1 Introduction

The plasmodium of Physarum polycephalum is a unicellular and multinuclear giant amoeba that has computational abilities. About 20 years ago, Nakagaki and colleagues demonstrated that the plasmodium is able to find the shortest path in a maze [1]. With that as the starting point of subsequent research, the plasmodium has gained the status of a model system in the field of bio-computing. Following the maze-solving study, various bio-computational experiments have also been performed using the plasmodium, and they include the optimization of networks [2, 3], the construction of logic gates [4, 5], solving the traveling salesman problem [6, 7], the duplication of real transport networks [8, 9], and other geometrical computing problems [10,11,12] (for the other representative studies in this field, see [8, 13, 14]). Furthermore, inspired by these studies, many models and solvers have been developed to imitate the search and optimization mechanisms of the plasmodium [9, 15,16,17,18,19,20]. It is certain that plasmodium research has greatly inspired computational research.

However, the results of the previous computational studies involving plasmodia have been somewhat non-biological. For example, many of the experimental studies introduced above [1,2,3, 6,7,8,9,10,11,12] were performed in closed spaces; as the basic strategy of these experiments, the plasmodia first searched the entire experimental spaces, and then some form of optimization was performed. In contrast, in the context of bio-computing, we believe that it is more important to study the ability of an organism to adapt and respond to open and unknown situations. Furthermore, we also believe that the most important goal of bio-computing is to develop a new framework of computation based on the ability of biological entities to deal with open and unknown conditions. Indeed, with such an objective, we performed an experiment to test whether plasmodia can learn under a new situation [21], and we performed experiments to observe the behavior of the plasmodium in open spaces [22,23,24]. Furthermore, in another previous study, we conducted an experiment presenting plasmodia with a contradictory situation, in which there was no single optimal solution [25]. In this previous experiment, we gave the plasmodium bimodal stimuli, consisting of a mixture of attractant and repellent stimuli, and we observed diverse responses of the plasmodia to the stimuli that could not be explained in terms of a simple stimulus–response system.

In this study, we performed a further experiment based on this finding, enhancing the biological characteristics of the system with relevance to engineering. In the experiment, we confined each plasmodium in a closed circular space surrounded by repellent and observed how the plasmodium escaped from the situation. Based on our previous result [25], the timing of the plasmodium’s escape from the field was expected to be determined by the choice of the plasmodium itself, and indeed, the results in this study suggested that this is the case. In other words, the analysis revealed that the escape phenomenon was not induced deterministically by starvation caused by being trapped for a long time or the repellent response caused by the accumulation of secretions, nor did it occur stochastically. In the context of logic, the significance of our results is that they suggest ways in which organisms may use to overcome logical deadlocks. In our experiment, the plasmodium is in a state equivalent to a logical deadlock, which continues to apply the rule of avoiding repellent in the situation surrounded by repellents. Our results give an implication on a rational way of how to get out of deadlocks, and by further exploring the mechanism of the phenomenon of autonomous switching from the normal rule of avoiding repellent substances to the rule of overcoming repellent substance and escaping, novel engineering applications may be developed, for example, the development of a new and more versatile machine learning search algorithm that switches problem spaces autonomously. Finally, it should be noted that an experiment similar to ours has already been partly conducted in a previous study [26]. However, the present research revisits the phenomenon from a completely different perspective, and our results are also qualitatively different.

2 Materials and Methods

2.1 Culture of Plasmodia

All the samples of plasmodia in this study were cultivated according to the method developed by Camp [27]. Glass Petri dishes with a 9-cm diameter were closely arranged in a plastic box having dimension of 20 \(\times \) 30 cm to minimize the gaps between dishes. Pieces of paper towel moistened with ion-exchanged water were placed on the arranged Petri dishes. The space under the paper towels was filled with water to maintain the moisture inside the box and on the paper towels. The box was placed in darkness, and the plasmodia were cultured on the pieces of paper towel and maintained at 23 \(^\circ \)C. Oatmeal was used as food for the plasmodia, and food was given daily and once within 24 h before the experiment. The fresh tips of the plasmodia were scraped off and used in the experiments.

