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Computa(onal principles of synap(c memory

Stefano Fusi Columbia University

Defini(on of memory

Defini(on of memory (for variable x)

time

Event A Present: x=f(A)

What is x?

Long-‐term memory

Declara0ve (Explicit): consciously accessible

memories

Non-‐declara0ve (Implicit, Procedural)

Episodic: memories of specific events + (me and situa(on in which they occurred (e.g. autobiographical memories)

Seman0c: informa(on about general knowledge (facts or concepts). E.g. the capital of Italy

Skill learning e.g. riding a bicycle

Priming: previous exposure to a sensory s(mulus affects our reac(on (mes to a later s(mula(on

Condi0oning:e.g. saliva(ng when you see your favorite food

Memory consolida(on

Long-‐term memory

Short-‐term memory

time

ξ1 ξ2 ξ3 ξ4 ξ5

Formalizing the memory problem

x=wij (weight of the synapse connecting neuron j to i)

time

Memory strength

Δwij1

wij

ξ1 ξ2 ξ3 ξ4 ξ5

Example: the Hopfield model

time

wij=0

Δwijµ

ξ1 ξµ ... ξp ...

Tracking memory 1

time

wij=0

Δwij1

ξ1 ξ2 ξ3 ... ξp

wij=0 wij

ξ1 ξ2 ξ3 ... ξp

wij=0 wij

Memories are random and uncorrelated

ξ1 ξ2 ξ3 ... ξp

wij=0 wij


1

ξ1 ξ2 ξ3 ... ξp

ξ1 ξ2 ξ3 ... ξp

wij=0 wij


0 1

Memory noise

Memory noise

Noise increases with the number of memories

Hopfield 1982, Amit Gutfreund Sompolinsky, 1985

p (# of memories or time)

Hopfield 1982, Amit Gutfreund Sompolinsky 1985

Blackout Catastrophe

p (# of memories or time)

Why this complexity?

… toward more realis(c synapses

Unbounded


Unbounded

Bounded (binary), offline

learning Sompolinsky 1986


Unbounded

Bounded (binary), offline

learning

Bounded (binary), online

learning

Sompolinsky 1986

time

w

Forgetfulness

time

time

Forgetfulness

Δwij1=+1

wij=0

time

time

Forgetfulness

Δwij1=+1

wij=0 wij=1

time

time

Forgetfulness

Δwij1=+1

wij=0

wij=1

wij=1

wij=1

pre post

Δw=+1

Δw=-1

A learning rule for binary synapses

+1 +1

-1 +1

Δw=-1

Δw=+1

+1 -1

-1 -1

with probability q

Initial signal (starting from equilibrium)


Probability that Δwij1=wij by chance


Probability that Δwij1=wij by chance

Probability that initially Δwij1≠wij (by chance, 1/2) and

the weight is flipped by learning (q)

Signal after t memories

Probability that the synapse is NOT modified


…

Noise (at equilibrium)

Signal after p memories

Tsodyks 1988, Amit Fusi 1992, Amit Fusi 1994

Blackout Catastrophe

Bounded synapses Unbounded unrealistic synapses

Fast (large q)

Slow (small q)

SN

R

p

SNR(0) Number of memories

FAST

SLOW

Learning rate

~ 1

0

Multi-state synapses

2 states

Multi-state synapses

2 states

… m states

Fusi, Abbott, Nature Neurosci. 2007


FAST

SLOW

Learning rate

~ 1

0

MULTI-STATE SYNAPSE (BALANCED LTP/LTD)

MULTI-STATE SYNAPSE (IMBALANCED LTP/LTD)

Sparse memories

ξ=1 with probability f

ξ=0 with probability 1-f

pre post

Δw=+1 (LTP) with probability q+

Δw=-1 (LTD) with probability q-

Learning rule

Tsodyks Feigelman 1989, Amit Fusi 1994

if q+=q- then: strong imbalance

Sparse memories

ξ=1 with probability f

ξ=0 with probability 1-f

pre post

Δw=+1 (LTP) with probability q+

Δw=-1 (LTD) with probability q-

Learning rule

q+=q

q-= qf

Tsodyks Feigelman 1989, Amit Fusi 1994


SLOW (DENSE)

Learning rate SPARSE

Amit Fusi , Neural Computation, 1994

A significant improvement, but… 1) Not robust to noise (ξ=0 must be exactly 0). In the presence of noise: p ~ Nsyn

2) The amount of informa(on per memory is significantly smaller (it scales like f) 3) Not scalable (for large Nsyn it is very difficult to readout the relevant info)

Ben Dayan Rubin, Fusi, Frontiers in Comp. Neuroscience 2007

Learning systems with mul(ple (mescales

Fast

Slow

SN

R

p

…

m groups

FAST SLOW

q1 qm-1 q2 qm=qs

qk=(qs)(k-1)/(m-1)

Nsyn/m Nsyn/m Nsyn/m Nsyn/m

t=p (memories are stored at a constant rate)

~1/t

~e-qst

SNR

qs√N

√N log(1/qs)

qs√N e-qst qs√N √m e-qst

SNR


FAST

SLOW

Learning rate

SNR

HETEROGENEOUS

Fusi, Abbott, Neuron 2005; Roxin Fusi, PLoS Comp Biol. 2013

Hom

ogenous q=q1

SN

R

(=p)

Heterogeneous systems with memory transfer

A. Roxin, S. Fusi, PLoS Comp. Biology 2013

BEFORE SURGERY

AFTER BEFORE SURGERY

AFTER

…

m groups

FAST SLOW

q1 qm-1 q2 qm=qs

qk=(qs)k/m

N/m N/m N/m N/m

…

FAST SLOW

q1 qm-1 q2 qm=qs

INPUT

Memory transfer

Foster, Wilson 2006

…

FAST SLOW

q1 qm-1 q2 qm=qs

INPUT

Memory transfer

…

FAST SLOW

q1 qm-1 q2 qm=qs

S1 S2 Sm-1 Sm

Readout: S = max {S1 ,…,Sm }

√N log(1/qs)

qs√N

√N √m

~1/qs ~m/qs

~m1/4

(=p)


FAST

SLOW

Learning rate

HETEROGENEOUS

HETEROGENEOUS with MEMORY TRANSFER

…

m groups

FAST SLOW

q1 qm-1 q2 qm=qs

qk=(qs)k/m

N/m N/m N/m N/m

The cascade model

1/ ~Signal Noise t N−

Fusi, Drew, AbboY, Neuron (2005) Strong Weak

… … METAPLASTICITY

PLASTICITY

SNR(0) Number of memories Learning rate

HETEROGENEOUS


VARIABLE (metaplasticity)

SNR(0) Number of memories Learning rate

HETEROGENEOUS


VARIABLE (metaplasticity)

Tomorrow… p ~ Nsyn

Conclusions

Synapses that are bounded and can modified with limited precision require special machinery for preventing catastrophic forgetting Two important principles to improve performance: 1)  Heterogeneity (multiple timescales) 2)  Efficient memory transfer

People Theory of memory Daniel Amit Nicolas Brunel Francesco Battaglia Francesco Carusi Walter Senn Larry Abbott Daniel Ben Dayan Rubin Alex Roxin Srdjan Ostojic Marcus Benna

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