We reformulate the Equal Coding Theorem in sensory neural encoding with ON- and OFF-neurons as a channel capacity problem. We then present a capacity-based proof of the Equal Coding Theorem, and generalize it to neurons with different firing probabilities. We also briefly discuss the biological implications of this generalization.

We study rate-distortion problems of a Poisson process using a group theoretic approach. By describing a realization of a Poisson point process with either point timings or inter-point intervals and by choosing appropriate distortion measures, we establish rate-distortion problems of a homogeneous Poisson process as ball-or sphere-covering problems for realizations of the hyperoctahedral group in $\mathbb{R}^{n}$. Specifically, the realizations we investigate are a hypercube and a hyperoctahedron. Thereby we unify three known rate-distortion problems of a Poisson process (with different distortion measures, but resulting in the same rate-distortion function) with the Laplacian-$\ell_{1}$ rate-distortion problem.

The geometric interpretation of sphere covering describing the rate distortion problem of a Gaussian source with the squared-error distortion measure is generalized to a Laplacian source and the ℓ1-distortion measure. Using additional constraints on the distortion measure, sphere covering is further generalized to exponential sources and to Poisson point processes.