A Physicist’s Brain J. C. Sprott Department of Physics University of Wisconsin - Madison Presented...
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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
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Transcript of A Physicist’s Brain J. C. Sprott Department of Physics University of Wisconsin - Madison Presented...
A Physicist’s Brain
J. C. SprottDepartment of PhysicsUniversity 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
jN
jjxax
1tanhout
Ninputs
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
La
rge
st
Ly
ap
un
ov
Ex
po
ne
nt
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 LandscapeR
elat
ive
Err
or
-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/ l
ectures/
brain.ppt (this talk)
(contact me)
