Comparing Theories for the Maintenance of Late LTP and Long-Term Memory: Computational Analysis of the Roles of Kinase Feedback Pathways and Synaptic Reactivation

Introduction

Long-term memory (LTM) lasting days or more necessitates synaptic structural changes and strengthening, and de novo protein synthesis (Davis and Squire, 1984; Kandel, 2001; Mayford et al., 2012). Long-term potentiation (LTP), and the invertebrate analog long-term synaptic facilitation (LTF), are likely necessary for many types of memory formation (Lynch, 2004; Nabavi et al., 2014; Mirisis et al., 2016; Asok et al., 2019). Late LTP (L-LTP, lasting > ~2 h) requires protein synthesis (Stanton and Sarvey, 1984; Frey et al., 1998; Abraham and Williams, 2008) and can last at least a year (Abraham et al., 2002). Thus a fundamental concern is to determine mechanisms that preserve LTM for months or even a lifespan (Vanyushin et al., 1974; Crick, 1984; Schwartz, 1993; Holliday, 1999; Roberson and Sweatt, 1999).

One hypothesized mechanism involves persistent upregulation of local translation, or enhanced incorporation, of synaptic proteins termed plasticity-related proteins (PRPs; Figure 1) (Fonkeu et al., 2019). Increased translation or incorporation would be based on positive feedback loops specific to synapses strengthened to participate in LTM storage (Smolen et al., 2019). As illustrated in models, a common element of such positive feedback is a bistable switch in which the strength or weight of specific synapses, and biophysical variables such as kinase activities, switch post-stimulus from basal to persistently increased values (Lisman, 1985; Bhalla and Iyengar, 1999; Song et al., 2007; Zhang et al., 2010, 2016). Although synapse-specific mechanisms for maintenance of LTM may be supplemented by cell-wide mechanisms, such as upregulation of specific genes (e.g., Kyrke-Smith and Williams, 2018) or neuronal excitability (Mozzachiodi and Byrne, 2010), here we explore the necessity, sufficiency, and interactions among proposed synapse-specific positive feedback loops to maintain LTP. Synapse-specific feedback loops are necessary. In their absence, given protein turnover time scales of hours to days (Eden et al., 2011; Cohen et al., 2013), homeostatic processes (Turrigiano et al., 1998; Abbott and Nelson, 2000) and relaxation would average out synaptic strengths, erasing LTM, LTP, or LTF. We focus on proposed mechanisms for maintenance of NMDA-dependent, excitatory LTP. Mechanisms for long-term plasticity at other synapse types may differ (Freeman, 2015; Hawkins and Byrne, 2015; Hige, 2018; Lamsa and Lau, 2019).

Essential kinases for LTP induction include protein kinase A (PKA) (Abel et al., 1997; Matsushita et al., 2001), the atypical PKC denoted protein kinase M ζ (PKMζ) (Sacktor et al., 1993), the MAP kinase (MAPK) isoform ERK (Rosenblum et al., 2002), and CaM kinase II (CaMKII) (Malenka et al., 1989; Otmakhov et al., 1997). Smolen et al. (2019) discuss hypothesized positive feedback involving ERK and other kinases. Here we limit our focus to CaMKII and PKMζ.

CaMKII

This feedback loop relies on self-sustaining autophosphorylation of CaMKII holoenzymes resulting in autonomous activation (Lisman and Goldring, 1988), which has generated, in models, a bistable switch of CaMKII activity and synaptic strength (Zhabotinsky, 2000; Miller et al., 2005; Aslam et al., 2009). Active CaMKII can enhance translation (Atkins et al., 2004, 2005). Persistence of CaMKII activation, and the necessity of CaMKII activation for maintaining L-LTP and LTM, has been challenged (Lengyel et al., 2004; see Smolen et al., 2019 for review). A light-inducible inhibitor of CaMKII was used by Murakoshi et al. (2017) who argued persistent CaMKII activity is not necessary to maintain LTP. However, this study did not specifically examine whether CaMKII inhibition blocks L-LTP at late times (>2 h post-stimulus) or late LTM (~24 h post-training). Thus, it is still possible that autonomously active CaMKII, perhaps in a small postsynaptic zone, maintains L-LTP and LTM. Saneyoshi et al. (2019) delineated a positive feedback loop that could contribute to localized, persistent CaMKII activation. CaMKII phosphorylates Tiam1, a Rac guanine nucleotide exchange factor. Binding of Tiam1 to CaMKII is thereby enhanced, and Tiam1, in turn, occludes the autoinhibitory domain of CaMKII in a manner that enhances autonomous activity of CaMKII. A caveat is that the activity of CaMKII was not monitored for more than 40 min post-stimulus, preventing direct application of this study to L-LTP maintenance. A study using dominant negative (dn) CaMKII also suggests persistent CaMKII activity is required to maintain spatial LTM (Rossetti et al., 2017), although with a caveat that the molecular mechanism by which dnCaMKII disrupts LTM is not known and may not depend on activity per se. Considering all these data, it remains plausible that persistently enhanced CamKII activity is important to maintain some forms of L-LTP and LTM, although further study is needed.

