the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

nearest neighbor classification with dependent training sequences the asymptotic classification risk for nearest neighbor procedures is well understood in the case of i i d training sequences in this article we generalize these results to a class of dependent models including hidden markov models in the case where the observed patterns have lebesgue densities the asymptotic risk takes the same expression as in the i i d case for discrete distributions we show that the asymptotic risk depends on the rule used for breaking ties of equal distances

connected simple systems transition matrices and heteroclinic bifurcations given invariant sets a b and c and connecting orbits a b and b c we state very general conditions under which they bifurcate to produce an a c connecting orbit in particular our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits and for which the conley index of the invariant sets is the same as that of a hyperbolic critical point our proof depends on the connected simple system associated with the conley index for isolated invariant sets

finite size effects in a field theoretic model with long range exchange interaction we present a systematic approach to the calculation of finite size fs effects for ano n field theoretic model with both short range sr and long range lr exchange interactions the lr exchange interaction decays at large distances as 1 rd 2 2 0 0 renormalization group calculations ind du e are performed for a system with a fully finite block geometry under periodic boundary conditions we calculate the fs shift of the critical temperature and the fs renormalized coupling constant of the model to one loop order the universal scaling variable is obtained and the fs scaling hypothesis is verified

directed percolation in a two dimensional stochastic fire diffuse fire model in this paper we establish from extensive numerical experiments that the two dimensional stochastic fire diffuse fire model belongs to the directed percolation universality class this model is an idealized model of intracellular calcium release that retains the both the discrete nature of calcium stores and the stochastic nature of release it is formed from an array of noisy threshold elements that are coupled only by a diffusing signal the model supports spontaneous release events that can merge to form spreading circular and spiral waves of activity the critical level of noise required for the system to exhibit a non equilibrium phase transition between propagating and non propagating waves is obtained by an examination of the textit local slope delta t of the survival probability pi t propto exp delta t for a wave to propagate for a time t

the effect of noise on β cell burst period bursting electrical behavior is commonly observed in a variety of nerve and endocrine cells including that in electrically coupled cells located in intact pancreatic islets however individual cells usually display either spiking or very fast bursting behavior and the difference between isolated and coupled cells has been suggested to be due to stochastic fluctuations of the plasma membrane ion channels which are supposed to have a stronger effect on single cells than on cells situated in clusters the channel sharing hypothesis this effect of noise has previously been studied using numerical simulations we show here how the application of two recent methods allows an analytic treatment of the stochastic effects on the location of the saddle node and homoclinic bifurcations which determine the burst period thus the stochastic system can be analyzed similarly to the deterministic system but with a quantitative description of the effect of noise this approach supports previous investigations

choice of neighbor order in nearest neighbor classification the kth nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure however application of this method is inhibited by lack of knowledge about its properties in particular about the manner in which it is influenced by the value of k and by the absence of techniques for empirical choice of k in the present paper we detail the way in which the value of k determines the misclassification error we consider two models poisson and binomial for the training samples under the first model data are recorded in a poisson stream and are assigned to one or other of the two populations in accordance with the prior probabilities in particular the total number of data in both training samples is a poisson distributed random variable under the binomial model however the total number of data in the training samples is fixed although again each data value is assigned in a random way although the values of risk and regret associated with the poisson and binomial models are different they are asymptotically equivalent to first order and also to the risks associated with kernel based classifiers that are tailored to the case of two derivatives these properties motivate new methods for choosing the value of k

field theory of directed percolation with long range spreading it is well established that the phase transition between survival and extinction in spreading models with short range interactions is generically associated with the directed percolation dp universality class in many realistic spreading processes however interactions are long ranged and well described by l evy flights char22 i e by a probability distribution that decays in d dimensions with distance r as r ensuremath d ensuremath ensuremath sigma we employ the powerful methods of renormalized field theory to study dp with such long range l evy flight spreading in some depth our results unambiguously corroborate earlier findings that there are four renormalization group fixed points corresponding to respectively short range gaussian l evy gaussian short range and l evy dp and that there are four lines in the ensuremath sigma d plane which separate the stability regions of these fixed points when the stability line between short range dp and l evy dp is crossed all critical exponents change continuously we calculate the exponents describing l evy dp to second order in an ensuremath epsilon expansion and we compare our analytical results to the results of existing numerical simulations furthermore we calculate the leading logarithmic corrections for several dynamical observables

