Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

Avitable, Daniele; Wedgwood, Kyle C. A.

doi:10.1007/s00285-016-1070-9

Cited by 13 publications

(12 citation statements)

References 76 publications

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“…The corresponding eigenvalues of the linearisation of equation (19), which involves the flow over time P, are thus clustered around minus unity and the Krylov iterations tend to converge rapidly. Our application of pseudo-spectral discretisation and Newton-Krylov continuation to a dynamical system with spatial interactions described by convolutions is similar to the work of Avitabile et al on a neural mass model [51,6]. In their model, the convolution represents the integration of synaptic signals originating from a neighbourhood of a given neuron, weighed by the connection density.…”

Section: Continuation Algorithmmentioning

confidence: 95%

Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

Buono

Eftimie

Kovacic

2019

Active Particles, Volume 2

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In this study we start by reviewing a class of 1D hyperbolic/kinetic models (with two velocities) used to investigate the collective behaviour of cells, bacteria or animals. We then focus on a restricted class of nonlocal models that incorporate various inter-individual communication mechanisms, and discuss how the symmetries of these models impact the various types of spatially-heterogeneous and spatially-homogeneous equilibria exhibited by these nonlocal models. In particular, we characterise a new type of equilibria that was not discussed before for this class of models, namely a relative equilibria. Then we simulate numerically these models and show a variety of spatio-temporal patterns (including classic equilibria and relative equilibria) exhibited by these models. We conclude by introducing a continuation algorithm (which takes into account the models symmetries) that allows us to track the solutions bifurcating from these different equilibria. Finally, we apply this algorithm to identify a D 3 -symmetric steady-state solution.

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Section: Continuation Algorithmmentioning

confidence: 95%

Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

Buono

Eftimie

Kovacic

2019

Active Particles, Volume 2

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“…(ii) For all u ∈ E there exists the second Frechet Derivative D (2) f (u) ∈ L(H × H, H) such that for all v, w ∈ H,…”

Section: General Assumptionsmentioning

confidence: 99%

“…It can be shown (see [77,Section 3.1] or [35]) that there exists (again under some conditions on the parameters) a smooth function 2 and speed c ∈ R such that U is a solution to (3.9). Moreover û and v are both smooth functions whose derivatives are all bounded and in L 2 (R).…”

Section: Traveling Pulses In Neural Fieldsmentioning

confidence: 99%

“…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%

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Metastability of Waves and Patterns Subject to Spatially-Extended Noise

MacLaurin

2020

Preprint

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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.

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“…The low dimensional attractor is the wave function of the cobit -the "shape" of the oscillations that represent the probability of the cobit being observed in these states. This cortical unit of information allows for representation of amplitudes in single bumps, multiple bumps or as a regular field of bumps with the potential for a rich spectrum of dynamics (Wang et al, 2015;Avitable, 2017). Remember that underlying the normalized probabilities of the system are the positive and negative complex number amplitudes input to the dendritic computations.…”

Section: Part 5: a Sketch Of The Logical Primitivementioning

confidence: 99%

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Jones¹

2020

Preprint

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A model of computation describes how units of information and an organizational architecture transform input to output. In the cerebral cortex the unit of information is a normalized collection of neuronal membrane potentials representing a vector of complex amplitudes. The organizational architecture of the cortex allows for programming operator matrices in dendritic arborizations on vectors of complex amplitudes. Three well-documented facts are sufficient to define the cortical model of computation: first, cortical dendritic inputs to compute membrane potentials are, in general, complex numbers (amplitudes) that may be configured as state vectors in complex Hilbert spaces; second, normalization is a canonical neural computation, specifically, for vectors of cortical membrane amplitudes representing orthogonal states; and, third, dendritic arborizations are programmable devices in a universal sense for mathematical and logical functions. These three facts, well studied and accepted for years, are all that is needed for a simple, well-defined, and well-known model of computation. A great deal of research is presently devoted to understanding particular neural algorithms, circuits, and programming. Understanding the algorithm that a particular circuit implements is not to be confused with identifying the much more basic underlying model of computation. The analogy for classical computing is that the classical model of computing stores and manipulates information fundamentally in the form of bits – essentially a finite set of possible states, typically, {0,1}. The primitive classical container of information is analogous to a two-state switch. Nevertheless, sophisticated computational circuitry and algorithms are developed on that simple classical realization. In contrast to the finite set of states found in the fundamental unit of classical information, the fundamental units of information in the cerebral cortex form a model of computation over the complex field in normalized sets of mutually inhibitory amplitudes. This means that the cortex represents information in a fundamental container of information (a vector of amplitudes in membrane potentials) that can encode discrete states over a continuum of values (no doubt down to some constant.) A well-known result from probability theory and standard neuroimaging analysis argue that these mutually inhibitory fundamental units of information may normalize complex amplitudes under the 2-norm. Sufficiently sophisticated computational means are available in the cortex to program unitary operators, state reduction and expansion operators and tensor products that operate on vectors of normalized membrane amplitudes. This model of computation is typically viewed as a probabilistic or predictive model of computation. Again, the model of computation is not a particular circuit or implementation of any one algorithm. The model of computation is the arena in which cortical circuits and algorithms are implemented. The fundamental arena then for cortical computations is a field of complex amplitudes normalized for probabilistic computation.

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Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

Cited by 13 publications

References 76 publications

Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

Metastability of Waves and Patterns Subject to Spatially-Extended Noise

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

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