The role of noise in self-organized decision making by the true slime mold Physarum polycephalum

Self-organized mechanisms are frequently encountered in nature and known to achieve flexible, adaptive control and decision-making. Noise plays a crucial role in such systems: It can enable a self-organized system to reliably adapt to short-term changes in the environment while maintaining a generally stable behavior. This is fundamental in biological systems because they must strike a delicate balance between stable and flexible behavior. In the present paper we analyse the role of noise in the decision-making of the true slime mold Physarum polycephalum, an important model species for the investigation of computational abilities in simple organisms. We propose a simple biological experiment to investigate the reaction of P. polycephalum to time-variant risk factors and present a stochastic extension of an established mathematical model for P. polycephalum to analyze this experiment. It predicts that—due to the mechanism of stochastic resonance—noise can enable P. polycephalum to correctly assess time-variant risk factors, while the corresponding noise-free system fails to do so. Beyond the study of P. polycephalum we demonstrate that the influence of noise on self-organized decision-making is not tied to a specific organism. Rather it is a general property of the underlying process dynamics, which appears to be universal across a wide range of systems. Our study thus provides further evidence that stochastic resonance is a fundamental component of the decision-making in self-organized macroscopic and microscopic groups and organisms.

Self-organization enables even simple organisms to solve surprisingly complex tasks, specifically optimization tasks essential for survival [1]. Prominent examples are ant colonies which optimize their foraging choices among multiple food patches [2] taking a variety of criteria into account [3] and slime molds, which optimize path choices even in complex mazes [4].

In the past, the self-organized behavior of such organisms has mostly been investigated in unchanging, static environments. While this seems a natural starting point for such investigations, dynamic settings are much more relevant to the behavior of organisms in the real world, where change is ubiquitous. This is why recently the focus of research has been shifting towards dynamic environments where the properties of the environment change over time. The question addressed is “can species x efficiently adapt its behavioral patterns to the environmental changes?”

Such a dynamic setting imposes additional burdens on a systematic investigation. First, the notion of optimality becomes even more slippery than it is in the static case already [5]. Second, the corresponding experimental set-ups are more complex, leading to an increase in the number of parameters governing numerical studies. Thus, theoretical research guiding the design of meaningful (and realistically feasible) experiments becomes very important. Here we present a theoretical study that analyzes fundamental properties of dynamic decision making by the true slime mold Physarum polycephalum, an important model species for the study of information processing in biological systems [6, 7]. Our study directly suggests foraging experiments that will allow us to verify these properties for the real system.

One of the recent advances in research into dynamic foraging was the finding that noise in the decision making process is one of the crucial factors enabling self-organized insect societies to adapt their foraging patterns to changes in the environment [8, 9]. This interesting and counter-intuitive result suggests that noise is not a disturbance in self-organized systems. On the contrary, noise serves an important functional role. The studies [8, 9] are based on experiments with mass foraging ant colonies, one of the prototypical model systems in the study of self-organization. Interestingly, the fact that noise enables adaptive decision making is not due to any specific physical details of the ant foraging mechanism. Instead, it arises from very general mathematical properties of the underlying self-organized processes [10]. This suggests that the same should also apply to other similar types of self-organized collective decision making in organisms such as slime molds and bees.

In the present paper we investigate the assessment of time-variant risk for the true slime mold P. polycephalum. We show that noise can enable P. polycephalum to correctly assess time-variant risk factors in dynamic environments and, as a consequence, to make near-optimal foraging choices. We extend a deterministic phenomenological model developed by Tero et al. [11] for the foraging behavior of P. polycephalum to explicitly capture effects of noise. Numerical and analytical investigation of the resulting stochastic model shows that a well-attuned level of noise can enable P. polycephalum to integrate variable risk factors correctly over time. This is not the case if there is little or no noise in the system. We suggest comparatively simple biological experiments that will allow us to test these predictions.

Our results hold interest beyond their immediate relevance for the study of P. polycephalum. They demonstrate with a concrete case that it is possible to transfer insights about collective behavior from one species (ants) to another species (slime molds). Ants and slime molds are physically entirely different systems. Yet, the self-organized mechanisms that govern their fundamental behavior are so similar that both species share essential behavioral characteristics.

