Many social or gregarious living organisms are effective decision-makers, in the sense that they are able to select the best of several available options [1]–[9]. Extensive experimental work and mathematical modelling suggest that a basic feature underlying this phenomenon is a symmetry-breaking bifurcation. That is, there is a transition from a “homogeneous” exploitation of the resources (all options are equally exploited) to an “inhomogeneous” mode where a focus on a particular option is occurring after a certain critical value of a parameter, typically the number of individuals [3], [10]–[13]. A key factor in the emergence of such patterns of exploitation is the amplification of an initial asymmetry arising through a fluctuation. For example in social insects, an individual discovering a food source will produce a signal that will be followed and reinforced by recruited individuals [14]. If the number of individuals is large enough, a slight initial imbalance of the fraction of individuals visiting one or the other source will entrain the majority of foragers to focus on a particular food source resulting in a collective decision. Such collective decision-making has been seen in predator avoidance [15], shelter selection [16] and has even been interpreted in terms of rationality [17], [18]. The idea of a symmetry-breaking depending on the number of individuals have also inspired other fields of research focusing on human behaviour [19]–[21] or economics [22], [23]. In all these examples, symmetry is broken when a critical number of individuals is exceeded.

While symmetry breaking is important, we also know that symmetry can be restored when the system size (e.g. number of individuals) becomes very large. For example, direct contacts resulting from crowding in foraging ants lead to the exploitation of two routes to food, despite the fact that only one route is chosen when there is no crowding [24]. More intricate situations can arise in, for example, ant species using two pheromones [25] or in social caterpillars Malacosoma disstria displaying behavioural polymorphism [26], [27]. Here exploitation patterns are shown to arise in which past a first symmetry-breaking transition there is coexistence of inhomogeneous and homogeneous modes the latter becoming even the rule under certain conditions. The interplay between symmetry breaking and symmetry restoring is also a basic issue in statistical and condensed matter physics [28], [29] and in high energy physics [30] when more than two phases of matter can coexist.

In this paper, we analyse decision-making at the cellular level, on the paradigmatic case of the true slime mold Physarum polycephalum. We show that non-trivial decision patterns, including a symmetry restoring bifurcation may arise depending on the mass of the slime mold.

P. polycephalum is a unicellular, multinucleate protist. Its vegetative phase is a multi-nucleate plasmodium. It is during this stage that the organism searches for food. Depending on the strains of the organism considered, the plasmodium sets out pseudopodia in all directions for a certain distance and then builds one or few extended search fronts (Fig. 1) during exploration. The plasmodium is able to sense various stimuli from a distance and move toward them via chemotaxis [31]. When the plasmodium comes into contact with a food source, it completely surrounds it and resumes exploration while remaining in physical contact with the initial food source. The plasmodium can grow to cover large area (up to 900 cm), and is capable of moving at relative high speed (up to 5 cm/hr) [32] and of building efficient transportation networks [33]. In most of these studies reported in the literature, a single strain was used to reveal the decision-making patterns of P. polycephalum. In this paper we develop a model based on [34] and [35], along with experiments carried out on three strains of the true slime mould P. polycephalum to test our predictions and reveal the differences between these three strains in decision-making outcome.

The model describes how commitment to two identical options evolves in time. We let and be the number of units within the system committed to options 1 and 2, respectively. We further assume a pool of uncommitted units of size where is the system size. We express the build up of commitment to the options with respect to time as (1)

Here is the rate per individual unit time to choose between one of the options, accounts for the feedbacks present in the decision-making and is the rate at which commitment decays.

A number of authors [1], [13], [24] have analysed a similar model – particularly in the context of foraging in ants – under the hypothesis that rate of decision-making is constant, i.e. is replaced by a constant . This hypothesis is reasonable in the limit where the initial system size is large and the number of units committed to the options remains small. However, in many natural systems the initial mass is significantly depleted as time goes on. This is certainly the case in our current experiment on foraging by P. polycephalum where a substantial part of the initial mass ends up covering one or both the food sources. The system we study here can thus be viewed as having a “passive” negative feedback of , , whereby depletion of units reduces the rate of recruitment.

