detection of couplings in ensembles of stochastic oscillators the problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed we suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint moreover we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series the technique applies to weakly coupled nonsynchronized oscillators it is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies efficiency of the technique is demonstrated in numerical experiments

delay independence of mutual information rate of two symbolic sequences introduced by shannon as a rate of actual transmission mutual information rate mir is an extension of mutual information to a pair of dynamical processes we show a delay independence theorem according to which mir is not sensitive to a time shift between the two processes numerical studies of several benchmark situations confirm that this theoretical asymptotic property remains valid for realistic finite sequences estimations based on block entropies and a causal state machine algorithm perform better than an estimation based on a lempel ziv compression algorithm provided that block length and maximum history length respectively can be chosen larger than the delay mir is thus a relevant index for measuring nonlinear correlations between two experimental or simulated sequences when the transmission delay in input output devices or dephasing in coupled systems is variable or unknown

resonances of third order differential operators abstract we consider resonances for third order ordinary differential operators with compactly supported coefficients on the real line resonance are defined as zeros of a fredholm determinant on a non physical sheet of three sheeted riemann surface we determine upper bounds of the number of resonances in complex discs at large radius we express the trace formula in terms of resonances only

information based detection of nonlinear granger causality in multivariate processes via a nonuniform embedding technique we present an approach framed in information theory to assess nonlinear causality between the subsystems of a whole stochastic or deterministic dynamical system the approach follows a sequential procedure for nonuniform embedding of multivariate time series whereby embedding vectors are built progressively on the basis of a minimization criterion applied to the entropy of the present state of the system conditioned to its past states a corrected conditional entropy estimator compensating for the biasing effect of single points in the quantized hyperspace is used to guarantee the existence of a minimum entropy rate at which to terminate the procedure the causal coupling is detected according to the granger notion of predictability improvement and is quantified in terms of information transfer we apply the approach to simulations of deterministic and stochastic systems showing its superiority over standard uniform embedding effects of quantization data length and noise contamination are investigated as practical applications we consider the assessment of cardiovascular regulatory mechanisms from the analysis of heart period arterial pressure and respiration time series and the investigation of the information flow across brain areas from multichannel scalp electroencephalographic recordings

momentary information transfer as a coupling measure of time series we propose a method to analyze couplings between two simultaneously measured time series our approach is based on conditional mutual sorting information it is related to other concepts for detecting coupling directions the old idea of marko for directed information and the more recent concept of schreiber s transfer entropy by setting suitable conditions we first of all consider momentary information in both time series this enables the detection not only of coupling directions but also delays sorting information refers to ordinal properties of time series which makes the analysis robust with respect to strictly monotonous distortions and thus very useful in the analysis of proxy data in climatology fortunately ordinal analysis is easy and fast to compute we consider also the problem of reliable estimation from finite time series

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble a method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail mutual nonlinear prediction also provides information on the directionality of the coupling between systems furthermore the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony numerical examples studied include three classes of unidirectionally coupled systems systems with identical parameters nonidentical parameters and stochastic driving of a nonlinear system this technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool the interrelationships examined include single neuron unit firing the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex and intracellular measurements of integrated excitatory postsynaptic potentials epsp s dynamical interdependence perhaps generalized synchrony was identified in this neuronal network between simultaneous single unit firings between units and the population and between units and intracellular epsp s

examples of non kahler hamiltonian torus actions an important question with a rich history is the extent to which the symplectic category is larger than the kahler category many interesting examples of non kahler symplectic manifolds have been constructed t m g however sufficiently large symmetries can force a symplectic manifolds to be kahler d kn in this paper we solve several outstanding problems by constructing the first symplectic manifold with large non trivial symmetries which does not admit an invariant kahler structure the proof that it is not kahler is based on the atiyah guillemin sternberg convexity theorem at gs using the ideas of this paper c woodward shows that even the symplectic analogue of spherical varieties need not be kahler w

