Information processing in the brain takes place in a dense network of neurons connected through synapses. The collaborative work between these two components (Synapses and Neurons) allows basic brain functions such as learning and memorization. In this work we shall study the reaction-diffusion model arising in biology. First we shall approximate this model with two dimensional Cellular Nonlinear Network (CNN) with memristor synapses. The stability of the obtained system will be investigated and a bifurcation analysis provided. Furthermore, device parameter variations will be taken into account in order to determine admissible variation intervals ensuring local activity in CNN cells. Large variety of complex phenomena will be studied which will demonstrate that the domain of edge of chaos exhibits all of the well-known complexities such as—Turing patterns, spiral waves, spatio-temporal chaos, as well as new information processing capabilities. Applications in EEG signal propagation will be shown in order to predict the epileptic seizures.