Stochastic Interactive Activation andInteractive Activation in the Brain

PDP ClassJanuary 20, 2010

A Problem For the Interactive Activation Model

• Bayes Rule, Massaro’s Fuzzy Logical model, and the logistic activation function all give rise to a pattern of data we will call ‘logistic additivity’.

• And data from many experiments exhibits this pattern• Unfortunately, the original interactive activation model

did not exhibit this pattern.• Does this mean that the interactive activation model is

fundamentally wrong – i.e. processing is strictly feedforward (as Massaro believed)?

Joint Effect of Context and Stimulus Information in Phoneme Identification (/l/ or /r/)

From Massaro & Cohen (1991)

Massaro’s Model

• Joint effects of context and stimulus obey the fuzzy logical model of perception:

• ti is the stimulus support for r given input i and cj is the contextual support for r given context j, and pij(r) is the probability of responding r in this situation.

• The model is equivalent to the Bayesian model in which stimulus and context are assumed to be conditionally independent.

• Massaro’s model and the Bayesian model imply ‘logistic additivity’:

log(pij(r)/(1-pij(r))) = log(ti/(1-ti)) + log(cj/(1-cj))

Evaluate stimulus

Evaluate contextIntegration Decision

pij(r)

Ideal logistic-additive pattern (upper right)vs. mini-IA simulation results (lower right).

Failure of Logistic Additivity in the IA Model

What was wrong with the Interactive Activation model?

• The original interactive activation model ‘tacked the variability on at the end’ but neural activity is intrinsically stochastic.

• McClelland (1991) incorporated that intrinsic variability in the computation of the net input:

• Now we choose the alternative with the highest activation after settling.

• Logistic additivity is observed in simulations (shown at right).

• The result holds up in full-scale models and can be proven to hold under certain constraints on network architecture (Movellan & McClelland, 2001).

i i j ij ij

net bias a w

Intrinsic Variability

The Boltzmann Machine

Units have binary states [0,1], Update is asynchronous. The activation function is:

In the Boltzmann machine, if you run a network long enough, the distribution over states reaches an equilibrium (gradually reducing temperature can help).

We saw last time that at equilibrium, the following relationships hold:

More generally, at equilibrium we have the Probability-Goodness Equation:

or

Why logistic additivity holds in a Boltzmann machine version of the IA model

• Suppose the task is to identify the letter in position 2 in the IA model. The model is allowed to run to equilibrium… then states of the position 2 units are sampled until a state is found when one and only one of the letters in this position is active. The probability of this is given by:

• Define:

This decomposes into:

Why logistic additivity holds in the IA Model

• This reduces to:

• This consists of a factor for the bias, a factor for the stimulus (like Massaro’s ti) and a factor for the context (like Massaro’s cj).

Conditions on Logistic Additivity In Stochastic Interactive Models (Movellan & McClelland, 2001)

• Logistic additivity holds in a stochastic, bi-directionally connected neural network when two sources of input do not interact with each other except via the set of units that are the basis for specifying the response.

• This applies to the two sources of input to the identification of the letter in any position in the interactive activation model.

• Simulations suggest that the exact details of the activation function and source of variability are unimportant.

• Would the effects of two context letters on a third letter exhibit logistic additivity?

Interactivity in the Brain

• Bidirectional Connectivity• Interactions between V5 (MT) and V1/V2:

Bullier• Subjective Contours in V1:

Lee and Nguyen

Hupe, James, Payne, Lomber, Girard & Bullier (Nature, 1998, 394, 784-787)

• Investigated effects of cooling V5 (MT) on neuronal responses in V1, V2, and V3 to a bar on a background grid of lower contrast.

• MT cooling typically produces a reversible reduction in firing rate to V1/V2/V3 cells’ optimal stimulus (figure)

• Top down effect is greatest for stimuli of low contrast. If the stimulus is easy to see when it is not moving, top-down influences from MT have little effect.

• Concept of ‘inverse effectiveness’ arises here and in many other related cases.

*

Lee & Nguyen (PNAS, 2001, 98, 1907-1911)

• They asked the question:Do V1 neurons participate in the formation of a representation of the illusory contour seen in the upper panel (but not in the lower panel)?

• They recorded from neurons in V1 tuned to the illusory line segment, and varied the position of the illusory segment with respect to the most responsive position of the neuron.

Response to the illusory contour is found at

precisely the expected location.

Temporal Response to Real and Illusory Contours

Neuron’s receptive field falls rightover the middle of the real or illusoryline defining the bottom edge of the square

Figure shows a V1/V2 neuron that showed strong modulation in firing around epochs in which the monkey perceives the cell’s preferred stimulus.

From Leopold and Logothetis, 1996.

Top: psth’s show strong orientation preference.

Bottom: When both stimuli are presented simultaneously, neuron is silent just before a response indicating perception of the null direction, but quite active just before a response (t < 0) indicating perception of the preferred direction.

Leopold and Logothetis (Nature, 1996, 379, 549-553) found that some neurons in V1/V2 as well as V4 modulate their responses in concert with Monkey’s percept, as if participating in a massively distributed constraint-satisfaction process. However, some neurons in all areas do not modulate their responses. Thus the conscious percept appears to be correlated with the activity of only a subset of neurons. The fraction of neurons that covary with perception is greater in higher areas.