## Abstract

Behavioral diversity is an essential feature of living systems, enabling them to exhibit adaptive behavior in hostile and dynamically changing environments. However, traditional engineering approaches strive to avoid, or suppress, the behavioral diversity in artificial systems to achieve high performance in specific environments for given tasks. The goals of this research include understanding how living systems exhibit behavioral diversity and using these findings to build lifelike robots that exhibit truly adaptive behaviors. To this end, we have focused on one of the most primitive forms of intelligence concerning behavioral diversity, namely, a plasmodium of true slime mold. The plasmodium is a large amoeba-like unicellular organism that does not possess any nervous system or specialized organs. However, it exhibits versatile spatiotemporal oscillatory patterns and switches spontaneously between these. Inspired by the plasmodium, we built a mathematical model that exhibits versatile oscillatory patterns and spontaneously transitions between these patterns. This model demonstrates that, in contrast to coupled nonlinear oscillators with a well-designed complex diffusion network, physically interacting mechanosensory oscillators are capable of generating versatile oscillatory patterns without changing any parameters. Thus, the results are expected to shed new light on the design scheme for lifelike robots that exhibit amazingly versatile and adaptive behaviors.

## 1 Introduction

Classical engineering approaches strive to avoid, or suppress, the behavioral diversity of artificial systems to achieve high performance in specific environments for given tasks. Living systems, in contrast, exhibit qualitatively versatile behaviors and spontaneously switch among these behaviors in response to the situation encountered. Because of this behavioral diversity, living systems are able to exhibit adaptive behavior in hostile and dynamically changing environments. The goals of the present research include understanding how living systems generate this behavioral diversity and then using these findings to build lifelike robots that exhibit truly adaptive behavior.

To this end, we have employed the so-called back-to-basics approach. More specifically, we have focused on one of the most primitive living organisms with behavioral diversity: a plasmodium of the true slime mold (see Figure 1).^{1} This plasmodium is a large amoebalike multinucleated unicellular organism, whose motion is driven by spatially distributed biochemical oscillators in the body [3]. These oscillators induce rhythmic mechanical contractions, leading to a pressure increase in the protoplasm, which in turn generates protoplasmic streaming according to the pressure gradient [3]. Hence, the interaction between the homogeneous elements (i.e., the biochemical oscillators) induces global behavior in the plasmodium in the absence of a central nervous system or specialized organs. Yet, despite such a decentralized system, the plasmodium exhibits versatile spatiotemporal oscillatory patterns and, more interestingly, switches between these versatile oscillatory patterns spontaneously [5–7]. By exploiting this behavioral diversity, the plasmodium is thought to be capable of exhibiting adaptive behavior in response to the situation encountered. Thus, the plasmodium is the simplest model of a living organism that can be used to investigate the key mechanism that induces the underlying versatile behavior in a fully decentralized manner.

There are two important factors that help the plasmodium exhibit such oscillatory patterns: phase modification of the mechanosensory oscillators, and physical communication (i.e., morphological communication) stemming from protoplasmic streaming. The oscillators of the plasmodium, which are similar to a central pattern generator (CPG) [7], are physically coupled with tubes filled with protoplasm. By generating protoplasmic streaming through these tubes, long-distance physical interaction is induced between the oscillators. This physical interaction leads to phase modification based on thepressure (i.e., mechanosensory information) applied by the protoplasm [13]. In light of biological knowledge, the versatile oscillatory patterns are attributed mainly to the synergistic effect of the phase modification and morphological communication stemming from the protoplasmic streaming.

The purpose of this study was to build an autonomous decentralized system that induces various oscillatory patterns and to understand the design principle behind such intelligence. To this end, we developed a mathematical model of a robot inspired by the plasmodium of the true slime mold (Figure 2) thatconsists of several homogeneous modules filled with protoplasm. The significant features of this model are threefold: (i) the modules are physically connected by tubes to induce long-distance physical interaction among the modules; (ii) soft actuators, so-called real-time tunable springs (RTSs), are embedded in the modules to aid the flow of protoplasm between the modules; and (iii) phase modification exploits the mechanosensory information of the soft actuators. Simulation results show that the model is capable of exhibiting surprisingly versatile oscillatory patterns and transitioning between them, in a fully decentralized manner, without changing any of the parameters. The results obtained are expected to shed new light on the design scheme for a new type of lifelike robot with the capability of switching behaviors (e.g., taxiing, exploratory, and escape behaviors) spontaneously based on the situation encountered, without modifying any of the robot's parameters, as shown in Figure 3.

