hebb rule

Hebb Rule

• Linear neuron

• Hebb rule

• Similar to LTP (but not quite…)

Tv w u

1, w t tw

d tv v

dt w

u w w u

Hebb Rule

• Average Hebb rule= correlation rule

• Q: correlation matrix of u

w

T Tw

T T

dv

dtd

vdt

wu

wu w u u u w u

u u w uu w Qw

Hebb Rule

• Hebb rule with threshold= covariance rule

• C: covariance matrix of u• Note that <(v-< v >)(u-< u >)> would be unrealistic because it predicts

LTP when both u and v are low

Tw

T T

dv

dtC

C

wu u w u u u

w

u u u u u u u

Hebb Rule

• Main problem with Hebb rule: it’s unstable… Two solutions:

1. Bounded weights

2. Normalization of either the activity of the postsynaptic cells or the weights.

BCM rule

• Hebb rule with sliding threshold

• BCM rule implements competition because when a synaptic weight grows, it raises by v2, making more difficult for other weights to grow.

2

v

v

w v

vv

w

dv v

dtd

vdt

wu

Weight Normalization

• Subtractive Normalization:

1

, 11 11

1

1 0

u

wu

Ni

w i kku

wu

u

vdv

dt N

dwvu vu

dt N

vdv

dt N

vN

n u nwu n

n u nn wn u

n nn u

1

Const.uN

ii

w

n w

Weight Normalization

• Multiplicative Normalization:

• Norm of the weights converge to 1/

2

2

222

2

2 1

w

w w

dv v

dt

d ddt dt

dv

dt

wu w

w ww

ww

2

1

Const.uN

ii

w

• Convergence properties:

• Use an eigenvector decomposition:

where e are the eigenvectors of Q

Hebb Rule

w

ddt

wQw

1

uN

t c t

w e

Hebb Rule

e2e1

1>2

Hebb Rule

1

, uN

w

w

w

w

d tt t c t

dt

dc tc t

dtdc t

c tdt

dc tc t

dt

wQw w e

eQ e

e Qe

e e

Equations decouple because

e are the eigenvectors of Q

Hebb Rule

1

1 1

exp 0 exp 0

exp 0

for large , ,

u

w

w

w w

N

w

dc tc t

dtdc t

c tdt

t tc t c w

tt w

t t v

e e

e

w e e

w e e u

Hebb Rule

• The weights line up with first eigenvector and the postsynaptic activity, v, converges toward the projection of u onto the first eigenvector (unstable PCA)

Hebb Rule

• Non zero mean distribution: correlation vs covariance

Hebb Rule

• Limiting weights growth affects the final state

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

1

w1/wmax

w2/w

max

0.8 First eigenvector: [1,-1]

Hebb Rule

• Normalization also affects the final state. • Ex: multiplicative normalization. In this case,

Hebb rule extracts the first eigenvector but keeps the norm constant (stable PCA).

• Normalization also affects the final state. • Ex: subtractive normalization.

Hebb Rule

1

111

if

0

wu

wu u

ddt N

ddt N N

w Qn nwQw

e n

e Qn n n Qn neQe Qn

Hebb Rule

1

1

1 1

if

0, 1

wu

u

u

c tdc tc t

dt N

c tc t

N

c tc t

N

c t

e n

e n

e Qn ne Q e

e Qe nQ e

e e nQ e

Q e

• The constrain does not affect the other eigenvector:

• The weights converge to the second eigenvector (the weights need to be bounded to guarantee stability…)

Hebb Rule

1 12

0 exp 0uN

w

tt w w

w e e e e

Ocular Dominance Column

• One unit with one input from right and left eyes

R R L L

s dR R L R

d sR L L L

v w u w u

q qu u u u

q qu u u u

Q uu

s: same eye

d: different eyes


• The eigenvectors are:

s dR R L RT

d sR L L L

q qu u u u

q qu u u u

Q uu

1 1

2 2

1,1 / 2,

1, 1 / 2,

s d

s d

q q

q q

e

e


• Since qd is likely to be positive, qs+qd>qs-qd. As a result, the weights will converge toward the first eigenvector which mixes the right and left eye equally. No ocular dominance...

