On the properties of nonlinear nonlocal operators arising in neural field models

Oleynik, Anna; Ponosov, Arcady; Wyller, John

doi:10.1016/j.jmaa.2012.08.063

Cited by 17 publications

(31 citation statements)

References 21 publications

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“…where M n and L n are 2 × 2 matrices given by (29) and (30), respectively. Standard theory shows that the eigenvalues of the problem (36) are the roots of the quadratic equations…”

Section: General Frameworkmentioning

confidence: 99%

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Malyutina

Wyller

Ponosov

2014

Physica D: Nonlinear Phenomena

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We study existence and stability of 2-bump solutions of the one-population homogenized Wilson-Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2-bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. Numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2-bump solution which is stable for all admissible values of the heterogeneity parameter.

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“…where M n and L n are 2 × 2 matrices given by (29) and (30), respectively. Standard theory shows that the eigenvalues of the problem (36) are the roots of the quadratic equations…”

Section: General Frameworkmentioning

confidence: 99%

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Malyutina

Wyller

Ponosov

2014

Physica D: Nonlinear Phenomena

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“…While this conjecture is supported by numerical simulations (see for example [25]), there are few and far between the works addressing this problem in a rigorous mathematical way. We believe, however, that this problem can be dealt with methods of nonlinear functional analysis and degree theory in a way analogous to Oleynik et al [36] and singular perturbation analysis in a similar way as in Yousaf et al [37]. We do not pursue this problem here, however, but will return to it in future investigations.…”

Section: Discussionmentioning

confidence: 98%

“…The connectivity functions and the external input functions are given as gaussians (5), (6) and (7), respectively. The input parameters used here are given by set A in Table 1 and hence the pulse width coordinates of the initial condition are given by (36). Fig.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Effect of localized input on bump solutions in a two-population neural-field model

Yousaf¹,

Wyller²,

Tetzlaff³

et al. 2013

Nonlinear Analysis: Real World Applications

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We investigate a two-population Wilson-Cowan model extended with stationary and spatially dependent localized external inputs and study the existence and stability of localized stationary (bump) solutions. The generic situation for this model in the absence of external inputs corresponds to two bump pairs, one narrow and one broad pair. For spatially wide localized external inputs we find this generic picture to be unchanged. However, for strongly localized external inputs we find that three and even four bump pairs, all with symmetric activity profiles around the center of the localized external input, may coexist. We next investigate the stability of these bump pairs using two different techniques: a simplified phase-space reduction (Amari) technique and full stability analysis. Examples of models, i.e., choices of model parameters, exhibiting up to three stable bump pairs are found. The Amari technique is further found to be a poor predictor of stability in the case of strongly localized external inputs. The bump-pair states are also probed numerically using a fourth order Runge-Kutta method, and an excellent agreement is found between numerical simulations and the analytical predictions from full stability analysis.

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“…In a recent work by Oleynik et al [27] the justification of the unit step approximation of firing rate functions has been investigated by means of methods that are different from classic SPA (i.e. by nonlinear functional analysis and the degree theory).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…by nonlinear functional analysis and the degree theory). In a future investigation we intend to combine the methods of SPA with some ideas from Oleynik et al [27].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Neural firing rate model with a steep firing rate function

Yousaf

Ponosov

Wyller

et al. 2013

Nonlinear Analysis: Real World Applications

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In this study we justify rigorously the approximation of the steep firing rate functions with a unit step function in a two-population neural firing rate model with steep firing rate functions. We do this justification by exploiting the theory of switching dynamical systems. It has been demonstrated that switching dynamics offer a possibility of simplifying the dynamical system and getting approximations of the solution of the system for any specific choice of parameters. In this approach the phase space of the system is divided into regular and singular domains, where the limit dynamics can be written down explicitly. The advantages of this method are illustrated by number of numerical examples for different cases of the singular domains (i.e. for black, white and transparent walls) and for specific choices of parameters involved. General conditions have been formulated on these parameters to give black, white and transparent walls. Further, the existence and stability of regular and singular stationary points have been investigated. It has been shown that the regular stationary points (i.e. stationary points inside the regular domains) are always stable and this property is preserved for smooth and sufficiently steep activation functions. In the most technical part of the paper we have provided conditions on the existence and stability of singular stationary points (i.e. those belonging to the singular domains). For the * Corresponding Author: Tel. +47 64965489; fax.+47 64965401.Email existence results, the implicit function theorem has been used, whereas the stability of singular stationary points is addressed by applying the singular perturbation analysis and the Tikhonov theorem.

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On the properties of nonlinear nonlocal operators arising in neural field models

Cited by 17 publications

References 21 publications

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Effect of localized input on bump solutions in a two-population neural-field model

Neural firing rate model with a steep firing rate function

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