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There is a growing interest towards implementation of maze solving in spatially-extended physical, chemical and living systems. Several reports of prototypes attracted great publicity, e.g. maze solving with slime mould and epithelial cells, maze navigating droplets. We show that most prototypes utilise one of two phenomena: a shortest path in a maze is a path of the least resistance for fluid and current flow, and a shortest path is a path of the steepest gradient of chemoattractants. We discuss that substrates with so-called maze-solving capabilities simply trace flow currents or chemical diffusion gradients. We illustrate our thoughts with a model of flow and experiments with slime mould. The chapter ends with a discussion of experiments on maze solving with plant roots and leeches which show limitations of the chemical diffusion maze-solving approach., Comment: This is a preliminary version of the chapter to be published in Adamatzky A. (Ed.) Shortest path solvers. From software to wetware. Springer, 2018

The slime mould Physarum polycephalum has been used in developing unconventional computing devices for in which the slime mould played a role of a sensing, actuating, and computing device. These devices treated the slime mould rather as an active living substrate yet the slime mould is a self-consistent living creature which evolved for millions of years and occupied most part of the world, but in any case, that living entity did not own true cognition, just automated biochemical mechanisms. To "rehabilitate" the slime mould from the rank of a purely living electronics element to a "creature of thoughts" we are analyzing the cognitive potential of P. polycephalum. We base our theory of minimal cognition of the slime mould on a bottom-up approach, from the biological and biophysical nature of the slime mould and its regulatory systems using frameworks suh as Lyon's biogenic cognition, Muller, di Primio-Lengeler\'s modifiable pathways, Bateson's "patterns that connect" framework, Maturana's autopoetic network, or proto-consciousness and Morgan's Canon.

Liquid marbles are microlitre droplets of liquid, encapsulated by self-organised hydrophobic particles at the liquid/air interface. They offer an efficient approach for manipulating liquid droplets and compartmentalising reactions in droplets. Digital fluidic devices employing liquid marbles might benefit from having embedded computing circuits without electronics and moving mechanical parts (apart from the marbles). We present an experimental implementation of a collision gate with liquid marbles. Mechanics of the gate follows principles of Margolus' soft-sphere collision gate. Boolean values of the inputs are given by the absence (FALSE) or presence (TRUE) of a liquid marble. There are three outputs: two outputs are trajectories of undisturbed marbles (they only report TRUE when just one marble is present at one of the inputs), one output is represented by trajectories of colliding marbles (when two marbles collide they lose their horizontal momentum and fall), this output reports TRUE only when two marbles are present at inputs. Thus the gate implements AND and AND-NOT logical functions. We speculate that by merging trajectories representing AND-NOT output into a single channel one can produce a one-bit half-adder. Potential design of a one-bit full-adder is discussed, and the synthesis of both a pure nickel metal and hybrid nickel/polymer liquid marble is reported.

We construct logical gates via topology optimisation (aimed to solve a station problem of heat conduction) of a conductive material layout. Values of logical variables are represented high and low values of a temperature at given sites. Logical functions are implemented via the formation of an optimum layout of conductive material between the sites with loading conditions. We implement AND and XOR gates and a one-bit binary half-adder.

We represent a filamentous actin molecule as a graph of finite-state machines (F-actin automaton). Each node in the graph takes three states --- resting, excited, refractory. All nodes update their states simultaneously and by the same rule, in discrete time steps. Two rules are considered: threshold rule --- a resting node is excited if it has at least one excited neighbour and narrow excitation interval rule --- a resting node is excited if it has exactly one excited neighbour. We analyse distributions of transient periods and lengths of limit cycles in evolution of F-actin automaton, propose mechanisms for formation of limit cycles and evaluate density of information storage in F-actin automata.

The Microbial Fuel Cell (MFC) is a bio-electrochemical transducer converting waste products into electricity using microbial communities. Cellular Automaton (CA) is a uniform array of finite-state machines that update their states in discrete time depending on states of their closest neighbors by the same rule. Arrays of MFCs could, in principle, act as massive-parallel computing devices with local connectivity between elementary processors. We provide a theoretical design of such a parallel processor by implementing CA in MFCs. We have chosen Conway's Game of Life as the 'benchmark' CA because this is the most popular CA which also exhibits an enormously rich spectrum of patterns. Each cell of the Game of Life CA is realized using two MFCs. The MFCs are linked electrically and hydraulically. The model is verified via simulation of an electrical circuit demonstrating equivalent behaviors. The design is a first step towards future implementations of fully autonomous biological computing devices with massive parallelism. The energy independence of such devices counteracts their somewhat slow transitions - compared to silicon circuitry - between the different states during computation.

