Abstract
Changes in the efficacies of synapses are thought to be the neurobiological basis of learning and memory. The efficacy of a synapse depends on its current number of neurotransmitter receptors. Recent experiments have shown that these receptors are highly dynamic, moving back and forth between synapses on time scales of seconds and minutes. This suggests spontaneous fluctuations in synaptic efficacies and a competition of nearby synapses for available receptors. Here we propose a mathematical model of this competition of synapses for neurotransmitter receptors from a local dendritic pool. Using minimal assumptions, the model produces a fast multiplicative scaling behavior of synapses. Furthermore, the model explains a transient form of heterosynaptic plasticity and predicts that its amount is inversely related to the size of the local receptor pool. Overall, our model reveals logistical tradeoffs during the induction of synaptic plasticity due to the rapid exchange of neurotransmitter receptors between synapses.
https://doi.org/10.7554/eLife.37836.001Introduction
Simple mathematical models of Hebbian learning exhibit an unconstrained growth of synaptic efficacies. To avoid runaway dynamics, some mechanism for limiting weight growth needs to be present. There is a long tradition of addressing this problem in neural network models using synaptic normalization rules (von der Malsburg, 1973; Oja, 1982; Miller and MacKay, 1994; Wu and Yamaguchi, 2006; Lazar et al., 2009). Obviously, in order to keep up with the pace of synaptic changes due to Hebbian plasticity, normalization mechanisms must act sufficiently fast. Slow homeostatic synaptic scaling mechanisms (Turrigiano et al., 1998) may therefore be illsuited for ensuring stability (Wu and Yamaguchi, 2006; Zenke et al., 2013; Chistiakova et al., 2015). A particularly interesting fast normalization rule scales synapses multiplicatively such that the sum of synaptic weights remains constant. Attractive features of such a rule, next to its conceptual simplicity, are that the relative strength of synapses is maintained and that in combination with Hebbian mechanisms it naturally gives rise to lognormallike weight distributions as observed experimentally (Song et al., 2005; Loewenstein et al., 2011; Zheng et al., 2013; Miner and Triesch, 2016). While such normalization mechanisms are not considered biologically implausible, their link to neurobiological experiments has been tenuous.
In a recent review, Chistiakova et al. (2015) argue that socalled heterosynaptic plasticity (Lynch et al., 1977; Bailey et al., 2000; Jedlicka et al., 2015; Antunes and SimoesdeSouza, 2018) may be a prime candidate for such a fast synaptic normalization scheme. The term ‘heterosynaptic’ plasticity is used in contrast to the much more widely studied ‘homosynaptic’ plasticity, where changes occur in a stimulated synaptic pathway. In contrast, heterosynaptic plasticity refers to changes in synaptic efficacies that occur in an unstimulated pathway after the stimulation of a neighboring pathway. The most common form of heterosynaptic plasticity has a homeostatic nature: if synapses in stimulated pathways potentiate, then this is accompanied by a depression of unstimulated pathways. Conversely, if synapses in stimulated pathways depress, this is accompanied by a potentiation of unstimulated pathways. A classic example of this has been observed in intercalated neurons of the amygdala (Royer and Paré, 2003).
Interestingly, such homeostatic regulation is also consistent with findings at the ultrastructural level. The physical size of a synapse, in particular the surface area of the postsynaptic density (PSD), is commonly used as a proxy for a synapse’s efficacy (Chen et al., 2015; Bartol et al., 2015). Bourne and Harris (2011) have observed coordinated changes in PSD surface areas of dendritic spines in the hippocampus after LTP induction. They report that increases in the PSD surface areas of some synapses or the creation of new synapses are balanced by decreases of PSD surface areas of other synapses or their complete elimination such that the total amount of PSD surface area stays approximately constant. Recent findings support the idea that such regulation may occur at the level of individual dendritic branches (Barnes et al., 2017).
A proxy of synaptic efficacy that is more precise than PSD surface area is the number of AMPA receptors (AMPARs) inside the PSD. AMPARs are glutamategated ion channels responsible for most fast excitatory transmission in the vertebrate brain. During various forms of plasticity the number of these receptors at synapses is modified, leading to changes in synaptic efficacies, reviewed by Chater and Goda (2014). Therefore, a full understanding of synaptic plasticity requires a careful description of the mechanisms that regulate AMPAR numbers in synapses.
Here we show how the behavior of keeping the sum of synaptic efficacies approximately constant on short time scales naturally arises from a generic model in which individual synapses compete for a limited supply of synaptic building blocks such as AMPARs or other protein complexes that are necessary to stabilize AMPARs inside the PSD. We assume that there is a local dendritic store of these building blocks and that they enter and leave dendritic spines in a stochastic fashion. The model predicts that the redistribution of synaptic efficacies should act multiplicatively, as is often assumed in ad hoc normalization models. We also show that this model naturally gives rise to a homeostatic form of heterosynaptic plasticity, where synapses grow at the expense of other synapses. To this end, we introduce a model of homosynaptic LTP describing the time course of the incorporation of new receptors and slots during LTP induction. Finally, we quantify the scale of spontaneous synaptic efficacy fluctuations due to the fast stochastic exchange of AMPARs between the dendritic pool and postsynaptic receptor slots. We show that small synapses exhibit relatively stronger efficacy fluctuations, which are further accentuated if the local receptor pool is small. Overall, the model reveals how the dynamic behavior of neurotransmitter receptors plays an important role in shaping synaptic plasticity.
Results
Formulation of the model
The architecture of the model is shown in Figure 1. We consider a piece of dendrite with $N\in \mathbb{N}$ synaptic inputs. Each synapse is characterized by two variables. First, each synapse $i\in 1,\dots ,N$ has a number of slots ${s}_{i}\in {\mathbb{R}}^{\ge 0}$ for neurotransmitter receptors. Second, at any time a certain number of slots ${w}_{i}\in {\mathbb{R}}^{\ge 0}$ actually contain a receptor. ${w}_{i}$ determines the current weight or efficacy of a synapse. We assume that the PSD cannot hold more functional receptors than there are slots, that is, ${w}_{i}\le {s}_{i}$. At biological synapses AMPARs are clustered inside PSDs into nanodomains of about 70 nm that contain on average 20 receptors (Nair et al., 2013). Interestingly, those postsynaptic nanodomains are aligned with presynaptic release sites forming socalled nanocolunms. It is noteworthy that AMPARs have low affinity for glutamate such that receptors outside of nanodomains are unlikely to participate in synaptic transmission (Liu et al., 1999; Biederer et al., 2017). Next to receptors in the synapses, the neuron maintains a pool of receptors freely diffusing at the neuron surface and ready to be stabilized inside nanodomains. The size of this pool is denoted $p\in {\mathbb{R}}^{\ge 0}$. Note that for mathematical convenience we here consider the ${s}_{i}$, ${w}_{i}$ and $p$ to be real numbers that can take noninteger values. In the stochastic version of the model introduced below these will be natural numbers.
Receptors can transition from the pool to empty slots in a synapse or detach from such a slot and return into the pool with rates $\alpha \in {\mathbb{R}}^{>0}$ and $\beta \in {\mathbb{R}}^{>0}$, respectively. Receptors in the pool are removed with a rate $\delta \in {\mathbb{R}}^{>0}$ corresponding to internalization of the receptors from the cell surface (endocytosis). To counteract this loss, new receptors are added at a rate $\gamma \in {\mathbb{R}}^{>0}$ and injected into the pool corresponding to externalization of the receptors to the cell surface (exocytosis). In the limit of large receptor numbers, the dynamics of the system can be described by the following system of coupled ordinary nonlinear differential equations:
In the first equation, $\beta {w}_{i}$ describes the return of receptors from synapse $i$ into the pool. The term $\alpha p({s}_{i}{w}_{i})$ describes the binding of receptors from the pool to empty slots in synapse $i$, which is assumed to be proportional to both the number of receptors in the pool and the number of free slots in the synapse. In the second equation, $\delta p$ describes the deletion of receptors from the pool, $\gamma $ represents the gain of new receptors, ${\sum}_{i}}\text{\hspace{0.17em}}\beta {w}_{i$ describes the return of receptors from the synapses into the pool, and finally ${\displaystyle {\sum}_{i}}\text{\hspace{0.17em}}\alpha p({s}_{i}{w}_{i})$ describes the loss of receptors from the pool which bind to free slots in the synapses. Together, this is a system of $N+1$ coupled ordinary nonlinear differential equations. It is nonlinear, because the equations contain product terms of the state variables, in particular the $p{w}_{i}$ terms.
The model can be interpreted in different ways. Its generic interpretation is that the ‘receptors’ of the model are AMPA receptor (AMPAR) complexes composed of AMPARs and transmembrane AMPAR regulatory proteins (TARPs) such as stargazin. The ‘slots’ are postsynaptic density structures comprising membraneassociated guanylate kinase (MAGUK) proteins such as PSD95 attached to the postsynaptic membrane, which stabilize AMPARs in the postsynaptic density (PSD) (Hafner et al., 2015; Schnell et al., 2002; Sumioka et al., 2010). Inside the synapses PSD95 proteins are highly packed (roughly 300 molecules per PSD) (Kim and Sheng, 2004) and largely immobile (Sturgill et al., 2009). When a receptor enters a synapse binding to one or more immobile PSD95 proteins results in receptor immobilization. In this generic interpretation of the model, the pool of receptors is the set of AMPARs that diffuse in the plasma membrane and that are captured by the slots. Addition of receptors to the pool then subsumes (some or all of) the processes that assemble AMPARs and prepare them for the insertion into slots: assembly of the receptors from the component subunits, trafficking, attachment of TARPs, externalization, and potentially phosphorylation. Removal from the pool similarly subsumes the set of reverse processes. Several variations of this generic interpretation are possible depending on what exactly we would like to associate with the ‘receptors’ in the model: AMPARs, AMPAR+TARP complexes, AMPAR+TARP complexes that have already been exocytosed, phosphorylated, etc. Essentially, our model is a twostep model (production and insertion), but we leave it open for interpretation, what steps in the full chain of events are considered the ‘production’ (subsumed in rate $\gamma $) and which steps are considered the ‘insertion’ (subsumed in rate $\alpha $).
Evidently, receptor slots themselves must also be stabilized inside the PSD somehow. A second, maybe somewhat counterintuitive, interpretation of the model is therefore that it describes the binding and unbinding of receptor slots to what one might consider a slot for a receptor slot or simply slotforaslot. In this interpretation of the model, the ‘receptors’ in the description above are actually the PSD95 slot proteins and the ‘slots’ are slotsforaslot to which the PSD95 proteins can attach. The model then describes the trafficking of PSD95 into and out of the PSD, assuming that available AMPAR complexes are quickly redistributed among PSD95 slots (compared to the time scale of addition and removal of these PSD95 slots to the PSD). This interpretation may be particularly useful if the supply of PSD95 is the limiting factor in determining the number of functional AMPARs bound inside the PSD (Schnell et al., 2002). We leave open the question what exactly the slotsforaslot might be. It is clear however, that PSD95 molecules can form stable lattices inside the PSD such that PSD95 proteins could act as slots for other PSD95 proteins.
Interestingly, the analysis of the model presented in the following does not depend on which interpretation is chosen. The only additional assumption we will make is a separation of time scales between the fast trafficking of the ‘receptors’ into and out of the ‘slots’ and the slow addition and removal of receptors to the pool. Our main results only depend on this qualitative feature of the model. For the first generic interpretation of the model the assumption of a separation of time scales appears justified. If we interpret the receptor pool of the model to comprise AMPARs that have been exocytosed and diffuse in the cell membrane, then the halflife of an AMPAR in the pool is of the order of 10 min suggesting ${\delta}^{1}=10\phantom{\rule{mediummathspace}{0ex}}min/\mathrm{ln}2\approx 14\phantom{\rule{mediummathspace}{0ex}}min$ (Henley and Wilkinson, 2013; Henley and Wilkinson, 2016). In contrast, the time an AMPAR stays inside the PSD, which we interpret as the time the AMPAR is bound to a slot, appears to be of the order of maybe 30 s (Ehlers et al., 2007), suggesting $\beta}^{1}=30\phantom{\rule{mediummathspace}{0ex}}\text{s}/\mathrm{ln}2\approx 43\phantom{\rule{mediummathspace}{0ex}}\text{s$. We summarize these and other parameters of the model in Table 1. Regarding the second, slotsforaslot, interpretation of the model, we note that the halflife of PSD95 residing inside the synapse is of the order of 5 h (Sturgill et al., 2009), implying $\beta}^{1}\approx 5\phantom{\rule{mediummathspace}{0ex}}\text{h}/\mathrm{ln}2\approx 7\phantom{\rule{mediummathspace}{0ex}}\text{h$. In contrast, the global halflife of PSD95 has been estimated to be 3.67 days (Cohen et al., 2013), implying $\delta}^{1}=3.67\text{d}/\mathrm{ln}2\approx 5.30\text{d$. In either case, the assumption of a separation of time scales appears justified.
