formulae for cycles in the mandelbrot set ii abstract the analytical solutions to the parts of the boundary of the mandelbrot set associated with cycles of order 7 and 8 are obtained by an arithmetic method in the form of polynomials the inverse orbit method of calculating the values of cycle centres is extended to complex cases and applied to cycles up to order 10 some geometrical aspects of the various cardioids and bubbles are explored

modeling distributed axonal delays in mean field brain dynamics the range of conduction delays between connected neuronal populations is often modeled as a single discrete delay assumed to be an effective value averaging over all fiber velocities this paper shows the effects of distributed delays on signal propagation a distribution acts as a linear filter imposing an upper frequency cutoff that is inversely proportional to the delay width distributed thalamocortical and corticothalamic delays are incorporated into a physiologically based mean field model of the cortex and thalamus to illustrate their effects on the electroencephalogram eeg the power spectrum is acutely sensitive to the width of the thalamocortical delay distribution and more so than the corticothalamic distribution because all input signals must travel along the thalamocortical pathway this imposes a cutoff frequency above which the spectrum is overly damped the positions of spectral peaks in the resting eeg depend primarily on the distribution mean with only weak dependences on distribution width increasing distribution width increases the stability of fixed point solutions a single discrete delay successfully approximates a distribution for frequencies below a cutoff that is inversely proportional to the delay width provided that other model parameters are moderately adjusted a pair of discrete delays together having the same mean variance and skewness as the distribution approximates the distribution over the same frequency range without needing parameter adjustment delay distributions with large fractional widths are well approximated by low order differential equations

fractional kinetic equations and universal decoupling of a memory function in mesoscale region it is proved that kinetic equations containing non integer integrals and derivatives appear in the result of reduction of a set of micromotions to some averaged collective motion in the mesoscale region in other words it means that after a proper statistical average the microscopic dynamics is converted into a collective motion in the mesoscopic regime a fractal medium containing weakly and strongly correlated relaxation units has been considered it is shown that in most cases the original of the memory function recovers the riemann liouville integral for a strongly correlated fractal medium a generalization of the riemann liouville integral is obtained for the fractal branched processes one can derive the stretched exponential law of relaxation that is widely used for description of relaxation phenomena in disordered media it is shown that the generalized stretched exponential function describes the averaged collective motion in the fractal branched complex systems it is confirmed that the generalized stretched exponential law can be definitely applicable for description of a set of spontaneous evoked postsynaptic signals in the living matter the application of these fractional kinetic equations for the description of the dielectric relaxation phenomena is also discussed these kinetic equations and their generalizations can be applicable for the description of different relaxation or diffusion processes in the intermediate mesoscale region

age related alterations of relaxation processes and non markov effects in stochastic dynamics of r r intervals variability from human ecgs in this paper we consider the age related alterations of heart rate variability on the basis of the study of non markovian effects the age dynamics of relaxation processes is quantitatively described by means of local relaxation parameters calculated by the specific localization procedure we offer a quantitative informational measure of non markovity to evaluate the change of statistical effects of memory local relaxation parameters for young and elderly people differ by 3 3 times and quantitative measures of non markovity differ by 4 2 times the comparison of quantitative parameters allows to draw conclusions about the reduction of relaxation rate with ageing and the higher degree of the markovity of heart rate variability of elderly people

variational geometric and statistical methods for modeling brain anatomy and function we survey the recent activities of the odyssee laboratory in the area of the application of mathematics to the design of models for studying brain anatomy and function we start with the problem of reconstructing sources in meg and eeg and discuss the variational approach we have developed for solving these inverse problems this motivates the need for geometric models of the head we present a method for automatically and accurately extracting surface meshes of several tissues of the head from anatomical magnetic resonance mr images anatomical connectivity can be extracted from diffusion tensor magnetic resonance images but in the current state of the technology it must be preceded by a robust estimation and regularization stage we discuss our work based on variational principles and show how the results can be used to track fibers in the white matter wm as geodesics in some riemannian space we then go to the statistical modeling of functional magnetic resonance imaging fmri signals from the viewpoint of their decomposition in a pseudo deterministic and stochastic part that we then use to perform clustering of voxels in a way that is inspired by the theory of support vector machines and in a way that is grounded in information theory multimodal image matching is discussed next in the framework of image statistics and partial differential equations pdes with an eye on registering fmri to the anatomy the paper ends with a discussion of a new theory of random shapes that may prove useful in building anatomical and functional atlases

entropy pattern entropy and related methods for the analysis of data on the time intervals between heartbeats from 24 h electrocardiograms sequences of the time intervals between heartbeats char22 medically termed rr intervals char22 extracted from 24 h electrocardiogram recordings are examined as three dimensional return map images the recordings were made in humans by means of the medically widely used portable electrocardiograph holter system a time window measured in the number of heartbeats is used and different types of behavior are classified bifurcations between the types of dynamics of the heart are noted and a form of intermittency is found an alternative quantitative measure char22 a form pattern entropy of the return map image char22 is defined that characterizes the dynamics of the rr interval sequence it is shown that this is a measure of the degree of ordering of the rr interval sequence and as such it is a good novel medical diagnostic tool for analyzing heart rate variability which distinguishes between illness and health where other diagnostics fail

statistical mechanics of neocortical interactions canonical momenta indicatorsof electroencephalography a series of papers has developed a statistical mechanics of neocortical interactions smni deriving aggregate behavior of experimentally observed columns of neurons from statistical electrical chemical properties of synaptic interactions while not useful to yield insights at the single neuron level smni has demonstrated its capability in describing large scale properties of short term memory and electroencephalographic eeg systematics the necessity of including nonlinear and stochastic structures in this development has been stressed sets of eeg and evoked potential data were fit collected to investigate genetic predispositions to alcoholism and to extract brain signatures of short term memory adaptive simulated annealing asa a global optimization algorithm was used to perform maximum likelihood fits of lagrangians defined by path integrals of multivariate conditional probabilities canonical momenta indicators cmi are thereby derived for an individual s eeg data the cmi give better signal recognition than the raw data and can be used to advantage as correlates of behavioral states these results give strong quantitative support for an accurate intuitive picture portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or behavioral phenomena i e describing interactions among neurons columns of neurons and regional masses of neurons

statistical spectrum of the non markovity parameter for simple model systems the spectra of the non markovity parameter for three model systems are found an ideal gas weakly non ideal gas and a medium in hydrodynamic limits it is shown that the non markovian effects play the dominant role in all three above mentioned cases the effects of molecular memory are connected with the statistical appearance of intermolecular interactions in statistical systems

