Synchronization scenarios induced by delayed communication in arrays of diffusively coupled autonomous chemical oscillators

We study the impact of delayed feedbacks in the collective synchronization of ensembles of identical and autonomous micro-oscillators. To this aim, we consider linear arrays of Belousov-Zhabotinsky (BZ) oscillators confined in micro-compartmentalised systems, where the delayed feedback mimics natural lags that can arise due to the confinement properties and mechanisms driving the inter-oscillator communication. The micro-oscillator array is modeled as a set of Oregonator-like kinetics coupled via mass exchange of the chemical messengers. Changes in the synchronization patterns are explored by varying the delayed feedback introduced in the messenger species Br2. A direct transition from anti-phase to in-phase synchronization and back to the initial anti-phase scheme is observed by progressively increasing the time delay from zero to the value T₀, which is the oscillation period characterising the system without any delayed coupling. The route from anti- to in-phase oscillations (and back) consists of regimes where windows of in-phase oscillations are periodically broken by anti-phase beats. Similarities between these phase transition dynamics and synchronization scenarios characterising the coordination of oscillatory limb movements are finally discussed.

Oscillatory microcells connected on a ring by chemical waves

The dynamics of four coupled microcells with the oscillatory Belousov-Zhabotinsky (BZ) reaction in them is analyzed with the aid of partial differential equations. Identical BZ microcells are coupled in a circle via identical narrow channels containing all the components of the BZ reaction, which is in the stationary excitable state in the channels. Spikes in the BZ microcells generate unidirectional chemical waves in the channels. A thin filter is put in between the end of the channel and the cell. To make coupling between neighboring cells of the inhibitory type, hydrophobic filters are used, which let only Br₂ molecules, the inhibitor of the BZ reaction, go through the filter. To simulate excitatory coupling, we use a hypothetical filter that let only HBrO₂ molecules, the activator of the BZ reaction, go through it. New dynamic modes found in the described system are compared with the "old" dynamic modes found earlier in the analogous system of the "single point" BZ oscillators coupled in a circle by pulses with time delay. The "new" and "old" dynamic modes found for inhibitory coupling match well, the only difference being much broader regions of multi-rhythmicity in the "new" dynamic modes. For the excitatory type of coupling, in addition to four symmetrical modes of the "old" type, many new asymmetrical modes coexisting with the symmetrical ones have been found. Asymmetrical modes are characterized by the spikes occurring any time within some finite time intervals.

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.

Experimental verification of an opto-chemical "neurocomputer"

A theoretically predicted hierarchical network of pulse coupled chemical micro-oscillators and excitable micro-cells that we call a chemical "neurocomputer" (CN) or even a chemical "brain" is tested experimentally using the Belousov-Zhabotinsky reaction. The CN consists of five functional units: (1) a central pattern generator (CPG), (2) an antenna, (3) a reader for the CPG, (4) a reader for the antenna unit, and (5) a decision making (DM) unit. A hybrid CN, in which such chemical units as readers and DM units are replaced by electronic units, is tested as well. All these variations of the CN respond intelligently to external signals, since they perform an automatic transition from a current to a new dynamic mode of the CPG, which is similar to the antenna dynamic mode that in turn is induced by external signals. In other words, we show for the first time that a network of pulse coupled chemical micro-oscillators is capable of intelligent adaptive behavior.

Hierarchical network of pulse coupled chemical oscillators with adaptive behavior: Chemical neurocomputer

We consider theoretically a network of pulse coupled oscillators with time delays. Each oscillator is described by the Oregonator-like model for the Belousov-Zhabotinsky (BZ) reaction. Different groups of oscillators constitute five functional units: (1) a central pattern generator (CPG), (2) a "reader" unit that can identify dynamical modes of the CPG, (3) an antenna (A) unit that receives external signals and responds on them by generating different dynamical modes, (4) another reader unit for identification of the dynamical modes in the A unit, and (5) a decision making unit that switches the current dynamical mode of the CPG to the mode that is similar to the current mode in the A unit. We call this network a chemical neurocomputer, since chemical BZ reaction occurs in each micro-oscillator, while pulse connectivity of these cells is inspired by the brain.

"Cognitive" modes in small networks of almost identical chemical oscillators with pulsatile inhibitory coupling

The Lavrova-Vanag (LV) model of the periodical Belousov-Zhabotinsky (BZ) reaction has been investigated at pulsed self-perturbations, when a sharp spike of the BZ reaction induces a short inhibitory pulse that perturbs the BZ reaction after some time τ since each spike. The dynamics of this BZ system is strongly dependent on the amplitude C_inh of the perturbing pulses. At C_inh > C_cr, a new pseudo-steady state (SS) emerges far away from the limit cycle of the unperturbed BZ oscillator. The perturbed BZ system spends rather long time in the vicinity of this pseudo-SS, which serves as a trap for phase trajectories. As a result, the dynamics of the BZ system changes qualitatively. We observe new modes with packed spikes separated by either long "silent" dynamics or small-amplitude oscillations around pseudo-SS, depending on C_inh. Networks of two or three LV-BZ oscillators with strong pulsatile coupling and self-inhibition are able to generate so-called "cognitive" modes, which are very sensitive to small changes in C_inh. We demonstrate how the coupling between the BZ oscillators in these networks should be organized to find "cognitive" modes.

