SYNCHRONOUS CHAOS IN HIGH-DIMENSIONAL MODULAR NEURAL NETWORKS
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The relationship between certain types of high-dimensional neural networks and low-dimensional prototypical equations (neuromodules) is investigated. The high-dimensional systems consist of finitely many pools containing identical, dissipative and nonlinear single-units operating in discrete time. Under the assumption of random connections inside and between pools, the system can be reduced to a set of only a few equations, which — asymptotically in time and system size — describe the behavior of every single unit arbitrarily well. This result can be viewed as synchronization of the single units in each pool. It is stated as a theorem on systems of nonlinear coupled maps, which gives explicit conditions on the single unit dynamics and the nature of the random connections. As an application we compare a 2-pool network with the corresponding two-dimensional dynamics. The bifurcation diagrams of both systems become very similar even for moderate system size (N=50) and large disorder in the connection strengths (50% of mean), despite the fact, that the systems exhibit fairly complex behavior (quasiperiodicity, chaos, coexisting attractors).