estimation of gradients and coordinate covariation in classification we introduce an algorithm that simultaneously estimates a classification function as well as its gradient in the supervised learning framework the motivation for the algorithm is to find salient variables and estimate how they covary an efficient implementation with respect to both memory and time is given the utility of the algorithm is illustrated on simulated data as well as a gene expression data set an error analysis is given for the convergence of the estimate of the classification function and its gradient to the true classification function and true gradient

discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

noise in integrate and fire neurons from stochastic input to escape rates we analyze the effect of noise in integrate and fire neurons driven by time dependent input and compare the diffusion approximation for the membrane potential to escape noise it is shown that for time dependent subthreshold input diffusive noise can be replaced by escape noise with a hazard function that has a gaussian dependence on the distance between the noise free membrane voltage and threshold the approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics

control of cardiac function and noise from a decaying power spectrum evidence is presented that adds to the debate surrounding the question to what extent does neural control of cardiac output exploit noise the transduction capability of cardiac afferent neurons situated in and adjacent to the heart is vital to feedback in control of cardiac function an analysis of in situ cardiac afferent activity shows evidence of independent and exponentially distributed interspike intervals an anatomical basis for such memoryless interspike intervals ultimately derives from the fact that each afferent neuron is associated with a field of sensory neurites or bare nerve endings that transduce local chemical and mechanical stimuli in a many to one fashion as such cardiac afferent neurons and their sensory neurite inputs are respectively modeled here by the hodgkin huxley equations forced by red noise decaying power spectrum perturbing an otherwise constant subthreshold input a variable barrier competition model is derived from these equations in order to address the question how are noisy inputs being processed by sensory neurons to cause each spike it is found that ion channels are responsible for significant input whitening increased spectral power at higher frequency through differentiation of the inputs such whitening is a means to distinguish low frequency control signals from otherwise red noise fluctuations furthermore spiking occurs when backward moving averages of the whitened inputs over a window of the order of the sodium activation time scale exceed an approximately constant barrier

competition model for aperiodic stochastic resonance in a fitzhugh nagumo model of cardiac sensory neurons regional cardiac control depends upon feedback of the status of the heart from afferent neurons responding to chemical and mechanical stimuli as transduced by an array of sensory neurites emerging experimental evidence shows that neural control in the heart may be partially exerted using subthreshold inputs that are amplified by noisy mechanical fluctuations this amplification is known as aperiodic stochastic resonance asr neural control in the noisy subthreshold regime is difficult to see since there is a near absence of any correlation between input and the output the latter being the average firing spiking rate of the neuron this lack of correlation is unresolved by traditional energy models of asr since these models are unsuitable for identifying cause and effect between such inputs and outputs in this paper the competition between averages model is used to determine what portion of a noisy subthreshold input is responsible on average for the output of sensory neurons as represented by the fitzhugh nagumo equations a physiologically relevant conclusion of this analysis is that a nearly constant amount of input is responsible for a spike on average and this amount is approximately independent of the firing rate hence correlation measures are generally reduced as the firing rate is lowered even though neural control under this model is actually unaffected

mechanism of stochastic resonance enhancement in neuronal models driven by 1 f noise noise can assist neurons in the detection of weak signals via a mechanism known as stochastic resonance sr in a previous study phys lett a 243 281 1998 we showed that when colored noise with 1 f ensuremath beta spectrum is added to the fitzhugh nagumo fhn neuronal model the optimal noise variance for sr could be minimized with ensuremath beta ensuremath approx 1 in this study we investigate analytically how the noise color ensuremath beta affects the sr profile in a linearized version of the fhn model we demonstrate that the aforementioned effect of 1 f noise is related to the dynamical characteristics of the model neuron i e the refractory period the low pass filtering effect of the membrane capacitance and the high pass filtering effect of the recovery variable

effects of noise near a homoclinic bifurcation in an electrochemical system the interaction of external noise with an electrochemical oscillator anodic dissolution of iron has been studied experimentally on both sides of the homoclinic bifurcation in the oscillatory regime the regularity of the limit cycle behavior was destroyed with increasing noise amplitude until no periodicity was observable any more in contrast the response of the excitable state fixed point regime became more regular at intermediate noise levels

learning coordinate covariances via gradients we introduce an algorithm that learns gradients from samples in the supervised learning framework an error analysis is given for the convergence of the gradient estimated by the algorithm to the true gradient the utility of the algorithm for the problem of variable selection as well as determining variable covariance is illustrated on simulated data as well as two gene expression data sets for square loss we provide a very efficient implementation with respect to both memory and time

