Characterizing spontaneous Ca²⁺ local transients in OPCs using computational modeling

Abstract

Spontaneous ${\text{Ca}}^{2+}$ local transients (SCaLTs) in isolated oligodendrocyte precursor cells are largely regulated by the following fluxes: store-operated ${\text{Ca}}^{2+}$ entry (SOCE), Na⁺/Ca²⁺ exchange, ${\text{Ca}}^{2+}$ pumping through ${\text{Ca}}^{2+}$-ATPases, and ${\text{Ca}}^{2+}$-induced ${\text{Ca}}^{2+}$-release through ryanodine receptors and inositol-trisphosphate receptors. However, the relative contributions of these fluxes in mediating fast spiking and the slow baseline oscillations seen in SCaLTs remain incompletely understood. Here, we developed a stochastic spatiotemporal computational model to simulate SCaLTs in a homogeneous medium with ionic flow between the extracellular, cytoplasmic, and endoplasmic-reticulum compartments. By simulating the model and plotting both the histograms of SCaLTs obtained experimentally and from the model as well as the standard deviation of inter-SCaLT intervals against inter-SCaLT interval averages of multiple model and experimental realizations, we revealed the following: (1) SCaLTs exhibit very similar characteristics between the two data sets, (2) they are mostly random, (3) they encode information in their frequency, and (4) their slow baseline oscillations could be due to the stochastic slow clustering of inositol-trisphosphate receptors (modeled as an Ornstein-Uhlenbeck noise process). Bifurcation analysis of a deterministic temporal version of the model showed that the contribution of fluxes to SCaLTs depends on the parameter regime and that the combination of excitability, stochasticity, and mixed-mode oscillations are responsible for irregular spiking and doublets in SCaLTs. Additionally, our results demonstrated that blocking each flux reduces SCaLTs’ frequency and that the reverse (forward) mode of Na⁺/Ca²⁺ exchange decreases (increases) SCaLTs. Taken together, these results provide a quantitative framework for SCaLT formation in oligodendrocyte precursor cells.

Significance

Myelin sheaths in the central nervous system allow for fast signal transmission and metabolic support for neurons. Although neuronal activity is known to help regulate myelination through ${\text{Ca}}^{2+}$ transients, it remains unclear how such transients are generated spontaneously and locally in isolated oligodendrocyte precursor cells. Here, we provide a computational framework to quantify the contributions of various key ${\text{Ca}}^{2+}$ fluxes in generating the transients and the impact they exert upon blocking them. Our statistical and bifurcation analyses show that ${\text{Ca}}^{2+}$ signals are randomly generated, mainly due to excitability, stochasticity, and mixed-mode oscillations, and that information encoding is embedded in their frequency rather than their amplitude pattern. These results provide new insights into activity-dependent myelin plasticity.

Introduction

The oligodendrocyte lineage of cells is responsible for myelinating axonal fibers in the central nervous system (1). Oligodendrocyte progenitor cells (OPCs; also known as NG2 cells) develop into immature oligodendrocytes (OLs) and finally into myelinating mature OLs, with each developmental class having distinct roles in the creation of white matter (2,3). Evidence suggests that some control of myelination must depend on neuronal activity (4,5,6,7). This means that during development, strict energy budgets dictate how much myelin is allocated to each neuron (8).

According to the adaptive myelination hypothesis, it has been suggested that if neural electrical activity regulates subcellular events in OPCs necessary for myelin elaboration, then myelin would form preferentially on the electrically active axons (9,10). Evidence in favor of this hypothesis was first found in the electrical-activity-induced vesicular release by neurons, leading to increases in the number of rat optic nerve OPCs (11); it was later experimentally induced by stimulating cortical layer V projection neurons using optogenetics, causing OPCs in the deep layers of the premotor cortex and projections through the corpus callosum to proliferate (6). Interestingly, within the stimulated premotor areas, both the number of OLs and myelin thickness were increased.

${\text{Ca}}^{2+}$ signaling is the main pathway by which neural electrical activity mediates changes in myelination in OPCs. It is the key signaling molecule implicated in activity-dependent myelin growth in OPCs (12,13,14) and in controlling Stim1 and Golli, a set of two major proteins in the myelin basic protein family interactome (15). Inhibition of Ca²⁺ transients in OPCs impairs not only their process elaboration and branching but also myelin sheath size (16) and growth cone control (17). Indeed, slow, high-amplitude somatic Ca²⁺ currents cause greater diffusion of somatic Ca²⁺ to myelinating processes (18).

The frequency of Ca²⁺ transients appears more important than the absolute amplitude of fluctuations. Rate modulating experiments showed that Ca²⁺ induced myelin sheath elongation (in immature OLs) is promoted by high-frequency Ca²⁺ transients (19). On the other hand, low-frequency Ca²⁺ transients (or long Ca²⁺ bursts) promote shortening (19). However, the average rate of transients depends on cell maturity. Immature OLs have a high frequency Ca²⁺ transients, and the frequency of such transients increases as OL processes grow and elaborate (20).

Interestingly, spontaneous Ca²⁺ local transients (SCaLTs) are also present in isolated OPCs (i.e., in the absence of synaptic inputs from neurons), with profiles that exhibit both fast spiking along with underlying slow oscillations (16). This highlights the ability of OPCs in spontaneously generating such activity intrinsically. Evidence suggests that SCaLTs are produced by Ca²⁺ mobilization due to 1) Ca²⁺ entry into the cytosol governed by three main fluxes including store-operated Ca²⁺ entry (SOCE) using three primary SOCE channel proteins: ORAI1, STIM1, and STIM2 (15,16), Na/Ca²⁺ exchangers (NCXs) (20,21) that exchange three Na⁺ ions for one Ca²⁺ ion, and Ca²⁺ release events from the endoplasmic reticulum (ER) through inositol-trisphosphate receptors (IP₃Rs) (22) and ryanodine receptors (RyRs) (16,23,24), and 2) Ca²⁺ efflux through sarco/endoplasmic reticulum Ca²⁺-ATPase (SERCA) pumps (25) that transfer Ca²⁺ from the cytosol of the cell to the lumen of the ER and plasma membrane ${\text{Ca}}^{2+}$-ATPase (PMCA) pumps (26) that transfer Ca²⁺ from the cytosol to the extracellular medium.

The impact of manipulating some of these fluxes has been studied. For example, it was previously shown that applying ryanodine, a blocker of RyR, significantly reduces the frequency of SCaLTs (16,20), while blocking the SERCA pump to prevent reuptake into the ER by thapsigargin also diminishes SCaLTs. Additionally, applying KB-R7943 (an NCX reverse mode blocker) decreases SCaLT frequency, highlighting its significance in generating them (20). While these results elucidate the role of some of the fluxes in affecting SCaLTs, the contribution of other fluxes remain not fully understood, especially in the context of how they collectively produce the fast and slow components of SCaLTs.

