Determination of the mechanism behind the formation of spatio-temporal hexagonal activity patterns in a model neuronal network

Abstract

In this report I present the results from the mathematical analysis of a model neuronal network introduced by Ermentrout and Curtu, extended to two space dimensions. The model can be seen as a simple model of a cortical sheet that describes the firing rate activity of two populations of neurons coupled together, one excitatory that displays linear adaptation and the other one inhibitory. The dynamics of the activity is described by a system of nonlocal partial integro-differential equations in which a nonlinear sigmoidal shaped firing rate function is included. The (synaptic) coupling between the neurons is characterized by local excitation and long range inhibition. Running numerical simulations with this model shows that a travelling hexagonal activity pattern is a stable solution of the model given certain sets of values for the parameters of the model. During this project I have investigated the mechanism behind the formation of these patterns as solutions of the model. The results from this investigation are presented in this report. In order to investigate the mechanism I have used methods from nonlinear dynamical systems, pattern formation and bifurcation theory to analyze the model. I started by performing a linear stability analysis around the uniform quiescent state of the model (which is stable before the bifurcation points). From this analysis I could deduce the different types of bifurcation that could take place in the model. I decided to focus on the analysis of the Hopf bifurcation in order to investigate which types of spatio-temporal hexagonal pattern can be obtained in the model as a consequence of this type of bifurcation. Subsequently I performed a singular perturbation analysis to obtain the general solution of the model corresponding to spatio-temporal hexagonal activity patterns. Then I determined equilibrium solutions of this general solution which correspond with a travelling hexagonal activity pattern and a stationary, regularly oscillating hexagonal activity pattern. By finding these solutions through this analysis, it has become clear what the overall mechanism is behind their formation and why these solutions behave the way they do. Due to the fact that the stability of the trivial solution is lost at pure imaginary pairs of eigenvalues, spatio-temporal patterns can start bifurcating from the uniform quiescent state of the model. The exact shape of the function used to describe the coupling between the neurons in the model determines the shape of the activity patterns that can form in the model and the nonlinear sigmoidal shaped firing rate function prevents the solutions from growing without bounds. The exact values of the parameters of the model determine the stability of the activity patterns. It is interesting to note that the oscillating hexagonal activity pattern is a solution I've never found in numerical simulations. Due to a lack of time at present and the complexity of the problem at hand I have not been able to determine the exact stability conditions on the parameters of the model for each of the two found equilibrium solutions. However, an algorithm is presented to determine these conditions. With these stability conditions it must be possible to also find the oscillating pattern as a stable solution in numerical simulations with the model.