Implementation of Hebb's rules in a network of excitable chemical cells coupled by pulses.

A network of four excitable cells with the Belousov-Zhabotinsky (BZ) reaction is considered both theoretically and experimentally. All cells are coupled by pulses with time delays τnj between the moment of a spike in cell #n and the moment of the corresponding perturbation of an addressee (cell #j). The coupling strengths of all connections except the coupling strength C12 between cells #1 and #2 are constant. Cell #1 is periodically perturbed (with period Tex) and sends pulses to cell #2. The value of C12 is controlled by pulses from two other cells (with indexes #5 and #6; cells with indexes #3 and #4 are absent in the considered network), provided the pulses from cell #5 increase C12, while the pulses from cell #6 decrease C12. Cells #5 and #6 are mutually coupled by inhibitory pulses. Depending on the relations between the values of τnj, there are three dynamic modes in the network: (i) the coupling strength C12 increases stepwise, which is the "Hebb mode", (ii) the C12 decreases stepwise, which is the "anti-Hebb mode", and (iii) the C12 remains almost unchanged within some small adjustable range, which is the meander mode. The ability to tune the C12via "Hebb" and "anti-Hebb" modes introduces memory in the chemical network and, consequently, a mechanism of learning can be realized. The theoretical network is implemented experimentally using four microcells with the BZ reaction provided the pulse coupling between microcells is realized using optical links.

Plasticity in networks of active chemical cells with pulse coupling.

A method for controlling the coupling strength is proposed for pulsed coupled active chemical micro-cells. The method is consistent with Hebb's rules. The effect of various system parameters on this "spike-timing-dependent plasticity" is studied. In addition to networks of two and three coupled active cells, the effect of this "plasticity" on the dynamic modes of a network of four pulse-coupled chemical micro-cells unidirectionally coupled in a circle is studied. It is shown that the proposed adjustment of the coupling strengths leads to spontaneous switching between network eigenmodes.

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. A catalyst for the Belousov-Zhabotinsky (BZ) reaction is immobilized in the micro-spheres, which are called BZ micro-oscillators (BZ MOs). The ME is loaded with all the reagents of the BZ reaction, except the catalyst. Fresh BZ reagents in the ME constantly feed BZ MOs. Diffusively coupled BZ MOs in this system demonstrate excitatory coupling at short gaps and inhibitory coupling at long gaps between BZ MOs. The critical gap, at which the transition between excitatory and inhibitory types of coupling occurs, depends on both the BZ concentration and the size of the ME nanodroplets and can range from a few to tens of micrometers. Different oscillatory patterns of BZ MOs with in-phase and anti-phase oscillations of neighboring BZ MOs have been found. Turing patterns are found as well. A network of such BZ MOs can be used for the creation of a chemical computer.

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.

Size- and position-dependent bifurcations of chemical microoscillators in confined geometries.

The present theoretical study deals with microparticles (beads) that contain an immobilized Belousov-Zhabotinsky (BZ) reaction catalyst. In the theoretical experiment, a BZ bead is immersed in a small water droplet that contains all of the BZ reaction reagents but no catalyst. Such heterogeneous reaction-diffusion BZ systems with the same BZ reactant concentrations demonstrate various dynamic modes, including steady state and low-amplitude, high-amplitude, and mixed-mode oscillations (MMOs). The emergence of such dynamics depends on the sizes of the bead and water droplet, as well as on the location of the bead inside the droplet. MMO emergence is explained by time-delayed positive feedback in combination with a canard phenomenon. If two identical BZ beads are immersed in the same droplet, many different dynamic modes including chaos are observed.

Fabrication of New Belousov-Zhabotinsky Micro-Oscillators on the Basis of Silica Gel Beads.

We have fabricated and examined silica gel beads loaded with a catalyst of the Belousov-Zhabotinsky (BZ) reaction, tris(2,2'-bipyridyl)ruthenium(II) chloride, Ru(bpy)3Cl2, the BZ beads. The abilities of silica gel and the widely used ion-exchange resin (Dowex 50WX2) BZ beads to oscillate in a catalyst-free BZ solution are compared. The period of the BZ oscillations increases with an increase in the diameter (40-250 µm) of both types of the BZ beads. The ability of the ion-exchange resin beads to hold catalyst molecules significantly decreases with a decrease in pH, while this value is less dependent on pH for the silica gel BZ beads. The diffusive coupling of two BZ beads separated by either aqueous or oil gaps is studied as well. For the aqueous gap less than 100 µm, the BZ beads of both types demonstrate in-phase synchronization. For the oil phase, the Dowex BZ beads are unable to oscillate for a long enough time and therefore cannot be synchronized, while the silica gel BZ beads are able to oscillate for several hours and demonstrate anti-phase synchronization for gaps less than 40 µm.