## 2 Model

### 2.1 Mechanical System

The model consists of several homogeneous modules that are physically connected by tubes. As shown in Figure 2, each module consists of a soft and deformable outer skin, with fluid (i.e., protoplasm) below this outer skin. The outer skin consists of four mass particles and four RTSs that can actively alter their resting lengths (i.e., the unstretched length of the spring). This mechanical passivity of the RTS enables the outer skin to be soft and deformable. Furthermore, altering the resting length of each RTS causes the protoplasm to be pushed and pulled competitively. In this article, this physical interaction is simulated by a potential constraint on the area surrounded by the mass particles on each module (which will be explained in Section 2.1.2). As a control system, module *i* contains two decoupled phase oscillators, each of which controls the resting lengths of two RTSs, RTS_{i,n,m} (*m* = 0, 1), according to phase ϕ_{i,n} (*n* = *v*, *h*) (see Figure 2a).

#### 2.1.1 RTS

_{i,n}), of RTS

_{i,n,m}changes according to ϕ

_{i,n}, and it is given bywhere

*a*is constant in space and time (0 <

*a*< 1) and represents the mean length. Depending on its resting length, (ϕ

_{i,n}), the spring constant of RTS

_{i,n,m}varies as follows:where α is a constant given by the material and geometric properties of the elastic material.

#### 2.1.2 Protoplasm and Protoplasmic Streaming

*l*

_{i,n}is defined as

*l*

_{i,n,0}and

*l*

_{i,n,1}since we assume that

*l*

_{i,n,0}and

*l*

_{i,n,1}have the same values. Using the ideal gas law, the pressure of module

*i*can be calculated aswhere

*N*

_{i}is the number of moles inside module

*i*,

*V*

_{i}(=

*l*

_{i,v}

*l*

_{i,h}

*d*) is the volume of module

*i*,

*R*is a physical constant,

*K*is the temperature inside the module, and

*d*is the thickness of the modules. Based on this equation, the potential that keeps the volume of the module constant can be written aswhere

*p*

_{ex}is the external pressure.

^{2}The first term on the right-hand side is the potential stemming from the ideal gas inside the module, and the second term is derived from the potential caused by the external pressure,

*p*

_{ex}. As seen in the equation, the potential, φ

_{i}, has a concave shape along

*l*

_{i,v}and

*l*

_{i,h}, which provides volume conservation according to

*N*

_{i}. This in turn induces the intra- and inter-module physical interactions.

*D*

_{i,j}is the fluid conductance between module

*i*and module

*j*. Note that the total volume of the protoplasm inside all the modules and tubes is conserved. Because of this, long-distance physical interaction is induced among the modules.

### 2.1 Control System

_{n}is the intrinsic frequency of the phase oscillator and the second term is the local sensory feedback, which can be calculated from the discrepancy function

*I*

_{i,n}. This function is based on the mechanosensory information of the soft actuator, as mentioned in Section 2.1.1. Note that the phase oscillators only interact with each other through the mechanical system (i.e., the protoplasm).

To define the discrepancy function, let us discuss the possible cytological processes that play roles in the discrepancy function. It is well known that the so-called stretch-activated Ca^{2+} channel is regulated by the mechanical deformation of the cell shape [2]. This means that the channel can detect the local cell curvature and open or close in response. This type of channel can relax rhythmic contractions, because a high Ca^{2+} concentration leads to the relaxation of the actomyosin contraction in a plasmodium of the true slime mold [13]. When a local oscillator is pushed strongly by the protoplasm under high pressure, it stretches even if it is trying to contract. The amountof stretching is related to the increase in the Ca^{2+} concentration, which leads to the relaxationof the oscillator. This is one possible cytological scenario for the role of the discrepancyfunction.

*T*

_{i,n,m}is the tension in the RTS. This function is designed to increase in value when the absolute value of

*T*

_{i,n,m}increases. It should be noted that this mechanosensory information can only be produced by the mechanical softness of the actuator.

*I*

_{i,n}, and it can be calculated using only locally available variables, which include the discrepancy between the controlled value and its actual value.

## 3 Simulation Results

### 3.1 Numerical Experiment with One Module

To confirm the validity of the local sensory feedback based on the discrepancy function, a numerical experiment with one module was conducted, and the results are presented in Figure 5. The experiment began from the nearly in-phase condition. Hence, in-phase resting length contraction occurred in the beginning, which caused a high value of ∑∑*I*_{i,n} (see Figure 5c). However, at around 2600 s, a transition from the in-phase condition to the out-of-phase coordination inside the module occurred as a result of the decreasing total value of the discrepancy function. This simulation result demonstrates that each oscillator modifies its own phase to decrease the value of its own discrepancy function.

