Synchronization of oscillators with random nonlocal connectivity

In this paper we study the existing observation in literature about synchronization of a large number of coupled maps with random nonlocal connectivity [Chate and Manneville, Chaos 2, 307 (1992)]. These connectivities which lack any spatial significance can be realized in neural nets and electrical circuits. It is quite interesting and of practical importance to note that a huge number of maps can be synchronized with this connectivity. We show that this synchronization stems from the fact that the connectivity matrix has a finite gap in the eigenvalue spectrum in the macroscopic limit. We give a quantitative explanation for the gap. We compare the analytic results with the ones quoted in the above reference. We also study the departures from this highly collective behavior in the low connectivity limit and show that the behavior is almost statistical for very low connectivity.