The map with no predetermined firing order for the network of oscillators with time-delayed pulsatile coupling

The network of the pulse-coupled oscillators is studied in the presence of coupling delays. Because of the delays the past activity of the network is capable to influence the future network dynamics. In general case this leads to the infinite dimension of the corresponding dynamical system. We prove the Theorem that states that under certain conditions (weak coupling and appropriate initial conditions) the network can be fully characterized by a finite dimensional state vector. We construct the return map describing the evolution of this state vector over time. This map does not need any presupposed activity pattern in the network and works for any initial conditions.