In this study, we analyze experimental recordings (previously published, (16)) from targeted regions of interest in OPC processes. We adopted a computational approach by developing a stochastic spatiotemporal model (SSM) that simulates ${\text{Ca}}^{2+}$ dynamics in a one-dimensional homogeneous OPC process in order to systematically dissect the contributions of different fluxes to SCaLT formation. The model provides important insights into how irregular spiking and slow baseline oscillations are generated.

Materials and methods

Experimental data

${\text{Ca}}^{2+}$ fluorescence data obtained from rat OPCs, which were previously published (16), were used in this study. Briefly, cerebral cortices from postnatal day 0–2 rat pups were dissected and mechanically dissociated, with the final culture containing approximately 85%–90% OPCs and being devoid of neurons. The cultured OPCs were induced to differentiate and immunostained to allow ${\text{Ca}}^{2+}$ to be tracked at several sites on one cell. Regions of interest were targeted within OPC processes and imaged using fluo-4 at 0.5 Hz. In total 110 time series of $\Delta \text{F}/{\text{F}}_0$, Ca²⁺ fluorescence data were obtained in four groups of 27 independent recordings.

Mathematical model

We developed a flux-balanced based spatiotemporal Ca²⁺ spiking model to simulate intrinsic ${\text{Ca}}^{2+}$ dynamics of OPCs. The model is a combination of 1) the Li-Rinzel IP₃R/SERCA model of ${\text{Ca}}^{2+}$ handling (27), 2) the Levine-Keizer RyR model (28), 3) the Croisier et al. SOCE model (29), and 4) the Weber et al. NCX (and PMCA) model (30). It consists of seven equations, with one incorporating a second-order spatial term to represent cytosolic ${\text{Ca}}^{2+}$ diffusion and another incorporating noise to provide random input to the slow inactivation variable of IP₃ flux.

Descriptions and values for all the parameters of the model are given in Table 1. The equations describing the dynamics of ${\text{Ca}}^{2+}$ concentration in the cytosol $\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right)$ and ER $\left({\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}\right)$ are given by

$$\begin{array}{c}\frac{\partial {\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{\partial t}=\rho \frac{{\partial }^2{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{\partial x^2}+f_i(J_{\text{IP}3}\left(h\right)-J_{\text{SERCA}}+J_{\text{L}}+J_{\text{RyR}}\left(w\right)+J_{\text{SOCE}}\left(s\right)\\ +J_{\text{NCX}}({\left[{\text{Na}}^{+}\right]}_{\text{i}},\chi )-J_{\text{PMCA}})\end{array}$$ (1)

and

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}}{dt}=-\gamma f_e(J_{\text{IP}3}\left(h\right)-J_{\text{SERCA}}+J_{\text{L}}+J_{\text{RyR}}\left(w\right)).$$ (2)

Table 1.

Parameter values used in the SSM and DTM model simulations

Parameter	Value	Description	Source
IP3R flux (27)
$v_{\text{IP}3}$	0.88 s−1	max IP3R flux	(27)
$\left[{\text{IP}}_3\right]$	0.2 μM	IP3 concentration	(27)
$d_1$	0.13 μM	dissociation constant of IP3	(27)
$d_5$	0.08234 μM	dissociation constant of ${\text{Ca}}^{2+}$ (activation)	(27)
$a_2$	0.2 s	binding constant of ${\text{Ca}}^{2+}$ (inhibition)	(27)
$d_2$	1.049 μM	dissociation constant of ${\text{Ca}}^{2+}$ (inhibition)	(27)
$d_3$	0.9434 μM	dissociation constant of IP3	(27)
SERCA flux (27)
$v_{\text{SERCA}}$	120 μM s−1	max rate of SECRA pump	(28)
$k_3$	0.3 μM	dissociation constant for SERCA	(28,29,31)
Leak flux
$v_2$	0.5 s−1	rate constant for store leak	(28,32)
RyR flux (28)
$v_r$	18 s−1	max RyR flux	fitted
$k_a^4$	0.0192 μM4	ratio of kinetic constants	(28)
$k_b^3$	0.2573 μM3	ratio of kinetic constants	(28)
$k_c$	0.0571	ratio of kinetic constants	(28)
$k_c^{-}$	0.1 s−1	kinetic constants	(28)
SOCE flux (29)
$v_s$	0.0301 μM s−1	max SOCE flux	fitted
$k_s$	50 μM	STIM ER ${\text{Ca}}^{2+}$ affinity	(29)
${\tau }_s$	30 s	SOCE timescale	(29)
NCX flux (30)
${\left[{\text{Ca}}^{2+}\right]}_{\text{o}}$	12 mM	calcium concentration outside of the cell	(33)
${\left[{\text{Na}}^{+}\right]}_{\text{o}}$	14 mM	sodium concentration outside of the cell	(33)
$v_{\text{NCX}}$	1.4 A/F	max NCX flux	fitted
$\mu$	0.35	position of the energy barrier	(30)
$V$	−80 mV	transmembrane potential	(34)
$F$	96.5 mC/mol	Faraday’s constant	standard
$R$	8.314 JK−1 mol−1	gas constant	standard
$T$	300 K	temperature	standard
$k_{\text{sat}}$	0.25	saturation factor at negative potential	(30)
$k_{MC{a}_o}$	1.3 mM	Extracellular ${\text{Ca}}^{2+}$ dissociation constant	(30)
$k_{MN{a}_o}$	97.63 mM	extracellular Na+ dissociation constant	(30)
$k_{MC{a}_i}$	0.0026 mM	intracellular ${\text{Ca}}^{2+}$ dissociation constant	(30)
$k_{MN{a}_i}$	12.3 mM	intracellular Na+ dissociation constant	(30)
$k_n$	0.1 μM	calcium activation	(30)
PMCA flux (30)
$k_{\text{PMCA}}$	0.8 μM	dissociation constant for PMCA pump	(29,35,36)
$v_{\text{PMCA}}$	0.6 μM  s−1	max rate of PMCA pump	fitted
Ornstein-Uhlenbeck process
${\tau }_c$	0.01 s	characterisitc time constant	fitted
$D$	1.2	noise intensity	fitted
Other parameters
$f_i$	0.01	fraction of free calcium in cytosol	(37)
$f_e$	0.025	fraction of free calcium in ER	(37)
$\gamma$	9	ratio of cystolic to ER volume	(32,37)
$\rho$	50 μm2 s−1a	diffusion coefficient	(38)

^aEquivalent to 5 × 10⁻⁵ mm² s⁻¹.

