Biophysical Model of AMPA Receptor Trafficking and Its Regulation during Long-Term Potentiation/Long-Term Depression

Receptor trafficking under basal conditions

The parameter values chosen for receptor trafficking under basal conditions are listed in Table 1. As some of the model parameters have yet to be determined experimentally (e.g., the rate at which AMPA receptors bind to and are released from scaffolding proteins), we select values for these parameters that produce results consistent with the known experimental data. Our choice for the rate of exocytosis is based on the work of Passafaro et al. (2001), which suggests time constants for exocytosis of ∼10–30 min. Given a basal rate of exocytosis σ ≈ κS, where S is the steady-state number of receptors in the intracellular pool, we take κ to be the reciprocal of the Passafaro et al. (2001) time constant. For GluR2/3 receptors, we choose the rate of endocytosis approximately to balance the flux attributable to exocytosis, which yields the constitutive recycling of GluR2/3 heteromers at the dendritic spine. We take endocytic rates to be consistent with those suggested by Ehlers (2000). We choose the hopping rate h of receptors flowing between the PSD and ESM so that approximately one-half of all synaptic receptors are mobile, consistent with the data of Groc et al. (2004) and Ashby et al. (2006). Because GluR2/3 receptors are inserted directly into the PSD, the majority of basal free receptors found in the PSD in our model are of this type. The steady state is thus maintained primarily by the constitutive recycling of GluR2/3 receptors, which involves a constant flux of receptors from the PSD to the ESM, where they are endocytosed and either reinserted into the membrane surface or degraded. We select the basal number of binding sites to approximately match the number of free receptors, and choose basal binding and release rates so that (1) nearly all of the binding sites are filled by GluR2/3 receptors, and (2) they are consistent with other known systems (Lauffenburger and Linderman, 1993). Note that taking the synapse to be unsaturated, that is, to have a significant fraction of free binding sites, would require unrealistically low binding affinities. This, in turn, makes it difficult to match the range of experimental data presented below. Estimates of receptor concentrations within the PSD range from 100 to 1000 receptors μm⁻², whereas within the ESM they range from 1 to 20 receptors μm⁻² (Nusser et al., 1998; Cottrell et al., 2000; Tanaka et al., 2005). The total number of receptors in the PSD depends on the size of the PSD and can vary from 1 to 200. We choose a parameter regimen in which there are ∼40 synaptic receptors under basal conditions and the concentration of extrasynaptic receptors is ∼25 μm⁻². Finally, the background concentration of GluR1/2 receptors in the dendrite is taken to be R̄_I = 10 μm⁻², whereas the corresponding background concentration of GluR2/3 receptors is taken to be zero. This is based on the assumption that GluR2/3 receptor trafficking is local to the synapse, whereas trafficking of GluR1/2 receptors occurs extrasynaptically. It should be possible to determine R̄_I self-consistently by considering a multisynapse version of our model, in which extrasynaptic receptors diffuse between synapses distributed on the surface of a dendritic cable.

In Figure 2, we show how the steady-state number of GluR1/2 and GluR2/3 receptors in the PSD depends on various trafficking parameters as determined by Equation 5–8 in Materials and Methods. Figures 2A–D show how total receptor number varies with the rates of exocytosis and endocytosis. As one expects, the number of receptors is an increasing (decreasing) function of the rate of exocytosis (endocytosis). However, this dependence is weak unless free receptors tend to be confined within their compartment, that is, the hopping rates h_j, Ω₁₂₇ have to be sufficiently small. Note that the number of free and bound receptors within the PSD are approximately equal. In Figure 2E, we show how the PSD binding sites become saturated with GluR2/3 receptors as the ratio of binding to unbinding rates α_II/β_II increases.

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Figure 2.

