arXiv subjects

Hideo Hasegawa

Publications and source records attributed to Hideo Hasegawa.

At least 55 records · Page 3Linked to original sources

arXiv2007

A moment approach to non-Gaussian colored noises

The Langevin system subjected to non-Gaussian noise has been discussed, by using the second-order moment approach with two kinds of models for generating the noise. We have derived the effective differential equation (DE) for a variable $x$, from which the stationary probability distribution $P(x)$ has been calculated with the use of the Fokker-Planck equation. The result of $P(x)$ calculated by the moment method is compared to several expressions obtained by different methods such as the universal colored noise approximation (UCNA) [Jung and Hänggi, Phys. Rev. A {\bf 35}, 4464 (1987)] and the functional-integral method. It has been shown that our $P(x)$ is in good agreement with that of direct simulations (DSs). We have also discussed dynamical properties of the model with an external input, solving DEs in the moment method.

cond-mat.stat-mech↗

arXiv2007

A Generalized Rate Model for Neuronal Ensembles

There has been a long-standing controversy whether information in neuronal networks is carried by the firing rate code or by the firing temporal code. The current status of the rivalry between the two codes is briefly reviewed with the recent studies such as the brain-machine interface (BMI). Then we have proposed a generalized rate model based on the {\it finite} $N$-unit Langevin model subjected to additive and/or multiplicative noises, in order to understand the firing property of a cluster containing $N$ neurons. The stationary property of the rate model has been studied with the use of the Fokker-Planck equation (FPE) method. Our rate model is shown to yield various kinds of stationary distributions such as the interspike-interval distribution expressed by non-Gaussians including gamma, inverse-Gaussian-like and log-normal-like distributions. The dynamical property of the generalized rate model has been studied with the use of the augmented moment method (AMM) which was developed by the author [H. Hasegawa, J. Phys. Soc. Jpn. 75 (2006) 033001]. From the macroscopic point of view in the AMM, the property of the $N$-unit neuron cluster is expressed in terms of {\it three} quantities; $μ$, the mean of spiking rates of $R=(1/N) \sum_i r_i$ where $r_i$ denotes the firing rate of a neuron $i$ in the cluster: $γ$, averaged fluctuations in local variables ($r_i$): $ρ$, fluctuations in global variable ($R$). We get equations of motions of the three quantities, which show $ρ\sim γ/N$ for weak couplings. This implies that the population rate code is generally more reliable than the single-neuron rate code. Our rate model is extended and applied to an ensemble containing multiple neuron clusters.

cond-mat.dis-nn↗

arXiv2007

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 {\it finite} $N$ neurons. Calculations using the Fokker-Planck equation (FPE) have shown that owing to the multiplicative noise, our rate model yields various kinds of stationary non-Gaussian distributions such as gamma, inverse-Gaussian-like and log-normal-like distributions, which have been experimentally observed. 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 with a macroscopic point of view for finite-unit stochastic systems. In the AMM, original $N$-dimensional stochastic differential equations (DEs) are transformed into three-dimensional deterministic DEs for means and fluctuations of local and global variables. 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. Variabilities of the firing rate and of the interspike interval (ISI) are shown to increase with increasing the magnitude of multiplicative noise, which may be a conceivable origin of the observed large variability in cortical neurons.

cond-mat.dis-nn↗

arXiv2006

Stationary and dynamical properties of finite $N$-unit Langevin models subjected to multiplicative noises

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. Jpn. {\bf 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.

cond-mat.dis-nn↗

arXiv2006

Effects of multiplicative noises on synchronization in finite $N$-unit stochastic ensembles: Augmented moment approach

We have studied the synchronization in finite $N$-unit FitzHugh-Nagumo neuron ensembles subjected to additive and multiplicative noises, by using the augmented moment method (AMM) which is reformulated with the use of the Fokker-Planck equation. It has been shown that for diffusive couplings, the synchronization may be enhanced by multiplicative noises while additive noises are detrimental to the synchronization. In contrast, for sigmoid coupling, both additive and multiplicative noises deteriorate the synchronization. The synchronization depends not only on the type of noises but also on the kind of couplings.

