The Hodgkin-Huxley Equations and Analytical Approximations for them Peter Schuster Institut 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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Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling, ... The human brain 10 11 neurons connected by � 10 13 to 10 14 synapses
Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling, ... A single neuron signaling to a muscle fiber
B A 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, collective properties, nonlinear dynamics, signalling, ... 1 d V = − − − − − − 3 4 ( ) ( ) ( ) I g m h V V g n V V g V V Na Na K K l l d t C M dm = α − − β Hogdkin-Huxley OD equations ( 1 ) m m m m dt dh = α − − β ( 1 ) h h h h dt dn = α − − β ( 1 ) n n n n dt A single neuron signaling to a muscle fiber
Gating functions of the Hodgkin-Huxley equations
Temperature dependence of the Hodgkin-Huxley equations
1 d V = − − − − − − 3 4 ( ) ( ) ( ) I g m h V V g n V V g V V Na Na K K l l d t C M dm = α − − β ( 1 ) m m m m dt dh = α − − β ( 1 ) h h h h dt dn = α − − β ( 1 ) n n n n dt Hogdkin-Huxley OD equations Hhsim.lnk Simulation of space independent Hodgkin-Huxley equations: Voltage clamp and constant current
∂ ∂ 2 1 V V = + − + − + − π 3 4 ( ) ( ) ( ) 2 C g m h V V g n V V g V V r L ∂ ∂ 2 Na Na K K l l R x t ∂ m = α − − β ( 1 ) m m Hodgkin-Huxley PDEquations ∂ m m t ∂ h = α − − β ( 1 ) Travelling pulse solution: V ( x,t ) = V ( � ) with h h ∂ h h t � = x + � t ∂ n = α − − β ( 1 ) n n ∂ n n t Hodgkin-Huxley equations describing pulse propagation along nerve fibers
[ ] 2 1 d V d V = θ + − + − + − π 3 4 ( ) ( ) ( ) 2 C g m h V V g n V V g V V r L ξ ξ 2 M Na Na K K l l R d d Hodgkin-Huxley PDEquations d m θ = α − − β ( 1 ) m m ξ m m d Travelling pulse solution: V ( x,t ) = V ( � ) with d h θ = α − − β ( 1 ) h h � = x + � t ξ h h d d n θ = α − − β ( 1 ) n n ξ n n d Hodgkin-Huxley equations describing pulse propagation along nerve fibers
100 50 ] V m [ V 0 -50 1 2 3 4 5 6 � [cm] 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
40 30 20 ] V m [ 10 V 0 -10 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
2 X 1 0 -1 -2 1.0 0.5 0 -0.5 -1.0 s
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 a 0.1 0.005 0.2 0.3 b 0.010 0.6 0.015 0.4 0.6 � 0.020 0.2 0.4 0.020 � 0.015 0 0.2 0.010 0 b 0 0.005 0.1 0.2 a 0 0.3
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 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 Universität Wien der Wissenschaften Acknowledgement of support Österreichische Akademie der Wissenschaften, Universität Wien and University of Colorado
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
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