Steady-state behavior of synaptic receptor number as a function of trafficking parameters. All parameters are at their basal values as listed in Table 1 unless specified otherwise. A, B, Variation of receptor number with rate of endocytosis k_j, j = I, II. The solid curves show number of receptors in the PSD at the basal hopping rate between the ESM and dendrite, denoted by Ω̄_j, whereas the dashed curves show corresponding receptor numbers when Ω̄_j is increased by a factor of 10. Receptor number decreases with increasing endocytosis, and dependence becomes weaker with increasing hopping rate between the ESM and dendrite. C, Variation of receptor number with rate of exocytosis σ_I. The solid curves show number of receptors in the PSD at basal hopping rates, whereas dashed curves show corresponding receptor numbers when Ω̄_I is increased by a factor of 10. Receptor number increases with increasing exocytosis, and dependence becomes weaker with increasing hopping rate between the PSD and ESM. D, Variation of receptor number with rate of exocytosis σ_II. The solid curves show number of receptors in the PSD at basal hopping rate between PSD and ESM, denoted by h̄_II, whereas the dashed curves show corresponding receptor numbers when h̄_II is increased by a factor of 10. Receptor number increases with increasing exocytosis, and dependence becomes weaker with increasing hopping rate between the PSD and ESM. The nonlinear dependence of the total number of receptors near σ_II = 0 represents the transition from unsaturated to saturated binding sites. In A–D, the point corresponding to basal conditions is indicated by a filled circle on the solid black curve. Note that the number of free receptors is approximately equal to one-half the total number of receptors. E, Variation of receptor number with the ratio of type II binding rate to unbinding rate. Total receptor number increases with increasing α_II/β_II and saturates near α_II/β_II = 1, indicating that all binding sites are full. This is attributable to the large number of free GluR2/3 receptors in the PSD, which fill available binding sites and replace bound GluR1/2 receptors. Dependence is weak near the basal value of α_II/β_II = 10 (data not shown), indicating a strong affinity of GluR2/3 receptors for binding sites. Receptor numbers are relatively insensitive to α_I/β_I (data not shown).

The dependence of synaptic strength on exocytosis/endocytosis has been investigated experimentally by pharmacologically blocking the insertion or internalization of receptors in a CA1 hippocampal cell (Luscher et al., 1999). For example, loading a cell with Botox disrupts exocytosis by inactivating v-SNAREs (vesicle-soluble N-ethylmaleimide-sensitive factor attachment protein receptors), and causes a 40% reduction in AMPA receptor EPSCs over 20 min. However, a targeted inhibition of endocytosis results in a twofold increase in the AMPA receptor EPSCs over a similar timescale. If we assume that the amplitude of a recorded EPSP is approximately proportional to the number of AMPA receptors from which the recording is made, then the time course of the number of AMPA receptors should approximately follow the time course of the recorded EPSPs. Thus, the Luscher et al. (1999) data suggest that, when exocytosis is blocked, the number of AMPA receptors at the PSD should approximately halve, and when endocytosis is blocked, the number of receptors should approximately double.

We can reproduce the results of Luscher et al. (1999) in our model by setting to zero either exocytosis (σ_I = σ_II = 0) or endocytosis (k_I = k_II = 0) and determining the resulting time-dependent decrease or increase in the number of synaptic receptors. We find that blocking exocytosis without changing any other parameters of the model leads to a loss of about one-half of the receptors in the PSD (Fig. 3A), whereas we find a doubling of receptors in the PSD after endocytosis is blocked (Fig. 3B). Note that, in the latter case, one might expect blocking of endocytosis within the dendrite of a cell to raise the background concentrations R̄_j of AMPA receptors. However, the results shown in Figure 3B are insensitive to increases in background concentration, at least at the given basal hopping rates. Interestingly, the time courses predicted by our model are also similar to those found by Luscher et al. (1999), although the reduction in response to exocytic blockage is slightly faster in our model and the increase in response to endocytic blockage is slightly slower. It should be noted that, over the time course shown in Figure 3A, the receptor concentration has not yet reached a steady state, that is, the asymptotic value shown in Figure 3A is larger than the steady-state value shown in Figure 2D for zero exocytosis. Our model thus predicts that when exocytosis is completely blocked the number of synaptic receptors should continue to decrease at a slow rate over several hours to reach steady state. This slower component represents the unbinding of receptors from scaffolding proteins and their ultimate escape from the PSD to the ESM boundary.

