Engineering Devices with Slime Mold

Slimeware: Engineering Devices Andrew Adamatzky* with Slime Mold UWE Bristol

Keywords Parallel biological computers, amorphous computers, living technology, slime mold Abstract The plasmodium of the acellular slime mold Physarum A version of this paper with color figures polycephalum is a gigantic single cell visible to the unaided eye. The is available online at http://dx.doi.org/10.1162/ cell shows a rich spectrum of behavioral patterns in response to artl_a_00110. Subscription required. Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 environmental conditions. In a series of simple experiments we demonstrate how to make computing, sensing, and actuating devices from the slime mold. We show how to program living slime mold machines by configurations of repelling and attracting gradients and demonstrate the workability of the living machines on tasks of computational geometry, logic, and arithmetic.

1 Introduction

Since its inception in the early 1980s, the field of unconventional computation [16] has been dominated by theoretical research, including quantum computation, membrane computing, and dynamical-systems computing. Just a few experimental laboratory prototypes have been designed so far [14, 39], such as chemical reaction-diffusion processors [11], extended analog computers [26], microfluidic circuits [20], gas-discharge systems [32], chemotactic droplets [24], enzyme-based logical circuits [23, 31], crystallization computers [2], geometrically constrained chemical computers [21, 22, 27, 34, 41], and molecular logical gates and circuits [25, 38]. In contrast, there are hundreds if not thousands of published articles on quantum computation, membrane computing, and artificial immune systems. Such a weak representation of laboratory experiments in the field of unconventional computation may be due to technical difficulties and the costs of prototyping. If there were a substrate that was simple to maintain and required minimal equipment to experiment on, then progress in designing novel computing devices would be much more visible. The slime mold Physarum polycephalum could play the role of such a “dream” substrate for implementing prototypes of novel computers. We exemplify how some modes of computation can be executed by the plasmodium of P. polycephalum (Figure 1) and show how simple slime mold computers—Physarum machines— can be built. The devices described in this article are instances of wetware of a secondary class of living technologies [15]. Rephrasing Bedau et al. [15], we can say that Physarum machines are artificial in that they are created by our intentional activities, yet they are also natural in that they grow, respond to environmental stimuli, and adapt by their own biological laws.

* Corresponding author. UWE, Bristol, BS16 1QY, UK. E-mail: [email protected]

2 Physarum polycephalum

Physarum polycephalum belongs to a species of order Physarales,subclassMyxogastromycetidae,class Myxomycetes, division Myxostelida. It is commonly known as a true, acellular, or multiheaded slime mold. The plasmodium is a vegetative phase, a single cell with myriad diploid nuclei. The plasmodium is visible to the naked eye, and looks like an amorphous yellowish mass with networks of protoplasmic tubes. It behaves and moves as a giant amoeba, feeding on bacteria, spores, and other microbial creatures and microparticles [37]. P. polycephalum has quite a rich life cycle [37]: fruiting bodies, spores, single-cell amoebas, and syncytium. In its plasmodium stage, P. polycephalum consumes microscopic particles, and during its foraging behavior the plasmodium spans scattered sources of nutrients with a network of protoplasmic tubes (Figure 2). The plasmodium optimizes its protoplasmic network, which covers all Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 sources of nutrients and guarantees robust and quick distribution of nutrients in the plasmodiumʼs body. The plasmodiumʼs foraging behavior can be interpreted as a computation [28–30]: data are represented by spatial configurations of attractants and repellents, and the results are represented by the structure of the protoplasmic network [6]. The plasmodium can solve computational problems with natural parallelism, such as those related to shortest paths [29] and hierarchies of planar proximity graphs [1], computation of plane tessellations [36], execution of logical computing schemes [4, 40], and natural implementation of spatial logic and process algebra [33].

