Impact of β-Amyloids Induced Disruption of Ca²⁺ Homeostasis in a Simple Model of Neuronal Activity

Abstract

Alzheimer’s disease is characterized by a marked dysregulation of intracellular Ca²⁺ homeostasis. In particular, toxic β-amyloids (Aβ) perturb the activities of numerous Ca²⁺ transporters or channels. Because of the tight coupling between Ca²⁺ dynamics and the membrane electrical activity, such perturbations are also expected to affect neuronal excitability. We used mathematical modeling to systematically investigate the effects of changing the activities of the various targets of Aβ peptides reported in the literature on calcium dynamics and neuronal excitability. We found that the evolution of Ca²⁺ concentration just below the plasma membrane is regulated by the exchanges with the extracellular medium, and is practically independent from the Ca²⁺ exchanges with the endoplasmic reticulum. Thus, disruptions of Ca²⁺ homeostasis interfering with signaling do not affect the electrical properties of the neurons at the single cell level. In contrast, the model predicts that by affecting the activities of L-type Ca²⁺ channels or Ca²⁺-activated K⁺ channels, Aβ peptides promote neuronal hyperexcitability. On the contrary, they induce hypo-excitability when acting on the plasma membrane Ca²⁺ ATPases. Finally, the presence of pores of amyloids in the plasma membrane can induce hypo- or hyperexcitability, depending on the conditions. These modeling conclusions should help with analyzing experimental observations in which Aβ peptides interfere at several levels with Ca²⁺ signaling and neuronal activity.

1. Introduction

Ca²⁺ ions play a key role in intracellular signaling by controlling vital physiological processes such as secretion, gene expression, and apoptosis [1]. In neurons, they also act as a charge carrier. Thus, during neuronal activity, Ca²⁺ concentration and plasma membrane (PM) voltage changes are intimately related: Ca²⁺ entry through voltage gated Ca²⁺ channels can affect Ca²⁺ signaling, while voltage independent Ca²⁺ changes are able to alter PM excitability [2].

This dual role of Ca²⁺ as an ion involved in signaling and in neuronal activity is highly relevant for the understanding of the development of neurodegenerative diseases and, in particular, Alzheimer’s disease (AD). Together with the hyperphosphorylation of τ proteins and the accumulation of amyloid-β peptides (Aβ) inside neurons, the progression of AD is indeed characterized by a perturbation of intracellular Ca²⁺ homeostasis [3]. Ca²⁺ dysregulation associated with neurodegenerative diseases is multifactorial. Aβ can indeed affect Ca²⁺ exchanges with the extracellular medium and with internal Ca²⁺ stores [4]. These modifications result in an increase in cytosolic Ca²⁺. It is now widely accepted that disruption of ER calcium mediates the most significant signaling pathways that are associated with Alzheimer’s disease, and is a main driving force in the development of the disease [5,6,7]. Among many consequences, excessive Ca²⁺ increase can lead to apoptotic neuronal death. Ca²⁺ also stimulates the activity of γ-secretase, a key enzyme involved in the production of toxic Aβ peptides. Such a vicious circle between amyloids and Ca²⁺ has been proposed to play a key role in the progressive and irreversible character of the disease [4,8].

At the level of neuronal activity, extracellular plaques of Aβ peptides are reported to induce electrical hyperexcitability [9]. Such an excitatory effect relies on a network effect associated with the disruption of synaptic function [6], but similar observations have been found on cultured neurons [10] or in brain slices [11,12]. How Aβ peptides, Ca²⁺, and neuronal excitability are related to each other at the single cell level is not well understood and is difficult to assess experimentally. Intuitively, increases in Ca²⁺ are expected to be associated with a decrease in neuronal activity via the activation by Ca²⁺ of a hyperpolarizing K⁺ current mediated by SK or BK channels, which represent the major link between Ca²⁺ concentration and the electrical activity of the plasma membrane. The question thus arises as to how AD’s progression can be associated at the same time with the increase in intracellular Ca²⁺ that follows the calcium hypothesis of AD, and with an increase in neuronal electrical activity. Computational modeling is well suited to gain insight into this paradox, because it allows for individually assessing the consequence of Aβ peptides on their various targets. It can also predict their relative importance.

From a mathematical point of view, interactions between membrane electrical activity and Ca²⁺ signaling have been described in specific cell types, such as astrocytes [13,14], pancreatic cells [15,16], GnRH neurons [17] motoneurons [18], and cardiac cells [19,20]. To the best of our knowledge, there are, however, very few general computational models describing electrical activity and intracellular Ca²⁺ dynamics in a prototypical neuronal cell. An early generic model of a neuron shows that less current is necessary to generate an AP when Aβ peptides are considered to block the fast-activating potassium current [21]. In a previous study, we used a simple model of the electrical activity of granule cells, considering its interplay with membrane Ca²⁺ dynamics, to study the impact of tonic NMDA receptor activity on neuronal excitability [22]. Modeling could show that the constant Ca²⁺ influx induced by this tonic activity can lock previously active neurons in a hyperactive state. This result agrees with the observed hyperactivation of neurons in presence of Aβ-induced glutamate spill-over [23,24].

