Double-blind deconvolution: the analysis of post-synaptic currents in nerve cells D. S. Poskitt, K. Dog Æanc Ëay and Shin-Ho Chung Australian National University, Canberra, Australia [Received April 1996. Final revision February 1998] Summary. This paper is concerned with the analysis of observations made on a system that is being stimulated at ®xed time intervals but where the precise nature and effect of any individual stimulus is unknown. The realized values are modelled as a stochastic process consisting of a random signal embedded in noise. The aim of the analysis is to use the data to unravel the unknown structure of the system and to ascertain the probabilistic behaviour of the stimuli. A method of parameter estimation based on quasi-pro®le likelihood is presented and the statistical properties of the estimates are established while recognizing that there will be a discrepancy between the model and the true data-generating mechanism. A method of model validation and determination is also advanced and kernel smoothing techniques are proposed as a basis for identifying the amplitude distribution of the stimuli. The data processing techniques described have a direct application to the investigation of excitatory post-synaptic currents recorded from nerve cells in the central nervous system and their use in quantal analysis of such data is illustrated. Keywords: Blind deconvolution; Excitatory post-synaptic currents; Finite algorithm; Gaussian estimator; Gauss±Newton recursions; Initial estimates; Kernel smoothing; Quantal analysis; Quasi-pro®le likelihood 1. Introduction Consider a situation where T data values are derived from a stochastic process fytg de®ned by the signal plus noise relationship yt st, u t, t  0, :::, T  1, 1 where fst, ug denotes an unknown signal that is generated by passing an input process futg through an observed system and ftg denotes a zero-mean, white Gaussian noise disturbance with variance  2  , independent of the signal, that represents such features as background noise and measurement error. The input futg, like the individual signal and noise components, is not directly observable. It is made up of a periodic pulse train where the time interval between pulses is given but the amplitude of each pulse is random. We express this algebraically as ut P N r1 ft r  1Lga r 2 where t is the Kronecker delta function, Address for correspondence: D. S. Poskitt, Department of Statistics, Australian National University, Canberra, ACT 0200, Australia. E-mail: Don.Poskitt@anu.edu.au & 1999 Royal Statistical Society 1369±7412/99/61191 J. R. Statist. Soc. B (1999) 61, Part 1, pp.191^212