Autowave processes in diffusion neuron systems (Q2284250)

The authors consider the following system of delay-differential equations \(\dot u_1 = D(u_2 -u_1) +\lambda f(u_1(t-1))u_1\), \(\dot u_2 = D(u_1 -u_2) +\lambda f(u_2(t-1))u_2\), where \( D\) is the coupling weighs and \( \lambda\gg 1\). The paper provides conditions for the existence of periodic ``pulse-like'' solutions with large period of the order of \(\lambda\). The exponential orbital stability of the obtained solutions is proven. The results are also generalized to the case of a chain of diffusively coupled systems \[ \dot u_j = D(u_{j+1} -2u_{j+1}+u_{j-1}) +\lambda f(u_j(t-1))u_j. \]