An Introduction to the Mathematical Modeling of...

An Introduction to the Mathematical

Modeling of Neurons: The Hodgkin-

Huxley Equations

Louis Tao 陶乐天 taolt @ mail.cbi.pku.edu.cn, letaotao @ pku.edu.cn

北京大学生命科学学院生物信息中心定量生物中心

Center for Bioinformatics, College of Life Sciences

Center for Quantitative Biology

Peking University

交大致远学院自然科学研究院计算神经科学短期班

17 December 2012

Neuronal Networks Are Complex

~1011 neurons & 1015 connections

104 cells & 1 km wiring in 1 mm3 of cortex

Neurons 神经元

• Information processing units 信息处理加工单元

• 1010-1013 neurons in mammalian brains 哺乳动物大脑

• 104 cell bodies and roughly 1 km of ‘wiring’ per mm3

• Different shapes, sizes, functions, … 不同形状、大小、功能

• Spiking vs. Analog neurons 锋电位（动作电位） (“analog neurons,” e.g., bipolar and amacrine cells in retina,

sensory-motor neurons of invertebrates, …

视网膜里的双极细胞与无长突细胞，无脊椎动物的感觉-运动神经元，等)

• Many other cells (e.g., glia cells 胶质细胞) in cortex: to supply

energy, to provide structural stability, …, and not directly

involved in information processing

Computational Neuroscience

• What “computations” are performed by neurons & neuronal

networks?

• How are these computations done?

• What?

– Feature detection (visual systems, olfactory system, …)

– Coincidence / timing (auditory system)

– Memory (hippocampus)

– Sensory-motor (eye saccades, …)

– Neural Code: firing rate, spike timing

• How?

– Cell level: molecular and biophysical

– Network & systems level

• What & How to Study?

– Cellular: membrane potential, ion channels, synaptic mechanisms

– Extracellular: firing rates, spike times, statistics of spike trains, …

– Systems: fMRI, optical imaging, …

A cell at ‘rest’:

• ionic currents in dynamical equilibrium: net zero current

across membrane potential

• maintenance requires huge amounts of energy, it has

been estimated that half the energy consumed by brain is

used to maintain ionic charge gradients

• Vrest ~ -30 mV to -90 mV (depends on concentration

ratios of ions, e.g. Cl- )

Action Potentials 动作电位

• Action potential (spikes): voltage pulses responding to ‘strong’

input

• All-or-None electrical events, initiated near the cell body,

propagates along axons (at roughly constant velocity and

amplitude)

• Hodgkin & Huxley (1950s) studied various ionic currents

individually (using pharmacological blocks); used squid giant

axon (1/2 mm diameter)

• Action Potential generation involves 2 major, voltage-

dependent currents: sodium and potassium; individual ionic

current obeys Ohm’s law

,

,

1 ln

ion inside

ion ion ion syn ion

ion ion outside

cRTI G V V G V

R zF c

2

2

m mm Na K Leak syn inj

V VdC I I I I I

t R x

Hodgkin Huxley Neuron Model

1963 Nobel Prize

Squid

Giant Axon

2

2


V VdC I I I I I

t R x

3 4 Na Na Na K K KI g m h V V I g n V V


2

2

2 2

55 mV, 115 mS/cm

90 mV, 36 mS/cm

65 mV, 0.1 mS/cm , 1 F/cm

/ 10 msec

Na Na

K K

Leak Leak m

m m Leak

V g

V g

V g C

C g

syn syn m synI G t V V

leak leak m restI G V V

Synaptic current: induced by

action potentials of other neurons

Leak current:

2

2


V VdC I I I I I

t R x


Voltage Clamp

Delcomyn (1997)

Device to measure the current necessary to keep the desired V

2

2


V VdC I I I I I

t R x


+25mV Vrest

voltage step command

KK

K

Ig

V E

nndt

dn

ngg

n

KK

4

Estimate n and n

from time course


Vrest

Estimate n(V) and n(V)

for different voltage steps

n

0

1

V

Modeling the K-current

2

2


V VdC I I I I I

t R x


3

Na Na

m

h

g g m h

dmm m

dt

dhh h

dt

Estimate m(V), m(V), h(V), h(V)

for different voltage steps V

Modeling the Na-current

2

2


V VdC I I I I I

t R x


3

Na Na

m

h

g g m h

dmm m

dt

dhh h

dt

Estimate m(V), m(V), h(V), h(V)

for different voltage steps V

m

h

V

V 0

1

0

1 Modeling the Na-current

0.1 25

exp 0.1 25 1

4exp18

m

m

VV

V

VV

0.07exp20

1.0

exp 0.1 30 1

h

h

VV

VV

0.01 10

exp 0.1 10 1

0.125exp80

n

n

VV

V

VV

Gating Variable Kinetics:

• Action Potential generation involves 2 major, voltage-dependent currents

• Functional forms guessed by Hodgkin & Huxley to fit experimental data!!!

1 1 1m m h h n n

dm dh dnV m V m V h V h V n V n

dt dt dt

m 1 - m m

m


K-channel, activation Na-channel, activation & inactivation

mm Na K Leak inj

VC I I I I

t



2

2

2 2

55 mV, 115 mS/cm

90 mV, 36 mS/cm

65 mV, 0.1 mS/cm , 1 F/cm

/ 10 msec

Na Na

K K

Leak Leak m

m m Leak

V g

V g

V g C

C g

syn syn m synI G t V V

leak leak m restI G V V

Synaptic current: induced by

action potentials of other neurons

Leak current:

Action Potentials 动作电位

• Weak input, no action potential, voltage deflection linear in

applied current

• Strong input, action potential generated

• Each ‘action potential’ is characterized by a stereotypical V-

trajectory

• Absolute and relative refractory periods

• Time for HH Numerical Simulations

Reference: F.C. Hoppensteadt & C.S. Peskin, Modeling and

Simulation in Medicine and the Life Sciences, Springer-Verlag,

2002, Chap 3 Matlab files at http://www.math.nyu.edu/faculty/peskin/ModSimPrograms/ch3/

HH Model Phenomenology

Weak applied current: membrane potential deflection linear in current

Current step applied between t=10 and 11


Existence of Threshold in Action Potential (AP) Generation:

Applied current has to be sufficiently strong


8.28

ThresI

A


Absolute Refractory Period after Action Potentials:

Time during which no amount of current can lead to another AP



Relative Refractory Period after Action Potentials:

Time during which a larger than threshold current is needed for AP


8.28

ThresI

A

Recall


Repetitive firing in response to constant current

Constant current

Some References:

• A. L. Hodgkin and A. Huxley, J. Physiol. (1952) series of

papers

• Jane Cronin, “Mathematical Theory of the Hodgkin-Huxley

Equations”

• Christof Koch, “Biophysics of Computation”

• Peter Dayan and Lawrence Abbott, “Theoretical

Neuroscience: Computational and Mathematical Modeling of

Neural Systems”

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