Testing Turing's theory of morphogenesis in chemical cells

Capillary-assisted printing of droplets at a solid-like liquid-liquid interface

HYPOTHESIS

Nanoparticle surfactants ('NPS') assemble at the oil-water interface to form solid-like films. Aqueous droplets placed on these solid-like oil-water interfaces are expected to be stable against coalescence with the underlying water phase. These droplets will deform the solid-like interface at which they are placed, leading to capillary forces and the assembly of large, multi-droplet structures.

EXPERIMENTS

Aqueous droplets were placed on a solid-like film of cellulose nanocrystals surfactants ('CNCS') assembled at an oil-water interface. Droplet dynamics were quantified using single-particle tracking. A custom-made droplet printer was used to control initial droplet positions to guide droplet assembly into large structures. The composition of both the droplets and the NPS assembly was modified to produce heterogeneous droplet structures and light-responsive oil-water interfaces.

FINDINGS

The droplets could be placed at the solid-like oil-water interface for extended periods of time. Microlitre-sized droplets attracted each other over millimetric scales. System dynamics differed from theoretical predictions for pristine interfaces and were captured by a simple model. This inter-droplet capillary attraction facilitated the printing of self-building droplet structures. Embedding gold nanoparticles in the NPS assembly allowed us to generate local temperature gradients by illuminating the system with a laser and manipulate the droplets via plasmon-assisted optofluidics.

Inferring networks of chemical reactions by curvature analysis of kinetic trajectories

Quantifying interaction networks of chemical reactions allows description, prediction, and control of a range of phenomena in chemistry and biology. The challenge lies in unambiguously assigning contributions to changes in rates from different interactions. We propose that the curvature change of kinetic trajectories due to a systematic perturbation of a node in a network can identify the coupling strength and topology. Specifically, the coupling strength can be calculated as the ratio of the curvature change measured from the coupled node and the rate change of a perturbed node. We verified the methodology in numerical simulations with a network with complex ordinary differential equations and experiments with electrochemical networks. The experiments show excellent network inference (without false positive or negative links) of various systems with large heterogeneity in local dynamics and network structure without any a priori knowledge of the kinetics. The theory and the experiments also clarify the influence of local perturbations on response amplitude and timing through network-wide up-regulation. A major advantage of our technique is its independence from hidden/unobserved nodes. This makes our method highly suitable for applications with high temporal and low spatial resolution data from interacting chemical and biochemical systems including neuronal activity monitoring with multi-electrode arrays.

Unraveling the myths and mysteries of photon avalanching nanoparticles

Photon avalanching (PA) nanomaterials exhibit some of the most nonlinear optical phenomena reported for any material, allowing them to push the frontiers of applications ranging from nanoscale imaging and sensing to optical computing. But PA remains shrouded in mystery, with its underlying physics and limitations misunderstood. Photon avalanching is not, in fact, an avalanche of photons, at least not in the same way that snowballs beget more snowballing in an actual avalanche. In this focus article, we dispel these and other common myths surrounding PA in lanthanide-based nanoparticles and unravel the mysteries of this unique nonlinear optical effect. We hope that removing the misconceptions surrounding avalanching nanoparticles will inspire new interest and applications that harness the giant nonlinearity of PA across a broad range of scientific fields.

Turing patterns with cellular computers

Turing patterns are a key theoretical foundation for understanding organ development and organization. While they have been found to occur in natural systems, implementing new biological systems that form Turing patterns has remained challenging. To address this, Tica et al.1 used synthetic genetic networks to engineer living cellular computers that successfully generate Turing patterns within growing bacterial populations.

Turing covalent organic framework membranes via heterogeneous nucleation synthesis for organic solvent nanofiltration

Although covalent organic frameworks (COFs) demonstrate notable potential for developing advanced separation membranes, contemporary COF membranes still lack controlled stacking and highly efficient sieving. Here, Turing-architecture COF membranes were constructed by engineering a reaction-diffusion assembly process via heterogeneous nucleation synthesis with tannic acid (TA). TA covalently binds with amine monomers to form a composite precursor with increased reactivity and decreased diffusivity. This altered the pathway of Schiff base reactions with aldehyde monomers, fulfilling suitable reaction-diffusion conditions, and ultimately formed the labyrinthine stripe or spot-patterned Turing COF film with controlled stacking and uniform pore structure. This endows our COF membrane with highly efficient molecule sieving ability for organic solvent nanofiltration while exhibiting a flux that is 621% greater than that of commercial membranes. Thus, this study provides a paradigm for the in situ synthesis of highly efficient COF membranes for diversely sustainable separations.

Turing patterns on rotating spiral growing domains

We investigate the emergence of Turing patterns in a system growing as a rotating spiral in two dimensions, utilizing the photosensitivity of the chlorine dioxide-iodine-malonic acid (CDIMA) reaction to control the growth process. We observe the formation of single and multiple (double and triple) stationary spiral patterns as well as transitional patterns. From numerical simulations of the Lengyel-Epstein model with an additional term to account for the effects of illumination on the reaction, we analyze the relationship between the final morphologies and the radial and angular growth velocities, identify conditions conducive to the formation of transitional structures, examine the importance of the size of the initial nucleation site in determining the spiral's multiplicity, and evaluate the stability and robustness of these Turing patterns. Our results indicate how inclusion of rotational degrees of freedom in the growth process may lead to the formation of a diverse new class of patterns in chemical and biological systems.

