Computational investigation of Amyloid-β-induced location- and subunit-specific disturbances of NMDAR at hippocampal dendritic spine in Alzheimer’s disease

In Alzheimer’s disease (AD), dysregulation of intracellular Ca2+ signalling has been observed as an early event prior to the presence of clinical symptoms and is believed to be a crucial factor contributing to AD pathogenesis. Amyloid-β oligomers (AβOs) disturb the N-methyl-D-aspartate receptor (NMDAR)-mediated postsynaptic Ca2+ signalling in response to presynaptic stimulation by increasing the availability of extracellular glutamate as well as directly disturbing the NMDARs. The abnormal Ca2+ response can further lead to impairments in long-term potentiation (LTP), an important process in memory formation. In this study, we develop a mathematical model of a CA1 pyramidal dendritic spine and conduct computational experiments. We use this model to mimic alterations by AβOs under AD conditions to investigate how they are involved in the Ca2+ dysregulation in the dendritic spine. The alterations in glutamate availability, as well as NMDAR availability and activity, are studied both individually and globally. The simulation results suggest that alterations in glutamate availability mostly affect the synaptic response and have limited effects on the extrasynaptic receptors. Moreover, overactivation of extrasynaptic NMDARs in AD is unlikely to be induced by presynaptic stimulation, but by upregulation of the resting level of glutamate, possibly resulting from these alterations. Furthermore, internalisation of synaptic NR2A-NMDAR shows greater damage to the postsynaptic Ca2+ response in comparison with the internalisation of NR2B-NMDARs; thus, the suggested neuroprotective role of the latter is very limited during synaptic transmission in AD. We integrate a CaMKII state transition model with the Ca2+ model to further study the effects of alterations of NMDARs in the CaMKII state transition, an important downstream event in the early phase of LTP. The model reveals that cooperation between NR2A- and NR2B-NMDAR is required for LTP induction. Under AD conditions, internalisation of membrane NMDARs is suggested to be the cause of the loss of synapse numbers by disrupting CaMKII-NMDAR formation.

Introduction transmission and reductions in memory formation. However, to some extent, the loss of synaptic NMDARs may also be a neuroprotective mechanism against the glutamate-induced neurotoxicity and excessive influx of Ca 2+ [32,33]. The effects of these alterations are studied individually in transgenic animal models or by injecting a high concentration of Aβ into healthy animals or cells [32,[34][35][36][37][38][39][40]. However, the use of different experiment materials or experimental protocols from different research groups can lead to controversial results for AβO disturbances (see reviews in [41][42][43]). A mathematical model of Ca 2+ dynamics in the dendritic spine with presynaptic stimulations as inputs, will provide useful insights into the effects of the above disturbances at different levels in AD, both individually and collectively.
In this paper, we present an integrated mathematical model of an average CA1 pyramidal dendritic spine that includes location-and NR2 subunit-specific characteristics of NMDARs to understand the AβO-induced dysregulation of intracellular Ca 2+ and its relationship to the balance between synaptic and extrasynaptic NMDARs in the dendritic spine of pyramidal neurons. We use the model to mimic several types of disturbances of AβO that have been proposed to be critical to the pathogenesis of AD (see Section 2 of S1 Appendix). The model shows that AβO-dependent disturbances on synaptic glutamatergic transmission mainly affect Ca 2+ signalling in the dendritic spine and have only minor effects on Ca 2+ signalling in the whole cell. In contrast, AβO-induced non-synaptic glutamate release and elevation in extrasynaptic glutamate concentration mainly affect the Ca 2+ dynamics of the whole cell. Furthermore, the simulation results suggest that among all disturbances tested, the presynaptic release  (2) and (3) AβO promotes glutamate vesicle release from the presynaptic terminal and ambient astrocytes, respectively; and (4) AβO also mediates in the internalisation of the surface receptors at the synaptic site. (1), (2) and (3) result in an increase in the extracellular glutamate concentration and, ultimately, may lead to the over activation of synaptic glutamate receptors or of receptors at distant locations from the release site. In contrast, (4) leads to a decrease in functional synaptic receptors that may depress synaptic activity. This figure is produced using of glutamate is the most sensitive disturbance for NMDAR activity and Ca 2+ response in the postsynaptic neurons.
