SLIDE 1
The Hodgkin-Huxley Equations and Analytical Approximations for them - - PowerPoint PPT Presentation
The Hodgkin-Huxley Equations and Analytical Approximations for them - - PowerPoint PPT Presentation
The Hodgkin-Huxley Equations and Analytical Approximations for them Peter Schuster Institut fr Theoretische Chemie und Molekulare Strukturbiologie der Universitt Wien Seminar of the MPI for Mathematics in Science Leipzig, 22.07.2004
SLIDE 2
SLIDE 3
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
SLIDE 4
Neurobiology
Neural networks, collective properties, nonlinear dynamics, signalling, ...
The human brain 1011 neurons connected by 1013 to 1014 synapses
SLIDE 5
Neurobiology
Neural networks, collective properties, nonlinear dynamics, signalling, ...
A single neuron signaling to a muscle fiber
SLIDE 6
B A
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
SLIDE 7
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
SLIDE 8
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
SLIDE 9
Neurobiology
Neural networks, collective properties, nonlinear dynamics, signalling, ...
) ( ) ( ) ( 1
4 3 l l K K Na Na M
V V g V V n g V V h m g I C t d V d − − − − − − =
m m dt dm
m m
β α − − = ) 1 ( h h dt dh
h h
β α − − = ) 1 ( n n dt dn
n n
β α − − = ) 1 (
Hogdkin-Huxley OD equations
A single neuron signaling to a muscle fiber
SLIDE 10
Gating functions of the Hodgkin-Huxley equations
SLIDE 11
Temperature dependence of the Hodgkin-Huxley equations
SLIDE 12
) ( ) ( ) ( 1
4 3 l l K K Na Na M
V V g V V n g V V h m g I C t d V d − − − − − − =
m m dt dm
m m
β α − − = ) 1 ( h h dt dh
h h
β α − − = ) 1 ( n n dt dn
n n
β α − − = ) 1 (
Hogdkin-Huxley OD equations
Hhsim.lnk
Simulation of space independent Hodgkin-Huxley equations: Voltage clamp and constant current
SLIDE 13
L r V V g V V n g V V h m g t V C x V R
l l K K Na Na
π 2 ) ( ) ( ) ( 1
4 3 2 2
− + − + − + ∂ ∂ = ∂ ∂ m m t m
m m
β α − − = ∂ ∂ ) 1 ( h h t h
h h
β α − − = ∂ ∂ ) 1 ( n n t n
n n
β α − − = ∂ ∂ ) 1 (
Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V() with = x + t
Hodgkin-Huxley equations describing pulse propagation along nerve fibers
SLIDE 14
Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V() with = x + t
[ ]
L r V V g V V n g V V h m g d V d C d V d R
l l K K Na Na M
π ξ θ ξ 2 ) ( ) ( ) ( 1
4 3 2 2
− + − + − + =
m m d m d
m m
β α ξ θ − − = ) 1 ( h h d h d
h h
β α ξ θ − − = ) 1 ( n n d n d
n n
β α ξ θ − − = ) 1 (
Hodgkin-Huxley equations describing pulse propagation along nerve fibers
SLIDE 15
50
- 50
100 1 2 3 4 5 6 [cm] V [ m V ]
T = 18.5 C; θ = 1873.33 cm / sec
SLIDE 16
T = 18.5 C; θ = 1873.3324514717698 cm / sec
SLIDE 17
T = 18.5 C; θ = 1873.3324514717697 cm / sec
SLIDE 18
- 10
10 20 30 40 V [ m V ] 6 8 10 12 14 16 18 [cm]
T = 18.5 C; θ = 544.070 cm / sec
SLIDE 19
T = 18.5 C; θ = 554.070286919319 cm/sec
SLIDE 20
T = 18.5 C; θ = 554.070286919320 cm/sec
SLIDE 21
Propagating wave solutions of the Hodgkin-Huxley equations
SLIDE 22
SLIDE 23
FitzHugh-Nagumo model of the Hodgkin-Huxley equations
V ...... potential ; Y ...... refractory variable
SLIDE 24
FitzHugh-Nagumo model and ist approximations
SLIDE 25
FitzHugh-Nagumo equation: reduced model
SLIDE 26
FitzHugh-Nagumo equation: reduced model
SLIDE 27
FitzHugh-Nagumo model and ist approximations
SLIDE 28
0.5 1.0
- 0.5
- 1.0
2 1
- 1
- 2
s X
SLIDE 29
FitzHugh-Nagumo equation: broken linear model
SLIDE 30
SLIDE 31
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
SLIDE 32
SLIDE 33
FitzHugh-Nagumo pulse propagation
SLIDE 34
Reduced Hodgkin-Huxley equations V , m ...... fast variables, n , h ...... slow variables
SLIDE 35
- V
SLIDE 36
SLIDE 37
0.005 0.010 0.020 0.015 b 0.1 0.2 0.3 a 0.6 0.4 0.2
- 0.005
0.010 0.020 0.015 b 0.1 0.2 0.3 a 0.6 0.4 0.2
SLIDE 38
SLIDE 39
SLIDE 40
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 forced relaxation oscillations, Int.J.Bifurcation and Chaos 12:1295-1307, 2002 Paul E. Phillipson, Peter Schuster, An analytic picture of neuron
- scillations, 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
SLIDE 41
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
SLIDE 42
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
SLIDE 43