Competition for synaptic building blocks induces multiplicative scaling
We begin our analysis by finding the stationary solution of the system of coupled differential equations defined by Equations 1 and 2. First, it is convenient to introduce the total number of synaptic slots $S\equiv {\displaystyle {\sum}_{i}}\text{\hspace{0.17em}}{s}_{i}$ and the total number of docked receptors or total synaptic weight $W\equiv {\displaystyle {\sum}_{i}}\text{\hspace{0.17em}}{w}_{i}$ and note that its time derivative is $\dot{W}=\sum _{i}\text{}{\dot{w}}_{i}$. This allows us to rewrite Equation 2 as:
To find the fixed point solution ${p}^{\infty},{w}_{i}^{\infty}$ with ${W}^{\infty}={\displaystyle {\sum}_{i}}\text{\hspace{0.17em}}{w}_{i}^{\infty}$, we set the time derivatives to zero, that is, we require ${\dot{w}}_{i}=0\text{}\mathrm{\forall}i$ and $\dot{p}=0$ above. Inserting the first condition into Equation 1 and summing over $i$ yields:
Similarly, setting $\dot{p}=0$ in Equation 3 gives:
Adding Equation 4 to Equation 5 then gives the solution for ${p}^{\infty}$:
The simple and intuitive result is therefore that the total number of receptors in the pool in the steady state is given by the ratio of the externalization rate $\gamma $ and the internalization rate $\delta $. Specifically, the presence of many receptors in the pool requires $\gamma \gg \delta $.
We now solve for the steady state solutions ${w}_{i}^{\infty}$ of the ${w}_{i}$ by again setting ${\dot{w}}_{i}=0$ in Equation 1 and using Equation 6 to give:
Importantly, we find ${w}_{i}^{\infty}\propto {s}_{i}$, that is, in the steady state the weights of synapses are proportional to the numbers of slots they have. The constant of proportionality is a filling fraction and we denote it by $F$. Interestingly, the filling fraction $F$ is independent of the number of receptor slots. Figure 2A plots $F$ as a function of the ratio of the four rate constants $\left(\beta \delta \right)/\left(\alpha \gamma \right)$. We refer to this quantity as the removal ratio, because it indicates the rates of the processes that remove receptors from the slots relative to the rates of the processes that add them to slots. Note that a filling fraction close to one requires $\beta \delta \ll \alpha \gamma $.
Summing Equation 7 over $i$ reveals that ${W}^{\infty}=FS$, so we can also write:
where ${s}_{i}/S$ is the relative contribution of synapse $i$ to the total number of slots. Note that if the filling fraction changes, say, due to an increase in receptor externalization or a change in any of the other parameters, the relative strength of two synapses in the steady state is unaffected:
Therefore, all synaptic efficacies will be scaled multiplicatively by the same factor.
Thus, the analysis so far reveals a first prediction of the model (compare Table 2, Filling Fraction): Under basal conditions synapses in the local group have identical filling fractions. A first corollary from this prediction is that the ratio of two synapses’ efficacies in the steady state is given by the ratio of their numbers of receptor slots. A second corollary from this prediction is that when one (or more) of the transition rates changes, all synaptic efficacies are scaled multiplicatively.
To illustrate the effect of multiplicative scaling of synaptic efficacies, we consider a piece of dendrite with $N=100$ afferent synapses. The number of receptor slots ${s}_{i}$ in these synapses are drawn from a lognormal distribution with mean 1.0 and standard deviation 0.2 and subsequently scaled such that there are 100 slots per synapse on average. We consider three different filling fractions $F\in \{0.5,0.7,0.9\}$. The empirical cumulative distribution functions (CDFs) of the common (decadic) logarithms of the ${w}_{i}$ are shown in Figure 2B. The horizontal shifting of the empirical CDFs illustrates the multiplicative scaling of the individual synaptic efficacies.
The total number of receptors in the system in the steady state ${R}^{\infty}$ is given by the sum of the number of receptors in the pool and the number of receptors attached to slots. Combining the above results, we find:
In particular, the total number of receptors in the steady state depends on the total number of slots.
In the case of AMPARs, the total number of surface receptors, receptor density, or the number of slots per synapse still remain unknown. Moreover, it is likely that those numbers will vary depending on neuron type and developmental state. However, single particle tracking experiments from the laboratory of Antoine Triller performed on mature hippocampal cultured neurons provide valuable insights into the proportion of exocytosed receptors immobilized in dendritic spines in this particular system. Specifically, recent data suggest that 28% of surface AMPARs are immobilized at synapses while the remaining 72% reside in the pool of extrasynaptic receptors (Marianne Renner, personal communication). Since mature hippocampal cultured neurons are known to exhibit homeostatic and longterm plasticity, we decided to use those numbers for our simulations. Thus, we define the relative pool size $\varphi $ as:
The relative pool size $\varphi $ together with the filling fraction $F$ determine the unknown externalization rate $\gamma $ and the rate of binding to receptor slots $\alpha $. Specifically, using ${W}^{\infty}=FS$ and ${p}^{\infty}=\gamma /\delta $, we find:
Furthermore, by combining this with the implicit definition of $F$ from Equation 7 we can solve for $\alpha $ to obtain:
We can identify the term $S(1F)$ as the total number of empty receptor slots in the system. The intuitive interpretation of the result is therefore that the binding rate $\alpha $ will be big compared to the unbinding rate $\beta $ if the number of empty slots and the relative pool size are small. Using the definition of $\varphi $ we can also rewrite the expression for the total number of receptors as ${R}^{\infty}=(1+\varphi )FS$.
The above results fully describe the system after it had sufficient time to reach its equilibrium. On a shorter time scale, however, the system may transiently assume different quasistationary states, because receptor addition and removal are slow compared to receptor binding and unbinding to and from slots. In the following, we consider the shortterm behavior of the model on time scales where the total number of receptors is approximately constant. This will allow us to reveal, among other things, a transient form of heterosynaptic plasticity.
Fast redistribution of receptors between synapses is multiplicative
To study the redistribution of receptors on a fast time scale, we exploit the fact that the processes of receptor externalization and internalization are slow compared to the attaching and detaching of receptors to and from slots. For instance, the time that an AMPAR remains in the cell membrane is of the order of ten minutes while the time it resides inside the PSD is of the order of half a minute. A reasonable approximation on short times scales is therefore to neglect the production and removal terms in Equation 2. In this case, the total number of receptors $R\equiv W+p$ is constant, as can be seen by removing the $\delta p$ and $+\gamma $ terms from Equation 2, and adding Equation 1, summed over all $i$, which gives $\dot{p}+\dot{W}=\dot{R}=0$. In the Methods we show that the steady state solution on the fast time scale is then given by:
where we have introduced $\rho \equiv \beta /\alpha $ as a short hand for the ratio of the rates through which receptors leave and enter the synaptic slots. We define the corresponding shortterm steadystate filling fraction as ${F}^{\ast}={W}^{\ast}/S$. Importantly, the shortterm filling fraction ${F}^{\ast}$ is identical for all synapses. ${F}^{\ast}$ can also be expressed as a function of the steady state pool size ${p}^{\ast}=R{W}^{\ast}$ on the fast time scale, leading to a simple expression for the steady state efficacy ${w}_{i}^{\ast}$ of synapse $i$ on the fast time scale:
In the full model, this solution is assumed only transiently, because receptors can still enter and leave the system. If the number of receptors were held constant ($\gamma =0$ and $\delta =0$), then ${F}^{\ast}$, ${p}^{\ast}$, and the ${w}_{i}^{\ast}$ would describe the solution on long time scales.
The finding that the shortterm steadystate filling fraction is identical for all synapses is analogous to the solution for the long term filling fraction $F$ derived in Equation 7, which is also the same for all synapses. This implies a second prediction of the model (compare Table 2, Pool Size): When the size of the local receptor pool is manipulated, all synaptic efficacies are scaled multiplicatively.
In Figure 3 we show the behavior of ${F}^{\ast}$ as a function of $\rho $ for different combinations of total number of slots $S$ and total number of receptors $R$. For high values of $\rho $ the filling fraction ${F}^{\ast}$ always goes to zero. For $\rho $ approaching zero, ${F}^{\ast}$ achieves a maximum value which depends on whether there are fewer or more receptors than slots in the system. If there are more receptors than slots then ${F}^{\ast}$ approaches one. If there are fewer receptors than slots then ${F}^{\ast}$ approaches the ratio of receptors to slots in the system. In general, we find that the maximum shortterm filling fraction for $\rho \to 0$ is given by ${F}_{\text{max}}^{\ast}=min\{1,R/S\}$. In particular, a high filling fraction can only be achieved if $R\text{}\text{}S$, that is, there must be more receptors than slots in the system. On the other hand, ${F}^{\ast}$ is most sensitive to changes in $\rho $ when $R=S$. This can be seen by the steep negative slope of the black curves in Figure 3 for small values of $\rho $. In fact, for $R=S$ the derivative diverges, that is, ${F}^{\ast}$ reacts extremely sensitively to changes in $\rho $ (see Methods for details). We therefore note another prediction (compare Table 2, Sensitivity): On short time scales the filling fraction reacts most sensitively to changes in binding/unbinding rates if the total number of receptors matches the total number of receptor slots.
To illustrate the fast redistribution of receptors, we consider a sudden change in the pool size. In our generic interpretation of the model, this corresponds to the sudden externalization or internalization of AMPARs. To study the effect of such a manipulation, we discretize the full dynamic equations using the Euler method and solve them numerically. For illustration, we consider a piece of dendrite with just three synapses with 40, 60, and 80 slots, whose pool size is changed abruptly (Figure 4). Parameters are set to achieve a filling fraction of $F=0.9$ and a relative pool size $\varphi =2.67$. After 2 min, the number of receptors in the pool is either doubled (solid lines) or set to zero (dotted lines). In response, all synapses are rapidly scaled up or down multiplicatively. The new equilibrium is only transient, however. On a slower time scale the system returns to its starting point as the slow externalization and internalization processes drive the system back to its steady state solution ${w}_{i}^{\infty},{p}^{\infty}$.
The fast equilibration process to a transient steady state also naturally gives rise to a homeostatic form of heterosynaptic plasticity. When, e.g., the number of receptor slots in some synapses is quickly increased, then receptors are redistributed such that the efficacies of synapses with an increased number of receptor slots will grow, while the efficacies of other synapses will shrink, as we discuss in the following.
Competition for receptors induces transient heterosynaptic plasticity
During LTP and LTD, the number of PSD95 proteins in the synapse, which we assume to form the slots for AMPARs, is increased and decreased, respectively (Colledge et al., 2003; Lisman and Raghavachari, 2006; Ehrlich et al., 2007; Meyer et al., 2014). Importantly, these changes in slot numbers are mirrored by corresponding adjustments of synaptic AMPAR numbers leading to long lasting changes in synaptic efficacies. This suggests such modifications in AMPAR slot numbers as a central mechanism for memory storage. Therefore, we now investigate how the addition or removal of receptor slots in some synapses alters the efficacies of other synapses in the local group. We find that the model gives rise to a form of heterosynaptic plasticity, since all synapses are competing for a limited number of receptors inside the extrasynaptic receptor pool.