We also extend our research on synaptic Ca 2+ signalling to the downstream CaMKII state transition to investigate the disturbances from alterations in synaptic NMDAR in the emergence of LTP in AD: we integrate the Ca 2+ model with a state transition model of CaMKII to study the subunit-specific roles of NMDAR in the postsynaptic response and the formation of NMDARmediated LTP. The model reveals that cooperation between NR2A-and NR2B-NMDAR is necessary for NMDAR-LTP. Simulation results suggest that the internalisation of synaptic NR2B-NMDAR observed in the experiments [31,39] does not severely affect the synaptic Ca 2+ responses, thus, it is not enough to protect against the glutamate-induced neurotoxicity in AD. Instead, loss of synaptic membrane NR2B-NMDAR will inhibit LTP induction by disrupting CaMKII-NMDAR formation.

Model overview
We construct a mathematical model of Ca 2+ dynamics (denoted as Ca 2+ -Model) in the dendritic spine in response to presynaptic stimulation, which is carefully developed according to the characteristics of the CA1 pyramidal dendritic spine. Ca 2+ -Model integrates three aspects: (1) SubModel 1: action potential (AP)-induced presynaptic glutamate release and diffusion; (2) SubModel 2: glutamate receptor activation; and (3) SubModel 3: Ca 2+ dynamics within a single dendritic spine and its parent dendrite shaft (Fig 2A-2C). To further study the effects of NMDAR alterations in their roles in downstream events, we extend Ca 2+ -Model with a CaM-KII state transition model (denoted as CaMKII-ST-Model) by He et al. [44] (Fig 2D). All simulations are performed using Matlab software, with the built-in solver ode15.

Ca 2+ -Model development
SubModel 1: A model of glutamate release, uptake and diffusion. We simulate the glutamate release from the presynaptic terminal, and its diffusion inside the synaptic cleft and in the extrasynaptic space, based on the model by Rusakov and Kullmann [45]. A schematic of SubModel 1 in two-dimensions is given in Figure A in S2 Appendix. The dendritic spine head and the presynaptic terminal are configured as two opposite hemispheres with the same radius and there is no glutamate diffusion within them. The synaptic cleft is a flat cylinder between these two hemispheres, with a height of 20 nm. We take 0.1 μm 3 as the volume of the spine head to represent the average size of spine [46], which gives a radius of 363 nm for the synaptic cleft and the two opposite hemispheres. The extrasynaptic space is a spherically isotropic porous medium surrounded by the two hemispheric obstacles.
The concentration of glutamate in a glutamate vesicle is about 100 mM [47], which corresponds to about 1500 glutamate molecules. Therefore, we assume that a presynaptic stimulation pulse can induce the release of 1500 glutamate molecules from a point site in the centre of the presynaptic terminal surface (see Section 1 of S2 Appendix). Once released, glutamate molecules diffuse through the flat cylindrical cleft with an effective glutamate diffusion coefficient (D Glu ), and then escape from the cleft into the extrasynaptic space, where D Glu is reduced by a tortuosity factor λ (D Glu Ã ¼ D Glu l 2 ). The uptake of glutamate is governed by glutamate transporters in astrocytes in the extrasynaptic space. In the extrasynaptic space, glutamate transporters are distributed homogeneously at a concentration (B total ) of 0.5 mM [48] (see Section 1 of S2 Appendix). To track the glutamate after release at different locations, we define PSD or synaptic, perisynaptic and extrasynaptic sites as follows: the surface of PSD region and perisynaptic zone are set from 0 to 150 nm and 365 nm away from the centre of the postsynaptic surface, respectively [49] (Figure A (1) and (2) in S2 Appendix). The extrasynaptic space is set beyond the outside border of the perisynaptic zone and the extrasynaptic receptors are located at the dendritic shaft.
SubModel 2: A model of NMDAR and AMPAR activation. Glutamate receptors are located at the synaptic, perisynaptic and extrasynaptic membrane surfaces with varying numbers (Table B in S2 Appendix). Each receptor is independent of each other and the local glutamate concentration received by each receptor depends on its distance from the release site and is calculated by SubModel 1. We simulate the state transition of a single NMDAR and AMPAR using an NR2 subtype specific, eight-state kinetic model [50] and a seven-state model [51], respectively ( Figure B in S2 Appendix). The reaction rate constants are given in Table B in S2 Appendix.
