2. Comparing the ODE and PDE Models of the Hodgkin-Huxley Equation Sarah Arvey, Haley Rosehill Calculus 114

3. 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

4. Physical shape of a Neuron Dendrites Nucleus Cell body Myelin Axon –Variety of gates Synaptic Terminal

5. 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)

6. Action Potential Taken http://artsci-ccwin.concordia.ca/psychology/psyc358/Lectures/figures/act_pot1/s_ociloAP.gif

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

8. 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.

9. ODE of Hodgkin-Huxley Measures action potential at a given time Membrane potential –Based on sodium, potassium and leakage –Clamp method

10. Action Potential Taken http://artsci-ccwin.concordia.ca/psychology/psyc358/Lectures/figures/act_pot1/s_ociloAP.gif

11. The Model I = (m^3)(h) G Na (E Na - E ) + (n^4) G K (E K - E ) + G L (E L - 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?

12. The Variable Functions Dm/dt= a m (1-m)-b m m Dh/dt= a h (1-h)-b h h Dn/dt= a n (1-n)-b n n All ODE’s thus Hodgkin and Huxley is a system of ODE’s

13. 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

14. The Actual Model

15. 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

16. +/- 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

17. +/- 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

18. Which model is better? WE LIKE THE PDE!!!!

19. References http://www.math.niu.edu/~rusin/known-math/index/34- XX.html http://artsciccwin.concordia.ca/psychology/psyc358/Lecture s/figures/act_pot1/s_ociloAP.gif Segel, 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/~cgboeree http://tutorial.math.lamar.edu/AllBrowsers/2415/HighOrderP artialDerivs.asp