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

harmonic oscillator with multiplicative noise nonmonotonic dependence on the strength and the rate of dichotomous noise the output signal of a undamped linear oscillator with a random frequency subject to a periodic force shows nonmonotonic dependence on the strength and the rate of color noise stochastic resonance the effect is absent for white noise

on the distribution of the nodal sets of random spherical harmonics we study the volume of the nodal set of eigenfunctions of the laplacian on the m dimensional sphere it is well known that the eigenspaces corresponding to en n n m 1 are the spaces en of spherical harmonics of degree n of dimension n we use the multiplicity of the eigenvalues to endow en with the gaussian probability measure and study the distribution of the m dimensional volume of the nodal sets of a randomly chosen function the expected volume is proportional to en one of our main results is bounding the variance of the volume to be o en n in addition to the volume of the nodal set we study its leray measure we find that its expected value is n independent we are able to determine that the asymptotic form of the variance is const n

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

stochastic resonance in neuron models periodically stimulated sensory neurons typically exhibit a kind of statistical phase locking to the stimulus they tend to fire at a preferred phase of the stimulus cycle but not at every cycle hence the histogram of interspike intervals isih i e of times between successive firings is multimodal for these neurons with peaks centered at integer multiples of the driving period a particular kind of residence time histogram for a large class of noisy bistable systems has recently been shown to exhibit the major features of the neural data in the present paper we show that an excitable cell model the fitzhugh nagumo equations also reproduces these features when driven by additive periodic and stochastic forces this model exhibits its own brand of stochastic resonance as the peaks of the isih successively go through a maximum when the noise intensity is increased further the presence of a noise induced limit cycle introduces a third time scale in the problem this limit cycle is found to modify qualitatively the phase locking picture e g by suppressing certain peaks in the isih finally the role of noise and possibly of stochastic resonance sr in the neural encoding of sensory information is discussed

parameter induced aperiodic stochastic resonance in the presence of multiplicative noise and additive noise abstract parameter induced aperiodic stochastic resonance sr in the presence of multiplicative and additive noise is investigated in detail using theoretical and numerical analysis we evaluate the dynamical probability density by which the bit error rate is defined as a quantity to optimize in the aperiodic sr problem via tuning the system parameters tuning the system parameters includes two cases one is parameter optimized and the other is parameter induced sr to distinguish them we use the conventional method in which sr is realized by varying the noise intensity essentially parameter induced sr is based on the cooperative effect between the stochastic subjected dynamical system and the input signal by changing the intrinsic characteristic of the nonlinear system via tuning the system parameters rather than by adding noise the theory is tested against numerical simulations

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

information capacity of stochastic pooling networks is achieved by discrete inputs stochastic pooling networks spn are sensor networks where multiple sensors make independently noisy and compressed measurements of the same information source which are combined via pooling examples of spns range from nanoelectronics to biological sensory neurons here it is shown that optimal information transmission in spns with nodes that quantize to a finite number of states requires the input signal distribution to be discrete this is illustrated numerically for a simple spn consisting of n binary quantizing sensors the resultant information capacity is shown to be independent of the noise distribution when the signal distribution can be freely chosen but to imply an optimal noise distribution if the signal distribution is fixed while larger than the best performance of previously studied continuously valued input signals the capacity does not scale faster than the previous best result of text log 2 sqrt n bits per channel use it is also shown that a plot of the optimal input distribution contains bifurcations as n increases and that suprathreshold stochastic resonance occurs when the mutual information is determined for a suboptimal noise distribution

nodal length fluctuations for arithmetic random waves using the spectral multiplicities of the standard torus we endow the laplace eigenspaces with gaussian probability measures this induces a notion of random gaussian laplace eigenfunctions on the torus arithmetic random waves we study the distribution of the nodal length of random eigenfunctions for large eigenvalues and our primary result is that the asymptotics for the variance is nonuniversal our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy

stochastic resonance in a tunnel diode we study stochastic resonance in a fast bistable electronic system a tunnel diode we investigate the phenomenon in a higher frequency regime than that studied in previous experiments detailed measurements of the output signal are reported for two values of the frequency of the periodic signal mathit f mathit s 1 khz and mathit f mathit s 10 khz we observe in one case mathit f mathit s 1 khz a nonmonotonic behavior characterized by a sharp dip in the output noise level measured at the frequency of the driving signal

