performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

block adaptive filters with deterministic reference inputs for event related signals blms and brls adaptive estimation of the linear coefficient vector in truncated expansions is considered for the purpose of modeling noisy recurrent signals two different criteria are studied for block wise processing of the signal the mean square error mse and the least squares ls error the block lms blms algorithm being the solution of the steepest descent strategy for minimizing the mse is shown to be steady state unbiased and with a lower variance than the lms algorithm it is demonstrated that blms is equivalent to an exponential averager in the subspace spanned by the truncated set of basis functions the block recursive least squares brls solution is shown to be equivalent to the blms algorithm with a decreasing step size the brls is unbiased at any occurrence number of the signal and has the same steady state variance as the blms but with a lower variance at the transient stage the estimation methods can be interpreted in terms of linear time variant filtering the performance of the methods is studied on an ecg signal and the results show that the performance of the block algorithms is superior to that of the lms algorithm in addition measurements with clinical interest are found to be more robustly estimated in noisy signals

multilevel large deviations and interacting diffusions let n be a sequence of random variables with values in a topological space which satisfy the large deviation principle for eachm and eachn let m n denote the empirical measure associated withm independent copies of n as a main result we show that m n also satisfies the large deviation principle asm n we derive several representations of the associated rate function these results are then applied to empirical measure processes m n t m 1 i 1n in t 0 t t where 1n mn t is a system of weakly interacting diffusions with noise intensity 1 n this is a continuation of our previous work on the mckean vlasov limit and related hierarchical models 4 5

large deviations for finite state markov jump processes with mean field interaction via the comparison principle for an associated hamilton jacobi equation we prove the large deviation principle ldp for the trajectory of a broad class of finite state mean field interacting markov jump processes via a general analytic approach based on viscosity solutions examples include generalized ehrenfest models as well as curie weiss spin flip dynamics with singular jump rates the main step in the proof of the ldp which is of independent interest is the proof of the comparison principle for an associated collection of hamilton jacobi equations additionally we show that the ldp provides a general method to identify a lyapunov function for the associated mckean vlasov equation

power law statistics and universal scaling in the absence of criticality critical states are sometimes identified experimentally through power law statistics or universal scaling functions we show here that such features naturally emerge from networks in self sustained irregular regimes away from criticality in these regimes statistical physics theory of large interacting systems predict a regime where the nodes have independent and identically distributed dynamics we thus investigated the statistics of a system in which units are replaced by independent stochastic surrogates and found the same power law statistics indicating that these are not sufficient to establish criticality we rather suggest that these are universal features of large scale networks when considered macroscopically these results put caution on the interpretation of scaling laws found in nature

moderate deviation principles for weakly interacting particle systems moderate deviation principles for empirical measure processes associated with weakly interacting markov processes are established two families of models are considered the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump markov processes with a countable state space for both cases the moderate deviation principle is formulated in terms of a large deviation principle ldp with an appropriate speed function for suitably centered and normalized empirical measure processes for the first family of models the ldp is established in the path space of an appropriate schwartz distribution space whereas for the second family the ldp is proved in the space of l 2 the hilbert space of square summable sequences valued paths proofs rely on certain variational representations for exponential functionals of brownian motions and poisson random measures

spatiotemporal dynamics of continuum neural fields we survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields neural fields model the large scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections they provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and turing like patterns

an adaptive estimation of periodic signals using a fourier linear combiner presents an adaptive algorithm for estimating from noisy observations periodic signals of known period subject to transient disturbances the estimator is based on the lms algorithm and works by tracking the fourier coefficients of the data the estimator is analyzed for convergence noise misadjustment and lag misadjustment for signals with both time invariant and time variant parameters the analysis is greatly facilitated by a change of variable that results in a time invariant difference equation at sufficiently small values of the lms step size the system is shown to exhibit decoupling with each fourier component converging independently and uniformly detection of rapid transients in data with low signal to noise ratio can be improved by using larger step sizes for more prominent components of the estimated signal an application of the fourier estimator to estimation of brain evoked responses is included

