Fast Spatiotemporal Smoothing of Calcium Measurements in Dendritic Trees

We discuss methods for fast spatiotemporal smoothing of calcium signals in dendritic trees, given single-trial, spatially localized imaging data obtained via multi-photon microscopy. By analyzing the dynamics of calcium binding to probe molecules and the effects of the imaging procedure, we show that calcium concentration can be estimated up to an affine transformation, i.e., an additive and multiplicative constant. To obtain a full spatiotemporal estimate, we model calcium dynamics within the cell using a functional approach. The evolution of calcium concentration is represented through a smaller set of hidden variables that incorporate fast transients due to backpropagating action potentials (bAPs), or other forms of stimulation. Because of the resulting state space structure, inference can be done in linear time using forward-backward maximum-a-posteriori methods. Non-negativity constraints on the calcium concentration can also be incorporated using a log-barrier method that does not affect the computational scaling. Moreover, by exploiting the neuronal tree structure we show that the cost of the algorithm is also linear in the size of the dendritic tree, making the approach applicable to arbitrarily large trees. We apply this algorithm to data obtained from hippocampal CA1 pyramidal cells with experimentally evoked bAPs, some of which were paired with excitatory postsynaptic potentials (EPSPs). The algorithm recovers the timing of the bAPs and provides an estimate of the induced calcium transient throughout the tree. The proposed methods could be used to further understand the interplay between bAPs and EPSPs in synaptic strength modification. More generally, this approach allows us to infer the concentration on intracellular calcium across the dendritic tree from noisy observations at a discrete set of points in space. Author Summary: Spatiotemporal dendritic imaging data, through fluorescent calcium indicators, opens an exciting window on computations performed by single neurons at a subcellular level. However, the analysis and interpretation of such data is challenging. The measurements are noisy, intermittent in space and/or time, and depend critically on the choice of the fluorescent indicator. Consequently, analysis is typically limited to a specific branch of the dendritic tree, neglects spatiotemporal correlations between neighboring compartments, and requires averaging over multiple trials. Here we derive a model for the spatiotemporal concentration of calcium bound probe molecules. Using state-space and optimization tools we derive a fast algorithm for estimating the most likely concentration based on the given measurements obtained from a single trial, and argue that it can provide an estimate of the fast transients of the underlying calcium concentration. In particular, our algorithm estimates the timing and amplitude of calcium transients due to backpropagating action potentials. It provides a flexible approach to inferring the structure of dendritic dynamics that are important in neural computation, but are inaccessible to direct measurement with current experimental techniques.