estimating a state space model from point process observations a widely used signal processing paradigm is the state space model the state space model is defined by two equations an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time inspired by neurophysiology experiments in which neural spiking activity is induced by an implicit latent stimulus we develop an algorithm to estimate a state space model observed through point process measurements we represent the latent process modulating the neural spiking activity as a gaussian autoregressive model driven by an external stimulus given the latent process neural spiking activity is characterized as a general point process defined by its conditional intensity function we develop an approximate expectation maximization em algorithm to estimate the unobservable state space process its parameters and the parameters of the point process the em algorithm combines a point process recursive nonlinear filter algorithm the fixed interval smoothing algorithm and the state space covariance algorithm to compute the complete data log likelihood efficiently we use a kolmogorov smirnov test based on the time rescaling theorem to evaluate agreement between the model and point process data we illustrate the model with two simulated data examples an ensemble of poisson neurons driven by a common stimulus and a single neuron whose conditional intensity function is approximated as a local bernoulli process

rate limitations of unitary event analysis unitary event analysis is a new method for detecting episodes of synchronized neural activity riehle gruun diesmann aertsen 1997 it detects time intervals that contain coincident firing at higher rates than would be expected if the neurons fired as independent inhomogeneous poisson processes all coincidences in such intervals are called unitary events ues changes in the frequency of ues that are correlated with behavioral states may indicate synchronization of neural firing that mediates or represents the behavioral state we show that ue analysis is subject to severe limitations due to the underlying discrete statistics of the number of coincident events these limitations are particularly stringent for low 0 10 spikes s firing rates under these conditions the frequency of ues is a random variable with a large variation relative to its mean the relative variation decreases with increasing firing rate and we compute the lowest firing rate at which the 95 confidence interval around the mean frequency of ues excludes zero this random variation in ue frequency makes interpretation of changes in ues problematic for neurons with low firing rates as a typical example when analyzing 150 trials of an experiment using an averaging window 100 ms wide and a 5ms coincidence window firing rates should be greater than 7 spikes per second

neural coding higher order temporal patterns in the neurostatistics of cell assemblies recent advances in the technology of multiunit recordings make it possible to test hebb s hypothesis that neurons do not function in isolation but are organized in assemblies this has created the need for statistical approaches to detecting the presence of spatiotemporal patterns of more than two neurons in neuron spike train data we mention three possible measures for the presence of higher order patterns of neural activation coef ficients of log linear models connected cumulants and redundancies and present arguments in favor of the coefficients of loglinear models we present test statistics for detecting the presence of higher order interactions in spike train data by parameterizing these interactions in terms of coefficients of log linear models we also present a bayesian approach for inferring the existence or absence of interactions and estimating their strength the two methods the frequentist and the bayesian one are shown to be consistent in the sense that interactions that are detected by either method also tend to be detected by the other a heuristic for the analysis of temporal patterns is also proposed finally a bayesian test is presented that establishes stochastic differences between recorded segments of data the methods are applied to experimental data and synthetic data drawn from our statistical models our experimental data are drawn from multiunit recordings in the prefrontal cortex of behaving monkeys the somatosensory cortex of anesthetized rats and multiunit recordings in the visual cortex of behaving monkeys

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

a flexible method to measure synchrony in neuronal firing neurons can transmit information about the characteristics of a stimulus through the spike rate of neurons and synchronization of the neurons various association measure can be used to describe how synchronous two spike trains are we propose a new measure of synchrony the conditional synchrony measure which is the probability of firing together given that at least one of the two neurons is active focus is on the specification of a flexible marginal model for multivariate correlated binary data together with a pseudolikelihood estimation approach to adequately and directly describe the measures of interest a joint model must allow different time and covariate dependent firing rates for each neuron and also must account for the association between them the association between neurons might depend on covariates as well

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

function gradient and hessian recovery using quadratic edge bump functions an approximate error function for the discretization error on a given mesh is obtained by projecting via the energy inner product the functional residual onto the space of continuous piecewise quadratic functions which vanish on the vertices of the mesh conditions are given under which one can expect this hierarchical basis error estimator to give efficient and reliable function recovery asymptotically exact gradient recovery and convergent hessian recovery in the square norms one does not find similar function recovery results in the literature the analysis given here is based on a certain superconvergence result which has been used elsewhere in the analysis of gradient recovery methods numerical experiments are provided which demonstrate the effectivity of the approximate error function in practice

