Mathematical Neuroscience

The main subject of this research area is new models in mathematical neuroscience: artificial neural networks, which have many similarities with the structure of human brain and the functions of cells by electronic circuits. The dynamics of the networks are presented by differential equations with discontinuities of different types:
• Differential equations with piecewise constant argument of generalized type;
• Impulsive Differential Equations at fixed and variable moments of time;
• Both impulses at fixed moments and piecewise constant argument.
Qualitative analysis of existence and uniqueness of solutions, global asymptotic stability, uniform asymptotic stability and global exponential stability of equilibria, existence of periodic solutions, and their global asymptotic stability for these networks are obtained.
It is well known that the studies on neural dynamical systems not only involve stability and periodicity, but also involve other dynamic behaviors such as:
• Neural networks with state-dependent moments of time;
• Existence of almost periodic solutions;
• Synchronization;
• Chaos.
Furthermore, definitely, these results, if not confirmed by immediate biological observations, give a strong belief to the development of experiments and research to satisfy obtained new mathematical observations, as it has been done already for other real world problems.