PKMζ

This positive feedback loop relies on a self-sustaining increase in translation, and thus activity, of PKMζ. Inhibiting PKMζ blocks maintenance of L-LTP and several forms of LTM (Serrano et al., 2005, 2008; Pastalkova et al., 2006; Shema et al., 2007). Following spatial memory formation, PKMζ activity is increased for at least a month (Hsieh et al., 2017). PKMζ upregulates its own synthesis (Kelly et al., 2007), forming a positive feedback loop. Modeling has suggested this loop can generate a bistable switch (Smolen et al., 2012; Jalil et al., 2015; Helfer and Shultz, 2018). As output, PKMζ promotes AMPAR trafficking to synapses (Migues et al., 2010), which could maintain increased synaptic strength. PKMζ also phosphorylates the histone acetyltransferase denoted CREB binding protein, which may enhance transcription required for late maintenance of LTM (Ko et al., 2016). However, the requirement for persistently enhanced PKMζ activity has been questioned by observations of normal LTP and LTM following constitutive (Lee et al., 2013) or conditional (Volk et al., 2013) PKMζ knockouts. But another study (Wang et al., 2016) found, in contrast, impaired maintenance of LTM and LTP following PKMζ knockdown. Thus, it is plausible that persistently enhanced PKMζ activity is necessary for the maintenance of some forms of L-LTP and LTM. But if PKMζ activity is persistent, is it due to autonomous positive feedback, or is it maintained due to another process such as synaptic reactivation or persistent CamKII activity?

Reactivation

Specific assemblies of neurons, or memory engrams, are activated upon learning and during memory retrieval (Zhou et al., 2009; Park et al., 2016) and specific patterns of functional connectivity among neurons within an engram, based on potentiated synapses as well as neuronal excitability, are important for memory storage (Mozzachiodi and Byrne, 2010; Liu et al., 2012; Ryan et al., 2015; Josselyn et al., 2017; Choi et al., 2018; Tonegawa et al., 2018; Josselyn and Tonegawa, 2020). Recurrent reactivation of neurons within engrams encoding recent experience, commonly termed replay, appears generic (Ikegaya et al., 2004; Miller et al., 2014; Wu and Foster, 2014) and correlates with improved LTM (Schapiro et al., 2018).

Such reactivation of engram synapses may be necessary to maintain late LTM. Inducible forebrain-restricted knockdown of the NR1 subunit of the NMDA receptor, several months post-training, disrupts retention of remote cued and contextual fear memories and of taste memory (see also Shimizu et al., 2000; Cui et al., 2004, 2005). An implication of NR1 necessity is that recurrent reactivation of engram neurons and synapses connecting them induces Ca²⁺ influx through NMDA receptors and voltage-dependent Ca²⁺ channels, thereby reactivating processes, such as kinase activation, that originally induced LTP. By these processes, reactivation could generate recurring “rounds” of LTP, reinforcing and preserving L-LTP and LTM (Fauth and van Rossum, 2019). Recently, disrupting reactivation during sleep of hippocampal neurons associated with a specific environment was found to impair memory for goal location within that environment (Gridchyn et al., 2020).

In this scenario kinases would be persistently or recurrently active, and make essential contributions in maintaining LTM. However, in contrast to the feedback loops discussed previously, persistent activation of CaMKII, PKMζ, or other kinases would not be self-sustaining. Synaptic reactivation, and consequent kinase activation, would also likely induce transcription via stimulus-responsive factors such as cAMP response element binding protein (CREB) (Smolen et al., 2019). Resultant global increases in synaptic proteins may be a required permissive factor for reinforcement of L-LTP.

Summing up the above, evidence suggests self-sustaining kinase activation, in particular persistently elevated activities of CaMKII or PKMζ, may be necessary for the maintenance of LTM. There is also evidence that synaptic reactivation may be necessary. But a further question is, are any of these positive feedback mechanisms also sufficient for maintenance? A problem in addressing sufficiency is that these feedback loops are interlocked such that manipulation of one process affects others. One way to address this issue is to develop a mathematical model of the processes and modify the parameters of each process, and thereby examine conditions under which each process is necessary and sufficient for LTM.

