Competition for synaptic building blocks shapes synaptic plasticity

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.


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Simple mathematical models of Hebbian learning exhibit an unconstrained growth of synaptic 21 efficacies. To avoid runaway dynamics, some mechanism for limiting weight growth needs to be 22 present. There is a long tradition of addressing this problem in neural network models using synaptic 23 normalization rules (Malsburg, 1973;Oja, 1982 the neuron maintains a pool of receptors freely diffusing at the neuron surface and ready to be 89 stabilized inside nanodomains. The size of this pool is denoted ∈ ℝ ≥0 . Note that for mathematical 90 convenience we here consider the , and to be real numbers that can take non-integer values. 91 In the stochastic version of the model introduced below these will be natural numbers. 92 Receptors can transition from the pool to empty slots in a synapse or detach from such a slot 93 and return into the pool with rates ∈ ℝ >0 and ∈ ℝ >0 , respectively. Receptors in the pool are 94 removed with a rate ∈ ℝ >0 corresponding to internalization of the receptors from the cell surface 95 (endocytosis). To counteract this loss, new receptors are added at a rate ∈ ℝ >0 and injected into 96 the pool corresponding to externalization of the receptors to the cell surface (exocytosis). In the 97 limit of large receptor numbers, the dynamics of the system can be described by the following 98 system of coupled ordinary nonlinear differential equations: In the first equation, − describes the return of receptors from synapse into the pool. The term 100 ( − ) describes the binding of receptors from the pool to empty slots in synapse , which is 101 assumed to be proportional to both the number of receptors in the pool and the number of free 102 4 of 39 slots in the synapse. In the second equation, − describes the deletion of receptors from the pool, 103 represents the gain of new receptors, ∑ describes the return of receptors from the synapses 104 into the pool, and finally − ∑ ( − ) describes the loss of receptors from the pool which bind to 105 free slots in the synapses. Together, this is a system of + 1 coupled ordinary nonlinear differential that available AMPAR complexes are quickly redistributed among PSD-95 slots (compared to the 134 time scale of addition and removal of these PSD-95 slots to the PSD). This interpretation may 135 be particularly useful if the supply of PSD-95 is the limiting factor in determining the number of 136 functional AMPARs bound inside the PSD (Schnell et al., 2002). We leave open the question what 137 exactly the slots-for-a-slot might be. It is clear however, that PSD-95 molecules can form stable 138 lattices inside the PSD such that PSD-95 proteins could act as slots for other PSD-95 proteins. 139 Interestingly, the analysis of the model presented in the following does not depend on which 140 interpretation is chosen. The only additional assumption we will make is a separation of time scales 141 between the fast trafficking of the "receptors" into and out of the "slots" and the slow addition and To find the fixed point solution ∞ , ∞ with ∞ = ∑ ∞ , we set the time derivatives to zero, i.e., 161 we requirė = 0 ∀ anḋ = 0 above. Inserting the first condition into (1) and summing over yields: Similarly, settinġ = 0 in (3) gives: Adding (4) to (5) then gives the solution for ∞ : The simple and intuitive result is therefore that the total number of receptors in the pool in 166 the steady state is given by the ratio of the externalization rate and the internalization rate . 167 Specifically, the presence of many receptors in the pool requires ≫ . 168 We now solve for the steady state solutions ∞ of the by again settinġ = 0 in (1) and using 169 (6) to give: Importantly, we find ∞ ∝ , i.e. in the steady state the weights of synapses are proportional to 171 the numbers of slots they have. The constant of proportionality is a filling fraction and we denote it 172 by . Interestingly, the filling fraction is independent of the number of receptor slots. Figure 2A   173 plots as a function of the ratio of the four rate constants ( )∕( ). We refer to this quantity as 174 the removal ratio, because it indicates the rates of the processes that remove receptors from the 175 slots relative to the rates of the processes that add them to slots. Note that a filling fraction close to 176 one requires ≪ .

Summing (7) over reveals that ∞ =
, so we can also write: where ∕ is the relative contribution of synapse to the total number of slots. Note that if the 179 filling fraction changes, say, due to an increase in receptor externalization or a change in any of the 180 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. 182 Thus, the analysis so far reveals a first prediction of the model (compare Table 2 The relative pool size together with the filling fraction determine the unknown externalization 210 rate and the rate of binding to receptor slots . Specifically, using ∞ = and ∞ = ∕ , we find: Furthermore, by combining this with the implicit definition of from (7) we can solve for to obtain: We can identify the term (1 − ) as the total number of empty receptor slots in the system. The 215 intuitive interpretation of the result is therefore that the binding rate will be big compared to 216 the unbinding rate if the number of empty slots and the relative pool size are small. Using 217 the definition of we can also rewrite the expression for the total number of receptors as ∞ =

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(1 + ) . 219 Filling Fraction Synapses in a local group have identical filling fractions in the basal state.
Pool Size Manipulation of local pool size scales synapses multiplicatively.

