long memory continuous time models this paper presents a new family of long memory models the continuous time moving average fractional process the continuous time framework allows to reconcile two competitive types of modelling fractional integration of arma processes and fractional brownian motion a comparison with usual discrete time arfima models is lead some well known empirical evidence on macroeconomic and financial time series such as variability of forward rates aggregation of responses across heterogeneous agents are well captured by this continuous time modelling moreover the usual statistical tools for long memory series and for stochastic differential equations can be jointly applied in this setting

new reductions of the kadomtsev petviashvili and two dimensional toda lattice hierarchies via symmetry constraints new types of reductions of the kadomtsev petviashvili kp hierarchy and the two dimensional toda lattice 2dtl hierarchy are considered on the basis of sato s approach within this approach these hierarchies are represented by infinite sets of equations for potentials u1 u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp and the 2dtl hierarchies are studied under constraints of the following type n 1n nsn u1 u2 u3 x where sn are symmetries for these hierarchies n are arbitrary constants and is an arbitrary linear functional of the quantity it is shown that for the kp hierarchy these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 u3 un many known systems and several new ones are among them symmetry reductions for the 2dtl hierarchy give rise both to finite dimensional dynamical systems and 1 1 dimensional discrete systems some few results for the modified kp hierarc

variability in noise driven integrator neurons neural variability in the presence of noise has been studied mainly in resonator neurons such as hodgkinhuxley or fitzhugh nagumo models here we investigate this variability for integrator neurons whose excitability is due to a saddle node bifurcation of the rest state instead of a hopf bifurcation using simple theoretical expressions for the interspike times distributions we obtain coefficients of variation in good agreement with numerical calculations in realistic neuron models the main features of this coefficient as a function of noise depend on the refractory period and on the presence of bistability the bistability is responsible for the existence of two different time scales in the spiking behavior giving an antiresonance effect

relationship between the moyal kp and the sato kp hierarchies the formula describing the isomorphism between the moyal kp and the sato kp hierarchies is given we obtain the conserved densities of the dispersionless kp hierarchy by taking the limit k to 0 of the conserved quantities derived in the sato approach

conserved quantities and symmetries of kp hierarchy conserved quantities and symmetries of the kp equation from the point of view of the sato theory that provides a unifying approach to soliton equations is studied conserved quantities are derived from the generalized lax equations some reductions of the kp hierarchy such as kdv boussinesq a coupled kdv and sawada kotera equation are also considered by expansion of the squared eigenfunctions of the lax equations in terms of the function symmetries of the kp equations are obtained the relationship of this procedure to the two dimensional recursion operator newly found by fokas and santini is discussed

explorations of the extended nckp hierarchy a recently obtained extension xnckp of the moyal deformed kp hierarchy nckp hierarchy by a set of evolution equations in the moyal deformation parameters is further explored formulae are derived to compute these equations efficiently reductions of the xnckp hierarchy are treated in particular to the extended nckdv and ncboussinesq hierarchies furthermore a good part of the sato formalism for the kp hierarchy is carried over to the generalized framework in particular the well known bilinear identity theorem for the kp hierarchy expressed in terms of the formal baker akhiezer function extends to the xnckp hierarchy moreover it is demonstrated that n soliton solutions of the nckp equation are also solutions of the first few deformation equations this is shown to be related to the existence of certain families of algebraic identities

differential polynomial expressions related to the kadomtsev petviashvili and korteweg de vries hierarchies an integrable nonlinear partial differential equation typically extends to a hierarchy of integrable equations there exist several recursive schemes for obtaining these hierarchies recently explicit expressions for the kdv hierarchy have been found we derive explicit expressions for the hierarchy associated with the kp equation the main tools are sato s theory hirota s formalism and bell s polynomials