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

information measures quantifying aperiodic stochastic resonance aperiodic stochastic resonance asr is a phenomenon in which the response of a nonlinear system to a subthreshold information bearing signal is optimized by the presence of noise we have previously characterized this effect by the use of cross correlation based measures here we apply a measure transinformation that directly quantifies the rate of information transfer from stimulus to response and show that the presence of noise optimizes the information transfer rate by considering a nonlinear system the fitzhugh nagumo model that captures the functional dynamics of neuronal firing we demonstrate that sensory neurons could in principle harness asr to optimize the detection and transmission of weak stimuli s1063 651x 96 51309 7

regularization on graphs with function adapted diffusion processes harmonic analysis and diffusion on discrete data has been shown to lead to state of the art algorithms for machine learning tasks especially in the context of semi supervised and transductive learning the success of these algorithms rests on the assumption that the function s to be studied learned interpolated etc are smooth with respect to the geometry of the data in this paper we present a method for modifying the given geometry so the function s to be studied are smoother with respect to the modified geometry and thus more amenable to treatment using harmonic analysis methods among the many possible applications we consider the problems of image denoising and transductive classification in both settings our approach improves on standard diffusion based methods

stochastic resonance of electrochemical aperiodic spike trains aperiodic stochastic resonance in an electrochemical system with excitable dynamics is characterized in experiments and simulations two different spike trains one with stochastic and the other with chaotic interspike intervals are imposed on the system as subthreshold aperiodic signals information transmission is quantified by the cross correlation between the subthreshold input signal and the noise induced system response a maximum is exhibited in the input output correlation as a function of the noise amplitude numerical simulations with an electrochemical model are in excellent agreement with the experimental observations

stochastic resonance in models of neuronal ensembles two recently suggested mechanisms for the neuronal encoding of sensory information involving the effect of stochastic resonance with aperiodic time varying inputs are considered it is shown using theoretical arguments and numerical simulations that the nonmonotonic behavior with increasing noise of the correlation measures used for the so called aperiodic stochastic resonance asr scenario does not rely on the cooperative effect typical of stochastic resonance in bistable and excitable systems rather asr with slowly varying signals is more properly interpreted as linearization by noise consequently the broadening of the resonance curve in the multineuron stochastic resonance without tuningscenario can also be explained by this linearization computation of the input output correlation as a function of both signal frequency and noise for the model system further reveals conditions where noise induced firing with aperiodic inputs will benefit from stochastic resonance rather than linearization by noise thus our study clarifies the tuning requirements for the optimal transduction of subthreshold aperiodic signals it also shows that a single deterministic neuron can perform as well as a network when biased into a suprathreshold regime finally we show that the inclusion of a refractory period in the spike detection scheme produces a better correlation between instantaneous firing rate and input signal s1063 651x 97 01102 1

riemannian manifolds with maximal eigenfunction growth on any compact riemannian manifold m g of dimension n the l sp 2 normalized eigenfunctions phi sb lambda satisfy lvert rvert phi sb lambda lvert rvert sb infty leq c lambda sp n 1 2 where delta phi sb lambda lambda sp 2 phi sb lambda the bound is sharp in the class of all m g since it is obtained by zonal spherical harmonics on the standard n sphere s sp n but of course it is not sharp for many riemannian manifolds for example for flat tori mathbb r sp n gamma we say that s sp n but not mathbb r sp n gamma is a riemannian manifold with maximal eigenfunction growth the problem that motivates this paper is to determine the m g with maximal eigenfunction growth our main result is that such an m g must have a point x where the set mathscr l sb x of geodesic loops at x has positive measure in s sp ast sb xm we show that if m g is real analytic this puts topological restrictions on m for example only m s sp 2 or m mathbb r p sp 2 topologically in dimension 2 can possess a real analytic metric of maximal eigenfunction growth we further show that generic metrics on any m fail to have maximal eigenfunction growth in addition we construct an example of m g for which mathscr l sb x has positive measure for an open set of x but which does not have maximal eigenfunction growth thus it disproves a naive converse to the main result

stochastic resonance for nonlinear sensors with saturation we analyze the transmission of a noisy signal by sensor devices which are linear for small inputs and saturate at large inputs large information carrying signals are thus distorted in their transmission we demonstrate conditions where addition of noise to such large input signals can reduce the distortion that they undergo in the transmission this is established for periodic as well as aperiodic and random information carrying signals various measures characterizing the transmission such as signal to noise ratio input output cross correlation and mutual information are shown improvable by addition of noise these results constitute another instance of the nonlinear phenomenon of stochastic resonance where addition of noise enhances the signal

multifractal characterization of stochastic resonance we use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals to characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence we use wavelet transform modulus maxima method for the singularity spectrum computations it is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic stochastic or chaotic input signals we show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization this fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also moreover our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence we use also the proposed approach to characterize the noise enhanced dynamics of a coupled stochastic neurons model

transitions induced by cross correlated bounded noises and time delay in a genotype selection model abstract we present a numerical investigation of occurrence of transitions in a genotype selection model with time delay where two bounded noises are cross correlated stationary probability distribution spd function is obtained it is found that the multiplicative bounded noise can facilitate the gene separation and it plays a constructive role in the genetic selection progress while the additive bounded noise suppresses the gene separation the strong correlation between noises gives a big chance to one type haploid out of the group besides what is more interesting is that the correlation time can induce a new transitions i e the curve of the spd changes from unimodal to bimodal and then to four peaks as the correlation time increases

