Narrow Results

We implement a Kolmogorov-Uspensky machine on the Plasmodium of the slime mold Physarum polycephalum. We provide experimental findings on realization of the machine instructions, illustrate basic operations, and elements of programming.

Plasmodium of Physarum polycephalum spans sources of nutrients and constructs varieties of protoplasmic networks during its foraging behavior. When the plasmodium is placed on a substrate populated with sources of nutrients, it spans the sources with protoplasmic network. The plasmodium optimizes the network to deliver efficiently the nutrients to all parts of its body. How exactly does the protoplasmic network unfold during the plasmodium's foraging behavior? What types of proximity graphs are approximated by the network? Does the plasmodium construct a minimal spanning tree first and then add additional protoplasmic veins to increase reliability and through-capacity of the network? We analyze a possibility that the plasmodium constructs a series of proximity graphs: nearest-neighbour graph (NNG), minimum spanning tree (MST), relative neighborhood graph (RNG), Gabriel graph (GG) and Delaunay triangulation (DT). The graphs can be arranged in the inclusion hierarchy (Toussaint hierarchy): NNG⊆MST⊆RNG⊆GG⊆DT. We aim to verify if graphs, where nodes are sources of nutrients and edges are protoplasmic tubes, appear in the development of the plasmodium in the order NNG→MST→RNG→GG→DT, corresponding to inclusion of the proximity graphs.

We define a kind of simple actions of labelled transition systems. These actions cannot be atomic; consequently, their compositions cannot be inductive. Their informal meaning is that in one simple action we can suppose the maximum of its modifications. Such actions are called hybrid. Then we propose two formal theories on hybrid actions (the hybrid actions are defined there as non-well-founded terms and non-well-founded formulas): group theory and Boolean algebra. Both theories possess many unusual properties such as the following one: the same member of this group theory behaves as multiplicative zero in respect to one members and as multiplicative unit in respect to other members.

We experimentally demonstrate that both Voronoi diagram and its dual graph Delaunay triangulation are simultaneously constructed — for specific conditions — in cultures of plasmodium, a vegetative state of Physarum polycephalum. Every point of a given planar data set is represented by a tiny mass of plasmodium. The plasmodia spread from their initial locations but, in certain conditions, stop spreading when they encounter plasmodia originated from different locations. Thus space loci not occupied by the plasmodia represent edges of Voronoi diagram of the given planar set. At the same time, the plasmodia originating at neighboring locations form merging protoplasmic tubes, where the strongest tubes approximate Delaunay triangulation of the given planar set. The problems are solved by plasmodium only for limited data sets, however the results presented lay a sound ground for further investigations.

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In this paper, I theoretically summarize, which behavioral possibilities the plasmodium of Physarum polycephalum has in order to be considered the medium of computation. I show that plasmodia can be represented as a natural implementation of different abstract automata: cellular automata, Kolmogorov–Uspensky machines, Schönhage’s storage modification machines, random-access machines. As a programming language for simulating Physarum plasmodium behavior, process calculus can be used.

Plasmodium of Physarum polycephalum is a single cell visible by unaided eye, which spans sources of nutrients with its protoplasmic network. In a very simple experimental setup we recorded electric potential of the propagating plasmodium. We discovered a complex interplay of short range oscillatory behavior combined with long range, low frequency oscillations which serve to communicate information between different parts of the plasmodium. The plasmodium's response to changing environmental conditions forms basis patterns of electric activity, which are unique indicators of the following events: plasmodium occupies a site, plasmodium functions normally, plasmodium becomes "agitated" due to drying substrate, plasmodium departs a site, and plasmodium forms sclerotium. Using a collective particle approximation of Physarum polycephalum we found matching correlates of electrical potential in computational simulations by measuring local population flux at the node positions, generating trains of high and low frequency oscillatory behavior. Motifs present in these measurements matched the response "grammar" of the plasmodium when encountering new nodes, simulated consumption of nutrients, exposure to simulated hazardous illumination and sclerotium formation. The distributed computation of the particle collective was able to calculate beneficial network structures and sclerotium position by shifting the active growth zone of the simulated plasmodium. The results show future promise for the non-invasive study of the complex dynamical behavior within — and health status of — living systems.