conservative modified cranknicolson and time splitting wavelet methods for modeling boseeinstein condensates in delta potentials this paper explores two wavelet based energy conserving algorithms for the grosspitaevskii equation with delta potentials in boseeinstein condensates named modified cranknicolson wavelet method and time splitting wavelet method respectively both proposed methods can preserve the intrinsic properties of original problems as much as possible meanwhile the rigorous error estimates and some conservative properties are investigated they are proved to preserve the charge conservation exactly the global energy conservation laws can be preserved under several conditions in practical computations to avoid a large drift in energy values caused by discontinuous potential well an improved discrete delta function is implemented numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis

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

insights from a simple expression for linear fisher information in a recurrently connected population of spiking neurons a simple expression for a lower bound of fisher information is derived for a network of recurrently connected spiking neurons that have been driven to a noise perturbed steady state we call this lower bound linear fisher information as it corresponds to the fisher information that can be recovered by a locally optimal linear estimator unlike recent similar calculations the approach used here includes the effects of nonlinear gain functions and correlated input noise and yields a surprisingly simple and intuitive expression that offers substantial insight into the sources of information degradation across successive layers of a neural network here this expression is used to 1 compute the optimal i e information maximizing firing rate of a neuron 2 demonstrate why sharpening tuning curves by either thresholding or the action of recurrent connectivity is generally a bad idea 3 show how a single cortical expansion is sufficient to instantiate a redundant population code that can propagate across multiple cortical layers with minimal information loss and 4 show that optimal recurrent connectivity strongly depends on the covariance structure of the inputs to the network

double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem this paper presents a preconditioner for non overlapping schwarz methods applied to the helmholtz problem starting from a simple analytic example we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain the preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains and can be interpreted as a coarse grid for the domain decomposition method the resulting algorithm is shown to converge very fast independently of the number of subdomains and frequency the preconditioner has the advantage that like the original schwarz algorithm it can be implemented as a matrix free routine with no additional preprocessing

memory capacity in neural networks with spatial correlations between attractors abstract we consider the neural network model miyashita 1988 griniasty et al 1993 proposed to describe neurophysiological experiments in which structurally uncorrelated patterns are converted into spatially correlated attractors for such a network of n neurons and for values of the coupling constant a between succeeding patterns taken in the interval 0 1 2 we prove the existence of a threshold storage capacity c a such that there exists a local minima in the energy function near each of the m n c a n stimuli patterns

intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators a theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit cycle oscillators with distributed native frequencies in particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above this peculiar feature is demonstrated by numerical simulations mainly through finite size scaling analyses in the course of exploring its origin we encounter a new concept termed a correlation frequency which provides a natural interpretation of the finite size scaling laws a comment is given on a recent theory by kuramoto and nishikawa to clarify why it contradicts our results

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

mean field approximations for coupled populations of generalized linear model spiking neurons with markov refractoriness there has recently been a great deal of interest in inferring network connectivity from the spike trains in populations of neurons one class of useful models that can be fit easily to spiking data is based on generalized linear point process models from statistics once the parameters for these models are fit the analyst is left with a nonlinear spiking network model with delays which in general may be very difficult to understand analytically here we develop mean field methods for approximating the stimulus driven firing rates in both the time varying and steady state cases auto and cross correlations and stimulus dependent filtering properties of these networks these approximations are valid when the contributions of individual network coupling terms are small and hence the total input to a neuron is approximately gaussian these approximations lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct monte carlo simulation of the network activity these approximations also provide an analytical way to evaluate the linear input output filter of neurons and how the filters are modulated by network interactions and some stimulus feature finally in the case of strong refractory effects the mean field approximations in the generalized linear model become inaccurate therefore we introduce a model that captures strong refractoriness retains all of the easy fitting properties of the standard generalized linear model and leads to much more accurate approximations of mean firing rates and cross correlations that retain fine temporal behaviors

influence of external inputs and asymmetry of connections on information geometric measures involving up to ten neuronal interactions the investigation of neural interactions is crucial for understanding information processing in the brain recently an analysis method based on information geometry ig has gained increased attention and the property of the pairwise ig measure has been studied extensively in relation to the two neuron interaction however little is known about the property of ig measures involving more neuronal interactions in this study we systematically investigated the influence of external inputs and the asymmetry of connections on the ig measures in cases ranging from 1 neuron to 10 neuron interactions first the analytical relationship between the ig measures and external inputs was derived for a network of 10 neurons with uniform connections our results confirmed that the single and pairwise ig measures were good estimators of the mean background input and of the sum of the connection weights respectively for the ig measures involving 3 to 10 neuronal interactions we found that the influence of external inputs was highly nonlinear second by computer simulation we extended our analytical results to asymmetric connections for a network of 10 neurons the simulation showed that the behavior of the ig measures in relation to external inputs was similar to the analytical solution obtained for a uniformly connected network when the network size was increased to 1000 neurons the influence of external inputs almost disappeared this result suggests that all ig measures from 1 neuron to 10 neuron interactions are robust against the influence of external inputs in addition we investigated how the strength of asymmetry influenced the ig measures computer simulation of a 1000 neuron network showed that all the ig measures were robust against the modulation of the asymmetry of connections our results provide further support for an information geometric approach and will provide useful insights when these ig measures are applied to real experimental spike data