A cellular non-linear network (CNN) is a uniform regular array of locally connected continuous-state machines, or nodes, which update their states simultaneously in discrete time. A microbial fuel cell (MFC) is an electro-chemical reactor using the metabolism of bacteria to drive an electrical current. In a CNN model of the MFC, each node takes a vector of states which represent geometrical characteristics of the cell, like the electrodes or impermeable borders, and quantify measurable properties like bacterial population, charges produced and hydrogen ions concentrations. The model allows the study of integral reaction of the MFC, including temporal outputs, to spatial disturbances of the bacterial population and supply of nutrients. The model can also be used to evaluate inhomogeneous configurations of bacterial populations attached on the electrode biofilms.

We discuss possible designs and prototypes of computing systems that could be based on morphological development of roots, interaction of roots, and analog electrical computation with plants, and plant-derived electronic components. In morphological plant processors data are represented by initial configuration of roots and configurations of sources of attractants and repellents; results of computation are represented by topology of the roots' network. Computation is implemented by the roots following gradients of attractants and repellents, as well as interacting with each other. Problems solvable by plant roots, in principle, include shortest-path, minimum spanning tree, Voronoi diagram, $\alpha$-shapes, convex subdivision of concave polygons. Electrical properties of plants can be modified by loading the plants with functional nanoparticles or coating parts of plants of conductive polymers. Thus, we are in position to make living variable resistors, capacitors, operational amplifiers, multipliers, potentiometers and fixed-function generators. The electrically modified plants can implement summation, integration with respect to time, inversion, multiplication, exponentiation, logarithm, division. Mathematical and engineering problems to be solved can be represented in plant root networks of resistive or reaction elements. Developments in plant-based computing architectures will trigger emergence of a unique community of biologists, electronic engineering and computer scientists working together to produce living electronic devices which future green computers will be made of., Comment: The chapter will be published in "Inspired by Nature. Computing inspired by physics, chemistry and biology. Essays presented to Julian Miller on the occasion of his 60th birthday", Editors: Susan Stepney and Andrew Adamatzky (Springer, 2017)

A thin-layer Belousov-Zhabotinsky (BZ) medium is a powerful computing device capable for implementing logical circuits, memory, image processors, robot controllers, and neuromorphic architectures. We design the reversible logical gates --- Fredkin gate and Toffoli gate --- in a BZ medium network of excitable channels with sub-excitable junctions. Local control of the BZ medium excitability is an important feature of the gates' design. A excitable thin-layer BZ medium responds to a localised perturbation with omnidirectional target or spiral excitation waves. A sub-excitable BZ medium responds to an asymmetric perturbation by producing travelling localised excitation wave-fragments similar to dissipative solitons. We employ interactions between excitation wave-fragments to perform computation. We interpret the wave-fragments as values of Boolean variables. A presence of a wave-fragment at a given site of a circuit represents logical truth, absence of the wave-fragment --- logical false. Fredkin gate consists of ten excitable channels intersecting at eleven junctions eight of which are sub-excitable. Toffoli gate consists of six excitable channels intersecting at six junctions four of which are sub-excitable. The designs of the gates are verified using numerical integration of two-variable Oregonator equations.

In the Gray-Scott reaction-diffusion system one reactant is constantly fed in the system, another reactant is reproduced by consuming the supplied reactant and also converted to an inert product. The rate of feeding one reactant in the system and the rate of removing another reactant from the system determine configurations of concentration profiles: stripes, spots, waves. We calculate the generative complexity --- a morphological complexity of concentration profiles grown from a point-wise perturbation of the medium --- of the Gray-Scott system for a range of the feeding and removal rates. The morphological complexity is evaluated using Shannon entropy, Simpson diversity, approximation of Lempel-Ziv complexity, and expressivity (Shannon entropy divided by space-filling). We analyse behaviour of the systems with highest values of the generative morphological complexity and show that the Gray-Scott systems expressing highest levels of the complexity are composed of the wave-fragments (similar to wave-fragments in sub-excitable media) and travelling localisations (similar to quasi-dissipative solitons and gliders in Conway's Game of Life).