stability and bifurcations in neural fields with finite propagation speed and general connectivity a stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic proper ties sufficient conditions are given for the stability of equilibrium solutions it is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria whereas the stationary bifurcations depend only on the connectivity kernel in the case of nonstationary bifurca tions bounds are determined on the frequencies of the resulting oscillatory solutions a perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns oscillations and traveling waves for high propagation speeds a simple method is derived that allows the determi nation of the bifurcation type by visual inspection of the fourier transforms of the kernel and its first moment results are numerically illustrated on a class of neurologically plausible systems with combinations of gaussian excitatory and inhibitory connections

determination of effective brain connectivity from activity correlations effective connectivity embodied in transfer functions is derived from symmetric network activity correlations under task free conditions via a recent causal spectral factorization method this generalizes previous covariance based analyses to include frequency dependencies and time delays results are verified against analytic solutions of equations that have reproduced many aspects of experimental brain dynamics and against cases of more complex connectivity robustness to noise is also demonstrated

synchronous oscillations in the cerebral cortex the dynamics of a cortex driven by a finite number of white noise point sources is studied using a recently developed wave equation formulation green s functions power spectra fluctuation levels and two point correlation functions are computed analytically and numerically it is shown that a range of observed properties of so called synchronous oscillations in the cerebral cortex can be correctly reproduced using the wave equation that involves only excitatory interactions between neurons in particular the observed existence of a maximal correlation at zero temporal lag between spatially separated points is reproduced and explained for a cortex driven by two white noise sources

neural field dynamics under variation of local and global connectivity and finite transmission speed abstract spatially continuous networks with heterogeneous connections are ubiquitous in biological systems in particular neural systems to understand the mutual effects of locally homogeneous and globally heterogeneous connectivity we investigate the stability of the steady state activity of a neural field as a function of its connectivity the variation of the connectivity is implemented through manipulation of a heterogeneous two point connection embedded into the otherwise homogeneous connectivity matrix and by variation of the connectivity strength and transmission speed detailed examples including the ginzburg landau equation and various other local architectures are discussed our analysis shows that developmental changes such as the myelination of the cortical large scale fiber system generally result in the stabilization of steady state activity independent of the local connectivity non oscillatory instabilities are shown to be independent of any influences of time delay

diffeomorphisms groups and pattern matching in image analysis in a previous paper it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group we show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images using lie group ideas we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub optimal gradient pattern matching its links with fluid models are established

wave number spectrum of electroencephalographic signals a recently developed physiologically based continuum model of corticothalamic electrodynamics is used to derive the theoretical form of the electroencephalographic wave number spectrum and its projection onto a one dimensional recording array the projected spectrum is found to consist of a plateau followed by regions of power law decrease with various exponents which are dependent on both model parameters and temporal frequency the theoretical spectrum is compared with experimental results obtained in other studies showing good agreement the model provides a framework for understanding the nature of the spatial power spectrum by linking the underlying physiology with the large scale dynamics of the brain

intensity approximation of random fluctuation in complex systems abstract the markov and non markov processes in complex systems are examined with the help of dynamical information shannon entropy method here we consider the essential role of two mutually independent channels of entropy involving creation of correlation and annihilation of correlation the developed method has been used to analyze the intensity fluctuation of the complex systems of various nature in psychology to analyze numerical and pattern short time human memory to study the effect of stress on the parameters of the dynamical taping test and in cardiology to analyze the random dynamics of rr intervals in human ecg s and to diagnose various diseases of human cardiovascular systems the received results show that the application of intensity approximation allows to improve essentially the diagnostics of parameters in the evolution of human dynamic states

fluctuation noise spectroscopy and a universal fitting function of amplitudes of random sequences abstract a universal fitting function has been recognized that makes use of the eigen coordinates method physica a 285 2000 547 to accurately describe the distribution of ordered amplitudes within random sequences taken from a diversity of sources it is shown that sequences with a discrete structure can be described in terms of specific distributions of relative frequencies with respect to the number of quantum levels an investigation of these quantum distributions leads to an increase in both sensitivity and selectivity when attempting the statistical detection of various predominant factors of the hidden signals the physical meaning of this new function is discussed and a proposal is made as the basis of a new fluctuation noise spectroscopy in which the recognized function can be used for the detection of small signals and or the forerunners of strong signals that are hidden within the random sequences analyzed

quantification of heart rate variability by discrete nonstationary non markov stochastic processes we develop the statistical theory of discrete nonstationary non markov random processes in complex systems the objective of this paper is to find the chain of finite difference non markov kinetic equations for time correlation functions tcf in terms of nonstationary effects the developed theory starts from careful analysis of time correlation through nonstationary dynamics of vectors of initial and final states and nonstationary normalized tcf using the projection operators technique we find the chain of finite difference non markov kinetic equations for discrete nonstationary tcf and for the set of nonstationary discrete memory functions mf s the last one contains supplementary information about nonstationary properties of the complex system on the whole another relevant result of our theory is the construction of the set of dynamic parameters of nonstationarity which contains some information of the nonstationarity effects the full set of dynamic spectral and kinetic parameters and kinetic functions tcf short mf s statistical spectra of non markovity parameter and statistical spectra of nonstationarity parameter has made it possible to acquire the in depth information about discreteness non markov effects long range memory and nonstationarity of the underlying processes the developed theory is applied to analyze the long time holter series of rr intervals of human ecg s we had two groups of patients the healthy ones and the patients after myocardial infarction in both groups we observed effects of fractality standard and restricted self organized criticality and also a certain specific arrangement of spectral lines the received results demonstrate that the power spectra of all orders n 1 2 mf m n t exhibit the neatly expressed fractal feat ures we have found out that the full sets of non markov discrete and nonstationary parameters can serve as reliable and powerful means of diagnosis of the cardiovascular system states and can he used to distinguish healthy data from pathologic data

statistical mechanics of neocortical interactions path integral evolution of short term memory previous papers in this series of statistical mechanics of neocortical interactions smni have detailed a development from the relatively microscopic scales of neurons up to the macroscopic scales as recorded by electroencephalography eeg requiring an intermediate mesocolumnar scale to be developed at the scale of minicolumns approx 10 sup 2 neurons and macrocolumns approx 10 sup 5 neurons opportunity was taken to view smni as sets of statistical constraints not necessarily describing specific synaptic or neuronal mechanisms on neuronal interactions on some aspects of short term memory stm e g its capacity stability and duration a recently developed c language code pathint provides a non monte carlo technique for calculating the dynamic evolution of arbitrary dimension subject to computer resources nonlinear lagrangians such as derived for the two variable smni problem here pathint is used to explicitly detail the evolution of the smni constraints on stm