Multiscale Approach for Tuning Communication among Chemical Oscillators Confined in Biomimetic Microcompartments

Inspired by the biological world, new cross-border disciplines and technologies have emerged. Relevant examples include systems chemistry, which offers a bottom-up approach toward chemical complexity, and bio/chemical information and communication technology (bio/chemical ICT), which explores the conditions for propagating signals among individual microreactors separated by selectively permeable membranes. To fabricate specific arrays of microreactors, microfluidics has been demonstrated as an excellent method. In particular, droplet-based microfluidics is a powerful tool for encapsulating biological entities and chemical reagents in artificial microenvironments, mostly water-in-oil microdroplets. In these systems, the interfaces are liquid-liquid, and their physicochemical properties are key factors for tuning the coupling between molecular diffusion. Simple and double emulsions, where aqueous domains are in equilibrium with oil domains through boundary layers of amphiphilic molecules, are organized assemblies with high interfacial-area-to-volume ratios. These membranes can be engineered to obtain different surface charges, single- or multilayer stacking, and a variable degree of defects in molecular packing. Emulsions find application in many fields, including the food industry, pharmaceutics, and cosmetics. Furthermore, micro- and nanoemulsions can be used to model the propagation of chemical species through long distances, which is not only vital for cell signaling but also significant in molecular computing. Here we present in-depth research on the faceted world of solutions confined in restricted environments. In particular, we focused on the multiscale aspects of structure and dynamics from molecular to micro and macro levels. The Belousov-Zhabotinsky chemical reaction, known for its robustness and well-documented oscillatory behavior, was selected to represent a generic signal emitter/receiver confined within microenvironments separated by liquid-liquid interfaces. In this pulse generator, the temporal and spatial progressions are governed by periodic fluctuations in the concentration of chemical species, which act as activatory or inhibitory messengers over long distances. When organized into "colonies" or arrays, these micro-oscillators exhibit emergent dynamical behaviors at the population level. These behaviors can be finely tuned by manipulating the geometrical distribution of the oscillators and the properties of the interfaces at the nanoscale. By carefully selecting the membrane composition, it is possible to drive the system toward either in-phase, antiphase, or mixed synchronization regimes among individual oscillators, depending on messenger molecules. This relatively simple lab-scale model replicates some of the communication strategies commonly found in biological systems, particularly those based on the passive diffusion of chemical and electrical signals. It can help shed light on fundamental life processes and inspire new implementations in molecular computing and smart materials.

A programmable hybrid digital chemical information processor based on the Belousov-Zhabotinsky reaction

The exponential growth of the power of modern digital computers is based upon the miniaturization of vast nanoscale arrays of electronic switches, but this will be eventually constrained by fabrication limits and power dissipation. Chemical processes have the potential to scale beyond these limits by performing computations through chemical reactions, yet the lack of well-defined programmability limits their scalability and performance. Here, we present a hybrid digitally programmable chemical array as a probabilistic computational machine that uses chemical oscillators using Belousov-Zhabotinsky reaction partitioned in interconnected cells as a computational substrate. This hybrid architecture performs efficient computation by distributing information between chemical and digital domains together with inbuilt error correction logic. The efficiency is gained by combining digital logic with probabilistic chemical logic based on nearest neighbour interactions and hysteresis effects. We demonstrated the computational capabilities of our hybrid processor by implementing one- and two-dimensional Chemical Cellular Automata demonstrating emergent dynamics of life-like entities called Chemits. Additionally, we demonstrate hybrid probabilistic logic as a viable logic for solving combinatorial optimization problems.

Turing Instability of Liquid-Solid Metal Systems

The classical Turing morphogenesis often occurs in nonmetallic solution systems due to the sole competition of reaction and diffusion processes. Here, this work conceives that gallium (Ga) based liquid metals (LMs) possess the ability to alloy, diffuse, and react with a range of solid metals (SMs) and thus should display Turing instability leading to a variety of nonequilibrium spatial concentration patterns. This work discloses a general mechanism for obtaining labyrinths, stripes, and spots-like stationary Turing patterns in the LM-SM reaction-diffusion systems (GaX-Y), taking the gallium indium alloy and silver substrate (GaIn-Ag) system as a proof of concept. It is only when Ga atoms diffuse over Y much faster than X while X reacts with Y preferentially, that Turing instability occurs. In such a metallic system, Ga serves as an inhibitor and X as an activator. The dominant factors in tuning the patterning process include temperature and concentration. Intermetallic compounds contained in the Turing patterns and their competitive reactions have also been further clarified. This LM Turing instability mechanism opens many opportunities for constructing microstructure systems utilizing condensed matter to experimentally explore the general morphogenesis process.