In the present study, we extend the same minimal model of granule cell electrical activity and membrane Ca²⁺ dynamics to consider a more accurate description of the evolution of intracellular Ca²⁺, including exchanges with the ER. The model involves three compartments: a fictious, sub-plasmalemmal shell that controls PM electrical activity, the cytosol, and the ER. It considers Ca²⁺ fluxes through the plasma membrane Ca²⁺ ATPase (PMCA), the sodium calcium exchanger (NCX), the sarco- or endoplasmic reticulum Ca²⁺ ATPase (SERCA) pumps, and the inositol 1,4,5-trisphosphate receptor (IP₃) and ryanodine receptors (IP₃R and RYR), grouped together in a single phenomenological flux. In this first approach, we indeed do not incorporate Ca²⁺ fluxes resulting from stimulation by IP₃-inducing agonists. Our aim is to investigate how long-term alterations in Ca²⁺ fluxes modify the resting concentrations of Ca²⁺ in the different compartments and neuronal excitability. The model is then used to systematically investigate the various effects of Aβ peptides that have been reported in the literature. We found a strong independence between changes in cytosolic and subplasmalemmal Ca²⁺ concentrations, allowing cytosolic Ca²⁺ to vary without affecting neuronal excitability. In contrast, the latter is much affected by changes in subplasmalemmal Ca²⁺, with decreases in PMCA activity provoking neuronal hypo-excitability. Hyperexcitability is however predicted by the model when simulating the Aβ peptides reported changes in the L-type Ca²⁺ current [25] and the experimentally reported effect of Aβ peptides on Ca²⁺-sensitive K⁺ channels [26]. Lastly, an influx of Ca²⁺ flux through the pores of Aβ oligomers [27,28] can also increase the frequency of the simulated spikes. Altogether, the model helps understanding the apparent paradoxical dual role of Aβ peptides in increasing intracellular Ca²⁺ and provoking neuronal hyperexcitability. It also provides a simple description of intracellular Ca²⁺ dynamics and membrane excitability that can be further adapted to describe Ca²⁺ dysregulation in neurodegenerative diseases in a finely tuned manner.

2. Description of the Model

The core model used in this study is a previously proposed simple mathematical description of the electrical activity of cerebellar granule cells [2,29]. This model has been experimentally validated and can be seen as a good description of the electrical activity of a prototypical neuron. In the context of Alzheimer’s disease, we extended the model to consider tonic activity of NMDA receptors [22], which results from the Aβ-induced spill-over of glutamate [30].

The initial core model describes the evolution of the membrane voltage and Ca²⁺ concentration in a sub-plasmalemmal compartment that can be seen as a thin shell just below the plasma membrane (Ca). The currents included in the model, schematized in black in Figure 1, are a voltage-dependent Na⁺ current (I_Na(V)), a delayed rectifier K⁺ current (I_K(V)), a high-threshold voltage dependent Ca²⁺ current (I_Ca(V)), a Ca²⁺-activated K⁺ current (I_K(Ca)), and I_inj is the current injected into the cell that allows for studying its excitability properties. The mathematical expressions used for the currents, together with the evolution equations for the gating variables, are given in the Supplementary Material. Only one type of I_K(Ca) current is considered explicitly. In reality, most excitable cells express both large-conductance (BK) and small-conductance (SK) calcium-activated potassium channels. The former is sensitive to Ca²⁺ only, while the latter is sensitive to both voltage and calcium. For simplicity, Equations (S7), (S18) and (S19) empirically describe the two types of channels. The equations and parameter values that govern the dynamics of opening/closing of the ion channels are the same as in the previous studies [2], and are given in Supplementary Material and

Table 1. List of the parameters of the model. Values indicated have been used for all figures, except if mentioned in the legend. Parameters associated with the dynamics of the currents are given directly in the equations listed in the Supplementary Material.

