Temperature-modulated synchronization transition in coupled neuronal oscillators.

We study two firing properties to characterize the activities of a neuron: frequency-current (f-I) curves and phase response curves (PRCs), with variation in the intrinsic temperature scaling parameter (µ) controlling the opening and closing of ionic channels. We show a peak of the firing frequency for small µ in a class I neuron with the I value immediately after the saddle-node bifurcation, which is entirely different from previous experimental reports as well as model studies. The PRC takes a type II form on a logarithmic f-I curve when µ is small. Then, we analyze the synchronization phenomena in a two-neuron network using the phase-reduction method. We find common µ-dependent transition and bifurcation of synchronizations, regardless of the values of I. Such results give us helpful insight into synchronizations tuned with a sinusoidal-wave temperature modulation on neurons.

Modeling asset price processes based on mean-field framework.

We propose a model of the dynamics of financial assets based on the mean-field framework. This framework allows us to construct a model which includes the interaction among the financial assets reflecting the market structure. Our study is on the cutting edge in the sense of a microscopic approach to modeling the financial market. To demonstrate the effectiveness of our model concretely, we provide a case study, which is the pricing problem of the European call option with short-time memory noise.

Generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena.

The integrate-and-fire (IF) neuron model and the piecewise-linear version of FitzHugh-Nagumo (PL) neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena. Although the two models are regarded to be different type, we show a certain equivalence between them by deriving the coupled IF model of an improved version with a firing duration from the recovery variable coupled system of the PL model, under taking the limit of mu-->0 without a loss of any coupling properties. In the coupled IF model with the duration time, the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model (synaptic strength, decaying relaxation rate of the synaptic coupling, and the parameter exhibiting the firing duration). We find some irregularly synchronous behavior with or without constant firing order alternations. We show that the duration of an impulse plays an important role in synchronization phenomena.

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.

Statistical mechanics of stochastic neural networks: Relationship between the self-consistent signal-to-noise analysis, Thouless-Anderson-Palmer equation, and replica symmetric calculation approaches.

We study the statistical mechanical aspects of stochastic analog neural network models for associative memory with correlation type learning. We take three approaches to derive the set of the order parameter equations for investigating statistical properties of retrieval states: the self-consistent signal-to-noise analysis (SCSNA), the Thouless-Anderson-Palmer (TAP) equation, and the replica symmetric calculation. On the basis of the cavity method the SCSNA can be generalized to deal with stochastic networks. We establish the close connection between the TAP equation and the SCSNA to elucidate the relationship between the Onsager reaction term of the TAP equation and the output proportional term of the SCSNA that appear in the expressions for the local fields.

Stability analysis of mean-field-type nonlinear Fokker-Planck equations associated with a generalized entropy and its application to the self-gravitating system.

Multidimensional nonlinear Fokker-Planck equations of mean-field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions. Two types of eigenvalue equations are studied. The nonlinear Fokker-Planck equations are shown to exhibit an H theorem with a Liapunov functional that takes the form of a free energy involving generalized entropies of Tsallis. The second-order variation of the Liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean-field coupling potential. Assuming quasiequilibrium for the velocity distribution, the reduced eigenvalue equation with space coordinates alone is also obtained. The alternative type of eigenvalue equation based on the linearization of the nonlinear Fokker-Planck equations is presented. Taking the mean-field coupling potential to be the gravitational one, the nonlinear Fokker-Planck equation in terms of three-dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean-field model of self-gravitating system. By solving the eigenvalue equation for the eigenfunction with 0 eigenvalue, the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed.

Spiking neuron models with excitatory or inhibitory synaptic couplings and synchronization phenomena.

We investigate synchronization phenomena in a system of two piecewise-linear-type model neurons with excitatory or inhibitory synaptic couplings. Employing the phase plane analysis and a singular perturbation approach to split the dynamics into slow and fast ones, we construct analytically the Poincaré map of the solution to the piecewise-linear equations. We investigate conditions for the occurrence of synchronized oscillations of in phase as well as of antiphase in terms of parameters representing the strength of the synaptic coupling and the decaying relaxation rate of the synaptic dynamics. We present the results of numerical simulations that agree with our theoretical ones.