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# Mathematical Details of the Model

In this section we show how Contrastive Hebbian Learning (CHL) [Movellan, 1990] needs to be modified to accommodate units with relative phase angles. We follow the derivation in Movellan closely. Movellan defines a continuous Hopfield Energy function where E reflects the constraints imposed by the weights in the network and S the tendency to drive the activations to a resting value. For our network S is the same as for a network with no phase angles: where n is the number of units in the network, is the activation of unit i, is the activation function for unit i, and .

However, E becomes where is the weight connecting units i and j and is the coupling function associated with units i and j. In what follows we will abbreviate as .

The coupling function must be differentiable and satisfy the following:  When the network is stable, the inverse of the activation function for each unit is equal to the input into that unit: where ( ) represents equilibrium and is the input to unit i. Furthermore, when the network is stable, the phase angle of each unit no longer changes: Movellan defines the contrastive function J as and shows that the CHL rule minimizes J. We follow his derivation for the case where units have phase angles.

The energy of the network E at equilibrium is Extracting the terms with a term, Differentiating with respect to a single weight and considering that is the only weight depending on , From Equation 10, we have and Substituting these into Equation 16, From 11 and 12, we have the following for the case where . Since there are no self-recurrent connections in our network, we need only consider this case. From 12, the last term is 0, and we have From Equation 7, and from Equation 11, we have making which shows that the modified CHL rule descends in the J function.   Next: References Up: Playpen: Toward an Architecture Previous: Conclusions and Future Work

eliana colunga-leal
Mon Jun 23 04:27:19 EST 1997