The PhD course was led by Simona Olmi from ISC, CNR and consisted in the recruitment and supervision of the PhD student Halgurd Taher. The grant amounts to 105.000 euros.
A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain functions arise from the collective dynamics of cortical circuits. Often reported collective phenomena include oscillations, synchronous dynamics or more general rhythms, characteristic of various neural circuits . Oscillations of neural activity are ubiquitous in the brain in many frequency bands and it has been argued that they play a functional role in cortical processing . In particular, the mechanisms underlying the coupling between neural oscillations at different timescales have recently received much attention from experimentalists and theoreticians . The most studied example of this phenomenon, usually termed cross-frequency coupling (CFC), concerns the coupling between θ and γ oscillatory activity in the rodent hippocampus . Furthermore, CFC appears to be implicated in cognitive operations: multi-item representation, long-distance communication, and stimulus parsing .
It has been recently suggested that simple neural architectures involving few excitatory and inhibitory populations are able to reproduce the mechanisms underlying CFC . These and other evidences suggest that information is encoded in the population response and hence can be captured via macroscopic measures of the network activity . The collective behavior is particularly relevant given that current brain measurement techniques, such as EEG or fMRI, provide data averaged over the activity of a large number of neurons. So far the analysis of spiking neural circuits has been mainly addressed through numerical simulations, with limitations in the maximal affordable number of neurons due to the available numerical resources. Alternatively, researchers have formulated effective mean-field representation of the neural dynamics at the level of populations, in terms of neural mass models . However an extremely powerful exact method recently developed, is the Ott-Antonsen (OA) Ansatz , which allows to rewrite the dynamics of fully-coupled networks of phase oscillators in terms of few collective variables in the thermodynamic limit. Only recently a few studies, published in statistical physics journals, revealed the possibility, by applying and extending the OA Ansatz, to
derive exact neural field models starting from microscopic spiking neural circuits [9,10]. This novel exact reduction methodology (ERM) allows to reproduce the collective evolution of a population of spiking Θ-neurons  with a few collective variables representing the firing rate and the mean membrane potential of the neuronal populations  or their level of synchronization .
These results are particularly relevant for the neuroscience community, since the Θ-neuron is a paradigmatic model employed, for example, for the study of working memory and for the reproduction of θ-γ coupled rhythms emerging in the auditory cortex . However, these theoretical results have not been yet applied to address relevant issues in neuroscience.
 G. Buzsaki, Rhythms of the Brain (Oxford University Press, USA, 2006), 1st ed., ISBN 0195301064.
 T. Womelsdorf et al., Science 316, 1609 (2007); X.-J. Wang, Physiological reviews 90, 1195 (2010).
 A. Hyafil, A.-L. Giraud, L. Fontolan, and B. Gutkin, Trends in Neurosciences 38, 725 (2015).
 J. E. Lisman and O. Jensen, Neuron 77, 1002 (2013).
 L. Fontolan, M. Krupa, A. Hyafil, and B. Gutkin, The Journal of Mathematical Neuroscience 3, 1 (2013).
 B. B. Averbeck, P. E. Latham, and A. Pouget, Nature Reviews Neuroscience 7, 358 (2006).
 G. Deco, V. K. Jirsa, P. A. Robinson, M. Breakspear, and K. Friston, PLoS Comput Biol 4, e1000092 (2008); J. Touboul, F. Wendling, P. Chauvel, O. Faugeras, Neural computation, 23(12), 3232-86 (2011).
 E. Ott and T. M. Antonsen, Chaos, 18, 037113 (2008).
 D. Pazó and E. Montbrió, Phys. Rev. X 4, 011009 (2014); P. So, T. B. Luke, and E. Barreto, Physica D 267, 16 (2014); C. R. Laing, Phys. Rev. E 90, 010901 (2014).
 E. Montbrió, D. Pazó, and A. Roxin, Phys. Rev. X 5, 021028 (2015).
 G. Ermentrout and N. Kopell, SIAM Journal on Applied Mathematics 46, 233 (1986).
 M. Dipoppa and B. S. Gutkin, PNAS 110, 12828 (2013); M. Krupa, S. Gielen, and B. Gutkin, J. Comput. Neurosci. 37, 357 (2014)