Decision-making ability of Physarum polycephalum enhanced by its coordinated spatiotemporal oscillatory dynamics

A plasmodium of the true slime mold Physarum polycephalum is a single-celled multi-nucleated organism. It has a single gel layer composed of crosslinked actomyosin proteins and an intracellular sol, called protoplasm. The gel layer rhythmically contracts and relaxes with a characteristic period of approximately 1–2 min. This contraction-relaxation oscillation of the gel layer causes the flow of the intracellular sol, leading to the deformation of the macroscopic shape of the organism. As a result, the vertical thickness oscillates with the same period as the gel layer does. The mathematical models of the relation between this oscillation and amoeboid locomotion have been developed [1, 2]. It is known that the oscillation of the vertical thickness of the organism exhibits various spatiotemporal patterns, such as rotating waves, chaotic patterns, and transitions between these patterns [3, 4]. Such oscillatory patterns of the organism can be characterized as a time series generated by a set of coupled chaotic oscillators [5]. Hence, there are some deterministic chaotic dynamics controlling the oscillatory behavior of the organism.

Recently, many researchers have demonstrated that the Physarum plasmodium exhibits the computational abilities, such as searching for the optimal solutions to geometric path planning problems [6–13], construction of Voronoi diagram, Delaunay triangulation, and collision-free path [14, 15], and forming regular graphs [16], where the organism exhibited its ability to connect the optimal paths among food sources. Moreover, the organism was shown to be capable of anticipation of periodic events [17], associative learning [18], logical operations [19], robot control [20], and implementation of a programmable computer [21, 22]. Aono et al showed that an 'amoeba-based computer (ABC)' [23], which consists of the Physarum plasmodium and an optical feedback control unit, finds a high-optimality solution to the traveling salesman problem (TSP) with a high probability [24].

The TSP, one of the best-studied combinatorial optimization problems, is stated as follows: given a map of n cities that defines the travel distance from any city to any other city, find the shortest route for visiting each city exactly once and returning to the starting city [25]. With an increase in the number of cities n, the number of possible routes (valid solutions) grows exponentially as (n − 1)!/2. When the ABC searches for a solution to the TSP, the optical feedback control is applied depending on the fitness of the current shape of the organism with respect to the current route length. A local stimulation to a branch of the organism by visible light induces shrinkage of the branch owing to the photoavoidance (negative phototaxis) of the organism. In the ABC, the organism attempts to maximize its body area to obtain more nutrients from beneath the agar plate while minimizing the potential risk of being illuminated. When the shape of the organism represents the shortest TSP route, it maximizes the stability against its own perturbing movements under the optical feedback control [24]. Hence, using the ABC, we can quantitatively measure the optimality of the decision made by the organism as the shortness of the TSP route found by the ABC. (For more detailed discussions, see [24].)

Unlike the previous studies that reported the ability of the organism to minimize its body area in the physical space by connecting the optimal paths among food sources [6, 8–10, 13], the ABC evaluates the ability to find the optimal combination of n branches out of (n − 1)! feasible combinations, where the body area of the organism, which represents a feasible TSP route, converges to an equal value independent of the shortness of the route. One of the reasons why Aono et al [23] did not implement the TSP solution in the physical space but instead, orienting the combinatorial space, was that the TSP maps defined in the physical space are restricted to satisfy special conditions such as triangle inequality, whereas the ABC allows tackling of general TSP instances. Another reason was that optical feedback control enables us to explicitly evaluate the contribution of spatiotemporal oscillatory dynamics of the organism to the performance of the TSP solution, since the shape of the organism can be monitored at a temporal resolution that is high enough to reflect the difference in the spatiotemporal patterns.

In the spatiotemporal oscillatory dynamics while solving the TSP, the organism generates the traveling waves of its vertical thickness, and a number of waves sometimes coexist. Traveling waves result from the collective oscillation of the branches of the organism. Hence, the larger number of coexisting traveling waves indicates that local branches of the organism oscillate more independently and randomly in an uncorrelated manner. On the other hand, a lower number of coexisting traveling waves indicates more branches oscillating synchronously in a highly coordinated manner. Thus, we can consider the number of coexisting traveling waves as an index of uncoordinated oscillations of the branches of the organism.

In this study, to reveal the relationship between the spatiotemporal patterns and computational abilities of the organism, specifically whether or not the coordinated spatiotemporal patterns of the organism contribute to computation, we focused on the number of coexisting traveling waves of volumes during the solving of the TSP.

