an improved method of voltage clamp experiment parameter estimation

8/12/2019 An improved method of voltage clamp experiment parameter estimation

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Journal of Theoretical Biology 242 (2006) 123134

HodgkinHuxley type ion channel characterization: An improved

method of voltage clamp experiment parameter estimation

Jack Lee, Bruce Smaill, Nicolas Smith

Bioengineering Institute, Level 6, 70 Symonds Street, University of Auckland, Auckland, New Zealand

Received 27 October 2005; received in revised form 7 February 2006; accepted 10 February 2006

Available online 24 March 2006

Abstract

The HodgkinHuxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological

models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the

assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows

that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach

to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior

accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time

constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A

sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably

amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel

parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Parameter estimation; Fitting; HodgkinHuxley; Cell model; Voltage clamp

1. Introduction

Since its introduction in 1952 (Hodgkin and Huxley,

1952c), the HodgkinHuxley (HH) formalism has had

profound impact on the development of integrated ionic

current models of cell excitability. It has been used to

model electrical activation in a range of cell types where

increasingly detailed representations of whole cell electro-

physiology have been developed by combining HH

channels. Specific examples include models of electrical

activation in ventricular myocytes (Beeler and Reuter,1977; Luo and Rudy, 1991, 1994), Purkinje fibers (Di

Francesco and Noble, 1985;McAllister et al., 1975;Noble,

1962), neurons (Plant and Kim, 1976; Shorten and Wall,

2000) and smooth muscles (Lang and Rattray-Wood, 1996;

Miftakhov et al., 1999). In the HH formalism, the flux of

ions through a membrane channel is driven by the net

electro-chemical potential for that ion, while the time-

dependence of channel conductance is associated with

molecular gates incorporated into the ion channel. An

individual channel may have several independently behav-

ing activation and inactivation gates arranged in series and

it is their combined behavior that governs its conductance.

Each gate is modeled as a first order kinetic process, with a

time constant that is assumed to be a function of

membrane potential.

Recently, there has been increasing use of Markov

models to represent membrane channel conductance. Thisapproach extends and generalizes the HH formalism to

accommodate multiple-state gating kinetics and it enables

macroscopic currents to be consolidated from experimental

single channel data. However, the HH formalism is still the

approach-of-choice in the emerging field of multi-scale

modeling of organ function (Hunter et al., 2003; Winslow

et al., 2000). The integration of models from cell to organ

level is computationally intensive and the use of Markov

models within this context is limited by the substantial

additional demands that they impose. Reflecting this,

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doi:10.1016/j.jtbi.2006.02.006

Corresponding author. Tel.: +6421 825010 (mob),

+649 3737599x83055 (work).

E-mail address: [email protected] (J. Lee).
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biophysically based models of cellular electrophysiology

incorporated into large-scale cardiac tissue models have

almost without exception utilized HH kinetics. Moreover,

it seems certain that the HH models will continue to be

used in this subset of integrative computational biology for

the foreseeable future. Although some attempts have been

made to adapt the HH formalism to better fit newerexperimental data while retaining its computationally

efficient structurein squid axon, for example (Clay,

1998, 2005)little attention has been paid to developing

a systematic method to aid this process. The efficacy of

such models to represent membrane channel behavior will

benefit from searching not only for improved empirical

expressions, but also by constructing methods that will

estimate best-fit parameters accurately and efficiently from

experimental data.

The voltage clamp technique (Cole, 1949) is widely used

to characterize membrane ion channel activation and

inactivation parameters in the HH formulation. In a

typical scenario, the time-course of ionic current following

a voltage step is observed, and from this, the number of

gates that best reproduces the current is determined. At

each of the clamp voltages, steady-state activation vari-

ables are estimated from peak currents and activation time

constants are also calculated. Both are then approximated

as smooth functions of membrane potential. The conven-

tional approach to parameter identification, within this

setting, has two significant flaws. Firstly, it is generally

assumed that, because inactivation occurs relatively slowly,

its influence on activation can be discounted, which makes

it possible to estimate activation and inactivation variables

independently. However, the time-course of inactivationcan affect the rise in ionic current in the later phases of

activation even when activation and inactivation time

constants are well separated. This will lead to lower

apparent rates of activation and reduced peak currents,

and simple simulations employing HH gating kinetics

indicate that the errors introduced may be substantial.

Secondly, inactivation variables are not characterized from

the single step protocol used to identify activation

parameters. Instead, these parameters are obtained in a

related fashion using two pulse experiments typically

performed after the single step protocol. Despite half a

century of widespread usage of HH formalism, there has

been no analysis of either the errors introduced by

neglecting the effects of inactivation when estimating

activation variables, or whether it is possible to character-

ize both activation and inactivation parameters using data

from a single experimental protocol.

