continuous attractors of lotka volterra recurrent neural networks with infinite neurons continuous attractors of lotka volterra recurrent neural networks lv rnns with infinite neurons are studied in this brief a continuous attractor is a collection of connected equilibria and it has been recognized as a suitable model for describing the encoding of continuous stimuli in neural networks the existence of the continuous attractors depends on many factors such as the connectivity and the external inputs of the network a continuous attractor can be stable or unstable it is shown in this brief that a lv rnn can possess multiple continuous attractors if the synaptic connections and the external inputs are gussian like in shape moreover both stable and unstable continuous attractors can coexist in a network explicit expressions of the continuous attractors are calculated simulations are employed to illustrate the theory

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

mutual information of population codes and distance measures in probability space we studied the mutual information between a stimulus and a system consisting of stochastic statistically independent elements that respond to a stimulus using statistical mechanical methods the properties of the mutual information mi in the limit of a large system size n are calculated for continuous valued stimuli the mi increases logarithmically with n and is related to the log of the fisher information of the system for discrete stimuli the mi saturates exponentially with n we find that the exponent of saturation of the mi is the chernoff distance between response probabilities that are induced by different stimuli

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

a unified theory of full order and low order observers based on singular system theory the standard construction of the minimal order state observer for a linear time invariant plant is shown to be a degenerate form of the ordinary full order observer the naive state estimation technique of successively differentiating the plant output is also shown to be a limiting case of the same full order structure various results concerning the set of all degenerate forms of the full order observer are presented in particular conditions are established under which shifting observer eigenvalues far to the left in the complex plane implies proximity to a successively differentiating system in addition it is shown that such a scheme can never yield satisfactory estimation singular perturbation and singular system theory are used throughout the analysis

subdiffusive dynamics of bump attractors mechanisms and functional roles bump attractors are localized activity patterns that can self sustain after stimulus presentation and they are regarded as the neural substrate for a host of perceptual and cognitive processes one of the characteristic features of bump attractors is that they are neutrally stable so that noisy inputs cause them to drift away from their initial locations severely impairing the accuracy of bump location dependent neural coding previous modeling studies of such noise induced drifting activity of bump attractors have focused on normal diffusive dynamics often with an assumption that noisy inputs are uncorrelated here we show that long range temporal correlations and spatial correlations in neural inputs generated by multiple interacting bumps cause them to drift in an anomalous subdiffusive way this mechanism for generating subdiffusive dynamics of bump attractors is further analyzed based on a generalized langevin equation we demonstrate that subdiffusive dynamics can significantly improve the coding accuracy of bump attractors since the variance of the bump displacement increases sublinearly over time and is much smaller than that of normal diffusion furthermore we reanalyze existing psychophysical data concerning the spread of recalled cue position in spatial working memory tasks and show that its variance increases sublinearly with time consistent with subdiffusive dynamics of bump attractors based on the probability density function of bump position we also show that the subdiffusive dynamics result in a long tailed decay of firing rate greatly extending the duration of persistent activity

correlation and independence in the neural code the decoding scheme of a stimulus can be different from the stochastic encoding scheme in the neural population coding the stochastic fluctuations are not independent in general but an independent version could be used for the ease of decoding how much information is lost by using this unfaithful model for decoding there are discussions concerning loss of information nirenberg latham 2003 schneidman bialek berry 2003 we elucidate the nirenberg latham loss from the point of view of information geometry

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

population coding with correlation and an unfaithful model this study investigates a population decoding paradigm in which the maximum likelihood inference is based on an unfaithful decoding model umli this is usually the case for neural population decoding because the encoding process of the brain is not exactly known or because a simplified decoding model is preferred for saving computational cost we consider an unfaithful decoding model that neglects the pair wise correlation between neuronal activities and prove that umli is asymptotically efficient when the neuronal correlation is uniform or of limited range the performance of umli is compared with that of the maximum likelihood inference based on the faithful model and that of the center of mass decoding method it turns out that umli has advantages of decreasing the computational complexity remarkably and maintaining high level decoding accuracy moreover it can be implemented by a biologically feasible recurrent network pouget zhang deneve latham 1998 the effect of correlation on the decoding accuracy is also discussed

