effect of intermodular connection on fast sparse synchronization in clustered small world neural networks we consider a clustered network with small world subnetworks of inhibitory fast spiking interneurons and investigate the effect of intermodular connection on the emergence of fast sparsely synchronized rhythms by varying both the intermodular coupling strength j inter and the average number of intermodular links per interneuron m syn inter in contrast to the case of nonclustered networks two kinds of sparsely synchronized states such as modular and global synchronization are found for the case of modular sparse synchronization the population behavior reveals the modular structure because the intramodular dynamics of subnetworks make some mismatching on the other hand in the case of global sparse synchronization the population behavior is globally identical independently of the cluster structure because the intramodular dynamics of subnetworks make perfect matching we introduce a realistic cross correlation modularity measure representing the matching degree between the instantaneous subpopulation spike rates of the subnetworks and examine whether the sparse synchronization is global or modular depending on its magnitude the intermodular coupling strength j inter seems to play dual roles for the pacing between spikes in each subnetwork for large j inter due to strong inhibition it plays a destructive role to spoil the pacing between spikes while for small j inter it plays a constructive role to favor the pacing between spikes through competition between the constructive and the destructive roles of j inter there exists an intermediate optimal j inter at which the pacing degree between spikes becomes maximal in contrast the average number of intermodular links per interneuron m syn inter seems to play a role just to favor the pacing between spikes with increasing m syn inter the pacing degree between spikes increases monotonically thanks to the increase in the degree of effectiveness of global communication between spikes furthermore we employ the realistic sub and whole population order parameters based on the instantaneous sub and whole population spike rates to determine the threshold values for the synchronization unsynchronization transition in the sub and whole populations and the degrees of global and modular sparse synchronization are also measured in terms of the realistic sub and whole population statistical mechanical spiking measures defined by considering both the occupation and the pacing degrees of spikes it is expected that our results could have implications for the role of the brain plasticity in some functional behaviors associated with population synchronization

optimized double sweep schwarz method by complete radiation boundary conditions we present an optimized double sweep nonoverlapping schwarz method for solving the helmholtz equation in semi infinite waveguides the domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high order absorbing boundary conditions are solved forward and backward sequentially for communication between subdomains neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and dirichlet data are exploited in the backward direction the complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find neumann data without introducing errors that would be produced if finite difference formulas were used for computing neumann data the convergence of the double sweep schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

variety and generality of clustering in globally coupled oscillators abstract clustering behavior in globally coupled identical oscillators is studied by using the phase model any pair of oscillators is assumed to interact through their phase difference without specifying the coupling function the stability of symmetric cluster states in analyzed where symmetric means that the clusters contain the same number of oscillators it is found that higher harmonics in the coupling function are essential to clustering examples show that various cluster states appear depending on the coupling form including those with oscillating intervals anti phase states and chaotic long transient apart from the phase model we also study a general model for globally coupled limit cycle oscillators in the vicinity of the hopf bifurcation and find that such a system does not exhibit clustering within the lowest order approximation

complete radiation boundary conditions for the helmholtz equation i waveguides we consider the use of complete radiation boundary conditions for the solution of the helmholtz equation in waveguides a general analysis of well posedness convergence and finite element approximation is given in addition methods for the optimization of the boundary condition parameters are considered the theoretical results are illustrated by some simple numerical experiments

