Andrey A. Polezhaev

This paper presents a generalization of the correlation measure in space, which has been commonly used in the analysis of partial synchronization regimes, in particular chimera states, in systems of interacting oscillators. The chosen approach is based on a generalization of classical centered differences to their fractional analogue. The presented spectrum of fractional measures of spatial correlation makes it possible to classify partial synchronization modes developed at the level of total synchronization and at the level of phase waves of certain scales in discrete systems. The latter case is essential for neuroscience applications and mathematical study of the dynamic behavior of networks of interacting neurons, due to the importance of both partial synchronization states and large-scale cortical waves in sensory information processing and memory consolidation.

The partial synchronization states of collective activity, as well as the spike avalanches realization in systems of interacting neurons, are extremely important distinguishing features of the neocortical circuits that have multiple empirical validations. However, at this stage, there is a limited number of studies highlighting their potential interrelationship at the level of nonlinear mathematical models. In this study, we investigate the development of chimera states and the emergence of spike avalanches in superdiffusive neural networks, as well as analyze the system's approach to quasicriticality. The analysis of the available ideas suggests that partial synchronization states, spike avalanches, and quasicritical neuronal dynamics are all directly implicated in core cognitive functions such as information processing, attention, and memory. Given this fundamental role, the results presented in this work could have significant implications for both theoretical neuroscience and applied machine learning, particularly in the development of reservoir computing systems.

Partial synchronization states exhibit remarkable diversity in their manifestations, many of which are not yet fully understood. In the present study, a recently discovered type called “shadowed waves,” characterized by the formation of large-scale order emerging against a background of developed disorder, has been analyzed in detail through numerical simulations and analytical tools applied to a three-component reaction-diffusion system. Our work provides conceptual clarity by positioning these novel partial synchronization states in the broader hierarchy of dynamic regimes. Specifically, this study demonstrates that the organized emergence of structures within a heterogeneous dynamics is inherently linked to the intermediate nature of this dynamic regime between chimera states and stationary patterns formation. Based on the dynamic features of large-scale and small-scale manifestations, we have also introduced an extended classification for shadowed waves. Similarly, it is shown that even minimal deviations from a single-immobile-component approximation, which simplifies system complexity, can disrupt shadowed waves entirely. This finding highlights their unique fragility compared to other partial synchronization states. Given their novelty and potential implications for understanding collective behavior across various systems, including neuroscience and ecology, this research offers foundational insights into the robustness and classification of emergent patterns, in particular partial synchronization states, in coupled oscillator networks.

The development of synchronization and chimera states plays an extremely important role in the collective activity of systems of interacting neurons. For this reason, it is crucial from both fundamental and applied perspectives to investigate empirically relevant (in a biological sense) configuration models that support analytical studies of the development of partial synchronization states. Herein we consider a system of superdiffusively coupled neurons from the limiting case perspective of the linear stability of two-component reaction-superdiffusion equations. It is found that extended linear analysis in the parameter space of the fractional Laplace operator exponents is able to characterize the configuration features of networks of superdiffusively coupled neurons supporting both the development of full and partial synchronization, as well as states of full incoherence. The possibility of not only numerical but also analytical study in a continuous perspective of superdiffusion neural networks opens new opportunities for problems of chimera state control, as well as providing extended possibilities to design functional zones on the basis of these systems.

In this paper, a network of interacting neurons based on a two-component system of reaction-superdiffusion equations with fractional Laplace operator responsible for the coupling configuration and nonlinear functions of the Hindmarsh–Rose model is considered. The process of synchronization transition in the space of the fractional Laplace operator exponents is studied. This parametric space contains information about both the local interaction strength and the asymptotics of the long-range couplings for both components of the system under consideration. It is shown that in addition to the homogeneous transition, there are regions of inhomogeneous synchronization transition in the space of the fractional Laplace operator exponents. Weak changes of the corresponding exponents in inhomogeneous zones are associated with the significant restructuring of the dynamic modes in the system. The parametric regions of chimera states, solitary states, phase waves, as well as dynamical modes combining them, are determined. The development of filamentary structures associated with the manifestation of different partial synchronization modes has been detected. In view of the demonstrated link between changes in network topology and internal dynamics, the data obtained in this study may be useful for neuroscience tasks. The approaches used in this study can be applied to a wide range of natural science disciplines.

Two- and three-component systems of superdiffusion equations describing the dynamics of action potential propagation in a chain of non-locally interacting neurons with Hindmarsh–Rose nonlinear functions have been considered. Non-local couplings based on the fractional Laplace operator describing superdiffusion kinetics are found to support chimeras. In turn, the system with local couplings, based on the classical Laplace operator, shows synchronous behavior. For several parameters responsible for the activation properties of neurons, it is shown that the structure and evolution of chimera states depend significantly on the fractional Laplacian exponent, reflecting non-local properties of the couplings. For two-component systems, an anisotropic transition to full incoherence in the parameter space responsible for non-locality of the first and second variables is established. Introducing a third slow variable induces a gradual transition to incoherence via additional chimera states formation. We also discuss the possible causes of chimera states formation in such a system of non-locally interacting neurons and relate them with the properties of the fractional Laplace operator in a system with global coupling.

Precipitation patterns are commonly concentric rings forming in a Petri dish or parallel bands appearing in a test tube (Liesegang phenomenon). The rings frequently consist of a number of convex segments that are separated from each other by spaces devoid of precipitate resulting in small gaps (dislocations). Along these gaps, the so-called zig-zag structures can form, which connect one side of a gap with its opposite side. We observe that the occurrence of zig-zags requires a minimum thickness of the reactive layer (≥ 0.8 mm). This fact together with microscopic evidence indicates their three-dimensional character. One finds that at the very beginning of the precipitation reaction a curling process starts in the corresponding contour lines. These observations suggest structures of a helicoid with the axis perpendicular to the plane of the reaction–diffusion front to pass through the layer. Zig-zags are not parallel to the reaction plane, i.e., they are not formed periodically, but evolve continuously as a rotating spiral wave. Thus, their topology is closely related to helices in a test tube.