Slime mould solves maze in one pass ... assisted by gradient of chemo-attractants
SSlime mould solves maze in one pass . . . assistedby gradient of chemo-attractants
Andrew Adamatzky ∗ August 26, 2011
Abstract
Plasmodium of
Physarum polycephalum is a large cell, visible by un-aided eye, which exhibits sophisticated patterns of foraging behaviour.The plasmodium’s behaviour is well interpreted in terms of computation,where data are spatially extended configurations of nutrients and obsta-cles, and results of computation are networks of protoplasmic tubes formedby the plasmodium. In laboratory experiments and numerical simulationwe show that if plasmodium of
P. polycephalum is inoculated in a maze’speripheral channel and an oat flake (source of attractants) in a the maze’scentral chamber then the plasmodium grows toward target oat flake andconnects the flake with the site of original inoculation with a pronouncedprotoplasmic tube. The protoplasmic tube represents a path in the maze.The plasmodium solves maze in one pass because it is assisted by a gra-dient of chemo-attractants propagating from the target oat flake.
Keywords: slime mould, Physarum computing, maze, shortest path
A typical strategy for a maze-solving is to explore all possible passages, whilemarking visited parts, till the exit or a central chamber is found. This is whatShannon’s electromagnetic mouse Theseus was doing in first ever laboratoryexperiment on solving maze by physical means [11]. The task of maze search istime consuming for a single mobile computing device. Therefore with advent ofunconventional computing paradigm scientists focused on uncovering physical,chemical and biological substrates which can solve maze in parallel. Thus in [10]it is experimentally demonstrated that detecting superposition of excitationwave fronts propagating from source to destination, and from destination tosource, in a maze filled with Belousov-Zhabotinsky (BZ) reaction mixture allowsus to approximate a shortest path in the maze. Travelling excitation waves ∗ University of the West of England, Bristol BS16 1QY, United Kingdom Email: [email protected] a r X i v : . [ n li n . PS ] A ug ropagate to all channels of the maze thus examining the maze’s structure infull. The BZ maze solver does not represent path by the medium’s physicalcharacteristics — an external observer is required to reconstruct a shortest pathfrom chemical waves’ dynamics.In a gas-discharge maze-solver [9] an electric field, generated between sourceand destination electrodes, explores a maze in parallel. The field is strongeralong the shortest path between source- and destination-electrodes thereforechannels of the shortest path glow with high intensity. Such maze-solver has avolatile memory because the shortest path is visible as long as voltage to theelectrodes is applied. A slime mould maze-solver reported in [8] is a fusion ofBZ and gas-discharge approaches. Plasmodium of P. polycephalum is placedin several sites of maze at once. Initially the plasmodium develops a networkof protoplasmic tubes spanning all channels of the maze, thus representing allpossible solutions. Then oat flakes are placed in source and destination sitesand the plasmodium enhances the tube connecting source and destination alongthe shortest path. A selection of shortest protoplasmic tube is implemented viainteraction of propagating bio-chemical, electric potential and contractile wavesin plasmodium’s body. Not shortest or cul-de-sac branches are abandoned. Thuswe can call prototype [8] as ’pruning plasmodium’. The largest protoplasmictube (exactly its walls), which represents the shortest path, remains visibleeven when plasmodium ceases functioning. Therefore we can consider the slimemould maze-solver as having non-volatile memory (analogous to precipitatingreaction-diffusion chemical processors).In laboratory prototypes of a mobile droplet [6] and hot ice computer [1] acomputation of shortest path in a maze is separated on two stages: computationof many-sources-one-destination set of paths and extraction of a shortest pathfrom the set of paths. Many-sources-one-destination paths are computed byspreading acidity in [6] and propagating crystallisation pattern in [1]. A shortestpath is selected and traced by a droplet travelling along pH gradient [6] or avirtual robot traversing crystallisation pattern [1]. In present paper we applythe two-stage computation in design of slime mould maze-solver which navigatesa maze in one go, without exploring all possible options because the options are‘explored’ by chemo-attractants emitted by destination oat flake.
