Model Description

We consider a Wilson-Cowan neural population model. In particular the simplest model of neural oscillator consists of an excitatory neural population  and an inhibitory neural population . Let us suggest that identical oscillators are arranged in 2D grid with local connections between neighboring oscillators of the first and second order. The dynamics of the network is described by the equations:

 

(1)

 

Here  are the coupling strengths between the populations inside the oscillator; Pn is the value external input to excitatory population; ;  is the monotonically increasing sigmoid-type function given by formula: .

Parameter values are:

,  and  describe influences of neighbors of the first and second order respectively. A choice of these particular parameter values corresponds to the oscillatory regime of a single neural oscillator (Wilson & Cowan, 1973). If connection strengths are zero, then a single neural oscillator has only a limit cycle attractor in two-dimensional phase space. Coupling between oscillators is described by the terms Vn and Wn which are determined by the type of connections. For example, for excitatory-to-excitatory local connections these terms are:

(2)

 

Here  are sets of the first and second order neighbors of the nth population,  and  are strengths of homogeneous connections of the first and second order respectively.

We consider boundary conditions on the opposite sides of the grid as identical and the resulting surface is a torus.

 Wave Propagation in a Network of Locally Coupled Neural Oscillators

We study spatio-temporal patterns of neural activity both in 1D network with local connections (chain of oscillators) and in 2D network on the toroidal surface. The regime of propagating waves was found in case of excitatory to excitatory connections only. In case of other local connection types (excitatory to inhibitory, inhibitory to excitatory and inhibitory to inhibitory) the regime of propagating waves was not found. We suppose here that a wave propagates in a stationary environment where all oscillators are in a low activity stationary mode (the parameter value of the external input P=0.8 corresponds to a stable steady-state of a single oscillator  with a low background activity) except oscillators in the middle of grid (or chain), which are in the oscillatory mode with external parameter value  [6].
     A speed and length of wave propagation depends on strengths of local connections. For example, in case of weak coupling, a  wave propagates within a short distance, decays, and disappears. Other  connection strength values can significantly reduce a speed of wave propagation and in this case a bump-like structure of neural activity exists for a long time. This kind of persistent activity is traditionally used for modelling of a short-term  memory.
     Propagating wave of neural activity can be appear as a result of constant stimulation or a short-time stimulus presentation. It is interesting to note that a short-term stimulation can cause a persistent wave propagation. Thus, even if the media is non-oscillating (low level stationary activity), a short  presentation of a local stimulus can result in persistent wave propagation. 
     Fig. 1 shows the regime of wave propagation over 2D grid on the torus surface. The grid contains neural oscillators with local connections of first and second order (equations (1) and (2)). All oscillators are in a low activity steady-state except a group of oscillators inside of square region in the centre of the figure. These oscillators are in oscillatory regime but due to excitatory-to-excitatory connections their mutual excitation is large enough to keep them in a stationary state of high activity and they form a permanent source of propagating waves. Here, as usual red colors mean high activity and blue colors mean a low activity level. It is interesting to note that inside of propagating wave neural activity has a complex structure (see left-bottom frame) which constantly changes. On the boundary, the wave interacts with itself and that results in a complex spatial pattern of neural activity. Fig. 1 shows activities of excitatory populations at 6 sequential times.

 

            

Fig. 1. 2D wave propagation. There is a constant source of oscillatory activity- a square region at the centre contains oscillator in oscillatory mode. All other oscillators are in the regime of low stationary activity. Six time cuts of wave propagation on the torus (opposite sides of the grid are considered to be identical) are shown. The last frame shows a complex and irregular pattern of spatial activity. To watch a video clip please click here.
 
Fig. 2 shows interaction of two propagating waves. Similar to Fig. 1, we consider two sources of propagating waves over 2D grid of locally coupled oscillator on the torus. The waves demonstrate non-linear interaction resulting in an appearance of complex spatial pattern of neural activity. Fig. 2 shows activities of excitatory populations progressing with time. Two propagating waves interact and cooperate merging to a ring-like structure (see right-top frame in Fig.2) which surrounds both sources of propagating waves. This combined source of propagating waves can be considered as a new complex source of large wave propagation. The wave reaches boundaries and interacts with itself giving rise to a complex spatio-temporal pattern.
            
Fig. 2. 2D waves propagation and interaction. There are two identical sources of propagating waves. Wave interaction results in a complex pattern of spatial activity. Different frames show spatial activity patterns of excitatory populations at sequential times. To watch a video clip please click here.