Minimum-Risk Path Finding by an Adaptive Amoebal Network

When two food sources are presented to the slime mold Physarum in the dark, a thick tube for absorbing nutrients is formed that connects the food sources through the shortest route. When the light-avoiding organism is partially illuminated, however, the tube connecting the food sources follows a different route. Defining risk as the experimentally measurable rate of light-avoiding movement, the minimum-risk path is exhibited by the organism, determined by integrating along the path. A model for an adaptive-tube network is presented that is in good agreement with the experimental observations.

Figure 1

Schematic illustration of the procedure for measuring relative levels of risk in photoavoidance behavior. (a) Top view of the organism that extends linearly. The organism (gray) extends from left to right in the time sequences from $t_0$ to $t_4$. PL: plasmodium, $X_i$: position of the plasmodium tip. (b) Side view of the experimental setup. VC (video camera for video image analysis), FL (filter to eliminate visible light), WL (cold white light to stimulate the posterior of the plasmodium), IR (infrared light to observe the plasmodium).

Figure 2

Photographs of the connecting paths between two food sources (FS). (a) The rectangular sheetlike morphology of the organism immediately before the presentation of two FS and illumination of the region indicated by the dashed white lines. (b),(c) Examples of connecting paths in the control experiment in which the field was uniformly illuminated. A thick tube was formed in a straight line (with some deviations) between the FS. (d)–(f) Typical connecting paths in a nonuniformly illuminated field ($95\mathrm{K}\mathrm{lx}$). Path length was reduced in the illuminated field, although the total path length increased. Note that fluctuations in the path are exhibited from experiment to experiment.

Figure 3

Comparison between the measured connection path and the two routes based on minimum-risk estimates. Mean paths, averaged over ca. 10 experiments, are shown by the filled circles (with error bars) at light intensities of 0 lx (a), 15 000 lx (b), 35 000 lx (c), 55 000 lx (d), 75 000 lx (e), and 95 000 lx (f). Each point indicates the tube position on the vertical axis at the corresponding horizontal position. The dotted lines and the dashed lines indicate the minimum-risk paths calculated according to the relative rate of decrease in thickness ${\alpha }_1$ and the relative migration velocity ${\alpha }_2$, respectively. The $\alpha$ values were $({\alpha }_1,{\alpha }_2)=(1.14,1.03)$, ($1.61,1.04$), ($1.50,1.40$), ($2.07,1.49$), and ($2.14,1.62$) at 15, 35, 55, 75, and $95\mathrm{K}\mathrm{lx}$, respectively. (The fluctuations in the ${\alpha }_1$ values are likely due to the difficulty of preparing each organism in exactly the same state.)

Figure 4

Simulation of path formation. (a) Schematic illustration of the mathematical model. The tube network of Physarum is represented by a random mesh lattice. Between junctions $i$ and $j$ the tube has fixed length $L_{ij}$, variable radius $r_{ij}$, and the conductance $D_{ij}\propto r_{ij}^4$. All tubes initially have a similar small value of $r_{ij}$ with random fluctuations around the mean value. (b)–(e) Results of simulation. A constant current flows into the source in the upper left-hand corner and out of the sink on the lower right hand. Some paths including the minimum-risk path appear at an intermediate stage (c),(d) and finally only the minimum-risk path remains (e). (f) The minimum-risk path obtained by a conventional path finding algorithm, Dijkstra’s algorithm [10], where the length of a lattice edge was scaled by the relative level of risk. Above the dashed line $a=2$ (corresponding to the illuminated region) and below the dashed line $a=1$. The simulations utilized 20 000 nodes in order to generate a sufficiently fine meshwork. No change in the qualitative behavior was found when the number of nodes was increased.