Existence and Stability of Traveling Pulses in a Neural Field Equation with Synaptic Depression

Faye, Grégory

doi:10.1137/130913092

Cited by 26 publications

(47 citation statements)

References 44 publications

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“…Using a specific connectivity kernel can sound very restrictive, but it has proven in other contexts its efficiency to overcome the difficulty of the nonlocal nature of the problem while still gaining some general insights. In particular, we refer to some recent works on the existence and stability of traveling pulses in neural field equations with synaptic depression or on some pinning and unpinning phenomena in nonlocal systems [3,15] where kernels with rational Fourier transform have been used to reduced the problem to a high-order system of partial differential equations. Figure 1: A modulated traveling front obtained from direct numerical simulation of (1.1) with kernel (1.8) for a = 0.7 and µ = 32.…”

Section: Introductionmentioning

confidence: 99%

Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach

Faye

Holzer

2015

Journal of Differential Equations

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We consider a nonlocal generalization of the Fisher-KPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are two-fold. First, we prove the existence of a two-parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions. We also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co-periodic perturbations. Secondly, we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation. In this context, we prove the existence of modulated traveling fronts near the Turing bifurcation that describe the invasion of the Turing unstable homogeneous state by the periodic pattern established in the first part. Both results rely on a center manifold reduction to a finite dimensional ordinary differential equation.

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Section: Introductionmentioning

confidence: 99%

Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach

Faye

Holzer

2015

Journal of Differential Equations

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“…The main setback in our problem is that the Heaviside firing rate function creates phase space dynamics with discontinuities; it is important that this issue is properly dealt with in order to apply such a technical result. Some promising partial results have been obtained in [22,23] for related models with smooth firing rate functions.…”

Section: Existence Of Traveling Pulsesmentioning

confidence: 99%

Traveling wave solutions to a neural field model with oscillatory synaptic coupling types

Dyson

2019

Mathematical Biosciences and Engineering

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In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. In turn, we hope to analyze traveling fronts and pulses with novel shapes. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits; hence, our work answers open problems and provides insight into new ones.

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“…We note that because of the symmetry of u 0 (x), the functions, H i (θ) have a similar symmetry which we will exploit in the analysis of Equation (8).…”

Section: Figurementioning

confidence: 99%

Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation

Park¹,

Ermentrout²

2018

SIAM J. Appl. Dyn. Syst.

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We study a deterministic version of a one-and two-dimensional attractor neural network model of hippocampal activity first studied by Itskov et al 2011. We analyze the dynamics of the system on the ring and torus domain with an even periodized weight matrix, assuming weak and slow spike frequency adaptation and a weak stationary input current. On these domains, we find transitions from spatially localized stationary solutions ("bumps") to (periodically modulated) solutions ("sloshers"), as well as constant and non-constant velocity traveling bumps depending on the relative strength of external input current and adaptation. The weak and slow adaptation allows for a reduction of the system from a distributed partial integro-differential equation to a system of scalar Volterra integro-differential equations describing the movement of the centroid of the bump solution. Using this reduction, we show that on both domains, sloshing solutions arise through an Andronov-Hopf bifurcation and derive a normal form for the Hopf bifurcation on the ring. We also show existence and stability of constant velocity solutions on both domains using Evans functions. In contrast to existing studies, we assume a general weight matrix of Mexican-hat type in addition to a smooth firing rate function.

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Existence and Stability of Traveling Pulses in a Neural Field Equation with Synaptic Depression

Cited by 26 publications

References 44 publications

Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach

Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach

Traveling wave solutions to a neural field model with oscillatory synaptic coupling types

Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation

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