Spiking at the edge: Excitability at interfaces in reaction-diffusion systems

Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spikes arise due to local bistabilities, which require a fine-tuned ratio between local amplification and suppression strengths. But, in nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, we show how boundaries can generate and protect spiking when the reacting components can spread out: Even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking arises due to a global bistability, which can occur even if amplification and suppression strengths do not allow spiking when mixed. We analytically derive a spiking phase diagram that depends on two parameters: i) the ratio between the system size and the characteristic diffusive length-scale and ii) the ratio between the amplification and suppression strengths. Our analysis explains recent experimental observations of action potentials at the interface between two non-excitable bioelectric tissues. Beyond electrophysiology, we highlight how edge spiking emerges in predator-prey dynamics and in oscillating chemical reactions. Our findings provide a theoretical blueprint for a class of interfacial excitations in reaction-diffusion systems, with potential implications for spatially controlled chemical reactions, nonlinear waveguides and neuromorphic computation, as well as spiking instabilities, such as cardiac arrhythmias, that naturally occur in heterogeneous biological media.

Diffusiophoresis-enhanced Turing patterns

Turing patterns are fundamental in biophysics, emerging from short-range activation and long-range inhibition processes. However, their paradigm is based on diffusive transport processes that yield patterns with shallower gradients than those observed in nature. A complete physical description of this discrepancy remains unknown. We propose a solution to this phenomenon by investigating the role of diffusiophoresis, which is the propulsion of colloids by a chemical gradient, in Turing patterns. Diffusiophoresis enables robust patterning of colloidal particles with substantially finer length scales than the accompanying chemical Turing patterns. A scaling analysis and a comparison to recent experiments indicate that chromatophores, ubiquitous in biological pattern formation, are likely diffusiophoretic and the colloidal Péclet number controls the pattern enhancement. This discovery suggests that important features of biological pattern formation can be explained with a universal mechanism that is quantified straightforwardly from the fundamental physics of colloids.

The emergence of spatial patterns for compartmental reaction kinetics coupled by two bulk diffusing species with comparable diffusivities

Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns can typically only occur from a spatially uniform state when a slowly diffusing 'activator' species reacts with a much faster diffusing 'inhibitor' species. However, from a modelling perspective, this large diffusivity ratio requirement for pattern formation is often unrealistic in biological settings since different molecules tend to diffuse with similar rates in extracellular spaces. As a result, one key long-standing question is how to robustly obtain pattern formation in the biologically realistic case where the time scales for diffusion of the interacting species are comparable. For a coupled one-dimensional bulk-compartment theoretical model, we investigate the emergence of spatial patterns for the scenario where two bulk diffusing species with comparable diffusivities are coupled to nonlinear reactions that occur only in localized 'compartments', such as on the boundaries of a one-dimensional domain. The exchange between the bulk medium and the spatially localized compartments is modelled by a Robin boundary condition with certain binding rates. As regulated by these binding rates, we show for various specific nonlinearities that our one-dimensional coupled PDE-ODE model admits symmetry-breaking bifurcations, leading to linearly stable asymmetric steady-state patterns, even when the bulk diffusing species have equal diffusivities. Depending on the form of the nonlinear kinetics, oscillatory instabilities can also be triggered. Moreover, the analysis is extended to treat a periodic chain of compartments. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Multiscale Biofabrication: Integrating Additive Manufacturing with DNA-Programmable Self-Assembly

Structure and hierarchical organization are crucial elements of biological systems and are likely required when engineering synthetic biomaterials with life-like behavior. In this context, additive manufacturing techniques like bioprinting have become increasingly popular. However, 3D bioprinting, as well as other additive manufacturing techniques, show limited resolution, making it difficult to yield structures on the sub-cellular level. To be able to form macroscopic synthetic biological objects with structuring on this level, manufacturing techniques have to be used in conjunction with biomolecular nanotechnology. Here, a short overview of both topics and a survey of recent advances to combine additive manufacturing with microfabrication techniques and bottom-up self-assembly involving DNA, are given.

Spatiotemporal Patterning Enabled by Gene Regulatory Networks

Spatiotemporal pattern formation plays a key role in various biological phenomena including embryogenesis and neural network formation. Though the reaction-diffusion systems enabling pattern formation have been studied phenomenologically, the biomolecular mechanisms behind these processes have not been modeled in detail. Here, we study the emergence of spatiotemporal patterns due to simple, synthetic and commonly observed two- and three-node gene regulatory network motifs coupled with their molecular diffusion in one- and two-dimensional space. We investigate the patterns formed due to the coupling of inherent multistable and oscillatory behavior of the toggle switch, toggle switch with double self-activation, toggle triad, and repressilator with the effect of spatial diffusion of these molecules. We probe multiple parameter regimes corresponding to different regions of stability (monostable, multistable, oscillatory) and assess the impact of varying diffusion coefficients. This analysis offers valuable insights into the design principles of pattern formation facilitated by these network motifs, and it suggests the mechanistic underpinnings of biological pattern formation.

Cellular micromasonry: biofabrication with single cell precision

In many tissues, cell type varies over single-cell length-scales, creating detailed heterogeneities fundamental to physiological function. To gain understanding of the relationship between tissue function and detailed structure, and eventually to engineer structurally and physiologically accurate tissues, we need the ability to assemble 3D cellular structures having the level of detail found in living tissue. Here we introduce a method of 3D cell assembly having a level of precision finer than the single-cell scale. With this method we create detailed cellular patterns, demonstrating that cell type can be varied over the single-cell scale and showing function after their assembly.

