Akihisa Ichiki

Thermal effects on phase response curves and synchronization transition

We study temperature modulated synchronization phenomena in the Morris-Lecar (ML) models with synaptic couplings. Little has been known about the thermal effects on synchronization in a real nervous system. Dynamical mechanisms on such synchronization are investigated by linear stability analysis with phase descriptions for the ML type, in order to understand the effects of temperature on the phase response curve (PRC). We find two types of PRC shape modulation induced by changes in temperature that depend on an injected current amplitude: (1) the PRC shape switch between the type-I and type-II, and (2) the almost unchanged appearance of a type-II PRC. A large variety of synchronization is demonstrated with these changes in the PRC shapes.

Phase transitions driven by Lévy stable noise: Exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations

Phase transitions and effects of external noise on many-body systems are one of the main topics in physics. In mean-field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of Levy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with Levy noise have attracted much attention and behaviors of systems with double-well potential subjected to Levy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an analytically solvable model to study the occurrence of phase transitions driven by Levy stable noise.

The Thouless–Anderson–Palmer equation for an analogue neural network with temporally fluctuating white synaptic noise

Effects of synaptic noise on the retrieval process of associative memory neural networks are studied from the viewpoint of neurobiological and biophysical understanding of information processing in the brain. We investigate the statistical mechanical properties of stochastic analogue neural networks with temporally fluctuating synaptic noise, which is assumed to be white noise. Such networks, in general, defy the use of the replica method, since they have no energy concept. The self-consistent signal-to-noise analysis (SCSNA), which is an alternative to the replica method for deriving a set of order parameter equations, requires no energy concept and thus becomes available in studying networks without energy functions. Applying the SCSNA to stochastic networks requires the knowledge of the Thouless–Anderson–Palmer (TAP) equation which defines the deterministic networks equivalent to the original stochastic ones. The study of the TAP equation which is of particular interest for the case without energy concept is very less, while it is closely related to the SCSNA in the case with energy concept. This paper aims to derive the TAP equation for networks with synaptic noise together with a set of order parameter equations by a hybrid use of the cavity method and the SCSNA.

Effects of noise on synchronization phenomena exhibited by mean-field coupled limit cycle oscillators with two natural frequencies

AbstractRelationships between inter-cluster synchronization phenomena and external noise are studied on the basis of noise level-freeanalysis. We consider a mean-field model of ensembles of coupled limit cycle oscillators with two natural frequencies, which aresubjected to external white Gaussian noise. Using a nonlinear Fokker-Planck equation approach, weanalytically derive the orderparameter equations associated with nonequilibrium phase transitions in the thermodynamic limit. Solving them numerically, wesystematically investigate the model parameter dependences of the appearance and disappearance of synchronization phenomena.Demonstrating bifurcations from chaotic attractors in the deterministic limit to limit cycle attractors with increasing noise intensity,we confirm the occurrence of nonequilibrium phase transitio ns including inter-cluster synchronization induced by external whiteGaussian noise.Keywords: Noise induced synchronization, Mean-field model, Nonlinear Fokker-Planck equation, Nonequilibrium phasetransitions, Stochastic limit cycle oscillators1. IntroductionEffects of noise on synchronization phenomena in oscilla-tory systems have recently attracted much attention from manyresearchers. Synchronization phenomena are ubiquitous onesobserved in various fields of natural sciences [1]. In neuro-sciences, neurons in the basal ganglia exhibit more synchronyin Parkinson’sdisease than in normal state, suggesting that neu-ral information coding is closely related to the synchronizationphenomena [2]. For such a reason, to study how the noise ex-erts its influence on the structure of synchronization will b e ofparamount importance from the viewpoint of nonlinear dynam-ical control involving changes in synchrony of oscillatory sys-tems.On one hand, someone would intuitively suppose that thepresence of noise might deteriorate the degree of synchro-nization of oscillatory systems. Breakdown of synchroniza-tion due to external noise of an ensemble of limit cycle os-cillators was reported [3]. On the other hand, the oppositephenomena of noise-induced synchronization are becomingan active field of the study of nonlinear dynamical systems[4, 5, 6, 7, 8, 9]. Noise induced synchronization in coupled ex-citable systems/active rotators is investigated both analytically[4, 5] and numerically [6]. Among analytical studies on syn-chronization phenomena of ensemble of limit cycle oscillatorsis the phase reduction analysis. Such type of studies revealedthe effects of common noise on synchronization of uncoupledoscillators [7] and uncoupled two populations of oscillators [8].These studies, however, are restricted to the case with weaknoise. Numerical approaches to noise induced synchronizationof coupled limit cycle oscillators might include the possibili-ties that what are so obtained are subjected to a finite size e ffectwithout taking the thermodynamic limit as in Ref. [9].To avoid such limitations, one may take nonlinear Fokker-Planck equation approaches, which are very closely related tothe study of (thermal) equilibrium phase transitions. Present-ing the validity of an H theorem for a nonlinear Fokker-Planckequation for a stochastic system of mean-field coupled over-damped oscillators, Shiino studied statistical behaviors of thesystem exhibiting equilibrium phase transitions [10] (see alsoRef. [11, 12]). Furthermore, in the case of nonequilibriumphase transitions, one can take advantage of using nonlinearFokker-Planck equations to exactly derive the time evolution ofthe order-parameter equations in the thermodynamic limit forsystems of nonlinearly mean-field coupled oscillators. There,the nonlineardynamic aspects of phase transitions were investi-gated, yielding the occurrence of chaos-nonchaos phase transi-tions and those including intra-cluster synchronization inducedby external noise [13]. Purely noise induced phase transitionsinvolving chaos-nonchaos phase transitions in an ensembleoflimit cycle oscillators were also explored [14]. A nonlinearlycoupled system with time delay was found to exhibit varioustypes of nonequilibrium phase transitions [15].In the present study, we apply the nonlinear Fokker-Planckapproach to noise induced inter-cluster synchronization phe-nomena of coupled limit cycle oscillators. Dealing with a solv-able model based on the mean-field concept to derive order pa-rameter equations [13, 14, 15], we study the effects of noiseon synchronizationin the thermodynamiclimit and nonequilib-rium stationary states, for which numerical approaches to solv-ing stochastic differential equations do not properly work. Ourresults showtheappearanceofnonequilibriumphasetransitionsinvolving inter-cluster synchronization. Since the modelof os-cillators is based on that of analog neural networks, behaviors

