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Neuron Models

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: www.millennium.berkeley.edu/ ganglia/images/neuron.gif

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Neuron Models

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

  3. Nerve Impulse Generation Source: http://faculty.washington.edu/chudler/ap3.gif Source:www.biology.eku.edu/RITCHISO/ nervous_depolarization.gif Source:www.biology.eku.edu/RITCHISO/ nervous_repolarization.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)*[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 Potassium (K+) Ion Conductance Sodium (Na+) Ion Conductance

  6. 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)

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

  8. Action PotentialH-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 4 0 2 3 1 3 2 1 3 0 4 0 4

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

  11. Fitzhugh-Nagumo Neuron ModelLow Stimulation

  12. Fitzhugh-Nagumo Neuron ModelModerate Stimulation – Limit Cycle

  13. Fitzhugh-Nagumo Neuron ModelModerate Stimulation - Bursting

  14. Fitzhugh-Nagumo Neuron ModelHigh 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.

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