stochastic resonance in continuous and spiking neuron models with levy noise levy noise can help neurons detect faint or subthreshold signals levy noise extends standard brownian noise to many types of impulsive jump noise processes found in real and model neurons as well as in models of finance and other random phenomena two new theorems and the ito calculus show that white levy noise will benefit subthreshold neuronal signal detection if the noise process s scaled drift velocity falls inside an interval that depends on the threshold values these results generalize earlier forbidden interval theorems of neuronal stochastic resonance sr or noise injection benefits global and local lipschitz conditions imply that additive white levy noise can increase the mutual information or bit count of several feedback neuron models that obey a general stochastic differential equation sde simulation results show that the same noise benefits still occur for some infinite variance stable levy noise processes even though the theorems themselves apply only to finite variance levy noise the appendix proves the two ito theoretic lemmas that underlie the new levy noise benefit theorems

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

coherence resonance induced by non gaussian noise in a deterministic hodgkin huxley neuron in this paper we report a novel effect of a particular kind of non gaussian noise ngn especially its deviation from gaussian noise on the spiking activity in a deterministic hodgkin huxley hh neuron it is found via numerical simulations that the coefficient of variation cv characterizing the spiking regularity nonlinearly changes with varying deviation and passes through a minimum at an optimal deviation value representing the presence of the coherence resonance cr this phenomenon shows that when the ngn is optimal the neuron can exhibit the most regular firing behavior this result provides a constructive role of the ngn in the timing precision of information processing and signal transduction in neural systems

effects of signal spectrum varying on signal processing by parameter induced stochastic resonance abstract the effects of signal spectrum varying on signal processing by the method of parameter induced stochastic resonance psr are investigated for a binary signal with a smooth power spectral density psd when the psd curve becomes sharper and narrower the performance of the nonlinear system via psr is better for a multi frequency signal formed by sine waves with different frequencies the larger the signal spectral density is the lower the ability of the psr system processing signal is and the signal to noise ratio snr gain of the psr system is increased with the increasing height of the spectral line moreover with the method of psr the stochastic signal the combination of sine waves and noise improvement is obvious the results obtained via this method are superior to those with a linear filter

adaptive stochastic resonance in noisy neurons based on mutual information noise can improve how memoryless neurons process signals and maximize their throughput information such favorable use of noise is the so called stochastic resonance or sr effect at the level of threshold neurons and continuous neurons this work presents theoretical and simulation evidence that 1 lone noisy threshold and continuous neurons exhibit the sr effect in terms of the mutual information between random input and output sequences 2 a new statistically robust learning law can find this entropy optimal noise level and 3 the adaptive sr effect is robust against highly impulsive noise with infinite variance histograms estimate the relevant probability density functions at each learning iteration a theorem shows that almost all noise probability density functions produce some sr effect in threshold neurons even if the noise is impulsive and has infinite variance the optimal noise level in threshold neurons also behaves nonlinearly as the input signal amplitude increases simulations further show that the sr effect persists for several sigmoidal neurons and for gaussian radial basis function neurons

stochastic resonance in a hodgkin huxley neuron in the absence of external noise we study numerically nonlinear responses of a periodically forced hodgkin huxley neuron the coherence of the system in the absence of external noise namely the intrinsic stochastic resonance is evidenced by the multimodal aperiodic firing pattern a bell shaped curve in the signal to noise ratio and the statistical features of the mean firing rate the subthreshold intrinsic oscillations enhance the signal transduction in a manner different from that in models studied previously

stochastic resonance in a statistical model of a time integrating detector we study an optimal nonparametric regression model for a threshold detector exposed to a noisy subthreshold signal the problem of recovering the signal is similar to that faced by neurons in nervous systems although our model is intended to be normative rather than realistic in our approach the time integrating activity of the neuron is modeled by kernel regression several aspects of the performance of the model are studied including the existence of an optimal amount of noise stochastic resonance we construct a sequential data driven procedure for estimating the subthreshold signal the performance of our model for threshold data is compared with kernel estimation for fully observed data finally we discuss differences between our estimator and the best estimator for a constant signal

robust gradient learning with applications this paper addresses the robust gradient learning rgl problem gradient learning models aim at learning the gradient vector of some target functions in supervised learning problems which can be further used to applications such as variable selection coordinate covariance estimation and supervised dimension reduction however existing gl models are not robust to outliers or heavy tailed noise this paper provides an rgl framework to address this problem in both regression and classification this is achieved by introducing a robust regression loss function and proposing a robust classification loss moreover our rgl algorithm works in an instance based kernelized dictionary instead of some fixed reproducing kernel hilbert space which may provide more flexibility to solve the proposed nonconvex model a simple computational algorithm based on gradient descent is provided and the convergence of the proposed method is also analyzed we then apply the proposed rgl model to applications such as nonlinear variable selection and coordinate covariance estimation the efficiency of our proposed model is verified on both synthetic and real data sets

