Neuron Models Math 451 Final Project April 29, 2002 Randy Voland

Neuron Structure Cell Body Dendrites Synapses on Cell Body and Dendrites (Input) Axon and Axon Branches (Output) Source: ganglia/images/neuron.gifSource: neuro/40x%20neuron.JPG

Nerve Impulse Generation Source: nervous_depolarization.gifSource: nervous_repolarization.gif Source:

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)*[g Na *m 3 *h*(v-v Na )+g K *n 4 *(v-v K )+g L *(v-v L )] dn/dt = α n (v)*(1-n)- β n (v)*n dm/dt = α m (v)*(1-m)- β m (v)*m dh/dt = α h (v)*(1-h)- β h (v)*h Sodium (Na +) Ion Conductance Potassium (K +) Ion Conductance

Hodgkin-Huxley Neuron Model c=1.0 g Na =120.0 g K =36.0 g L =0.3 v Na = v K =12.0 v L = α n = 0.01*(v+10)/(exp((v+10)/10)-1) α m = 0.1*(v+25)/(exp((v+25)/10)-1) α h = 0.07*exp(v/20) β n = 0.125*exp(v/80) β m = 4*exp(v/18) β h = 1/(exp((v+30)/10)+1)

Variation in Ion Conductance H-H Model vs. Nerve Source:

Action Potential H-H Model vs. Nerve Source:

H-H Model in the v, m Phase Plane

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

Fitzhugh-Nagumo Neuron Model Low Stimulation

Fitzhugh-Nagumo Neuron Model Moderate Stimulation – Limit Cycle

Fitzhugh-Nagumo Neuron Model Moderate Stimulation - Bursting

Fitzhugh-Nagumo Neuron Model High 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, Hodgkin, A.L. and Huxley, A.F. (1952) J. Physiol., 117, 500 – 544. Fitzhugh, R. (1960) J. Gen. Physiol., 43, Fitzhugh, R. (1961) Biophys. J., 1, Feng, J. (2001) Neural Networks, 14,