neuronal populations with reciprocal inhibition and rebound currents effects of synaptic and threshold noise the analysis of networks of time summating binary neural networks is relevant to the study of coherent oscillatory behaviour in neuronal populations a class of networks based on a discrete time version of leaky integrator networks has recently been extended to include the effects of hyperpolarisationactivated inward currents s coombes and s h doole dynamics and stability of systems 11 36 1996 such rebound currents are important for central pattern generation in neuronal circuits with reciprocal inhibition in this paper we incorporate models of intrinsic synaptic and threshold noise into the above neural system the macroscopic behaviour of time summating networks with rebound currents and random thresholds is analysed in the thermodynamic limit mean field equations are derived for the average network activity in a homogeneous network with inhibitory synaptic connections periodic and chaotic solutions are shown to exist together with hysteretic transitions between periodic orbits this hysteresis is observed between particular periodic orbit branches as well as more globally with respect to variations in external input or threshold noise moreover rebound currents are shown to suppress chaotic network response to external input in favour of low order periodic responses which in turn define well ordered coherent macroscopic oscillatory 1 states for the system the response characteristic of a single neuron in the presence of synaptic multiplicative noise is also considered and compared to its zero noise limit in this latter case the dynamics is reduced to a piecewise linear discontinuous circle map whilst the former is expressed in terms of a random iterated function system 87 10 e 02 50 ey typeset using revtex 2

the finite element method with nonuniform mesh sizes for unbounded domains the finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains consider the following model problem du f in w sup c u g on partialw partialu partialr 1 r u o 1 r as r vertical barxvertical bar infinity where w sup c is the complement in r sup 3 three dimensional eucidean space of a bounded set w with smooth boundary partialw f and g are smooth functions and f has bounded support this problem is approximately solved by introducing an artifical bound g sub r near infinity e g a sphere of sufficiently large radius r the intersection of this sphere with w sup c is denoted by w sub r sup c and the given problem is replaced by du sub r f in w sup c sub r u sub r g on partialw partialu sub r partialr 1 r u sub r 0 on g sub r this problem is then solved approximately by the finite element method resulting in an approximate solution u sub r sup h for each h 0 in order to obtain a reasonably small error for u u sub r sup h u u sub r u sub r u sub r sup h it is necessary to make r large this necessitates the solution of amore large number of linear equations so that this method is often not very good when a uniform mesh size h is employed it is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by ch sup 3 with c independent of h and r less

the direct method of lines for the problem of infinite elastic foundation we consider the numerical simulation for the problem of infinite elastic foundation a polygonal artificial boundary is introduced and a discrete artificial boundary condition on it is presented by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain which is solved by the finite element method in addition we prove an optimal a priori error bound for the displacement in the bounded computational domain finally numerical example shows that the discrete artificial boundary condition given in this paper is very effective and more accurate than neumann boundary condition which is often used in engineering literatures

low dimensional firing rate dynamics for populations of renewal type spiking neurons the macroscopic dynamics of large populations of neurons can be mathematically analyzed using low dimensional firing rate or neural mass models however these models fail to capture spike synchronization effects and nonstationary responses of the population activity to rapidly changing stimuli here we derive low dimensional firing rate models for homogeneous populations of neurons modeled as time dependent renewal processes the class of renewal neurons includes integrate and fire models driven by white noise and has been frequently used to model neuronal refractoriness and spike synchronization dynamics the derivation is based on an eigenmode expansion of the associated refractory density equation which generalizes previous spectral methods for fokker planck equations to arbitrary renewal models we find a simple relation between the eigenvalues characterizing the timescales of the firing rate dynamics and the laplace transform of the interspike interval density for which explicit expressions are available for many renewal models retaining only the first eigenmode already yields a reliable low dimensional approximation of the firing rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset we explicitly demonstrate the validity of our model for a large homogeneous population of poisson neurons with absolute refractoriness and other renewal models that admit an explicit analytical calculation of the eigenvalues the eigenmode expansion presented here provides a systematic framework for alternative firing rate models in computational neuroscience based on spiking neuron dynamics with refractoriness

