the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

classification of irreducible weight modules over higher rank virasoro algebras let g be a rank n additive subgroup of c and vir g the corresponding virasoro algebra of rank n in the present paper irreducible weight modules with finite dimensional weight spaces over vir g are completely determined there are two different classes of them one class consists of simple modules of intermediate series whose weight spaces are all 1 dimensional the other is constructed by using intermediate series modules over a virasoro subalgebra of rank n 1 the classification of such modules over the classical virasoro algebra was obtained by o mathieu in 1992 using a completely different approach

dynamics of moments of fitzhugh nagumo neuronal models and stochastic bifurcations for the study of the behavior of noisy neuronal models rodriguez and tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means variances and covariance of the state variables r rodriguez and h c tuckwell phys rev e 54 5585 1996 in this work we first report a modification of their method in the case of the fitzhugh nagumo model which enhances the accuracy of the approximation without including higher order moments this method is then combined with a self consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto and cross correlation functions of the state variables finally we argue that the moments equations can also reveal the existence of stochastic bifurcations i e qualitative changes in the dynamics of stochastic systems

variational superposed gaussian approximation for time dependent solutions of langevin equations we propose a variational superposed gaussian approximation vsga for dynamical solutions of langevin equations subject to applied signals determining time dependent parameters of superposed gaussian distributions by the variational principle we apply the proposed vsga to systems driven by a chaotic signal where the conventional fourier method cannot be adopted and calculate the time evolution of probability density functions pdfs and moments both white and colored gaussian noises terms are included to describe fluctuations our calculations show that time dependent pdfs obtained by vsga agree excellently with those obtained by monte carlo simulations the correlation between the chaotic input signal and the mean response are also calculated as a function of the noise intensity which confirms the occurrence of aperiodic stochastic resonance with both white and colored noises

modules of highest weight type for the algebra of derivations over a rational quantum torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus

new solutions of distinct physical structures to high dimensional nonlinear evolution equations abstract in this work we employ the tanh coth method for a reliable treatment on nonlinear evolution equations three models namely the 3 1 dimensional jimbo miwa equation the 2 1 dimensional calogero bogoyavlenskii schiff equation and the 3 1 dimensional potential ytsf equation are use as vehicles to conduct the analysis new travelling wave solutions are formally derived the work shows the power of the proposed scheme and the variety of the obtained solutions

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

augmented variational superposed gaussian approximation for fokker planck equations with rational polynomial functions abstract reliable methods for obtaining time dependent solutions of fokker planck equations are in high demand in the field of non equilibrium theory in this paper we present a new method based on variational superposed gaussian approximation vsga and pade approximants the vsga obtains time dependent probability density functions as a superposition of multiple gaussian distributions however a limitation of the vsga is that the expectation of the drift term with respect to the gaussian distribution needs to be calculated analytically which is typically satisfied when the drift term is a polynomial function when this condition is not met the vsga must rely on the numerical integration of the expectation at each step resulting in huge computational cost we propose an augmented vsga a vsga method that effectively overcomes the limitation of the vsga by approximating non linear functions with the pade approximant we apply the a vsga to two systems driven by chaotic input signals a stochastic genetic regulatory system and a soft bistable system whose drift terms are a rational polynomial function and a hyperbolic tangent function respectively the numerical calculations show that the proposed method provides accurate results with less computational time than that required for monte carlo simulation

statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks dynamical stochastic models of single neurons and neural networks often take the form of a system of n 2 coupled stochastic differential equations we consider such systems under the assumption that third and higher order central moments are relatively small in the general case a system of 1 n n13 generally nonlinear coupled ordinary differential equations holds for the approximate means variances and covariances for the general linear system the solutions of these equations give exact results this is illustrated in a simple case generally the moment equations can be solved numerically results are given for a spiking fitzhugh nagumo model neuron driven by a current with additive white noise differential equations are obtained for the means variances and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise s1063 651x 96 03511 8 pacs number s 87 10 1e 05 40 1j 02 50 2r

a reduction theorem for highest weight modules over toroidal lie algebras let open image in new window be a toroidal lie algebra corresponding to a semisimple lie algebra open image in new window we describe all borel subalgebras of open image in new window which contain the cartan subalgebra open image in new window where open image in new window is a fixed cartan subalgebra of open image in new window we show that each such borel subalgebra open image in new window determines a parabolic decomposition open image in new window where open image in new window is a proper toroidal subalgebra of open image in new window and open image in new window our first main result is that for any weight which does not vanish on open image in new window an arbitrary subquotient of the verma module open image in new window is induced from its open image in new window submodule of open image in new window invariant vectors this reduces the study of subquotients of open image in new window to the study of subquotients of verma modules over open image in new window we then introduce categories open image in new window and open image in new window and their respective blocks open image in new window and open image in new window corresponding to a central charge which is nonzero on open image in new window our second main result is that the functors of induction and open image in new window invariants are mutually inverse equivalences of the category open image in new window and the full subcategory of open image in new window whose objects are generated by their open image in new window invariants

