model selection criteria in multivariate models with multiple structural changes abstract this paper considers the issue of selecting the number of regressors and the number of structural breaks in multivariate regression models in the possible presence of multiple structural changes we develop a modified akaike information criterion aic a modified mallows c p criterion and a modified bayesian information criterion bic the penalty terms in these criteria are shown to be different from the usual terms we prove that the modified bic consistently selects the regressors and the number of breaks whereas the modified aic and the modified c p criterion tend to overfit with positive probability the finite sample performance of these criteria is investigated through monte carlo simulations and it turns out that our modification is successful in comparison to the classical model selection criteria and the sequential testing procedure robust to heteroskedasticity and autocorrelation

estimating and testing structural changes in multivariate regressions this paper considers issues related to estimation inference and computation with multiple structural changes that occur at unknown dates in a system of equations changes can occur in the regression coefficients and or the covariance matrix of the errors we also allow arbitrary restrictions on these parameters which permits the analysis of partial structural change models common breaks that occur in all equations breaks that occur in a subset of equations and so forth the method of estimation is quasi maximum likelihood based on normal errors the limiting distributions are obtained under more general assumptions than previous studies for testing we propose likelihood ratio type statistics to test the null hypothesis of no structural change and to select the number of changes structural change tests with restrictions on the parameters can be constructed to achieve higher power when prior information is present for computation an algorithm for an efficient procedure is proposed to construct the estimates and test statistics we also introduce a novel locally ordered breaks model which allows the breaks in different equations to be related yet not occurring at the same dates

the birth of e8 out of the spinors of the icosahedron e8 is prominent in mathematics and theoretical physics and is generally viewed as an exceptional symmetry in an eight dimensional 8d space very different from the space we inhabit for instance the lie group e8 features heavily in 10d superstring theory contrary to that point of view here we show that the e8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3d geometry the 240 roots of e8 arise in the 8d clifford algebra of 3d space as a double cover of the 120 elements of the icosahedral group generated by the root system h3 as a by product by restricting to even products of root vectors spinors in the 4d even subalgebra of the clifford algebra one can show that each 3d root system induces a root system in 4d which turn out to also be exactly the exceptional 4d root systems the spinorial point of view explains their existence as well as their unusual automorphism groups this spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions which opens up a novel interpretation of these phenomena in terms of spinorial geometry

level shift estimation in the presence of non stationary volatility with an application to the unit root testing problem this paper focuses on the estimation of the location of level breaks in time series whose shocks display non stationary volatility permanent changes in unconditional volatility we propose a new feasible weighted least squares wls estimator based on an adaptive estimate of the volatility path of the shocks we show that this estimator belongs to a generic class of weighted residual sum of squares which also contains the ordinary least squares ols and wls estimators the latter based on the true volatility process for fixed magnitude breaks we show that the consistency rate of the generic estimator is unaffected by non stationary volatility we also provide local limiting distribution theory for cases where the break magnitude is either local to zero at some polynomial rate in the sample size or is exactly zero the former includes the pitman drift rate which is shown via monte carlo experiments to predict well the key features of the finite sample behaviour of both the ols and our feasible wls estimators the simulations highlight the importance of the break location break magnitude and the form of non stationary volatility for the finite sample performance of these estimators and show that our proposed feasible wls estimator can deliver significant improvements over the ols estimator under heteroskedasticity we discuss how these results can be applied by using level break fraction estimators on the first differences of the data when testing for a unit root in the presence of trend breaks and or non stationary volatility methods to select between the break and no break cases using standard information criteria and feasible weighted information criteria based on our adaptive volatility estimator are also discussed simulation evidence suggests that unit root tests based on these weighted quantities can display significantly improved finite sample behaviour under heteroskedasticity relative to their unweighted counterparts an empirical illustration to u s and u k real gdp is also considered

associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise the memory patterns are assumed to be given by uniformly distributed random numbers on 0 2 ensuremath pi so that the patterns encode the phase differences of the oscillators to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns we introduce an effective transfer function by assuming a fixed point equation of the form of the thouless anderson palmer equation which describes the time averaged output as a function of the effective time averaged local field properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations and are shown to be in good agreement with the results of numerical simulations two types of retrieval states are found to occur with respect to the degree of synchronization when the size of the width of the natural frequency distribution is changed