Based on the experimental observations [24,25,49,52,53], the distributions of NMDARs are: 12 synaptic NR2A-NMDARs, 8 synaptic NR2B-NMDAR, 3 perisynaptic NR2B-NMDAR, and 8 extrasynaptic NR2B-NMDAR. The number of AMPARs depends on the spine geometry and is positively correlated with PSD size [49,54]. We assume that the functional AMPARs are homogeneously located in the membrane of PSD and at the dendritic shaft with different densities, and there is no AMPAR in the rest of the spine membrane [55]. The number of AMPARs per spine is critical for the generation of the temporary depolarisation of the postsynaptic membrane potential after stimulation; this is also called the excitatory postsynaptic potential (EPSP). Based on the experiments [56,57], the EPSP amplitude at the synaptic site is lower than 5 mV after a single synaptic stimulation. We estimate the number of synaptic AMPARs based on the established experimental data [58,59], to generate the expected EPSP amplitude (see Section 5 of S2 Appendix). The neck conductance we use in the simulation is 157 MO, representing the resistance of a medium sized spine neck of CA1 pyramidal neurons [59]. The number of extrasynaptic AMPARs (eAMPARs) at the dendritic shaft is low and stable, and is calculated based on the membrane surface area (see Section 5 of S2 Appendix) [55]. The removal of the Mg 2+ blockage of NMDARs and the activation of voltage-dependent calcium channel (VDCC) depend on the membrane depolarisation after stimulation. We build an electrical model of a single spine and its adjacent dendritic shaft to capture the dynamics of the membrane potential (see Section 3 of S2 Appendix for the model details).
SubModel 3: A compartmental model of a dendritic spine and shaft. We construct a four-compartment model to represent a pyramidal neuron dendritic spine and its adjacent dendritic shaft. The four compartments are PSD, cytosol, spine neck and dendritic shaft ( Figure D in S2 Appendix). The geometry of the spine is consistent with that of SubModel 1 and SubModel 2 (Table A in S2 Appendix). The spine head is assumed to be a hemisphere and is divided into PSD and cytosol compartments. The PSD compartment is a cylinder attached to the postsynaptic membrane and the rest of the spine head is in the cytosol compartment. PSD occupies 10% of the total volume of the spine head [60,61]. The spine neck is represented as a long thin cylinder, which is coaxial with the spine head. The dendritic shaft is another cylinder that is attached to the bottom of the spine neck with a radius of 0.5 μm and a length of 1 μm (details of this compartmental model are in Section 4 of S2 Appendix).

Parameter calibration and estimation
The simulation temperature used in this paper is 34˚C, a near-physiological temperature widely used in experiments and computational simulations [46,62]. All temperature-dependent rate constants are adjusted according to their temperature coefficients (Q10), as listed in Table 1.
The values of the ten uncertain parameters of Ca 2+ -Model are estimated using Markov Chain Monte Carlo (MCMC) [64,65]: the density for VDCC, the densities for two membrane Ca 2+ pumps, PMCA and NCX, the concentration of the endogenous immobile buffer and its binding and unbinding rates in the dendritic and spine locations, respectively. The parameter estimation is based on the experimental observations by Sabatini, Oertner et al. [46]. They estimated the Ca 2+ dynamics at the dendritic spines of the CA1 pyramidal neurons and their parent dendrites by a single backpropagation of action potentials (bAP) in the absence of exogenous buffers (Ca 2+ indicator) and the washout of mobile buffers. The results correspond to spines and small dendrites with surface-to-volume ratios of 4-20 μm -1 and 1-4 μm -1 , respectively [46]. The geometry of our model lies well within these ranges. The details of MCMC estimation are discussed in Section 6 of S2 Appendix. Model performance. The control condition is defined as the condition without AβO-disturbances and is simulated using standard values of parameters listed in Section 1-6 of S2 Appendix. We investigate the model performance in response to presynaptic stimulation (see Section 7 of S2 Appendix for a detailed explanation) based on glutamate profile, the open fraction of receptors at different locations and, consequently, the Ca 2+ dynamics in the spine head and its parent dendritic shaft. The model performance under control conditions in response to various stimulation protocols is given in Section 8 of S2 Appendix.

Integration of Ca 2+ -Model and CaMKII-ST-Model
CaMKII-ST-Model developed by He et al. [44] simulates the formation of CaMKII-NMDAR complex in PSD in response to presynaptic stimulation. It consists of a series of key events induced by the Ca 2+ influx through NMDAR after stimulation. Ca 4 CaM complex and postsynaptic NR2B-NMDAR are two linking factors between Ca 2+ -Model and CaMKII-ST-Model ( Fig 2D). The details of the model integration are in Section 9 of S2 Appendix.