causality analysis of neural connectivity critical examination of existing methods and advances of new methods granger causality gc is one of the most popular measures to reveal causality influence of time series and has been widely applied in economics and neuroscience especially its counterpart in frequency domain spectral gc as well as other granger like causality measures have recently been applied to study causal interactions between brain areas in different frequency ranges during cognitive and perceptual tasks in this paper we show that 1 gc in time domain cannot correctly determine how strongly one time series influences the other when there is directional causality between two time series and 2 spectral gc and other granger like causality measures have inherent shortcomings and or limitations because of the use of the transfer function or its inverse matrix and partial information of the linear regression model on the other hand we propose two novel causality measures in time and frequency domains for the linear regression model called new causality and new spectral causality respectively which are more reasonable and understandable than gc or granger like measures especially from one simple example we point out that in time domain both new causality and gc adopt the concept of proportion but they are defined on two different equations where one equation for gc is only part of the other for new causality thus the new causality is a natural extension of gc and has a sound conceptual theoretical basis and gc is not the desired causal influence at all by several examples we confirm that new causality measures have distinct advantages over gc or granger like measures finally we conduct event related potential causality analysis for a subject with intracranial depth electrodes undergoing evaluation for epilepsy surgery and show that in the frequency domain all measures reveal significant directional event related causality but the result from new spectral causality is consistent with event related time frequency power spectrum activity the spectral gc as well as other granger like measures are shown to generate misleading results the proposed new causality measures may have wide potential applications in economics and neuroscience

multiplicity free hamiltonian actions need not be kahler multiplicity free actions are symplectic manifolds with a very high degree of symmetry delzant 2 showed that all compact multiplicity free torus actions admit compatible kahler structures and are therefore toric varieties in this note we show that delzant s result does not generalize to the non abelian case our examples are constructed by applying u 2 equivariant symplectic surgery to the flag variety u 3 t3 we then show that these actions fail a criterion which tolman 9 shows is necessary for the existence of a compatible kahler structure

estimating coupling direction between neuronal populations with permutation conditional mutual information to further understand functional connectivity in the brain we need to identify the coupling direction between neuronal signals recorded from different brain areas in this paper we present a novel methodology based on permutation analysis and conditional mutual information for estimation of a directionality index between two neuronal populations first the reliability of this method is numerically assessed with a coupled mass neural model the simulations show that this method is superior to the conditional mutual information method and the granger causality method for identifying the coupling direction between unidirectional or bidirectional neuronal populations that are generated by the mass neuronal model the method is also applied to investigate the coupling direction between neuronal populations in ca1 and ca3 in the rat hippocampal tetanus toxin model of focal epilepsy the propagation direction of the seizure events could be elucidated through this coupling direction estimation method all together these results suggest that the permutation conditional mutual information method is a promising technique for estimating directional coupling between mutually interconnected neuronal populations

horizontal visibility graph transfer entropy hvg te a novel metric to characterize directed connectivity in large scale brain networks we propose a new measure horizontal visibility graph transfer entropy hvg te to estimate the direction of information flow between pairs of time series hvg te quantifies the transfer entropy between the degree sequences of horizontal visibility graphs derived from original time series twenty one rossler attractors unidirectionally coupled in the posterior to anterior direction were used to simulate 21 channel electroencephalography eeg brain networks and validate the performance of the hvg te we showed that the hvg te is robust to different levels of coupling strengths between the coupled rossler attractors a wide range of time delays different sample sizes the effects of noise and linear mixing and the choice of reference for eeg data we also applied hvg te to eeg data in 20 healthy controls and compared its performance to a recently introduces phase based te measure pte we found that compared with pte hvg te consistently detected stronger posterior to anterior information flow patterns in the alpha band 8 13hz eeg brain networks for three different references moreover in contrast to pte hvg te does not require an assumption on the periodicity of input signals therefore it can be more widely applicable even for non periodic signals this study shows that the hvg te is a directed connectivity measure to characterise the direction of information flow in large scale brain networks