driven diffusive system with non local perturbations we investigate the impact of non local perturbations on driven diffusive systems two different problems are considered here in one case we introduce a non local particle conservation along the direction of the drive and in another case we incorporate a long range temporal correlation in the noise present in the equation of motion the effect of these perturbations on the anisotropy exponent or on the scaling of the two point correlation function is studied using renormalization group analysis

a period adding phenomenon a simple geometrical explanation is given for the period adding phenomenon observed in the 1927 experiment of van der pol and van der mark and refined recently by chua and kennedy the problem is reduced to a simple circle map in the second part of the paper some previous results are outlined to show how the same phenomenon arises in a class of van der pol type systems with periodic forcing and in conclusion some recent topological results are applied to such systems

properties of a bursting model with two slow inhibitory variables models for certain excitable cells such as the pancreatic beta cell must reproduce bursting oscillations of the membrane potential this has previously been done using one slow variable to drive bursts the dynamics of such models have been analyzed however new models for the beta cell often include additional slow variables and therefore the previous analysis is extended to two slow variables using a simplified version of a beta cell model some unusual time courses of this model motivated a geometric singular perturbation analysis and the application of averaging to reduce the dynamics to the slow variable phase plane a geometric understanding of the solutionstructure and of transitions between various modes of behavior was then developed a novel use of the bifurcation code auto finds nullclines for the slow variables when the fast variables are periodic by averaging over the fast oscillations in contrast with the parabolic neuronal burster this model requires bistability in th

on the inequality of cover and hart in nearest neighbor discrimination when x1 1 xn n are independent identically distributed random vectors from ird x 0 1 distributed as x and when is estimated by its nearest neighbor estimate 1 then cover and hart have shown that p 1 n 2e x 1 x 2r 1 r where r is the bayes probability of error and x p 1 x x they have conditions on the distribution of x we give two proofs one due to stone and a short original one of the same result for all distributions of x if ties are carefully taken care of we also show that p 1 x1 1 xn n converges in probability to a constant for all distributions of x thereby strengthening results of wagner and fritz

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

renormalisation of random walks with memory the renormalisation of field theoretic models of the true self avoiding random walk is analysed for short range interactions the field theory is shown to be renormalisable only in a special case which corresponds to the problem of a random walk in a random environment and non renormalisable otherwise the long range version of the field theory is shown to be renormalisable in a larger region of the parameter space in the case which corresponds to the original long range true self avoiding random walk the renormalisation group analysis is carried out to two loop order anomalous dimensions of various composite operators are calculated and the crossover between scaling regimes described by short range and long range models is discussed

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

two dimensional sir epidemics with long range infection we extend a recent study of susceptible infected removed epidemic processes with long range infection referred to as i in the following from 1 dimensional lattices to lattices in two dimensions as in i we use hashing to simulate very large lattices for which finite size effects can be neglected in spite of the assumed power law p x x 2 for the probability that a site can infect another site a distance vector x apart as in i we present detailed results for the critical case for the supercritical case with 2 and for the supercritical case with 0 2 for the latter we verify the stretched exponential growth of the infected cluster with time predicted by m biskup for 2 we find generic power laws with dependent exponents in the supercritical phase but no kosterlitz thouless kt like critical point as in 1 d instead of diverging exponentially with the distance from the critical point the correlation length increases with an inverse power as in an ordinary critical point finally we study the dependence of the critical exponents on in the regime 0 2 and compare with field theoretic predictions in particular we discuss in detail whether the critical behavior for slightly less than 2 is in the short range universality class as conjectured recently by f linder et al as in i we also consider a modified version of the model where only some of the contacts are long range the others being between nearest neighbors if the number of the latter reaches the percolation threshold the critical behavior is changed but the supercritical behavior stays qualitatively the same

