Nonlocal coupling can prevent the collapse of spatiotemporal chaos

Alternating activity patterns and a chimeralike state in a network of globally coupled excitable Morris-Lecar neurons

Spatiotemporal chaos collapses to either a rest state or a propagating pulse in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Adding global varying synaptic coupling to the ring network reveals complex transient behavior. Spatiotemporal chaos collapses into a transient pulse that reinitiates spatiotemporal chaos to allow sequential pattern switching until a collapse to the rest state. A domain of irregular neuron activity coexists with a domain of inactive neurons forming a transient chimeralike state. Transient spatial localization of the chimeralike state is observed for stronger synapses.

Transient chaos and associated system-intrinsic switching of spacetime patterns in two synaptically coupled layers of Morris-Lecar neurons

Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a single layer of diffusively coupled, excitable Morris-Lecar neurons. Weak synaptic coupling of two such layers reveals system intrinsic switching of spatiotemporal activity patterns within and between the layers at irregular times. Within a layer, switching sequences include spatiotemporal chaos, erratic and regular pulse propagation, spontaneous network wide neuron activity, and rest state. A momentary substantial reduction in neuron activity in one layer can reinitiate transient spatiotemporal chaos in the other layer, which can induce a swap of spatiotemporal chaos with a pulse state between the layers. Presynaptic input maximizes the distance between propagating pulses, in contrast to pulse merging in the absence of synapses.

Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model

We examine the prototypical Gray-Scott model, which mimics cubic autocatalytic reaction with linear decay of the autocatalyst, to model the kinetics of a reaction-diffusion system subjected to advective streamline flow. For a proper choice of boundary conditions and parameter space, the system admits wave-induced spatiotemporal chaos in the absence of flow. We show that flow above a critical value leads to an arrest of the spatiotemporal chaos due to a change in the instability from absolute to convective type. Furthermore, stationary spatial structures are borne out of a second successive bifurcation for yet another critical flow value. The theoretical formulations are corroborated by extensive numerical simulation of the full reaction-diffusion-advection system in one dimension.

Impact of weak excitatory synapses on chaotic transients in a diffusively coupled Morris-Lecar neuronal network

Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Weak excitatory synapses can increase the Lyapunov exponent, expedite the collapse, and promote the collapse to the rest state rather than the pulse state. A single traveling pulse solution may no longer be asymptotic for certain combinations of network topology and (weak) coupling strengths, and initiate spatiotemporal chaos. Multiple pulses can cause chaos initiation due to diffusive and synaptic pulse-pulse interaction. In the presence of chaos initiation, intermittent spatiotemporal chaos exists until typically a collapse to the rest state.

Transient spatiotemporal chaos in the Morris-Lecar neuronal ring network

Transient behavior is thought to play an integral role in brain functionality. Numerical simulations of the firing activity of diffusively coupled, excitable Morris-Lecar neurons reveal transient spatiotemporal chaos in the parameter regime below the saddle-node on invariant circle bifurcation point. The neighborhood of the chaotic saddle is reached through perturbations of the rest state, in which few initially active neurons at an effective spatial distance can initiate spatiotemporal chaos. The system escapes from the neighborhood of the chaotic saddle to either the rest state or to a state of pulse propagation. The lifetime of the chaotic transients is manipulated in a statistical sense through a singular application of a synchronous perturbation to a group of neurons.

Origin of chaotic transients in excitatory pulse-coupled networks

We develop an approach to understanding long chaotic transients in networks of excitatory pulse-coupled oscillators. Our idea is to identify a class of attractors, sequentially active firing (SAF) attractors, in terms of the temporal event structure of firing and receipt of pulses. Then all attractors can be classified into two groups: SAF attractors and non-SAF attractors. We establish that long transients typically arise in the transitional region of the parameter space where the SAF attractors are collectively destabilized. Bifurcation behavior of the SAF attractors is analyzed to provide a detailed understanding of the long irregular transients. Although demonstrated using pulse-coupled oscillator networks, our general methodology may be useful in understanding the origin of transient chaos in other types of networked systems, an extremely challenging problem in nonlinear dynamics and complex systems.

Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol oscillators near a codimension-two bifurcation point

Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincaré map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves.

