Wandering bumps in a stochastic neural field: A variational approach

MacLaurin, James; Bressloff, Paul C.

doi:10.1016/j.physd.2020.132403

Cited by 17 publications

(25 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This behaviour resembles the phenomena appearing in the classical Kuramoto model [23,34,1,10,13] for synchronisation and other neural field models [39] in the computational neuroscience literature. It is shown that the homogeneous in space stationary state is linearly unstable for small noise strength, similarly to basic ring and neural field models [25,21,8]. We numerically analysed the bifurcation diagram of stationary patterns showing the appearance of different branches identified by their symmetries, see Fig.…”

Section: Discussionmentioning

confidence: 91%

See 1 more Smart Citation

Noise-driven bifurcations in a neural field system modelling networks of grid cells

Carrillo¹,

Holden²,

Solem³

2021

Preprint

View full text Add to dashboard Cite

The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by upscaling a noisy grid cell model to a system of partial differential equations in order to analyse the robustness of network activity patterns with respect to noise. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We prove that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.

show abstract

Section: Discussionmentioning

confidence: 91%

“…In general, works on the understanding of noisy neural field models have been lacking [4,Sec. 6] until very recently [21,20,25,7,39,38,5,8]. However, theoretical foundations are still sparse.…”

Section: Introductionmentioning

confidence: 99%

Noise-driven bifurcations in a neural field system modelling networks of grid cells

Carrillo¹,

Holden²,

Solem³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…Furthermore biology is typically very noisy, and thus it is of great importance to understand the effect of stochasticity on these patterns and waves [75,66,79]. The literature on stochastic patterns and waves includes general Turing patterns [10], the Allen-Cahn / Cahn-Hilliard equation [43], waves and patterns in the stochastic Brusselator [8,9], patterns in neural fields [49,38,56,41,85,2,62,16,70], interfaces in the Ginzburg-Landau equation [13,48], the stochastic burger's equation [12] and the effect of spatially-distributed noise on traveling waves [72,1,23], such as the FKPP traveling waves [29,20], invasion waves in ecology [64], the stochastic Nagumo equation [59,47,39], geometric waves [90] and numerical methods for stochastic traveling waves [68]. Good reviews of the literature on the effect of noise on traveling waves can be found in [75,79,60].…”

Section: Introductionmentioning

confidence: 99%

“…• The bound on the growth of the error in [50,Corollary 6.3] is suboptimal. This bound was greatly improved in [70] for 'bumps' of activity in stochastic neural fields. In this work we use a similar method to [70] to show that the probability of the system leaving the manifold of translated bump solutions over an exponentially long period of time (i.e.…”

Section: Introductionmentioning

confidence: 99%

Metastability of Waves and Patterns Subject to Spatially-Extended Noise

MacLaurin

2020

Preprint

View full text Add to dashboard Cite

In this paper we present a general framework in which one can rigorously study the effect of spatio-temporal noise on traveling waves, stationary patterns and oscillations that are invariant under the action of a finite-dimensional set of continuous isometries (such as translation or rotation). This formalism can accommodate patterns, waves and oscillations in reactiondiffusion systems and neural field equations. To do this, we define the phase by precisely projecting the infinite-dimensional system onto the manifold of isometries. Two differing types of stochastic phase dynamics are defined: (i) a variational phase, obtained by insisting that the difference between the projection and the original solution is orthogonal to the nondecaying eigenmodes, and (ii) an isochronal phase, defined as the limiting point on manifold obtained by taking t → ∞ in the absence of noise. We outline precise stochastic differential equations for both types of phase. The variational phase SDE is then used to show that the probability of the system leaving the attracting basin of the manifold after an exponentially long period of time (in ǫ −2 , the magnitude of the noise) is exponentially unlikely. In the case that the manifold is periodic (such as for spiral waves, spatially-distributed oscillations, or neural-field phenomena on a compact domain), the isochronal phase SDE is used to determine asymptotic limits for the average occupation times of the phase as it wanders in the basin of attraction of the manifold over very long times. In particular, we find that frequently the correlation structure of the noise biases the wandering in a particular direction, such that the noise induces a slow oscillation that would not be present in the absence of noise.

show abstract

Wandering bumps in a stochastic neural field: A variational approach

Cited by 17 publications

References 33 publications

Noise-driven bifurcations in a neural field system modelling networks of grid cells

Noise-driven bifurcations in a neural field system modelling networks of grid cells

Stochastic Rotating Waves

Metastability of Waves and Patterns Subject to Spatially-Extended Noise

Contact Info

Product

Resources

About