2.2 Experimental Setup

A 1.5% (w/w) agar solution was prepared by adding 6.0 g of agar powder to 400 ml of ion-exchanged water and boiling it sufficiently in a microwave oven. After that, 1.5% agar plates were prepared by pouring the agar solution into a plastic box having dimensions of 20 \(\times \) 30 cm, where it was allowed to solidify. We also created circular templates with an inner diameter of 2 cm and a width of 5 mm intended to confine the plasmodia with plastic film (a substance that is repellant to plasmodia), which was placed on the agar plate. To prevent the plasmodia from interfering with each other, a 5-mm ditch was created between the samples.

All experiments were performed in the dark at a temperature of 23 \(^\circ \)C and 80% humidity or higher. A custom-made surface light source consisting of light emitting diodes (Aitec System Co., Ltd., Yokohama, Japan) producing 600-nm wavelength light was used as the photographic light source because this wavelength is outside the range that induces phototaxis in plasmodia [28]. The plastic box with the agar plates and circular templates was placed on the surface light. A portion of the plasmodial cell body (20–50 mg) was scraped off the plasmodium culture and used to inoculate the center of each circular template. The plasmodia in this setup were photographed using a digital camera (EOS 5D Mk-II; Canon, Tokyo, Japan).

2.3 Experiment 1: Escape Time Measurement

Experiment 1 was performed using the experimental setup described in Sect. 2.2 with 100 plasmodium samples of 20, 30, 40, and 50 mg each. (Plasmodium samples of less than 20 mg cannot cover all of the given field, whereas plasmodium samples greater than 50 mg can escape the field immediately.) The plasmodia were photographed at 1-min intervals for 1200 min. The escape time was defined and measured as the time when the plasmodium came into contact with the outside of the template.

2.4 Experiment 2: Analysis of Plasmodial Internal State

Experiments were performed using the experimental setup described in Sect. 2.2 and two plasmodium samples of 20, 30, 40, and 50 mg each. The plasmodia were photographed at 5-sec intervals for 1200 min. In the image analysis, the space in the circular template was divided into a hexagonal grid, and the thickness of the plasmodium in each grid section was calculated based on the intensity of transmitted light, that is, the brightness of the image. In this way, the oscillation in thickness of the plasmodium within each grid section was analyzed, and the amount of mutual information was measured between the grid section near the center (i.e., the inoculation site) and all other grid sections.

Fig. 1
figure 1

An example result from the escape time measurement experiment. (a) The plasmodium immediately after inoculation. (b) The plasmodium spreading in the field, 200 min after inoculation. (c) The plasmodium covering the whole space inside the template, 330 min after inoculation. (d) The plasmodium completing its escape, 410 min after inoculation

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3 Results

3.1 Experiment 1 Results

To investigate the mechanism by which the plasmodium determines how to escape from a closed space, we first tried to clarify the distribution of escape times. An example experimental result is shown in Fig. 1, and the results from the experiment based on 400 plasmodium samples are summarized in Figs. 2 and 3.

These results indicate that the distribution of escape times was almost random (Fig. 2), and the distribution is much different than that predicted by assuming the escape occurs with a certain probability at regular intervals (Fig. 3).

Fig. 2
figure 2

Escape time for the plasmodia of each weight. Each datapoint indicates the result from one sample. Samples that did not escape within 1200 min are excluded from this dataset

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Fig. 3
figure 3

Number of samples remaining in the experimental field at each time, counted at 120-min intervals. As the data indicate, the number of plasmodia remaining in the experimental field decreased almost linearly over time, irrespective of their weight. The numerical values of the stochastic model are calculated based on the assumption that the plasmodia escape with a certain probability within a certain time

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Furthermore, in comparing plasmodia immediately after the full search of the field and immediately before their escape, the former spread homogeneously in the field, whereas the latter actively formed new pseudopodia inside (Fig. 4). That is, just before the start of escape, an evident change in the behavior of the plasmodium was observed that might indicate a precursor of the escape.