population dynamics variance and the sigmoid activation function this paper demonstrates how the sigmoid activation function of neural mass models can be understood in terms of the variance or dispersion of neuronal states we use this relationship to estimate the probability density on hidden neuronal states using non invasive electrophysiological eeg measures and dynamic casual modelling the importance of implicit variance in neuronal states for neural mass models of cortical dynamics is illustrated using both synthetic data and real eeg measurements of sensory evoked responses

markov and non markov processes in complex systems by the dynamical information entropy abstract we consider the markov and non markov processes in complex systems by the dynamical information shannon entropy dise method the influence and important role of the two mutually dependent channels of entropy alternation creation or generation of correlation and anti correlation destroying or annihilation of correlation have been discussed the developed method has been used for the analysis of the complex systems of various natures slow neutron scattering in liquid cesium psychology short time numeral and pattern human memory and effect of stress on the dynamical taping test random dynamics of rr intervals in human ecg problem of diagnosis of various disease of the human cardio vascular systems chaotic dynamics of the parameters of financial markets and ecological systems

modeling brain activation patterns for the default and cognitive states we argue that spatial patterns of cortical activation observed with eeg meg and fmri might arise from spontaneous self organisation of interacting populations of excitatory and inhibitory neurons we examine the dynamical behavior of a mean field cortical model that includes chemical and electrical gap junction synapses focusing on two limiting cases the slow soma limit with slow voltage feedback from soma to dendrite and the fast soma limit in which the feedback action of soma voltage onto dendrite reversal potentials is instantaneous for slow soma dendrite feedback we find a low frequency approximately 1 hz dynamic hopf instability and a stationary turing instability that catalyzes formation of patterned distributions of cortical firing rate activity with pattern wavelength approximately 2 cm turing instability can only be triggered when gap junction diffusion between inhibitory neurons is strong but patterning is destroyed if the tonic level of subcortical excitation is raised sufficiently interaction between the hopf and turing instabilities may describe the non cognitive background or default state of the brain as observed by bold imaging in the fast soma limit the model predicts a high frequency hopf approximately 35 hz instability and a traveling wave gamma band instability that manifests as a 2 d standing wave pattern oscillating in place at approximately 30 hz small levels of inhibitory diffusion enhance and broaden the definition of the gamma antinodal regions by suppressing higher frequency spatial modes but gamma emergence is not contingent on the presence of inhibitory gap junctions higher levels of diffusion suppress gamma activity fast soma instabilities are enhanced by increased subcortical stimulation prompt soma dendrite feedback may be an essential component of the genesis and large scale cortical synchrony of gamma activity observed at the point of cognition

1 f α noise from self organized critical models with uniform driving using the well known olami feder christensen model as our paradigm we show how to modify uniform driven self organized critical models to generate 1 f ensuremath alpha noise this model can reproduce all the main features of 1 f ensuremath alpha noise 1 ensuremath alpha is close to one and does not depend on the dimension of the system 2 the 1 f ensuremath alpha behavior is found for very low frequencies 3 the spatial correlations do not obey a power law that proves that spatially extended systems based on activation deactivation processes do not have to be point driven to produce 1 f ensuremath alpha noise the essential ingredient is a local memory of the activation deactivation process

quantitative theory of driven nonlinear brain dynamics abstract strong periodic stimuli such as bright flashing lights evoke nonlinear responses in the brain and interact nonlinearly with ongoing cortical activity but the underlying mechanisms for these phenomena are poorly understood at present the dominant features of these experimentally observed dynamics are reproduced by the dynamics of a quantitative neural field model subject to periodic drive model power spectra over a range of drive frequencies show agreement with multiple features of experimental measurements exhibiting nonlinear effects including entrainment over a range of frequencies around the natural alpha frequency f subharmonic entrainment near 2 f and harmonic generation further analysis of the driven dynamics as a function of the drive parameters reveals rich nonlinear dynamics that is predicted to be observable in future experiments at high drive amplitude including period doubling bistable phase locking hysteresis wave mixing and chaos indicated by positive lyapunov exponents moreover photosensitive seizures are predicted for physiologically realistic model parameters yielding bistability between healthy and seizure dynamics these results demonstrate the applicability of neural field models to the new regime of periodically driven nonlinear dynamics enabling interpretation of experimental data in terms of specific generating mechanisms and providing new tests of the theory

multiscaled randomness a possible source of 1 f noise in biology we evaluate the possibility that the 1 f fluctuations observed in many biological time series result simply from the fact that biological processes have many inputs with differing time scales we present a stochastic model whose output is the summation of multiple random inputs i e different regulatory mechanisms we derive the conditions under which the model reproduces the complex fluctuations and 1 f scaling observed in biological systems simulations demonstrate that if model parameters are unconstrained the likelihood of generating 1 f noise is quite small thus while the model can be used to generate 1 mathit f mathrm ensuremath beta noise with various scaling exponents it is unlikely that the 1 f behavior observed in many biological systems is due only to the fact that these systems are regulated by many different inputs acting on different time scales textcopyright 1996 the american physical society

long range memory and non markov statistical effects in human sensorimotor coordination in this paper the non markov statistical processes and long range memory e ects in human sensorimotor coordination are investigated the theoretical basis of this study is the statistical theory of non stationary discrete non markov processes in complex systems phys rev e 62 6178 2000 the human sensorimotor coordination was experimentally studied by means of standard dynamical tappingtest on the g roup of 32 youngpeoples with tap numbers up to 400 this test was carried out separately for the right and the left hand according to the degree of domination of each brain hemisphere the numerical analysis of the experimental results was made with the help of power spectra of the initial time correlation function the memory functions of low orders and the 7rst three points of the statistical spectrum of non markovity parameter our observations demonstrate that with the regard to results of the standard dynamic tapping test it is possible to divide all examinees into 7ve di erent dynamic types we have introduced the con ict coe cient to estimate quantitatively the order disorder e ects underlying life systems the last one re ects the existence of disbalance between the nervous and the motor human coordination the suggested classi7cation of the neurophysiological activity represents the dynamic generalization of the well known neuropsychological types and provides the new approach in a modern neuropsychology c

interrelating anatomical effective and functional brain connectivity using propagators and neural field theory it is shown how to compute effective and functional connection matrices ecms and fcms from anatomical cms acms and corresponding strength of connection matrices scms using propagator methods in which neural interactions play the role of scatterings this analysis demonstrates how network effects dress the bare propagators the scms to yield effective propagators the ecms that can be used to compute the covariances customarily used to define fcms the results incorporate excitatory and inhibitory connections multiple structures and populations asymmetries time delays and measurement effects they can also be postprocessed in the same manner as experimental measurements for direct comparison with data and thereby give insights into the role of coarse graining thresholding and other effects in determining the structure of cms the spatiotemporal results show how to generalize cms to include time delays and how natural network modes give rise to long range coherence at resonant frequencies the results are demonstrated using tractable analytic cases via neural field theory of cortical and corticothalamic systems these also demonstrate close connections between the structure of cms and proximity to critical points of the system highlight the importance of indirect links between brain regions and raise the possibility of imaging specific levels of indirect connectivity aside from the results presented explicitly here the expression of the connections among acms scms ecms and fcms in terms of propagators opens the way for propagator theory to be further applied to analysis of connectivity