framework to study dynamic dependencies in networks of interacting processes the analysis of dynamic dependencies in complex systems such as the brain helps to understand how emerging properties arise from interactions here we propose an information theoretic framework to analyze the dynamic dependencies in multivariate time evolving systems this framework constitutes a fully multivariate extension and unification of previous approaches based on bivariate or conditional mutual information and granger causality or transfer entropy we define multi information measures that allow us to study the global statistical structure of the system as a whole the total dependence between subsystems and the temporal statistical structure of each subsystem we develop a stationary and a nonstationary formulation of the framework we then examine different decompositions of these multi information measures the transfer entropy naturally appears as a term in some of these decompositions this allows us to examine its properties not as an isolated measure of interdependence but in the context of the complete framework more generally we use causal graphs to study the specificity and sensitivity of all the measures appearing in these decompositions to different sources of statistical dependence arising from the causal connections between the subsystems we illustrate that there is no straightforward relation between the strength of specific connections and specific terms in the decompositions furthermore causal and noncausal statistical dependencies are not separable in particular the transfer entropy can be nonmonotonic in dependence on the connectivity strength between subsystems and is also sensitive to internal changes of the subsystems so it should not be interpreted as a measure of connectivity strength altogether in comparison to an analysis based on single isolated measures of interdependence this framework is more powerful to analyze emergent properties in multivariate systems and to characterize functionally relevant changes in the dynamics

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

asymptotics of resonances for a schrodinger operator with vector values abstract we show that the resonance counting function for a schrodinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

learning driver response relationships from synchronization patterns we test recent claims that causal driver response relationships can be deduced from interdependencies between simultaneously measured time series we apply two recently proposed interdependence measures that should give results similar to cross predictabilities used by previous authors the systems that we study are asymmetrically coupled simple models lorenz roessler and henon models the couplings being such that they lead to generalized synchronization if the data were perfect noise free infinitely long we should be able to detect at least in some cases which of the coupled systems is the driver and which the response this might no longer be true if the time series has finite length instead estimated interdependencies depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them and causal relationships are more difficult to detect we also show that slightly different variants of the interdependence measure can have quite different sensitivities

resonances for euler bernoulli operator on the half line abstract we consider resonances for fourth order differential operators on the half line with compactly supported coefficients we determine asymptotics of a counting function of resonances in complex discs at large radius describe the forbidden domain for resonances and obtain trace formulas in terms of resonances we apply these results to the euler bernoulli operator on the half line the coefficients of this operator are positive and constants outside a finite interval we show that this operator does not have any eigenvalues and resonances iff its coefficients are constants on the whole half line

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

exploring transient transfer entropy based on a group wise ica decomposition of eeg data this paper presents a data driven pipeline for studying asymmetries in mutual interdependencies between distinct components of eeg signal due to volume conductance estimating coherence between scalp electrodes may lead to spurious results a group based independent component analysis ica which is conducted across all subjects and conditions simultaneously is an alternative representation of the eeg measurements within this approach the extracted components are independent in a global sense while short lived or transient interdependencies may still be present between the components in this paper functional roles of the ica components are specified through a partial least squares pls analysis of task effects within the time course of the derived components functional integration is estimated within the information theoretic approach using transfer entropy analysis based on asymmetries in mutual interdependencies of reconstructed phase dynamics a secondary pls analysis is performed to assess robust task specific changes in transfer entropy estimates between functionally specific components