For illustration purposes we consider a piece of dendrite with four synaptic inputs (Figure 5). At the beginning of the simulation, the number of slots in the four synapses are 20, 40, 60, and 80. We start the system in its steady state with a filling fraction ${F}^{}=0.5$ and a relative pool size of $\varphi =2.67$. After 2 min we instantaneously increase the number of slots in the first (blue) and third (red) synapse by 100%. Subsequently, the system settles into a new (transient) equilibrium (Figure 5A). While ${w}_{1}$ and ${w}_{3}$ increase, the number of receptors in synapses 2 and 4 slightly decrease, although their numbers of slots have not changed. This behavior corresponds to a form of heterosynaptic plasticity where synapses grow at the expense of other synapses and is due to the approximately constant number of receptors on a fast time scale. Note that the sum of synaptic efficacies is not perfectly constant, however. The increase in synaptic efficacies ${w}_{1}$ and ${w}_{3}$ is bigger than the decrease of synaptic efficacies ${w}_{2}$ and ${w}_{4}$. The bigger the size of the pool, the stronger is this effect. Close to perfect balancing of synaptic weights would require $p\ll W$. Figure 5B shows the relative changes of efficacies of the synapses undergoing homosynaptic LTP (blue curve corresponding to ${w}_{1}$ and ${w}_{3}$ in A) vs. heterosynaptic LTD (green curve corresponding to ${w}_{2}$ and ${w}_{4}$ in A) and the pool (black curve).
What determines the magnitude of the heterosynaptic change? We can calculate this analytically by using the above shortterm approximation ${F}^{\ast}$ for constant receptor number. Before plasticity induction, the synaptic efficacy of a synapse in equilibrium is given by ${w}_{i}=F{s}_{i}=({s}_{i}/S)W$. The induction of homosynaptic plasticity in other synapses changes the total number of available receptor slots and we denote the new number of slots ${S}^{\prime}$. Shortly after homosynaptic plasticity induction the synaptic efficacies of a synapse that did not undergo homosynaptic plasticity will be approximately ${w}_{i}^{\ast}={F}^{\ast}{s}_{i}=({s}_{i}/{S}^{\prime}){W}^{\ast}$ as receptors are redistributed through the system. Therefore, the relative heterosynaptic change of such a synapse is given by $({w}_{i}^{\ast}{w}_{i})/{w}_{i}=({F}^{\ast}F)/F$ as long as the total number of receptors has not changed much.
In Figure 5C we plot this relative change in synaptic efficacy due to heterosynaptic plasticity as a function of the total number of receptor slots following homosynaptic plasticity induction for different filling fractions. The relative pool size is assumed to be $\varphi =2.67$. First, we can observe that reductions in the total number of slots due to homosynaptic LTD cause heterosynaptic LTP. Conversely, increases in the total number of slots due to homosynaptic LTP cause heterosynaptic LTD. Second, the amount of heterosynaptic plasticity depends on the filling fraction prior to plasticity induction. Specifically, a high filling fraction of 0.9 leads to weaker heterosynaptic plasticity.
Figure 5D shows the analogous solution for the case of a smaller receptor pool. Here we set the relative pool size to $\varphi =1.0$. Everything else is identical to Figure 5C. The scarcely filled pool strongly amplifies the heterosynaptic plasticity effect. When, e.g., new slots are added in this case, the synapses can recruit fewer receptors from the small receptor pool and the heterosynaptic effect on other synapses becomes bigger. A large receptor pool essentially functions as a buffer shielding synapses from heterosynaptic plasticity. Consistent with C, larger filling fractions again lead to less heterosynaptic plasticity.
Importantly, these effects are inherently transient. Over a sufficiently long time, the system will settle into a new (true) equilibrium, where every synapse has the same filling fraction $F$ determined by the rate constants $\alpha ,\beta ,\gamma ,\delta $ as described above. The approach towards this new true equilibrium can be seen most easily in Figure 5B, where the relative change of synaptic efficacy due to heterosynaptic plasticity (green curve) slowly decays towards zero. The new equilibrium will be stable unless the numbers of slots in the synapses change again. In the particular example of Figure 5A,B, synapses 2 and 4 slowly return to their original efficacies, while synapses 1 and 3 remain permanently strengthened due to their increased number of slots. This effect might explain the often transient nature of heterosynaptic plasticity observed in experiments, e.g., Abraham and Goddard (1983).
We therefore note the following additional predictions of the model (compare Table 2, Heterosynaptic Plasticity I, II): First, the amount of heterosynaptic plasticity is inversely related to the size of the local receptor pool and the filling fraction. Second, heterosynaptic plasticity is only transient.
Another mechanism for producing a heterosynaptic effect is changing the transition rates $\alpha $ and $\beta $ in a synapsespecific fashion. For example, increasing $\alpha $ for some synapses will attract additional receptors to these synapses and lead to a heterosynaptic removal of receptors from the remaining synapses and the receptor pool. A more complete model of homosynaptic LTP that includes a transient synapsespecific change in $\alpha $ and induces heterosynaptic LTD is discussed next.
Time course of homosynaptic LTP and accompanying heterosynaptic LTD
The assumption of a sudden increase in slot numbers from the last section is helpful for mathematical analysis but does not reflect biological reality well. Receptor slots need to be trafficked and integrated into the PSD, which cannot happen instantaneously. In fact, modifications in PSD95 protein number after plasticity induction are known to take several minutes (Colledge et al., 2003; Ehrlich et al., 2007; Meyer et al., 2014). In general, the induction of LTP is a complex process unfolding across multiple time scales. Here we propose and analyze a more realistic model of homosynaptic LTP and the accompanying heterosynaptic LTD. The model incorporates a synapsespecific transient increase in the insertion rate $\alpha $ of a potentiating synapse and a rapid and pronounced increase of its number of slots followed by a gradual decay back to a sustained elevated level. Thus, both the insertion rates ${\alpha}_{i}$ and the slot numbers ${s}_{i}$ are now considered a function of time. Formally, in order to do so we replace Equations 1 and 2 by:
where we have introduced synapse specific insertion rates ${\alpha}_{i}$ and made the time dependence of the various quantities explicit.
We model the transient increase in $\alpha $ as a linear increase to four times the original value within 17 s followed by a linear decrease back to the original value over two minutes. This time course roughly corresponds to the one reported for calcium calmodulin kinase II (CaMKII) activation by Lee and colleagues (Lee et al., 2009), essential for LTP induction and maintenance (Malenka et al., 1989). CaMKII activation leads to the phosphorylation of many synaptic target proteins including the AMPAR auxiliary protein Stargazin which in turn increases the number of stabilized receptors in the synapse (Opazo et al., 2010). Thus, we assume here that increased CaMKII activation observed experimentally drives up the insertion rate $\alpha $. For the time course of the insertion of receptor slots, no direct measurements exist to our knowledge. Therefore, we make the simplifying assumption that the number of slots is related to the change in size of the dendritic spine, which was also measured by Lee and colleagues (Lee et al., 2009). We model their data as a sigmoidal increase to five times the original spine volume over the course of two minutes followed by an exponential decay to two times the original spine volume over the course of around twenty minutes (time constant of 5 min). We model the change in the number of receptor slots to scale with the $2/3$ power of the change in spine volume, assuming scaling with the surface area rather than the volume of the spine. The filling fraction was set to $F=0.9$ and the relative pool size to $\varphi =2.67$. The results are shown in Figure 6.
Figure 6A shows the time course of the relative change of the insertion rate $\alpha $ and the number of receptor slots of the potentiated synapses. Figure 6B shows the time course of synaptic efficacies. At around 4 min the number of slots of the stimulated synapses peaks. Thereafter, the number of slots and the synaptic efficacies of the stimulated synapses decay to their new equilibrium values and the size of the receptor pool slowly recovers. In this example with a high filling fraction of $F=0.9$ and a relative pool size of $\varphi =2.67$ the heterosynaptic effect is very small. This can also be seen in Figure 6C, which shows the relative changes in the synaptic efficacies and the pool size as a function of time. As in the previous section, the amount of heterosynaptic plasticity depends on the filling fraction and the relative size of the receptor pool, however. This is illustrated in Figure 6D, where we consider a smaller filling fraction of $F=0.5$ and a smaller relative pool size of $\varphi =1.0$. This leads to a strong depletion of the receptor pool and a large heterosynaptic depression effect.
To quantify this effect, we systematically vary the relative pool size $\varphi $ and filling fraction $F$ and observe the peak relative changes in synaptic efficacies during homosynaptic LTP and heterosynaptic LTD (Figure 6E,F). We find that a small pool size strongly reduces the peak homosynaptic LTP and greatly increases the peak heterosynaptic LTD. Furthermore, both homosynaptic LTP and heterosynaptic LTD tend to be reduced by a high filling fraction. These results are consistent with those from Figure 5.
In addition to these already noted effects on heterosynaptic plasticity, this implies another prediction of the model regarding homosynaptic plasticity (compare Table 2, Homosynaptic): The amount of shortterm homosynaptic plasticity expression is modulated by the pool size and the filling fraction.
The changes in efficacies of synapses whose number of receptor slots are unaltered in Figures 4, 5 and 6 are only transient. In the following, we will study the longterm behavior of the model on the time scale associated with receptor externalization and internalization to determine how long it takes for the system to reach its (new) stable fixed point given by ${w}_{i}^{\infty}$ and ${p}^{\infty}$.
Approach to the steady state is governed by externalization and internalization rates
To study the system’s approach to its longterm steady state we again make use of the separation of time scales argument. Specifically, we assume that the fast dynamics of receptor exchanges between the pool and the synapses quickly reaches its equilibrium before the total number of receptors can change much due to receptor externalization and internalization. For this analysis we return to the original formulation of the model. Specifically, the change in the total receptor number from Equation 1 and Equation 2 is approximated by:
where we have replaced the current pool size $p$ with its steady state value ${p}^{\ast}\left(R\right)=R{W}^{\ast}\left(R\right)$ for a constant number of receptors in the system. Using ${F}^{\ast}\left(R\right)\equiv {W}^{\ast}\left(R\right)/S$ we arrive at:
For small numbers of receptors in the system, that is, $R$ close to zero, the steady state filling fraction ${F}^{\ast}\left(R\right)$ will be close to zero so that $\dot{R}\approx \gamma$. In contrast, for high numbers of receptors and the filling fraction close to its longterm steadystate value $F$ we find:
indicating that $R$ will exponentially approach its steady state value of $\gamma /\delta +FS$ with the time constant $1/\delta $. This behavior is illustrated in Figure 7. The simulated piece of dendrite has a total of 10 000 receptor slots and is initialized with different receptor numbers. We plot the numerical solution of Equation 19 for different initial numbers of receptors in the system. For low receptor numbers, the growth rate of $R$ is approximately $\gamma $ (compare dotted line). For a filling fraction close to its final steadystate value, $R$ exponentially converges to its steady state of $\gamma /\delta +FS$ with time constant ${\delta}^{1}$.
Smaller spontaneous synaptic efficacy fluctuations in larger synapses
Our analysis of the differential equation model above is suitable for studying the average behavior of the system for large numbers of receptors. However, small synapses may only have a few receptors inside them and the effects of stochastic fluctuations may become substantial. To quantify the size of such fluctuations of bound receptor numbers we have developed a stochastic version of the model that explicitly simulates the stochastic binding and unbinding, internalization and externalization of individual receptors (see Materials and methods). We use the model to study the fluctuations of synaptic efficacies under basal steady state conditions. This allows us to quantify the size of synaptic efficacy fluctuations due to the fast exchange of AMPARs between synapses and the receptor pool.
For illustration, we consider a local group of 7 synapses with 1, 2, 5, 10, 20, 50, and 100 slots, respectively. We quantify the size of fluctuations of synaptic efficacies using the coefficient of variation (CV), which is defined as the standard deviation of the fluctuating number of receptors bound inside a synapse divided by the time average of the number of receptors bound inside this synapse. A high CV indicates strong relative fluctuations of the synapse’s efficacy.
Figure 8A shows the numbers of receptors bound in each synapse as a function of time in one example simulation of 10 min. Parameters are as given in Table 1 with the filling fraction set to $F=0.5$ and the relative pool size set to $\varphi =2.67$. Figure 8B shows an example for a much higher filling fraction of $F=0.9$. Fluctuations are greatly attenuated.