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

establishing a direct connection between detrended fluctuation analysis and fourier analysis to understand methodological features of the detrended fluctuation analysis dfa using a higher order polynomial fitting we establish the direct connection between dfa and fourier analysis based on an exact calculation of the single frequency response of the dfa the following facts are shown analytically 1 in the analysis of stochastic processes exhibiting a power law scaling of the power spectral density psd s f f a higher order detrending in the dfa has no adverse effect in the estimation of the dfa scaling exponent which satisfies the scaling relation 1 2 2 the upper limit of the scaling exponents detectable by the dfa depends on the order of polynomial fit used in the dfa and is bounded by m 1 where m is the order of the polynomial fit 3 the relation between the time scale in the dfa and the corresponding frequency in the psd are distorted depending on both the order of the dfa and the frequency dependence of the psd we can improve the scale distortion by introducing the corrected time scale in the dfa corresponding to the inverse of the frequency scale in the psd in addition our analytical approach makes it possible to characterize variants of the dfa using different types of detrending as an application properties of the detrending moving average algorithm are discussed

the constraint on potential and decomposition for 2 1 dimensional integrable systems besides a sufficient condition a necessary condition is given to determine some kind of consistent constraint on the potential of a 2 1 dimensional integrable system which cannot be obtained from the sufficient condition under this kind of constraint on the potential a 2 1 dimensional integrable system can be decomposed into two commuting 1 1 dimensional integrable systems and can be further decomposed into three commuting finite dimensional integrable hamiltonian systems the bkp hierarchy is taken as an illustrative example

on 1 f noise and other distributions with long tails abstract it is shown following shockley shockley w 1957 proc ire 45 279 290 that when a population is engaged in tasks whose completion requires the successful conclusion of many independent subtasks the distribution function for successes in the primary task is log normal it is also shown that when the dispersion of the log normal distribution is large the distribution is mimicked by a 1 x distribution over a wide range of x this argument provides a generic set of processes that yields the much observed 1 x distribution and will also lead to a 1 f noise spectrum it is commonly found that distributions that seem to be log normal over a broad range say to the 95th percentile of a population change to an inverse fractional power pareto distribution for the last few percentile annual income distributions are examples with this structure the very wealthy generally achieve their superwealth through amplification processes that are not available to most we have introduced a simple amplification model to characterize the transition from a log normal distribution to an inverse power pareto tail

signal detection in fractional gaussian noise several results related to the reproducing kernel hilbert space of fractional brownian motion are presented to facilitate the study of signal detection in additive fractional gaussian noise this hilbert space is completely characterized and an alternative characterization for the restriction of this class of functions to a compact interval 0 t is given infinite and finite interval whitening filters for fractional brownian motion are also developed the application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional gaussian noise a formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed

wronskian and rational solutions of the differential difference kp equation we present the wronskian form of the n soliton solutions of the differential difference kadomtsev petviashvilli dkp equation also a wider class of rational solutions are derived using semi discrete analogue of schur polynomials and its generalization our approach is based on sato theory formalism which gives all these solutions naturally

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

long memory via networking many time series exhibit long memory their autocorrelation function decays slowly with lag this behavior has traditionally been modeled via unit roots or fractional brownian motion and explained via aggregation of heterogeneous processes nonlinearity learning dynamics regime switching or structural breaks this paper identifies a different and complementary mechanism for long memory generation by showing that it can naturally arise when a large number of simple linear homogeneous economic subsystems with short memory are interconnected to form a network such that the outputs of the subsystems are fed into the inputs of others this networking picture yields a type of aggregation that is not merely additive resulting in a collective behavior that is richer than that of individual subsystems interestingly the long memory behavior is found to be almost entirely determined by the geometry of the network while being relatively insensitive to the specific behavior of individual agents

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

the spectra and periodograms of anti correlated discrete fractional gaussian noise discrete fractional gaussian noise dfgn has been proposed as a model for interpreting a wide variety of physiological data the form of actual spectra of dfgn for frequencies near zero varies as f 1 2h where 0 h 1 is the hurst coefficient however this form for the spectra need not be a good approximation at other frequencies when h approaches zero dfgn spectra exhibit the 1 2h power law behavior only over a range of low frequencies that is vanishingly small when dealing with a time series of finite length drawn from a dfgn process with unknown h practitioners must deal with estimated spectra in lieu of actual spectra the most basic spectral estimator is the periodogram the expected value of the periodogram for dfgn with small h also exhibits non power law behavior at the lowest fourier frequencies associated with a time series of n values sampled from a dfgn process the expected value of the periodogram for h approaching zero varies as f 0 rather than f 1 2h for finite n and small h the expected value of the periodogram can in fact exhibit a local power law behavior with a spectral exponent of 1 2h at only two distinct frequencies

