Brain stimulation can modulate the activity of neural circuits impaired by Alzheimer’s disease (AD), having promising clinical benefit. However, all individuals with the same condition currently receive identical brain stimulation, with limited theoretical basis for this generic approach. In this study, we introduce a control theory framework for obtaining exogenous signals that revert pathological electroencephalographic activity in AD at a minimal energetic cost, while reflecting patients’ biological variability. We used anatomical networks obtained from diffusion magnetic resonance images acquired by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) as mediators for the interaction between Duffing oscillators. The nonlinear nature of the brain dynamics is preserved, given that we extend the so-called state-dependent Riccati equation control to reflect the stimulation objective in the high-dimensional neural system. By considering nonlinearities in our model, we identified regions for which control inputs fail to correct abnormal activity. There are changes to the way stimulated regions are ranked in terms of the energetic cost of controlling the entire network, from a linear to a nonlinear approach. We also found that limbic system and basal ganglia structures constitute the top target locations for stimulation in AD. Patients with highly integrated anatomical networks–namely, networks having low average shortest path length, high global efficiency–are the most suitable candidates for the propagation of stimuli and consequent success on the control task. Other diseases associated with alterations in brain dynamics and the self-control mechanisms of the brain can be addressed through our framework.

We studied the interactions between different temporal scales of diffusion processes on complex networks and found them to be stronger in scale-free (SF) than in Erdos-Renyi (ER) networks, especially for the case of phase-amplitude coupling (PAC)-the phenomenon where the phase of an oscillatory mode modulates the amplitude of another oscillation. We found that SF networks facilitate PAC between slow and fast frequency components of the diffusion process, whereas ER networks enable PAC between slow-frequency components. Nodes contributing the most to the generation of PAC in SF networks were non-hubs that connected with high probability to hubs.Additionally, brain networks from healthy controls (HC) and Alzheimer's disease (AD) patients presented a weaker PAC between slow and fast frequencies than SF, but higher than ER. We found that PAC decreased in AD compared to HC and was more strongly correlated to the scores of two different cognitive tests than what the strength of functional connectivity was, suggesting a link between cognitive impairment and multi-scale information flow in the brain.interactions correspond to the phenomenon known as cross-frequency coupling (CFC) 13 . We focus on three types of CFC: phase-amplitude coupling (PAC), the phenomenon where the instantaneous phase of a low frequency oscillation modulates the instantaneous amplitude of a higher frequency oscillation 14 15 ; amplitude-amplitude coupling (AAC), which measures the co-modulation of the instantaneous amplitudes of two oscillations 16 ; and phase-phase coupling (PPC), which corresponds to the synchronization between two instantaneous phases 17 . Results Diffusion of simulated ER and SF networksWe start by considering an unweighted network consisting of nodes. We place a large number ( ≫ ) of random walkers onto this network. At each time step, the walkers move randomly between the nodes that are directly linked to each other. We allow the walkers to perform time steps. As a walker visits a node, we record the fraction of walkers present at it, which we term node activity. Thus, after time steps, we obtain time series reflecting different realizations of the flow of information in the network.Two types of simulated complex networks are considered here, ER and SF networks. An ER network is a random graph where each possible edge has the same probability of existing. The degree of a node i ( ) is defined as the number of connections it has to other nodes. The degree distribution ( ) of an ER network is a binomial distribution, which decays exponentially for large degrees , allowing only very small degree fluctuations 18 . On the other hand, SF networks

Resting-state functional connectivity MRI (rs-fcMRI) is a common method for mapping functional brain networks. However, estimation of these networks is affected by the presence of a common global systemic noise, or global signal (GS). Previous studies have shown that the common preprocessing steps of removing the GS may create spurious correlations between brain regions. In this paper, we decompose fMRI signals into 5 spatial and 3 temporal intrinsic mode functions (SIMF and TIMF, respectively) by means of the empirical mode decomposition (EMD), which is an adaptive data-driven method widely used to analyze non-linear and non-stationary phenomena. For each SIMF, functional connectivity matrices were computed by means of Pearson correlation between TIMFs of different brain areas. Thus, instead of a single connectivity matrix, we obtained 5 × 3 = 15 functional connectivity matrices. Given the high correlation and global efficiency values of the connectivity matrices related to the low spatial maps (SIMF3, SIMF4, and SIMF5), our results suggest that these maps can be considered as spatial global signal masks. Thus, by summing up the first two SIMFs extracted from the fMRI signals, we have automatically excluded the GS which is now voxel-specific. We compared the performance of our method with the conventional GS regression and to the results when the GS was not removed. While the correlation pattern identified by the other methods suffers from a low level of precision in identifying the correct brain network connectivity, our approach demonstrated expected connectivity patterns for the default mode network and task-positive network.