Neuron Models

Math 451 Final ProjectApril 29, 2002

Randy Voland

Neuron Structure

• Cell Body • Dendrites • Synapses on Cell Body and Dendrites (Input) • Axon and Axon Branches (Output)

Source: www.millennium.berkeley.edu/ ganglia/images/neuron.gif Source: www.gsu.edu/~wwwbgs/bgsa/ neuro/40x%20neuron.JPG

Nerve Impulse Generation

Source:www.biology.eku.edu/RITCHISO/ nervous_depolarization.gif Source:www.biology.eku.edu/RITCHISO/ nervous_repolarization.gif

Source: http://faculty.washington.edu/chudler/ap3.gif

Hodgkin-Huxley Neuron Model

• Studied giant squid axons– Electrical stimulation

– Measurements of ion currents

• Mathematical model of action potential– Equivalent electric circuit of transmembrane processes

– Four first order differential equations• Voltage rate of change

• Rate of change of Na and K ion conductance

Hodgkin-Huxley Neuron Model

dv/dt = (-1/c)*[gNa*m3*h*(v-vNa)+gK*n4*(v-vK)+gL*(v-vL)]

dn/dt = αn(v)*(1-n)- βn(v)*n

dm/dt = αm(v)*(1-m)- βm(v)*m

dh/dt = αh(v)*(1-h)- βh(v)*h

Sodium (Na+) Ion Conductance

Potassium (K+) Ion Conductance

Hodgkin-Huxley Neuron Model

c=1.0 gNa=120.0 gK=36.0 gL=0.3

vNa=-115.0 vK=12.0 vL=-10.5989

αn = 0.01*(v+10)/(exp((v+10)/10)-1)

αm = 0.1*(v+25)/(exp((v+25)/10)-1)

αh = 0.07*exp(v/20)

βn = 0.125*exp(v/80)

βm = 4*exp(v/18)

βh = 1/(exp((v+30)/10)+1)

Variation in Ion ConductanceH-H Model vs. Nerve

Source: http://courses.washington.edu/biophys/homework/hw6_files/image003.jpg

Action PotentialH-H Model vs. Nerve

Source: http://courses.washington.edu/biophys/homework/hw6_files/image003.jpg

H-H Model in the v, m Phase Plane

1

1

2

23

3

3

4

4

4

0

0

0

Fitzhugh’s Reduced H-H Model in the v, m Phase Plane

Fitzhugh-Nagumo Neuron ModelLow Stimulation

Fitzhugh-Nagumo Neuron ModelModerate Stimulation – Limit Cycle

Fitzhugh-Nagumo Neuron ModelModerate Stimulation - Bursting

Fitzhugh-Nagumo Neuron ModelHigh Stimulation – No Recovery

Summary

• Hodgkin-Huxley Model– Models physical processes– Complex

• Fitzhugh-Nagumo Model– Simpler/less physical– Models neuron bursting

• Many other models in literature many based on Hodgkin-Huxley or Fitzhugh-Nagumo

Further Reading

• Edelstein-Keshet, E. (1988) Mathematical Models in Biology, McGraw-Hill, 311-341.

• Hodgkin, A.L. and Huxley, A.F. (1952) J. Physiol., 117, 500 – 544.

• Fitzhugh, R. (1960) J. Gen. Physiol., 43, 867-896.

• Fitzhugh, R. (1961) Biophys. J., 1, 445-466.

• Feng, J. (2001) Neural Networks, 14, 955-975.