Results
We mimic the AβO-dependent disturbances on glutamatergic transmission on the following aspects: the availability of glutamate to glutamate receptors and the distribution of synaptic NMDARs. Based on the experimental protocols [44,66], we apply three types of presynaptic stimulation patterns as inputs to Ca 2+ Model: (1) a single stimulus (1 pulse); (2) a low frequency stimulation (LFS) at 10 Hz; and a (3) high frequency stimulation (HFS) at 100 Hz (see Section 7 of S2 Appendix for the details of the stimulation protocols). To investigate the effects of these alterations on the postsynaptic response, we compare the activities of NMDAR and the dynamics of Ca 2+ at different locations with those under the control condition.

Simulation of AβO-dependent disturbances on glutamate transmission
Effects of increases in presynaptic release. We first investigate the effects of the AβOinduced release of glutamate vesicles from the presynaptic terminal [67][68][69][70][71]. We assume every vesicle contains the same number of glutamate molecules, therefore, the number of vesicles per release is represented by the total number of glutamate molecules. We vary the number of glutamate molecules per release from 500 to 10000 and keep other parameters at the standard values.
The simulation results show that glutamate concentrations ([Glu] peak ) at three locations all increase linearly with the number of glutamate molecules released (S1 Fig). To examine the contribution of an increase in the presynaptic release of glutamate to the NMDAR transition, we calculate the open time of each receptor and the number of Ca 2+ ions entering through each receptor after stimulation. We define the additional time for a receptor staying in a state (t add ) as the difference between the total time in this state under the current condition and the time under control condition. Similarly, the additional number of Ca 2+ ions entering (Ca 2+ add ) through a receptor is the difference between the total number of the Ca 2+ ion flux under the current condition and the one under control condition.
The simulation results show that multiple-vesicle releases have a negligible effect on the total open time of synaptic NR2A-NMDARs ( Fig 3A). Even though, Ca 2+ add through all synaptic NR2A-NMDARs still has made a large contribution to the total Ca 2+ add into the spine head ( Fig 3B). This is because the number of synaptic NR2A-NMDARs is more than that of the other receptors.
For synaptic and perisynaptic NR2B-NMDARs, multiple vesicle releases lead to increases in t add in the open state under a 1 pulse stimulation and LFS. Under HFS, in contrast, only the perisynaptic NR2B-NMDARs show an increase in open time ( Fig 3A). Consequently, total Ca 2+ add through all perisynaptic NR2B-NMDARs increases under LFS and HFS, whereas total Ca 2+ add through all synaptic NR2B-NMDARs only increases under LFS ( Fig 3B). Increases in open time and Ca 2+ influx of synaptic and perisynaptic receptors lead to a higher peak in Ca 2+ concentration ([Ca 2+ ] peak ) in the spine head ( Fig 3C). Specifically under HFS, Ca 2+ level rises to a peak of about 4.8 μM in 55 ms (almost a three-fold increase in Ca 2+ level over the control condition) upon increasing the glutamate molecules per release to 10,000 ( Fig 3C). After reaching the peak, Ca 2+ levels then rapidly decrease and stay on a plateau after around 40 pulses until the end of the stimulation. The decrease in Ca 2+ level is because of the desensitisation of NMDARs by repetitive stimulation. The plateau in Ca 2+ level is in a range of 0.5 to 1.2 μM, and increases with the number of glutamate molecules per release.
The extrasynaptic NR2B-NMDARs are not (1 pulse stimulation and LFS) or slightly (HFS) affected by the number of glutamate molecules released in response to presynaptic stimulation (Fig 3A and 3B). In Fig 3C, the elevations in [Ca 2+ ] peak in the dendritic shaft by high releases of glutamate are largely the result of Ca 2+ diffusion from the spine head, rather than Ca 2+ influx through extrasynaptic NMDARs.