steady state mse convergence of lms adaptive filters with deterministic reference inputs with applications to biomedical signals we analyze the steady state mean square error mse convergence of the lms algorithm when deterministic functions are used as reference inputs a particular adaptive linear combiner is presented where the reference inputs are any set of orthogonal basis functions the adaptive orthogonal linear combiner aolc several authors have applied this structure always considering in the analysis a time average behavior over one signal occurrence we make a more precise analysis using the deterministic nature of the reference inputs and their time variant correlation matrix two different situations are considered in the analysis orthogonal complete expansions and incomplete expansions the steady state misadjustment is calculated using two different procedures with equivalent results the classical one analyzing the transient behavior of the mse and as the residual noise at the output of the equivalent time variant transfer function of the system the latter procedure allows a very simple formalism being valid for colored noise as well the derived expressions for steady state misadjustment are contrasted with experimental results in electrocardiographic ecg signals giving exact concordance for any value of the step size

generalising the kuramoto model for the study of neuronal synchronisation in the brain abstract in this article we have generalised the kuramoto model to allow one to model neuronal synchronisation more appropriately the generalised version allows for different connective arrangements time varying natural frequencies and time varying coupling strengths to be realised within the framework of the original kuramoto model by incorporating the above mentioned features into the original kuramoto model one can allow for the adaptive nature of neurons in the brain to be accommodated extensive tests using the generalised kuramoto model were performed on a n 4 coupled oscillator network examination of how different connective arrangements time varying natural frequencies and time varying coupling strengths affected synchronisation separately and in combination are reported the effects on synchronisation for large n are also reported

a certain class of diffusion processes associated with nonlinear parabolic equations we introduce a martingale problem to associate diffusion processes with a kind of nonlinear parabolic equation then we show the existence and uniqueness theorems for solutions to the martingale problem

the complex lms adaptive algorithm transient weight mean and covariance with applications to the ale the transient and steady state mean and covariance of the complex valued lms adaptive element weights are investigated when the inputs are samples from circularly normal processes it is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix this result permits describing the complete transient behavior of the adaptive line enhancer ale weight covariance matrix in closed form for the case of multiple equal power narrow band statistically independent orthogonal rayleigh fading sinusoids in broad band noise based on these transient covariance results it is shown that for any stage of adaptation there exists an optimum ale gain constant that minimizes the excess mean squared prediction error this result is of particular significance when processing time limited data

adaptive cancellation of periodic synchronously sampled interference a special case of the adaptive noise canceller is considered in which the interference is harmonic and synchronously sampled the implementation of the lms algorithm in this case can be made very efficient and its behavior can be represented exactly by that of a linear time invariant comb filter

sinusoidal interference rejection analysis of an lms adaptive feedforward controller with a noisy periodic reference this paper applies a new vector subspace model to determine the non wiener solutions of the lms algorithm when the reference input is an arbitrary noisy periodic signal the lms weights are modeled as a deterministic time varying mean plus a zero mean fluctuating part for the mean weight each harmonic component of the periodic reference is shown to excite only a two dimensional 2 d subspace of the n dimensional tap weight space the fluctuating part of the weight is due to the reference noise that excites the full n dimensional space the power spectrum of the error is computed using the correlation function of the weight fluctuations the error is shown to be primarily the sum of a white noise and the canceled periodic residual for an independent noise vector model when the effects of the tapped delay line are incorporated in the model the noise becomes colored with a spectrum centered about the canceled sinusoidal frequency this weight model significantly simplifies the understanding of the non wiener behavior and can be applied to active noise cancellation problems when filters appear in the cancellation loop

kb spaces and subinvariant elements of positive operators our purpose in the note is to obtain a characterization of the kb spaces among the order continuous banach lattices the characterization is stated in terms of subinvariant elements of positive power bounded operators and two ergodic decompositions which we have defined in previous papers