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

direction of coupling from phases of interacting oscillators an information theoretic approach a directionality index based on conditional mutual information is proposed for application to the instantaneous phases of weakly coupled oscillators its abilities to distinguish unidirectional from bidirectional coupling as well as to reveal and quantify asymmetry in bidirectional coupling are demonstrated using numerical examples of quasiperiodic chaotic and noisy oscillators as well as real human cardiorespiratory data

directionality of coupling from bivariate time series how to avoid false causalities and missed connections we discuss some problems encountered in inference of directionality of coupling or in the case of two interacting systems in inference of causality from bivariate time series we identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results

functional connectivity in resting state fmri is linear correlation sufficient functional connectivity fc analysis is a prominent approach to analyzing fmri data especially acquired under the resting state condition the commonly used linear correlation fc measure bears an implicit assumption of gaussianity of the dependence structure if only the marginals but not all the bivariate distributions are gaussian linear correlation consistently underestimates the strength of the dependence to assess the suitability of linear correlation and the general potential of nonlinear fc measures we present a framework for testing and estimating the deviation from gaussianity by means of comparing mutual information in the data and its gaussianized counterpart we apply this method to 24 sessions of human resting state fmri for each session matrix of connectivities between 90 anatomical parcel time series is computed using mutual information and compared to results from its multivariate gaussian surrogate that conserves the correlations but cancels any nonlinearity while the group level tests confirmed non gaussianity in the fc the quantitative assessment revealed that the portion of mutual information neglected by linear correlation is relatively minor on average only about 5 of the mutual information already captured by the linear correlation the marginality of the non gaussianity was confirmed in comparisons using clustering of the parcels the disagreement between clustering obtained from mutual information and linear correlation was attributable to random error we conclude that for this type of data practical relevance of nonlinear methods trying to improve over linear correlation might be limited by the fact that the data are indeed almost gaussian

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

frankel s theorem in the symplectic category we prove that if an n 1 dimensional torus acts symplectically on a 2n dimensional symplectic manifold then the action has a fixed point if and only if the action is hamiltonian one may regard it as a symplectic version of frankel s theorem which says that a kahler circle action has a fixed point if and only if it is hamiltonian the case of n 2 is the well known theorem by mcduff

explicit granger causality in kernel hilbert spaces granger causality gc is undoubtedly the most widely used method to infer cause effect relations from observational time series several nonlinear alternatives to gc have been proposed based on kernel methods we generalize kernel granger causality by considering the variables cross relations explicitly in hilbert spaces the framework is shown to generalize the linear and kernel gc methods and comes with tighter bounds of performance based on rademacher complexity we successfully evaluate its performance in standard dynamical systems as well as to identify the arrow of time in coupled r ossler systems and it is exploited to disclose the el ni no southern oscillation phenomenon footprints on soil moisture globally

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

assessing causality from multivariate time series in this work we propose a general nonparametric test of causality for weakly dependent time series more precisely we study the problem of attribution i e the proper comparison of the relative influence that two or more external dynamics trigger on a given system of interest we illustrate the possible applications of the proposed methodology in very different fields like physiology and climate science

differentiable generalized synchronization of chaos we consider simple lyapunov exponent based conditions under which the response of a system to a chaotic drive is a ital smooth function of the drive state we call this ital differentiable generalized synchronization dgs when dgs does not hold we quantify the degree of nondifferentiability using the h umlt o lder exponent we also discuss the consequences of dgs and give an illustrative numerical example copyright ital 1997 ital the american physical society

detecting the harmonics of oscillations with time variable frequencies a method is introduced for the spectral analysis of complex noisy signals containing several frequency components it enables components that are independent to be distinguished from the harmonics of nonsinusoidal oscillatory processes of lower frequency the method is based on mutual information and surrogate testing combined with the wavelet transform and it is applicable to relatively short time series containing frequencies that are time variable where the fundamental frequency and harmonics of a process can be identified the characteristic shape of the corresponding oscillation can be determined enabling adaptive filtering to remove other components and nonoscillatory noise from the signal thus the total bandwidth of the signal can be correctly partitioned and the power associated with each component then can be quantified more accurately the method is first demonstrated on numerical examples it is then used to identify the higher harmonics of oscillations in human skin blood flow both spontaneous and associated with periodic iontophoresis of a vasodilatory agent the method should be equally relevant to all situations where signals of comparable complexity are encountered including applications in astrophysics engineering and electrical circuits as well as in other areas of physiology and biology

convex analysis on cartan subspaces in 1937 von neumann 31 gave a famous characterization of unitarily invariant matrix norms that is norms f on cp q satisfying f uxv f x for all unitary matrices u and v and matrices x in cp q his result states that such norms are those functions of the form g where the map x cp q 7 x r has components the singular values 1 x 2 x p x of x assuming p q and g is a norm on rp invariant under sign changes and permutations of components furthermore he showed the respective dual norms satisfy