Turing instability in quantum activator–inhibitor systems

Turing instability is a fundamental mechanism of nonequilibrium self-organization. However, despite the universality of its essential mechanism, Turing instability has thus far been investigated mostly in classical systems. In this study, we show that Turing instability can occur in a quantum dissipative system and analyze its quantum features such as entanglement and the effect of measurement. We propose a degenerate parametric oscillator with nonlinear damping in quantum optics as a quantum activator–inhibitor unit and demonstrate that a system of two such units can undergo Turing instability when diffusively coupled with each other. The Turing instability induces nonuniformity and entanglement between the two units and gives rise to a pair of nonuniform states that are mixed due to quantum noise. Further performing continuous measurement on the coupled system reveals the nonuniformity caused by the Turing instability. Our results extend the universality of the Turing mechanism to the quantum realm and may provide a novel perspective on the possibility of quantum nonequilibrium self-organization and its application in quantum technologies.

Synchronization of Belousov-Zhabotinsky oscillators with electrochemical coupling in a spontaneous process

A passive electrochemical coupling approach is proposed to induce spontaneous synchronization between chemical oscillators. The coupling exploits the potential difference between a catalyst redox couple in the Belousov-Zhabotinsky (BZ) reaction, without external feedback, to induce surface reactions that impact the kinetics of the bulk system. The effect of coupling in BZ oscillators under batch condition is characterized using phase synchronization measures. Although the frequency of the oscillators decreases nonlinearly over time, by a factor of 2 or more within 100 cycles, the coupling is strong enough to maintain synchronization. In such a highly drifting system, the Gibbs-Shannon entropy of the cyclic phase difference distribution can be used to quantify the coupling effect. We extend the Oregonator BZ model to account for the drifting natural frequencies in batch condition and for electrochemical coupling, and numerical simulations of the effect of acid concentration on synchronization patterns are in agreement with the experiments. Because of the passive nature of coupling, the proposed coupling scheme can open avenues for designing pattern recognition and neuromorphic computation systems using chemical reactions in a spontaneous process.

Bridging scales in a multiscale pattern-forming system

Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.

Quantitative models for building and growing fated small cell networks

Small cell clusters exhibit numerous phenomena typically associated with complex systems, such as division of labour and programmed cell death. A conserved class of such clusters occurs during oogenesis in the form of germline cysts that give rise to oocytes. Germline cysts form through cell divisions with incomplete cytokinesis, leaving cells intimately connected through intercellular bridges that facilitate cyst generation, cell fate determination and collective growth dynamics. Using the well-characterized Drosophila melanogaster female germline cyst as a foundation, we present mathematical models rooted in the dynamics of cell cycle proteins and their interactions to explain the generation of germline cell lineage trees (CLTs) and highlight the diversity of observed CLT sizes and topologies across species. We analyse competing models of symmetry breaking in CLTs to rationalize the observed dynamics and robustness of oocyte fate specification, and highlight remaining gaps in knowledge. We also explore how CLT topology affects cell cycle dynamics and synchronization and highlight mechanisms of intercellular coupling that underlie the observed collective growth patterns during oogenesis. Throughout, we point to similarities across organisms that warrant further investigation and comment on the extent to which experimental and theoretical findings made in model systems extend to other species.

Turing patterns by supramolecular self-assembly of a single salphen building block

In the 1950s, Alan Turing showed that concerted reactions and diffusion of activating and inhibiting chemical species can autonomously generate patterns without previous positional information, thus providing a chemical basis for morphogenesis in Nature. However, access to these patterns from only one molecular component that contained all the necessary information to execute agonistic and antagonistic signaling is so far an elusive goal, since two or more participants with different diffusivities are a must. Here, we report on a single-molecule system that generates Turing patterns arrested in the solid state, where supramolecular interactions are used instead of chemical reactions, whereas diffusional differences arise from heterogeneously populated self-assembled products. We employ a family of hydroxylated organic salphen building blocks based on a bis-Schiff-base scaffold with portions responsible for either activation or inhibition of assemblies at different hierarchies through purely supramolecular reactions, only depending upon the solvent dielectric constant and evaporation as fuel.

Science, serendipity, coincidence, and the Oregonator at the University of Oregon, 1969-1974

This historical review of the development of the Oregonator model of the Belousov-Zhabotinsky reaction is based on a lecture Dick Field presented during IrvFest2015-Celebrating a founding father of chaos!, a meeting in commemoration of Irving R. Epstein's 70 th birthday. For Dick's 80 th birthday festschrift, we focus here on the five papers in the series named "Oscillations in chemical systems," published in 1972 [Noyes et al., J. Am. Chem. Soc. 94, 1394-1395 (1972); Field et al., J. Am. Chem. Soc. 94, 8649-8664 (1972); Field and Noyes, Nature 237, 390-392 (1972)] and 1974 [Field and Noyes, J. Chem. Phys. 60, 1877-1884 (1974); Field and Noyes, J. Am. Chem. Soc. 96, 2001-2006 (1974)].

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.

Morphogenesis in Synthetic Chemical Cells

We present an efficient model for describing morphogenesis and the emergence of spatiotemporal structures in synthetic chemical cells. This work is motivated by an experimental setup used for testing Turing's theory of morphogenesis. The model developed is based on the general theory of chemically active droplets, which combines the classical theory of phase separation with reaction-diffusion systems. Through the 2D calculations, we find the six spatiotemporal structures predicted by Turing in 1952 and experimentally observed, in a 1D array of droplets. Moreover, under Turing instability, with a determined chemical wavelength, the system undergoes morphogenesis. This theoretical approach provides a useful tool for understanding the physical differentiation through the direct calculation of the osmotic pressure in each cell as the chemical reaction occurs.