on subinvariant elements in banach lattices in the paper we obtain an ergodic decomposition generated by a positive operator t e e e being a banach lattice having the projection property if t is a power bounded operator and if e is a kb space then the decomposition permits us to describe sufficient conditions for the existence of nonzero t subinvariant elements in e we also study the relationship between the hopf decomposition in banach lattices and the one defined in the paper in order to obtain sufficient conditions for the existence of a t invariant weak order unit

large deviations dynamics and phase transitions in large stochastic and disordered neural networks neuronal networks are characterized by highly heterogeneous connectivity and this disorder was recently related experimentally to qualitative properties of the network the motivation of this paper is to mathematically analyze the role of these disordered connectivities on the large scale properties of neuronal networks to this end we analyze here large scale limit behaviors of neural networks including for biological relevance multiple populations random connectivities and interaction delays due to the randomness of the connectivity usual mean field methods e g coupling cannot be applied but similarly to studies developed for spin glasses we will show that the sequences of empirical measures satisfy a large deviation principle and converge towards a self consistent non markovian process from a mathematical viewpoint the proof differs from previous works in that we are working in infinite dimensional spaces interaction delays and consider multiple cell types the limit obtained formally characterizes the macroscopic behavior of the network we propose a dynamical systems approach in order to address the qualitative nature of the solutions of these very complex equations and apply this methodology to three instances in order to show how non centered coefficients interaction delays and multiple populations networks are affected by disorder levels we identify a number of phase transitions in such systems upon changes in delays connectivity patterns and dispersion and particularly focus on the emergence of non equilibrium states involving synchronized oscillations

on the hopf decomposition abstract in the paper we extend the hopf decomposition to positive operators in banach lattices let e be a banach lattice let e be the dual of e let t e e be a positive operator and let t be the dual of t we define two conservative ideals ic bc in e e respectively and two dissipative ideals id bd in e e respectively we prove that bc and bd are projection bands in e and that if e has order continuous norm then icand id are projection bands as well the major result of the paper is that if e has order continuous norm and if e has a t subinvariant weak order unit e then the projection of e on bd is also t subinvariant we conclude the paper by proving that our extension is a natural one and by applying our construction to positive contractions of order complete am spaces with unit

truncated orthogonal expansions of recurrent signals equivalence to a linear time variant periodic filter we show that orthogonal expansions of recurrent signals like electrocardiograms ecgs with a reduced number of coefficients is equivalent to a linear time variant periodic filter instantaneous impulse and frequency responses are analyzed for two classical ways of estimating the expansion coefficients inner product and adaptive estimation with the lms algorithm the obtained description as a linear time variant periodic filter is a useful tool in order to quantify the distortion produced by the effect of using a reduced number of coefficients in the expansion and to give frequency criteria to select the appropriate number of functions moreover the misadjustment of the lms algorithm can be explained as a distortion of the instantaneous frequency response experimental results are illustrated with the karhunen loeve transform of ecg signals but this approach can also be applied to any orthogonal transform

non wiener solutions for the lms algorithm a time domain approach a time domain analysis of the lms algorithm is presented for a sinusoidal deterministic reference input for the sinusoidal reference input only the n dimensional time varying linear matrix recursion for the weight vector is solved using a 2 d orthogonal subspace decomposition using this weight vector solution it is shown that there exists a linear time invariant relationship between the desired input and the filter output

mean field linear quadratic gaussian lqg games for stochastic integral systems in this technical note we formulate and investigate a class of mean field linear quadratic gaussian lqg games for stochastic integral systems unlike other literature on mean field games where the individual states follow the controlled stochastic differential equations sdes the individual states in our large population system are characterized by a class of stochastic volterra type integral equations we obtain the nash certainty equivalence nce equation and hence derive the set of associated decentralized strategies the epsilon nash equilibrium properties are also verified due to the intrinsic integral structure the techniques and estimates applied here are significantly different from those existing results in mean field lqg games for stochastic differential systems for example some fredholm equation in the mean field setup is introduced for the first time as for applications two types of stochastic delayed systems are formulated as the special cases of our stochastic integral system and relevant mean field lqg games are discussed