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

dynamics of two mutually coupled slow inhibitory neurons abstract inhibition in oscillatory networks of neurons can have apparently paradoxical effects sometimes creating dispersion of phases sometimes fostering synchrony in the network we analyze a pair of biophysically modeled neurons and show how the rates of onset and decay of inhibition interact with the timescales of the intrinsic oscillators to determine when stable synchrony is possible we show that there are two different regimes in parameter space in which different combinations of the time constants and other parameters regulate whether the synchronous state is stable we also discuss the construction and stability of nonsynchronous solutions and the implications of the analysis for larger networks the analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow flows and fast jumps

fast global oscillations in networks of integrate and fire neurons with low firing rates we study analytically the dynamics of a network of sparsely connected inhibitory integrate and fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate in the limit when the number of neurons n 1 the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized the activity becomes oscillatory when the inhibitory feedback is strong enough the period of the global oscillation is found to be mainly controlled by synaptic times but depends also on the characteristics of the external input in large but finite networks the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold their characteristics are determined as functions of systems parameters in these two different regimes the results are found to be in good agreement with numerical simulations

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

finite element methods applied to nearly one way wave propagation abstract a numerical method is developed for approximately solving stationary wave propagation problems for which most of the energy propagates in an angle band of less than 90 about a fixed direction this property often occurs in realistic physical models the numerical method is based on factoring out the principal propagation direction in the solution and then applying the finite element method the method is applied to wave propagation in a duct or waveguide and to the exterior scattering problem it is shown both theoretically and by means of numerical examples that this method requires considerably less grid points and computational cost than standard discretization methods

a finite element method for laplace and helmholtz type boundary value problems with singularities laplace and helmholtz type boundary value problems with singularities are considered a sequence of approximations to the exact boundary conditions at an artificial boundary is given then the original problem is reduced to a boundary value problem in a domain away from the singularities furthermore finite element approximations are applied to this problem and error estimates are obtained finally some numerical examples show the effectiveness of this method

the number of synaptic inputs and the synchrony of large sparse neuronal networks the prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons we study synchronization in models of large networks of spiking neurons with random sparse connectivity synchrony occurs only when the average number of synapses m that a cell receives is larger than a critical value mc below mc the system is in an asynchronous state in the limit of weak coupling assuming identical neurons we reduce the model to a system of phase oscillators that are coupled via an effective interaction in this framework we develop an approximate theory for sparse networks of identical neurons to estimate mc analytically from the fourier coefficients of our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it we apply this theory to compute mc for a model of inhibitory networks of integrate and fire i f neurons as a function of the intrinsic neuronal properties e g the refractory period tr the synaptic time constants and the strength of the external stimulus iext the number mc is found to be nonmonotonous with the strength of iext for tr 0 we estimate the minimum value of mc over all the parameters of the model to be 363 8 above mc the neurons tend to fire in smeared one cluster states at high firing rates and smeared two or more cluster states at low firing rates refractoriness decreases mc at intermediate and high firing rates these results are compared to numerical simulations we show numerically that systems with different sizes n behave in the same way provided the connectivity m is such that 1 meff 1 m 1 n remains constant when n varies this allows extrapolating the large n behavior of a network from numerical simulations of networks of relatively small sizes n 800 in our case we find that our theory predicts with remarkable accuracy the value of mc and the patterns of synchrony above mc provided the synaptic coupling is not too large we also study the strong coupling regime of inhibitory sparse networks all of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity above a critical coupling strength the network activity is asynchronous we point out a fundamental limitation for the mechanisms of synchrony relying on inhibition alone if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account