Theoretical constructs of logical gates implemented with plant roots are morphological computing asynchronous devices. Values of Boolean variables are represented by plant roots. A presence of a plant root at a given site symbolises the logical {\sc True}, an absence the logical {\sc False}. Logical functions are calculated via interaction between roots. Two types of two-inputs-two-outputs gates are proposed: a gate $\langle x, y \rangle \rightarrow \langle xy, x+y \rangle$ where root apexes are guided by gravity and a gate $\langle x, y \rangle \rightarrow \langle \overline{x}y, x \rangle$ where root apexes are guided by humidity. We propose a design of binary half-adder based on the gates.

A cellular automaton collider is a finite state machine build of rings of one-dimensional cellular automata. We show how a computation can be performed on the collider by exploiting interactions between gliders (particles, localisations). The constructions proposed are based on universality of elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and computing on rings., Comment: 39 pages, 32 figures, 3 tables

A fluidic one-bit half-adder is made of five channels which intersect at a junction. Two channels are inputs, two channels are outputs and one channel is the drain. The channels direct fluid from input fragments to output fragments and the streams of fluid interact at the junctions. Binary signals are represented by water droplets introduced in the input channels: presence of a droplet in an input or output segments symbolises logical {\sc True}, absence --- {\sc False}. The droplets travel along channels by following a path of least resistance unless deflected at the junction. We demonstrate the function of the half-adder in both computer modelling and laboratory experiments, and propose a design of a one-bit full adder based on simulation.

We overview experimental laboratory prototypes of maze solvers. We speculate that all maze solvers implement Lee algorithm by first developing a gradient of values showing a distance from any site of the maze to the destination site and then tracing a path from a given source site to the destination site. All prototypes approximate a set of many-source-one-destination paths using resistance, chemical and temporal gradients. They trace a path from a given source site to the destination site using electrical current, fluidic, growth of slime mould, Marangoni flow, crawling of epithelial cells, excitation waves in chemical medium, propagating crystallisation patterns. Some of the prototypes visualise the path using a stream of dye, thermal camera or glow discharge; others require a computer to extract the path from time lapse images of the tracing. We discuss the prototypes in terms of speed, costs and durability of the path visualisation., Comment: Final version of the paper will be published in "Emergent Computation. Festschrift for Selim Akl" (Springer, 2016)

Actin is a globular protein which forms long polar filaments in eukaryotic. The actin filaments play roles of cytoskeleton, motility units , information processing and learning. We model actin filament as a double chain of finite state machines, nodes, which take states `0' and `1'. The states are abstractions of absence and presence of a sub-threshold charge on an actin units corresponding to the nodes. All nodes update their state in parallel in discrete time. A node updates its current state depending on states of two closest neighbours in the node chain and two closest neighbours in the complementary chain. Previous models of actin automata considered momentary state transitions of nodes. We enrich the actin automata model by assuming that states of nodes depends not only on the current states of neighbouring node but also on their past states. Thus, we assess the effect of memory of past states on the dynamics of acting automata. We demonstrate in computational experiments that memory slows down propagation of perturbations, decrease entropy of space-time patterns generated, transforms travelling localisations to stationary oscillators, and stationary oscillations to still patterns., Comment: Accepted for publication in "International Journal of Bifurcation and Chaos"

Slime mould \emph{Physarum polycephalum} is a large single cell capable for distributed sensing, concurrent information processing, parallel computation and decentralised actuation. The ease of culturing and experimenting with Physarum makes this slime mould an ideal substrate for real-world implementations of unconventional sensing and computing devices. In the last decade the Physarum became a swiss knife of the unconventional computing: give the slime mould a problem it will solve it. We provide a concise summary of what exact computing and sensing operations are implemented with live slime mould. The Physarum devices range from morphological processors for computational geometry to experimental archeology tools, from self-routing wires to memristors, from devices approximating a shortest path to analog physical models of space exploration., Comment: this is a draft of the chapter to appear in 'Advances in Unconventional Computing' (Sprineer, 2016)

