Deterministic Limit of Intracellular Calcium Spikes

In nonexcitable cells, global ${\mathrm{Ca}}^{2+}$ spikes emerge from the collective dynamics of clusters of ${\mathrm{Ca}}^{2+}$ channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin, while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Figure 1

Model describing the dynamics of a cluster of ${\mathrm{IP}}_3\mathrm{Rs}$ from Calabrese et al. [22].

Figure 2

(a) Puff sequence showing the local $[{\mathrm{Ca}}^{2+}]$ averaged in 1 fL around an isolated cluster, corresponding to experimental observations in the presence of EGTA. ${\mathrm{\Sigma }}_{\mathrm{MF}}=5\mu \mathrm{M}{\mathrm{s}}^{-1}$ ($\mathrm{\Sigma }=500\mu \mathrm{M}{\mathrm{s}}^{-1}$ and ${\overline{\rho }}_c=0.01$). (b) Spike sequence showing the global $[{\mathrm{Ca}}^{2+}]$ averaged over the whole system, corresponding to experimental observations in the absence of EGTA. ${\mathrm{\Sigma }}_{\mathrm{MF}}=50\mu \mathrm{M}{\mathrm{s}}^{-1}$ ($\mathrm{\Sigma }=500\mu \mathrm{M}{\mathrm{s}}^{-1}$ and ${\overline{\rho }}_c=0.1$). The mean ISI is $(102.1\pm 50.4)\mathrm{s}$. (a),(b) The whole system size is $5\times 5\mu {\mathrm{m}}^2$, $k_{co}=20\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $v_p=0.9\mu \mathrm{M}{\mathrm{s}}^{-1}$, and the other parameters are given in Table 1.

Figure 3

Bifurcation diagram showing the stationary $[{\mathrm{Ca}}^{2+}]$ (in black) and the maximal and minimal $[{\mathrm{Ca}}^{2+}]$ reached during oscillations (in red) for $k_{co}=20\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $v_p=0.9\mu \mathrm{M}{\mathrm{s}}^{-1}$, and the other parameter values given in Table 1. Stable and unstable branches are in plain and dashed lines, respectively. HB and LP stand for Hopf bifurcation and fold, respectively.

Figure 4

${\sigma }_{\mathrm{ISI}}-T_{\mathrm{av}}$ plot obtained by numerical simulations with $\mathrm{\Sigma }=500\mu \mathrm{M}{\mathrm{s}}^{-1}$, ${\overline{\rho }}_c=0.1$ (average intercluster distance of $1\mu \mathrm{m}$), and (a) $k_{co}=20\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ and $v_p=0.9\mu \mathrm{M}{\mathrm{s}}^{-1}$, (b) $k_{co}=5\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ and $v_p=0.9\mu \mathrm{M}{\mathrm{s}}^{-1}$, and (c) $k_{co}=20\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ and $v_p=0.7\mu \mathrm{M}{\mathrm{s}}^{-1}$. The other parametric values are given in Table 1. Each dot corresponds to a given spike sequence of at least ten spikes and the solid line is the linear fit through these points. $T_{\min }$ is equal to (a) 6.6 s, (b) 41.8 s, and (c) 12.59 s. The dashed line corresponds to the case $\alpha =1$.

Figure 5

Comparison between the deterministic period (plain line) and the average ISI (dots) as a function of ${\mathrm{\Sigma }}_{\mathrm{MF}}$ in the range between HB1 and HB2. Black and red dots correspond to ${\rho }_c=0.1$ and ${\rho }_c=1.0$, respectively. Error bars indicate variations of $\pm {\sigma }_{\mathrm{ISI}}$.