wide sense stationary solutions of linear systems with additive noise let z be a wide sense stationary process with spectral measure f z acting as noise on qquad dot x ax bz quad y gx the aim of this paper is to describe the set of stationary outputs y a stationary solution x is called hard if it satisfies as a stochastic process its spectral measure f x then satisfies qquad i lambda a df x lambda i lambda a df z lambda any solution of is called a soft stationary solution of if i lambda in sigma a lambda is called critical for b g i there exists a soft stationary solution if and only if there exists a hard stationary solution on the original probability space if and only if i lambda a 1 in l 2 f z and operatorname image delta f 2 lambda subset operatorname image i lambda a for critical lambda the intuitive meaning of this condition is that the critical frequencies of the undisturbed system have to be missing in the noise spectrum in a certain s

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

theoretical and computational aspects of the stochastic boundary element method basic concepts of the stochastic boundary element computational methodology are presented stochastic boundary integral equations are discretized into a system of random algebraic equations using the midpoint and the boundary averaging methods the influence of the chosen boundary mesh geometry on covariances is tested applications of stochastic boundary elements to static and dynamic problems with random boundary conditions stochastic media stochastic sensitivity analysis and identification of internal defects are presented

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

multiple solutions and eigenvalues for third order right focal boundary value problems abstract we are concerned with the right focal boundary value problem x t f t x t t 1 t t 3 x t 1 x t 2 x t 3 0 and the associated eigenvalue problem x t a t f x t with the same boundary conditions under various assumptions on f a and we establish intervals of the parameter which yield the existence of a positive solution of the eigenvalue problem by placing certain restrictions on the nonlinearity we prove the existence of at least one at least two at least three and infinitely many positive solutions of the boundary value problem by applying some known fixed point theorems as well as some recent generalizations of these fixed point theorems

firing statistics of a neuron model driven by long range correlated noise we study the statistics of the firing patterns of a perfect integrate and fire neuron model driven by additive long range correlated ornstein uhlenbeck noise using a quasistatic weak noise approximation we obtain expressions for the interspike interval isi probability density the power spectral density and the spike count fano factor we find unimodal long tailed isi densities lorenzian power spectra at low frequencies and a minimum in the fano factor as a function of counting time the implications of these results for signal detection are 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

eigenvalue for a singular third order three point boundary value problem in this paper the existence of positive solutions for a singular third order three point boundary value problem is investigated by using krasnoselskii s fixed point theorem several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained

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

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

positive solutions for third order boundary value problems with change of signs abstract in this paper we study the third order triple point nonlinear boundary value problems with sign changing nonlinearities by employing krasnoselkii guo fixed point theorem and avery henderson fixed point theorem and establish results on the existence of at least one or two positive solutions to the boundary value problems

on the existence of nonnegative solutions to an integral equation with applications to boundary value problems abstract by the use of a guo krasnoselskii theorem in cones existence of positive eigenvalues yielding nonnegative or positive solutions to an integral equation is studied the results are applied to a variety of boundary value problems concerning ordinary differential equations

existence of triple positive solutions for a third order three point boundary value problem in this paper we investigate the existence of triple positive solutions for the nonlinear third order three point boundary value problem u t a t f t u t u t u t 0 0 q 0 1 0 are continuous first green s function for the associated linear boundary value problem is constructed and then by using a fixed point theorem due to avery and peterson we establish results on the existence of triple positive solutions to the boundary value problem

third order boundary value problems with sign changing solutions we consider the third order nonlinear differential equation u t f t u t u t u t a e t 0 1 satisfying u 0 u 0 u 1 0 and u 0 u 1 u 1 0 where f 0 1 r3 r is lp caratheodory 1 p 1 and p 1

existence of solutions for a class of third order nonlinear boundary value problems abstract in this paper we are concerned with the following third order ordinary differential equation x t f t x t x t x t 0 0 with the nonlinear boundary conditions x 0 0 g x 0 x 0 a h x 1 x 1 b where a b r f 0 1 r 3 r is continuous g h r 2 r are continuous the existence result is given by using a priori estimate nagumo condition upper and lower solutions and leray schauder degree and we give an example to demonstrate our result

the occurrence of large values in stationary sequences a direct proof is given of the tanny 1974 result that for certain non decreasing sequences a n it is true that lim supa n 1xn 0 or with probability one for all ergodic stationary sequences x n the condition on a n is shown to be necessary for all non decreasing a n and stationary x n lim sup an 1xn lim sup an 1x n a s similar continuous time theorems are also given