a moving bump in a continuous manifold a comprehensive study of the tracking dynamics of continuous attractor neural networks understanding how the dynamics of a neural network is shaped by the network structure and consequently how the network structure facilitates the functions implemented by the neural system is at the core of using mathematical models to elucidate brain functions this study investigates the tracking dynamics of continuous attractor neural networks canns due to the translational invariance of neuronal recurrent interactions canns can hold a continuous family of stationary states they form a continuous manifold in which the neural system is neutrally stable we systematically explore how this property facilitates the tracking performance of a cann which is believed to have clear correspondence with brain functions by using the wave functions of the quantum harmonic oscillator as the basis we demonstrate how the dynamics of a cann is decomposed into different motion modes corresponding to distortions in the amplitude position width or skewness of the network state we then develop a perturbation approach that utilizes the dominating movement of the network s stationary states in the state space this method allows us to approximate the network dynamics up to an arbitrary accuracy depending on the order of perturbation used we quantify the distortions of a gaussian bump during tracking and study their effects on tracking performance results are obtained on the maximum speed for a moving stimulus to be trackable and the reaction time for the network to catch up with an abrupt change in the stimulus

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

decorrelation of spiking variability and improved information transfer through feedforward divisive normalization response variability is often positively correlated in pairs of similarly tuned neurons in the visual cortex many authors have considered correlated variability to prevent postsynaptic neurons from averaging across large groups of inputs to obtain reliable stimulus estimates however a simple average of variability ignores nonlinearities in cortical signal integration this study shows that feedforward divisive normalization of a neuron s inputs effectively decorrelates their variability furthermore we show that optimal linear estimates of a stimulus parameter that are based on normalized inputs are more accurate than those based on nonnormalized inputs due partly to reduced correlations and that these estimates improve with increasing population size up to several thousand neurons this suggests that neurons may possess a simple mechanism for substantially decorrelating noise in their inputs further work is needed to reconcile this conclusion with past evidence that correlated noise impairs visual perception

local likelihood tracking of fault lines and boundaries summary we suggest locally parametric methods for estimating curves such as boundaries of density supports or fault lines in response surfaces in a variety of spatial problems the methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane as a function of that point the local likelihood might be a regular likelihood computed locally with kernel weights e g in the case of support boundary estimation or a local version of a likelihood ratio statistic e g in fault line estimation in either case the local likelihood surface represents a function which is relatively large near the target curve and relatively small elsewhere therefore the curve may be estimated as a ridge line of the surface we require only a numerical algorithm for tracking the projection of a ridge into the plane this approach offers several potential advantages over alternative methods first the local log likelihood surface can be graphed and the degree of ridginess assessed visually to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately secondly the local likelihood surface does not need to be computed in anything like its entirety once we have a reasonable approximation to a point on the curve we may track it by numerically walking along the ridge line thirdly the method is appropriate without change for many different types of spatial explanatory vanables gridded stochastic or otherwise three examples are explored in detail fault lines in response surfaces and in intensity or density surfaces and boundaries of supports of probability densities

correlated neural variability in persistent state networks neural activity that persists long after stimulus presentation is a biological correlate of short term memory variability in spiking activity causes persistent states to drift over time ultimately degrading memory models of short term memory often assume that the input fluctuations to neural populations are independent across cells a feature that attenuates population level variability and stabilizes persistent activity however this assumption is at odds with experimental recordings from pairs of cortical neurons showing that both the input currents and output spike trains are correlated it remains unclear how correlated variability affects the stability of persistent activity and the performance of cognitive tasks that it supports we consider the stochastic long timescale attractor dynamics of pairs of mutually inhibitory populations of spiking neurons in these networks persistent activity was less variable when correlated variability was globally distributed across both populations compared with the case when correlations were locally distributed only within each population using a reduced firing rate model with a continuum of persistent states we show that when input fluctuations are correlated across both populations they drive firing rate fluctuations orthogonal to the persistent state attractor thereby causing minimal stochastic drift using these insights we establish that distributing correlated fluctuations globally as opposed to locally improves network s performance on a two interval delayed response discrimination task our work shows that the correlation structure of input fluctuations to a network is an important factor when determining long timescale persistent population spiking activity