Parameter Name	Parameter Meaning	Value
${\mathrm{C}}_{\mathrm{m}}$	Membrane capacity	3.14 pF
${\mathrm{V}}_{\mathrm{shell}}$	Volume of the shell	$26.378 {\mathsf{\mu }\mathrm{m}}^3$
F	Faraday constant	$96.485 {\mathrm{C} \mathrm{mol}}^{-1}$
f	Calcium buffering capacity of the cytoplasm	0.01
${\mathrm{I}}_{\mathrm{inj}}$	Injected Current	[0.150] pA
${\mathrm{k}}_{\mathrm{in}}$	$\mathrm{Rate} \mathrm{constant} \mathrm{of} \mathrm{unstimulated} \mathrm{C}{\mathrm{a}}^{2+}$ release from the ER	$0.015 {\mathrm{ms}}^{-1}$
${\mathrm{Ca}}_{\mathrm{ER}}$	$\mathrm{ER} {\mathrm{Ca}}^{2+}$ concentration	$500 \mathsf{\mu }\mathrm{M}$
γ	Shell-Cytoplasm Diffusion Coefficient	$0.001{ \mathrm{ms}}^{-1}$
${\mathrm{V}}_{{\mathrm{Na}}^{+}}$	$\mathrm{Equilibrium} \mathrm{Potential} \mathrm{for} {\mathrm{Na}}^{+}$ ions	55 mV
${\mathrm{V}}_{{\mathrm{K}}^{+}}$	$\mathrm{Equilibrium} \mathrm{Potential} \mathrm{for} {\mathrm{K}}^{+}$ ions	−90 mV
${\mathrm{V}}_{{\mathrm{Ca}}^{2+}}$	$\mathrm{Equilibrium} \mathrm{Potential} \mathrm{for} \mathrm{C}{\mathrm{a}}^{2+}$ ions	80 mV
${\mathrm{V}}_{\mathrm{NCX}}$	Equilibrium Potential for the NCX channel	−20 mV
${\mathrm{g}}_{{\mathrm{Na}}^{+}}$	$\mathrm{Conductance} \mathrm{of} \mathrm{the} \mathrm{voltage-gated} {\mathrm{Na}}^{+}$ channel	172 nS
${\mathrm{g}}_{{\mathrm{K}}^{+}}$	$\mathrm{Conductance} \mathrm{of} \mathrm{the} \mathrm{voltage-gated} {\mathrm{K}}^{+}$ channel	28 nS
${\mathrm{g}}_{{\mathrm{Ca}}^{2+}}$	$\mathrm{Conductance} \mathrm{of} \mathrm{the} \mathrm{voltage-gated} {\mathrm{Ca}}^{2+}$ channel	58 nS
${\mathrm{g}}_{\mathrm{K}-{\mathrm{Ca}}^{2+}}$	$\mathrm{Conductance} \mathrm{of} \mathrm{the} {\mathrm{K}}^{+}$$\mathrm{activated} {\mathrm{Ca}}^{2+}$ channel	56.5 nS
${\mathrm{K}}_{\mathrm{NCX}}$	Half saturation constant of the NCX channel	$1 \mathsf{\mu }\mathrm{M}$
${\mathrm{K}}_{\mathrm{PMCA}}$	Half saturation constant of the PMCA	$1 \mathsf{\mu }\mathrm{M}$
${\mathrm{K}}_{\mathrm{SERCA}}$	Half saturation constant of the SERCA	$0.2 \mathsf{\mu }\mathrm{M}$
${\mathrm{V}}_{\mathrm{PMCA}}$	Maximal velocity of the PMCA	$75 {\mathsf{\mu }\mathrm{M} \mathrm{ms}}^{-1}$
${\mathrm{V}}_{\mathrm{SERCA}}$	Maximal velocity of the SERCA	$195 {\mathsf{\mu }\mathrm{M} \mathrm{ms}}^{-1}$
${\mathrm{V}}_{\mathrm{Abeta}}$	$\mathrm{Maximal} \mathrm{rate} \mathrm{of} {\mathrm{Ca}}^{2+}$ release through pores of Aβ peptides	$0 {\mathsf{\mu }\mathrm{M} \mathrm{ms}}^{-1}$
${\mathrm{q}}_1$	$\mathrm{Voltage} \mathrm{dependence} \mathrm{of} {\mathrm{Ca}}^{2+}$ entry through the pores of Aβ peptides	−30 mV
${\mathrm{q}}_2$	$\mathrm{Voltage} \mathrm{dependence} \mathrm{of} {\mathrm{Ca}}^{2+}$ entry through the pores of Aβ peptides	23 mV

. Temporal evolution of the membrane voltage is as follows:

$$C_m\frac{dV}{dt}=-I_{Na\left(V\right)}-I_{K\left(V\right)}-I_{Ca\left(V\right)}-I_{K\left(Ca\right)}+I_{Inj}$$

(1)

where C_m is the cell capacitance. When active, the I_Ca(V) current mediates the entry of Ca²⁺ into the cell and hence leads to an increase in Ca²⁺ concentration in the sub-plasmalemmal compartment. Thus,

$$\frac{dCa}{dt}=f\left[\frac{-I_{Ca\left(V\right)}}{2F{V}_{shell}}-\beta Ca\right]$$

(2)

where f is the buffering capacity of the cytoplasm, F the Faraday constant, V_shell the volume of the sub-plasmalemmal compartment, and β is a first order rate constant that empirically gathers all the processes that remove Ca²⁺ from the shell.

