Mesocopic comparison of complex networks based on periodic orbits

Abstract

Complex noiseless dynamical systems can be represented in a compressed manner by unstable periodic orbits. It is unknown, however, how to use this technique to obtain a suitable notion of similarity among them, how to extend such an approach to more general complex networks, and how to apply such a method in the important case of noisy systems. Our approach provides a solution to these questions. For a proof-of-concept, we consider Drosophila's precopulatory courtship, where our method reveals the existence of a complex grammar (similar to those found in complex physical systems and in language), leading to the conclusion that the observed grammar is very unlikely the product of chance. Complex noiseless dynamical systems can be represented in a compressed manner by unstable periodic orbits. It is unknown, however, how to use this technique to obtain a suitable notion of similarity among them, how to extend such an approach to more general complex networks, and how to apply such a method in the important case of noisy systems. Our approach provides a solution to these questions. For a proof-of-concept, we consider Drosophila's precopulatory courtship, where our method reveals the existence of a complex grammar (similar to those found in complex physical systems and in language), leading to the conclusion that the observed grammar is very unlikely the product of chance. Complex networks are usually described in terms of abstract network structures, where a node often comes equipped with a rule characterizing its local activity, or by using Markov random walks models. For their understanding, however, intermediate scales may play a distinguished role. This is the case if measures characterizing the network as a whole are too crude for providing the desired insights and if the local node dynamics do not provide an overview of the network. We formulate here in precise mathematical terms such as intermediate, mesoscopic, description of the network, based on periodic orbits as the mesoscopic observables. From these observables, we develop a method for measuring the similarity of graphs, a challenge of great importance that seems not to have been solved in the generality put forward here. An obvious typical application of our approach would be, e.g., the characterization of behavior, with robotics motion planning or the characterization of animal courtship behavior as prominent examples. Animal courtship, which we chose as the real-world testing case of our approach, is a puzzling phenomenon. Its function and purpose have largely remained unexplained, not least because of the generally large and complex underlying network structure and the lack of methods able to deal with such situations. The driving biological questions that we wish to answer here are: Could it be that during courtship, nontrivial information about a prospective mate is conveyed? And if so, how is this information organized and what purpose could it serve? The proposed approach provides partial answers to these questions. We verify that Dro...

Deviation from Criticality in Functional Biological Networks

“…Power-law like bursts of cortical activity were also observed on a more macroscopic scale in sleep experiments [13,14]. The issue of functional relevance is avoided in this paper by working instead with functional output itself: the body language actions of Drosophila melanogaster during its pre-copulatory courtship [15,16] provides an opportunity to code and study distinct, relevant, neural states, in terms of a clearly defined and finite set of fundamental actions. Sequences of these fundamental actions naturally define a network, which gives a glimpse of the topology of transitions between relevant neural states expressable by the system.…”

“…In this paper, the sequence types are studied in 3 groups: male behaviour ; female behaviour ; and male and female behaviour ( Table 1). We use behaviour in the sense defined in [15,16], so that the female behaviour includes the gender switching observed in male vs. fruitless (however sequences produced by fruitless as protagonist are excluded).…”

“…Sequences of these fundamental actions naturally define a network, which gives a glimpse of the topology of transitions between relevant neural states expressable by the system. Further, these sequences have been shown to convey important and specific information with a broad range of grammatical sophistication [15,16,17].…”

At Grammatical Faculty of Language, Flies Outsmart Men

“…Behavior is characterized by rituals that consist of well-chosen sequences of individual actions. Since it is in the nature of these rituals that they need to be repeated if required, we characterize behavior by sequences of indecomposable closed cycles of indecomposable individual actions, so-called irreducible cycles of irreducible acts [11], [12]. This approach is also motivated by the theory of complex dynamical systems, where it has been shown that such systems can be reduced to a minimal set of closed sequences of actions (there called ‘irreducible closed orbits’).…”

“…Using a decomposition of courtship data into irreducible cycles, it has been found that with high confidence during D. ’s precopulatory courtship, individual information is transmitted to the prospective partner, i.e. genuine communication with essential information exchange takes place [11], [12] (Fig. S2).…”