In laboratory experiments we used plastic mazes (Tesco’s Toy Mazes, TescoPlc), 70 mm diameter with 4 mm wide and 3 mm deep channels (Fig. 1a).We filled channels with 2% agar gel (Select agar, Sigma Aldrich) as a non-nutrient substrate. We smeared top of channel walls with strawberry flavouredChasptick (Pfizer Consumer Healthcare Ltd) to deter plasmodium from making’illegal’ shortcuts over the walls separating channels. A rolled oat was placed inthe central chamber of the maze and an oat flake colonised by plasmodium of
P. polycephalum was placed in the most peripheral channel of the maze. Mazeswith plasmodium were kept in the dark in 20-23 C o temperature. Images of2 a) (b)(c) (d) Figure 1: Experimental maze-solving with plasmodium of
P. polycephalum :(a) maze used in experiments on slime mould maze-solving; (bc) plasmodiumis inoculated in peripheral channel, east part of the maze, and a virgin oatflake is place in central chamber; (b) scanned image of the experimental maze,protoplasmic tubes are yellow (light gray); (c) binarised, based on red and greencomponents, images, major protoplasmic tubes are thick black lines; (d) schemeof plasmodium propagation, arrows symbolise velocity vectors of propagatingactive zone. See experimental laboratory videos at mazes were scanned in Epson Perfection 4490. Photos are taken using FujiPix6000 camera. 3 typical experiment is illustrated in Fig. 1. We placed an oat flake in thecentral chamber and inoculated plasmodium of
P. polycephalum in a peripheralchannels. The plasmodium started exploring its vicinity and at first generatedtwo active zones propagating clock- and contra-clockwise. By the time diffus-ing chemo-attractants reached distant channels one of the active zone alreadybecame dominant and suppressed another active zone. In example shown inFig. 1ab active zone travelling contra-clockwise dominated and ’extinguished’active zone propagating clockwise. The dominating active zone then followedgradient of chemo-attractants inside the maze, navigated along intersections ofthe maze’s channels and solved the maze by entering its central chamber.
A plasmodium of
P. polycephalum can be seen as a network of coupled bio-chemical oscillators [12, 14]. Interactions between the oscillators determinespace-time dynamics of contractile activity and protoplasmic streaming in pro-toplasmic tubes [3,5] and ultimately shape of plasmodium’s cell [7]. Laboratoryexperiments show that oscillators in plasmodium network interact similarly toneurons in a simple neural networks [13]. We can speculate that plasmod-ium’s active zones (analogs of growth cones of maturing neuroblast) establishmutually inhibiting relationships. An active zone proximal to a source of chemo-attractants sends biochemical and electrical signals which suppress activity ofactive zones distal to the source.A profile of plasmodium’s active zone on a non-nutrient substrate is isomor-phic to shapes of and behaves analogously to wave-fragments in sub-excitablemedia [2]. When active zone propagates two processes occur simultaneously —movement of the wave-shaped tip of the pseudopodium and formation of a trailof protoplasmic tubes. We simulate the tactic traveling of plasmodium growthfront using two-variable Oregonator equation [4]: ∂u x ∂t = 1 (cid:15) ( u x − u x − ( f v x + φ x ) u x − qu x + q ) + D u ∇ u∂v x ∂t = u x − v x . The variable u x is abstracted as a local density of plasmodium’s protoplasmat site x and v x reflects local concentration of metabolites and nutrients. Pa-rameters q and f are inherited from model of Belousov-Zhabotinsky medium.Parameter φ x characterises excitability of medium in the Oregonator model andcan be seen as analog of a degree of plasmodium’s isometric tension responseto chemo-attractants’ concentration [16]. We integrate the system using Eulermethod with five-node Laplacian, time step ∆ t = 5 · − and grid point spacing∆ x = 0 . (cid:15) = 0 . f = 1 . q = 0 . a) (b)(c) Figure 2: Simulating maze-solving in Oregonator model: (a) gradient of chemo-attractants originated in central chamber, intensity of black is proportionalto concentration of the attractants; impassable walls of the maze are white;(b) time-lapse images of propagating active zones, active zones are recordedevery 400th step of simulation as sites with u x > . Let c x be a concentration of chemo-attractant at site x . At moment t = 0only site occupied by oat flake has concentration c x = 1, all other sites have zero5oncentration. At every moment t of simulation a site x updates its state c x bythe following rule. If c tx = 0 and there is at least one immediate neighbour y suchthat c y > c t +1 x = t − . Such rude approximation of diffusion is enough tobuild satisfactory gradient of chemo-attractants for guiding propagating wave-fronts. The gradient developed in simulation is shown in Fig. 2a.Initially all sites have the same excitability parameter φ x = 0 . φ x the higher is excitability of x . The excitability is revised asfollows. At every σ th step of simulation we detect excited, i.e. u z ≥ .
1, site z with maximum, amongst all sites excited at this moment, concentration ofchemo-attractants c z . Then for every site x we revise its excitability as follows:if c x < c z then φ x = 0 .