We developed four variants of a simplified model, to compare qualitative dynamics of kinase activities and synaptic strength predicted by these positive feedback loops. Two interrelated but contrasting hypotheses were continued: (1) autonomous activation of one or more kinases such as PKMζ or CaMKII is necessary and sufficient to maintain L-LTP and LTM, and (2) persistent kinase activation is necessary, but it depends on synaptic reactivation, which is sufficient. Simulations of these mechanisms suggest experiments that can examine dynamics predicted by these hypotheses. In two model variants, bistable switches of kinase activation are generated, respectively by (1) persistently increased PKMζ synthesis and activity, or (2) persistent CaMKII activation. In a third variant bistability in synaptic weight is generated by positive feedback in which synaptic reactivation preferentially occurs at strengthened synapses. L-LTP maintenance requires recurrent kinase activation downstream of reactivation. A fourth variant illustrates that synaptic reactivation and autonomous kinase activation could synergistically maintain L-LTP.

Model Development

The model describes dynamics of kinases necessary for L-LTP induction, and represents kinase-dependent setting of a synaptic “tag” that has been shown to be necessary for localized “capture” of plasticity-related proteins necessary for L-LTP (Frey and Morris, 1997, 1998). This model is simplified from those we have previously published (Smolen et al., 2006, 2014) in that it focuses on synapse-specific positive feedback loops. It does not represent gene induction and transcriptional or epigenetic positive feedback, although these have been hypothesized as permissive factors for maintaining memory (Zhang et al., 2016; Smolen et al., 2019).

Figure 1 schematizes the model. Stimuli, such as a tetanus, are represented by concurrent brief elevations of synaptic Ca²⁺ and cAMP, and brief activation of Raf kinase upstream of ERK. Stimulus parameters, including durations, are provided in Methods and are similar to Smolen et al. (2014). Some intermediate processes, such as interaction of Ca²⁺ with calmodulin or calmodulin with CaMKII, and Ras activation upstream of Raf, are not included. CaMKII, PKA, and ERK activate a synaptic tag to place a synapse in a labile state capable of “capturing” plasticity factors (proteins or mRNAs) to increase synaptic strength. Parameters describing kinetics of enzyme activation are given in Methods. Data support the importance of these kinases for tag setting. At least one tag modification appears to be mediated by PKA (Barco et al., 2002). Postsynaptic CAMKII activity is required for L-LTP, and tag setting is blocked by inhibition of CaMKII (Redondo et al., 2010). CaMKII stimulates translation via phosphorylation of cytoplasmic polyadenylation element binding protein, a candidate for Tag-1 (Atkins et al., 2004, 2005). ERK is also necessary for LTP (English and Sweatt, 1997; Rosenblum et al., 2002). In the model, tagging requires phosphorylation of three sites; Tag-1, Tag-2, and Tag-3; respectively, substrates of CAMKII, PKA, and synaptic ERK. We assume multiplicative dependence of synaptic tagging on these phosphorylations, thus the tag (variable TAG) is the product of amounts of phosphorylated Tag-1–Tag-3. The molecular nature of the synaptic tag is not characterized. However, assumed multiplicative dependence is consistent with block of tag setting by inhibition of a single kinase, CaMKII (Redondo et al., 2010), and with the requirement of ERK activity for LTP (Rosenblum et al., 2002).

CaMKII and ERK are assumed to cooperate to induce PKMζ synthesis, because inhibition of these kinases impairs PKMζ translation (Kelly et al., 2007). PKMζ translation is increased by LTP induction (Hernandez et al., 2003), and PKMζ concentrates at potentiated synapses (Hsieh et al., 2017). Stimulus-induced increases in TAG and PKMζ are assumed to converge, with these variables multiplying to determine the rate of increase of synaptic weight, denoted W. Increased W corresponds to L-LTP. Without positive feedback, W decays to baseline over several hours. We include as a parameter the level of a generic plasticity-related protein (PRP) required for LTP. A multiplicative dependence for the rate of increase of W is assumed, with this rate proportional to PRP, TAG, and PKMζ. This multiplicative dependence is an assumption and a phenomenological representation of the net effect of numerous processes. It is chosen to represent the separate necessity of each of the components (TAG, PRP, and PKMζ) which must act in concert to support L-LTP induction. There is also a small basal rate of increase in W, to yield positive basal W in the absence of stimulus.