cond-mat.dis-nn↗

arXiv2006

$N$-dependent Multiplicative-Noise Contributions in Finite $N$-unit Langevin Models: Augmented Moment Approach

Finite $N$-unit Langevin models with additive and multiplicative noises have been studied with the use of the augmented moment method (AMM) previously proposed by the author [H. Hasegawa, Phys. Rev E {\bf 67}, 041903 (2003)]. Original $N$-dimensional stochastic equations are transformed to the three-dimensional deterministic equations for means and fluctuations of local and global variables. Calculated results of our AMM are in good agreement with those of direct simulations (DS). We have shown that although the effective strength of the additive noise of the $N$-unit system is scaled as $β(N)=β(1)/\sqrt{N}$, it is not the case for multiplicative noise [$α(N) \neq α(1)/\sqrt{N}$], where $α(N)$ and $β(N)$ denote the strength of multiplicative and additive noises, respectively, for the size-$N$ system. It has been pointed out that the naive assumption of $α(N) = α(1)/\sqrt{N}$ leads to result which violates the central-limit theorem and which does not agree with those of DS and AMM.

cond-mat.stat-mech↗

arXiv2005

Nonextensive thermodynamics of a cluster consisting of $M$ Hubbard dimers ($M=1,2,3$ and $\infty$)

The thermodynamical property of a small cluster including $M$ Hubbard dimers, each of which is described by the two-site Hubbard model, has been discussed within the nonextensive statistics (NES). We have calculated the temperature dependence of the energy, entropy, specific heat and susceptibility for $M = 1, 2, 3$ and $\infty$, assuming the relation between the entropic index $q$ and the cluster size $N$ given by $q=1+2/N$ ($N = 2 M$ for $M$ dimers), which was previously derived by several methods. For relating the physical temperature $T$ to the Lagrange multiplier $β$, two methods have been adopted: $T=1/k_B β$ in the method A [Tsallis {\it et al.} Physica A {\bf 261} (1998) 534], and $T=c_q/k_B β$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281} (2001) 126], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, and $p_i$ the probability distribution of the $i$th state. The susceptibility and specific heat of spin dimers described by the Heisenberg model have been discussed also by using the NES with the methods A and B. A comparison between results calculated by the two methods suggests that the method B may be better than the method A for small-scale systems.

cond-mat.stat-mech↗

arXiv2005

Synchronizations in small-world networks of spiking neurons: Diffusive versus sigmoid couplings

By using a semi-analytical dynamical mean-field approximation previously proposed by the author [H. Hasegawa, Phys. Rev. E, {\bf 70}, 066107 (2004)], we have studied the synchronization of stochastic, small-world (SW) networks of FitzHugh-Nagumo neurons with diffusive couplings. The difference and similarity between results for {\it diffusive} and {\it sigmoid} couplings have been discussed. It has been shown that with introducing the weak heterogeneity to regular networks, the synchronization may be slightly increased for diffusive couplings, while it is decreased for sigmoid couplings. This increase in the synchronization for diffusive couplings is shown to be due to their local, negative feedback contributions, but not due to the shorten average distance in SW networks. Synchronization of SW networks depends not only on their structure but also on the type of couplings.

cond-mat.dis-nn↗

arXiv2005

Nonextensive aspects of small-world networks

Nonextensive aspects of the degree distribution in Watts-Strogatz (WS) small-world networks, $P_{SW}(k)$, have been discussed in terms of a generalized Gaussian (referred to as {\it $Q$-Gaussian}) which is derived by the three approaches: the maximum-entropy method (MEM), stochastic differential equation (SDE), and hidden-variable distribution (HVD). In MEM, the degree distribution $P_Q(k)$ in complex networks has been obtained from $Q$-Gaussian by maximizing the nonextensive information entropy with constraints on averages of $k$ and $k^2$ in addition to the normalization condition. In SDE, $Q$-Gaussian is derived from Langevin equations subject to additive and multiplicative noises. In HVD, $Q$-Gaussian is made by a superposition of Gaussians for random networks with fluctuating variances, in analogy to superstatistics. Interestingly, {\it a single} $P_{Q}(k)$ may describe, with an accuracy of $\mid P_{SW}(k)-P_Q(k)\mid \siml 10^{-2} $, main parts of degree distributions of SW networks, within which about 96-99 percents of all $k$ states are included. It has been demonstrated that the overall behavior of $P_{SW}(k)$ including its tails may be well accounted for if the $k$-dependence is incorporated into the entropic index in MEM, which is realized in microscopic Langevin equations with generalized multiplicative noises.