texture roughness analysis and synthesis via extended self similar ess model the 2d fractional brownian motion fbm model provides a useful tool to model textured surfaces whose roughness is scale invariant to represent textures whose roughness is scale dependent we generalize the fbm model to the extended self similar ess model in this research we present an estimation algorithm to extract the model parameters from real texture data furthermore a new incremental fourier synthesis algorithm is proposed to generate the 2d realizations of the ess model finally the estimation and rendering methods are combined to synthesize real textured surfaces

varieties of long memory models long memory is defined as a series having a slowly declining correlogram or equivalently an infinite spectrum at zero frequency fractional integrated processes have such properties but here it is pointed out that a number of other processes can also be long memory including generalized fractionally integrated models arising from aggregation time changing coefficient models and possibly nonlinear models it seems that there are many classes of processes that deserve further study the relevance of long memory is illustrated using absolute returns from a daily stock market index

on the structure and properties of the singularity manifold equations of the kp hierarchy this paper is concerned with the painleve expansion and the singularity manifold sm equation of the kadomtsev petviashvili kp equation several aspects of the interrelation between the sm equation and the kp auxiliary linear system are studied it is shown that the simultaneous painleve expansion for the kp potential u and the kp eigenfunction can be treated as a backlund gauge transformation two methods for the derivation of the sm equation based on this treatment are proposed and their equivalence is proved the interrelation between the sm equation and the vertical hierarchy of the kp eigenfunction equations is discussed the explanation of the coincidence of the kp eigenfunction equation of the second level and the kp sm equation is given compact forms of the hierarchy of sm equations of the kp hierarchy are presented the connection between the kp singularity manifold function and the kp eigenfunctions and the adjoint kp eigenfunctions is derived the bilinear bilocal description of th

negative semi discrete kp and bkp hierarchies via nonlocal symmetries in this paper nonlocal symmetries for the bilinear semi discrete kadomtsev petviashvili kp and bilinear semi discrete b type kp bkp equations are derived by expanding these nonlocal symmetries in power series of each of two parameters we have derived bilinear negative semi discrete kp and bkp hierarchies an interesting observation is that bilinear positive and negative semi discrete kp hierarchies may be derived from the same nonlocal symmetry for the semi discrete kp equation

darboux theorems and wronskian formulas for integrable systems i constrained kp flows abstract generalizations of the classical darboux theorem are established for pseudo differential scattering operators of the form l i 0 n u i i i 1 m i 1 i i iteration of the darboux transformations leads to a gauge transformed operator with coefficients given by wronskian formulas involving a set of eigenfunctions of l nonlinear integrable partial differential equations are associated with the scattering operator l which arise as a symmetry reduction of the multicomponent kp hierarchy with a suitable linear time evolution for the eigenfunctions the darboux transformation is used to obtain solutions of the integrable equations in terms of wronskian determinants

the conserved quantities and symmetries of the two dimensional toda lattice hierarchy the conserved quantities and symmetries of the soliton equations included in the 2dtl two dimensional toda lattice hierarchy are considered by means of the theory of the function a typical example is presented for the 2dtl equation that is the simplest nontrivial equation in the hierarchy then by applying suitable reductions the cases of the toda lattice equation and the sine gordon equation are discussed

physiological gain leads to high isi variability in a simple model of a cortical regular spiking cell to understand the interspike interval isi variability displayed by visual cortical neurons softky koch 1993 it is critical to examine the dynamics of their neuronal integration as well as

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

the reduced self dual yang mills equation binary and infinitesimal darboux transformations the self dual yang mills equation in the 2 1 dimensional space time is considered binary darboux transformation of a new kind is applied to obtain the infinite hierarchy of solutions expressed through ones of corresponding lax pairs on an initial solution of the equation studied sufficient conditions are given for the solutions built to take values in the class of unimodular positive definite matrices a new infinitesimal darboux transformation is introduced and a particular solution of the linearization of the self dual yang mills equation is given two hierarchies of infinitesimal symmetries depending on a solution of the nonlinear equation are extracted