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Figure 3.

Time course of AMPA receptors after blocking exocytosis/endocytosis. A, Blocking exocytosis. With receptors at basal steady state at time t < 0, exocytosis is blocked by setting σ_j = 0 (j = 1,2) at t = 0. The number of AMPA receptors in the PSD almost halves in <10 min (because of the loss of free receptors) and decreases to ∼1 over ∼10 d (data not shown). B, Blocking endocytosis. Endocytosis is blocked by setting k_j = 0 (j = 1,2) at t = 0. The number of AMPA receptors in the PSD nearly doubles within 1 h (because of the addition of free receptors) and reaches a new steady-state value of ∼84. These results are consistent with those of Luscher et al. (1999).

Increase of synaptic receptor concentration during LTP

There is growing experimental evidence to suggest that a major contribution to the expression of early-phase LTP is the trafficking of GluR1/2 receptors into the PSD (Song and Huganir, 2002; Bredt and Nicoll, 2003; Choquet and Triller, 2003; Collingridge et al., 2004). This is likely to involve an activity-dependent increase in the rate of exocytosis of GluR1/2 receptors into the extrasynaptic membrane, and a corresponding increase in the number and/or affinity of receptor binding sites within the PSD. A number of PSD-95/Discs large/zona occludens-1 (PDZ)-domain-containing proteins have been identified as playing a role in the regulation of AMPA receptor trafficking. PDZ domains are protein–protein interaction motifs that bind to the intracellular C-terminal domain of their target proteins. One example of a PDZ-domain-containing protein that binds directly to GluR1/2 is SAP-97 (synapse-associated protein 97), which has been implicated in the trafficking of AMPA receptors into dendritic spines after phosphorylation by calcium/calmodulin-dependent protein kinase II (CaMKII) (Hayashi et al., 2000; Shi et al. 2001; Wu et al., 2002; Mauceri et al., 2004). CaMKII is itself activated by an activity-dependent rise in intracellular Ca²⁺ concentration under LTP stimulus protocols (Lisman 2003; Malenka and Bear, 2004).

GluR1/2 receptors also associate with TARPs (transmembrane AMPA receptor regulatory proteins) such as stargazin. Interestingly, stargazin binds with the PDZ-domain-containing protein PSD-95, a major scaffolding protein of the PSD (Chen et al., 2000; Schnell et al., 2002). Disrupting the ability of stargazin to interact with PSD-95 leads to a massive decrease in synaptic AMPA receptors and an increase in extrasynaptic receptors. Moreover, an overexpression of PSD-95 enhances the number of synaptic AMPA receptors (El-Husseini et al., 2000; Schnell et al., 2002), whereas removal of PSD-95 from the synapse by depalmitoylation depletes synaptic AMPA receptors (El-Husseini et al., 2002). However, increasing the expression of stargazin without changing the level of PSD-95 increases the number of extrasynaptic AMPA receptors without changing the strength of a synapse (Schnell et al., 2002). It is thus hypothesized that the interaction between stargazin and AMPA receptor subunits is important for the surface delivery of AMPA receptors, whereas the interaction between stargazin and PSD-95 is important for the synaptic targeting of the receptors. Recent evidence suggests that phosphorylation of stargazin by CaMKII facilitates the interaction with PSD-95 and is a critical component of LTP (Tomita et al., 2005).

Motivated by the above experimental findings, we numerically solve the kinetic Equations 1–4 of our receptor trafficking model to determine the time-dependent variation in the GluR1/2 receptor concentration in response to increases in the rate of exocytosis σ_I, the affinity α_I of binding sites in the PSD, and the hopping rate h_I between the PSD and ESM (under the assumption that the interaction between PSD-95 and stargazin facilitates the entry of receptors into the PSD) (see Materials and Methods). We assume that such changes occur rapidly relative to the time course associated with the redistribution of AMPA receptors. This is based on experimental data indicating that CaMKII, one of the crucial components of the signaling pathways involved in the induction of LTP, acts like a rapid molecular switch (Zhabotinsky, 2000; Lisman and Zhabotinsky, 2001; Lisman et al., 2002). It is important to emphasize, however, that the detailed molecular mechanisms underlying the trafficking of AMPA receptors during LTP are still far from clear. One of the useful features of our mathematical model is that it allows us to explore a given hypothesis regarding the regulation of receptor trafficking during LTP.