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

fast subsystem bifurcations in a slowly varying lie nard system exhibiting bursting a perturbed lienard differential system is examined using local stability and hopf bifurcation analyses asymptotic techniques and melnikov s method the results of these analyses are applied to a simple cubic model that exhibits a variety of different oscillatory behaviors for different parameter values for a bounded region in fast parameter space the model exhibits square wave bursting patterns analogous to the bursting electrical activity observed in pancreatic beta cells under certain hypotheses solutions of the cubic model are known to have square wave patterns by using the theory developed for the more general lienard system each hypothesis is shown to correspond to a curve in parameter space together the curves bound a region in which the model exhibits square wave bursting patterns since the model is simple the curves that bound this region can all be determined analytically

voltage interval mappings for activity transitions in neuron models for elliptic bursters abstract we performed a thorough bifurcation analysis of a mathematical elliptic bursting model using a computer assisted reduction to equationless one dimensional poincare mappings for a voltage interval using the interval mappings we were able to examine in detail the bifurcations that underlie the complex activity transitions between tonic spiking and bursting bursting and mixed mode oscillations and finally mixed mode oscillations and quiescence in the fitzhugh nagumo rinzel model we illustrate the wealth of information qualitative and quantitative that was derived from the poincare mappings for the neuronal models and for similar electro chemical systems

critical behaviour of the long range ϕ2 2 model in the short range limit the relation between scaling regimes described by 2 2 field theories with long range and short range exchange is analysed it is shown that the crossover from the long range regime to the short range regime takes place due to the singular behaviour of three loop and higher order contributions to the beta function of the renormalization group which controls the scaling behaviour of the long range model the critical exponents eta and nu are shown to be continuous functions of the parameter alpha which characterises the power like fall off of the exchange interaction

the slow passage through a hopf bifurcation delay memory effects and resonance this paper explores analytically and numerically in the context of the fitzhugh nagumo model of nerve membrane excitability an interesting phenomenon that has been described as a delay or memory effect it can occur when a parameter passes slowly through a hopf bifurcation point and the system s response changes from a slowly varying steady state to slowly varying oscillations on quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation this delay and its dependence on the speed of the parameter variation are described the model involves several parameters and particular singular limits are investigated one in particular is the slow passage through a low frequency hopf bifurcation where the system s response changes from a slowly varying steady state to slowly varying relaxation oscillations we find in this case the onset o

neighborhood size selection in the k nearest neighbor rule using statistical confidence the k nearest neighbor rule is one of the most attractive pattern classification algorithms in practice the choice of k is determined by the cross validation method in this work we propose a new method for neighborhood size selection that is based on the concept of statistical confidence we define the confidence associated with a decision that is made by the majority rule from a finite number of observations and use it as a criterion to determine the number of nearest neighbors needed the new algorithm is tested on several real world datasets and yields results comparable to the k nearest neighbor rule however in contrast to the k nearest neighbor rule that uses a fixed number of nearest neighbors throughout the feature space our method locally adjusts the number of nearest neighbors until a satisfactory level of confidence is reached in addition the statistical confidence provides a natural way to balance the trade off between the reject rate and the error rate by excluding patterns that have low confidence levels we believe that this property of our method can be of great importance in applications where the confidence with which a decision is made is equally or more important than the overall error rate

choice of the smoothing parameter and efficiency of k nearest neighbor classification abstract a simulation study was performed to investigate the sensitivity of the k nearest neighbor nn k rule of classification to the choice of k the optimal choice of k was found to be a function of the dimension of the sample space the size of the space the covariance structure and the sample proportions the nearest neighbor rules chosen using the k suggested by the simulations had correct classification rates at least as high as those rates for the linear discriminant function and the logistic regression method in particular the rule became more efficient as the difference in the covariance matrices increased and also when the difference in sample proportion was large an adaptive rule which selects k by iteratively maximizing the local mahalanobis distance is shown to be efficient thus abrogating the need to know the underlying population variance covariance structure

