Sparse Spatiotemporal Codes to explain complex cells

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Sparse Spatiotemporal Codes to explain complex
cells
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V1 simple cells
Selective For
Orientation
Position
Frequency
Phase
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V1 linear model
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Complex cells
Selective For
Orientation
Position
Frequency
Phase Invariant
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Quadratic forms
½ xT A x
g(x)
bT x
½ xT E D ET x
½ (ET x)T D (ET x)
½ ?i di (eiT x) 2
2
2
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1
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Functional explanation
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Simple Cells

Complex Cells
Our Question
Why?
To code our visual world efficiently.
metabolic cost
fast processing
useful representation
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Slow Feature Analysis
V1 ? V2 ? V4 ? IT
face selective cells
simple/complex cells
Larger, more selective RFs, but slower change in
response
Phase changes quickly
So SFA would force phase invariant

• Minimize f (x) 2 with constraints
• f(x) has zero mean
• f(x) has unit variance
• fa and fb are uncorrelated

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Sparse coding and ICA
One Goal effectively represent the image
Sparse coding
Have fewest neurons highly active at a time (as
opposed to compact coding)
Independent Components Analysis
p(f1, f2) p(f1) p(f2)
Make the neural responses as statistically
independent as possible
For practical purposes these approaches are
fairly similar
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Quadratic forms functional approaches
Slow Feature Analysis
Successful phase invariant forms (Wiskott)
Biologically plausible hebbian mechanism suggested
(Kording, Kayser 2004)
Questionable objective function
Sparse coding/ICA
Hyvarinen Hoyer 2002 Local pooled energies
assumed, not derived
Hashimoto 2003 used general quadratic
forms, unsuccessful required SFA
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Spatiotemporal quadratic forms
All previous models have dynamic learning
methods, but assume static filters
Filters are not static, for example
A sparse static code is inefficient over time.
A sparse spatiotemporal code?
Difficulty for quad forms a 4x4x4 (width,
height, time) patch requires 2000 learned
coefficients