Most importantly, the fundamental mathematical structure of the self-organized decision making mechanisms in these systems is similar to those in a broad variety of other organisms, such as bees [1] and bacteria [12], to those in human social decision making [13, 14], and even to those in bio-inspired engineering solutions, such as swarm robots [15]. In conjunction with earlier work on similar phenomena in ant colonies our study thus provides further evidence that noise may play a crucial role in the decision making of a broad range of self-organized systems beyond those investigated so far.

P. polycephalum is a slime mold that spends most of its life cycle as a plasmodium, a uni-cellular multinucleate amoeboid. The plasmodium is an aggregate of protoplasm with a network of tubular elements. The protoplasm is differentiated into two phases: a gel phase (ectoplasm) that makes up the walls of the tubular structures, and a sol phase (endoplasm) that flows within the tubes. The motion of the sol, so-called shuttle streaming, is driven by organized rhythmic contractions of the gel with a period of ca. two minutes. The sol serves as a circulation system for the cell transporting nutrients and chemical signals. The tubes act as pseudopodia and enable the organism to navigate around its environment [4, 16]. The organism can reconfigure the tube network within a few hours in response to changes in external conditions. As it moves over a surface, the plasmodium changes its shape and if food is placed at different points, it will put out tubes that connect these food sources [6].

It has been shown that P. polycephalum, despite its extremely simple morphology, is able to solve computational problems of surprising complexity. About a decade ago, a seminal experiment [4] demonstrated that P. polycephalum can solve the shortest path problem in mazes. Since then, these studies have been extended and it has been demonstrated that it can solve (or approximately solve) a variety of other network optimization problems [7, 17, 18] even when taking multiple objectives into account [19]. It has also been shown that P. polycephalum possesses a memory and is able to anticipate periodic events [20]. These capabilities in combination with its simple morphology and comparatively large size make P. polycephalum an almost ideal model system for the study of information processing and problem solving in biological systems.

The most fundamental demonstration of its computational abilities is the maze experiment [4]. In this experiment an agar surface is masked with plastic film, such that the accessible surface forms a maze. Food (oat flakes) is placed at the entrance and exit points of the maze, and pieces of the plasmodium are distributed in the maze. These pieces spread and coalesce to form a single plasmodium that fills the agar maze and avoids the dry surface of the plastic film. Subsequently the plasmodium shrinks and only leaves a single thick tube behind which traces the shortest path between the two food sources (see Fig 1).

It appears this is the result of the organism’s attempt to simultaneously optimize two different goals, namely to (1) maintain sufficient connectivity of the entire plasmodium in order to maintain chemical communication, and (2) to maximize food absorption [19]. The maximization of both goals causes almost all body mass to cover the two food sources while only a minimal connection between these areas is maintained.

While a full explanation of these abilities from first bio-physical principles is currently still beyond reach, Tero et al. have proposed a simple phenomenological model that describes the development of the tube network [11]. We use this model as the departure point for our analysis of risk evaluation by P. polycephalum.

In the Tero–Kobayashi model, the shape of the cell body is represented by a graph: the edges correspond to tubes and nodes correspond to the junctions between tubes. Fig 1(d) shows the example graph for the maze-solving experiment. The two nodes with food-sources are labeled N1 and N2 and the other nodes are numbered N3, N4, N5, …. Edges (tubes) between node i and j are labeled Mi,j. Suppose that the fluid (sol) pressure at nodes i and j is pi and pj, respectively, and that the tubes are idealized as cylinders of length Li,j and radius ri,j. Assuming a Poiseuille flow, the flux through the tube is Qi,j=8πr4(pi-pj)ηLi,j=Di,jLi,j(pi-pj)(1) where η is the viscosity of the fluid (sol), and Di,j=8πr4η is a measure of the conductivity of the tube. Although the tube walls are not rigid and the radius changes over time, the dynamics of tube adaptation are slow enough (10-20 minutes) for the flow to be taken as steady in time. The amount of fluid at internal nodes must be conserved, while the nodes that correspond to food sources drive the flow through the network by changing their volume, so that ∑jQi,j={0ifi≠1,2Siotherwise(2) with S1 + S2 = 0, because the total volume of fluid in the network is conserved. The source terms Si could be periodic in time and drive shuttle streaming through the network. However, because the time scale of network adaptation is an order of magnitude longer than the time scale of shuttle streaming, the sources are taken to be constant in the following.