Three forms of positive feedback are represented in distinct model variants, I–III, respectively. Each type of feedback is able, on its own, to generate robust bistability in W, maintaining L-LTP. Simulations consider one type of feedback at a time. These feedback loops (Figure 1) are: (1) Self-sustaining synthesis of PKMζ. Persistent activity is implemented as a Hill function in which active PKMζ feeds back to increase the rate of its own synthesis, with activity necessary and sufficient for maintaining increased W. This feedback could serve as a proxy for more detailed mechanisms, such as repression by PKMζ of the translation inhibitor Pin1 (Westmark et al., 2010). (2) Self-sustaining CaMKII autoactivation. Persistent activity is implemented as a Hill function in which active CaMKII feeds back to increase the rate of its own activation. This feedback could serve as a proxy for mechanisms including incorporation and activation of new subunits in a CaMKII holoenzyme (Miller et al., 2005) or reciprocal activation of CaMKII and Tiam1 (Saneyoshi et al., 2019). (3) Spontaneous reactivation in which a potentiated synapse is subject to brief, repeated elevations of Ca²⁺ and cAMP. Long-lasting kinase activation requires synaptic reactivation. Reactivation amplitudes are assumed to be greater at stronger synapses (i.e., the amplitudes of these brief Ca²⁺ and cAMP elevations increase with synaptic weight W). Therefore, reactivation is only effective at increasing W, counteracting its spontaneous decay, at synapses that are already strong. Reactivation stimuli, which only need to generate minor increases in W to counteract slow passive decay, are assumed weaker than the stimulus that induces L-LTP. Equations and parameters are given in Methods.

To maintain L-LTP in the model, TAG must remain elevated to some extent, either periodically (due to synaptic reactivation) or constitutively. TAG elevation allows a slow ongoing incorporation of plasticity-related protein, increasing synaptic weight to counter passive weight decay. The elevation of TAG required during LTP maintenance remains well below the peak value of TAG during LTP induction.

Results

Simulated L-LTP Maintenance Due to Self-Sustaining Kinase Activation

L-LTP Induction

We first turned off all positive feedback loops, simulating only induction (Figures 2A1,A2). We simulated a standard induction protocol—three 1-s tetani with interstimulus intervals of 5 min. CaMKII responds rapidly to each Ca²⁺ increase and deactivates in ~1 min, similar to the time course in dendritic spines (Lee et al., 2009), whereas synaptic ERK activity, and PKA activity, accumulate over tetani. The synaptic tag variable TAG increases by a large relative factor from basal to peak, due to the multiplicative convergence of ERK, PKA, and CaMKII activities to increase TAG. In alternate simulations in which an additive kinase interaction was assumed, the relative increase in TAG was much lower, resulting in excessive basal levels and in a lack of effective block of LTP upon single kinase inhibition (not shown). The synaptic weight W increases for > 1 h, consistent with data that L-LTP takes >1 h to develop when induced by chemical stimuli that bypass early LTP (Ying et al., 2002). The magnitude of L-LTP was assessed 2 h after the last tetanus, as the increase in W relative to pre-stimulus. Standard values for all parameters yielded L-LTP of 131%. Empirical excitatory postsynaptic potential increases after three or four tetani have similar values (English and Sweatt, 1997, Woo et al., 2000). Without positive feedback, L-LTP decayed with W returning to near basal ~10 h post-stimulus. This time scale for decay of W is set by the time constant τ_ltp (Methods, Equation 21), and was chosen to allow comparison of model dynamics when L-LTP was maintained by the feedback mechanisms proposed above.

Self-Sustained Elevation of PKMζ Synthesis

In Figures 2B1,B2, this positive feedback loop sustains a bistable switch, with PKMζ persistently active. All other dynamic variables, including TAG and W, decay to basal levels post-stimulus. L-LTP is not maintained. This lack of maintenance follows from the lack of dependence of TAG on PKMζ phosphorylation. Without elevated TAG, PRP cannot be incorporated. Thus, an important implication, and topic for future empirical investigation, is that if self-sustained activity of PKMζ is sufficient to maintain LTP, then either: (1) the tag is not needed for maintenance, or (2) PKMζ activation participates in and is sufficient for persistent synaptic tagging, or (3) there is a constitutive basal level of the synaptic tag, independent of kinase activation and sufficient to maintain L-LTP. In cases (2) and (3), as noted above, the low level of TAG required for maintenance may not conflict with empirical observations of synaptic tag decay. Activation of metabotropic glutamate receptors could allow PKMζ to participate in synaptic tagging (Li et al., 2014).