We turn next to the positive feedback functions . Several possible forms have been proposed for these (see [35] and [36] for recent reviews). One of the dominant ideas has been that an individual bases its decision on previous decisions made by others, i.e., on the numbers of units having already committed the different options. For example, [13] use (2a)while, [37] argue, on the basis of Bayesian estimation, that the form (2b)gives a form of optimal decision-making, being a sensitivity parameter.

Both the above forms assume that information about commitment to both of the options is available to the decision-making units. An alternative view is the quorum model [34]. Here one assumes that the probability of accepting an option is simply an increasing function of the number of units that have already accepted this particular option independent of the number of units choosing the other option. In this paper we model this phenomenon using (2c)

This form follows [34], albeit without an additional spontaneous probability of adopting an option. A similar form has also been derived by [36] within a Bayesian framework. They found that (2d)provided a good match to decisions made by zebrafish. As it turns out the use of either (2c) or (2d) is not essential in what follows. Both these functions have the same sigmoidal form, which produces the sequence of bifurcations we now describe.

In the case of P. polycephalum feedback is in the form of the growth of tubes as a result of protoplasmic flow. There is evidence that there is an upper limit of tube thickness in real organisms [38]. The parameter then stands for the threshold beyond which this feedback becomes effective or, alternatively the threshold flow for tube construction. Nakagaki et al. [39] consider the sigmoidal function of the form to account for these effects but, again, the results reported below are not affected qualitatively by the choice of exponents greater than 2.

Summarising, model (1) for two equal food sources can be written as(3)

It captures two essential properties of a class of decision-making systems of which Physarum polycephalum constitutes a prototypical example. First, decisions are local in the sense that each of the two positive feedback functions depends only on the fraction of system's mass attracted to the particular option . In particular, in P. polycephalum tubes are being built to food sources on the basis of only local information. Second, for any given value of initial mass the portion of the system not yet committed to the options is decreasing as , are increasing.

In order to investigate the role of randomness in the model and to fit it to data, we also implemented a Monte Carlo version of this model. See Materials and Methods for details.

Fig. 2 shows the bifurcation diagram of the steady-state solutions of eqs. (3), i.e., how the steady-sate level of commitment to an option changes for initial system sizes. Three bifurcation points can be identified. Before the first bifurcation there is one stable steady state corresponding to no decision (trivial steady state). After the first bifurcation point (see Material and Methods, eq. (6)) the system has three stable states, one corresponding to no decision and the other two corresponding to the exclusive exploitation of one or the other of the two options (semi-trivial steady state). In terms of the behavior of Physarum polycephalum, the trivial steady state describes a situation where the plasmodium did not find food or never moved from the starting point. The semi-trivial steady state describes the situation where the plasmodium exploits just one option.

For larger initial mass values a second bifurcation occurs and unstable homogeneous solutions appear. In terms of the decision-making of Physarum polycephalum the instability of these symmetric solutions means that the plasmodium does not have enough mass to exploit two options at the same time and thus moves to just one. After a critical value (see Materials and Methods, eq. (10)), corresponding to a third bifurcation, the upper branch of the homogeneous solutions becomes stable. This corresponds to the plasmodium equally exploiting both food sources. We label the bifurcation at a symmetry restoring bifurcation, since a stable, nontrivial symmetric solution appears at this point.

This stabilisation coincides with the appearance of two non-homogeneous (asymmetric) unstable solutions, characteristic of a subcritical pitchfork bifurcation. Here we have tristability such that, depending on initial conditions, the plasmodium will exploit either none of the options, one of the two or both. The asymptotic analysis of these solutions for shows that the distance between the inhomogeneous solutions and the stable upper branch of the semi-trivial steady state decreases as increases. This means that for large mass these two solutions are approximately equal. As a result, the stable upper branch of the semi-trivial solution(see Material and Methods, eq. (5)) can never be reached in the sense that the set of initial conditions in its attraction basin decreases in size with . The biological conclusion is that a plasmodium of very large mass nearly always spreads between two options rather than moving to one.

