Essential tremor (ET) is the most common movement disorder. Its neural underpinnings remain unclear. Here, we quantified structural covariance between cortical thickness (CT), surface area (SA), and mean curvature (MC) estimates in patients with ET before and 1 year after ventro-intermediate nucleus stereotactic radiosurgical thalamotomy, and contrasted the observed patterns with those from matched healthy controls. For SA, complex rearrangements within a network of motion-related brain areas characterized patients with ET. This was complemented by MC alterations revolving around the left middle temporal cortex and the disappearance of positive-valued covariance across both modalities in the right fusiform gyrus. Recovery following thalamotomy involved MC readjustments in frontal brain centers, the amygdala, and the insula, capturing nonmotor characteristics of the disease. The appearance of negative-valued CT covariance between the left parahippocampal gyrus and hippocampus was another recovery mechanism involving high-level visual areas. This was complemented by the appearance of negative-valued CT/MC covariance, and positive-valued SA/MC covariance, in the right inferior temporal cortex and bilateral fusiform gyrus. Our results demonstrate that different morphometric properties provide complementary information to understand ET, and that their statistical cross-dependences are also valuable. They pinpoint several anatomical features of the disease and highlight routes of recovery following thalamotomy. Author Summary Doubts remain regarding the anatomical alterations underlying essential tremor, partly owing to heterogeneity in symptoms’ severity and response to medication. Here, we studied drug-resistant patients clinically assessed and imaged before as well as 1 year after stereotactic radiosurgical thalamotomy, which significantly lowered tremor intensity. We extracted morphometric estimates of volume (subcortex and cerebellum), cortical thickness, surface area, and mean curvature (cortex), and quantified cross-regional statistical dependences across subjects (i.e., structural covariance or SC) for each measure, as well as cross-measure relationships for each region. Compared to matched healthy controls, patients showed altered surface area structural covariance within motion-related areas. Thalamotomy modulated mean curvature SC in frontal and subcortical centers. In both comparisons, SC and cross-measure relationship differences were also observed in visual areas.

Graph spectral analysis can yield meaningful embeddings of graphs by providing insight into distributed features not directly accessible in nodal domain. Recent efforts in graph signal processing have proposed new decompositions—for example, based on wavelets and Slepians—that can be applied to filter signals defined on the graph. In this work, we take inspiration from these constructions to define a new guided spectral embedding that combines maximizing energy concentration with minimizing modified embedded distance for a given importance weighting of the nodes. We show that these optimization goals are intrinsically opposite, leading to a well-defined and stable spectral decomposition. The importance weighting allows us to put the focus on particular nodes and tune the trade-off between global and local effects. Following the derivation of our new optimization criterion, we exemplify the methodology on the C. elegans structural connectome. The results of our analyses confirm known observations on the nematode’s neural network in terms of functionality and importance of cells. Compared with Laplacian embedding, the guided approach, focused on a certain class of cells (sensory neurons, interneurons, or motoneurons), provides more biological insights, such as the distinction between somatic positions of cells, and their involvement in low- or high-order processing functions.