In P. polycephalum the radii of the tubes change in response to the sol flux: while generally all tubes have a tendency to shrink, tubes with a large flow expand in response to the flow. Thus the evolution of tube conductivities can be modelled as: ∂Di,j∂t=f(|Qi,j|)-δDi,j(3) With initial conditions Di,j(t = 0) to be specified. This establishes a self-limiting feedback system in which positive feedback is counterbalanced by negative feedback (shorter and larger tubes attract more flow, which in turn expands the tubes, while longer and less used tubes shrink and thus attract even less flow). The evolution of the tubular network according to this model agrees well with biological experiments [11]. A biologically plausible choice for f is a sigmoidal function. To make our results comparable with earlier literature we choose the following form which has been used to analyse the Tero-Kobayashi model in previous work [11]: f[α,γ,μ](Q)=(γ+α)Qμγ+αQμ(4) While the forcing function f appears as a three-parametric function (parameters are α, γ and μ) of the dependent variable Q, it is only the linear combination γ/α which has an impact on the forcing. This symmetry is exploited by rewriting f as a two-parametric function reducing the number of parameters on which the system depends, by one: f[ϵ,μ](Q)=(1+ϵ)Qμϵ+Qμ;withϵ≡γ/α.(5) Hence, a specific value of ϵ incorporates all combinations of α and γ which yield this specific value.

Note that this is a saturating feedback function. While it complicates analytical investigation of the system, a saturating function is biologically more realistic. In the following we use Eq (4) with μ = 2 and ϵ = 0.2 unless stated otherwise. The choice of μ = 2 is motivated by comparability with previous work. The parameter ε is chosen such that the system is well within its tristable regime. Backed by the extensive analytical investigation of the system in Appendix A, we are confident that the dynamics of the system depend on the value of ε only in quantitative details, but not qualitatively.

To gain an understanding of the system dynamics, we consider the simplest possible decision network consisting of only two different paths between two food sources. Dropping subscripts i, j for nodes and simply numbering the two tubes as i ∈ {1, 2}, the system becomes ∂Di∂t=-δDi+fϵ(Di/LiD1/L1+D2/L2)(6) with initial conditions D1(t = 0) ≡ D1,0 and D2(t = 0) ≡ D2,0. In Appendix A we show that dynamics of this system critically depend on ϵ. For ϵ < 1/4 it can be shown that the system has two unstable and three stable equilibria (Fig 2). Two of the stable equilibria correspond to full convergence to a single tube (i.e. a single foraging path) (D1 = 0 or D2 = 0) and the third one corresponds to equal utilization of both paths D1 = D2 ≠ 0. As ϵ → 1/4 the basin of attraction for the third fixpoint vanishes and the fixpoint becomes unstable for ϵ < 1/4. Thus, depending on the parameter ϵ, the system is either a binary or a ternary decision model.

Our fundamental concern is whether self-organized decision making enables P. polycephalum to successfully react to dynamically changing environments. The maze experiment reviewed above investigates a static scenario. The question arises whether it can be adequately modified into a dynamic version. A commonly used set-up for dynamic foraging experiments is to change the type or location of food sources or the paths to these. However, as the body of P. polycephalum actually covers the food sources and the paths, this is difficult to achieve without disturbing the organism too much. The possibility of an alternative set-up arises as P. polycephalum exhibits phototaxis, being photophobic at some wavelengths in the visible range of EM radiation [21]. The organism experiences bright light as a “risk” factor and consequentially tends to withdraw its tubes in more brightly lit areas. In the model this can be captured by increasing the thinning factor δ. It has been shown experimentally that P. polycephalum is able to select paths in inhomogeneously lit fields such that the risk imposed by light and path lengths, respectively, are balanced [19]. This photophobic behavior of the organism suggests a different set-up for a dynamic foraging experiment: instead of changing the spatial arrangement we can change the lighting on different parts of the set-up, thus avoiding to disturb the organism in the process. Toxic light has also been used to induce a time variant risk in [22]. However, the focus of [22] was on spatial search rather than on the role of noise in a fundamental binary decision task, which we investigate here.

