A Physicist’s Brain
J. C. Sprott
Department of Physics
University of Wisconsin Madison
Presented at the
Chaos and Complex Systems
Seminar
In Madison, Wisconsin
On October 18, 2005
Collaborators


David Albers,
Max Planck
Institute (Leipzig,
Germany)
Matt Sieth, Univ
Wisc - Undergrad
A Physicist’s Neuron
N
N
xout  tanh  a j x j
j1
inputs
tanh x
x
Architecture
1
N neurons
3
2
4
N
x (t )  tanh  a x (t 1)
i
ij
j
j1
Artificial Neural Network (P-Brain)



Nonlinear, discrete-time, complex,
dynamical system
“Universal” approximator (?)
aij chosen from a random
Gaussian distribution with mean
zero and standard deviation s

Two parameters: N and s

Arbitrary (large) N  infinity

Initial conditions random in the
range -1 to +1.
Probability of Chaos
A Physicist’s EEG
Strange Attractor
Artist’s Brain
Airhead
Dumbbell
Featherbrain
Egghead
Scatterbrain
Attractor Dimension
DKY = 0.46 N
N
Route to Chaos at Large N (=64)
1.5
Largest Lyapunov Exponent
1
0.5
0
0.01
-0.5
0.1
1
-1
-1.5
-2
-2.5
-3
s
10
Animated Route to Chaos
Summary of High-N Dynamics

Chaos is the rule

Maximum attractor dimension is
of order N/2

Quasiperiodic route is usual

Attractor is sensitive to
parameter perturbations, but
dynamics are not
P-Brain Artist

Train a neural network to
produce art

Choose N = 6

Find “good” regions of the 36-D
parameter space

Randomly explore a
neighborhood of that region
Automatic Preselection

Must be chaotic (positive
Lyapunov exponent)

Not too “thin” (fractal
dimension > 1)

Not too small or too large

Not too off-centered
Training on an Image
Relative Error
Problem – Rugged Landscape
-5%
0
+5%
Hurricane Rita
Robin Chapman
Information Content

Robin: 244 x 340 x 3 x 8 = 2 Mbits
Compresses (gif) to 283 kbits
Compresses (jpeg) to 118 kbits
Compresses (png) to 1.8 Mbits

P-Brain: 36 x 5 = 180 bits

 Cannot expect a good replica
Future Directions

More biological realism

More neurons

More realistic architecture

Training on real EEG data
or task performance
References

http://sprott.physics.wisc.edu/
lectures/brain.ppt (this talk)

sprott@physics.wisc.edu
(contact me)