3 Physarum Machines

We developed a concept and designed a series of experimental laboratory prototypes of computing devices—Physarum machines [6]—based on P. polycephalum.APhysarum machine is a programmable amorphous biological computing device experimentally implemented in the plasmodium of P. polycephalum.It is programmed by configurations of repelling and attracting gradients; see the detailed analysis in [6]. The mechanics of Physarum machines is based on the following unique features of P. polycephalum:

• Physarum is a living, dynamical reaction-diffusion pattern formation mechanism. • Physarum may be considered as equivalent to a membrane-bound subexcitable system (excitation stimuli provided by chemoattractants and chemorepellents). • Physarum may be regarded as a highly efficient and living micromanipulation and microfluidic transport device.

Figure 1. Slime mold P. polycephalum growing from several sources of inoculation.

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Figure 2. Plasmodium of P. polycephalum on a data set on an agar gel. (a) Virgin oat flakes. (b) Oat flakes colonized by the plasmodium. (c) Protoplasmic tubes. (d) Active zones, growing parts of the plasmodium.

• The induction of pattern types is determined partly by the environment: specifically, nutrient quality and substrate hardness, dryness, and the like. • Physarum is sensitive to illumination and ac electric fields, and therefore allows for parallel and nondestructive input of information. • Physarum represents results of computation by configuration of its body.

Physarum is thus a computational material based on modification of protoplasm transport by the presence of external stimuli. Plasmodia can be cultivated on a non-nutrient (e.g., Select agar, Sigma Aldrich) or a nutrient agar (e.g., corn meal agar). While grown on a nutrient agar, the plasmodium propagates as an omnidirectional wave. On a non-nutrient agar it propagates as a traveling finite localization, and behaves like a wave fragment in a subexcitable medium [3, 12]; most implementations discussed in this article are done on a non-nutrient agar. Thus by an active zone we mean either an omnidirectional growing pattern (on nutrient substrate) or—in the majority of examples—a localized growing pattern (on non-nutrient substrate). While presented with a configuration of attractants (e.g., oat flakes (Figure 2a)) on a non- nutrient substrate, the plasmodium develops active zones (Figure 2d) that explore the substrate and propagate toward the oat flakes. Neighboring oat flakes colonized by a plasmodium (Figure 2b) are usually connected by protoplasmic tubes (Figure 2c). The distribution of chemoattractants and the position of initial inoculation of the plasmodium are input data for Physarum machines. The structures of the protoplasmic networks and/or domains occupied by plasmodia are the results of computation in Physarum machines. Propagating active zones can be considered as elementary processors of Physarum machines.

4 Routing Signals

In the Physarum machine, computation is implemented by an active zone or several active zones. To make the computation process programmable, one needs to find ways to sensibly and purposefully manipulate the active zones. In [13] we experimentally demonstrated how active zones can be manipu- lated by dynamical addition of attractants. Programming with chemoattractants is not really efficient, because once the source is placed in the computing space, it irreversibly changes the configuration of

Artificial Life Volume 19, Number 3 & 4 319 A. Adamatzky Slimeware: Engineering Devices with Slime Mold attracting fields. Light inputs allow for an online reconfiguration of obstacles and thus provide increased opportunities for embedding complex programs in Physarum machines [5]. P. polycephalum exhibits photoavoidance. Thus we expect a plasmodium to change its velocity after entering an illuminated domain. Basic operations of routing active zones with illuminated domains are shown in Figure 3a–d. In an ideal situation the plasmodium propagates as a wave fragment and—unless it encounters an obstacle on its way—keeps its shape and velocity vectors con- served. If a proximal part of the plasmodium wave comes upon a highly illuminated domain, the frequency of protoplasm oscillations in this domain increases. Due to a difference in the protoplasm oscillation frequency, the plasmodium wave slightly turns to the side with less-oscillating protoplasm. Experimental implementation of the operation RIGHT(A)isshowninFigure3c–h. The active zone propagates northward (Figure 3e). The plasmodium hits a light triangle with its western

side (Figure 3f ). The light increases the frequency of oscillations in the illuminated part of the Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 plasmodium, and the plasmodium wave then turns eastward (Figure 3g). A propagating plasmodium wave or a pseudopodium can be split by a suitably shaped domain of illumination. In some situations the propagating plasmodiumʼs active zone hits a light obstacle that is small enough to divert the whole active zone. If parts of the active zone remain outside the illuminated shape, these parts continue to travel as independent plasmodium waves. Thus the active zone splits into two independent active zones [6].