a network of oscillators a model of neuronal behaviour capable of accounting for the oscillatory plateau and rebound properties of biological neurons is derived discussed and analysed the model is based on a piecewise linear form of the fitzhugh nagumo equations but reduces to a set of maps very similar to those of the hopfield model 1982 in particular the binary description of individual neurons and the well studied form of the synaptic current sigma jijsj are preserved although the model is capable of reproducing behaviours on the slow timescales characteristic of plateau and oscillation by coupling two model cells together a mutually inhibitory or half centred oscillator and an oscillator fixed phase follower pairs are constructed

fast global oscillations in networks of integrate and fire neurons with low firing rates we study analytically the dynamics of a network of sparsely connected inhibitory integrate and fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate in the limit when the number of neurons n 1 the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized the activity becomes oscillatory when the inhibitory feedback is strong enough the period of the global oscillation is found to be mainly controlled by synaptic times but depends also on the characteristics of the external input in large but finite networks the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold their characteristics are determined as functions of systems parameters in these two different regimes the results are found to be in good agreement with numerical simulations

travelling waves in chains of pulse coupled integrate and fire oscillators with distributed delays abstract we analyse travelling waves in a chain of pulse coupled integrate and fire oscillators with nearest neighbour coupling and delayed interactions this is achieved by approximating the equations for phase locking in terms of a singularly perturbed two point continuum boundary value problem the latter has a solution provided that a self consistent value for the collective frequency of oscillations can be found we investigate how the qualitative behaviour of travelling waves depends on the distribution of natural frequencies across the chain and the form of delayed interactions a linear stability analysis of phase locked solutions is carried out in terms of perturbations of the firing times of the oscillators it is shown how travelling waves destabilize when the detuning between oscillators or the strength of the coupling becomes too large

catalan solids derived from three dimensional root systems and quaternions catalan solids are the duals of the archimedean solids the vertices of which can be obtained from the coxeter dynkin diagrams a3 b3 and h3 whose simple roots can be represented by quaternions the respective weyl groups w a3 w b3 and w h3 acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries vertices of the platonic and archimedean solids result from the orbits derived from fundamental weights the platonic solids are dual to each other however the duals of the archimedean solids are the catalan solids whose vertices can be written as the union of the orbits up to some scale factors obtained by applying the above weyl groups on the fundamental highest weights 100 010 and 011 for each diagram the faces are represented by the orbits derived from the weights 010 110 101 011 and 111 which correspond to the vertices of the archimedean solids representations of the weyl groups w a3 w b3 and w h3 by the quaternions sim

quaternionic root systems and subgroups of the aut f4 cayley dickson doubling procedure is used to construct the root systems of some celebrated lie algebras in terms of the integer elements of the division algebras of real numbers complex numbers quaternions and octonions starting with the roots and weights of su 2 expressed as the real numbers one can construct the root systems of the lie algebras of so 4 sp 2 so 5 so 8 so 9 f4 and e8 in terms of the discrete elements of the division algebras the roots themselves display the groups structures besides the octonionic roots of e8 which form a closed octonion algebra the automorphism group aut f4 of the dynkin diagram of f4 of order 2304 the largest crystallographic group in four dimensional euclidean space is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of f4 the weyl groups of many lie algebras such as g2 so 7 so 8 so 9 su 3 xsu 3 and sp 3 su 2 have been constructed as the subgroups of aut f4 we have also class

tests of the co integration rank in var models in the presence of a possible break in trend at an unknown point in this paper we consider the problem of testing for the co integration rank of a vector autoregressive process in the case where a trend break may potentially be present in the data it is known that un modelled trend breaks can result in tests which are incorrectly sized under the null hypothesis and inconsistent under the alternative hypothesis extant procedures in this literature have attempted to solve this inference problem but require the practitioner to either assume that the trend break date is known or to assume that any trend break cannot occur under the co integration rank null hypothesis being tested these procedures also assume the autoregressive lag length is known to the practitioner all of these assumptions would seem unreasonable in practice moreover in each of these strands of the literature there is also a presumption in calculating the tests that a trend break is known to have happened this can lead to a substantial loss in finite sample power in the case where a trend break does not in fact occur using information criteria based methods to select both the autoregressive lag order and to choose between the trend break and no trend break models using a consistent estimate of the break fraction in the context of the former we develop a number of procedures which deliver asymptotically correctly sized and consistent tests of the co integration rank regardless of whether a trend break is present in the data or not by selecting the no break model when no trend break is present these procedures also avoid the potentially large power losses associated with the extant procedures in such cases