refined generalization bounds of gradient learning over reproducing kernel hilbert spaces gradient learning gl initially proposed by mukherjee and zhou 2006 has been proved to be a powerful tool for conducting variable selection and dimensional reduction simultaneously this approach presents a nonparametric version of a gradient estimator with positive definite kernels without estimating the true function itself so that the proposed version has wide applicability and allows for complex effects between predictors in terms of theory however existing generalization bounds for gl depend on capacity independent techniques and the capacity of kernel classes cannot be characterized completely thus this letter considers gl estimators that minimize the empirical convex risk we prove generalization bounds for such estimators with rates that are faster than previous results moreover we provide a novel upper bound for rademacher chaos complexity of order two which also plays an important role in general pairwise type estimations including ranking and score problems

array enhancement of stochastic synchronization and signal to noise ratio gain in the nonlinear regime of signal transmission the nonlinear transformation of an external noisy signal by an array of noninteracting elements with internal noise is considered the array simulates a neuronal system that processes spike trains it is shown that increasing the number of array elements entails significant extending of the stochastic synchronization region and improvement of the signal to noise ratio snr the effects are demonstrated for arrays of triggers overdamped bistable oscillators and fitzhugh nagumo systems the interrelation between snr improvement and the efficiency of information processing is discussed

local universality for zeros and critical points of monochromatic random waves this paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves phi lambda of frequency lambda on a compact smooth riemannian manifold m g as lambda rightarrow infty we prove global variance estimates for the measures of integration over the zeros and critical points of phi lambda these global estimates hold for a wide class of manifolds for example when m g has no conjugate points and rely on new local variance estimates on zeros and critical points of phi lambda in balls of radius approx lambda 1 around a fixed point our local results hold under conditions about the structure of geodesics that are generic in the space of all metrics on m

probability densities for noisy delay bifurcations the delay of a transition in a nonlinear system due to a slowly varying control parameter can be significantly reduced by very small noise a new asymptotic approximation for the time dependent probability density function gives a complete description of the process into the transition region and is easily interpreted in terms of the noisy dynamics it is also used to calculate mean transition times the method is applied to two nonlinear systems with noise a one dimensional canonical model for a steady bifurcation and the noisy fitzhugh nagumo model

dimensionality reduction of multimodal labeled data by local fisher discriminant analysis reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high dimensional data analysis fisher discriminant analysis fda is a traditional technique for supervised dimensionality reduction but it tends to give undesired results if samples in a class are multimodal an unsupervised dimensionality reduction method called locality preserving projection lpp can work well with multimodal data due to its locality preserving property however since lpp does not take the label information into account it is not necessarily useful in supervised learning scenarios in this paper we propose a new linear supervised dimensionality reduction method called local fisher discriminant analysis lfda which effectively combines the ideas of fda and lpp lfda has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem we demonstrate the practical usefulness and high scalability of the lfda method in data visualization and classification tasks through extensive simulation studies we also show that lfda can be extended to non linear dimensionality reduction scenarios by applying the kernel trick

effects of small time delay on a bistable system subject to levy stable noise l vy stable noise is often used to describe impulsive noise bursting in communication systems this paper investigates the effects of small time delay on a bistable system driven by an aperiodic bipolar pulse signal and l vy stable noise we obtain the dynamical probability density of the system response by solving the approximated time delayed fractional fokker planck equation ffpe via an implicit finite difference method a new approach to evaluate the system response time is presented the bit error rate ber is employed to measure the performance of the bistable system in detecting binary signals the theoretical ber is validated by the monte carlo simulation we find that the existence of time delay can change both the drift term and the diffusion coefficient in time delayed ffpe for small noise intensity the time delay extends the system response time and thus reduces the detection performance however effects of this kind will fade away with the increase of noise intensity

high dimensional sparse factor modeling applications in gene expression genomics we describe studies in molecular profiling and biological pathway analysis that use sparse latent factor and regression models for microarray gene expression data we discuss breast cancer applications and key aspects of the modeling and computational methodology our case studies aim to investigate and characterize heterogeneity of structure related to specific oncogenic pathways as well as links between aggregate patterns in gene expression profiles and clinical biomarkers based on the metaphor of statistically derived factors as representing biological subpathway structure we explore the decomposition of fitted sparse factor models into pathway subcomponents and investigate how these components overlay multiple aspects of known biological activity our methodology is based on sparsity modeling of multivariate regression anova and latent factor models as well as a class of models that combines all components hierarchical sparsity priors address questions of dimension reduction and multiple co

