waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

balanced truncation for temporal and spatial lpv interconnected systems based on the full block s procedure abstractthis paper presents a technique for model order reduction of exponentially stable temporal and spatial lpv linear parameter varying interconnected systems represented in linear fractiona

model reduction of nonstationary lpv systems this paper focuses on the model reduction of nonstationary linear parameter varying nslpv systems we provide a generalization of the balanced truncation procedure for the model reduction of stable nslpv systems along with a priori error bounds then for illustration purposes this method is applied to reduce the model of a two mass translational system furthermore we give an approach for the model reduction of stabilizable and detectable systems which requires the development and use of coprime factorizations for nslpv models for the general class of eventually periodic lpv systems which includes periodic and finite horizon systems as special cases our results can be explicitly computed using semidefinite programming

coupling layers regularizes wave propagation in stochastic neural fields we explore how layered architectures influence the dynamics of stochastic neural field models our main focus is how the propagation of waves of neural activity in each layer is affected by interlaminar coupling synaptic connectivities within and between each layer are determined by integral kernels of an integrodifferential equation describing the temporal evolution of neural activity excitatory neural fields with purely positive connectivities support traveling fronts in each layer whose speeds are increased when coupling between layers is considered studying the effects of noise we find coupling reduces the variance in the position of traveling fronts as long as the noise sources to each layer are not completely correlated neural fields with asymmetric connectivity support traveling pulses whose speeds are decreased by interlaminar coupling again coupling reduces the variance in traveling pulse position asymptotic analysis is performed using a small noise expansion assuming interlaminar connectivity scales similarly

two bump solutions of amari type models of neuronal pattern formation abstract we study a partial integro differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks for a one dimensional domain we develop criteria for the existence and stability of equal width 2 bump solutions under the assumption that the firing rate function is the heaviside function we apply these criteria to an example for which the connectivity is of lateral inhibition type i e the coupling function has one positive zero and find that families of 2 bump solutions exist but none of the solutions are stable extensive numerical searches suggest that this is true for all coupling functions of this form however for a large class of coupling functions which have three positive zeros we find the coexistence of both stable and unstable 2 bump solutions we also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1 bump and 2 bump solutions our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity

feature combination and the knn framework in object classification in object classification feature combination can usually be used to combine the strength of multiple complementary features and produce better classification results than any single one while multiple kernel learning mkl is a popular approach to feature combination in object classification it does not always perform well in practical applications on one hand the optimization process in mkl usually involves a huge consumption of computation and memory space on the other hand in some cases mkl is found to perform no better than the baseline combination methods this observation motivates us to investigate the underlying mechanism of feature combination with average combination and weighted average combination as a result we empirically find that in average combination it is better to use a sample of the most powerful features instead of all whereas in one type of weighted average combination the best classification accuracy comes from a nearly sparse combination we integrate these observations into the k nearest neighbors knns framework based on which we further discuss some issues related to sparse solution and mkl finally by making use of the knn framework we present a new weighted average combination method which is shown to perform better than mkl in both accuracy and efficiency in experiments we believe that the work in this paper is helpful in exploring the mechanism underlying feature combination

neural activity measures and their dynamics we provide an asymptotically justified derivation of activity measure evolution equations amees for a finite size neural network the approach takes into account the dynamics for each isolated neuron in the network being modeled by a biophysical model i e hodgkin huxley equations or their reductions by representing the interacting network as self and pairwise interactions we propose a general definition of spatial projections of the network called activity measures that quantify the activity of a network we show that the evolution equations that govern the dynamics of the activity measure shadow the activity measure of the network i e the two quantities stay close to each other for all times for general interactions and various asymptotic dynamics the amees effectively serve as a dimensionality reduction technique for the complex network when spatial synchrony and coherence are present and allow us to a priori predict network dynamics that would not be guessed from individual neuron behav

