Symmetrized importance samplers for stochastic differential equations

We study a class of importance sampling methods for stochastic differential equations (SDEs). A small-noise analysis is performed, and the results suggest that a simple symmetrization procedure can significantly improve the performance of our importance sampling schemes when the noise is not too large. We demonstrate that this is indeed the case for a number of linear and nonlinear examples. Potential applications, e.g., data assimilation, are discussed.Comment: Added brief discussion of Hamilton-Jacobi equation. Also made various minor corrections. To appear in Communciations in Applied Mathematics and Computational Scienc

Small-noise analysis and symmetrization of implicit Monte Carlo samplers

Implicit samplers are algorithms for producing independent, weighted samples from multi-variate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algo- rithms that leads to improved (implicit) sampling schemes at a rel- atively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems

Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator

The Hodgkin-Huxley model describes action potential generation in certain types of neurons and is a standard model for conductance-based, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking Hodgkin-Huxley neuron model to a periodic pulsatile drive. The response as a function of drive period and amplitude is systematically characterized. A wide range of qualitatively distinct responses are found, including entrainment to the input pulse train and persistent chaos. These observations are consistent with a theory of kicked oscillators developed by Qiudong Wang and Lai-Sang Young. In addition to general features predicted by Wang-Young theory, it is found that most combinations of drive period and amplitude lead to entrainment instead of chaos. This preference for entrainment over chaos is explained by the structure of the Hodgkin-Huxley phase resetting curve.Comment: Minor revisions; modified Fig. 3; added reference

Modeling temporal networks using random itineraries

We propose a procedure to generate dynamical networks with bursty, possibly repetitive and correlated temporal behaviors. Regarding any weighted directed graph as being composed of the accumulation of paths between its nodes, our construction uses random walks of variable length to produce time-extended structures with adjustable features. The procedure is first described in a general framework. It is then illustrated in a case study inspired by a transportation system for which the resulting synthetic network is shown to accurately mimic the empirical phenomenology

Revisiting chaos in stimulus-driven spiking networks: signal encoding and discrimination

Highly connected recurrent neural networks often produce chaotic dynamics, meaning their precise activity is sensitive to small perturbations. What are the consequences for how such networks encode streams of temporal stimuli? On the one hand, chaos is a strong source of randomness, suggesting that small changes in stimuli will be obscured by intrinsically generated variability. On the other hand, recent work shows that the type of chaos that occurs in spiking networks can have a surprisingly low-dimensional structure, suggesting that there may be "room" for fine stimulus features to be precisely resolved. Here we show that strongly chaotic networks produce patterned spikes that reliably encode time-dependent stimuli: using a decoder sensitive to spike times on timescales of 10's of ms, one can easily distinguish responses to very similar inputs. Moreover, recurrence serves to distribute signals throughout chaotic networks so that small groups of cells can encode substantial information about signals arriving elsewhere. A conclusion is that the presence of strong chaos in recurrent networks does not prohibit precise stimulus encoding.Comment: 8 figure