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

fractional brownian motion a maximum likelihood estimator and its application to image texture fractals have been shown to be useful in characterizing texture in a variety of contexts use of this methodology normally involves measurement of a parameter h which is directly related to fractal dimension in this work the basic theory of fractional brownian motion is extended to the discrete case it is shown that the power spectral density of such a discrete process is only approximately proportional to f a instead of in direct proportion as in the continuous case an asymptotic cramer rao bound is derived for the variance of an estimate of h subsequently a maximum likelihood estimator mle is developed to estimate h it is shown that the variance of this estimator nearly achieves the minimum bound a generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the mle when the discrete fractional brownian process is contaminated with additive gaussian noise the results show that even at signal to noise ratios of 30 db significant errors in estimation of h can result when noise is present the mle is then applied to x ray images of the human calcaneus to demonstrate how the line to line formulation can be applied to the two dimensional case these results indicate that it has strong potential for quantifying texture

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

a note on the discrete wavelet transform of second order processes some properties of second order random processes such as stationarity and scaling are characterized via corresponding multiscale properties of their discrete wavelet transform the single scale characterizations valid in the continuous setting do not extend to the discrete setting some extensions of these results are also indicated

an algebraic scheme associated with the noncommutative kp hierarchy and some of its extensions a well known ansatz trace method for soliton solutions turns the equations of the non commutative kp hierarchy and those of certain extensions into families of algebraic sum identities we develop an algebraic formalism in particular involving a mixable shuffle product to explore their structure more precisely we show that the equations of the non commutative kp hierarchy and its extension xnckp in the case of a moyal deformed product as derived in previous work correspond to identities in this algebra furthermore the moyal product is replaced by a more general associative product this leads to a new even more general extension of the non commutative kp hierarchy relations with rota baxter algebras are established

characteristics of natural scenes related to the fractal dimension many objects in images of natural scenes are so complex and erratic that describing them by the familiar models of classical geometry is inadequate in this paper we exploit the power of fractal geometry to generate global characteristics of natural scenes in particular we are concerned with the following two questions 1 can we develop a measure which can distinguish between different global backgrounds e g mountains and trees and 2 can we develop a measure that is sensitive to change in distance or scale we present a model based on fractional brownian motion which will allow us to recover two characteristics related to the fractal dimension from silhouettes the first characteristic is an estimate of the fractal dimension based on a least squares linear fit we show that this feature is stable under a variety of real image conditions and use it to distinguish silhouettes of trees from silhouettes of mountains next we introduce a new theoretical concept called the average holder constant and relate it mathematically to the fractal dimension it is shown that this measurement is sensitive to scale in a predictable manner and hence provides the potential for use as a range indicator corroborating experimental results are presented

bi differential calculi and integrable models the existence of an infinite set of conserved currents in completely integrable classical models including chiral and toda models as well as the kp and self dual yang mills equations is traced back to a simple construction of an infinite chain of closed respectively covariantly constant 1 forms in a gauged bi differential calculus the latter consists of a differential algebra on which two differential maps act in a gauged bi differential calculus these maps are extended to flat covariant derivatives

behavior of integrate and fire and hodgkin huxley models with correlated inputs we assess both numerically and theoretically how positively correlated poisson inputs affect the output of the integrate and fire and hodgkin huxley models for the integrate and fire model the variability of efferent spike trains is an increasing function of input correlation and of the ratio between inhibitory and excitatory inputs interestingly for the hodgkin huxley model the variability of efferent spike trains is a decreasing function of input correlation and for fixed input correlation it is almost independent of the ratio between inhibitory and excitatory inputs in terms of the signal to noise ratio of efferent spike trains the integrate and fire model works better in an environment of asynchronous inputs but the hodgkin huxley model has an advantage for more synchronous correlated inputs in conclusion the integrate and fire and hodgkin huxley models respond to correlated inputs in totally opposite ways

commuting flows and conservation laws for noncommutative lax hierarchies we discuss commuting flows and conservation laws for lax hierarchies on noncommutative spaces in the framework of the sato theory on commutative spaces the sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the lax hierarchies in terms of pseudodifferential operators noncommutative extension of the sato theory has been already studied by the author and toda and the existence of various noncommutative lax hierarchies are guaranteed in this paper we present conservation laws for the noncommutative lax hierarchies with both space space and space time noncommutativities and prove the existence of infinite number of conserved densities we also give the explicit representations of them in terms of lax operators our results include noncommutative versions of kp kdv boussinesq coupled kdv sawada kotera modified kdv equation and so on

