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The simplified models of neural networks based on biophysical Hodgkin–Huxley neurons are studied with a focus on coherent-phase dynamics. In our approach, each neuron is considered as a nonlinear oscillator, and collective dynamics of a mesoscopic network of neural oscillators are studied using the methods of nonlinear dynamics. We explore the mechanisms for synchrony, clustering and their breakup in the synaptic parameter space and discuss implications to temporal aspects of neural-information processing.

The idea of phase dynamics originally considered for regular non-equilibrium patterns is applied to irregular patterns. In the dissipation-dominated regime, a slow modulation of irregular pattern is shown to decay in time according to the diffusion law. Possible relevance of this result for the ultra-slow diffusion modes observed in glass-forming systems is discussed. In this connection we speculate about a possibility of appearance of some sorts of soft structures resembling quasi-periodic microcrystalline states in glass-forming systems.

Soap froths as archetypal disordered cellular structures, exhibiting spatial and temporal evolution, have been predominantly studied in terms of their topological properties. Recently, constrained geometries in froth have attracted attention, since these arise both from existing or natural structure or, more usually, can be artificially configured. Using a direct simulation method, which retains detailed information on bubble (cell) mechanisms, we have investigated dynamics in 2D froths with various initial structures corresponding to controlled disorder. In particular, we examine the special case of a defect ring surrounding a central inclusion in a uniform froth, for different number of defects and ring radius. This geometry permits comparison with shell theory, as well as insight on dynamics of a virtual phase (as defined for persistence and non-equilibrium processes in general). It appears that defect location and pattern of inclusion in the virtual phase cause considerable variation in the evolutionary behaviour, leading to non-universal exponents for the phase dynamics. This is probably explained by the fact that the froth is still in the transient period over simulation time scales, rather than achieving the final stage of persistence. However, distinctive patterns of response can be identified for the different froth regions, despite system size limitations, with topological and other properties indicating that a quasi-equilibrium results.

In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamics is well characterized by means of the variance of the CEF phase component.

A brief review is presented about our recent work on computer studies of a continuum model describing spinodal decomposition of binary fluid, which quantitatively accounts for corresponding experimental results. We then switch over to discussing problems of glass transitions. First we consider the ultraslow mode observed in fragile glass-formers, where the diffusive behavior of this mode is explained in terms of phase dynamics describing slow relaxation of spatial heterogeneous structure resulting from spontaneous break-down of continuous translational symmetry. Next we present a stochastic model for density fluctuation in supercooled liquids, which is mapped onto a certain kinetic lattice gas model which is convenient for Monte Carlo simulation studies.

The dynamics of many-fermion systems is analyzed in terms of a mean-field expansion obtained from a functional integral representation of quantum amplitudes proposed by Kerman, Levit and Troudet. The dynamical problem is formulated as an initial conditions problem. Lowest order corrections to the mean-field (stationary phase) dynamics, including exchange effects, involve dynamical two-body correlations. The effective dynamics of the one-body density is analyzed in this framework.