Alzheimer’s disease (AD) is currently the most common neurodegenerative disease and is the major dementia type, which accounts for 60–70% of cases (Prince, 2015; Canter et al., 2016). The Alzheimer Disease International estimated in 2019 that over 50 million people were living with dementia and this number is expected to exceed 150 million by 2050 (Lynch, 2020). The disease has already caused incalculable losses worldwide. Little is known about the complex pathophysiology of AD, and thus, there is no cure. Pathological characteristics of accumulation of amyloid β-peptide (Aβ), is considered to be drivers in AD pathogenesis (Hardy and Selkoe, 2002; Birch, 2014).

Astrocytes were historically considered to provide support for neurons (Carmignoto and Gómez-Gonzalo, 2010). Since they were found to be involved in many brain functions and neurodegenerative diseases such as AD and Parkinson’s disease etc., astrocytes have become a hot topic in neuroscience research over the past few decades (Tewari and Majumdar, 2012; Tewari and Parpura, 2013; Bazargani and Attwell, 2016). In cultures of mixed neurons and astrocytes treated with Aβ, astrocytes always exhibit pathological alterations before neuronal death suggesting that astrocytes appear to be the primary target of Aβ (Abramov et al., 2003). Their role as protector and housekeeper in central nervous system is universally acknowledged, however, Aβ impairs important supportive astrocyte functions in AD cases (Verkhratsky and Nedergaard, 2018).

Astrocytes do not generate electrical signals like neurons (Wu et al., 2015). However, the concept of “cellular excitability” in astrocytes has been recently formalized to describe the changes in cytosolic Ca²⁺ concentration in response to chemical or mechanical stimulation (Verkhratsky, 2019). For example, they encode synaptic information via the modulation of intracellular calcium dynamics in response to synaptic activity (De Pittà et al., 2009). However, this kind of Ca²⁺ homeostasis can be disrupted by Aβ, whether in neurons or astrocytes, especially its soluble oligomeric form is more harmful (Demuro et al., 2010). In an AD mouse model, astrocytes displayed higher basal astrocyte Ca²⁺ levels and increased transient Ca²⁺ signals (Kuchibhotla et al., 2009).

Aβ is known to interfere with multiple calcium fluxes in astrocytes. Aβ oligomers deposition not only form pores in the lipid bilayer permeable to cationic ions, but also directly or indirectly activate L-type Ca_V, which increases the concentration of intracellular Ca²⁺ (Alves et al., 2019). Additionally, the expression of astroglial mGluR₅ is up-regulated by exposure to Aβ (Lim et al., 2013). Inside astrocytes, Aβ can induce endoplasmic reticulum (ER) Ca²⁺ release through ryanodine receptors (RyRs) and inositol triphosphate receptors (IP₃Rs) (Alberdi et al., 2013). How to combine these different findings to understand the big picture is crucial for further research (Markowetz, 2017). However, it is experimentally very difficult to investigate the interactions of many mechanisms of Ca²⁺ dyshomeostasis (Cutsuridis and Moustafa, 2017), e.g., the limitations of Ca²⁺ indicators and imaging techniques.

Simulation based on the mathematical model has been an invaluable tool to investigate complex interactions. There are hundreds of computational models on astrocyte Ca²⁺ dynamics and homeostasis either in a single astrocyte or in astrocyte networks or in neuron-astrocyte synapses (Manninen et al., 2018). Here, we focus primarily on modeling efforts in single astrocyte. Most of them studied Ca²⁺ oscillations, while a small part of them modeled spontaneous Ca²⁺ activity (Lavrentovich and Hemkin, 2008; Riera et al., 2011b); some assessed neurotransmitter-evoked Ca²⁺ excitability (De Pittà et al., 2009; Dupont et al., 2011). These computational studies are mostly based on classic models, including components for calcium-induced calcium release (CICR) and the sarco-endoplasmic Ca²⁺ ATPase pump (SERCA). In astrocytes, intracellular Ca²⁺ oscillations depend mainly on CICR, while Ca²⁺ influx from extracellular space via receptors or channels on membrane such as voltage-gated calcium channels (VGCCs) has also been linked with Ca²⁺ oscillations (Lavrentovich and Hemkin, 2008; Zeng et al., 2009). Taheri et al. (2017) modeled capacitive Ca²⁺ entry, which is mediated via store-operated Ca²⁺ channels. Ding et al. (2018) constructed two stochastic models, one describing the VGCC channel noise and the other describing the stochastic IP₃R dynamics. Recently, some modeling work has addressed the effect of Aβ on Ca²⁺ dynamics in generic cells (Latulippe et al., 2018). We established an Aβ-mediated calcium signaling model in astrocytes for the pilot study (Gao et al., 2020). All the models have useful implications for understanding Ca²⁺ signaling. However, no model has integrated the abovementioned putative mechanisms (Abramov et al., 2003; Alberdi et al., 2013; Lim et al., 2013; Alves et al., 2019) to simulate Ca²⁺ dynamics in astrocytes. The synergistic effect of the different Ca²⁺ fluxes on Ca²⁺ dynamics is not well understood so far. In particular, the current development of the theme lacks results regarding restoring the Ca²⁺ homeostasis in astrocytes during the progression of AD.