1 1

2 2

1,1 / 2,

1, 1 / 2,

s d

s d

q q

q q

e

e


• To get ocular dominance we need subtractive normalization.

1 1

2 2

1,1 / 2,

1, 1 / 2,

s d

s d

q q

q q

e

e


• Note that the weights will be proportional to e2 or –e2 (i.e. the right and left eye are equally likely to dominate at the end). Which one wins depends on the initial conditions.

1 1

2 2

1,1 / 2,

1, 1 / 2,

s d

s d

q q

q q

e

e


• Ocular dominance column: network with multiple output units and lateral connections.


• Simplified model

u L uR

B


• If we use subtractive normalization and no lateral connections, we’re back to the one cell case. Ocular dominance is determined by initial weights, i.e., it is purely stochastic. This is not what’s observed in V1.

• Lateral weights could help by making sure that neighboring cells have similar ocular dominance.


• Lateral weights are equivalent to feedforward weights

0?

ii iR R iL L

R R L L

dvv w u w u

dtd

u udt

ddt

+ Mv

vv w w + Mv

v


• Lateral weights are equivalent to feedforward weights

1

i iR R iL L

R R L L

RR L

L

v w u w u

u u

u

u

= + Mv

v = w w + Mv

v = I M w w

v = KWu


w

w

w

ddt

ddt

ddt

Wvu

v = KWu

WKWuu

WKWQ


• We first project the weight vectors of each cortical unit (wiR,wiL) onto the eigenvectors of Q.

1

w

w

w

ddt

ddt

ddt

WKWQ

WKWPΛP

WPKWPΛ


• There are two eigenvectors, w+ and w-, with eigenvalues qs+qd and qs-qd:

=

=

R L

R L

R L R L

w w w

w w w

WP w w w w

w w


0

0

w

s dw

s d

s dw

s d

ddtd

q qdtq qd

dtd

q qdtq qd

dt

WPKWPΛ

w

K w ww

wK w

K ww


• Ocular dominance column: network with multiple output units and lateral connections.

w s d

w s d

dq q

dtd

q qdt

wKw

wKw


• Once again we use a subtractive normalization, which holds w+ constant. Consequently, the equation for w- is the only one we need to worry about.

w s d

dq q

dt

wKw


• If the lateral weights are translation invariant, Kw- is a convolution. This is easier to solve in the Fourier domain.

( )*

w s d

s d

kw s d k k

dq q

dtq q x

d wq q K w

dt

wKw

K w


• The sine function with the highest Fourier coefficient (i.e. the fundamental) growth the fastest.

exp 0

kw s d k k

s d kk k

w

d wq q K w

dt

q q K tw t w


• In other words, the eigenvectors of K are sine functions and the eigenvalues are the Fourier coefficients for K.

2cosa

v

ae

N


• The dynamics is dominated by the sine function with the highest Fourier coefficients, i.e., the fundamental of K(x) (note that w- is not normalized along the x dimension).

• This results is an alternation of right and left columns with a periodicity corresponding to the frequency of the fundamental of K(x).


• If K is a Gaussian kernel, the fundamental is the DC term and w ends up being constant, i.e., no ocular dominance columns (one of the eyes dominate all the cells).

• If K is a mexican hat kernel, w will show ocular dominance column with the same frequency as the fundamental of K.

• Not that intuitive anymore…


• Simplified model

-0 .6 -0 .4 -0 .2 0 0.2 0.4 0.6-1

-0 .5

0

0.5

1

0 20 40 600

0.2

0.4

0.6

c o rtic a l d ista nc e (m m ) k (1/m m )

K, e K~

A B


• Simplified model: weights matrices for right and left eyes

W L W R W R W LW W W - W

hebb rule

Documents

similar ocular dominance

ocular dominance columnto

ocular dominance columnnote

ocular dominance columnthere

initial weights

hebb rulenormalization

eigenvector decomposition

bcm rulehebb rule