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

inferring the directionality of coupling with conditional mutual information uncovering the directionality of coupling is a significant step in understanding drive response relationships in complex systems in this paper we discuss a nonparametric method for detecting the directionality of coupling based on the estimation of information theoretic functionals we consider several different methods for estimating conditional mutual information the behavior of each estimator with respect to its free parameter is shown using a linear model where an analytical estimate of conditional mutual information is available numerical experiments in detecting coupling directionality are performed using chaotic oscillators where the influence of the phase extraction method and relative frequency ratio is investigated

from granger causality to long term causality application to climatic data quantitative characterization of interaction between processes from time series is often required in different fields of natural science including geophysics and biophysics typically one estimates short term influences e g the widely used granger causality is defined via one step ahead predictions such an approach does not reveal how strongly the long term behavior of one process under study is affected by the others to overcome this problem we introduce the concept of long term causality which extends the concept of granger causality the long term causality is estimated from data via empirical modeling and analysis of model dynamics under different conditions apart from mathematical examples we apply both approaches to find out how strongly the global surface temperature gst is affected by variations in carbon dioxide atmospheric content solar activity and volcanic activity during the last 150 years influences of all the three factors on gst are detected with the granger causality however the long term causality shows that the rise in gst during the last decades can be explained only if the anthropogenic factor co 2 is taken into account in a model

renyi s information transfer between financial time series a b s t r a c t in this paper we quantify the statistical coherence between financial time series by means of the renyi entropy with the help of campbell s coding theorem we show that the renyi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others this accentuation is controlled with renyi s parameter q to tackle the issue of the information flow between time series we formulate the concept of renyi s transfer entropy as a measure of information that is transferred only between certain parts of underlying distributions this is particularly pertinent in financial time series where the knowledge of marginal events such as spikes or sudden jumps is of a crucial importance we apply the renyian information flow to stock market time series from 11 world stock indices as sampled at a daily rate in the time period 02 01 1990 31 12 2009 corresponding heat maps and net information flows are represented graphically a detailed discussion of the transfer entropy between the dax and s p500 indices based on minute tick data gathered in the period 02 04 2008 11 09 2009 is also provided our analysis shows that the bivariate information flow between world markets is strongly asymmetric with a distinct information surplus flowing from the asia pacific region to both european and us markets an important yet less dramatic excess of information also flows from europe to the us this is particularly clearly seen from a careful analysis of renyi information flow between the dax and s p500 indices

conditional lyapunov exponents from time series a method for estimating conditional lyapunov exponents from time series of two unidirectionally coupled chaotic systems is developed it uses two scalar data sets one taken from the driving and the other from the response system and enables one to detect a generalized synchronization in an experiment without recourse to an auxiliary response system the method is illustrated on coupled maps as well as coupled chaotic flow models

a wall crossing formula for the signature of symplectic quotients we use symplectic cobordisrn and the localization result of ginzburg guillemin and karshon to find a wall crossing formula for the signature of regular symplectic quotients of hamiltonian torus actions the formula is recursive depending ultimately on fixed point data in the case of a circle action we obtain a formula for the signature of singular quotients as well we also show how formulas for the poincar6 polynomial and the euler characteristic equivalernt to those of kirwan can be expressed in the same recursive manner