Figure 8C plots the logarithm of the CV of the number of receptors as a function of the logarithm of the average number of receptors per synapse, which is given by the product of the theoretical filling fraction $F$ and the number of receptor slots ${s}_{i}$ of the synapse $i$. Data are shown for three different filling fractions obtained by increasing $\alpha $, the rate of receptors binding to receptor slots, while setting $\gamma $ to maintain a constant relative pool size of $\varphi =2.67$. The linear relationships evident in the loglog plot indicate a power law scaling. We fit power law functions of the form $\text{CV}=a{\left(Fs\right)}^{b}$ to the data (solid lines). Parameters of the fits are given in Table 3 and indicate slopes of around $1/2$, i.e. the CV declines approximately with one over the square root of the average number of bound receptors. Specifically, small synapses exhibit substantial fluctuations of their efficacies with CVs of up to 100%, while fluctuations are greatly attenuated in strong synapses. For different filling fractions, the curves are shifted vertically such that fluctuations are particularly strong for a filling fraction of 0.5 and are reduced for higher filling fractions.
Figure 8D considers the case where the rates $\alpha ,\beta ,\delta $ are held constant and the externalization rate $\gamma $ is varied to achieve different relative pool sizes $\varphi \in \{1.0,2.67,5.0\}$. Specifically, to achieve a particular relative pool size $\varphi $ we set:
The change in $\gamma $ also leads to different filling fractions in the three cases, see Equation 7. The results in Figure 8D show that an increased pool size will dampen spontaneous fluctuations of synaptic efficacies, while a reduced pool size promotes stronger fluctuations. We again fit power law functions to the data. Parameters of the fits are given in Table 4. Taken together, these results imply another prediction of the model (compare Table 2, Fluctuations): Small synapses undergo relatively larger spontaneous efficacy fluctuations, which are attenuated by a large pool size.
In conclusion, the spontaneous exchange of synaptic building blocks between synapses and dendritic pool leads to substantial fluctuations in synaptic efficacies. This finding is reminiscent of the surprisingly large spontaneous fluctuations in spine sizes in the absence of activitydependent synaptic plasticity observed recently (Dvorkin and Ziv, 2016; Shomar et al., 2017; Ziv and Brenner, 2018).
Discussion
The detailed molecular mechanisms underlying different forms of synaptic plasticity are complex. Recent years have seen enormous progress in identifying many of the relevant molecules and signaling pathways. This rapid development is in stark contrast to the simplistic and often purely phenomenological descriptions of synaptic plasticity used in most neural network models. While highly simplified mathematical models have been essential for relating synaptic plasticity ‘rules’ to learning processes at the network level, a full understanding of synaptic plasticity requires the development of more elaborate models that do justice to the complexities of synaptic plasticity at the molecular scale (Bhalla, 2011; Bhalla, 2014; Tsodyks et al., 1998; Urakubo et al., 2008). Here we have taken a step in this direction.
Hebbian learning tends to lead to runaway growth of synaptic efficacies if not counteracted by competitive or homeostatic mechanisms. To be effective, these compensatory mechanisms must act fast enough so they can catch up with changes induced by Hebbian learning (Zenke et al., 2013; Chistiakova et al., 2015). Prominent candidate mechanisms are synaptic normalization and heterosynaptic plasticity (Lynch et al., 1977). The idea has a long history. Synapses on the dendritic tree compete for a limited supply of synaptic building blocks such that when some synapses grow, they have to do so at the expense of other synapses (von der Malsburg, 1973; Lynch et al., 1977; Antunes and SimoesdeSouza, 2018). However, until recently, the lack of knowledge on the nature and the timescales of the molecular processes taking place at synapses did not allow for realistic modeling of such a competition for synaptic resources. Here we have presented a concrete model with a fast normalization of the efficacies of a neuron’s afferent synapses based on this competition for synaptic resources such as AMPAtype glutamate receptors.
Our model makes several contributions. First, it formalizes the idea of a fast synaptic normalization based on a competition for dendritic resources in an abstract and analytically tractable model. Second, analysis of the model reveals that under the given assumptions, normalization should act multiplicatively, such that relative strengths of synapses are maintained. Multiplicative normalization rules have been used in neural network models for a long time but usually in an ad hoc fashion. Our model supports the idea that a fast multiplicative normalization may in fact be biologically plausible. Third, the model naturally gives rise to a transient form of homeostatic heterosynaptic plasticity where synapses grow in efficacy at the expense of other synapses. Fourth, the model quantifies how the amount of heterosynaptic plasticity depends on the size of the local receptor pool and the filling fraction of receptor slots. It thereby reveals a fundamental tradeoff: the smaller the pool of available receptors, the more pronounced the heterosynaptic plasticity. In other words, neurons can limit heterosynaptic plasticity effects, but this comes at the price of having to maintain a big receptor pool. Similarly, the model predicts that a larger receptor pool attenuates spontaneous fluctuations in synaptic efficacies, which are particularly strong for small synapses. In the following we discuss how this prediction and others summarized in Table 2 could be tested.
How to test the model’s predictions
The first prediction of the model is that synapses in a local group have identical filling fractions, see Equation 7. That is, under basal conditions the same percentage of receptor slots should be filled in these synapses on average. Testing this prediction requires measuring both the number of receptor slots and the number of filled receptor slots for a local group of individual synapses. This could be achieved using a quantitative super resolution approach such as dualcolor direct stochastic optical reconstruction microscopy (dSTORM). For a given dendrite, one would have to quantify the number of AMPARs and PSD95 proteins in a local group of synapses under basal conditions. Our prediction is that the ratio of AMPARs to PSD95 proteins should be similar in all the synapses. As a corollary, we predict that the relative efficacies of two synapses from a local group should be identical to their relative slot numbers. Testing this hypothesis requires measuring the slot numbers and synaptic efficacies of two synapses from a local group. Specifically, the efficacies of a group of synapses could initially be measured using local glutamate uncaging. Subsequently the number of PSD95 could be assessed using dSTORM. This second approach seems rather challenging, however, as one would have to find in the fixed sample the exact dendrite and specific spines that were stimulated during liveimaging. A second corollary of the model’s first prediction is that if any of the transition rates changes, e.g., the rate at which receptors unbind from receptor slots, the filling fractions and synaptic efficacies are scaled by the same factor. Testing this prediction can be achieved by interventions that alter the transition rates. Activation of CaMKII leads to the phosphorylation of the AMPAR auxiliary subunit Stargazin increasing its affinity to PSD95 (Hafner et al., 2015). Thus one could induce a global activation of CaMKII in the neurons (chemicalLTP), fix the cells immediately after, and perform dualcolor dSTORM for PSD95 proteins and AMPARs. When comparing basal state to chemicalLTP, the ratio of PSD95 proteins to AMPARs should decrease by the same factor for all synapses.
The model’s second prediction can be tested in a similar way. We predict that manipulating the size of the local receptor pool leads to a multiplicative rescaling of the efficacies of the local group of synapses. To test this prediction, the size of the local receptor pool has to be altered, e.g., by triggering externalization of additional receptors. This could be achieved by treating neurons with TNF$\alpha $ for instance (Zhao et al., 2010). Subsequently the efficacies of the local group of synapses have to be monitored. These efficacies should scale by the same factor.
Another prediction of the model is that the amount of heterosynaptic plasticity is inversely related to the size of the local receptor pool. The most direct way of testing this prediction is to manipulate the local receptor pool as suggested above while inducing homosynaptic plasticity in a subset of synapses and measuring the amount of heterosynaptic plasticity in other synapses of the local group. In fact, this set of experiments could resemble the ones performed by Oh and colleagues but adding a TNF$\alpha$ condition (Oh et al., 2015).
The transient nature of heterosynaptic plasticity predicted by the model can be tested more easily. It merely requires the induction of homosynaptic plasticity in a subset of synapses in the local group while monitoring the time course of heterosynaptic plasticity in the remaining synapses. Specifically, the time course of recovery from heterosynaptic plasticity should be close to the internalization rate $\delta $ of AMPARs.
The model’s prediction of an influence of the size of the receptor pool on the expression of homosynaptic plasticity requires manipulating the size of the local receptor pool and subsequently inducing homosynaptic plasticity. For example, the peak change in synaptic efficacy during LTP induction should be bigger when the receptor pool has been increased than when it has been depleted prior to LTP induction.
Finally, the model predicts that synapses exhibit spontaneous fluctuations in synaptic efficacies due to the dynamic exchange of receptors with the local receptor pool. It predicts that these fluctuations, as measured by the coefficient of variation (CV), scale approximately as one over the square root of the synapses’ average efficacies. Testing this prediction can be achieved by repeated measurements of the synaptic efficacy of single synapses in the absence of any plasticity induction using glutamate uncaging.
Dendritic morphology and local production
We have assumed that the basal transition rates for receptors attaching and detaching to and from slots are identical for all synapses and that the receptors are distributed homogeneously inside the pool. These assumptions are essential for the multiplicative behavior of the model. If, in contrast, the distribution of receptors across the dendritic tree were very inhomogeneous, this would, all else being equal, correspond to different pool sizes in different parts of the dendritic tree, leading to different filling fractions across the dendritic tree.
Properly distributing synaptic building blocks across the dendritic tree is a formidable task (Williams et al., 2016). Specifically, if receptors were only produced at a single site corresponding to the neuron cell body (or soma) and spreading from this point source according to slow transport processes then one would expect a high concentration of receptors close to the soma and a low concentration far away from it. This would, all else being equal, lead to large receptor pools close to the soma and small receptor pools far away from it. Earnshaw and Bressloff (2008) have presented such a model. They consider a long dendrite and diffusion of receptors from the soma along this dendrite leading to a high concentration of receptors close to the soma and a small concentration far away from it. In contrast, our model considers a local piece of dendrite, where the concentration of receptors can be assumed to be approximately constant. Therefore, our model does not attempt to make predictions regarding scaling of synaptic efficacies at the global level of a neuron’s entire dendritic tree. Earnshaw and Bressloff conclude from their model that ‘it does not appear possible to obtain a global multiplicative scaling’ of synaptic efficacies just by changing reaction rates. This conclusion rests on the fact that the distribution of receptors along their simulated dendrite is inhomogeneous. Specifically, Earnshaw and Bressloff assume that protein synthesis occurs mostly at the soma, which leads to an approximately exponential decay of the concentration of receptors towards the distal end of the dendrite. This assumption failed to be confirmed experimentally and has in fact been contradicted by Tao–Cheng and colleagues who found a homogeneous distribution of AMPARs at the neuron surface along the dendritic arbor of hippocampal cultured neurons (TaoCheng et al., 2011). Earnshaw and Bressloff also cite a study by Adesnik et al. (2005) to support the idea of an inhomogeneous distribution of AMPARs. They used ANQX (a modified version of DNQX) known at that time as an AMPAR antagonist (Chambers et al., 2004) to monitor synaptic AMPAR exchange after specific inactivation of the surface population. They measured a significantly slower recovery of AMPAR current in dendrites compared to the somatic region. Thus, they concluded that AMPARs are mainly exocytosed at the somatic extracellular membrane and trafficked distally through lateral diffusion. However, since then DNQX has also been shown to act on kainate and NMDA receptors. Additionally, DNQX effects on AMPARs appear to depend on the composition of AMPAR complexes and in particular the type of auxiliary subunits associated with those receptors (Maclean and Bowie, 2011; Greger et al., 2017). Since the concentration of receptors between somatic and dendritic membranes appears to be fairly homogeneous, it might be that the actual composition of the receptors varies between those two compartments. In this case, global multiplicative scaling is to be expected in the model of Earnshaw and Bressloff as well. Hence, we believe that our model using minimal assumptions and being restricted to a single dendritic segment with multiple dendritic spines is in good accordance with the recent literature on AMPAR trafficking.