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

fisher information and temporal correlations for spiking neurons with stochastic dynamics population coding accuracy can be studied using fisher information here the fisher information and correlation functions are determined analytically for a network of coupled spiking neurons with a more general than poisson stochastic dynamics it is shown that stimulus driven temporal correlations between neurons always increase the fisher information whereas stimulus independent correlations need not do so additionally we find that for subthreshold stimuli there is some nonzero level of noise for which network coding is optimal we also find that the fisher information is larger for purely excitatory than for purely inhibitory networks but only in a limited range of values of synaptic coupling strengths in most cases the dependence of the fisher information on time is linear except for excitatory networks with strong synaptic couplings and for strong stimuli in the latter case this dependence shows two distinct regimes fast and slow for excitatory networks short term synaptic depression can improve the coding accuracy significantly whereas short term facilitation can lower the coding accuracy for inhibitory networks coding accuracy is insensitive to short term synaptic dynamics

associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise the memory patterns are assumed to be given by uniformly distributed random numbers on 0 2 ensuremath pi so that the patterns encode the phase differences of the oscillators to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns we introduce an effective transfer function by assuming a fixed point equation of the form of the thouless anderson palmer equation which describes the time averaged output as a function of the effective time averaged local field properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations and are shown to be in good agreement with the results of numerical simulations two types of retrieval states are found to occur with respect to the degree of synchronization when the size of the width of the natural frequency distribution is changed

spatiotemporal dynamics of continuum neural fields we survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields neural fields model the large scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections they provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and turing like patterns

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

fundamental limits on persistent activity in networks of noisy neurons neural noise limits the fidelity of representations in the brain this limitation has been extensively analyzed for sensory coding however in short term memory and integrator networks where noise accumulates and can play an even more prominent role much less is known about how neural noise interacts with neural and network parameters to determine the accuracy of the computation here we analytically derive how the stored memory in continuous attractor networks of probabilistically spiking neurons will degrade over time through diffusion by combining statistical and dynamical approaches we establish a fundamental limit on the network s ability to maintain a persistent state the noise induced drift of the memory state over time within the network is strictly lower bounded by the accuracy of estimation of the network s instantaneous memory state by an ideal external observer this result takes the form of an information diffusion inequality we derive some unexpected consequences despite the persistence time of short term memory networks it does not pay to accumulate spikes for longer than the cellular time constant to read out their contents for certain neural transfer functions the conditions for optimal sensory coding coincide with those for optimal storage implying that short term memory may be co localized with sensory representation

correlated attractors from uncorrelated stimuli analytic results for a neural network model proposed to account for neurophysiological experiments in which attractors for uncorrelated patterns are found to be correlated are presented exact expressions for the associative attractors and their correlations are described

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

finite size effects in separable recurrent neural networks we perform a systematic analytical study of finite size effects in separable recurrent neural network models with sequential dynamics away from saturation we find two types of finite size effects thermal fluctuations and disorder induced frozen corrections to the mean field laws the finite size effects are described by equations that correspond to a time dependent ornstein uhlenbeck process we show how the theory can be used to understand and quantify various finite size phenomena in recurrent neural networks with and without detailed balance

analytical investigation of the effects of lateral connections on the accuracy of population coding we studied how lateral connections affect the accuracy of a population code by using a model of orientation selectivity in the primary visual cortex investigating the effects of lateral connections on population coding is a complex problem because these connections simultaneously change the shape of tuning curves and correlations between neurons both of these changes caused by lateral connections have to be taken into consideration to correctly evaluate their effects we propose a theoretical framework for analytically computing the fisher information which measures the accuracy of a population code in stochastic spiking neuron models with refractory periods within our framework we accurately evaluated both the changes in tuning curves and correlations caused by lateral connections and their effects on the fisher information we found that their effects conflicted with each other and the answer to whether or not the lateral connections increased the fisher information strongly depended on the intrinsic properties of the model neuron by systematically changing the coupling strengths of excitations and inhibitions we found the parameter regions of lateral connectivities where sharpening of tuning curves through mexican hat connectivities led to an increase in information which is in contrast to some previous findings

a domain decomposition method for discontinuous galerkin discretizations of helmholtz problems with plane waves and lagrange multipliers a nonoverlapping domain decomposition dd method is proposed for the iterative solution of systems of equations arising from the discretization of helmholtz problems by the discontinuous enrichment method this discretization method is a discontinuous galerkin finite element method with plane wave basis functions for approximating locally the solution and dual lagrange multipliers for weakly enforcing its continuity over the element interfaces the primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface lagrange multipliers only as in the feti h and feti dph dd methods for continuous galerkin discretizations this system of lagrange multipliers is iteratively solved by a krylov method equipped with both a local preconditioner based on subdomain data and a global one using a coarse space numerical experiments performed for two and three dimensional acoustic scattering problems suggest that the proposed dd based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains copyright 2009 john wiley sons ltd

