Oscillations and variability in neuronal systems: interplay of autonomous transient dynamics and fast deterministic fluctuations

Abstract Neuronal systems are subject to rapidly fluctuations both intrinsically and externally. In mathematical models, these fluctuations are typically incorporated as stochastic noise (e.g., Gaussian white or colored noise). Noise can be both disruptive and constructive, for example, by creating irregularities and variability in otherwise regular patterns or by creating oscillatory patterns and increasing the signal coherence, respectively. The dynamic mechanisms underlying the interactions between rapidly fluctuating signals and the intrinsic properties of the target cells to produce variable and/or coherent responses are not fully understood. In particular, it is not clear what properties of the target cell’s intrinsic dynamics control these interactions and whether the generation of this phenomena requires stochasticity of the input signal and, if yes, to what degree. In this paper we investigate these issues by using linearized and non-linear conductance-based models and piecewise constant (PWC) inputs with short duration pieces and variable amplitudes, which are arbitrarily, but not necessarily stochastically distributed. The amplitude distributions of the constant pieces consist of arbitrary permutations of a baseline PWC function with monotonically increasing amplitudes. In each trial within a given protocol we use one of these permutations and each protocol consists of a subset of all possible permutations, which is the only source of uncertainty in the protocol. We show that sustained oscillatory behavior can be generated in response to additive and multiplicative PWC inputs in both linear and nonlinear systems, independently of whether the stable equilibria of the corresponding unperturbed systems are foci (exhibiting damped oscillations) or nodes (exhibiting overshoots). The oscillatory responses are amplified by the model nonlinearities and attenuated for conductance-based PWC inputs as compared to current-based PWC inputs, consistent with previous theoretical and experimental work. In addition, the responses to PWC inputs exhibited variability across trials, which is reminiscent of the variability generated by stochastic noise (e.g., Gaussian white noise). This variability was modulated by the model parameters and the type of cellular intrinsic dynamics. Our analysis demonstrates that both oscillations and variability are the result of the interaction between the PWC input and the autonomous transient dynamics with little to no contribution from the dynamics around the steady-state. The generation of oscillations and variability does not require input stochasticity, but rather the sequential activation of the transient responses to abrupt changes in constant inputs. Each piece with the same amplitude evokes different responses across trials due to the differences in initial conditions in the corresponding regime. These initial conditions are determined by the value of the voltage at the end of the previous regime, which is different for different trials.The predictions made in this papers are amenable for experimental testing both in vitro and in vivo.