The objectives of the research outlined in this paper are

to (i) analyze the errors introduced in estimating gating

variables caused by the assumption that the time-scales of

activation and inactivation are so different, thus they can

be decoupled, and (ii) to develop numerical techniques that

will enable more accurate estimates of the variables to be

made from more compact experimental data sets. In the

first of the following sections, a simplified case (when

steady-state activation equals 1 and inactivation equals 0)

will be considered for a generic ion channel to obtain the

worst-case bounds of error that could be generated from

using traditional parameter estimation methods. A simple

analytic correction is proposed to reduce this error

significantly. Then in the next section, a general analysis

is carried out leading to the formulation of the minimiza-tion problem and its numerical solution. Using this method

the performance of both the new approach and the

traditional parameter estimation technique are contrasted

in a set of simulated voltage clamp experiments. Following

this, a sensitivity analysis for the new technique is carried

out to assess its robustness under conditions where noisy

experimental input parameters are provided.

2. HodgkinHuxley approach

In this section, we define the parameters involved in the

HH description of current flow through a membrane

channel and outline the processes of experimental identi-

fication of channel gating parameters. Net transmembrane

current flow I for a given ion is described below:

I gmPhQVm E, (1)

where m and h represent the proportions of activation and

inactivation gates, respectively, that are open. Pand Q are

the numbers of independent gates required to account

for the observed time-course of activation or inactivation.

Vm is membrane potential, while E denotes the Nernst

potential for the ion under consideration. The variable g

scales for maximum channel conductance. The general

description above can be applied for different ion species.Gating variables are modeled as first-order kinetic pro-

cesses and therefore

dx

dt a1 x bx

x1 x

tx, (2)

where x denotes either m or h in Eq. (1). a and b are rate

constants for gate opening and closing, respectively, and

both are functions ofVm. Note that the final expression in

Eq. (2) presents an equivalent form in which the steady-

state activation/inactivation parameter (xN) and time

constant (tx) are used.

Results for an HH simulation of the current flowing

through a sodium channel are given in Fig. 1. The gating

variables used are those identified byHodgkin and Huxley

(1952c) under experimental conditions identical to those

simulated. A step increase in Vm of 60 mV was imposed

from a holding potential in which the membrane is

hyperpolarized for a sufficient period to ensure that both

activation and inactivation gates are completely reset.

There is a sigmoidal rise in m3 with time and a slower

exponential fall in h. Because time constants for activation

and inactivation at this membrane potential differ only by

a factor of 4 at this voltage (0.27 and 1.05 ms fortmand th,

respectively), the combined gating variable m3his less than

40% of the steady-state activation under these conditions.

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J. Lee et al. / Journal of Theoretical Biology 242 (2006) 123134124


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In addition, the rate of rise of the combined gating variable

is significantly slower than would be predicted on the basis

of activation time constant alone. Typically, when the ratio

of activation and inactivation time constants is in the range

1050, it is assumed that inactivation can be ignored when

estimating gating parameters. Our preliminary results

showed that the errors introduced by this assumption

may be on the order of 1030%. The analysis which

follows seeks to characterize this source of error in a more

systematic fashion.

3. Analysis for simplified case

3.1. Steady-state activation

We begin by considering a simple hypothetical ion

channel, with one activation gate (P 1) and one

inactivation gate (Q 1), denoted by dand f, respectively.

From Eq. (2), the status of these gates is given by

d d1 d0 d1et=td andf f1 f0 f1e

t=tf .

(3)

It may be presumed that activation and in-

activation gates are both fully reset due to strong

conditioning hyperpolarization, i.e. d0 0 and f0 1. In

the simplified expression below, it is assumed that thevoltage step leads to d1 1 and f1 0. This is also the

worst-case scenario for producing error in the peak

conductance:

d f 1 et=td

et=tf . (4)

Apart from the variables d and f, all other terms ( g, V

andE) that are multiplied to giveIare constant during the

voltage clamp. To find its maximum value we differentiate

Eq. (4) with respect to time

d df

dt

et=tf 1

tf

td tf

tdtfet=td (5)

and set this to zero, to find the time t* at which peak

current occurs

t td ln tf

td 1

. (6)

Substituting Eq. (6) back into Eq. (4), the maximum

value ofdf is

dfmax tf

tf td

td

tf td

td=tf

1 1

t 1

1 t1=t, 7

where the non-dimensional parametert tf=tdis the ratioof activation and inactivation time constants.

Fig. 2displays (df)max, as a function oft. Sinced1 1 in

this case, 1(df)max is the error introduced by assuming

thatd1can be estimated directly from the peak current on

activation. As foreshadowed previously, this error can be

substantial when inactivation and activation time constants

are not well separated.

3.2. Activation time constant

Using a similar approach, we will now consider the

errors introduced in estimating td when the effects ofinactivation are assumed to be negligible. Typically, td is

evaluated by fitting a single exponential function to the

voltage clamp trace. Provided that activation and inactiva-

tion gates are both fully reset (i.e. d0 0 and f0 1), this

process can be described as finding tcdsuch that

et=tf 1 et=td

ffi C 1 et=tcd

; tot, (8)

where gVm E on both sides of the equation has been

cancelled out.

In this case the superscript c is used to denote

conventional methods of estimating gating variables. This

nomenclature will be used hereinafter.

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0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (ms)

Gatingvariables

Activation

Inactivation

Relative conductance

Fig. 1. Time-dependent change in activation and inactivation variables

following a voltage step and the resulting relative conductance, calculated

as the product of the gating variables.