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

multiple tests based on a gaussian approximation of the unitary events method with delayed coincidence count the unitary events ue method is one of the most popular and efficient methods used over the past decade to detect patterns of coincident joint spike activity among simultaneously recorded neurons the detection of coincidences is usually based on binned coincidence count grun 1996 which is known to be subject to loss in synchrony detection grun diesmann grammont riehle aertsen 1999 this defect has been corrected by the multiple shift coincidence count grun et al 1999 the statistical properties of this count have not been further investigated until this work the formula being more difficult to deal with than the original binned count first we propose a new notion of coincidence count the delayed coincidence count which is equal to the multiple shift coincidence count when discretized point processes are involved as models for the spike trains moreover it generalizes this notion to nondiscretized point processes allowing us to propose a new gaussian approximation of the count since unknown parameters are involved in the approximation we perform a plug in step where unknown parameters are replaced by estimated ones leading to a modification of the approximating distribution finally the method takes the multiplicity of the tests into account via a benjamini and hochberg approach benjamini hochberg 1995 to guarantee a prescribed control of the false discovery rate we compare our new method mtgaue multiple tests based on a gaussian approximation of the unitary events and the ue method proposed in grun et al 1999 over various simulations showing that mtgaue extends the validity of the previous method in particular mtgaue is able to detect both profusion and lack of coincidences with respect to the independence case and is robust to changes in the underlying model furthermore mtgaue is applied on real data

discretisation and control of polytopic systems with uncertain sampling rates and network induced delays this paper investigates the problems of uncertain sampling rate discretisation and the networked control of uncertain time invariant continuous time linear systems in polytopic domains the sampling period is assumed to be unknown but belonging to a given interval to avoid the difficulty of dealing with the exponential of uncertain matrices a discrete time model is obtained by applying a taylor series expansion of degree l to the original system the resulting discrete time model is composed of homogeneous polynomial matrices with parameters lying in the cartesian product of simplexes called a multi simplex plus an additive norm bounded term representing the discretisation residual error the original continuous time system is controlled through a communication network that introduces a time delay in the process linear matrix inequality relaxations that include a scalar parameter search are proposed for the design of a digital robust state feedback controller that guarantees the closed loop stability

asymptotically exact a posteriori error estimators part i grids with superconvergence in part i of this work we develop superconvergence estimates for piecewise linear finite element approximations on quasi uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in particular we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant ui we then analyze a postprocessing gradient recovery scheme showing that q h nabla u h is a superconvergent approximation to nabla u here qh is the global l2 projection in part ii we analyze a superconvergent gradient recovery scheme for general unstructured shape regular triangulations this is the foundation for an a posteriori error estimate and local error indicators

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

mesh smoothing using a posteriori error estimates we develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations the algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations in this paper we derive the method present some numerical illustrations and give a brief analysis of the issue of uniqueness

asymptotically exact functional error estimators based on superconvergent gradient recovery the use of dual adjoint problems for approximating functionals of solutions of pdes with great accuracy or to merely drive a goal oriented adaptive refinement scheme has become well accepted and it continues to be an active area of research the traditional approach involves dual residual weighting drw in this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact the first is of drw type and is derived for meshes in which most triangles satisfy an inlinemediaobject not available see fulltext approximate parallelogram property the second functional estimator involves dual error estimate weighting dew using any superconvergent gradient recovery technique for the primal and dual solutions several experiments are done which demonstrate the asymptotic exactness of a dew estimator which uses a gradient recovery scheme proposed by bank and xu and the effectiveness of refinement done with respect to the corresponding local error indicators

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

correlations without synchrony peaks in spike train correlograms are usually taken as indicative of spike timing synchronization between neurons strictly speaking however a peak merely indicates that the two spike trains were not independent two biologically plausible ways of departing from independence that are capable of generating peaks very similar to spike timing peaks are described here covariations over trials in response latency and covariations over trials in neuronal excitability since peaks due to these interactions can be similar to spike timing peaks interpreting a correlogram may be a problem with ambiguous solutions what peak shapes do latency or excitability interactions generate when are they similar to spike timing peaks when can they be ruled out from having caused an observed correlogram peak these are the questions addressed here the previous article in this issue proposes quantitative methods to tell cases apart when latency or excitability covariations cannot be ruled out

a posteriori error estimates based on the polynomial preserving recovery superconvergence of order o h 1 rho for some rho 0 is established for the gradient recovered with the polynomial preserving recovery ppr when the mesh is mildly structured consequently the ppr recovered gradient can be used in building an asymptotically exact a posteriori error estimator

function value recovery and its application in eigenvalue problems function value recovery techniques for linear finite elements are discussed using the recovered function and its gradient we are able to enhance the eigenvalue approximation and increase its convergence rate to h 2 alpha where alpha 1 is the superconvergence rate of the recovered gradient this is true in both symmetric and nonsymmetric eigenvalue problems

superconvergence analysis for the explicit polynomial recovery method a new recovery technique explicit polynomial recovery epr is analyzed for finite element methods epr reconstructs the value at edge centers by solving a local problem in combination with the finite element solution at the vertex a quadratic approximation is constructed besides improving the accuracy it can also be applied in building the epr based error estimator for the poisson equation the element center is a superconvergent point of the gradient of the epr recovered function on an equilateral triangulation numerical examples are presented to verify the theoretical results and to show the performance of the epr in the adaptive finite element method

superconvergence of quadratic finite elements on mildly structured grids superconvergence estimates are studied in this paper on quadratic finite element discretizations for second order elliptic boundary value problems on mildly structured triangular meshes for a large class of practically useful grids the finite element solution u h is proven to be superclose to the inter polant u i and as a result a postprocessing gradient recovery scheme for u h can be devised the analysis is based on a number of carefully derived identities in addition to its own theoretical interests the result in this paper can be used for deriving asymptotically exact a posteriori error estimators for quadratic finite element methods