A uniform distribution of synaptic building blocks across the entire dendritic tree could be facilitated by local production of these building blocks across the dendritic tree. Local protein synthesis may therefore be essential for global multiplicative scaling behavior observed in biological experiments (Turrigiano et al., 1998). More specifically, synthesis of AMPAR subunits happens inside the cell at the endoplasmic reticulum membrane (ER). This synthesis of proteins seems to occur in a burst fashion in local ‘hot spots’ distributed across the dendritic tree (Katz et al., 2016). Importantly, however, newly synthesized receptors are not necessarily immediately trafficked to the cell surface and in fact a large fraction are distributed across and maintained inside the ER compartment constituting an intracellular pool of receptors waiting to be exocytosed (Greger et al., 2002). Thus, the distribution of receptors in the ER may already be more homogeneous than hot spot synthesization would suggest. Furthermore, once released from the ER into the cytoplasm, fast distribution of receptors along microtubules could lead to a rather homogeneous distribution inside the cytoplasm, from where the receptors would be trafficked to the surface. Thus, bursty translation at hotspots inside the ER may still allow for a homogeneous distribution of receptors at the cell surface. We therefore predict that local production of synaptic building blocks across the dendritic tree contributes to their uniform distribution, which in turn might allow global multiplicative scaling behavior and the maintenance of relative strengths of synapses. This could be tested, for example, by specifically blocking local production of synaptic building blocks, which should make their distribution across the dendritic tree less homogeneous and lead to systematic inhomogeneities in synaptic efficacies across the dendritic tree.
In this context it is also interesting to note that at least one form of heterosynaptic plasticity tends to be induced locally (De Roo et al., 2008; Losonczy et al., 2008; Li et al., 2016b), that is, at neighboring synapses. Such local action is readily expected in our model if competition for synaptic building blocks is restricted to a local pool such as a section of a dendritic branch, with comparatively slow trafficking of building blocks between adjacent pools.
Detailed descriptions of AMPAR trafficking and diffusion
The stochastic version of our model describes individual binding and unbinding events of AMPARs to receptor slots, but it does not describe in detail the paths taken by individual AMPARs during their diffusion in the cell membrane. This is a gross simplification, but it facilitates mathematical analysis. More elaborate models were conceived to describe the movement of individual receptors inside the dendritic branch and the PSD (Earnshaw and Bressloff, 2006; Czöndör et al., 2012; Li et al., 2016a). Such models can incorporate, e.g., the detailed spine geometry or effects of protein crowding.
Control of transition rates
Apart from our experiments on modeling LTP, where we introduced a transient and synapsespecific increase of the rate at which receptors bind to slots, we have kept all transition rates constant throughout this paper. In reality, we expect the various transition rates to be flexibly controlled to allow for robust and efficient functioning of the neuron, allowing it to cope with various perturbations. Indeed, constructing a model to describe these various regulatory processes will be an important challenge for the future. Furthermore, AMPARs can be in different states expected to have different transition rates. Specifically, AMPAR complexes containing various sets of auxiliary subunits are very likely to coexist at the neuron surface (Schwenk et al., 2012). Since only a couple of auxiliary subunits have binding domains for PSD95, multiple types of AMPARs with different $\alpha $ and $\beta $ parameters could be considered. These topics are left for future work.
Slot production and removal
In future work, it will also be interesting to consider changes to slot numbers in more detail. We simulated increases in slot numbers of individual synapses in the context of LTP. Obviously, however, the building blocks of these ‘slots’ also have to be produced, transported, and inserted into synapses, which could be based on similar mechanisms as we have postulated for receptors. Furthermore, slots are also degraded and have to be replaced. In fact, the alternative interpretation of our model discussed in the beginning of the Results section already describes how PSD95 slots are produced (or degraded) and bind to (or detach from) slots for these receptor slots (‘slotsforaslot’ interpretation). Future work should aim for a model that more fully describes the interactions of AMPARs (and other types of receptors), various TARPs such as stargazin, MAGUK proteins such as PSD95, and neuroligins as well as their production and trafficking. Along these lines, it will also be interesting to consider the mechanisms underlying different stages of LTP and LTD in more detail.
Modeling slow homeostatic synaptic scaling
The model could also be extended to capture slow homeostatic synaptic scaling processes (Turrigiano et al., 1998; Ibata et al., 2008). In the simplest case, a sensor for the average neural activity of the neuron would drive the production of receptors and/or slots in a homeostatic fashion, such that if, e.g., the average neural activity falls below a target level or range, then receptor and/or slot production are increased to drive up excitatory synaptic efficacies. Such a model would naturally explain the multiplicative behavior of homeostatic synaptic scaling (Turrigiano et al., 1998). Obviously, the activity sensor could also sense the average activity in a local neighborhood through a diffusive mechanism (Sweeney et al., 2015). Furthermore, instead of homeostatically regulating firing rates, the amount of afferent drive to the neuron or to the local population could be controlled (Savin et al., 2009), or even other measures of neural and synaptic activity could be used. Finally, all these ideas are not mutually exclusive. It seems likely that neurons control both their firing rate distributions and their amounts of excitatory and inhibitory afferent drive through a combination of different intrinsic and synaptic plasticity mechanisms on different time scales.
Receptor subunit composition
Finally, not all AMPARs are created equal. Depending on the composition of subunits, AMPARs have distinct properties in terms of, e.g., calcium permeability and trafficking (see Henley and Wilkinson, 2016) for a recent review). A more complete model should incorporate the diversity of AMPARs (or even other receptor types) and their properties.
Conclusion
In conclusion, our model offers a parsimonious explanation for a transient form of homeostatic heterosynaptic plasticity and fast local synaptic normalization, which it predicts to be multiplicative. It therefore supports the use of such rules in neural network models. The model also reveals a fundamental tradeoff between the size of the local receptor pool and the amount of heterosynaptic plasticity. This tradeoff is akin to a common logistics problem: how much to produce and store of a particular resource in order to (a) minimize production costs and storage space while (b) limiting the risk of running out of this resource? Arguably, efficient neural functioning requires solving a plethora of related logistics problems with respect to production, transport, and storage of various ‘goods’ and supply of the necessary energy for all these processes. We feel that the time is ripe for a concerted effort to study individual neurons and the entire nervous system from such a neurologistics perspective.
Materials and methods
Simulation software
Request a detailed protocolThe simulation software was written in Python and is available at: https://github.com/triesch/synapticcompetition (Triesch and Vo, 2018); copy archived at https://github.com/elifesciencespublications/synapticcompetition).
Differential equations were discretized with the Euler method.
The stochastic version of the model was simulated using the Gillespie algorithm (Gillespie, 1976). Stochastic reactions were defined for receptors entering or leaving each of the seven synapses and for being added or removed from the receptor pool. This gave rise to a total of 16 possible ‘reactions’ occurring with different probabilities per unit time depending on the current state of the system, that is, how many receptors are bound in each synapse and reside in the pool. Stochastic simulations were validated against the differential equation model to verify that their average behavior matched that of the differential equation model in different situations.
Calculation of the shortterm filling fraction
Request a detailed protocolWe exploit the separation of time scales between fast receptor binding and unbinding from slots and slow externalization and internalization of receptors. On the fast time scale, the processes of internalization and externalization can be ignored. Removing the corresponding terms in Equation 2, we again look for the steady state solution by setting the time derivatives of ${w}_{i}$ and $p$ to zero and summing over $i$. This leads to the following quadratic equation for ${W}^{\ast}$, the steady state number of bound receptors in the shortterm approximation (which must not be confused with the longterm steady state solution ${W}^{\infty}$ of the full system):
We introduce $\rho \equiv \beta /\alpha $ as the ratio of the rates through which receptors leave and enter the synaptic slots. Using this, the two solutions of Equation 22 are given by:
The ‘+’ solution is not biologically meaningful, since it leads to ${W}^{\ast}\ge S$ or ${W}^{\ast}\ge R$ (see Appendix), so that the desired steady state solution of the shortterm approximation is given by:
and the corresponding shortterm steadystate filling fraction is ${F}^{\ast}={W}^{\ast}/S$. In the full system, this solution is assumed only transiently, because receptors can still enter and leave the system. If the number of receptors were held constant, then ${F}^{\ast}$ and ${W}^{\ast}$ would describe the stable solution on long time scales.
Sensitive reaction of the shortterm filling fraction to changes in reaction rates when number of receptors matches number of slots
Request a detailed protocolWe are interested in how the shortterm filling fraction ${F}^{\ast}$ changes, when the reaction rates $\alpha $ and $\beta $ or their ratio $\rho \equiv \beta /\alpha $ change. Formally, we consider the partial derivative of the shortterm filling fraction ${F}^{\ast}={W}^{\ast}/S$ with respect to $\rho $. Using Equation 24 we find:
As can be seen in Figure 3C,D, the most extreme slope is obtained at $\rho =0$. There the derivative simplifies to:
For $R=S$ the slope diverges, that is, the short term filling fraction reacts extremely sensitively to small changes in $\rho $ when $\rho $ is close to zero.
Appendix 1
The '+' solution from Equation 23 is not biologically meaningful.
We show that the '+' solution from Equation 23 is not biologically meaningful. To see this, first note that $W}_{1}\le {W}_{2$. Furthermore, any meaningful solution $W$ must fulfill $W\le R$ and $W\le S$, that is, the number of receptors bound to slots cannot be bigger than the total number of receptors or the total number of slots. If the smaller solution $W}_{1$ does not meet both criteria, then the larger $W}_{2$ cannot meet them either. So we assume in the following that $W}_{1$ meets both these criteria so that $W}_{1}\le min\left\{R,S\right\$. Our argument uses Vieta’s formulas for the quadratic Equation 22:
Using the second formula we can write:
from which follows that:
In the case that $R\text{}\text{}S$, this leads to ${W}_{2}\ge R$. The only biologically meaningful solution to this is the equality ${W}_{2}=R$. This is the extreme case where all receptors are bound in slots and no receptors remain in the pool. With Vieta’s second formula we see that in this case ${W}_{1}=S$. Plugging both results into Vieta’s first formula, we see that this solution requires $\rho =0$, which in turn requires $\beta =0$. In this case, no receptors would ever leave synapses.
The case $R\text{}\text{}S$ leads to ${W}_{2}\ge S$. The only biologically meaningful solution to this is the equality ${W}_{2}=S$. This is the extreme case where all slots are filled with receptors. Using Vieta’s formulas again leads to the uninteresting requirement $\rho =0$ for this solution.
Finally, the case $R=S$ leads to $R=S={W}_{1}={W}_{2}$ and also requires $\rho =0$. In summary, the ‘+' solution in Equation 23 only admits the extreme solutions $W=S$ or $W=R$ requiring $\rho =0$ (and therefore $\beta =0$), which are not biologically meaningful.
References
 1
 2
 3