the lorenz fermi pasta ulam experiment we consider a chain of lorenz 63 systems connected through a local nearest neighbour coupling we refer to the resulting system as the lorenz fermi pasta ulamlattice because of its similarity to the celebrated experiment conducted by fermi pasta and ulam at large coupling strengths the systems synchronize to a global chaotic orbit of the lorenz attractor for smaller coupling the synchronized state loses stability instead steady spatially structured equilibrium states are observed these steady states are related to the heteroclinic orbits of the system describing stationary solutions to the partial differential equation that emerges on taking the continuum limit of the lattice notably these orbits connect saddle foci suggesting the existence of a multitude of such equilibria in relatively wide systems on lowering the coupling strength yet further the steady states lose stability in what appear to be always subcritical hopf bifurcations this can lead to a variety of time dependent states with fixed time averaged spatial structure such solutions can be limit cycles tori or possibly chaotic attractors cluster states can also occur though with less regularity consisting of lattices in which the elements are partitioned into families of synchronized subsystems ultimately for very weak coupling the lattice loses its time averaged spatial structure at this stage the properties of the lattice are probably chaotic and approximately scale with the lattice size suggesting that the system is essentially an ensemble of elements that evolve largely independent of one another the weak interaction however is sufficient to induce widespread coherent phases these are ephemeral states in which the dynamics of one or more subsystems takes a more regular form we present measures of the complexity of these incoherent lattices and discuss the concept of a dynamical horizon that is the distance along the lattice that one subsystem can effectively influence another and error propagation how the introduction of a disturbance in one subsystem becomes spread throughout the lattice 2000 elsevier science b v all rights reserved

mixed finite element method and high order local artificial boundary conditions for exterior problems of elliptic equation the mixed finite element method is used to solve the exterior elliptic problem with high order local artificial boundary conditions new unknowns are introduced to reduce the order of the derivatives to two this leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary error estimates are obtained for some local artificial boundary conditions with prescribed order numerical examples are presented and the results demonstrate the effectiveness of this method

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

the simplified hybrid combined methods for laplace s equation with singularities abstract in li and liang 1983 the simplified hybrid combined method is presented for combining the ritz galerkin method and the finite element method in this paper we will apply this method to solve singularity problems of laplace s equation error bounds and stability analyses will be provided while taking into account the integration approximation along the coupling boundary a significant coupling relation between the ritz galerkin and the finite element method has been found for the laplace equation with singularities an optimal rate of convergence has also been achieved numerical experiments have been carried out for solving the benchmark problem motz s problem to verify the theoretical results

synchrony and clustering in an excitatory neural network model with intrinsic relaxation kinetics synchronized firing of a population of neurons in the hippocampus has been studied as a model for epilepsy this paper formulates an abstract model of an excitatory neural network which is qualitatively derived from a biophysical model of hippocampal neurons in this model the phase space of an isolated neuron is a cylinder theta z an additional variable s conveys the effects of excitatory coupling between cells a small parameter epsilon is introduced which represents the ratio of two time scales intrinsic to each neuron it is proved that under certain conditions this model will possess stable approximately synchronous solutions these are periodic or quasi periodic solutions in which all cells fire approximately simultaneously i e within a time interval of duration o epsilon relative to the intrinsic cell period these approximately synchronous solutions persist in the face of cell heterogeneity of magnitude o 1 in the case where the network is homogenous stable approximat

solution of helmholtz equation by differential quadrature method the polynomial based differential quadrature pdq and the fourier expansion based differential quadrature fdq methods are applied in this work to solve the two dimensional helmholtz equation it is demonstrated through some sample examples that accurate numerical solution can be obtained by using only 2 to 3 mesh points per wavelength it was found that the fdq approach is more suitable for solving the helmholtz equation

finite element analysis of a scattering problem a finite element method for the solution of a scattering problem for the reduced wave equation is formulated and analyzed the method involves a reformulation of the problem on a bounded domain with a nonlocal boundary condition the space of trial functions includes piecewise polynomial functions and functions arising from spherical harmonics

numerical simulation for the problem of infinite elastic foundation in this paper we consider the numerical simulation for the problem of infinite elastic foundation a half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain furthermore finite element approximation of this boundary value problem defined in the bounded domain is considered the numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the neumann boundary condition which is often used in engineering literatures