To illustrate maintenance of L-LTP dependent on activation of the synaptic tag by PKMζ, we repeated the simulation of Figure 2B with the equation giving the level of TAG (Equation 17, Methods) modified. A positive term was added to the right-hand side, k_TPKMPKM. This is the simplest plausible dependence. It assumes the level of TAG and the activity of PKMζ equilibrate rapidly compared to changes in PKMζ. As illustrated in Figures 2C1,C2, this PKMζ dependence suffices for persistent maintenance of L-LTP.

During maintenance, minor but persistent elevation of TAG above its pre-stimulus baseline is evident. This elevation is minor because the other kinases that contribute to tag setting (CaMKII, ERK, PKA) are not persistently activated. Minor TAG elevation suffices to allow continued slow incorporation of plasticity-related protein to counter passive synaptic weight decay. This TAG elevation is much less than the peak of TAG during L-LTP induction. It may not be readily evident empirically, and thus may not conflict with data suggesting a tag lifetime of 2–3 h (Frey and Morris, 1997, 1998). In this simulation, for bistability in W and TAG to be evident, it is essential to assume that PKMζ activity increases TAG additively, with the k_TPKMPKM term independent of CaMKII, ERK, and PKA. If, instead, PKMζ activity simply multiplies together with the activities of CaMKII, ERK, and PKA, then the persistent elevation of TAG and W is negligible, because of the return of CaMKII activity to its very low basal value.

The model can also simulate maintenance of L-LTP for the case in which a low, basal level of synaptic tagging is assumed to persist, independent of the activity of all modeled kinases (not shown). In this case, the maintained increase in synaptic weight W is entirely due to the persistent increase in PKMζ activity.

Self-Sustained Autoactivation of CaMKII

To simulate maintenance of L-LTP due to self-sustained activation of CaMKII, PKMζ autoactivation was removed and feedback mechanism two was implemented (Methods, Equation 23). In Figures 3A1,A2, this feedback sustains a bistable switch, with CaMKII activity and W persistently active. ERK and PKA activities still decay. Similarly to Figure 2C2, TAG remains persistently elevated. This elevation is again well below the peak during induction. Intriguingly, PKMζ activity also remains elevated. CaMKII increases synthesis and activity of PKMζ (Kelly et al., 2007). Thus, in this model variant, PKMζ activity depends on CaMKII-catalyzed phosphorylation (Equations 18 and 20, Methods), so that both activities are concurrently enhanced. Persistent PKMζ activity is not autonomously self-sustaining. However, as noted in the Introduction, the hypothesis that autonomous CaMKII activity sustains L-LTP has substantial empirical challenges.

Simulated Synaptic Reactivation Maintains L-LTP by Reactivating KINASES and Tagging

Maintaining L-LTP by reactivation of a strengthened synapse (feedback mechanism 3) yields qualitatively different dynamics. As illustrated in Figures 3B1,3B2, recurrent increases of cAMP, Ca²⁺, and Raf activity are induced by reactivations (Methods, Equations 25–27). These increases activate ERK, PKA, and CaMKII, synthesize PKMζ, and elevate TAG. These increases were adjusted by trial and error to maintain a stable, although fluctuating, state of increased synaptic strength. During reactivation, peak elevations of kinase activities excepting CaMKII, and of the synaptic tag, are substantially less than during LTP induction, and TAG relaxes to low levels between reactivations. If, as assumed here, reactivations are not very frequent, then these dynamics may be compatible with empirically reported decay of the tag within ~1–2 h after LTP induction. These simulations illustrate a model property more general than this specific reactivation loop. Sustained or recurrent activation of all simulated kinases is necessary and sufficient to maintain a degree of synaptic tagging and maintain L-LTP.

The simulation of Figures 3B1,B2 thus indicates a way in which LTP could be maintained in the absence of autonomous biochemical feedback loops. A qualitative difference between this maintenance mechanism and those based on autonomous kinase activation is that here, the synaptic weight can exhibit significant oscillations depending on the frequency of reactivation, whereas weights may be more stable with automous kinase activation.

The Model Is Not Unduly Sensitive to Most Parameter Variations

The sensitivity of the stimulus response—the amount of L-LTP—to relatively minor changes (±15%) of all parameter values including stimulus amplitudes was analyzed, perturbing from the standard values (Methods). There are 43 parameters, yielding 86 parameter variations for each model variant (note for each variant, 2–3 parameters governing strength of positive feedback are set to zero, thus these parameter variations are zero). We used a standard analysis defining a set of relative sensitivities S_i, with i ranging over all parameters p_i (Frank, 1978). R denotes the magnitude of stimulus response. For each p_i, a change is made, and the resulting change in R is determined. Relative sensitivity S_i is defined as the absolute value of the relative, or fractional, change in R divided by the relative change in p_i,

R is the magnitude of L-LTP after tetani. The sensitivity of four simulations was assessed—without feedback (Figure 2A), with kinase feedback (Figures 2B, 3A), and with reactivation feedback (Figure 3B). For Figures 2A,B, which were not bistable in W, L-LTP was quantified 2 h post-stimulus, whereas for the bistable simulations of Figures 3A,B, L-LTP was quantified 50 h post-stimulus.