population dynamics under the laplace assumption in this paper we describe a generic approach to modelling dynamics in neuronal populations this approach models a full density on the states of neuronal populations but finesses this high dimensional problem by re formulating density dynamics in terms of ordinary differential equations on the sufficient statistics of the densities considered c f the method of moments the particular form for the population density we adopt is a gaussian density c f the laplace assumption this means population dynamics are described by equations governing the evolution of the population s mean and covariance we derive these equations from the fokker planck formalism and illustrate their application to a conductance based model of neuronal exchanges one interesting aspect of this formulation is that we can uncouple the mean and covariance to furnish a neural mass model which rests only on the populations mean this enables us to compare equivalent mean field and neural mass models of the same populations and evaluate quantitatively the contribution of population variance to the expected dynamics the mean field model presented here will form the basis of a dynamic causal model of observed electromagnetic signals in future work

population rate codes carried by mean fluctuation and synchrony of neuronal firings abstract a population of firing neurons is expected to carry information not only by the mean firing rate but also by fluctuation and synchrony among neurons in order to examine this possibility we have studied responses of neuronal ensembles to three kinds of inputs mean fluctuation and synchrony driven inputs the generalized rate code model including additive and multiplicative noise h hasegawa phys rev e 75 2007 051904 has been studied by direct simulations dss and the augmented moment method amm in which equations of motion for mean firing rate fluctuation and synchrony are derived results calculated by the amm are in good agreement with those by dss the independent component analysis ica of our results has shown that mean firing rate fluctuation or variability and synchrony may carry independent information in the population rate code model the input output relation of mean firing rates is shown to have higher sensitivity for larger multiplicative noise as recently observed in prefrontal cortex a comparison is made between results obtained by the integrate and fire if model and our rate code model

lowest energy representations of non centrally extended diffeomorphism algebras we describe a class of non central extensions of the diffeomorphism algebra in n dimensional spacetime and construct lowest energy modules thereof thus generalizing work of eswara rao and moody there is one representation for each representation of an extension of the semi direct product similar modules are constructed for gauge algebras

some integrable hierarchies in 2 1 dimensions and their twistor description the twistor space construction due to mason and sparling for solutions of the nonlinear schrodinger and korteweg devries hierarchies is generalized resulting in families of integrable models in 2 1 dimensions soliton solutions to one such hierarchy associated with the gauge group su 2 are constructed by adapting the methods used to construct instanton and monopole solutions these integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants in contrast to the closely related davey stewartson hierachy

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

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

controlling statistical moments of stochastic dynamical networks we consider a general class of stochastic networks and ask which network nodes need to be controlled and how to stabilize and switch between desired metastable target states in terms of the first and second statistical moments of the system we first show that it is sufficient to directly interfere with a subset of nodes which can be identified using information about the graph of the network only then we develop a suitable method for feedback control which acts on that subset of nodes and preserves the covariance structure of the desired target state finally we demonstrate our theoretical results using a stochastic hopfield network and a global brain model our results are applicable to a variety of model networks and further our understanding of the relationship between network structure and collective dynamics for the benefit of effective control

integrable representations for generalized virasoro toroidal lie algebras abstract in this paper we give the structure of irreducible integrable modules of generalized virasoro toroidal lie algebras with finite dimensional weight spaces and non zero central charge

differential equations in vertex algebras and simple modules for the lie algebra of vector fields on a torus we study irreducible representations for the lie algebra of vector fields on a 2 dimensional torus constructed using the generalized verma modules we show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the lie algebra of vector fields we prove this result by studying vertex algebras associated with the lie algebra of vector fields on a torus and solving non commutative differential equations that we derive using the vertex algebra technique

dynamics of a modified hindmarsh rose neural model with random perturbations moment analysis and firing activities abstract in this paper an attempt has been made to understand the activity of mean membrane voltage and subsidiary system variables with moment equations i e mean variance and covariance s under noisy environment we consider a biophysically plausible modified hindmarsh rose h r neural system injected by an applied current exhibiting spiking bursting phenomenon the effects of predominant parameters on the dynamical behavior of a modified h r system are investigated numerically it exhibits period doubling period halving bifurcation and chaos phenomena further a nonlinear system has been analyzed for the first and second order moments with additive stochastic perturbations it has been solved using fourth order runge kutta method and noisy systems by euler s scheme it has been demonstrated that the firing properties of neurons to evoke an action potential in a certain parameter space of the large exact systems can be estimated using an approximated model strong stimulation can cause a change in increase or decrease of the firing patterns corresponding to a fixed set of parameter values the firing behavior and dynamical differences of the collective variables of a large exact and approximated systems are investigated

classification of category j modules for divergence zero vector fields on a torus abstract we consider a category of modules that admit compatible actions of the commutative algebra of laurent polynomials and the lie algebra of divergence zero vector fields on a torus and have a weight decomposition with finite dimensional weight spaces we classify indecomposable and irreducible modules in this category

