# The BCM theory of synaptic plasticity. The BCM BCM theory of synaptic plasticity. The BCM theory of cortical plasticity BCM stands for Bienestock Cooper and Munro, it dates back to

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• The BCM theory of synaptic plasticity.

The BCM theory of cortical plasticity

BCM stands for Bienestock Cooper and Munro, it dates back to 1982. It was designed in order to account for experiments which demonstrated that the development of orientation selective cells depends on rearing in a patterned environment.

• BCM Theory (Bienenstock, Cooper, Munro 1982; Intrator, Cooper 1992)

LTD LTP

dw jdt

=x j(y,m )

(y,m )

m

y

y

w1 w2 w3 w4 w5

x1

x2

x3

x4

x5

y = w jj x j

For simplicity linear neuron

1) The learning rule:

• BCM Theory

Bidirectional synaptic modification LTP/LTD Sliding modification threshold The fixed points depend on the environment, and in a patterned environment only selective fixed points are stable.

LTD LTP

Requires

dw jdt

=x j(y,m )

(y,m )

m E y2[ ] = 1

y 2

t

( t )e( t t ) /d t

m

y

2) The sliding threshold

1)

• Is equivalent to this differential form: The integral form of the average:

Note, it is essential that m is a superlinear function of the history of C, that is:

with p>1

Note also that in the original BCM formulation (1982) rather then

m E y2[ ] = 1

y 2

t

( t )e( t t ) /d t

dmdt

=1(y 2 m )

dmdt

=1(y p m )

m E y[ ]2

m E y2[ ]

• What is the outcome of the BCM theory?

Assume a neuron with N inputs (N synapses), and an environment composed of N different input vectors.

An N=2 example:

What are the stable fixed points of W in this case?

x1

x2

x1 =1.00.2

x2 =

0.10.9

• (Notation: )

What are the fixed points? What are the stable fixed points?

Note:"Every time a new input is presented, W changes, and so does m"

x1

x2

yi = wT xi

(Show matlab)

• Two examples with N= 5

Note: The stable FP is such that for one pattern yi=wTxi=m while for the others"y(ij)=0.

(note: here c=y)

Show movie

• BCM Theory Stability

One dimension Quadratic form

Instantaneous limit

dwdt

=y y M( )x

dwdt

= y y y 2( )x= y 2(1 y)x

• BCM Theory Selectivity

Two dimensions Two patterns

Quadratic form Averaged threshold

dwdt

= yk yk M( )x k

M = E y2[ ]patterns

= pk (yk )2

k=1

2

,

Fixed points

• BCM Theory: Selectivity

Learning Equation Four possible fixed points

, , , , (unselective)

(unselective) (Selective)* (Selective)

Threshold*

dwdt

=yk yk M( )x k

• Stability of Fixed point

Assume w* is a fixed point:

Assume that w is the deviation from this fixed point. w=w*+w.

Then:

If then

dwdt

=dw*

dt+dwdt

=dwdt

dwdt

= F(w)

dwdt

= F(w* + w) F w*( ) + w FW w*

F(w*) = 0

• Consider a selective F.P (w1) where:

and

So that

for a small pertubation from the F.P such that

The two inputs result in:

m1 = E[y 2] = 1

2(w1 x1)2 + (w1 x 2)2[ ] = 12 m

1[ ]2

• So that

m =12(w1 x1 + w x1)2 + (w1 x 2 + w x 2 )2{ }

m* (1+ w x1) +O(w2) = m

* + 2w x1

• 2y

1 y m( )

At y0 and at ym we make a linear approximation

In order to examine whether a fixed point is stable we examine if the average norm of the perturbation ||w|| increases or decreases.