Modern perspectives on near-equilibrium analysis of Turing systems

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Insights from chemical systems into Turing-type morphogenesis

In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction-diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237, 37-72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction-diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

The design principles of discrete turing patterning systems

The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-species networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find these Turing pattern topologies to be substantially more robust to changes in the parameters of the model. We also find that diffusion-driven instabilities are substantially weaker predictors for Turing patterns in our discrete modelling framework in comparison to the continuous case, in the sense that the presence of an instability does not guarantee a pattern emerging in simulations. We show that a more refined criterion constitutes a stronger predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.

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’.

The bibliography includes 250 references.

Disappearance of Temporal Collinearity in Vertebrates and Its Eventual Reappearance

Simple Summary

In 1999 T. Kondo and D. Duboule performed excisions of posterior upstream DNA domains in mouse embryos and they observed that for an extended excision (including Evx gene) the Hox genes of the cluster were simultaneously expressed with the first Hoxd1 gene ‘as if’ Temporal Collinearity (TC) had disappeared. According to a Biophysical Model (BM) during Hox gene expression, Hox clusters behave similar toexpanding elastic springs. For the extended upstream DNA excision, BM predicts the TC disappearance and an experiment is proposed to test this BM prediction. In the chick limb bud C. Tickle et al. observed that the excision of the apical ectodermal ridge (AER) caused the inhibition of HoxA13 expression. However, the implantation of FGF soaked beads at the tip of the limb could surprisingly rescue HoxA13 expression after 24 h so that TC is restored.Brachyury transcription factor (TF) is essential in identifying the targets of this transcription and a chromatin immunoprecipitation microarray chip (ChIP-chip) was produced which can be inserted in the mouse embryonic cells. It is here proposed to insert this chip in the mutant cells where TC has disappeared and compare it to the limb bud case.Is TC restored? It is an important issue worth exploring.

Abstract

It was observed that a cluster of ordered genes (Hox1, Hox2, Hox3…) in the genome are activated in the ontogenetic units (1, 2, 3 …) of an embryo along the Anterior/Posterior axis following the same order of the Hox genes. This Spatial Collinearity (SC) is very strange since it correlates events of very different spatial dimensions. It was later observed in vertebrates, that, in the above ordering, first is Hox1expressed in ontogenetic unit 1, followed later by Hox2 in unit 2 and even later Hox3 in unit 3. This temporal collinearity (TC) is an enigma and even to-day is explored in depth. In 1999 T. Kondo and D. Duboule, after posterior upstream extended DNA excisions, concluded that the Hox cluster behaves ‘as if’ TC disappears. Here the consideration of TC really disappearing is taken face value and its repercussions are analyzed. Furthermore, an experiment is proposed to test TC disappearance. An outcome of this experiment could be the reappearance (partial or total) of TC.

Probing β-Cell Biology in Space and Time

β-Cells in the islet of Langerhans have a central role in maintaining energy homeostasis. Understanding the physiology of β-cells and other islet cells requires a deep understanding of their structural and functional organization, their interaction with vessels and nerves, the layout of paracrine interactions, and the relationship between subcellular compartments and protein complexes inside each cell. These elements are not static; they are dynamic and exert their biological actions at different scales of time. Therefore, scientists must be able to investigate (and visualize) short- and long-lived events within the pancreas and β-cells. Current technological advances in microscopy are able to bridge multiple spatiotemporal scales in biology to reveal the complexity and heterogeneity of β-cell biology. Here, I briefly discuss the historical discoveries that leveraged microscopes to establish the basis of β-cell anatomy and structure, the current imaging platforms that allow the study of islet and β-cell biology at multiple scales of resolution, and their challenges and implications. Lastly, I outline how the remarkable longevity of structural elements at different scales in biology, from molecules to cells to multicellular structures, could represent a previously unrecognized organizational pattern in developing and adult β-cells and pancreas biology.

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.

Emergent Bioanalogous Properties of Blockchain-based Distributed Systems

We apply a novel definition of biological systems to a series of reproducible observations on a blockchain-based distributed virtual machine (dVM). We find that such blockchain-based systems display a number of bioanalogous properties, such as response to the environment, growth and change, replication, and homeostasis, that fit some definitions of life. We further present a conceptual model for a simple self-sustaining, self-organizing, self-regulating distributed 'organism' as an operationally closed system that would fulfill all basic definitions and criteria for life, and describe developing technologies, particularly artificial neural network (ANN) based artificial intelligence (AI), that would enable it in the near future. Notably, such systems would have a number of specific advantages over biological life, such as the ability to pass acquired traits to offspring, significantly improved speed, accuracy, and redundancy of their genetic carrier, and potentially unlimited lifespans. Public blockchain-based dVMs provide an uncontained environment for the development of artificial general intelligence (AGI) with the capability to evolve by self-direction.