stochastic resonance reverse resonance and resonant activation induced by a multi state noise abstract in this paper we investigate the periodic response for a linear system driven by a multiplicative multi state noise which is composed of the multiplication of two dichotomous noises to an input temporal oscillatory signal and the escape of brownian particles over the fluctuating potential barrier for a system with a piece wise linear potential and driven by an additive multi state noise which is also composed of the multiplication of two dichotomous noises for the first system we get the stochastic resonance phenomenon for the amplitude of the periodic response vs the two dichotomous noise strengths and the phenomenon of reverse resonance for the amplitude of the periodic response vs k which represents the asymmetry degree of the dichotomous noises for the second system we obtain the resonant activation phenomenon for which the mean first passage time of the brownian particles over the fluctuating potential barrier shows a minimum as the function of the transition rates of the multi state noise

predicting the coherence resonance curve using a semianalytical treatment emergence of noise induced regularity or coherence resonance in nonlinear excitable systems is well known we explain theoretically why the normalized variance v n of interspike time intervals which is a measure of regularity in such systems has a unimodal profile our semianalytic treatment of the associated spiking process produces a general yet simple formula for v n which we show is in very good agreement with numerics in two test cases namely the fitzhugh nagumo model and the chemical oscillator model

robust stochastic resonance for simple threshold neurons simulation and theoretical results show that memoryless threshold neurons benefit from small amounts of almost all types of additive noise and so produce the stochastic resonance or sr effect input output mutual information measures the performance of such threshold systems that use subthreshold signals the sr result holds for all possible noise probability density functions with finite variance the only constraint is that the noise mean must fall outside a forbidden threshold related interval that the user can control a new theorem shows that this condition is also necessary a corollary and simulations show that the sr effect occurs for right sided beta and weibull noise as well these sr results further hold for the entire uncountably infinite class of alpha stable probability density functions alpha stable noise densities have infinite variance and infinite higher order moments and often model impulsive noise environments the stable noise densities include the special case of symmetric bell curve densities with thick tails such as the cauchy probability density the sr result for alpha stable noise densities shows that the sr effect in threshold and thresholdlike systems is robust against occasional or even frequent violent fluctuations in noise regression analysis reveals both an exponential relationship for the optimal noise dispersion as a function of the alpha bell curve tail thickness and an approximate linear relationship for the sr maximal mutual information as a function of the alpha bell curve tail thickness

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

analytic expressions for rate and cv of a type i neuron driven by white gaussian noise we study the one dimensional normal form of a saddle node system under the influence of additive gaussian white noise and a static bias current input parameter a model that can be looked upon as the simplest version of a type i neuron with stochastic input this is in contrast with the numerous studies devoted to the noise driven leaky integrate and fire neuron we focus on the firing rate and coefficient of variation cv of the interspike interval density for which scaling relations with respect to the input parameter and noise intensity are derived quadrature formulas for rate and cv are numerically evaluated and compared to numerical simulations of the system and to various approximation formulas obtained in different limiting cases of the model we also show that caution must be used to extend these results to the neuron model with multiplicative gaussian white noise the correspondence between the first passage time statistics for the saddle node model and the neuron model is obtained only in the stratonovich interpretation of the stochastic neuron model while previous results have focused only on the ito interpretation the correct stratonovich interpretation yields cvs that are still relatively high although smaller than in the ito interpretation it also produces certain qualitative differences especially at larger noise intensities our analysis provides useful relations for assessing the distance to threshold and the level of synaptic noise in real type i neurons from their firing statistics we also briefly discuss the effect of finite boundaries finite values of threshold and reset on the firing statistics

gradient particle solutions of fokker planck equations for noisy delay bifurcations a gradient particle method is used to compute probability densities for processes with delay bifurcations that are sensitive to very small noise the method is particularly useful for computing the probability density in the transition region where asymptotic approximations may not be valid for a one dimensional steady bifurcation problem and a two dimensional fitzhugh nagumo model we solve the fokker planck equation fpe and compare the results with direct simulations and asymptotic approximations

classical harmonic oscillator with multiplicative noise abstract the first two moments of an underdamped harmonic oscillator were found for additive and multiplicative noises a detailed analysis is performed of the case of a random damping which supplements the case of a random frequency which has received the most study in both cases the second moments are unstable for the sufficient large noise strength the difference appears in the stability conditions of the first moments which can diverge already for white noise for the random damping and only for color noise for the random frequency the difference between these two types of multiplicative noises shows up also in the response to an external periodic force namely white noise influences the first moment only in the case of a random damping however the second moments in both cases show the non monotonic dependence on the noise strength and correlation rate stochastic resonance in linear systems

learning gradients predictive models that infer geometry and statistical dependence the problems of dimension reduction and inference of statistical dependence are addressed by the modeling framework of learning gradients the models we propose hold for euclidean spaces as well as the manifold setting the central quantity in this approach is an estimate of the gradient of the regression or classification function two quadratic forms are constructed from gradient estimates the gradient outer product and gradient based diffusion maps the first quantity can be used for supervised dimension reduction on manifolds as well as inference of a graphical model encoding dependencies that are predictive of a response variable the second quantity can be used for nonlinear projections that incorporate both the geometric structure of the manifold as well as variation of the response variable on the manifold we relate the gradient outer product to standard statistical quantities such as covariances and provide a simple and precise comparison of a variety of supervised dimensionality reduction methods we provide rates of convergence for both inference of informative directions as well as inference of a graphical model of variable dependencies