on several types of convergence and divergence in archimedean riesz spaces abstract in the paper we define and study the basic properties of several concepts of convergence divergence and unboundedness in archimedean riesz spaces the notions which we introduce will be used in order to obtain an extension of the ornstein ratio ergodic theorem in a forthcoming paper

large deviations for infinite dimensional stochastic dynamical systems the large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation the construction and justification of the approximations can be onerous especially in the case where the process state is infinite dimensional in this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of brownian noise the approach is based on a variational representation for functionals of brownian motion proofs of large deviations properties are reduced to demonstrating basic qualitative properties existence uniqueness and tightness of certain perturbations of the original process

mean field equations for stochastic firing rate neural fields with delays derivation and noise induced transitions abstract in this manuscript we analyze the collective behavior of mean field limits of large scale spatially extended stochastic neuronal networks with delays rigorously the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro differential mckean vlasov equation that remain impenetrable leaving the stochastic collective dynamics of such networks poorly understood in order to study these macroscopic dynamics we analyze networks of firing rate neurons i e with linear intrinsic dynamics and sigmoidal interactions in that case we prove that the solution of the mean field equation is gaussian hence characterized by its two first moments and that these two quantities satisfy a set of coupled delayed integro differential equations these equations are similar to usual neural field equations and incorporate noise levels as a parameter allowing analysis of noise induced transitions we identify through bifurcation analysis several qualitative transitions due to noise in the mean field limit in particular stabilization of spatially homogeneous solutions synchronized oscillations bumps chaotic dynamics wave or bump splitting are exhibited and arise from static or dynamic turing hopf bifurcations these surprising phenomena allow further exploring the role of noise in the nervous system

averaged and quenched propagation of chaos for spin glass dynamics we study the asymptotic behaviour for both asymmetric and symmetric spin glass dynamics in a sherrington kirkpatrick model as proposed by sompolinsky zippelius we prove without any condition on time and temperature averaged propagation of chaos results extending this result to replicated systems we conclude that the law of a single spin converges to a non markovian probability measure in law with respect to the random interaction

simplified analysis of the lms adaptive filter using a transfer function approximation a simple approximation to the behavior of the lms adaptive filter as a discrete transfer function is developed this representation is a valid description for both deterministic inputs and for the expected results with random inputs including correlated inputs the results are shown to be exact for some classes of input including periodic signals one result of this analysis is the demonstration that the lms filter can produce results which are biased from the least squares solution under the combined conditions of a nonzero mean primary and a correlated reference input

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

mixing for positive operators in banach lattices abstract in this paper we extend m lin s definition of mixing for positive contractions in l 1 x m with m x 1 to positive operators in banach lattices with weak order units and we generalize lin s theorem 2 1 z wahrsch verw gebiete 19 1971 231 249 to the case of power bounded positive operators in kb spaces in the particular case of weakly compact power bounded positive operators the same theorem is extended to banach lattices with order continuous norms

large deviation properties of weakly interacting processes via weak convergence methods we study large deviation properties of systems of weakly interacting particles modeled by ito stochastic differential equations sdes it is known under certain conditions that the corresponding sequence of empirical measures converges as the number of particles tends to infinity to the weak solution of an associated mckean vlasov equation we derive a large deviation principle via the weak convergence approach the proof which avoids discretization arguments is based on a representation theorem weak convergence and ideas from stochastic optimal control the method works under rather mild assumptions and also for models described by sdes not of diffusion type to illustrate this we treat the case of sdes with delay

on large deviations for particle systems associated with spatially homogeneous boltzmann type equations we consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous boltzmann type equation as the number of particles tends to infinity these laws of large numbers were proved for the maxwellian molecules by h tanaka tal and for the hard spheres by a s sznitman szl in the present paper we investigate the corresponding large deviations the large deviation upper bound is obtained and using convex analysis a non variational formulation of the rate function is given our results hold for maxwellian molecules with a cutoff potential and for hard spheres

