transient turing patterns in a neural field model we investigate turing bifurcations in a neural field model with one spatial dimension for some parameter values the resulting turing patterns are stable while for others the patterns appear transiently we show that this difference is due to the relative position in parameter space of the saddle node bifurcation of a spatially periodic pattern and the turing bifurcation point by varying parameters we are able to observe transient patterns whose duration scales in the same way as type i intermittency similar behavior occurs in two spatial dimensions

waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

two bump solutions of amari type models of neuronal pattern formation abstract we study a partial integro differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks for a one dimensional domain we develop criteria for the existence and stability of equal width 2 bump solutions under the assumption that the firing rate function is the heaviside function we apply these criteria to an example for which the connectivity is of lateral inhibition type i e the coupling function has one positive zero and find that families of 2 bump solutions exist but none of the solutions are stable extensive numerical searches suggest that this is true for all coupling functions of this form however for a large class of coupling functions which have three positive zeros we find the coexistence of both stable and unstable 2 bump solutions we also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1 bump and 2 bump solutions our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity

parallel exponential rosenbrock methods exponential rosenbrock integrators were shown to be very efficient in solving large systems of stiff ordinary differential equations so far such exponential methods have been derived up to order 5 the aim of this paper is to construct new integrators of orders 4 5 and 6 in contrast to the existing schemes the new schemes which are called parallel exponential rosenbrock integrators can be implemented on a multi processor system or parallel computers the new schemes of orders 4 and 5 require the same number of stages as the old schemes of the same orders of accuracy however while the parallel integrator of order 4 can be implemented with the same cost as a 2 stage method the ones of orders 5 and 6 can be implemented at the cost of a 3 stage method only this offers a significant improvement over the old schemes in terms of computational time when implemented in parallel the numerical experiments show the efficiency of the new integrators as well as the comparative performance with the old ones

nonlocal ginzburg landau equation for cortical pattern formation we show how a nonlocal version of the real ginzburg landau gl equation arises in a large scale recurrent network model of primary visual cortex we treat cortex as a continuous two dimensional sheet of cells that signal both the position and orientation of a local visual stimulus the recurrent circuitry is decomposed into a local part which contributes primarily to the orientation tuning properties of the cells and a long range part that introduces spatial correlations we assume that a the local network exists in a balanced state such that it operates close to a point of instability and b the long range connections are weak and scale with the bifurcation parameter of the dynamical instability generated by the local circuitry carrying out a perturbation expansion with respect to the long range coupling strength then generates a nonlocal coupling term in the gl amplitude equation we use the nonlocal gl equation to analyze how axonal propagation delays arising from the slow conduction velocities of the long range connections affect spontaneous pattern formation

piecewise homotopy methods for nonlinear ordinary differential equations abstract piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations these methods are based on the introduction of an artificial or book keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere three piecewise adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size are presented and applied to eight nonlinear ordinary differential equations it is shown that piecewise adaptive homotopy perturbation methods predict essentially the same solutions as matlab s variable step variable order solvers and variable order transition matrix techniques provided that five term approximations of the decomposition method are applied to both the displacement and the velocity it is also shown that piecewise homotopy perturbation techniques that use three term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second order accurate time linearization technique when the same step is employed in both schemes

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

the visual cortex as a crystal a theory of pattern formation in primary visual cortex v1 is presented that takes into account its crystalline like structure the cortex is partitioned into fundamental domains or hypercolumnsof a lattice describing the distribution of singularities or pinwheels in the orientation preference map each hypercolumn is modelled as a network of orientation and spatial frequency selective cells organised around a pair of pinwheels which are associated with high and low spatial frequency domains respectively the network topology of the hypercolumn is taken to be a sphere with the pinwheels located at the poles of the sphere interactions between hypercolumns are mediated by anisotropic long range lateral connections that link cells with similar feature preferences using weakly nonlinear analysis we investigate the spontaneous formation of cortical activity patterns through the simultaneous breaking of an internal o 3 symmetry and a discrete lattice symmetry the resulting patterns are characterised by states in which each hypercolumn exhibits a tuned response to both orientation and spatial frequency and the distribution of optimal responses across hypercolumns is doubly periodic or quasi periodic with respect to the underlying lattice 2002 published by elsevier science b v

traveling fronts and wave propagation failure in an inhomogeneous neural network we use averaging and homogenization theory to study the propagation of traveling wavefronts in an inhomogeneous excitable neural medium motivated by the functional architecture of primary visual cortex we model the inhomogeneity as a periodic modulation in the long range neuronal connections we derive an expression for the effective wavespeed and show that propagation failure can occur if the speed is too slow or the degree of inhomogeneity is too large we find that there are major qualitative differences in the wavespeed for different choices of the homogenized weight distribution