fast numerical integration of relaxation oscillator networks based on singular limit solutions relaxation oscillations exhibiting more than one time scale arise naturally from many physical systems when relaxation oscillators are coupled in a way that resembles chemical synapses we propose a fast method to numerically integrate such networks the numerical technique called the singular limit method is derived from analysis of relaxation oscillations in the singular limit in such a limit system evolution gives rise to time instants at which fast dynamics take place and intervals between them during which slow dynamics take place a full description of the method is given for a locally excitatory globally inhibitory oscillator network legion where fast dynamics characterized by jumping which leads to dramatic phase shifts is captured in this method by iterative operation and slow dynamics is entirely solved the singular limit method is evaluated by computer experiments and it produces remarkable speedup compared to other methods of integrating these systems the speedup makes it possible to simulate large scale oscillator networks

asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons we investigate theoretically the conditions for the emergence of synchronous activity in large networks consisting of two populations of extensively connected neurons one excitatory and one inhibitory the neurons are modeled with quadratic integrate and fire dynamics which provide a very good approximation for the subthreshold behavior of a large class of neurons in addition to their synaptic recurrent inputs the neurons receive a tonic external input that varies from neuron to neuron because of its relative simplicity this model can be studied analytically we investigate the stability of the asynchronous state as of the network with given average firing rates of the two populations first we show that the as can remain stable even if the synaptic couplings are strong then we investigate the conditions under which this state can be destabilized we show that this can happen in four generic ways the first is a saddle node bifurcation which leads to another state with different average firing rates this bifurcation which occurs for strong enough recurrent excitation does not correspond to the emergence of synchrony in contrast in the three other instability mechanisms hopf bifurcations which correspond to the emergence of oscillatory synchronous activity occur we show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations we also show that besides these codimension 1 bifurcations the system can display several codimension 2 bifurcations takens bogdanov gavrielov guckenheimer and double hopf bifurcations

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

collatz wielandt numbers in general partially ordered spaces abstract it has been shown by forster and nagy that the convergence of the collatz wielandt numbers to bound the spectral radius of a k nonnegative linear operator t in a partially ordered bannach space essentially depends upon the starting vector of the iteration process in this paper necessary and sufficient conditions for convergence are presented under rather general hypotheses e g the emptiness of the interior of the order cone k is admitted we also present min sup and max inf characterizations of the spectral radius in the spirit of earlier work

partial synchronization in a network of neural oscillators a network with three types of model neurons is considered these are designated as excitatory fast inhibitory and slow inhibitory cells each excitatory cell is modeled as a relaxation oscillator while for simplicity the inhibitory cells are modeled by a single equation the coupling between cells is modeled in a way that mimics chemical synapses this network is introduced as a simplification of the model in traub and miles neuronal networks of the hippocampus cambridge university press new york 1991 for oscillations in the ca3 region of the hippocampus a geometric approach for analyzing the solution structure of the network is developed the analysis clarifies the role that each type of cell and each parameter plays in obtaining partially synchronized rhythms determines how the parameters should be chosen for this collective behavior and indicates possible instabilities which arise as parameters are varied a symbolic scheme is derived which accurately determines the firing patterns for a l

error estimates for the finite element approximation of linear elastic equations in an unbounded domain in this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain the finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one in fact there are a family of nonlocal approximate boundary conditions with increasing accuracy and computational cost and a family of local ones for a given artificial boundary our error estimates show how the errors of the finite element approximations depend on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary a numerical example for navier equations outside a circle in the plane is presented numerical results demonstrate the performance of our error estimates

partial synchronization in populations of pulse coupled oscillators i study the long term behavior of populations of nonlinear oscillators with all to all noninstantaneous pulse coupling with fast enough excitatory coupling both the fully synchronized and the asynchronous state are unstable in this case individual units fire quasiperiodically even though the network as a whole shows a periodic firing pattern the behavior of networks with three or more units is different in this regard from that of two unit networks with inhibitory coupling the network can break up into a variable number of fully synchronized clusters for fast inhibition the number of clusters tends to be large while the number of clusters is smaller for slow inhibition textcopyright 1996 the american physical society

