Supervised Synaptic Weight Adaptation for a Spiking Neuron

A novel algorithm named Spike-LMS is described that adapts the synaptic weights of an artificial spiking neuron to produce a desired response. The derivation of Spike-LMS follows from the derivation of the Least-Mean Squares (LMS) algorithm used in adaptive filter theory. Spike-LMS works directly in the domain of spike trains, and therefore makes no assumptions about any particular neural encoding method. This algorithm is able to identify the synaptic weights of a spiking neuron given the pre-synaptic and post-synaptic spike trains. INTRODUCTION Spiking neural networks have received a good deal of attention in the past few years. A key difficultly in applying them to engineering applications is that application data is not often described by spike trains. Many methods of encoding continuous-valued, discrete-time data into spike trains have been proposed, usually taking the form of rate encoding or temporal encoding. These two encoding methods have problems: Rate encoding regards the individual arrival times as unimportant, and thus underutilizes the information capacity of the spike train. Using only rates is known to be insufficient for many time-sensitive processing tasks, unless the output of a large number of neurons is considered in aggregation (population encoding). On the other hand, temporal encoding methods invariably require some form of synchronization, and the resulting firing patterns are not consistent with many biological firing patterns [1]. When using any formally described encoding method, it is possible to develop supervised learning rules for training a spiking neuron by finding the local minima of a cost function. The cost function should be defined in terms of the error of the pre-encoded data, and the local minima can be found using gradient descent. However, this method will only ensure that the pre-encoded error will be minimized. If a rate code is used, there is no guarantee that the output spike train will match the desired spike train; only their rates will match. If instead a temporal code is used then a synchronization signal must be incorporated into the model to provide the relative timing information [1]. An example of supervised learning with synchronized inputs is described in [2]. The Spike-LMS learning method proposed in this paper does not suffer from these ailments. The algorithm assumes no form of encoding and thus works directly with the spike trains. It is difficult to define a performance criterion in terms of only spike trains (an attempt is made in the derivation described here, but it is not a true performance criterion), and This work was partially supported by NSF EIA 0135946. such a performance measure is necessary in order to use gradient techniques typically used in artificial neural network models. Instead of attempting to optimize some measure performance, we formulate the supervised learning problem as a system identification problem. We then describe an algorithm that adapts a neural model to produce the same output as some unknown neural model. If we are able to identify the parameters of the unknown system accurately, then output of the two systems should be nearly identical. To perform system identification, we create an adaptive model identical to the given model as shown in Fig. 1. We then adapt the weights so that the outputs of the two models converge. In this figure, the variables describing the unknown fixed-weight neuron are identified by a superscript . All inputs and the output are spike trains. SPIKE RESPONSE MODEL OF A NEURON We are using the Spike Response Model (SRM) of a spiking neuron. This model is shown to be a generalization of the Integrate-and-Fire model used throughout the literature. This section is provided to give the reader a basic understanding of the model, and to introduce quantities used in subsequent sections in this paper. A more complete treatise of the SRM can be found in [3].