Oscillations in systems of equations with delay and difference diffusion that model local neuron nets (Q1363272)

A local net is a population of similar neurons, each of which is connected only with close neighbors. The term ``local'' characterizes the architecture of the connections. The elements, or formal neurons, can be the Pitz-McCulloch neurons, operational amplifiers, oscillators, etc. We will consider formal neurons that are described by equations with delay. The mechanism of interaction of elements is one more characteristic of a net. The most natural hypothesis is that equations are connected by diffusion. Two problems usually need to be set up for neuron nets. The first problem is to develop algorithms that make it possible to choose forces of connection in such a way that attractors of a prescribed structure are present in the phase space. The second problem is to study attractors for the chosen forces of connection. In the case of homogeneous connections, we will solve the second problem. We show that, for any curve on the plane, the neurons that lie on it can function synchronously. If we interpret a curve as an image, then the net can preserve it for an infinitely long period.