To model changes in the rate of exocytosis, we set σ_I = κ_IS_I, where S_I is the total number of GluR1/2 receptors in the intracellular pool and κ_I is the rate of exocytosis per receptor (see Eq. 4). We assume that, in steady state, the rate of exocytosis is equal to the rate of receptor synthesis, which we denote by δ_I. A sudden increase in κ_I, and hence σ_I, results in the rapid insertion of intracellular receptors into the extrasynaptic membrane, and a corresponding reduction in S_I so that, after an initial transient, the rate of exocytosis returns to the steady-state value δ_I. (Increasing the rate of receptor synthesis δ_I would lead to a persistent change in the rate of receptor insertion, but does not produce realistic time courses for LTP.) Therefore, to maintain an increase in the number of synaptic receptors, we further assume that LTP involves an increase in the concentration L of binding sites. One proposal for how this could occur is that AMPA receptors delivered to the synapse bring with them so-called “slot” proteins that provide the additional binding sites (Malinow, 2003). We model this by supplementing Equations 1–4 with the following dynamical equation for the concentration L of binding sites: That is, the rate of increase in L is taken to be proportional to the rate at which the intracellular store of GluR1/2 receptors is depleted, with c a dimensionless constant. Solving Equations 4 and 11 shows that L increases asymptotically to a new steady-state value. It is important to note that L only satisfies Equation 11 while the rate of exocytosis per receptor κ_I is maintained above basal levels. When the latter returns to its basal value, perhaps because of deactivation of the CaMKII switch, the number of intracellular receptors S_I recovers according to Equation 4, whereas L is now held fixed. [Over longer timescales of hours and days, additional mechanisms are needed to stabilize L with respect to the constitutive recycling of scaffolding proteins (see below and Discussion).]

In Figure 4, A and B, we plot the resulting time courses for the total number of receptors in the PSD and ESM, respectively. Also shown in Figure 4A is the increase in the number of scaffolding proteins within the PSD, which asymptotically approaches a new steady-state value over the time course of a few minutes. The corresponding depletion of the intracellular pool is shown in Figure 4B. If we assume that the number of synaptic AMPA receptors is proportional to the size of EPSPs, then the profile shown in Figure 4A is consistent with recordings from single synapses (Petersen et al., 1998; O'Connor et al., 2005) and field EPSPs (Bliss and Lomo, 1973; Hanse and Gustafsson, 1992). That is, typical EPSPs recorded during LTP show a sharp, initial rise that peaks in ∼30–60 s at ∼200–300% of the baseline response, and then settles at a slower rate to ∼150–200% of baseline response. In Figure 4, C and D, we show the time-dependent variation in the number of synaptic and extrasynaptic receptors in response to changes in the rate of exocytosis alone, without a corresponding increase in binding sites, binding affinity, or hopping rate. It can be seen that there is a large increase in the number of extrasynaptic receptors but only a small transient increase in the number of synaptic receptors. This is consistent with what happens when there is an overexpression of stargazin without a corresponding increase in PSD-95 (Schnell et al., 2002). That is, stargazin can facilitate transport of AMPA receptors to the surface but is not able to target synapses unless it can interact with PSD-95.

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Figure 4.

