Necessary and sufficient conditions are given for the existence of a graph decomposition of the Kneser Graph $K{{G}_{n,2}}$ and of the Generalized Kneser Graph $GK{{G}_{n,3,1}}$ into paths of length three.

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Certain limit theorems due to Berman involve the total time spent by Brownian motion with positive drift below an independent exponentially distributed level. Imhof has calculated the density function and shown that this random variable has two interesting probabilistic properties. We give sample path arguments which explain these two facts.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.