Reliability of signal transmission in stochastic nerve axon equations

Sauer, Martin; Stannat, Wilhelm

doi:10.1007/s10827-015-0586-0

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…A very important future direction for research will be to connect waves/patterns for SPDEs more closely to applications [62]. Stochastic wave-like structures in SPDEs have already appeared in an extremely diverse set of modelling contexts such as neuroscience [161,180,181], spin glasses [30], biological invasions [170], predator-prey systems [89,169], directed polymers [27], evolutionary biology [38,73], and epidemics [115,185]. Although small fluctuations in the modelling context are sometimes just neglected, this is generally a false hypothesis near instability, e.g., when we are close to propagation failure or when the deterministic PDE part undergoes a bifurcation.…”

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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Section: Discussionmentioning

confidence: 99%

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Kuehn

2019

Jahresber. Dtsch. Math. Ver.

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“…Also the SODE variant for ν = 0 is quite well studied, mainly due to a flurry of activity since the mid 1990s; see, for example, [3,7,9,63,64,72]. Yet, the full SPDE variant (1.1) has only attracted major attention quite recently, including a large number of numerical studies [67,80,81,83,[86][87][88] as well as analytical studies regarding existence, regularity, invariant measures and attractors [4,8,12,61,62,90]. The question regarding stochastic stability of pulses for additive noise is far less studied.…”

Section: Introductionmentioning

confidence: 99%

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

2022

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We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat (Nonlinear Anal. 162 (2017) 197-223) for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications. CONTENTS

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“…The αβ dimer polymerizes at the end of microtubules and has a certain dynamic nature that allows Tubulin to extend forward and produce a characteristic tubular fiber structure that confers cell polarity. Because of this characteristic, Tubulin produces isolated domains in cells, mediates the process of nerve differentiation, such as axonal growth, synaptic activity, and cell polarity [6], and plays a guiding role in intracellular material transport [7]. Proteomics and animal depression model studies have also shown that there are changes in the scaffold microtubule system in MDD animal models.…”

Section: Introductionmentioning

confidence: 99%

Long-term fluoxetine administration induces changes in microtubule plasticity mediated by CRMP2 in differentiated PC12 cells

Wu¹,

Wang²,

Xiao³

et al. 2019

Preprint

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Backgroud Depression is a neuropsychiatric disease with a high risk of relapse. Clinical guidelines advocate whole course of antidepressant treatment, but the relapse rate of MDD is still high. Methods We intended to study the changes in microtubule scaffold plasticity induced by long-term antidepressant administration in PC12 cell to explore the possible mechanism of depression relapse. Results The cell activity of NC group decreased after 3 days of culture. The results of immunofluorescence showed the extension of processes was obvious on the first day in Flu group and NC group, but neurite connections were inhibited on the third day in Flu group.CRMP2 and Tubulin were co-located by IF, and Co-IP confirmed the interaction between CRMP2 and Tubulin. RT-PCR showed the mRNA expression of CRMP2 and Tubulin increased on the first and second day in Flu group, but decreased on the third day. Similarly, Western Blotting showed the contents of CRMP2 and Tubulin protein in Flu group increased on the first day, but decreased on the third day. The expression of CRMP2 and Tubulin was affected by changes in the activity of CRMP2. Conclusions The long-term fluoxetine administration can affect the microtubule scaffold plasticity in PC12 cell, and CRMP2 is involved in this process. This may be one of the biological bases of the high risk for depression relapse induced by long-term antidepressant administration.

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Reliability of signal transmission in stochastic nerve axon equations

Cited by 9 publications

References 14 publications

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

Long-term fluoxetine administration induces changes in microtubule plasticity mediated by CRMP2 in differentiated PC12 cells

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