Transcript Epilepsy as a dynamic disease: Musings by a clinical computationalist

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
dx
 f (x)
dt
= hypothesis
= “Prediction”
Variables versus parameters
dV0
RC
 V0  Vi
dt
• 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
dx
0
dt
• 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
dx
 f (x)
dt
x
F( x )    f ( s)ds
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
Q  CV
dQ
dV
 C
dt
dt
dV
C
 I
dt
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
dx
 f (x)
dt
• Neurodynamics
– Neurons are “pulsecoupled”
– 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