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

multiclass classification with potential function rules margin distribution and generalization motivated by the potential field of static electricity a binary potential function classifier views each training sample as an electrical charge positive or negative according to its class label the resulting potential field divides the feature space into two decision regions based on the polarity of the potential in this paper we revisit potential function classifiers in their original form and reveal their connections with other well known results in the literature we derive a bound on the generalization performance of multiclass potential function classifiers based on the observed margin distribution of the training data a new model selection criterion using a normalized margin distribution is then proposed to learn good potential function classifiers in practice

multimodal regimes in a compartmental model of the dopamine neuron we study chains of strongly electrically coupled relaxation oscillators modeling dopamine neurons when individual oscillators are in the regime close to an andronov hopf bifurcation ahb the coupled system exhibits a variety of oscillatory behavior we show that the proximity of individual oscillators to the ahb has a significant impact on the system dynamics in a wide range of parameters it manifests itself through a family of stable multimodal periodic solutions that are composed out of large amplitude relaxation oscillations and small amplitude oscillations this family of solutions has a rich bifurcation structure the waveform and the period vary greatly across the family the structure and bifurcations of the stable periodic solutions of the coupled system are investigated using numerical and analytic techniques

fractional langevin equations of distributed order distributed order fractional langevin like equations are introduced and applied to describe anomalous diffusion without unique diffusion or scaling exponent it is shown that these fractional langevin equations of distributed order can be used to model the kinetics of retarding subdiffusion whose scaling exponent decreases with time and the strongly anomalous ultraslow diffusion with mean square displacement which varies asymptotically as a power of logarithm of time

on bifurcations of the lorenz attractor in the shimizu morioka model abstract bifurcations of lorenz like attractors are studied in the shimizu morioka model two scenarios for their destruction are suggested the first is connected with the appearance of a lacuna the second with a transition to a quasi attractor the effects of codimension two bifurcations are considered

bayes error estimation using parzen and k nn procedures the use of k nearest neighbor k nn and parzen density estimates to obtain estimates of the bayes error is investigated under limited design set conditions by drawing analogies between the k nn and parzen procedures new procedures are suggested and experimental results are given which indicate that these procedures yield a significant improvement over the conventional k nn and parzen procedures we show that by varying the decision threshold many of the biases associated with the k nn or parzen density estimates may be compensated and successful error estimation may be performed in spite of these biases experimental results are given which demonstrate the effect of kernel size and shape parzen the size of k k nn and the number of samples in the design set

crossover between field theories with short range and long range exchange or correlations the relation between scalar field theories with short range and long range exchange or correlations is studied it is shown to all orders in perturbation theory that the critical exponents of fields and composite operators are continuous functions of the parameters alpha xi characterising the decay rate of the long range exchange correlations with power like falloff 1 rd 2 2 alpha 1 rd 2 xi the scaling law d 2 xi v 2 where v is the correlation length exponent as well as the harris criteria 1970 are shown to be exact to all orders in perturbation theory a discrepancy between two widely used approaches to the crossover problem is pointed out

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

directed percolation with incubation times we introduce a model for directed percolation with a long range temporal diffusion while the spatial diffusion is kept short ranged in an interpretation of directed percolation as an epidemic process this non markovian modification can be understood as incubation times which are distributed accordingly to a levy distribution we argue that the best approach to find the effective action for this problem is through a generalization of the cardy sugar method adding the non markovian features into the geometrical properties of the lattice we formulate a field theory for this problem and renormalize it up to one loop in a perturbative expansion we solve the various technical difficulties that the integrations possess by means of an asymptotic analysis of the divergences we show the absence of field renormalization at one loop order and we argue that this would be the case to all orders in perturbation theory consequently in addition to the characteristic scaling relations of directed percolation we find a scaling relation valid for the critical exponents of this theory in this universality class the critical exponents vary continuously with the levy parameter

synchronization of elliptic bursters periodic bursting behavior in neurons is a recurrent transition between a quiescent state and repetitive spiking when the transition to repetitive spiking occurs via a subcritical andronov hopf bifurcation and the transition to the quiescent state occurs via fold limit cycle bifurcation the burster is said to be of elliptic type also known as a subhopf fold cycle burster here we study the synchronization dynamics of weakly connected networks of such bursters we find that the behavior of such networks is quite different from the behavior of weakly connected phase oscillators and resembles that of strongly connected relaxation oscillators as a result such weakly connected bursters need few usually one bursts to synchronize and synchronization is possible for bursters having quite different quantitative features we also find that interactions between bursters depend crucially on the spiking frequencies namely the interactions are most effective when the presynaptic interspike frequency matches the frequency of postsynaptic oscillations finally we use the fitzhugh rinzel morris lecar and hodgkin huxley models to illustrate our major results

