exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

bounds on colength and maximal multiplicity sequences let a be a characteristic zero p i algebra with cocharacters n a m we obtain upper and lower bounds on max n m and n m and compute the exact rates of growth when a is a prime product algebra

a six generalized squares theorem with applications to polynomial identity algebras the theories of superalgebras and of p i algebras lead to a natural z2 graded extension of the integers for these generalized integers a six generalized squares theorem is proved which can be considered as a z2 graded analogue of the classical four squares theorem for the natural numbers this theorem was conjectured by a berele and a regev exponential growth of some p i algebras br2 and has applications to p i algebras

testing for measurement invariance with respect to an ordinal variable researchers are often interested in testing for measurement invariance with respect to an ordinal auxiliary variable such as age group income class or school grade in a factor analytic context these tests are traditionally carried out via a likelihood ratio test statistic comparing a model where parameters differ across groups to a model where parameters are equal across groups this test neglects the fact that the auxiliary variable is ordinal and it is also known to be overly sensitive at large sample sizes in this paper we propose test statistics that explicitly account for the ordinality of the auxiliary variable resulting in higher power against monotonic violations of measurement invariance and lower power against non monotonic ones the statistics are derived from a family of tests based on stochastic processes that have recently received attention in the psychometric literature the statistics are illustrated via an application involving real data and their performance is studied via simulation

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

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

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

response mixture modeling accounting for heterogeneity in item characteristics across response times in item response theory modeling of responses and response times it is commonly assumed that the item responses have the same characteristics across the response times however heterogeneity might arise in the data if subjects resort to different response processes when solving the test items these differences may be within subject effects that is a subject might use a certain process on some of the items and a different process with different item characteristics on the other items if the probability of using one process over the other process depends on the subject s response time within subject heterogeneity of the item characteristics across the response times arises in this paper the method of response mixture modeling is presented to account for such heterogeneity contrary to traditional mixture modeling where the full response vectors are classified response mixture modeling involves classification of the individual elements in the response vector in a simulation study the response mixture model is shown to be viable in terms of parameter recovery in addition the response mixture model is applied to a real dataset to illustrate its use in investigating within subject heterogeneity in the item characteristics across response times

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

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

a constrained formulation of maximum likelihood estimation for normal mixture distributions in this context the normal densities are sometimes referred to as component densities the log likelihood function l zy corresponding to a random sample xl xnj is defined by l ty ek 1 10g pm xk y the first problem associated with maximum likelihood estimation arises from the unboundedness of l qy on r day 1969 a global maximum likelihood estimate always fails to exist in addition the unboundedness of l y causes failures of optimization algorithms of both the em redner and walker 1984 and quasi newton fowlkes 1979 types in spite of the unboundedness of l 4y statistical theory kiefer 1978 guarantees that a particular local maximizer of l y is strongly consistent and asymptotically efficient several local maximizers can exist for a given sample and the other major maximum likelihood difficulty is in determining when the correct one has been found day 1969 noted that spurious maximizers corresponding to parameter points having some component standard deviations very small relative to others are generated by any small number of sample points grouped sufficiently close together the spurious maximizers like the unboundedness of l zy can create difflculties when using the em or quasi newton algorithms the constrained maximum likelihood formulation presented in the next section avoids the unconstrained problems of singularities and spurious maximizers of the type noted by day section 3 contains the statement and proof of a consistency result for the corresponding constrained estimator

exponential growth for codimensions of some p i algebras abstract by the giambruno zaicev theorem for associative p i algebras the exponential rate of growth of the codimensions of such a p i algebra is always a positive integer here we calculate that integer for various generic p i algebras which are given by a single identity these include capelli type identities and the various powers of the standard polynomials

