Bistability between synchrony and incoherence in limit-cycle oscillators with coupling strength inhomogeneity

Sub-Second Fluctuation between Top-Down and Bottom-Up Modes Distinguishes Diverse Human Brain States

Information continuously flows between regions of the human brain, exhibiting distinct patterns that dynamically shift across states of consciousness, cognitive modes, and neuropsychiatric conditions. In this study, we introduce Relative Phase Analysis (RPA), a method that leverages phase-lead/lag relationships to reveal the real-time dynamics of dominant directional patterns and their rapid transitions. We demonstrate that the human brain switches on a sub-second timescale between a top-down mode-where anterior regions drive posterior activity-and a bottom-up mode, characterized by reverse directionality. These dynamics are most pronounced during full consciousness and gradually become less distinct as awareness diminishes. Furthermore, we find from simultaneous EEG-fMRI recordings that the top-down mode is expressed when higher-order cognitive networks are more active while the bottom-up mode is expressed when sensory systems are more active. Moreover, comparisons of an attention deficit hyperactivity disorder (ADHD) inattentive cohort with typically developing individuals reveal distinct imbalances in these transition dynamics, highlighting the potential of RPA as a diagnostic biomarker. Complementing our empirical findings, a coupled-oscillator model of the structural brain network recapitulates these emergent patterns, suggesting that such directional modes and transitions may arise naturally from inter-regional neural interactions. Altogether, this study provides a framework for understanding whole-brain dynamics in real-time and identifies sub-second fluctuations in top-down versus bottom-up directionality as a fundamental mechanism underlying human information processing.

Synchronization in a system of Kuramoto oscillators with distributed Gaussian noise

We consider a system of globally coupled phase-only oscillators with distributed intrinsic frequencies and evolving in the presence of distributed Gaussian white noise, namely, a Gaussian white noise whose strength for every oscillator is a specified function of its intrinsic frequency. In the absence of noise, the model reduces to the celebrated Kuramoto model of spontaneous synchronization. For two specific forms of the mentioned functional dependence and for a symmetric and unimodal distribution of the intrinsic frequencies, we unveil the rich long-time behavior that the system exhibits, which stands in stark contrast to the case in which the noise strength is the same for all the oscillators, namely, in the studied dynamics, the system may exist in either a synchronized, or an incoherent, or a time-periodic state; interestingly, all these states also appear as long-time solutions of the Kuramoto dynamics for the case of bimodal frequency distributions, but in the absence of any noise in the dynamics.

First-order like phase transition induced by quenched coupling disorder

We investigate the collective dynamics of a population of X Y model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value and subject to thermal noise controlled by temperature T. We find that the system at T = 0 exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state; when thermal noise is present ( T > 0 ), the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case ( T = 0 ). We derive an exact formula for the critical transition from incoherent to coherent oscillations for the deterministic and stochastic case based on both stability analysis for finite oscillators as well as for the thermodynamic limit ( N → ∞) based on a rigorous mean-field theory using graphons, valid for heterogeneous graph structures. Our theoretical results are supported by extensive numerical simulations. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies, which consider uniform input or output strengths for each oscillator node [H. Hong and S. H. Strogatz, Phys. Rev. E 84(4), 046202 (2011); Phys. Rev. Lett. 106(5), 054102 (2011)], which suggests that these systems display a "universal" character for the onset of synchronization.

Phase and amplitude dynamics of coupled oscillator systems on complex networks

We investigated locking behaviors of coupled limit-cycle oscillators with phase and amplitude dynamics. We focused on how the dynamics are affected by inhomogeneous coupling strength and by angular and radial shifts in coupling functions. We performed mean-field analyses of oscillator systems with inhomogeneous coupling strength, testing Gaussian, power-law, and brain-like degree distributions. Even for oscillators with identical intrinsic frequencies and intrinsic amplitudes, we found that the coupling strength distribution and the coupling function generated a wide repertoire of phase and amplitude dynamics. These included fully and partially locked states in which high-degree or low-degree nodes would phase-lead the network. The mean-field analytical findings were confirmed via numerical simulations. The results suggest that, in oscillator systems in which individual nodes can independently vary their amplitude over time, qualitatively different dynamics can be produced via shifts in the coupling strength distribution and the coupling form. Of particular relevance to information flows in oscillator networks, changes in the non-specific drive to individual nodes can make high-degree nodes phase-lag or phase-lead the rest of the network.

