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A Physicist’s Brain
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Transcript of A Physicist’s Brain
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A Physicist’s Brain
J. C. SprottDepartment of PhysicsUniversity of Wisconsin - Madison
Presented at theChaos and Complex Systems SeminarIn Madison, WisconsinOn October 18, 2005
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Collaborators
David Albers, Max Planck Institute (Leipzig, Germany)
Matt Sieth, Univ Wisc - Undergrad
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A Physicist’s Neuron
jN
j jxax 1
tanhoutN
inputs
tanh x
x
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2 4
1
3
Architecture
)1(tanh)(1
txatxj
N
j iji
N neurons
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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.
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Probability of Chaos
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A Physicist’s EEG
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Strange Attractor
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Artist’s Brain
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Airhead
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Dumbbell
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Featherbrain
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Egghead
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Scatterbrain
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Attractor Dimension
DKY = 0.46 N
N
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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
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Animated Route to Chaos
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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
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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
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Automatic Preselection Must be chaotic (positive
Lyapunov exponent)
Not too “thin” (fractal
dimension > 1)
Not too small or too large
Not too off-centered
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Training on an Image
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Problem – Rugged LandscapeRe
lativ
e Er
ror
-5% +5%0
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Hurricane Rita
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Robin Chapman
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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
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Future Directions
More biological realism
More neurons
More realistic architecture
Training on real EEG data
or task performance
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References
http://sprott.physics.wisc.edu/ lectures/
brain.ppt (this talk)
(contact me)