how a nonconvergent recovered hessian works in mesh adaptation hessian recovery has been commonly used in mesh adaptation for obtaining the required magnitude and direction information of the solution error unfortunately a recovered hessian from a linear finite element approximation is nonconvergent in general as the mesh is refined it has been observed numerically that adaptive meshes based on such a nonconvergent recovered hessian can nevertheless lead to an optimal error in the finite element approximation this also explains why hessian recovery is still widely used despite its nonconvergence in this paper we develop an error bound for the linear finite element solution of a general boundary value problem under a mild assumption on the closeness of the recovered hessian to the exact one numerical results show that this closeness assumption is satisfied by the recovered hessian obtained with commonly used hessian recovery methods moreover it is shown that the finite element error changes gradually with the closeness of the recovered hessian this provides a

unitary events in multiple single neuron spiking activity i detection and significance it has been proposed that cortical neurons organize dynamically into functional groups cell assemblies by the temporal structure of their joint spiking activity here we describe a novel method to detect conspicuous patterns of coincident joint spike activity among simultaneously recorded single neurons the statistical significance of these unitary events of coincident joint spike activity is evaluated by the joint surprise the method is tested and calibrated on the basis of simulated stationary spike trains of independently firing neurons into which coincident joint spike events were inserted under controlled conditions the sensitivity and specificity of the method are investigated for their dependence on physiological parameters firing rate coincidence precision coincidence pattern complexity and temporal resolution of the analysis in the companion article in this issue we describe an extension of the method designed to deal with nonstationary firing rates

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

a posteriori error estimation of hierarchical type for the schrodinger operator with inverse square potential we develop an a posteriori error estimate for mixed boundary value problems of the form delta fancyscript v u f v u f where the potential fancyscript v v may possess inverse square singularities at finitely many points in the domain we prove that our error estimate can be efficiently computed and is asymptotically identical to the actual error in the energy norm on a family of geometrically graded meshes appropriate for singular solutions of such problems therefore our estimate can be used for a practical stopping criterion a variety of numerical experiments support our theoretical results we also offer a direct convergence and effectivity comparison between the geometrically graded meshes which are based on a priori knowledge of possible singularities in the solution and adaptively refined meshes driven by local error indicators associated with our a posteriori error estimate

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

information geometric measure for neural spikes this study introduces information geometric measures to analyze neural firing patterns by taking not only the second order but also higher order interactions among neurons into account information geometry provides useful tools and concepts for this purpose including the orthogonality of coordinate parameters and the pythagoras relation in the kullback leibler divergence based on this orthogonality we show a novel method for analyzing spike firing patterns by decomposing the interactions of neurons of various orders as a result purely pairwise triple wise and higher order interactions are singled out we also demonstrate the benefits of our proposal by using several examples

a new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates a new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented it generates anisotropic adaptive meshes as quasi uniform ones in some metric space with the metric tensor being computed based on hierarchical a posteriori error estimates a global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution instead of solving the global error problem exactly which is costly in general we solve it iteratively using the symmetric gaus seidel method numerical results show that a few gs iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation the new method is compared with several strategies using local error estimators or recovered hessians numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients

asymptotically exact a posteriori error estimators part ii general unstructured grids in part i of this work siam j numer anal 41 2003 pp 2294 2312 we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in this work we consider superconvergence for general unstructured but shape regular meshes we develop a postprocessing gradient recovery scheme for the finite element solution uh inspired in part by the smoothing iteration of the multigrid method this recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact next we use the superconvergent gradient to approximate the hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms we provide several numerical examples illustrating the effectiveness of our procedures

on estimators for eigenvalue eigenvector approximations we consider a large class of residuum based a posteriori eigen value eigenvector estimates and present an abstract framework for proving their asymptotic exactness equivalence of the estimator and the error is also established to demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for laplace eigenvalue problems in planar polygonal regions to this end we develop new error analysis for the galerkin approximation which avoids the use of the strengthened cauchy schwarz inequality and the saturation assumption and gives reasonable and explicitly computable upper bounds on the discretization error a brief discussion is also given concerning the design of estimators which are in the same spirit but are based on different a posteriori techniques notably those of gradient recovery type

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

open loop discretization methods for control systems design in this paper a unified theory is presented which addresses essential aspects of the open loop discretization procedure initially factors which affect the generation and propagation of discretization errors are identified by analytical heuristic and experimental arguments following this a discretization algorithm is presented which takes these factors into account the fundamental idea of the discretization method is the replacement of the analogue integrators of the prototype continuous time system by discrete time approximations this is done in such a way as to optimize a given cost function with respect to a given input unlike many discretization schemes presented in the literature the procedure developed attempts to first determine the magnitude of the expected discretization errors and then discretize with an appropriate complexity giving the designer control over the order of the discrete time system this appears to be an effective means of designing filters with low complexity which sti