To date, the treatment of AD has remained a challenge. Although some promising drugs are under continuous development, clinical trials in recent years fail to get satisfied results (Qian et al., 2015). The currently approved anti-AD drugs fall into two types: cholinesterase inhibitors (donepezil, rivastigmine and galantamine) and N-methyl-D-aspartic acid receptor antagonists (memantine) (Aupperle, 2006). On the one hand, these drugs can only relieve symptoms but do not prevent the progression of AD, suggesting that their targets might not be the disease origin (Liu et al., 2019). On the other hand, memantine can prevent NMDAR-mediated Ca²⁺ flux indicating that other Ca²⁺ mechanisms may be a potential target for the treatment of AD when they have been demonstrated to play a proximal role in AD.

Here, a comprehensive model integrating multiple Aβ-affected Ca²⁺ pathways is proposed. This model of astrocytes describes Ca²⁺ signals as individual Ca²⁺ transport pathways rather than a macroscopic flow of Ca²⁺, including both intracellular release and extracellular influx. With this computational model, we can begin to study how Aβ affects each source of Ca²⁺ through various pathways. Recent wet-lab data are also incorporated into modeling work to obtain insights into AD treatment (Sadick and Liddelow, 2019). The explicit intention is to test the specific treatment strategy for AD.

Our simulations show that Aβ can trigger disruptions of cytosolic Ca²⁺ levels through interactions between various mechanisms, or, the single components. Each pathway exhibits different characteristics with the influence of Aβ. Changes in astrocyte resting membrane potential (RMP) were considered and incorporated into our model. With the intention of restoring dysregulated calcium signals, some therapeutic measures were tested, and the multi-pathway involved “combination therapy” gained effective recovery.

With a rapidly aging population, AD has become a major public health concern. However, the causation of AD remains unclear. Dysregulation of astrocytic Ca²⁺ has been widely regarded as an important component of AD. While controversy still remains, because some researches have not found acute [Ca²⁺]_i responses to Aβ (Casley et al., 2009; Toivari et al., 2011), this could possibly be attributed to the variability of Aβ species (monomers, oligomers, fibrils) and astrocyte types (from different brain areas) (Lim et al., 2014). More experiments reported that Aβ triggered transient [Ca²⁺]_i increases or [Ca²⁺]_i oscillations in astrocytes (Jalonen et al., 1997; Abramov et al., 2003; Alberdi et al., 2013; Lim et al., 2014). Unfortunately, the interactions of various mechanisms make it difficult to precisely understand how Aβ impacts cytosolic Ca²⁺ levels and individual fluxes. Decoupling various components and integrating their contributions into a full view may help us better understand the complexity of Aβ mechanisms in intracellular calcium dysregulation.

Computational modeling is a powerful approach that provides great opportunities to study complex mechanisms since it is experimentally difficult to isolate each component for separate studies. Our previous work on bioRxiv (Gao et al., 2020) has proposed a model related to AD focusing on the effect of Aβ through four different pathways on cytosolic Ca²⁺. Although we have considered a multi-pathway model, there is still room for more in-depth study of the individual contribution of Aβ on each pathway and methods of restoring the disrupted signals. There are also some limitations to the model itself. Specifically, for example, our prior study has only included the PLCδ-dependent production while IP₃ is regulated by two entirely different pathways in astrocytes. Also, differences in volume of cytosol and ER, and changes in membrane potential were neglected. These all deserve rigorous consideration and continued refinement.

In this paper, we modified and extended the previous studies in conjunction with experimental data and a comprehensive model was proposed. Not only all the issues mentioned are considered, but also the model is more physiologically reasonable. Just as important, the brevity and universality of the modeling method are worth noting. We previously used Lavrentovich’s model (Lavrentovich and Hemkin, 2008) to describe CICR dynamics depended on [IP₃] and [Ca²⁺]_i from both the ER and cytosol. However, a form simplified from De Young’s model (De Young and Keizer, 1992) by Li (Li and Rinzel, 1994) based on the gating of IP₃R provides good agreement with experimental recordings of channel opening and is more suitable for biophysical modeling. These, then, make a model more compatible with physiological data, providing good foundations for simulation studies of AD-induced dysregulation of astrocytic Ca²⁺.

