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Memory. Hopfield Network. Content addressable Attractor network. Hopfield Network. Hopfield Network. General Case: Lyapunov function. Neurophysiology. Mean Field Approximation. Null Cline Analysis. What are the fixed points?. E. I. C I. C E. Null Cline Analysis.

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Memory

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Memory


Hopfield Network

  • Content addressable

  • Attractor network


Hopfield Network


Hopfield Network

  • General Case:

  • Lyapunov function


Neurophysiology


Mean Field Approximation


Null Cline Analysis

  • What are the fixed points?

E

I

CI

CE


Null Cline Analysis

  • What are the fixed points?


I

I

E

Null Cline Analysis

Unstable fixed point

E

Stable fixed point


I

Null Cline Analysis

E

E


Null Cline Analysis

I

E

E


Null Cline Analysis

I

E

E


Null Cline Analysis

I

E

E


Null Cline Analysis

Stable branches

I

Unstable branch

E

E


Null Cline Analysis

I

E

E


Null Cline Analysis

I

Stable fixed point

I

E


E

Null Cline Analysis

I

I

E


E

Null Cline Analysis

I

I

E


Null Cline Analysis

Inhibitory null cline

I

Excitatory null cline

E

Fixed points


E

I

CI

CE

Binary Memory

I

E


E

I

CI

CE

Binary Memory

Storing

I

Decrease inhibition (CI)

E


E

I

CI

CE

Binary Memory

Storing

I

Back to rest

E


E

I

CI

CE

Binary Memory

Reset

I

Increase inhibition

E


E

I

CI

CE

Binary Memory

Reset

I

Back to rest

E


Networks of Spiking Neurons

  • Problems with the previous approach:

    • Spiking neurons have monotonic I-f curves (which saturate, but only at very high firing rates)

    • How do you store more than one memory?

    • What is the role of spontaneous activity?


Networks of Spiking Neurons


Networks of Spiking Neurons

Ij

R(Ij)


Networks of Spiking Neurons


Networks of Spiking Neurons

  • A memory network must be able to store a value in the absence of any input:


Networks of Spiking Neurons


Networks of Spiking Neurons

cR(Ii)

Ii

Iaff


Networks of Spiking Neurons

  • With a non saturating activation function and no inhibition, the neurons must be spontaneously active for the network to admit a non zero stable state:

cR(Ii)

I2*

Ii


Networks of Spiking Neurons

  • To get several stable fixed points, we need inhibition:

Unstable fixed point

Stable fixed points

I2*

Ii


Networks of Spiking Neurons

  • Clamping the input: inhibitory Iaff

Ii

Iaff


Networks of Spiking Neurons

  • Clamping the input: excitatory Iaff

cR(Ii)

Ii

I2*

Iaff


Networks of Spiking Neurons

Ij

R(Ij)


Networks of Spiking Neurons

  • Major Problem: the memory state has a high firing rate and the resting state is at zero. In reality, there is spontaneous activity at 0-10Hz and the memory state is around 10-20Hz (not 100Hz)

  • Solution: you don’t want to know (but it involves a careful balance of excitation and inhibition)…


Line Attractor Networks

  • Continuous attractor: line attractor or N-dimensional attractor

  • Useful for storing analog values

  • Unfortunately, it’s virtually impossible to get a neuron to store a value proportional to its activity


Line Attractor Networks

  • Storing analog values: difficult with this scheme….

cR(Ii)

Ii


Line Attractor Networks

Implication for transmitting rate and integration…

cR(Ii)

Ii


Line Attractor Networks

  • Head direction cells

DH

100

80

60

Activity

40

20

0

-100

0

100

Preferred Head Direction (deg)


Line Attractor Networks

  • Attractor network with population code

  • Translation invariant weights

DH

100

80

60

Activity

40

20

0

-100

0

100

Preferred Head Direction (deg)


Line Attractor Networks

  • Computing the weights:


Line Attractor Networks

  • The problem with the previous approach is that the weights tend to oscillate. Instead, we minimize:

  • The solution is:


Line Attractor Networks

  • Updating of memory: bias in the weights, integrator of velocity…etc.


Line Attractor Networks

  • How do we know that the fixed points are stable? With symmetric weights, the network has a Lyapunov function (Cohen, Grossberg 1982):


Line Attractor Networks

  • Line attractor: the set of stable points forms a line in activity space.