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

the artificial boundary conditions for computing the flow around a submerged body in this paper we consider the numerical simulations of the steady linearized potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below a free surface two vertical segments are introduced as the artificial boundary to limit the unbounded physical domain to a bounded computational domain a series of approximate artificial boundary conditions at the given artificial boundary are designed by imposing the continuity of the solution and its normal derivative then the original problem is reduced to a problem defined in a bounded computational domain which is solved numerically by the finite element method the numerical examples show that our artificial boundary conditions are very effective

the artificial boundary conditions for incompressible materials on an unbounded domain in this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in frak r 2 a series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given then the original problem is reduced to a problem on a bounded domain which be solved numerically by a mixed finite element method the numerical example shows that our artificial boundary conditions are very effective

interplay between excitability type and distributions of neuronal connectivity determines neuronal network synchronization while the interplay between neuronal excitability properties and global properties of network topology is known to affect network propensity for synchronization it is not clear how detailed characteristics of these properties affect spatiotemporal pattern formation here we study mixed networks composed of neurons having type i and or type ii phase response curves with varying distributions of local and random connections and show that not only average network properties but also the connectivity distribution statistics significantly affect network synchrony namely we study networks with fixed networkwide properties but vary the number of random connections that nodes project we show that varying node excitability type i vs type ii influences network synchrony most dramatically for systems with long tailed distributions of the number of random connections per node this indicates that a cluster of even a few highly rewired cells with a high propensity for synchronization can alter the degree of synchrony in the network as a whole we show this effect generally on a network of coupled kuramoto oscillators and investigate the impact of this effect more thoroughly in pulse coupled networks of biophysical neurons

the finite element method with non uniform mesh sizes applied to the exterior helmholtz problem the finite element method with non uniform mesh sizes is employed to approximately solve helmholtz type equations in unbounded domains the given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary this approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases this results in a great reduction in the number of equations to be solved it is proved that optimal error estimates hold inl 2 h 1 andl provided that certain relationships hold between the frequency mesh size and outer radius

clustering in globally coupled inhibitory neurons abstract a model of a large population of identical excitable neurons with a global slowly decaying inhibitory coupling is studied and its patterns of synchrony are examined in addition to converging to a homogeneous fixed point and a homogeneous limit cycle the system exhibits cluster states in which it breaks spontaneously into a few macroscopically big clusters each of which is fully synchronized a method for calculating the stability of cluster states is described and used for investigating the dynamical behavior of the network versus the parameters that describe the neurons and synapses effects of stochastic noise on the network dynamics are discussed at large enough noise the system goes to a globally stationary state low levels of noise preserve the cluster like neuron trajectories in the regime where a noiseless system converges to a fully synchronized periodic state relatively low noise levels cause neurons to burst only every two or more consecutive time periods

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

finite difference analysis of borehole flows involving domain contractions around three dimensional anomalies a finite difference analysis in cylindrical coordinates of borehole flows governed by poisson s equation yields an infinite cylindrical electrical grid as a model the customary technique of truncating the medium in order to obtain a finite grid can be avoided at least to some extent by using the recently developed theory of infinite electrical networks when the medium around the borehole is either uniform or cylindrically layered except for a three dimensional anomalous region near the borehole infinite electrical network theory allows one to represent the medium outside any cylinder s containing the anomaly and centered on the borehole by an infinite continued fraction of circulant laurent operators the continued fraction in turn can be represented by scalar conductances connected to the grid s nodes on s by choosing s just large enough to contain the anomaly one can replace the large system of equations resulting from the usual medium truncations far removed from the anomaly by a much smaller systems of equations for the grid within and on s an operational calculus for the circulant laurent operators and the fast fourier transform expedites the computations and moreover allows a rapidly computed determination of the solution outside s

generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena the integrate and fire if neuron model and the piecewise linear version of fitzhugh nagumo pl neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena although the two models are regarded to be different type we show a certain equivalence between them by deriving the coupled if model of an improved version with a firing duration from the recovery variable coupled system of the pl model under taking the limit of mu 0 without a loss of any coupling properties in the coupled if model with the duration time the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model synaptic strength decaying relaxation rate of the synaptic coupling and the parameter exhibiting the firing duration we find some irregularly synchronous behavior with or without constant firing order alternations we show that the duration of an impulse plays an important role in synchronization phenomena