regular and stochastic behavior of parkinsonian pathological tremor signals abstract regular and stochastic behavior in the time series of parkinsonian pathological tremor velocity is studied on the basis of the statistical theory of discrete non markov stochastic processes and flicker noise spectroscopy we have developed a new method of analyzing and diagnosing parkinson s disease pd by taking into consideration discreteness fluctuations long and short range correlations regular and stochastic behavior markov and non markov effects and dynamic alternation of relaxation modes in the initial time signals the spectrum of the statistical non markovity parameter reflects markovity and non markovity in the initial time series of tremor the relaxation and kinetic parameters used in the method allow us to estimate the relaxation scales of diverse scenarios of the time signals produced by the patient in various dynamic states the local time behavior of the initial time correlation function and the first point of the non markovity parameter give detailed information about the variation of pathological tremor in the local regions of the time series the obtained results can be used to find the most effective method of reducing or suppressing pathological tremor in each individual case of a pd patient generally the method allows one to assess the efficacy of the medical treatment for a group of pd patients

possibility between earthquake and explosion seismogram differentiation by discrete stochastic non markov processes and local hurst exponent analysis the basic scientific point of this paper is to draw the attention of researchers to new possibilities of differentiation of similar signals having different nature one example of such kinds of signals is presented by seismograms containing recordings of earthquakes eq s and technogenic explosions te s eq s are among the most dramatic phenomena in nature we propose here a discrete stochastic model for possible solution of a problem of strong eq forecasting and differentiation of te s from the weak eq s theoretical analysis is performed by two independent methods by using statistical theory of discrete non markov stochastic processes phys rev e 62 6178 2000 and the local hurst exponent the following earth states have been considered among them before ib and during i strong eq during weak eq ii and during te iii and in a calm state of earth s core iv the estimation of states i ii and iii has been made on the particular examples of turkey 1999 eq s state iv has been taken as an example of earth s state before underground te time recordings of seismic signals of the first four dynamic orthogonal collective variables six various planes of phase portrait of four dimensional phase space of orthogonal variables and the local hurst exponent have been calculated for the dynamic analysis of states of systems i char21 iv the analysis of statistical properties of seismic time series i char21 iv has been realized with the help of a set of discrete time dependent functions time correlation function and first three memory functions their power spectra and the first three points in the statistical spectrum of non markovity parameters in all systems studied we have found a bizarre combination of the following spectral characteristics the fractal frequency spectra adjustable by phenomena of usual and restricted self organized criticality spectra of white and color noises and unusual alternation of markov and non markov effects of long range memory detected earlier j phys a 27 5363 1994 only for hydrodynamic systems our research demonstrates that discrete non markov stochastic processes and long range memory effects play a crucial role in the behavior of seismic systems i char21 iv the approaches permitting us to obtain an algorithm of strong eq forecasting and to differentiate te s from weak eq s have been developed

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

a robust variational approach for simultaneous smoothing and estimation of dti abstract estimating diffusion tensors is an essential step in many applications such as diffusion tensor image dti registration segmentation and fiber tractography most of the methods proposed in the literature for this task are not simultaneously statistically robust and feature preserving techniques in this paper we propose a novel and robust variational framework for simultaneous smoothing and estimation of diffusion tensors from diffusion mri our variational principle makes use of a recently introduced total kullback leibler tkl divergence for dti regularization tkl is a statistically robust dissimilarity measure for diffusion tensors and regularization by using tkl ensures the symmetric positive definiteness of tensors automatically further the regularization is weighted by a non local factor adapted from the conventional non local means filters finally for the data fidelity we use the nonlinear least squares term derived from the stejskal tanner model we present experimental results depicting the positive performance of our method in comparison to competing methods on synthetic and real data examples

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

stochastic dynamics of time correlation in complex systems with discrete time in this paper we present the concept of description of random processes in complex systems with discrete time it involves the description of kinetics of discrete processes by means of the chain of finite difference non markov equations for time correlation functions tcfs we have introduced the dynamic time dependent information shannon entropy s i t where i 0 1 2 3 ellipsis as an information measure of stochastic dynamics of time correlation i 0 and time memory i 1 2 3 ellipsis the set of functions s i t constitute the quantitative measure of time correlation disorder i 0 and time memory disorder i 1 2 3 ellipsis in complex system the theory developed started from the careful analysis of time correlation involving dynamics of vectors set of various chaotic states we examine two stochastic processes involving the creation and annihilation of time correlation or time memory in details we carry out the analysis of vectors dynamics employing finite difference equations for random variables and the evolution operator describing their natural motion the existence of tcf results in the construction of the set of projection operators by the usage of scalar product operation harnessing the infinite set of orthogonal dynamic random variables on a basis of gram shmidt orthogonalization procedure tends to creation of infinite chain of finite difference non markov kinetic equations for discrete tcfs and memory functions mfs the solution of the equations above thereof brings to the recurrence relations between the tcf and mf of senior and junior orders this offers new opportunities for detecting the frequency spectra of power of entropy function s i t for time correlation i 0 and time memory i 1 2 3 ellipsis the results obtained offer considerable scope for attack on stochastic dynamics of discrete random processes in a complex systems application of this technique on the analysis of stochastic dynamics of rr intervals from human ecg s shows convincing evidence for a non markovian phenomemena associated with a peculiarities in short and long range scaling this method may be of use in distinguishing healthy from pathologic data sets based in differences in these non markovian properties

propagation and stability of waves of electrical activity in the cerebral cortex nonlinear equations are introduced to model the behavior of the waves of cortical electrical activity that are responsible for signals observed in electroencephalography these equations incorporate nonlinearities axonal and dendritic lags excitatory and inhibitory neuronal populations and the two dimensional nature of the cortex while rendering nonlinear features far more tractable than previous formulations both analytically and numerically the model equations are first used to calculate steady state levels of cortical activity for various levels of stimulation dispersion equations for linear waves are then derived analytically and an analytic expression is found for the linear stability boundary beyond which a seizure will occur the effects of boundary conditions in determining global eigenmodes are also studied in various geometries and the corresponding eigenfrequencies are found numerical results confirm the analytic ones which are also found to reproduce existing results in the relevant limits thereby elucidating the limits of validity of previous approximations