We set the initial value of the area of the module as 1.0. The parameters of the model are as follows: α = 5.0; σ = 0.1; *a* = 0.2; = 1.0; η = 1.0; *RK* = 100.0; *p*_{ex} = 100.0; *d* = 1.0; *dt* = 0.001; ω_{v} = ω_{h} = 1.0; ϕ_{0,v} (*t* = 0) = 0.001; ϕ_{0,h} (*t* = 0) = 0.0.

### 3.2 Numerical Experiment with Two Modules

Now, let us describe what happens when two modules are connected, as shown in Figure 4. To investigate the behavior based on fluid conductance, we perform two numerical experiments: when the modules are connected by a thick tube (large fluid conductance) and when they are connected by a thin tube (small fluid conductance). The results are presented in Figures 6 and 7.

At the beginning of these simulations, all the phase oscillators start with an almost in-phase condition, which leads to a balance between the competitive pushing of the protoplasm (note that the phase oscillators controlled the resting lengths, not the actual lengths) and a high value of ∑ ∑ *I*_{i,n}. Therefore, the areas have values of 1.0, as seen in Figures 6a and 7a. As a result, very little of the protoplasm (gas) is exchanged between the modules.

However, when the fluid conductance is high, the volumes of the two modules eventually oscillate in an out-of-phase manner as a result of the phase modification mechanism stemming from thelocal sensory feedback. This means that the two modules are exchanging protoplasm, as seen in Figure 6. On the other hand, when the fluid conductance is low, it becomes difficult for the two modules to exchange protoplasm. In this condition, the modules decrease the total value of the discrepancy function “inside” each module, which produces out-of-phase oscillation between the two phase oscillators in each module, as shown in Figure 7. Note that this behavioral change is produced as a result of decreasing the total value of the discrepancy function in a fully decentralized manner.

We set the initial value of the area of each module as 1.0. The parameters of the model are as follows: ω_{v} = ω_{h} = 1.0; ϕ_{0,v} (*t* = 0) = 0*.*0; ϕ_{0,h} (*t* = 0) = 0.001; ϕ_{1,v} (*t* = 0) = 0.01; ϕ_{1,h} (*t* = 0) = 0.011; *D*_{i,j} = 1.0 (when fluid conductance is high); *D*_{i,j} = 0*.*1 (when fluid conductance is low). The remaining parameters are the same as those in Section 3.1.

### 3.3 Numerical Experiment with Three Modules

We now describe what happens when three modules are connected, as shown in Figure 2b. The results are shown in Figures 8 and 9. As seen in Figure 8, we confirmed the existence of four oscillatory patterns during one consecutive simulation run *without* any change in the parameters: (a) a rotation mode, (b) partial in-phase mode, (c) out-of-phase and half-period mode, and (d) intra-oscillation mode.^{4} Furthermore, the model spontaneously switched between these four oscillation modes during one observation without any change in the parameters, as shown in Figure 9.

Figure 8a shows the rotation mode, where the rotating wave is observed in the module order 0, 2, and 1. The phase difference between neighboring oscillators is approximately 2π/3. Figure 8b shows the partial in-phase mode, where the volumes of two modules (1 and 2) are in phase, and the other module (0) is out of phase with the first two modules. Figure 8c shows the out-of-phase and half-period mode, where the volumes of two modules (1 and 2) are out of phase, and module 0 oscillates twice while the first two oscillate once. Figure 8d shows the intra-oscillation mode, where the volumes of all themodules barely oscillate. In this oscillation mode, two phase oscillators are out of phase inside each module, which does not require a volume-change oscillation. A schematic diagram of the phase relationships between the three oscillators is given at the upper right corner of each plot. The relationships between two modules are indicated by → for phase shift, ↔ for out of phase, and = for in phase.

We set the initial value of the area of each module as 1.0. The parameters of the model are as follows: ω_{v} = 1.0; ω_{h} = 1.004; ϕ_{0,v} (*t* = 0) = 0.0; ϕ_{0,h} (*t* = 0) = 0.01; ϕ_{1,v} (*t* = 0) = 3.14; ϕ_{1,h} (*t* = 0) = 3.15; ϕ_{2,v} (*t* = 0) = 3.16; ϕ_{2,h} (*t* = 0) = 3.17; *D*_{i,j} = 1.0; The remaining parameters are the same as in Section 3.1.

