A General Framework for Stochastic Traveling Waves and Patterns, with Application to Neural Field Equations

Inglis, James; MacLaurin, James

doi:10.1137/15m102856x

Cited by 35 publications

(58 citation statements)

References 35 publications

(65 reference statements)

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“…Perturbation theory can then be used to derive an explicit stochastic differential equation (SDE) for the diffusive-like wandering of the bump in the weak noise regime. (A more rigorous mathematical treatment that provides bounds on the size of transverse fluctuations has also been developed [15,45].) In this paper, we apply the theory of wandering bumps in stochastic neural fields in order to characterize various features of stimulus-dependent variability in cortical neurons.…”

Section: Wandering Bumps and Neural Variabilitymentioning

confidence: 99%

Stochastic neural field model of stimulus-dependent variability in cortical neurons

Bressloff

2019

Preprint

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We use stochastic neural field theory to analyze the stimulus-dependent tuning of neural variability in ring attractor networks. We apply perturbation methods to show how the neural field equations can be reduced to a pair of stochastic nonlinear phase equations describing the stochastic wandering of spontaneously formed tuning curves or bump solutions. These equations are analyzed using a modified version of the bivariate von Mises distribution, which is well-known in the theory of circular statistics. We first consider a single ring network and derive a simple mathematical expression that accounts for the experimentally observed bimodal (or M-shaped) tuning of neural variability. We then explore the effects of inter-network coupling on stimulus-dependent variability in a pair of ring networks. These could represent populations of cells in two different layers of a cortical hypercolumn linked via vertical synaptic connections, or two different cortical hypercolumns linked by horizontal patchy connections within the same layer. We find that neural variability can be suppressed or facilitated, depending on whether the inter-network coupling is excitatory or inhibitory, and on the relative strengths and biases of the external stimuli to the two networks. These results are consistent with the general observation that increasing the mean firing rate via external stimuli or modulating drives tends to reduce neural variability.A topic of considerable current interest concerns the neural mechanisms underlying the suppression of cortical variability following the onset of a stimulus. Since trial-by-trial variability and noise correlations are known to affect the information capacity of neurons, such suppression could improve the accuracy of population codes. One of the main candidate mechanisms is the suppression of noise-induced transitions between multiple attractors, as exemplified by ring attractor networks. The latter have been used to model experimentally measured stochastic tuning curves of directionally selective middle temporal (MT) neurons. In this paper we show how the stimulus-dependent tuning of neural variability in ring attractor networks can be analyzed in terms of the stochastic wandering of spontaneously formed tuning curves or bumps in a continuum neural field model. The advantage of neural fields is that one can derive explicit mathematical expressions for the second-order statistics of neural activity, and explore how this depends on important model parameters, such as the level of noise, the strength of recurrent connections, and the input contrast. PLOS1/32 PLOS 2/32 Coupled ring modelConsider a pair of mutually coupled ring networks labeled j = 1, 2. Let u j (θ, t) denote the activity at time t of a local population of cells with stimulus preference θ ∈ [−π, π) in network j. Here θ could represent the direction preference of neurons in area-middle temporal cortex (MT) [63], the orientation preference of V1 neurons, after rescaling θ → θ/2 [8,9], or a coordinate in spatial working memory [17,47,54]...

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Section: Wandering Bumps and Neural Variabilitymentioning

confidence: 99%

Stochastic neural field model of stimulus-dependent variability in cortical neurons

Bressloff

2019

Preprint

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“…Indeed, perturbations are not able to grow even on short timescales, but always decay exponentially fast back to the wave. We do not use this property here, but it has played an essential role in many previous studies on stochastic waves [19,26].…”

Section: Introductionmentioning

confidence: 99%

“…Proof. The computation(5.25) shows thatb(Φ + v, γ) ,R) = χ h (Φ + V, γ) 2 T C (Φ + v A , γ A ) of (A 19…”

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confidence: 99%

Travelling waves for reaction–diffusion equations forced by translation invariant noise

Hamster

Hupkes

2020

Physica D: Nonlinear Phenomena

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Inspired by applications, we consider reaction-diffusion equations on R that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [16,17] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.

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“…The minimization then implicitly defines the position of the wave p = p(t) by computing a minimizer y for each t ≥ 0 and setting y =: p(t). However, one first has to guarantee that a minimizer exists [83]. Although it does exist under reasonable conditions, it does not have to be unique.…”

Section: Stochastic Wavesmentioning

confidence: 99%

“…vanishes, the local minimizer becomes degenerate, so we have a criterion to test for potential jumps. Now, we have essentially given another implicit definition [117,83,100] of the wave speed for a stochastic travelling wave and we have chosen p = p(t) as moving-frame coordinates for its measurement. Next, one may ask, whether there is a differential equation for p(t)?…”

Section: Stochastic Wavesmentioning

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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A General Framework for Stochastic Traveling Waves and Patterns, with Application to Neural Field Equations

Cited by 35 publications

References 35 publications

Stochastic neural field model of stimulus-dependent variability in cortical neurons

Stochastic neural field model of stimulus-dependent variability in cortical neurons

Travelling waves for reaction–diffusion equations forced by translation invariant noise

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

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