a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the nonrelativistic limit regime a multiscale time integrator fourier pseudospectral mti fp method is proposed and analyzed for solving the klein gordon schrodinger kgs equations in the nonrelativistic limit regime with a dimensionless parameter 0 varepsilon le 1 0 1 which is inversely proportional to the speed of light in fact the solution of the kgs equations propagates waves with wavelength at o varepsilon 2 o 2 and o 1 in time and space respectively when 0 varepsilon ll 1 0 1 which brings significantly numerical burden in practical computation the mti fp method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives respectively we rigorously establish two independent error bounds for the mti fp at o tau 2 varepsilon 2 h m 0 o 2 2 hm0 and o varepsilon 2 h m 0 o 2 hm0 for varepsilon in 0 1 0 1 with tau time step size h mesh size and m 0 ge 4 m0 4 an integer depending on the regularity of the solution which imply that the mti fp converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau o for varepsilon in 0 1 0 1 in addition the mti fp method converges optimally with quadratic convergence rate at o tau 2 o 2 in the regime when 0 tau lesssim varepsilon 2 0 2 and the error is at o varepsilon 2 o 2 independent of tau in the regime when 0 varepsilon lesssim tau 1 2 0 1 2 thus the meshing strategy requirement or varepsilon scalability of the mti fp is tau o 1 o 1 and h o 1 h o 1 for 0 varepsilon ll 1 0 1 which is significantly better than that of classical methods numerical results demonstrate that our error bounds are optimal and sharp finally the mti fp method is applied to study numerically convergence rates of the kgs equations to its limiting models in the nonrelativistic limit regime

information geometric measures as robust estimators of connection strengths and external inputs information geometry has been suggested to provide a powerful tool for analyzing multineuronal spike trains among several advantages of this approach a significant property is the close link between information geometric measures and neural network architectures previous modeling studies established that the first and second order information geometric measures corresponded to the number of external inputs and the connection strengths of the network respectively this relationship was however limited to a symmetrically connected network and the number of neurons used in the parameter estimation of the log linear model needed to be known recently simulation studies of biophysical model neurons have suggested that information geometry can estimate the relative change of connection strengths and external inputs even with asymmetric connections inspired by these studies we analytically investigated the link between the information geometric measures and the neural network structure with asymmetrically connected networks of n neurons we focused on the information geometric measures of orders one and two which can be derived from the two neuron log linear model because unlike higher order measures they can be easily estimated experimentally considering the equilibrium state of a network of binary model neurons that obey stochastic dynamics we analytically showed that the corrected first and second order information geometric measures provided robust and consistent approximation of the external inputs and connection strengths respectively these results suggest that information geometric measures provide useful insights into the neural network architecture and that they will contribute to the study of system level neuroscience

statistical mechanics for neural networks with continuous time dynamics the present paper is intended to summariz our rent knowledge about the long time behaviour of nehwrks of graded response neurons with continuous time dynamics we demonstrate the workings of our previously developed statistical mechanical approach to continuous time dynamics by applying it to networks with various forms of synaptic organization leaming rules and neural composition neuron types as encoded in gain functions as well as to networks varying with respect to the ensemble of stored data unbiased and low activity patterns we present phase diagrams and compute distributions of local fields for a variety of examples local field distributions are found to deviate from the gaussian form obtained for stochastic neumns in the wntexi of the replica approach a solution to the low firing rates problem within the framework of nets of analogue neurons is also briefly discussed finally the statisticalmechanical approach to the analysis of continuowtime dynamics is extended to include effects of fast stochastic noise detailed balance solutions are shown to be unique and of canonical form governed by hamiltonians which exhibit a reciprocity relation between potential dynamics and fuing rate dynamics for the potential dynamics the hamiltonian is given by the lyapounw function of the system expressed in terms of the fuing rates and it generates a gibbs distribution over fuing rate space for the noisy firinerate dynamics the same lyapounov function now expressed in terms of neural potentials generates a gibbs distribution over the space of these potentials as a consequence the firing rate dynamics will freeze in configurations saturating the neural input xtput relations whenever such saturation levels exist both types of stationary distribution are shown to exist only under unrealistic assumptions about the noise in the system

domain decomposition method for maxwell s equations scattering off periodic structures abstract we present a domain decomposition approach for the computation of the electromagnetic field within periodic structures we use a schwarz method with transparent boundary conditions at the interfaces of the domains transparent boundary conditions are approximated by the perfectly matched layer method pml an adaptive strategy to determine optimal pml parameters is developed thus we can treat wood anomalies appearing in periodic structures we focus on the application to typical euv lithography line masks light propagation within the multilayer stack of the euv mask is treated analytically this results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer

analysis of spike statistics in neuronal systems with continuous attractors or multiple discrete attractor states attractor networks are likely to underlie working memory and integrator circuits in the brain it is unknown whether continuous quantities are stored in an analog manner or discretized and stored in a set of discrete attractors in order to investigate the important issue of how to differentiate the two systems here we compare the neuronal spiking activity that arises from a continuous line attractor with that from a series of discrete attractors stochastic fluctuations cause the position of the system along its continuous attractor to vary as a random walk whereas in a discrete attractor noise causes spontaneous transitions to occur between discrete states at random intervals we calculate the statistics of spike trains of neurons firing as a poisson process with rates that vary according to the underlying attractor network since individual neurons fire spikes probabilistically and since the state of the network as a whole drifts randomly the spike trains of individual neurons follow a doubly stochastic poisson point process we compare the series of spike trains from the two systems using the autocorrelation function fano factor and interspike interval isi distribution although the variation in rate can be dramatically different especially for short time intervals surprisingly both the autocorrelation functions and fano factors are identical given appropriate scaling of the noise terms since the range of firing rates is limited in neurons we also investigate systems for which the variation in rate is bounded by either rigid limits or because of leak to a single attractor state such as the ornstein uhlenbeck process in these cases the time dependence of the variance in rate can be different between discrete and continuous systems so that in principle these processes can be distinguished using second order spike statistics

