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.
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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.
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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)
• 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)