neuron models math 451 final project april 29, 2002 randy voland
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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
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0
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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.