SYNCHRONIZATION OF OSCILLATORS WITH RANDOM NONLOCAL CONNECTIVITY

In this paper we study the existing observation in literature about sy nchronization of a large number of coupled maps with random nonlocal c onnectivity [Chate and Manneville, Chaos 2, 307 (1992)]. Theses connec tivities which lack any spatial significance can be realized in neural nets and electrical circuits. It is quite interesting and of practica l importance to note that a huge number of maps can be synchronized wi th 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 t he above reference. We also study the departures from this highly coll ective behavior in the low connectivity limit and show that the behavi or is almost statistical for very low connectivity.