Fig. 6: Data-driven control of functional brain networks.

Panel (a) provides a schematic of the experimental setup. A set of external stimuli represented by m different task commands induce brain activity. Functional magnetic resonance (fMRI) blood oxygen level-dependent (BOLD) signals are measured and recorded at different times and converted into p time series, one for each brain region. The top and center heatmaps of the panel (b) show the inputs and outputs, respectively, for the first 110 measurements of one subject of the HCP dataset. The inputs are divided into m = 6 channels corresponding to different task conditions, i.e., CUE (a visual cue preceding the occurrence of other task conditions), LF (squeeze left toe), LH (tap left fingers), RF (squeeze right toe), RH (tap right finger), and T (move tongue). As in ref. ⁵⁸, each input is a binary 0–1 signal taking the value 1 when the corresponding task condition is issued and 0 otherwise. The outputs represent the BOLD signals of the p = 148 brain regions obtained from and enumerated according to the Destrieux 2009 atlas⁵⁹. The bottom heatmap of the panel (b) displays the simulated outputs obtained by exciting the approximate low-dimensional linear model of ref. ⁵⁸ with the input sequence of the top plot. In panel (c), we compare the performance of the data-driven and model-based strategy, assuming that the dynamics obey the above-mentioned approximate linear model. We set the control horizon to T = 100 and generate the data matrices by sliding a time window of size T across the data samples. The target state y_f,i is the eigenvector associated with the i-th eigenvalue of the empirical Gramian matrix \({\hat{{\boldsymbol{{\cal{W}}}}}}_{T}={\hat{{\boldsymbol{{\cal{C}}}}}}_{T}^{{\mathsf{T}}}{\hat{{\boldsymbol{{\cal{C}}}}}}_{T}\), where \({\hat{{\boldsymbol{{\cal{C}}}}}}_{T}={{\bf{Y}}}_{T}{{\bf{U}}}_{0:T-1}^{\dagger }\). The left plot shows the error to reach the targets \({\{{{\bf{y}}}_{\text{f},i}\}}_{i = 1}^{20}\) using the data-driven minimum-energy input in Eq. (5) and the model-based one. The right plot shows the norm of the two inputs. The colored bars denote the mean over 100 unrelated subjects and the error bars are the 95% confidence intervals around the mean.