Flux due to IP₃R $\left(J_{\text{IP}3}\right)$ is given by

$$J_{\text{IP}3}=v_{\text{IP}3}{m}_{\infty }^3{n}_{\infty }^3{h}^3({\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}),$$ (3)

$$m_{\infty }=\frac{\left[{\text{IP}}_3\right]}{\left[{\text{IP}}_3\right]+d_1},$$ (4)

$$n_{\infty }=\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}+d_5},$$ (5)

$$\frac{dh}{dt}=\frac{h_{\infty }-h}{{\tau }_h}+\eta \left(t\right),$$ (6)

$$h_{\infty }=\frac{Q_2}{Q_2+{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}},$$ (7)

$${\tau }_h=\frac{1}{a(Q_2+{\left[{\text{Ca}}^{2+}\right]}_{\text{i}})},$$ (8)

$$Q_2=d_2\frac{\left[{\text{IP}}_3\right]+d_1}{\left[{\text{IP}}_3\right]+d_3},$$ (9)

and

$$\frac{d\eta }{dt}=\frac{1}{{\tau }_c}\left(-\eta +\sqrt{2D{\tau }_c}{W}_t\right),$$ (10)

where $m_{\infty }$ and $n_{\infty }$ are the steady-state activation functions (indicating fast receptor activation) and $h\left(t\right)$ is a slow inactivation variable with an exponentially correlated noise term $\eta \left(t\right)$ added it in a manner similar to that used in (39,40). The noise $\eta \left(t\right)$ was generated by an Ornstein-Uhlenbeck process to represent slow clustering of IP₃R necessary for reproducing the slow oscillations in ${\text{Ca}}^{2+}$ signals. The parameter ${\tau }_c$ is the characteristic correlation time of the noise, $D$ is the noise intensity, and $W_t$ is a Gaussian white noise process with $\mu =0$ and $\sigma =1$.

Flux due to SERCA pumps $\left(J_{\text{SERCA}}\right)$ is given by

$$J_{\text{SERCA}}=\frac{v_{\text{SERCA}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}{k_3^2+{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}.$$ (11)

The constant ${\text{Ca}}^{2+}$ leak going into the cytosol from the ER through unspecified channels $\left(J_{\text{L}}\right)$ is given by

$$J_{\text{L}}=v_2\left({\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right).$$ (12)

The ${\text{Ca}}^{2+}$ flux due to RyR $\left(J_{\text{RyR}}\right)$ is given by

$$\begin{array}{l}{J}_{\text{RyR}}=v_{r}w\left(1+{\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{k_b}\right)}^3\right)/({\left(\frac{k_a}{1+{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}\right)}^4\hfill \\ +{\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{k_b}\right)}^3)({\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}})\hfill \end{array},$$ (13)

$$\frac{dw}{dt}=\frac{w_{\infty }-w}{{\tau }_w},$$ (14)

$$w_{\infty }=\frac{\left(1+{\left(k_a/{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right)}^4+{\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}/k_b\right)}^3\right)}{\left(1/k_c+{\left(k_a/{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right)}^4+1+{\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^3/k_b\right)}^3\right)}$$ (15)

and

$${\tau }_w=\frac{w_{\infty }}{k_c^{-}},$$ (16)

where $w\left(t\right)$ is a dynamic slow inactivation variable.

The flux of ${\text{Ca}}^{2+}$ through the SOCE channel $\left(J_{\text{SOCE}}\right)$ is given by

$$J_{\text{SOCE}}=V_{s}s,$$ (17)

$$\frac{ds}{dt}=\frac{s_{\infty }-s}{{\tau }_s},$$ (18)

and

$$s_{\infty }=\frac{k_s^4}{(k_s^4+{\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}^4)},$$ (19)

where $s\left(t\right)$ is a slow activation variable.

The reversible, polarization-dependant Ca²⁺ flux between the cytosol to the extracellular medium via the NCX $\left(J_{\text{NCX}}\right)$ is given by

$$J_{\text{NCX}}=\chi n_{\infty }\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right)J_{\Delta E},$$ (20)

$$\begin{gathered} k_{\chi }=k_{MC{a}_o}{\left[{\text{Na}}^{+}\right]}_{\mathrm{i}}^3+k_{MN{a}_o}^3{\left[{\text{Ca}}^{2+}\right]}_{\text{i}} \\ +k_{MN{a}_i}^3{\left[{\text{Ca}}^{2+}\right]}_{\mathrm{o}}\left(1+\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{k_{MC{a}_i}}\right) \\ +k_{MC{a}_i}{\left[{\text{Na}}^{+}\right]}_{\mathrm{o}}^3\left(1+{\left(\frac{{\left[{\text{Na}}^{+}\right]}_{\mathrm{i}}}{k_{MN{a}_i}}\right)}^3\right) \\ +{\left[{\text{Na}}^{+}\right]}_{\mathrm{i}}^3{\left[{\text{Ca}}^{2+}\right]}_{\mathrm{o}}+{\left[{\text{Na}}^{+}\right]}_{\mathrm{o}}^3{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}, \end{gathered}$$ (21)

$$J_{\Delta E}=v_{\text{NCX}}\frac{f\left(\mu \right){\left[{\text{Na}}^{+}\right]}_{\mathrm{i}}^3{\left[{\text{Ca}}^{2+}\right]}_{\mathrm{o}}-f\left(\mu -1\right){\left[{\text{Na}}^{+}\right]}_{\mathrm{o}}^3{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{k_{\chi }\left(1+k_{sat}f\left(\mu -1\right)\right)},$$ (22)

$$n_{\infty }=\frac{1}{\left(1+{\left(\frac{k_n}{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}\right)}^2\right)},$$ (23)

$$\frac{d\chi }{dt}=\frac{{\chi }_{\infty }-\chi }{{\tau }_{\chi }},$$ (24)

$${\chi }_{\infty }=1-\frac{1}{\left(\left(1+{\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}/k_{Ca}\right)}^2\right)\left(1+{\left(k_{Na}/{\left[{\mathrm{Na}}^{+}\right]}_{\mathrm{i}}\right)}^2\right)\right)}$$ (25)

$${\tau }_{\chi }=0.25+\frac{{\tau }_0}{\left(1+\left({\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{i}}/k_{\tau }\right)\right)},$$ (26)

and

$$\frac{d{\left[{\text{Na}}^{+}\right]}_{\text{i}}}{dt}=-3J_{\text{NCX}}\left({\left[{\text{Na}}^{+}\right]}_{\mathrm{i}},\chi \right),$$ (27)

where $\chi \left(t\right)$ is the slow inactivation variable of the NCX and $f\left(\mu \right)=\exp \frac{\mu VF}{RT}$.