Partition, Reaction, and Diffusion Coefficients of Bromine in Elastomeric Polydimethylsiloxane

Experiments and models were used to determine the extent to which aqueous bromine permeated into, and reacted with, the elastomer polydimethylsiloxane (PDMS). Thin films of PDMS were immersed in bromine water, and the absorbance of bromine in the aqueous phase was measured as a function of time. Kinetics were studied as a function of mass and thickness of the immersed PDMS films. We attribute the decrease of bromine in solution to permeation into PDMS, followed by a combination of diffusion, reversible binding, and an irreversible reaction with PDMS. In order to decouple the irreversible reaction from the reversible processes, kinetics were also studied for bromine-passivated PDMS films. Fits of the models to a variety of experiments yielded the partition coefficient of bromine between the water and PDMS phases, the diffusion constant of bromine in PDMS, the irreversible reaction constant between bromine and PDMS, the molar concentration of the reactive sites within PDMS, and the on and off rates of reversible binding of bromine to PDMS. Developing a quantitative reaction-diffusion model accounting for the transport of bromine through PDMS is necessary for the design of microfluidic devices fabricated using PDMS, which are used in experimental studies of the nonlinear dynamics of reaction-diffusion networks containing Belousov-Zhabotinsky chemical oscillators.

A theory of pattern formation for reaction–diffusion systems on temporal networks

Networks have become ubiquitous in the modern scientific literature, with recent work directed at understanding ‘temporal networks’—those networks having structure or topology which evolves over time. One area of active interest is pattern formation from reaction–diffusion systems, which themselves evolve over temporal networks. We derive analytical conditions for the onset of diffusive spatial and spatio-temporal pattern formation on undirected temporal networks through the Turing and Benjamin–Feir mechanisms, with the resulting pattern selection process depending strongly on the evolution of both global diffusion rates and the local structure of the underlying network. Both instability criteria are then extended to the case where the reaction–diffusion system is non-autonomous, which allows us to study pattern formation from time-varying base states. The theory we present is illustrated through a variety of numerical simulations which highlight the role of the time evolution of network topology, diffusion mechanisms and non-autonomous reaction kinetics on pattern formation or suppression. A fundamental finding is that Turing and Benjamin–Feir instabilities are generically transient rather than eternal, with dynamics on temporal networks able to transition between distinct patterns or spatio-temporal states. One may exploit this feature to generate new patterns, or even suppress undesirable patterns, over a given time interval.

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.

Impact of PDMS-Based Microfluidics on Belousov-Zhabotinsky Chemical Oscillators

Sub-nanoliter volumes of the Belousov-Zhabotinsky (BZ) reaction are sealed in microfluidic devices made from polydimethylsiloxane (PDMS). Bromine, which is a BZ reaction intermediate that participates in the inhibitory pathway of the reaction, is known to permeate into PDMS, and it has been suggested that PDMS and bromine can react ( J. Phys. Chem. A. 108, 2004, 1325-1332). We characterize the extent to which PDMS affects BZ oscillations by varying the volume of the PDMS surrounding the BZ reactors. We measure how the oscillation period varies with PDMS volume and compare with a theoretical reaction-diffusion model, concluding that bromine reacts with PDMS. We demonstrate that minimizing the amount of PDMS by making the samples as thin as possible maximizes the number of oscillations before the BZ reaction reaches equilibrium and ceases to oscillate. We also demonstrate that the deleterious effects of the PDMS-BZ interactions are somewhat mitigated by imposing constant chemical boundary conditions through using a light-sensitive catalyst, ruthenium, in combination with patterned illumination. Furthermore, we show that light can modulate the frequency and phase of the BZ oscillators contained in a PDMS matrix by 20-30%.

Light-Activated Signaling in DNA-Encoded Sender-Receiver Architectures

Collective decision making by living cells is facilitated by exchange of diffusible signals where sender cells release a chemical signal that is interpreted by receiver cells. A variety of nonliving artificial cell models have been developed in recent years that mimic various aspects of diffusion-based intercellular communication. However, localized secretion of diffusive signals from individual protocells, which is critical for mimicking biological sender-receiver systems, has remained challenging to control precisely. Here, we engineer light-responsive, DNA-encoded sender-receiver architectures, where protein-polymer microcapsules act as cell mimics and molecular communication occurs through diffusive DNA signals. We prepare spatial distributions of sender and receiver protocells using a microfluidic trapping array and set up a signaling gradient from a single sender cell using light, which activates surrounding receivers through DNA strand displacement. Our systematic analysis reveals how the effective signal range of a single sender is determined by various factors including the density and permeability of receivers, extracellular signal degradation, signal consumption, and catalytic regeneration. In addition, we construct a three-population configuration where two sender cells are embedded in a dense array of receivers that implement Boolean logic and investigate spatial integration of nonidentical input cues. The results offer a means for studying diffusion-based sender-receiver topologies and present a strategy to achieve the congruence of reaction-diffusion and positional information in chemical communication systems that have the potential to reconstitute collective cellular patterns.