finite mixtures in confirmatory factor analysis models in this paper various types of finite mixtures of confirmatory factor analysis models are proposed for handling data heterogeneity under the proposed mixture approach observations are assumed to be drawn from mixtures of distinct confirmatory factor analysis models but each observation does not need to be identified to a particular model prior to model fitting several classes of mixture models are proposed these models differ by their unique representations of data heterogeneity three different sampling schemes for these mixture models are distinguished a mixed type of the these three sampling schemes is considered throughout this article the proposed mixture approach reduces to regular multiple group confirmatory factor analysis under a restrictive sampling scheme in which the structural equation model for each observation is assumed to be known by assuming a mixture of multivariate normals for the data maximum likelihood estimation using the em expectation maximization algorithm and the as approximate scoring method are developed respectively some mixture models were fitted to a real data set for illustrating the application of the theory although the em algorithm and the as method gave similar sets of parameter estimates the as method was found computationally more efficient than the em algorithm some comments on applying the mixture approach to structural equation modeling are made

tests of measurement invariance without subgroups a generalization of classical methods the issue of measurement invariance commonly arises in factor analytic contexts with methods for assessment including likelihood ratio tests lagrange multiplier tests and wald tests these tests all require advance definition of the number of groups group membership and offending model parameters in this paper we study tests of measurement invariance based on stochastic processes of casewise derivatives of the likelihood function these tests can be viewed as generalizations of the lagrange multiplier test and they are especially useful for i identifying subgroups of individuals that violate measurement invariance along a continuous auxiliary variable without prespecified thresholds and ii identifying specific parameters impacted by measurement invariance violations the tests are presented and illustrated in detail including an application to a study of stereotype threat and simulations examining the tests abilities in controlled conditions

fitting multivariage normal finite mixtures subject to structural equation modeling this paper is about fitting multivariate normal mixture distributions subject to structural equation modeling the general model comprises common factor and structural regression models the introduction of covariance and mean structure models reduces the number of parameters to be estimated in fitting the mixture and enables one to investigate a variety of substantive hypotheses concerning the differences between the components in the mixture within the general model individual parameters can be subjected to equality nonlinear and simple bounds constraints confidence intervals are based on the inverse of the hessian and on the likelihood profile several illustrations are given and results of a simulation study concerning the confidence intervals are reported

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

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

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

analysis of time series subject to changes in regime abstract this paper introduces an em algorithm for obtaining maximum likelihood estimates of parameters for processes subject to discrete shifts in autoregressive parameters with the shifts themselves modeled as the outcome of a discrete valued markov process the simplicity of the em algorithm permits potential application of the approach to large vector systems

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

pi algebras and their cocharacters abstract all pi algebras r satisfy identities of the form f x y x 1 y 1 x 2 y 2 y n 1 x n theorem a the existence of these identities imply also that cocharacters x n r lie in a hook shaped strip of width depending on the degree of the minimal identity of r theorem c this extends a characterization of rings satisfying a capelli identity theorem b

on the relation between the linear factor model and the latent profile model the relationship between linear factor models and latent profile models is addressed within the context of maximum likelihood estimation based on the joint distribution of the manifest variables although the two models are well known to imply equivalent covariance decompositions in general they do not yield equivalent estimates of the unconditional covariances in particular a 2 class latent profile model with gaussian components underestimates the observed covariances but not the variances when the data are consistent with a unidimensional gaussian factor model in explanation of this phenomenon we provide some results relating the unconditional covariances to the goodness of fit of the latent profile model and to its excess multivariate kurtosis the analysis also leads to some useful parameter restrictions related to symmetry

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

using bootstrap likelihood ratios in finite mixture models statistical inference using the likelihood ratio statistic for the number of components in a mixture model is complicated when the true number of components is less than that of the proposed model since this represents a non regular problem the true parameter is on the boundary of the parameter space and in some cases the true parameter is in a non identifiable subset of the parameter space the maximum likelihood estimator is shown to converge to the subset characterized by the same density function and connection is made to the bootstrap method proposed by aitkin and co workers and mclachlan for testing the number of components in a finite mixture and deriving confidence regions in a finite mixture

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

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

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

a bayesian analysis of finite mixtures in the lisrel model in this paper we propose a bayesian framework for estimating finite mixtures of the lisrel model the basic idea in our analysis is to augment the observed data of the manifest variables with the latent variables and the allocation variables the gibbs sampler is implemented to obtain the bayesian solution other associated statistical inferences such as the direct estimation of the latent variables establishment of a goodness of fit assessment for a posited model bayesian classification residual and outlier analyses are discussed the methodology is illustrated with a simulation study and a real example

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

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