existence and stability of limit cycles in a macroscopic neuronal population model abstract we present rigorous results concerning the existence and stability of limit cycles in a macroscopic model of neuronal activity the specific model we consider is developed from the ki set methodology popularized by walter freeman in particular we focus on a specific reduction of the kii sets denoted rkii sets we analyse the unfolding of supercritical hopf bifurcations via consideration of the normal forms and centre manifold reductions subsequently we analyse the global stability of limit cycles on a region of parameter space and this is achieved by applying a new methodology termed global analysis of piecewise linear systems the analysis presented may also be used to consider coupled systems of this type a number of macroscopic mean field approaches to modelling human eeg may be considered as coupled rkii networks hence developing a theoretical understanding of the onset of oscillations in models of this type has important implications in clinical neuroscience as limit cycle oscillations have been demonstrated to be critical in the onset of certain types of epilepsy

non markov stochastic dynamics of real epidemic process of respiratory infections abstract the study of social networks and especially of stochastic dynamics of diseases spread in human population has recently attracted considerable attention in statistical physics in this work we present a new statistical method of analyzing the spread of epidemic processes of grippe and acute respiratory track infections arti by means of the theory of discrete non markov stochastic processes we use the results of our last theory phys rev e 65 2002 046107 to study statistical memory effects long range correlation and discreteness in real data series describing the epidemic dynamics of human arti infections and grippe we have carried out the comparative analysis of the data of the two infections grippe and arti in one of the industrial districts of kazan one of the largest cities of russia the experimental data are analyzed by the power spectra of the initial time correlation function and the memory functions of junior orders the phase portraits of the four first dynamic variables the three first points of the statistical non markov parameter and the locally averaged kinetic and relaxation parameters the received results give an opportunity to provide a strict quantitative description of regular and stochastic components in epidemic dynamics of social networks taking into account their time discreteness and effects of statistical memory they also allow to reveal the degree of randomness and predictability of the real epidemic process in the specific social network

a note on traveling fronts and pulses in a firing rate model of a neuronal network we study the activity of a one dimensional synaptically coupled neural network by means of a firing rate model developed by coombes et al physica d 178 2003 their approach incorporates the biologically motivated finite conduction velocity of action potentials into a neural field equation of wilson and cowan type kybernetik 13 1973 the resulting integro differential equation with a space depending delay term under the convolution may exhibit a variety of traveling and stationary patterns in this paper we construct traveling wave solutions for the case of a firing rate given by the heaviside step function exponential synaptic kernel and exponential synaptic footprint in contrast to coombes et al where the model equation is first reduced to an equivalent system of partial differential equations we make the traveling pattern ansatz into the initial integro differential equation we analyse two types of traveling patterns fronts and pulses for which we derive shape and speed we further determine necessary conditions for the linear stability of the traveling waves

modeling electrocortical activity through improved local approximations of integral neural field equations neural field models of firing rate activity typically take the form of integral equations with space dependent axonal delays under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation pde model that properly treats the axonal delay terms of the integral formulation our analysis avoids the so called long wavelength approximation that has previously been used to formulate pde models for neural activity in two spatial dimensions direct numerical simulations of this pde model show instabilities of the homogeneous steady state that are in full agreement with a turing instability analysis of the original integral model we discuss the benefits of such a local model and its usefulness in modeling electrocortical activity in particular we are able to treat patchy connections whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion in this case the emergence of a lattice directed traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled pdes

isotropy properties of the multi step markov symbolic sequences abstract a new object of the probability theory the two sided chain of symbols introduced in ref arxiv physics 0306170 is used to study isotropy properties of binary multi step markov chains with the long range correlations established statistical correspondence between the markov chains and certain two sided sequences allows us to prove the isotropy properties of three classes of the markov chains one of them is the important class of weakly correlated additive markov chains which turned out to be equivalent to the additive two sided sequences

subthreshold dynamics of a single neuron from a hamiltonian perspective we use hamilton s equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential we use a two component dynamic model of a single neuron due to wilson with added noise inputs we derive a lagrangian for the system from which we construct hamilton s equations the conjugate momenta are found to be linear combinations of the noise input to the system we use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event we find that the firing of a neuron is a result of a drop in inhibition due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation moreover we demonstrate through theory and simulation that on average the bias in the noise increases in an exponential manner on the approach to an action potential in the hamiltonian description an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state into a nonlinear regime

dynamics of the information entropy in random processes abstract the problem of the description of temporal evolution of the random variables in terms of the information shannon entropy of correlation processes has been considered the influence and important role of the two mutually dependent channels of the entropy alternation correlation creation or the generation of correlations and anti correlation decay or the annihilation of correlation have been discussed the method developed is used for the analysis of chaotic dynamics of the long range fluctuation in the short time human memory

high order cycles in the logistic map or centers of cardioids in the mandelbrot set asymptotic expressions for the positions of the centres along the real axis of high order cycle cardioids closest to the limit point 2 in the mandelbrot set are obtained by a combination of numerical and analytical methods

the quantified histograms detection of the hidden unsteadiness abstract for a given level of digitization a number of relative frequencies presented by a histogram constructed for a short sample defined as a segment of a random series related in a certain way to both the length of the sampling l and the number of decompositions k may form a set of some levels possessing a discrete spectrum this spectrum can be defined as a quantified histogram qh the structure of parameters of a certain qh can be analyzed via both the number of quantum levels nql and the proper collection of relative frequencies on every level these parameters form kind of specific fingerprints which characterizes statistical properties of the segment of random series treated in this way qhs allow us to introduce the concept of a standard noise sn which can be used as a noise ruler to compare random series of distinct nature this standard noise turns out to be very useful in the detection of both a hidden unsteadiness and superweak signals existing in random series in the capacity of an example of the application of this new method a comparison of statistically sensible distinctions between noises accompanying either earthquakes or technogenic explosions is given distinctions thus obtained called earthquake forerunner can be used for earthquakes forecasting