Finally, ${\text{Ca}}^{2+}$ flux due to PMCA pump $\left(J_{\text{PMCA}}\right)$ is given by

$$J_{\text{PMCA}}=v_{\text{PMCA}}\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2+k_{\text{PMCA}}^2}.$$ (28)

To solve this system of differential equations, we implemented the forward time-centered space scheme with Neumann boundary conditions $\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\mathrm{i}}}{dt}=0$ at both ends. The Ornstein-Uhlenbeck equation was solved using the Euler-Murayama scheme. The time was discretized in intervals of 0.01 s and the space in intervals of 0.01 μm. To ensure numerical stability, the inequality $\rho \frac{\Delta t}{\Delta x^2}\le \frac{1}{2}$ was checked prior to running the simulations (41).

Data analysis

SCaLTs in both simulated and experimental data were identified by detecting peaks greater than 20% from the baseline, or 1 standard deviation above the mean. For all analyses, comparing simulated and experimental data time series were z-scored. To analyze the variability of ${\text{Ca}}^{2+}$ signals, we applied the method of Skupin et al. (42). For each realization of both data and simulation, the inter-SCaLT interval standard deviation $\sigma$ was plotted against the mean inter-SCaLT interval $⟨ISI⟩$ and fitted with the linear equation

$$\sigma =\alpha (⟨ISI⟩-IS{I}_{\min })+{\sigma }_{\text{min}},$$ (29)

where $IS{I}_{\min }$ and ${\sigma }_{\min }$ are the minimum inter-SCaLT interval and standard deviation, respectively. The intercept of the equation corresponds to an absolute refractory period in which no more SCaLTs can occur. The slope $\alpha$ is the coefficient of variation (CV) that indicates the rate of recovery of the Ca²⁺ signal from negative feedback (43). Generally, it is also a measure of the contribution of stochasticity. A CV of 1 is a Poisson process, while a CV of 0 indicates a deterministic process (44).

Hardware and software

Computer simulations of the SSM and the deterministic temporal model (DTM) obtained by ignoring diffusion and noise in the SSM were run on a workstation with an Intel CORE i7 12700k running at 3.6 Ghz and 32 GB of DDR5 RAM. Solutions to the SSM system were computed using a custom Python script optimized with the Numba JIT compiler (45). Versions of the SSM solver code are available for download at https://github.com/loprea91/Ca2SSM in Python, MATLAB, and Julia. The DTM was numerically solved in python; XPPAUT (a freeware available online at http://www.math.pitt.edu/bard/xpp/xpp.html) was used to compute the bifurcation diagrams. The DTM code and the XPPAUT function file are also available for download at the link above.

Results

Spatiotemporal model displays slow and fast oscillations

In order to replicate the SCaLTs in OPCs (16), we developed a homogeneous SSM (Fig. 1 A; materials and methods) of several microns of an OPC process that incorporated seven different fluxes, including those associated with SOCE, NCX, PMCA, IP₃R, RyR, leak, and SERCA. Although actin dynamics are strongly coupled to SCaLTs (16), we do not feature them in this flux-balance-based simplified model.

Figure 1.

Framework and outcome of the stochastic spatiotemporal model (SSM) of spontaneous Ca²⁺ local transients (SCaLTs). (A) Schematic of the two-compartment SSM showing all fluxes across the membrane of the cell and ER included in the model. The two compartments represent cytosolic and ER ${\text{Ca}}^{2+}$ concentrations. (B) Spatiotemporal heatmap of ${\text{Ca}}^{2+}$ amplitude over time produced by the SSM. (C) Time courses of z-scored ${\text{Ca}}^{2+}$ signals at a random spatial location along an OPC process obtained using the SSM (top) or a recorded ${\text{Ca}}^{2+}$ fluorescence signal (bottom). A threshold of 1 SD (20% greater than baseline [black]) was used to detect the presence of SCaLTs (green). Black arrows indicate doublets or SCaLTs that occur <20 s apart. To see this figure in color, go online.

Only cytosolic Ca²⁺ concentration $\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right)$ was assumed to diffuse freely. Stochasticity in the model was obtained by adding an exponentially correlated noise term, generated using an Ornstein-Uhlenbeck process, to the slow inactivation variable of IP₃R flux. The noise process with its slow time constant was interpreted to represent the slow clustering of IP₃R. SCaLTs were defined as events with amplitude greater than 1 standard deviation from the mean.

Simulating the SSM, using the parameter values reported in Table 1 (unless otherwise noted), over time for 900 s produced irregular spiking events distributed uniformly within the OPC process (Fig. 1 B). These SCaLTs show a wave pattern explained by Ca²⁺-induced Ca²⁺-release with a scale of coupling up to $\approx 1$ μm long. The length of this coupling could be calibrated to produce processes with higher or lower spatial correlation. To further examine these events, a random spatial location along the process was selected, and the time course of the ${\text{Ca}}^{2+}$ concentration ${\left[{\text{Ca}}^{2+}\right]}_{\text{i}}$ was then tracked (Fig. 1 C, top). Note that in Rui et al. (16), fluorescence recordings were obtained from larger regions of interest than the ones considered in this modeling study, where each signal was generated at a specific spatial location x obtained from discretizing the one spatial domain representing a process. However, we expect this to be a reasonable approximation of what would be observed in one pixel of an experimental region of interest. Furthermore, the fact that the width and height of a process are significantly smaller than its length makes the use of a one-spatial domain to represent a process also reasonable. This type of approximation has been previously applied to spatiotemporal models of protein-protein interactions (46).

To allow for direct comparison with ${\text{Ca}}^{2+}$ fluorescence recordings taken from rat pup OPCs and measured in terms of the ratio $\Delta \text{F}/\text{F}$ (Fig. 1 C, bottom), experimental and simulated traces were both z-scored and plotted with the same sampling rate of 0.5 Hz. The resulting simulated signal was strikingly similar to the ${\text{Ca}}^{2+}$ recording. They both exhibited irregular SCaLTs superimposed on a baseline that displayed slow oscillations (Fig. 1 C, black trace). The SCaLTs had faster upstrokes than downstrokes, a hallmark of ${\text{Ca}}^{2+}$ signals in OPCs, and sporadically appeared as doublets (Fig. 1 C, red arrowheads). These results suggest that the combination of fluxes included in the model are sufficient for producing the intrinsic ${\text{Ca}}^{2+}$ signals seen in OPCs and that the slow timescale of the stochastic IP₃R clustering could be underlying the slow baseline oscillations.