0 100 200 3000.2

0.4

0.6

0.8

1

dfmax

Fig. 2. Observed peak relative conductancedfmaxplotted as a function of

t, the ratio of inactivation time constant to activation time constant. Since

dN

is 1 in this case,dfshown here may be interpreted as the experimentally

derived value ofdN

, by the conventional approach.

J. Lee et al. / Journal of Theoretical Biology 242 (2006) 123134 125


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The constant C on the right-hand side is an arbitrary

scaling term which will reflect g and (VE), as the

equation is commonly fitted to the current-time trace

rather thandf-time trace. Interestingly, the inclusion of this

constant in the fitting process may provide some

compensation for error introduced by neglecting the

inactivation process. As this problem is solved as a

nonlinear least-squares fit and because the time range of

fit commonly varies anywhere up to t*, the exact value

oftcd and C can be specified only on a case-by-case basis.

We have therefore used the following example. The

activation time constant td was set to 10 ms, and tfwas varied so that 1oto300. Least-squares fits were

performed for 0otot for each tf, with an RMSerror p108. Fig. 3 presents the ratio of tcd=td, a s afunction oft and it shows a nearly identical trend toFig. 2.

That is, while errors in estimatingtdmay be substantial for

to30, there is still significant residual error (410%) when

t 30. This is intuitive once we realize that the most

rapid decrease in inactivation variable occurs in the early

phase of the rise in transmembrane current, due to its

exponential nature.

4. General analysis

For a channel with activation and inactivation gates, we

identify three characteristic features in the experimentally

measured membrane current generated by a voltage step.

These are (i) peak current (ii) time to peak current and

(iii) steady-state current. It is possible to derive general

expressions from the HH formulation which relate to each

of these. We again assume that d0 0 and f0 1, but the

values ofd1andf1 are no longer constrained to be 1 and

0, respectively. Thus,

df d1 1 et=td f1 1 f1et=tfh i. (9)

To determine the time to peak current t*, we calculate

the rate of change ofdf

ddf

dt

d1f1td

et=td d11 f1

tfet=tf

d11 f1td tf

tdtf

ettd

ttf . 10

Setting Eq. (10) equal to 0 and multiplying by tdtf=d1we obtain

f1tfet=td 1 f1tde

t=tf 1 f1td tfet

tdt

tf 0.

(11)

From Eqs. (9) and (11), we can derive equations for the

experimentally measured inputs t* and (df)max. These will

be nonlinear functions of the variables td, tf, fN and dN.

In addition, it is possible to construct another equation for

the measuredd1f1in a similar manner. Because there are

four parameters to determine, an additional equation must

be prescribed in order to yield a complete system. As this isa system of nonlinear equations, a simultaneous numerical

solution scheme is appropriate. The formulation and the

solution of this system of equations are dealt with in the

following section.

4.1. Numerical solution for the general case

To solve the set of equations derived in the above

analysis, we have employed a multidimensional nonlinear

minimization procedure. The first component of the

objective function is calculated as the squared residual of

Eq. (11), using experimentally determined t* as the input:

R1 f1tfet=td 1 f1tde

t=tf

1 f1td tfet

tdt

tf

2. 12

The second component can be obtained from Eq. (9),

using the experimental measurement of (df)max:

R2 d1 1 et=td

f1 1 f1e

t=tfh i

dfmax

h i2.

(13)

The third equation d1f1exp d1f1is not formulatedexplicitly as a residual term, but is used to modify R2 so

thatd1can be removed from the minimization process via

Eq. (14):

d1 d1f1exp

f1. (14)

One further component is necessary in the objective

function in order to solve for td, tf, fN and the following

residual is used:

R3 d1 1 e2t=td

f1 1 f1e

2t=tf

h i df

2t

h i2

,

(15)

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0 100 200 3000.2

0.4

0.6

0.8

1

cd/d

Fig. 3. Estimated time constant tcd normalized against the actual time

constant td as a function oft.



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where d1 can be replaced using Eq. (14) as before. Our

experience suggests that choosing a point on the dfcurve

where t4t provides a system of equations most likely to

converge to the correct solution. However, if tbt the

problem will likely be ill-defined, since dfapproachesdN

fN

asymptotically. For this reason, we have elected to use

t 2t

. There is no reason why R3 must be constructedfrom a single data point. It is possible, and sometimes

advantageous, to use several points or the entire trace data

set and this is discussed further in a later section.

The full objective function can then be assembled as

Rf1; td; tf R1

t

R2

dfmax

R3

df

2t

. (16)

Here, the residual terms are normalized to ensure that

their contributions to the objective function are compar-

able. Strictly, the denominators should be squared, but in

practice we have found Eq. (16) to perform better. The

minimization problem was solved using the NelderMead

simplex algorithm (Nelder and Mead, 1965). This is an

unconstrained, derivative-free minimization method suita-

ble for objective functions in multiple dimensions. Unlike

gradient descent methods, this algorithm does not involve

differentiation of the objective function and typically

requires only one or two function evaluations per iteration

(Lagarias et al., 1998). Although it is highly efficient,

convergence is not guaranteed with this method. In our

case, this is not a problem because an approximate starting

solution is always known.