Time course of AMPA receptors during LTP. A, B, With receptors at basal steady state for t < 0, LTP is induced at time t = 0 by making the following changes to the basal GluR1/2 parameter values listed in Table 1 and numerically solving Equations 1–4: binding rate α_I = 0.001 μm² s⁻¹, exocytic rate per receptor κ_I = 0.0556 s⁻¹, and hopping rate h_I = 0.01 μm s⁻¹. Binding site trafficking is also activated according to Equation 11 with c = 0.65. A shows the variation in the total number of receptors (solid black curve) and the number of binding sites (solid green curve) within the PSD, whereas B shows the corresponding variation in the total number of receptors in the ESM (solid black curve) and the number of intracellular receptors (solid gray curve). The contributions from the various receptor types are also shown in A. The number of receptors in the ESM rises transiently because of the exocytosis of intracellular GluR1/2 receptors. Some of these newly exocytosed receptors enter the PSD and are immobilized by the newly available binding sites. These results are consistent with experimentally recorded EPSPs after LTP induction (Hanse and Gustafsson, 1992; O'Connor et al., 2005). C, D, Time course of synaptic receptors (C) and extrasynaptic receptors (D) without synaptic targeting. Labeling of various curves is as in A and B. With receptors at basal steady state at time t < 0, the rate of GluR1/2 exocytosis is increased by setting κ_I = 0.0556 s⁻¹ at time t = 0. However, the hopping rate and binding affinity of GluR1/2 receptors and the number of binding sites remain at basal levels. The concentration in the ESM rises transiently as GluR1/2 receptors from the intracellular pool are exocytosed there, as in the case of LTP, but now there is only a small transient rise in the number of synaptic receptors. These results illustrate that both exocytosis and synaptic targeting are required for LTP. This is consistent with the suggestion that stargazin plays a role in transporting GluR1/2 receptors to the membrane surface, whereas its interaction with PSD-95 is required for synaptic targeting (Schnell et al., 2002). E, Exchange of GluR1/2 and GluR2/3 AMPA receptors. After 1 h of maintaining LTP parameters, all parameters are returned to their basal values except the binding site concentration L, at which time GluR2/3 receptors begin to replace GluR1/2 receptors at the binding sites. These results are consistent with those of McCormack et al. (2006).

It is important to emphasize that the distribution of receptors has not reached a steady state over the time course of a few minutes shown in Figure 4, A and B. For during this period, the rate of exocytosis σ_I has returned to its basal level, which implies that there are not enough free GluR1/2 receptors to maintain equilibrium with the receptors bound to the newly activated binding sites within the PSD. Thus, over a longer time period of several hours, the GluR1/2 receptors are slowly exchanged with GluR2/3 receptors through the process of constitutive recycling (Fig. 4E). Such an exchange has been observed experimentally and has been suggested as a mechanism for maintaining bidirectional synaptic plasticity (McCormack et al., 2006). However, to stabilize the strength of a synapse over these longer timescales, additional mechanisms are necessary. In particular, the slow turnover of scaffolding proteins within the PSD (Okabe et al., 1999; Inoue and Okabe, 2003) suggests that the increase in the concentration L of binding sites through the trafficking of slot proteins is only temporary and that L eventually returns to basal levels. One possible way of maintaining the increased number of binding sites is through structural changes in the dendritic spine (see Discussion).

Decrease of synaptic receptor concentration during LTD

Many synapses that exhibit LTP also exhibit LTD under appropriate stimulus conditions. A common stimulus protocol for inducing LTD is a prolonged repetitive synaptic stimulation at 0.5–5 Hz involving ∼900 stimuli (Dudek and Bear, 1992, 1993). Whether LTP or LTD occurs depends on the spatiotemporal properties of the intracellular Ca²⁺ signal (Malenka and Bear, 2004; Rubin et al., 2005). LTP is induced by a large, fast increase of intracellular Ca²⁺ concentration in the dendritic spine, whereas LTD is induced by a moderate, slow increase that may be accompanied by Ca²⁺ release from intracellular stores (Franks and Sejnowski, 2002). The LTD induction signal triggers signaling cascades that involve the activation of enzymes such as PKC (protein kinase C), PP1 (protein phosphatase 1), and calcineurin (Winder and Sweatt, 2001; Song and Huganir, 2002; Steinberg et al., 2006). Just as LTP is associated with an increase in the number of synaptic AMPA receptors because of the influx of receptors from the extrasynaptic membrane, LTD appears to involve a loss of receptors from the PSD because of modifications in constitutive recycling (Carroll, et al., 1999a,b; Beattie et al., 2000; Lin et al., 2000; Man et al., 2000). One triggering mechanism is thought to be phosphorylation of GluR2/3 synaptic receptors, which disrupts the interactions with the stabilizing scaffolding protein GRIP/ABP and allows for association with PICK1 (Chung et al., 2000; Matsuda et al., 2000; Kim et al., 2001; Perez et al., 2001; Lu and Ziff, 2005). PICK1 mediates the loss of AMPA receptors at the PSD, because the overexpression of PICK1 at synapses is correlated with a decrease in membrane expression of AMPA receptors. It has been suggested that the switch from association with GRIP/ABP to PICK1 plays a role in untethering receptors from PSD binding sites, escorting them out of the PSD, and then facilitating internalization of the receptors once they have reached the ESM (Song and Huganir, 2002; Steinberg et al., 2006). Another possible mechanism for reducing the number of synaptic AMPA receptors during LTD is through the removal of scaffolding proteins from the PSD. There is some indirect experimental evidence for this, namely, that NMDA receptor activation can lead to the ubiquitination and subsequent degradation of the scaffolding protein PSD-95 (Colledge et al., 2003). The removal of a scaffolding protein releases the associated bound receptor, which can then diffuse out of the PSD and be internalized through endocytosis.

