The science of complexity is a new and extremely trans-disciplinary field of research. Many definitions have been given so far and yet the concept has resisted categorization, but, nevertheless becomes more and more attractive for many researchers in different fields. One of the points of agreement among the scientists is that complex systems are those composed of a large number of interacting elements, so that the collective behaviour of those elements goes far beyond the simple sum of the individual behaviours [1].

Initially the concept of complex systems was mainly associated with the temporal evolution of systems made up of many interacting units each characterized by highly nonlinear dynamics, prominent examples are represented by spatio-temporal chaos or pattern forming systems [2]. In the last decadet he interest has moved towards an even more intriguing subject: the emergence of   nontrivial collective dynamics in networks composed of elements whose evolution is extremely simple, like oscillators with periodic dynamics. However, the interaction of identical oscillators can lead, in spatially homogeneous system, to the emergence of nontrivial macroscopic dynamics ranging from quasi-periodic to chaotic [3, 4]. More recently the introduction of nonlocal interaction in networks of identical oscillators has led to the discovery of states with broken spatial symmetry (the so-called “chimera states” [5-7]).

The research activity of the Computational Neuroscience Lab in Florence focuses on interacting sub-populations of oscillators, with the goal of characterizing the dynamics both at the macroscopic (collective) level, as well as at the level of each sub-population (mesoscopic level). Neural systems represent an important research field where the mesoscopic evolution of sub-networks is extremely relevant. Several recent studies have demonstrated experimentally and numerically that neuronal groups or ensembles (cliques), rather than individual neurons, are the emergent functional units of cortical activity [8-11]. Furthermore, mammalian cortical neurons form transient oscillatory assemblies supporting temporal representation and long-term consolidation of information in the brain [12]. Theoretical methods and techniques developed for coupled oscillators can be profitably applied in the context of neural networks, since these systems can be mathematically modelized as networks of coupled phase oscillators [13]. Along this direction, exact macroscopic dynamical equations (analogous to neural field models) have been recently derived for networks of theta-neurons [14-16] and extended to neural circuits presenting both chemical and electrical synapses[17,18] and modelling working memory [19].

A second important research line of the Computational Neuroscience Lab deals with the analysis of electrophysiological recordings such as EEG and neuronal spike trains. Here a recent focus of interest is neuronal population coding, i.e., the study of how the sensory world is represented in the action potentials of neuronal networks in the brain [20].

[1] M. E. J. Newman, Am. J. Phys. 79, 800 (2011)
[2] A. Politi, ed. G. Fraser,Cambridge University Press, (2006).
[3] P. C. Matthews, R. E. Mirollo and S. H. Strogatz, Physica D 52, 293 (1991)
[4] N. Nakagawa and Y. Kuramoto, Physica D 80, 307 (1995)
[5] Y. Kuramoto and D. Battogtokh, Nonlin. Phen. in Complex Sys. 5, 380 (2002)
[6] D.M. Abrams and S.H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004)
[7] S. Olmi, A. Politi and A. Torcini, EPL (Europhysics Letters) 92, 60007 (2010)
[8] E. Fuchs, A. Ayali, E. Ben-Jacob and S. Boccaletti, Phys. Biol. 6, 036018 (2009)
[9] S. Luccioli, E. Ben-Jacob, A. Barzilai, P. Bonifazi, and A. Torcini, PLOS Computational Biology 10(9): e1003823 (2014)
[10] J.E.K Miller, I. Ayzenshtat, L. Carrillo-Reid, and R. Yuste, Proceedings of the National Academy of Sciences 111, E4053-E4061 (2014)
[11] T.A. Seabrook, and A. D. Huberman, Neuron 86, 1113 (2015); S.J. Barnes, R.P. Sammons, R.I.
Jacobsen, J. Mackie, G.B. Keller, and T. Keck, Neuron 86, 1290 (2015)
[12] G. Buzsaki, and A. Draguhn, Science 304(5679), 1926-1929 (2004)
[13] S. H. Strogatz, Nature 410(6825), 268-276 (2001)
[14] P. So, T. B. Luke and E. Barreto, Physica D: Nonlinear Phenomena 267, 16-26 (2014)
[15] C. R. Laing, Phys. Rev. E 90, 010901(R) (2014).
[16] E. Montbrio’, D. Pazo’, and A. Roxin, Phys. Rev. X 5, 021028 (2015)
[17] I. Ratas, K. Pyragas, Phys. Rev. E 94, 032215 (2016)
[18] S. Coombes, A. Byrne, In Nonlinear Dynamics in Computational Neuroscience, eds. F. Corinto and A. Torcini (Springer). 1-16 (2019)
[19] H. Taher, A. Torcini, and S. Olmi, PLoS Computational Biology 16, e1008533 (2020)
[20] T. Kreuz, C. Houghton, and J. D. Victor, Journal: Encyclopedia of Computational Neuroscience, 1-14 (2019).