University of São Paulo: Mathematical theory explains complex systems through triplets rather than pairs
To understand how complex systems like the human brain work, one is little, two is good, three is much better. This is what a survey by the Center for Mathematical Sciences Applied to Industry (Cepid-CeMEAI), based at the Institute of Mathematics and Computer Sciences (ICMC) at USP, in São Carlos, demonstrates. The study created a mathematical theory that makes it possible to describe the interactions within complex systems, such as between the cells that make up the brain, through trios, and not just through pairs, as was previously proposed. The research is described in the paper Emergent hypernetworks in weakly coupled oscillators , published in the scientific journal Nature Communications on Aug. 17.
In addition to biology, the research results can be applied to study the structure of compounds formed by various chemical elements. “A complex system is made up of many parts and shows behaviors generated by the interaction of these parts. The pairs appear in the description of the physical problem”, explains Professor Tiago Pereira, from ICMC, who participated in the research , to Jornal da USP . “For example, the Sun influences the Earth. The Moon influences the Earth. The total influence is the sum of the two influences. That is, essentially, the influence, to be explained, is given as a description of pair, both Sun-Earth and Moon-Earth. The same is true in neural networks and in various chemistry experiments.”
According to the professor, several experimental results of complex systems in which the parts theoretically interacted in pairs showed that the best description of the data from the experiments was by trios. “The open problem was explaining why trios were great descriptions when they weren’t present in the physical structure, and how to mathematically predict them,” he points out.
Trios
Pereira points out that, normally, the equations that theoretically describe the interior of complex systems showed the influences happening in pairs. “When researchers found interactions that occurred in trios in the experimental phase, they imagined that they could be wrong”, he points out. “We created a mathematical theory that allows us to make small perturbations in the equations and prove that only the important terms remain.”
“When this theory is applied to certain systems with peer interaction, the peer contributions disappear and only the triples and quadruples are stable”, explains the professor. “The theory makes it possible to predict which trios will be stable and also a description of the new system, which also applies to sets with a greater number of units.”
According to Pereira, the results can contribute to biology research, such as studies on brain functioning and chemistry, to analyze the interactions between different substances. “Theory allows controlling the experimental system based on the observation of data”, he emphasizes. “And it sheds light on why threesome interactions are so common.” In addition to the ICMC professor, the research has the participation of Eddie Nijholt, from CeMEAI, Jorge Ocampo-Espindola and István Kiss, from the University of Saint Louis (United States), and Deniz Eroglu, from Kadir Haz University (Turkey).