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

Abstract In electrically coupled networks, the coupling coefficient (CC) quantifies the strength of the connectivity between pairs of nodes. The CC is typically measured by computing the relative stationary responses to constant inputs of the indirectly activated (post-J) and the directly activated (pre-J) nodes. The natural extension of the CC to time-dependent inputs is frequency-dependent and has two components reflecting the contributions of the amplitude and phase frequency-dependent profiles (curves of these quantities as a function of the frequency $$ f $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> ) of the participating nodes: the quotient of amplitudes $$ K(f) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and the phase-difference $$ \Delta \Phi (f)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>Φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> profiles. The properties and mechanisms of generation of these frequency-dependent CCs (FD-CCs) are largely unknown beyond electrically coupled passive cells and their electrical linear circuit equivalents. For passive cells, $$ K(f) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is monotonically decreasing (low-pass filter) and $$ \Delta \Phi (f) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>Φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is monotonically increasing and positive. Moreover, for linear systems, the FD-CCs depend on the properties of the post-J cell and the connectivity and are independent of the properties of the pre-J cell and the input amplitude. It remains largely unclear how the FD-CCs are shaped by the presence of (i) intrinsic cellular positive and negative feedback currents (resonance and amplification), and (ii) 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 profiles 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 responses as the input frequency increases. The formalism and tools we develop and use in this paper are amenable to 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.