We now study the role of the threshold and flux parameters and . Fig. 3 depicts critical values of parameter as function of parameter for the fixed mass . The three lines correspond to the three types of bifurcations identified above. The bold solid line corresponds to the condition of the first bifurcation to occur and thus, to the existence of semi trivial solutions (see eq.(6) in Materials and Methods). The solid line corresponds to the condition of the second bifurcation to occur and to existence of a non-trivial unstable homogeneous solution (see eq. (8) in Material and Methods) Finally, if parameters and are chosen under the dashed line in Fig. 3 the existence of all types solutions and all bifurcation points is secured.

We next turn to the experimental results. Fig. 4a,b,c shows the probability to move to a food source as a function of the size of plasmodium. For very small size, there is a non-negligible probability to select none of the food sources. Plasmodia of small masses exploit more often only one source, while larger ones exploit both food sources at the same time. For example, the smallest Japanese plasmodia ( cm) exploit only one food in 95 of the cases while for the largest size ( cm), this frequency decreases to 54 (see Fig. 4a). Similar results are observed for the other strains (see Fig. 4b, c). We notice however that there are some quantitative differences of exploitation patterns between strains: The largest Australian plasmodia ( cm) exploit two food sources in 75 of the cases, a value which is larger than for the two other strains. Decision making by Physarum polycephalum depends thus on the size of the plasmodium as well as on the different exploration patterns of the strains.

The experimental results are qualitatively consistent with the model predictions. Indeed, for small values of the parameter , there is no option chosen. For larger , there is coexistence between a state where one option is chosen and a state where no option is selected. Finally, for still larger , there is coexistence between three states corresponding to the selection of one option, to the simultaneous selection of two options and no selection at all. In terms of Physarum polycephalum, an organism with a small mass exploits one or no options, while a large mass endows it with the possibility to select simultaneously two options, one option or none.

In order to compare the predictions of the model to the experimental outcome, we identified the best fit model in terms of the parameters and for each strain. For each mass used in the experiment we performed a Monte Carlo simulation of the model (Materials and Methods) for different parameter combinations. We run the Monte Carlo simulation 1000 times for each pair of ranging with the step 0.1 from 0.5 to 3.5 and ranging from 0.5 to 2, and identified the best fit parameters (see eq. (12) in Materials and Methods for details of model fitting).

The best-fit parameters identified for each strain of plasmodium are given in Table 1, along with the goodness of fit parameter (see Model fitting in Materials and Methods section). The fitting parameter can be roughly interpreted as the proportion of data explained by our Monte Carlo simulation model. It varies for each strain between 0.84 and 0.92, indicating that the simulation model accounts for the large majority of observed variation, supporting the validity of the inferred values of and .

Fig. 4d,e,f shows the probability of selecting an option as a function of the mass, resulting from an average of 1000 realisations for every value of the mass considered and from the best fit parameters shown in Table 1. This is to be compared with the experimental probabilities (Fig. 4a,b,c). The model captures adequately the different patterns of exploitation for the masses and the strains considered in the experiment.

Fig. 5 shows the bifurcation diagrams corresponding to the best-fit parameters for each of the three strains. We now identify the positions of the symmetry-restoring bifurcation point beyond which a simultaneous exploitation of the two options becomes possible. We notice that the critical value of the mass is different for the three strains, the Japanese one occurring at (cf. Fig. 5a) while the Australian and American ones occur at smaller values ( and respectively, Fig. 5b,c).

These differences can be explained in biological terms and the exploration patterns of the slime mold (Fig. 1). The exploratory pattern of the Japanese strain is directional, forming thick tubes during its displacement. A larger mass is then needed to be able to exploit two options. In contrast, the Australian strain explores its environment more uniformly by forming thin tubes. A smaller mass is then needed to be able to exploit two options. As for the American strain, its exploration pattern combines both Japanese and Australian ones and an intermediate value of the mass is then needed.