dsets dbscan a parameter free clustering algorithm clustering image pixels is an important image segmentation technique while a large amount of clustering algorithms have been published and some of them generate impressive clustering results their performance often depends heavily on user specified parameters this may be a problem in the practical tasks of data clustering and image segmentation in order to remove the dependence of clustering results on user specified parameters we investigate the characteristics of existing clustering algorithms and present a parameter free algorithm based on the dsets dominant sets and dbscan density based spatial clustering of applications with noise algorithms first we apply histogram equalization to the pairwise similarity matrix of input data and make dsets clustering results independent of user specified parameters then we extend the clusters from dsets with dbscan where the input parameters are determined based on the clusters from dsets automatically by merging the merits of dsets and dbscan our algorithm is able to generate the clusters of arbitrary shapes without any parameter input in both the data clustering and image segmentation experiments our parameter free algorithm performs better than or comparably with other algorithms with careful parameter tuning

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

existence and uniqueness of travelling waves for a neural network a one dimensional scalar neural network with two stable steady states is analysed it is shown that there exists a unique monotone travelling wave front which joins the two stable states some additional properties of the wave such as the direction of its velocity are discussed

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

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

interactive image segmentation based on synthetic graph coordinates in this paper we propose a framework for interactive image segmentation the goal of interactive image segmentation is to classify the image pixels into foreground and background classes when some foreground and background markers are given the proposed method minimizes a min max bayesian criterion that has been successfully used on image segmentation problem and it consists of several steps in order to take into account visual information as well as the given markers without any requirement of training first we partition the image into contiguous and perceptually similar regions superpixels then we construct a weighted graph that represents the superpixels and the connections between them an efficient algorithm for graph clustering based on synthetic coordinates is used yielding an initial map of classified pixels this method reduces the problem of graph clustering to the simpler problem of point clustering instead of solving the problem on the graph data structure as most of the known algorithms from literature do finally having available the data modeling and the initial map of classified pixels we use a markov random field mrf model or a flooding algorithm to get the image segmentation by minimizing a min max bayesian criterion experimental results and comparisons with other methods from the literature are presented on lhi gulshan and zhao datasets demonstrating the high performance and accuracy of the proposed scheme

traveling fronts and wave propagation failure in an inhomogeneous neural network we use averaging and homogenization theory to study the propagation of traveling wavefronts in an inhomogeneous excitable neural medium motivated by the functional architecture of primary visual cortex we model the inhomogeneity as a periodic modulation in the long range neuronal connections we derive an expression for the effective wavespeed and show that propagation failure can occur if the speed is too slow or the degree of inhomogeneity is too large we find that there are major qualitative differences in the wavespeed for different choices of the homogenized weight distribution

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

a simple feature combination method based on dominant sets abstract feature combination is a popular method for improving object classification performances in this paper we present a simple and effective weighting scheme for feature combination based on the dominant set notion of a cluster specifically we use dominant sets clustering to evaluate how accurate a kernel matrix is expected to be for a svm classifier this expected kernel accuracy reflects the discriminative power of the kernel matrix and thus used in weighting the kernel matrix in feature combination our method is simple intuitive memory and computation efficient and performs comparably to the popular and sophisticated optimization based methods we conduct experiments with several datasets of diverse object types and validate the effectiveness of the proposed method in fact in five out of the six datasets used in our experiments we obtained the best results until now in our knowledge

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

multiple bumps in a neuronal model of working memory we study a partial integro differential equation defined on a spatially extended domain that arises from the modeling of working or short term memory in a neuronal network the equation is capable of supporting spatially localized regions of high activity which can be switched on and off by transient external stimuli we analyze the effects of coupling between units in the network showing that if the connection strengths decay monotonically with distance then no more thanon e regionof high activity canpersist whereas if they decay inanoscillatory fashion then multiple regions can persist

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

structure preserving model reduction for spatially interconnected systems with experimental validation on an actuated beam abstracta technique for model reduction of exponentially stable spatially interconnected systems is presented where the order of the reduced model is determined by the number of truncated small generalised singular values of the structured solutions to a pair of lyapunov inequalities for parameter invariant spatially interconnected systems the technique is based on solving a pair of lyapunov inequalities in continuous time and space domain with a rank constraint using log det and cone complementarity methods an improved error bound can be obtained the approach is extended to spatially parameter varying systems and a balanced truncation approach using parameter dependent gramians is proposed to reduce the conservatism caused by the use of constant gramians this is done by considering two important operators which can be used to represent multidimensional systems temporal and spatial linear parameter varying interconnected systems the results are illustrated with their application to an experi