We propose the following dynamic foraging experiment to clarify how the self-organized organism can successfully cope with changing environments. P. polycephalum is presented with a minimal “maze” of only two alternative paths connecting two food sources. Both paths have the same length and are illuminated with light sources of the same calibrated luminosity. The reduction to only two possible paths is made to ensure that the mathematical model remains tractable. So far there is no reason for the organism to prefer either path. From experiments and the analysis of the basic model [11] we know that the organism will select one of the two paths depending on which one is initially set up with the thicker tube (note that we can regulate the initial tube thickness according to requirements by preceding the experiment with an initialization phase in which targeted lighting is used to thin selected tubes). Now consider the use of intermittent lighting instead of continuous lighting. Conceptually, if both paths are lit with the same luminosity, but with different light-dark periods, one of them should be selected preferentially, because it represent the lower total risk (integrated over time). Number the two paths 1 and 2 and let the light-to-dark ratio on these be ri=lidi, where li and di are the durations of the light and dark period on path i, respectively. If r1 < r2, the organism should select path 1 provided it assesses the time-variant risk factors correctly. Of course, the time-scale of the light-dark cycles must be significantly faster than the time scale of adaptation, as the organism could otherwise simply adapt to every period separately and would not need to integrate the time-variant signal.

An interesting question is whether we potentially need to account for directional bias of the organism. Recently, P. polycephalum was found to exhibit chirality [23] in the search phase of the foraging, i.e. a directional preference when the plasmodium is expanding. In contrast to this, our suggested experiment investigates the contraction phase of foraging, i.e. the shrinking of the tube network. Previous studies [16] have not found a bias in the contraction phase in very similar set-ups. However, we cannot categorically exclude that it may occur in some experimental settings. We thus suggest to account for this possibility in the following way. In a preliminary phase our experiment is conducted without lighting to check for any directional bias. If, contrary to our assumptions, bias is found we can proceed in two ways. Firstly, we can modify the experiment to use a different geometry that eliminates directional differences: Three food sources are arranged such that a central source is located in the middle between two other sources. The plasmodium is restricted in the usual way to use two paths from the central source to the outer sources. The paths are shaped such that together they form a symmetric ‘S’-shape. Since both paths have the same left-right curvatures, we would expect this to eliminate bias. Any bias that cannot be eliminated can easily be accommodated in the numerical model by using a different thinning factor δi for each branch (Eq (5)) and fitting these to the experiment. We emphasise that it is unlikely that this is necessary, since previous studies have not found bias in the contraction phase [16].

The proposed experiment corresponds to the following variation of the Tero-Kobayashi model restricted to two paths as given in Eq (6). As outlined above, a variation of lighting can be captured as a variation of δ. For intermittent lighting we introduce a forcing function Φ(t,Di)={-βiDiif(ωtmod2π)>bri0otherwise(7) where ω is the frequency of the forcing, bri is the length of the darkness period on path i, and βi is a measure of the intensity of the lighting on path i. The lighting function is shown in Fig 3A.

The different behaviours of a system that remains forever in one of the three forcing regimes are shown in Fig 3B, which also gives the corresponding equilibria. The filled squares at (1/0), (0, 1), and ∼(0.7, 0.7) are the stable equilibria of a system in which both branches are continuously dark. Thus these are the states that such a system would attain in the long-run. Equivalently, circles mark the equilibria for an unchanging system in which only a single path is lit. Stars mark the equilibria for an unchanging system in which both paths are continuously lit. We direct the reader’s attention to how the equilibrium point that corresponds to equal use of both paths in the unlit regime moves away from the lit path(s). The corresponding basins of attraction (delineated with red lines for the light/dark regime and blue lines for the light/light regime) shift together with the stable equilibrium from their position in the completely unlit regime (grey shading). For reference, Fig 3 also shows with hollow markers the corresponding unstable equilibria. These are, however, not relevant to the long-run behaviour.

