0 , paranodes1−1 {

MYSA[ i ] {nseg=1diam=fiberDL=paralength1Ra=rhoa * ( 1 / ( paraD1 / fiberD ) ^2) /10000cm=2*paraD1 / fiberDinsert pasg_pas=g_pas_mysa*paraD1 / fiberDe_pas=v_ in i t

252

insert extrace l lu lar xraxial=Rpn1 xg=mygm/ ( nl *2) xc=mycm/ ( nl *2)}

}for i =0 , paranodes2−1 {

FLUT[ i ] {nseg=1diam=fiberDL=paralength2Ra=rhoa * ( 1 / ( paraD2 / fiberD ) ^2) /10000cm=2*paraD2 / fiberDinsert pasg_pas=g_pas_f lut *paraD2 / fiberDe_pas=v_ in i tinsert extrace l lu lar xraxial=Rpn2 xg=mygm/ ( nl *2) xc=mycm/ ( nl *2)}

}for i =0 , axoninter−1 {

STIN[ i ] {nseg=1diam=fiberDL=interlengthRa=rhoa * ( 1 / ( axonD / fiberD ) ^2) /10000cm=2*axonD / fiberDinsert pasg_pas=g_pas_stin *axonD / fiberDe_pas=v_ in i tinsert extrace l lu lar xraxial=Rpx xg=mygm/ ( nl *2) xc=mycm/ ( nl *2)}

}for i =0 , axonnodes−2 {

connect MYSA[2* i ] ( 0 ) , node [ i ] ( 1 )connect FLUT[2* i ] ( 0 ) , MYSA[2* i ] ( 1 )connect STIN[6* i ] ( 0 ) , FLUT[2* i ] ( 1 )connect STIN[6* i +1] (0 ) , STIN[6* i ] ( 1 )connect STIN[6* i +2] (0 ) , STIN[6* i +1] (1 )connect STIN[6* i +3] (0 ) , STIN[6* i +2] (1 )connect STIN[6* i +4] (0 ) , STIN[6* i +3] (1 )connect STIN[6* i +5] (0 ) , STIN[6* i +4] (1 )connect FLUT[2* i +1] (0 ) , STIN[6* i +5] (1 )connect MYSA[2* i +1] (0 ) , FLUT[2* i +1] (1 )connect node [ i +1] (0 ) , MYSA[2* i +1] (1 )}

}i n i t i a l i z e ( )

Listing B.3: AXNODE.mod

TITLE Motor Axon Node channels

: 2/02

: Cameron C. McIntyre

:

: Fast Na+ , Pers is tant Na+ , Slow K+ , and Leakage currents

: responsib le for nodal action po t en t ia l

: I t e r a t i v e equations H−H notation r e s t = −80 mV

:

: This model i s described in de ta i l in :

:

: McIntyre CC, Richardson AG, and Gri l l WM. Modeling the e x c i t a b i l i t y o f

: mammalian nerve f i b e r s : in f luence of a f t e r p o t e n t i a l s on the recovery

253

: c y c l e . Journal o f Neurophysiology 87:995−1006, 2002.

INDEPENDENT { t FROM 0 TO 1 WITH 1 (ms) }

NEURON {SUFFIX axnodeNONSPECIFIC_CURRENT inaNONSPECIFIC_CURRENT inapNONSPECIFIC_CURRENT ikNONSPECIFIC_CURRENT i lRANGE gnapbar , gnabar , gkbar , gl , ena , ek , e lRANGE mp_inf , m_inf , h_inf , s_ in fRANGE tau_mp , tau_m , tau_h , tau_sRANGE ampA, ampB, ampCRANGE bmpA, bmpB, bmpCRANGE amA, amB, amCRANGE bmA, bmB, bmCRANGE ahA, ahB, ahCRANGE bhA, bhB, bhCRANGE use_schwarz_ksRANGE asA , asB , asCRANGE bsA , bsB , bsC