Effects of inhibitions of the glutamate transporter. Experimental evidence suggests that AβO may disturb glutamate clearance mechanisms by reducing the number of glutamate transporters [34][35][36][37][38].To examine if the down-regulation of glutamate transporters contributes to the abnormal opening of NMDARs, we simulate Ca 2+ -Model a total concentration of glutamate transporters (B total ) that ranges from 0.5 mM (0% reduction) to 0 mM (100% reduction). Fig 4 shows that there is no effect on [Glu] peak at the postsynaptic site from reducing B total . In contrast, at the perisynaptic and extrasynaptic sites, the reduction of B total to 0 mM (100% reduction) increases [Glu] peaks by about 3 μM and 1.5 μM, respectively, while causing a slower decay to baseline (Fig 4A). These increases lead to higher Ca 2+ adds through perisynaptic NR2B-NMDARs and extrasynaptic NR2B-NMDARs, especially under HFS (Fig 4B). Consequently, inhibition in the glutamate uptake results in increases in [Ca 2+ ] peak in the dendritic shaft but not in the spine head ( Fig 4C).
We then increase the glutamate numbers per release from 1500 to 5000 to investigate if there are any significant changes in the multi-vesicle releases. The results (S2 Fig) are consistent with the single-vesicle release in the experiment above. Therefore, the down-regulation in glutamate uptake does not directly affect the postsynaptic spine in response to the presynaptic stimulations. In contrast, this down-regulation promotes a Ca 2+ influx through the extrasynaptic receptors in the dendritic shaft, especially under HFS.
Effects of increases in the resting glutamate levels in the extrasynaptic space. Elevations in the resting level of extrasynaptic glutamate ([Glu] rest ) have been observed in the hippocampus of AD transgenic mice that overexpress the human amyloid precursor protein [72]. In contrast to a [Glu] rest of 0.25 μM in the control mice, Talantova et al. [72] reported 0.8 and 3.3 μM in 12-month-old and 22 to 24-month-old mice, respectively. To examine how [Glu] rest affects the receptors at the resting state, we simulate SubModel 2 with [Glu] rests from 0.01 μM to 100 μM. The results show both NR2A-and NR2B-NMDAR reach the maximum open fractions (0.08 and 0.02, respectively) when glutamate increases to about 10 μM (Fig 5A). The fraction of NR2A-NMDAR and NR2B-NMDAR in the desensitised state approaches a maximum of 0.81 at the same time ( Fig 5B). The background opening of NR2B-NMDAR by [Glu] rest causes a persistent inward current and Ca 2+ influx. When increasing [Glu] rest to 10 μM, the background Ca 2+ influx reaches a maximum of 580 (NR2A-NMDAR) and 160 (NR2B-NMDAR) Ca 2+ ions per second, which is 5-10 Ca 2+ ions per second lower control conditions (Fig 5C).
When applying different presynaptic stimulations to Ca 2+ -Model, elevation in [Glu] rest slightly reduces the open time of the synaptic and perisynaptic NMDARs but not extrasynaptic Computational investigation of Amyloid-β-induced disturbances in Alzheimer's disease NMDARs (Fig 6A). Consequently, the elevation in [Glu] rest leads to fewer Ca 2+ ions entering ( Fig 6C) and decreases [Ca 2+ ] peak in the spine head (Fig 6D).
In contrast, the elevation in [Glu] rest leads to a large increase in the desensitisation time of the perisynaptic and extrasynaptic NR2B-NMDARs (Fig 6B). The high level of [Glu] rest results  in a high fraction of background opening of the extrasynaptic NR2B-NMDAR and small increases in response to presynaptic stimulation (Fig 6E).
Effects of non-synaptic release of glutamate. AβO has been found to induce glutamate release from astrocytes in AD transgenic mice [72][73][74][75]. The astrocyte stays close to the dendritic shaft, which can potentially activate extrasynaptic glutamate receptors. We simulate the astrocytic release of glutamate by applying a brief pulse (1 to 20 ms) of 1 mM glutamate to the extrasynaptic sites. The glutamate concentration in the synaptic cleft and presynaptic site will not be affected [47]. Fig 7A shows that the stimulation causes large numbers of Ca 2+ to enter the dendritic shaft, which increases with the length of the stimulation pulse. [Ca 2+ ] peak in the dendritic shaft ranges from 0.083 to 0.087 μM (Fig 7B). The dendritic shaft has larger volume and fewer receptors than the spine head. Ca 2+ ions entering through the extrasynaptic NR2B-NMDAR are largely diluted in the dendritic shaft; therefore, [Ca 2+ ] peak is much less than that in the spine head in response to presynaptic stimulation. Moreover, extrasynaptic AMPARs activated by the astrocytic glutamate release create a 4-6 mV depolarisation (Fig 7C). Even when increasing the stimulation time to 20 ms, it still fails to create a larger depolarisation to activate other voltage-dependent Ca 2+ channels in the dendritic shaft membrane. This is understandable because of the low AMPAR density in the extrasynaptic site (20 receptor/μm 2 ; in synaptic location up to 1000 receptor/μm 2 ) [55].