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

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

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

stimulus locked traveling waves and breathers in an excitatory neural network we analyze the existence and stability of stimulus locked traveling waves in a one dimensional synaptically coupled excitatory neural network the network is modeled in terms of a nonlocal integro differential equation in which the integral kernel represents the spatial distribution of synaptic weights and the output firing rate of a neuron is taken to be a heaviside function of activity given an inhomogeneous moving input of amplitude i0 and velocity v we derive conditions for the existence of stimulus locked waves by working in the moving frame of the input we use this to construct existence tongues in v i0 parameter space whose tips at i0 0 correspond to the intrinsic waves of the homogeneous network we then determine the linear stability of stimulus locked waves within the tongues by constructing the associated evans function and numerically calculating its zeros as a function of network parameters we show that as the input amplitude is reduced a stimulus locked wave within the tongue of

coprime factors reduction of distributed nonstationary lpv systems abstractthis paper is on the coprime factors reduction of distributed systems formed by discrete time heterogeneous nonstationary linear parameter varying subsystems the subsystems are represent

brain behaviour and mathematics are we using the right approaches mathematics are used in biological sciences mostly as a quantifying tool for it is the science of numbers after all there is a long standing interest in the application of mathematical methods and concepts to neuroscience in attempts to decipher brain activity while there has been a very wide use of mathematical physical methodologies less effort has been made to formulate a comprehensive and integrative theory of brain function this review concentrates on recent developments uses and abuses of mathematical formalisms and techniques that are being applied in brain research particularly the current trend of using dynamical system theory to unravel the global collective dynamics of brain activity it is worth emphasising that the theoretician neuroscientist eager to apply mathematical analysis to neuronal recordings has to consider carefully some crucial anatomo physiological assumptions that may not be as accurate as the specific methods require on the other hand the experimentalist neuro physicist with an inclination to implement mathematical thoughts in brain science has to make an effort to comprehend the bases of the theoretical concepts that can be used as frameworks or as analysis methods of brain electrophysiological recordings and to critically inspect the accuracy of the interpretations of the results based on the neurophysiological ground it is hoped that this brief overview of anatomical and physiological presumptions and their relation to theoretical paradigms will help clarify some particular points of interest in current trends in brain science and may provoke further reflections on how certain or uncertain it is to conceptualise brain function based on these theoretical frameworks if the physiological and experimental constraints are not as accurate as the models prescribe

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

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

stochastic parameter estimation in nonlinear time delayed vibratory systems with distributed delay abstract the stochastic estimation of parameters and states in linear and nonlinear time delayed vibratory systems with distributed delay is explored the approach consists of first employing a continuous time approximation to approximate the delayed integro differential system with a large set of ordinary differential equations having stochastic excitations then the problem of state and parameter estimation in the resulting stochastic ordinary differential system is represented as an optimal filtering problem using a state augmentation technique by adapting the extended kalman bucy filter to the augmented filtering problem the unknown parameters of the time delayed system are estimated from noise corrupted possibly incomplete measurements of the states similarly the upper bound of the distributed delay can also be estimated by the proposed technique as an illustrative example to a practical problem in vibrations the parameter delay upper bound and state estimation from noise corrupted measurements in a distributed force model widely used for modeling machine tool vibrations in the turning operation is investigated

influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks abstract the main purpose of this paper is to apply mathematical analysis to investigate the influence of sodium currents on the speeds of traveling wave fronts the authors use speed index functions to investigate the speeds of traveling wave fronts of some scalar integral differential equations arising from synaptically coupled neuronal networks the mathematical model equation is u t f u r k x y h u y t 1 c x y d y where 0 c 0 and 0 are constants satisfying the condition 0 2 f the function f u represents sodium currents the function k denotes synaptic coupling in a neuronal network and the heaviside step function h is defined by h u 0 for all u 0 h 0 1 2 and h u 1 for all u 0

noncommuting multidimensional realization theory minimality reachability and observability a generalization of the notions of minimality reachability and observability for parameter dependent and multidimensional systems modeled by linear fractional transformations over noncommuting operators is presented the multidimensional system results developed herein are formally similar to and directly specialize to the standard results when one dimensional systems are considered