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

coarse grained entropy rates for characterization of complex time series abstract a method for classification of complex time series using coarse grained entropy rates cer s is presented the cer s which are computed from information theoretic funcionals redundancies are relative measures of regularity and predictability and for data generated by dynamical systems they are related to kolmogorov sinai entropy a deterministic dynamical origin of the data under study however is not a necessary condition for the use of the cer s since the entropy rates can be defined for stochastic processes as well sensitivity of the cer s to changes in data dynamics and their robustness with respect to noise are tested by using numerically generated time series resulted from both deterministic chaotic and stochastic processes potential application of the cer s in analysis of physiological signals or other complex time series is demonstrated by using examples from pharmaco eeg and tremor classification

centered complexity one hamiltonian torus actions we consider symplectic manifolds with hamiltonian torus actions which are almost but not quite completely integrable the dimension of the torus is one less than half the dimension of the manifold we provide a complete set of invariants for such spaces when they are centered and the moment map is proper in particular this classifies the preimages under the moment map of all sufficiently small open sets which is an important step towards global classification as an application we construct a full packing of each of the grassmannians gr 2 r5 and gr 2 r6 by two equal symplectic balls

detection of synchronization between chaotic signals an adaptive similarity based approach we present an adaptive similarity based approach to detect generalized synchronization gs with n m phase synchronization ps where n and m are integers and one of them is 1 this approach is based on the similarity index si and gaussian mixture model with the minimum description length criterion the clustering method which is shown to be superior to the closeness and connectivity of a continuous function is employed in this study to detect the existence of gs with n m ps we conducted a computer simulation and a finger lifting experiment to illustrate the effectiveness of the proposed method in the simulation of a rossler lorenz system our method outperformed the conventional si and gs with 2 1 ps within the coupled system was found in the experiment of self paced finger lifting movement cortico muscular gs with 1 2 and 1 3 ps was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area the gs with n m ps n or m 1 has been simultaneously resolved from both simulation and experiment the proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science

detecting couplings between interacting oscillators with time varying basic frequencies instantaneous wavelet bispectrum and information theoretic approach in the natural world the properties of interacting oscillatory systems are not constant but evolve or fluctuating continuously in time thus the basic frequencies of the interacting oscillators are time varying which makes the system analysis complex for studying their interactions we propose a complementary approach combining wavelet bispectral analysis and information theory we show how these methods uncover the interacting properties and reveal the nature strength and direction of coupling wavelet bispectral analysis is generalized as a technique for detecting instantaneous phase time dependence for the case of two or more coupled nonlinear oscillators whereas the information theory approach can uncover the directionality of coupling and extract driver response relationships in complex systems we generate bivariate time series numerically to mimic typical situations that occur in real measured data apply both methods to the same time series and discuss the results the approach is applicable quite generally to any system of coupled nonlinear oscillators

kernel granger causality and the analysis of dynamical networks we propose a method of analysis of dynamical networks based on a recent measure of granger causality between time series based on kernel methods the generalization of kernel granger causality to the multivariate case here presented shares the following features with the bivariate measures i the nonlinearity of the regression model can be controlled by choosing the kernel function and ii the problem of false causalities arising as the complexity of the model increases is addressed by a selection strategy of the eigenvectors of a reduced gram matrix whose range represents the additional features due to the second time series moreover there is no a priori assumption that the network must be a directed acyclic graph we apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network it is shown that the underlying topology of the network can be reconstructed from time series of node s dynamics provided that a sufficient number of samples is available considering a linear dynamical network built by preferential attachment scheme we show that for limited data use of the bivariate granger causality is a better choice than methods using l1 minimization finally we consider real expression data from hela cells 94 genes and 48 time points the analysis of static correlations between genes reveals two modules corresponding to well known transcription factors granger analysis puts in evidence 19 causal relationships all involving genes related to tumor development