Disturbances in NMDARs in AD
Effects of reductions in the surface expression of synaptic NMDAR. We next investigate how inhibitions of AβO on the membrane surface expression of different receptors affect the postsynaptic response. We mimic the reduction in surface expression by decreasing the receptor number from the standard value (Table B in S2 Appendix) to zero.
The reduction in synaptic NR2A-NMDAR numbers shows negligible effects on [Ca 2+ ] peak under a 1 pulse stimulus and LFS, whereas, under HFS, the peak gradually decreases from 3 μM to nearly 0 μM (Fig 8A). The reduction in synaptic NR2B-NMDAR number only affects the [Ca 2+ ] peak in the spine head under HFS. There is about a 1 μM reduction in [Ca 2+ ] peak when the synaptic NR2B-NMDAR is fully removed (Fig 8B).
Effects of reductions in desensitisation of synaptic receptors. Under control conditions, NMDARs undergo desensitisation in response to the prolonged presence of their agonists, to prevent an excess Ca 2+ influx [76]. A slow NMDA receptor desensitisation has been observed in transgenic mice that overexpress large amounts of Aβ within neurons [77,78]. To mimic the inhibition of receptor desensitisation by AβO, we decrease the desensitisation related parameters (k d1+ and k d2+ in Table B in S2 Appendix) of each receptor by 0% to 100% of the standard. Computational investigation of Amyloid-β-induced disturbances in Alzheimer's disease The inhibition of desensitisation of synaptic NR2A-NMDAR and NR2B-NMDAR shows no obvious effects on Ca 2+ transients in the spine head under a 1 pulse stimulus and LFS. Under HFS, [Ca 2+ ] peak increases significantly from 2.9 μM to 17.9 μM, with an increase in the inhibition level of synaptic NR2A-NMDAR desensitisation (Fig 8C). When desensitisation of the synaptic NR2A-NMDAR is fully inhibited, the Ca 2+ concentration in the spine head maintains a high level after reaching a peak during the stimulation period. The inhibition of synaptic NR2B-NMDAR desensitisation also positively affect the Ca 2+ transients in the spine head under HFS (Fig 8D). This leads to about a 2 μM increase in [Ca 2+ ] peak when the synaptic NR2B-NMDAR desensitisation is fully inhibited. However, the cytosolic Ca 2+ concentration fails to maintain a high level after reaching a peak.

Global sensitivity analysis
To identify the key players in AβO-induced Ca 2+ dysregulation among the above factors, we apply global sensitivity analysis to Ca 2+ -Model. We select 15 factors that have been tested in previous sections (Table 2A) and eight outputs (Table 2B), which represent the postsynaptic neuron responses to presynaptic stimulation. We use a partial rank correlation coefficient (PRCC) to identify the most important or sensitive factors [79] (see Section 10 of S2 Appendix for details).  In the spine head, [Ca 2+ ] peak is the most sensitive to #sNR2A followed by #sNR2B. Under HFS, the correlation to #sNR2B decreases, which suggests a relatively lower activity of synaptic NR2B-NMDAR in HFS than under LFS. Ca 2+ by sNR2A, but not Ca 2+ by sNR2B, correlates with #sAMPAR, which infers that only synaptic NR2A-NMDAR is sensitive to membrane depolarisation. #sAMPAR becomes more important in HFS in comparison to LFS, indicating that a larger depolarisation by HFS brings more Ca 2+ ions into the spine head.
