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Fundamentals of ComputationalNeuroscience 2e
Thomas Trappenberg
January 16, 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. B.
D. E.
C.
Axon
Nucleus
Ion channels
+
+
+
++
+
++
+
+
A. Leakage
channel
+
+
+
+
++
+
++
+
+
+
D. Ionotropic
+
+
+
++
+
+
+
+
B. Voltage-gated
ion channel
+
+
+
+
+
++
+
+
C. Ion pump
+
+
+
+
++
+
++
+
+
+
E. Metabotropic
(second messenger)
Neurotransmitter-gated ion channels
Synapse
A. Schematic synapse B. Microscope image of synapse
CaCa
Neurotransmitter
Synaptic vescicleVoltage-gated
Ca channel2+
Neurotransmitter
receptor
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
mg
L
C
Time
I (t)/5syn
g (t)*5
V (t)
A. Electric circuit of basic synapse
Capacitor
Battery
Resistor
B. Time course of variables
0 2 4 6 8 10−2
0
2
4
m
syn
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
0
Resting potential(leakage channels)
Hyperpolarization
Spike
Depolarization due
to sodium channelsInactivation 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 )]
0 50 100 50
0
50
100
150
Time [ms]
Me
bra
ne
po
ten
tia
l [m
V]
0 5 10 150
20
40
60
80
100
External current [mA/cm ]
Firi
ng
fre
qu
en
cy [
Hz]
A. Spike train with constant input B. Frequency-current plot
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.
