Relaxation oscillator networks with time delays

We study relaxation oscillators with couplings that mimic excitatory chemical synapses. Such oscillator networks have been shown to synchronize quickly without time delays. We present analytic results for a pair of oscillators showing that loose synchrony occurs for a wide range of initial conditions and time delays. Simulations indicate that locally coupled networks in one and two dimensions also exhibit loose synchrony. To characterize loose synchrony we introduce a measure of synchrony, the maximum time difference. We obtain histograms of this measure for one and two dimensional oscillator networks. Also, we conjecture that there is a range of initial conditions for which the maximum time difference remains bounded as the system evolves.