09. In simulation illustrated in Fig. 2b σ = 500. Themedium is perturbed by an initial excitation, where a 5 × u = 1 . u in matrix L , which is processed at the end of simulation. For any site x and time step t if u x > . L x = 0 then L x = 1. The matrix L represents time lapse su-perposition of propagating wave-fronts. The simulation is considered completedwhen propagating pattern reaches destination site (central chamber in scenarioillustrated in Fig. 2) and halts any further motion. At the end of simulationwe repeatedly apply the erosion operation [2] to L . This operation symbolises astretch-activation effect [5] necessary for formation of plasmodium tubes. Theresultant protoplasmic network provides a good phenomenological match fornetworks recorded in laboratory experiments (Fig. 2c). We experimentally demonstrated that plasmodium of
P. polycephalum solvesmaze in one-pass, i.e. without exploring all possible solutions, if a source ofchemo-attractants is placed at destination site. In our experiments we inocu-lated slime mould in a peripheral channel and a target oat flake in central cham-ber. Positions do not matter, if there is an obstacle-free pass from a source to adestination the slime mould will trace it. For example, slime mould inoculatedin a central chamber finds exit out of a maze. In Fig. 3 we placed an oat flakecolonised by plasmodium in the central chamber and put an attracting oat flakein the west part of an outer channel. The plasmodium navigated thought firstjunction. It branched at the second junction. Active zone travelling clock-wisedetected higher concentration of chemo-attractants than concentration detectedby contra-clockwise propagating active zone. Thus active zone moving clock-wise became dominating. Eventually it reached its destination site (oat flake)along the shortest path (Fig. 3).Is our prototype better — in terms of complexity or real-life speed andcosts — then other laboratory prototypes of maze-solvers (Tab. 1)? All existingprototypes have the same computational time complexity O ( L ), where L is a6 a) (b) Figure 3: Experimental maze-solving with plasmodium of
P. polycephalum : plasmodium is inoculated in central chamber and a virgin flake is place in themost peripheral channel, west of the maze. (a) photo of experimental maze,(b) binarised image.Table 1: Brief comparison of laboratory prototypes of maze-solvers implementedin spatially extended physical, chemical or biological media. Two stages ofmaze-solving are outlined: maze exploration, or computation of many-sources-one-destination paths, and path tracing, or following from source site to specifieddestination site.Prototype Maze is explored by Path is traced byBZ medium [10] Excitation waves ComputerPruning plasmodium [7] Plasmodium PlasmodiumGas-discharge [9] Electrical field Electrical fieldHot ice [1] Crystallisation pattern ComputerMobile droplet [6] Diffusing chemicals DropletSingle-pass plasmodium Diffusing chemo-attractants Plasmodiumlength of a worst-case scenario shortest path in a maze (a worst case is whenshortest path spans all channels of a maze, as e.g. Chartres and Remis mazesshown in Fig. 4).The time complexity is determined by time taken by waves, electricity, plas-modium or diffusing chemicals to explore the maze. Al these substances exploremaze in parallel by propagating from their original locations to all other sitesof maze, thus it takes them O ( L ) steps to span the maze. Tracing of a pathfrom source to destination is rather a menial tasks — to follow gradients orother physical or chemical characteristics of the medium. It takes O ( L ) steps7 a) (b) Figure 4: Solving Chartres (a) and Reims (b) mazes in Oregonator modelof slime mould. Time-lapse images of an active zones (analogs of excitationwave-fronts in sub-excitable chemical medium) following a gradient of chemo-attractants emitted by central chamber of mazes. At the beginning of experi-ments simulated plasmodium is inoculated at the mazes’ entrances, south edgesof mazes.as well. Space complexity of all prototypes is the same O ( L ): it takes O ( L )instances (molecules, charges, micro-volumes of mixture or protoplasm) to fillin all channels during exploration stage.In terms of a real time gas-discharge [9] is the fastest maze-solver, it takesjust hundred of milliseconds to solve the maze. Hot ice computer [1] is thesecond fastest one, a maze of 0.1 m in diameter can be solved in few seconds.Gas-discharge and hot-ice solvers are followed by BZ-medium [10] and mobile-droplet [6] solvers: solution time is measured in minutes and hours. Slimemould based solvers are the slowest ones: it usually takes them a couple ofdays to solve a maze like one we used in present experiments. The prototypescan be arranged in the following real-costs (which includes consumables andlaboratory equipment) descending order: gas-discharge, mobile droplets, BZ-medium, hot ice, and slime mould. Gas-discharge maze solver is most expensivewhile P. polycephalum solver can run literally for free.Accuracy of slime mould computing is far from ideal. Situation illustratedin Fig. 1 is a typical one: slime computes almost shortest path, compare withthe shortest path from source to destination computed in Oregonator model inFig. 2. In some situations, e.g. the one shown in Fig. 5, plasmodium chooses thelongest path from source to destination. Said that in neither of 35 experimentswe undertook plasmodium failed to solve the maze, it solves builds a path fromsource to destination. 8 a) (b)
Figure 5: Plasmodium of
P. polycephalum chooses longest path to maze’scentral chamber: plasmodium is inoculated in peripheral channel, east partof the maze, and a virgin oat flake is place in central chamber; (a) scannedimage of the experimental maze, protoplasmic tubes are yellow (light gray);(b) binarised, based on red and green components, images, major protoplasmictubes are thick black lines.Our final ’disclaimer’ is that neither of slime mould maze-solvers can suc-cessfully compete with existing conventional computer architectures. However,
P. polycephalum is an ideal biological substrate which represents all essentialfeatures of reaction-diffusion chemical computers yet encapsulated in an elasticgrowing membrane [2]. It can be treated as a meso-scale prototype of futuregrowing nano-scale circuits. We envisage slime mould maze-solvers can also con-tribute toward designs of parallel amorphous drug delivery systems, or smartneedles, which grow towards the target tissue.
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