generalized fisher information matrix in nonextensive systems with spatial correlation by using the q gaussian distribution derived by the maximum entropy method for spatially correlated n unit nonextensive systems we have calculated the generalized fisher information matrix of g ensuremath theta n ensuremath theta m for ensuremath theta 1 ensuremath theta 2 ensuremath theta 3 ensuremath mu q ensuremath sigma q 2 s where ensuremath mu q ensuremath sigma q 2 and s denote the mean variance and degree of spatial correlation respectively for a given entropic index q it has been shown from the cram er rao theorem that 1 an accuracy of an unbiased estimate of ensuremath mu q is improved degraded by a negative positive correlation s 2 that of ensuremath sigma q 2 is worsen with increasing s and 3 that of s is much improved for s ensuremath simeq ensuremath 1 n ensuremath 1 or s ensuremath simeq 1 0 though it is worst at s n ensuremath 2 2 n ensuremath 1 our calculation provides a clear insight to the long standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles we discuss also a calculation of the q gaussian distribution applying the superstatistics to the langevin model subjected to spatially correlated inputs

theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback a network of leaky integrate and fire neurons with global inhibitory feedback and under the influence of spatially correlated noise is studied we calculate the spectral statistics of the network power spectrum of the population activity cross spectrum between spike trains of different neurons as well as of a single neuron power spectrum of spike train cross spectrum between external noise and spike train within the network as shown by comparison with numerical simulations our theory works well for arbitrary network size if the feedback is weak and the amount of external noise does not exceed that of the internal noise by means of our analytical results we discuss the quality of the correlation induced oscillation in a large network as a function of the transmission delay and the internal noise intensity it is shown that the strongest oscillation is obtained in a system with zero internal noise and adiabatically long delay i e the delay period is longer than any other time scale in the system for a neuron with a strong intrinsic frequency the oscillation becomes strongly anharmonic in the case of a long delay time we also discuss briefly the kind of synchrony introduced by the feedback induced oscillation

recovering convex boundaries from blurred and noisy observations we consider the problem of estimating convex boundaries from blurred and noisy observations in our model the convolution of an intensity function f is observed with additive gaussian white noise the function f is assumed to have convex support g whose boundary is to be recovered rather than directly estimating the intensity function we develop a procedure which is based on estimating the support function of the set g this approach is closely related to the method of geometric hyperplane probing a well known technique in computer vision applications we establish bounds that reveal how the estimation accuracy depends on the ill posedness of the convolution operator and the behavior of the intensity function near the boundary short title recovering convex boundaries

singular observer based compensators and the deterministic separation principle the concept of observer based compensators obcs and the deterministic separation principle have long been cornerstones of modern control theory in our paper we extend these ideas via singular system theory to encompass a wider variety of rational compensators we show that the separation principle generalizes in a natural way to obcs of all orders including the static case we pay particular attention to the problems of properness and conjugate symmetry our intention is to provide an introductory account of this framework and to suggest possible avenues of further development

computing with continuous attractors stability and online aspects two issues concerning the application of continuous attractors in neural systems are investigated the computational robustness of continuous attractors with respect to input noises and the implementation of bayesian online decoding in a perfect mathematical model for continuous attractors decoding results for stimuli are highly sensitive to input noises and this sensitivity is the inevitable consequence of the system s neutral stability to overcome this shortcoming we modify the conventional network model by including extra dynamical interactions between neurons these interactions vary according to the biologically plausible hebbian learning rule and have the computational role of memorizing and propagating stimulus information accumulated with time as a result the new network model responds to the history of external inputs over a period of time and hence becomes insensitive to short term fluctuations also since dynamical interactions provide a mechanism to convey the prior knowledge of stimulus that is the information of the stimulus presented previously the network effectively implements online bayesian inference this study also reveals some interesting behavior in neural population coding such as the trade off between decoding stability and the speed of tracking time varying stimuli and the relationship between neural tuning width and the tracking speed

