Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps

What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically-based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can either appear chaotic (with provably positive Lyaponov exponent for the associated circle maps), or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noiseComment: 17 figures, 40 page

Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles

We would like to thank the partial support of this work by the Brazilian agencies FAPESP (processes: 2011/19296-1, 2013/26598-0, and 2015/20407-3), CNPq and CAPES. MSB acknowledges EPSRC Ref. EP/I032606/1.Peer reviewedPublisher PD

Mini-Workshop: Dynamics of Stochastic Systems and their Approximation

The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area

From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns

We quantify the effect of Gaussian white noise on fast--slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincar\'e section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. These results yield predictions on the observed random mixed-mode oscillation patterns. Our analysis shows that there is an intricate interplay between the number of small-amplitude oscillations and the global return mechanism. In combination with a local saturation phenomenon near the folded node, this interplay can modify the number of small-amplitude oscillations after a large-amplitude oscillation. Finally, sufficient conditions are derived which determine when the noise increases the number of small-amplitude oscillations and when it decreases this number.Comment: 56 pages, 14 figures; revised versio

Data-driven reduction strategies for Bayesian inverse problems

A persistent central challenge in computational science and engineering (CSE), with both national and global security implications, is the efficient solution of large-scale Bayesian inverse problems. These problems range from estimating material parameters in subsurface simulations to estimating phenomenological parameters in climate models. Despite recent progress, our ability to quantify uncertainties and solve large-scale inverse problems lags well behind our ability to develop the governing forward simulations. Inverse problems present unique computational challenges that are only magnified as we include larger observational data sets and demand higher-resolution parameter estimates. Even with the current state-of-the-art, solving deterministic large-scale inverse problems is prohibitively expensive. Large-scale uncertainty quantification (UQ), cast in the Bayesian inversion framework, is thus rendered intractable. To conquer these challenges, new methods that target the root causes of computational complexity are needed. In this dissertation, we propose data-driven strategies for overcoming this “curse of di- mensionality.” First, we address the computational complexity induced in large-scale inverse problems by high-dimensional observational data. We propose a randomized misfit approach (RMA), which uses random projections—quasi-orthogonal, information-preserving transformations—to map the high-dimensional data-misfit vector to a low-dimensional space. We provide the first theoretical explanation for why randomized misfit methods are successful in practice with a small reduced data-misfit dimension (n = O(1)). Next, we develop the randomized geostatistical approach (RGA) for Bayesian sub- surface inverse problems with high-dimensional data. We show that the RGA is able to resolve transient groundwater inverse problems with noisy observed data dimensions up to 107, whereas a comparison method fails due to out-of-memory errors. Finally, we address the solution of Bayesian inverse problems with spatially localized data. The motivation is CSE applications that would gain from high-fidelity estimation over a smaller data-local domain, versus expensive and uncertain estimation over the full simulation domain. We propose several truncated domain inversion methods using domain decomposition theory to build model-informed artificial boundary conditions. Numerical investigations of MAP estimation and sampling demonstrate improved fidelity and fewer partial differential equation (PDE) solves with our truncated methods.Computational Science, Engineering, and Mathematic