Frequency-dependent communication of information innetworks of non-oscillatory neurons in response to oscillatory inputs

Understanding how neuronal networks process oscillatory inputs is key for deciphering the brains information processing dynamics. Neuronal filters describe the frequency-dependent relationship of neuronal outputs (e.g., membrane potential amplitude, firing rate) and their inputs for the level of neuronal organization (e.g., cellular, network) considered. Band-pass filters are associated to the notion of resonance and reflect the systems ability to respond maximally to inputs at a nonzero (resonant) frequency or a limited (resonant) frequency band. The complementary notion of phasonance refers to the ability of a system to exhibit a zero-phase response for a nonzero (phasonant) input frequency. The biophysical and dynamic mechanisms that shape neuronal filters and give raise to preferred frequency responses to oscillatory inputs are poorly understood beyond single cells. Moreover, the mechanisms that control the frequency-dependent communication of information across cells in a network remain unclear. Here, we use mathematical modeling, analytical calculations, computational simulations and dynamical systems tools to investigate how the complex and nonlinear interaction of the systemss biophysical properties and interacting time scales shape neuronal filters in minimal network models receiving oscillatory inputs with frequencies (f) within some range. The minimal networks consist of one directly stimulated cell (cell 1) connected to another (not directly stimulated) cell (cell 2) via graded chemical synapses. Individual cells are either passive or resonators and chemical synapses are either excitatory or inhibitory. The network outputs consist of the voltage peak envelopes and the impedance amplitude and phase profiles (as a function of f) for the two cells. We introduce the frequency-dependent amplitude K(f) and phase {Delta}{Phi}(f) communication coefficients, defined as the ratio of the amplitude responses of the indirectly and directly stimulated cells and the phase difference between these two cells, respectively. Extending previous work, we also introduce the K-curve, parametrized by f, in the phase-space diagram for the voltage variables of the two participating cells. This curve joins the peak voltage values of the two cells in response to the oscillatory inputs and is a geometric representation of the communication coefficient. It allows to interpret the results and explain the dependence of the properties of the communication coefficient in terms of the biophysical and dynamic properties of the participating cells and synaptic connectivity when analytical calculations are not possible. We describe the conditions under which one or the two cells in the network exhibit resonance and phasonance and the conditions under which the network exhibits K-resonance and {Delta}{Phi}-phasonance and more complex network responses depending as the complexity of the participating cells increases. For linear networks (linear nodes and linear connectivity), K is proportional to the impedance of the indirectly activated cell 2 and {Delta}{Phi} is equal to the phase of the indirectly stimulated cell 2, independent of the directly stimulated cell 1 in both cases. We show that the presence of nonlinear connectivity in the network creates (nonlinear) interactions between the two cells that give rise to K-resonance, {Delta}{Phi}-phasonance and more complex responses that are absent in the corresponding linear networks. The results and methods developed in this paper have implications for the processing of information in more complex networks.