5 Solving the Maze

A typical strategy for maze solving with a single device is to explore all possible passages while marking visited parts, until the exit or a central chamber is found. With the advent of unconventional computing, several attempts have been made to outperform Shannonʼs electronic mouse Theseus [35] using propagation of disturbances in spatially extended nonlinear media, including excitable chemical systems, gas discharge, and crystallization. Most experimental prototypes were successful, yet suffered from their computing substratesʼ specific drawbacks [9].

Figure 3. Routing in a Physarum machine. (a–d) Scheme of routing operations. (e–g) Example of the signal deviation RIGHT(A), turning of the active zone by a light triangle. Plasmodium is inoculated in the southern part of the Petri dish; oat flakes, acting as attractants, are placed in the northern part of the dish. A triangular illuminated domain is arranged between the start and attracting sites (the domain is shown as a white triangle in the illustrations).

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Figure 4. Experimental maze solving with a plasmodium of P. polycephalum. The plasmodium is inoculated in the peripheral channel on the east part of the maze, and a virgin oat flake is placed in the central chamber. (a) Scanned image of the experimental maze; protoplasmic tubes are light-colored. (b) Binarized image; major protoplasmic tubes are thick black lines. (c) Scheme of plasmodium propagation; arrows symbolize velocity vectors of propagating active zones. (d) Locations of active growing zones, sprouted by plasmodium during exploration of the maze.

In laboratory experiments we used plastic mazes, 70 mm in diameter with 4-mm-wide and 3-mm- deep channels (Figure 4a). We filled channels with agar gel as a non-nutrient substrate. An oat flake was placed in the central chamber of the maze, and the plasmodium was inoculated in the most peripheral channel of the maze. A typical experiment is illustrated in Figure 4. After its inoculation the plasmodium started exploring its vicinity and first generated two active zones propagating clockwise and counterclockwise (Figure 4a, b). Several active zones were developed to explore the maze (Figure 4d). By the time diffusing chemoattractants reached distant channels, one of the active zones had already become dominant and suppressed other active zones (Figure 4c). In the example shown, the active zone traveling counterclockwise inhibited active zones traveling clockwise. The dominating active zone then followed the gradient of chemoattractants inside the maze, navigated along intersections of the mazeʼs channels, and solved the maze by entering its central chamber (Figure 4c).

6 Approximating Planar Hulls

The convex hull of a finite set P of planar points is the smallest convex polygon that contains all points of P (Figure 5a). The a-hull of P is the intersection of the complements of all closed disks of

Artificial Life Volume 19, Number 3 & 4 321 A. Adamatzky Slimeware: Engineering Devices with Slime Mold radius 1/a that include no points of P [18, 19]. The a-shape becomes the convex hull when a → ∞. With a decrease of a the shapes may shrink, develop holes, and become disconnected; they collapse to P when a → 0. A concave hull is a nonconvex polygon representing the area occupied by P.A concave hull is a connected a-shape without holes (Figure 5b). The plasmodium of P. polycephalum approximates the connected a-hull without holes of a finite planar set whose points are represented by sources of long-distance attractants and short-distance repellents. Given a planar set P represented by physical objects, the plasmodium of P. polycephalum must approximate a concave hull of P by its largest protoplasmic tube. The slime mold does not compute concave or convex hulls of a set represented by attracting sources [8]. Using repellents only (e.g.) as a set of obstacles between attractants and the inoculation site will not help with this computation, because the plasmodium will just pass around the repellents [6]. The only solution would be to employ “ ” P attractants to pull the plasmodium toward the planar set and to use repellents to prevent it Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 from spanning the points of P. The strength of repellents should be proportional to a,andthuswill determine the exact shape of the constructed hull. Such an approach fits well with the original de- finition [18] of the a-hull of P as the intersection of all closed disks with radius 1/a that contain all the points of P. We represented points of P by the somniferous herbal pill Kalms Tablets [8]. In laboratory experiments we arranged 4–8 half pills (a representation of P) in a random fashion near the center of a Petri dish and inoculated an oat flake colonized by a plasmodium 2–4 cm away from the set P. A typical experiment is illustrated in Figure 5d, e. Twelve hours after inoculation, the plasmodium propagates toward P and starts enveloping the set with its body and a network of protoplasmic tubes (Figure 5d).

