Abstract
Collective behaviours are known to be the result of diverse dynamics and are sometimes likened to living systems. Although many studies have revealed the dynamics of various collective behaviours, their main focus has been on the information processing performed by the collective, not on interactions within the collective. For example, the qualitative difference between three and four elements in a system has rarely been investigated. Tononi et al. proposed integrated information theory (IIT) to measure the degree of consciousness Φ. IIT postulates that the amount of information loss caused by the minimum information partition is equivalent to the degree of information integration in the system. This measure is not only useful for estimating the degree of consciousness but can also be applied to more general network systems. Here, we obtained two main results from the application of IIT (in particular, IIT 3.0) to the analysis of real fish schools (Plecoglossus altivelis). First, we observed that the discontinuity on 〈Φ(N)〉 distributions emerges for a school of four or more fish. This transition was not observed by measuring the mutual information or the sum of the transfer entropy. We also analysed the IIT on Boids simulations with respect to different coupling strengths; however, the results of the Boids model were found to be quite different from those of real fish. Second, we found a correlation between this discontinuity and the emergence of leadership. We discriminate leadership in this paper from its traditional meaning (e.g. defined by transfer entropy) because IIT-induced leadership refers not to group behaviour, as in other methods, but the degree of autonomy (i.e. group integrity). These results suggest that integrated information Φ can reveal the emergence of a new type of leadership which cannot be observed using other measures.
Highlights
Collective behaviours in nature are emergent properties produced through the local interactions of self-organising individuals
We tracked the trajectories of ayu fish schools of N = 2, N = 4, and N = 5 with three samples each, and N = 3 with four samples (10–15 minutes recording length; see Materials and methods: Experimental settings for details)
An ON state means some interaction will occur in a given context
Summary
Collective behaviours in nature are emergent properties produced through the local interactions of self-organising individuals. Such behaviours include swarming [1,2,3,4,5,6], fish schooling. [7,8,9,10,11], bird flocking [12,13,14,15,16,17], or high-level cognitive functions arising from ‘bottom-up’ neural networks [18,19,20,21] These systems of many interacting elements can achieve optimal information processing capabilities when poised at the critical boundary separating chaos from order [22,23,24]. The unity of collective behaviours remains an unsolved question of nature [30,31,32], because the interactions are hidden, whereas the resultant actions are observable
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