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)

• 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

€

dw j

dt= ηx jφ(y,θM )

€

φ(y,θM )

€

θM ∝ E y 2[ ] =

1

τy 2

−∞

t

∫ ( ′ t )e−( t− ′ t ) /τ d ′ t

€

θM

€

y

Is equivalent to this differential form:

The integral form of the average:

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

with p>0

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

€

θM ∝ E y 2[ ] =

1

τy 2

−∞

t

∫ ( ′ t )e−( t− ′ t ) /τ d ′ t

€

dθm

dt=

1

τ(y 2 −θm )

€

dθm

dt=

1

τ(y1+ p −θm )

€

θM ∝ E y[ ]2

€

θM ∝ E y 2[ ]

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 W in this case?

x1

x2 €

x1 =1.0

0.2

⎛

⎝ ⎜

⎞

⎠ ⎟ x2 =

0.1

0.9

⎛

⎝ ⎜

⎞

⎠ ⎟

(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

x1

x2

€

y i = wT ⋅x i

Two examples with N= 5Note: The stable FP is such that for one pattern yi=wTxi=θm while for the othersy(i≠j)=0.

(note: here c=y)

Show movie

BCM TheoryStability

•One dimension

•Quadratic form

•Instantaneous limit

Ty xw

xyydt

dwMθ

2yM θ

€

dw

dt= y y − y 2

( )x

= y 2(1− y)x

y10

y

)(c

BCM TheorySelectivity

•Two dimensions

•Two patterns

•Quadratic form

•Averaged threshold

Txwxwy xw 2211

€

dw

dt= μ y k y k −θM( )x

k

€

θM = E y 2[ ]

patterns

= pk (y k )2

k=1

2

∑

11 xwy 22 xwy,

2x

2x

•Fixed points 0dt

dw

BCM Theory: Selectivity

•Learning Equation

•Four possible fixed points

M

M

y

y

y

y

θ

θ

1

1

1

1

0

0 ,

M

M

y

y

y

y

θθ

2

2

2

2

0

0

,,,(unselective)

(unselective)(Selective)(Selective)

•Threshold21

122

221

1 )()()( ypypypM θ

11 /1 py

€

dw

dt= y k y k −θM( )x

k

1w 1x

2x

2w

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:

So that

021

111

xw

xw mθ

€

θm1 = E[y 2] =

1

2(w1 ⋅ x1)2 + (w1 ⋅ x 2)2

[ ] =1

2θm

1[ ]

2

www *

22

111

xwxw

xwxw m

θ

))((2 211 wOxwmm θθ

21 mθ

y2 my θ 1

At y≈0 and at y≈θm we make a linear approximation

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

Decrease ≡ Stable Increase ≡ Unstable

Note: for a small perturbation θm changes such that:

For the preferred input x1:

(show form here up to end of proof + bonus 50 pt)

For the non preferred input x2

€

θm =1

2(w1 ⋅ x1 + Δw ⋅ x1)2 + (w1 ⋅ x 2 + Δw ⋅ x 2 )2

{ }

≈ θm* (1+ Δw ⋅ x1) + O(Δw2) = θm

* + 2Δw ⋅ x1

1111 )()( xxwyw m

θ

2222 )( xxwyw

Note: for a small perturbation θm changes such that:

For the preferred input x1:

(show from here up to end of proof + bonus 25 pt)

For the non preferred input x2

)(2)()(2

1

)()(2

1

21*221*

22212111

wOxwwOxw

xwxwxwxw

mm

m

θθ

θ

1111 )()( xxwyw m

θ

2222 )( xxwyw

(Note O(Δw2) is very small)

Use trick:

And

Insert previous to show that:

€

d

dtΔw[ ]

2=

d

dtΔw ⋅Δw[ ] = 2Δw ⋅Δw

•

€

Ed

dtΔw[ ]

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

1

2Δw ⋅Δw

• ⎡ ⎣ ⎢

⎤ ⎦ ⎥x1

+1

2Δw ⋅Δw

• ⎡ ⎣ ⎢

⎤ ⎦ ⎥x 2

€

Ed

dtΔw[ ]

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥= − ε1(Δw ⋅ x1)2 + ε2(Δw ⋅ x 2)2

[ ] < 0

Phase plane analysis of BCM in 1D

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

Will the stability be altered?

Look at 1D example €

dθm

dt=

1

τ(y 2 −θm )

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, θm=0, and y=1, θm=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 θm

y 1

0.5

0

?

€

dy

dt= ηy(y −θm )

dθ m

dt=

1

τ(y 2 −θm )

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 θ be?

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

Homework 2: due in 10 days

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), θ(t) and their trajectories in the m θ plane for different parameters. Compare stability to analytical results (Key parameters, η τ)

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

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

Monocular DeprivationHomosynaptic model (BCM)

High noise

Low noise

Monocular DeprivationHeterosynaptic model (K2)

High noise

Low noise

Noise Dependence of MDTwo families of synaptic plasticity rules

Hom

osyn

ap

tic

Hete

rosyn

ap

tic

Blais, Shouval, Cooper. PNAS, 1999

QBCM K1 S1

Noise std Noise std Noise std

S2

K2PCA

Noise std Noise std Noise std

N

orm

alize

d

Tim

e

N

orm

alize

d

Tim

e

Intraocular injection of TTX reduces activityof the "deprived-eye" LGN inputs to cortex

05

1015202530

0 21

Eye open

0

5

10

15

0 21

Sp

ikes

/sec

0

5

10

15

0 21

Lid closed

Time (sec)

RightRetina

LGN

TTX

Experiment design

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

Dark rearingto allow TTXto wear off

Quantitative measurements of ocular dominance

)()(

)()(OD

SRERSLER

SRERSLER

Res

pons

e ra

te (

spik

es/s

ec)

Rittenhouse, Shouval, Paradiso, Bear - Nature 1999

MS= Monocular lid sutureMI= 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?