Figure 5. Basics of approximation of planar shapes by the Physarum machine. Examples of convex (a) and concave (b) hulls; points of the planar set P are empty disks. (c) Proposed distribution of attracting and repelling gradients that may force a plasmodium to approximate a concave hull (arrows aiming toward the set of disks are attractive forces; arrows originating in data points (disks) are repelling forces). (d, e) Computation of the concave hull of point set P: snapshots of the experimental setup taken (d) 12 h and (e) 24 h after inoculation.

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Figure 6. Physarum logical gates. (a, b) Geometrical structure of Physarum gates (a) P1 and (b) P2: x and y are inputs; p and q are outputs. (c–e) Experimental examples of the transformation〈x, y〉→〈p, q〉implemented by Physarum gate P1.(c)〈0, 1〉→ 〈0, 1〉.(d)〈1, 0〉→〈0, 1〉.(e)〈1, 1〉→〈1, 1〉.(f–i) Experimental examples of the transformation〈x, y〉→〈p, q〉implemented by Physarum gate P2.(f)〈0, 1〉→〈1, 0〉.(g)〈1, 0〉→〈0, 1〉. (h, i) Two snapshots (taken with 11 h interval) of the transformation〈1, 1〉→〈0, 1〉.

The plasmodium completes approximation of a shape by entirely enveloping P in the next 12 h (Figure 5e). The plasmodium does not propagate inside the configuration of pills.

7 Boolean Gates and Adder

Given a cross-junction of agar channels and a plasmodium inoculated in one of the channels, the plasmodium propagates straight through the junction [4]; the speed of propagation may increase if sources of chemoattractants are present (however, the presence of nutrients does not affect the direction of propagation). An active zone, or growing tip, of the plasmodium propagates in the initially chosen direction, as if it has some kind of inertia. Based on this phenomenon, we designed two Boolean gates with two inputs and two outputs; see Figure 6a, b. The input variables are x and y,

Artificial Life Volume 19, Number 3 & 4 323 A. Adamatzky Slimeware: Engineering Devices with Slime Mold and the outputs are p and q. The presence of a plasmodium in a given channel indicates TRUE,and the absence indicates FALSE. Each gate implements a transformation 〈x, y〉 → 〈p, q〉. Experimental examples of the transformations are shown in Figure 6. 〈〉 → Plasmodia of P. polycephalum implement the two-input, two-output Boolean gate P1: x, y 〈xy, x + y〉. A plasmodium inoculated in input y of P1 propagates along the channel yq and appears in the output q (Figure 6c). Plasmodia inoculated in input x of P1 propagate until the junction of x and y, collide with the impassable edge of channel yq, and appear in output q (Figure 6d). When plasmodia are inoculated in both inputs x and y of P1, they collide with each other, and the plasmodium that originated in x continues along the route xp. Thus, the plasmodia appear in both outputs p and q (Figure 6e). 〈〉 → 〈〉 Plasmodia of P. polycephalum also implement the two-input, two-output gate P2: x, y x, xȳ. Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 If input x is empty, a plasmodium placed in input y of P2 propagates directly toward output p (Figure 6f ). A plasmodium inoculated in input x of P2 (when input y is empty) travels directly toward output q (Figure 6g). Thus transformations 〈0, 1〉 → 〈1, 0〉 and 〈1, 0〉 → 〈0, 1〉 are implemented. The gateʼs structure is asymmetric; the x channel is shorter than the y channel. Therefore, the plasmodium placed in input x of P2 usually passes the junction by the time the plasmodium originated in input y arrives at the junction (Figure 6h). The y plasmodium merges with the x plasmodium, and they both propagate toward output q (Figure 6i). Exten- sion of the gel substrate beyond output q usually facilitates implementation of the transformation 〈1, 1〉 → 〈0, 1〉. A one-bit half adder is a logical circuit that takes two inputs x and y and produces two outputs: sum x̄y + xȳand carry xy. To construct a one-bit half adder with Physarum gates we need two copies of gate P1 and two copies of gate P2. Cascading the gates into the adder is shown in Figure 7a. Signals x and y are input in P2 gates. Outputs of P2 gates are connected to inputs of P1 gates. We did not manage to realize a one-bit half adder in experiments with a living plasmodium, because the plasmodium behaved differently in the assembly of the gates than in isolated gates. Therefore, we simulated the adder using the Oregonator model; see details in [8]. To simulate inputs x = 0andy = 1 we initiated the plasmodiumʼs active zones near the entrances to the channels, marked with y and an arrow in Figure 7a. The active zones propagated along their channels (Figure 7b). For input values x = 1andy = 0 active zones are originated at sites marked x and arrow in Figure 7a. The active zone started in the left x input channel propagated toward the x + y output of the adder. The active zone originated in the right x input channel traveled toward x̄y + xȳ (Figure 7c). When both inputs are activated (x = 1andy = 1), an active zone originated in the left y input channel is blocked by an active zone originated in the left x input channel. The plasmodium traveling in the right x input channel is blocked by the active zone traveling in the right y input channel. The active zones representing x = 1andy = 1 enter the top right gate P1 and emerge at its outputs xy and x + y (Figure 7d). The Physarum adder has also been implemented in chemical laboratory experiments with an excitable chemical system employing the Belousov-Zhabotinsky reaction [17].