Model validation and statistical analysis of SCaLTs

In order to validate the SSM against experimental recordings, simulated and experimental data sets consisting of 110 independent simulations of 900 s with a sampling rate of 0.5 Hz were used. We selected a random spatial location not at the boundary of the process to be clear of edge effects. The magnitude of the z-scored experimental and simulated Ca²⁺ signals were binned, and histograms representing the number of points in each bin were plotted (Fig. 2, A and B, left). Although the profiles of experimentally obtained histograms were fairly heterogeneous, the majority of them (65%) were typically unimodal, peaking around zero with a relatively thin right-tail skew (Fig. 2 B, left). We believe these variations to be the result of recordings from different cells. The parameters reported in Table 1 replicated these data very closely (Fig. 2 A, left). The remaining 35% of histograms obtained from experimental recordings exhibited profiles that were roughly bimodal and centered around zero (Fig. 2 B, right). We surmised that varying the maximum rate of ${\text{Ca}}^{2+}$ efflux from the cytosolic compartment (including $v_{\text{PMCA}}$, $v_3$) could explain the heterogeneity in the data. To recapitulate this, we assigned 35% of simulations a randomly chosen ${\text{Ca}}^{2+}$ outflow rate between 50% and 100% of their default values (Fig. 2 A, right). Simulations with lower efflux generated nearly bimodal distributions centered at zero, in agreement with the smaller population of experimental recordings.

Figure 2.

Comparing the outcomes of the SSM to experimentally recorded heterogeneous fluorescence ${\text{Ca}}^{2+}$ signals. (A) Histograms of two z-scored simulated ${\text{Ca}}^{2+}$ signals produced by the SSM obtained using the default parameter values listed in Table 1 (left) or after randomly reducing the maximum outward ${\text{Ca}}^{2+}$ fluxes from the cytosolic compartment by 50%–100% of their default values (right). (B) Histograms of two z-scored recorded ${\text{Ca}}^{2+}$ signals showcasing two typical distributions obtained from a heterogeneous data: a left-skewed one clustered around zero and seen in 65% of the recordings (left), and a bimodal one centered around zero and seen in 35% of the recordings (right). (C) Linear fits of the standard deviation of ISIs $\sigma$ of both experimental (solid, black) and simulated (dashed, gray) recordings (65% group) versus average ISIs $⟨ISI⟩$, as defined by Eq. 29. Each circle represents one single experimental (black solid) and simulated (gray open) recording.

To further explore how simulations compare with fluorescence recordings, we analyzed two key properties of SCaLTs in the majority 65% group: the mean inter-SCaLT intervals (ISIs) $⟨ISI⟩$ and the ISI standard deviation $\sigma$. In general, different cells of the same type, when subjected to the same protocol, are not randomly scattered across the $⟨ISI⟩$–$\sigma$ plane but rather form a linear relation with slope $\alpha$ (42,47). We used this fact to “fingerprint” and thus compare the experimental data with simulations. The plot of $\sigma$ against average ISI $⟨ISI⟩$ for experimental recordings showed that the two properties are of the same magnitude $(\alpha =0.93\approx 1)$ (Fig. 2 C). This was well recapitulated in the simulated recordings, exhibiting variability very similar to that seen in the data $(\alpha =0.88)$. The linearity and the near-unity slope indicate that SCaLTs are mainly spontaneous transient events, not generated by a regular oscillatory mechanism perturbed by noise (42,43). Additionally, the intercepts for both simulation and data were almost zero, indicating there is no absolute refractory period (42).

Interestingly, despite the parameter variation in $v_{\text{PMCA}}$ and $v_{\text{SERCA}}$ when generating the linear fit (Fig. 2 C), the slope $\alpha$ remained robust without showing variability, which indicates that these SCaLTs are generated by the same stochastic mechanism (48). These robust linear trends with slopes less than 1 are indicative of frequency encoding (43). We verified this hypothesis by increasing the IP₃ concentration parameter $\left[{\text{IP}}_{\text{3}}\right]$, which led to an increase in SCaLT frequency (Fig. S6 A). This effect saturated when $\left[{\mathrm{IP}}_3\right]$ exceeded 1 μM (Fig. S6 B). These results thus validate the model and highlight the conclusion that information coding is actually embedded in the frequency of SCaLTs rather than in their amplitude patterns.

Contributions of fluxes to SCaLTs

After model validation, we went on to explore how fluxes included in the SSM contribute to producing fast spiking and slow oscillations in ${\left[{\text{Ca}}^{2+}\right]}_{\text{i}}$ (Fig. 3 A). To do this, we first tested the effects of blocking the SERCA pump by setting the J_SERCA term in the SSM to zero (Fig. 3 B). Our results showed that, at first, ${\left[{\text{Ca}}^{2+}\right]}_{\text{i}}$ rapidly increased as the ER depleted, then steadily decayed to zero (due to PMCA flux) in a manner similar to that previously seen when blocking SERCA pump with thapsigargin (20); fast spiking and slow oscillations were both abolished in this case. We then explored the role of RyR in generating SCaLTs by setting the J_RyR term to zero in the SSM. The removal of this flux significantly reduced SCaLTs and lowered the overall baseline ${\text{Ca}}^{2+}$ level (Fig. 3 C); the reduced number of SCaLTs was further confirmed when computing the average frequency of these SCaLTs over multiple realizations of the model (110 in total) in the presence and absence of this flux term (Fig. 3 C, inset). Such an outcome was in agreement with experimental observations obtained when blocking RyR with ryanodine (16). Similar results were predicted by the SSM when removing the IP₃R flux $\left(J_{\text{IP}3}\right)$ and flux due to PMCA pumps $\left(J_{\text{PMCA}}\right)$. In the former case, the SCaLTs and slow oscillations completely disappeared (Fig. 3 D), an expected outcome in view of the fact that the stochastic term is embedded in the inactivation variable of this flux term, while in the latter case, SCaLTs were significantly reduced (Fig. 3 E). Computing the average frequency of these SCaLTs over multiple realizations (110 in total) in the presence and absence of this latter flux confirmed the result (Fig. 3 E, inset).

Figure 3.

Effects of blocking ${\text{Ca}}^{2+}$ fluxes on SCaLTs in the SSM. Cytosolic ${\text{Ca}}^{2+}$ concentration ${\left[{\text{Ca}}^{2+}\right]}_{\text{i}}$ when (A) all fluxes were left unaltered (i.e., maximum flux rates were left at their default values, labeled wild-type), (B) maximum SERCA flux rate $\left(v_3\right)$ was set to zero, (C) maximum RyR flux rate $\left(v_r\right)$ was set to zero, (D) maximum IP₃R flux rate $\left(v_{\text{IP}3}\right)$ was set to zero, (E) maximum PMCA flux rate $\left(v_{\text{PMCA}}\right)$ was set to zero, and (F) direct and reverse modes of NCX were blocked individually or collectively. The trace in (F) shows a representative time series when NCX flux is without mode 2 (reverse mode). Insets in (C), (E), and (F) show average frequencies of SCaLTs obtained from 110 different simulations (realizations) of the SSM for each condition (mean ± SEM). WT, wild-type; RYR KO, RYR flux removed; PMCA KO, PMCA flux removed; M1 KO, mode 1 of NCX flux removed; M2 KO, mode 2 of NCX flux removed; none, NCX flux removed.