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

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

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

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

on the balanced truncation of ltv systems this note furthers results on the balanced truncation of stable linear time varying discrete time systems the main result contributes better error bounds than the currently available ones in certain model reduction scenarios this is demonstrated in an example of a four mass translational system also this note gives new finite error bounds for the balanced truncation of stable eventually periodic models

dominant sets and pairwise clustering we develop a new graph theoretic approach for pairwise data clustering which is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices a notion introduced here which generalizes that of a maximal complete subgraph to edge weighted graphs we establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex thereby allowing the use of straightforward and easily implementable continuous optimization techniques from evolutionary game theory numerical examples on various point set and image segmentation problems confirm the potential of the proposed approach

a novel sequence representation for unsupervised analysis of human activities formalizing computational models for everyday human activities remains an open challenge many previous approaches towards this end assume prior knowledge about the structure of activities using which explicitly defined models are learned in a completely supervised manner for a majority of everyday environments however the structure of the in situ activities is generally not known a priori in this paper we investigate knowledge representations and manipulation techniques that facilitate learning of human activities in a minimally supervised manner the key contribution of this work is the idea that global structural information of human activities can be encoded using a subset of their local event subsequences and that this encoding is sufficient for activity class discovery and classification in particular we investigate modeling activity sequences in terms of their constituent subsequences that we call event n grams exploiting this representation we propose a computational framework to automatically discover the various activity classes taking place in an environment we model these activity classes as maximally similar activity cliques in a completely connected graph of activities and describe how to discover them efficiently moreover we propose methods for finding characterizations of these discovered classes from a holistic as well as a by parts perspective using such characterizations we present a method to classify a new activity to one of the discovered activity classes and to automatically detect whether it is anomalous with respect to the general characteristics of its membership class our results show the efficacy of our approach in a variety of everyday environments

the link between integral equations and higher order odes we study a class of partial integro differential equations defined on a spatially extended domain that arise in the modeling of neuronal networks in previous studies the fourier transform was used to derive associated fourth order ordinary differential equations when the solution of the time independent integral equation is a homoclinic orbit this gives rise to the question of whether solutions of the ode whose fourier transform is not well defined are also solutions of the time independent integral equation we address this question and show that any solution of the ode that satisfies a fairly relaxed growth condition is also a solution of the integral equation applications to two specific examples are given

two bump solutions of a homogenized wilson cowan model with periodic microstructure abstract we study the existence and stability of 2 bump solutions of the one population homogenized wilson cowan model where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale the existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a heaviside function a framework for analyzing the stability of 2 bumps is formulated based on spectral theory for fredholm integral operators the stability method deforms to the standard evans function approach for the translationally invariant case in the limit of no heterogeneity in a way analogous to the single bump case for the homogenized model the numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function for the damped oscillating connectivity kernel we give a concrete example of a 2 bump solution which is stable for all admissible values of the heterogeneity parameter

towards parameter independent data clustering and image segmentation while there are a large amount of clustering algorithms proposed in the literature the clustering results of existing algorithms usually depend on user specified parameters heavily and it is usually difficult to determine the optimal parameters with the pairwise data similarity matrix as the input dominant sets clustering has been shown to be an effective data clustering and image segmentation approach partly due to its ability to find out the underlying data structure and determine the number of clusters automatically however we find that the original dominant sets algorithm is sensitive to the similarity measures used in building the similarity matrix this means that parameter tuning is required to generate satisfactory clustering results and dominant sets clustering results are also parameter dependent in order to remove the dependence on the user specified parameter we study how the similarity measures influence the dominant sets clustering results as a result we propose to transform similarity matrices by histogram equalization before clustering while this transformation is shown to remove the sensitiveness to similarity measures effectively it also results in over segmentation therefore in the next step we present a cluster extension method to overcome the over segmentation effect and generate more reasonable clustering results we test the enhanced clustering algorithm in both data clustering and image segmentation experiments and comparisons with the state of the art algorithms validate the effectiveness of our algorithm highlightswe study how similarity measures influence the dominant sets clustering results we use histogram equalization to remove the dependence on similarity parameters we present a density based cluster extension method to overcome over segmentation experiments validate the effectiveness of our algorithm

