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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.gif

Movellan defines a continuous Hopfield Energy function

  equation577

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:

  equation580

where n is the number of units in the network, tex2html_wrap_inline3022 is the activation of unit i, tex2html_wrap_inline3014 is the activation function for unit i, and tex2html_wrap_inline3018 .

However, E becomes

equation592

where tex2html_wrap_inline2961 is the weight connecting units i and j and tex2html_wrap_inline2967 is the coupling function associated with units i and j. In what follows we will abbreviate tex2html_wrap_inline3038 as tex2html_wrap_inline3040 .

The coupling function must be differentiable and satisfy the following:

  equation612

  equation617

When the network is stable, the inverse of the activation function for each unit is equal to the input into that unit:

  equation622

where ( tex2html_wrap_inline2999 ) represents equilibrium and tex2html_wrap_inline1874 is the input to unit i. Furthermore, when the network is stable, the phase angle of each unit no longer changes:

  equation640

Movellan defines the contrastive function J as

equation656

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

equation662

Extracting the terms with a tex2html_wrap_inline2961 term,

equation679

Differentiating with respect to a single weight tex2html_wrap_inline2961 and considering that tex2html_wrap_inline2961 is the only weight depending on tex2html_wrap_inline2961 ,

  eqnarray706

From Equation 10, we have

eqnarray789

and

equation828

Substituting these into Equation 16,

equation855

From 11 and 12, we have the following for the case where tex2html_wrap_inline3062 . Since there are no self-recurrent connections in our network, we need only consider this case.

equation891

From 12, the last term is 0, and we have

equation917

From Equation 7,

equation935

and from Equation 11, we have

equation948

making

equation958

which shows that the modified CHL rule

equation980

descends in the J function.


next up previous
Next: References Up: Playpen: Toward an Architecture Previous: Conclusions and Future Work

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