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

replica symmetric theory of nonlinear analogue neural networks on the basis of the theory of the naive mean field model of spin glasses analogue neural networks of the hopfield type are investigated in the saturation limit the saddle point equations for the order parameters describing retrieval and spin glass phases of the networks are obtained by means of a statistical mechanical analysis within the framework of the replica symmetric theory and are shown to undergo a modification to that of ags theory due to the absence of the onsager reaction term in the tap equation based on the equations the memory storage capacity of the networks is analysed as a function of the analogue gain beta a small increase in the critical storage capacity is found for finite values of beta compared with that of the ising model networks with corresponding inverse temperature although the qualitative nature of the phase diagram is unchanged

a quasi optimal non overlapping domain decomposition algorithm for the helmholtz equation this paper presents a new non overlapping domain decomposition method for the helmholtz equation whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the dirichlet to neumann operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions

a uniformly accurate multiscale time integrator spectral method for the kleingordonzakharov system in the high plasma frequency limit regime a multiscale time integrator sine pseudospectral mti sp method is presented for discretizing the kleingordonzakharov kgz system with a dimensionless parameter 0 1 which is inversely proportional to the plasma frequency in the high plasma frequency limit regime i e 0 1 the solution of the kgz system propagates waves with amplitude at o 1 and wavelength at o 2 in time and o 1 in space which causes significantly numerical burdens due to the high oscillation in time the main idea of the numerical method is to carry out a multiscale decomposition by frequency mdf to the electric field component of the solution at each time step and then apply the sine pseudospectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the mdf and the equation of the ion density component the method is explicit and easy to be implemented extensive numerical results show that the mti sp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o for 0 1 with time step size and optimally with quadratic convergence rate at o 2 in the regime when either o 1 or 0 thus the meshing strategy requirement or scalability of the mti sp for the kgz system in the high plasma frequency limit regime is o 1 and h o 1 for 0 1 which is significantly better than classical methods in the literatures finally we apply the mti sp method to study the convergence rates of the kgz system to its limiting models in the high plasma frequency limit and the interactions of bright solitons of the kgz system and to identify certain parameter regimes that the solution of the kgz system will be blow up in one dimension

cross correlations in high conductance states of a model cortical network neuronal firing correlations are studied using simulations of a simple network model for a cortical column in a high conductance state with dynamically balanced excitation and inhibition although correlations between individual pairs of neurons exhibit considerable heterogeneity population averages show systematic behavior when the network is in a stationary state the average correlations are generically small correlation coefficients are of order 1 n where n is the number of neurons in the network however when the input to the network varies strongly in time much larger values are found in this situation the network is out of balance and the synaptic conductance is low at times when the strongest firing occurs however examination of the correlation functions of synaptic currents reveals that after these bursts balance is restored within a few milliseconds by a rapid increase in inhibitory synaptic conductance these findings suggest an extension of the notion of the balanced state to include balanced fluctuations of synaptic currents with a characteristic timescale of a few milliseconds

conversion of temporal correlations between stimuli to spatial correlations between attractors it is shown that a simple modification of synaptic structures of the hopfield type constructed to produce autoassociative attractors produces neural networks whose attractors are correlated with several learned patterns used in the construction of the matrix the modification stores in the matrix a fixed sequence of uncorrelated patterns the network then has correlated attractors provoked by the uncorrelated stimuli thus the network converts the temporal order or temporal correlation expressed by the sequence of patterns into spatial correlations expressed in the distributions of neural activities in attractors the model captures phenomena observed in single electrode recordings in performing monkeys by miyashita et al the correspondence is as close as to reproduce the fact that given uncorrelated patterns as sequentially learned stimuli the attractors produced are significantly correlated up to a separation of 5 five in the sequence this number 5 is universal in a range of parameters and requires essentially no tuning we then discuss learning scenarios that could lead to this synaptic structure as well as experimental predictions following from it finally we speculate on the cognitive utility of such an arrangement

stochastic dynamics of a finite size spiking neural network we present a simple markov model of spiking neural dynamics that can be analytically solved to characterize the stochastic dynamics of a finite size spiking neural network we give closed form estimates for the equilibrium distribution mean rate variance and autocorrelation function of the network activity the model is applicable to any network where the probability of firing of a neuron in the network depends on only the number of neurons that fired in a previous temporal epoch networks with statistically homogeneous connectivity and membrane and synaptic time constants that are not excessively long could satisfy these conditions our model completely accounts for the size of the network and correlations in the firing activity it also allows us to examine how the network dynamics can deviate from mean field theory we show that the model and solutions are applicable to spiking neural networks in biophysically plausible parameter regimes