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

wave number spectrum of electrocorticographic signals a physiologically based continuum model of corticothalamic electrodynamics is generalized and used to derive the theoretical form of the electrocorticographic ecog wave number spectrum a one dimensional projection of the spectrum is derived as is the azimuthally averaged two dimensional spectrum for isotropic and anisotropic cortices the predicted spectra are found to consist of a low k plateau followed by three regions of power law decrease which result from filtering of the electrical activity through physical structures at different scales in the cortex the magnitude of the maximum theoretical power law exponent is larger for the two dimensional 2d spectrum than for its 1d counterpart the predicted spectra agree well with experimental data obtained from 1d and 2d recording arrays on the cortical surface enabling the structures in the brain that are important in determining spatial cortical dynamics to be identified the cortical dispersion relation predicted by our model is also investigated providing insight into the relationships between temporal and spatial brain dynamics

competition between two kinds of correlations in literary texts a theory of additive markov chains with long range memory is used to describe the correlation properties of coarse grained literary texts the complex structure of the correlations in the texts is revealed anti persistent correlations at small distances l approximately 300 define this non trivial structure for some concrete examples of literary texts the memory functions are obtained and their power law behavior at long distances is disclosed this property is shown to be a cause of self similarity of texts with respect to the decimation procedure

derivation and analysis of an ordinary differential equation mean field model for studying clinically recorded epilepsy dynamics in this paper we describe how an ordinary differential equation model of corticothalamic interactions may be obtained from a more general system of delay differential equations we demonstrate that transitions to epileptic dynamics via changes in system parameters are qualitatively the same as in the original model with delay as well as demonstrating that the onset of epileptic activity may arise due to regions of bistability hence the model presents in one unique framework two competing theories for the genesis of epileptiform activity similarities between model transitions and clinical data are presented and we argue that statistics obtained from and a parameter estimation of this model may be a potential means of classifying and predicting the onset and offset of seizure activity

alignment by maximization of mutual information a new information theoretic approach is presented for finding the pose of an object in an image the technique does not require information about the surface properties of the object besides its shape and is robust with respect to variations of illumination in our derivation few assumptions are made about the nature of the imaging process as a result the algorithms are quite general and may foreseeably be used in a wide variety of imaging situations experiments are presented that demonstrate the approach registering magnetic resonance mr images aligning a complex 3d object model to real scenes including clutter and occlusion tracking a human head in a video sequence and aligning a view based 2d object model to real images the method is based on a formulation of the mutual information between the model and the image as applied here the technique is intensity based rather than feature based it works well in domains where edge or gradient magnitude based methods have difficulty yet it is more robust than traditional correlation additionally it has an efficient implementation that is based on stochastic approximation

learning dynamics from nonstationary time series analysis of electroencephalograms we propose an empirical modeling technique for a nonstationary time series analysis proposed methods include a high dimensional n 3 dynamical model construction in the form of delay differential equations a nonparametric method of respective time delay calculation the detection of quasistationary regions of the process by recurrence analysis in the space of model coefficients and final fitting of the model to quasistationary segments of observed time series we also demonstrate the effectiveness of our approach for nonstationary signal classification in the space of model coefficients applying the empirical modeling technique to electroencephalogram eeg records analysis we find evidence of high dimensional nonlinear dynamics in quasistationary eeg segments recurrence analysis of model parameters reveals long term correlations in nonstationary eeg records using the dynamical model as a nonlinear filter we find that different emotional states of subjects can be clearly distinguished in the space of model coefficients

well posedness of two nonrigid multimodal image registration methods image registration methods may be designed as solutions of minimization problems on a set of geometric deformations in the nonrigid case solving these problems often means computing the steady state of a system of evolution equations involving the gradient of the error criterion defined from the euler lagrange equations of the corresponding minimization problem the well posedness of the registration methodrequires showing the existence of a solution to the minimization problem as well as that of a stable solution of the evolution equations derived from it we provide such proofs in the case where the error criterion is derived from two different statistical similarity measures global mutual information and local cross covariance we also describe our numerical implementation for solving the corresponding evolution equations and show examples of registrations of real 2d and 3d images achieved with these algorithms the proofs are quite general and can be applied to most of the known nonrigid image

a topology preserving level set method for geometric deformable models active contour and surface models also known as deformable models are powerful image segmentation techniques geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior parameterization independence and ease of implementation however a long claimed advantage of geometric deformable models the ability to automatically handle topology changes turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved we present a new class of geometric deformable models designed using a novel topology preserving level set method which achieves topology preservation by applying the simple point concept from digital topology these new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces moreover since the topology preserving constraint is enforced efficiently through local computations the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm

deviations from uniform power law scaling in nonstationary time series a classic problem in physics is the analysis of highly nonstationary time series that typically exhibit long range correlations here we test the hypothesis that the scaling properties of the dynamics of healthy physiological systems are more stable than those of pathological systems by studying beat to beat fluctuations in the human heart rate we develop techniques based on the fano factor and allan factor functions as well as on detrended fluctuation analysis for quantifying deviations from uniform power law scaling in nonstationary time series by analyzing extremely long data sets of up to n 10 5 beats for 11 healthy subjects we find that the fluctuations in the heart rate scale approximately uniformly over several temporal orders of magnitude by contrast we find that in data sets of comparable length for 14 subjects with heart disease the fluctuations grow erratically indicating a loss of scaling stability

spirals in the mandelbrot set ii abstract the alpha function is used to quantify the asymptotic structure of the various branches and embedded spirals around the left hand side of the main cardioid in the mandelbrot set

dynamical shannon entropy and information tsallis entropy in complex systems abstract in this work we offer a new approach to the concept of the information entropy for the description of the dynamic behaviour of complex systems on the basis of the unification of the known information approach to the shannon entropy submitted in works of authors phys rev e 62 5 2000 6178 physica a 303 2002 427 and generalization of the boltzman gibbs entropy offered by tsallis braz j phys 29 1 1999 1 we have received a new representation of the dynamic information entropy here we present concrete applications of the received equations to the study of complex systems related to the electric signals of ecgs of healthy people and patients with myocardial infarction mi as an example various modifications of the non markovity parameter are also submitted they were received with the help of the new approach to the information entropy we have received detailed information about markov quasi markov and non markov characteristics of the rr interval fluctuations of the ecgs with the help of the chain of zwanzig mori s discrete kinetic equations and dynamical tsallis entropy

a solution to the dynamical inverse problem of eeg generation using spatiotemporal kalman filtering we present a new approach for estimating solutions of the dynamical inverse problem of eeg generation in contrast to previous approaches we reinterpret this problem as a filtering problem in a state space framework for the purpose of its solution we propose a new extension of kalman filtering to the case of spatiotemporal dynamics the temporal evolution of the distributed generators of the eeg can be reconstructed at each voxel of a discretisation of the gray matter of brain by fitting linear autoregressive models with neighbourhood interactions to eeg time series new classes of inverse solutions with improved resolution and localisation ability can be explored for the purposes of model comparison and parameter estimation from given data we employ a likelihood maximisation approach both for instantaneous and dynamical inverse solutions we derive estimators of the time dependent estimation error at each voxel the performance of the algorithm is demonstrated by application to simulated and clinical eeg recordings it is shown that by choosing appropriate dynamical models it becomes possible to obtain inverse solutions of considerably improved quality as compared to the usual instantaneous inverse solutions

controlling limit cycle behaviors of brain activity the limit cycles of brain activity are studied using a compact continuum model that reproduces the main features of electroencephalographic signals including bifurcations of fixed points and limit cycles in seizures frequencies and amplitudes are predicted analytically and related to physiology gaussian stimuli yield two distinct evoked responses in the linearly stable zone consistent with experiment limit cycles can be initiated or suppressed by control signals or stimuli