stabilization of pulse waves through inhibition in a feedforward neural network we consider a firing rate model of a neural network of excitatory and inhibitory populations with an excitatory feedforward connectivity we analyze traveling wave solutions and determine the conditions for their existence and stability our study demonstrates the role of inhibition in stable pulse propagation in a purely excitatory network pulse waves are unstable because of the existence of stable front wave and back wave with different velocities pulse waves can propagate stably in the network where excitation is appropriately balanced by inhibition analytical results on the wave speeds and the shape of waves are obtained

robust path based spectral clustering spectral clustering and path based clustering are two recently developed clustering approaches that have delivered impressive results in a number of challenging clustering tasks however they are not robust enough against noise and outliers in the data in this paper based on m estimation from robust statistics we develop a robust path based spectral clustering method by defining a robust path based similarity measure for spectral clustering under both unsupervised and semi supervised settings our proposed method is significantly more robust than spectral clustering and path based clustering we have performed experiments based on both synthetic and real world data comparing our method with some other methods in particular color images from the berkeley segmentation data set and benchmark are used in the image segmentation experiments experimental results show that our method consistently outperforms other methods due to its higher robustness

the evolution of synaptically generated waves in one and two dimensional domains we study the onset and evolution of waves in an integrate and fire network of synaptically coupled neurons in one and two dimensional domains which are restricted to firing one spike we determine the critical size of initial excitation of the network necessary for the onset of propagation we also determine the ignition time we derive an integro differential equation for the evolution of the firing time as a function of spatial position we use the evolution equation to understand propagation failure we compare the behavior of the integrate and fire model with a biophysically based model finally we show that weak heterogeneity has similarly weak effects on the firing times

on the controllability and observability of networked dynamic systems some necessary and sufficient conditions are obtained for the controllability and observability of a networked system with linear time invariant lti dynamics the topology of this system is fixed but arbitrary and every subsystem is permitted to have different dynamics these conditions essentially depend only on transmission zeros of every subsystem and the subsystem connection matrix which makes them attractive in the analysis and synthesis of a large scale networked system as an application these conditions are utilized to characterize systems whose steady estimation accuracy with the distributed predictor of zhou 2013 is equal to that of the lumped kalman filter it has been made clear that to guarantee this equivalence the steady update gain matrix of the kalman filter must be block diagonal

numerical analysis of bump solutions for neural field equations with periodic microstructure frameworks for studying nonexistence of y dependent solutions using a bumps width technique frameworks for studying nonexistence of y dependent solutions using iteration schemes and two scale convergence technique frameworks for studying and constructing of single bump solutions of homogenized equation we study numerically single bump solutions of a homogenized amari equation with periodic microvariation two attempts are made to detect single bumps that depend on the microvariable the first attempt which is based on a pinning function technique is applicable in the heaviside limit of the firing rate function in the second attempt we develop a numerical scheme which combines the two scale convergence theory and an iteration procedure for the corresponding heterogeneous amari equation the numerical simulations in both attempts indicate the nonexistence of single bump solutions that depend on the microvariable motivated by this result we finally develop a fixed point iteration scheme for the construction of single bump solutions that are independent of the microvariable when the firing rate function is given by a sigmoidal firing rate function

natural image segmentation based on tree equipartition bayesian flooding and region merging we propose a general purpose image segmentation framework which involves feature extraction and classification in feature space followed by flooding and merging in spatial domain region growing is based on the computed local measurements and distances from the distribution of features describing the different classes using the properties of the label dependent distances spatial coherence is ensured since the image features are described globally the distribution of the features for the different classes are obtained by block wise unsupervised clustering based on the construction of the minimum spanning tree of the blocks grid using the mallows distance and the equipartition of the resulting tree the final clustering is obtained by using the k centroids algorithm with high probability and under topological constraints connected components of the maximum likelihood classification map are used to compute a map of initially labelled pixels an efficient flooding algorithm is introduced namely priority multi class flooding algorithm pmcfa that assign pixels to labels using bayesian dissimilarity criteria a new region merging method which incorporates boundary information is introduced for obtaining the final segmentation map therefore the merging stage is based on region features and edge localization segmentation results on the berkeley benchmark data set demonstrate the effectiveness of the proposed methods