Various synchronous states due to coupling strength inhomogeneity and coupling functions in systems of coupled identical oscillators

We study the effects of coupling strength inhomogeneity and coupling functions on locking behaviors of coupled identical oscillators, some of which are relatively weakly coupled to others while some are relatively strongly coupled. Through the stability analysis and numerical simulations, we show that several categories of fully locked or partially locked states can emerge and obtain the conditions for these categories. In this system with coupling strength inhomogeneity, locked and drifting subpopulations are determined by the coupling strength distribution and the shape of the coupling functions. Even the strongly coupled oscillators can drift while weakly coupled oscillators can be locked. The simulation results with Gaussian and power-law distributions for coupling strengths are compared and discussed.

Synchronization, non-linear dynamics and low-frequency fluctuations: analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

In this paper, the topographical relationship between functional connectivity (intended as inter-regional synchronization), spectral and non-linear dynamical properties across cortical areas of the healthy human brain is considered. Based upon functional MRI acquisitions of spontaneous activity during wakeful idleness, node degree maps are determined by thresholding the temporal correlation coefficient among all voxel pairs. In addition, for individual voxel time-series, the relative amplitude of low-frequency fluctuations and the correlation dimension (D2), determined with respect to Fourier amplitude and value distribution matched surrogate data, are measured. Across cortical areas, high node degree is associated with a shift towards lower frequency activity and, compared to surrogate data, clearer saturation to a lower correlation dimension, suggesting presence of non-linear structure. An attempt to recapitulate this relationship in a network of single-transistor oscillators is made, based on a diffusive ring (n = 90) with added long-distance links defining four extended hub regions. Similarly to the brain data, it is found that oscillators in the hub regions generate signals with larger low-frequency cycle amplitude fluctuations and clearer saturation to a lower correlation dimension compared to surrogates. The effect emerges more markedly close to criticality. The homology observed between the two systems despite profound differences in scale, coupling mechanism and dynamics appears noteworthy. These experimental results motivate further investigation into the heterogeneity of cortical non-linear dynamics in relation to connectivity and underline the ability for small networks of single-transistor oscillators to recreate collective phenomena arising in much more complex biological systems, potentially representing a future platform for modelling disease-related changes.

Three synaptic components contributing to robust network synchronization

Robust synchronized activity is widely observed in real neural systems. However, a mechanism for robust synchronization that can be understood analytically, and has a clear physical meaning, remains elusive. This paper considers such a mechanism by formalizing three synaptic components contributing to robust synchronization in networks with heterogeneous external drive currents and conductance-based synapses. The first component arises from the assumption that the aggregate post-synaptic potential of a neuron decays more if it fires later within a spike volley. The second component results because neurons with smaller drives have reached a lower membrane potential at the time when the volley of inputs arrives than that reached by neurons with larger drives. The third component arises from the assumption that neurons firing later in the previous volley have had less time to integrate their drives than neurons firing earlier have had, again causing a lower membrane potential at the time when the volley of inputs arrives. Because of the voltage-dependent properties of synaptic currents, either of the last two components will cause smaller inhibitions for the later-firing neurons if the synapses are inhibitory. This smaller inhibition causes the later-firing neurons to fire earlier in the next cycle, thereby forcing them toward synchrony. With these three synaptic components, we discuss the relationship between the robustness of the synchrony and the parameters, search for the optimal parameter set for the robust network synchronization of a certain frequency band, and demonstrate the key role of the voltage-dependent properties of synaptic currents in robust or stable synchronization.

Synchronization transition of identical phase oscillators in a directed small-world network

We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation.

Solvable model of spiral wave chimeras

Spiral waves are ubiquitous in two-dimensional systems of chemical or biological oscillators coupled locally by diffusion. At the center of such spirals is a phase singularity, a topological defect where the oscillator amplitude drops to zero. But if the coupling is nonlocal, a new kind of spiral can occur, with a circular core consisting of desynchronized oscillators running at full amplitude. Here, we provide the first analytical description of such a spiral wave chimera and use perturbation theory to calculate its rotation speed and the size of its incoherent core.

Perturbation analysis of complete synchronization in networks of phase oscillators

The behavior of weakly coupled self-sustained oscillators can often be well described by phase equations. Here we use the paradigm of Kuramoto phase oscillators which are coupled in a network to calculate first- and second-order corrections to the frequency of the fully synchronized state for nonidentical oscillators. The topology of the underlying coupling network is reflected in the eigenvalues and eigenvectors of the network Laplacian which influence the synchronization frequency in a particular way. They characterize the importance of nodes in a network and the relations between them. Expected values for the synchronization frequency are obtained for oscillators with quenched random frequencies on a class of scale-free random networks and for a Erdös-Rényi random network. We briefly discuss an application of the perturbation theory in the second order to network structural analysis.