Lilypad aggregation: localised self-assembly and metal sequestration at a liquid-vapour interface

Spatially resolved soft materials, such as vesicles and microgels, have shown promise as selective adsorbents and microscale reaction vessels. However, spatiotemporal control of aggregation can be difficult to achieve. In this study, nickel(ii) chloride and a dipyridyl oligo(urea) ligand were combined in a vapour-diffusion setup to produce a localised spheroidal aggregate at the liquid-vapour interface. This aggregate forms via the self-assembly and fusion of monodisperse colloids and grows until its weight is no longer counterbalanced by surface tension. A simple physical model reveals that this process, termed lilypad aggregation, is possible only for surface energies that favour neither bulk aggregation nor the growth of an interfacial film. These surface energies dictate the final size and shape of the aggregate and may be estimated through visual monitoring of its changing morphology. Lilypad aggregates sequester metal from the surrounding sol and can be collected manually from the surface of the liquid.

Hanging droplets from liquid surfaces

Natural and man-made robotic systems use the interfacial tension between two fluids to support dense objects on liquid surfaces. Here, we show that coacervate-encased droplets of an aqueous polymer solution can be hung from the surface of a less dense aqueous polymer solution using surface tension. The forces acting on and the shapes of the hanging droplets can be controlled. Sacs with homogeneous and heterogeneous surfaces are hung from the surface and, by capillary forces, form well-ordered arrays. Locomotion and rotation can be achieved by embedding magnetic microparticles within the assemblies. Direct contact of the droplet with air enables in situ manipulation and compartmentalized cascading chemical reactions with selective transport. Applications including functional microreactors, motors, and biomimetic robots are evident.

Vitamin A and its natural derivatives

Vitamin A and derivatives, the natural retinoids, underpin signaling pathways of cellular differentiation, and are key chromophores in vision. These functions depend on transfer across membranes, and carrier proteins to shuttle retinoids to specific cell compartments. Natural retinoids, ultimately derived from plant carotenoids by metabolism to all-trans retinol, are lipophilic and consist of a cyclohexenyl (β-ionone) moiety linked to a polyene chain. This structure constrains the orientation of retinoids within lipid membranes. Cis-trans isomerization at double bonds of the polyene chain and s-cis/s-trans rotational isomerization at single bonds define the functional dichotomy of retinoids (signaling/vision) and specificities of interactions with specific carrier proteins and receptors. Metabolism of all-trans retinol to 11-cis retinal, transfer to photoreceptors, and removal and recycling of all-trans retinal generated by photoreceptor irradiation, is the key process underlying vision. All-trans retinol transferred into cells is metabolized to all-trans retinoic acid and shuttled to the cell nucleus to regulate gene expression controlling organ, tissue and cell differentiation, and cellular homeostasis. Research methods need to address the potential of photoisomerization in vitro to confound research results, and data should be interpreted in the context of membrane-association properties of retinoids and physiological concentrations in vivo. Despite a century of research, there are many fundamental questions of retinoid cellular biochemistry and molecular biology still to be answered. Computational modeling techniques will have an important role for understanding the nuances of vitamin A signaling and function.

Membrane Structure Drives Synchronization Patterns in Arrays of Diffusively Coupled Self-Oscillating Droplets

Networks of diffusively coupled inorganic oscillators, confined in nano- and microcompartments, are effective for predicting and understanding the global dynamics of those systems where the diffusion of activatory or inhibitory signals regulates the communication among different individuals. By taking advantage of a microfluidic device, we study the dynamics of arrays of diffusively coupled Belousov-Zhabotinsky (BZ) oscillators encapsulated in water-in-oil single emulsions. New synchronization patterns are induced and controlled by modulating the structural and chemical properties of the phospholipid-based biomimetic membranes via the introduction of specific dopants. Doping molecules do not alter the membrane basic backbone, but modify the lamellarity (and, in turn, the permeability) or interact chemically with the reaction intermediates. A transition from two-period clusters showing 1:2 period-locking to one-period antiphase synchronization is observed by decreasing the membrane lamellarity. An unsynchronized scenario is found when the dopant is able to interfere with chemical communication by reacting with the chemical messengers.

Phenylalanine roles in the seed-to-seedling stage: Not just an amino acid

Phenylalanine is an aromatic amino acid that provides the carbon skeleton for the phenylpropanoid pathway, making many diverse chemicals used for structure, defense, and yet undiscovered functions. The identification of the arogenate dehydratase (ADT) enzymes in the genetic model Arabidopsis thaliana provided a platform to explore the roles of phenylalanine in all stages of life: germination, in the seed-to-seedling transition stage, organelle function, and in generation of defense mechanisms, enabling further studies in other plants. From the literature, data indicate that phenylalanine produced by ADT may have direct roles in organellar and tissue development. Recent studies implicate ADTs in cell division and protection from Reactive Oxygen Species, and in signaling and growth. Research in phenylalanine and subsequent phenylpropanoids also point to a role of phenylalanine as a purveyor of C and N nutrients. The understanding of phenylalanine action in plant cells is enhanced by recent research on phenylalanine in animal cells.

Dynamics of Reaction-Diffusion Oscillators in Star and other Networks with Cyclic Symmetries Exhibiting Multiple Clusters

We experimentally and theoretically investigate the dynamics of inhibitory coupled self-driven oscillators on a star network in which a single central hub node is connected to k peripheral arm nodes. The system consists of water-in-oil Belousov-Zhabotinsky ∼100  μm emulsion drops contained in storage wells etched in silicon wafers. We observed three dynamical attractors by varying the number of arms in the star graph and the coupling strength: (i) unlocked, uncorrelated phase shifts between all oscillators; (ii) locked, arm hubs synchronized in phase with a k-dependent phase shift between the arm and central hub; and (iii) center silent, a central hub stopped oscillating and the arm hubs oscillated without synchrony. We compare experiment to theory. For case (ii), we identified a logarithmic dependence of the phase shift on star degree, and were able to discriminate between contributions to the phase shift arising from star topology and oscillator chemistry.

Plasticity facilitates pattern selection of networks of chemical oscillations

Rotating wave synchronization patterns are explored with a ring of 20 electrochemical oscillators during nickel electrodissolution in sulfuric acid. With desynchronized initial states, coupling alone yields predominance of nonrotating solutions, i.e., in-phase synchronization. An experimental technique is presented in which, through a combination of temporary alterations in topology, the application of global feedback provides rotational solutions. With phase repulsive global feedback, the in-phase synchronization is destabilized and a rotating wave is obtained. This feedback induced rotating wave can be employed to establish an initial condition for the rotating wave with coupling only. Higher order rotating solutions with 2, 3, and 4 waves corotating around the ring are observed, where the initial conditions are generated by temporary network rewiring to a structure with 2, 3, and 4 loops, respectively, and by global feedback. The experimental observations are supported by numerical simulations with a phase model. The results indicate that while network plasticity is thought to be significant in the operation of neural systems, it can also play a role in pattern selection of chemical systems.

Redox Is a Global Biodevice Information Processing Modality

Biology is well-known for its ability to communicate through (i) molecularly-specific signaling modalities and (ii) a globally-acting electrical modality associated with ion flow across biological membranes. Emerging research suggests that biology uses a third type of communication modality associated with a flow of electrons through reduction/oxidation (redox) reactions. This redox signaling modality appears to act globally and has features of both molecular and electrical modalities: since free electrons do not exist in aqueous solution, the electrons must flow through molecular intermediates that can be switched between two states - with electrons (reduced) or without electrons (oxidized). Importantly, this global redox modality is easily accessible through its electrical features using convenient electrochemical instrumentation. In this review, we explain this redox modality, describe our electrochemical measurements, and provide four examples demonstrating that redox enables communication between biology and electronics. The first two examples illustrate how redox probing can acquire biologically relevant information. The last two examples illustrate how redox inputs can transduce biologically-relevant transitions for patterning and the induction of a synbio transceiver for two-hop molecular communication. In summary, we believe redox provides a unique ability to bridge bio-device communication because simple electrochemical methods enable global access to biologically meaningful information. Further, we envision that redox may facilitate the application of information theory to the biological sciences.

The Cognitive Lens: a primer on conceptual tools for analysing information processing in developmental and regenerative morphogenesis

Brains exhibit plasticity, multi-scale integration of information, computation and memory, having evolved by specialization of non-neural cells that already possessed many of the same molecular components and functions. The emerging field of basal cognition provides many examples of decision-making throughout a wide range of non-neural systems. How can biological information processing across scales of size and complexity be quantitatively characterized and exploited in biomedical settings? We use pattern regulation as a context in which to introduce the Cognitive Lens-a strategy using well-established concepts from cognitive and computer science to complement mechanistic investigation in biology. To facilitate the assimilation and application of these approaches across biology, we review tools from various quantitative disciplines, including dynamical systems, information theory and least-action principles. We propose that these tools can be extended beyond neural settings to predict and control systems-level outcomes, and to understand biological patterning as a form of primitive cognition. We hypothesize that a cognitive-level information-processing view of the functions of living systems can complement reductive perspectives, improving efficient top-down control of organism-level outcomes. Exploration of the deep parallels across diverse quantitative paradigms will drive integrative advances in evolutionary biology, regenerative medicine, synthetic bioengineering, cognitive neuroscience and artificial intelligence. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Multivariate statistical analysis of chemical and electrochemical oscillators for an accurate frequency selection

The effect of experimental parameters on the frequency of chemical oscillators has been systematically studied since the first observations of clock reactions. The approach is mainly based on univariate changes in one specific parameter while others are kept constant. The frequency is then monitored and the effect of each parameter is discussed separately. This type of analysis, however, does not take into account the multiple interactions among the controllable parameters and the synergic responses on the oscillation frequency. We have carried out a multivariate statistical analysis of chemical (BZ-ferroin catalyzed reaction) and electrochemical (Cu/Cu₂O cathodic deposition) oscillators and identified the contributions of the experimental parameters on frequency variations. The BZ reaction presented a strong dependence on the initial concentration of sodium bromate and temperature, resulting in a frequency increase. The concentration of malonic acid, the organic substrate, affects the system but with lower intensity compared with the combination of sodium bromate and temperature. On the other hand, the Cu/Cu₂O electrochemical oscillator was shown to be less sensitive to changes in the temperature. The applied current density and pH were the two parameters which most perturbed the system. Interestingly, the frequency behaved nonmonotonically with a quadratic dependence. The multivariate analysis of both oscillators exhibited significant differences - while the homogenous oscillator displayed a linear dependence with the factors, the heterogeneous one revealed a more complex dependence with quadratic terms. Our results may contribute, for instance, in the synthesis of self-organized materials in which an accurate frequency selection is required and, depending on its value, different physicochemical properties are obtained.

"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.