For a test case in which td 10 ms, tf 300 ms,

f1 0:22, d1 0:85 and where initial parameter esti-

mates were varied, solutions were obtained to within104% in all cases where correct convergence was achieved.

Failure of convergence was rarely observed, and occurred

only when initial estimates for activation and inactivation

variables differed substantially (in the order of7100%)

from the correct values.

4.2. Simulated experiment: Activation protocol

The conventional method of determining activation

parameters from voltage clamp experiments is compared

here with the method outlined above. A hypothetical ion

channel

d1 1

1 exp Vm1010

; f1 11 exp Vm10

10

,

td 3 exp Vm 10

20

2 !

2:5; tf 10 exp Vm 5

80

2 ! 50,

g 0:5 mS=cm2 and E 0 mV

was used to generate voltage clamp data in silico and

estimates of td and dN were extracted using both

approaches. A series of voltage steps (inset, Fig. 4) was

used to estimatedc1as a function ofVmin the conventional

manner. Following from Eq. (1), g dc1 was evaluated as

g dc1 ffi Ipeak=Vm E (17)

assuming that inactivation is negligible at peak current i.e.

f 1. The maximum conductance gwas evaluated directly

at a large depolarization wheredc1approaches 1 (Hodgkin

and Huxley, 1952a). Note that estimating g from the

straight portion in the positive limb of the IpeakVm plot

returns very similar results.In the case above, g was estimated at 120 mV as 0.445,

or 88.8% of the correct value. The significant under-

estimation of gis a direct result of neglecting the effects of

inactivation. As one might expect, the variation ofdc1with

Vm is reproduced with reasonable accuracy given that the

data are normalized to ensure that dc1 approaches 1 in the

upper plateau (seeFig. 4). On the other hand, the method

proposed in the previous section determined dn1 over the

entire range of membrane potential accurate to within

0.02% using the idealized simulated experimental data.

However, in obtaining this result it was assumed, for now,

that correct value of gd1 I1exp=f1V E wasknown. This assumption is properly addressed in a latersection. The superscript n, which will be used from this

point on, denotes the variables determined using the new or

nonlinear minimization method.

The activation time constant tdwas also estimated in the

conventional manner by least squares fitting to the current

time course data:

It C 1 et=td

. (18)

Time constants determined as above are compared with

those obtained with our numerical technique in Fig. 5. The

error in tcd increased with Vm to a maximum of around

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-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1

Transmembrane potential (mV)

Gating

variables

d

f

dc

dn

Fig. 4. Steady-state activation variable (dN

) determined using two

different approaches. With the conventional method, normalization of

peak I/(VE) with respect to that obtained at 50 mV yielded accurate

(o2% error) estimates. The minimization method produced very little

error (o0.02%). Voltage step protocol used for the conventional

approach is shown on right.



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20%, as inactivation became more significant. In contrast,

tnddetermined with the new method were all within 0.1% of

the true value.

4.3. Simulated experiment: inactivation protocol

Using the ion channel defined in the previous section, we

now compare the effectiveness of new and conventional

techniques for parameter estimation with data obtained

using the inactivation protocol. The standard procedure

for estimating fc1 is the double-pulse protocol described in

(Hodgkin and Huxley, 1952b) where a voltage step to a

common test potential is applied from a range of varying

conditioning potentials (see inset in Fig. 6). If we assume

instantaneous activation, then the peak current measured

will represent the degree of inactivation such that

Ipeakffi g 1 fc1Vm E. (19)

Estimation offc1on this basis produced significant error

(410%), when the correct value of g was used. However,

when these estimates were normalized with respect to the

value obtained at 50 mV, the correspondence between

estimated and actual fN

was much closer (seeFig. 6). This

is to be expected since neglecting the time course of

inactivation will produce very similar effects on peak

current in both protocols. While there were substantialpercentage errors at more positive test potentials, absolute

error was negligible because the magnitude of fN

was

small.

tcf is often estimated by using some form of the double-

pulse protocol, similar to that used to determine f1 with

an extension being that duration of the conditioning

potential (Dt) is varied (Hodgkin and Huxley, 1952b).

The inactivation time constants can be estimated by fitting

an exponential curve to the conditioning potential dura-

tion-peak current relationship. 20 different values of Dt

were used (between 0 and 360 ms) for each conditioning

voltage. As shown in Fig. 7, the results of double-pulse

protocol were very accurate with up to 2% error. However,

when the number ofDtin each voltage step was reduced to

10 (between 0 and 320 ms), the maximum and average

error doubled. With the nonlinear minimization method,

fN

and tf were determined to within 0.002% and 0.03%,

respectively.

4.4. Estimating maximum conductance g

Obtaining an accurate estimate ofgis vital to the success

of the nonlinear minimization method formulated in

Eqs. (12)(16) because df, the input to the objective

function, must be calculated from the observed ionic

ARTICLE IN PRESS

-100 -50 0 50 1002

2.5

3

3.5

4

4.5

5

5.5


Activationtim

econstants(ms)

d

cd

nd

Fig. 5. A comparison of the activation time constants (td) determined by

least-square fitting and the minimization method. The error involved in

the conventional method increased as t decreased and the difference

between dN

and fN

became larger.