In the dendritic shaft, [Ca 2+ ] peak is sensitive to #eNR2B under LFS and, surprisingly, it is also sensitive to #sNR2A under HFS. This suggests that under HFS increasing synaptic NR2A-NMDAR expression allows large numbers of Ca 2+ ion influx and leads to more Ca 2+ ions diffusing into the dendritic shaft. Computational investigation of Amyloid-β-induced disturbances in Alzheimer's disease

Effect of disturbances of synaptic NMDAR numbers on CaMKII state transition
Synaptic NMDARs are suggested to have a dual role in the formation of NMDAR-mediated-LTP by acting as Ca 2+ channels (NR2A-and NR2B-NMDAR) and as a scaffold in PSD (NR2B-NMDAR) to anchor CaMKII [16,17]. To further investigate how disturbances in synaptic NMDARs affect such a dual role, we integrate our Ca 2+ -Model with CaMKII-ST-Model by He et al. [44] (see Methods for model integration). We simulate the degree of reduction in the availability of a particular NMDAR type in PSD by decreasing the receptor number from the standard value (Table B in S2 Appendix) to zero. The stimulation protocols applied for the computational experiments are 1 s of pairing HFS [66] and 4 trains of pairing theta-burst stimulation (4 TBS), respectively, which are used for triggering LTP in vivo [80] (see Section 7 of S2 Appendix for a detailed explanation). We investigate the Ca 2+ elevation in the spine head and four chosen outputs: (1) numbers of Ca 4 CaM complexes; (2) numbers of autophosphorylated CaMKII subunits; (3) numbers of CaMKII in PSD; and (4) numbers of CaMKII-NMDAR complex in PSD. These outputs are the key factors from downstream events that determine the amount of CaMKII to be activated, autophosphorylated, translocated and anchored in PSD [81].
The reduction in the numbers of NR2A-NMDAR in PSD greatly reduces [Ca 2+ ] peak in response to pairing HFS ( Fig 10A) and 4 TBS (S3A Fig). The decrease in [Ca 2+ ] peak further reduces all four outputs. During both stimulation protocols, a 50% reduction (6 NR2A-NM-DAR left in PSD) can block all downstream events (Fig 10A and S3A Fig). In particular, the formation of CaMKII-NMDAR is mostly sensitive to the NR2A-NMDAR reduction. Even an In contrast, a reduction in NR2B-NMDAR numbers in PSD affects the Ca 2+ responses and all outputs less than the reduction in NR2A-NMDAR numbers (Fig 10B and S3B Fig). Even when reducing the NR2B-NMDAR from 8 to 1 (88% reduction), [Ca 2+ ] peak only decreases by about 35%. This decrease is not able to fully block most chosen downstream events, except for the formation of CaMKII-NMDAR complex. The formation of CaMKII-NMDAR complex is largely reduced by the reduction in NR2B-NMDAR numbers. A reduction of the NR2B-NMDAR numbers by 50% leads to over a 75% reduction in the final level CaMKII-NMDAR complex at t = 300 s.
Experimental evidence suggests that NR2A-and NR2B-NMDAR may contribute differently to the postsynaptic Ca 2+ response and CaMKII activation. Both receptors are required in NMDAR-induced LTP [17]. The ratio of synaptic NR2A-to NR2B-NMDAR (NR2A/NR2B ratio) is precisely regulated by the production, trafficking and degradation of NMDARs. Different ratios regulate the preferences in the induction of different types of synaptic plasticity [82]. Therefore, we investigate the effects of disturbances on the NR2A/NR2B ratio in the CaMKII-NMDAR complex formation. We simulate the disturbance by varying the NR2A/ NR2B ratio from 1: 19 to 19:1 while keeping the total number of NMDARs in PSD constant (20 NMDARs). Fig 10C and S3C Fig show that [Ca 2+ ] peak stimulated by both pairing HFS and 4 TBS increases with the NR2A/NR2B ratio. The productions of the other four outputs are blocked when the NR2A/NR2B ratio is below 6:14. When a further increase in the NR2A/NR2B ratio occurs, the normalised peak levels of all outputs, except the production of CaMKII-NMDAR Computational investigation of Amyloid-β-induced disturbances in Alzheimer's disease complex, rise with different slopes in response to a pairing HFS (Fig 10C and S3C Fig). In contrast, the normalised final levels of CaMKII-NMDAR after stimulation increase to maximum levels of 1.69 (pairing HFS) and 2.04 (pairing TBS), respectively, at the NR2A/NR2B ratio of 15:5, and decreases afterwards (Fig 10D). Therefore, our results indicate the existence of an optimal NR2A/NR2B ratio in the generation of the CaMKII-NMDAR complex.

Discussion
In this work, we present computational models of Ca 2+ dynamics in the dendritic spine and its parent dendrite shaft to investigate the effects of AβO-dependent disturbances on synaptic transmission. These disturbances are related to the availability of both glutamate and receptors. In particular, we include NMDARs with different subunit compositions and at different locations. Using simulations under different conditions, we study the activation patterns of specific NMDARs and the Ca 2+ response at different locations.
