circuit models of neurons

Bo DengUniversity of Nebraska-Lincoln

Outlines: Hodgkin-Huxley Model Circuit Models --- Elemental Characteristics --- Ion Pump Dynamics Examples of Dynamics --- Bursting Spikes --- Metastability and Plasticity --- Chaos --- Signal Transduction

AMS Regional Meeting at KU 03-30-12

Hodgkin-Huxley Model (1952)

Pros: The first system-wide model for excitable membranes. Mimics experimental data. Part of a Nobel Prize work. Fueled the theoretical neurosciences for the last 60 years and counting.

Hodgkin-Huxley Model (1952)

Pros: The first system-wide model for excitable membranes. Mimics experimental data. Part of a Nobel Prize work. Fueled the theoretical neurosciences for the last 60 years and counting. Cons: It is not entirely mechanistic but phenomenological. Different, ad hoc, models can mimic the same data. It is ugly. Fueled the theoretical neurosciences for the last 60 years and counting.

Hodgkin-Huxley Model --- Passive vs. Active Channels

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Vn

nn

KKK

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Hodgkin-Huxley Model

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Hodgkin-Huxley Model

)( LLL VVgI

Hodgkin-Huxley Model

dtdVCIC

-I (t)CThe only mechanistic part ( by Kirchhoff’s Current Law)

+

Hodgkin-Huxley Model --- A Useful Clue

VIdt

dIK

K

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• Morris, C. and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical J., 35(1981), pp.193--213.

• Hindmarsh, J.L. and R.M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. Lond. B. 221(1984), pp.87--102.

• Chay, T.R., Y.S. Fan, and Y.S. Lee Bursting, spiking, chaos, fractals, and universality in biological rhythms, Int. J. Bif. & Chaos, 5(1995), pp.595--635.

H-H Type Models for Excitable Membranes

Our Circuit Models Elemental Characteristics -- Resistor

gVI

Our Circuit Models Elemental Characteristics -- Diffusor

funcitons decreasingboth )(or )( IhVVfI

Our Circuit Models Elemental Characteristics -- Ion Pump

pumpway -onefor

AVdtdA

Dynamics of Ion Pump as Battery Charger

Na

K

' with

0:LawCurrent sKirchhoff’

C

extCApK,pNa,

CVCI

IIIII

Equivalent IV-Characteristics --- for parallel channels

Passive sodium current can be explicitly expressed as

2

Na

NaNa2

12

21NaNaNa

21

NaNa

Na12

11NaNa

NapNa,

)(||2

))((||)('

gintegratin from derived

2 ,

||2with

tantan

)(

m

m

mm

vVg

dgvvvVvVdgVf

vvvdg

gvv

μvvVdVg

VfI

Equivalent IV-Characteristics --- for serial channels

Passive potassium current can be implicitly expressed as

A standard circuit technique to represent the hysteresis is to turn it into a singularly perturbed equation

0

By Ion Pump Characteristics

with substitution and assumption

to get

Equations for Ion Pump

2 ,

||||

2

tantan11)(

and

2 ,

||2

tantan)(

21

KK

K12

11

KKK

21

NaNa

Na12

11NaNaNa

iiidg

dii

iiId

Ig

Ih

vvvdg

gvv

μvvVdVgVf

m

mm

m

mm

I Na = fNa (VC – ENa)

VK = hK (IK,p)

Examples of Dynamics --- Bursting Spikes --- Chaotic Shilnikov Attractor --- Metastability & Plasticity --- Signal Transduction

Geometric Method of Singular Perturbation

Small Parameters: 0 < e << 1 with ideal hysteresis at e = 0 both C and have independent time scales

C = 0.005

Bursting Spikes

C = 0.005

C = 0.5

Neural ChaosC = 0.5 = 0.05 g = 0.18 e =

0.0005I

in = 0

gK = 0.1515

dK

= -0.1382i1 = 0.14

i2 = 0.52

EK

= - 0.7

gNa

= 1d

Na = - 1.22

v1 = - 0.8

v2 = - 0.1

ENa

= 0.6

Griffith et. al. 2009

Metastability and Plasticity

Terminology: A transient state which behaves like a steady state is referred to as metastable.

A system which can switch from one metastable state to another metastable state is referred to as plastic.

Metastability and Plasticity

Metastability and Plasticity

All plastic and metastable states are lost with only one ion pump. I.e. when ANa = 0 or AK = 0 we have either Is = IA or Is = -IA and the two ion pump equations are reduced to one equation, leaving the phase space one dimension short for the coexistence of multispike burst or periodic orbit attractors.

With two ion pumps, all neuronal dynamics run on transients, which represents a paradigm shift from basing neuronal dynamics on asymptotic properties, which can be a pathological trap for normal physiological functions.

Metastability and Plasticity

Saltatory Conduction along Myelinated Axon with Multiple Nodes

Inside the cell

Outside the cell

Joint work with undergraduate and graduate students: Suzan Coons, Noah Weis, Adrienne Amador, Tyler Takeshita, Brittney Hinds, and Kaelly Simpson

Coupled Equations for Neighboring Nodes

• Couple the nodes by adding a linear resistor between them

1 2 11 1 1 1

11

1 1 1

11 1 1

11 1 1

2 2 12 2 2 2

12

2 2 2

2

C C Cext Na K KC A

AS C A

SA C A

NaNa Na NaC

C C CNa K KC A

AS C A

S

dV V VC I I f V E I RdtdI

I V IdtdI

I V IdtdI V E h IdtdV V V

C I f V E I RdtdI

I V IdtdI

g

g

e

g

2 2 2

22 2 2

A C A

NaNa Na NaC

I V IdtdI V E h I

dt

g

e

Current between the

nodes

The General Case for N Nodes

This is the general equation for the nth node

In and out currents are derived in a similar manner:

1n

n n n n n nCout inNa K KC A

nn n nAS C A

nn n nSA C A

nn n nNa

Na Na NaC

dVC I I f V E I I

dtdI

I V Idt

dII V I

dtdI

V E h Idt

g

g

e

1 1

1

1

if 1

if 1

if 1

0 if

extn n nout C C

n

n nC Cn nin

I nI V V

nR

V Vn NI R

n N

C=.1 pF C=.7 pF

(x10 pF)

Transmission SpeedC=.01 pFC=.1 pF

Closing Remarks: The circuit models can be further improved by dropping the serial connectivity assumption of the passive electrical and diffusive currents. Existence of chaotic attractors can be rigorously proved, including junction-fold, Shilnikov, and canard attractors.

Can be easily fitted to experimental data.

Can be used to build real circuits.

References: [BD] A Conceptual Circuit Model of Neuron, Journal of Integrative

Neuroscience, 8(2009), pp.255-297. Metastability and Plasticity of Conceptual Circuit Models of

Neurons, Journal of Integrative Neuroscience, 9(2010), pp.31-47.

• Kandel, E.R., J.H. Schwartz, and T.M. Jessell Principles of Neural Science, 3rd ed., Elsevier, 1991.• Zigmond, M.J., F.E. Bloom, S.C. Landis, J.L. Roberts, and L.R. Squire Fundamental Neuroscience, Academic Press, 1999.