Quantifying the uncertainty of spontaneous Ca2+ oscillations in astrocytes: particulars of Alzheimer's disease

Abstract

The quantification of spontaneous calcium (Ca(2+)) oscillations (SCOs) in astrocytes presents a challenge because of the large irregularities in the amplitudes, durations, and initiation times of the underlying events. In this article, we use a stochastic context to account for such SCO variability, which is based on previous models for cellular Ca(2+) signaling. First, we found that passive Ca(2+) influx from the extracellular space determine the basal concentration of this ion in the cytosol. Second, we demonstrated the feasibility of estimating both the inositol 1,4,5-trisphosphate (IP(3)) production levels and the average number of IP(3) receptor channels in the somatic clusters from epifluorescent Ca(2+) imaging through the combination of a filtering strategy and a maximum-likelihood criterion. We estimated these two biophysical parameters using data from wild-type adult mice and age-matched transgenic mice overexpressing the 695-amino-acid isoform of human Alzheimer β-amyloid precursor protein. We found that, together with an increase in the passive Ca(2+) influx, a significant reduction in the sensitivity of G protein-coupled receptors might lie beneath the abnormalities in the astrocytic Ca(2+) signaling, as was observed in rodent models of Alzheimer's disease. This study provides new, to our knowledge, indices for a quantitative analysis of SCOs in normal and pathological astrocytes.

Figure 1

Examples of the relative fluorescent changes ΔF/F₀ obtained from astrocytes of both WT and Tg2576 (APP) mice. In both types of mice, we are able to distinguish three kinds of Ca²⁺ signaling states: inactive, a stable Ca²⁺ baseline; active, irregular Ca²⁺ events; and hyperactive, a highly fluctuating Ca²⁺ baseline.

Figure 2

Procedure to characterize single Ca²⁺ events. (A) Examples of fitting a generalized-Gaussian function to a particular Ca²⁺ event in both WT (top) and Tg2576 (APP) (bottom) mice. The respective values for location τ_i, shape κ_i, scale α_i, amplitude A_i, and baseline c_i are also shown. (B) A least-square criterion was used to fit the generalized-Gaussian function to the actual ΔF/F₀ waveforms. Ca²⁺ events with fitting errors >0.1 (solid arrows) were considered doubtful, and therefore, excluded from the analysis. (C) The normalized Ca²⁺ events (solid, single trials; open, mean). The total numbers of well-fitted events for WT and Tg2576 (APP) mice, which additionally were required to have amplitudes >15%, were n = 147 and n = 121, respectively.

Figure 3

Dependency of the basal Ca²⁺ concentration in the cytosol C_basal (A) and the baseline for the total free Ca²⁺ ions per astrocyte volume c_o,b (B) with the physiological parameters $X_{I{P}_3}$ and j_in. (Dashed lines) Corresponding mean levels of these parameters for the experimentally observed C_basal in WT (0.081 μM) and APP(PS1) (0.149 μM) mice.

Figure 4

Statistical inference for a particular cell of a WT (left) and an APP (right) mouse. We used the most likely value of C_basal for each mouse type, i.e., j_in = 0.036 μM/s (WT mice) and j_in = 0.043 μM/s (Tg2576 mice). (Upper panel) Relative fluorescent changes ΔF/F₀ (solid) and the resulting innovations η (shaded) after model fitting for each cell. The estimated hidden state variables $\left\{\widehat{\left[I{P}_3\right]},\widehat{h},{\widehat{c}}_o\right\}$ are shown in three separated panels below.

Figure 5

Summary of parameter estimation and statistical tests for all cells in both WT (n = 67, solid bars) and Tg2576 (APP) (n = 84, shaded bars) mice. Though we provided an empirical estimation in this study, C_basal is experimentally unknown for Tg2576 mice. Therefore, the statistical inference in the particular case of using data from our APP transgenic mice was performed for different values of C_basal (i.e., mice with no β-amyloid accumulation, 81 nM; mice with a slight number of plaques, Tg2576, 92 nM; mice with a moderate number of plaques, 110 nM; and mice with a severe number of plaques, APP(PS1), 149 nM). The IP₃ production levels $X_{I{P}_3}$ were significantly higher in WT mice. While raising the value of C_basal, the estimators of variance ${\widehat{\sigma }}_h^2$ and the average number of IP₃R channels in a single cluster ${\widehat{N}}_{cluster}$ reduced and increased, respectively. For the predicted C_basal in Tg2576 mice, there were no significant differences with the corresponding estimators of these two parameters for WT mice. A bar represents the mean value of the estimated parameter and the 1.96 × SE.