Introduction

Transitions from rest to oscillatory behavior can occur in mathematical models of neural networks in a variety of ways. One way is through the disappearance of an impediment to oscillation, such as coalescence and disappearance of stable and unstable rest states lying on a stable limit cycle, resulting in a stable oscillation. This phenomenon is referred to as being a saddle-node bifurcation on a limit cycle (SNLC) [9]. The SNLC and the Hopf bifurcation are the simplest ones exhibiting the onset of oscillations in the sense that they have codimension one; i.e., they have the fewest restrictions.

SNLC bifurcations are observed in the neurobiological models [1], and a canonical model for a SNLC bifurcation is the VCON model developed in [2, 3]. (VCON denotes voltage controlled oscillator neuron model.) The integrate-and-fire model [14] also exhibits this type of bifurcation, and in this sense it is closely related to the VCON.

We describe here some interesting aspects of wave propagation in networks that are near multiple SNLCs by studying networks of VCONs. This work is related to modeling regions of the neocortex and other brain structures [5,7], and these models are similar to many others that have been derived for parts of the neocortex using neurooscillators [8,15]. An advantage of the VCON model is that it enables one to study information flow in networks using routine frequency domain methods.

A VCON network is described by the model [2,3]

$$\dot{\theta}(s,t)= \omega +\int_{\cal D}K(s-s') f\left(\theta(s,t)-\theta(s',t)-\psi(s-s')\right)ds'$$

where

$s\in E^n$ for $n=1,2,\mbox{ or } 3$ addresses sites.

$\theta$ is a vector of phase variables taking values in E^N.

$\dot\theta=\partial\theta/\partial t$.

The connection matrix kernel ${K}\in E^{N\times N}$ is L periodic in each dimension of s. We suppose here that K describes isotropic connections: K=K(s-s'). It describes the amplitude of connections from s' to s.

$\psi$ describes the polarity of connections from s' to s. It is also L-periodic.

${\cal D}$ is a domain in the space of interest. In our case, we consider this set to be ${\cal D}=[-L,L]^n$ for $n=1,2,\mbox{ or } 3$.

The vector of functions $f(\phi)$ is $2\pi$-periodic in each component of $\phi\in E^N$.

$\omega$ is the vector of center frequencies of the network elements.

This model can be derived from general networks that are operating near a multiple SNLC bifurcation [7]. It has a rich structure of wavelike solutions, and it will be studied in greater detail elsewhere. We wish here to identify steady progressing wave solutions of this system, describe their stability properties, and illustrate them through computer simulations.