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

group actions homeomorphisms and matching a general framework this paper constructs metrics on the space of images cal i defined as orbits under group actions cal g the groups studied include the finite dimensional matrix groups and their products as well as the infinite dimensional diffeomorphisms examined in trouve 1999 quaterly of applied math and dupuis et al 1998 quaterly of applied math left invariant metrics are defined on the product cal g times cal i thus allowing the generation of transformations of the background geometry as well as the image values examples of the application of such metrics are presented for rigid object matching with and without signature variation curves and volume matching and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another

life space foam a medium for motivational and cognitive dynamics abstract general stochastic dynamics developed in a framework of feynman path integrals have been applied to lewinian field theoretic psychodynamics k lewin field theory in social science university of chicago press chicago 1951 k lewin resolving social conflicts and field theory in social science american psychological association washington 1997 m gold a kurt lewin reader the complete social scientist american psychological association washington 1999 resulting in the development of a new concept of life space foam lsf as a natural medium for motivational and cognitive psychodynamics according to lsf formalisms the classic lewinian life space can be macroscopically represented as a smooth manifold with steady force fields and behavioral paths while at the microscopic level it is more realistically represented as a collection of wildly fluctuating force fields loco motion paths and local geometries and topologies with holes a set of least action principles is used to model the smoothness of global macro level lsf paths fields and geometry to model the corresponding local micro level lsf structures an adaptive path integral is used defining a multi phase and multi path multi field and multi geometry transition process from intention to goal driven action application examples of this new approach include but are not limited to information processing motivational fatigue learning memory and decision making

characterising the dynamics of eeg waveforms as the path through parameter space of a neural mass model application to epilepsy seizure evolution in this paper we propose that the dynamic evolution of eeg activity during epileptic seizures may be characterised as a path through parameter space of a neural mass model reflecting gradual changes in underlying physiological mechanisms previous theoretical studies have shown how boundaries in parameter space of the model so called bifurcations correspond to transitions in eeg waveforms between apparently normal spike and wave and subsequently poly spike and wave activity in the present manuscript we develop a multi objective genetic algorithm that can estimate parameters of an underlying model from clinical data recordings a standard approach to this problem is to transform both clinical data and model output into the frequency domain and then choose parameters that minimise the difference in their respective power spectra instead in the present manuscript we estimate parameters in the time domain their choice being determined according to the best fit obtained between the model output and specific features of the observed eeg waveform this results in an approximate path through the bifurcation plane of the model obtained from clinical data we present comparisons of such paths through parameter space from separate seizures from an individual subject as well as between different subjects differences in the path reflect subtleties of variation in the dynamics of eeg which at present appear indistinguishable using standard clinical techniques

robust nonparametric tests of general linear model coefficients a comparison of permutation methods and test statistics statistical inference in neuroimaging research often involves testing the significance of regression coefficients in a general linear model in many applications the researcher assumes a model of the form y x z e where y is the observed brain signal and x and z contain explanatory variables that are thought to be related to the brain signal the goal is to test the null hypothesis h0 0 with the nuisance parameters included in the model several nonparametric permutation methods have been proposed for this problem and each method uses some variant of the f ratio as the test statistic however recent research suggests that the f ratio can produce invalid permutation tests of h0 0 when the e terms are heteroscedastic i e have non constant variance which can occur for a variety of reasons this study compares the classic f test statistic to the robust w wald test statistic using eight different permutation methods the results reveal that permutation tests using the f ratio can produce accurate results when the errors are homoscedastic but high false positive rates when the errors are heteroscedastic in contrast permutation tests using the w test statistic produced valid results when the errors were homoscedastic and asymptotically valid results when the errors were heteroscedastic in the situation with homoscedastic errors permutation tests using the w statistic showed slightly reduced power compared to the f statistic but the difference disappeared as the sample size n increased consequently the w test statistic is recommended for robust nonparametric hypothesis tests of regression coefficients in neuroimaging research

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

activity dynamics in nonlocal interacting neural fields we study the activity of a synaptically coupled neuronal network consisting of an excitatory and an inhibitory layer with isotropic connections and nonlinear interactions using the mathematical model of wilson and cowan in two spatial dimensions we first discuss a spatial hysteresis phenomenon then we analyze special traveling wave solutions with stationary shape we establish existence conditions derive analytic expressions of the particular solutions and their velocity and finally present numerical simulations

time coherent expansion of meg eeg cortical sources in this study we estimated the spatial extent of cortical areas of time coherent activity using the inverse problem in magneto electroencephalography meeg the model discussed here uses classical regularization tools in order to force the inverse solution to be piecewise coherent first the cortex was seeded by focal dipolar sources then a time coherent expansion tce onto the cortical surface was performed in order to obtain surface source models composed of patches with uniform current density patches represent extended cortical regions with one single time course per active area results obtained from synthetic data show that using the tce method is relevant even with a low signal to noise ratio although the final estimation is often slightly biased we applied the tce method to evoked magnetic fields obtained after electrical stimulation of fingers in order to estimate the somatotopic cortical maps of the primary somatosensory cortex

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract this work studies the stability of equilibria in spatially extended neuronal ensembles we first derive the model equation from statistical properties of the neuron population the obtained integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities the work derives conditions for stationary and nonstationary instabilities for both kernel types in addition a nonlinear analysis for general kernels yields the order parameter equation of the turing instability to compare the results to findings for partial differential equations pdes two typical pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed kernels the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges as a novel finding we obtain turing instabilities in fields with local inhibition lateral excitation while wave instabilities occur in fields with local excitation and lateral inhibition numerical simulations support the analytical results

diffusion based spatial priors for imaging we describe a bayesian scheme to analyze images which uses spatial priors encoded by a diffusion kernel based on a weighted graph laplacian this provides a general framework to formulate a spatial model whose parameters can be optimized the application we have in mind is a spatiotemporal model for imaging data we illustrate the method on a random effects analysis of fmri contrast images from multiple subjects this simplifies exposition of the model and enables a clear description of its salient features typically imaging data are smoothed using a fixed gaussian kernel as a pre processing step before applying a mass univariate statistical model e g a general linear model to provide images of parameter estimates an alternative is to include smoothness in a multivariate statistical model penny w d trujillo barreto n j friston k j 2005 bayesian fmri time series analysis with spatial priors neuroimage 24 350 362 the advantage of the latter is that each parameter field is smoothed automatically according to a measure of uncertainty given the data in this work we investigate the use of diffusion kernels to encode spatial correlations among parameter estimates nonlinear diffusion has a long history in image processing in particular flows that depend on local image geometry romeny b m t 1994 geometry driven diffusion in computer vision kluwer academic publishers can be used as adaptive filters this can furnish a non stationary smoothing process that preserves features which would otherwise be lost with a fixed gaussian kernel we describe a bayesian framework that incorporates non stationary adaptive smoothing into a generative model to extract spatial features in parameter estimates critically this means adaptive smoothing becomes an integral part of estimation and inference we illustrate the method using synthetic and real fmri data