Decrease Stable Increase Unstable

• Remember:

For the preferred input x1:

Similarly, for the non preferred input x2:

m = m* + 2w x1

dwdt

= 1(y m )x1 = 1(m

* + w x1 m )x1

1(m* + w x1 m

* 2w x1)x = 1(w x1)x1

w

= 2y x2 = 2(w x

2)x 2

• Use trick:

And average over two input patterns

Insert previous result to show that:

For the non selective F.P we get:

ddt

w[ ]2 = ddt

w w[ ] = 2w w

E ddt

w[ ]2

=12w w

x1

+12w w

x 2

E ddt

w[ ]2

= 1(w x

1)2 + 2(w x2)2[ ] < 0

E ddt

w[ ]2

0

• Phase plane analysis of BCM in 1D

Previous analysis assumed that m=E[y2] exactly. If we use instead the dynamical equation

Will the stability be altered?

Look at 1D example

dmdt

=1(y 2 m )

• Phase plane analysis of BCM in 1D

Assume x=1 and therefore y=w. Get the two BCM equations:

There are two fixed points y=0, m=0, and y=1, m=1. The previous analysis shows that the second one is stable, what would be the case here? How can we do this? (supplementary homework problem)

0 0.5 1 m

y 1

0.5

0

?

dydt

=y(y m )

dmdt

=1(y 2 m )

• Linear stability analysis:

• Summary

The BCM rule is based on two differential equations, what are they?

When there are two linearly independent inputs, what will be the BCM stable fixed points? What will be?

When there are K independent inputs, what are the stable fixed points? What will be?

• Homework 2: due on Feb 1

1. Code a single BCM neuron, apply to case with 2 linearly independent inputs with equal probability

2. Apply to 2 inputs with different probabilities, what is different?

3. Apply to 4 linearly indep. Inputs with same prob.

Extra credit 25 pt 4. a. Analyze the f.p in 1D case, what are the stable

f.p as a function of the systems parameters. b. Use simulations to plot dynamics of y(t), (t) and their trajectories in the m plane for different parameters. Compare stability to analytical results (Key parameters, )

• Natural Images, Noise, and Learning

image retinal activity

present patches update weights

Patches from retinal activity image

Patches from noise

Retina

LGN

Cortex

• Resulting receptive fields Corresponding tuning curves (polar representation)

• BCM neurons can develop both orientation selectivity and varying degrees of Ocular Dominance

Shouval et. al., Neural Computation, 1996

Left Eye

Right Eye

Right Synapses Left

Synapses

0 50 100

0 50 100

0 20 40

1 2 3 4 5 0 50 100

Bin

No. o

f Cel

ls

• The distinction between homosynaptic and heterosynaptic models

Examples:

BCM homosynaptic

PCA (Oja) - heterosynaptic

dw jdt

=x j(y,m )

• Monocular Deprivation Homosynaptic model (BCM)

High noise

Low noise

• Monocular Deprivation Heterosynaptic model (K2)

High noise

Low noise

• Noise Dependence of MD Two families of synaptic plasticity rules

Hom

osyn

aptic

He

tero

syna

ptic

Blais, Shouval, Cooper. PNAS, 1999

QBCM K1 S1

Noise std Noise std Noise std

S2 K2 PCA

Noise std Noise std Noise std

Nor

mali

zed

Tim

e

N

orm

alize

d T

ime

• IntraocularinjectionofTTXreducesactivityofthe"deprived-eye"LGNinputstocortex

051015202530

0 21

Eyeopen

051015

0 21

Sp

ike

s /s

ec

051015

0 21

Lidclosed

Time(sec)

iLGN

TTX

• Experiment design

Blind injection of TTX and lid- suture (P49-61)

Dark rearing to allow TTX to wear off

Quantitative measurements of ocular dominance

• Res

pons

e ra

te (s

pike

s/se

c)

• Rittenhouse, Shouval, Paradiso, Bear - Nature 1999

MS= Monocular lid suture MI= Monocular inactivation (TTX)

Cumulative distribution Of OD

• Why is Binocular Deprivation slower than Monocular Deprivation?

Monocular Deprivation Binocular Deprivation

• What did we learn up to here?

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