-50 0 500

0.2

0.4

0.6

0.8

1


Gatingvariables

d

f

fc

, unscaled

fc

, scaled

fn

Fig. 6. A comparison of estimated steady-state inactivation variable (fN

).

When using the conventional approach, normalization with respect to the

peak I/(VE) (labeled scaled) yielded accurate estimates of fN

, whereas

greater that 10% error were found without scaling, using the correct value

of maximum conductance. Voltage step protocol used is shown on the

right and it was found that conditioning potential must be held for at least

300ms to yield accurate results. The new method produced less than

0.002% error.

-60 -40 -20 0 20 40 6060

62

64

66

68

70


Inactivationtimeconstant(ms) f

cf

nf

Fig. 7. Inactivation time constant (tf) calculated using the double-pulse

protocol and the minimization algorithm. The conventional method

yielded accurate results generally. Their performance worsened signifi-

cantly if maximum Dt(conditioning potential duration) was reduced from

around 300 ms. The new method produced negligible errors (o0.03%).



7/12

current using g. We have shown that conventional

techniques for determining g typically lead to substantial

underestimation of this parameter. Moreover, g cannot be

estimated reliably, by introducing it as a further unknown

in an extended minimization formulation, because it would

necessarily be incorporated into the residual equations as

an arbitrary scaling factor. Changing the objective functionfrom R 0 to gR 0 would not fundamentally alter the

solution and gwould therefore be non-unique. This section

describes modifications to the problem formulation which

allow ionic current to be used directly as the input to the

objective function, and which in turn, enable accurate

estimates of g to be obtained.

Without loss of generality, the terms inside the square

brackets in both R2 and R3 were multiplied by g (VE) so

that dfcan be replaced by ionic current. For example

R2 gV Ed1 1 eti=td

f1 1 f1e

ti=tfh ih

gV Edf

max

i2

I1exp

f11 eti=td

f1 1 f1eti=tf

h i

Ijmax

2. 20

Note that thed1terms in the residual has been replaced

by

d1d1f1exp

f1. (21)

With this modified objective function it was possible to

solve for td, tfand f1 as before.

Once fN

was determined for the particular voltage step

from the minimization, we could calculate

gd1I1exp

f1V E. (22)

This expression does not provide a means of estimating g

independently. However, provided that the depolarizing

step is sufficiently large, it may be reasonably assumed that

d1 1. This yields a much more accurate estimate of g

than assuming (df)maxE1, as in Eq. (17).

4.5. Analysis and synthesis of HH sodium channel:

comparative evaluation of new technique

We have used the classical model of squid axon sodium

channel (Hodgkin and Huxley, 1952c) to assess the

effectiveness of our numerical technique in a more complex

setting. The HH equations used and associated residual

formulation are given in Appendix A. Note that in

(Hodgkin and Huxley, 1952c) V is defined as the

displacement from resting potential and depolarization is

taken to be negative. Note that standard nomenclature is

used for steady-state activation and inactivation variables

(m and h, respectively) and m is raised to a power of 3. R3

was expanded to a sum of 5 data points at it*,

(i 2,3,y,6). The initial estimates for td, tfand f1 were

obtained by randomly perturbing each variable by up to

100% of their actual value.

AsFig. 8shows, the minimization technique converged

with very high accuracy (71% or better for all variables)

while the conventional methods suffered up to 2030%

error due to the fact that the time constant ratio t

was smaller for the sodium channel than in the previous

example. Activation parameters mc1 and tcm were

least accurately predicted while inactivation variables hc1

and tch were obtained with greatest accuracy. It can beseen inFig. 8 that, even with scaling, there is still 410%

error in mc1 in the physiological range of membrane

potential. It is noteworthy that the new method gave

substantially more accurate estimates of gNa than the

conventional approach. Thus, at V 120mV, gnNa was

within 0.2% of the correct value, 120 mS/cm2, whereas gcNawas 75.7 mS/cm2.

Current traces reproduced using the actual parameters

and also with those determined using the conventional

approach (c-), are compared in Fig. 9a and b. Currents

simulated using the parameters from new method are

omitted, since they overlap the actual currents almost

exactly. The poor reproduction of the original current-time

behavior could not be accounted for by the 37% error in

gcNa alone. Inaccuracies in gating kinetics parameters

also contribute and this is reflected in the errors evident

inFig. 9b, in which the actual gNa was used together with

conventional estimates of all other parameters.

4.6. Sensitivity analysis

It is possible now, to determine all of the parameters in

the mathematical model from a single minimization

procedure using the modifications described. The sensitiv-

ity of the outlined numerical technique to the error in

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-100 -50 00

0.2

0.4

0.6

0.8

1

Potential V-Erest(mV)

Gating

variablesor

Timeconstant(ms)

m

h

m

h

/10

Fig. 8. Estimated gating variables and time constants for HodgkinHux-

ley sodium channel. Gating variables calculated using the conventional

methods were normalized. The inactivation time constants and variables

were estimated with high accuracy using all results in contrast to mc1 an d

tm, which displayed 410% and 430% errors, respectively.