The NCX channel normally operates in its “forward” mode (M1), using the electrochemical gradient of Na⁺ to remove ${\text{Ca}}^{2+}$ from the cell, but switches to its “reverse” mode (M2) when internal Na⁺ levels are elevated, fluxing ${\text{Ca}}^{2+}$ inward. The reverse mode of the NCX channel tends to increase cell excitability and aid in the repletion of ER and buffered ${\text{Ca}}^{2+}$ stores (49). In order to study the effects of NCX flux, we removed either M1, M2, or both modes by discarding one or both terms in the numerator of Eq. 22 (Fig. 3 F). Removal of the reverse model M2 showed a significant decrease in SCaLTs, in agreement with experimental observations obtained when blocking this mode pharmacologically with KB-R7943 (20). In contrast, the SSM predicted that the removal of the direct mode M1 increases SCaLTs due to trapping of ${\text{Ca}}^{2+}$ within the cytosol and the subsequent activation of RyR and IP₃R. Interestingly, the model also predicted that the net effect of removing both modes decreases SCaLTs but not to the same extent as when removing M2 only. These differences were further corroborated by computing the average frequency of these SCaLTs over multiple realizations (110 in total) in the presence and absence of these modes (Fig. 3 F, inset).

Characterizing the regimes of behavior exhibiting specific patterns of SCaLTs

To analyze the underlying dynamic of the SSM, we first simplified the SSM into a deterministic temporal model (DTM). This was done by removing the stochastic term from the inactivation variable of IP₃ flux (Eq. 6), as well as discarding the diffusion term in the cytosolic term (Eq. 1). Using the continuation method in AUTO, we computed the bifurcation diagram of ${\left[{\text{Ca}}^{2+}\right]}_{\text{i}}$ in this simplified model with respect to $v_s$, the maximum SOCE flux (Fig. 4 A). The resulting dynamic structure was highly complex with a series of bifurcations that altered the behavior of the model significantly.

Figure 4.

Deterministic temporal model (DTM) displays diverse ${\text{Ca}}^{2+}$ signals when altering the maximum rate of SOCE flux. (A) One-parameter bifurcation of intracellular ${\text{Ca}}^{2+}$ concentration ${\left[{\text{Ca}}^{2+}\right]}_{\text{i}}$ with respect to maximal SOCE conductance $v_s$, highlighting the various regimes of behavior. Line colors indicate branch type: magenta (black) = branch of stable (unstable) equilibria, and solid/dotted green (solid blue) = envelopes of stable (unstable) periodic orbits. Bifurcations are indicated by their labels. HB, Hopf; PD, period doubling; HM, homoclinic; TR, torus; SNP, saddle-node of periodics. (B) Time series of DTM with the maximal flux rate of SOCE (v_s) set to various values as reported in the legend of each panel. The following profiles have been showcased (clockwise from top left): quiescent/excitable, tonic spiking, mixed-mode oscillations (doublets), and burst-like. Inset in the top left panel: single SCaLT generated by injecting a 1 s long 0.03 $\mu$M ${\text{Ca}}^{2+}$ pulse at 2,000 s (red arrow). To see this figure in color, go online.

Specifically, we found that for small maximum SOCE flux $v_s$ (Fig. 4 A), there was a branch of stable equilibria (magenta line) that lost stability at a Hopf bifurcation (HB1; black line). This branch then regained stability at HB2 (magenta line), lost it again at HB3 (black line), and finally regained it at HB4 (magenta line). At HB1, two envelopes of stable limit cycles (solid/dotted green lines) emerged. These envelopes initially displayed torus bifurcation very close to HB1 (Fig. 4 A) and then period-doubling bifurcation (PD1) close to $v_s\approx 0.05$ μMs⁻¹; this was followed by another period-doubling bifurcation (PD2) where the limit cycles became unstable (blue lines) before terminating at a homoclinic bifurcation (HM2). At HB2 and HB4, only unstable envelopes of periodic orbits (blue lines) emerged and terminated at homoclinic bifurcations (only HM1 is shown), whereas at HB3 (see inset in Fig. 4 A), very small envelopes of stable limit cycles (green lines) emerged initially before becoming unstable (blue lines) at period-doubling bifurcations and then at a saddle-node of periodics bifurcation. According to this configuration, two key observations can be made: 1) a regime of bistability between the stable equilibria and stable envelopes of limit cycles exists between HB2 and HB3, and 2) between HB3 and HB4, there are isolas of stable limit cycles (data not shown).

To understand the implications of how these bifurcations affect dynamics, the DTM was simulated in each regime with the purpose of determining how the profiles of their time series appear. In the regime before HB1 (e.g., at $v_s=0.001$ μMs⁻¹), the model was quiescent but excitable (Fig. 4 B, top left). Excitability was verified by showing that the DTM was able to generate a single SCaLT upon stimulation with a very brief (1 s) and small (0.03 μM) suprathreshold pulse (inset). This feature allowed the model to be sensitive to the Ornstein-Uhlenbeck noise process added to the slow inactivation variable of IP₃R and caused it to produce random SCaLTs. In the regime between torus bifurcation and PD1 (e.g., at $v_s=0.0301$ μMs⁻¹), on the other hand, the model produced mixed-mode oscillations with two pronounced peaks in each cycle in the form of doublets (Fig. 4 B, bottom left). Mixed-mode oscillations are characterized by alternations between small- and large-amplitude oscillations (50). Both fluctuations in ${\text{Ca}}^{2+}$ can be seen in the DTM (Fig. 4 B, bottom left, the wavy baseline and doublet), but the small-amplitude oscillations are likely hidden in the SSM and experimental data. The fact that the default value of maximum SOCE flux $\left(v_s\right)$ is very close to where HB1 provides a rationale as to why the SSM was intrinsically able to exhibit random SCaLTs and occasional doublets in simulated ${\text{Ca}}^{2+}$ signals in the presence of noise (Fig. 1 C, top), as well as suggests that the doublets seen in experimentally recorded ${\text{Ca}}^{2+}$ signals (Fig. 1 C, bottom) are an inherent aspect of SCaLTs in OPCs. Note that all fluxes included in the DTM contributed to generating the single SCaLTs and doublets, albeit to different levels (Figs. S1–S4).