## 4 Discussion

One of the most significant features of this model is that two physically interacting phase oscillators are embedded to control one module, which allows the module to induce the intra-oscillation mode. The intra-oscillation mode is not possible with a single phase oscillator. Because of this module design, competitive pushing of the protoplasm can be induced between the *intra*-module and *inter*-module interactions.

Based on the module design, there are two possible factors that induce a transition between the oscillatory patterns: a small difference between ω_{v} and ω_{h}, and the magnitude of σ. The small differencebetween ω_{v} and ω_{h} is plausible from the perspective of the biological variability of the true slime mold (such as the spatiotemporal thickness variability of the true slime mold). In addition, the difference is thought to prevent the convergence to the intra-oscillation mode when more than two modules are connected by the tube. The magnitude of σ is another factor because the phase oscillators only interact physically through the protoplasm, and this parameter specifies the strength of the phase modification.

To investigate this, we examine the transitions between the oscillatory modes in relation to σ and the difference between ω_{v} and ω_{h}. The results are presented in Figure 10. When ω_{v} = ω_{h} (i.e., ω_{h}/ω_{v} − 1 = 0), a transition between the oscillation modes can be confirmed. However, the oscillation eventually converges to the intra-oscillation mode. When there is a slight difference between ω_{v} and ω_{h}, various transitions can be confirmed: oscillation that eventually converges to the rotation mode, continuous transitions between the rotation and intra-oscillation modes, and continuous transitions between all the modes.

## 5 Conclusions

We have presented a mathematical model of a robot that exhibits versatile oscillatory patterns and spontaneous transitions between them without changing any parameters. The two main results of this work are as follows. The first concerns the emphasis on the physical interaction between body parts stemming from protoplasmic streaming, instead of designing complex diffusion interactions between nonlinear coupled oscillators, to generate behavioral diversity. The second is related to the design scheme of the phase modification originating from the mechanosensory information, which is based on the discrepancy function. The simulation results clearly indicate that the synergistic effect of the phase modification and the morphological communication provided by the protoplasmic streaming successfully generates versatile oscillatory patterns and spontaneous transitions between them without relying on complex oscillators such as chaos oscillators.

Our future work will focus on investigating the essential mechanism that induces such versatile behavior and on implementing this mechanism in a real physical robot (e.g., the robot we developed in [10]) to generate locomotion. To investigate the essential mechanism of the behavioral diversity, we will analyze this model mathematically from the perspective of dynamical system theory. In addition to this, we will implement such a mechanism in a real physical robot constructed with more than three modules, as in [5, 6], which will endow the robot with the capability of spontaneously switching between versatile behaviors (e.g., taxiing, exploratory, and escape behaviors), depending on the situation encountered, without increasing the complexity of the control system.

## Acknowledgments

The authors are deeply indebted to Ryo Kobayashi, Professor of Mathematical and Life Sciences at Hiroshima University, for his valuable advice and considerable effort in the development of the robot. This research is partially supported by a Grant-in-Aid for Challenging Exploratory Research (No. 23656171) and the Tateishi Science and Technology Foundation (No. 2021005).

## Notes

The vessel-like tube filled with protoplasm is left behind after the front part proceeds. Additional detailed information about the plasmodium of the true slime mold can be found in [1, 3, 4, 8, 9].

In this model, the volume conservation is the essential characteristic, whether it is of a gas or a liquid. To simplify this, we use the ideal gas equation. The potential can be calculated from the work involved in the volume change as follows: *W* = −∫ *p*_{i}*dV*_{i} + ∫ *p*_{ex}*dV*_{i}.

We assume that the variation in the resting length is slow enough to neglect the inertial force.

The names of the oscillatory patterns in (a), (b), and (c) are based on Takamatsu's work [5–7], although out of phase has been used for antiphase.

## References

## Author notes

Contact author.

Department of Mathematical and Life Sciences, Hiroshima University, 1-3-1 Kagamiyama, Higashi Hiroshima 739-8526, Japan. E-mail: takuya.umedachi@gmail.com (T.U.); kentaro@hiroshima-u.ac.jp (K.I.)

Japan Society for the Promotion of Science, Sumitomo-Ichibancho Chiyoda-ku, Tokyo 102-8472, Japan.

Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aobaku, Sendai 980-8577, Japan. E-mail: idei@riec.tohoku.ac.jp (R.I.); ishiguro@riec.tohoku.ac.jp (A.I.)

CREST, Japan Science and Technology Agency, 4-1-8, Honcho, Kawaguchi-shi, Saitama 332-0012, Japan.