Here, we extend the model to consider a cytosolic and an endoplasmic reticulum (ER) Ca²⁺ compartment, and to provide a more accurate description of the processes that remove Ca²⁺ out of the shell. Note that while the subplasmalemmal space represents a fictious compartment, the cytosol and ER correspond to real cellular spaces delimited by membranes. The model is schematized in Figure 1. From the sub-plasmalemmal compartment, Ca²⁺ is extruded towards the extracellular medium via an ATPase and via the Na⁺/Ca²⁺ exchanger (NCX). The rate of Ca²⁺ transport through the ATP-consuming plasma membrane Ca²⁺ ATPase (PMCA) is given by:

$$J_{PMCA}=V_{PMCA}\frac{C{a}^2}{K_{PMCA}^2+C{a}^2}$$

(3)

in which V_PMCA and K_PMCA stand for the maximal velocity and the half saturation constant of the PMCA, respectively. Note that export of Ca²⁺ by the PMCA is accompanied by a concomitant influx of H⁺ [31], in such a way that the activity of this pump does not influence the membrane voltage. The Na⁺/Ca²⁺ exchanger removes Ca²⁺ from the cytoplasm at the expense of Na⁺ entry, with a 1:3 stoichiometry. It is thus electrogenic. Following Gall et al. (1999) [16], we describe this current by

$$I_{NCX}=g_{NCX}\frac{C{a}^2}{K_{NCX}^2+C{a}^2}\left(V-V_{NCX}\right)$$

(4)

where g_NCX is the NCX conductance, K_NCX the half saturation constant for NCX activation by Ca²⁺, and V_NCX the resting potential for this channel calculated using resting Na⁺ and Ca²⁺ concentrations. This current is added (with a negative sign) to the right-hand side of Equation (1) and with the appropriate factors to the evolution equation for Ca (see below).

To introduce Ca²⁺ diffusion from the sub-membrane region to the cytosol, we constructed a compartmental model wherein the cytoplasm and the shell are assumed to be homogeneous compartments. In such an approach, the corresponding efflux of Ca²⁺ from the cytosol considers the Ca²⁺ diffusion coefficient and the respective volumes of the compartments. It can be written as:

$$J_{diff}=-\gamma \left(Ca-C{a}_{cyt}\right), \mathrm{with} \gamma =\frac{D}{h^2}$$

(5)

where D stands for the effective Ca²⁺ diffusion in the cytosol and h for the distance between the shell and the cytoplasmic compartment. Thus, for a typical granule cell with a radius of ~4 μm, γ is around 0.001 ms⁻¹ considering that D = 15 μm²s⁻¹ [32]. It should be noted that this flux indirectly depends on the buffering capacity of the cytoplasm, because D decreases together with f.

Assuming that the volume of the shell is 1/10 of the volume of the cytosol, the corresponding change of Ca²⁺ concentration in the cytosol (Ca_cyt) is given by $-J_{diff}/9$.

The cytosolic compartment also exchanges Ca²⁺ with the ER, in which the Ca²⁺ concentration is much larger, around 500 μM. Ca²⁺ exchanges between these two compartments have been much studied, both experimentally and by modelling, because they can produce cytosolic Ca²⁺ oscillations [19,33]. It is known that relative changes in Ca²⁺ concentration are much smaller in the ER than in the cytoplasm. In the present study where only basal, non-stimulated Ca²⁺ release from the ER is simulated, we consider that the ER Ca²⁺ concentration remains constant [34,35] and fixed Ca_ER at 500 μM [36]. Basal Ca²⁺ efflux from the ER is given by

$$J_{ER}=k_{in}\left(C{a}_{ER}-C{a}_{cyt}\right)$$

(6)

in which k_in gathers basal ER Ca²⁺ release through the IP₃R and the RyR, as well as unspecific leaks. The opposite flux, i.e., Ca²⁺ pumping from the cytosol to the ER, is mediated by the sarco- or endoplasmic reticulum Ca²⁺ ATPase (SERCA) and is modelled as

$$J_{SERCA}=V_{SERCA}\frac{C{a}_{cyt}^2}{K_{SERCA}^2+C{a}_{cyt}^2}$$

(7)

in which V_SERCA and K_SERCA stand for the maximal velocity and the half saturation constant of SERCA, respectively.

When considering these additional processes, the core model originally described by Equations (1) and (2) takes the form:

$$C_m\frac{dV}{dt}=-I_{Na\left(V\right)}-I_{K\left(V\right)}-I_{Ca\left(V\right)}-I_{K\left(Ca\right)}-I_{NCX}+I_{Inj}$$

(8)

$$\frac{dCa}{dt}=f\left[\frac{-I_{Ca\left(V\right)}+2I_{NCX}}{2F{V}_{shell}}-J_{PMCA}\right]-J_{diff}$$

(9)

$$\frac{dC{a}_{cyt}}{dt}=\frac{J_{diff}}{9}+k_{in}\left(C{a}_{ER}-C{a}_{cyt}\right)-J_{SERCA}$$

(10)

Together with the equations for the currents (Equations (S1)–(S4)) and for the fluxes (Equations (3)–(7)), the evolution Equations (8)–(10) define the model used throughout the whole study. The values of parameters are given in Table 1, unless indicated in the figure legends. Numerical integration and bifurcation diagrams have been obtained using XPPAUT and its AUTO package [37]. The code is available on github (https://github.com/genedupont/amyloids, accessed on 26 November 2021).