chaos and asymptotical stability in discrete time neural networks abstract this paper aims to theoretically prove by applying marotto s theorem that both transiently chaotic neural networks tcnn and discrete time recurrent neural networks drnn have chaotic structure a significant property tcnn and drnn is that they have only one bounded fixed point when absolute values of the self feedback connection weights in tcnn and the difference time in drnn are sufficiently large we show that this unique fixed point can actually evolve into a snap back repeller which generates chaotic structure if several conditions are satisfied on the other hand by using the lyapunov functions we also derive sufficient conditions on asymptotical stability for symmetrical versions of both tcnn and drnn under which tcnn and drnn asymptotically converge to a fixed point furthermore related bifurcations are also considered in this paper since both tcnn and drnn are not special but simple and general the obtained theoretical results hold for a wide class of discrete time neural networks to demonstrate the theoretical results of this paper better several numerical simulations are provided as illustrating examples

responses of a hodgkin huxley neuron to various types of spike train inputs numerical investigations have been made of responses of a hodgkin huxley hh neuron to spike train inputs whose interspike interval isi is modulated by deterministic semi deterministic chaotic and stochastic signals as deterministic one we adopt inputs with the time independent isi and with time dependent isi modulated by sinusoidal signal the rossler and lorentz models are adopted for chaotic modulations of isi stochastic isi inputs with the gamma distribution are employed it is shown that distribution of output isi data depends not only on the mean of isis of spike train inputs but also on their fluctuations the distinction of responses to the three kinds of inputs can be made by return maps of input and output isis but not by their histograms the relation between the variations of input and output isis is shown to be different from that of the integrate and fire if model because of the refractory period in the hh neuron

tests for parameter instability and structural change with unknown change point this paper considers tests for parameter instability and structural change with unknown change point the results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures the asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant the tests are found to perform quite well in a monte carlo experiment reported elsewhere copyright 1993 by the econometric society

dynamics of encoding in neuron populations some general mathematical features the use of a population dynamics approach promises efficient simulation of large assemblages of neurons depending on the issues addressed and the degree of realism incorporated in the simulated neurons a wide range of different population dynamics formulations can be appropriate here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common this underlying structure serves as a guide toward efficient means of simulation as an example we derive the general population firing rate frequency response and show how it may be used effectively to address a broad range of interacting population response and stability problems a few specific cases will be worked out a summary of this work appears at the end before the appendix

likelihood ratio tests for multiple structural changes abstract this paper proposes a likelihood ratio type test for multiple structural changes in regression models the model allows for lagged dependent variables and trending regressors the limiting distribution of the test is derived we show that asymptotic critical values can be obtained analytically in addition the number and the locations of change points can be consistently determined via the test procedure the method is straightforward to implement

maximal subgroups of the coxeter group w h4 and quaternions abstract the largest finite subgroup of o 4 is the non crystallographic coxeter group w h 4 of order 14 400 its derived subgroup is the largest finite subgroup w h 4 z 2 of so 4 of order 7200 moreover up to conjugacy it has five non normal maximal subgroups of orders 144 two 240 400 and 576 two groups w h 2 w h 2 z 4 and w h 3 z 2 possess non crystallographic structures with orders 400 and 240 respectively the groups of orders 144 240 and 576 are the extensions of the weyl groups of the root systems of su 3 su 3 su 5 and so 8 respectively we represent the maximal subgroups of w h 4 with sets of quaternion pairs acting on the quaternionic root systems

grand antiprism and quaternions vertices of the four dimensional 4d semi regular polytope the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm it follows from the icosian representation of the e8 root system which decomposes into two copies of the root system of h4 the symmetry of the grand antiprism is a maximal subgroup of the coxeter group w h4 it is the group aut h2 h 2 which is constructed in terms of 20 quaternionic roots of the coxeter diagram h2 h 2 the root system of h4 represented by the binary icosahedral group i of order 120 constitutes the regular 4d polytope 600 cell when its 20 quaternionic vertices corresponding to the roots of the diagram h2 h 2 are removed from the vertices of the 600 cell the remaining 100 quaternions constitute the vertices of the grand antiprism we give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions the dual polytope of the grand antiprism has also been constructed

