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To obtain explicit understanding of the behavior of dynamical systems, geometrical methods and slow-fast analysis have proved to be highly useful. Such methods are standard for smooth dynamical systems, and increasingly used for continuous, non-smooth dynamical systems. However, they are much less used for random dynamical systems, in particular for hybrid models with discrete, random dynamics. Indeed, the analysis of such systems has typically been done by studying the corresponding deterministic system and considering how noise perturbs the deterministic geometrical structures. Here we propose a geometrical method that works directly with the hybrid system. We illustrate our approach through an application to a hybrid pituitary cell model in which the stochastic dynamics of very few active large-conductance potassium (BK) channels is coupled to a deterministic model of the other ion channels and calcium dynamics. To employ our geometric approach, we exploit the slow-fast structure of the model. The random fast subsystem is analyzed by considering discrete phase planes, corresponding to the discrete number of open BK channels, and stochastic events correspond to jumps between these planes. The evolution within each plane can be understood from nullclines and limit cycles, and the overall dynamics, e.g., whether the model produces a spike or a burst, is determined by the location at which the system jumps from one plane to another. Our approach is generally applicable to other scenarios to study discrete random dynamical systems defined by hybrid stochastic-deterministic models., Comment: 15 pages, 8 figures

Word embeddings provide an unsupervised way to understand differences in word usage between discursive communities. A number of recent papers have focused on identifying words that are used differently by two or more communities. But word embeddings are complex, high-dimensional spaces and a focus on identifying differences only captures a fraction of their richness. Here, we take a step towards leveraging the richness of the full embedding space, by using word embeddings to map out how words are used differently. Specifically, we describe the construction of dialectograms, an unsupervised way to visually explore the characteristic ways in which each community use a focal word. Based on these dialectograms, we provide a new measure of the degree to which words are used differently that overcomes the tendency for existing measures to pick out low frequent or polysemous words. We apply our methods to explore the discourses of two US political subreddits and show how our methods identify stark affective polarisation of politicians and political entities, differences in the assessment of proper political action as well as disagreement about whether certain issues require political intervention at all.

In this paper, we use geometric singular perturbation theory and blowup, as our main technical tool, to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. In particular, we demonstrate that the MMOs in this model are not due to generic folded singularities, but rather due to singularities at a cusp -- not a fold -- of the critical manifold. Using blowup, we determine the number of SAOs analytically, showing -- as for the folded nodes -- that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a (symmetric) saddle-node bifurcation in the desingularized reduced problem, which -- although resembling a folded saddle-node (type II) at this level -- also occurs on a cusp, and not a fold. We demonstrate that this bifurcation is associated with the emergence of an invariant cylinder, the onset of SAOs, as well as SAOs of increasing amplitude. We relate our findings with numerical computations and find excellent agreement.