In the experimental studies, Jalonen et al. (1997) exposed astrocytes to 1 μM Aβ and found that transient increase in [Ca²⁺]_i rose from about 0.6 to 0.9 μM. Takano et al. (2007) reported that after the injection of Aβ in both Dutch/Iowa mice and controls, the frequency of astrocytic Ca²⁺ oscillations increased significantly and became approximately three times as the control group. In our model, similar results were observed. The [Ca²⁺]_i and frequency are about 0.83 μM and 0.12 Hz when a = 0.6, compared to 0.58 μM and 0.034 Hz when a = 0. This means that our model has a good physiological agreement. Besides, we also found that the accumulation of Aβ not only leads to the increase of [Ca²⁺]_i and frequency in astrocytes, but also makes a transition from oscillation to a high steady state after exceeding a certain threshold, in which, the intracellular [Ca²⁺]_i will remain at a very high level, and this level will continue to rise rapidly with the increase of Aβ. In AD, the high intracellular [Ca²⁺]_i directly impacts memory formation and consolidation (Berridge, 2014).

Upregulation of astrocytic VGCCs expression in astrocytes has been indicated in pathological conditions, especially the L-Type in AD. Our model shows that VGCCs may contribute less to the dysregulation of cytosolic Ca²⁺ levels. This is because [Ca²⁺]_i is very robust either under high strength of the effect of Aβ or a large amount of Aβ. Even when there is a clear change in frequency, the sensitivity is low relative to the changes in Aβ. In response to the disruptions, Anekonda et al. (2011) have demonstrated that L-type Ca²⁺ current blockers can protect cells from the inductive effect of Aβ. We did confirm the efficacy of blocking VGCCs, but the recovery was not significant enough.

Aβ has been shown to impair the membrane permeability, due to the Aβ-formed pores resulting in the increase of Ca²⁺ influx through the membrane (Demuro et al., 2010). This additional influx shows damage to astrocytic homeostasis in our simulations. Especially, when oscillations disappear and the astrocyte calcium level is at a high steady state, Aβ can cause dramatic increases in [Ca²⁺]_i. In fact, channel formation has been already proposed as a molecular mechanism for Aβ toxicity in the early 1990s (Arispe et al., 1993), and our model, in the light of physiology, has well reflected how ionic leakage strongly affects and rapidly disrupts the cellular homeostasis. This intense pathology can be treated with Zn²⁺ (Abramov et al., 2003). We demonstrated that blocking the channel achieves desirable effects and may serve as a promising therapeutic approach.

Although disruptions on the membrane by Aβ are believed to be an important mechanism, the intracellular signaling pathways also deserve attention. In this study, we reflect the toxic effects of Aβ by increasing the channel open probability of RyRs. Other studies also suggested that Aβ can directly increase the RyR expression (Supnet et al., 2006). The increasing [Ca²⁺]_i and unchanged low steady state were observed. Unlike the other pathways, Aβ caused a decrease in frequency, which was more rapid in the presence of large amounts of Aβ because of the occurrence of mixed oscillations. Research has reported that RyR-mediated Ca²⁺ release can be reduced after treatment with RyR inhibitor (Oulès et al., 2012). Results showed a significant therapeutic effect on abnormal high [Ca²⁺]_i in our study.

Accumulated evidence indicates that Aβ is involved in the regulation of IP₃ production in AD (Demuro and Parker, 2013; Jensen et al., 2013). In this paper, we considered both Aβ-mediated channels or receptors, as well as non-directly acting intermediates such as IP₃, which embodies the concept of “multi-pathway” in a true sense and illustrates the complexity of our model. According to the results of the simulation, increasing IP₃ levels can lead to the decrease of [Ca²⁺]_i but the increase of frequency. The recovery effect is stronger than the blockage of VGCCs but not as effective as the other two pathways. Reducing the intracellular IP₃ level has a certain effect on frequency recovery, but the recovery effect on [Ca²⁺]_i is not ideal. This may reveal that IP₃ mainly regulates the frequency of intracellular calcium signals.