The above role of GluR2/3 receptor trafficking in LTD is consistent with data suggesting that, under basal conditions, the majority of receptors within the PSD are of this type (Shi et al., 2001; McCormack et al., 2006). However, experimental studies in knock-out mice provide evidence that dephosphorylation of GluR1 subunits is an essential component of hippocampal LTD (Lee et al., 1998, 2000, 2003). This suggests that there may be a number of distinct mechanisms for the removal of synaptic receptors during LTD (Bredt and Nicoll, 2003; Malenka and Bear, 2004). As in the case of LTP, we can use our model to explore various hypotheses regarding how changes in receptor trafficking generate responses that are consistent with those observed during LTD. Here, we will focus on the role of GluR2/3 under the assumption that basal levels of GluR1/2 within the PSD are low. To proceed, we extend our basic model by assuming that GluR2/3 receptors within the PSD exist in two distinct states corresponding to association with GRIP/ABP and PICK1, respectively (see Materials and Methods). Suppose that under basal conditions the transition rate μ from the GRIP-associated state to the PICK-associated state is negligible (μ = 0) so that the number of receptors in the PICK-associated state is approximately zero. The receptor concentrations then satisfy the original set of Equations 1–4. We now assume that, during the induction phase of LTD, the transition rate μ increases, leading to the conversion of some bound receptors to the PICK-associated state. We further assume that the change in μ is rapid compared with the timescale of receptor trafficking, and is maintained during the presentation of the LTD stimulus. (μ rapidly returns to zero once the stimulus is removed.) The combined system of GRIP-associated and PICK-associated GluR2/3 receptors within the PSD now evolves according to Equations 9 and 10. In particular, PICK-associated receptors rapidly untether from their binding sites and hop to the ESM where they are endocytosed, resulting in a reduction in the number of receptors within the PSD. However, on its own, this mechanism would not maintain LTD once the low-frequency stimulus is removed, because there is currently no evidence for a bistable switch analogous to CaMKII that would allow levels of phosphorylation to persist. Therefore, receptors would convert back to the GRIP-associated state and the synapse would recover. To have a persistent reduction in synaptic strength, we assume that, as receptors untether from binding sites, these sites are removed at some rate γ. We thus supplement Equations 1a, 2a, 9, and 10 of the extended model by the following equation for the concentration L of binding sites: where Q_I is the concentration of GluR1/2 receptors and Q_II,a and Q_II,b are, respectively, the concentrations of GRIP-associated and PICK-associated GluR2/3 receptors bound to scaffolding proteins within the PSD. This equation, which takes the rate of decrease of L to be proportional to the concentration of free binding sites within the PSD, only holds during the presentation of the LTD stimulus; once the stimulus is removed, L stops decreasing. Under basal conditions, the binding sites are almost fully occupied. However, during LTD, GRIP-associated receptors are converted to PICK-associated receptors so that Q_II,a decreases and Q_II,b increases. Because PICK-associated receptors are more likely to unbind from scaffolding proteins, we find that Q_II,a decreases faster than Q_II,b increases. The net result is that L decreases because of the freeing of binding sites. Once the LTD stimulus is removed, all receptors convert back to the GRIP-associated state and all of the remaining binding sites become reoccupied.