simple model for bursting dynamics of neurons neuronal cells in isolation or as an assembly exhibit bursting behavior on two different time scales we introduce a simple one dimensional model which requires only one phase variable to describe the phenomenon of parabolic bursting the analysis in the continuum limit reveals that for any unimodal distribution of frequencies the qualitative properties of the full and the reduced model are identical further we derive analytically an exact low dimensional description of a globally coupled network of bursting oscillators for our model study of the stability for this low dimensional model reveals different dynamical signatures in the parameter space we demonstrate that the structure of the parameter space remains independent of the number of spikes per burst

a shocking display of synchrony abstract this article explores the kuramoto model describing the synchronization of a population of coupled oscillators two versions of this model are considered a discrete version suitable for a population with a finite number of oscillators and a continuum model found in the limit of an infinite population when the strength of the coupling between the oscillators exceeds a threshold the oscillators partially synchronize we explore the transition in the continuum model which takes the form of a bifurcation of a discrete mode from a continuous spectrum we use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition and compare with the synchronization predicted by the discrete model there are similarities with instabilities in ideal plasmas and inviscid fluids but these are superficial

the hopf decomposition and the individual divergence to on a band abstract recently we have extended a ratio ergodic theorem of ornstein to archimedean riesz spaces and in doing so we have defined several types of convergence and divergence in riesz spaces one of them being the individual divergence to on a band in the note we use the notion of individual divergence to on a band in order to obtain a new approach to the hopf decomposition and to extend one of our previous results concerning the hopf decomposition in ordered vector spaces

stochastic neural field theory and the system size expansion we analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of n identical neurons the state of the network is specified by the fraction of active or spiking neurons in each population and transition rates are chosen so that in the thermodynamic or deterministic limit n rightarrow infty we recover standard activity based or voltage based rate models we derive the lowest order corrections to these rate equations for large but finite n using two different approximation schemes one based on the van kampen system size expansion and the other based on path integral methods both methods yield the same series expansion of the moment equations which at mathcal o 1 n can be truncated to form a closed system of equations for the first and second order moments taking a continuum limit of the moment equations while keeping the system size n fixed generates a system of integrodifferential equations for the

uniform monotonicity of norms and the strong zero two law abstract recently rainer wittmann proved a strong zero two law for positive contractions of lp spaces 1 p

the modulus of a regular linear operator and the zero two law in lp spaces 1 p p 2 abstract using the linear modulus of a regular operator we extend the zero two law for positive contractions of l p spaces 1 p p 2 the proof uses the approximate point spectrum of positive contractions and a procedure obtained by foguel

coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps we present a computer assisted approach to coarse graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a fixed network structure the existence of a spectral gap for the coupling network graph laplacian suggests that the graph dynamics may quickly become low dimensional our first choice of coarse variables consists of the components of the oscillator states their complex phase angles along the leading eigenvectors of this laplacian we then use the equation free framework circumventing the derivation of explicit coarse grained equations to perform computational tasks such as coarse projective integration coarse fixed point and coarse limit cycle computations in a second step we explore an approach to incorporating oscillator heterogeneity in the coarse graining process the approach is based on the observation of fast developing correlations between oscillator state and oscillator intrinsic properties and establishes a connection with tools developed in the context of uncertainty quantification

the mean field schrodinger problem ergodic behavior entropy estimates and functional inequalities we study the mean field schrodinger problem mfsp that is the problem of finding the most likely evolution of a cloud of interacting brownian particles conditionally on the observation of their initial and final configuration its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent brownian particles we undertake a fine study of the dynamics of its solutions including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger as well as a novel class of functional inequalities involving the mean field entropic cost i e the optimal value in mfsp our strategy unveils an interesting connection between forward backward stochastic differential equations and the riemannian calculus on the space of probability measures introduced by otto which is of independent interest

community detection in sparse networks via grothendieck s inequality we present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs the method is based on grothendieck s inequality unlike the previous uses of this inequality that lead to constant relative accuracy we achieve any given relative accuracy by leveraging randomness we illustrate the method with the problem of community detection in sparse networks those with bounded average degrees we demonstrate that even in this regime various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices the method is general it can be applied to a variety of stochastic models of networks and semidefinite programs

