The Physics of the Brain

1
Learning, memory, development and their cellular
basis synaptic plasticity.
2
Classical Conditioning
3
Neuron cell body

Action potentials
Axon output


Synapse
Dendrite input
4
Classical Conditioning Hebbs rule
Ear
A
salivation
tone
Nose
c
d
B
Tongue
 
 
When an axon in cell A is near enough to excite
cell B and repeatedly and persistently takes
part in firing it, some growth process or
metabolic change takes place in one or both
cells such that As efficacy in firing B is
increased
D. O. Hebb (1949)
5
Neuron cell body

Action potentials
Axon output


Synapse
Dendrite input
6
Various different protocols for inducing
bidirectional synaptic plasticity

• Presynaptic rate induced

• Pairing induced
• (postsynaptic voltage clamp)

• Spike time dependent plasticity
• (STDP)

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The generalized Hebb rule where di are the
inputs and c the output is assumed
linear Results in 2D
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Example of Hebb in 2D
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Mathematics of the generalized Hebb rule
The change in synaptic weight m is where d are
input vectors and c is the neural
response. Assume for simplicity a linear
neuron So we get
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Averaging over the input environment
using and In matrix form where
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Often use the substitution
If k1 is not zero, this has a fixed point,
however it is not stable.
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If k1 0 And the solution has the form What
are these elements?
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The Hebb rule is unstable how can it be
stabilized while preserving its properties? The
stabilized Hebb (Oja) rule.
This is has a fixed point at Where The only
stable fixed point is for ?max
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Therefore a stabilized Hebb (Oja neuron) carries
out Eigen-vector, or principal component analysis
(PCA).
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Homework 5 (due 4/19) 5a) Implement a simple
Hebb neuron with random 2D input, tilted at an
angle, ?30o with variances 1 and 3 and mean 0.
Show the synaptic weight evolution. (200 patterns
at least) 5b) Calculate the correlation matrix of
the input data. Find the eigen-values,
eigen-vectors of this matrix. Compare to 5a. 5c)
Repeat 5a for an Oja neuron, compare to 5b.
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Visual Pathway
Visual Cortex
Receptive fields are

• Binocular
• Orientation Selective

Area 17
LGN
Receptive fields are

• Monocular
• Radially Symmetric

Retina
17
Right
Left
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Orientation Selectivity
Binocular Deprivation
Normal
Adult
Response (spikes/sec)
Response (spikes/sec)
Adult
angle
angle
Eye-opening
Eye-opening
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Monocular Deprivation
Normal
Left
Right
Right
Left
Response (spikes/sec)
angle
angle
20
30
of cells
15
10
1 2 3 4 5 6 7
1 2 3 4 5 6 7
Rittenhouse et. al.
group
group
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Aim get selective neurons using a Hebb/PCA
rule Simple exmple
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Show examples with plasticity package for q1 and
q4 Why?
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The eigen-value equation has the form Can be
rewritten in the equivalent form And a
possible solution can be written as the sum
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Inserting, and by orthogonality get So
for get three solutions and with
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Orientation selectivity from a natural
environment The Images
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Raw images (fig 5.8) Show how to do
this in plasticity.
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Preprocessed images (fig 5.9) Show how
to do this in plasticity.
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Monocular Deprivation
Normal
Left
Right
Right
Left
Response (spikes/sec)
angle
angle
20
30
of cells
15
10
1 2 3 4 5 6 7
1 2 3 4 5 6 7
Rittenhouse et. al.
group
group
29
Binocularity simple examples. Q is a 2-eye
correlation function. What is the solution of
the eigen-value equation
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In a higher dimensional case Qll, Qlr etc.
are now matrixes. And QlrQrl. The eigen-vectors
now have the form
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In a simpler case This implies QllQrr, that
is eyes are equivalent. And the cross eye
correlation is a scaled version of the one eye
correlation. If
then with
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Positive correlations (?0.2)
Hebb with lower saturation at 0
Negative correlations (?-0.2)
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Lets now assume that Q is as above for the
1D selectivity example.
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With 2D space included
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2 partially overlapping eyes using natural images
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Orientation selectivity and Ocular Dominance
PCA
Right
Left
No. of Cells
1
2
3
Bin
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