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58
Evans functions for integral neural field equations with Heaviside firing rate function
 SIAM Journal on Applied Dynamical Systems
, 2004
"... In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of s ..."
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Cited by 68 (7 self)
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In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.
Breathing Pulses in an Excitatory Neural Network
, 2004
"... In this paper we show how a local inhomogeneous input can stabilize a stationarypulse solution in an excitatory neural network. A subsequent reduction of the input amplitude can then induce a Hopf instability of the stationary solution resulting in the formation of a breather. The breather can it ..."
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Cited by 47 (8 self)
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In this paper we show how a local inhomogeneous input can stabilize a stationarypulse solution in an excitatory neural network. A subsequent reduction of the input amplitude can then induce a Hopf instability of the stationary solution resulting in the formation of a breather. The breather can itself undergo a secondary instability leading to the periodic emission of traveling waves. In one dimension such waves consist of pairs of counterpropagating pulses, whereas in two dimensions the waves are circular target patterns.
Existence and Stability of Standing Pulses in Neural Networks
 I. Existence. SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse soluti ..."
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Cited by 37 (2 self)
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Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse solutions coexist, the small pulse is unstable, and the large pulse is stable. The large single pulse is bistable with the “alloff ” state. This bistable localized activity may have strong implications for the mechanism underlying working memory. We show that dimple pulses have similar stability properties to large pulses but double pulses are unstable.
Spiral waves in nonlocal equations
 SIAM Journal on Applied Dynamical Systems
, 2005
"... Abstract. We present a numerical study of rotating spiral waves in a partial integrodifferential equation defined on a circular domain. This type of equation has been previously studied as a model for large scale pattern formation in the cortex and involves spatially nonlocal interactions through a ..."
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Cited by 36 (7 self)
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Abstract. We present a numerical study of rotating spiral waves in a partial integrodifferential equation defined on a circular domain. This type of equation has been previously studied as a model for large scale pattern formation in the cortex and involves spatially nonlocal interactions through a convolution. The main results involve numerical continuation of spiral waves that are stationary in a rotating reference frame as various parameters are varied. We find that parameters controlling the strength of the nonlinear drive, the strength of local inhibitory feedback, and the steepness and threshold of the nonlinearity must all lie within particular intervals for stable spiral waves to exist. Beyond the ends of these intervals, either the whole domain becomes active or the whole domain becomes quiescent. An unexpected result is that the boundaries seem to play a much more significant role in determining stability and rotation speed of spirals, as compared with reactiondiffusion systems having only local interactions.
Local/global analysis of the stationary solutions of some neural field equations
 SIAM J. Applied Dynamical Systems
"... Abstract. Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integrodifferential equations. The solutions of these ..."
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Cited by 29 (8 self)
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Abstract. Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integrodifferential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighbouring brain areas. Understanding the properties of these solutions is essential in advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviours of the solutions. We also provide a generic finite dimensional approximation of these equations which allows us to study in great details two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model [40]. The second model is a general neural field model where the spatial connectivity is described by heterogeneous Gaussianlike functions. 1. Introduction. Neural
Localized hexagon patterns of the planar SwiftHohenberg equation
 SIAM J. Appl. Dyn. Syst
"... We investigate stationary spatially localized hexagon patterns of the twodimensional Swift–Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized ..."
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Cited by 27 (8 self)
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We investigate stationary spatially localized hexagon patterns of the twodimensional Swift–Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the oneparameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region as expected from heuristic arguments.
Waves, bumps, and patterns in neural field theories
, 2005
"... Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integrodifferential equations. Their nonlocal nature has led to the development of a set of analytica ..."
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Cited by 24 (2 self)
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Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integrodifferential equations. Their nonlocal nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axodendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.
Analysis of nonlocal neural fields for both general and gammadistributed connectivities
, 2005
"... This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integrodifferential equation includes synaptic and spacedependent transmission delay for both general and ga ..."
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Cited by 22 (7 self)
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This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integrodifferential equation includes synaptic and spacedependent transmission delay for both general and gammadistributed synaptic connectivities. The latter connectivity type reveals infinite, finite, and vanishing selfconnectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDEtypes are derived from the examined model equation, namely the general reaction–diffusion equation and the Swift–Hohenberg equation. Hence, the discussed integrodifferential equation generalizes these PDEs. In the case of the gammadistributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition–lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results.
Neural fields with distributed transmission speeds and longrange feedback delays
, 2006
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Twodimensional bumps in piecewise smooth neural fields with synaptic depression
, 2011
"... We analyze radially symmetric bumps in a twodimensional piecewisesmooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights betwe ..."
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Cited by 14 (1 self)
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We analyze radially symmetric bumps in a twodimensional piecewisesmooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, sufficiently strong synaptic depression can destabilize a stationary bump solution that would be stable in the absence of depression. Numerically it is found that the resulting instability leads to the formation of a traveling spot. The local stability of a bump is determined by solutions to a system of pseudolinear equations that take into account the sign of perturbations around the circular bump boundary.