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 n David Albers, Max Planck Institute (Leipzig, Germany) n Matt Sieth, Univ Wisc - Undergrad

A Physicist’s Neuron N inputs tanh x x

Architecture N neurons

Artificial Neural Network (P-Brain) n Nonlinear, discrete-time, complex, dynamical system n “Universal” approximator (?) a ij chosen from a random Gaussian distribution with mean zero and standard deviation s Two parameters: N and s Arbitrary (large) N  infinity n 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 D KY = 0.46 N N

Route to Chaos at Large N (=64)

Animated Route to Chaos

Summary of High- N Dynamics n Chaos is the rule Maximum attractor dimension is of order N /2 n Quasiperiodic route is usual n Attractor is sensitive to parameter perturbations, but dynamics are not

P-Brain Artist n Train a neural network to produce art Choose N = 6 n Find “good” regions of the 36-D parameter space n Randomly explore a neighborhood of that region

Automatic Preselection n Must be chaotic (positive Lyapunov exponent) n Not too “thin” (fractal dimension > 1) n Not too small or too large n Not too off-centered

Training on an Image

Problem – Rugged Landscape Relative Error -5%+5%0

Hurricane Rita

Robin Chapman

Information Content n 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 n P-Brain: 36 x 5 = 180 bits n  Cannot expect a good replica

Future Directions n More biological realism n More neurons n More realistic architecture n Training on real EEG data or task performance

References n lectures/brain.ppt (this talk) lectures/brain.ppt n (contact me)