8/12

experimental data is considered below. First, we chose to

simulate the voltage clamp data again using the channel

described in Appendix A. Then, to the input parameters t*,

Ipeakand IN extracted from the simulated current, a known

level of perturbation was introduced. Then, its effects on

the output variables were examined against the known

actual solution. This procedure allowed the true robustness

of the numerical scheme to be assessed independently of

the experimental factors. In the following investigations,

each input parameter was perturbed in turn, from its exact

value by up to 20%. This was done at voltages of 20,

40, 60 and 80 mV. In all cases, gnNa was estimated at

120 mV and percentage errors in the calculated variables

were compared.

The results for perturbations in t*, Ipeak and IN are

shown inFig. 10. The principal observation that could be

drawn is that for moderate (p15%) levels of perturbation,

the resulting percentage errors in the output variables are

approximately equal to or less than the magnitude of the

applied perturbation. This shows that the scheme does not

amplify small errors in an unstable fashion. On a less

significant note, the sensitivity of each output variable was

dependent on the input parameter being perturbed as well

as the transmembrane potential.

4.7. Combined error in inputs and initial estimates

As errors were introduced into the input parameters, the

solution became more sensitive to the initial estimate of the

variables (dN

, td and tf) and a modest shift in the initial

values often led to the selection of incorrect local minima.

In order to analyse the potential error in the experimentalsituation, where all inputs include noise and initial

estimates are unlikely to be accurate, the inputs t*, Ipeak,

IN

and initial estimates for hN

, tm and th were all

simultaneously perturbed and resultant errors were

evaluated. Monte Carlo approach was used in which,

between 5% and 15% random perturbation was imposed

to each input parameter. The average errors in the

estimated variables were then calculated from 100,000

minimizations carried out. These were found to be

remarkably consistent for each parameter, across the test

potentials in the range 80 to 20 mV. The following

results are mean7sd percentage errors for the representa-

tive voltage 60 mV.tnm(12.378.8%) was most sensitive to

input noise, followed by hn1 (9.577.3%). For tnh, the error

(6.674.7%) was comparable to the lowest magnitude of

the perturbation, whereas mn1 (2.671.8%) was least

sensitive to noise.

5. Discussion

In this manuscript, we present an improved method for

estimating HH style cell model parameters. This was

motivated by the observation that the methods conven-

tionally used to characterize ion channels may produce

large errors under physiological conditions. We show thatthese errors result from the assumption that inactivation

may be neglected within the time scale of activation and

vice versa. We describe a novel numerical approach which

markedly improves the accuracy of channel parameters

estimates. Unlike conventional techniques, both activation

and inactivation parameters can be obtained from the same

set of voltage clamp recordings. As a result, the volume of

experimental data required in the fitting process is

markedly reduced.

The starting point of our analysis was to examine the

efficacy of the conventional methods and to determine the

worst-case error bounds. In the first instance, errors were

related to the ratio of inactivation time constant to

activation time constant (t). As expected, the assumption

underlying the conventional approach introduces signifi-

cant error when activation and inactivation time constants

are of comparable magnitude. In the theoretically worst

case, error of 100% was approached when t-1. For a

typical sodium channel, the minimum t is on the order of 5

(Ebihara and Johnson, 1980) and errors encountered with

such experimental data were generally 2030%. Surpris-

ingly, though, both steady-state activation variable and

activation time constants exhibited in excess of 10% error

for t 50. Dokos and Lovell (2004) have shown that

errors of as little as 10% can cause significant variation in

ARTICLE IN PRESS

0 1 2 3 40

0.5

1

1.5

2

2.5

Time (ms)

Curre

nt(A/cm2)

Current(A

/cm2)

Actual

c-parameters

0 1 2 3 40

0.5

1

1.5

2

2.5

Time (ms)

Actual

g=120 mS/ cm2

(a)

(b)

Fig. 9. (a) Voltage clamp current reproduced using the parameters

determined from conventional methods, for clamp voltages ranging

between 100 and 40 mV. (b) Currents reproduced with same parameters

as in a), except gNa of 120 mS/cm2 was used.



9/12

action potentials generated with the Beeler and Reuter

model (1977). Secondly, we have shown that maximum

membrane channel conductance may be significantly

underestimated if the effects of inactivation on peak

current are neglected. An unfortunate consequence of

these errors is that HH models incorporating parameters

fitted using the conventional simplifications commonly fail

to reproduce the experimental data from which the

parameters were estimated.

Hodgkin and Huxley were clearly aware of these

problems. The parameter estimation techniques originally

employed byHodgkin and Huxley (1952c) did not involve

the assumption that t is infinite. Instead, the experimental

curves were plotted on double log paper and a family of

theoretical curves was fitted graphically while varying tmand th. Although steady-state activation and inactivation

variables were determined independently, this approach

sought a simultaneous solution and achieved a very good

fit to the experimental current recordings. The results of

present study suggest that the error in the steady-state

variables estimated in this way is likely to have been small.

Hodgkin and Huxley (1952c) also acknowledged that an

approximate correction for maximum channel conductance

could be made by guessing the level of inactivation at peak

current.