a renormalization approach to the universality of scaling in phyllotaxis abstract phyllotaxis i e the arrangement of plant organs like leaves florets scales bracts etc around a shoot stem or cone is often highly regular across the plant kingdom phyllotaxis shows not only qualitatively but also quantitatively identical features like the occurrence of divergence angles close to noble irrationals in a previous study reick 2012 a mechanism has been identified that explains the selection of these particular divergence angles on the basis of self similarity and scaling numerically found in the bifurcation diagrams of simple dynamical models of phyllataxis in the present paper by constructing a renormalization theory the universality of this scaling is proved for a whole class of models prototypically represented by thornley s model of phyllotaxis thornley 1975 the renormalization is constructed from another self similarity found numerically for the fourier transform of the abstract potential governing the mutual inhibition of primordia surprisingly the resulting renormalization transformation is already known from the treatment of the quasiperiodic transition to chaos but operates here on a different function space it turns out that the fixed points of the renormalization transformation are characterized by divergences of the form 1 where written as continued fraction n to show the universality of the scaling it is demonstrated that the fixed points are unstable and that the associated scaling factors 2 and are exactly those that were numerically found in reick 2012 to rule the selfsimilarity of the bifurcation structure thereby the present paper puts forward an explanation for the universal appearance of certain phyllotactic patterns that is independent of physiological detail of plant growth

fluctuation theorem uncertainty relation the fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations such as the second law of thermodynamics and the jarzynski equality recently the thermodynamic uncertainty relation was discovered which states that the fluctuation of observables is lower bounded by the entropy production in the present letter we derive a thermodynamic uncertainty relation from the fluctuation theorem we refer to the obtained relation as the fluctuation theorem uncertainty relation and it is valid for arbitrary dynamics stochastic as well as deterministic and for arbitrary antisymmetric observables for which a fluctuation theorem holds we apply the fluctuation theorem uncertainty relation to an overdamped langevin dynamics for an antisymmetric observable we demonstrate that the antisymmetric observable satisfies the fluctuation theorem uncertainty relation but does not satisfy the relation reported for current type observables in continuous time markov chains moreover we show that the fluctuation theorem uncertainty relation can handle systems controlled by time symmetric external protocols in which the lower bound is given by the work exerted on the systems

stationary and dynamical properties of finite n unit langevin models subjected to multiplicative noises abstract we have studied the finite n unit langevin model subjected to multiplicative noises by using the augmented moment method amm as a continuation of our previous paper h hasegawa j phys soc japan 75 2006 033001 effects of couplings on stationary and dynamical properties of the model have been investigated the difference and similarity between the results of diffusive and sigmoid couplings are studied in details time dependences of average and fluctuations in local and global variables calculated by the amm are in good agreement with those of direct simulations dss we also discuss stationary distributions of local and global variables with the use of the fokker planck equation fpe method and dss it is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account

stochastic bifurcation in fitzhugh nagumo ensembles subjected to additive and or multiplicative noises abstract we have studied the dynamical properties of finite n unit fitzhugh nagumo fn ensembles subjected to additive and or multiplicative noises reformulating the augmented moment method amm with the fokker planck equation fpe method h hasegawa j phys soc japan 75 2006 033001 in the amm original 2 n dimensional stochastic equations are transformed to eight dimensional deterministic ones and the dynamics is described in terms of averages and fluctuations of local and global variables the stochastic bifurcation is discussed by a linear stability analysis of the deterministic amm equations the bifurcation transition diagram of multiplicative noise is rather different from that of additive noise the former has the wider oscillating region than the latter the synchronization in globally coupled fn ensembles is also investigated results of the amm are in good agreement with those of direct simulations dss

a differential difference system related to toroidal lie algebra we present a novel differential difference system in 2 1 dimensional space time one discrete two continuum arising from bogoyavlensky s 2 1 dimensional korteweg de vries hierarchy our method is based on the bilinear identity of the hierarchy which is related to the vertex operator representation of the toroidal lie algebra tor2

mean field approximation for noisy delay coupled excitable neurons abstract mean field approximation of a large collection of fitzhugh nagumo excitable neurons with noise and all to all coupling with explicit time delays modelled by n 1 stochastic delay differential equations is derived the resulting approximation contains only two deterministic delay differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system

moment analysis of the hodgkin huxley system with additive noise abstract we consider a classical space clamped hodgkin huxley hh model neuron stimulated by a current which has a mean together with additive gaussian white noise of amplitude a system of 14 deterministic first order nonlinear differential equations is derived for the first and second order moments means variances and covariances of the voltage v and the subsidiary variables n m and h the system of equations is integrated numerically with a fourth order runge kutta method as long as the variances as determined by these deterministic equations remain small the latter accurately approximate the first and second order moments of the stochastic hodgkin huxley system describing spiking neurons on the other hand for certain values of when rhythmic spiking is inhibited by larger amplitude noise the solutions of the moment equation strongly overestimate the moments of the voltage a more refined analysis of the nature of such irregularities leads to precise insights about the effects of noise on the hodgkin huxley system for suitable values of which enable rhythmic spiking we analyze by numerical examples from both simulation and solutions of the moment equations the three factors which tend to promote its cessation namely the increasing variance the nature and shape of the basins of attraction of the limit cycle and stable equilibrium point and the speed of the process