4
Is heterosynaptic modulation essential for stabilizing Hebbian plasticity and memory?Nature Reviews Neuroscience 1:11–20.https://doi.org/10.1038/35036191
 5
 6
 7

8
Molecular computation in neurons: a modeling perspectiveCurrent Opinion in Neurobiology 25:31–37.https://doi.org/10.1016/j.conb.2013.11.006
 9
 10

11
Photochemically knocking out glutamate receptors in vivoJournal of the American Chemical Society 126:13886–13887.https://doi.org/10.1021/ja048331p

12
The role of AMPA receptors in postsynaptic mechanisms of synaptic plasticityFrontiers in Cellular Neuroscience, 8, 10.3389/fncel.2014.00401, 25505875.
 13

14
Homeostatic role of heterosynaptic plasticity: models and experimentsFrontiers in Computational Neuroscience 9:89.https://doi.org/10.3389/fncom.2015.00089
 15
 16
 17
 18
 19
 20

21
Modeling the role of lateral membrane diffusion in AMPA receptor trafficking along a spiny dendriteJournal of Computational Neuroscience 25:366–389.https://doi.org/10.1007/s1082700800848
 22
 23

24
A general method for numerically simulating the stochastic time evolution of coupled chemical reactionsJournal of Computational Physics 22:403–434.https://doi.org/10.1016/00219991(76)900413
 25
 26
 27

28
AMPA receptor trafficking and the mechanisms underlying synaptic plasticity and cognitive agingDialogues in clinical neuroscience 15:11–27.

29
Synaptic AMPA receptor composition in development, plasticity and diseaseNature Reviews Neuroscience 17:337–350.https://doi.org/10.1038/nrn.2016.37
 30
 31
 32

33
PDZ domain proteins of synapsesNature Reviews Neuroscience 5:771–781.https://doi.org/10.1038/nrn1517

34
SORN: a selforganizing recurrent neural networkFrontiers in Computational Neuroscience 3:23.https://doi.org/10.3389/neuro.10.023.2009
 35

36
Protein Crowding within the Postsynaptic Density Can Impede the Escape of Membrane ProteinsJournal of Neuroscience 36:4276–4295.https://doi.org/10.1523/JNEUROSCI.315415.2016
 37
 38
 39
 40
 41
 42

43
Transmembrane AMPA receptor regulatory protein regulation of competitive antagonism: a problem of interpretationThe Journal of Physiology 589:5383–5390.https://doi.org/10.1113/jphysiol.2011.219485
 44
 45

46
The Role of Constraints in Hebbian LearningNeural Computation 6:100–126.https://doi.org/10.1162/neco.1994.6.1.100
 47
 48
 49

50
A simplified neuron model as a principal component analyzerJournal of Mathematical Biology 15:267–273.https://doi.org/10.1007/BF00275687
 51
 52

53
Epileptogenesis due to gliamediated synaptic scalingJournal of The Royal Society Interface 6:655–668.https://doi.org/10.1098/rsif.2008.0387
 54
 55

56
Cooperative stochastic binding and unbinding explain synaptic size dynamics and statisticsPLOS Computational Biology 13:e1005668.https://doi.org/10.1371/journal.pcbi.1005668
 57
 58
 59

60
A Diffusive Homeostatic Signal Maintains Neural Heterogeneity and Responsiveness in Cortical NetworksPLOS Computational Biology 11:e1004389.https://doi.org/10.1371/journal.pcbi.1004389

61
Trafficking of AMPA receptors at plasma membranes of hippocampal neuronsJournal of Neuroscience 31:4834–4843.https://doi.org/10.1523/JNEUROSCI.474510.2011
 62

63
Neural networks with dynamic synapsesNeural Computation 10:821–835.https://doi.org/10.1162/089976698300017502
 64

65
Requirement of an allosteric kinetics of NMDA receptors for spike timingdependent plasticityJournal of Neuroscience 28:3310–3323.https://doi.org/10.1523/JNEUROSCI.030308.2008
 66
 67
 68

69
Synaptic plasticity in neural networks needs homeostasis with a fast rate detectorPLoS Computational Biology 9:e1003330.https://doi.org/10.1371/journal.pcbi.1003330
 70
 71