3. Results

3.1. Model Behavior in the Absence of Aβ Peptides

We first checked if the extended model described in Section 2 provides an adequate description of the neuronal activity in unperturbed conditions [2,29]. For a sufficiently large value of the injected current, I_inj, the model displays repetitive action potentials (Figure 2A), which are accompanied by periodic variations of the Ca²⁺ concentration in the sub-plasmalemmal compartment (Ca, Figure 2B). These changes in Ca are due to the Ca²⁺ entry through the voltage sensitive Ca²⁺ channels ($I_{Ca\left(V\right)})$ and extrusion from this compartment via three mechanisms. The most significant is the PMCA (Figure 2C), with a flux that reaches ~6 μMms⁻¹. The current mediated by NCX reaches, at most, −5 pA (Figure 2D), which corresponds to a Ca²⁺ flux of about 0.1 μMms⁻¹. These values are in the order of experimental observations [38]. The third mechanism by which Ca²⁺ leaves the sub-plasmalemmal Ca²⁺ compartment is diffusion into the cytoplasm. This term oscillates between 5 × 10⁻⁷ and 2.6 × 10⁻⁶ μMs⁻¹. Such modest fluxes are due to the small value of γ (Equation (5)), which reflect the slow diffusion of Ca²⁺ in the cytoplasm because of buffering. Because sub-plasmalemmal Ca²⁺ is mainly removed by the PMCA, when the cell is electrically active the Ca²⁺ concentration in the cytoplasm remains practically constant at 40 nM. This value of Ca_cyt is determined by the exchanges with the ER.

The bifurcation diagram (Figure 2E) of the extended model resembles that of the original model. After a threshold value of injected current, the stable steady state (plain black line) loses its stability and action potentials (red lines) develop. The Hopf bifurcation is sub-critical, but the zone of coexistence between the stable steady state and the stable limit cycle is very limited [22]. Ca²⁺ diffusion potentially affects the firing threshold of the injected current (Figure 2F). When diffusion is fast, I_inj at the bifurcation point decreases. Indeed, Ca²⁺ entering the cell is removed more efficiently, reducing the activation of the K(Ca) current. Membrane polarization is therefore attenuated, which decreases the value of the depolarizing current that must be injected to initiate an action potential.

The model described in Section 2 provides a simple and realistic description of neuronal and Ca²⁺ dynamics. In the following, we use this model to investigate how changes in Ca²⁺ dynamics influence neuronal excitability. In particular, we focus on how known targets of Aβ peptides modify (1) the threshold value of injected current triggering repetitive action potentials, i.e., the value of I_inj at the bifurcation point, and (2) the frequency of repetitive APs. A decrease of the threshold value of I_inj or an increase in frequency from the control situation (Figure 2E) will be interpreted as neuronal hyperexcitability.

3.2. Aβ Peptides Induced Changes in the Rates of Ca²⁺ Pumps and Pores

3.2.1. Ca²⁺ ATPases

PMCA are dysregulated in AD in an Aβ-dependent manner [39]. The potential influence of this dysregulation deserves investigation, as we have seen in the previous section that active pumping of Ca²⁺ out of the cell by the PMCA plays the most important role in the extrusion of Ca²⁺ from the sub-plasmalemmal space (Figure 2). Thus, changes in the kinetic parameters of the PMCA are expected to have a strong influence on the sub-membrane Ca²⁺ dynamics and hence on neuronal activity. A decrease in the maximal rate of the PMCA (V_PMCA) indeed provokes an increase in the amplitude of the current that must be injected to initiate neuronal firing (Figure 3A). Simulations indicate that the increase of I_K(Ca) with decreasing PMCA activity is responsible for the observed change in I_inj at the bifurcation point. Figure 3E shows that the I_K(Ca) current gets larger when V_PMCA decreases. Below a threshold value (red line in Figure 3A), action potentials disappear. Changing the half saturation constant of the PMCA from its default value (K_PMCA = 1 μM) also affects the domain of repetitive firing (Figure 3B), although in a less abrupt manner.