}

UNITS {(mA) = ( milliamp )(mV) = ( m i l l i v o l t )

}

PARAMETER {

gnapbar = 0.01 (mho/ cm2)gnabar = 3.0 (mho/ cm2)gkbar = 0.08 (mho/ cm2)gl = 0.007 (mho/ cm2)ena = 50.0 (mV)ek = −90.0 (mV)e l = −90.0 (mV)ce l s ius ( degC )dt (ms)v (mV)vtraub=−80ampA = 0.01ampB = 27ampC = 10.2bmpA = 0.00025bmpB = 34bmpC = 10amA = 1.86amB = 21.4amC = 10.3bmA = 0.086bmB = 25.7bmC = 9.16ahA = 0.062ahB = 114.0ahC = 11.0bhA = 2.3bhB = 31.8bhC = 13.4

254

: whether to use Schwarz ’ s equations for slow K

use_schwarz_ks = 0asA = 0.3asB = −27asC = −5bsA = 0.03bsB = 10bsC = −1

}

STATE {mp m h s

}

ASSIGNED {inap (mA/ cm2)ina (mA/ cm2)ik (mA/cm2)i l (mA/ cm2)mp_infm_infh_infs_ in ftau_mptau_mtau_htau_sq10_1q10_2q10_3

}

BREAKPOINT {SOLVE states METHOD cnexpinap = gnapbar * mp*mp*mp * ( v − ena )ina = gnabar * m*m*m*h * ( v − ena )ik = gkbar * s * ( v − ek )i l = gl * ( v − e l )

}

DERIVATIVE states { : exact Hodgkin−Huxley equations

evaluate_fct ( v )mp’= ( mp_inf − mp) / tau_mpm’ = ( m_inf − m) / tau_mh ’ = ( h_inf − h) / tau_hs ’ = ( s_ in f − s ) / tau_s

}

UNITSOFF

INITIAL {:

: Q10 adjustment

:

q10_1 = 2.2 ^ ( ( ce ls ius −20) / 10 )q10_2 = 2.9 ^ ( ( ce ls ius −20) / 10 )q10_3 = 3.0 ^ ( ( ce ls ius −36) / 10 )

evaluate_fct ( v )mp = mp_infm = m_inf

255

h = h_infs = s_ in f

}

PROCEDURE evaluate_fct ( v (mV) ) { LOCAL a , b , v2

a = q10_1*vtrap1 ( v )b = q10_1*vtrap2 ( v )tau_mp = 1 / ( a + b )mp_inf = a / ( a + b )

a = q10_1*vtrap6 ( v )b = q10_1*vtrap7 ( v )tau_m = 1 / ( a + b )m_inf = a / ( a + b )

a = q10_2*vtrap8 ( v )b = q10_2*bhA / (1 + Exp(−(v+bhB) /bhC) )tau_h = 1 / ( a + b )h_inf = a / ( a + b )

i f ( use_schwarz_ks ) {a = q10_3*asA*( v+asB ) /(1−exp ((−asB−v ) / asC ) )b = q10_3*bsA*(−bsB−v ) /(1−exp ( ( v+bsB ) / bsC ) )

} e lse {v2 = v − vtraub : convert to traub convention

a = q10_3*asA / (Exp ( ( v2+asB ) / asC ) + 1)b = q10_3*bsA / (Exp ( ( v2+bsB ) / bsC ) + 1)

}tau_s = 1 / ( a + b )s_ in f = a / ( a + b )

}

FUNCTION vtrap ( x ) {i f ( x < −50) {

vtrap = 0} e lse {

vtrap = bsA / (Exp ( ( x+bsB ) / bsC ) + 1)}

}

FUNCTION vtrap1 ( x ) {i f ( fabs ( ( x+ampB) /ampC) < 1e−6) {

vtrap1 = ampA*ampC} e lse {

vtrap1 = (ampA*( x+ampB) ) / (1 − Exp(−(x+ampB) /ampC) )}

}

FUNCTION vtrap2 ( x ) {i f ( fabs ( ( x+bmpB) /bmpC) < 1e−6) {

vtrap2 = bmpA*bmpC : Ted Carnevale minus sign bug f i x

} e l se {vtrap2 = (bmpA*(−(x+bmpB) ) ) / (1 − Exp ( ( x+bmpB) /bmpC) )

