Epilepsy as a dynamic disease: Musings by a clinical computationalist

John Milton, MD, PhDWilliam R. Kenan, Jr. Chair

Computational NeuroscienceThe Claremont Colleges

Computational neuroscience?

Variables as a function of time

Differential equations

= hypothesis

= “Prediction”

)x(fdt

dx

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.

iVVdt

dVRC 0

0

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”

0dt

dx

Qualitative theory: pictorial approach

• Potential, F(x), where

x

ds)s(f)x(F

)x(fdt

dx

0

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

Idt

dVC

dt

dVC

dt

dQ

CVQ

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

)x(fdt

dx

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