Our goal in this paper is to leverage the potential of the topological signal processing (TSP) framework for analyzing brain networks. Representing brain data as signals over simplicial complexes allows us to capture higher-order relationships within brain regions of interest (ROIs). Here, we focus on learning the underlying brain topology from observed neural signals using two distinct inference strategies. The first method relies on higher-order statistical metrics to infer multiway relationships among ROIs. The second method jointly learns the brain topology and sparse signal representations, of both the solenoidal and harmonic components of the signals, by minimizing the total variation along triangles and the data-fitting errors. Leveraging the properties of solenoidal and irrotational signals, and their physical interpretations, we extract functional connectivity features from brain topologies and uncover new insights into functional organization patterns. This allows us to associate brain functional connectivity (FC) patterns of conservative signals with well-known functional segregation and integration properties. Our findings align with recent neuroscience research, suggesting that our approach may offer a promising pathway for characterizing the higher-order brain functional connectivities.

Permutation entropy (PE) has been widely used to assess the complexity of time series and, in this context, has been applied to the analysis of various types of signals. In this paper, we advance the theory related to PE by defining joint entropy, mutual information, and quadratic dependence measures based on so-called graph permutation entropies. Similarly to their original counterparts commonly employed in information theory, the proposed measures aim to quantify the interdependence between time series. This is validated by applying the measures to the modeling of brain functional connectivity, using electromagnetic resonance imaging data selected from the Human Connectome Project. Our results are supported by performance scores that evaluate the precision of clustering operations through which known brain subsystems are detected in healthy subjects. In this context, we conclude that the introduced measures may achieve better performance than those obtained using previously proposed measures.