large deviations and mean field theory for asymmetric random recurrent neural networks in this article we study the asymptotic dynamics of a noisy discrete time neural network with random asymmetric couplings and thresholds more precisely we focus our interest on the limit behaviour of the network when its size grows to infinity with bounded time in the case of gaussian connection weights we use the same techniques as ben arous and guionnet see 3 to prove that the image law of the distribution of the neurons activation states by the empirical measure satisfies a temperature free large deviation principle moreover we prove that if the connection weights satisfy a general condition of domination by gaussian tails then the distribution of the activation potential of each neuron converges weakly towards an explicit gaussian law the characteristics of which are contained in the mean field equations stated by cessac doyon quoy samuelides see 4 6 furthermore under this hypothesis we obtain a law of large numbers and a propagation of chaos result finally we show that many classical distributions on the couplings fulfill our general condition thus this paper provides rigorous mean field results for a large class of neural networks which is currently investigated in neural network literature

stochastic approximation with correlated data new almost sure convergence results for a special form of the multidimensional robbins monro stochastic approximation procedure are developed the results are applicable to cases where the training data is heavily correlated no conditional expectation properties or boundedness assumptions are required to apply the new results for example when the data sequence is normal and i m dependent ii autoregressive moving average or iii band limited the results can be used to establish the almost sure convergence of each algorithm treated the special form of the robbins monro procedure considered is motivated by a consideration of several algorithms that have been proposed for discrete time adaptive signal processing applications most of these algorithms can also be viewed as stochastic gradient following algorithm the ease with which the new results can be applied is illustrated

some applications of dubinin s lemma to rational functions with prescribed poles abstract by developing a new technique some results for the rational functions with prescribed poles and restricted zeros have been obtained the estimated results strengthen many well known inequalities concerning the derivative and polar derivative of polynomials

systematic fluctuation expansion for neural network activity equations population rate or activity equations are the foundation of a common approach to modeling for neural networks these equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity the shortcoming of these equations is that they take into account only the average firing rate while leaving out higher order statistics like correlations between firing a stochastic theory of neural networks that includes statistics at all orders was recently formulated we describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms each level of the approximation yields closed equations they depend only on the mean and specific correlations of interest without an ad hoc criterion for doing so we show in an example of an all to all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone

schilder theorem for the brownian motion on the diffeomorphism group of the circle abstract we prove a large deviation principle for flows associated to stochastic differential equations with non lipschitz coefficients as an application we establish a schilder theorem for the brownian motion on the group of diffeomorphisms of the circle

convergence analysis of lms filters with uncorrelated gaussian data statistical analysis of the least mean squares lms adaptive algorithm with uncorrelated gaussian data is presented exact analytical expressions for the steady state mean square error mse and the performance degradation due to weight vector misadjustment are derived necessary and sufficient conditions for the convergence of the algorithm to the optimal wiener solution within a finite variance are derived it is found that the adaptive coefficient which controls the rate of convergence of the algorithm must be restricted to an interval significantly smaller than the domain commonly stated in the literature the outcome of this paper therefore places fundamental limitations on the mse performance and rate of convergence of the lms adaptive scheme

stochastic lohe matrix model on the lie group and mean field limit we propose a lohe matrix model in a random environment where each oscillator can be regarded as an element of a general matrix lie group g in order to make the stochastic system stays on g for all time we introduce suitable noise terms so that the underlying manifold g is positively invariant under the stochastic system then we formally derive the fokker planck type equation defined on g times mathfrak g in which mathfrak g denotes the lie algebra corresponding to g after identifying the target fokker planck equation we especially consider the unitary group g mathbf u d and show that the equation on mathbf u d admits a global unique solution and that it can be rigorously derived using a stochastic mean field limit procedure with a convergence rate of order o 1 sqrt n finally we restrict our concern to g mathbf su 2 to provide explicit calculation and present the nonlinear stability of an incoherent state for the fokker planck equation depending on the relation between parameters

