ORCID Profile
0000-0002-5577-9846
Current Organisations
Inria Centre de Recherche Sophia Antipolis Méditerranée
,
New Jersey Institute of Technology
,
University of Utah
,
University of Oxford
,
University of Sydney
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Publications
A variational method for analyzing limit cycle oscillations in stochastic hybrid systems
Publisher: AIP Publishing
Date: 06-2018
DOI: 10.1063/1.5027077
Abstract: Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic ex le is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ−1. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (−Ca/ϵ).
Synchronization of stochastic hybrid oscillators driven by a common switching environment
Publisher: AIP Publishing
Date: 12-2018
DOI: 10.1063/1.5054795
Abstract: Many systems in biology, physics, and chemistry can be modeled through ordinary differential equations (ODEs), which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However, the discrete nature of the Markov jump process raises some difficulties: in fact, we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapunov exponent is more accurate.
Phase reduction of stochastic biochemical oscillators
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2020
DOI: 10.1137/18M1221205
Asymptotic description of stochastic neural networks. I. Existence of a large deviation principle
Publisher: Elsevier BV
Date: 10-2014
Asymptotic Description of Neural Networks with Correlated Synaptic Weights
Publisher: MDPI AG
Date: 06-07-2015
DOI: 10.3390/E17074701
Asymptotic description of stochastic neural networks. II. Characterization of the limit law
Publisher: Elsevier BV
Date: 10-2014
Mean field dynamics of a Wilson–Cowan neuronal network with nonlinear coupling term
Publisher: World Scientific Pub Co Pte Ltd
Date: 29-10-2018
DOI: 10.1142/S0219493718500466
Abstract: In this paper we prove the propagation of chaos property for an ensemble of interacting neurons subject to independent Brownian noise. The propagation of chaos property means that in the large network size limit, the neurons behave as if they are probabilistically independent. The model for the internal dynamics of the neurons is taken to be that of Wilson and Cowan, and we consider there to be multiple different populations. The synaptic connections are modeled with a nonlinear “electrical” model. The nonlinearity of the synaptic connections means that our model lies outside the scope of classical propagation of chaos results. We obtain the propagation of chaos result by taking advantage of the fact that the mean-field equations are Gaussian, which allows us to use Borell’s Inequality to prove that its tails decay exponentially.
Stochastic rotating waves
Publisher: World Scientific Pub Co Pte Ltd
Date: 11-2022
Phase Reduction of Waves, Patterns, and Oscillations Subject to Spatially Extended Noise
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 12-06-2023
DOI: 10.1137/21M1451221
The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis
Publisher: Elsevier BV
Date: 08-2013
On uniform propagation of chaos
Publisher: Informa UK Limited
Date: 17-04-2018
Theory of corticothalamic brain activity in a spherical geometry: Spectra, coherence, and correlation
Publisher: American Physical Society (APS)
Date: 21-11-2017
An emergent autonomous flow for mean-field spin glasses
Publisher: Springer Science and Business Media LLC
Date: 07-05-2021
Determination of effective brain connectivity from activity correlations
Publisher: American Physical Society (APS)
Date: 15-04-2019
A Large Deviation Principle and an Expression of the Rate Function for a Discrete Stationary Gaussian Process
Publisher: MDPI AG
Date: 22-12-2014
DOI: 10.3390/E16126722
Stochastic Hybrid Systems in Cellular Neuroscience
Publisher: Springer Science and Business Media LLC
Date: 22-08-2018
A large deviation principle for networks of rate neurons with correlated synaptic weights
Publisher: Springer Science and Business Media LLC
Date: 07-2013
Synchronization in Stochastic Biochemical Oscillators Subject to Common Multiplicative Extrinsic Noise
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2021
DOI: 10.1137/20M1332402
A variational method for analyzing stochastic limit cycle oscillators
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2018
DOI: 10.1137/17M1155235
A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator
Publisher: MDPI AG
Date: 22-12-2014
DOI: 10.3390/E16126705
The buckling of capillaries in solid tumours
Publisher: The Royal Society
Date: 10-10-2012
Abstract: We develop a model of the buckling (both planar and axial) of capillaries in cancer tumours, using nonlinear solid mechanics. The compressive stress in the tumour interstitium is modelled as a consequence of the rapid proliferation of the tumour cells, using a multiplicative decomposition of the deformation gradient. In turn, the tumour cell proliferation is determined by the oxygen concentration (which is governed by the diffusion equation) and the solid stress. We apply a linear stability analysis to determine the onset of mechanical instability, and the Liapunov–Schmidt reduction to determine the postbuckling behaviour. We find that planar modes usually go unstable before axial modes, so that our model can explain the buckling of capillaries, but not as easily their tortuosity. We also find that the inclusion of anisotropic growth in our model can substantially affect the onset of buckling. Anisotropic growth also results in a feedback effect that substantially affects the magnitude of the buckle.
A general framework for stochastic traveling waves and patterns, with application to neural field equations
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2016
DOI: 10.1137/15M102856X
Wandering bumps in a stochastic neural field: A variational approach
Publisher: Elsevier BV
Date: 05-2020
Related Organisations
Inria Centre De Recherche Sophia Antipolis Méditerranée
Location: France
New Jersey Institute Of Technology
Location: United States of America
University Of Oxford
Location: United Kingdom of Great Britain and Northern Ireland
Related Funding Activities
No related grants have been discovered for James MacLaurin.