Understanding the relation between the structure of brain networks and its functions is a fundamental open question. Simple models of neural activity based on real anatomical networks have proven effective in describing features of whole-brain spontaneous activity when tuned at their critical point. In this work, we show that indeed structural networks are a crucial ingredient in the emergence of synchronized oscillations in a whole-brain stochastic model at criticality. We study such model in the mean-field limit, providing an analytical understanding of the associated first-order phase transition, arising from the presence of a bistable region in the parameters space. Then, we derive the power spectrum in the linear noise approximation and we show that, in the mean-field limit, no global oscillations emerge. Finally, by adding back an underlying brain network structure with homeostatic normalization, we numerically show how the bi-stability region is disrupted and concomitantly a synchronized phase with maximal dynamic range is observed. Hence, both the structure of brain networks and criticality are fundamental in driving the collective coordinated responses and maximal sensitivity of whole-brain stochastic models.

By characterizing the time evolution of COVID-19 in term of its ‘velocity’ (log of the new cases per day) and its rate of variation, or ‘acceleration’, we show that in many countries there has been a deceleration even before lockdowns were issued. This feature, possibly due to the increase of social awareness, can be rationalized by a susceptible-hidden-infected-recovered model introduced by Barnes, in which a hidden (isolated from the virus) compartment H is gradually populated by susceptible people, thus reducing the effectiveness of the virus spreading. By introducing a partial hiding mechanism, for instance due to the impossibility for a fraction of the population to enter the hidden state, we obtain a model that, although still sufficiently simple, faithfully reproduces the different deceleration trends observed in several major countries.

Understanding the relation between the structure of brain networks and its functions is a fundamental open question. Simple models of neural activity based on real anatomical networks have proven effective in describing features of whole-brain spontaneous activity when tuned at their critical point. In this work, we show that indeed structural networks are a crucial ingredient in the emergence of synchronized oscillations in a whole-brain stochastic model at criticality. We study such model in the mean-field limit, providing an analytical understanding of the associated first-order phase transition, arising from the presence of a bistable region in the parameters space. Then, we derive the power spectrum in the linear noise approximation and we show that, in the mean-field limit, no global oscillations emerge. Finally, by adding back an underlying brain network structure with homeostatic normalization, we numerically show how the bi-stability region is disrupted and concomitantly a synchronized phase with maximal dynamic range is observed. Hence, both the structure of brain networks and criticality are fundamental in driving the collective coordinated responses and maximal sensitivity of whole-brain stochastic models.