We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
Generalized and partial synchronization of coupled neural networks
Frank Pasemann and Thomas Wennekers
Synchronization of neural signals has been proposed as a temporal coding scheme representing cooperated computation in distributed cortical networks. Previous theoretical studies in that direction mainly focused on the synchronization of coupled oscillatory subsystems and neglected more complex dynamical modes, that already exist on the single-unit level. In the present work we study the parameterized time-discrete dynamics of two coupled recurrent networks of graded neurons. Conditions for the existence of partially synchronized dynamics of these systems are derived, referring to a situation where only subsets of neurons in each sub-network are synchronous. The coupled networks can have different architectures and even a different number of neurons. Periodic as well as quasiperiodic and chaotic attractors constrained to a manifold M of synchronized components are observed. Examples are discussed for coupled 3-neuron networks having different architectures, and for coupled 2-neuron and 3-neuron networks. Partial synchronization of different degrees is demonstrated by numerical results for selected sets of parameters. In conclusion, the results show that synchronization phenomena far beyond completely synchronized oscillations can occur even in simple coupled networks. The type of the synchronization depends in an intricate way on stimuli, history and connectivity as well as other parameters of the network. Specific inputs can further switch between different operational modes in a complex way, suggesting a similarly rich spatio-temporal behavior in real neural systems.