Theory of transient chimeras in finite Sakaguchi-Kuramoto networks

Chimera states are a phenomenon in which order and disorder can co-exist within a network that is fully homogeneous. Precisely how transient chimeras emerge in finite networks of Kuramoto oscillators with phase-lag remains unclear. Utilizing an operator-based framework to study nonlinear oscillator networks at finite scale, we reveal the spatiotemporal impact of the adjacency matrix eigenvectors on the Sakaguchi-Kuramoto dynamics. We identify a specific condition for the emergence of transient chimeras in these finite networks: the eigenvectors of the network adjacency matrix create a combination of a zero phase-offset mode and low spatial frequency waves traveling in opposite directions. This combination of eigenvectors leads directly to the coherent and incoherent clusters in the chimera. This approach provides two specific analytical predictions: (1) a precise formula predicting the combination of connectivity and phase-lag that creates transient chimeras, (2) a mathematical procedure for rewiring arbitrary networks to produce transient chimeras

Composed solutions of synchronized patterns in multiplex networks of Kuramoto oscillators

Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to investigate these networks, obtaining mathematical descriptions for the dynamics of multilayer and multiplex systems is still an open problem. Here, we combine ideas and concepts from linear algebra and graph theory with nonlinear dynamics to offer a novel approach to study multiplex networks of Kuramoto oscillators. Our approach allows us to study the dynamics of a large, multiplex network by decomposing it into two smaller systems: one representing the connection scheme within layers (intra-layer), and the other representing the connections between layers (inter-layer). Particularly, we use this approach to compose solutions for multiplex networks of Kuramoto oscillators. These solutions are given by a combination of solutions for the smaller systems given by the intra and inter-layer system and, in addition, our approach allows us to study the linear stability of these solutions

Small changes at single nodes can shift global network dynamics

Understanding the sensitivity of a system's behavior with respect to parameter changes is essential for many applications. This sensitivity may be desired - for instance in the brain, where a large repertoire of different dynamics, particularly different synchronization patterns, is crucial - or may be undesired - for instance in power grids, where disruptions to synchronization may lead to blackouts. In this work, we show that the dynamics of networks of phase oscillators can acquire a very large and complex sensitivity to changes made in either their units' parameters or in their connections - even modifications made to a parameter of a single unit can radically alter the global dynamics of the network in an unpredictable manner. As a consequence, each modification leads to a different path to phase synchronization manifested as large fluctuations along that path. This dynamical malleability occurs over a wide parameter region, around the network's two transitions to phase synchronization. One transition is induced by increasing the coupling strength between the units, and another is induced by increasing the prevalence of long-range connections. Specifically, we study Kuramoto phase oscillators connected under either Watts-Strogatz or distance-dependent topologies to analyze the statistical properties of the fluctuations along the paths to phase synchrony. We argue that this increase in the dynamical malleability is a general phenomenon, as suggested by both previous studies and the theory of phase transitions.Comment: 14 pages, 8 figure

Spatial permutation entropy distinguishes resting brain states

We use ordinal analysis and spatial permutation entropy to distinguish between eyes-open and eyes-closed resting brain states. To do so, we analyze EEG data recorded with 64 electrodes from 109 healthy subjects, under two one-minute baseline runs: One with eyes open, and one with eyes closed. We use spatial ordinal analysis to distinguish between these states, where the permutation entropy is evaluated considering the spatial distribution of electrodes for each time instant. We analyze both raw and post-processed data considering only the alpha-band frequency (8–12 Hz) which is known to be important for resting states in the brain. We conclude that spatial ordinal analysis captures information about correlations between time series in different electrodes. This allows the discrimination of eyes closed and eyes open resting states in both raw and filtered data. Filtering the data only amplifies the distinction between states. Importantly, our approach does not require EEG signal pre-processing, which is an advantage for real-time applications, such as brain-computer interfaces.B.R.R.B. and E.E.N.M. acknowledge support of São Paulo Research Foundation (FAPESP), Brazil, Proc. 2018/03211-6 and 2021/09839-0; and Financiadora de Estudos e Projetos (FINEP), Brazil. R.C.B. acknowledges support of Western Institute for Neuroscience Clinical Research Postdoctoral Fellowship and Western Academy for Advanced Research. K.L.R. acknowledges supported of German Academic Exchange Service (DAAD). C.M. acknowledges support of Ministerio de Ciencia, Innovación ������ Universidades (PID2021-123994NB-C21), Spain and Institució Catalana de Recerca i Estudis Avançats (ICREA), Spain.Peer ReviewedPostprint (published version

Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems

Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems

Image segmentation with traveling waves in an exactly solvable recurrent neural network

We study image segmentation using spatiotemporal dynamics in a recurrent neural network where the state of each unit is given by a complex number. We show that this network generates sophisticated spatiotemporal dynamics that can effectively divide an image into groups according to a scene's structural characteristics. Using an exact solution of the recurrent network's dynamics, we present a precise description of the mechanism underlying object segmentation in this network, providing a clear mathematical interpretation of how the network performs this task. We then demonstrate a simple algorithm for object segmentation that generalizes across inputs ranging from simple geometric objects in grayscale images to natural images. Object segmentation across all images is accomplished with one recurrent neural network that has a single, fixed set of weights. This demonstrates the expressive potential of recurrent neural networks when constructed using a mathematical approach that brings together their structure, dynamics, and computation

An exact mathematical description of computation with transient spatiotemporal dynamics in a complex-valued neural network

We study a complex-valued neural network (cv-NN) with linear, time-delayed interactions. We report the cv-NN displays sophisticated spatiotemporal dynamics, including partially synchronized ``chimera'' states. We then use these spatiotemporal dynamics, in combination with a nonlinear readout, for computation. The cv-NN can instantiate dynamics-based logic gates, encode short-term memories, and mediate secure message passing through a combination of interactions and time delays. The computations in this system can be fully described in an exact, closed-form mathematical expression. Finally, using direct intracellular recordings of neurons in slices from neocortex, we demonstrate that computations in the cv-NN are decodable by living biological neurons. These results demonstrate that complex-valued linear systems can perform sophisticated computations, while also being exactly solvable. Taken together, these results open future avenues for design of highly adaptable, bio-hybrid computing systems that can interface seamlessly with other neural networks

Investigation of Details in the Transition to Synchronization in Complex Networks by Using Recurrence Analysis

The study of synchronization in complex networks is useful for understanding a variety of systems, including neural systems. However, the properties of the transition to synchronization are still not well known. In this work, we analyze the details of the transition to synchronization in complex networks composed of bursting oscillators under small-world and scale-free topologies using recurrence quantification analysis, specifically the determinism. We demonstrate the existence of non-stationarity states in the transition region. In the small-world network, the transition region denounces the existence of two-state intermittency