A Physicist’s Brain

J. C. SprottDepartment of PhysicsUniversity of Wisconsin - Madison

Presented at theChaos and Complex Systems SeminarIn Madison, WisconsinOn October 18, 2005

Collaborators

David Albers, Max Planck Institute (Leipzig, Germany)

Matt Sieth, Univ Wisc - Undergrad

A Physicist’s Neuron

jN

j jxax 1

tanhoutN

inputs

tanh x

x

2 4

1

3

Architecture

)1(tanh)(1

txatxj

N

j iji

N neurons

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)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0.01 0.1 1 10

s

Larg

est L

yapu

nov

Expo

nent

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

Problem – Rugged LandscapeRe

lativ

e Er

ror

-5% +5%0

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)