Sensitivity to most of the 86 parameter variations was not large, with S_i below 3. There were 13 exceptions with S_i between 3.0 and 9.9. Interestingly, 10 of these exceptions were for Raf-ERK pathway parameters (Methods, Equations 6–13), and these 10 were >3 for all four simulations. These exceptions were: 15% increases and decreases in the total concentration of Raf kinase ([Raf_tot], Equation 6 in Methods), in the rate constant for Raf activation (k_fbasRaf, first paragraph of Methods and Equation 7), in the rate constants for MEK activation (k_fMEK) and inactivation (k_bMEK), and in the inactivation rate constant for Raf (k_bRaf). Two other exceptions were for ERK rate constants (+k_fERK, -k_bERK). Finally, the simulation with PKMζ autoactivation (Figure 2B) had S_i = 7.0 for a decrease in the corresponding Hill constant (K_pkm, Equation 24). This lack of large sensitivity for most parameter variations (73 out of 86) suggests model variants I—III are relatively robust and can be regarded as reasonable qualitative models of synaptic feedback mechanisms. Further studies might seek to add buffering mechanisms, especially within the Raf-ERK pathway, to reduce sensitivity to the specific parameters noted.

Simulations Suggest a Test to Distinguish Feedback Mechanisms for Maintenance of L-LTP

Further simulations suggest the consequences of inhibition of key kinases, in particular PKMζ or CaMKII, may provide an empirical test to distinguish between feedback mechanisms. Figure 4A illustrates that if self-sustained activation of CaMKII is sufficient to maintain L-LTP, then transient inhibition of a different kinase, in this case PKMζ, would not abolish L-LTP maintenance. W repotentiates after inhibition is removed, because CaMKII autoactivation does not require PKMζ activity and persists throughout inhibition. In contrast to this simulation, empirical inhibition of PKMζ does appear to inhibit L-LTP and LTM (Ling et al., 2002; Shema et al., 2007; Serrano et al., 2008), suggesting self-sustained CaMKII activation is not sufficient. However CaMKII autoactivation could still be necessary as some data suggests (Sanhueza et al., 2007; Rossetti et al., 2017) and as we simulated (model variant II).

Figure 4B illustrates that if recurrent synaptic reactivation maintains L-LTP, then inhibition of PKMζ can permanently reverse L-LTP. W declines while PKMζ is inhibited, because PKMζ activity is needed to maintain elevated synaptic weight. Therefore, if W declines sufficiently during PKMζ inhibition, selective reactivation of a strong synapse is no longer present, and LTP cannot be maintained. Even if inhibition of PKMζ is then removed, W continues to decline toward its basal value (Figure 4B). For standard parameter values (Methods), PKMζ inhibition does need to last ~ 50 h in order for W to decrease sufficiently, so that after inhibition removal reactivation is not able to re-potentiate W. Empirically, PKMζ inhibition is likely to reduce synaptic weight more rapidly than would a simple loss of reactivation, in which case a shorter inhibition duration would suffice. This simulation argues that empirically, inhibition of PKMζ during the maintenance phase of L-LTP does not enable discrimination between: (A) the sufficiency of a feedback loop based on autonomous, activity-independent PKMζ activation, vs. (B) the necessity, or even sufficiency, of a reactivation feedback loop involving PKMζ as an essential component.

Synaptic Reactivation and Autonomous Kinase Activation Could Synergistically Maintain L-LTP

The preceding model variants contrast qualitative dynamics that characterize maintenance of L-LTP and LTM due to one of two plausible forms of positive feedback, autonomous kinase activation and synaptic reactivation. It is also possible that multiple positive feedback loops act in concert to sustain bistability in synaptic weight, maintaining L-LTP and LTM. To explore dynamics associated with this scenario, we constructed a fourth model variant (Methods). We modified model variant III, based on synaptic reactivation, to include self-sustaining enhancement of PKMζ translation as in variant I. Parameters determining the strength of both forms of feedback were reduced from their standard values, so that neither reactivation nor PKMζ activation alone could maintain L-LTP. However, when both forms of positive feedback operated concurrently, bistability resulted in PKMζ activity and in W, and L-LTP was maintained.