stimulus dependent suppression of chaos in recurrent neural networks neuronal activity arises from an interaction between ongoing firing generated spontaneously by neural circuits and responses driven by external stimuli using mean field analysis we ask how a neural network that intrinsically generates chaotic patterns of activity can remain sensitive to extrinsic input we find that inputs not only drive network responses but they also actively suppress ongoing activity ultimately leading to a phase transition in which chaos is completely eliminated the critical input intensity at the phase transition is a nonmonotonic function of stimulus frequency revealing a resonant frequency at which the input is most effective at suppressing chaos even though the power spectrum of the spontaneous activity peaks at zero and falls exponentially a prediction of our analysis is that the variance of neural responses should be most strongly suppressed at frequencies matching the range over which many sensory systems operate

a note on dynamical models on random graphs and fokker planck equations we address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model i e a model on the complete graph with a suitably renormalized interaction parameter examples include erdos renyi graphs with edge probability p n n is the number of vertices such that lim n to infty p n n infty the purpose of this note it twofold 1 to establish this proximity on finite time horizon by exploiting the fact that both systems are accurately described by a fokker planck pde or equivalently by a nonlinear diffusion process in the n infty limit 2 to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system

interacting diffusions on random graphs with diverging average degrees hydrodynamics and large deviations we consider systems of mean field interacting diffusions where the pairwise interaction structure is described by a sparse and potentially inhomogeneous random graph examples include the stochastic kuramoto model with pairwise interactions given by an erd s renyi graph our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions for a broad class of interaction functions we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit which is given by a mckean vlasov diffusion moreover we also prove matching behavior of the two systems at the level of large deviations our results extend classical results of dai pra and den hollander and provide the first examples of ldps for systems with sparse random interactions

existence and robustness of phase locking in coupled kuramoto oscillators under mean field feedback motivated by the recent development of deep brain stimulation dbs for neurological diseases we study a network of interconnected oscillators under the influence of mean field feedback and analyze the robustness of its phase locking with respect to general inputs under standard assumptions this system can be reduced to a modified version of the kuramoto model of coupled nonlinear oscillators in the first part of the paper we present an analytical study on the existence of phase locked solutions under generic interconnection and feedback configurations in particular we show that in general no oscillating phase locked solutions can co exist with any non zero proportional mean field feedback in the second part we prove some robustness properties of phase locked solutions namely total stability this general result allows in particular to justify the persistence of practically phase locked states if sufficiently small feedback gains are applied and to give explicit necessary conditions on the intensity of a desynchronizing mean field feedback furthermore the lyapunov function used in the analysis provides a new characterization of the robust phase locked configurations in the kuramoto system with symmetric interconnections

quenched large deviations for interacting diffusions in random media the aim of the paper is to establish a large deviation principle ldp for the empirical measure of mean field interacting diffusions in a random environment the point is to derive such a result once the environment has been frozen quenched model the main theorem states that a ldp holds for every sequence of environment satisfying appropriate convergence condition with a rate function that does not depend on the disorder and is different from the rate function in the averaged model similar results concerning the empirical flow and local empirical measures are provided

on power bounded operators abstract the main result asserts that for any contraction t on an arbitrary banach space x t n t n 1 0 as n if and only if the spectrum of t has no points on the unit circle except perhaps z 1 this theorem is extended for t t n where is a function of spectral synthesis on the unit circle as an application we generalize the so called zero two law of ornstein and sucheston and zaharopol to positive contraction on a very large class of banach lattices

on the maximal ergodic theorem abstract in the paper we extend hopf s maximal ergodic theorem to banach lattices the extension is obtained by using garsia s approach to the topic

spatiotemporal dynamics of the kuramoto sakaguchi model with time dependent connectivity we study the dynamics of the paradigmatic kuramoto sakaguchi model of identical coupled phase oscillators with various kinds of time dependent connectivity using eulerian discretization we explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities namely the strength of incoherence and the discontinuity measure in the quasistatic limit of the changing of coupling range we observe how the system relaxes from one state to another and identify a few interesting collective dynamical states along the way under a sinusoidal change of the coupling range the global order parameter characterizing the degree of synchronization in the system is shown to undergo a hysteresis with the coupling range we also study the low dimensional spatiotemporal dynamics of the local order parameter in the continuum limit using the recently developed ott antonsen ansatz and justify some of our numerical results in particular we identify an intrinsic time scale of the kuramoto system and show that the simulations exhibit two distinct kinds of qualitative behavior in two cases when the time scale associated with the switching of the coupling radius is very large compared to the intrinsic time scale and when it is comparable to the intrinsic time scale