8 Physarum Nose

Another potential application of the slime mold is in hybrid bio-inspired sensors. The slime mold P. polycephalum has proved to be very sensitive to aromatic substances [7]. We found that the plasmodium not only is strongly attracted to herbal somniferous tablets, but can differentiate between various of types of plants with sedative properties. To select the principal chemoattractant in the tablets, we undertook laboratory experiments on the plasmodiumʼs binary choice between samples of dried plants: Valeriana officinalis, Humulus lupulus, Passiflora incarnate, Lactuca virosa, Gentiana lutea,andVerbena officinalis. Valerian root dominates in the hierarchy of chemoattractive forces, or

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Physarum preferences, shown in Figure 8a. Possible molecular mechanisms linking the sedative activity of valerian and its chemoattraction via relaxation of contractile activities of slime mold are outlined in [10]. Contractile activity of the plasmodium is closely associated with its electrical potential. We believe that in the future it will be possible to build a mapping between types of aromatic substances and patterns of electrical activity exhibited by the slime mold. This mapping will lay a foundation for manufacturing of slime-mold-based sensors. There remain many un- answered questions in this area; for example, why P. polycephalum prefers Cannabis sativa to tobacco, and tobacco to oat flakes (Figure 8b).

9 Discussion: The Future of Physarum Machines Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 We have introduced a concept of Physarum machines and exemplified experimental prototypes of slime mold computing devices. Physarum machines are instances of parallelism on demand. The number of elementary processors (a plasmodiumʼs active zones) is determined by the computational problem the slime mold computer is solving. Elementary processors—active growing zones—can be pro- duced when necessary and halted when they are no longer required. Examples of constant yet data-defined numbers of processors are plane tessellation and planar shape computation. When a slime

Figure 7. Physarum one-bit half adder. (a) Scheme of one-bit half adder made of gates P1 and P2. Inputs are indicated by arrows. Outputs x̄y + xȳand xy are sum and carry values. Outputs 0 and x + y are by-products. (b-d) Time-lapse images of the plasmodiumʼs active zones traveling in channels of the one-bit half adder. Dynamics of growth is shown for input values (b) x = 0 and y = 1, (c) x = 1 and y = 0, (d) x = 1 and y = 1.