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.

Experimental Investigation of the Dynamical Modes of Four Pulse-Coupled Chemical Micro-Oscillators.

We present an experimental system of four identical microreactors (MRs) in which the photosensitive oscillatory Belousov-Zhabotinsky (BZ) reaction occurs. The inhibitory coupling of these BZ MRs is organized via pulses of light coming to each MR from a computer projector. These pulses are induced by spike(s) in other MR(s) of the same network. Time delay between the spike in one BZ MR and the pulsed perturbation of the other BZ MR(s), the amplitude of light pulses, their duration, and the connectivity of the MRs are controlled by the LabVIEW software. Recording the dynamics of the BZ reaction in the MRs via a microscope equipped with a CCD camera, we observe all the main dynamical modes of our network of MRs, which are the IP (in-phase), AP (anti-phase), W (walk), and WR (walk reverse) for the unidirectional coupling, and the IP, two-cluster, three-cluster, and splay modes for the all-to-all coupling. Our software detects all the modes of the network automatically and makes it possible to switch between them on demand using a few special "switching" pulses. As the result of the present work, the experimental implementation of the adaptive behaviour of the pulse-coupled chemical micro-oscillator networks becomes available.

"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 Cinh of the perturbing pulses. At Cinh > Ccr, 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 Cinh. 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 Cinh. 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 coupled in a ring via unidirectional inhibitory pulsatile coupling with a time delay is analysed computationally and experimentally. There are five stable modes in this network: in-phase, anti-phase, walk, walk reverse, and three-cluster modes. Transitions between the modes are carried out by short external pulses applied to one or several oscillators. We consider three types of switching between the modes: (i) forced switching, when the phases of oscillators of an initial mode reset in such a way that they correspond to the phases of the final mode; internal pulses of the network play no role in this resetting; (ii) "specific" switching, when the phase of only one oscillator is changed by an external perturbation which induces a chain of phase changes in other oscillators due to internal coupling between oscillators; and (iii) multistep switching through intermediate modes, which can be either stable or unstable attractors. All these types of switching have been found in simulations and verified in laboratory experiments.

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. GNF is established by homogeneous illumination of the 1D array with the light intensity proportional to the number of BZ droplets in the oxidized state with the coefficient of proportionality ge. MDs are immersed in the continuous oil phase and diffusively coupled with the neighboring droplets via inhibitor Br2 which is soluble in the oil phase. At chosen concentrations of the BZ reactants, illumination suppresses the BZ oscillators. Without GNF, or at a very small ge < 0.29, local inhibitory coupling leads to out-of-phase oscillations of the neighboring BZ droplets with an almost constant phase shift Δφ between them, which makes a space-time plot of the BZ MDs look like a staircase. At 0.3 < ge < 0.6, regular oscillatory clusters consisting of distant BZ MDs (mostly 5-6 phase clusters) emerge. At 0.6 ≤ ge ≤ 1.0, chaotic clusters are observed. At 1.2 < ge < 1.8, regular (mostly 3-4-phase) clusters emerge again. At 1.8 < ge < (3-4), complex clusters with different (but multiple) periods of oscillation are observed. At the same time, some droplets stop oscillating. At large enough ge (>4), in the region of two-phase clusters (with several suppressed BZ MDs), final patterns seem to resemble the initial patterns. Intensive computer simulations with the ordinary differential equations support experimental results.

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

The dynamical regimes of two almost identical Belousov-Zhabotinsky oscillators with both pulsatile (with time delay) and diffusive coupling have been studied theoretically with the aid of ordinary differential equations for four combinations of these types of coupling: inhibitory diffusive and inhibitory pulsatile (IDIP); excitatory diffusive and inhibitory pulsatile; inhibitory diffusive and excitatory pulsatile; and finally, excitatory diffusive and excitatory pulsatile (EDEP). The combination of two types of coupling creates a condition for new feedback, which promotes new dynamical modes for the IDIP and EDEP coupling.

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 Cinh-τ, where Cinh 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 Cinh. The C and OS modes observed at larger Cinh intertwine in the Cinh-τ plane. In a relatively narrow range of Cinh, the dynamics of the C modes are very sensitive to small changes in Cinh 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 Cinh, the C modes lose sensitivity to small changes in the parameters and finally vanish, while the OS modes survive.