The strong influence of PMCA on electrical activity contrasts with the lack of effect of changing the kinetic parameters of the SERCA visible in Figure 3C,D. It can be seen that the value of the injected current leading to repetitive action potentials is not influenced by the values of V_SERCA and K_SERCA. Although changing these parameters modifies the level of Ca²⁺ in the cytoplasm, the influence of this change on sub-plasmalemmal Ca²⁺, and hence on electrical activity, is nearly nil, as shown in Figure 3F. Thus, changes in the activity of PMCA and SERCA pumps have a strong influence on the concentrations of Ca²⁺ in the sub-plasmalemmal compartment and in the bulk of the cytosol, respectively. These changes are however not transmitted to the other compartment because the transport of Ca²⁺ by active pumping by far exceeds diffusion. It should be kept in mind that the average cytosolic Ca²⁺ concentration, which corresponds to (Ca+9Cac)/10, is influenced in both cases. As far as neuronal activity is concerned, excitability decreases together with the activity of the PMCA’s, while it is not influenced by the SERCAs.

3.2.2. Ryanodine Receptor- and IP₃ Receptor-Mediated Basal Release of ER Ca²⁺

In neurons, Ca²⁺ is released from the ER via ryanodine and IP₃ receptors and there is compelling evidence that Aβ peptides mobilize internal Ca²⁺ via these two receptors [40]. Moreover, it has been shown that the injection of Aβ42 oligomers stimulates the production of IP₃ [41]. Thereby, Aβ peptides not only exacerbate stimulus-induced cytosolic Ca²⁺ responses, but also provoke a sustained increase in the basal rate of ER Ca²⁺ release that can be modeled by an increase in the value of parameter k_in that appears in Equation (6). Bifurcation analysis of the influence of this parameter (Figure 4A) indicates that it does not affect the PM membrane electrical activity up to a point where cytosolic Ca²⁺ concentration becomes so high that the cell can only be electrically silent. However, this situation is not appropriately represented by the present model, which does not take the Ca²⁺-induced Ca²⁺ release regulation of these two receptors into account. For more modest changes in k_in, the conclusions are qualitatively the same as for the SERCA pumps, pointing to an independence between cytoplasmic Ca²⁺ changes on one hand and sub-plasmalemmal Ca²⁺ and electrical activity on the other hand. That the increase in cytosolic Ca²⁺ does barely not affect sub-membrane Ca²⁺ is visible in Figure 4A,B.

Simulations thus predict that an increase in the rate of basal Ca²⁺ release from the ER does not change the levels of Ca²⁺ in the subplasmalemmal space, and thus also not neuronal excitability.

3.2.3. Pores of β-Amyloids in the Plasma Membrane

Aβ peptides can form Ca²⁺ permeable pores in the PM [28,41]. We investigated the effect of such pores in the model defined in Section 2 by considering the following rate of Ca²⁺ entry in the sub-plasmalemmal compartment:

$$J_{Abeta}=V_{Abeta}\frac{1}{1+e^{\left(V-q_1\right)/q_2}}$$

(11)

which reflects a passive, electrogenic flux [42]. V_Abeta, the maximal rate of Ca²⁺ entry through the pores, reflects both the single pore permeability and the number of pores in the PM. q₁ and q₂ characterize the voltage-dependence of Ca²⁺ entry through the pores. It reproduces a steep decrease of the flux of positively charged Ca²⁺ ions through the pores when the membrane voltage increases.

The evolution equations for sub-plasmalemmal Ca²⁺ and membrane voltage are modified to consider this flux, i.e.,

$$\frac{dCa}{dt}=f\left[\frac{-I_{Ca\left(V\right)}+2I_{NCX}}{2F{V}_{shell}}-J_{PMCA}+J_{Abeta}\right]-J_{diff}$$

(12)

$$C_m\frac{dV}{dt}=-I_{Na\left(V\right)}-I_{K\left(V\right)}-I_{Ca\left(V\right)}-I_{K\left(Ca\right)}-I_{NCX}+I_{Inj}+2F{V}_{Shell} J_{Abeta}$$

(13)

where F stands for the Faraday constant and V_shell for the volume of the sub-plasmalemmal compartment, as defined above.

Simulations of the model predict that for increasing values of V_Abeta, larger currents must be injected to induce a repetitive action potential (Figure 5A). Again, this is due to the increase in subplasmalemmal Ca²⁺ concentration (Figure 5B) that stimulates the Ca²⁺-dependent K⁺ current, which hyperpolarizes the cell. Ca²⁺ concentration in the cytosol is not affected (dashed lines in Figure 5B). These consequences of the presence of pores of amyloids in the plasma membrane are qualitatively similar to the Aβ peptides induced reduction of PMCA activity. However, these two targets differ by the fact that a flux of Ca²⁺ through the pores induces a change in membrane voltage (Equation (13)), while PMCAs are electroneutral [31]. This difference is responsible for the possible increase in the frequency of repetitive AP’s with V_Abeta (Figure 5C) that is not observed upon PMCA inhibition. Thus, although the presence of pores increases the threshold value of current that must be injected to induce neuronal repetitive firing, once active, the frequency is higher in the presence of Aβ peptide pores. As expected intuitively, the spiking frequency increases faster with V_Abeta if the conductance of the Ca²⁺-activated K⁺ channels decreases (compare blue and red curve in Figure 5C). When V_Abeta is very large, the shape of the action potential is modified drastically and irregular spiking occurs (Figure 5D).