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

stochastic resonance of ensemble neurons for transient spike trains wavelet analysis by using the wavelet transformation wt i have analyzed the response of an ensemble of n 1 10 100 and 500 hodgkin huxley neurons to transient m pulse spike trains m i to 3 with independent gaussian noises thecross correlation between the input and output signals is expressed in terms of the wt expansion coefficients the signal to noise ratio snr is evaluated by using the denoising method within the wt by which the noise contribution is extracted from the output signals although the response of a single n 1 neuron to subthreshold transient signals with noises is quite unreliable the transmission fidelity assessed by the cross correlation and snr is shown to be much improved by increasing the value of n a population of neurons plays an indispensable role in the stochastic resonance sr for transient spike inputs it is also shown that in a large scale ensemble the transmission fidelity for suprathreshold transient spikes is not significantly degraded by a weak noise which is responsible to sr for subthreshold inputs

experiments on coherence resonance noisy precursors to hopf bifurcations experimental and numerical evidence of coherence resonance in an electrochemical system is reported external noise with a gaussian distribution is superimposed on the system when the anodic current is exhibiting stationary fixed point dynamics below a supercritical hopf bifurcation the amplitude of the added stochastic perturbations is increased monotonically and the induced oscillatory behavior is analyzed it is observed both in experiments and in simulations that the regularity of the noise induced current oscillations reaches a maximum value for an optimum noise level this is indicative of coherence resonance and can be explained with a mechanism based on noisy precursors to a hopf bifurcation

noise can speed convergence in markov chains a new theorem shows that noise can speed convergence to equilibrium in discrete finite state markov chains the noise applies to the state density and helps the markov chain explore improbable regions of the state space the theorem ensures that a stochastic resonance noise benefit exists for states that obey a vector norm inequality such noise leads to faster convergence because the noise reduces the norm components a corollary shows that a noise benefit still occurs if the system states obey an alternate norm inequality this leads to a noise benefit algorithm that requires knowledge of the steady state an alternative blind algorithm uses only past state information to achieve a weaker noise benefit simulations illustrate the predicted noise benefits in three well known markov models the first model is a two parameter ehrenfest diffusion model that shows how noise benefits can occur in the class of birth death processes the second model is a wright fisher model of genotype drift in population genetics the third model is a chemical reaction network of zeolite crystallization a fourth simulation shows a convergence rate increase of 64 for states that satisfy the theorem and an increase of 53 for states that satisfy the corollary a final simulation shows that even suboptimal noise can speed convergence if the noise applies over successive time cycles noise benefits tend to be sharpest in markov models that do not converge quickly and that do not have strong absorbing states

slicing regression a link free regression method slicing regression a link free regression m e t h o d author s naihua duan and k e r c h a u l i s o u r c e the annals of statistics v o l 19 n o 2 t u n 1991 p p 5 0 5 5 3 0 p u b l i s h e d b y institute of mathematical statistics s t a b l e u r l http www jstor org stable 2242072 accessed 16 05 2011 your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available at http www jstor org page info about policies terms jsp jstor s terms and conditions of use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained at http www jstor org action showpublisher publishercode ims each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jstor is a not for profit service that helps scholars researchers and students discover use and build upon a wide range of content in a trusted digital archive we use information technology and tools to increase productivity and facilitate new forms of scholarship for more information about jstor please contact support jstor org http www jstor org

estimating and improving the signal to noise ratio of time series by symbolic dynamics we investigate the effect of symbolic encoding applied to times series consisting of some deterministic signal and additive noise as well as time series given by a deterministic signal with randomly distributed initial conditions as a model of event related brain potentials we introduce an estimator of the signal to noise ratio snr of the system by means of time averages of running complexity measures such as shannon and r enyi entropies and prove its asymptotical equivalence with the linear snr in the case of shannon entropies of symbol distributions a snr improvement factor is defined exhibiting a maximum for intermediate values of noise amplitude in analogy to stochastic resonance phenomena we demonstrate that the maximum of the snr improvement factor can be shifted toward smaller noise amplitudes by using higher order r enyi entropies instead of the shannon entropy for a further improvement of the snr a half wave encoding of noisy time series is introduced finally we discuss the effect of noisy phases on the linear snr as well as on the snr defined by symbolic dynamics it is shown that longer symbol sequences yield an improvement of the latter

