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P-spline models have achieved great popularity both in statistical and in applied research. A possible drawback of P-spline is that they assume a smooth transition of the covariate effect across its whole domain. In some practical applications, however, it is desirable and needed to adapt smoothness locally to the data, and adaptive P-splines have been suggested. Yet, the extra flexibility afforded by adaptive P-spline models is obtained at the cost of a high computational burden, especially in a multidimensional setting. Furthermore, to the best of our knowledge, the literature lacks proposals for adaptive P-splines in more than two dimensions. Motivated by the need for analysing data derived from experiments conducted to study neurons' activity in the visual cortex, this work presents a novel locally adaptive anisotropic 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. The practical performance of the proposal is evaluated through simulations, and comparisons with alternative methods are reported. In addition to the spatio-temporal analysis of the data that motivated this work, we also discuss an application in two dimensions on the absenteeism of workers.