Relaxation oscillators with time delay coupling

We study networks of relaxation oscillators coupled with time delays synapses. A pair of oscillators is analyzed and shown to attain loosely synchronous solutions for a wide range of initial conditions and time delays. Simulations of one-and two-dimensional oscillator networks indicate that locally coupled oscillators are also loosely synchronous. Desynchronous solutions are possible when system parameters are varied. To characterize lossely synchronous networks, we introduce a measure of synchrony, the maximum time difference between any two oscillators. In locally excitatory globally inhibitory oscillator networks with time delays, we find that desynchronous solutions for different groups of oscillators are maintained, and the number of groups that can be segregated is related to the maximum time difference within each group. To examine the maximum time difference, we display its histograms for oscillator networks in one and two dimensions. Also, a range of initial conditions is given so that the maximum time difference is contained as the system evolves.