Investigation on Amari’s dynamical neural field with global constant inhibition

Abstract

In this paper, the properties of Amari’s dynamical neural field with global constant inhibition induced by its kernel are investigated. Amari’s dynamical neural field illustrates many neurophysiological phenomena successfully and has been applied to unsupervised learning like data clustering in recent years. In its applications, the stationary solution to Amari’s dynamical neural field plays an important role that the underlying patterns being perceived are usually presented as the excited region in it. However, the type of stationary solution to dynamical neural field with typical kernel is often sensitive to parameters of its kernel that limits its range of application. Different from dynamical neural field with typical kernel that have been discussed a lot, there are few theoretical results on dynamical neural field with global constant inhibitory kernel that has already shown better performance in practice. In this paper, some important results on existence and stability of stationary solution to dynamical neural field with global constant inhibitory kernel are obtained. All of these results show that such kind of dynamical neural field has better potential for missions like data clustering than those with typical kernels, which provide a theoretical basis of its further extensive application.

Introduction

Dynamical neural field models are presented during 1970s, which describe large scale activation of cortical neurons as a continuous neural field (Amari, 1977, Ermentrout and Cowan, 1979, Feldman and Cowan, 1975, Kishimoto and Amari, 1979, Wilson and Cowan, 1972, Wilson and Cowan, 1973). Among these models, an important one is Amari’s dynamical neural field model (Amari, 1977). Amari’s dynamical neural field model has successfully interpreted large amounts of important phenomena and problems (Giese, 1999, Simmering et al., 2008), so that it has been extensively applied to psychophysics, neurophysiology, machine vision and cognition (Engels and Schöner, 1995, Erlhagen and Bicho, 2006, Faubel and Schöner, 2008, Schöner et al., 1995). In these application, Amari’s dynamical neural field is usually employed to describe the fundamental interaction function among neurons in the same cortical layer such as primary visual cortex V1, while some other models, for example, the neural model of Favorov and Kursun (2011), are employed to approximate more complex functions. In recent years, Amari’s dynamical neural field has been introduced to the fields like data clustering and obtained good results (Jin, Peng, & Li, 2011).

In the research and applications of Amari’s dynamical neural field model, especially for practical applications such as data clustering, the stationary solution with local excited region to Amari’s dynamical neural field model that are usually considered as the patterns been perceived. As a result, The existence and stability of stationary solution to dynamical neural field have been discussed a lot. The existence of a local excitation pattern solution as well as its waveform stability in 1-dimensional Euclid space $\mathbb{R}$ is proved in the absent of external input, which shows that the field can keep short-term memory (Amari, 1977, Kishimoto and Amari, 1979, Laing and Troy, 2003, Owen et al., 2007, Wennekers, 2002). Kubota et al. prove that with time-invariant external input there can be bistable local excitation solutions with different lengths (Kubota, Hamaguchi, & Aihara, 2009). The properties of stationary solution to Amari’s dynamical neural field in 2-dimensional Euclid space ${\mathbb{R}}^2$ are studied, which show that some different conditions are required for the existence and stability for local excitation solutions (Taylor, 1999, Taylor, 2003, Werner and Richter, 2001). For high dimensional space ${\mathbb{R}}^n$, some results on the existence and stability of solution and stationary solution to Amari’s dynamical neural field are obtained (Jin, Liang et al., 2011, Potthast and Graben, 2010).

In these studies of the stationary solution to Amari’s dynamical neural field model, most discussions focus on dynamical neural field with typical “Mexican Hat” shape kernel. For dynamical neural field with such kind of kernel, the type of its stationary solution is sensitive to its parameters. When it is applied to missions like data clustering, sometimes it is difficult to find proper parameters, especially in some cases for data set with complex structures in high dimensional space (Jin, Peng et al., 2011). As a result, though dynamical neural field with “Mexican Hat” shape kernel has been studied a lot, it is not quite convenient for practical issues. Another kind of kernels of Amari’s dynamical neural field are those which generate global constant inhibition. Dynamical neural field with such global constant inhibitory kernel has also been employed to pattern recognition and data clustering (Faubel and Schöner, 2008, Jin and Huang, 2013), which shows good performance. However, the properties of dynamical neural field with global constant inhibitory kernel are rarely discussed.

In this paper, the existence and stability of stationary solution to dynamical neural field with global constant inhibitory kernel are discussed. Some existence conditions for three important types of stationary solution: “$\varphi$”-solution, “$\infty$”-solution and “bubble”-solution for Heaviside Step and Sigmoid threshold functions, as well as unbounded and bounded perceive field, are presented, which provide the theoretical basis for further extensive application of dynamical neural field with global constant inhibitory kernel.

The remainder of this paper is arranged as follows. In Section  2, Amari’s dynamical neural field model is briefly described, and the clustering result using dynamical neural field with“Mexican Hat” shape kernel and global constant inhibitory kernel are given for comparison. In Section  3, some general results on stationary solution to dynamical neural field with global constant inhibitory kernel are given. In Section  4, when the threshold function is a discontinuous Heaviside Step function, the properties of dynamical neural field with global constant inhibitory kernel is discussed, in unbounded and bounded perceive field $\Omega$ respectively. In Section  5, when the threshold function is a continuous Sigmoid function, the properties of dynamical neural field with global constant inhibitory kernel is discussed, in unbounded and bounded perceive field $\Omega$ respectively. Section  6 is conclusion.