quaternionic representation of the coxeter group w h4 and the polyhedra the vertices of the four dimensional polytope 3 3 5 and its dual 5 3 3 admitting the symmetry of the non crystallographic coxeter group w h4 of order 14 400 are represented in terms of quaternions with unit norm where the polytope 3 3 5 is represented by the elements of the binaryicosahedral group of quaternions of order 120 we projected the polytopes to three dimensional euclidean space where the quaternionic vertices are the orbits of the coxeter group w h3 icosahedral group with inversion where w h3 z2 is one of the maximal subgroups of the coxeter group w h4 the orbits of the icosahedral group w h3 in the polytope 3 3 5 are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons dodecahedrons and one icosidodecahedron in three dimensions the 15 orbits of the icosahedral group w h3 in the polytope 5 3 3 represent the dodecahedrons icosidodecahedrons small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry one of the convex solids with 60 vertices is very similar to the truncated icosahedron soccer ball but with two different edge lengths which can be taken as a realistic model of the c60 molecule at extreme temperature and pressure

self organizing dual coding based on spike time dependent plasticity it has been a matter of debate how firing rates or spatiotemporal spike patterns carry information in the brain recent experimental and theoretical work in part showed that these codes especially a population rate code and a synchronous code can be dually used in a single architecture however we are not yet able to relate the role of firing rates and synchrony to the spatiotemporal structure of inputs and the architecture of neural networks in this article we examine how feedforward neural networks encode multiple input sources in the firing patterns we apply spiketime dependent plasticity as a fundamental mechanism to yield synaptic competition and the associated input filtering we use the fokker planck formalism to analyze the mechanism for synaptic competition in the case of multiple inputs which underlies the formation of functional clusters in downstream layers in a self organizing manner depending on the types of feedback coupling and shared connectivity clusters are independently engaged in population rate coding or synchronous coding or they interact to serve as input filters classes of dual codings and functional roles of spike time dependent plasticity are also discussed

symmetries of the octonionic root system of e8 octonionic root system of e8 is decomposed as the 9 sets of hurwitz integers each set satisfying the binary tetrahedral group structure and the 12 sets of quaternionic units each set obeying the dicylic group structure of order 12 this fact is used to prove that the automorphism group of the octonionic root system of e7 is the finite subgroup of g2 of order 12 096 an explicit 7 7 matrix realization of which is constructed possible use of the octonionic representation of the e6 root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested

structural break estimation for nonstationary time series models this article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive ar processes the number and locations of the piecewise ar segments as well as the orders of the respective ar processes are assumed unknown the minimum description length principle is applied to compare various segmented ar fits to the data the goal is to find the best combination of the number of segments the lengths of the segments and the orders of the piecewise ar processes such a best combination is implicitly defined as the optimizer of an objective function and a genetic algorithm is implemented to solve this difficult optimization problem numerical results from simulation experiments and real data analyses show that the procedure has excellent empirical properties the segmentation of multivariate time series is also considered assuming that the true underlying model is a segmented autoregression this procedure is shown to be consistent for estimating the location of

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

automorphism groups of pure integral octonions a matrix representation of the automorphism group of pure integral octonions constituting the root system of e7 is constructed it is shown that it is a finite subgroup of the exceptional group of g2 of order 12096 called the adjoint chevalley group g2 2 its four maximal subgroups of orders 432 192 192 and 336 preserve respectively the octonionic root systems of e6 so 12 su 2 3 so 8 and su 8 it is also shown explicitly that the full automorphism group of the pure octonions or ei i 1 7 constituting the roots of su 2 7 is a group of order 1344 possible implications in physics are discussed

quaternionic roots of so 8 so 9 f4 and the related weyl groups the root systems of so 8 so 9 and f4 are constructed by quaternions triality manifests itself as permutations of pure quaternion units e1 e2 and e3 it is shown that the automorphism groups of the associated root systems are the finite subgroups of o 4 generated by left right actions of unit quaternions on the root systems the relevant finite groups of quaternions the binary tetrahedral and binary octahedral groups play essential roles in the construction of the weyl groups and their conjugacy classes the relations between the dynkin indices standard orthogonal vector and the quaternionic weights are obtained