a stochastic maximum principle for delayed mean field stochastic differential equations and its applications in this technical note we discuss the stochastic optimal control problems of mean field stochastic differential delayed equations mfsddes which arise naturally from various backgrounds including economics finance engineering and physics etc to this end some new estimates are used to handle the complex structure of our controlled system due to the presence of both delay and mean field characters as the main result a stochastic maximum principle for the mean field stochastic optimal control with delay mfsocd is derived in terms of necessary and sufficient conditions in particular applying the convex variation and duality relation we obtain the necessary condition for optimality see theorem 1 in addition the sufficient condition of the optimality is also obtained under some convex condition see theorem 2 based on our maximum principle the related mean field linear quadratic delayed mflqd optimal control problems are also investigated the optimal control is derived and its existence is also verified refer theorem 3 as illustration an example is also proposed and its explicit optimal control is derived

large deviations for langevin spin glass dynamics we study the asymptotic behaviour of asymmetrical spin glass dynamics in a sherrington kirkpatrick model as proposed by sompolinsky zippelius we prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime we study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureq which is not markovian we deduce that the quenched law of the empirical measure converges to q extending then the preceeding results to replicated dynamics we investigate the quenched behavior of a single spin we get quenched convergence toq in the case of a symmetric initial law and even potential for the free spin

equations de type de boltzmann spatialement homogenes we study the construction of a nonhomogeneous n valued markov process whose laws at fixed times evolve according to a boltzmann type equation spatially homogeneous we then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles

emergent behavior of a second order lohe matrix model on the unitary group we study a second order extension to the first order lohe matrix model on the unitary group which can be reduced to the second order kuramoto model with inertia as a special case for the proposed second order model we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters for the identical hamiltonians we show that the complete synchronization emerges asymptotically in contrast for the non identical hamiltonians the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small

exponential synchronization of the high dimensional kuramoto model with identical oscillators under digraphs abstract for the kuramoto model and its variations it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry in this paper for the high dimensional kuramoto model of identical oscillators a matrix riccati differential equation mrde is proposed to describe the error dynamics based on the mrde the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting directed spanning trees

dynamics of phase slips in systems with time periodic modulation the adler equation with time periodic frequency modulation is studied a series of resonances between the period of the frequency modulation and the time scale for the generation of a phase slip is identified the resulting parameter space structure is determined using a combination of numerical continuation time simulations and asymptotic methods regions with an integer number of phase slips per period are separated by regions with noninteger numbers of phase slips and include canard trajectories that drift along unstable equilibria both high and low frequency modulation is considered an adiabatic description of the low frequency modulation regime is found to be accurate over a large range of modulation periods

mean field equations bifurcation map and route to chaos in discrete time neural networks abstract we investigate the dynamical behaviour of neural networks with asymmetric synaptic weights in the presence of random thresholds we inspect low gain dynamics before using mean field equations to study the bifurcations of the fixed points and the change of regime that occurs when varying control parameters we infer different areas with various regimes summarized by a bifurcation map in the parameter space we numerically show the occurence of chaos that arises generically by a quasi periodicity route we then discuss some features of our system in relation with biological observations such as low firing rates and refractory periods

a variational representation for certain functionals of brownian motion in this paper we show that the variational representation log ee f w inf v e 1 2 int 0 1 parallel v s parallel 2 ds f w int 0 cdot v s ds holds where w is a standard d dimensional brownian motion f is any bounded measurable function that maps c 0 1 mathbb r d into mathbb r and the infimum is over all processes v that are progressively measurable with respect to the augmentation of the filtration generated by w an application is made to a problem concerned with large deviations and an extension to unbounded functions is given