Length scale of interaction in spatiotemporal chaos

Extensive systems have no long scale correlations and behave as a sum of their parts. Various techniques are introduced to determine a characteristic length scale of interaction beyond which spatiotemporal chaos is extensive in reaction-diffusion networks. Information about network size, boundary condition, or abnormalities in network topology gets scrambled in spatiotemporal chaos, and the attenuation of information provides such characteristic length scales. Space-time information flow associated with the recovery of spatiotemporal chaos from finite perturbations, a concept somewhat opposite to the paradigm of Lyapunov exponents, defines another characteristic length scale. High-precision computational studies of asymptotic spatiotemporal chaos in the complex Ginzburg-Landau system and transient spatiotemporal chaos in the Gray-Scott network show that these different length scales are comparable and thus suitable to define a length scale of interaction. Preliminary studies demonstrate the relevance of these length scales for stable chaos.

Diverse self-sustained oscillatory patterns and their mechanisms in excitable small-world networks

Diverse self-sustained oscillatory patterns and their mechanisms in small-world networks (SWNs) of excitable nodes are studied. Spatiotemporal patterns of SWNs are sensitive to long-range connection probability P and coupling intensity D . By varying P in wide range with fixed D , we observe totally six types of asymptotic states: pure spiral waves, pure self-sustained target waves, patterns of mixtured spirals and target waves, pseudospiral turbulence, synchronizing oscillations, and rest state. The parameter conditions for all these states are specified, and the mechanisms of these states are heuristically explained. In particular, the mechanism of emergence and annihilation of synchronizing oscillations is explained by using the shortest path length analysis.

Structure and control of self-sustained target waves in excitable small-world networks

Small-world networks describe many important practical systems among which neural networks consisting of excitable nodes are the most typical ones. In this paper we study self-sustained oscillations of target waves in excitable small-world networks. A dominant phase-advanced driving (DPAD) method, which is generally applicable for analyzing all oscillatory complex networks consisting of nonoscillatory nodes, is proposed to reveal the self-organized structures supporting this type of oscillations. The DPAD method explicitly explores the oscillation sources and wave propagation paths of the systems, which are otherwise deeply hidden in the complicated patterns of randomly distributed target groups. Based on the understanding of the self-organized structure, the oscillatory patterns can be controlled with an extremely high efficiency.

Transient spatiotemporal chaos is extensive in three reaction-diffusion networks

Extensive (asymptotic) spatiotemporal chaos is comprised of statistically similar subsystems that interact only weakly. A systematic study of transient spatiotemporal chaos reveals extensive system behavior in all three reaction-diffusion networks for various boundary conditions. The Lyapunov dimension, the sum of positive Lyapunov exponents, and the logarithm of the transient lifetime grow linearly with the system size. The unstable manifold of the chaotic saddle has nearly the same dimension as the saddle itself, and the stable manifold is nearly space filling.

Master stability analysis in transient spatiotemporal chaos

The asymptotic stability of spatiotemporal chaos is difficult to determine, since transient spatiotemporal chaos may be extremely long lived. A master stability analysis reveals that the asymptotic state of transient spatiotemporal chaos in the Gray-Scott system and in the Bär-Eiswirth system is characterized by negative transverse Lyapunov exponents on the attractor of the invariant synchronization manifold. The average lifetime of transient spatiotemporal chaos depends on the number of transverse directions that are unstable along a typical excitation cycle.

Emergence of self-sustained patterns in small-world excitable media

Motivated by recent observations that long-range connections (LRCs) play a role in various brain phenomena, we have observed two distinct dynamical transitions in the activity of excitable media where waves propagate both between neighboring regions and through LRCs. When the LRC density p is low, single or multiple spiral waves are seen to emerge and cover the entire system. This state is self-sustaining and robust against perturbations. At p=p(c)(l) , the spirals are suppressed, and there is a transition to a spatially homogeneous, temporally periodic state. Finally, above p=p(c)(u) , activity ceases after a brief transient.

Noise can delay and advance the collapse of spatiotemporal chaos

Spatiotemporal chaos on a regular ring network of excitable Gray-Scott dynamical elements collapses to a stable asymptotic state. We find that the addition of dynamical noise clearly influences the spatiotemporal pattern and the transient lifetime of spatiotemporal chaos. Spatially uniform noise significantly decreases the average lifetime of spatiotemporal chaos due to an enlargement of regions of local collapse. For spatially inhomogeneous noise the collapse is maximally delayed at an intermediate noise level, but drastically advanced for larger noise levels.