Second-order agent-based models of emergent behaviour of Dictyostelium discoideum and their inspiration for swarm robotics

Some mathematical models study the slug migration phase of D. discoideum [5, 31] by underlining the significance of the mechanical interactions between cells. These works examine how wave generation and cell migration together result in slug migration and slug form variations. They present a mixture of the hydrodynamic phenomenon of cell flows and a fixed pattern based on the concentration of cAMP, which is propagated from the tip of the slug.

In most recent works using cellular automata or cellular Potts approach a cell is described by several smaller automatons [11, 13]. The authors consider the cell boundaries continuous, and study the cell’s size and cell’s shape. These models do not consider the physical forces of cells and neighbours’ interactions. For instance, Mare et al. [13, 15] analyse the slug tactic behaviour in a 2D lattice, in the absence and in the presence of a temperature gradient. They succeed to reproduce the same behaviour using cellular automata for modeling deformable individual cells and partial differential equation for cAMP simulation. The motive force plays an important role in the combination of cAMP concentration and cell adhesion. The tip oscillatory amoebae and the follower cells only transmit the signals.

Palsson et al. [17, 18] introduce a biologically realistic model for slug’s movement that considers: cell’s shape as deformable viscoelastic ellipsoids, the cells active forces. In this off-lattice model, cells communicate via surface molecules, and they measure and react to different chemotactic signals. The model considers both internal and external forces. A 3D model expresses the association of cell–cell adhesion and cell-signalling with collective cell movements. The movement of the cell depends on the cell’s internal state and external parameters, such as the neighbouring cells and chemical signals.

Kilobots [25, 26], with a diameter of 3 cm, were developed at Harvard University’s Self-organising Systems Research Lab. They serve the need for actual experimentation of collective algorithms in self-organising systems.

The Swarmanoid [8] project developed a swarm robotic system made of many robots of different types. The system uses different coordination mechanisms to provide complicated assignments in three-dimensional environments. Examples of such assignments include finding and retrieving a book on a shelf. Such a system possesses different robotic components with diverse abilities and different actuating modalities. The system is made from eye-bots, equipped with vision and autonomous flight capabilities, foot-bots, moving around on the floor, and hand-bots able to seize or grip objects.

Mathews et al. [16] introduce a new concept of robots with mergeable nervous systems (MNS robots). When robots physically dock, they share a joint architecture for sensing the environment and for the decision-making process, called the robot’s nervous systems. In such an order, one robot will act as a brain unit to make the decisions.

Valentini et al. [29, 30] propose an iterative strategy to tackle decision problems in a swarm of robots. Each robot examines all available choices, measures the quality of each option. Then, the robot makes an autonomous decision and broadcasts the decision to its neighbours. The proposed strategy is implemented through a swarm of 100 Kilobots, involving the motors, the light sensor, and the infrared system of the Kilobots.

Research in swarm-robotics investigates various organisations, from centralised or hierarchic, to decentralised ones, with homogeneous or heterogeneous robots, aiming at various activities (solving specific tasks or decision problems). We are interested in the full process from identifying first- and second-order emergent behaviour in natural systems and how to translate them into artificial ones, e.g. swarm robotics.

Phototaxis [4, 9, 14]: Moving toward the light sources—phototactic behaviour. In our desired model, individual amoebae are not able to measure the direction from which the light comes, and differences in light intensity do not lead to differentiation in motion velocity. Nevertheless, the whole slug orientates itself toward the light. The tip of the slug, formed by pre-stalk cells, leads the slug toward light.

Thermotaxis [4, 9]: Slugs move toward the heat as a result of positive thermotaxis. D. discoideum slugs exhibit negative and positive thermotaxis, both of which promote migration of the slug to the soil surface for optimal spore dispersal [9]. Basically, in daylight time, slugs move upwards to the surface, towards higher temperatures. Similarly to phototaxis, thermotaxis is a capability of the slug, but not of the individual amoeba.

Slug’s locomotion, interactions, and coordination [9]: In the absence of external thermal or chemical gradients, slugs migrate uniformly in random directions to find an appropriate place for culmination. Additionally, depending on their size, they can merge or split.