dynamical mean field theory of noisy spiking neuron ensembles application to the hodgkin huxley model a dynamical mean field approximation dma previously proposed by the present author h hasegawa phys rev e 67 041903 2003 has been extended to ensembles described by a general noisy spiking neuron model ensembles of n unit neurons each of which is expressed by coupled k dimensional differential equations des are assumed to be subject to spatially correlated white noises the original mathrm kn dimensional stochastic des have been replaced by k k 2 dimensional deterministic des expressed in terms of means and the second order moments of local and global variables the fourth order contributions are taken into account by the gaussian decoupling approximation our dma has been applied to an ensemble of hodgkin huxley hh neurons k 4 for which effects of the noise the coupling strength and the ensemble size on the response to a single spike input have been investigated numerical results calculated by the dma theory are in good agreement with those obtained by direct simulations although the former computation is about a thousand times faster than the latter for a typical hh neuron ensemble with n 100

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

statistical mechanics for a network of spiking neurons we show that a simple statistical mechanics model can capture the collective behavior of large networks of spiking neurons qualitative arguments suggest that regularly firing neurons should be described by a planar spin of unit length we extract these spins from spike trains and then measure the interaction hamiltonian using simulations of small clusters of cells correlations among spike trains obtained from simulations of large arrays of cells are in quantitative agreement with the predictions from these hamiltonians we comment on the novel computational abilities of these xy networks

synchronization induced by periodic inputs in finite n unit bistable langevin models the augmented moment method we have studied the synchronization induced by periodic inputs applied to a finite n unit coupled bistable langevin model which is subjected to cross correlated additive and multiplicative noises the effects on the synchronization of the system size n the coupling strength and the cross correlation between additive and multiplicative noise have been investigated using the semi analytical augmented moment method amm which is a second order moment approximation for local and global variables h hasegawa phys rev e 67 2003 041903 a linear analysis of the stationary solution of the amm equations shows that the stability is improved degraded by positive negative couplings results of the nonlinear bistable langevin model are compared to those of the linear langevin model

dynamical mean field theory of spiking neuron ensembles response to a single spike with independent noises a semianalytical dynamical mean field theory has been developed for a study of dynamics of an ensemble of n unit fitzhugh nagumo neurons subject to white noises assuming weak noises and the gaussian distribution of state variables we have driven equations of motions for moments of local and global variables dynamical mean field approximation dma has replaced original 2n dimensional stochastic differential equations des by eight dimensional deterministic des whereas the conventional moment method yields n 2n 3 dimensional deterministic des for local variables we have discussed the dependence of the spike firing precision and the synchronization on the noise intensity synaptic coupling and the size of the neuron ensemble the spike timing precision is shown to be improved by increasing the size of the neuron ensemble even when there are no couplings among neurons when the coupling is introduced neurons in ensembles respond to an input spike with a partial synchronization results calculated by our dma theory are in good agreement with those obtained by direct simulations dma theory is extended to a large cluster which can be divided into multiple subclusters according to their functions a model calculation has demonstrated that when the noise intensity is moderate the spike propagation with a fairly precise timing is possible among noisy subclusters with feed forward couplings as in the synfire chain we have compared dma theory with the conventional moment method showing that the former may be alternatively derived from the latter by a reduction in the number of moments with the mean field approximation

analyzing the combined multi waves polynomial solutions in a two layer liquid medium abstract in this work we investigate the behavior of mixed wave solutions in two layer liquid with dispersive waveguide these combined multi wave solutions were obtained in polynomial types for the 3 1 dimensional yu toda sasa fukuyama ytsf equation with variable coefficients by virtue of the generalized unified method and symbolic computations we formally derive polynomial solutions for this equation the obtained solutions include multi soliton solutions multi periodic solutions and multi elliptic solutions furthermore the physical insight and the movement role of the waves in the two layers are discussed and analyzed graphically for different selections of the arbitrary functions of the obtained solutions

two extensions of 1d toda hierarchy the extended toda hierarchy of carlet dubrovin and zhang is reconsidered in light of a 2 1d extension of the 1d toda hierarchy constructed by ogawa these two extensions of the 1d toda hierarchy turn out to have a very similar structure and the former may be thought of as a kind of dimensional reduction of the latter in particular this explains an origin of the mysterious structure of the bilinear formalism proposed by milanov