Controllable switching between stable modes in a small network of pulse-coupled chemical oscillators

Switching between stable oscillatory modes in a network of four Belousov–Zhabotinsky oscillators unidirectionally coupled in a ring analysed computationally and experimentally.

Travelling waves in somitogenesis: Collective cellular properties emerge from time-delayed juxtacrine oscillation coupling

The sculpturing of the vertebrate body plan into segments begins with the sequential formation of somites in the presomitic mesoderm (PSM). The rhythmicity of this process is controlled by travelling waves of gene expression. These kinetic waves emerge from coupled cellular oscillators and sweep across the PSM. In zebrafish, the oscillations are driven by autorepression of her genes and are synchronized via Notch signalling. Mathematical modelling has played an important role in explaining how collective properties emerge from the molecular interactions. Increasingly more quantitative experimental data permits the validation of those mathematical models, yet leads to increasingly more complex model formulations that hamper an intuitive understanding of the underlying mechanisms. Here, we review previous efforts, and design a mechanistic model of the her1 oscillator, which represents the experimentally viable her7;hes6 double mutant. This genetically simplified system is ideally suited to conceptually recapitulate oscillatory entrainment and travelling wave formation, and to highlight open questions. It shows that three key parameters, the autorepression delay, the juxtacrine coupling delay, and the coupling strength, are sufficient to understand the emergence of the collective period, the collective amplitude, and the synchronization of neighbouring Her1 oscillators. Moreover, two spatiotemporal time delay gradients, in the autorepression and in the juxtacrine signalling, are required to explain the collective oscillatory dynamics and synchrony of PSM cells. The highlighted developmental principles likely apply more generally to other developmental processes, including neurogenesis and angiogenesis.

Dynamic modes in a network of five oscillators with inhibitory all-to-all pulse coupling

The dynamic modes of five almost identical oscillators with pulsatile inhibitory coupling with time delay have been studied theoretically. The models of the Belousov-Zhabotinsky reaction and phase oscillators with all-to-all coupling have been considered. In the parametric plane C_inh-τ, where C_inh is the coupling strength and τ is the time delay between a spike in one oscillator and pulsed perturbations of all other oscillators, three main regimes have been found: regular modes, when each oscillator gives only one spike during the global period T, C (complex) modes, when the number of pulses of different oscillators is different, and OS (oscillations-suppression) modes, when at least one oscillator is suppressed. The regular modes consist of several cluster modes and are found at relatively small C_inh. The C and OS modes observed at larger C_inh intertwine in the C_inh-τ plane. In a relatively narrow range of C_inh, the dynamics of the C modes are very sensitive to small changes in C_inh and τ, as well as to the initial conditions, which are the characteristic features of the chaos. On the other hand, the dynamics of the C modes are periodic (but with different periods) and well reproducible. The number of different C modes is enormously large. At still larger C_inh, the C modes lose sensitivity to small changes in the parameters and finally vanish, while the OS modes survive.

Dynamics of a 1D array of inhibitory coupled chemical oscillators in microdroplets with global negative feedback

We have investigated the effect of global negative feedback (GNF) on the dynamics of a 1D array of water microdroplets (MDs) filled with the reagents of the photosensitive oscillatory Belousov–Zhabotinsky (BZ) reaction.

Dynamical modes of two almost identical chemical oscillators connected via both pulsatile and diffusive coupling

The dynamics of two almost identical chemical oscillators with mixed diffusive and pulsatile coupling are systematically studied.

A 'reader' unit of the chemical computer

We suggest the main principals and functional units of the parallel chemical computer, namely, (i) a generator (which is a network of coupled oscillators) of oscillatory dynamic modes, (ii) a unit which is able to recognize these modes (a 'reader') and (iii) a decision-making unit, which analyses the current mode, compares it with the external signal and sends a command to the mode generator to switch it to the other dynamical regime. Three main methods of the functioning of the reader unit are suggested and tested computationally: (a) the polychronization method, which explores the differences between the phases of the generator oscillators; (b) the amplitude method which detects clusters of the generator and (c) the resonance method which is based on the resonances between the frequencies of the generator modes and the internal frequencies of the damped oscillations of the reader cells. Pro and contra of these methods have been analysed.

Dynamical regimes of four oscillators with excitatory pulse coupling

Dynamics of four almost identical chemical oscillators pulse coupled via excitatory coupling with time delays are systematically studied.

Chemical memory with states coded in light controlled oscillations of interacting Belousov–Zhabotinsky droplets

The information storing potential of droplets, in which an oscillatory, photosensitive Belousov–Zhabotinsky (BZ) reaction proceeds, is investigated experimentally.