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Comparing the ODE and PDE Models of the Hodgkin-Huxley
Equation
Sarah Arvey, Haley Rosehill
Calculus 114
History of Hodgkin-Huxley Model
• Hodgkin and Huxley experimented on squid giant axon and discovered how the signal is produced within the neuron
• Model was published in Journal of Physiology (1952)
• Hodgkin and Huxley awarded the 1963 Nobel Prize for model
Physical shape of a Neuron
• Dendrites• Nucleus• Cell body• Myelin• Axon
– Variety of gates
• Synaptic Terminal
Brief Biology Background of a Neuron
• A message is sent down the axon• The axon membrane contains a variety of
gates.• The gates slowly and continually open so
sodium and potassium ions can get through the gates
• The rate at which the ions are pumped across the membrane establishes the “resting potential” (-70 mV)
Action Potential
Taken http://artsci-ccwin.concordia.ca/psychology/psyc358/Lectures/figures/act_pot1/s_ociloAP.gif
Ordinary Differential
Equations• Model phenomena that
evolve continuously in time
• Equations in which the unknown element is a function, rather than a number
• Involves one independent variable
Partial Differential Equations
• Involves two or more independent variables
• Can track a function over space and time
VS.
ODE of Hodgkin-Huxley
• Measures action potential at a given time
• Membrane potential– Based on sodium, potassium and leakage– Clamp method
Action Potential
Taken http://artsci-ccwin.concordia.ca/psychology/psyc358/Lectures/figures/act_pot1/s_ociloAP.gif
The Model
I = (m^3)(h) GNa (ENa - E ) + (n^4) GK (EK - E ) + GL (EL - E )The parameter names in bold are fixed variables. I : the total ionic current across the membrane m : the probability that 1 of the 3 required activation particles has contributed to the activation of the Na gate (m^3 : the probability that all 3 activation particles have produced an open channel) h : the probability that the 1 inactivation particle has not caused the Na gate to close G_Na : Maximum possible Sodium Conductance (about 120 mOhms^-1/cm2) E : total membrane potential (about -60 mV) E_Na : Na membrane potential (about 55 mV) n : the probability that 1 of 4 activation particles has influenced the state of the K gate. G_K : Maximum possible Potassium Conductance (about 36 mOhms^-1/cm2) E_K : K membrane potential (about -72 mV) G_L : Maximum possible Leakage Conductance (about .3 mOhms^-1/cm2) E_L : Leakage membrane potential (about -50 mV)
M, H, and N are variables. 3 variables? How is it an ODE?
The Variable Functions
• Dm/dt= am(1-m)-bmm
• Dh/dt= ah(1-h)-bhh
• Dn/dt= an(1-n)-bnn
• All ODE’s thus Hodgkin and Huxley is a system of ODE’s
PDE of Hodgkin-Huxley
• Analysis of a traveling pulse
• Measures the state of the action potential over time and space
• Can be taken in respect to m, h, or n
What is this?!?• a= radius of axon
• p= resistance of the intracellular space
• The x variable is that of space
- just as single variable functions have higher order derivative, so do multi- variable functions
+/- of ODE
Positive Aspects• Simple• Gives total ionic
current at a specific time
• Tracks excitability and conductance of a neuron
Negative Aspects• Does not give
membrane potential over space– No true idea of action
potential activity
+/- of PDE
Positive Aspects• More telling of the
action potential’s activity
-space and time• Tracks excitability and
conductance via wave pulse
Negative Aspects• Confusing
Referenceshttp://www.math.niu.edu/~rusin/known-math/index/34-XX.html http://artsciccwin.concordia.ca/psychology/psyc358/Lectures/
figures/act_pot1/s_ociloAP.gifSegel, Lee A. “Biological Waves.” Mathematical Models in
Molecular and Cellular Biology. New York: Cambridge University Press, 1980.
http://retina.anatomy.upenn.edu/~lance/modelmath/hogkin_huxley.html
Muratov, C.B. “A Quantative Approximation Scheme for the Traveling Wave Solutions in the Hogkin-Huxley Model.” Biophysical Journal. Newark, New Jersey: University Heights, 2000.
http://www.ship.edu/~cgboereehttp://tutorial.math.lamar.edu/AllBrowsers/2415/
HighOrderPartialDerivs.asp