Slime mould Physarum polycephalum is a single cell visible by unaided eye. Let the slime mould span two electrodes with a single protoplasmic tube: if the tube is heated to approximately 40{\deg}C, the electrical resistance of the protoplasmic tube increases from 3 M{\Omega} to approximatively 10'000 M{\Omega}. The organism's resistance is not proportional nor correlated to the temperature of its environment. Slime mould can therefore not be considered as a thermistor but rather as a thermic switch. We employ the P. polycephalum thermic switch to prototype hybrid electrical analog summator, NAND gates, and cascade the gates into Flip-Flop latch. Computing operations performed on this bio-hybrid computing circuitry feature high repeatability, reproducibility and comparably low propagation delays, Comment: 9 pages, 6 figures & 1 Table, submitted to Scientific Reports

The manipulation of biological substrates is becoming more popular route towards generating novel computing devices. Physarum polycephalum is used as a model organism in biocomputing because it can create `wires' for use in hybrid circuits; programmable growth by manipulation through external stimuli and the ability withstanding a current and its tolerance to hybridisation with a variety of nano/microparticles. Lettuce seedlings have also had previous interest invested in them for generating plant wires, although currently there is little information as to their suitability for such applications. In this study both P. polycephalum and Lettuce seedlings were hybridised with gold nanoparticles - functionalised and unfunctionalised - to explore their uptake, toxicological effects and, crucially, any alterations in electrical properties they bestow upon the organisms. Using various microscopy techniques it was shown that P. polycephalum and lettuce seedlings are able to internalize nanoparticles and assemble them in vivo, however some toxicological effects were observed. The electrical resistance of both lettuce seedlings and P. polycephalum was found to decrease, the most significant reduction being with lettuce seedlings whose resistance reduced from 3MOhms to 0.5MOhms. We conclude that gold is a suitable nanomaterial for biohybridisation specifically in creating conductive pathways for more efficient biological wires in self-growing hybrid circuitry.

Leeches are fascinating creatures: they have simple modular nervous circuitry m yet exhibit a rich spectrum of behavioural modes. Leeches could be ideal blue-prints for designing flexible soft robots which are modular, multi-functional, fault-tolerant, easy to control, capable for navigating using optical, mechanical and chemical sensorial inputs, have autonomous inter-segmental coordination and adaptive decision-making. With future designs of leech-robots in mind we study how leeches behave in geometrically constrained spaces. Core results of the paper deal with leeches exploring a row of rooms arranged along a narrow corridor. In laboratory experiments we find that rooms closer to ends of the corridor are explored by leeches more often than rooms in the middle of the corridor. Also, in series of scoping experiments, we evaluate leeches capabilities to navigating in mazes towards sources of vibration and chemo-attraction. We believe our results lay foundation for future developments of robots mimicking behaviour of leeches.

Safe evacuation of people from building and outdoor environments, and search and rescue operations, always will remain actual in course of all socio-technological developments. Modern facilities offer a range of automated systems to guide residents towards emergency exists. The systems are assumed to be infallible. But what if they fail? How occupants not familiar with a building layout will be looking for exits in case of very limited visibility where tactile sensing is the only way to assess the environment? Analogous models of human behaviour, and socio-dynamics in general, are provided to be fruitful ways to explore alternative, or would-be scenarios. Crowd, or a single person, dynamics could be imitated using particle systems, reaction-diffusion chemical medium, electro-magnetic fields, or social insects. Each type of analogous model offer unique insights on behavioural patterns of natural systems in constrained geometries. In this particular paper we have chosen leeches to analyse patterns of exploration. Reasons are two-fold. First, when deprived from other stimuli leeches change their behavioural modes in an automated regime in response to mechanical stimulation. Therefore leeches can give us invaluable information on how human beings might behave under stress and limited visibility. Second, leeches are ideal blueprints of future soft-bodied rescue robots. Leeches have modular nervous circuitry with a rich behavioral spectrum. Leeches are multi-functional, fault-tolerant with autonomous inter-segment coordination and adaptive decision-making. We aim to answer the question: how efficiently a real building can be explored and whether there any dependencies on the pathways of exploration and geometrical complexity of the building. In our case studies we use templates made on the floor plan of real building.