Analysis and approximation of stochastic nerve axon equations

Sauer, Martin; Stannat, Wilhelm

doi:10.1090/mcom/3068

Cited by 15 publications

(13 citation statements)

References 33 publications

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“…This follows directly from the discussion in Section 4, taking into account the decomposition (33) It is easy to verify that if (S ρ ) holds for F and G separately with excess regularity F ρ resp. G ρ , then a version of (S ρ ) holds for F + G as well, with excess regularity min( F ρ , G ρ ).…”

Section: Robustness Under Model Uncertaintymentioning

confidence: 67%

Drift estimation for stochastic reaction-diffusion systems

Pasemann¹,

Stannat²

2020

Electron. J. Statist.

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A parameter estimation problem for a class of semilinear stochastic evolution equations is considered. Conditions for consistency and asymptotic normality are given in terms of growth and continuity properties of the nonlinear part. Emphasis is put on the case of stochastic reaction-diffusion systems. Robustness results for statistical inference under model uncertainty are provided.Note that the derivation ofθ N is purely heuristic, so asymptotic properties of the estimator cannot be simply derived from the general theory of maximum likelihood estimation (as presented e.g. in [18]).

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Section: Robustness Under Model Uncertaintymentioning

confidence: 67%

Drift estimation for stochastic reaction-diffusion systems

Pasemann¹,

Stannat²

2020

Electron. J. Statist.

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“…However, there are additional new phenomena possible if we consider systems for d ≥ 2 such as spiral-like structures [167,44]. It is natural to conjecture that spiral-like waves can be found if we perturb the classical models for spiral waves such as the FitzHugh-Nagumo [57,134] equation, the Barkley model [11], or the Oregonator system by noise [15,17,110,160,167].…”

Section: Discussionmentioning

confidence: 99%

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Kuehn

2019

Jahresber. Dtsch. Math. Ver.

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In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.

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“…Of course, going beyond steady states is important and in the last few decades, the cases of travelling waves and moving interfaces for SPDEs have taken center stage. The research on this topic started in the early 1980s [60] and it has been growing quickly in recent years particularly for the Fisher-KPP equation [48,21,15,11,47], Nagumo-type SPDEs [35,14,27,23,28,44], neural field integro-differential equations [30,34,61,41,45], ecology [16] as well as regarding associated computational tools [42,62,59]. For more detailed surveys including a larger-scale view of the literature on the effect of noise on travelling waves we refer to [50,55,37].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Rotating Waves

Kuehn¹,

MacLaurin²,

Zucal³

2021

Preprint

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Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are travelling wave patterns occurring in stochastic partial differential equations (SPDEs), i.e., how deterministic travelling waves behave under stochastic perturbations. In this paper, we start the mathematical study of related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic dynamics PDE techniques with methods from stochastic analysis. We establish two different approaches, the variational phase and the approximated variational phase, for defining stochastic phase variables along the rotating wave, which track the effect of noise on neutral spectral modes associated to the special Euclidean symmetry group of rotating waves. Furthermore, we prove transverse stability results for rotating waves showing that over certain time scales and for small noise, the stochastic rotating wave stays close to its deterministic counterpart.

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Analysis and approximation of stochastic nerve axon equations

Cited by 15 publications

References 33 publications

Drift estimation for stochastic reaction-diffusion systems

Drift estimation for stochastic reaction-diffusion systems

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Stochastic Rotating Waves

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