retrieval properties of bidirectional associative memories we study a two layer neural network made of n and m n neurons producing a two way association search for a family of p n patterns where each pattern is a pair of two independent sub categories of information having respectively n and m n components in terms of the ratio limn m n n we study the retrieval capability of this network and show that there exists at least three regimes of association for which we determine the evolution of the threshold c of the storage capacity limn p n n

stimulus dependent correlations in threshold crossing spiking neurons we consider a threshold crossing spiking process as a simple model for the activity within a population of neurons assuming that these neurons are driven by a common fluctuating input with gaussian statistics we evaluate the cross correlation of spike trains in pairs of model neurons with different thresholds this correlation function tends to be asymmetric in time indicating a preference for the neuron with the lower threshold to fire before the one with the higher threshold even if their inputs are identical the relationship between these results and spike statistics in other models of neural activity is explored in particular we compare our model with an integrate and fire model in which the membrane voltage resets following each spike the qualitative properties of spike cross correlations emerging from the threshold crossing model are similar to those of bursting events in the integrate and fire model this is particularly true for generalized integrate and fire models in which spikes tend to occur in bursts as observed for example in retinal ganglion cells driven by a rapidly fluctuating visual stimulus the threshold crossing model thus provides a simple analytically tractable description of event onsets in these neurons

uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear schrodinger equation with wave operator we propose an exponential wave integrator sine pseudospectral ewi sp method for the nonlinear schrodinger equation nls with wave operator nlsw and carry out rigorous error analysis the nlsw is nls perturbed by the wave operator with strength described by a dimensionless parameter varepsilon in 0 1 as varepsilon to0 the nlsw converges to the nls and for the small perturbation i e 0 varepsilon ll1 the solution of the nlsw differs from that of the nls with a function oscillating in time with o varepsilon 2 wavelength at o varepsilon 2 and o varepsilon 4 amplitudes for ill prepared and well prepared initial data respectively this rapid oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in varepsilon in this work we show that the proposed ewi sp possesses the optimal uniform error bounds at o tau 2 and o tau in tau time step for well prepared initial data and ill prepared initial

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 master equation formalism for macroscopic modeling of asynchronous irregular activity states many efforts have been devoted to modeling asynchronous irregular ai activity states which resemble the complex activity states seen in the cerebral cortex of awake animals most of models have considered balanced networks of excitatory and inhibitory spiking neurons in which ai states are sustained through recurrent sparse connectivity with or without external input in this letter we propose a mesoscopic description of such ai states using master equation formalism we derive a second order mean field set of ordinary differential equations describing the temporal evolution of randomly connected balanced networks this formalism takes into account finite size effects and is applicable to any neuron model as long as its transfer function can be characterized we compare the predictions of this approach with numerical simulations for different network configurations and parameter spaces considering the randomly connected network as a unit this approach could be used to build large scale networks of such connected units with an aim to model activity states constrained by macroscopic measurements such as voltage sensitive dye imaging

self consistent formulations for stochastic nonlinear neuronal dynamics neural dynamics is often investigated with tools from bifurcation theory however many neuron models are stochastic mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals noise changes the dynamics with respect to the deterministic model in particular classical bifurcation theory cannot be applied we formulate the stochastic neuron dynamics in the martin siggia rose de dominicis janssen msrdj formalism and present the fluctuation expansion of the effective action and the functional renormalization group frg as two systematic ways to incorporate corrections to the mean dynamics and time dependent statistics due to fluctuations in the presence of nonlinear neuronal gain to formulate self consistency equations we derive a fundamental link between the effective action in the onsager machlup om formalism which allows the study of phase transitions and the msrdj effective action which is computationally advantageous these results in particular allow the derivation of an om effective action for systems with non gaussian noise this approach naturally leads to effective deterministic equations for the first moment of the stochastic system they explain how nonlinearities and noise cooperate to produce memory effects moreover the msrdj formulation yields an effective linear system that has identical power spectra and linear response starting from the better known loopwise approximation we then discuss the use of the frg as a method to obtain self consistency beyond the mean we present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the blaizot m endez and wschebor approximation from the derivative expansion to the vertex expansion the methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

stochastic neural field theory and the system size expansion we analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of n identical neurons the state of the network is specified by the fraction of active or spiking neurons in each population and transition rates are chosen so that in the thermodynamic or deterministic limit n rightarrow infty we recover standard activity based or voltage based rate models we derive the lowest order corrections to these rate equations for large but finite n using two different approximation schemes one based on the van kampen system size expansion and the other based on path integral methods both methods yield the same series expansion of the moment equations which at mathcal o 1 n can be truncated to form a closed system of equations for the first and second order moments taking a continuum limit of the moment equations while keeping the system size n fixed generates a system of integrodifferential equations for the

