Transcript A Physicist's Brain - University of Wisconsin

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
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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)
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
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Nonlinear, discrete-time, complex,
dynamical system
“Universal” approximator (?)
aij chosen from a random
Gaussian distribution with mean
zero and standard deviation s
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Two parameters: N and s
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Arbitrary (large) N  infinity
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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
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Chaos is the rule
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Maximum attractor dimension is
of order N/2
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Quasiperiodic route is usual
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Attractor is sensitive to
parameter perturbations, but
dynamics are not
P-Brain Artist
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Train a neural network to
produce art
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Choose N = 6
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Find “good” regions of the 36-D
parameter space
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Randomly explore a
neighborhood of that region
Automatic Preselection
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Must be chaotic (positive
Lyapunov exponent)
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Not too “thin” (fractal
dimension > 1)
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Not too small or too large
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Not too off-centered
Training on an Image
Relative Error
Problem – Rugged Landscape
-5%
0
+5%
Hurricane Rita
Robin Chapman
Information Content
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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
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P-Brain: 36 x 5 = 180 bits
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 Cannot expect a good replica
Future Directions
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More biological realism
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More neurons
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More realistic architecture
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Training on real EEG data
or task performance
References
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http://sprott.physics.wisc.edu/
lectures/brain.ppt (this talk)
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[email protected]
(contact me)