cond-mat.dis-nn↗

arXiv2005

Non-extensive thermodynamics of transition-metal nanoclusters

In recent years, much study has been made by applying the {\it non-extensive statistics} (NES) to various non-extensive systems where the entropy and/or energy are not necessarily proportional to the number of their constituent subsystems. The non-extensivity may be realized in many systems such as physical, chemical and biological ones, and also in small-scale nanosystems. After briefly reviewing the recent development in nanomagnetism and the NES, I have discussed, in this article, NES calculations of thermodynamical properties of a nanocluster containing noninteracting $M$ dimers. With bearing in mind a transition-metal nanocluster, each of the dimers is assume to be described by the two-site Hubbard model ({\it a Hubbard dimer}). The temperature and magnetic-field dependences of the specific heat, magnetization and susceptibility have been calculated by changing M=1, 2, 3 and $\infty$, results for $M=\infty$ corresponding to those of the conventional Boltzman-Gibbs statistics (BGS). It has been shown that the thermodynamical property of nanoclusters containing a small number of dimers is considerably different from that of macroscopic counterparts calculated within the BGS The specific heat and susceptibility of spin dimers described by the Heisenberg model have been discussed also by employing the NES.

cond-mat.mes-hall↗

arXiv2005

Nonextensive thermodynamics of the two-site Hubbard model

Thermodynamical properties of canonical and grand-canonical ensembles of the half-filled two-site Hubbard model have been discussed within the framework of the nonextensive statistics (NES). For relating the physical temperature $T$ to the Lagrange multiplier $β$, two methods have been adopted: $T=1/k_B β$ in the method A [Tsallis {\it et al.} Physica A {\bf 261} (1998) 534], and $T=c_q/k_B β$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281} (2001) 126], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, $p_i$ the probability distribution of the $i$th state, and $q$ the entropic index. Temperature dependences of specific heat and magnetic susceptibility have been calculated for $1 \lleq q \lleq 2$, the conventional Boltzman-Gibbs statistics being recovered in the limit of $q = 1$. The Curie constant $Γ_q$ of the susceptibility in the atomic and low-temperature limits ($t/U \to 0, T/U \to 0$) is shown to be given by $Γ_q=2 q 2^{2(q-1)}$ in the method A, and $Γ_q=2 q$ in the method B, where $t$ stands for electron hoppings and $U$ intra-atomic interaction in the Hubbard model. These expressions for $Γ_q$ are shown to agree with the results of a free spin model which has been studied also by the NES with the methods A and B. A comparison has been made between the results for canonical and grand-canonical ensembles of the model.

cond-mat.stat-mech↗

arXiv2004

Dynamical mean-filed 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 (DMA) previously proposed by the author [H. Hasegawa, Phys. Rev. E {\bf 67}, 41903 (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 (FN) neurons with couplings whose average coordination number $Z$ may change from local ($Z \ll N $) to global couplings ($Z=N-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 $2 N$-dimensional {\it stochastic} differential equations are transformed to 13-dimensional {\it 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 {\it worse} because of increased heterogeneous couplings, although the average network distance becomes shorter. Results calculated by out theory are in good agreement with those by direct simulations.

cond-mat.dis-nn↗

arXiv2004

An augmented moment method for stochastic ensembles with delayed couplings: I. Langevin model

By employing a semi-analytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E {\bf 67}, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an $N$-unit ensemble described by linear and nonlinear Langevin equations with delays. In AMM, original $N$-dimensional {\it stochastic} delay differential equations (SDDEs) are transformed to infinite-dimensional {\it 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 (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. The synchronization induced by an applied single spike is shown to be enhanced in the nonlinear Langevin ensemble with model parameters locating at the transition between oscillating and non-oscillating states. Results calculated by AMM6 are in good agreement with those obtained by direct simulations.

