Part of Advances in Neural Information Processing Systems 13 (NIPS 2000)

*Xiaohui Xie, Richard Hahnloser, H. Sebastian Seung*

It has long been known that lateral inhibition in neural networks can lead to a winner-take-all competition, so that only a single neuron is active at a steady state. Here we show how to organize lateral inhibition so that groups of neurons compete to be active. Given a collection of poten(cid:173) tially overlapping groups, the inhibitory connectivity is set by a formula that can be interpreted as arising from a simple learning rule. Our analy(cid:173) sis demonstrates that such inhibition generally results in winner-take-all competition between the given groups, with the exception of some de(cid:173) generate cases. In a broader context, the network serves as a particular illustration of the general distinction between permitted and forbidden sets, which was introduced recently. From this viewpoint, the computa(cid:173) tional function of our network is to store and retrieve memories as per(cid:173) mitted sets of coactive neurons.

In traditional winner-take-all networks, lateral inhibition is used to enforce a localized, or "grandmother cell" representation in which only a single neuron is active [1, 2, 3, 4]. When used for unsupervised learning, winner-take-all networks discover representations similar to those learned by vector quantization [5]. Recently many research efforts have focused on unsupervised learning algorithms for sparsely distributed representations [6, 7]. These algorithms lead to networks in which groups of multiple neurons are coactivated to represent an object. Therefore, it is of great interest to find ways of using lateral inhibition to mediate winner-take-all competition between groups of neurons, as this could be useful for learning sparsely distributed representations.

In this paper, we show how winner-take-all competition between groups of neurons can be learned. Given a collection of potentially overlapping groups, the inhibitory connectivity is set by a simple formula that can be interpreted as arising from an online learning rule. To show that the resulting network functions as advertised, we perform a stability analysis. If the strength of inhibition is sufficiently great, and the group organization satisfies certain conditions, we show that the only sets of neurons that can be coactivated at a stable steady state are the given groups and their subsets. Because of the competition between groups, only one group can be activated at a time. In general, the identity of the winning group depends on the initial conditions of the network dynamics. If the groups are ordered by the aggregate input that each receives, the possible winners are those above a cutoff that is set by inequalities to be specified.

1 Basic definitions

Let m groups of neurons be given, where group membership is specified by the matrix

fl = {I if the ith neuron is in the ath group , ° otherwise

(1)

We will assume that every neuron belongs to at least one group l, and every group contains at least one neuron. A neuron is allowed to belong to more than one group, so that the groups are potentially overlapping. The inhibitory synaptic connectivity of the network is defined in terms of the group membership,

Ji ' = lIm (1 _ ~a ~'!) = {o

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