the space clamped hodgkin huxley system with random synaptic input inhibition of spiking by weak noise and analysis with moment equations we consider a classical space clamped hodgkin huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by ornstein uhlenbeck processes using numerical solutions of the stochastic model system obtained by an euler method it is found that with excitation only there is a critical value of the steady state excitatory conductance for repetitive spiking without noise and for values of the conductance near the critical value small noise has a powerfully inhibitory effect for a given level of inhibition there is also a critical value of the steady state excitatory conductance for repetitive firing and it is demonstrated that noise in either the excitatory or inhibitory processes or both can powerfully inhibit spiking furthermore near the critical value inverse stochastic resonance was observed when noise was present only in the inhibitory input process the system of deterministic differential equations for the approximate first and second order moments of the model is derived they are solved using runge kutta methods and the solutions are compared with the results obtained by simulation for various sets of parameters including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex the mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels in the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells the assumptions underlying the moment equation approach are not satisfied the simulation results show that noisy synaptic input of either an excitatory or inhibitory character or both may lead to the suppression of firing in neurons operating near a critical point and this has possible implications for cortical networks although suppression of firing is corroborated for the system of moment equations there seem to be substantial differences between the dynamical properties of the original system of stochastic differential equations and the much larger system of moment equations

multiple addition theorem for discrete and continuous nonlinear problems the addition relation for the riemann theta functions and for its limits which leads to the appearance of exponential functions in soliton type equations is discussed the form of addition property presented resolves itself to the factorization of the n tuple product of the shifted functions and it seems to be useful for analysis of soliton type continuous and discrete processes in the n 1 space time a close relation with the natural generalization of bi and tri linear operators into multiple linear operators concludes the paper

augmented moment method for stochastic ensembles with delayed couplings ii fitzhugh nagumo model dynamics of fitzhugh nagumo fn neuron ensembles with time delayed couplings subject to white noises has been studied by using both direct simulations and a semianalytical augmented moment method amm which has been proposed in a preceding paper h hasegawa phys rev e 70 021911 2004 for n unit fn neuron ensembles amm transforms original 2n dimensional stochastic delay differential equations sddes to infinite dimensional deterministic des for means and correlation functions of local and global variables infinite order recursive des are terminated at the finite level m in the level m amm mathrm amm m yielding 8 m 1 dimensional deterministic des when a single spike is applied the oscillation may be induced if parameters of coupling strength delay noise intensity and or ensemble size are appropriate effects of these parameters on the emergence of the oscillation and on the synchronization in fn neuron ensembles have been studied the synchronization shows the fluctuation induced enhancement at the transition between nonoscillating and oscillating states results calculated by amm5 are in fairly good agreement with those obtained by direct simulations

reaction diffusion model for phyllotaxis abstract this paper presents a numerical study of a reaction diffusion model with three chemical substances the model makes possible the breaking of the bilateral symmetry and presents a new kind of solution with a two time periodic asymptotic behavior these solutions are of particular interest for the phyllotaxis because they correspond to propagating structures peaks which locally appear and disappear on a ring

extended method of moments for deterministic analysis of stochastic multistable neurodynamical systems the analysis of transitions in stochastic neurodynamical systems is essential to understand the computational principles that underlie those perceptual and cognitive processes involving multistable phenomena like decision making and bistable perception to investigate the role of noise in a multistable neurodynamical system described by coupled differential equations one usually considers numerical simulations which are time consuming because of the need for sufficiently many trials to capture the statistics of the influence of the fluctuations on that system an alternative analytical approach involves the derivation of deterministic differential equations for the moments of the distribution of the activity of the neuronal populations however the application of the method of moments is restricted by the assumption that the distribution of the state variables of the system takes on a unimodal gaussian shape we extend in this paper the classical moments method to the case of bimodal distribution of the state variables such that a reduced system of deterministic coupled differential equations can be derived for the desired regime of multistability

augmented moment method for stochastic ensembles with delayed couplings i langevin model by employing a semianalytical dynamical mean field approximation theory previously proposed by the author h hasegawa phys rev e 67 041903 2003 we have developed an augmented moment method amm in order to discuss dynamics of an n unit ensemble described by langevin equations with delays in an amm original n dimensional stochastic delay differential equations sddes are transformed to infinite dimensional deterministic des for means and correlations of local as well as global variables infinite order des arising from the non markovian property of sdde are terminated at the finite level m in the level m amm mathrm amm m which yields 3 m dimensional deterministic des model calculations have been made for linear and nonlinear langevin models the stationary solution of amm for the linear langevin model with n 1 is nicely compared to the exact result in the nonlinear langevin ensemble the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states results calculated by amm6 are in good agreement with those obtained by direct simulations

analysis of inverse stochastic resonance and the long term firing of hodgkin huxley neurons with gaussian white noise abstract in order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels as noise level increases inverse stochastic resonance isr in hodgkin huxley hh systems we analyze the underlying transitions from a stable equilibrium point to limit cycle and vice versa for a value of at which isr is pronounced properties of the corresponding stable equilibrium point are found a linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur a one dimensional diffusion is also constructed for small noise properties of the basin of attraction of the limit cycle spike are investigated heuristically long term trials of duration 500000 ms are carried out for values of from 0 to 2 0 the graph of mean spike count versus is divided into 4 regions r 1 r 4 where r 3 contains the minimum associated with isr in r 1 transitions to the basin of attraction of the rest point are not observed until a small critical value of c 1 is reached at the beginning of r 2 the sudden decline in firing rate when is just greater than c 1 implies that there is only a small range of noise levels 0 c 1 where repetitive spiking is safe from annihilation by noise the firing rate remains small throughout r 3 at a larger critical value c 2 which signals the beginning of r 4 the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing the quantitative scheme underlying the isr curve is outlined in terms of the properties of exit time random variables in the final subsection several statistical properties of the main random variables associated with long term spiking activity are given including distributions of exit times from the two relevant basins of attraction and the interspike interval

