neuron models
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
1
1
2
23
3
3
4
4
4
0
0
0
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.