Application of Levenberg-Marquardt method to the training of spiking neural networks Sergio M. Silva, Ant6nio E. Ruano Centre for Intelligent Systems, University of Algarve, Faro, Portugal E-mail: al8020@ualg.pt, aruano@ualg.pt Abstract- One of the basic aspects of some neural networks is their attempt to approximate as much as possible their biological counterparts. The goal is to achieve a simple and robust network, easy to comprehend and capable of simulating the human brain at a computational level. This paper presents improvements to the Spikepro algoritm, by introduting a new encoding scheme, and Mustrate the application of the Levenberg Marquardt algorithm to this third generation of neural network. I. INTRODUCTION The third generation of neural networks [1], the Spiking Neural Networks (SNN), have a stronger biological inspiration than those from the first and second generations. Most neural networks use analog values to communicate information between neurons which compute a non-linear function of their inputs. In SNN the time of a electrical pulse, or spike, is used to encode the information. The way by which a neuron of this type of neural networks computes its output is quite simple. All incoming spikes are integrated, resulting in a time decaying signal. When a spike arrives to the membrane its potential is altered; when the membrane potential crosses a prescribed threshold the neuron emits a spike at that exact time instant and the membrane potential is reverted to the initial value. Based on this general description much research effort has been put in the creation of models of spiking neurons [2] and on training algorithms for the various types of neurons. In the framework of supervised training, the Error Back Propagation (BP) algorithm is the most frequently found in literature. For the SNN this training algorithm is called Spikepro [3]. This algorithm uses the time of occurrence of a single spike to encode the inputs and outputs of the network. In order to apply the Levenberg-Marquardt (LM) algorithm to SNN, some parts of Spikepro need to be retained. The inputs encoding used is such that, to the 0 and 1 values correspond firing times of 0 and 6 ms, respectively. The bias neuron fires at 6 ms analogously to other neural networks where the bias value is 1, given the previously stated correspondence. This condition is not the same as presented in [3] where the bias input is considered at a reference time with a value of 0 ms. The outputs encoding used is such that, to the 0 and 1 values correspond fire times of 16 and 10 ms, respectively. The same network topology introduced in [3] is used, con- sisting in a feed forward neural network where each synapse terminal is composed by sub-connections that have a delay and weigth associated. The initial conditions employed are the same, except for the case where inhibitory neurons are used and where only positive weights are allowed. In our case, inhibitory neurons need not to be enforced given the fact that the weights can become positive or negative after the training. The only condition that must be fulfilled is that the weights must be arranged in such way that before the first weigth update all neurons must fire. This paper is organized as follows: in the following two Subsection the Spikepro and the LM algorithm are presented. In the next sections the methodology used, the results using Spikepro and Spikelm algorithms are presented. In the last section conclusions are drawn. A. Spikepro In order to explain the Spikepro algorithm it is necessary to review the spike neuron model used, in our case the SRM model defined in [2]. This model can be adapted using the spike-response function, reflecting the dynamics necessary for this case. The equation that reflects the dynamics of membrane po- tential is defined by: m xj(t) E k j(t)I iErj k=l (1) where Wk is the weigth of sub-connection k, and yk (t) is the function of sub-connection k, both between the neuron i and j. The function of sub-connection k is given by: (2) where ti is the firing time of neuron i, dk is the delay associated to the sub-connection k of neuron i and e is the response function of a neuron to one spike, which in turn is defined by an exponential function that decays with time: E(t) { e- t if t < 0 if t 0 (3) The model functionality is illustrated by figure 1. The Spikepro algorithm, like the BP algorithm for the second generation neural networks, tries to minimize the error between the current and desired output of the network. The error function used is the least mean square defined by: 0-7803-9422-4/05/$20.00 C2005 IEEE 1354 k k) YZ,j (t) e(t (ti + di