applications of hta and ehta to ytsf equation new approaches homoclinic test approach hta and extended homoclinic test approach ehta are proposed to seek solitary wave solution of high dimensional nonlinear wave system exact periodic solitary wave periodic soliton cross solitary wave and doubly periodic wave solutions for ytsf equation are obtained using hta and ehta respectively

new traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized big frac g prime g big expansion method in this paper we construct new traveling wave solutions of some nonlinear evolution equations in mathematical physics via the 3 1 dimensional potential ytsf equation the 3 1 dimensional modified kdv zakharov kuznetsev equation the 3 1 dimensional kadomtsev petviashvili equation and the 1 1 dimensional kdv equation by using a generalized expansion method where g g satisfies the jacobi elliptic equation g 2 p g here we assume that p g is a polynomial of fourth order many new exact solutions in terms of the jacobi elliptic functions are obtained

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

an extension of the korteweg de vries hierarchy arising from a representation of a toroidal lie algebra abstract in this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal lie algebras in the smallest example rank 2 toroidal cover of sl2 we obtain an extension of the kdv hierarchy we use the action of the corresponding infinite dimensional group to construct solutions for these non linear pdes

dynamical models of phyllotaxis leave patterns on plants phyllotaxis present remarkable universal features the underlying morphogenesis phenomenon can be modelled by a class of dynamical systems referred to as inhibitory field models numerical studies have shown that surprisingly often the fixed points of such systems closely correspond to noble numbers this is exactly what is observed in nature in the present paper a proof of this fact is given that applies in particular to the model of thornley but the analysis is general enough to embrace all models usually considered an explicit criterion is given to explain under which circumstances noble numbers are actually selected a stability analysis of the fixed points is also performed explaining observed bifurcations

painleve property auto backlund transformation lax pairs and reduction to the standard form for the korteweg de vries equation with nonuniformities it is demonstrated that the kdv equation with nonuniformities ut a t u b x t u x c t uux d t uxxx e x t 0 has the painleve property if the compatibility condition among the coefficients of it holds bt a lc b bbx dbxxx 2ah hl d c2 dh dt ce x 2a2 al d3 c4 da dt l d c l d c2 d dt l d c where l d dt lg and h t is an arbitrary function of t the auto backlund transformation and lax pairs for this equation are obtained by truncating the laurent expansion furthermore assuming the compatibility condition then the kdv equation with nonuniformities is transformable via suitable variable transformations to the standard kdv

exact periodic cross kink wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional potential ytsf equation abstract in this paper the 3 1 dimensional potential ytsf equation is investigated exact solutions with three wave form including periodic cross kink wave periodic two solitary wave and breather type of two solitary wave solutions are obtained using hirota s bilinear form and generalized three wave approach with the aid of symbolic computation moreover the properties for some new solutions are shown with some figures

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

representations of toroidal extended affine lie algebras abstract we show that the representation theory for the toroidal extended affine lie algebra is controlled by a voa which is a tensor product of four voas a sub voa v hyp of a hyperbolic lattice voa affine g and sl n voas and a virasoro voa a tensor product of irreducible modules for these voas admits the structure of an irreducible module for the toroidal extended affine lie algebra we also show that for n 12 v hyp becomes an exceptional irreducible module for the extended affine lie algebra of rank 0

highest weight modules over pre exp polynomial lie algebras abstract in this paper we introduce pre exp polynomial lie algebras which include loop algebras virasoro like algebras and some quantum torus lie algebras we study highest weight representations of these z n 1 graded lie algebras more precisely we show that non graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite dimensional weight spaces or not and they have all finite dimensional weight spaces if and only if the highest weight is an exp polynomial highest weight we also show that non graded and graded highest weight verma modules with the same highest weight are simultaneously irreducible or not and we determine necessary and sufficient conditions for a verma module to be irreducible

n soliton solutions to a dimensional integrable equation we give explicitly n soliton solutions of a 2c 1 dimensional equation xtc xxxz 4cxxzcxxz 2c 1 x zzz 4 d 0 this equation is obtained by unifying two directional generalizations of the potential kdv equation the closed ring with the potential kp equation and the calogero bogoyavlenskij schiff equation this equation is also a reduction of the kp hierarchy we also find the miura transformation which yields the same ring of the corresponding modified equations the study of higher dimensional integrable systems is one of the central themes in integrable systems a typical example of a higher dimensional integrable system is obtained by modifying the lax operators of a basic equation the potential kdv p kdv equation in this paper the lax pair of the p kdv equation have the form l x t d 2 xcx x t 1 t x t d l x t 3 2

