Table 1. Analogous components in three different inverse problems. Blind Deconvolution of Postsynaptic Currents Shape of excitatory postsynaptic current (EPSC) Point-spread function of telescope Point-spread function of imaging system Astronomy Position of a star Photograph Two Results Fluorescence microscopy Location of a fluorescence source Z-stack of images Three We describe a general framework for a blind deconvolution algorithm, in which the observed data x are the sum of the signal s filtered through a convolution with impulse response f and noise n: x~s fzn: Given observation x, we wish to infer the signal and impulse response. We can write the posterior using Bayes’ rule: Pðs,fDx,hÞ~ PðxDs,f,hÞPðs,fDhÞ , PðxDhÞ in which h represents the remaining parameters such as the noise variance and regularization constant described below. Because we wish to minimize the number of parameters that must be provided to the algorithm by the user, we intend to automatically estimate or marginalize over several of them. The likelihood PðxDs,f,hÞ describes the character of the noise n, which for the noise in electrophysiological experiments is typically additive and Gaussian. Although s and f appear symmetrically in the likelihood, they have different properties and priors and therefore require different approaches to their numerical solution. These differences and constraints are captured by the prior 1 ð Þ 2 ð Þ Signal Filter Data Dimensionality Electrophysiology Neurotransmitter release accompanying vesicle fusion Current trace One doi:10.1371/journal.pone.0038198.t001 containing multiple overlapping EPSCs pose a challenge for existing algorithms. We therefore approached EPSC deconvolu- tion using a Bayesian framework, constructing and then solving a generative model. We aimed to improve upon existing algorithms in several ways. First, we sought to distinguish individual events even as the temporal proximity of neighboring events ap- proached the risetime of the EPSC. Second, variability in experimental conditions demanded that the algorithm adapt automatically to each recording situation without the need for the user to supply precise EPSC profiles. Finally, we desired a blind algorithm that could estimate the shape of the unitary EPSC directly from the EPSCs detected in each recording. We expected that obtaining a good estimate of the unitary EPSC’s shape–the impulse-response function–would result in increased temporal acuity of the algorithm, permitting an improved analysis of multiphasic waveforms. We developed and imple- mented such an algorithm, then evaluated its performance on real and synthetic data. Methods The algorithm was tested on data obtained from paired whole- cell patch-clamp recordings performed at hair-cell synapses in the amphibian papilla of the American bullfrog (Rana catesbeiana) as previously described [1]. Figure 1. Structure and responses of a hair cell’s ribbon synapse. A, Fast exocytosis owing to the fusion of a synaptic vesicle with the presynaptic cell’s plasma membrane floods the synaptic cleft with neurotransmitter. Postsynaptic receptors, in particular AMPA receptors at the hair- cell synapse, open and allow the flow of current into the postsynaptic nerve terminal on the right. B, The impulse response of the system is the excitatory postsynaptic current (EPSC) of stereotyped amplitude and timecourse evoked by the release of transmitter from a single presynaptic vesicle. Postsynaptic AMPA receptor channels open and close stochastically; here the mean current is shown for 150 channels. Although the channels open quickly, the risetime of the measured current is limited by the time constant of the postsynaptic membrane. The reclosure of AMPA receptors is a slower process that is well approximated by a single exponential with a time constant of 1–2 ms. C, Postsynaptic-current recordings are modeled statistically in three steps. In the first row, peaks in the density of neurotransmitter in the synaptic cleft are caused by stochastic fusions of vesicles in several size classes. In the second row, these spikes of neurotransmitter produce in the postsynaptic cell bursts of current shaped by the kinetics of receptor activation and deactivation (panel B). In the third row, noise from processes in the cell and the experimental apparatus are added to the postsynaptic signal. The purpose of our algorithm is to reconstruct the first record from observation of the third record, inferring and using the response shown in panel B. doi:10.1371/journal.pone.0038198.g001 PLoS ONE | www.plosone.org 2 June 2012 | Volume 7 | Issue 6 | e38198