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

issues about retinex theory and contrast enhancement we present an interpretation of land s retinex theory that we show to be consistent with the original formulation the proposed model relies on the computation of the expectation value of a suitable random variable weighted with a kernel function thus the name kernel based retinex kbr for the corresponding algorithm kbr shares the same intrinsic characteristics of the original retinex it can reduce the effect of a color cast and enhance details in low key images but since it can only increase pixel intensities it is not able to enhance over exposed pictures comparing the analytical structure of kbr with that of a recent variational model of color image enhancement we are able to perform an analysis of the action of kbr on contrast showing the need to anti symmetrize its equation in order to produce a two sided contrast modification able to enhance both under and over exposed pictures the anti symmetrized kbr equations show clear correspondences with other existing color correction models in particular ace whose relationship with retinex has always been difficult to clarify finally from an image processing point of view we mention that both kbr and its antisymmetric version are free from the chromatic noise due to the use of paths in the original retinex implementation and that they can be suitably approximated in order to reduce their computational complexity from mathcal o n 2 to mathcal o n log n being n the number of input pixels

preconditioned implicit exponential integrators imexp for stiff pdes abstract we propose two new classes of time integrators for stiff des the implicit exponential imexp and the hybrid exponential methods in contrast to the existing exponential schemes the new methods offer significant computational advantages when used with preconditioners any preconditioner can be used with any of these new schemes this leads to a broader applicability of exponential methods the proof of convergence of these integrators and numerical demonstration of their efficiency are presented

multiple bumps in a neuronal model of working memory we study a partial integro differential equation defined on a spatially extended domain that arises from the modeling of working or short term memory in a neuronal network the equation is capable of supporting spatially localized regions of high activity which can be switched on and off by transient external stimuli we analyze the effects of coupling between units in the network showing that if the connection strengths decay monotonically with distance then no more thanon e regionof high activity canpersist whereas if they decay inanoscillatory fashion then multiple regions can persist

quantitative theory of driven nonlinear brain dynamics abstract strong periodic stimuli such as bright flashing lights evoke nonlinear responses in the brain and interact nonlinearly with ongoing cortical activity but the underlying mechanisms for these phenomena are poorly understood at present the dominant features of these experimentally observed dynamics are reproduced by the dynamics of a quantitative neural field model subject to periodic drive model power spectra over a range of drive frequencies show agreement with multiple features of experimental measurements exhibiting nonlinear effects including entrainment over a range of frequencies around the natural alpha frequency f subharmonic entrainment near 2 f and harmonic generation further analysis of the driven dynamics as a function of the drive parameters reveals rich nonlinear dynamics that is predicted to be observable in future experiments at high drive amplitude including period doubling bistable phase locking hysteresis wave mixing and chaos indicated by positive lyapunov exponents moreover photosensitive seizures are predicted for physiologically realistic model parameters yielding bistability between healthy and seizure dynamics these results demonstrate the applicability of neural field models to the new regime of periodically driven nonlinear dynamics enabling interpretation of experimental data in terms of specific generating mechanisms and providing new tests of the theory

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

stationary bumps in networks of spiking neurons we examine the existence and stability of spatially localized bumps of neuronal activity in a network of spiking neurons bumps have been proposed in mechanisms of visual orientation tuning the rat head direction system and working memory we show that a bump solution can exist in a spiking network provided the neurons fire asynchronously within the bump we consider a parameter regime where the bump solution is bistable with an all off state and can be initiated with a transient excitatory stimulus we show that the activity profile matches that of a corresponding population rate model the bump in a spiking network can lose stability through partial synchronization to either a traveling wave or the all off state this can occur if the synaptic timescale is too fast through a dynamical effect or if a transient excitatory pulse is applied to the network a bump can thus be activated and deactivated with excitatory inputs that may have physiological relevance