a new variable coefficient korteweg de vries equation based sub equation method and its application to the 3 1 dimensional potential ytsf equation with the aid of maple we present variable coefficient korteweg de vries equation based sub equation method the key idea of our method is to take advantage of the variable coefficient korteweg de vries equation and its various solutions to generate various solutions of nonlinear evolution equations the efficiency of the method can be demonstrated on the 3 1 dimensional potential ytsf equation and we construct successfully its new styles of solutions

plants and fibonacci the universality of many features of plant patterns and phyllotaxis has mystified and intrigued natural scientists for at least four hundred years it is remarkable that to date there is no widely accepted theory to explain the observations we hope that the ideas explained below lead towards increased understanding

new kink multi soliton solutions for the 3 1 dimensional potential yu toda sasa fukuyama equation by means of transformation of unknown function the non integrable system of 3 1 dimensional potential yu toda sasa fukuyama ytsf equation is converted into the combined equation of differently two bilinear forms by the darvishi s idea some new kink multi soliton solutions are obtained by using homoclinic test approach and three wave method respectively

concept of complexity in random dynamical systems we introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system in dynamical systems with small random perturbations this indicator coincides with the rate k of divergence of nearby trajectories evolving under two different noise realizations the meaning of k is discussed in the context of the information theory and it is shown that it can be determined from real experimental data in the presence of strong dynamical intermittency the value of k is very different from the standard lyapunov exponent ensuremath lambda mathrm ensuremath sigma computed considering two nearby trajectories evolving under the same realization of the randomness however the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models textcopyright 1996 the american physical society

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

phyllotaxis an interdisciplinary phenomenon abstract regular phyllotaxis is largely non uniform in the sense that 1 2 and the golden mean are the most frequently observed values of the characteristic parameter divergence the different approaches to this enigmatic phenomenon form a fascinating chapter in the history of science more recently different models have been independently suggested in an attempt to recreate the phenomenon these are dynamic systems in discrete time founded on a simple inhibiting hypothesis ih we analyse the divergences selected by these models and consequently we investigate the ih capability to account for regular phyllotaxis this capability is conditional it depends on the convex concave shape of the elementary inhibitions and involves considering the ih from a statistical standpoint our conclusions are essentially new and they partly conflict with previous predictions because of its generality our analysis applies to botanical as well as to physical systems such as vortex networks

extended diffeomorphism algebras and trajectories in jet space let the dro diffeomorphism reparametrization observer algebra dro n be the extension of diff n diff 1 by its four inequivalent virasoro like cocycles here diff n is the diffeomorphism algebra in n dimensional spacetime and diff 1 describes reparametrizations of trajectories in the space of tensor valued p jets dro n has a fock module for each p and each representation of gl n analogous representations for gauge algebras higher dimensional kac moody algebras are also given the reparametrization symmetry can be eliminated by a gauge fixing procedure resulting in previously discovered modules in this process two dro n cocycles transmute into anisotropic cocycles for diff n thus the fock modules of toroidal lie algebras and their derivation algebras are geometrically explained

dynamical mean field approximation to small world networks of spiking neurons from local to global and or from regular to random couplings by extending a dynamical mean field approximation previously proposed by the author h hasegawa phys rev e 67 041903 2003 we have developed a semianalytical theory which takes into account a wide range of couplings in a small world network our network consists of noisy n unit fitzhugh nagumo neurons with couplings whose average coordination number z may change from local z n to global couplings z n ensuremath 1 and or whose concentration of random couplings p is allowed to vary from regular p 0 to completely random p 1 we have taken into account three kinds of spatial correlations the on site correlation the correlation for a coupled pair and that for a pair without direct couplings the original 2n dimensional stochastic differential equations are transformed to 13 dimensional deterministic differential equations expressed in terms of means variances and covariances of state variables the synchronization ratio and the firing time precision for an applied single spike have been discussed as functions of z and p our calculations have shown that with increasing p the synchronization is worse because of increased heterogeneous couplings although the average network distance becomes shorter results calculated by our theory are in good agreement with those by direct simulations

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

n soliton solutions to the bogoyavlenskii schiff equation and a quest for the soliton solution in 3 1 dimensions we study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the hirota s bilinear form we explicitly discuss the bogoyavlenskii schiff equation in 2 1 dimensions its analytical proof of multisoliton solution and a new feature are given being guided by the strong symmetry we also propose a new equation in 3 1 dimensions