Our results demonstrate that the increased glutamate release from the presynaptic terminal will promote Ca 2+ responses mainly in the spine head. Global sensitivity analysis suggests a great sensitivity of the postsynaptic response to the number of glutamate molecules released during presynaptic stimulation. This confirms that the Aβ-induced increase in synaptic glutamate release plays a major role in the over-excitation and Ca 2+ overload of postsynaptic neurons. The glutamate spillover from the synaptic cleft shows a much lower effect on the activation of extrasynaptic NMDARs than perisynaptic NMDARs. This is because of the fast diffusion of glutamate molecules and their uptake by glutamate transporters in astrocytes before they reach extrasynaptic NMDAR [47]. Therefore, Aβ-induced multiple vesicle releases from the presynaptic terminal alone are not sufficient to cause an overactivation of the extrasynaptic receptors.
Inhibition of glutamate uptake by glutamate transporters only affects the peri-and extrasynaptic receptors. On a longer time scale, this inhibition will cause a glutamate accumulation in the extrasynaptic space and lead to a gradual increase in [Glu] rest . We have shown that elevation in [Glu] rest reduces the sensitivity of the postsynaptic neurons to the presynaptic signals, as a result of increased background opening of extrasynaptic NMDARs under the resting condition. Moreover, the AβO-induced astrocytic glutamate release also leads to a high-level Ca 2+ ion influx in the absence of presynaptic stimulation. This finding is consistent with experimental observations [83], which show excitotoxicity results from Aβ-induced over-activation of the extrasynaptic NMDAR, but not the synaptic NMDAR. The overactivation can, in turn, promote Aβ production [84]. Even though these abnormal Ca 2+ influxes have failed to induce large Ca 2+ transients in the dendritic shaft, they still can potentially induce downstream pathways by affecting proteins located close to the receptors. Over a long time, this will contribute to the Ca 2+ overload and neuronal death in AD [85]. Therefore, monitoring the extracellular glutamate concentration using precise measuring technology [86] could be useful for early diagnosis of AD, and therapeutic research can be carried out to investigate controlling the extracellular glutamate level to avoid excess activation of the extrasynaptic NMDARs.
Our simulation shows that synaptic NR2B-NMDAR contributes less to the synaptic Ca 2+ transient compared to synaptic NR2A-NMDAR, in agreement with the simulation results from [87,88]. Therefore, internalisation of synaptic NR2B-NMDAR disturbs the synaptic transmission [31, 39] not by affecting Ca 2+ entry but, possibly, by disturbing interactions with other key players. Specifically, synaptic NR2B-NMDARs bind to CaMKII and are involved in mediating synapse strength and plasticity [89].
Simulation of the internalisation of synaptic NR2A-and NR2B-NMDAR shows negative effects on the activation of CaMKII and the formation of CaMKII-NMDAR complexes, to different degrees. Both types of NMDAR are necessary for LTP formation but they contribute to LTP induction and maintenance in different ways. Specifically, the role of NR2A-NMDARs is to allow a sufficient Ca 2+ influx to trigger downstream Ca 2+ -CaM interactions, which determines the activation of CaMKII. NR2B-NMDAR contributes less as a Ca 2+ channel than NR2A-NMDAR; however, it is required to function as a scaffold to anchor CaMKII in PSD. This result is consistent with the experimental findings and the hypothesis that the opening of NR2B-NMDAR is not necessary for LTP induction [23,90]. Therefore, the internalisation of synaptic NR2B-NMDARs disturbs the synaptic functions without affecting the Ca 2+ dynamics.
Simulation with different NR2A/NR2B ratios provides a clearer picture showing that NMDAR-LTP requires cooperation between the NR2A-and NR2B-NMDAR. AβO-induced internalisation of synaptic NMDAR in AD [31,39] could underlie some of the critical alterations in the pathology of the disease. For instance, in AD transgenic mice, AβO has been observed to alter CaMKII distribution and reduce the synaptic CaMKII level [91]. Consequently, internalisation of synaptic NMDAR is suggested to contribute to the deficits of LTP and loss of synapses in AD [92,93]. Therefore, selective inhibition on the internalisation of synaptic NR2B-NMDARs in AD (such as modifying key proteins in the NMDAR trafficking pathway [94]) could be a useful therapeutic approach that may prevent the loss of synapses and memory decline.