On brain modeling in resting-state as a network of coupled oscillators Chiara Favaretto and Angelo Cenedese Abstract— The problem of emergent synchronization pat- terns in a complex network of coupled oscillators has caught scientists’ interest in a lot of different disciplines. In particular, from a biological point of view, considerable attention has been recently devoted to the study of the human brain as a network of different cortical regions that show coherent activity during resting-state. In literature, there can be found different large- scale models of resting-state dynamics in health and disease. In this context, the Kuramoto model, a classical model apt to describe oscillators’ dynamics, has been extended to capture the spatial displacement and the communication conditions in such brain network. Starting from a previous work in this field [1], we analyze this modified model and compare it with other existing large-scale models. In doing so, our aim is to promote a set of mathematical tools useful to better understand real experimental data in neuroscience and estimate brain dynamics. I. I NTRODUCTION Over the past decade, considerable attention has been devoted to the problem of emergence of synchronization patterns in a network of coupled oscillators. Indeed this is a phenomenon that can be observed in various different fields, from engineering to biology. Among these, one of the most fascinating biological field that has recently focused on such issues is neuroscience, with the aim of clarifying the intrigu- ing dynamics of the brain at rest, observable thanks to the BOLD fMRI signals. BOLD (Blood oxygen level-dependent) signals describe changes in magnetic susceptibility and MRI tissue contrast that are indirectly indicative of underlying changes in spontaneous or experimentally controlled brain activation (a control system perspective is provided in [2]). Resting-state is defined as the condition achieved by complete bed rest for at least one hour. During this condition, spatial patterns of coherent activity across different brain areas can been identified [3]. This description of the cerebral cortex refers to a partition of the whole brain into a set of Resting-state networks (RSNs). Various studies have proved that resting-state activity is not stationary [4], [3], but it is characterized by spatio-temporal organized low-frequency fluctuations (< 0.1 Hz). How such type of behavior emerges from the interaction among neurons may be investigated through computational models, which include concepts of neurophysiology and physics. During the last decade, differ- ent large-scale models have been developed and the modified Kuramoto model is one of them. The Kuramoto model [5] is a classical model used to describe the coupled oscillators dynamics. In particular it has C. Favaretto and A. Cenedese are with the Department of Information Engineering, University of Padova, via Gradenigo 6/B, 35131 Padova, Italy. Authors’ contacts: chiara.favaretto@dei.unipd.it, angelo.cenedese@unipd.it. acquired a main role in this context thanks to its simplicity and the limited number of parameters. In order to model the brain resting-state activity, the classical Kuramoto model has to be extended, by considering the following aspects: first, depending on their different functions, not all the cortical regions communicate with the others with the same strength. As a consequence, we have to consider a network of agents, described by a weighted and not fully connected graph. Second, because of the spatial distance among the regions, the model has to consider the delay which attends during the communication. Third, the fluctuations in the resting behavior may be modeled with the addition of a term of noise. The need of a term that takes into account the spatial distances among the oscillators has been already discussed in a previous work [1] and other studies [4] [3] test a modified Kuramoto model, with these characteristics or with some of them. In this paper we study these aspects, in order to identify how a Kuramoto modeling approach can explain real brain dynamics data. Furthermore, we propose a comparison with other two classical large-scale models used to study the brain behavior in resting-state, from an oscillatory point of view. Since we are particularly interested on the synchronization analysis, we use a method to reduce each model to be described by only one variable, namely the angular phase. In order to compare the synthetic results with the real cortical behavior, we consider a real set of public fMRI time- series from 24 healthy subjects [4]. However, since the three examined models differ from each other for several aspects, e.g. the number of state-equations, the kind of coupling among the agents and the absence or presence of an external current, they give rise to different dynamics. Hence, we focus on whether and how they are able to reproduce two behaviors that are largely associated to the resting-state activity of the human brain. These requisites can be expressed as follows: R1 the functional correlation network (FC) has to keep a shape similar to that associated to the structural (anatomical) network (SC), characterized by a small- world structure, with the presence of hub-nodes that define the minimum spanning tree of the network. R2 The brain activity in resting state appears to be char- acterized by time-varying patterns which alternately activate (synchronize) and deactivate (desynchronize). In summary, the aim of this paper is to propose of a set of mathematical tools useful to model, estimate and better understand the real data of brain dynamics. We try to answer some questions about modeling this kind of dynamics: 1) Since previous studies [6] showed that both alpha