Complex behavior in chains of nonlinear oscillators

Complexity and transition to chaos in coupled Adler-type oscillators

Coupled nonlinear oscillators are ubiquitous in dynamical studies. A wealth of behaviors have been found mostly for globally coupled systems. From a complexity perspective, less studied have been systems with local coupling, which is the subject of this contribution. The phase approximation is used, as weak coupling is assumed. In particular, the so-called needle region, in parameter space, for Adler-type oscillators with nearest neighbors coupling is carefully characterized. The reason for this emphasis is that, in the border of this region to the surrounding chaotic one, computation enhancement at the edge of chaos has been reported. The present study shows that different behaviors within the needle region can be found and a smooth change of dynamics could be identified. Entropic measures further emphasize the region's heterogeneous nature with interesting features, as seen in the spatiotemporal diagrams. The occurrence of wave-like patterns in the spatiotemporal diagrams points to nontrivial correlations in both dimensions. The wave patterns change as the control parameters change without exiting the needle region. Spatial correlation is only achieved locally at the onset of chaos, with different clusters of oscillators behaving coherently while disordered boundaries appear between them.

Collective oscillations of coupled cell cycles

Problems with networks of coupled oscillators arise in multiple contexts, commonly leading to the question about the dependence of network dynamics on network structure. Previous work has addressed this question in Drosophila oogenesis, in which stable cytoplasmic bridges connect the future oocyte to the supporting nurse cells that supply the oocyte with molecules and organelles needed for its development. To increase their biosynthetic capacity, nurse cells enter the endoreplication program, a special form of the cell cycle formed by the iterated repetition of growth and synthesis phases without mitosis. Recent studies have revealed that the oocyte orchestrates nurse cell endoreplication cycles, based on retrograde (oocyte to nurse cells) transport of a cell cycle inhibitor produced by the nurse cells and localized to the oocyte. Furthermore, the joint dynamics of endocycles has been proposed to depend on the intercellular connectivity within the oocyte-nurse cell cluster. We use a computational model to argue that this connectivity guides, but does not uniquely determine the collective dynamics and identify several oscillatory regimes, depending on the timescale of intercellular transport. Our results provide insights into collective dynamics of coupled cell cycles and motivate future quantitative studies of intercellular communication in the germline cell clusters.

Computational capabilities at the edge of chaos for one dimensional systems undergoing continuous transitions

While there has been a keen interest in studying computation at the edge of chaos for dynamical systems undergoing a phase transition, this has come under question for cellular automata. We show that for continuously deformed cellular automata, there is an enhancement of computation capabilities as the system moves towards cellular automata with chaotic spatiotemporal behavior. The computation capabilities are followed by looking into the Shannon entropy rate and the excess entropy, which allow identifying the balance between unpredictability and complexity. Enhanced computation power shows an increase of excess entropy, while the system entropy density has a sudden jump to values near one. The analysis is extended to a system of non-linear locally coupled oscillators that have been reported to exhibit spatiotemporal diagrams similar to cellular automata.

Complex spatiotemporal behavior and coherent excitations in critically-coupled chains of neural circuits

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" M embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period τ , thus given by a unitary matrix U = exp ⁡ ( τ 2 M ) . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

Phenomenology of coupled nonlinear oscillators

A recently introduced model of coupled nonlinear oscillators in a ring is revisited in terms of its information processing capabilities. The use of Lempel-Ziv based entropic measures allows to study thoroughly the complex patterns appearing in the system for different values of the control parameters. Such behaviors, resembling cellular automata, have been characterized both spatially and temporally. Information distance is used to study the stability of the system to perturbations in the initial conditions and in the control parameters. The latter is not an issue in cellular automata theory, where the rules form a numerable set, contrary to the continuous nature of the parameter space in the system studied in this contribution. The variation in the density of the digits, as a function of time is also studied. Local transitions in the control parameter space are also discussed.