Persistent homology decomposition (PHD) for single-run model-free fMRI and its application to a task probing fluid...

Conventional linear analyses of task-evoked fMRI rely on group averaging and general linear model (GLM) with an assumed hemodynamic response function (HRF) to amplify weak BOLD signals. However, group-averaging obscures idiosyncratic brain responses, and HRFs capture only a portion of neurovascular dynamics. Here, a nonlinear alternative - persistent homology decomposition (PHD) - is introduced to overcome both limitations. In block-design paradigms, voxel-wise temporal periodicity is reconstructed as a loop manifold in high-dimensional phase space via Takens embedding, and its cyclicity is quantified using persistent homology. This framework yields two distinct SNR boosting mechanisms: (i) a m enhancement inherent to representing one-dimensional manifold in m-dimensional phase space, replacing group-averaging, and (ii) an additional nonlinear gain from topological unfolding, replacing the GLM. By identifying topological features in phase space, PHD eliminates the need for HRF and pre-whitening, reduces the high false positive and negative rates caused by ill-posed Moore-Penrose pseudo-inversion due to noisy signals in GLM, and enables a single-run, HRF-free analysis that matches or surpasses group-averaged GLM in both SNR and explained variances. Applied to a 20-participnat fluid-intelligence (gF) fMRI dataset, PHD reveals that idiosyncratic activation within a left-lateralized visual-conceptual pathway and the salience-control network complementarily drive individual gF performance - effects obscured by group-averaged analysis dominated by frontoparietal patterns. These findings demonstrate that PHD uncovers subject-specific neural computations inaccessible to conventional linear models.