Significance: Phenotypes of an isogenic cell population are determined by states of low and high gene expression. These are often associated with multiple steady states predicted by deterministic models of gene regulatory networks. Gene expression is, however, regulated via stochastic molecular interactions at transcriptional and translational levels. Here, we show that intrinsic noise can induce multimodality in a wide class of regulatory networks whose corresponding deterministic description lacks multistability, thus offering a plausible alternative mechanism for phenotypic switching without the need for ultrasensitivity or highly cooperative interactions. We derive a general analytical framework for quantifying this phenomenon, thereby elucidating how cells encode decisions, and how they retain memory of transient environmental signals, using common gene regulatory motifs.

Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) have been observed both experimentally and numerically in various prototypical systems in the natural sciences. In the present article, we propose a mathematical model problem which, though analytically simple, exhibits a wide variety of MMO patterns upon variation of a control parameter. One characteristic feature of our model is the presence of three distinct time-scales, provided a singular perturbation parameter is sufficiently small. Using geometric singular perturbation theory and geometric desingularization, we show that the emergence of MMOs in this context is caused by an underlying canard phenomenon. We derive asymptotic formulae for the return map induced by the corresponding flow, which allows us to obtain precise results on the bifurcation (Farey) sequences of the resulting MMO periodic orbits. We prove that the structure of these sequences is determined by the presence of secondary canards. Finally, we perform numerical simulations that show good quantitative agreement with the asymptotics in the relevant parameter regime.

Mixed-mode dynamics is a complex type of dynamical behavior that has been observed both numerically and experimentally in numerous prototypical systems in the natural sciences. The compartmental Wilson-Callaway model for the dopaminergic neuron is an example of a system that exhibits a wide variety of mixed-mode patterns upon variation of a control parameter. One characteristic feature of this system is the presence of multiple time scales. In this article, we study the Wilson-Callaway model from a geometric point of view. We show that the observed mixed-mode dynamics is caused by a slowly varying canard structure. By appropriately transforming the model equations, we reduce them to an underlying three-dimensional canonical form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popović, and N. Kopell (unpublished).