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

green s function for a third order generalized right focal problem abstract we determine the green s function for the third order three point generalized right focal boundary value problem x t 0 t 1 t t 3 x t 1 x t 2 0 x t 3 x t 3 0 where 0 and determine conditions on the coefficients and boundary points to ensure its positivity then we use the krasnoselskii and leggett and williams fixed point theorems to prove the existence of solutions to the nonlinear problem x t f t x t t 1 t t 3 x t 1 x t 2 0 x t 3 x t 3 0 for certain constraints placed on nonnegative f

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

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

sub gaussian tail bounds for the width and height of conditioned galton watson trees we study the height and width of a galton watson tree with offspring distribution satisfying e 1 0 var conditioned on having exactly n nodes under this conditioning we derive sub gaussian tail bounds for both the width largest number of nodes in any level and height greatest level containing a node the bounds are optimal up to constant factors in the exponent under the same conditioning we also derive essentially optimal upper tail bounds for the number of nodes at level k for 1 k n

positive solutions of singular third order three point boundary value problem abstract in this paper we investigate the problem of existence of positive solutions for the nonlinear singular third order three point boundary value problem u t a t f t u t 0 0 t 1 u 0 u u 1 0 where is a positive parameter and 1 2 1 is a constant by using a fixed point theorem of cone expansion compression type due to krasnosel skii we establish various results on the existence of single and multiple positive solutions to the boundary value problem under various assumptions on functions f and a we give explicitly the intervals for parameter in which the existence of positive solutions is guaranteed especially we allow the function a t of nonlinear term to have suitable singularities

kpp equation and supercritical branching brownian motion in the subcritical speed area application to spatial trees if r t is the position of the rightmost particle at time t in a one dimensional branching brownian motion u t x p r t x is a solution of kpp equation frac partial u partial t frac 1 2 frac partial 2 u partial x 2 f u where f u 1 u g 1 u g is the generating function of the reproduction law and the inverse of the mean lifetime if m g 1 1 and g 0 0 it is known that frac r t t xrightarrow p c 0 sqrt 2a m 1 text when t to infty for the general kpp equation we show limit theorems for u t ct c c 0 t large deviations for r t and probabilities of presence of particles for the branching process are deduced where z t denotes the random point measure of particles living at time t and a yaglom type theorem is proved the conditional distribution of the spatial tree given z t ct 0 is studied in the limit as t

existence of positive solutions for nonlinear third order three point boundary value problems abstract this paper is concerned with the following nonlinear third order three point boundary value problem u t a t f u t 0 t 0 1 u 0 u 0 0 u 1 u where 0 1 and 1 1 first the green s function for the associated linear boundary value problem is constructed and then some useful properties of the green s function are obtained by a new method finally existence results for at least one positive solution for the above problem are established when f is superlinear or sublinear

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

a cascade decomposition theory with applications to markov and exchangeable cascades a multiplicative random cascade refers to a positive t martingale in the sense of kahane on the ultrametric space t 0 1 b 1 a new approach to the study of multiplicative cascades is introduced the methods apply broadly to the problems of i non degeneracy criterion ii dimension spectra of carrying sets and iii divergence of moments criterion specific applications are given to cascades generated by markov and exchangeable processes as well as to homogeneous independent cascades 1 positive t martingales positive t martingales were introduced by jean pierre kahane as the general framework for independent multiplicative cascades and random coverings although originating in statistical theories of turbulence the general framework also includes certain spin glass and random polymer models as well as various other spatial distributions of interest in both probability theory and the physical sciences for basic definitions let t be a compact metric space with borel sigmafield b and let f p be a probability space together with an increasing sequence fn n 1 2 of sub sigmafields of f a positive t martingale is a sequence qn of b f measurable non negative functions on t such that i for each t t qn t n 0 1 is a martingale adapted to fn n 0 1 ii for p a s qn n 0 1 is a sequence of borel measurable non negative real valued functions on t let m t denote the space of positive borel measures on t and suppose that qn t is a positive t martingale for m t such that q t eqn t l let n qn denote the random measure defined by qn and dqn d t qn t t t then essentially by the martingale convergence theorem one obtains a random borel measure q such that for f c t 1 1 lim n t f t qn t dt t f t q dt a s received by the editors august 18 1994 1991 mathematics subject classification primary 60g57 60g30 60g42 secondary 60k35 60d05 60j10 60g09

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case