theory of correlations in stochastic neural networks one of the main experimental tools in probing the interactions between neurons has been the measurement of the correlations in their activity in general however the interpretation of the observed correlations is difficult since the correlation between a pair of neurons is influenced not only by the direct interaction between them but also by the dynamic state of the entire network to which they belong thus a comparison between the observed correlations and the predictions from specific model networks is needed in this paper we develop a theory of neuronal correlation functions in large networks comprising several highly connected subpopulations and obeying stochastic dynamic rules when the networks are in asynchronous states the cross correlations are relatively weak i e their amplitude relative to that of the autocorrelations is of order of 1 n n being the size of the interacting populations using the weakness of the cross correlations general equations that express the matrix of cross correlations in terms of the mean neuronal activities and the effective interaction matrix are presented the effective interactions are the synaptic efficacies multiplied by the gain of the postsynaptic neurons the time delayed cross correlation matrix can be expressed as a sum of exponentially decaying modes that correspond to the nonorthogonal eigenvectors of the effective interaction matrix the theory is extended to networks with random connectivity such as randomly dilute networks this allows for a comparison between the contribution from the internal common input and that from the direct interactions to the correlations of monosynaptically coupled pairs a closely related quantity is the linear response of the neurons to external time dependent perturbations we derive the form of the dynamic linear response function of neurons in the above architecture in terms of the eigenmodes of the effective interaction matrix the behavior of the correlations and the linear response when the system is near a bifurcation point is analyzed near a saddle node bifurcation the correlation matrix is dominated by a single slowly decaying critical mode near a hopf bifurcation the correlations exhibit weakly damped sinusoidal oscillations the general theory is applied to the case of a randomly dilute network consisting of excitatory and inhibitory subpopulations using parameters that mimic the local circuit of 1 mathrm mm 3 of the rat neocortex both the effect of dilution as well as the influence of a nearby bifurcation to an oscillatory state are demonstrated

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

recurrent biological neural networks the weak and noisy limit a perturbative method is developed for calculating the effects of recurrent synaptic interactions between neurons embedded in a network a series expansion is constructed that converges for networks with noisy membrane potential and weak synaptic connectivity the terms of the series can be interpreted as loops of interactions between neurons so the technique is called a loop expansion a diagrammatic method is introduced that allows for construction of analytic expressions for the parameter dependencies of the spike probability function and correlation functions an analytic expression is obtained to predict the effect of the surrounding network on a neuron during an intracellular current injection the analytic results are compared with simulations to test the range of their validity and significant effects of the the recurrent connections in network are accurately predicted by the loop expansion

two level domain decomposition methods with lagrange multipliers for the fast iterative solution of acoustic scattering problems we present two different but related lagrange multiplier based domain decomposition dd methods for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting feti method to indefinite or complex problems the first method employs a single lagrange multiplier field to glue the local solutions at the subdomain interface boundaries the second method employs two lagrange multiplier fields for that purpose the key ingredients of both of these feti methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary and the preconditioning of the global interface problem by a coarse second level problem constructed with planar waves we show numerically that both methods are scalable with respect to the mesh size the subdomain size and the wavenumber but that the feti method with a single lagrange multiplier field labeled feti h h for helmholtz in this paper delivers superior computational performances we apply the feti h method to the parallel solution on a 24 processor origin 2000 of an acoustic scattering problem with a submarine shaped obstacle and report performance results that highlight the unique efficiency of this dd method for the solution of high frequency acoustic scattering problems

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

condition number estimates for the nonoverlapping optimized schwarz method and the 2 lagrange multiplier method for general domains and cross points the optimized schwarz method and the closely related 2 lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations although these methods are known to work well in special cases e g when the domain is a square and the two subdomains are rectangles the problem has never been systematically stated nor analyzed for general domains with general subdomains the problem of cross points when three or more subdomains meet at a single vertex has been particularly vexing we introduce a 2 lagrange multiplier method for domain decompositions with cross points we estimate the condition number of the iteration and provide an optimized robin parameter for general domains we hope that this new systematic theory will allow broader utilization of optimized schwarz and 2 lagrange multiplier preconditioners

systematic fluctuation expansion for neural network activity equations population rate or activity equations are the foundation of a common approach to modeling for neural networks these equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity the shortcoming of these equations is that they take into account only the average firing rate while leaving out higher order statistics like correlations between firing a stochastic theory of neural networks that includes statistics at all orders was recently formulated we describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms each level of the approximation yields closed equations they depend only on the mean and specific correlations of interest without an ad hoc criterion for doing so we show in an example of an all to all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

convergence condition of the tap equation for the infinite ranged ising spin glass model it is shown that the power expansion of the gibbs potential of the sk model up to second order in the exchange couplings leads to the tap equation this result remains valid for the general including a ferromagnetic exchange sk model theorems of power expansions and resolvent techniques are employed to solve the convergence problem the convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations

a combination of multiscale time integrator and two scale formulation for the nonlinear schrodinger equation with wave operator abstract in this paper we consider the nonlinear schrodinger equation with wave operator nlsw which contains a dimensionless parameter 0 e 1 as 0 e 1 the solution of the nlsw propagates fast waves in time with wavelength o e 2 and the problem becomes highly oscillatory in time the oscillations come from two parts one part is from the equation and another part is from the initial data for the ill prepared initial data case as described in bao and cai 2014 which brings inconsistency in the limit regime standard numerical methods have strong convergence order reduction in time when e becomes small we review two existing methods to solve the nlsw an exponential integrator and a two scale method we comment on their order reduction issues then we derive a multiscale decomposition two scale method for solving the nlsw by first performing a multiscale decomposition on the nlsw which decomposes it into a well behaved part and an energy unbounded part and then applying an exponential integrator for the well behaved part and a two scale approach for the energy unbounded part numerical experiments are conducted to test the proposed method which shows uniform second order accuracy without significant order reduction for all 0 e 1 comparisons are made with the existing methods

