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

Action Potential

Taken from C. George Boeree: www.ship.edu/~cgboeree

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

The Actual Model

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

Which model is better?

WE LIKE THE PDE!!!!

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