Synchronization of heterogeneous oscillator populations in response to weak and strong coupling

The influence of synaptic plasticity on critical coupling estimates for neural populations

The presence or absence of synaptic plasticity can dramatically influence the collective behavior of populations of coupled neurons. In this work, we consider spike-timing dependent plasticity (STDP) and its resulting influence on phase cohesion in computational models of heterogeneous populations of conductance-based neurons. STDP allows for the influence of individual synapses to change over time, strengthening or weakening depending on the relative timing of the relevant action potentials. Using phase reduction techniques, we derive an upper bound on the critical coupling strength required to retain phase cohesion for a network of synaptically coupled, heterogeneous neurons with STDP. We find that including STDP can significantly alter phase cohesion as compared to a network with static synaptic connections. Analytical results are validated numerically. Our analysis highlights the importance of the relative ordering of action potentials emitted in a population of tonically firing neurons and demonstrates that order switching can degrade the synchronizing influence of coupling when STDP is considered.

Collective chemomechanical oscillations in active hydrogels

We report on the collective response of an assembly of chemomechanical Belousov-Zhabotinsky (BZ) hydrogel beads. We first demonstrate that a single isolated spherical BZ hydrogel bead with a radius below a critical value does not oscillate, whereas an assembly of the same BZ hydrogel beads presents chemical oscillation. A BZ chemical model with an additional flux of chemicals out of the BZ hydrogel captures the experimentally observed transition from oxidized nonoscillating to oscillating BZ hydrogels and shows this transition is due to a flux of inhibitors out of the BZ hydrogel. The model also captures the role of neighboring BZ hydrogel beads in decreasing the critical size for an assembly of BZ hydrogel beads to oscillate. We finally leverage the quorum sensing behavior of the collective to trigger their chemomechanical oscillation and discuss how this collective effect can be used to enhance the oscillatory strain of these active BZ hydrogels. These findings could help guide the eventual fabrication of a swarm of autonomous, communicating, and motile hydrogels.

Temporal signals drive the emergence of multicellular information networks

Coordinated responses to environmental stimuli are critical for multicellular organisms. To overcome the obstacles of cell-to-cell heterogeneity and noisy signaling dynamics within individual cells, cells must effectively exchange information with peers. However, the dynamics and mechanisms of collective information transfer driven by external signals are poorly understood. Here we investigate the calcium dynamics of neuronal cells that form confluent monolayers and respond to cyclic ATP stimuli in microfluidic devices. Using Granger inference to reconstruct the underlying causal relations between the cells, we find that the cells self-organize into spatially decentralized and temporally stationary networks to support information transfer via gap junction channels. The connectivity of the causal networks depends on the temporal profile of the external stimuli, where short periods, or long periods with small duty fractions, lead to reduced connectivity and fractured network topology. We build a theoretical model based on communicating excitable units that reproduces our observations. The model further predicts that connectivity of the causal network is maximal at an optimal communication strength, which is confirmed by the experiments. Together, our results show that information transfer between neuronal cells is externally regulated by the temporal profile of the stimuli and internally regulated by cell-cell communication.

Pattern formation in a four-ring reaction-diffusion network with heterogeneity

In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies.

Chemical micro-oscillators based on the Belousov–Zhabotinsky reaction

The results of studies on the development of micro-oscillators (MOs) based on the Belousov –Zhabotinsky (BZ) oscillatory chemical reaction are integrated and systematized. The mechanisms of the BZ reaction and the methods of immobilization of the catalyst of the BZ reaction in micro-volumes are briefly discussed. Methods for creating BZ MOs based on water microdroplets in the oil phase and organic and inorganic polymer microspheres are considered. Methods of control and management of the dynamics of BZ MO networks are described, including methods of MO synchronization. The prospects for the design of neural networks of MOs with intelligent-like behaviour are outlined. Such networks present a new area of nonlinear chemistry, including, in particular, the creation of a chemical ‘computer’.

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Distance dependent types of coupling of chemical micro-oscillators immersed in a water-in-oil microemulsion

A system of micro-spheres immersed in a water-in-oil microemulsion (ME) is studied both theoretically and experimentally.

Dynamics of hierarchical weighted networks of van der Pol oscillators

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.

Transition from spiral wave chimeras to phase cluster states

Photochemically coupled Belousov-Zhabotinsky micro-oscillators are studied in experiments and simulations. Generally good agreement between the experimental and simulated dynamical behavior is found, with spiral wave chimeras exhibited at small values of the time delay in the coupling between the oscillators, spiral wave core splitting at higher values, and phase cluster states replacing the spiral wave dynamics at the highest values of the time delay. Spiral wave chimera dynamics is exhibited experimentally for much of the time delay range, while spiral wave phase cluster states are exhibited more in the model simulations. In addition to comparing the experimental and simulation behavior, we explore the novel spiral wave phase cluster states and develop a mechanism for this new and unusual dynamical behavior.