Complex dynamics and the structure of small neural networks

The discrete-time dynamics of small neural networks is studied empirically, with emphasis laid on non-trivial bifurcation scenarios. For particular two- and three-neuron networks interesting dynamical properties like periodic, quasi-periodic and chaotic attractors are observed, many of them co-existing for one and the same set of parameters. An appropriate equivalence class of networks is defined, describing them as parametrized dynamical systems with identical dynamical capacities. Combined symmetries in phase space and parameter space are shown to generate different representatives of such a class. Moreover, conditions on the connectivity structure are suggested, which guarantee the existence of complex dynamics for a considered equivalence class of network configurations.