1. 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

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

3. A Physicist’s Neuron N inputs tanh x x

4. Architecture N neurons

5. 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.

6. Probability of Chaos

7. A Physicist’s EEG

8. Strange Attractor

9. Artist’s Brain

10. Airhead

11. Dumbbell

12. Featherbrain

13. Egghead

14. Scatterbrain

15. Attractor Dimension D KY = 0.46 N N

16. Route to Chaos at Large N (=64)

17. Animated Route to Chaos

18. 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

19. 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

20. 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

21. Training on an Image

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

23. Hurricane Rita

24. Robin Chapman

25. 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

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

27. References n http://sprott.physics.wisc.edu/ lectures/brain.ppt (this talk) http://sprott.physics.wisc.edu/ lectures/brain.ppt n sprott@physics.wisc.edu (contact me) sprott@physics.wisc.edu