A Bayesian approach to modelling heterogeneous calcium responses in cell populations

Abstract

Calcium responses have been observed as spikes of the whole-cell calcium concentration in numerous cell types and are essential for translating extracellular stimuli into cellular responses. While there are several suggestions for how this encoding is achieved, we still lack a comprehensive theory. To achieve this goal it is necessary to reliably predict the temporal evolution of calcium spike sequences for a given stimulus. Here, we propose a modelling framework that allows us to quantitatively describe the timing of calcium spikes. Using a Bayesian approach, we show that Gaussian processes model calcium spike rates with high fidelity and perform better than standard tools such as peri-stimulus time histograms and kernel smoothing. We employ our modelling concept to analyse calcium spike sequences from dynamically-stimulated HEK293T cells. Under these conditions, different cells often experience diverse stimulus time courses, which is a situation likely to occur in vivo. This single cell variability and the concomitant small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances. Our results therefore pave the way towards a statistical description of heterogeneous calcium oscillations in a dynamic environment.

A Statistical View on Calcium Oscillations

“…(1). A point in case is the Gamma distribution, which is consistent with Ca 2+ oscillations in HEK293 cells [67] and also with voltage spikes in neurons [68,69]. One common representation for the density of the Gamma distribution reads as…”

“…This will provide us e.g. with the most likely intensity function that is consistent with the data as well as confidence intervals, see [67]. A different approach is to try and sample from p(θ |y) without having to explicitly compute it.…”

“…which follows from the conservation of probability [67,77]. We illustrate a practical calculation for Eq.…”

“…An example for this is when cells are challenged with a time-varying stimulus as e.g. in [67]. Since the stimulus changes smoothly over time, a reasonable assumption is that the intensity function inherits this smoothness.…”

Deterministic Limit of Intracellular Calcium Spikes

“…Cell-level spikes can be surprisingly well described either by deterministic models with oscillatory dynamics [14] or by stochastic models, based on the coupling of clusters whose local dynamics is aperiodic [15] or on intensity functions describing spikes from a top-down perspective [16]. Experiments [5] strongly suggest that global spikes and waves emerge from a nucleation mechanism, which is reproduced by deterministic models considering excitable clusters [17].…”