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

2. Computational neuroscience?

3. Variables as a function of time

4. Differential equations
= hypothesis
= “Prediction”

5. 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.

6. 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”

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

8. Linear black boxes

9. 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

10. 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”

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

12. Potential surfaces and stability

13. Cubic nonlinearity: Bistability

14. Success story of computational neuroscience

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

16. Dynamics of RC circuit

17. Hodgkin-Huxley equations

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

19. Ion channel dynamics
Hypothesis

20. HH equations
Continuing in this way we obtain

21. Still too complicated: Fitzhugh-Nagumo equations

22. Graphical method: Nullcline
V nullcline
W nullcline

23. Neuron: Excitability

24. Neuron: Bistability

25. Neuron: Periodic spiking

26. Neuron: Starting & stopping oscillations

27. 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

28. The challenge …..

29. A -> B -> C -> D -> ?

30. Is the anatomy important?

31. What should we be modeling?

32. 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

33. Fundamental problem
Models
Measurements

34. 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