Inner-product operators, often referred to as kernels in statistical learning, de- fine a mapping from some input space into a feature space. The focus of this paper is the construction of biologically-motivated kernels for cortical ac- tivities. The kernels we derive, termed Spikernels, map spike count sequences into an abstract vector space in which we can perform various prediction tasks. We discuss in detail the derivation of Spikernels and describe an efficient al- gorithm for computing their value on any two sequences of neural population spike counts. We demonstrate the merits of our modeling approach using the Spikernel and various standard kernels for the task of predicting hand move- ment velocities from cortical recordings.