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

fast exact synthesis of gaussian and nongaussian long range dependent processes 1 f noise and statistically self similar random processes such as fractional brownian motion fbm and fractional gaussian noise fgn are fundamental models for a host of real world phenomena from network traffic to dna to the stock market synthesis algorithms play a key role by providing the feedstock of data necessary for running complex simulations and accurately evaluating analysis techniques unfortunately current algorithms to correctly synthesize these long range dependent lrd processes are either abstruse or prohibitively costly which has spurred the wide use of inexact approximations to fill the gap we develop a simple fast o n log n operations for a length n signal framework for exactly synthesizing a range of gaussian and nongaussian lrd processes as a bonus we introduce and study a new bi scaling fbm process featuring a kinked correlation function that exhibits distinct scaling laws at coarse and fine scales

statistical analysis of dwt coefficients of fgn processes using arfima p d q models abstract fractional gaussian noise fgn provides an important parametric representation for the data recorded from long memory processes also it has been well established in literature that the orthogonal wavelet transforms prove to be the optimal bases to represent the data as fgn or fbm fractional brownian motion models this paper highlights the statistical properties of discrete wavelet transform dwt coefficients in the wavelet expansion of fgn statistical analysis was carried out by analyzing the inter scale and intra scale correlations of the dwt coefficients for wavelets with varying vanishing moments two types of auto regressive moving average arma models were fit to the wavelet coefficients of fgn namely i arma p q and ii arfima p d q models the latter represents the arma models with fractional differencing using the akaike information criteria aic and the bayesian information criteria bic it has been shown that arfima models best represent the wavelet coefficients of fgn the above observation holds good when wavelets with increasing number of vanishing moments are used for obtaining dwt coefficients after estimating the optimal model and its parameters different properties pertaining to the inter scale and intra scale correlations were verified using these models

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence

dynamics of membrane excitability determine interspike interval variability a link between spike generation mechanisms and cortical spike train statistics we propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains the key lies in the nonlinear dynamics of cortical spike generation which are consistent with type i membranes where saddle node dynamics underlie excitability rinzel ermentrout 1989 we present a canonical model for type i membranes the neuron the neuron is a phase model whose dynamics reflect salient features of type i membranes this model generates spike trains with coefficient of variation cv above 0 6 when brought to firing by noisy inputs this happens because the timing of spikes for a type i excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses a noisy input current giving random amplitude kicks to the cell evokes highly irregular firing across a wide range of firing rates an intrinsically oscillating cell gives regular spike trains we corroborate the results with simulations of the morris lecar m l neural model with random synaptic inputs type i m l yields high cvs when this model is modified to have type ii dynamics periodicity arises via a hopf bifurcation however it gives regular spike trains cv below 0 3 our results suggest that the high cv values such as those observed in cortical spike trains are an intrinsic characteristic of type i membranes driven to firing by random inputs in contrast neural oscillators or neurons exhibiting type ii excitability should produce regular spike trains

statistical properties of spike trains universal and stimulus dependent aspects statistical properties of spike trains measured from a sensory neuron in vivo are studied experimentally and theoretically experiments are performed on an identified neuron in the visual system of the blowfly it is shown that the spike trains exhibit universal behavior over a short time modulated by a stimulus dependent envelope over a long time a model of the neuron as a nonlinear oscillator driven by noise and by an external stimulus is suggested to account for these results in the short time universal regime the main biophysical effect is refractoriness which can be described as a repulsive 1 x interaction law among spikes a universal distribution function for intervals is found defining a point process with special symmetry properties the long time modulations in the spike train are related in a simple way to the properties of the input stimulus as seen through the neuronal nonlinearity thus our model enables a separation of the effects of internal neuronal properties from the effect of external stimulus properties explicit formulas are derived for different statistical properties which are in very good agreement with the data in both the universal and the stimulus dependent regimes

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated