Logical and arithmetic circuits in Belousov-Zhabotinsky encapsulated disks

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.

Provenance of life: Chemical autonomous agents surviving through associative learning

We present a benchmark study of autonomous, chemical agents exhibiting associative learning of an environmental feature. Associative learning systems have been widely studied in cognitive science and artificial intelligence but are most commonly implemented in highly complex or carefully engineered systems, such as animal brains, artificial neural networks, DNA computing systems, and gene regulatory networks, among others. The ability to encode environmental information and use it to make simple predictions is a benchmark of biological resilience and underpins a plethora of adaptive responses in the living hierarchy, spanning prey animal species anticipating the arrival of predators to epigenetic systems in microorganisms learning environmental correlations. Given the ubiquitous and essential presence of learning behaviors in the biosphere, we aimed to explore whether simple, nonliving dissipative structures could also exhibit associative learning. Inspired by previous modeling of associative learning in chemical networks, we simulated simple systems composed of long- and short-term memory chemical species that could encode the presence or absence of temporal correlations between two external species. The ability to learn this association was implemented in Gray-Scott reaction-diffusion spots, emergent chemical patterns that exhibit self-replication and homeostasis. With the novel ability of associative learning, we demonstrate that simple chemical patterns can exhibit a broad repertoire of lifelike behavior, paving the way for in vitro studies of autonomous chemical learning systems, with potential relevance to artificial life, origins of life, and systems chemistry. The experimental realization of these learning behaviors in protocell or coacervate systems could advance a new research direction in astrobiology, since our system significantly reduces the lower bound on the required complexity for autonomous chemical learning.

Coexistence of oscillatory and reduced states on a spherical field controlled by electrical potential

The Belousov-Zhabotinsky (BZ) reaction was investigated to elucidate features of oscillations depending on the applied electrical potential, E. A cation-exchange resin bead loaded with the catalyst of the BZ reaction was placed on a platinum plate as a working electrode and then E was applied. We found that global oscillations (GO) and a reduced state coexisted on the bead at a negative value of E and that the source point of GO changed depending on E. The thickness of the reduced state was determined by a yellow colored region which corresponded to the distribution of Br₂. The present studies suggest that the distribution of the inhibitor, Br^-, which is produced from Br₂, plays an important role in the existence of the reduced state and GO, and the source point of GO.

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.

Implementing parallel arithmetic via acetylation and its application to chemical image processing

Chemical mixtures can be leveraged to store large amounts of data in a highly compact form and have the potential for massive scalability owing to the use of large-scale molecular libraries. With the parallelism that comes from having many species available, chemical-based memory can also provide the physical substrate for computation with increased throughput. Here, we represent non-binary matrices in chemical solutions and perform multiple matrix multiplications and additions, in parallel, using chemical reactions. As a case study, we demonstrate image processing, in which small greyscale images are encoded in chemical mixtures and kernel-based convolutions are performed using phenol acetylation reactions. In these experiments, we use the measured concentrations of reaction products (phenyl acetates) to reconstruct the output image. In addition, we establish the chemical criteria required to realize chemical image processing and validate reaction-based multiplication. Most importantly, this work shows that fundamental arithmetic operations can be reliably carried out with chemical reactions. Our approach could serve as a basis for developing more advanced chemical computing architectures.

How Does a Simple Network of Chemical Oscillators See the Japanese Flag?

Chemical computing is something we use every day (e.g., in the brain), but we can still not explore and master its potential in human-made experiments. It is expected that the maximum computational efficiency of a chemical medium can be achieved if information is processed in parallel by different parts of the medium. In this paper, we use computer simulations to explore the efficiency of chemical computing performed by a small network of three coupled chemical oscillators. We optimize the network to recognize the white and red regions of the Japanese flag. The input information is introduced as the inhibition times of individual oscillators, and the output information is coded in the number of activator maxima observed on a selected oscillator. We have used the Oregonator model to simulate the network time evolution and the evolutionary optimization to find the best network for the considered task. We have found that even a network of three interacting oscillators can recognize the color of a randomly selected point with 95% accuracy.

Advanced Chemical Computing Using Discrete Turing Patterns in Arrays of Coupled Cells

We examine dynamical switching among discrete Turing patterns that enable chemical computing performed by mass-coupled reaction cells arranged as arrays with various topological configurations: three coupled cells in a cyclic array, four coupled cells in a linear array, four coupled cells in a cyclic array, and four coupled cells in a branched array. Each cell is operating as a continuous stirred tank reactor, within which the glycolytic reaction takes place, represented by a skeleton inhibitor-activator model where ADP plays the role of activator and ATP is the inhibitor. The mass coupling between cells is assumed to be operating in three possible transport regimes: (i) equal transport coefficients of the inhibitor and activator (ii) slightly faster transport of the activator, and (iii) strongly faster transport of the inhibitor. Each cellular array is characterized by two pairs of tunable parameters, the rate coefficients of the autocatalytic and inhibitory steps, and the transport coefficients of the coupling. Using stability and bifurcation analysis we identified conditions for occurrence of discrete Turing patterns associated with non-uniform stationary states. We found stable symmetric and/or asymmetric discrete Turing patterns coexisting with stable uniform periodic oscillations. To switch from one of the coexisting stable regimes to another we use carefully targeted perturbations, which allows us to build systems of logic gates specific to each topological type of the array, which in turn enables to perform advanced modes of chemical computing. By combining chemical computing techniques in the arrays with glycolytic excitable channels, we propose a cellular assemblage design for advanced chemical computing.

Applications of Information Theory Methods for Evolutionary Optimization of Chemical Computers

It is commonly believed that information processing in living organisms is based on chemical reactions. However, the human achievements in constructing chemical information processing devices demonstrate that it is difficult to design such devices using the bottom-up strategy. Here I discuss the alternative top-down design of a network of chemical oscillators that performs a selected computing task. As an example, I consider a simple network of interacting chemical oscillators that operates as a comparator of two real numbers. The information on which of the two numbers is larger is coded in the number of excitations observed on oscillators forming the network. The parameters of the network are optimized to perform this function with the maximum accuracy. I discuss how information theory methods can be applied to obtain the optimum computing structure.

Modeling somatic computation with non-neural bioelectric networks

The field of basal cognition seeks to understand how adaptive, context-specific behavior occurs in non-neural biological systems. Embryogenesis and regeneration require plasticity in many tissue types to achieve structural and functional goals in diverse circumstances. Thus, advances in both evolutionary cell biology and regenerative medicine require an understanding of how non-neural tissues could process information. Neurons evolved from ancient cell types that used bioelectric signaling to perform computation. However, it has not been shown whether or how non-neural bioelectric cell networks can support computation. We generalize connectionist methods to non-neural tissue architectures, showing that a minimal non-neural Bio-Electric Network (BEN) model that utilizes the general principles of bioelectricity (electrodiffusion and gating) can compute. We characterize BEN behaviors ranging from elementary logic gates to pattern detectors, using both fixed and transient inputs to recapitulate various biological scenarios. We characterize the mechanisms of such networks using dynamical-systems and information-theory tools, demonstrating that logic can manifest in bidirectional, continuous, and relatively slow bioelectrical systems, complementing conventional neural-centric architectures. Our results reveal a variety of non-neural decision-making processes as manifestations of general cellular biophysical mechanisms and suggest novel bioengineering approaches to construct functional tissues for regenerative medicine and synthetic biology as well as new machine learning architectures.

A brief history of liquid computers

A substrate does not have to be solid to compute. It is possible to make a computer purely from a liquid. I demonstrate this using a variety of experimental prototypes where a liquid carries signals, actuates mechanical computing devices and hosts chemical reactions. We show hydraulic mathematical machines that compute functions based on mass transfer analogies. I discuss several prototypes of computing devices that employ fluid flows and jets. They are fluid mappers, where the fluid flow explores a geometrically constrained space to find an optimal way around, e.g. the shortest path in a maze, and fluid logic devices where fluid jet streams interact at the junctions of inlets and results of the computation are represented by fluid jets at selected outlets. Fluid mappers and fluidic logic devices compute continuously valued functions albeit discretized. There is also an opportunity to do discrete operation directly by representing information by droplets and liquid marbles (droplets coated by hydrophobic powder). There, computation is implemented at the sites, in time and space, where droplets collide one with another. The liquid computers mentioned above use liquid as signal carrier or actuator: the exact nature of the liquid is not that important. What is inside the liquid becomes crucial when reaction-diffusion liquid-phase computing devices come into play: there, the liquid hosts families of chemical species that interact with each other in a massive-parallel fashion. I shall illustrate a range of computational tasks, including computational geometry, implementable by excitation wave fronts in nonlinear active chemical medium. The overview will enable scientists and engineers to understand how vast is the variety of liquid computers and will inspire them to design their own experimental laboratory prototypes. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Thermal switch of oscillation frequency in Belousov-Zhabotinsky liquid marbles

External control of oscillation dynamics in the Belousov-Zhabotinsky (BZ) reaction is important for many applications including encoding computing schemes. When considering the BZ reaction, there are limited studies dealing with thermal cycling, particularly cooling, for external control. Recently, liquid marbles (LMs) have been demonstrated as a means of confining the BZ reaction in a system containing a solid-liquid interface. BZ LMs were prepared by rolling 50 μl droplets in polyethylene (PE) powder. Oscillations of electrical potential differences within the marble were recorded by inserting a pair of electrodes through the LM powder coating into the BZ solution core. Electrical potential differences of up to 100 mV were observed with an average period of oscillation ca 44 s. BZ LMs were subsequently frozen to -1°C to observe changes in the frequency of electrical potential oscillations. The frequency of oscillations reduced upon freezing to 11 mHz cf. 23 mHz at ambient temperature. The oscillation frequency of the frozen BZ LM returned to 23 mHz upon warming to ambient temperature. Several cycles of frequency fluctuations were able to be achieved.

Towards Functional Droplet Architectures: a Belousov-Zhabotinsky Medium for Networks

The confluence of droplet-compartmentalised chemical systems and architectures composed of interacting droplets points towards a novel technology mimicking core features of the cellular architecture that dominates biology. A key challenge to achieve such a droplet technology is long-term stability in conjunction with interdroplet communication. Here, we probed the parameter space of the Belousov-Zhabotinsky (BZ) medium, an extensively studied model for non-equilibrium chemical reactions, pipetted as 2.5 mm droplets in hexadecane oil. The presence of asolectin lipids enabled the formation of arrays of contacted BZ droplets, of which the wave patterns were characterised over time. We utilised laser-cut acrylic templates with over 40 linear oil-filled slots in which arrays are formed by pipetting droplets of the desired BZ composition, enabling parallel experiments and automated image analysis. Using variations of conventional malonic acid BZ medium, wave propagation over droplet-droplet interfaces was not observed. However, a BZ medium containing both malonic acid and 1,4-cyclohexanedione was found to enable inter-droplet wave propagation. We anticipate that the chemical excitation properties of this mixed-substrate BZ medium, in combination with the droplet stability of the networks demonstrated here for nearly 400 droplets in a template-defined topology, will facilitate the development of scalable functional droplet networks.

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.

Cancer classification with a network of chemical oscillators

We discuss chemical information processing considering dataset classifiers formed with a network of interacting droplets.