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Figure 8. Toward a Physarum nose. (a) Preferences of P. polycephalum toward pills and herbs with somniferous properties. (b) An experimental example of slime mold preferences regarding common substances: The slime mold is inoculated at the center of a Petri dish; equal-weight portions of oat flakes (southeast), dry leaves of Cannabis sativa (south-west), and dry leaves of tobacco (northwest) are placed in the corners of the dish. At first the slime mold propagates to the C. sativa, and then to the tobacco; it ignores the oat flakes despite their nutritional value.

mold approximates a Voronoi diagram of n planar points on a nutrient substrate, it has n processors, but when the Physarum machine computes a concave hull, it has just two processors. During implementation of logical gates, maze solving, or approximation of proximity graphs, the number of elementary processors changes dynamically depending on the structure of problem instances. Physarum machines have one significant disadvantage. They are very slow. For example, it takes several days for the slime mold to approximate a proximity graph or a Voronoi diagram in a standard Petri dish. The efficiency of Physarum machines can be improved by “hybridizing” the slime mold with conventional conductive materials (Figure 9a, b). Our future research focuses on the design and manufacture of a Physarum chip—a network of processing elements made of the slime moldʼs protoplasmic tubes coated with conductive substances populated by living slime mold. A living network of protoplasmic tubes acts as an active nonlinear transducer of information, while templates of tubes coated with conductor act as fast information channels (Figure 9c). The Physarum chip will have parallel inputs (optical, chemo-, and electro-based) and outputs (electrical and optical); see the

326 Artificial Life Volume 19, Number 3 & 4 A. Adamatzky Slimeware: Engineering Devices with Slime Mold scheme in Figure 10. The Physarum chip will solve a wide range of computational tasks, including optimization on graphs, computational geometry, robot control, logic, and arithmetical computing. The slime-mold-based implementation is a biophysical model of future nanochips based on biomorphic mineralization. We envisage that research and development centered on novel computing substrates like self- assembled and fault-tolerant fungal networks will lead to a revolution in the bioelectronics and computer industry. Combined with conventional electronic components in a hybrid chip, Physarum networks will radically improve the performance of digital and analog circuits. In the near future Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021

Figure 9. Physarum wires. (a) Protoplasmic tubes filled with magnetic nanoparticles (fluidMAG-D Starch, diameter 100 nm). (b) Protoplasmic tube with colloid gold crosses bare plastic. (c) Schematic interaction of Physarum networks: Living slime mold network is shown in dark gray; network coated with conductive material is striped.

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Figure 10. Scheme of multilayered Physarum chip: (a) optical I/O array, (b) living Physarum network, (c) biomorphic metal network, (d) multielectrode I/O array.

we are planning to deliver a proof of concept for a biomorphic information technology based on hybrid live and conductor-coated slime mold devices that are:

• analogous to reaction-diffusion chemical systems encapsulated in a growing elastic membrane, • combining dead (but coated with conductors) and living parts of slime mold in communication channels, • powered directly and efficiently by biochemical power, • fabricated using self-growth and self-organization, • controllably shaped into two- and three-dimensional structures, • interfaced with conventional technology (optically or electronically) as well as via chemical means, • responding to distinct activating and inhibiting stimuli, • robust to physical damage and exhibiting a degree of self-repair.

The fabricated Physarum chip will perform computation by classical means of electrical charge propagation, by traveling waves of contraction, and by physical propagation of structures. In terms of classical computing architectures, the following characteristics can be attributed to Physarum chips:

• Massive parallelism: There are thousands of elementary processing units (oscillatory bodies) in a slime mold colonized in a Petri dish. • Massive signal integration: The membrane of a plasmodium is able to integrate massive amounts of complex spatial and time-varying stimuli to effect local changes in contraction rhythm and, ultimately, global behavior of the plasmodium;

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• Local connections: Microvolumes and oscillatory bodies of cytoplasm change their states, due to diffusion and reaction, depending on the states or concentrations of reactants, shapes, and electrical charges in their closest neighbors. • Parallel input and output: By changing their shape, Physarum computers can record computation optically; Physarum is light sensitive, so data can be input by local illumination. • Fault tolerance: By constantly changing shape, Physarum chips can restore their architecture even after a substantial part of their protoplasmic network is removed.