dynamics and computation of continuous attractors continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems in a continuous attractor the stationary states of the neural system form a continuous parameter space on which the system is neutrally stable this property enables the neutral system to track time varying stimuli smoothly but it also degrades the accuracy of information retrieval since these stationary states are easily disturbed by external noise in this work based on a simple model we systematically investigate the dynamics and the computational properties of continuous attractors in order to analyze the dynamics of a large size network which is otherwise extremely complicated we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the noise components perpendicular to the attractor space very quickly we therefore project the network dynamics onto the tangent of the attractor space and simplify it successfully as a one dimensional ornstein uhlenbeck process based on this simplified model we investigate 1 the decoding error of a continuous attractor under the driving of external noisy inputs 2 the tracking speed of a continuous attractor when external stimulus experiences abrupt changes 3 the neural correlation structure associated with the specific dynamics of a continuous attractor and 4 the consequence of asymmetric neural correlation on statistical population decoding the potential implications of these results on our understanding of neural information processing are also discussed

population coding and decoding in a neural field a computational study this study uses a neural field model to investigate computational aspects of population coding and decoding when the stimulus is a single variable a general prototype model for the encoding process is proposed in which neural responses are correlated with strength specified by a gaussian function of their difference in preferred stimuli based on the model we study the effect of correlation on the fisher information compare the performances of three decoding methods that differ in the amount of encoding information being used and investigate the implementation of the three methods by using a recurrent network this study not only rediscovers main results in existing literatures in a unified way but also reveals important new features especially when the neural correlation is strong as the neural correlation of firing becomes larger the fisher information decreases drastically we confirm that as the width of correlation increases the fisher information saturates and no longer increases in proportion to the number of neurons however we prove that as the width increases further wider than 2 times the effective width of the turning function the fisher information increases again and it increases without limit in proportion to the number of neurons furthermore we clarify the asymptotic efficiency of the maximum likelihood inference mli type of decoding methods for correlated neural signals it shows that when the correlation covers a nonlocal range of population excepting the uniform correlation and when the noise is extremely small the mli type of method whose decoding error satisfies the cauchy type distribution is not asymptotically efficient this implies that the variance is no longer adequate to measure decoding accuracy

likelihood based confidence bands for fault lines in response surfaces we consider the problem of constructing asymptotic confidence bands both pointwise and simultaneous for a smooth fault line in a response surface when the design is represented by a point process either deterministic or stochastic with intensity n diverging to infinity the estimator of the fault line is defined as the ridge line on the likelihood surface which arises from locally fitting a model that employs a linear approximation to the fault line to a kernel smooth of the data the construction is based on analysis of the limiting behaviour of perpendicular distance from a point on the true fault line to the nearest point on the ridge we derive asymptotic properties of bias and the limiting distribution of stochastic error this distribution is given by the location of the maximum of a gaussian process with quadratic drift although the majority of attention is focused on the regression problem the limiting distribution is shown to have wider relevance to local likelihood approaches to fault line estimation for density or intensity surfaces

dynamics of moment neuronal networks a theoretical framework is developed for moment neuronal networks mnns within this framework the behavior of the system of spiking neurons is specified in terms of the first and second order statistics of their interspike intervals i e the mean the variance and the cross correlations of spike activity since neurons emit and receive spike trains which can be described by renewal but generally non poisson processes we first derive a suitable diffusion type approximation of such processes two approximation schemes are introduced the usual approximation scheme uas and the ornstein uhlenbeck scheme it is found that both schemes approximate well the input output characteristics of spiking models such as the if and the hodgkin huxley models the mnn framework is then developed according to the uas scheme and its predictions are tested on a few examples

on the parametrization of all regularly implementing and stabilizing controllers in this paper we deal with problems of controller parametrization in the context of behavioral systems given a full plant behavior a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior in such a way that the controller does not impose restrictions that are already present in the plant we establish a parametrization of all controllers that regularly implement a given behavior we also obtain a parametrization of all stabilizing controllers