}}

FUNCTION vtrap6 ( x ) {i f ( fabs ( ( x+amB) /amC) < 1e−6) {

vtrap6 = amA*amC} else {

vtrap6 = (amA*( x+amB) ) / (1 − Exp(−(x+amB) /amC) )

256

}}

FUNCTION vtrap7 ( x ) {i f ( fabs ( ( x+bmB) /bmC) < 1e−6) {

vtrap7 = bmA*bmC : Ted Carnevale minus sign bug f i x

} e l se {vtrap7 = (bmA*(−(x+bmB) ) ) / (1 − Exp ( ( x+bmB) /bmC) )

}}

FUNCTION vtrap8 ( x ) {i f ( fabs ( ( x+ahB) / ahC) < 1e−6) {

vtrap8 = ahA*ahC : Ted Carnevale minus sign bug f i x

} e l se {vtrap8 = (ahA*(−(x+ahB) ) ) / (1 − Exp ( ( x+ahB) / ahC) )

}}

FUNCTION Exp( x ) {i f ( x < −100) {

Exp = 0} e lse {

Exp = exp ( x )}

}

UNITSON

Listing B.4: mrg.py

import s i t es i t e . addsitedir ( ’ / usr / l o c a l / nrn / l i b / python ’ )

from neuron import himport numpy as np

def model ( globals = { } , dependents = { } , axnode = { } ) :# make a new model .

# overr ide values s e t in model_globals ( ) , dependent_var ( ) , or in AXNODE.

�→ mod.

# make sure no junk i s around from the l as t run

h( ’ f o r a l l de le te_sect ion ( ) ’ )

h . xopen ( ’ MRGglobals . hoc ’ )

# default independent globals

h . model_globals ( )for item in globals . items ( ) :

h ( ’%s = %s ’ % item )

# default dependents on the globals

h . dependent_var ( )for item in dependents . items ( ) :

h ( ’%s = %s ’ % item )

h . xopen ( ’MRGaxon. hoc ’ )

for (name, value ) in axnode . items ( ) :for i in range ( int (h . axonnodes ) ) :

257

exec ’ h . node [ i ].% s_axnode = value ’ % name

def point_ locs ( ) :# x coordinates for centre o f s e c t i ons

# into a s ing le vec tor sui tab le for calculat ing G−matrices

h( ’ f o r a l l pt3dclear ( ) ’ )h ( ’ f o r a l l define_shape ( ) ’ ) # ca lcu la t es an x−coord for every point

out = [ ]for sec in h . a l l s e c ( ) :

i f h . n3d ( sec=sec ) != 3 :raise IOError ( " Expected 3 3d points per section , too confused to

�→ continue " )out . append (h . x3d (1 , sec=sec ) )

return out

def simulate (G, stim_vec , tstep , tstop , node_record=None) :h . dt = tstep

def rec ( seg , val ) :v = h . Vector ( )v . record ( eval ( " seg . _re f_%s " % val ) , h . dt )return v

i_outs = [ ]i_areas = [ ]for sec in h . a l l s e c ( ) :

for seg in sec : # should only be 1

i _ rec = h . Vector ( )i _ rec . record ( seg . _ref_i_membrane , h . dt )# i : mA/cm^2

i_outs . append ( i _ rec )# area : um^2 −> cm^2

i_areas . append (h . area ( 0 . 5 , sec=sec ) / 1e8 )

rec_outs = Nonei f node_record is not None :

rec_outs = [ [ rec ( seg , name) for sec in h . node for seg in sec ] for

�→ name in node_record ]

h . f i n i t i a l i z e (h . v_ in i t )h . fcurrent ( )

while h . t < tstop :i f len ( stim_vec ) and h . t >= stim_vec [ 0 , 0 ] :