absolute stability and synchronization in neural field models with transmission delays abstract neural fields model macroscopic parts of the cortex which involve several populations of neurons we consider a class of neural field models which are represented by integro differential equations with transmission time delays which are space dependent the considered domains underlying the systems can be bounded or unbounded a new approach called sequential contracting instead of the conventional lyapunov functional technique is employed to investigate the global dynamics of such systems sufficient conditions for the absolute stability and synchronization of the systems are established several numerical examples are presented to demonstrate the theoretical results

snakes and ladders in an inhomogeneous neural field model continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity we analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes and ladders bifurcation structure in the case of heaviside firing rates we construct analytically symmetric and asymmetric states and hence derive closed form expressions for the corresponding bifurcation diagrams we show that the ideas proposed by beck and co workers to analyze snaking solutions to the swift hohenberg equation remain valid for the neural field model even though the corresponding spatial dynamical formulation is non autonomous we investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the heaviside limit

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

ltv model reduction with upper error bounds a new approach for computing upper error bounds for reduced order models of linear time varying systems is presented it is based on a transformation technique of the hankel singular values using positive real odd incremented functions by applying such time varying functions the singular values to be removed can be forced to become equal and constant so that they can be reduced two variations of this method are proposed one for finite time horizons and the other for infinite time problems including periodic systems

parameter independent clustering based on dominant sets and cluster merging abstract clustering is an important unsupervised learning approach with wide application in data mining pattern recognition and intelligent information processing however existing clustering algorithms usually involve one or more user specified parameters as input and their clustering results depend heavily on these parameters in order to solve this problem we present a parameter independent clustering algorithm based on the dominant sets algorithm and cluster merging in the first step histogram equalization transformation is applied to solve the parameter dependence problem of the dominant sets algorithm we provide the theoretic foundation of this method and discuss the implementation details the clustering result is then refined with a cluster merging method which is based on a new clustering quality evaluation criterion we use extensive experiments on several datasets to validate each step and the whole procedures of our algorithm it is shown that our parameter independent algorithm performs comparably to some existing clustering algorithms which benefit from user specified parameters

stimulus driven traveling solutions in continuum neuronal models with a general smooth firing rate function we examine the existence of traveling wave solutions for a continuum neuronal network modeled by integro differential equations first we consider a scalar field model with a general smooth firing rate function and a spatiotemporally varying stimulus we prove that a traveling front solution that is locked to the stimulus exists for a certain interval of stimulus speeds next we include a slow adaptation equation and obtain a formula which involves a certain adjoint solution for the stimulus speeds that induce locked traveling pulse solutions further we use singular perturbation analysis to characterize an approximation to the adjoint solution that we compare to a numerically computed adjoint numerical simulations are used to illustrate the traveling fronts and pulses that we study and to make comparisons with our analytically computed bounds for stimulus locked wave behavior

distributed control of spatially reversible interconnected systems with boundary conditions we present a class of spatially interconnected systems with boundary conditions that have close links with their spatially invariant extensions in particular well posedness stability and performance of the extension imply the same characteristics for the actual finite extent system in turn existing synthesis methods for control of spatially invariant systems can be extended to this class the relation between the two kinds of systems is proved using ideas based on the method of images of partial differential equations theory and uses symmetry properties of the interconnection as a key tool

neural gas network for vector quantization and its application to time series prediction a neural network algorithm based on a soft max adaptation rule is presented this algorithm exhibits good performance in reaching the optimum minimization of a cost function for vector quantization data compression the soft max rule employed is an extension of the standard k means clustering procedure and takes into account a neighborhood ranking of the reference weight vectors it is shown that the dynamics of the reference weight vectors during the input driven adaptation procedure are determined by the gradient of an energy function whose shape can be modulated through a neighborhood determining parameter and resemble the dynamics of brownian particles moving in a potential determined by the data point density the network is used to represent the attractor of the mackey glass equation and to predict the mackey glass time series with additional local linear mappings for generating output values the results obtained for the time series prediction compare favorably with the results achieved by backpropagation and radial basis function networks