population coding in neuronal systems with correlated noise neuronal representations of external events are often distributed across large populations of cells we study the effect of correlated noise on the accuracy of these neuronal population codes our main question is whether the inherent error in the population code can be suppressed by increasing the size of the population n in the presence of correlated noise we address this issue using a model of a population of neurons that are broadly tuned to an angular variable in two dimensions the fluctuations in the neuronal activities are modeled as gaussian noises with pairwise correlations that decay exponentially with the difference between the preferred angles of the correlated cells we assume that the system is broadly tuned which means that both the correlation length and the width of the tuning curves of the mean responses span a substantial fraction of the entire system length the performance of the system is measured by the fisher information fi which bounds its estimation error by calculating the fi in the limit of a large n we show that positive correlations decrease the estimation capability of the network relative to the uncorrelated population the information capacity saturates to a finite value as the number of cells in the population grows in contrast negative correlations substantially increase the information capacity of the neuronal population these results are supplemented by the effect of correlations on the mutual information of the system our analysis provides an estimate of the effective number of statistically independent degrees of freedom denoted n mathrm eff that a large correlated system can have according to our theory n mathrm eff remains finite in the limit of a large n estimating the parameters of the correlations and tuning curves from experimental data in some cortical areas that code for angles we predict that the number of effective degrees of freedom embedded in localized populations in these areas is less than or of the order of ensuremath approx 10 2

on global performance of approximations to smooth curves using gridded data approximating boundaries using data recorded on a regular grid induces discrete rounding errors in both vertical and horizontal directions in cases where grid points exhibit at least some degree of randomness an extensive theory has been developed for local polynomial boundary estimators it is inapplicable to regular grids however in this paper we impose strict regularity of the grid and describe the performance of local linear estimators in this context unlike the case of classical curve estimation problems pointwise convergence rates vary erratically along the boundary depending on number theoretic properties of the boundary s slope however average convergence rates expressed in the l 1 metric are much less susceptible to fluctuation we derive theoretical bounds to performance coming within no more than a logarithmic factor of the optimal convergence rate

correlated firing in a feedforward network with mexican hat type connectivity we report on deterministic and stochastic evolutions of firing states through a feedforward neural network with mexican hat type connectivity the prevalence of columnar structures in a cortex implies spatially localized connectivity between neural pools although feedforward neural network models with homogeneous connectivity have been intensively studied within the context of the synfire chain the effect of local connectivity has not yet been studied so thoroughly when a neuron fires independently the dynamics of macroscopic state variables a firing rate and spatial eccentricity of a firing pattern is deterministic from the law of large numbers possible stable firing states which are derived from deterministic evolution equations are uniform localized and nonfiring the multistability of these three states is obtained where the excitatory and inhibitory interactions among neurons are balanced when the presynapse dependent variance in connection efficacies is incorporated into the network the variance generates common noise then the evolution of the macroscopic state variables becomes stochastic and neurons begin to fire in a correlated manner due to the common noise the correlation structure that is generated by common noise exhibits a nontrivial bimodal distribution the development of a firing state through neural layers does not converge to a certain fixed point but keeps on fluctuating

tracking a smooth fault line in a response surface we suggest a sequential or tracking algorithm for estimating a smooth fault line in a response surface the method starts with an approximation to a point on the line and from there the line is tracked as it meanders through the plane the technique differs from recent approaches in that it does not require a large part of the plane to be searched for evidence of a fault line this offers potential computational savings and produces a method that is invariant under rotations of coordinate axes except insofar as a rotation might affect the estimated starting point and the relative orientation of the grid on which calculations are done that feature is important if design points are not located on a regular grid we investigate properties of the method under very general conditions on the design allowing poisson cluster processes jiggled grid processes and deterministic regular lattices uniform rates of convergence are derived in all these settings for the case of noisy data and shown to be within logarithmic factors of optimal pointwise convergence rates in the no noise setting

image denoising pointwise adaptive approach a new method of pointwise adaptation has been proposed and studied in spokoiny 1998 ann statist 26 1356 1378 in the context of estimation of piecewise smooth univariate functions the present paper extends that method to estimation of bivariate grey scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design the proposed estimator hat f x at a point x is simply the average of observations over a window widehat u x selected in a data driven way the theoretical properties of the procedure are studied for the case of piecewise constant images we present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixels n of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary it is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions we briefly discuss algorithmic aspects and the complexity of the procedure the numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues an example from satellite sar imaging illustrates the applicability of the method