Vext = np .sum( stim_vec [ 0 , 1 : ] *G, axis =1)for ( i , sec ) in enumerate (h . a l l s e c ( ) ) :

sec . e_extrace l lu lar = Vext [ i ]stim_vec = stim_vec [ 1 : , : ]

h . fadvance ( )

currents = np . array ( i_outs ) * np . array ( i_areas ) [ : , None]Vrec = np .sum( currents [ : , : , None]*G[ : , None , : ] , axis =0)i f rec_outs is not None :

rec_outs = map(lambda x : np . array ( x ) .T, rec_outs ) # column per node

�→ − more c i v i l i s e d

return ( Vrec , currents , rec_outs )

def calc_G ( dist , elec_pos , Rext=3e6 ) :# dis t = distance e l ec t rodes −f i b r e axis (um)

# e lec_pos = longitudinal pos i t ion of e l e c t r o d e s (um, can be l i s t )

# Rext = volume conductor r e s i s t i v i t y (ohm um)

258

node_pos = np . array ( po int_ locs ( ) )G = Rext / 4 / np . pi / np . sqrt ( ( np . array ( elec_pos ) [None , : ] − node_pos [ : , None ] )

�→ **2 + dis t **2)return G

def mk_stim_vec (pw, current , n_elec =2 , stim_pos =1 , stim_neg=None) :# pw ( us ) , current (mA)

# the " pos " e l e c has the same sign as ’ current ’ ; " neg " i s the opposi te

out = np . zeros ( ( 2 , n_elec +1) )out [1 ,0 ] = pw/1 e3out [0 , stim_pos ] = currenti f stim_neg is not None :

out [0 , stim_neg ] = −currentreturn out

i f __name__ == " __main__ " :g_params = { ’ axonnodes ’ : 101 , ’ f iberD ’ : 10 .0}d_params = { }

model ( g_params , d_params )

G = calc_G (3000 , [1000 , 35000])stim_vec = mk_stim_vec (40 , −20)

tstep = 0.001tstop = 5( Vrec , i_memb , x_memb) = simulate (G, stim_vec , tstep , tstop , [ ’ m_axnode ’

�→ ] )

from pylab import *# th i s i s in m i l l i v o l t s

plot ( arange (0 , tstop+tstep , tstep ) , Vrec [ : , 1 ] )x label ( ’Time (ms) ’ )y label ( ’ Amplitude (mV) ’ )show ( )

Listing B.5: mrg_validate.py

# Parameter s e t s used for validation .

import mrgimport numpy as np

def validation_model ( diam=10.0) :g_params = { ’ axonnodes ’ : 200 , ’ f iberD ’ : diam }d_params = { }a_params = { }

mrg . model ( g_params , d_params , a_params )

# f i b r e 3mm away , with e l e c t r o d e s 15mm and 50mm along

G_nrn = mrg . calc_G (3000 , [15000 , 50000])

# 40us , 1mA cathodic pulse ( unit stim )

unit_stim = mrg . mk_stim_vec (40 , −1)

# run for 3ms with 1us steps

( tstep , tstop ) = (0 .001 , 3)

f = mkfiber ( G_nrn )

return ( G_nrn , f , unit_stim , tstep , tstop )

259

def mkfiber ( G_nrn ) :# create an SCS model Fibre instance

import f i b e rimport f i b e r _ t o o l sfrom neuron import h(Gaxon , nodal_area ) = f i b e r _ t o o l s . e lec (h . deltax , h . fiberD )

G_mdl = G_nrn [ : h . axonnodes ]

f = f i b e r . Fiber (G_mdl , Gaxon , nodal_area )

return f

260