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

irreducible representations of the lie algebra of the diffeomorphisms of ad dimensional torus in this paper we investigate the irreducibility of certain modules for the lie algebra of diffeomorphisms of torus t the lie algebra of diffeomorw 1 1 x phisms can be described as the derivations of a s c t t see 1 d w x w x rss we denote this lie algebra by der a larsson l1 constructed a functor from gl modules to der a modules in this paper we prove that d the image of a finite dimensional irreducible gl module is most often d irreducible the only exceptions are fundamental modules and the onedimensional modules in these cases we describe the sub modules and the quotients surprisingly the same class of modules image of finite dimen w x sional gl module are also constructed in e without any reference to d gl modules they are motivated by the vertex operator constructions in d w x em more details near the end of this introduction we will now describe our results in more detail let gl be the d lie algebras of d d matrices over complex numbers c let gl s sl d d c id where sl is the finite dimensional simple lie algebra of trace zero d matrices and id the identity matrix which is central it is well known that finite dimensional irreducible modules and dominant integral weights are in one one correspondence for a finite dimensional simple lie algebra so let v c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c let id the central element act by a complex number b and denote the corresponding gl module by v c b d a d d larsson defined see 1 6 a functor f for a belonging to c rz from w x gl modules to der a modules in l1 these modules are weight modules d

generalized rate code model for neuron ensembles with finite populations we have proposed a generalized langevin type rate code model subjected to multiplicative noise in order to study stationary and dynamical properties of an ensemble containing a finite number n of neurons calculations using the fokker planck equation have shown that owing to the multiplicative noise our rate model yields various kinds of stationary non gaussian distributions such as ensuremath gamma inverse gaussian like and log normal like distributions which have been experimentally observed the dynamical properties of the rate model have been studied with the use of the augmented moment method amm which was previously proposed by the author from a macroscopic point of view for finite unit stochastic systems in the amm the original n dimensional stochastic differential equations des are transformed into three dimensional deterministic des for the means and fluctuations of local and global variables the dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the amm are in good agreement with those obtained by direct simulation the synchronization in the neuronal ensemble is discussed the variabilities of the firing rate and of the interspike interval are shown to increase with increasing magnitude of multiplicative noise which may be a conceivable origin of the observed large variability in cortical neurons

multiple soliton solutions for the calogero bogoyavlenskii schiff jimbo miwa and ytsf equations abstract in this work we employ the hirota s bilinear method to derive multiple front solutions for nonlinear evolution equations three models namely the 2 1 dimensional calogero bogoyavlenskii schiff equation the 3 1 dimensional jimbo miwa equation and the 3 1 dimensional potential ytsf equation are used as vehicles to conduct the analysis the work shows the power of the bilinear method and the variety of the obtained front solutions

integrate and fire and hodgkin huxley models with current inputs for the integrate and fire model and the hodgkin huxley model we consider how current inputs including wave and square wave affect their outputs firstly the usual approximation is employed to approximate the models with current inputs which quantitatively reveals the difference between instantaneous and noninstantaneous current inputs when the rising time of wave inputs is long or the ratio between the inhibitory and excitatory inputs is close to one the usual approximation fails to approximate the wave inputs in the integrate and fire model for the hodgkin huxley model the usual approximation in general gives an unsatisfying approximation a novel approach based upon a superposition of coloured and white noise is then proposed to replace the usual approximation numerical results show that the novel approach substantially improves the approximation within widely physiologically reasonable regions of the rising time of wave inputs

integrable representations for the twisted full toroidal lie algebras the purpose of this work is to classify irreducible integrable modules of the twisted full toroidal lie algebra with finite dimensional weight spaces and non zero central charges there are three families of such modules two of which are classified

lie symmetry analysis and exact solutions of 3 1 dimensional yutodasasafukuyama equation in mathematical physics in this paper the lie symmetry analysis method has been proposed for finding similarity reduction and exact solutions of nonlinear evolution equation here for illustrating the effectiveness and accuracy of proposed method we have taken 3 1 dimensional yutodasasafukuyama ytfs equation also by using symmetry reduction method we have reduced nonlinear partial differential equation npde to nonlinear ordinary differential equation which has advantage to provide semi analytical solution we have obtained infinitesimal generators the entire geometric vector field commutator table of lie algebra and symmetry group by using lie symmetry analysis method then we have used tanh method for finding new analytical exact solutions of some reduced transform equations

some analytical results about two physical models of phyllotaxis we present analytical results about a generalization of the variational problem of levitov and the experiment of douady and couder we show that the hierarchical selection of noble numbers in these systems occurs with a precision depending on the range of the interaction this precision can be infinite if the interaction is scale invariant

time to first spike in stochastic hodgkin huxley systems abstract the time to first spike is an experimentally observed quantity in laboratory experiments in the auditory somatic and visual sensory modalities the times of first spikes in the corresponding cortical neurons have been implicated as coding much of the information about stimulus properties we describe an analytical approach for determining the time to first spike from a given initial state which may be applied to a general nonlinear stochastic model neuron we illustrate with a standard hodgkin huxley model with a gaussian white noise input current whose drift parameter is and whose variance parameter is partial differential equations pdes of second order are obtained for the first two moments of the time taken for the depolarization to reach a threshold value from rest state as functions of the initial values simulation confirms that for small noise amplitudes a 2 component model is reasonably accurate for small values of the noise parameter including the deterministic case 0 perturbation methods are used to find the moments of the firing time and the results compare favorably with those from simulation the approach is accurate for almost all when is above threshold for action potentials in the absence of noise and over a considerable range of values of when is as small as 2 the same methods may be applied to models similar to hodgkin huxley which involve channels for additional or different ionic currents