an amplitude equation approach to contextual effects in visual cortex a mathematical theory of interacting hypercolumns in primary visual cortex v1 is presented that incorporates details concerning the anisotropic nature of long range lateral connections each hypercolumn is modeled as a ring of interacting excitatory and inhibitory neural populations with orientation preferences over the range 0 to 180 degrees analytical methods from bifurcation theory are used to derive nonlinear equations for the amplitude and phase of the population tuning curves in which the effective lateral interactions are linear in the amplitudes these amplitude equations describe how mutual interactions between hypercolumns via lateral connections modify the response of each hypercolumn to modulated inputs from the lateral geniculate nucleus such interactions form the basis of contextual effects the coupled ring model is shown to reproduce a number of orientation dependent and contrast dependent features observed in center surround experiments a major prediction of the model is that the anisotropy in lateral connections results in a nonuniform modulatory effect of the surround that is correlated with the orientation of the center

cortical spatiotemporal dimensionality reduction for visual grouping the visual systems of many mammals including humans are able to integrate the geometric information of visual stimuli and perform cognitive tasks at the first stages of the cortical processing this is thought to be the result of a combination of mechanisms which include feature extraction at the single cell level and geometric processing by means of cell connectivity we present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visual stimuli and show how they can be used to obtain low level object segmentation the main idea is to define a spectral clustering procedure with anisotropic affinities over data sets consisting of embeddings of the visual stimuli into higher dimensional spaces neural plausibility of the proposed arguments will be discussed

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

fourth order time stepping for stiff pdes on the sphere we present in this paper algorithms for solving stiff pdes on the unit sphere with spectral accuracy in space and fourth order accuracy in time these are based on a variant of the double fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones and implicit explicit time stepping schemes operating in coefficient space with these new matrices allows one to use a sparse direct solver avoids the coordinate singularity and maintains smoothness at the poles while implicit explicit schemes circumvent severe restrictions on the time steps due to stiffness a comparison is made against exponential integrators and it is found that implicit explicit schemes perform best implementations in matlab and chebfun make it possible to compute the solution of many pdes to high accuracy in a very convenient fashion

exponential runge kutta methods for stiff kinetic equations we introduce a class of exponential runge kutta integration methods for kinetic equations the methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of bgk type for boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes we give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times the methods also offer favorable properties such as nonnegativity of the solution and entropy inequality for this reason as we will show the methods are suitable both for deterministic as well as probabilistic numerical techniques

natural scene statistics and the structure of orientation maps in the visual cortex visual activity after eye opening influences feature map structure in primary visual cortex v1 for instance rearing cats in an environment of stripes of one orientation yields an over representation of that orientation in v1 however whether such changes also affect the higher order statistics of orientation maps is unknown a statistical bias of orientation maps in normally raised animals is that the probability of the angular difference in orientation preference between each pair of points in the cortex depends on the angle of the line joining those points relative to a fixed but arbitrary set of axes natural images show an analogous statistical bias however whether this drives the development of comparable structure in v1 is unknown we examined these statistics for normal stripe reared and dark reared cats and found that the biases present were not consistently related to those present in the input or to genetic relationships we compared these results with two computational models of orientation map development an analytical model and a hebbian model the analytical model failed to reproduce the experimentally observed statistics in the hebbian model while orientation difference statistics could be strongly driven by the input statistics similar to those seen in experimental maps arose only when symmetry breaking was allowed to occur spontaneously these results suggest that these statistical biases of orientation maps arise primarily spontaneously rather than being governed by either input statistics or genetic mechanisms

analytical substrate flux approximation for the monod boundary value problem we present an analytical approximation for the diffusive flux of a substrate into a reactive layer in which the substrate is degraded according to monod kinetics this problem is described by a nonlinear two point boundary value problem the approximation is derived based on a homotopy analysis method idea and verified computationally by comparison against a numerical solution of the problem the analytical approximation is easy to evaluate and depends only on model parameters

asymptotic expansions for the sturm liouville problem by homotopy perturbation method the asymptotic formulae for the eigenvalues and eigenfunctions of sturm liouville problem with the dirichlet boundary conditions when the potential is square integrable on 0 1 are obtained by using homotopy perturbation method

exponential krylov methods for ordinary differential equations this paper develops a new family of exponential time discretization methods called exponential krylov expk the new schemes treat the time discretization and the krylov based approximation of exponential matrix vector products as a single computational process the classical order conditions theory developed herein accounts for both the temporal and the krylov approximation errors unlike traditional exponential schemes expk methods require the construction of only a single krylov space at each timestep the number of basis vectors that guarantee the temporal order of accuracy does not depend on the application at hand numerical results show favorable properties of expk methods when compared to current exponential schemes