chaotic balanced state in a model of cortical circuits the nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are not well understood we consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells we study a network model with excitatory and inhibitory populations of simple binary units the internal feedback is mediated by relatively large synaptic strengths so that the magnitude of the total excitatory and inhibitory feedback is much larger than the neuronal threshold the connectivity is random and sparse the mean number of connections per unit is large though small compared to the total number of cells in the network the network also receives a large temporally regular input from external sources we present an analytical solution of the mean field theory of this model which is exact in the limit of large network size this theory reveals a new cooperative stationary state of large networks which we term a balanced state in thi

mean field dynamics of sequence processing neural networks with finite connectivity abstract a recent dynamic mean field theory for sequence processing in fully connected neural networks of hopfield type is extended and analyzed here for a symmetrically diluted network with finite connectivity near saturation equations for the dynamics and the stationary states are obtained for the macroscopic observables and the precise equivalence is established with the single pattern retrieval problem in a layered feed forward network with finite connectivity

absorbing interface conditions for domain decomposition methods a general presentation the continuity conditions and the transmission conditions involved in domain decomposition methods are of major importance for the fast and robust convergence of these algorithms in this review paper we study step by step the methodology to derive various continuity conditions and compatible transmission conditions for two different types of problems i e the laplace equation and the helmholtz equation an original homogeneous formulation is also presented both in the continuous and in the discrete analysis numerical experiments show the relative efficiency of these continuity conditions and of these transmission conditions on academic problems

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

coexistence of uncorrelated and correlated attractors in a nonmonotonic neural network a neural network with correlated attractors ala griniasty et al 1993 conversion of temporal correlations between stimuli to spatial correlations between attractors neural comp 5 1 17 is studied for a nonmonotonic response function in the cases of both finite and extensive memory loading in the finite loading case with symmetric synaptic connections the number of neighbouring memory patterns correlated with an attractor tends to be decreased as the degree of nonmonotonicity is increased to study the extensive loading case we generalize the previously proposed self consistent signal to noise analysis to associative memory models with a general class of local hebbian synaptic connections applying the results to the present model we derive analytic expressions for the equilibrium order parameters new finding includes the coexistence of uncorrelated and correlated attractors in the obtained phase diagram this fact which is confirmed by extensive numerical simulations may have significant implications for long term visual memory

efficient learning in boltzmann machines using linear response theory the learning process in boltzmann machines is computationally very expensive the computational complexity of the exact algorithm is exponential in the number of neurons we present a new approximate learning algorithm for boltzmann machines based on mean field theory and the linear response theorem the computational complexity of the algorithm is cubic in the number of neurons in the absence of hidden units we show how the weights can be directly computed from the fixed point equation of the learning rules thus in this case we do not need to use a gradient descent procedure for the learning process we show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role finally we apply the method to a pattern completion task and show good performance for networks up to 100 neurons

low firing rates an effective hamiltonian for excitatory neurons the authors analyse the behaviour of an attractor neural network which exhibits low mean temporal activity levels despite the fact that the intrinsic neuronal cycle time is very short 2 3 ms information and computation are represented on the excitatory neurons only the influence of inhibitory neurons which are assumed to react on a shorter timescale than the excitatory ones is expressed as an effective interaction of the excitatory neurons this leads to an effective model which describes the interplay of excitation and inhibition acting on excitatory neurons in terms of the excitatory neural variables alone the network operates in the presence of fast noise which is large relative to the frozen randomness induced by the stored patterns the overall fraction of active neurons is controlled by a single free parameter which expresses the relative strength of the effective inhibition associative retrieval is identified as usual with the breakdown of ergodicity in the dynamics of the network in particular with the presence of dynamical attractors corresponding to the retrieval of a given pattern in such an attractor the activity of neurons corresponding to active sites in the stored patterns increases at the expense of other neurons yet only a small fraction of the neurons active in the pattern are in the active state in each elementary time cycle and they vary from cycle to cycle in an uncorrelated fashion due to the noise hence the observed mean activity rate of any individual neuron is kept low this scenario is demonstrated by an analytical study based on the replica method and the results are tested by numerical simulations

temporal correlations in stochastic networks of spiking neurons the determination of temporal and spatial correlations in neuronal activity is one of the most important neurophysiological tools to gain insight into the mechanisms of information processing in the brain its interpretation is complicated by the difficulty of disambiguating the effects of architecture single neuron properties and network dynamics we present a theory that describes the contribution of the network dynamics in a network of spiking neurons for a simple neuron model including refractory properties we calculate the temporal cross correlations in a completely homogeneous excitatory fully connected network in a stable stationary state for stochastic dynamics in both discrete and continuous time we show that even for this simple network architecture the crosscorrelations exhibit a large variety of qualitatively different properties strongly dependent on the level of noise the decay constant of the refractory function and the network activity at the critical point the crosscorrelations oscillate with a frequency that depends on the refractory properties or decay exponentially with a diverging damping constant for weak refractory properties we also investigate the effect of the synaptic time constants it is shown that these time constants may apart from their influence on the asymmetric peak arising from the direct synaptic connection also affect the long term properties of the cross correlations

