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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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Hopfield network l.jpg
Hopfield Network

  • Content addressable

  • Attractor network



Hopfield network4 l.jpg
Hopfield Network

  • General Case:

  • Lyapunov function




Null cline analysis l.jpg
Null Cline Analysis

  • What are the fixed points?

E

I

CI

CE


Null cline analysis8 l.jpg
Null Cline Analysis

  • What are the fixed points?


Null cline analysis9 l.jpg

I

I

E

Null Cline Analysis

Unstable fixed point

E

Stable fixed point


Null cline analysis10 l.jpg

I

Null Cline Analysis

E

E





Null cline analysis14 l.jpg
Null Cline Analysis

Stable branches

I

Unstable branch

E

E



Null cline analysis16 l.jpg
Null Cline Analysis

I

Stable fixed point

I

E


Null cline analysis17 l.jpg

E

Null Cline Analysis

I

I

E


Null cline analysis18 l.jpg

E

Null Cline Analysis

I

I

E


Null cline analysis19 l.jpg
Null Cline Analysis

Inhibitory null cline

I

Excitatory null cline

E

Fixed points


Binary memory l.jpg

E

I

CI

CE

Binary Memory

I

E


Binary memory21 l.jpg

E

I

CI

CE

Binary Memory

Storing

I

Decrease inhibition (CI)

E


Binary memory22 l.jpg

E

I

CI

CE

Binary Memory

Storing

I

Back to rest

E


Binary memory23 l.jpg

E

I

CI

CE

Binary Memory

Reset

I

Increase inhibition

E


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E

I

CI

CE

Binary Memory

Reset

I

Back to rest

E


Networks of spiking neurons l.jpg
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?





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Networks of Spiking Neurons

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




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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


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Networks of Spiking Neurons

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

Unstable fixed point

Stable fixed points

I2*

Ii


Networks of spiking neurons34 l.jpg
Networks of Spiking Neurons

  • Clamping the input: inhibitory Iaff

Ii

Iaff


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Networks of Spiking Neurons

  • Clamping the input: excitatory Iaff

cR(Ii)

Ii

I2*

Iaff



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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)…


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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


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Line Attractor Networks

  • Storing analog values: difficult with this scheme….

cR(Ii)

Ii


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Line Attractor Networks

Implication for transmitting rate and integration…

cR(Ii)

Ii


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Line Attractor Networks

  • Head direction cells

DH

100

80

60

Activity

40

20

0

-100

0

100

Preferred Head Direction (deg)


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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)


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Line Attractor Networks

  • Computing the weights:


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Line Attractor Networks

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

  • The solution is:


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Line Attractor Networks

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


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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):


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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



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Memorized Saccades

+

+

R1

R2

T1

T2

S1

R2

S2

S1=R1 S2=R2-S1


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Memorized Saccades

+

+

R1

R2

T1

T2

S1

S2

S1

T1

T2

S2


Memorized saccades51 l.jpg
Memorized Saccades

+

+

R1

R2

T1

T2

S1

S2

T1

T2

S1

S2


Memorized saccades52 l.jpg

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 l.jpg
Neural Integrator

  • Oculomotor theory

  • Evidence integrator for decision making

  • Transmitting rates in multilayer networks

  • Maximum likelihood estimator


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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 memory55 l.jpg
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 memory56 l.jpg
Semantic Memory

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


Semantic memory57 l.jpg
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.


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Semantic Memory

  • Bayesian approach

  • Generative model for document

g

z

w

Gist

Topics

words


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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 memory60 l.jpg
Semantic Memory

  • Bayesian approach

  • Generative model for document

g

z

w

Gist

Topics

words


Semantic memory61 l.jpg
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…


Slide62 l.jpg


Semantic memory63 l.jpg
Semantic Memory varied from one sentence (or even word) to the next.

  • 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 memory64 l.jpg
Semantic Memory varied from one sentence (or even word) to the next.

Words in

Topic 1

Words in

Topic 3

Words in

Topic 2


Semantic memory65 l.jpg
Semantic Memory varied from one sentence (or even word) to the next.

  • Bayesian approach

  • Generative model for document

g

z

w

Gist

Topics

words


Semantic memory66 l.jpg
Semantic Memory varied from one sentence (or even word) to the next.

  • 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 memory67 l.jpg
Semantic Memory varied from one sentence (or even word) to the next.

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


Semantic memory68 l.jpg

g varied from one sentence (or even word) to the next.

z

w

Gist

Topics

words

Semantic Memory

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


Semantic memory69 l.jpg
Semantic Memory varied from one sentence (or even word) to the next.

  • 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.


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Countryside varied from one sentence (or even word) to the next.

Word being observed

Finance

Predicted next word

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