replica free evaluation of the neuronal population information with mixed continuous and discrete stimuli from the linear to the asymptotic regime recent studies have explored theoretically the ability of populations of neurons to carry information about a set of stimuli both in the case of purely discrete or purely continuous stimuli and in the case of multidimensional continuous angular and discrete correlates in the presence of additional quenched disorder in the distribution an analytical expression for the mutual information has been obtained in the limit of large noise by means of the replica trick here we show that the same results can actually be obtained in most cases without the use of replicas by means of a much simpler expansion of the logarithm fitting the theoretical model to real neuronal data we show that the introduction of correlations in the quenched disorder improves the fit suggesting a possible role of signal correlations actually detected in real data in a redundant code we show that even in the more difficult analysis of the asymptotic regime an explicit expression for the mutual information can be obtained without resorting to the replica trick despite the presence of quenched disorder both with a gaussian and with a more realistic thresholded gaussian model when the stimuli are mixed continuous and discrete we find that with both models the information seem to grow logarithmically to infinity with the number of neurons and with the inverse of the noise even though the exact general dependence cannot be derived explicitly for the thresholded gaussian model in the large noise limit lower values of information were obtained with the thresholded gaussian model for a fixed value of the noise and of the population size on the contrary in the asymptotic regime with very low values of the noise a lower information value is obtained with the gaussian model

spontaneous motion on two dimensional continuous attractors attractor models are simplified models used to describe the dynamics of firing rate profiles of a pool of neurons the firing rate profile or the neuronal activity is thought to carry information continuous attractor neural networks canns describe the neural processing of continuous information such as object position object orientation and direction of object motion recently it was found that in one dimensional canns short term synaptic depression can destabilize bump shaped neuronal attractor activity profiles in this article we study two dimensional canns with short term synaptic depression and spike frequency adaptation we found that the dynamics of canns with short term synaptic depression and canns with spike frequency adaptation are qualitatively similar we also found that in both kinds of canns the perturbative approach can be used to predict phase diagrams dynamical variables and speed of spontaneous motion

representational capacity of a set of independent neurons the capacity with which a system of independent neuron like units represents a given set of stimuli is studied by calculating the mutual information between the stimuli and the neural responses both discrete noiseless and continuous noisy neurons are analyzed in both cases the information grows monotonically with the number of neurons considered under the assumption that neurons are independent the mutual information rises linearly from zero and approaches exponentially its maximum value we find the dependence of the initial slope on the number of stimuli and on the sparseness of the representation

on the recovery of a function on a circular domain we consider the problem of estimating a function f x y on the unit disk f x y x sup 2 y sup 2 spl les 1 given a discrete and noisy data recorded on a regular square grid an estimate of f x y based on a class of orthogonal and complete functions over the unit disk is proposed this class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate for radial functions the orthogonal set has a particularly simple form being related to the classical legendre polynomials we give the statistical accuracy analysis of the proposed estimate of f x y in the sense of the l sub 2 metric it is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain this is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle in fact the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain

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

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

digitized circular arcs characterization and parameter estimation the digitization of a circular arc causes an inherent loss of geometrical information arcs with slightly different local curvature or position may lead to exactly the same digital pattern in this paper the authors give a characterization of all centers and radii of circular arcs yielding the same digitization pattern the radius of the arcs varies over the set however only one curvature or radius estimate can be assigned to the digital pattern the authors derive an optimal estimator and give expressions for the bound on the precision of estimation this bound due to digitization is the deterministic equivalent of the cramer rao bound known from parameter estimation theory consider the estimation of the local curvature and local radius of a smooth object typically such parameters are estimated by moving a window along the digital boundary methods in literature show a poor precision in estimating curvature values relative errors of over 40 are often found from the definition of curvature it follows that locally the curve can be considered a circular arc and hence the method presented in this paper can be applied to the pattern in the window giving estimates with optimal precision and a measure for the remaining error on the practical side the authors present examples of the residual error due to the discrete grid the estimation of the radius or curvature of a circular arc at random position with an estimation window containing 10 points coded with nine freemancodes has a relative deviation exceeding 2 for a full disk the deviation is below 1 when the radius r exceeds four grid units the presented method is particularly useful for problems where some prior knowledge on the distribution of radii is known and where there is a noise free sampling