the existence of periodic travelling waves for singularly perturbed predator prey equations via the conley index this paper is concerned with the existence of periodic travelling wave solutions of reactiondiffusion equations which arise as models of predator prey interactions in mathematical ecology the relevant equations are somewhat similar to the fitz hugh nagumo equations and our results apply to this latter system as well in studying the fitz hugh nagumo equations conley 3 i initiated a new topological approach to this problem his method depends on the construction of an isolating neighborhood cf section 2a about a singular periodic solution associated with such a neighborhood is a topological invariant the conley index which when nontrivial implies the existence of a smooth solution near the singular one the new aspects of the present discussion are in the method used to construct the isolating neighborhood in ifi4 and in the computation of the index with regard to the isolating neighborhood our construction is global in that a parameter taken to be small in conley s equations is now allowed to assume large values in order to compute the index we continue our equations to those studied by conley actually we go one step further and continue conley s equations to the van der pol equations crossed with a pair of linear equations admitting a repelling critical point the index for this latter system is easily computed and turns out to be non trivial it follows from the invariance of the index under continuation 21 that the desired

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

spike adding in parabolic bursters the role of folded saddle canards abstract the present work develops a new approach to studying parabolic bursting and also proposes a novel four dimensional canonical and polynomial based parabolic burster in addition to this new polynomial system we also consider the conductance based model of the aplysia r15 neuron known as the plant model and a reduction of this prototypical biophysical parabolic burster to three variables including one phase variable namely the baer rinzel carillo brc phase model revisiting these models from the perspective of slow fast dynamics reveals that the number of spikes per burst may vary upon parameter changes however the spike adding process occurs in an explosive fashion that involves special solutions called canards this spike adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques we find that the bifurcation structure persists across all considered systems that is spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit namely the true canard of a folded saddle singularity however there can be a difference in the spike adding transitions in parameter space from one case to another according to whether the process is continuous or discontinuous which depends upon the geometry of the folded saddle canard using these findings we construct a new polynomial approximation of the plant model which retains all the key elements for parabolic bursting including the spike adding transitions mediated by folded saddle canards finally we briefly investigate the presence of spike adding via canards in planar phase models of parabolic bursting namely the theta model by ermentrout and kopell

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

on the asymptotic properties of smoothed estimators of the classifiction error rate abstract a smoothed correction of the classification error rate counting estimate is examined the consistency of the estimate under very general conditions is established for normal distributions asymptotic theory reveals that the relative deficiency of the counting estimate with respect to the smoothed one can tend to infinity as the sample size increases

homoclinic chaos in chemical systems abstract we first focus on complex dynamical phenomena generated by chemical kinetics in homogeneous media we comment on the alternating sequences of periodic and chaotic states observed in some experiments on the belousov zhabotinsky reaction when conducted in a continuously stirred tank reactor we present numerical results obtained with a seven variable oregonator model which reproduce most of the features of the experimental sequences we discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity we construct poincare map models which reduce to either multi humped or multi branched 1d maps in the strong area contraction limit we report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of spiral type strange attractors satistifies the symbolic dynamics predicted by sil nikov theory of homoclinic chaos we then investigate spatio temporal pattern forming phenomena in a one dimensional reaction diffusion system with equal diffusion coefficients when imposing a concentration gradient through the system this model mimics the sustained stationary and periodically oscillating front structures observed in recent experiments conducted in an open couette flow reactor we emphasize the possibility that the oscillations of the spatial structure become chaotic we report on numerical simulations of a diffusion induced intermittent bursting phenomenon that is likely to be observed in bench experiments we elaborate on the interpretation of this intermittent occurrence of spatially localized structures in an extended system in terms of sil nikov s homoclinic chaos

critical dynamics in systems controlled by fractional kinetic equations abstract the field theory renormalization group is used for analyzing the fractional langevin equation with the order of the temporal derivative 0 1 fractional laplacian of the order and gaussian noise correlator the case of non linearity m with odd m 3 is considered it is proved that the model is multiplicatively renormalizable propagators were found in the momentum and coordinate representation expressed in terms of fox s h functions existence of the dissipative scaling regime in the framework of the e expansion for 2 1 l l 1 2 is proved requirement of the continuous dependence of the critical exponents on imposes the condition m 3 the main quantitative result is the calculation of the dynamical critical exponent z for 1 2 up to e 2 we have obtained for it the expression z 1 2 4 0 1555 e 2 o e 3