capacity of networks with correlated attractors we analyse the extensive loading of the neural network model proposed to describe neurophysiological experiments in which correlated attractors associated to uncorrelated patterns are found the phase diagram is obtained and discussed some generalizations of the original model are also considered in all the cases we demonstrate the existence of a region in the phase diagram with correlated attractors results from numerical simulations which confirm the mean field theory results are also presented

information geometric measures for estimation of connection weight under correlated inputs the brain processes information in a highly parallel manner determination of the relationship between neural spikes and synaptic connections plays a key role in the analysis of electrophysiological data information geometry ig has been proposed as a powerful analysis tool for multiple spike data providing useful insights into the statistical interactions within a population of neurons previous work has demonstrated that ig measures can be used to infer the connection weight between two neurons in a neural network this property is useful in neuroscience because it provides a way to estimate learning induced changes in synaptic strengths from extracellular neuronal recordings a previous study has shown however that this property would hold only when inputs to neurons are not correlated since neurons in the brain often receive common inputs this would hinder the application of the ig method to real data we investigated the two neuron ig measures in higher order log linear models to overcome this limitation first we mathematically showed that the estimation of uniformly connected synaptic weight can be improved by taking into account higher order log linear models second we numerically showed that the estimation can be improved for more general asymmetrically connected networks considering the estimated number of the synaptic connections in the brain we showed that the two neuron ig measure calculated by the fourth or fifth order log linear model would provide an accurate estimation of connection strength within approximately a 10 error these studies suggest that the two neuron ig measure with higher order log linear expansion is a robust estimator of connection weight even under correlated inputs providing a useful analytical tool for real multineuronal spike data

error estimates in the energy space for a gautschi type integrator spectral discretization for the coupled nonlinear klein gordon equations we propose and analyze an efficient and accurate numerical method for solving the coupled nonlinear klein gordon equations the method is based on the application of a gautschi type exponential integrator in time combined with sine spectral discretization in space the main results achieved in this paper are the rigorous error estimates in the energy space h 1 h 1 for the proposed scheme numerical tests are reported and agree with the error estimates quite well

constrained optimization for neural map formation a unifying framework for weight growth and normalization computational models of neural map formation can be considered on at least three different levels of abstraction detailed models including neural activity dynamics weight dynamics that abstract from the neural activity dynamics by an adiabatic approximation and constrained optimization from which equations governing weight dynamics can be derived constrained optimization uses an objective function from which a weight growth rule can be derived as a gradient flow and some constraints from which normalization rules are derived in this article we present an example of how an optimization problem can be derived from detailed nonlinear neural dynamics a systematic investigation reveals how different weight dynamics introduced previously can be derived from two types of objective function terms and two types of constraints this includes dynamic link matching as a special case of neural map formation we focus in particular on the role of coordinate transformations to derive different weight dynamics from the same optimization problem several examples illustrate how the constrained optimization framework can help in understanding generating and comparing different models of neural map formation the techniques used in this analysis may also be useful in investigating other types of neural dynamics

theory of correlation in a network with synaptic depression synaptic depression affects not only the mean responses of neurons but also the correlation of response variability in neural populations although previous studies have constructed a theory of correlation in a spiking neuron model by using the mean field theory framework synaptic depression has not been taken into consideration we expanded the previous theoretical framework in this study to spiking neuron models with short term synaptic depression on the basis of this theory we analytically calculated neural correlations in a ring attractor network with mexican hat type connectivity which was used as a model of the primary visual cortex the results revealed that synaptic depression reduces neural correlation which could be beneficial for sensory coding furthermore our study opens the way for theoretical studies on the effect of interaction change on the linear response function in large stochastic networks

investigation of possible neural architectures underlying information geometric measures a novel analytical method based on information geometry was recently proposed and this method may provide useful insights into the statistical interactions within neural groups the link between informationgeometric measures and the structure of neural interactions has not yet been elucidated however because of the ill posed nature of the problem here possible neural architectures underlying information geometric measures are investigated using an isolated pair and an isolated triplet of model neurons by assuming the existence of equilibrium states we derive analytically the relationship between the information geometric parameters and these simple neural architectures for symmetric networks the first and second order information geometric parameters represent respectively the external input and the underlying connections between the neurons provided that the number of neurons used in the parameter estimation in the log linear model and the number of neurons in the network are the same for asymmetric networks however these parameters are dependent on both the intrinsic connections and the external inputs to each neuron in addition we derive the relation between the information geometric parameter corresponding to the two neuron interaction and a conventional cross correlation measure we also show that the information geometric parameters vary depending on the number of neurons assumed for parameter estimation in the log linear model this finding suggests a need to examine the information geometric method carefully a possible criterion for choosing an appropriate orthogonal coordinate is also discussed this article points out the importance of a model based approach and sheds light on the possible neural structure underlying the application of information geometry to neural network analysis