statistical analysis of stochastic resonance in a simple setting a subthreshold signal may be detected if noise is added to the data we study a simple model consisting of a constant signal to which at uniformly spaced times independent and identically distributed noise variables with known distribution are added a detector records the times at which the noisy signal exceeds a threshold there is an optimal noise level called stochastic resonance we explore the detectability of the signal in a system with one or more detectors with different thresholds we use a statistical detectability measure the asymptotic variance of the best estimator of the signal from the thresholded data or equivalently the fisher information in the data in particular we determine optimal configurations of detectors varying the distances between the thresholds and the signal as well as the noise level the approach generalizes to nonconstant signals

theory of stochastic resonance in signal transmission by static nonlinear systems a deterministic periodic signal plus a stationary random noise is applied to a static nonlinearity taking the form of a monovariable arbitrary function on real numbers the property of noise enhanced signal transmission through stochastic resonance is studied for this class of static nonlinear systems a theory is developed that provides expressions for the output autocorrelation function power spectral density signal to noise ratio and input output phase shift in the presence of a periodic input a noise distribution and a static nonlinearity all three being arbitrary both white and colored input noises are successively considered for white input noise exact expressions are derived in a discrete time framework directly confrontable to simulations or experiments the theory is applied to describe stochastic resonance in various examples of static nonlinear systems for instance a diode nonlinearity in addition confrontations with experiments and simulations are given that support the theory in particular interesting effects are reported such as a signal to noise ratio larger at the output than at the input or stochastic resonance at zero frequency finally the validity of the theory is extended to dynamic nonlinear systems that can be decomposed into a static nonlinearity followed by an arbitrary dynamic linear system s1063 651x 97 12002 5

aperiodic stochastic resonance stochastic resonance sr is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise recently we presented a method and theory for characterizing sr type behavior in excitable systems with aperiodic i e broadband input signals phys rev e 52 r3321 1995 we coined the term aperiodic stochastic resonance asr to describe this general type of behavior in that earlier study we demonstrated asr in the fitzhugh nagumo neuronal model here we demonstrate asr in three additional systems a bistable well system an integrate and fire neuronal model and the hodgkin huxley hh neuronal model we present computational and theoretical results for each system in the context of the hh model we develop a general theory for asr in excitable membranes this work clearly shows that sr type behavior is not limited to systems with periodic inputs thus in general noise can serve to enhance the response of a nonlinear system to a weak input signal regardless of whether the signal is periodic or aperiodic textcopyright 1996 the american physical society

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

noise sensitivity of phase synchronization time in stochastic resonance theory and experiment recent numerical and heuristic arguments have revealed that the average phase synchronization time between the input and the output associated with stochastic resonance is highly sensitive to noise variation in particular there is evidence that this average time exhibits a cusplike behavior as the noise strength varies through the optimal value here we present an explicit formula for the average phase synchronization time in terms of the phase diffusion coefficient and the average frequency difference between the input and the output signals we also provide experimental evidence for the cusplike behavior by using a bistable microelectronic circuit system

nonstationary stochastic resonance it is by now established that remarkably the addition of noise to a nonlinear system may sometimes facilitate rather than hamper the detection of weak signals this phenomenon usually referred to as stochastic resonance was originally associated with strictly periodic signals but it was eventually shown to occur for stationary aperiodic signals as well however in several situations of practical interest the signal can be markedly nonstationary we demonstrate that the phenomenon of stochastic resonance extends to nonstationary signals as well and thus could be relevant to a wider class of biological and electronic applications building on both nondynamic and aperiodic stochastic resonance our scheme is based on a multilevel trigger mechanism which could be realized as a parallel network of differentiated threshold sensors we find that optimal detection is reached for a number of thresholds of order 10 and that little is gained by going much beyond that number we raise the question of whether this is related to the fact that evolution has favored some fixed numbers of precisely this order of magnitude in certain aspects of sensory perception

optimising threshold levels for information transmission in binary threshold networks independent multiplicative noise on each threshold abstract the problem of optimising the threshold levels in multilevel threshold system subject to multiplicative gaussian and uniform noise is considered similar to previous results for additive noise we find a bifurcation phenomenon in the optimal threshold values as the noise intensity changes this occurs when the number of threshold units is greater than one we also study the optimal thresholds for combined additive and multiplicative gaussian noise and find that all threshold levels need to be identical to optimise the system when the additive noise intensity is a constant however this identical value is not equal to the signal mean unlike the case of additive noise when the multiplicative noise intensity is instead held constant the optimal threshold levels are not all identical for small additive noise intensity but are all equal to zero for large additive noise intensity the model and our results are potentially relevant for sensor network design and understanding neurobiological sensory neurons such as in the peripheral auditory system

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m