Pm. Gade, SYNCHRONIZATION OF OSCILLATORS WITH RANDOM NONLOCAL CONNECTIVITY, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(1), 1996, pp. 64-70
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.