Study of a Sequence of Neural Oscillators
A B Alberto Herrera, G Alejandro Padrón · International Joint Conference on Neural Network · 2000
Nowadays, the main interest in neural computation is based on a Hopfield's suggestion stated in his now classical work of 1982 [1]. He proposed that the collective properties of an artificial neural system could be used to perform more complex non-linear operations than those that can be made with other kinds of computational systems. Hopfield stated this proposition from his analysis of the behavior of networks with symmetrical connections, based on very simple neuron models. Since the publication of Hopfield's work, other research groups had studied, with different kinds of mathematical and simulation tools the collective properties of other classes of artificial neural networks. This combination of efforts has led to establish the concept of ?computation with attractors? in the following sense [2]. A neural network, in particular one with recurrent connections, is said to perform a computation by analyzing the internal conditions reached by some of the network units, previously chosen, once the dynamic behavior of the network had settle down to a suitable chosen external perturbation. In this picture, the external perturbation is interpreted as the input data to be processed, the neural network itself is the machine that will perform the desired computation, and the conditions of the chosen units are considered as the output data. The easiest way to implement the previous idea, at least in the case of recurrent neural networks, is to use the equilibrium states of the network as the final conditions of the computation; because the equilibrium solutions are an especial kind of attractor, we have the name of the computing scheme. However, in this dynamic approximation to neural computation, it is, in principle, also possible to use other kinds of attractors, for example, periodic or even chaotic ones.In particular, the observation of non-stationary neural activity in visual cortex and olfactory cortex has stimulated interest in associative memory models with an oscillatory or chaotic dynamics. In this way, neural modeling of associative memory has aimed at exploring new associative recall principles and other novel forms of neural computation. Additionally, several research groups had recently proposed that oscillatory neural activity could be considered as a basic signal reflecting natural frequencies of the brain. In this approach relevant both to biology as to neuro-engineering, it is believed oscillatory activities govern the most general transformations or calculations in the brain or in a particular neuro-computer. Thus, it becomes of primary interest to understand the mechanisms responsible for the oscillatory activity of neuron models, both at the cellular and at the small and large ensembles level.The most common periodic behavior described in the literature correspond to limit-cycle oscillations. This kind of attractors is used to describe periodic non- linear and dissipative phenomena. Additionally, a limit-cycle oscillator comes to have a unique amplitude and waveform under stationary conditions, independently of initial conditions. However, limit-cycle oscillators are generally difficult to treat analytically, and, thus, a theoretical study of collective behavior of multi-oscillator system would be hopelessly difficult without resorting to some drastic simplifications of the model. The two most used simplifications are the phase model [3] and the weakly connected model [4]. However, we should not forget that the study of neural circuits with a few neurons is also important not only because of the application of these circuits, but mainly they can be used as building blocks to build and to understand bigger systems [5]. Therefore, in this work we present the study of the behavior of a homogeneous linear array of two neural oscillators, with the objective of determining the characteristics of the oscillatory state of the circuit, as well as the effect of some of the parameters of the circuit on the oscillatory behavior.