Taking into account the enormous and growing interest of research centers and commercial labora- tories in the recent experimental implementations of chemical, molecular, and biological computers, – we can predict that in the next 20 30 years, networks of slime mold coated with compound sub- Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 stances will become a widespread commodity and a very promising component of novel information- processing circuits.

Acknowledgments We acknowledge the European Commission grant “Physarum Chip: Growing Computers from Slime Mould,” program type “Seventh Framework Programme,” subprogram area “Unconventional Computation (UCOMP),” project reference 316366. References 1. Adamatzky, A. (2008). Developing proximity graphs by Physarum polycephalum: Does the plasmodium follow Toussaint hierarchy? Parallel Processing Letters, 19, 105–127. 2. Adamatzky, A. (2009). Hot ice computer. Physics Letters A, 374, 264–271. 3. Adamatzky, A. (2009). If BZ medium did spanning trees these would be the same trees as Physarum built. Physics Letters A, 373, 952–956. 4. Adamatzky, A. (2009). Slime mould logical gates. arXiv:1005.2301v1 [nlin.PS]. 5. Adamatzky, A. (2009). Steering plasmodium with light: Dynamical programming of Physarum machine. arXiv:0908.0850v1 [nlin.PS]. 6. Adamatzky, A. (2010). Physarum machines: Making computers from slime mould. Singapore: World Scientific. 7. Adamatzky, A. (2011). On attraction of slime mould Physarum polycephalum to plants with sedative properties. Available from Nature Proc., http://dx.doi.org/10.1038/npre.2011.5985.1 8. Adamatzky, A. (2012). Simulating strange attraction of acellular slime mould Physarum polycephalum to herbal tablets. Mathematics and Computer Modelling, 55(3–4), 884–900. 9. Adamatzky, A. (2012). Slime mould solves maze in one pass … assisted by gradient of chemo-attractants. IEEE Transactions on NanoBioscience, 11(2), 131–134. 10. Adamatzky, A., & De Lacy Costello, B. (2012). Physarum attraction: Why slime mold behaves as cats do? Communicative and Integrative Biology, 5(3), 297–299. 11. Adamatzky, A., De Lacy Costello, B., & Asai, T. (2005). Reaction-diffusion computers. Amsterdam: Elsevier. 12. Adamatzky, A., De Lacy Costello, B., & Shirakawa, T. (2008). Universal computation with limited resources: Belousov–Zhabotinsky and Physarum computers. International Journal of Bifurcation and Chaos, 18, 2373–2389. 13. Adamatzky, A., & Jones, J. (2010). Programmable reconfiguration of Physarum machines. Journal of Natural Computation, 9, 219–237. 14. Adamatzky, A., & Teuscher, C. (2006). From utopian to genuine unconventional computers. Bristol, UK: Luniver Press. 15. Bedau, M. A., McCaskill, J. S., Packard, N. H., & Rasmussen, S. (2010). Living technology: Exploiting lifeʼs principles in technology. Artificial Life, 16,89–97. 16. Calude, C. S., Casti, J., & Dinneen, M. (Eds.). (1998). Unconventional models of computation. Berlin: Springer-Verlag. 17. De Lacy Costello, B., Adamatzky, A., Jahan, I., & Zhang, L. (2011). Towards constructing one-bit binary adder in excitable chemical medium. Chemical Physics, 381,88–99.

