Fundamentals of ComputationalNeuroscience 2e

December 26, 2009

Chapter 2: Neurons and conductance-based model

Biological background

Inhibitory axon terminal

Excitatory axon terminal

Dendrites

Axon

Axon hillock

Myelin sheath

Ranvier node

Postsynaptic neurons

Synaptic cleft

Soma

A. Schematic neuron B. Pyramidal cell

D. Spiny cell

Axon

Nucleus

C. Granule cell

E. Purkinje cell

Ion channels

++

++ +

+

++

+

+

A. Leakagechannel

+

+

++

++

+

+ ++

+

+

D. Ionotropic

++

++ +

+

+

+

+

B. Voltage-gated ion channel

+

+

+

++

++

+

+

C. Ion pump

+

+

++

++

+

+ ++

+

+

E. Metabotropic (second messenger)

Neurotransmitter-gated ion channels

Synapse

CaCa

Neurotransmitter

Synaptic vescicleVoltage-gated Ca channel2+

Neurotransmitterreceptor

non-NMDA: GABA, AMPA

∆V non−NMDAm ∝ t e−t/tpeak

Conductance-based models

cmdV (t)

dt= −I (1)

I(t) = g0V (t)− g(t)(V (t)− Esyn) (2)

τsyndg(t)

dt= −g(t) + δ(t − tpre − tdelay) (3)

g

m

gL

C

Time

I (t)/5syn

g (t)*5

V (t)

A. Electric circuit of basic synapse

Cap

acito

r

Battery

Resistor

B. Time course of variables

0 2 4 6 8 10−2

0

2

4

m

syn

MATLAB Program

1 %% Synaptic conductance model to simulate an EPSP2 clear; clf; hold on;34 %% Setting some constants and initial values5 c_m=1; g_L=1; tau_syn=1; E_syn=10; delta_t=0.01;6 g_syn(1)=0; I_syn(1)=0; v_m(1)=0; t(1)=0;78 %% Numerical integration using Euler scheme9 for step=2:10/delta_t

10 t(step)=t(step-1)+delta_t;11 if abs(t(step)-1)<0.001; g_syn(step-1)=1; end12 g_syn(step)= (1-delta_t/tau_syn) * g_syn(step-1);13 I_syn(step)= g_syn(step) * (v_m(step-1)-E_syn);14 v_m(step) = (1-delta_t/c_m*g_L) * v_m(step-1) ...15 - delta_t/c_m * I_syn(step);16 end1718 %% Plotting results19 plot(t,v_m); plot(t,g_syn*5,’r--’); plot(t,I_syn/5,’k:’)

Hodgkin–Huxley model

+40

-70

0Resting potential(leakage channels)

Hyperpolarization

Spike

Depolarization due to sodium channels Inactivation of sodium channels

& opening of potassium channels

Closing of sodium& potassium channels

Figure: Typical form of an action potential; redrawn from an oscilloscopepicture from Hodgkin and Huxley (1939).

The minimal mechanisms

Resting potential

+Na +Na

+Na +Na+Na

+K

+K

+K

+K

+K

+K

+K +K

+K

+Na

Depolarization

+Na

+Na +Na+Na+Na

+Na+K

+K

+K

+K

+K

+K+K

+K

+K

+Na

Hyperpolarization

+Na+Na +Na

+Na+Na

+Na

+K

+K

+K+K

+K

+K

+K

+K

+K

+K

Hodgkin–Huxley equations and simulation

CdVdt

= −gKn4(V − EK)− gNam3h(V − ENa)− gL(V − EL) + I(t)

τn(V )dndt

= −[n − n0(V )]

τm(V )dmdt

= −[m −m0(V )]

τh(V )dhdt

= −[h − h0(V )]

Spike train with constant input

0 50 100 50

0

50

100

150

Time [ms]

Mem

bran

e po

tent

ial

[mV]

Activation function

0 5 10 150

20

40

60

80

100

External current [mA/cm ]

Firin

g fre

quen

cy [H

z]

2

Noise

Compartmental models

j + 1jj - 1

j

j + 1

j + 2

A. Chain of compartments C. Compartmental reconstruction

B. Branching compartments

Simulators

Further Readings

Mark F. Bear, Barry W. Connors, and Michael A. Paradiso (2006),Neuroscience: exploring the brain, Lippincott Williams & Wilkins ,3rd edition.

Eric R. Kandel, James H. Schwartz, and Thomas M. Jessell (2000),Principles of neural science, McGraw-Hill, 4th edition

Gordon M. Shepherd (1994), Neurobiology, Oxford University Press, 3rdedition.

Christof Koch (1999), Biophysics of computation; informationprocessing in single neurons, Oxford University Press

Christof Koch and Idan Segev (eds.) (1998), Methods in neuralmodelling, MIT Press, 2nd edition.

C. T. Tuckwell (1988), Introduction to theoretical neurobiology,Cambridge University Press.

Hugh R. Wilson (1999) Spikes, decisions and actions: dynamicalfoundations of neuroscience, Oxford University Press. See also hispaper in J. Theor. Biol. 200: 375–88, 1999.