causal conditioning and instantaneous coupling in causality graphs this study aims at providing the definitive links between massey and kramer s directed information theory and granger causality graphs recently formalized by eichler this naturally leads to consider two concepts of causality that can occur in physical systems dynamical causality and instantaneous coupling although it is well accepted that the former is assessed by conditional transfer entropy the latter is often overlooked even if it was clearly introduced and understood in seminal studies in the bivariate case we show that directed information is decomposed into the sum of transfer entropy and instantaneous information exchange in the multivariate case encountered for conditional graph modeling such a decomposition does not hold anymore we provide a new insight into the problem of instantaneous coupling and show how it affects the estimated structure of a graphical model that should provide a sparse representation of a complex system this result is discussed and analyzed for the inference of causality graphs two synthetic examples are developed to illustrate the theoretical concepts practical issues are also briefly discussed on these examples and an extension of leonenko s k nearest neighbors based entropy estimators is used to derive a nonparametric estimator of directed information

statistical validation of mutual information calculations comparison of alternative numerical algorithms given two time series x and y their mutual information i x y i y x is the average number of bits of x that can be predicted by measuring y and vice versa in the analysis of observational data calculation of mutual information occurs in three contexts identification of nonlinear correlation determination of an optimal sampling interval particularly when embedding data and in the investigation of causal relationships with directed mutual information in this contribution a minimum description length argument is used to determine the optimal number of elements to use when characterizing the distributions of x and y however even when using partitions of the x and y axis indicated by minimum description length mutual information calculations performed with a uniform partition of the xy plane can give misleading results this motivated the construction of an algorithm for calculating mutual information that uses an adaptive partition this algorithm also incorporates an explicit test of the statistical independence of x and y in a calculation that returns an assessment of the corresponding null hypothesis the previously published fraser swinney algorithm for calculating mutual information includes a sophisticated procedure for local adaptive control of the partitioning process when the fraser and swinney algorithm and the algorithm constructed here are compared they give very similar numerical results less than 4 difference in a typical application detailed comparisons are possible when x and y are correlated jointly gaussian distributed because an analytic expression for i x y can be derived for that case based on these tests three conclusions can be drawn first the algorithm constructed here has an advantage over the fraser swinney algorithm in providing an explicit calculation of the probability of the null hypothesis that x and y are independent second the fraser swinney algorithm is marginally the more accurate of the two algorithms when large data sets are used with smaller data sets however the fraser swinney algorithm reports structures that disappear when more data are available third the algorithm constructed here requires about 0 5 of the computation time required by the fraser swinney algorithm

examples of non kahler hamiltonian circle manifolds with the strong lefschetz property abstract in this paper we construct six dimensional compact non kahler hamiltonian circle manifolds which satisfy the strong lefschetz property themselves but nevertheless have a non lefschetz symplectic quotient this provides the first known counterexamples to the question whether the strong lefschetz property descends to the symplectic quotient we also give examples of hamiltonian strong lefschetz circle manifolds which have a non lefschetz fixed point submanifold in addition we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong lefschetz manifold we then use it to show the existence of lefschetz four manifolds with non lefschetz finite covering spaces

granger causality and information flow in multivariate processes the multivariate versus bivariate measures of granger causality were considered granger causality in the application to multivariate physiological time series has the meaning of the information flow between channels it was shown by means of simulations and by the example of experimental electroencephalogram signals that bivariate estimates of directionality in case of mutually interdependent channels give erroneous results therefore multivariate measures such as directed transfer function should be used for determination of the information flow

comparison of six methods for the detection of causality in a bivariate time series in this comparative study six causality detection methods were compared namely the granger vector autoregressive test the extended granger test the kernel version of the granger test the conditional mutual information transfer entropy the evaluation of cross mappings between state spaces and an assessment of predictability improvement due to the use of mixed predictions seven test data sets were analyzed linear coupling of autoregressive models a unidirectional connection of two h enon systems a unidirectional connection of chaotic systems of r ossler and lorenz type and of two different r ossler systems an example of bidirectionally connected two species systems a fishery model as an example of two correlated observables without a causal relationship and an example of mediated causality we tested not only 20 phantom rule 0 16em 0ex 000 points long clean time series but also noisy and short variants of the data the standard and the extended granger tests worked only for the autoregressive models the remaining methods were more successful with the more complex test examples although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation

