Pulsatile Localized Dynamics in Delayed Neural Field Equations in Arbitrary Dimension

Faye, Grégory; Touboul, Jonathan

doi:10.1137/140955458

Cited by 4 publications

(5 citation statements)

References 49 publications

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“…( 17) is heuristic, it has proven to be remarkably effective at describing the oscillatory dynamics of networks of spiking neurons with strong noise [7][8][9][10][11][12][13][14], and is a paradigmatic mean-field model to investigate the effect of various types of delays in neuronal networks, see e.g. [15][16][17][18][19][20][21][22][23][24][25].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…The study of large networks of spiking neurons with time delays is convoluted, and in this context the mathematical and numerical analysis of firing rate descriptions have been particularly productive, see e.g. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Yet, how much of the actual dynamics of a large network of spiking neurons with synaptic delays can be captured using traditional firing rate descriptions?…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of a large system of spiking neurons with synaptic delay

2018

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We analyze a large system of heterogeneous quadratic integrate-and-fire (QIF) neurons with time delayed, all-to-all synaptic coupling. The model is exactly reduced to a system of firing rate equations that is exploited to investigate the existence, stability and bifurcations of fully synchronous, partially synchronous, and incoherent states. In conjunction with this analysis we perform extensive numerical simulations of the original network of QIF neurons, and determine the relation between the macroscopic and microscopic states for partially synchronous states. The results are summarized in two phase diagrams, for homogeneous and heterogeneous populations, which are obtained analytically to a large extent. For excitatory coupling, the phase diagram is remarkably similar to that of the Kuramoto model with time delays, although here the stability boundaries extend to regions in parameter space where the neurons are not self-sustained oscillators. In contrast, the structure of the boundaries for inhibitory coupling is different, and already for homogeneous networks unveils the presence of various partially synchronized states not present in the Kuramoto model: Collective chaos, quasiperiodic partial synchronization (QPS), and a novel state which we call modulated-QPS (M-QPS). In the presence of heterogeneity partially synchronized states reminiscent to collective chaos, QPS and M-QPS persist. In addition, the presence of heterogeneity greatly amplifies the differences between the incoherence stability boundaries of excitation and inhibition. Finally, we compare our results with those of a traditional (Wilson Cowan-type) firing rate model with time delays. The oscillatory instabilities of the traditional firing rate model qualitatively agree with our results only for the case of inhibitory coupling with strong heterogeneity. arXiv:1809.00607v2 [nlin.AO]

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of a large system of spiking neurons with synaptic delay

2018

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“…In addition, we plan to explore the impact of delays on propagating patterns, such as traveling waves [45]. It is questionable whether or not delays will make wave propagation more reliable, since it may lead to instabilities, as in [12,13].…”

Section: Discussionmentioning

confidence: 99%

“…The fact that (12) holds can be seen by differentiating the system (7) and using integration by parts to rearrange the integral terms. Similar results have been founded in linear stability analyses of non-delayed neural field equations, and they typically imply that perturbations that translate solutions in precisely this way will neither grow nor decay [37,38,27].…”

Section: Impact Of Delays On Bump Stabilitymentioning

confidence: 99%

“…Many studies have explored the impact of delays on the resulting spatiotemporal solutions of these equations [8,9,10]. One common observation is that the inclusion of delays can lead to oscillations via a Hopf bifurcation in the linear system describing the local stability of solutions to the delay-free system: Turing patterns [10], stationary pulses [11,12], and traveling waves [6,13]. Thus, a major finding across many studies of delayed neural field equations is that delay will tend to contribute to instabilities in stationary states [14].…”

Section: Introductionmentioning

confidence: 99%

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Delay stabilizes stochastic motion of bumps in layered neural fields

Kilpatrick

2015

Physica D: Nonlinear Phenomena

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We study the effects of propagation delays on the stochastic dynamics of bumps in neural fields with multiple layers. In the absence of noise, each layer supports a stationary bump. Using linear stability analysis, we show that delayed coupling between layers causes translating perturbations of the bumps to decay in the noise-free system. Adding noise to the system causes bumps to wander as a random walk. However, coupling between layers can reduce the variability of this stochastic motion by canceling noise that perturbs bumps in opposite directions. Delays in interlaminar coupling can further reduce variability, since they couple bump positions to states from the past. We demonstrate these relationships by deriving an asymptotic approximation for the effective motion of bumps. This yields a stochastic delay-differential equation where each delayed term arises from an interlaminar coupling. The impact of delays is well approximated by using a small delay expansion, which allows us to compute the effective diffusion in bumps' positions, accurately matching results from numerical simulations.Comment: 21 pages, 7 figures; To appear in Physica

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Traveling waves and breathers in an excitatory-inhibitory neural field

Folias

2017

Phys. Rev. E

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We study existence and stability of traveling activity bump solutions in an excitatory-inhibitory (E-I) neural field with Heaviside firing rate functions by deriving existence conditions for traveling bumps and an Evans function to analyze their spectral stability. Subsequently, we show that these existence and stability results reduce, in the limit of wave speed c→0, to the equivalent conditions developed for the stationary bump case. Using the results for the stationary bump case, we show that drift bifurcations of stationary bumps serve as a mechanism for generating traveling bump solutions in the E-I neural field as parameters are varied. Furthermore, we explore the interrelations between stationary and traveling types of bumps and breathers (time-periodic oscillatory bumps) by bridging together analytical and simulation results for stationary and traveling bumps and their bifurcations in a region of parameter space. Interestingly, we find evidence for a codimension-2 drift-Hopf bifurcation occurring as two parameters, inhibitory time constant τ and I-to-I synaptic connection strength w[over ¯]_{ii}, are varied and show that the codimension-2 point serves as an organizing center for the dynamics of these four types of spatially localized solutions. Additionally, we describe a case involving subcritical bifurcations that lead to traveling waves and breathers as τ is varied.

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Pulsatile Localized Dynamics in Delayed Neural Field Equations in Arbitrary Dimension

Cited by 4 publications

References 49 publications

Dynamics of a large system of spiking neurons with synaptic delay

Dynamics of a large system of spiking neurons with synaptic delay

Delay stabilizes stochastic motion of bumps in layered neural fields

Traveling waves and breathers in an excitatory-inhibitory neural field

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