2.1. Amoeba-based computing

The amoeba-based computer consists of the organism and the optical feedback control applied according to a recurrent neural network model (figure 1). The organism was placed on a stellate chip, which was then placed on a nutrient-rich agar plate (figure 1(a)). There were N lanes representing the current state of the computation in the chip. The experimental setup is illustrated in figure 1(b). The current shape of the organism was recorded by a video camera (VC). A computer (PC) processed the recorded image and controlled optical feedback by a neural network algorithm [5, 26]. The optical feedback was implemented by projecting a grayscale image using a commercial PC projector (PJ).

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ABC searches for the optimal state with the change in the shape of the organism. The state ${s}_{i}(t)\in [0.0,1.0],(i=1,...,N)$ at time t is defined as the fraction of the area occupied by the branch of the organism in the ith lane. Let l_i(t) be the illumination status of the ith lane at time t. All illuminations l_i(t) are updated synchronously as determined by the modified Hopfield-Tank neural network dynamics [5, 26] at every interval of Δ t = 6 s as follows:

$$\begin{eqnarray}{l}_{i}(t+{\rm{\Delta }}t) & = & 1-F\left(\displaystyle \sum _{j}{w}_{{ij}}f({s}_{j}(t))\right),\\ F(s) & = & \displaystyle \frac{1}{1+\mathrm{exp}(-B(s-{\rm{\Theta }}))},\\ f(s) & = & \displaystyle \frac{1}{1+\mathrm{exp}(-b(s-\theta ))}.\end{eqnarray} \tag{ 1 }$$

Here, w_ij is the coupling weight from lane j to lane i. In this network, there are two sigmoidal functions: F(s) is defined with B = 1000 and Θ = −0.5 while f(s) is defined with b = 35 and θ = 0.6.

As in our previous studies [24], we solved the 8-city TSP using the ABC (figure 1). The 8-city map is shown in figure 2(a). This map provides the unique shortest and longest routes, whose values are 100 and 200, respectively (figure 2(b)). The median and average route lengths for all the valid solutions are 147 and 149.1, respectively.

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The organism was placed on the stellate chip with N = 64 lanes, where each lane is labeled by $i\in \{{Pn}| P\in \{A,B,...,H\},n\in \{1,2,...,8\}\}$. If the branch of the organism occupied a sufficiently large area in lane Pn, i.e., the corresponding state s_i(i = Pn) was sufficiently large, city P was visited on the nth visit. To solve the TSP, the coupling weights of the neural network are defined as

$$\begin{eqnarray}&&\begin{array}{l}{w}_{{ij}}\\ =\;\left\{\begin{array}{ll}-\alpha & {\rm{if}}\;i={Pn}\ \wedge \ j={Pm}\ \wedge \ n\ne m,\\ -\beta & {\rm{if}}\;i={Pn}\ \wedge \ j={Qn}\ \wedge \ P\ne Q,\\ -\gamma d(P,Q) & {\rm{if}}\;i={Pn}\ \wedge \ j={Qm}\ \wedge \ P\ne Q\\ & \wedge \ | n-m| =1,\\ 0 & {\rm{otherwise}}.\end{array}\right.\end{array}\end{eqnarray} \tag{ 2 }$$

Here, d(P, Q) denotes the distance between the cities P and Q. The parameter α = 0.5 inhibits the revisiting of the same city again. Similarly, β = 0.5 inhibits visiting to multiple cities simultaneously. To find shorter routes, the lane Pn is inhibited, depending on the distances between P and the cities which are visited on the (n − 1)th and (n + 1)th visits. The inhibition is proportional to the parameter γ = 0.0081.

Route ABCDEFGH, the shortest TSP route in figure 2, can be found when the organism elongates branches A1, B2, C3, D4, E5, F6, G7, and H8 exclusively. It may be supposed that the shortest route was more likely found because the lanes in the chip were labeled as city names in a clockwise manner, i.e., ABCDEFGH. However, this would not be the case, as no symmetric advantage is conferred to the organism in elongating the branches A1, B2, C3, D4, E5, F6, G7, and H8 in any other combination that represents an arbitrary feasible TSP route.

