The BCM theory of synaptic plasticity.

Simple Model of a Neuron

Inputs

Synaptic weights

Output

1m

1d 2d 3d

3m

2m

c

Neuron Activation

Inputs

Synaptic weights

Output

1m

1d 2d 3d

3m

2m

c

( )dm

dm×»

×=

÷ø

öçè

æ×= å

=

s

s i

n

ii dmc

1

( )dm×s

dm ×

Synaptic Modification

Input signal

Weight increase

Weight decrease

Output signal

Output increase

Output decrease

Synaptic weight

m

dmc

d

c

d d

c

m m

Hebbian Learning“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 A’s efficiency in firing B is increased.” - Hebb, 1949

“Those that fire together wire together”

•Mathematically:

ii cd

dt

dm

Stability and Behavior of Hebbian Learning

•Unstable as written: requires synaptic decrease•Finds correlations in environment

ii cd

dt

dm

jjij

ijj

j

ij

jj

ii

mC

ddm

ddm

cddt

dm

tenvironmen

tenvironmen

tenvironmentenvironmen

Hebbian Learning and Principal Components

•Matrix equivalent of Hebbian Learning

•Eigenvectors of C, the principle components:•Expand in terms of eigenvectors, :

•Component with largest eigenvalue wins

vCv

vm av

vCv aa

v

dt

da

dt

dmi

a

dt

da teata )0()(

j

jiji mC

dt

dmCm

mdt

d

Synaptic Stabilization

constant2 i

im

Synaptic weights

1m

2m

3m

Mathematical method implies Biological mechanism

•Saturation limits

•Normalization

•Decay terms

•Moving threshold

maxmin mmm i

iii mccd

dt

dm 2

(Linsker 1986;Miller 1994)

(Oja 1982, Blais et. al. 1998)

(BCM 1982, IC 1992; Blais et. al. 1999)

•For response increases

•For response decreases

•Yields selectivity…

•… but not stable

Mc

Mc

Combining Hebbian and Anti-Hebbian Learning

•A more general Hebbian-like rule

•Includes a decrease of weights in

ii dc

dt

dm)(

)(c

)(c

Hebb

Selective

c

M

BCM Theory

•Selectivity learning rule with moving threshold

2

),(

cE

dcdt

dm

M

iMi

)(c

BCM

c

M

(Bienenstock, Cooper, Munro 1982; Intrator, Cooper 1992)

2

0

2

lim c

cE

M

M

•Time average of the square of the neuron activity

Mcc

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

• 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.

LTDLTP

Requires

dm j

dtd j(c,M )

M E c 2

1 c 2

t

( t )e (t t )/d t

)(cBCM

c

M

Is equivalent to this differential form:

The integral form of the average:

tdetc ttt

/)(2 )(

1M

)1

( 2m

m cdt

d

BCM TheoryStability

•One dimension

•Quadratic form

•Instantaneous limit

mdc

dccdt

dmM

2cM

dcc

dcccdt

dm

)1(2

2

c10

c

)(c

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.

A N=2 example:

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

9.01.0

2.00.1 21 dd

(Notation: )

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

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

idmci ×=

Two examples with N= 5

Note: The stable FP is such that for one pattern ci=mdi=θm while for the othersC(i≠j)=0.

BCM TheoryStability

•One dimension

•Quadratic form

•Instantaneous limit

mdc

dccdt

dmM

2cM

dcc

dcccdt

dm

)1(2

2

c10

c

)(c

BCM TheorySelectivity

•Two dimensions

•Two patterns

•Quadratic form

•Averaged threshold

dm 2211 dmdmc

kMkk ccdt

dd

m

2

1

2

patterns2

kkk

M

cp

cE

11 dmc 22 dmc,

1d

2d

•Fixed points 0dt

dm

BCM Theory: Selectivity

•Learning Equation

•Four possible fixed points

M

M

c

c

c

c

1

1

1

1

0

0 ,

M

M

c

c

c

c

2

2

2

2

0

0

,,,(unselective)

(unselective)(Selective)(Selective)

•Threshold211

222

211 cpcpcpM

1/1 p

kMkk ccdt

dd

m

1m

2m

1d

2d

BCM Theory: Stability

•Learning Equation

•Four possible fixed points

M

M

c

c

c

c

1

1

1

1

0

0 ,

M

M

c

c

c

c

2

2

2

2

0

0

,,,(unstable)

(unstable)(stable)(stable)

•Threshold211

222

211 cpcpcpM

1/1 p

only selective fixed points are stable

kMkk ccdt

dd

m

1m

2m

1d

2d

Ex1 - Final Task

• Create a BCM learning rule which can go into the Fast ICA algorithm of Hyvarinen. – Run it on multi modal distributions as well

as other distributions.– Running should be as the regular fast ICA

but with a new option for the BCM rule.– Demonstrate how down in Fisher score

can you go to still get separation

Experimental vs. Theoretical Evidence

Re s

pon

se (

s pik

e s/s

e c)

