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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
Neuron Models

Math 451 Final Project

April 29, 2002

Randy Voland

neuron structure
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
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

hodgkin huxley neuron model
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 model5
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

hodgkin huxley neuron model6
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 conductance h h model vs nerve
Variation in Ion ConductanceH-H Model vs. Nerve

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

action potential h h model vs nerve
Action PotentialH-H Model vs. Nerve

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

summary
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
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.