Microfluidic platform for reproducible self-assembly of chemically communicating droplet networks with predesigned number and type of the communicating compartments

We report a microfluidic system for individually tailored generation and incubation of core-shell liquid structures with multiple cores that chemically communicate with each other via lipid membranes. We encapsulate an oscillating reaction-diffusion Belousov-Zhabotinsky (BZ) medium inside the aqueous droplets and study the propagation of chemical wave-fronts through the membranes. We further encapsulate the sets of interconnected BZ-droplets inside oil-lipid shells in order to i) chemically isolate the structures and ii) confine them via tunable capillary forces which leads to self-assembly of predesigned topologies. We observe that doublets (pairs) of droplets encapsulated in the shell exhibit oscillation patterns that evolve in time. We collect statistical data from tens of doublets all created under precisely controlled, almost identical conditions from which we conclude that the different types of transitions between the patterns depend on the relative volumes of the droplets within a chemically coupled pair. With this we show that the volume of the compartment is an important control parameter in designing chemical networks, a feature previously appreciated only by theory. Our system not only allows for new insights into the dynamics of geometrically complex and interacting chemical systems but is also suitable for generating autonomous chemically interconnected microstructures with possible future use, e.g., as smart biosensors or drug-release capsules.

Binary full adder, made of fusion gates, in a subexcitable Belousov-Zhabotinsky system

In an excitable thin-layer Belousov-Zhabotinsky (BZ) medium a localized perturbation leads to the formation of omnidirectional target or spiral waves of excitation. A subexcitable BZ medium responds to asymmetric local perturbation by producing traveling localized excitation wave-fragments, distant relatives of dissipative solitons. The size and life span of an excitation wave-fragment depend on the illumination level of the medium. Under the right conditions the wave-fragments conserve their shape and velocity vectors for extended time periods. I interpret the wave-fragments as values of Boolean variables. When two or more wave-fragments collide they annihilate or merge into a new wave-fragment. States of the logic variables, represented by the wave-fragments, are changed in the result of the collision between the wave-fragments. Thus, a logical gate is implemented. Several theoretical designs and experimental laboratory implementations of Boolean logic gates have been proposed in the past but little has been done cascading the gates into binary arithmetical circuits. I propose a unique design of a binary one-bit full adder based on a fusion gate. A fusion gate is a two-input three-output logical device which calculates the conjunction of the input variables and the conjunction of one input variable with the negation of another input variable. The gate is made of three channels: two channels cross each other at an angle, a third channel starts at the junction. The channels contain a BZ medium. When two excitation wave-fragments, traveling towards each other along input channels, collide at the junction they merge into a single wave-front traveling along the third channel. If there is just one wave-front in the input channel, the front continues its propagation undisturbed. I make a one-bit full adder by cascading two fusion gates. I show how to cascade the adder blocks into a many-bit full adder. I evaluate the feasibility of my designs by simulating the evolution of excitation in the gates and adders using the numerical integration of Oregonator equations.

Understanding Networks of Computing Chemical Droplet Neurons Based on Information Flow

In this paper, we present general methods that can be used to explore the information processing potential of a medium composed of oscillating (self-exciting) droplets. Networks of Belousov–Zhabotinsky (BZ) droplets seem especially interesting as chemical reaction-diffusion computers because their time evolution is qualitatively similar to neural network activity. Moreover, such networks can be self-generated in microfluidic reactors. However, it is hard to track and to understand the function performed by a medium composed of droplets due to its complex dynamics. Corresponding to recurrent neural networks, the flow of excitations in a network of droplets is not limited to a single direction and spreads throughout the whole medium. In this work, we analyze the operation performed by droplet systems by monitoring the information flow. This is achieved by measuring mutual information and time delayed mutual information of the discretized time evolution of individual droplets. To link the model with reality, we use experimental results to estimate the parameters of droplet interactions. We exemplarily investigate an evolutionary generated droplet structure that operates as a NOR gate. The presented methods can be applied to networks composed of at least hundreds of droplets.