Uniform distribution by ammonia as a gas gradient effect [4]: In the wild, ammonia (as a gas gradient) is implicated in different phases of D. discoideum life cycle [4], in particular, new centre prevention in the aggregation phase and increased slug migration in slug formation phase. Slugs orient themselves away from high concentrations of ammonia to bypass the starved cell crowd or other slugs [7]. This negative chemotaxis to ammonia is considerable when two slugs are very close to each other. They migrate away from each other even though they still orient themselves to the light source gradient. Thus, they follow the resultant of two gradients. It can be found using the simple parallelogram method.

Slug’s speed and size [3]: Each slug consists of an anterior (tip) and posterior part. The one which is shorter and fatter will move slower than the one which is thinner and longer.

During the transition phase, Fig. 2 (green part), a PST cell moves toward the centre of the slug if it finds itself surrounded by PSP cells. The aggregation centre (pacemaker) takes on a PST cell behaviour, changes behaviour, becomes sensitive to light (red part of Fig. 2) and leads the slug. The model of the slug follows a leader-follower pattern, where the cells at the tip of the slug act as leaders, the following cells follow the cells in front of them. In our model, a slug consists of five blocks; one red block (itself composed of several cells) act as supreme-leader and four blue blocks act as the followers, and also temporary leaders. The supreme-leader cells release cAMP to guide the follower cells (red part). Thus, the cAMP concentration will be displayed as a gradient inside the slug; its lowest concentration at the posterior part and highest at the anterior part.

The slug size plays a central role in the development process of D. discoideum. What happens when the slug is too small or the anterior part cannot lead the rest of the slug properly? Under these circumstances, small slugs need to merge to produce larger slugs or unite with bigger slugs. To that end, the cells in the anterior part monitor the total number of cells composing the slug, which is possible by the ability of PST cells to perceive and also respond to their micro-environment. To achieve this purpose, the cells in the anterior part use a chemical signal (released into the extracellular space) to estimate the whole population by measuring only PST cell number (since they are 20% of all population [12]). If the concentration is above the threshold, they will activate the merging process (blue part of Fig. 2).

All cells receive information from their neighbouring cells, their environment, and other slugs’ cells. We modeled a local signal “merge-flag” which serves specifically for small slugs to signal themselves in the environment. Individual PSP cells follow some simple rules to achieve a complex and coordinated movement. They will follow the internal signal provided by PST cells (phototaxis) unless they receive a “merge-flag” from another slug (which has priority over the PST cells signalling). In that case, they follow that signal and merge with the other slug. To achieve such a collective movement, the cells need to choose one direction over the other all together as a group.

Cells use the majority voting process to have an influential decision-making process of the direction. Our method is single-choice voting (simple plurality) to accomplish the collective directional change. Each voter (a cell inside a block) is allowed to vote for only one direction. Hence, the direction which pulls the most among their counterparts is chosen.

To reach a uniform distribution of the slugs, we use the ammonia effect, modeled as a repulsive gradient generated by all cells of a given slugs. The ammonia increases the slug migration and helps the slugs move away from each other using a negative chemotaxis [7]. In our model, we assume that every cell inside a slug can produce ammonia, but only PST cells can check its concentration.

Most of the slug properties are controlled by the tip. Regardless of the slug’s age, each cell contributes to the movement, with the tip giving the direction and the signal for movement. To model the slug movement, we used the formulated equation of the slug movement of Innouye’s work [10] and the results of Smith’s work [27]. There are some assumptions of slug movement in Innouye’s work [10]: Also, in Innouye’s work [10], we can find an equation between the slug’s length, width, and velocity. They have extracted Eq. (1) by making a multiple regression analysis on the data of 27 different slugs.where, v denotes the migrating velocity and L, \(\omega\) for length and width of the slug respectively. In this equation, the coefficient for the width of the slug is so small so that it can be regarded as zero. Thus, we can conclude the Eq. (2) based on the length of the slug:As shown in Fig. 2 (green part), in the beginning, each cell can be in one of the three cell types: PST (pre-stalk), PSP (pre-spore), or Pacemaker. If it is a PSP cell, it will just follow the tip cAMP signal. However, if it is a PST cell, it will follow the whole flowchart. The PST cells notice when they become a slug (identification of emergent property), i.e., when they realise that they are surrounded by the same type of cells (they are surrounded by other PST cells). Additionally, if they are pacemaker cells, they act as PST cells, with the ability to release cAMP signals spontaneously. In this model, we have two quorum sensing phenomena, which are indicated by red circles (one and two), used for the PST cells to collectively decide on which direction to follow. Additionally, we use a diffusion-based approach based on a chemical signal (DIF) that helps PST cells inside the anterior part to measure the population size, informing in this manner the cell about the slug length, its motive force and the velocity to adopt (Eq. 2). This is used both for slugs’ merging and during phototaxis.