Figures 5A1,A2 illustrates induction of L-LTP and its maintenance post-tetanus. These dynamics are similar to Figure 3B. During L-LTP maintenance, activities of CaMKII, ERK, and PKA, along with the synaptic tag, fluctuate above basal levels. At 2,000 min post-stimulus PKMζ positive feedback was switched off. W, and PKMζ activity, both decline to basal levels (with a slower time constant for W), illustrating that reactivation alone cannot sustain bistability. Figure 5B illustrates an analogous simulation, for which the PKMζ positive feedback remains on and reactivation is switched off. Bistability is again lost. PKMζ activity and W decline to basal levels, although the decline is quite slow for the simulated parameter values.

Discussion

We have considered two interrelated hypotheses: (1) autonomous self-sustaining activation of a kinase, focusing on PKMζ or CaMKII, is necessary and sufficient to maintain L-LTP; and (2) persistent activation of kinases is not autonomous, and is recurrently induced by synaptic reactivation, which maintains L-LTP. There are empirical data in favor of each hypothesis. Modeling investigated whether the dynamics of kinase activities, synaptic tagging, and synaptic weight predicted by these hypotheses could be discriminated. We identified dynamic differences, and make empirical predictions that could further discriminate these mechanisms.

Dynamics of Autonomous Kinase Activity

L-LTP was maintained by positive feedback based on autonomous CaMKII activation (Figure 2C), or based on synaptic reactivation (Figure 3A), or based on autonomously increased synthesis of active PKMζ (Figure 2C). With autonomous CaMKII activation, CaMKII activity remains high during maintenance, well above the other kinases. In contrast, when L-LTP is maintained by persistent PKMζ activation or by synaptic reactivation, CaMKII activity is maintained at a lower level, sufficient to sustain, in cooperation with other kinases, a low persistent level of TAG that allows continued necessary incorporation of plasticity-related proteins (PRPs). In these latter two cases, low CaMKII activity suffices because its cooperating kinase, PKMζ, is persistently or recurrently elevated.

Model variant I, based on autonomous, self-sustaining synthesis of active PKMζ did not initially maintain L-LTP (Figure 2B), because we first assumed there was no persistence of the synaptic tag considered essential for incorporation of plasticity related proteins. However, when PKMζ activity was assumed to elevate TAG, L-LTP was maintained (Figure 2C). With this model variant, L-LTP could also be maintained if a low basal level of TAG, independent of the kinase activities in the model, was assumed to persist, or if L-LTP maintenance was assumed to be independent of tagging.

If a positive feedback loop based on intrinsic activation of a kinase, such as CaMKII or PKMζ, is necessary and sufficient for maintenance of L-LTP, then the simulation of Figure 4A illustrates that inhibition of a kinase or process not within the feedback loop should fail to permanently reverse L-LTP. If sustained activity of PKMζ is both necessary and sufficient, then inhibiting CaMKII should have no effect, and also vice versa. Assuming inhibitor specificity, the empirical loss of established L-LTP and LTM upon PKMζ inhibition implies that no feedback loop (intrinsic or network based) independent of PKMζ can be both necessary and sufficient for maintenance.

We note additional positive feedback loops have been proposed to maintain LTM (Smolen et al., 2019). For example, in mammals and in Aplysia, isoforms of the translation activator denoted cytoplasmic polyadenylation element binding protein (CPEB) can upregulate translation at synapses that have undergone LTP or LTF. Aggregation of CPEB and increased translation has been hypothesized to be self-perpetuating and to maintain synaptic strength (Si et al., 2003, 2010; Drisaldi et al., 2015; Fioriti et al., 2015; Rayman and Kandel, 2017; Li et al., 2018). However, data suggesting that ongoing synaptic reactivation is necessary in mammals (Shimizu et al., 2000; Cui et al., 2004, 2005) imply that autonomous CPEB aggregation would not be sufficient.

We have not further explored the possible consequences of a low basal TAG independent of kinase activities. However, it is qualitatively evident that in this case, any persistent activation of a self-sustaining kinase or other process that increases PRP levels could maintain L-LTP, via enhanced PRP incorporation driven by basal TAG. For example, if CPEB aggregation increased translation of PRPs, then a basal TAG level could suffice to facilitate incorporation of these PRPs, maintaining L-LTP. Persistent CaMKII activity could upregulate PRP translation via CPEB phosphorylation (Atkins et al., 2004, 2005), maintaining L-LTP.