stationary solutions of linear stochastic delay differential equations applications to biological systems recently k uchler and mensch stochastics stochastics rep 40 23 1992 derived exact stationary probability densities for linear stochastic delay differential equations this paper presents an alternative derivation of these solutions by means of the fokker planck approach introduced by guillouzic phys rev e 59 3970 1999 61 4906 2000 applications of this approach which is argued to have greater generality are discussed in the context of stochastic models for population growth and tracking movements

universal approach to overcoming nonstationarity unsteadiness and non markovity of stochastic processes in complex systems abstract in the present paper we suggest a new universal approach to study complex systems by microscopic mesoscopic and macroscopic methods we discuss new possibilities of extracting information on nonstationarity unsteadiness and non markovity of discrete stochastic processes in complex systems we consider statistical properties of the fast intermediate and slow components of the investigated processes in complex systems within the framework of microscopic mesoscopic and macroscopic approaches separately among them theoretical analysis is carried out by means of local noisy time dependent parameters and the conception of a quasi brownian particle qbp mesoscopic approach as well as the use of wavelet transformation of the initial row time series as a concrete example we examine the seismic time series data for strong and weak earthquakes in turkey 1998 1999 in detail as well as technogenic explosions we propose a new possible solution to the problem of forecasting strong earthquakes besides we have found out that an unexpected restoration of the first two local noisy parameters in weak earthquakes and technogenic explosions is determined by exponential law in this paper we have also carried out the comparison and have discussed the received time dependence of the local parameters for various seismic phenomena

a neural mass model with direct and indirect excitatory feedback loops identification of bifurcations and temporal dynamics neural mass modeling is a part of computational neuroscience that was developed to study the general behavior of a neuronal population this type of mesoscopic model is able to generate output signals that are comparable to experimental data such as electroencephalograms classically neural mass models consider two interconnected populations excitatory pyramidal cells and inhibitory interneurons however many authors have included an excitatory feedback on the pyramidal cell population two distinct approaches have been developed a direct feedback on the main pyramidal cell population and an indirect feedback via a secondary pyramidal cell population in this letter we propose a new neural mass model that couples these two approaches we perform a detailed bifurcation analysis and present a glossary of dynamical behaviors and associated time series our study reveals that the model is able to generate particular realistic time series that were never pointed out in either simulated or experimental data finally we aim to evaluate the effect of balance between both excitatory feedbacks on the dynamical behavior of the model for this purpose we compute the codimension 2 bifurcation diagrams of the system to establish a map of the repartition of dynamical behaviors in a direct versus indirect feedback parameter space a perspective of this work is from a given temporal series to estimate the parameter value range especially in terms of direct versus indirect excitatory feedback

an elasticity based covariance analysis of shapes we introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects based on the notion of nonlinear elastic deformations from one shape to another a suitable linearization of geometric shape variations is introduced once such a linearization is available a principal component analysis can be investigated this requires the definition of a covariance metric an inner product on linearized shape variations the resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation the underlying elasticity concept represents an alternative to riemannian shape statistics in this paper we compare a standard l 2 type covariance metric with a metric based on the hessian of the nonlinear elastic energy furthermore we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity for the built in pairwise elastic registration a relaxed model formulation is employed which allows for a non exact matching shape contours are approximated by single well phase fields which enables an extension of the method to a covariance analysis of image morphologies the model is implemented with multilinear finite elements embedded in a multi scale approach the characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2d and 3d

compact dynamical model of brain activity a compact physiologically based mean field formulation of brain dynamics is proposed to model observed brain activity and electroencephalographic eeg signals in contrast to existing formulations which are more detailed and complicated our model is described by a single second order delay differential equation that encapsulates salient aspects of the physiology the model captures essential features of activity mediated by fast corticocortical connections and delayed feedbacks via extracortical pathways and external stimuli in the linear regime these features can be simply expressed by three coefficients derived from the properties of these physiological pathways and explicit nonlinear approximations are also derived this compact model successfully reproduces the main features of experimental eeg s and the predictions of previous models including resonance peaks in eeg spectra and nonlinear dynamics as an illustration key features of the dynamics of epileptic seizures are shown to be reproduced by the model due to its compact form the model will facilitate insight into nonlinear brain dynamics via standard nonlinear techniques and will guide analysis and investigation of more complex models it is thus a useful tool for analyzing complex brain activity especially when it exhibits low dimensional dynamics

relaxation and phase space singularities in time series of human magnetoencephalograms as indicator of photosensitive epilepsy to analyze the crucial role of fluctuation and relaxation effects for the function of the human brain we studied some statistical quantifiers that support the information characteristics of neuromagnetic brain responses magnetoencephalogram meg the signals to a flickering stimulus of different color combinations have been obtained from a group of control subjects which is then contrasted with those of a patient suffering photosensitive epilepsy pse we found that the existence of the specific stratification of the phase clouds and the concomitant relaxation singularities of the corresponding nonequilibrium dynamics of the chaotic behavior of the signals in separate areas in a patient provide likely indicators for the zones which are responsible for the appearance of pse

variational methods for multimodal image matching matching images of different modalities can be achieved by the maximization of suitable statistical similarity measures within a given class of geometric transformations handling complex nonrigid deformations in this context turns out to be particularly difficult and has attracted much attention in the last few years the thrust of this paper is that many of the existing methods for nonrigid monomodal registration that use simple criteria for comparing the intensities e g ssd can be extended to the multimodal case where more complex intensity similarity measures are necessary to this end we perform a formal computation of the variational gradient of a hierarchy of statistical similarity measures and use the results to generalize a recently proposed and very effective optical flow algorithm l alvarez j weickert and j sanchez 2000 technical report and ijcv 39 1 41 56 to the case of multimodal image registration our method readily extends to the case of locally computed similarity measures thus providing the flexibility to cope with spatial non stationarities in the way the intensities in the two images are related the well posedness of the resulting equations is proved in a complementary work o d faugeras and g hermosillo 2001 technical report 4235 inria using well established techniques in functional analysis we briefly describe our numerical implementation of these equations and show results on real and synthetic data