  • Limitations: Requires symmetric weights…

  • Neutrally stable along the attractor: unavoidable drift


Memorized Saccades

+

+

T1

T2


Memorized Saccades

+

+

R1

R2

T1

T2

S1

R2

S2

S1=R1 S2=R2-S1


Memorized Saccades

+

+

R1

R2

T1

T2

S1

S2

S1

T1

T2

S2


Memorized Saccades

+

+

R1

R2

T1

T2

S1

S2

T1

T2

S1

S2


A

B

-DE

Activity

Activity

Vertical Ret. Pos. (deg)

Vertical Ret. Pos. (deg)

Horizontal Ret. Pos. (deg)

Horizontal Ret. Pos. (deg)

Memorized Saccades


Neural Integrator

  • Oculomotor theory

  • Evidence integrator for decision making

  • Transmitting rates in multilayer networks

  • Maximum likelihood estimator


Semantic Memory

  • Memory of words is sensitive to semantic (not just spelling)

  • Experiment: Subjects are first trained to remember a list of words. A few hours later, they are presented with a list of words and they have to pick the ones they were supposed to remember. Many mistakes involve words semantically related to the remembered words.


Semantic Memory

  • Usual solution: semantic networks (nodes: words, links: semantic similarities) and spreading activation

  • Problem 1: The same word can have several meanings (e.g. bank). This is not captured by semantic network

  • Problem 2: some interaction between words are negative, even when they have no semantic relationship (e.g. doctor and hockey).


Semantic Memory

  • Usual solution: semantic networks (nodes: words, links: semantic similarities) and spreading activation


Semantic Memory

  • Bayesian approach (Griffiths, Steyvers, Tenenbaum, Psych Rev 06)

  • Documents are bags of words (we ignore word ordering).

  • Generative model for document. Each document has a gist which is a mixture of topics. A topic in turn defines a probability distribution over words.


Semantic Memory

  • Bayesian approach

  • Generative model for document

g

z

w

Gist

Topics

words


Semantic Memory

  • z = Topics = finance, english country side… etc.

  • Gist: mixture of topics. P(z|g) mixing proportions.

  • Some documents might be 0.9 finance, 0.1 english country side (e.g. wheat market).

    P(z=finance|g1)=0.9, P(z=engl country|g1)=0.1

  • Other might be 0.2 finance, 0.8 english country side (e.g. Lloyds CEO buys a mansion)

    P(z=finance|g1)=0.2, P(z=engl country|g1)=0.8


Semantic Memory

  • Bayesian approach

  • Generative model for document

g

z

w

Gist

Topics

words


Semantic Memory

  • Topic (z1)=finance

  • Words: P(w|z1)

  • 0.01 bank, 0.008 money, 0.0 meadow…

  • Topic (z2)=english country side

  • Words: P(w|z2)

  • 0.001 bank, 0.001 money, 0.002 meadow…


  • The gist is shared within a document but the topics can be varied from one sentence (or even word) to the next.


Semantic Memory

  • Problem: we only observe the words, not the topic of the gist…

  • How do we know how many topics and how many gists to pick to account for a corpus of words, and how do we estimate their probabilities?

  • To pick the number of topics and gist: Chinese restaurant process, Dirichlet process and hierarchical Dirichlet process. MCMC sampling.

  • Use techniques like EM to learn the probability for the latent variables (topics and gists).

  • However, a human is still needed to label the topics…


Semantic Memory

Words in

Topic 1

Words in

Topic 3

Words in

Topic 2


Semantic Memory

  • Bayesian approach

  • Generative model for document

g

z

w

Gist

Topics

words


Semantic Memory

  • Problems we may want to solve

  • Prediction P(wn+1|w).What’s the next word?

  • DisambiguationP(z|w). What are the mixture of topics in this document?

  • Gist extractionP(g|w). What’s the probability distribution over gists?


Semantic Memory

  • What we need is a representation of P(w,z,g)


g

z

w

Gist

Topics

words

Semantic Memory

  • P(w,z,g) is given by the generative model.


Semantic Memory

  • Explain semantic interferences in list

  • will tend to favor words that are semantically related through the topics and gists.

  • Capture the fact that a given word can have different meanings (topics and gists) depending on the context.


Countryside

Word being observed

Finance

Predicted next word

Money less likely to be seen if the topic is country side


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