estimating and testing linear models with multiple structural changes this paper develops the statistical theory for testing and estimating multiple change points in regression models the rate of convergence and limiting distribution for the estimated parameters are obtained several test statistics are proposed to determine the existence as well as the number of change points a partial structural change model is considered the authors study both fixed and shrinking magnitudes of shifts in addition the models allow for serially correlated disturbances mixingales an estimation strategy for which the location of the breaks need not be simultaneously determined is discussed instead the authors method successively estimates each break point

nonlinear neural networks i general theory a neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through anonlinear operation the original hopfield model is linear whereas for instance clipped synapses constitute a nonlinear model here a general theory is presented to obtain the statistical mechanics of a neural network with finitely many patterns and arbitrary symmetric nonlinearity the problem is reduced to minimizing a free energy functional over all solutions of a fixed point equation with synaptic kernelq the case of clipped synapses with bimodal and gaussian probability distribution is analyzed in detail to this end a simple theory is developed that gives the spectrum ofq and thereby all the solutions that bifurcate from the high temperature phase

dual information representation with stable firing rates and chaotic spatiotemporal spike patterns in a neural network model although various means of information representation in the cortex have been considered the fundamental mechanism for such representation is not well understood the relation between neural network dynamics and properties of information representation needs to be examined we examined spatial pattern properties of mean firing rates and spatiotemporal spikes in an interconnected spiking neural network model we found that whereas the spatiotemporal spike patterns are chaotic and unstable the spatial patterns of mean firing rates spmfr are steady and stable reflecting the internal structure of synaptic weights interestingly the chaotic instability contributes to fast stabilization of the spmfr findings suggest that there are two types of network dynamics behind neuronal spiking internally driven dynamics and externally driven dynamics when the internally driven dynamics dominate spikes are relatively more chaotic and independent of external inputs the spmfr are steady and stable when the externally driven dynamics dominate the spiking patterns are relatively more dependent on the spatiotemporal structure of external inputs these emergent properties of information representation imply that the brain may adopt a dual coding system recent experimental data suggest that internally driven and externally driven dynamics coexist and work together in the cortex

delay induced multistable synchronization of biological oscillators we analyze the dynamics of pulse coupled oscillators depending on strength and delay of the interaction for two oscillators we derive return maps for subsequent phase differences and construct phase diagrams for a broad range of parameters in phase synchronization proves stable for inhibitory coupling and unstable for excitatory coupling if the delay is not zero if the coupling strength is high additional regimes with marginally stable synchronization are found simulations with n 2 oscillators reveal a complex dynamics including spontaneous synchronization and desynchronization with excitatory coupling and multistable phase clustering with inhibitory coupling we simulate a continuous description of the system for n oscillators and demonstrate that these phenomena are independent of the size of the system phase clustering is shown to relate to stability and basins of attraction of fixed points in the return map of two oscillators our findings are generic in the sense that they qualitatively are robust with respect to modeling details we demonstrate this using also pulses of finite rise time and the more realistic model by hodgkin and huxley which exhibits multistable synchronization as predicted from our analysis as well s1063 651x 98 09902 4

polyhedra obtained from coxeter groups and quaternions we note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the coxeter dynkin diagrams the vertices of a regular or semiregular polytope are the weights obtained as the orbit of the coxeter weyl group acting on the highest weight representing a selected irreducible representation of the lie group this paper in particular deals with the determination of the vertices of the platonic and archimedean solids from the coxeter diagrams a3 b3 and h3 in the context of the quaternionic representations of the root systems and the coxeter weyl groups we use lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral octahedral and icosahedral symmetries are related to the coxeter weyl groups representing the symmetries of the diagrams of a3 b3 and h3 respectively this technique leads to the determination of the vertices of all platonic and archimedean solids except two chiral polyhedra snubcuboctahedr