Interestingly, two additional outcomes were also observed in other regimes of behavior, including tonic spiking and burst-like activity. Tonic spiking was seen in simulated ${\text{Ca}}^{2+}$ signals when the value of maximum SOCE flux was set to $v_s=0.25$ μMs⁻¹, a value that lies between PD1 and PD2 (Fig. 4 B, top right). Note that this behavior could be also generated in the bistable regime provided that the DTM is initiated away from the stable equilibria. In the other regime between PD2 and HB4 (e.g., at $v_s=0.36$ μMs⁻¹), however, Ca²⁺ signals exhibited a burst-like profile (Fig. 4 B, bottom right). Although ${\text{Ca}}^{2+}$ fluxes each contributed to spiking in this profile as well as the tonic spiking one (Figs. S1–S4), it is important to note that, experimentally, these two profiles would require a major level of SOCE potentiation to produce them. In other words, while it is theoretically possible to attain these parameter regimes through the DTM, physiologically, it is unlikely to find a pathway to induce such behaviors.

Excitability and mixed-mode oscillations underpin spiking in ${\text{Ca}}^{2+}$ signals produced by OPCs

In the previous section, we outlined the two most physiologically relevant dynamic behaviors associated with SCaLTs in OPCs, namely single spiking due to excitability and mixed-mode oscillations with doublets (Fig. 4). Here, we explored how these regimes define dynamics in the SSM, especially the source of irregularity seen in ${\text{Ca}}^{2+}$ spiking.

To do this, we first extended our previous analysis of the parameter regimes into performing a two-parameter bifurcation with respect to the maximum flux rates of IP₃R and SOCE $\left(v_s\right)$ (Fig. 5 A). This was done by tracing all Hopf bifurcations (obtained in Fig. 4 A) to generate the boundaries of the various regimes of behavior in this two-parameter space. The goal was to investigate the collective contribution of these two fluxes in defining DTM dynamics and to explore how perturbing them could alter dynamics of the SSM.

Figure 5.

Parameter regimes defined by the DTM dictate the outcomes of the SSM, including irregular ${\text{Ca}}^{2+}$ spiking and doublets. (A) Two-parameter bifurcation with respect to the maximum flux rates of IP₃R $\left(v_{\text{IP}3}\right)$ and SOCE $\left(v_s\right)$ obtained by tracing the Hopf bifurcations of Fig. 4 while changing these two parameters. Four different regimes of behavior were identified: quiescent (white), oscillatory containing mixed-mode oscillations and tonic spiking (yellow), bistable between quiescent and tonic spiking (green), and oscillatory containing tonic spiking and burst-like activity (blue). (B) Linear fits of the standard deviation of ISIs $\sigma$ versus average ISIs $⟨ISI⟩$, as defined by Eq. 29, for two simulated data sets: an original one generated using the default parameter values of $v_{\text{IP}3}$ and $v_s$ (black), identified by the i-labeled dot in (A) (see Table 1) and a new one generated by setting the values of these two parameters to $v_{\text{IP}3}=20$ s⁻¹ and $v_s=0.155$$\mu$M s⁻¹ (gray), identified by the ii-labeled dot in (A). Dotted lines represent the linear fits, while open circles represent simulated recordings. To see this figure in color, go online.

The resulting two-parameter bifurcation diagram obtained (Fig. 5 A) showed that there were four distinct regimes, two of which were oscillatory (yellow and blue), one was quiescent (white), and one that included both, i.e., was bistable (green). The yellow-colored oscillatory regime included two types of oscillatory behaviors (Fig. 4 B): mixed-mode oscillations with doublets close to the left boundary (i.e., in the vicinity of the i-labeled dot) and tonic spiking close to the right boundary (i.e., in the vicinity of the ii-labeled dot). The blue-colored oscillatory regime, on the other hand, included the tonic spiking and burst-like activity (Fig. 4 B). Finally, the bistable regime included the quiescent and tonic firing activity. Similar outcomes were also obtained when plotting the two-parameter bifurcation with respect to maximum RyR $\left(v_r\right)$, PMCA $\left(v_{\text{PMCA}}\right)$, and SERCA $\left(v_{\text{SERCA}}\right)$ flux rates (Figs. S5 and S7).

Although it was difficult to pinpoint the boundary of the excitable regime, the white-colored region to the left of the yellow-colored oscillatory regime happened to be excitable (Fig. 5 A). The default values of the IP₃R and SOCE maximum flux rates lie very close to the boundary of this region (at the i-labeled dot in Fig. 5 A). These default values allowed the SSM to generate a slope close to 1 for the linear fit of $\sigma$ versus $⟨ISI⟩$ (Fig. 2 C), as per Eq. 29. We postulated that moderately perturbing these two parameters by choosing values close to the right boundary of the yellow-colored oscillatory regime (at the ii-labeled dot in Fig. 5 A) would significantly alter the slope of the linear fit. To test this hypothesis, we generated 110 independent simulations of 900 s with a sampling rate of 0.5 Hz using this new parameter combination for IP₃R and SOCE maximum flux rates and performed the same linear fit defined by Eq. 29 (Fig. 5 B). The slope of this linear fit (gray dotted line) obtained from these new model realizations (gray open circles) was 0.51, a value substantially lower than the 0.88 slope of the linear fit (black dotted line) obtained from the original model realizations (black open circles) when default parameter values were used (the same data set as in Fig. 2 C). These results suggest that the combination of excitability and stochasticity are the necessary ingredients to generate irregularity in SCaLTs and that the mixed-mode oscillations are causing the system not only to exhibit doublets but also to make the slope of the linear fit slightly smaller than 1 due to its periodicity.

It is important to note that in the DTM model, the default value of $v_s$ (Table 1) is located just to the left of the Hopf bifurcation, lying within the type III excitability regime. In the deterministic spatiotemporal component of the SSM, however, the Hopf is significantly right shifted away from the default value of $v_s$ due to diffusion, a feature that can be verified using local perturbation analysis (46). Based on this, it follows that, in the SSM, the vast majority of SCaLTs were generated using type III excitability, whereupon noise occasionally shifts the system state sufficiently to the right, past the torus and Hopf bifurcations, to produce the doublets in the model.

Although our bifurcation analysis is described in detail for a subset of key maximum flux parameters, they showed significant robustness to modest variations away from their default parameter values. This robustness is also maintained by other maximum flux parameters, including $v_{\text{NCX}}$ and $v_{\mathrm{PMCA}}$ (data not shown). We expect other parameters to exhibit similar levels of robustness to such modest variations.

Discussion

SCaLTs in OPCs result from a complex interaction of multiple ${\text{Ca}}^{2+}$ fluxes that are produced or modulated by several ion channels, receptors, and pumps. These SCaLTs, through their coupling with the golli protein pathway in OPCs, suggest that they might play a key role in the development of myelin (51,52). In this study, we focused on the following key fluxes generated by those on the membrane of the cell: SOCE, NCXs, and PMCA pumps, as well as those on the membrane of the ER: IP₃Rs, RyRs, SERCA pumps, and leak. We used a data-driven computational modeling approach to investigate how these fluxes interact together to generate the SCaLTs seen OPCs.