on the rate of convergence of local averaging plug in classification rules under a margin condition the rates of convergence of plug in kernel partitioning and nearest neighbors classification rules are analyzed a margin condition which measures how quickly the a posteriori probabilities cross the decision boundary smoothness conditions on the a posteriori probabilities and boundedness of the feature vector are imposed the rates of convergence of the plug in classifiers shown in this paper are faster than previously known

pattern recognition for conditionally independent data in this work we consider the task of relaxing the i i d assumption in pattern recognition or classification aiming to make existing learning algorithms applicable to a wider range of tasks pattern recognition is guessing a discrete label of some object based on a set of given examples pairs of objects and labels we consider the case of deterministically defined labels traditionally this task is studied under the assumption that examples are independent and identically distributed however it turns out that many results of pattern recognition theory carry over a weaker assumption namely under the assumption of conditional independence and identical distribution of objects while the only assumption on the distribution of labels is that the rate of occurrence of each label should be above some positive threshold we find a broad class of learning algorithms for which estimations of the probability of the classification error achieved under the classical i i d assumption can be generalized to the similar estimates for case of conditionally i i d examples

on the asymptotic probability of error in nonparametric discrimination let x y x 1 y 1 cdots x n y n be independent identically distributed random vectors from r d times 0 1 and let hat y be the k nearest neighbor estimate of y from x and the x i y i s we show that for all distributions of x y the limit of l n p hat y neq y exists and satisfies lim n rightarrow infty l n leq 1 a k r ast a k frac alpha sqrt k k 3 25 big 1 frac beta sqrt k 3 big k text odd k geq 5 where r ast is the bayes probability of error and alpha 0 3399 cdots and beta 0 9749 cdots are universal constants this bound is shown to be best possible in a certain sense

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

complex dynamics of a single neuron model we study a mathematical model of a single neuron with self coupling the model is based on the fitzhugh nagumo oscillator and an equation describing synaptic properties of the neuron the analysis of the model is focused on its dynamics depending on parameters characterizing synaptic time constants and external signals that affect the neuron applying lyapunov exponents and bifurcation analysis we point out the occurrence of parameter regions with different behavior such as bursting chaotic or periodic spiking and multistable phenomena moreover we can describe the dynamics of the model using an analytical approximation of the one dimensional poincar e map extracted from the numerical simulations

convergence rate of the fuzzy generalized nearest neighbor rule tizzy k nearest neighbor rule k nnr has been applied in a variety of substantive areas yang and chen l described a fuzzy generalized k nn algorithm which is a unified approach to a variety of fuzzy k nnr s they created the strong consistency of posterior risk of the fuzzy generalized nnr in this paper we give their convergence rate that is the convergence rate of posterior risk of the fuzzy generalized nnr is exponentially fast

the nearest neighbor and the bayes error rates the k l nearest neighbor method of pattern classification is compared to the bayes method if the two acceptance rates are equal then the asymptotic error rates satisfy the inequalities ek l 1 e ek l de where d is a function of k l and the number of pattern classes and is the reject threshold for the bayes method an explicit expression for d is given which is optimal in the sense that for some probability distributions ek l and de are equal

bias of nearest neighbor error estimates the bias of the finite sample nearest neighbor nn error from its asymptotic value is examined expressions are obtained which relate the bias of the nn and 2 nn errors to sample size dimensionality metric and distributions these expressions isolate the effect of sample size from that of the distributions giving an explicit relation showing how the bias changes as the sample size is increased experimental results are given which suggest that the expressions accurately predict the bias it is shown that when the dimensionality of the data is high it may not be possible to estimate the asymptotic error simply by increasing the sample size a new procedure is suggested to alleviate this problem this procedure involves measuring the mean nn errors at several sample sizes and using our derived relationship between the bias and the sample size to extrapolate an estimate of the asymptotic nn error the results are extended to the multiclass problem the choice of an optimal metric to minimize the bias is also discussed