Sensitivity
Filling fraction is most sensitive when pool size matches slot numbers.
Heterosynaptic I High pool size and filling fraction reduce heterosynaptic plasticity.
Heterosynaptic II Heterosynaptic plasticity is only transient.

Homosynaptic
Pool size and filling fraction modulate homosynaptic plasticity.

Fluctuations
Spontaneous efficacy fluctuations are bigger for small synapses.

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To study the redistribution of receptors on a fast time scale, we exploit the fact that the processes 227 of receptor externalization and internalization are slow compared to the attaching and detaching of 228 receptors to and from slots. For instance, the time that an AMPAR remains in the cell membrane 229 is of the order of ten minutes while the time it resides inside the PSD is of the order of half a 230 minute. A reasonable approximation on short times scales is therefore to neglect the production 231 and removal terms in (2). In this case, the total number of receptors ≡ + is constant, as can 232 be seen by removing the − and + terms from (2), and adding (1), summed over all , which gives 233̇ +̇ =̇ = 0. In the Methods we show that the steady state solution on the fast time scale is then 234 given by: where we have introduced ≡ ∕ as a short hand for the ratio of the rates through which receptors 236 leave and enter the synaptic slots. We define the corresponding short-term steady-state filling 237 fraction as * = * ∕ . Importantly, the short-term filling fraction * is identical for all synapses. The finding that the short-term steady-state filling fraction is identical for all synapses is analo-244 gous to the solution for the long term filling fraction derived in (7), which is also the same for all 245 synapses. This implies a second prediction of the model (compare Table 2 fewer or more receptors than slots in the system. If there are more receptors than slots then * 251 approaches one. If there are fewer receptors than slots then * approaches the ratio of receptors 252 to slots in the system. In general, we find that the maximum short-term filling fraction for → 0 is 253 given by * max = min{1, ∕ }. In particular, a high filling fraction can only be achieved if > , i.e., there must be more receptors than slots in the system. On the other hand, * is most sensitive 255 to changes in when = . This can be seen by the steep negative slope of the black curves 256 in Fig. 3 for small values of . In fact, for = the derivative diverges, i.e., * reacts extremely 257 sensitively to changes in (see Methods for details). We therefore note another prediction (compare   258   Table 2 Goddard (1983). 335 We therefore note the following additional predictions of the model (compare Table 2 where we have introduced synapse specific insertion rates and made the time dependence of 356 the various quantities explicit. 357 We model the transient increase in as a linear increase to four times the original value within To quantify this effect, we systematically vary the relative pool size and filling fraction and 386 observe the peak relative changes in synaptic efficacies during homosynaptic LTP and heterosy-387 naptic LTD (Fig. 6 E,F). We find that a small pool size strongly reduces the peak homosynaptic 388 LTP and greatly increases the peak heterosynaptic LTD. Furthermore, both homosynaptic LTP and 389 heterosynaptic LTD tend to be reduced by a high filling fraction. These results are consistent with 390 those from Fig. 5. 391 In addition to these already noted effects on heterosynaptic plasticity, this implies another 392 prediction of the model regarding homosynaptic plasticity (compare Table 2 where we have replaced the current pool size with its steady state value * ( ) = − * ( ) for a 408 constant number of receptors in the system. Using * ( ) ≡ * ( )∕ we arrive at: For small numbers of receptors in the system, i.e. close to zero, the steady state filling fraction 410 * ( ) will be close to zero so thaṫ ≈ . In contrast, for high numbers of receptors and the filling 411 fraction close to its long-term steady-state value we find: 1∕ . This behavior is illustrated in Fig. 7