cond-mat.stat-mech↗

arXiv2004

An 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 semi-analytical augmented moment method (AMM) which has been proposed in a recent paper [H. Hasegawa, E-print: cond-mat/0311021]. For $N$-unit FN neuron ensembles, AMM transforms original $2N$-dimensional {\it stochastic} delay differential equations (SDDEs) to infinite-dimensional {\it 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 (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 {\it fluctuation-induced} enhancement at the transition between non-oscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.

cond-mat.dis-nn↗

arXiv2004

Graded persisting activity of heterogeneous neuron ensembles subject to white noises

Effects of distractions such as noises and parameter heterogeneity have been studied on the firing activity of ensemble neurons, each of which is described by the extended Morris-Lecar model showing the graded persisting firings with the aid of an included ${\rm Ca}^{2+}$-dependent cation current. Although the sustained activity of {\it single} neurons is rather robust in a sense that the activity is realized even in the presence of the distractions, the graded frequency of sustained firings is vulnerable to them. It has been shown, however, that the graded persisting activity of {\it ensemble} neurons becomes much robust to the distractions by the pooling (ensemble) effect. When the coupling is introduced, the synchronization of firings in ensemble neurons is enhanced, which is beneficial to firings of target neurons.

q-bio.NC↗

arXiv2003

Spike propagation for spatially correlated inputs through noisy multilayer networks

Spike propagation for spatially correlated inputs in layered neural networks has been investigated with the use of a semi-analytical dynamical mean-field approximation (DMA) theory recently proposed by the author [H. Hasegawa, Phys. Rev. E {\bf 67}, 041903 (2003)]. Each layer of the network is assumed to consist of FitzHugh-Nagumo neurons which are coupled by feedforward couplings. Applying single spikes to the network with input-time jitters whose root-mean-square (RMS) value and the spatial correlation are $σ_I$ and $s_I$, respectively, we have calculated the RMS value ($σ_{Om}$) and the correlation ($s_{Om}$) of jitters in output-firing times on each layer $m$. For all-to-all feedforward couplings, $s_{Om}$ gradually grows to a fairly large value as spikes propagate through the layer, even for inputs without the correlation. This shows that for the correlation to be in the range of observed value of 01-0.3, we have to take into account noises and more realistic feedforward couplings. Model calculations including local feedforward connection besides all-to-all feedforward couplings in multilayers subject to white noises, have shown that in a long multilayer, $σ_{Om}$ and $s_{Om}$ converge to fixed-point values which are determined by model parameters characterizing the multilayer architecture. Results of DMA calculations are in fairly good agreement with those of direct simulations although the computational time of the former is much smaller than that of the latter.

cond-mat.dis-nn↗

arXiv2003

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 {\bf 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 $KN$-dimensional {\it stochastic} DEs have been replaced by $K(K+2)$-dimensional {\it deterministic} DEs expressed in terms of means and the second-order moments of {\it local} and {\it 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. Results calculated by DMA theory are in good agreement with those obtained by direct simulations.

cond-mat.dis-nn↗

arXiv2002

Dynamical mean-field approximation to coupled active rotator networks subject to white noises

A semi-analytical dynamical mean-field approximation (DMA) has been developed for large but finite $N$-unit active rotator (AR) networks subject to individual white noises. Assuming weak noises and the Gaussian distribution of state variables, we have derived equations of motions for moments of local and global variables up to the {\it infinite} order. In DMA, the original $N$-dimensional {\it stochastic} differential equations (DEs) are replaced by three-dimensional {\it deterministic} DEs while the conventional moment method yields $(1/2)N(N+3)$ deterministic DEs for moments of local variables. We have discussed the characters of the stationary state, the time-periodic state and the random, disordered state, which are realized in excitable AR networks, depending on the model parameters. It has been demonstrated that although fluctuations of global variable vary as $1/\sqrt{N}$ when $N$ is increased, those of local variables remain finite even for $N \to \infty$. Results calculated with the use of our DMA are compared to those obtained by direct simulations and by the Fokker-Planck equation which is applicable to the $N=\infty$ AR model. The advantage and disadvantage of DMA are also discussed.

cond-mat.dis-nn↗