bifurcation diagram of the steady state solutions of eq. (1) as a function of $M$. Parameter values : see Zabzina et al.20141

How different slime mould strains make decisions.

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One of the best-known manifestations of collective behavior is decision-making, whereby a group composed of many individuals (an insect colony, a swarm of bees, a population of fish) chooses among different available options. In a recent paper [1] we investigated a new paradigm of decision-making, operating at the cellular level, which involves unicellular organisms like the plasmodia of Physarum polycephalum. Each plasmodium can be viewed as a network of nuclei communicating through the exchanges of mass, energy and information. It turned out these organisms are ideally suited for studying the flexibility and plasticity in decision-making as well as the emergence of cooperativity.

We investigated how individuals from three strains of Physarum polycephalum - the Australian, Japanese and American strains - exploit and explore food options in their environment. The first result is that there are significant differences in the 'behaviour' of these strains:

  • Individuals from an Australian strain slowly set out pseudopodia in all directions and then builds up several front search (Fig 1a) ;
  • Individuals from a Japanese strain are quicker and more directional in setting out one or two front search (Fig 1c) ;
  • Individuals from an American strain have a behaviour somewhere between Australian and Japanese strain (Fig 1b).
Figure 1
Figure 1 : Exploration patterns of three different strains of Physarum polycephalum. (a) Australian, (b) American and (c) Japanese.

 

 

We next addressed the question, how do these qualitative features influence how slime moulds choose between two identical food sources and how is this affected by changing with the mass of individuals? To this end we built a mathematical model inspired by previous studies on collective decision-making in ants. The model has the following form

 \frac{dx_i}{dt} = \phi \left(M-\sum_{i=1}^2 x_i\right) \frac{x_i^2 }{k+x_i ^2 } - \nu x_i \qquad i=1,2 \qquad (1)

x_i is being part of the initial mass M being on option i. The first positive term corresponds to the local growth of tubes as a result of protoplasmic flow that saturates when the mass of the individual at the initial position is decreasing. The negative term corresponds to the rate at which the mass is decaying. Parameters \phi and k are related to the speed of the decision-making process and to the threshold flow for tube construction, respectively.

The results are summarized in the bifurcation diagram of Figure 2 showing how the steady-state solutions of the model equations, associated to the long-time behavior of the system, change as a function of the mass M.

 

Figure 2
Figure 2 : bifurcation diagram of the steady state solutions of eq. (1) as a function of M. Parameter values : see Zabzina et al. 2014.

 

For small masses the only stable solution is the zero one (corresponding to no exploitation at all). After a first bifurcation a qualitative change takes place. Specifically the individuals exploit only one option or no option at all (though the latter one is never accessible due to the presence of a nearby unstable state), a behavior referred to as symmetry-breaking bifurcation. The second bifurcation point from the left is of less interest as no new stable states become available. But after a third bifurcation, when the mass is large enough, a new qualitative change occurs where the individual can select both options simultaneously, no option at all or focus all its activity on one option. We are now in the presence of a transition restoring the symmetry broken in the first bifurcation. Eventually for very large values of M the homogeneous solution becomes the dominant one as unstable states are very close to the inhomogeneous stable one.

These predictions compare well with experiments undertaken with the three different strains. Figure 3 displays the experimental (a-c) and theoretical (d-f) probabilities to choose zero, one or two options for the three different strains and for different values of initial masses. The latter have been obtained from a Monte Carlo version of the model.

 

Figure 3
Figure 3 : Probability to choose options with respect to the mass of plasmodium. The grey colour corresponds to the probability to move to the one option, the light grey shows the probability to move to two options at the same time and the black colour corresponds to probability to exploit zero option. Experiment outcomes a) Japanese strain, b) Australian strain and c) American strain. Model outcomes d) Japanese strain, e) Australian strain and f) American strain . Parameter values : see Zabzina et al. 2014.

 

In conclusion, in this paper we can see how decision-making depends on mass. We found very rich behaviors, even richer than in the case of ant colonies choosing between two options in a Y-maze. In particular the presence of different, multiple stable states in coexistence enhances flexibility. We expect that in nature , for large masses, individuals can cope efficiently with a dynamic environment where the location and the value of the resources vary in time.

 

[1] Zabzina, N., Dussutour, A., Mann, R. P., Sumpter, D. J., & Nicolis, S. C. (2014). Symmetry Restoring Bifurcation in Collective Decision-Making. PLoS computational biology, 10(12), e1003960.