72
Synaptic tenacity or lack thereof: spontaneous remodeling of synapsesTrends in Neurosciences 41:89–99.https://doi.org/10.1016/j.tins.2017.12.003
Decision letter
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
[Editors’ note: the authors were asked to provide a plan for revisions before the editors issued a final decision. What follows is the editors’ letter requesting such plan.]
Thank you for sending your article entitled "Competition for synaptic building blocks shapes synaptic plasticity" for peer review at eLife. Your article has been reviewed by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Eve Marder as the Senior Editor.
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
In particular, the relevance, novelty and main conclusions of the manuscript may be undermined by previous work that was not cited:
Earnshaw and Bressloff, 2006 and Earnshaw and Bressloff,.2008.
The second paper concludes, "even given a number of simplifying assumptions, it does not appear possible to obtain a global multiplicative scaling of synaptic receptor numbers along a dendrite from a simple up or down regulation of constitutive recycling." This contradicts the present study's main conclusion. See reviewer #2's comments for additional details.
Please examine these previous studies carefully and prepare a response, outlining whether these disparities can be addressed and whether the present study offers insights, results or an approach that is more relevant than this previous work.
Reviewer #1:
This paper proposes a novel mathematical model that explains the heterosynaptic plasticity in local dendrite after the induction of LTP and/or LTD. The model comprises four processes at the cellular and molecular level, which are the binding and unbinding of receptors to and from slots on postsynaptic membrane, and the addition and removal of free receptors to and from the local pool in the dendrite. The model is biophysically sound and analysed in detail to reveal how it predicts/accounts for a variety of higher level effects. For example, by quantifying the number of filled slots and the size of local pool on short timescales (i.e. shorter that biosynthesis), the model successfully predicts that:
1) On short time scales the redistribution of receptors between synapses is multiplicative, a phenomenon commonly known as synaptic scaling.
2) The amount of heterosynaptic plasticity is inversely related to the size of local receptor pool.
Transient heterosynaptic plasticity has rarely been modelled in systematically from the perspective of receptor trafficking, or in a way that highlights the role of timescale separation. This paper is accessible to experimentalists and will aid in forming experimental predictions. In particular, the model suggests that synaptic scaling can happen locally, and independently of biosynthesis on the timescales it is typically studied in experiments. In other words, synaptic scaling is simply an unavoidable consequence of having a pool of receptors with constitutive trafficking.
Issues:
To improve the presentation, the authors should reduce the number of predictions they made (some of them are almost tautologies!) to make the most important prediction stand out. These and other specific issues that should be clarified/addressed are outlined below:
 Subsection “Formulation of the model”: Need to briefly explain the physical meaning of involving p in the term αp(s_{i} – w_{i}).
 Table 2 Scaling II: The prediction is only true in steadystate (or on average).
 Subsection “Competition for Synaptic Building Blocks Induces Multiplicative Scaling”, fifth paragraph: Three predictions may be reduced to one otherwise it could be difficult to grasp your key point. They all come from the same Equation 7 in fact, so you could keep one most important prediction in Table 2 and leave others as part of your analysis in this section.
 Subsection “Fast redistribution of receptors between synapses is multiplicative”, first paragraph: should read "the steady state solution of fast time scale".
 Subsection “Fast redistribution of receptors between synapses is multiplicative”, second paragraph: "As above" is inaccurate. F* here is clearly different from F in Equation 7, as γ=0 can no longer serve as a denominator.
 Subsection “Fast redistribution of receptors between synapses is multiplicative”, second paragraph: Articulate more explicitly. This is one of the most important predictions and therefore deserves a detailed explanation. For example, add the mathematical relationship between the size of local receptor p and the synaptic efficacy w in the quasi steadystate:
w*=p*p+p*si
 Subsection “Competition for receptors induces transient heterosynaptic plasticity”, first paragraph: Motivate it better for changing the number of slots. It is a bit out of the blue here, because until now s has never been assumed to be a temporal variable in your model.
 Subsection “Time course of homosynaptic LTP and accompanying heterosynaptic LTD”, first paragraph: Explain/justify "not reflect biological reality well".
 Suggest a clear distinction between α modulation and s modulation.
Reviewer #2:
This study fleshes out the implications of an existing idea, that multiplicative synaptic scaling of synapses could be due to competition for shared synaptic resources, identified by the authors as e.g. AMPARs.
I very much like the approach but am wary of two of the main conclusions and predictions (and also the novelty of the findings), for the following reasons:
1) A similar model was studied by Earnshaw and Bressloff across two papers (not cited in the current study):
Earnshaw and Bressloff, 2006. and Earnshaw and, 2008. The 2008 paper is particularly relevant because they studied a model that included receptor diffusion along the dendrite, where synapses competed for the same shared pool of receptors. They concluded that "even given a number of simplifying assumptions, it does not appear possible to obtain a global multiplicative scaling of synaptic receptor numbers along a dendrite from a simple up or down regulation of constitutive recycling.". The current study may have not found this for two reasons: first the Bressloff model is nonlinear, and second the Bressloff model includes space. Can the authors comment on this discrepancy? Does it undermine their whole study?
2) I find it surprising that the model predicts (Equation 6) that the total number of receptors in the pool is independent of synapse number, slot number, and receptor binding and unbinding rates (α and β). The maths makes sense – due to linearity of the model – but it would be nice if the authors could comment on the likely validity of this prediction/assumption if nonlinearities were to be introduced. For example, there may be two types of receptor state within the synapse, trapped and not trapped.
Reviewer #3:
This paper represents an important intermediate level of modeling between purely abstract views of normalization and homeostasis of synapse strength and detailed biological modeling, something that is still impossible at this scale but which is being attempted in the context of the individual spine.
The authors do a good job of reaching up (topward) to the abstract formulation but make less of an effort to reach down (bottomward) to the biology. The downreach is more difficult and will necessarily be speculative but I would encourage the authors to make this effort. This represents the essence of such a levelofinvestigation bridging model: from the problem level (normalization), to algorithms, to implementations.
"In the limit of large receptor numbers" – please justify. What are the estimated numbers? What numbers are required to avoid substantial variability due to stochastics? There is brief discussion of this in the Discussion section that could be moved up.
Predictions are the eventual goal of modeling and should be given more attention. Please move the predictions of Table 2 into a major textual section and indicate how each prediction could currently (or with some imagined technical advance in the future) be tested experimentally.
Implementations determines algorithms determines problems (inverse of the Marr procedure). In this case, what is the role of catabolism of damaged receptors? Are most receptors actually returned to the pool or do many need to be replaced? What are estimates of the metabolic requirements for such replacement? How are receptors within the dendritic pool mobilized into spines? What is the involvement of endoplasmic reticulum, rough ER?
"Spreading from [nucleus by] slow diffusion process" is not really accurate; what is putative interplay of microtubules and actin vs. role of diffusion in bringing receptors into play? How might these factors relate to the role of pool mobilization in synaptic tagging and capture (STC)? Does the sudden increase in pool size in Figure 4 represent a 'capture' event?
[Editors’ note: formal revisions were requested, following approval of the authors’ plan of action.]
Thank you for sending a revision plan for your article entitled "Competition for synaptic building blocks shapes synaptic plasticity" for peer review at eLife. We are pleased to accept a final, revised version of your manuscript conditional on the essential changes outlined in the revision plan and the comments below.
Your revision plan for this article has been evaluated by 3 peer reviewers, and the evaluation is being overseen by a Reviewing Editor and Eve Marder as the Senior Editor. In addition to the revisions you have outlined, we request, in line with reviewer comments, that you address the following important issues, which were raised in the evaluation:
 "The authors should make more explicit throughout the manuscript that the analysis does not predict global synaptic scaling. This is especially important in the discussion, which links their findings to the global synaptic scaling as studied by Turrigiano et al. It seems another component of the model would need to be added to make this link; perhaps something linking the receptor pools between dendrites, or between dendrites and the soma."
 "One small point  Bressloff et al didn't assume a nonuniform distribution of synaptic receptors, that was a prediction from the model and indeed was one of the reasons that they claimed multiplicative scaling was tricky."
 "Finally there are empirical reasons to challenge the authors' assumption that synaptic receptor expression is flat within even a single dendrite. This might be OK as a simplifying assumption for a local analysis but it needs to be put in context of experimental evidence e.g. Spruston and Burrone labs have found that synaptic protein content seems to decrease from proximal to distal portions of single dendrites in hippocampal pyramidal neurons (Menon et al, Neuron, 2013; Bloss et al, Neuron, 2016; Walker et al, PNAS, 2017)."
Please resubmit a revised manuscript with the changes outlined in your revision plan as well as addressing the comments above.
https://doi.org/10.7554/eLife.37836.017Author response
[Editors’ notes: the authors’ response after being formally invited to submit a revised submission follows.]
In particular, the relevance, novelty and main conclusions of the manuscript may be undermined by previous work that was not cited:
Earnshaw and Bressloff, 2006 and Earnshaw and Bressloff, 2008.
The second paper concludes; "even given a number of simplifying assumptions, it does not appear possible to obtain a global multiplicative scaling of synaptic receptor numbers along a dendrite from a simple up or down regulation of constitutive recycling." This contradicts the present study's main conclusion. See reviewer #2's comments for additional details.
Please examine these previous studies carefully and prepare a response, outlining whether these disparities can be addressed and whether the present study offers insights, results or an approach that is more relevant than this previous work.
We were unaware of the papers by Earnshaw and Bressloff. Thank you for pointing them out to us and our apologies for this oversight. The 2006 paper is concerned with a detailed model of receptor trafficking inside a single dendritic spine. The 2008 paper simplifies the 2006 model (e.g. no distinction between different receptor types) but at the same time extends it to a long onedimensional dendrite and describes the surface diffusion of receptors along this dendrite. Our model is in fact intermediate in that we consider a local piece of dendrite, where the concentration of receptors in the local pool can be treated as approximately constant. Therefore, our model does not speak to the question of global multiplicative scaling, but it does predict a local multiplicative scaling. Importantly, research since 2008 suggests that some of the assumptions of Earnshaw and Bressloff (2008) need to be revisited, which could invalidate some of their conclusions. We elaborate on this in our detailed response to reviewer 2.
"The authors should make more explicit throughout the manuscript that the analysis does not predict global synaptic scaling. This is especially important in the Discussion, which links their findings to the global synaptic scaling as studied Okby Turrigiano et al. It seems another component of the model would need to be added to make this link; perhaps something linking the receptor pools between dendrites, or between dendrites and the soma."
We have made this clear throughout the paper now and specifically discuss local vs. global synaptic scaling in the Discussion (section “Dendritic morphology and local production”).
"One small point – Bressloff et al. didn't assume a nonuniform distribution of synaptic receptors, that was a prediction from the model and indeed was one of the reasons that they claimed multiplicative scaling was tricky."
Yes, we are aware of this, but admit that our wording did not make this clear. This has been corrected.
"Finally there are empirical reasons to challenge the authors' assumption that synaptic receptor expression is flat within even a single dendrite. This might be OK as a simplifying assumption for a local analysis but it needs to be put in context of experimental evidence e.g. Spruston and Burrone labs have found that synaptic protein content seems to decrease from proximal to distal portions of single dendrites in hippocampal pyramidal neurons (Menon et al., Neuron, 2013; Bloss et al., Neuron, 2016; Walker et al., PNAS, 2017)."
We thank the reviewer for pointing out this aspect of neuronal function. Indeed, we are aware that synapses along the full dendritic arbor of a neuron will differ in their composition. In fact, many labs have shown that ion channels in particular can be specifically expressed in definite regions of the dendritic arbor of CA1 pyramidal neurons. Local protein synthesis that we propose as a compensatory mechanism for the point source supply of synaptic proteins is unlikely to concern all synaptic proteins. Also, some proteins may exclusively be synthesized in specific dendritic regions. Thus, some elements of the synapse will show nonhomogeneous distribution along the dendritic arbor. This is the main reason, as the reviewer points out, why we consider a small portion of dendrite and not the full dendritic arbor in the current manuscript.
In the present study, what concerns us the most is the homogeneous distribution of surface AMPARs in a small portion of a dendrite. Toa–Cheng and colleagues have described such a distribution for GluA2containing AMPARs at the neuron surface along the dendritic arbor of cultured hippocampal neurons (ToaCheng et al., 2011).
In Menon et al., 2013 the authors describe in Figure 3 that the number of AMPARs per perforated synapses increases, as dendrites are further away from the soma. Interestingly, this effect is not observed in nonperforated synapses that are by far the most abundant type of synapse on CA1 pyramidal neurons (ratio perforated to nonperforated synapses range from 0.1 to 0.25 (Nicholson et al., 2006)). This suggests that perforated synapses might represent an intermediate plasticity state associated with a transient increase in AMPAR insertion rate α.
Reviewer #1:
This paper proposes a novel mathematical model that explains the heterosynaptic plasticity in local dendrite after the induction of LTP and/or LTD. The model comprises four processes at the cellular and molecular level, which are the binding and unbinding of receptors to and from slots on postsynaptic membrane, and the addition and removal of free receptors to and from the local pool in the dendrite. The model is biophysically sound and analysed in detail to reveal how it predicts/accounts for a variety of higher level effects. For example, by quantifying the number of filled slots and the size of local pool on short timescales (i.e. shorter that biosynthesis), the model successfully predicts that:
1) On short time scales the redistribution of receptors between synapses is multiplicative, a phenomenon commonly known as synaptic scaling.
2) The amount of heterosynaptic plasticity is inversely related to the size of local receptor pool.
Transient heterosynaptic plasticity has rarely been modelled in systematically from the perspective of receptor trafficking, or in a way that highlights the role of timescale separation. This paper is accessible to experimentalists and will aid in forming experimental predictions. In particular, the model suggests that synaptic scaling can happen locally, and independently of biosynthesis on the timescales it is typically studied in experiments. In other words, synaptic scaling is simply an unavoidable consequence of having a pool of receptors with constitutive trafficking.