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Our analysis of the differential equation model above is suitable for studying the average behavior of The change in also leads to different filling fractions in the three cases, see (7). The results in Fig. 8 452 D show that an increased pool size will dampen spontaneous fluctuations of synaptic efficacies, 453 while a reduced pool size promotes stronger fluctuations. We again fit power law functions to 454 the data. Parameters of the fits are given in Table 4. Taken together, these results imply another 455 prediction of the model (compare Table 2 (Lynch et al., 1977). The idea has a long history. Synapses on the dendritic 477 tree compete for a limited supply of synaptic building blocks such that when some synapses grow, could be assessed using dSTORM. This second approach seems rather challenging, however, as one 514 would have to find in the fixed sample the exact dendrite and specific spines that were stimulated 515 during live-imaging. A second corollary of the model's first prediction is that if any of the transition 516 rates changes, e.g., the rate at which receptors unbind from receptor slots, the filling fractions 517 and synaptic efficacies are scaled by the same factor. Testing this prediction can be achieved by 518 interventions that alter the transition rates. Activation of CaMKII leads to the phosphorylation of the 519 AMPAR auxiliary subunit Stargazin increasing its affinity to PSD-95 (Hafner et al., 2015). Dendritic morphology and local production. 553 We have assumed that the basal transition rates for receptors attaching and detaching to and from inside the ER may still allow for a homogeneous distribution of receptors at the cell surface. 608 We therefore predict that local production of synaptic building blocks across the dendritic tree 609 contributes to their uniform distribution, which in turn might allow global multiplicative scaling 610 behavior and the maintenance of relative strengths of synapses. This could be tested, for example, 611 by specifically blocking local production of synaptic building blocks, which should make their 612 distribution across the dendritic tree less homogeneous and lead to systematic inhomogeneities in 613 synaptic efficacies across the dendritic tree. 614 In this context it is also interesting to note that at least one form of heterosynaptic plasticity Slot production and removal. 639 In future work, it will also be interesting to consider changes to slot numbers in more detail. 640 We simulated increases in slot numbers of individual synapses in the context of LTP. Obviously, 641 however, the building blocks of these "slots" also have to be produced, transported, and inserted 642 into synapses, which could be based on similar mechanisms as we have postulated for receptors. 643 Furthermore, slots are also degraded and have to be replaced. In fact, the alternative interpretation 644 of our model discussed in the beginning of the Results section already describes how PSD-95 slots 645 are produced (or degraded) and bind to (or detach from) slots for these receptor slots ("slots-for-a-646 slot" interpretation). Future work should aim for a model that more fully describes the interactions 647 of AMPARs (and other types of receptors), various TARPs such as stargazin, MAGUK proteins such as 648 PSD-95, and neuroligins as well as their production and trafficking. Along these lines, it will also be 649 interesting to consider the mechanisms underlying different stages of LTP and LTD in more detail. 650 Modeling slow homeostatic synaptic scaling. 651 The model could also be extended to capture slow homeostatic synaptic scaling processes (Turri- The simulation software was written in Python and is available at: 684 https://github.com/triesch/synaptic-competition (Triesch and Vo, 2018). 685 Differential equations were discretized with the Euler method. 686 The stochastic version of the model was simulated using the Gillespie algorithm (Gillespie, 1976). 687 Stochastic reactions were defined for receptors entering or leaving each of the seven synapses We introduce ≡ ∕ as the ratio of the rates through which receptors leave and enter the synaptic 702 slots. Using this, the two solutions of (22) are given by: The "+" solution is not biologically meaningful, since it leads to * ≥ or * ≥ (see Appendix), 704 so that the desired steady state solution of the short-term approximation is given by: and the corresponding short-term steady-state filling fraction is * = * ∕ . In the full system, this 706 solution is assumed only transiently, because receptors can still enter and leave the system. If the 707 number of receptors were held constant, then * and * would describe the stable solution on 708 long time scales.

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Sensitive reaction of the short-term filling fraction to changes in reaction rates 710 when number of receptors matches number of slots 711 We are interested in how the short-term filling fraction * changes, when the reaction rates and 712 or their ratio ≡ ∕ change. Formally, we consider the partial derivative of the short-term filling 713 fraction * = * ∕ with respect to . Using (24) we find: As can be seen in Fig. 3C,D, the most extreme slope is obtained at = 0. There the derivative 715 simplifies to: For = the slope diverges, i.e., the short term filling fraction reacts extremely sensitively to small 717 changes in when is close to zero. = 1∕43 s −1 , = 1∕14 min −1 . The desired relative pool size was set to = 2.67 and the desired filling fraction to = 0.9. The production rate and attachment rate were calculated according to (12) and (13), respectively.
The steady-state total number of receptors in this example is given by ∞ = (1 + ) = 33 030. The "+" solution from (23) is not biologically meaningful. 875 We show that the "+" solution from (23) is not biologically meaningful. To see this, first note that 1 ≤ 2 . Furthermore, any meaningful solution must fulfill ≤ and ≤ , i.e., 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 1 does not meet both criteria, then the larger 2 cannot meet them either. So we assume in the following that 1 meets both these criteria so that 1 ≤ min{ , }. Our argument uses Vieta's formulas for the quadratic equation (22): 1 + 2 = + + and 1 2 = .
Using the second formula we can write: from which follows that: In the case that > , this leads to 2 ≥ . The only biologically meaningful solution to this is the equality 2 = . 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 1 = .
Plugging both results into Vieta's first formula, we see that this solution requires = 0, which in turn requires = 0. In this case, no receptors would ever leave synapses. The case < leads to 2 ≥ . The only biologically meaningful solution to this is the equality 2 = . This is the extreme case where all slots are filled with receptors. Using Vieta's formulas again leads to the uninteresting requirement = 0 for this solution.