In each of the forcing regimes, the lighting can be taken into account as a modification of the tube thinning δ. If forcings are on a significantly faster time scale than the relaxation time of the system (ω2π≫1δ), we may ignore the phase of the signal. Taking intensities into account, the time-averaged ratio of risks (ratio of forcings) is ρ=β1β2·2π-br12π-br2=⟨Φ1(t)⟩⟨Φ2(t)⟩(8) where the second path is preferred for ρ < 1 and the first one for ρ > 1.

As our aim is to analyze the role of noise in the assessment of time-variant risk, we introduce an explicit noise term ξi(t) into the model. Here ξi(t) is a Gaussian white noise process with an expected value of 〈ξi(t)〉 = 0, unit variance 〈ξi(t)ξi(t)〉 ≡ σ2 = 1, and uncorrelated in time. This is also in line with the experimental finding that tube thicknesses fluctuate randomly to some degree.

The full model for the two path experiment with intermittent lighting becomes ∂Di∂t={F(Di),ifDi>00,otherwise(9) F(Di)=-δDi+fϵ(Di/li∑j=12Dj/lj)+Φ(t,Di)+σξi(t)(10) where we propose additive noise representing random perturbation originating from external sources.

We use the parameters in Table 1 with l ≡ L1 = L2 and D1,0 ≡ D1(t = 0) and similarly for D2. The risk ratio ρ is not a parameter, but is calculated from br1, br2, β1, β2. With ω ≫ δ, this establishes a process in which the forcing frequency is significantly faster than the relaxation time, but not so fast that—in the corresponding biological experiment—time averaging would happen on the sensory level of the organism. This combination of parameters implies a week preference of path 1 since ρ = 5/6 ≈ 1 and we expect that it is hard for the system to decide correctly.

Numerical integration of the modified model (Eqs (9) and (10)) with parameters as listed in Table 1 confirms that a well-attuned level of noise allows the system to decide correctly in respect to the time-variant risk. As a one-dimensional measure for the correctness of the decision we define c=D1-D2D1+D2.(11) For these parameters and assuming time-averaged risks, Path 1 is preferable to Path 2, because ρ=0.250.6·2-12-3/2=56.(12) Note that this risk ratio corresponds well to the ones found in biological experiments [19]. Thus, if the system decides “correctly” we expect limt → ∞c → 1.

Disregarding forcing and noise, the system has three fixpoints (steady-states). Initialization determines which steady-state is attained. As we are interested in the question whether noise improves the decision behavior of the system, we initialize biased towards the wrong steady state with (D10=0.5,D20=1,c=-1/3). It turns out that the system decides correctly if a noisy process is assumed, while it will not be able to do so if noise is eliminated or reduced to a very low level.

The evolution of the noise-free deterministic process (Eqs (9) and (10)) for the parameters given above and σ2 = 0 is plotted in Fig 4A with a solid gray line. It is clearly visible that the process does not reach the correct decision c = 1, but instead assumes the third fixpoint (D1 ≈ D2 with c ≈ 0 but slightly shifted due to the forcing).

If noise is introduced into the system it does, however, decide correctly. To show this we compute 5000 sample paths of the stochastic process (Eqs (9) and (10)) with σ2 = 0.05 for over 2000 forcing cycles (200π) with 16 time steps per forcing cycle (Δt = 2π/(16ω) ≈ 0.04) using the method of Milstein forward integration [24]. Fig 4B shows the mean and standard deviation for this process (refer to the bold continuous and bold dotted lines). It is clearly visible that the vast majority of samples decides correctly (c → 0.76). A further inspection of individual sample paths (not shown) reveals that none of them develops towards the state of no decision D1 ≈ D2 such that 88% of the realizations end up on the path imposing lower risk. This is due to the fact that noise destabilizes this fixpoint of the deterministic system.