the hodgkin-huxley equations and analytical approximations ...pks/presentation/leipzig-04.pdf ·...
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The Hodgkin-Huxley Equations andAnalytical Approximations for them
Peter SchusterInstitut für Theoretische Chemie und Molekulare
Strukturbiologie der Universität Wien
Seminar of the MPI for Mathematics in Science
Leipzig, 22.07.2004
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Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
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Neurobiology
Neural networks, collectiveproperties, nonlinear
dynamics, signalling, ...
The human brain
1011 neurons connected by 1013 to 1014 synapses
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Neurobiology
Neural networks, collectiveproperties, nonlinear
dynamics, signalling, ...
A single neuron signaling to a muscle fiber
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BA
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
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Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
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Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
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Neurobiology
Neural networks, collectiveproperties, nonlinear
dynamics, signalling, ...
)()()(1 43llKKNaNa
M
VVgVVngVVhmgICtd
Vd−−−−−−=
mmdtdm
mm βα −−= )1(
hhdtdh
hh βα −−= )1(
nndtdn
nn βα −−= )1(
Hogdkin-Huxley OD equations
A single neuron signaling to a muscle fiber
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Gating functions of the Hodgkin-Huxley equations
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Temperature dependence of the Hodgkin-Huxley equations
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)()()(1 43llKKNaNa
M
VVgVVngVVhmgICtd
Vd−−−−−−=
mmdtdm
mm βα −−= )1(
hhdtdh
hh βα −−= )1(
nndtdn
nn βα −−= )1(Hogdkin-Huxley OD equations
Hhsim.lnk
Simulation of space independent Hodgkin-Huxley equations:Voltage clamp and constant current
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LrVVgVVngVVhmgtVC
xV
R llKKNaNa π2)()()(1 432
2
−+−+−+∂∂
=∂∂
mmtm
mm βα −−=∂∂ )1(
hhth
hh βα −−=∂∂ )1(
nntn
nn βα −−=∂∂ )1(
Hodgkin-Huxley PDEquations
Travelling pulse solution: V(x,t) = V( ) with
= x + t
Hodgkin-Huxley equations describing pulse propagation along nerve fibers
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Hodgkin-Huxley PDEquations
Travelling pulse solution: V(x,t) = V( ) with
= x + t
[ ] LrVVgVVngVVhmgd
VdCd
VdR llKKNaNaM π
ξθ
ξ2)()()(1 43
2
2
−+−+−+=
mmd
mdmm βα
ξθ −−= )1(
hhd
hdhh βα
ξθ −−= )1(
nnd
ndnn βα
ξθ −−= )1(
Hodgkin-Huxley equations describing pulse propagation along nerve fibers
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50
0
-50
100
1 2 3 4 5 6 [cm]
V [
mV
]
T = 18.5 C; θ = 1873.33 cm / sec
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T = 18.5 C; θ = 1873.3324514717698 cm / sec
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T = 18.5 C; θ = 1873.3324514717697 cm / sec
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-10
0
10
20
30
40
V
[mV
]
6 8 10 12 14 16 18 [cm]
T = 18.5 C; θ = 544.070 cm / sec
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T = 18.5 C; θ = 554.070286919319 cm/sec
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T = 18.5 C; θ = 554.070286919320 cm/sec
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Propagating wave solutions of the Hodgkin-Huxley equations
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FitzHugh-Nagumo model of the Hodgkin-Huxley equations
V ...... potential ; Y ...... refractory variable
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FitzHugh-Nagumo model and ist approximations
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FitzHugh-Nagumo equation: reduced model
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FitzHugh-Nagumo equation: reduced model
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FitzHugh-Nagumo model and ist approximations
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0
0.5
1.0
-0.5
-1.0210-1-2
s
X
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FitzHugh-Nagumo equation: broken linear model
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V, dV/d
Close-up of the relaxation oscillation as used in the calculations of period and pulse amplitude in the Reduced Broken-Linear Model
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FitzHugh-Nagumo pulse propagation
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Reduced Hodgkin-Huxley equations
V , m ...... fast variables, n , h ...... slow variables
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V
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0
0.005
0.010
0.020
0.015
b0
0.10.2
0.3a
0.60.40.2
0
0
0.005
0.010
0.020
0.015
b
00.1
0.20.3
a
0.60.40.20
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References
Paul E. Phillipson, Peter Schuster, Dynamics of relaxation oscillations, Int.J.Bifurcation and Chaos 11:1471-1481, 2001
Paul E. Phillipson, Peter Schuster, Bistability of harmonically forcedrelaxation oscillations, Int.J.Bifurcation and Chaos 12:1295-1307, 2002
Paul E. Phillipson, Peter Schuster, An analytic picture of neuron oscillations, Int.J.Bifurcation and Chaos 14:1539-1548, 2004
Paul E. Phillipson, Peter Schuster, A comparative study of the Hodgkin-Huxley and FitzHugh-Nagumo models of neuron pulse propagation, Int.J.Bifurcation and Chaos, submitted 2004
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Coworker
Paul Phillipson, Department of Physics, University of Colorado, Boulder, CO
Österreichische Akademie der Wissenschaften
Universität Wien
Acknowledgement of support
Österreichische Akademie der Wissenschaften,
Universität Wien and University of Colorado
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Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
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