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

distributed control design for spatially interconnected systems this paper deals with analysis synthesis and implementation of distributed controllers designed for spatially interconnected systems we develop a state space framework for posing problems of this type and focus on systems whose model is spatially discrete in this paper analysis and synthesis results are developed for this class of systems using the l sub 2 induced norm as the performance criterion the results are stated in terms of linear matrix inequalities and are thus readily amenable to computation a special implementation of the resulting controllers is presented which is particularly attractive for distributed operation of the controller several examples are provided to further illustrate the application of the results

continuation of localised coherent structures in nonlocal neural field equations we study localised activity patterns in neural field equations posed on the euclidean plane such models are commonly used to describe the coarse grained activity of large ensembles of cortical neurons in a spatially continuous way we employ matrix free newton krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation this opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent pde formulation we present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking the regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function the results presented form a basis for the general study of localised cortical activity with inputs and more specifically for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input

robust stability analysis of sparsely interconnected uncertain systems in this paper we consider robust stability analysis of large scale sparsely interconnected uncertain systems by modeling the interconnections among the subsystems with integral quadratic constraints we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities we also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness the sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem

coprime factors model reduction of spatially distributed ltv systems over arbitrary graphs this technical note is on the model reduction of distributed systems formed by discrete time linear time varying heterogeneous subsystems interconnected over arbitrary directed graphs and subjected to communication latency we give two procedures to construct a strongly stable coprime factorization for a strongly stabilizable and strongly detectable system one of the procedures ensures the contractiveness of the resulting factorization then we apply the structure preserving balanced truncation method for distributed systems we illustrate the proposed methods through an example

contour based object detection as dominant set computation contour based object detection can be formulated as a matching problem between model contour parts and image edge fragments we propose a novel solution by treating this problem as the problem of finding dominant sets in weighted graphs the nodes of the graph are pairs composed of model contour parts and image edge fragments and the weights between nodes are based on shape similarity because of high consistency between correct correspondences the correct matching corresponds to a dominant set of the graph consequently when a dominant set is determined it provides a selection of correct correspondences as the proposed method is able to get all the dominant sets we can detect multiple objects in an image in one pass moreover since our approach is purely based on shape we also determine an optimal scale of target object without a common enumeration of all possible scales both theoretic analysis and extensive experimental evaluation illustrate the benefits of our approach

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

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

a necessary and sufficient minimality condition for uncertain systems a necessary and sufficient condition is given for the exact reduction of systems modeled by linear fractional transformations lfts on structured operator sets this condition is based on the existence of a rank deficient solution to either of a pair of linear matrix inequalities which generalize lyapunov equations the notion of gramians is thus also generalized to uncertain systems as well as kalman like decomposition structures a related minimality condition the converse of the reducibility condition may then be inferred from these results and the equivalence class of all minimal lft realizations defined these results comprise the first stage of a complete generalization of realization theory concepts to uncertain systems subsequent results such as the definition of and rank tests on structured controllability and observability matrices are also given the minimality results described are applicable to multidimensional system realizations as well as to uncertain systems connections to formal powers series representations also exist

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

induced l 2 norm model reduction of polytopic uncertain linear systems abstract in this paper we study an induced l 2 norm model reduction problem for polytopic uncertain linear systems the polytopic system has its state space data contained in a convex polytope a situation which often arises the problem we address is to approximate this type of system by a lower order polytopic uncertain linear system with a guaranteed induced l 2 norm error a sufficient solvability condition is provided in terms of lmis with one extra coupling rank constraint which generally leads to a nonconvex feasibility problem we use an improved computational scheme based on the alternating projection method to obtain solutions to the problem but only local convergence is guaranteed the reduced order uncertain system with its associated polytope is given explicitly by the solution of the feasibility problem