2.2. Preparation for analysis

2.3. Number of traveling waves

To estimate the number of coexisting traveling waves automatically, we extracted the edges of traveling waves as borderlines between the areas in which the volume of the organism increased and those in which it decreased, and counted the number of connected components in those borderlines. In figure 3, red and blue pixels are sparsely distributed. To make a clear distinction between areas of increase and decrease in the volume of the organism, we smoothed the distributions of red and blue pixels as follows:

$$\begin{eqnarray}&&I(x,y)\quad ={\displaystyle \sum }_{({r}_{x},{r}_{y})\in R}\\ &&\phantom{\rule{4em}{0ex}}\times \mathrm{exp}\left(-\displaystyle \frac{{({r}_{x}-x)}^{2}+{({r}_{y}-y)}^{2}}{{\sigma }^{2}}\right),\\ &&D(x,y)\quad ={\displaystyle \sum }_{({b}_{x},{b}_{y})\in B}\\ &&\phantom{\rule{4em}{0ex}}\times \mathrm{exp}\left(-\displaystyle \frac{{({b}_{x}-x)}^{2}+{({b}_{y}-y)}^{2}}{{\sigma }^{2}}\right).\end{eqnarray} \tag{ 3 }$$

Here, σ is the smoothing parameter and $({r}_{x},{r}_{y})\in R\;{\text{}}{\rm{\text{}}}{\rm{\text{}}}{\rm{and}}\;\;({b}_{x},{b}_{y})\in B$ denote the positions of red and blue pixels in the image, respectively. We set σ = 15. The larger value of I(x, y) (D(x, y)) indicates the higher possibility of an increase (decrease) of the volume of the organism at (x, y). Examples of I(x, y) and D(x, y) obtained by smoothing of figure 3 are illustrated in figure 4(a). We infer that the volume of the organism increased at the pixel (x, y) if I(x, y) is larger than D(x, y) and decreased otherwise. Figure 4(b) shows the areas of increase and decrease of figure 3. Areas of increase and decrease are indicated as red and blue areas, respectively.

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Next, we extracted the edges of traveling waves, namely the borderlines between the areas of increase and decrease in the volume. When the sign of U(x, y) = I(x, y) − D(x, y) at one pixel (x, y) differs from that of neighboring pixels (x + 1, y) or (x, y + 1), we consider that the pixel (x, y) is part of the borderlines. That is, the set of pixels composing the borderlines is defined as

$$\begin{eqnarray}B & = & \{(x,y)| U(x,y)U(x+1,y)\lt 0\}\\ & & \times \cup \{(x,y)| U(x,y)U(x,y+1)\lt 0\}.\end{eqnarray} \tag{ 4 }$$

Hereafter, we refer to such pixels as 'border pixels'.

Finally, we estimated the number of connected components of borderlines rather than the number of coexisting traveling waves. We considered the network of border pixels, where neighboring border pixels are connected with each other. The adjacency matrix of the network of border pixels is defined as

$$\begin{eqnarray}{A}_{{ij}}=\left\{\begin{array}{ll}1 & | {x}_{i}-{x}_{j}| \lt 1\wedge | {y}_{i}-{y}_{j}| \lt 1,\\ 0 & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ 5 }$$

where x_i and y_i denote horizontal and vertical coordinates for the position of the ith border pixel. It is known that the number of connected components of the network corresponds to the number of zero eigenvalues of the graph Laplacian matrix [27], which is defined as,

$$\begin{eqnarray}{L}_{{ij}}=\left\{\begin{array}{ll}{\displaystyle \sum }_{k}{A}_{{ik}} & i=j,\\ -{A}_{{ij}} & i\ne j.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

In practice, we cannot obtain exact values for the eigenvalues. Hence, we counted up the eigenvalues of the graph Laplacian matrix that are less than 10⁻⁵.

We performed 15 experiment trials to solve the 8-city TSP. In each trial, the computation was started up to 12 hours after we detached the organism from food sources (oat flakes) to put the organism under 'starved' conditions. The organism used for a trial was never used again for another trial. In all of the trials, a valid solution was found. The best route length found was 117 and the worst one was 165. The route lengths of the solutions in 13 out of 15 trials were shorter than 147, which is the median of the route lengths of all possible solutions. This result means that the ABC could find significantly short routes (p = 0.0037; binomial test).

The fact that ABC could find significantly short routes indicates that ABC found better solutions rather than expected by randomly selecting one from all possible solutions. Thus, we can consider that the organism made the right decisions on which branches should expand through the interactions with the optical feedback. In other words, we can quantify the optimality of the decision made by the organism by the shortness of the TSP route found by the ABC.