Thouless-Anderson-Palmer equation and self-consistent signal-to-noise analysis for the Hopfield model with three-body interaction.

The self-consistent signal-to-noise analysis (SCSNA) is an alternative to the replica method for deriving the set of order parameter equations for associative memory neural network models and is closely related with the Thouless-Anderson-Palmer equation (TAP) approach. In the recent paper by Shiino and Yamana the Onsager reaction term of the TAP equation has been found to be obtained from the SCSNA for Hopfield neural networks with two-body interaction. We study the TAP equation for an associative memory stochastic analog neural network with three-body interaction to investigate the structure of the Onsager reaction term, in connection with the term proportional to the output characteristic to the SCSNA. We report on the SCSNA framework for analog networks with three-body interactions as well as provide a recipe based on the cavity concept that involves two cavities and the hybrid use of the SCSNA to obtain the TAP equation.

A phase reduction method for weakly coupled stochastic oscillator systems

We progressively propose a generalized phase reduction method for a stochastic system of weakly coupled oscillators, regardless of the noise intensity, and analyze dynamical behavior in such a system. This is because noise effects on a phase space were so far described for an uncoupled single stochastic oscillator, subjected only to weak noise. Our method is established with definition of a phase of the distribution of state variables, rather than by defining distributions on a phase space. The stochastic system of weakly coupled oscillators can then be reduced straightforward to one dimensional phase dynamics. It is also confirmed that our method can be applied into the deterministic system without any noise intensity.

Computational analysis on temperature dependent synchronization transition in neurons

We study temperature dependent synchronization phenomena in a pair of synaptically coupled neuron models, because it is not so well-known about thermal effects on synchronization in a real nervous system. In the pair system, we employ various different types of single spiking models such as the Morris-Lecar (ML), the Hodgkin-Huxley (HH), the Wang-Buszaki (WB) and the FitzHugh-Nagumo (FHN) type models. In addition, the model of chemical synapses used here is given by alpha-function. To consider the effects of temperature on the pair system, we assume that a common temperature factor controls the time constants of gating variables of ionic and synaptic dynamics. Dynamical mechanisms of synchronization transitions are investigated by the linear stability analysis using phase description for each type. The phase description is closely related with the so-called phase reduction method. We employ the phase reduction method to understand the effects of temperature on the phase response curves (PRCs). In a single neuron, we classify the temperature modulation of the PRC into the two main types, depending on an injected current amplitude: (1) the PRC shape switch from type-I to type-II, and (2) the almost unchanged appearance of a type-II PRC. In a pair of the coupled neurons, a variety of synchronization transitions are found within a very narrow range of temperature. We try to explain how synchronous behaviors are dependent on or independent of model types. In particular, when a temperature scaling factor is small, we find a common property on bifurcation phenomena related to synchronization, regardless of the neuron model types such as the ML, HH, WB and FHN.

Stochastic phenomena of synchronization in ensembles of mean-field coupled limit cycle oscillators with two native frequencies

We study effects of independent white noise on synchronization phenomena in ensembles of coupled limit cycle oscillators with different native frequencies. We consider a simple model where the ensemble consists of two inter-connected clusters with own native frequencies and mean-field couplings are introduced between intra- and inter-clusters. Taking advantage of the nonlinear mean-field coupling concept together with the law of large numbers valid in the thermodynamic limit, we employ a nonlinear Fokker-Planck equation approach that turns out to be a noise level-free analysis, to analytically derive the time evolution of the order parameters. Showing the occurrence of bifurcations from chaotic attractors in the deterministic limit to limit cycle ones with increasing noise intensity, we confirm the occurrence of nonequilibrium phase transitions including inter-cluster synchronization induced by external noise.

Thouless-Anderson-Palmer Equation for Associative Memory Neural Network Models with Fluctuating Couplings

We derive Thouless-Anderson-Palmer (TAP) equations and order parameter equations for stochastic analog neural network models with fluctuating synaptic couplings. Such systems with finite number of neurons originally have no energy concept. Thus they defy the use of the replica method or the cavity method, which require the energy concept. However for some realizations of synaptic noise, the systems have the effective Hamiltonian and the cavity method becomes applicable to derive the TAP equations.