Transcript Theo Neuro II

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
dw j
dt
(Bienenstock, Cooper, Munro 1982;
Intrator, Cooper 1992)
 x j  (y, M )
 M  E y 2 
1
t
y
 
2
( t )e
Requires
( t t ) / 
 (y, M )
M
dt 

• 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.
y

LTD

LTP
The integral form of the average:
Is equivalent to this differential form:
M  E y 
2
1
t

 
y 2 ( t )e(t t )/  dt 
dm 1 2
 (y  m )
dt 
Note, it is essential that θm is a superlinear function
of the history of C, that is:
dm 1 1 p
 (y  m )
dt 
with p>0
Note also that in the original BCM formulation
(1982) M  E y2 rather then M  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:
x1
x2

1.0  2 0.1
x    x   
0.2
0.9
1
What are the stable
fixed points of W in
this case?
y i  wT  x i )
(Notation:

x1
x2
Note:
Every time a new
input is presented,
m changes, and so
does θm
What are the fixed points? What
are the stable fixed points?
Two examples
with N= 5
Note: The stable FP
is such that for one
pattern yi=wTxi=θm
while for the others
y(i≠j)=0.
(note: here c=y)
Show movie
BCM Theory
Stability
•One dimension
y  w  xT
dw
•Quadratic form
 y  y   M x
dt
•Instantaneous limit  M  y 2
dw
dt

0
 (c )
 y y  y 2 x
y
 y 2 (1 y)x
1
y
BCM Theory
Selectivity
•Two dimensions
y  w1x1  w2 x2  w  xT
1
1
2
2
,
y

w

x
y

w

x
•Two patterns
dw
k
k
k


y
y


x
•Quadratic form
 M
dt
•Averaged threshold M  E y 2 patterns


2
k 2
p
(y
 k )
k1
•Fixed points

dw
0
dt
x2
x2
BCM Theory: Selectivity
dw
•Learning Equation
 y k y k  M x k
dt
•Four possible fixed points
1
2
,
y

0
y
(unselective)
(Selective)  y1   M , y 2
(Selective) y1  0 , y 2
(unselective) y1   M , y 2
 0
 0
 M
 M
1 2
2 2
1 2


p
(
y
)

p
(
y
)

p
(
y
)
•Threshold M
1
2
1
 y1  1 / p1
w1
x1
x2
w2
Consider a selective F.P (w1) where:
w1  x1   m1
w1  x 2  0
1
1 1 2
1
1 2
1
2 2
and   E[y ]  (w  x )  (w  x )  m 
2
2
So that  m1  2
1
m

2
*
w

w
 w
for a small pertubation from the F.P such that
The two inputs result in:
w  x1   m1  w  x1
w  x 2  w  x 2
So that
m  m1  2w  x1  O (( w)2 )
At y≈0 and at y≈θm we make a linear approximation
   2 y
  1  y   m 
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:
1
m  (w1  x1  w  x1 ) 2  (w1  x 2  w  x 2 ) 2 
2
 m* (1 w  x1 )  O(w 2 )   m*  2w  x1
For the preferred input x1:

w  1 ( y   m )  1 ( w  x1 ) x1
(show form here up to end of proof + bonus 50 pt)
For the non preferred input x2

w   2 y   2 ( w  x 2 ) x 2
Note: for a small perturbation θm changes such that:
1
 m  ( w1  x1  w  x1 ) 2  ( w1  x 2  w  x 2 ) 2 
2
1
 ( m*  w  x1 ) 2  O ( w2 )   m*  2w  x1  O ( w2 )
2
(Note O(Δw2) is very small)
For the preferred input x1:

w  1 ( y   m )  1 ( w  x1 ) x1
(show from here up to end of proof + bonus 25 pt)
For the non preferred input x2

w   2 y   2 ( w  x 2 ) x 2
Use trick:

d
d
2
w  w  w  2w  w
dt
dt
And

 
 
d
1 
1 
2 
E  w  w  w  w  w
x1 2 
x 2
dt
 2 
Insert previous to show that:
d
2 
E  w  1(w  x1 ) 2  2 (w  x 2 ) 2  0
dt

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

dm 1 2
 (y  m )
dt 
Phase plane analysis of BCM in 1D
Assume x=1 and therefore y=w. Get the two BCM
equations:
y
1
?
dy
 y(y   m )
dt
d m
1 2
 (y   m )
dt

0.5
0
0
0.5
1
θm
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)
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
Retina
LGN
Cortex
•Patches from retinal activity image
•Patches from noise
Resulting receptive fields
Corresponding tuning curves
BCM neurons can develop both orientation
selectivity and varying degrees of Ocular Dominance
100
50
Right Eye
0
100
Left Eye
Left
Synapses
Right
Synapses
No. of Cells
50
0
40
20
0
100
50
0
12345
Bin
Shouval et. al., Neural Computation, 1996
Monocular Deprivation
Homosynaptic model (BCM)
Low noise
High noise
Monocular Deprivation
Heterosynaptic model (K2)
Low noise
High noise
Noise Dependence of MD
Two families of synaptic plasticity rules
QBCM
Noise std
PCA
Noise std
K1
Noise std
K2
Noise std
S1
Noise std
S2
Noise std
Blais, Shouval, Cooper. PNAS, 1999
Intraocular injection of TTX reduces activity
of the "deprived-eye" LGN inputs to cortex
LGN
Spikes/sec
Right
Retina
30
25
20
15
10
5
0
Eye open
0
1
Lid clos e d
15
2
10
10
5
5
0
0
1
Time (sec)
TTX
15
2
0
0
1
2
Experiment design
Blind injection of
TTX and lidsuture (P49-61)
Dark rearing
to allow TTX
to wear off
Quantitative
measurements of
ocular dominance
Response rate (spikes/sec)
OD 
( LER  S )  ( RER  S )
( LER  S )  ( RER  S )
Cumulative distribution
Of OD
MS= Monocular lid suture
MI= Monocular inactivation (TTX)
Rittenhouse, Shouval, Paradiso, Bear - Nature 1999
Why is Binocular Deprivation slower
than Monocular Deprivation?
Monocular Deprivation
Binocular Deprivation
What did we learn up to here?