Epilepsy as a dynamic disease: Musings by a clinical computationalist John Milton, MD, PhD William R. Kenan, Jr. Chair Computational Neuroscience The Claremont Colleges

Computational neuroscience?

Variables as a function of time

Differential equations = hypothesis = “Prediction”

Variables versus parameters Variable: Anything that can be measured Parameter: A variable which in comparison to other variables changes so slowly that it can be regarded to be constant.

Scientific Method Math/computer modeling Make better predictions Make better comparisons between observation and prediction In other words, essential scientific tools to enable science to “mature”

Inputs and outputs Measure outputs in response to inputs to figure out “what is inside the black box”

Linear black boxes

Neurons behave both as linear and nonlinear black boxes Linear aspects Graded potentials at axonal hillock sum linearly Nonlinear aspects Action potential Problem Cannot solve nonlinear problem with paper and pencil Qualitative methods

Qualitative theory of differential equations Consider system at equilibrium or steady state Assume for very small perturbations systems behaves linearly “If all you have is a hammer, then everything looks like a nail”

Qualitative theory: pictorial approach Potential, F(x), where

Potential surfaces and stability

Cubic nonlinearity: Bistability

Success story of computational neuroscience

Ionic pore behaves as RC circuit Membrane resistance Value intermediate between ionic solution and lipid bilayer Value was variable Membrane noise “shot noise”

Dynamics of RC circuit

Hodgkin-Huxley equations

HH equations (continued) “Linear” membrane hypothesis So equation looks like Problem: g is a variable not a parameter

Ion channel dynamics Hypothesis

HH equations Continuing in this way we obtain

Still too complicated: Fitzhugh-Nagumo equations

Graphical method: Nullcline V nullcline W nullcline

Neuron: Excitability

Neuron: Bistability

Neuron: Periodic spiking

Neuron: Starting & stopping oscillations

Dynamics and parameters Dynamics change as parameters change Not a continuous relationship Bifurcation: Abrupt qualitative change in dynamics as parameter passes through a bifurcation point

The challenge …..

A -> B -> C -> D -> ?

Is the anatomy important?

What should we be modeling?

Are differential equations appropriate? Physical Science Neurodynamics Neurons are “pulse-coupled” Such models meet requirement for low spiking frequency Models are not based on differential equations but instead focus on spike timing

Fundamental problem Models Measurements

Need for interdisciplinary teams Questions like these can only be answered using scientific method Epilepsy physicians are the only investigators who legally can investigate the brain of patient’s with epilepsy