a dynamical theory of spike train transitions in networks of integrate and fire oscillators a dynamical theory of spike train transitions in networks of pulse coupled integrate and fire if neural oscillators is presented we begin by deriving conditions for 1 1 frequency locking in a network with noninstantaneous synaptic interactions this leads to a set of phase equations determining the relative firing times of the oscillators and the self consistent collective period we then investigate the stability of phase locked solutions by constructing a linearized map of the firing times and analyzing its spectrum we establish that previous results concerning the stability properties of if oscillator networks are incomplete since they only take into account the effects of weak coupling instabilities we show how strong coupling instabilities can induce transitions to nonphase locked states characterized by periodic or quasi periodic variations of the interspike intervals on attracting invariant circles the resulting spatio temporal pattern of network activity is compatible with the behavior of a

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

numerical analysis of diffraction by periodic structures tm polarization the diffraction of a time harmonic wave in tm polarization by some periodic inhomogeneous material is studied in this paper the diffraction problem is modeled by a generalized helmholtz equation with transparent boundary conditions existence and uniqueness are proved for the model problem uniform error estimates for the finite element approximation of the solution are established error estimates are also obtained when the truncation of the nonlocal boundary operators takes place

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

application of a complete radiation boundary condition for the helmholtz equation in locally perturbed waveguides abstract this paper deals with an application of a complete radiation boundary condition crbc for the helmholtz equation in locally perturbed waveguides the crbc one of efficient high order absorbing boundary conditions has been analyzed in straight waveguides in hagstrom and kim 2019 in this paper we apply crbc to the helmholtz equation posed in locally perturbed waveguides and establish the well posedness of the problem and convergence of crbc approximate solutions the new crbc proposed in this paper improves the one studied in hagstrom and kim 2019 in two aspects the first one is that the new crbc involves more damping parameters with the same computational cost as that of crbc in hagstrom and kim 2019 which results in 50 smaller reflection errors the second one is that the new crbc takes a neumann terminal condition of three term recurrence relations of auxiliary variables instead of a dirichlet terminal condition used in hagstrom and kim 2019 so that it can treat cutoff modes effectively finally we present numerical experiments illustrating the convergence theory

error estimates of the dtn finite element method for the exterior helmholtz problem a priori error estimates are established for the dtn dirichlet to neumann finite element method applied to the exterior helmholtz problem the error estimates include the effect of truncation of the dtn boundary condition as well as that of the finite element discretization a property of the hankel functions which plays an important role in the proof of the error estimates is introduced

non reflecting boundary conditions for waveguides new non reflecting boundary conditions are introduced for the solution of the helmholtz equation in a waveguide these boundary conditions are perfectly transparent for all propagating modes they do not require the determination of these propagating modes but only their propagation constants a quasi local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes related convergence properties to the exact solution and optimal error estimates are established

population dynamics of spiking neurons fast transients asynchronous states and locking an integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate and fire type is discussed it is analytically shown that transients from a state of incoherent firing can be immediate the stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived the response of a population of noisy integrate and fire neurons to an input current of small amplitude is calculated and characterized by a linear filter l the stability of perfectly synchronized locked solutions is analyzed

the weighted finite element method for elasticity equations on unbounded domains abstract by introducing the weighted riesz galerkin method with a special weight function oh jou 23 made it possible to apply mam the method of auxiliary mapping introduced by babuska oh to general elliptic boundary value problems on unbounded domains in this paper this method is further generalized to yield highly accurate approximate solutions for linear elasticity problems on unbounded domains with bounded or unbounded boundaries it is shown that in the framework of the p version of the fem this method does efficiently handle elasticity problems on unbounded domains and yields accurate solutions at low cost