what geometric visual hallucinations tell us about the visual cortex many observers see geometric visual hallucinations after taking hallucinogens such as lsd cannabis mescaline or psilocybin on viewing bright flickering lights on waking up or falling asleep in near death experiences and in many other syndromes kluver organized the images into four groups called form constants i tunnels and funnels ii spirals iii lattices including honeycombs and triangles and iv cobwebs in most cases the images are seen in both eyes and move with them we interpret this to mean that they are generated in the brain here we summarize a theory of their origin in visual cortex area v1 based on the assumption that the form of the retino cortical map and the architecture of v1 determine their geometry a much longer and more detailed mathematical version has been published in philosophical transactions of the royal society b 356 2001 we model v1 as the continuum limit of a lattice of interconnected hypercolumns each comprising a number of interconnected iso orientation columns based on anatomical evidence we assume that the lateral connectivity between hypercolumns exhibits symmetries rendering it invariant under the action of the euclidean group e 2 composed of reflections and translations in the plane and a novel shift twist action using this symmetry we show that the various patterns of activity that spontaneously emerge when v1 s spatially uniform resting state becomes unstable correspond to the form constants when transformed to the visual field using the retino cortical map the results are sensitive to the detailed specification of the lateral connectivity and suggest that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges contours surfaces and textures

an analytical study for fisher type equations by using homotopy perturbation method in this paper homotopy perturbation method is applied to fisher type equations the solutions introduced in this study are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required the method is tested on various examples which are revealing the effectiveness and the simplicity of the method

a new class of split exponential propagation iterative methods of runge kutta type sepirk for semilinear systems of odes abstract in this paper the framework of the exponential propagation iterative methods of runge kutta type epirk is extended to construct split epirk sepirk integrators for semilinear systems of odes the structure of the sepirk methods possesses the flexibility and generality that allows construction of very efficient schemes we demonstrate how bicolored trees based b series can be used to derive the order conditions for the new integrators an algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed the numerical results illustrate the advantages offered by the new class of integrators the experiments also address the comparative performance of split vs non split epirk methods and the question of improving efficiency by optimizing coefficients of the sepirk schemes it is shown that specific schemes can be custom built to improve computational efficiency for a particular problem

kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train we present a kinetic theory for all to all coupled networks of identical linear integrate and fire excitatory point neurons in which a fast and a slow excitatory conductance are driven by the same spike train in the presence of synaptic failure the maximal entropy principle guides us in deriving a set of three 1 1 dimensional kinetic moment equations from a boltzmann like equation describing the evolution of the one neuron probability density function we explain the emergence of correlation terms in the kinetic moment and boltzmann like equations as a consequence of simultaneous activation of both the fast and slow excitatory conductances and furnish numerical evidence for their importance in correctly describing the coarse grained dynamics of the underlying neuronal network

symplectic exponential runge kutta methods for solving nonlinear hamiltonian systems symplecticity is an important property for exponential runge kutta erk methods when the underlying problem y t my t f y t is a hamiltonian system the main theme of this chapter is to present symplectic exponential runge kutta methods using the fundamental analysis of geometric integrators we first derive and analyse the symplectic conditions for erk methods these conditions reduce to the conventional ones when m rightarrow mathbf 0 furthermore revised stiff order conditions are proposed and investigated in detail this chapter is also accompanied by numerical results that demonstrate the potential of the symplectic erk methods

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract 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 integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities 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 pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed 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

explicit exponential runge kutta methods of high order for parabolic problems exponential runge kutta methods constitute efficient integrators for semilinear stiff problems so far however explicit exponential runge kutta methods are available in the literature up to order 4 only the aim of this paper is to construct a fifth order method for this purpose we make use of a novel approach to derive the stiff order conditions for high order exponential methods this allows us to obtain the conditions for a method of order 5 in an elegant way after stating the conditions we first show that there does not exist an explicit exponential runge kutta method of order 5 with less than or equal to 6 stages then we construct a fifth order method with 8 stages and prove its convergence for semilinear parabolic problems finally a numerical example is given that illustrates our convergence bound

persistence of spatial patterns produced by neural field equations abstract we study an integro differential equation defined on a spatially extended domain proposed in the literature as a model of short term memory previous researchers have established the necessary conditions for the existence of two bump stationary solutions of this equation when considering a heaviside firing rate function and a coupling function of mexican hat type one such condition concerns the positiveness of the external stimulus which is quite difficult to establish under general assumptions we address this problem and determine the necessary conditions for the existence of two bump stationary solutions that assure this requirement is met furthermore we prove the existence of symmetric two bump stationary solutions for a class of coupling functions and establish their linear stability