Thus, ions permeable pore of Aβ peptides provoke an increase in sub-plasmalemmal Ca²⁺. Because this influx is associated with PM depolarization, it can lead to an increase in spiking frequency for moderate values of the conductance of the K(Ca) channels. However, the threshold value of injected current is not affected by the pores.

3.3. Aβ Peptides Induced Changes in Voltage-Sensitive Ca²⁺ Currents

3.3.1. Voltage-Gated Ca²⁺ Channels

The L-type Ca²⁺ channel has attracted much attention in the field of Alzheimer’s disease. Over-activation of these channels by Aβ peptides has been observed in rat cultured cortical and hippocampal neurons [43]. Injection of Aβ peptides upregulates L-type Ca²⁺ channels in rat hippocampal neurons [25], a phenomenon that is also observed in AD transgenic mice [44]. In particular, the maximal activation of these channels is changed from 0 to about −15 mV in the presence of Aβ [45]. Such a change can be modelled by modifying the values of parameters in the activation and inactivation functions of the L-type Ca²⁺ current, noted α_s and β_s in Equations (S18) and (S19). When considering

$${\alpha }_S=\frac{150}{1+ex{p}^{-0.072\left(V-5\right)}}$$

(14)

and

$${\beta }_S=\frac{0.1\left(V+8.9V\right)}{ex{p}^{0.02\left(V+8.9\right)}-1}$$

(15)

instead of the default function defined by Equations (S18) and (S19), the maximal current indeed corresponds to a ~−15 mV voltage, as shown in Figure 6A.

Simulations indicate that repetitive APs occur in the absence of injected current with these modified values of parameters for the L-type channel (Figure 6B). Moreover, for a given value of I_inj, the frequency of action potentials is larger than in the control situation. As can be seen in Figure 6C, APs become significantly broader, which results in high levels of Ca²⁺ in the sub-plasmalemmal compartment (Figure 6D). As for the preceding Aβ peptides targets, these changes do not significantly affect the Ca²⁺ level in the bulk of the cytoplasm, because most of this Ca²⁺ is extruded to the extracellular medium by the PMCA and the NCX (not shown).

Thus, Aβ peptides induced up-regulation of the L-type Ca²⁺ channel results into marked neuronal hyperexcitability. This is reflected by an increase in both the firing frequency and the propensity of the cell to become electrically active. It also significantly raises the concentration of the sub-plasmalemmal Ca²⁺ during activity.

3.3.2. Ca²⁺-Sensitive Potassium Channels

The activity of the Ca²⁺-sensitive potassium channels has also been shown to be modified by the presence of Aβ peptides [26]. In mouse neurons, intracellular injection of these amyloids suppresses BK channels [46]. The same effect can be observed in cortical pyramidal cells from 3xTg mice [47]. More indirectly, Sosulina et al. (2020) [11] observed an increased intrinsic excitability of CA1 neurons in the absence of synaptic changes in hippocampal slices from a rat model of AD at the early stages following Aβ deposition. Because of the simultaneous decrease in the resting membrane potential and increase in AP half-width, this increase in intrinsic excitability was hypothesized to originate from a decrease in the activity of the K(Ca) channels.

In agreement with these observations, a decrease in the conductance of the K(Ca) channels provokes a marked decrease in the threshold value of current that must be injected to induce repetitive firing (Figure 7A). As visible in Figure 5C, the frequency of repetitive APs decreases if K(Ca) channels are less active, because they have a marked hyperpolarizing effect that delays the onset of the next spike. The minimal value of voltage between two spikes indeed decreases with the value of g_K(Ca) (Figure 7B). Simulations also show that the half-width of APs increases with decreasing values of g_K(Ca) (Figure 7B), as observed in the neurons where these channels are down-regulated by Aβ peptides.

Thus, a reduction in the conductance of K(Ca) points to electrical hyperexcitability at all levels: reduction of the threshold current intensity required to induce repetitive AP’s, and an increase in the frequency and broadening of the AP’s.

4. Discussion

Mathematical modeling provides a unique tool to derive insight into the possible molecular mechanisms that may be relevant in AD. We therefore used a modeling approach to systematically investigate the various effects of Aβ peptides on calcium dynamics and neuronal excitability. We extended a previously proposed mathematical description of the electrical activity of cerebellar granule cells [2] to propose a generic minimal model of neuronal and calcium dynamics. To take spatial heterogeneity into account, we considered a compartmental model, which is a standard approach that is largely used to model Ca²⁺ microdomains, for example [19].