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.

The dynamical regimes of four almost identical oscillators with pulsatile excitatory coupling have been studied theoretically with two models: a kinetic model of the Belousov-Zhabotinsky reaction and a phase-reduced model. Unidirectional coupling on a ring and all-to-all coupling have been considered. The time delay τ between the moments of a spike in one oscillator and a pulse perturbation of the other(s) plays a crucial role in the emergence of the dynamical modes, which are classified as regular, complex, and OS (oscillation-suppression)-modes. The regular modes, in which each oscillator gives only one spike during the period T, consist of the modes in which the period T is linearly dependent on τ and modes in which T is almost independent of τ. The τ-dependent and τ-independent modes alternate if τ increases. A unique sequence of modes observed at growing τ is the same for all types of connectivity and even for both excitatory and inhibitory coupling. For unidirectional coupling, the analytical dependence of T on τ is found for all regular modes. Multirhythmicity is observed at large values of the coupling strength Cex. The effect of small frequency dispersion (within a few percents) on the stability of the regular modes has been studied. Unusual modes like bursting or heteroclinic switching are found in narrow regions of the Cex-τ plane between the regular modes.

Self-Organization Induced by Self-Assembly in Microheterogeneous Reaction-Diffusion System.

When acrylamide (AA) monomers are added to the Belousov-Zhabotinsky (BZ) reaction incorporated into nanodroplets of water-in-oil aerosol OT (AOT) microemulsion (the BZ-AOT system), free radicals produced in the BZ reaction initiate polymerization of AA monomers and polyacrylamide particles are formed. These particles change the microstructure of the AOT microemulsion thus inducing the transition from Turing patterns to new dissipative patterns which can be either stationary "black" spots or waves.

Dynamical regimes of four almost identical chemical oscillators coupled via pulse inhibitory coupling with time delay.

The dynamic regimes in networks of four almost identical spike oscillators with pulsatile coupling via inhibitor are systematically studied. We used two models to describe individual oscillators: a phase-oscillator model and a model for the Belousov-Zhabotinsky reaction. A time delay τ between a spike in one oscillator and the spike-induced inhibitory perturbation of other oscillators is introduced. Diagrams of all rhythms found for three different types of connectivities (unidirectional on a ring, mutual on a ring, and all-to-all) are built in the plane C(inh)-τ, where C(inh) is the coupling strength. It is shown analytically and numerically that only four regular rhythms are stable for unidirectional coupling: walk (phase shift between spikes of neighbouring oscillators equals the quarter of the global period T), walk-reverse (the same as walk but consecutive spikes take place in the direction opposite to the direction of connectivity), anti-phase (any two neighbouring oscillators are anti-phase), and in-phase oscillations. In the case of mutual on the ring coupling, an additional in-phase-anti-phase mode emerges. For all-to-all coupling, two new asymmetrical patterns (two-cluster and three-cluster modes) have been found. More complex rhythms are observed at large C(inh), when some oscillators are suppressed completely or generate smaller number of spikes than others.

Inhibitory and excitatory pulse coupling of two frequency-different chemical oscillators with time delay.

Dynamical regimes of two pulse coupled non-identical Belousov-Zhabotinsky oscillators have been studied experimentally as well as theoretically with the aid of ordinary differential equations and phase response curves both for pure inhibitory and pure excitatory coupling. Time delay τ between a spike in one oscillator and perturbing pulse in the other oscillator plays a significant role for the phase relations of synchronous regimes of the 1:1 and 1:2 resonances. Birhythmicity between anti-phase and in-phase oscillations for inhibitory pulse coupling as well as between 1:2 and 1:1 resonances for excitatory pulse coupling have also been found. Depending on the ratio of native periods of oscillations T2/T1, coupling strength, and time delay τ, such resonances as 1:1 (with different phase locking), 2:3, 1:2, 2:5, 1:3, 1:4, as well as complex oscillations and oscillatory death are observed.

New type of excitatory pulse coupling of chemical oscillators via inhibitor.

We introduce a new type of pulse coupling between chemical oscillators. A constant inflow of inhibitor in one reactor is interrupted shortly after a time delay after a sharp spike of activity in the other reactor. We proved experimentally and theoretically that this reversed inhibitory coupling is analogous to excitatory coupling. We did this by analyzing phase response curves, dependences of different synchronous regimes of the 1 : 1 resonance on time delay, and other resonances of two coupled chemical oscillators. Dynamical rhythms of two Belousov-Zhabotinsky oscillators coupled via "negative" inhibitory pulses were investigated.