In our new method, the only assumptions required are

that the steady-state activation and inactivation variables

reach 0 and 1, respectively, upon hyperpolarization, and

that steady-state activation variable approaches 1 when the

cell membrane experiences a large depolarization. In order

to remove the dependency of the extracted parameters on t,

our technique incorporated the whole transmembrane

current equation into the problem formulation with no

linearized terms. This approach has numerous advantages,

including the improved accuracy of the calculated para-

meters. With a simulated experimental protocol, it was

shown that the method consistently produced less than 1%

error in all calculated variables witht as low as 3, provided

that accurate input parameters were used. While such

accuracy in the input parameters would be difficult to

achieve in actual experimental settings, it shows that the

method introduces negligible additional error. Under

comparable circumstances, the theoretical worst-case error

for conventional method is in excess of 70%. It is known

that in some cases single channel recordings display

flickering even at large depolarizing potentials, which leads

ARTICLE IN PRESS

0.8 0.9 1 1.1 1.2

-30

-20

-10

0

10

20

30

Perturbation

%

Error

0.8 0.9 1 1.1 1.2

-30

-20

-10

0

10

20

30

Perturbation

%

Error

0.8 0.9 1 1.1 1.2

-30

-20

-10

0

10

20

30

Perturbation

%Error

0.8 0.9 1 1.1 1.2

-30

-20

-10

0

10

20

30

Perturbation

%Error

(a) (b)

(c) (d)

Fig. 10. Percentage errors inmN

(J),hN

(D),tm(&) andth(+) as a result of perturbing the input parameter t* (),Ipeak(?) andIN (). From (a) to

(d): Depolarizations of 80, 60, 40 and 20 mV, respectively. abscissa: scale factor for perturbations. ordinate: % error in the estimated variables using

minimization method. The key observation is that, within 15% perturbation, the errors involved in estimated variables were generally of the same order or

less.



10/12

to steady-state activation being less than 1. For these

channels, using single channel studies to first determine the

correct steady-state activation value will improve the

accuracy of the calculated parameters.

Another major strength of the proposed technique is that

all of the model parameters can be determined from a

single experimental protocol. This reduces the total numberof experiments required to characterize an ion channel.

Take the two-pulse protocol (Hille, 2001) for example,

which is commonly used to determine inactivation time

constantsin this alternative protocol, two voltage pulses

separated by varying inter-pulse interval are applied and

peak currents during each pulse are observed. In order to

obtain the inactivation time constant, the relative peak

current magnitudes must be determined for a number of

different inter-pulse intervals. In a typical experiment, this

would involve around 10 different interval lengths and such

protocols are usually repeated several times on different

samples to provide an average measure. With the new

method, for each sample only one (average) recording

needs to be made from a voltage step for each membrane

potential and this data can be used to determine both

inactivation and activation parameters.

Although in the description of the methods, R3 was

constructed using only a few data points, it was only to

illustrate the minimum requirements of the technique. In

practice, where noise and error are involved, more stable

results may be expected from constructing this residual by

summing over the entire data set. In this case, the

minimization technique would revert back toward a least-

squares fit of the entire trace.

The simulated activation and inactivation experimentsrevealed that steady-state gating variables (d

N and f

N)

were insensitive to the method of calculation. Because these

parameters usually follow Boltzmanns curve, scaling

generally ensured that accurate estimates were obtained

using conventional methods. One exception to this

observation was the sodium channel activation variable

(see Appendix A), which was underestimated at higher

depolarizations. This variable is raised to the power of 3,

but, as shown inFig. 8, compensating for this by applying

a cube root did not correct the problem. Upon closer

examination it is observed that t is less than 5 over the

range of membrane potentials while activation and

inactivation variable approached 1 and 0, respectively,

for which the error is most marked and thus the worst case

scenario described earlier is approached. In general,

though, the performance of the conventional method was

better than expected for the sodium channel. The

sensitivity of the conventional methods to noise in the

data has not been examined here and should be investi-

gated in future work.

The analyses presented in this paper did not use real

experimental data, but this should not be viewed as a

limitation of this study. Rather, it presents a number of

advantages. With simulated data, we are able to quantify

exactly the component of error arising from the numerical

approach and obviate the effects of experimental noise.

When experimental data are used, we do not know what

the exact solution is and therefore the error cannot be

estimated. Furthermore, any failure of the empirical model

to reproduce the experimental data will introduce further

error that is unrelated to the fitting process. The issue

regarding whether HH formalism constitutes a soundmodel for a membrane ion channel, is outside the scope

of this work. However, more recent concerns (Patlak, 1991)

notwithstanding, it is likely that HH approach will

continue to be used in the multi-scale modeling of organ

function.

A limiting feature of the minimization technique devel-

oped here is the increased probability of failure when input

parameters are noisy and simultaneously when initial

estimates are inaccurate. Once error is introduced into

the input parameters, the global minimum of the problem

becomes different from the exact parameters which were

used to generate the current traces. Thus, even if the

numerical algorithm converges successfully and remains

faithful to the input parameters, the solution will not be

reliable. This could be addressed only by improving the

experimental accuracy, as we cannot reasonably expect the

parameter estimation technique to correct for experimental

sources of error.

