Supporting Materials: Photon shot noise limits on optical detection of neuronal spikes and estimation of spike timing

a linear combination of the Poisson-distributed variables, fn. We rewrite the log-likelihood ratio in terms of sn ≡ (A/F0) τν ( e − 1 ) e−n/(τν). Most calculations in this paper rely on taking series approximations in sn and performing the summations to leading order. This approach is valid as long as A < F0, since τν ( e − 1 ) e−n/(τν) is bounded above by 1 for n ≥ 1, and the interval of convergence for the series expansion of log (1 + x) is −1 < x ≤ 1. The mean, μ, and variance, σ, of the distribution of L(f) under the null hypothesis H of no spike having occurred are given by