According to the computational theory of cognition (CTC), cognitive capacities are explained by inner computations, which in biological organisms are realized in the brain. Computational explanation is so popular and entrenched that it's common for scientists and philosophers to assume CTC without argument. But if we presuppose that neural processes are computations before investigating, we turn CTC into dogma. If, instead, our theory is to be genuinely empirical and explanatory, it needs to be empirically testable. To bring empirical evidence to bear on CTC, we need an appropriate notion of computation. In order to ground an empirical theory of cognition, as CTC was designed to be, a satisfactory notion of computation should satisfy at least two requirements: it should employ a robust notion of computation, such that there is a fact of the matter as to which computations are performed by which systems, and it should not be empirically vacuous, as it would be if CTC could be established a priori. In order to satisfy these requirements, the computational theory of cognition should be grounded in a mechanistic account of computation. Once that is done, I evaluate the computational theory of cognition on empirical grounds in light of our best neuroscience. I reach two main conclusions: cognitive capacities are explained by the processing of spike trains by neuronal populations, and the processing of spike trains is a kind of computation that is interestingly different from both digital computation and analog computation.