Synthesis, Design, and Synchronization Analysis of Coupled Linear Electrical Networks

Synchronization has been associated with fundamental brain functions, especially learning behavior. Electrical circuits are candidates to mimic the tremendous information processing ability of simple brain structures due to their inherent massive parallelism. This paper gives an electrical interpretation of synchronization behavior in linear, identical subsystems with diffusive couplings. We consider a general linear state-space model for which we synthesize a minimal, generic electrical circuit. A conductance models the couplings between subsystems to form the overall electrical system. To investigate the synchronization behavior, we show how a beneficial placement of transformers decouples the overall circuit and consequently obtain many smaller circuits that are easier to examine. It is shown that the asymptotic stability in these decoupled circuits leads to a vanishing synchronization error over time. Based on this observation, we are able to formulate a synchronization condition that is entirely dependent on electrical quantities. One benefit, among others, is that by the notion of passivity, the asymptotic stability in some electrical circuits becomes evident without any further calculation. Lastly, we apply these insights to a network of interconnected Chua circuits mimicking neuron populations and their synaptic coupling structure. We investigate different topologies, such as a two ring-topology with a bridge-synapse connection.