It is worthwhile to speculate the likely magnitude of

error in each of the input variables. Some variables are

more likely to contain error than others. The error in t*

would depend on the sampling rate capability of the

recording system and for a very fast channel, such as

sodium, a small absolute error would lead to a significant

relative error. Under circumstances where gating dynamicsare slow, however, it may be difficult to determine an exact

value of t* due to a prolonged plateau. However, it is

unlikely that the error in Ipeak would be as large as 20%

except when the current is very small. On the other hand,

because its magnitude is generally small, IN

may well

contain errors at this level due to noise. Introducing

perturbations into the input parameters resulted in shifts in

the output variables in an intuitive cause and effect fashion.

The most closely related variables can be grouped

togetheraltering IN has the most significant influence

overhN

whiletmwas most influenced byt*. The sensitivity

analyses showed that for up to 1015% error in input

parameters, the average error in output variables is of

similar magnitude or less.

The estimation of parameters for a summed series of

exponential functions is a notoriously ill-conditioned

problem (Petersson and Holmstro m, 1998). In the case of

Eqs. (9) and (11), however, the powers and coefficients of

the exponential terms are not arbitrarily independent, but

are constrained by a priori informationthe model itself

which reduces the possible solution set and forms a well-

posed problem. This raises a further important point

that the choice of functional forms for activation and

inactivation variables must be seen as part of the process

of channel characterization. If the fitted function is

ARTICLE IN PRESS



11/12

particularly poor in representing the experimental data,

this will become immediately apparent in the minimization,

whereas in traditional methods this may not be revealed

until both the activation and inactivation parts are

assembled into the full model. Consequently, the use of

the minimization technique should not be limited to fitting

parameters for an ion channel, but also provides a methodto quantitatively assess the suitability of the chosen

function forms for activation and inactivation variables.

Although this study focuses specifically on the HH

formalism, it should be recognized that this is a special case

of the more general Markov channel modeling approach.

This can be illustrated using the sodium channel as an

example. A sodium channel comprises four subunits, three

m-type and one h-type. It is assumed that each subunit can

exist only in open or closed states. The resulting state space

for the channel therefore consists of 24 16 different

states, but there is only one open statewhen all four

subunits are open. The number of states can be reduced

from 16 to 8 if, as is the case in the HH approach, it is

assumed that the transition rates of the subunits are

uniform regardless of the current state and that the rates of

all m-subunits behave identically. The kinetic properties of

the channel can be described by a system of 8 ODEs, one

of which can be removed since the probabilities of all states

add to one. The full time-course of the channel, represented

by the sum of exponentials obtained by solving this system,

is equivalent to the expanded form of HHs m3h kinetics

(Hille, 2001). In the HH approach, it is assumed further

that subunits behave independently. This means the

problem can be reduced to 4 two-state models (with a

total of two distinct time constants), which conferssignificant computational advantage. Similar analyses

apply for other types of channel.

The work presented in this manuscript has taken the first

steps in reviewing and formalizing the procedures for

characterization of experimental data and assessment of

the suitability of chosen gating kinetics functions, via the

widely applied HH modeling paradigm. Further, we

emphasize that the key principal behind the outlined

techniquei.e. of avoiding partial fittings to linearized

forms of the modelcan lead to significantly higher

accuracy when applied at whole channel level and is easily

extendable to more complicated channel equations. In

doing so we seek to further enhance the application of

mathematical modeling in integrating measured single cell

electrophysiological data with multi-scale computations of

whole organ activation.

Acknowledgments

This work was supported by the Tertiary Education

Commission (TEC) of New Zealand, New Zealand Vice-

Chancellors Committee (NZVCC) and New Zealand

Institute of Mathematics and its Applications (NZIMA).

Appendix A

The equations for the sodium channel given in Hodgkin

and Huxley (1952c)are

gNa m3hgNa 120 mS=cm

2,

am 0:1V 25exp V25

10

1

,

bm 4 exp V

18

,

ah 0:07 exp V

20

,

bh 1

exp V3010

1

,

INa gNaV VNa and VNa 115 mV A:1

from these the steady-state variables and the time constants

can be determined using

m1 am

am bm,

tm 1

am bmA:2

and similarly for h. Note that in (Hodgkin and Huxley,

1952c), Vis defined to be the displacement of membrane

potential from its resting value and is negative compared to

the usual convention (i.e. depolarization is negative) as

explained in the text.

The residuals for the parameter estimation problem were

constructed as

R1

3h1tm

et=tm 2e2t

=tm e3t=tm

h11

th

et

=th 3h11tm

3h11th

e

t

tmt

th

6h11tm

3h11th

e

2t

tmt

th

3h11tm

h11th

e

3t

tmt

th

2666666664

3777777775

2

,

(A.3)

R2

I1exp

h1 1 3et=tm

3e2t=tm

e3t=tm

h1 h1 1et=th

Ipeak

i2, A:4

R3 X

i

1

Ii

I1exp

h11 3eti=tm 3e2ti=tm e3ti=tm

h1 h1 1eti=th

Ii

2, A:5

where all gm13V E term is replaced by I1exp

.h1.

R3here is expanded to include multiple data points ( ti,Ii) in

the problem.

ARTICLE IN PRESS



12/12

The terms were summed up as before, to give

R R1

t

R2

Ipeak R3. (A.6)

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