Fig. 4
figure 4

A plasmodium (a) immediately after the full search of the field at 330 min after inoculation, and (b) immediately before the start of escape at 370 min after inoculation. The white arrows indicate the newly formed pseudopodia

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Fig. 5
figure 5

The amount of mutual information, represented by the color of each grid section. (a) Immediately after the full search of the space. (b) Immediately after the start of escape. The strength of the signals are indicated by the color table in (b). The red arrow indicates the grid section that was the inoculation site

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3.2 Experiment 2 Results

We further analyzed the thickness oscillation of plasmodia before and after the start of their escape as an indicator of the internal state of each plasmodium [29]. Here, the space inside the experimental field was divided according to a hexagonal grid, and the thickness oscillation of the plasmodium in each grid was analyzed. Thus, we observed that the synchrony of oscillation was enhanced in the whole cell after the start of the escape behavior of the plasmodium. We calculated the mutual information between the grid section corresponding to the inoculation site and all other grid sections in order to numerically evaluate the increase in synchrony. As shown in Figs. 5 and 6, the amount of mutual information within the cell body was increased when the escape was started, although there was not a strong associated between the start of the escape event and the amount of mutual information. As shown in Fig. 6a, the increase in the amount of mutual information is clearly associated with the start of escape. However, as shown in Fig. 6b, although an increase in the amount of mutual information is observed at the start of the escape, unrelated changes in mutual information are also large.

Fig. 6
figure 6

The amount of mutual information averaged over all grid sections, from two different samples. The red arrows show the start time of escape behavior

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4 Discussion

In this study, to observe the phenomenon of plasmodia changing their response to the environment, we confined each plasmodium in a closed space and analyzed what happens when it escapes from the situation and what mechanism is used to achieve its escape.

In experiment 1, the length of time required for the plasmodium to escape was random (Fig. 2). This means, foremost, that the escape is not triggered by a stimulus exceeding some threshold. If starvation or accumulation of secretions (repellent substances) were the triggers of escape, the escape time would be normally distributed. Additionally, it is not possible that the randomness of observed escape times is derived from individual genetic differences among samples because the 400 samples used in this study were all from the same clone. In addition, the distribution of the escape times differed greatly from the distribution when escape is assumed to occur stochastically at regular intervals (Fig. 3). In other words, escape does not appear to be triggered by a stochastic mechanism. Furthermore, pseudopods were observed to be actively formed inside the cells when escape behavior was initiated (Fig. 4). This indicates that some kind of behavioral switching occurs during the transition to escape. In other words, during escape, the previous behavior is not continued, and the behavioral rules of the plasmodium change. Based on the above results and those of our previous studies [25], we assume that the escape by the plasmodium is achieved by some intrinsic mechanism and determined autonomously.

In experiment 2, we then observed how the thickness oscillation of the plasmodium, an indicator of the internal state of the plasmodium, differed before and after the onset of escape in order to identify and explore the mechanism responsible for the phenomenon. Thus, we found that the synchrony of the thickness oscillation tended to increase just before the escape (Figs. 5 and 6). However, the relationship between escape time and the increase in synchrony was not completely correlated within the scope of the analysis in this study, so further analysis of the relationship between the two is warranted.

In light of the above, it is expected that the plasmodium escape phenomenon observed in this study is realized by plasmodia autonomously changing the rules of their behavioral selection, likely through an emergent property realized by increased synchrony within the plasmodia. In addition, elucidating the mechanism underlying this phenomenon and examining it in a computable form may reveal critical insights that, for example, enable current specialized AI to autonomously change its own behavioral rules, change its own interpretation of the problem space, and realize more versatile capabilities. In other words, further development of the findings of this research has the potential to promote the construction of a new computational framework, which is a critical goal of bio-computing.