We numerically solve the extended receptor trafficking model given by equations 1a, 2a, 9, 10, and 12 to determine the time-dependent variation in synaptic receptor concentration during LTD. Figure 5A shows the time course of the total number of receptors for the various receptor types. The synapse is assumed to be in steady state under basal conditions for t < 0, with a negligible concentration of PICK-associated receptors (μ = 0). We assume that a low-frequency stimulus is applied for 900 s during which μ > 0. It can be seen that there is a conversion of GRIP to PICK during the presentation of the stimulus, with a partial recovery after the stimulus is removed. The steady-state receptor concentration has decreased, however, because of the removal of binding sites. Interestingly, our model can reproduce a variety of experimental results. For example, Dudek and Bear (1992) showed that increasing the frequency of the stimulus from 3 to 10 Hz, say, can lead to a transient reduction in synaptic strength rather than LTD. One way to generate this in our model is to assume that the stimulus induces the conversion of GRIP to PICK, but the number of binding sites is not reduced (Fig. 5B). In another experiment, Dudek and Bear (1993) showed how a sequence of low-frequency stimulations each separated by ∼45 min can induce a saturating sequence of LTD. This result can also be reproduced by our model, with the saturation arising from the fact that, even if most binding sites are removed, there are still free receptors present.

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Figure 5.

Time course of AMPA receptors during and after LTD. The variation in the total number of receptors (solid black curves) and binding sites (solid green curves) in the PSD are shown together with contributions from the various receptor types. A, With receptors at basal steady state for t < 0, LTD is induced at time t = 0 by increasing from zero the transition rate μ from GluR2/3–GRIP to GluR2/3–PICK and maintaining this for 900 s. We numerically solve the LTD model in which GluR2/3 receptors within the PSD evolve according to 9 and 10 and the concentration of binding sites decreases according to Equation 12. LTD parameter values are μ = 0.01 s⁻¹, ν = 0.01 s⁻¹, β_II* = 0.1 s⁻¹, k_II* = k_II = 0.1667 s⁻¹, and γ = 0.001 s⁻¹. All other parameters are as listed in Table 1. GluR2/3–GRIP is rapidly converted to GluR2/3–PICK during the first few minutes of LTD; afterward, this conversion occurs at a slower rate. Bound GluR2/3–PICK quickly releases from binding sites, and free GluR2/3-PICK exits the PSD and is endocytosed. Notice how the number of binding sites follows the loss of bound receptors. LTD induction ends at 900 s such that μ and γ return to zero, and GluR2/3–PICK rapidly converts back to GluR2/3–GRIP at the rate ν. However, the new steady-state number of receptors in the PSD is much lower because of the loss of binding sites. The variation in the number of synaptic receptors is consistent with typical recordings of EPSPs during LTD [Dudek and Bear (1992), their Fig. 2A]. B, Time course of receptors in the PSD during LTD with moderate frequency stimulus. LTD is induced as in A, except that γ = 0 s⁻¹ throughout. Although these time courses are similar to those in A, the number of binding sites remains unchanged, so that the number of AMPA receptors in the PSD returns to its initial steady-state value after LTD induction terminates. This result is consistent with Dudek and Bear (1992), their Fig. 2B. C, Saturation of LTD. LTD is induced as in A, except that it is followed by 45 min of basal activity, and this 1 h epoch is repeated three times, followed by the induction of LTP. (LTP is induced as in Fig. 4, except c = 0.325.) Notice that the loss of PSD receptors decreases in each consecutive epoch, proportional to the number of binding sites at the beginning of each epoch. Saturation occurs because only bound, and not free, receptors are lost during LTD. The variation in the number of synaptic receptors is consistent with Dudek and Bear (1993), their Fig. 7.