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

testing for common breaks in a multiple equations system abstract the issue addressed in this paper is that of testing for common breaks across or within equations of a multivariate system our framework is very general and allows integrated regressors and trends as well as stationary regressors the null hypothesis is that breaks in different parameters occur at common locations and are separated by some positive fraction of the sample size unless they occur across different equations under the alternative hypothesis the break dates across parameters are not the same and also need not be separated by a positive fraction of the sample size whether within or across equations the test considered is the quasi likelihood ratio test assuming normal errors though as usual the limit distribution of the test remains valid with non normal errors of independent interest we provide results about the rate of convergence of the estimates when searching over all possible partitions subject only to the requirement that each regime contains at least as many observations as some positive fraction of the sample size allowing break dates not separated by a positive fraction of the sample size across equations simulations show that the test has good finite sample properties we also provide an application to issues related to level shifts and persistence for various measures of inflation to illustrate its usefulness

random site spin glass models a simple general method is presented for solving mean field spin glass models where the bond randomness is expressible in terms of an underlying site randomness both separable and non separable models can be solved

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates

chaotic balanced state in a model of cortical circuits the nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are not well understood we consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells we study a network model with excitatory and inhibitory populations of simple binary units the internal feedback is mediated by relatively large synaptic strengths so that the magnitude of the total excitatory and inhibitory feedback is much larger than the neuronal threshold the connectivity is random and sparse the mean number of connections per unit is large though small compared to the total number of cells in the network the network also receives a large temporally regular input from external sources we present an analytical solution of the mean field theory of this model which is exact in the limit of large network size this theory reveals a new cooperative stationary state of large networks which we term a balanced state in thi

asynchronous states in networks of pulse coupled oscillators we use a mean field approach to analyze the stability of the asynchronous state in a population of all to all pulse coupled nonlinear oscillators we determine the conditions that must be satisfied by the time constants and phase dependence characterizing the coupling between the oscillators in order for the asynchronous state to be stable we also consider the effects of noise this work complements results on synchronous states in similar models and allows us to study the validity of firing rate models commonly used for neural networks

group theoretical analysis of 600 cell and 120 cell 4d polytopes with quaternions cell 3 3 5 and 120 cell 5 3 3 four dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the coxeter group w h4 the maximal subgroups w su 5 z2 and w h3 z2 of w h4 play important roles in the analysis of cell structures of the dual polytopes in particular the weyl coxeter group w su 4 is used to determine the tetrahedral cells of the polytope 3 3 5 and the coxeter group w h3 is used for the dodecahedral cells of 5 3 3 using the lie algebraic techniques in terms of quaternions we explicitly construct cell structures forming the vertices of the 4d polytopes

what matters in neuronal locking exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons under standard conditions a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire if the postsynaptic potential is decreasing oscillations are bound to be unstable this is a kind of locking theorem and boils down to a subtle interplay of axonal delays postsynaptic potentials and refractory behavior the theorem also allows for mixtures of excitatory and inhibitory interactions on the basis of the locking theorem we present a simple geometric method to verify the existence and local stability of a coherent oscillation

collective synchronization of pulse coupled oscillators and excitable units abstract a phase model is proposed and studied for a population of integrate and fire type oscillators and excitable units with pulsatile coupling the units are subject to external noise and their mutual connection is all to all an evolution equation is derived for the phase distribution in a closed form its steady state analysis in the limit of weak noise shows the possibility of collective bistability the evolution equation is simplified drastically when the coupling as well as noise is sufficiently weak this reduction of equation enables analytical calculation of various quantities among others the stability of the steady distribution is analyzed explicity and the onset of collective oscillations is described in terms of the hopf bifurcation it is also shown that the present model is improved in some respects by introducing absolute refractoriness

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

alternating oscillatory and stochastic dynamics in a model for a neuronal assembly abstract in recent neurophysiological experiments stimulus related neuronal oscillations were discovered in various species the oscillations are not persistent during the whole time of stimulation but instead seem to be restricted to rather short periods interrupted by stochastic periods in this contribution we argue that these observations can be explained by a bistability in the ensemble dynamics of coupled integrate and fire neurons this dynamics can be cast in terms of a high dimensional map for the time evolution of a phase density which represents the ensemble state a numerical analysis of this map reveals the coexistence of two stationary states in a broad parameter regime when the synaptic transmission is nonlinear the one state corresponds to a stochastic firing of individual neurons the other state describes a periodic activation we demonstrate that under the influence of additional external noise the system can switch between these states in this way reproducing the experimentally activity we also investigate the connection between the nonlinearity of the synaptic transmission function and the bistability of the dynamics to this purpose we heuristically reduce the high dimensional assembly dynamics to a one dimensional map which in turn yields a simple explanation for the relation between nonlinearity and bistability in our system