Membrane potential resonance in non-oscillatory neurons interacts with synaptic connectivity to produce network oscillations

Abstract

Several neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or it plays a functional role for the generation of neuronal network oscillations and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (band-pass filter) with 2D dynamics, (ii) a passive cell (lowpass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics. We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonators resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. Our results have direct implications for network models of firing rate type and other biological oscillatory networks (e.g, biochemical, genetic). * Corresponding Investigator, CONICET, Argentina Author SummaryBiological oscillations are ubiquitous in living systems and underlie fundamental processes in healthy and diseased individuals. Understanding how the intrinsic oscillatory properties of the participating nodes interact with the network connectivity is key for the mechanistic description of biological network oscillations. In several cases these intrinsic oscillatory properties are hidden and emerge only in the presence of external oscillatory inputs in the form of preferred amplitude responses to these inputs. This phenomenon is referred to as resonance and may occur in systems that do not exhibit intrinsic oscillations. Resonance has been primarily measured in neuronal systems, but their role in the generation of neuronal network oscillations remains largely an open question. We have identified a minimal network model consisting of a resonator (a node that exhibits resonance, but not intrinsic oscillations), a low-pass filter (no resonance and no intrinsic oscillations) and nonlinear connectivity with no dynamics. This network is able to produce oscillations, even in the absence of intrinsic oscillatory components. These oscillations crucially depend on the presence of the resonator. Moreover, the resonant frequency, a dynamic property of the interaction between the resonator and oscillatory inputs, controls the network frequency in a...

Inhibition-based relaxation oscillations emerge in resonator networks

“…To study the effects of f res on f ntw we follow [4,12] and change the model parameters in such a way that f res changes but Z max remains constant. In particular, we consider a fixed value of g L,1 , and various balanced combinations of values of g 1 and τ 1 so as to increase f res and maintain Z max constant.…”

“…In addition to the single-cell mechanisms of generation of oscillations described above, oscillations can be generated at the network level in mutually connected cells that do not exhibit sustained oscillations when disconnected, but they are rather damped oscillators or resonators [4,48,88,93]. The graded synaptic connectivity used in these studies has been assumed to be instantaneously fast and to have no dynamics.…”

“…In [4] we have identified a minimal (3D) network model consisting of a two-dimensional resonator and a one-dimensional passive cell mutually connected through graded inhibition. Oscillations are also obtained in self-excited resonators (2D), mutually excited resonators (4D) and mutually inhibited resonators (4D), also using graded synaptic connectivity.…”

“…In self-excited networks, self-excitation provides the amplifying process necessary for the generation of oscillations. In fact, self-excited resonators have the same mathematical structure as single neurons having both resonant and amplifying ionic processes whose dynamics are also 2D [4]. Mutually excited resonators exhibit synchronized in-phase patterns and therefore their dynamics are captured by those of self-excited resonators.…”

“…Subsequently, we show how the resonant frequency of the individual 2D resonator affects the network oscillation frequency. Finally, we review some results obtained in [4] for smooth connectivity functions, discuss the existence of relaxation oscillations and the associated canard phenomenon, and compare the oscillatory properties of networks having smooth and PWL connectivity functions. We conclude in Section 4 by discussing our results, their implications for network dynamics and future work.…”