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18. Edelsbrunner, H. D., Kirkpatrick, D., & Seidel, R. (1983). On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29, 551–559. 19. Edelsbrunner, H., & Muke, E. P. (1994). Three-dimensional alpha shapes. ACM Transactions on Graphics, 13,43–72. 20. Fuerstman, M. J., Deschatelets, P., Kane, R., Schwartz, A., Kenis, P. J. A., Deutsch, J. M., & Whitesides, G. M. (2003). Solving mazes using microfluidic networks. Langmuir, 19,4714–4722. 21. Górecki, J., & Górecka, J. N. (2006). Information processing with chemical excitations—From instant machines to an artificial chemical brain. International Journal of Unconventional Computing, 2, 321–336. 22. Górecki, J., Górecka, J. N., & Igarashi, Y. (2009). Information processing with structured excitable medium. Natural Computing, 8, 473–492. 23. Katz, E., & Privman, V. (2010). Enzyme-based logic systems for information processing. Chemical Society Reviews, 39, 1835–1857. Downloaded from http://direct.mit.edu/artl/article-pdf/19/3_4/317/1663945/artl_a_00110.pdf by guest on 27 September 2021 24. Lagzi, I., Soh, S., Wesson, P. J., Browne, K. P., & Grzybowski, B. J. (2010). Maze solving by chemotactic droplets. Journal of the American Chemical Society, 132, 1198–1199. 25. Macdonald, J., Li, Y., Sutovic, M., Lederman, H., Pendri, H., Lu, W., Andrews, B. L., Stefanovic, D., & Stojanovic, M. N. (2006). Medium scale integration of molecular logic gates in an automaton. Nano Letters, 6, 2598–2603. 26. Mills, J. (2008). The nature of the extended analog computer. Physica D, 237, 1235–1256. 27. Motoike, I. N., & Yoshikawa, K. (2003). Information operations with multiple pulses on an excitable field. Chaos, Solitons & Fractals, 17, 455–461. 28. Nakagaki, T., Yamada, H., & Toth, A. (2000). Maze-solving by an amoeboid organism. Nature, 407(6803), 470. 29. Nakagaki, T., Yamada, H., & Toth, A. (2001). Path finding by tube morphogenesis in an amoeboid organism. Biophysical Chemistry, 92,47–52. 30. Nakagaki, T., Iima, M., Ueda, T., Nishiura, Y., Saigusa, T., Tero, A., Kobayashi, R., & Showalter, K. (2007). Minimum-risk path finding by an adaptive amoeba network. Physical Review Letters, 99, 068104. 31. Privman, V., Pedrosa, V., Melnikov, D., Pita, M., Simonian, A., & Katz, E. (2009). Enzymatic AND-gate based on electrode-immobilized glucose-6-phosphate dehydrogenase: Towards digital biosensors and biochemical logic systems with low noise. Biosensors and Bioelectronics, 25, 695–701. 32. Reyes, D. R., Ghanem, M. G., & George, M. (2002). Glow discharge in micro fluidic chips for visible analog computing. Lab on a Chip, 1,113–116. 33. Schumann, A., & Adamatzky, A. (2011). Physarum spatial logic. New Mathematics and Natural Computation, 7,483–498. 34. Sielewiesiuk, J., & Górecki, J. (2001). Logical functions of a cross junction of excitable chemical media. Journal of Physical Chemistry A, 105, 8189. 35. Shannon, C. (1951). Presentation of a maze solving machine. In Transactions of the 8th Conference on Cybernetics: Circular, causal and feedback mechanisms in biological and social systems (pp. 169–181). 36. Shirakawa, T., Adamatzky, A., Gunji, Y.-P., & Miyake, Y. (2009). On simultaneous construction of Voronoi diagram and Delaunay triangulation by Physarum polycephalum. International Journal of Bifurcation and Chaos, 9, 3109–3117. 37. Stephenson, S. L., & Stempen, H. (2000). Myxomycetes: A handbook of slime molds. Portland, OR: Timber Press. 38. Stojanovic, M. N., Mitchell, Y. E., & Stefanovic, D. (2002). Deoxyribozyme-based logic gates. Journal of the American Chemical Society, 124, 3555–3556. 39. Teuscher, C., & Adamatzky, A. (Eds.) (2005). Unconventional Computing 2005: From cellular automata to wetware. Bristol, UK: Luniver Press. 40. Tsuda, S., Aono, M., & Gunji, Y. P. (2004). Robust and emergent Physarum-computing. BioSystems, 73, 45–55. 41. Yoshikawa, K., Motoike, I. M., Ichino, T., Yamaguchi, T., Igarashi, Y., Gorecki, J., & Gorecka, J. N. (2009). Basic information processing operations with pulses of excitation in a reaction-diffusion system. International Journal of Unconventional Computing, 5,3–37.

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