Multidimensional adaptive P‐splines with application to neurons' activity studies

The receptive field (RF) of a visual neuron is the region of the space that elicits neuronal responses. It can be mapped using different techniques that allow inferring its spatial and temporal properties. Raw RF maps (RFmaps) are usually noisy, making it difficult to obtain and study important features of the RF. A possible solution is to smooth them using P‐splines. Yet, raw RFmaps are characterized by sharp transitions in both space and time. Their analysis thus asks for spatiotemporal adaptive P‐spline models, where smoothness can be locally adapted to the data. However, the literature lacks proposals for adaptive P‐splines in more than two dimensions. Furthermore, the extra flexibility afforded by adaptive P‐spline models is obtained at the cost of a high computational burden, especially in a multidimensional setting. To fill these gaps, this work presents a novel anisotropic locally adaptive P‐spline model in two (e.g., space) and three (space and time) dimensions. Estimation is based on the recently proposed SOP (Separation of Overlapping Precision matrices) method, which provides the speed we look for. Besides the spatiotemporal analysis of the neuronal activity data that motivated this work, the practical performance of the proposal is evaluated through simulations, and comparisons with alternative methods are reported.