The Hodgkin-Huxley Equations and Analytical Approximations ...pks/Presentation/leipzig-04.pdf ·...
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Transcript of The Hodgkin-Huxley Equations and Analytical Approximations ...pks/Presentation/leipzig-04.pdf ·...
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
Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
Neurobiology
Neural networks, collectiveproperties, nonlinear
dynamics, signalling, ...
The human brain
1011 neurons connected by 1013 to 1014 synapses
Neurobiology
Neural networks, collectiveproperties, nonlinear
dynamics, signalling, ...
A single neuron signaling to a muscle fiber
BA
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
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
Gating functions of the Hodgkin-Huxley equations
Temperature dependence of the Hodgkin-Huxley equations
)()()(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
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
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
50
0
-50
100
1 2 3 4 5 6 [cm]
V [
mV
]
T = 18.5 C; θ = 1873.33 cm / sec
T = 18.5 C; θ = 1873.3324514717698 cm / sec
T = 18.5 C; θ = 1873.3324514717697 cm / sec
-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
T = 18.5 C; θ = 554.070286919319 cm/sec
T = 18.5 C; θ = 554.070286919320 cm/sec
Propagating wave solutions of the Hodgkin-Huxley equations
FitzHugh-Nagumo model of the Hodgkin-Huxley equations
V ...... potential ; Y ...... refractory variable
FitzHugh-Nagumo model and ist approximations
FitzHugh-Nagumo equation: reduced model
FitzHugh-Nagumo equation: reduced model
FitzHugh-Nagumo model and ist approximations
0
0.5
1.0
-0.5
-1.0210-1-2
s
X
FitzHugh-Nagumo equation: broken linear model
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
FitzHugh-Nagumo pulse propagation
Reduced Hodgkin-Huxley equations
V , m ...... fast variables, n , h ...... slow variables
V
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
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
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
Web-Page for further information:
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