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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.

Memory

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Memory

- Content addressable
- Attractor network

- General Case:
- Lyapunov function

Mean Field Approximation

- What are the fixed points?

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CI

CE

- What are the fixed points?

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Unstable fixed point

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Stable fixed point

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E

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E

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E

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E

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Stable branches

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Unstable branch

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Stable fixed point

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E

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I

E

E

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E

Inhibitory null cline

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Excitatory null cline

E

Fixed points

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CI

CE

I

E

E

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CI

CE

Storing

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Decrease inhibition (CI)

E

E

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CI

CE

Storing

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Back to rest

E

E

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CI

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Reset

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Increase inhibition

E

E

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CI

CE

Reset

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Back to rest

E

- 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?

Ij

R(Ij)

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

cR(Ii)

Ii

Iaff

- 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

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

Unstable fixed point

Stable fixed points

I2*

Ii

- Clamping the input: inhibitory Iaff

Ii

Iaff

- Clamping the input: excitatory Iaff

cR(Ii)

Ii

I2*

Iaff

Ij

R(Ij)

- 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)…

- 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

- Storing analog values: difficult with this scheme….

cR(Ii)

Ii

Implication for transmitting rate and integration…

cR(Ii)

Ii

- Head direction cells

DH

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60

Activity

40

20

0

-100

0

100

Preferred Head Direction (deg)

- Attractor network with population code
- Translation invariant weights

DH

100

80

60

Activity

40

20

0

-100

0

100

Preferred Head Direction (deg)

- Computing the weights:

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

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

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

- 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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T1

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R1

R2

T1

T2

S1

R2

S2

S1=R1 S2=R2-S1

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R1

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T1

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R1

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S2

T1

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S2

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-DE

Activity

Activity

Vertical Ret. Pos. (deg)

Vertical Ret. Pos. (deg)

Horizontal Ret. Pos. (deg)

Horizontal Ret. Pos. (deg)

- Oculomotor theory
- Evidence integrator for decision making
- Transmitting rates in multilayer networks
- Maximum likelihood estimator

- 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.

- 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).

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

- 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.

- Bayesian approach
- Generative model for document

g

z

w

Gist

Topics

words

- 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

- Bayesian approach
- Generative model for document

g

z

w

Gist

Topics

words

- 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.

- 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…

Words in

Topic 1

Words in

Topic 3

Words in

Topic 2

- Bayesian approach
- Generative model for document

g

z

w

Gist

Topics

words

- 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?

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

g

z

w

Gist

Topics

words

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

- 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