These exploration pattern differences are taken into account by the differences between two parameters that we used in our model. can be viewed as the speed of displacement of the plasmodium while reflects a threshold beyond which a tube can be built, and therefore is related to the way the different strains are moving: a small value of means that a tube is more easily constructed, even with a low mass. The function in eq. (4a) saturates therefore more quickly and favours the homogeneous solution. On the contrary, a large value of this parameter implies that a large mass will be needed to build a tube and that saturates more slowly, favouring the semi-trivial inhomogeneous solution (6).

We have presented a generic mathematical model for how different patterns of exploitation of two identical resources depend on the size of the system. The model takes into account two important features. Firstly, owing to the finite size of the system, the number of uncommitted units is limited by that already committed to the feeding sites. Secondly, the amplification process is local in that no direct comparison is made between the two options. The combination between local positive feedback and regulation of the traffic revealed a symmetry restoring bifurcation beyond which the system was able to select simultaneously two options, one of two options, or none of them. Past this bifurcation point, for increasingly larger initial system sizes, this tristability was still present but the symmetric solutions had an increasing basin of attraction. This was due to the existence of nearby unstable inhomogeneous states masking the other stable states.

Most of the studies investigating decision-making patterns in Physarum polycephalum were conducted using a single strain (the Australian strain obtained from Southern Biological Supplies: [40]–[42]; the Japanese strain: [43]). In order to test the model, we conducted experiments on three strains of Physarum polycephalum, each of them having different pattern of exploration. In our experimental set-up we took single individuals of different masses and let them choose between two identical food sources on a Petri dish. The different types of exploitation patterns obtained were similar to those predicted by the model, with the model capturing around 90% of the data.

Symmetry-restoring is a generic phenomenon resulting from the coexistence of positive and negative (regulatory) feedbacks. In addition to the case considered in this work, it is also encountered in social insect foraging [24], [25]. Beyond the case of decision-making in biological organisms, symmetry restoring is known to be also present in physical sciences, including phase transitions [29] and pattern formation in reaction-diffusion systems [44].

Our study highlights an important difference between local and global information in decision making. In slime mould, flow is a function of the thickness of the tube between the organism and a specific food source [38], [39]. As a result, tube growth is a local process in the sense that tubes oriented along different directions are not inhibiting growth. In many experiments on ant and fish decision-making there is a predetermined decision point, at for example a Y-shaped branch [3], [15], where animals compare the two options directly. This choice point provides global information. It would be interesting to investigate situations in ants and other social organisms in a natural environment where groups are still offered two options (two food sources) but there is no pre-determined choice point. A setup of this kind for ants could consist of colonies of variable sizes connected to an open arena containing two identical food sources placed equidistant to the nest. The traffic that will eventually be established will still privilege paths leading to the food sources, but the information held by individuals will be purely local. In these conditions, we predict that beyond a critical size of the colony, individuals will display the three exploitation patterns seen in this paper. In particular, we predict a symmetry restoration at large colony sizes. Notice that symmetry restoring should be possible even in a maze type experiment provided that returns to the main branch of the maze can occur. A full analysis of this problem would require to incorporate in the description the navigation strategies employed. This is beyond the scope of the present work.

The coexistence of multiple steady states in our model is expected to enhance flexibility. In nature, food sources are not constantly available and colonies focussing on one source can take a long time to switch to another [45]. However, in the region of coexistence between many solutions, a colony may quickly switch to another option [35]. Previously this was shown to be the case in the presence of crowding [24] or in the presence of more than two options [18]. We suggest that this may also happen in an open environment in the presence of only two options and a regulation of traffic of the kind considered in this paper.

We studied the model presented in the Introduction in two ways, as a system of coupled differential equations as defined by eqs. (3) and via a Monte Carlo simulation.