Figure 10 shows the transition phase, i.e. after the aggregation but before slugs are formed. Figure 10a shows five groups of four Kilobots, representing five aggregation of cells, leading to the corresponding five slugs. Figure 10b shows the Kilobots progressively organising themselves as a chain with a robot in front standing for the tip and the others following it. Progressively, the various slugs are formed and start moving (Fig. 10c, d). To achieve this behaviour, the Kilobot acting as the tip follows the light, while the following robot reproduces the behaviour of the Kilobot just in front of it. Each one of them acts as a leader for the next one in line, and sends its movement to its follower to reproduce.

A robot is able to operate in two states, move and pause. Imagining the form of a chain, a robot in its pause state, it waits until both Kilobots in front and behind have settled and transferred to the pause state. Next, the Kilobot changes to the moving state. The Kilobots, in the moving state, proceed toward the direction given by the acquired strength message from their leaders [1]. Each Kilobot transfers again to the pause state when the distance to its relative leader is less than one centimeter. The leader-follower approach is inspired by Rubenstein et al. work [24] and we provided it as a design pattern for decentralised systems [22].

In real D. discoideum, cells have the capability of pulling and pushing each other, which is not possible with Kilobots. We define the Kilobot gradient, which has the same effect of cAMP molecules among the original cell. We choose a robot as the head of the slug. This robot is responsible for finding a good direction and for leading the whole slug. This robot acts as the root of the gradient. We call the head Kilobot as the “source” or “supreme-leader”. The gradient for the source is zero. The nearest robot after the source has the gradient one and the second nearest robot is two, and so on. In each time-step, the robots send the signals and share the value of their gradients with each other as well. Therefore, the robots can replace their positions in the chain. In this way, the Kilobots recognise their leader and follower at each time-step. Figure 11, shows the case of a slug made of one robot as the leader and three robots as the followers.

In this section, we show a more complex behaviour involving two slugs and implement it with Kilobots, in two different scenarios Fig. 12): The left column is dedicated to the first experiment, and the right column is dedicated to the second experiment. Slugs are labeled by green and red color. Each slug has one leader with ID zero and three followers without any given ID (initially). The leaders always move straight. We do not consider yet robot merging. Figure 12 (\(a_1\)), (\(b_1\)) show the beginning and the end of the experiment, respectively. In Fig. 12 (\(a_1\)), the leaders start to move, and followers follow them one by one in each time step, with a dynamic, new ID. So, at the end of this experiment, Fig. 12 (\(b_1\)), we see the green slug making a turn at some point, and the two slugs are far from each other. Figure 12 (\(c_1\)) shows the trace of each robot. In the second experiment on the right side of Fig. 12, we consider the slug merging process. Again, Fig. 12 (\(a_2\)) shows the first frame of the experiment before the slugs move. Figure 12 (\(b_2\)) shows the last frame of the experiment when two slugs approach each other and merge. Two slugs consolidate and make one bigger slug with only one leader. The leader of the green slug became a normal entity with a random ID. A dead Kilobot in the experiment stopped for an unknown reason. Figure 12 (\(c_2\)) illustrates the track of the individual Kilobots with different colors.

Figure 13 shows the case of three slugs moving towards a light source positioned on the right and slightly at the back of the three slugs. For each slug, the robot at the front moves towards the light (Fig. 13a–c). The three other robots, forming the slug, follow the robot in front of them, according to the same schema as detailed above in Sect. 8.1. We see progressively how the three robots at the head of the slugs turn to the right in the direction of the light (Fig. 13d, e). The robots at the front use an environmental sensor to detect and move towards the light.

Figure 14 shows a similar case of phototaxis, this time with an additional behaviour integrated in the robots at the front. Sensing their neighbours, the robots at the head move away from each other, leading the whole slug with them. This is particularly visible in third frame (Fig. 14c), where the slug at the back approaches the slug in the middle. In the next two frames (Fig. 14d, e), we see the robots at the head of these two slugs moving away from each other: one towards the left (slug at the back) and the other towards the right (slug in the middle), de facto avoiding each other and leading the two slugs away from each other. To achieve repulsion, leaders broadcast a gradient. When sensing the gradient of another leader, the robot at the head of the slug makes the opposite movement it just took (go to the left if it had made a move to the right).