bursting oscillations induced by small noise we consider a model of a square wave bursting neuron residing in the regime of tonic spiking upon introduction of small stochastic forcing the model generates irregular bursting the statistical properties of the emergent bursting patterns are studied in the present work in particular we identify two principal statistical regimes associated with the noise induced bursting in the first case type i bursting oscillations are created mainly due to the fluctuations in the fast subsystem in the alternative scenario type ii bursting the random perturbations in the slow dynamics play a dominant role we propose two classes of randomly perturbed slow fast systems that realize type i and type ii scenarios for these models we derive the poincare maps the analysis of the linearized poincare maps of the randomly perturbed systems explains the distributions of the number of spikes within one burst and reveals their dependence on the small and control parameters present in the models the mathematical analy

zeta functions periodic trajectories and the conley index abstract we use the conley index to produce a simple sufficient condition stated in terms of the betti numbers of the cohomology conley index for the existence of periodic orbits for flows which admit a cross section

hammerstein system identification by non parametric regression estimation a discrete time multiple input non linear hammerstein system is identified the dynamical subsystem is recovered using the standard correlation method the main results concern estimation of the non linear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a non parametric kernel estimate of the regression function calculated from the dependent data it is shown that the algorithm converges to the characteristic of the subsystem in the pointwise as well as the global sense for sufficiently smooth characteristics the rate of convergence is o n 1 2 d in probability where d is the dimension of the input variable

voltage interval mappings for an elliptic bursting model we employed poincare return mappings for a parameter interval to an exemplary elliptic bursting model the fitzhugh nagumo rinzel model using the interval mappings we were able to examine in detail the bifurcations that underlie the complex activity transitions between tonic spiking and bursting bursting and mixed mode oscillations and finally mixed mode oscillations and quiescence in the fitzhugh nagumo rinzel model we illustrate the wealth of information qualitative and quantitative that was derived from the poincare mappings for the neuronal models and for similar electro chemical systems

the conley index for fast slow systems ii multidimensional slow variable we develop a qualitative theory for fast slow systems with a one dimensional slow variable using conley index theory for singularity perturbed systems conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0 these orbits persist when the derivative of the slow variable is small and nonzero

return map characterizations for a model of bursting with two slow variables various physiological systems display bursting electrical activity bea there exist numerous three variable models to describe this behavior however higher dimensional models with two slow processes have recently been used to explain qualitative features of the bea of some experimentally observed systems t chay and d cook math biosci 90 1988 pp 139 153 p smolen and j keizer j memb biol 127 1992 pp 9 19 r bertram et al biophys j 79 2000 pp 2880 2892 r bertram et al biophys j 68 1995 pp 2323 2332 j keizer and p smolen proc nat acad sci usa 88 1991 pp 3897 3901 in this paper we present a model with two slow and two fast variables for some parameter values the system has stable equilibria while for other values there exist bursting solutions singular perturbation methods are used to define a one dimensional return map wherein fixed points correspond to singular bursting solutions we analytically demonstrate that bursting solutions may exist

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

classical diffusion in random fields with long range correlations the effect of long range random static fields on classical diffusion is investigated it is shown that the long range component of random fields leads to the appearance of a new compared to the white noise case fixed point in the renormalisation group equations the critical exponents and oscillating corrections corresponding to the new fixed point are calculated

a tight upper bound on the bayesian probability of error in this paper we present a new upper bound on the minimum probability of error of bayesian decision systems for statistical pattern recognition this new bound is continuous everywhere and is shown to be tighter than several existing bounds such as the bhattacharyya and the bayesian bounds numerical results are also presented

reduction of a model of an excitable cell to a one dimensional map we use qualitative methods for singularly perturbed systems of differential equations and the principle of averaging to compute the first return map for the dynamics of a slow variable calcium concentration in the model of an excitable cell the bifurcation structure of the system with continuous time endows the map with distinct features it is a unimodal map with a boundary layer corresponding to the homoclinic bifurcation in the original model this structure accounts for different periodic and aperiodic regimes and transitions between them all parameters in the discrete system have biophysical meaning which allows for precise interpretation of various dynamical patterns our results provide analytical explanation for the numerical studies reported previously