Chemical computing with reaction-diffusion processes

Chemical reactions are responsible for information processing in living organisms. It is believed that the basic features of biological computing activity are reflected by a reaction-diffusion medium. We illustrate the ideas of chemical information processing considering the Belousov-Zhabotinsky (BZ) reaction and its photosensitive variant. The computational universality of information processing is demonstrated. For different methods of information coding constructions of the simplest signal processing devices are described. The function performed by a particular device is determined by the geometrical structure of oscillatory (or of excitable) and non-excitable regions of the medium. In a living organism, the brain is created as a self-grown structure of interacting nonlinear elements and reaches its functionality as the result of learning. We discuss whether such a strategy can be adopted for generation of chemical information processing devices. Recent studies have shown that lipid-covered droplets containing solution of reagents of BZ reaction can be transported by a flowing oil. Therefore, structures of droplets can be spontaneously formed at specific non-equilibrium conditions, for example forced by flows in a microfluidic reactor. We describe how to introduce information to a droplet structure, track the information flow inside it and optimize medium evolution to achieve the maximum reliability. Applications of droplet structures for classification tasks are discussed.

Routing of Physarum polycephalum "signals" using simple chemicals

In previous work the chemotaxis toward simple organic chemicals was assessed. We utilize the knowledge gained from these chemotactic assays to route Physarum polycephalum “signals” at a series of junctions. By applying chemical inputs at a simple T-junction we were able to reproducibly control the path taken by the plasmodium of P. Polycephalum. Where the chemoattractant farnesene was used at one input a routed signal could be reproducibly generated i.e., P. Polycephalum moves toward the source of chemoattractant. Where the chemoattractant was applied at both inputs the signal was reproducibly split i.e., at the junction the plasmodium splits and moves toward both sources of chemoattractant. If a chemorepellent was used then the signal was reproducibly suppressed i.e., P. Polycephalum did not reach either output and was confined to the input channel. This was regardless of whether a chemoattractant was used in combination with the chemorepellent showing a hierarchy of inhibition over attraction. If no chemical input was used in the simple circuit then a random signal was generated, whereby P. Polycephalum would move toward one output at the junction, but the direction was randomly selected.

Information coding with frequency of oscillations in Belousov-Zhabotinsky encapsulated disks

Information processing with an excitable chemical medium, like the Belousov-Zhabotinsky (BZ) reaction, is typically based on information coding in the presence or absence of excitation pulses. Here we present a new concept of Boolean coding that can be applied to an oscillatory medium. A medium represents the logical TRUE state if a selected region oscillates with a high frequency. If the frequency fails below a specified value, it represents the logical FALSE state. We consider a medium composed of disks encapsulating an oscillatory mixture of reagents, as related to our recent experiments with lipid-coated BZ droplets. We demonstrate that by using specific geometrical arrangements of disks containing the oscillatory medium one can perform logical operations on variables coded in oscillation frequency. Realizations of a chemical signal diode and of a single-bit memory with oscillatory disks are also discussed.

Chemical computing based on Turing patterns in two coupled cells with equal transport coefficients

Two diffusively coupled reaction cells with a nonlinear reaction are used to perform chemical computing based on targeted perturbations switching between two Turing patterns defining two states of a logical device.

Time-dependent wave selection for information processing in excitable media

We demonstrate an improved technique for implementing logic circuits in light-sensitive chemical excitable media. The technique makes use of the constant-speed propagation of waves along defined channels in an excitable medium based on the Belousov-Zhabotinsky reaction, along with the mutual annihilation of colliding waves. What distinguishes this work from previous work in this area is that regions where channels meet at a junction can periodically alternate between permitting the propagation of waves and blocking them. These valvelike areas are used to select waves based on the length of time that it takes waves to propagate from one valve to another. In an experimental implementation, the channels that make up the circuit layout are projected by a digital projector connected to a computer. Excitable channels are projected as dark areas and unexcitable regions as light areas. Valves alternate between dark and light: Every valve has the same period and phase, with a 50% duty cycle. This scheme can be used to make logic gates based on combinations of or and and-not operations, with few geometrical constraints. Because there are few geometrical constraints, compact circuits can be implemented. Experimental results from an implementation of a four-bit input, two-bit output integer square root circuit are given.