We extracted the areas of increase and decrease in the volume and their boundaries. Examples of the original images, the areas of increase and decrease, and the borderlines between those areas are shown in figure 5. We estimated the number of traveling waves at each frame and obtained the average number of traveling waves in each trial. The minimum and maximum numbers of traveling waves were 0 and 7, respectively. The time evolution and the distributions of the number of traveling waves in each trial is illustrated in figure 6. In most trials, the number of traveling waves fluctuated around 1 or 2 and distributions were unimodal.

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The average numbers of traveling waves and the route length of the TSP solution are plotted in each trial in figure 7. The correlation coefficient between the average number of traveling waves and the route length was 0.52 (p = 0.023; t-test). In trials, when a lower number of traveling waves appeared on average, the ABC was likely to find solutions with shorter route lengths.

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In this paper, we explored the relationship between the complex spatiotemporal oscillatory dynamics of the amoeboid organism Physarum and its decision-making ability by using the amoeba-based computer, which leads the organism to search for a solution to the traveling salesman problem under optical feedback control.

In this study, we used the organism under starved conditions. It is expected that the growth velocity of the branches of the organism under well-fed conditions would be slower than that under starved condition, suggesting that a longer time would be spent for finding a feasible TSP solution. It would be important to examine if the organism finds a shorter TSP route under well-fed conditions by spending a longer time than that under starved conditions. It would also be interesting to see if the organism exhibits the learning capacity to enhance its decision-making ability if an organism used in a trial were used repeatedly in subsequent trials.

The changes in the number of traveling waves over time indicate that the oscillation patterns in the amoeba-based computer were unstable. We can consider that the increase and decrease in traveling waves correspond to the appearance and disappearance of rotating spiral waves found in the large plasmodium [4]. They found that the spiral waves act to break down vein structure. Hence, fluctuation in the number of traveling waves might reconstruct the vein structure of the organization in the amoeba-based computer, which facilitates information exchange among the branches.

When the number of traveling waves is 0, either red or blue pixels rarely appear in the original image (see the leftmost panel of figure 5(a)). This observation suggests that the volume of the organism increases (or decreases) simultaneously almost everywhere in the central circular area. In the second panel of figure 5(a), where the number of traveling waves was 1, red and blue pixels, indicating an increase or decrease in the vertical thickness at the corresponding points, were separately distributed. This separation indicates that the oscillations' branches proximal to each other were correlated. On the other hand, the red and blue pixels were shuffled in the rightmost panel of figure 5(a), indicating that local branches oscillated almost independently. Therefore, coordinated oscillations of a larger number of branches resulted in lower numbers of coexisting traveling waves. The average number of traveling waves was significantly correlated with the TSP route length. This significant correlation means that the ABC found a better solution in the trials with a lower number of connected components, namely coexisting traveling waves. As mentioned above, the lower number of coexisting traveling waves indicates that a larger number of branches oscillated synchronously in a highly coordinated manner. Therefore, we can conclude that the globally coordinated oscillation of the organism is positively correlated with its decision-making ability.

We counted up the number of traveling waves appearing in the spatiotemporal patterns of the oscillatory dynamics of the organism. The number of traveling waves was significantly correlated with the shortness of the TSP routes found. Because coordinated oscillations of the branches of the organism result in lower numbers of traveling waves, this result indicates that when the organism oscillates in a more coordinated manner, the organism can find better shapes that maximize the nutrient absorption while minimizing the potential risk of being exposed to aversive light stimuli. This result hints that high solution-searching abilities of biological systems can be produced in biological systems from highly coordinated spatiotemporal oscillatory dynamics, and not from uncorrelated random fluctuations. It is an important future subject to establish the theory that accounts for the reason why highly coordinated dynamics is more advantageous than uncorrelated randomness. At this moment, we consider that information exchange among the branches by means of protoplasmic streaming and phase wave propagation may be more efficient when the branches exhibit coordinated dynamics than when the branches are dominated by uncorrelated randomness. For verifying this theory, our findings will provide inspirations to develop new physics-based computing devices for solving computationally demanding problems, which utilize non-random fluctuations generated by intrinsic physical dynamics of the devices [28, 29].

This research is supported by the Innovative Mathematical Modelling Project, the Japan Society for the Promotion of Science (JSPS) through the 'Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program),' initiated by the Council for Science and Technology Policy (CSTP). This work was partially undertaken when LZ, MA and MH belonged to RIKEN Advanced Science Institute, which was reorganized and integrated into RIKEN as of the end of March, 2013. This research is also partially suported by Platform for Dynamic Approaches to Living System from MEXT, Japan, and CREST, JST, Japan.