bounds on scattering poles in one dimension for the class of super exponentially decaying potentials on the real line sharp upper bounds on the counting function of the poles in discs are derived and the density of the poles in strips is estimated in the case of nonnegative potentials explicit estimates for the width of a pole free strip are obtained

testing for causality in reconstructed state spaces by an optimized mixed prediction method in this study a method of causality detection was designed to reveal coupling between dynamical systems represented by time series the method is based on the predictions in reconstructed state spaces the results of the proposed method were compared with outcomes of two other methods the granger var test of causality and the convergent cross mapping we used two types of test data the first test example is a unidirectional connection of chaotic systems of rossler and lorenz type the second one the fishery model is an example of two correlated observables without a causal relationship the results showed that the proposed method of optimized mixed prediction was able to reveal the presence and the direction of coupling and distinguish causality from mere correlation as well

the inverse problem for ordinary differential operators on the line on etudie le probleme inverse sur la ligne pour l operateur p d n q n 2 d n 2 q 0 d d idx pour tout ordre n 2 et sans hypothese de propriete d auto adjoint

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

on the inverse resonance problem for schrodinger operators we consider schrodinger operators on 0 with compactly supported possibly complex valued potentials in l1 0 it is known at least in the case of a real valued potential that the location of eigenvalues and resonances determines the potential uniquely from the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data in this paper we prove the validity of this statement i e we show conditional stability for finite data as a by product we also obtain a uniqueness result for the inverse resonance problem for complex valued potentials

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

gkm theory and hamiltonian non kahler actions in dimension 6 abstract using the classification of 6 dimensional manifolds by wall jupp and ubr we observe that the diffeomorphism type of simply connected compact 6 dimensional integer gkm t 2 manifolds is encoded in their gkm graph as an application we show that the 6 dimensional manifolds on which tolman and woodward constructed hamiltonian non kahler t 2 actions with finite fixed point set are diffeomorphic to eschenburg s twisted flag manifold su 3 t 2 in particular they admit a noninvariant kahler structure

network discovery with dcm this paper is about inferring or discovering the functional architecture of distributed systems using dynamic causal modelling dcm we describe a scheme that recovers the dynamic bayesian dependency graph connections in a network using observed network activity this network discovery uses bayesian model selection to identify the sparsity structure absence of edges or connections in a graph that best explains observed time series the implicit adjacency matrix specifies the form of the network e g cyclic or acyclic and its graph theoretical attributes e g degree distribution the scheme is illustrated using functional magnetic resonance imaging fmri time series to discover functional brain networks crucially it can be applied to experimentally evoked responses activation studies or endogenous activity in task free resting state fmri studies unlike conventional approaches to network discovery dcm permits the analysis of directed and cyclic graphs furthermore it eschews implausible markovian assumptions about the serial independence of random fluctuations the scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability relative to other networks the networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed and reciprocal effective connectivity between connected nodes or regions we envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting state studies

on nodes and modes in resting state fmri this paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fmri timeseries and simulations of neuronal fluctuations based upon the connectome its particular focus is on patterns or modes of distributed activity that underlie functional connectivity we first demonstrate that the eigenmodes of functional connectivity or covariance among regions or nodes are the same as the eigenmodes of the underlying effective connectivity provided we limit ourselves to symmetrical connections this symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling crucially the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory technically these modes have small negative lyapunov exponents that approach zero from below interestingly the superposition of modes whose exponents are sampled from a power law distribution produces classical 1 f scale free spectra we conjecture that the emergence of dynamical instability that underlies intrinsic brain networks is inevitable in any system that is separated from external states by a markov blanket this conjecture appeals to a free energy formulation of nonequilibrium steady state dynamics the common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes we use this as the basis of a dynamic causal model dcm of resting state fluctuations as measured in terms of their complex cross spectra in this model effective connectivity is parameterised in terms of eigenmodes and their lyapunov exponents that can also be interpreted as locations in a multidimensional scaling space model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space here we focus on conceptual issues with simulated fmri data and provide an illustrative application using an empirical multi region timeseries