We agree and have combined some predictions.
Issues:
To improve the presentation, the authors should reduce the number of predictions they made (some of them are almost tautologies!) to make the most important prediction stand out. These and other specific issues that should be clarified/addressed are outlined below:
 Subsection “Formulation of the model”: Need to briefly explain the physical meaning of involving p in the term αp(s_{i} – w_{i}).
Explanation has been added.
 Table 2 Scaling II: The prediction is only true in steadystate (or on average).
Correct, has been clarified.
 Subsection “Competition for Synaptic Building Blocks Induces Multiplicative Scaling”, fifth paragraph: Three predictions may be reduced to one; otherwise it could be difficult to grasp your key point. They all come from the same Equation 7 in fact, so you could keep one most important prediction in Table 2 and leave others as part of your analysis in this section.
Agreed. We have combined the scaling predictions into one.
 Subsection “Fast redistribution of receptors between synapses is multiplicative”, first paragraph: should read "the steady state solution of fast time scale".
Agreed. Change has been made.
 Subsection “Fast redistribution of receptors between synapses is multiplicative”, second paragraph: "As above" is inaccurate. F* here is clearly different from F in Equation 7, as γ=0 can no longer serve as a denominator.
Has been rephrased. The “As above” was meant to refer to the property of all synapses having the identical filling fraction.
 Subsection “Fast redistribution of receptors between synapses is multiplicative”, second paragraph: Articulate more explicitly. This is one of the most important predictions therefore deserves a detailed explanation. For example, add the mathematical relationship between the size of local receptor p and the synaptic efficacy w in the quasi steadystate:
w*=p*p+p*si
We have elaborated on this.
 Subsection “Competition for receptors induces transient heterosynaptic plasticity”, first paragraph: Motivate it better for changing the number of slots. It is a bit out of the blue here, because until now s has never been assumed to be a temporal variable in your model.
We have added some sentences and made clear references to literature pointing to the tight structurefunction relationship between synaptic metrics and synaptic strength and plasticity (1, 2). In this context we believe modulating slot numbers (as a rough approximation of PSD size) will be justified as an initial attempt to simulate forms of homosynaptic plasticity.
 Subsection “Time course of homosynaptic LTP and accompanying heterosynaptic LTD”, first paragraph: Explain/justify "not reflect biological reality well".
Agreed, this deserves a more careful explanation and in particular references to the time scale of modification of PSD95 protein number in synapses during plasticity (1) as well as justifications of how CaMKII activation could impact α (insertion rate) while s (number of slots) may initially remain unchanged.
 Suggest a clear distinction between α modulation and s modulation.
We are not entirely sure what you mean here. Obviously, a time dependence could be introduced only for α or only for s. We feel that discussing (and simulating) all these cases separately is not likely to add important insights. However, we have rephrased to suggest that these two kinds of modulation can be considered independently of each other.
Reviewer #2:
This study fleshes out the implications of an existing idea, that multiplicative synaptic scaling of synapses could be due to competition for shared synaptic resources, identified by the authors as e.g. AMPARs.
I very much like the approach but am wary of two of the main conclusions and predictions (and also the novelty of the findings), for the following reasons:
1) A similar model was studied by Earnshaw and Bressloff across two papers (not cited in the current study):
Earnshaw and Bressloff, 2006. and Earnshaw and, 2008. The 2008 paper is particularly relevant because they studied a model that included receptor diffusion along the dendrite, where synapses competed for the same shared pool of receptors. They concluded that "even given a number of simplifying assumptions, it does not appear possible to obtain a global multiplicative scaling of synaptic receptor numbers along a dendrite from a simple up or down regulation of constitutive recycling.". The current study may have not found this for two reasons: first the Bressloff model is nonlinear, and second the Bressloff model includes space. Can the authors comment on this discrepancy? Does it undermine their whole study?
Thank you for pointing out these studies, which we had simply overlooked. The most important difference is that Earnshaw and Bressloff consider a long dendrite and the diffusion along this long dendrite, while our model considers a local piece of dendrite, where the concentration of receptors can be assumed to be approximately constant. Therefore, our model does not attempt to make predictions regarding scaling at the global level of synapses across a neuron's entire dendritic tree. Furthermore, Earnshaw and Bressloff's conclusion that “it does not appear possible to obtain a global multiplicative scaling” rests on the assumption that the distribution of receptors along their simulated dendrite is inhomogeneous. Specifically, Earnshaw and Bressloff assume that protein synthesis occurs mostly at the soma and that the concentration of receptors falls off approximately exponentially towards the distal end of their dendrite.
To the best of our knowledge, this assumption failed to be confirmed experimentally and has in fact been contradicted by a study from the lab of Thomas Reese. Toa–Cheng and colleagues found homogenous distribution of GluA2containing AMPARs at the neuron surface along the dendritic arbor of hippocampal cultured neurons (3). Earnshaw and Bressloff made the inhomogeneous distribution of AMPARs assumption based a study by Adesnik and colleagues (4). In this study, the authors used ANQX (a modify version of DNQX) known at that time as an AMPAR antagonist, to monitor synaptic AMPAR exchange after specific inactivation of the surface population (5). In their experiments, they measured a significantly slower recovery of AMPAR current in dendrites compare to the somatic region. Thus they concluded that AMPARs are mainly exocytosed at the somatic extracellular membrane and trafficked distally through lateral diffusion. However, DNQX has been shown since then not to be specific to AMPAR but to also act on kainate and NMDA receptors. Additionally DNQX and CNQX inhibition of AMPARs seems to be dependent on the composition of AMPAR complexes and in particular the type of auxiliary subunits associated with those receptors (6, 7). While the concentration of receptors between somatic and dendritic membranes appears to be fairly homogenous, it might be that the actual composition of the receptors varies between those two compartments. This could explain the observation by Adesnik and colleagues in the absence of an inhomogeneous receptor distribution. And in this case global multiplicative scaling is to be expected in the model of Earnshaw and Bressloff as well. Thus, we believe that our model using minimal assumptions and being restricted to a single dendritic segment with multiple dendritic spines is in better accordance with the recent literature on AMPAR trafficking. We now discuss all these matters in the manuscript.
2) I find it surprising that the model predicts (Equation 6) that the total number of receptors in the pool is independent of synapse number, slot number, and receptor binding and unbinding rates (α and β). The maths makes sense – due to linearity of the model – but it would be nice if the authors could comment on the likely validity of this prediction/assumption if nonlinearities were to be introduced. For example, there may be two types of receptor state within the synapse, trapped and not trapped.
Strictly speaking, the model is nonlinear, because there are product terms of the state variables (p x w_{i}) in Equations 1 and 2. Interestingly, a slightly modified model (still nonlinear), where receptors in slots can be internalized directly with the same rate δ without first going through the pool does not have the surprising property you mention. This model seemed less biologically plausible to us. Other, more complex, models with different receptor states are an interesting topic for future work (and Earnshaw and Bressloff have already looked at this to some extent). However, without having implemented and analyzed these models in detail it's hard to foresee how their predictions might differ. Similarly, AMPAR complexes containing various sets of auxiliary subunits are also very likely to coexist at the neuron surface (8). Since only a couple of auxiliary subunits have binding domains for PSD95, it might also be that multiple types of slots with different α and β parameters could be considered. Nevertheless, those topics are better left for future work. We have added a brief discussion of this.
Reviewer #3:
This paper represents an important intermediate level of modeling between purely abstract views of normalization and homeostasis of synapse strength and detailed biological modeling, something that is still impossible at this scale but which is being attempted in the context of the individual spine.
The authors do a good job of reaching up (topward) to the abstract formulation but make less of an effort to reach down (bottomward) to the biology. The downreach is more difficult and will necessarily be speculative but I would encourage the authors to make this effort. This represents the essence of such a levelofinvestigation bridging model: from the problem level (normalization), to algorithms, to implementations.
"In the limit of large receptor numbers" – please justify. What are the estimated numbers? What numbers are required to avoid substantial variability due to stochastics? There is brief discussion of this in the Discussion section that could be moved up.
We have implemented a stochastic version of the model that simulates stochastic transitions of individual receptors using the Gillespie algorithm. This has allowed us to quantify the size of synaptic efficacy fluctuations and precisely answer all these questions. We have added this material to the manuscript (one additional figure, one additional coauthor). A new insight (not really unexpected, though) coming from these simulations is that small synapses can have substantial fluctuations in their efficacies due to fast receptor trafficking even under basal conditions. In contrast, large synapses have comparatively stable efficacies. Another insight is that the size of these fluctuations depends on the filling fraction of the synapses.
Predictions are the eventual goal of modeling and should be given more attention. Please move the predictions of Table 2 into a major textual section and indicate how each prediction could currently (or with some imagined technical advance in the future) be tested experimentally.
Having a table summarizing the major predictions was actually our attempt to make them more prominent. We have kept the table but also added a text section to the Discussion elaborating how the various predictions could be tested.
Implementations determines algorithms determines problems (inverse of the Marr procedure). In this case, what is the role of catabolism of damaged receptors? Are most receptors actually returned to the pool or do many need to be replaced? What are estimates of the metabolic requirements for such replacement? How are receptors within the dendritic pool mobilized into spines? What is the involvement of endoplasmic reticulum, rough ER?
The current version of the model does not consider intracellular trafficking per se. Thus involvement of the endoplasmic reticulum is beyond the scope of this study. Regarding receptor damage it is known that AMPAR halflife is rather short (~2 days – while the average halflife of neuronal proteins is 57 days) (9, 10). So AMPARs are constantly degraded and replaced by newly synthetized proteins. But, in any case, damaged receptors would have to be endocytosed into intracellular compartments to undergo lysosomemediated degradation. We now briefly discuss the role of the ER in protein synthesis in the context of the hotspot translation events (see above).
The dendritic pool of AMPARs is composed of receptors that have been produced or recycled in intracellular compartments and externalized to the cell surface. After externalization AMPARs are diffusing at the neuron surface. AMPARs are mobilized into synapses mainly via binding to PSD95 (11–13). At postsynaptic sites PSD95 molecules are highly packed (~300 molecules per postsynaptic synaptic density) (14) and largely immobile (15). When a receptor enters a synapse binding to one or more immobile PSD95 results in receptor immobilization. We have added corresponding text to the manuscript.
"Spreading from [nucleus by] slow diffusion process" is not really accurate; what is putative interplay of microtubules and actin vs. role of diffusion in bringing receptors into play? How might these factors relate to the role of pool mobilization in synaptic tagging and capture (STC)? Does the sudden increase in pool size in Figure 4 represent a 'capture' event?
Even the transport along microtubules may be viewed as a kind of diffusion process, where one single random walk “step” in the largescale “diffusion” process is the transport along a long stretch of microtubule. Differing probabilities of forward vs. backward steps would lead to a biased random walk but a random walk nevertheless and therefore a form of diffusion. To avoid any confusion, we have rephrased to “slow transport process”. The sudden increase in pool size in Figure 4 was not intended to model a capture event as suggested in STC. However, the increase in slot numbers and binding rate to receptor slots used in Figure 6 could be interpreted as part of an STC event.
References:
1) Meyer D, Bonhoeffer T, Scheuss V. Article Balance and Stability of Synaptic Structures during Synaptic Plasticity. Neuron [Internet]. Elsevier; 2014;82(2):430–43. Available from: http://dx.doi.org/10.1016/j.neuron.2014.02.031
2) Bartol TM, Bromer C, Kinney J, Chirillo MA, Bourne JN, Harris KM, et al. Nanoconnectomic upper bound on the variability of synaptic plasticity. Elife [Internet]. 2015 [cited 2016 Jul 10];4:e10778.
3) ToaCheng JH, Crocker VT, Winters CA, Azzam R, Chludzinski J, Reese TS. Trafficking of AMPA Receptors at Plasma Membranes of Hippocampal Neurons. J Neurosci. 2011;31(13):4834–43.
4) Adesnik H, Nicoll RA, England PM. Photoinactivation of Native AMPA Receptors Reveals Their RealTime Trafficking. 2005;48:977–85.
5) Chambers JJ, Gouda H, Young DM, Kuntz ID, England PM. Photochemically Knocking Out Glutamate Receptors in Vivo. 2004;13886–7.
6) Maclean DM, Bowie D. Transmembrane AMPA receptor regulatory protein regulation of competitive antagonism : a problem of interpretation. 2011;22:5383–90.
7) Greger IH, Watson JF, Cullcandy SG. Review Structural and Functional Architecture of AMPAType Glutamate Receptors and Their Auxiliary Proteins. Neuron [Internet]. Elsevier Inc.; 2017;94(4):713–30.
8) Schwenk J, Harmel N, Brechet A, Zolles G, Berkefeld H, Müller CS, et al. Highresolution proteomics unravel architecture and molecular diversity of native AMPA receptor complexes. Neuron [Internet]. 2012 May 24
9) Cohen LD, Zuchman R, Sorokina O, Müller A, Dieterich DC, Armstrong JD, et al. Metabolic Turnover of Synaptic Proteins: Kinetics, Interdependencies and Implications for Synaptic Maintenance. PLoS One. 2013;8(5).
10) Dörrbaum AR, Kochen L, Langer JD, Schuman EM. Local and global influences on protein turnover in neurons and glia. Elife [Internet]. 2018;7:1–24.
11) Schnell E, Sizemore M, Karimzadegan S, Chen L, Bredt DS, Nicoll RA. Direct interactions between PSD
95 and stargazin control synaptic AMPA receptor number. Proc Natl Acad Sci U S A. 2002 Oct 15
12) Bats C, Groc L, Choquet D. The interaction between Stargazin and PSD95 regulates AMPA receptor surface trafficking. Neuron [Internet]. 2007 Mar 1 [cited 2014 Oct 26];53(5):719–34. Available from: http://www.ncbi.nlm.nih.gov/pubmed/17329211
13) Sumioka A, Yan D, Tomita S. TARP phosphorylation regulates synaptic AMPA receptors through lipid bilayers. Neuron [Internet]. 2010 Jun 10 [cited 2015 Feb 23];66(5):755–67.
14) Kim E, Sheng M. PDZ domain proteins of synapses. Nat Rev Neurosci [Internet]. 2004 Oct 1
15) Sturgill JF, Steiner P, Czervionke BL, Sabatini BL. Distinct domains within PSD95 mediate synaptic incorporation, stabilization, and activitydependent trafficking. J Neurosci. 2009;29(41):12845–54.
https://doi.org/10.7554/eLife.37836.018Article and author information
Author details
Funding
Johanna Quandt Foundation
 Jochen Triesch
European Molecular Biology Organization (ALTF 10952015)
 AnneSophie Hafner
Alexander von HumboldtStiftung (3.31184902FRAHFSTP)
 AnneSophie Hafner
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Publication history
 Received: April 24, 2018
 Accepted: September 14, 2018
 Accepted Manuscript published: September 17, 2018 (version 1)
 Version of Record published: October 11, 2018 (version 2)
Copyright
© 2018, Triesch et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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