In this seminar we shall present the derivation of the CNN implementations through spatial discretization, which suggests a methodology for converting the differential equations (DE) to CNN templates and vice versa. The CNN solution of the DE has four basic properties, which are:

i) continuous in time;

ii) continuous and bounded in value;

iii) continuous in interaction parameters;

iv) discrete in space.

First, we shall present some basic theory of CNN including the main types of equations that describe these networks, as well as some results about their dynamics and stability.

Then, we shall demonstrate how an autonomous CNN can serve as a unifying paradigm for active wave propagation and several well-known examples chosen from different disciplines will be modeled. Moreover, we shall show how the three basic types of DE: the diffusion equation, the Laplace equation, and the wave equation, can be solved via CNN.

Finally, several equations arising in biology, physics and ecology will be modeled through the CNN approach and their dynamics will be studied.