Frequency-dependent coupling in responses to oscillatory inputs in networks of electrically coupled nodes: Gap junction networks and spatially extended neurons

In electrically coupled networks, the coupling coefficient (CC) quantifies the strength of the connectivity or the degree to which two participating nodes are coupled in response to an external input to one of them. The CC is measured by computing the relative responses of the indirectly activated (post-J) and the directly activated (pre-J) nodes. In response to time-dependent inputs, the CC is frequency-dependent and has two components capturing the contributions of the amplitude and phase frequency profiles of the participating nodes (quotient of the amplitudes and phase-difference, respectively). The properties and mechanisms of generation of the frequency-dependent CCs (FD-CCs) are largely unknown beyond electrically coupled passive cells and their electrical circuit equivalents. Being linear and 1D, the FD-CCs for passive cells are relatively simple, consisting of low-pass filters (amplitude) and positive and monotonically increasing phase-difference profiles. In linear systems, the FD-CCs depend on the properties of the pre-J cell and the connectivity and are independent of the properties of the post-J cell and the input amplitude. There is a gap in our understanding of the FD-CCs are shaped by (i) how the presence of intrinsic cellular positive and negative feedback currents and the resulting amplification and resonance phenomena, and (ii) the presence of cellular nonlinearities that incorporates the dependence of the FD-CC on the post-J node in addition to the pre-J one. In this paper we address these issues by using biophysically plausible (conductance-based) mathematical modeling, numerical simulations, analytical calculations and dynamical systems tools. We conduct a systematic analysis of the properties of the FD-CC in networks of two electrically connected nodes receiving oscillatory inputs, which is the minimal network architecture that allows for a systematic study of the biophysical and dynamic mechanisms that shape the FD-CC profiles. The participating neurons are either passive cells (low-pass filters) or resonators (band-pass filter) and exhibit lagging or mixed leading-lagging phase-shift responses as the input frequency increases. The formalism and tools we develop and use in this paper can be extended to larger networks with an arbitrary number of nodes, to spatially extended multicompartment neuronal models, and to neurons having a variety of ionic currents. The principles that emerge from our study are directly applicable to these scenarios. Our results make experimentally testable predictions and have implications for the understanding of spike transmission, synchronized firing and coincidence detection in electrically coupled networks in the presence of oscillatory inputs. For clarity, the paper includes an extensive supplementary material section.