Our simulations point to a clear decoupling between the evolution of Ca²⁺ concentration just below the PM and in the bulk of the cytosol. Indeed, Ca²⁺ changes in the sub-plasmalemmal compartment are governed by Ca²⁺ exchanges with the extracellular medium that occur with a much larger rate than passive diffusion from and to the bulk of the cytosol. On the other hand, these changes have a moderate influence on the Ca²⁺ concentration in the much larger cytosolic space. In indirect agreement with this independence between sub-plasmalemmal and bulk cytosolic Ca²⁺, Yao et al. (2020) [48] reported that limiting RYR open time does not have a major effect on the afterhyperpolarization current in hippocampal CA1 neurons, which agrees with the lack of effect of raising the bulk cytoplasmic Ca²⁺ on the activity of a Ca²⁺-dependent voltage gated channel located in the PM. In the context of AD, the decoupling between membrane and bulk cytosolic Ca²⁺ observed in the simulations suggests that disruptions of Ca²⁺ homeostasis that interfere with signaling and promote the progression of the disease are unrelated from changes in the electrical properties of the neurons. It should be kept in mind, however, that this conclusion only holds at the single cell level. At the network level, Ca²⁺ could affect electrical activity because AD-related synaptic loss may be partly due to Ca²⁺ dysregulations [49].

As a direct consequence of the decoupling between sub-PM and bulk cytosolic Ca²⁺, our results predict that the reported modifications induced by Aβ peptides on the activities of transporters affecting deep cytosolic Ca²⁺ have no impact on neuronal excitability in the absence of the ER-Ca²⁺ mobilizing agonist. The fact that we considered conditions corresponding to the absence of stimulation of IP₃ or RyR receptors is in contrast with the approaches followed by Latulippe et al. [13] and Liu et al. [14]. In the two studies, devoted to electrically non-excitable cells, a homogeneous model was considered to analyze the impact of Aβ peptides on Ca²⁺ dynamics. They concluded that amyloids affect the existence, frequency, and shape of agonist induced Ca²⁺ oscillations. The latter conclusion is significant in view of the observation that Aβ oligomers can stimulate IP₃ production [41], a point that is not investigated in our present study. The impact of this effect of amyloids on the changes in cytosolic Ca²⁺ and consequently on mitochondrial metabolism has also been investigated by modelling [35].

In contrast, when Aβ peptides affect Ca²⁺ fluxes across the PM, they have a significant influence on the membrane neuronal activity in the model. It was shown that Aβ peptides bind to the catalytic site of the PMCA, which corresponds to an increase in the value of K_PMCA appearing in Equation (3). As shown in Figure 3B, this induces a reduction of neuronal excitability in the model. Aβ peptides also tend to increase sub-membrane Ca²⁺ by forming pores in the PM. In the model, they have rather mixed consequences on neuronal activity (Figure 4). On the one hand, they increase the threshold value of current that must be injected in the neuron to induce repetitive action potential. On the other hand, they can provoke an increase in the spiking frequency. While the latter observation is associated with hyperexcitability, the former is associated with hypo-excitability. These apparently opposite consequences can be ascribed to the fact that the entry of Ca²⁺ through the pores stimulates the hyperpolarizing K(Ca) current, but also depolarizes the cell membrane. Indeed, hyperexcitability is more pronounced when decreasing the conductance of the K(Ca) channel. In addition, if Ca²⁺ influx through the pores is very large, aberrant spiking in the form of complex neuronal activity can also be observed in the model. As a perspective, it would be interesting to test if the same simulated behaviors can be recovered when considering a stochastic Markov model validated against experimental data [28] to describe the dynamics of pore opening and closing. Finally, the reported changes in the activities of L-type Ca²⁺ channels and Ca²⁺-sensitive K⁺ channels in the presence of Aβ peptides clearly point to hyperexcitability when considered in the model. These results are in agreement with the numerous studies reporting neuronal hyperexcitability in the presence of Aβ peptides. In particular, limiting the open time duration of RYR2 in the hippocampal CA1 neurons of an AD mouse model upregulates the surface expression of the K(Ca) channel and prevents electrical hyperactivity, but does not directly modulate the activity of the K(Ca) channels [48]. Altogether, our simulation results indicate that the reported consequences of Aβ peptides induced modifications of Ca²⁺ fluxes or Ca²⁺-regulated currents have qualitatively different outcomes. However, it should be noted that although Aβ peptides are most of the time associated with neuronal hyperexcitability, some studies however report opposite consequences [50,51].

The seemingly contradictory effects of Aβ peptides reported in this study may also be due to the limitations of the proposed model. For example, it does not consider the specificities of the various neuronal cell types. Additionally, we considered one target of Aβ peptides at a time, which is important to dissect the pathological consequences of the oligomers of amyloids, but does not correspond to a realistic situation. Because they are limited to a single neuron, simulations do not include network effects, which play a crucial role in the cognitive defects related to the disease [52]. It is clear that simplified modelling, such as the one performed here, does not provide a faithful view of the complexity of calcium signaling and neuronal activity corresponding to an in vivo situation. However, it allows for drawing some simple and unambiguous conclusions, which can help analyzing experimental observations when considered from a larger perspective.