The approach comprised of developing a SSM that included all the relevant fluxes and validating the model against ${\text{Ca}}^{2+}$ fluorescence data measured in terms of the ratio $\Delta \text{F}/{\text{F}}_0$ (16). The SSM was implemented over a one-dimensional OPC process (assuming the width and height of the process to be negligible compared with length) and incorporated an Ornstein-Uhlenbeck noise process that possessed a slow time constant into the inactivation variable of IP₃R flux. The latter was included due to the stochastic nature of ${\text{Ca}}^{2+}$ signaling that was quite prominent in our florescence recordings. Our implementation of adding noise to the slow inactivation variable of IP₃R resembled that used previously in a flux-balance based model for ${\text{Ca}}^{2+}$ (53). Indeed, this approach of implementing noise was also used when studying ion channel gating in Hodgkin-Huxley type models (54). We interpreted this noise to represent the slow clustering of IP₃R, leading to the stochastic dynamics. The Ornstein-Uhlenbeck noise implemented in our formalism is in the second timescale, concomitant with IP₃R clustering (55), rather than with IP₃R activation, which occurs in a millisecond timescale (56).

There are alternative formalisms that model the randomness of SCaLTs through randomness of single-molecule state transitions using stochastic Markov chain modeling coupled to deterministic diffusion equations (47,48). Because this approach is unlikely to produce the slow baseline oscillations seen in our SCaLTs recording, we opted not to apply such an approach. On the other hand, Voorsluijs et al. has described a stochastic model using mean-field equation formalism in which ${\text{Ca}}^{2+}$ spikes are generated by noise perturbation of limit cycles (57). There are likely several valid approaches that can be used to explain a given data set belonging to a specific cell line, but for the OPC cell recordings presented in this study, the formalism used here was sufficient to capture stochasticity seen in the data quite faithfully.

The model was then used to determine the contribution of each flux to SCaLTs and to decipher their underlying dynamics, making important predictions about how the irregular SCaLTs and slow baseline oscillations of the signal were generated. The model was able to reproduce results from several experimental protocols that involved RyR, SERCA, IP3R, and NCX channel blocking (see (20,58). Interestingly, each flux removal significantly reduced the frequency of SCaLTs to different levels, except for the forward mode of NCX, which did the opposite. The model suggests that the trapping of ${\text{Ca}}^{2+}$ inside the cell when blocking the forward mode of NCX was responsible; this represents a prediction that can be verified experimentally.

We showed, using the DTM, that the random nature of SCaLTs was a result of the excitable nature of the system in the most physiological regime (namely, the type III excitability regime) and that doublets were intrinsic to the system produced by noise-induced shifts to relevant parameter regimes. Specifically, in the type III excitability domain, noise (due to IP₃R clustering) pushes the system across a “firing threshold” and generates SCaLTs. If the noise is large enough, it could even perturbs the model sufficiently enough past the Hopf and the torus bifurcation, allowing the doublets to be produced. Using bifurcation analysis, we determined how the system depended on its current state in allowing a given ${\text{Ca}}^{2+}$ flux to exert its influence and demonstrated that the irregularity in SCaLTs would not persist if dynamics were shifted deep into the oscillatory regimes.

By plotting the linear fit of the standard deviation of ISIs of both experimental and simulated recordings against the average ISIs using Eq. 29, we demonstrated that the slope of the line was close to 1 and the intercept close to 0, indicating a highly stochastic process with no absolute refractory period. This also revealed that at its default parameter values (Table 1), the model could produce irregular Poissonian SCaLTs and that information coding was embedded in SCaLT frequency rather than SCaLT amplitude (42,43). However, the slope of the linear fit became significantly smaller than 1 when the model was simulated deeper in the tonic spiking regime. This indicates that there is a lower level of randomness in spiking and that excitability is the main driver of random SCaLT formation in OPCs, in agreement with previous observations (44,48). The effect of the recovery from the negative feedback described by the CV $\alpha$ has not yet been described analytically for such nonlinear systems, but a newer approach to this problem has been suggested recently (59).

Due to the stochastic nature of SCaLTs, we could not fit the model directly to the data. Indeed, experimental recordings displayed a great deal of statistical heterogeneity, thereby making it difficult to form any single metric to compare data and simulations. We resolved this issue by employing two approaches: one that compared the distributions of the experimental signals to those obtained by simulating the SSM (focusing on two representative distributions), and another that compared the linear fits. The parameter set in Table 1 effectively replicated the main characteristic features of the majority of the data using the two approaches. By doing so, we showed that scaling the total efflux from cytosol is necessary to replicate the bimodal distribution seen in 35% of the experimental recordings.

From a mathematical perspective, the dynamic structure of the DTM was not fully explored. There are likely interesting dynamics occurring at the regime bounded by PD2 and HB4 (Fig. 4 A). We presumed that this regime is likely occupied by an infinite number of isolas, each possessing a stable envelope of periodic orbits with specific number of SCaLTs associated with the burst-like activity. One can further investigate this behavior and apply slow-fast analysis (60) to determine the mechanism underlying this bursting behavior.

The spatiotemporal model in this article consists of one homogeneous process extending in space. The model could be further improved by the addition of coupled branches to mirror the dendritic nature of OPC processes. Moreover, the ER was assumed to be uniform across the process and in close proximity to the plasma membrane. One can develop a more realistic model by including a diffusion term into Eq. 2. Although this may provide an accurate framework to describe $\alpha$ dynamics in OPCs, we do not expect this additional complexity to alter to conclusions of this study.

Conclusion

Given that myelination in the central nervous system is likely coupled to the neuronal activity of nearby cells, developing this flux-balance based SSM in this study can provided a framework that can be further extended and analyzed to gain new insights about the intrinsic ${\text{Ca}}^{2+}$ transients underpinning neuro-oligo signaling. The investigation of this topic is critical for understanding myelin plasticity and the development of treatments for leukodystrophies and demyelinating diseases.

Author contributions

Conceptualization, N.D., X.J., and K.S.; formal analysis, N.D., X.J., L.O., K.S., and A.K.; investigation, N.D., X.J., and L.O.; methodology, N.D., X.J., L.O., and A.K.; software, N.D., X.J., and L.O.; validation, N.D., X.J., and L.O.; visualization, L.O.; writing – original draft, L.O. and A.K.; writing – review & editing, L.O. and A.K.; data curation, J.Q.Z.; funding acquisition, J.Q.Z. and A.K.; project administration, A.K.; resources, A.K.; supervision, A.K.

Editor: Arthur Sherman.