Left Right

Tuning curves

0 180 36090 270

RightLeft

Receptive field PlasticityOcular Dominance Plasticity (Mioche and Singer, 89)

Synaptic plasticity in Visual Cortex (Kirkwood and Bear, 94 )

S tim u la te R e c o rd

Visual Cortex Receptive Field PlasticityMioche and Singer, 1989

Monocular deprivation

Left eye response Right eye response

Initial state:

After 17 hours MD of left eye:

Reverse suture

Initial state (after prior MD of left eye):

After one day of RS:

After 2 days of RS:

Left eye response Right eye response

Left Eye Right Eye

3 01 50-1 55 0

1 0 0

1 5 0

2 0 0

% o

f b

asel

ine

Time (min)

LTP

HFS

Tim e from o n s e t o f LF S (m in )4 53 01 50-1 5-3 0

5 0

7 5

1 0 0

1 2 5

1 5 0

1 H z

% o

f b

asel

ine LTD

Visual Pathway

Area17

LGN

Visual Cortex

Retinalight electrical signals

•Monocular•Radially Symmetric

•Binocular•Orientation Selective

Receptive fields are:

Receptive fields are:

Model Architecture

Image plane

Left Retina

Right Retina

LGN

LGN

Cortex(single cell)

Left Synapses

Right Synapses

iiidmc

Output

Inputs

Synaptic weights

L[ id ]Rid id

L[ im ]Rim im

Orientation Selectivity

Binocular Binocular DeprivationDeprivation

NormalNormal

Adult

Eye-opening angle angle

Res

pon

se (

spik

es/s

ec)

Res

pon

se (

spik

es/s

ec)

Eye-opening

Adult

Monocular Monocular DeprivationDeprivation

NormalNormal

Left Right

group group

angleangleRes

pon

se (

spik

es/s

ec)

1 2 3 4 5 6 7

% o

f ce

lls

10

20

1 2 3 4 5 6 7

30

15

RightLeft

Rittenhouse et. al.

Natural Images, Noise, and Learning

image retinal activity

•present patches

•update weights

•Patches from retinal activity image

•Patches from noise

Cortical Properties and Synapses

•Synaptic weights output properties

•Binocularity– responds to both eyes– similar synapse configuration from each eye

•Orientation selectivity

– responds to bars of light at a particular orientation

– elongated regions of strong synapses

Left Both Right0

5

10

15

20

Num

ber

of

cells

N=33

(Mioche, Singer 1989)

Hebbian Learning and Orientation Selectivity

•Orientation selectivity

– responds to bars of light at a particular orientation

– elongated regions of strong synapses

experiment

simulation

BCM Learning and Orientation Selectivity

•Orientation selectivity

– responds to bars of light at a particular orientation

– elongated regions of strong synapses

experiment

simulation

RightLeft

RightLeft

Binocularity

Left Eye

Right Eye

Hebbian Learning

BCM Learning

Right SynapsesLeft

Synapses

Orientation selectivity and Ocular Dominance

Left Eye

Right Eye

Right SynapsesLeft

Synapses

RightLeft

PCA

0

50

100

0

50

100

0

50

100

0

50

100

1 2 3Bin

No

. of

Cel

ls

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

Shouval et. al., Neural Computation, 1996

Left Eye

Right Eye

Right SynapsesLeft

Synapses

0

50

100

0

50

100

0

20

40

1 2 3 4 50

50

100

Bin

No

. of

Cel

ls

Resulting receptive fields Corresponding tuning curves

Cortical Properties and Synapses

•Monocular deprivation (MD)

– in 12 hours, responds more strongly to open eye

– synapses from closed eye weaken

•Binocular deprivation (BD)

– in 3-4 days, responses are smaller from both eyes

– all synapses are weakened, but more slowly than MD

Left Both Right0

5

10

15

20

Num

ber

of

cells

N=33

(Mioche, Singer 1989)

0 1 2 3 4 5 6Days

Sel

ectiv

ity

N=42

(adapted from Freeman et. Al. 1981)

Observation• Loss of response during Monocular

Deprivation is much more rapid than during Binocular Deprivation. (Hubel and Wiesel, 1963, 1965)

• Therefore the two eyes compete for limited resources.

• Mechanism: Synaptic competition.

