Network resonance: Impedance interactions via a frequency response alternating map (FRAM)

We investigate how network resonance emerges as the result of the oscillatory interactions of the participating cells in networks where the individual cells need not be intrinsic oscillators. We develop a tool that we refer to as the frequency response alternating map (FRAM). The FRAM consists of decoupling the nonautonomous part of the coupling term and substituting the reciprocal coupling by a sequence of alternating one-directional forcing effects (cell 1 forces cell 2, which in turn forces cell 1, and so on). In this way, we break down the interaction between external forcing and network connectivity in a novel way that allows us to understand how oscillations are sequentially communicated among the nodes in the network. The end result is an expression of the network impedance for each node as a power series, each term involving the product of the extended impedances of the autonomous part of the individual oscillators. Each iteration in the map provides an approximation to the network impedance and phase. We use the FRAM to explain several results of network resonances in networks of both low- and band-pass filters with the same or different resonant frequencies.