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

2. 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.gifSource: www.gsu.edu/~wwwbgs/bgsa/ neuro/40x%20neuron.JPG

3. Nerve Impulse Generation Source:www.biology.eku.edu/RITCHISO/ nervous_depolarization.gifSource:www.biology.eku.edu/RITCHISO/ nervous_repolarization.gif Source: http://faculty.washington.edu/chudler/ap3.gif

4. 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

5. 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

6. Hodgkin-Huxley Neuron Model c=1.0 g Na =120.0 g K =36.0 g L =0.3 v Na =-115.0 v K =12.0 v L =-10.5989 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)

7. Variation in Ion Conductance H-H Model vs. Nerve Source: http://courses.washington.edu/biophys/homework/hw6_files/image003.jpg

8. Action Potential H-H Model vs. Nerve Source: http://courses.washington.edu/biophys/homework/hw6_files/image003.jpg

9. H-H Model in the v, m Phase Plane 1 1 2 2 3 3 3 4 4 4 0 0 0

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

11. Fitzhugh-Nagumo Neuron Model Low Stimulation

12. Fitzhugh-Nagumo Neuron Model Moderate Stimulation – Limit Cycle

13. Fitzhugh-Nagumo Neuron Model Moderate Stimulation - Bursting

14. Fitzhugh-Nagumo Neuron Model High Stimulation – No Recovery

15. 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

16. 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.