•Normalization implies competition

– for weights to increase, others decrease

•Monocular deprivation (MD)– open eye weights are driven up

– closed eye weights are driven down– more activity in closed eye reduces driving force

•No competition in binocular deprivation

Synaptic Competition and Monocular Deprivation

constant2 i

im

Left Both Right0

5

10

15

20

Num

ber

of

cells

N=33

(Mioche, Singer 1989)

time

resp

onse

closed eye

open eye

Heterosynaptic LTD

1m

m2

d 2

C

d1| || | || |

| | |

| | || | |

A stabilized Hebb rule: {

If Oja rule (PCA)

Many variants: Stent (73), von der Malsburg (73), Miller (89) ...

– ~ 0 for non-optimum patterns

– ~ for optimum patterns

•Temporal competition between incoming patterns•For a selective neuron, most responses are…

BCM Theory and Monocular Deprivation

time

resp

onse

closed eye

open eye

)(c

c

M

ii dc

dt

dm)(

c2 )(1 Mc

•Linear approximation of

)()(

)0(

1

2

MM c

c

)(c

M

•Pattern into open eye, •Noise into closed eye, •Output depends on pattern and noise•Two cases of patterns into the open eye

– non-optimum patterns

BCM Theory and Monocular Deprivation

closed2

2

closedopen2

2closed

i

ij

jjjj

ii

mn

nnmdm

cnm

ii dcm )(open

)(c

c

M

c2 )(1 Mc – optimum patterns

id

inii ncm )(closed

j

jjjj nmdmc closedopen

closed21

closedii mnm

•Two cases of patterns into the open eye– non-optimum patterns– optimum patterns

BCM Theory and Monocular Deprivation

closed22

closedii mnm

closed21

closedii mnm

optimum-nonoptimum NN

tNNntmioptimumoptimumnon2closed )(log

•For a selective neuron,

– closed eye weights decrease

– more activity in the closed eye increases the effect

)(c

c

M

c2 )(1 Mc

•Synaptic competition

– more activity into closed eye decreases shift in responses toward open eye

•BCM Theory

– more activity into closed eye increases shift in responses toward open eye

Summary of Theory

Right Both Left

Nu

mb

er

of

cells

Right Both Left

Right Both LeftRight Both Left

Nu

mb

er

of

cells

strong activity

weak activity

•Synaptic competition

Experiment and Theory

•BCM Theory

•Rittenhouse et. al. 1999•TTX in retina•consistent with BCM

strong activity

Right Both Left

Nu

mb

er

of

cells

Right Both Left

Right Both LeftRight Both Left

Nu

mb

er

of

cells

weak activity

Right Both Left0

40

80

120

160

N=273

Right Both Left0

40

80

120

160

Num

ber

of c

ells

N=238

Monocular DeprivationHomosynaptic model (BCM)

High noise

Low noise

Monocular DeprivationHeterosynaptic model (K2)

High noise

Low noise

Summary

• Heterosynaptic mechanisms: Loss of response in Monocular Inactivation is faster than in Monocular lid Suture

• Homosynaptic mechanisms: Loss of response in Monocular lid Suture is faster than in Monocular Inactivation

Theoretical predictions

Experimental results

MS faster than MI Homosynaptic

Networks of BCM Neurons

Shouval et. al., Vision Research, 1997

BCM Synaptic Plasticity.

Binocular natural image inputs.

Radially symmetric lateral connectivity.

distance

stre

ngt

h

Two identical networks with

different initial conditions

Summary• Both stabilized Hebb rules and BCM can account

for orientation selectivity.• BCM neurons show varying degrees of Ocular

Dominance.• Theoretical analysis and Experimental evidence

indicate that Homosynaptic LTD is the mechanism of ocular dominance plasticity.

• Structured long range connections, as observed in cortex, can account for the stability of orientation maps.

Conclusions

• Models of Synaptic Modification– differ by methods of synaptic stabilization– synaptic competition– BCM theory: moving threshold

• Reproduce deprivation experiments

• Dynamics of monocular deprivation– experiment to distinguish learning rules– Rittenhouse et. al. 1999 consistent with BCM

• Molecular

•Synaptic

•Cellular

•System/Maps

T im e f ro m o n s e t o f L F S (m in )

4 53 01 50- 1 5- 3 05 0

7 5

1 0 0

1 2 5

1 5 0

1 H z

% o

f b

asel

ine

LTD

Different levels of description

TheoreticalTheoretical

FrameworkFramework

Orientation Selectivity of Stabilized Hebb Neurons

Simulations

Using the Oja rule (PCA)

2)( akkQPower Spectrum:

Size and shape of retinal filter

Theory

Size of receptive field

Shouval and Liu. Network., 1996

)()()( '2' a

rmrdrmrrQ

PCA Neurons: Two-eye Parity

PCA Neurons are always binocular!

mQm

lR0rR0

Slr

rr

Monocular Deprivationom1

C| | | |

| | || | |

Open Eye(pattern vision)

Deprived Eye(noise)

od2| || | || | om2

od1

dd1dm1

dd2 dm2

| | |

| | | | | ||

BCM:

MD