The RMP of astrocytes is also a contributing factor within our consideration. Compared to their neuronal counterparts, astrocytes display a highly negative RMP (Bolton et al., 2006). Some studies have shown that [K⁺]_o is critical in establishing the RMP of astrocytes (Anderson et al., 1995; Bellot-Saez et al., 2017). However, defective extracellular K⁺ clearance mechanisms have also been observed in AD which led to the loss of astrocyte polarization (Olabarria et al., 2010). Our model shows that increasing [K⁺]_o contributes to astrocyte depolarization, which to some extent reflects the impairment of astrocyte polarization by abnormal accumulation of [K⁺]_o in AD. During the process of depolarization, the amplitude of transmembrane Ca²⁺ flow mediated by VGCCs, as well as frequency increases. The affected J_VGCC, in turn, will cause further damage to cytosolic Ca²⁺ dynamics, manifested primarily by a large increase in [Ca²⁺]_i at steady state and a marked increase in frequency. This finding suggests that the frequency of Ca²⁺ events can be increased by depolarization of astrocytes, through activating of VGCCs. On the other hand, the accumulation of Aβ can narrow the range of membrane potential where oscillations are triggered. This may indicate the destructive nature of Aβ to synchronization oscillation of astrocyte network. Besides, astrocyte RMP displays extensive heterogeneity in the central nervous system (McNeill et al., 2021). To a certain extent, our model also reflects the characteristics of astrocytes with different RMP.

Finding a cure for AD is one of the most urgent and difficult tasks in modern medicine. At present, the drugs that are used to treat AD can only relieve symptoms but cannot slow down or reverse the progression of the disease (Alzheimer’s Association, 2020). The recent development and FDA approval of the AD drug Aducanumab, which targets Aβ, caused a great controversy (Vaz and Silvestre, 2020). But there is no doubt that drug research targeting Aβ or Tau proteins has once fallen into a bottleneck (Congdon and Sigurdsson, 2018; Foroutan et al., 2019), so it is advisable to explore feasible therapies from the perspective of Ca²⁺ dynamics. Mounting evidence has demonstrated that calcium signals play an indispensable role in AD (Yu et al., 2009), but Ca²⁺-pathway therapeutics remain undeveloped. Our simulations suggested that therapy targeting a specific receptor, channel or product is efficacious but limited because of multiple targets of Aβ, particularly in the terminal stages of the disease. Meanwhile, combination therapy can perfectly compensate for the shortage of single therapy. Generally, restoring calcium homeostasis is useful and necessary (Lim et al., 2014). Memantine, approved for the treatment of AD, has been clinically proven to be effective in preventing NMDAR-mediated calcium flux for decades, and here, we have demonstrated that other Ca²⁺ pathways in astrocytes may also be potential therapeutic targets and unraveled which Ca²⁺ pathway is effective because it is also likely to bring significant side-effects if the pathway is not carefully chosen.

It is challenging to establish a model of Aβ-mediated multi-pathway calcium dynamics. Although several mechanisms by which Aβ affects astrocytes have been experimentally demonstrated, how some of these mechanisms occur remains unclear. But the ubiquitous Ca²⁺ regulatory fluxes used in our model make it easily applicable for studying various cell types with spatial components. In fact, astrocytes exhibit a very complex morphology suggesting the spatiotemporal characteristics of Ca²⁺ signals in different structures (Semyanov et al., 2020) while we considered a spatially homogeneous astrocyte for better quantifying. However, even by adopting some simplifications, our model includes a large number of parameters, some of which still lack the effectiveness of the verification experiment. Our deterministic model generated regular calcium oscillations, which is consistent with the situation in some reports (Parri and Crunelli, 2001; Tashiro et al., 2002), However, in most cases, the calcium signals observed in the experiments presented clear irregularities which revealed its stochastic nature, meaning that some level of stochasticity may be closer to the physiology. All the deficiencies can be altered in future follow-up work as we continue to improve our understanding of the effects of Aβ in the astrocytic system. Our codes are publicly available for reproducibility, assisting with fast, convenient, and accurate validation of the model. Moreover, continuous improvement of the model combined with experimental data helps to make the model more useful.

In general, we have presented a hypothetical AD-specific model regarding Ca²⁺ dysregulation in astrocytes. The proposed general model incorporates multiple critical individually modeled Ca²⁺ mechanisms into a single framework, which obtains a more comprehensive picture compared to present work, especially fewer models are containing VGCCs and RyRs in astrocytes. To our knowledge, this is one of the few computational models to investigate the contribution of various Ca²⁺ fluxes to Ca²⁺ dynamics including entry and release under the influence of Aβ. Furthermore, we tested methods of blocking affected pathways. The “combination therapy” was first proposed and showed the significant effects on restoring calcium homeostasis. This may provide factual predictions for future drug development. Our study can provide an in-depth understanding of AD and pave the way toward the development of much more effective treatment modalities.

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author/s.

SC, HG, and AZ conceived and designed the research. LL and HG conducted literature research and wrote MATLAB code. LL, SC, and HG performed simulations and analyzed data and discussed the results and wrote the article with input from AZ. SC supervised the study. LL plotted pictures. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.