Abstract
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.
Highlights
The problem of understanding the mechanisms that govern the change in strength of a synapse remains a key problem in cellular neuroscience
The development of fast scanning multi-photon microscopy techniques has revealed that intracellular calcium concentrations play an important role in the interplay between backpropagating action potentials and excitatory post-synaptic potentials (EPSPs) that mediate synaptic changes through spike-timing dependent plasticity (STDP)
Using statespace 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
Summary
The problem of understanding the mechanisms that govern the change in strength of a synapse remains a key problem in cellular neuroscience. Fluorescence microscopy provides a way to examine aspects of the structure and the function of living cells that are inaccessible to direct electrical recording. The experimenter performs optical recordings after delivering fluorescent probe molecules that translate a biological or biochemical signal into an optical output (for reviews see [1,2]). Calcium indicators are such fluorescent probes that, upon binding calcium ions, change the amount of emitted light, which can be measured with a photo detector. The available experimental techniques still lead to noisy and spatiotemporally-subsampled observations of the true underlying calcium signals. Optimal spatiotemporal smoothing of the calcium profile on a dendritic tree given localized noisy measurements remains a difficult computational problem due to the high dimensionality (in terms of number of compartments) and complex structure of dendritic trees
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