The shape of the PCA trajectories and the population neural coding of movement initiation in the basal ganglia

In this work we study the shape of the neural trajectories obtained from Principal Components Analysis (PCA) of neural activity, related to movement initiation, in the basal ganglia of rats. We focus on the relation between global and local aspects of the shape, and on the population code by means of the ensemble structure. We find that the structure of the principal components is intimately related to the ensembles in the population which, in turn, drive the evolution of the neural trajectories. From this point of view, the coding schemes in the input and output stages of the basal ganglia differ, being the output lower dimensional than the input. In the output stage we can identify specific ensembles that explain the main features (sharp points, singular points, etc.) of the shape, revealing novel aspects of the computations performed by these regions during movement. Also, based upon new measures of heterogeneity and sparseness, we conclude that the output stages are homogeneous but dense while the input stages are more heterogeneous. This work also contains novel mathematical results in relation with the restrictions on the shape imposed by the PCA and the by structure of the data. A novel relationship between the principal components and Catalan objects is proved.