Dynamics of Synaptic Reactivation

In model variants III-IV for which recurrent synaptic reactivation is essential, only potentiated synapses undergo substantial reactivation to reinforce L-LTP. This assumption needs to be empirically verified. If synaptic reactivation is stronger or more frequent at potentiated synapses in memory engrams, then reactivation is predicted to yield increased time-average activity of kinases, specific to potentiated synapses, and reinforcing LTP. As with LTP induction, these kinases include ERK, CaMKII, PKA, and plausibly PKMζ. Monitoring FRET-engineered substrates of these kinases over relatively long periods (h or more) might demonstrate increased activity. In addition, simulated dynamics of synaptic reactivation (Figure 3B) differ from those of autonomous kinase activation (Figures 2C, 3A) in that reactivation events generate substantial fluctuations in activities of synaptic CaMKII, ERK, and the level of TAG (Figure 3B). Such fluctuations may be observable empirically. Our model has not addressed whether the frequency of reactivation events would be increased at potentiated synapses. To model reactivation frequency would require a neuronal network model within which a synaptic memory engram can be stored and maintained, and reactivation events can be self-consistently generated.

It would be highly desirable to strengthen the empirical case for the necessity of synaptic reactivation by examining the consequences, for maintenance of L-LTP and LTM, of additional pharmacological or genetic methods for prolonged synaptic activity reduction. Because reported timescales on which reactivation appears necessary are long—weeks to months—such studies will need to be done in vivo rather than with in vitro preparations of limited duration. In vitro, over hours, synaptic activity patterns may not recapitulate patterns in vivo, reducing the likelihood of positive feedback loop based on reactivation of specific synapses. Biochemical positive feedback, such as persistent kinase activation, may contribute to maintain L-LTP over hours in vitro.

Although the present study has focused on engram-level reactivation, another form of reactivation could be mediated by autocrine actions. For example, brain-derived neurotrophic factor (BDNF) released by synaptic activity can act, via TrkB receptors, to reactivate biochemical pathways necessary for LTP, including the Ras → Raf → MEK → ERK signaling cascade (Panja and Bramham, 2014). These events can upregulate transcription of bdnf, generating a positive feedback loop (Bambah-Mukku et al., 2014). We note that recurrent engram-level reactivation would be expected to upregulate BDNF secretion (Kolarow et al., 2007), with secreted BDNF reinforcing LTP (Kovalchuk et al., 2002; Barco et al., 2005). A modeling study does suggest positive feedback in which BDNF upregulates bdnf expression may play a role in maintaining L-LTP (Zhang et al., 2016).

Predictions to Help Discriminate Between Proposed Maintenance Mechanisms

Figure 3A2 illustrates sustained activation can depend on a separate process in a way that complicates determination of which process is necessary. Here, persistent activation of PKMζ occurs, but it is not autonomous. Persistent PKMζ activity depends on self-sustaining activation of CaMKII, which enhances PKMζ synthesis. These dynamics suggest that if CaMKII activation maintains L-LTP, this maintenance could be due in part to downstream PKMζ activation. If so, a prediction is that if CaMKII is inhibited, PKMζ levels at potentiated synapses would decay. In contrast, if PKMζ activity is elevated due to autonomous biochemical feedback, elevated synthesis and activity of synaptic PKMζ may not be reversed by inhibition of CaMKII, or by inhibition of synaptic reactivation.

Simulations of the model variant for which selective reactivation of strong synapses maintains L-LTP (Figure 4B) predict that for this mechanism, inhibition of any kinase will permanently reverse L-LTP only if inhibition lasts long enough to decrease synaptic weight such that synapse-specific reactivation no longer occurs, and LTP is not reinforced. In this simulation PKMζ was inhibited, but the same argument applies for inhibition of CaMKII or any other kinase necessary for L-LTP. Prolonged, neuron-specific inhibition of kinases such as ERK, PKA, or CaMKII may provide a test of this prediction.

When we modeled L-LTP maintenance as dependent on synaptic reactivation, elevation of PKMζ activity was still assumed necessary for maintenance, but this elevation was due to synaptic reactivation and was not autonomous. For this case, without autonomy, empirical inhibition of synaptic activity would be predicted to knock down PKMζ levels.

It appears difficult to empirically separate a model in which PKMζ positive feedback and synaptic reactivation are both necessary to sustain L-LTP (Figure 5) from reliance on reactivation alone (Figure 3B). The difficulty occurs because for both variants, inhibiting either PKMζ or synaptic reactivation abolishes bistability and maintenance of L-LTP.

Our model is limited in what it can infer concerning the dependence of L-LTP maintenance on protein turnover rates. For CaMKII, our assumed fast deactivation time constant (1 min) is consistent with observed rapid kinetics of CaMKII deactivation (Murakoshi et al., 2017), but far slower than protein turnover. Half-lives of synaptic CaMKII α, δ, and γ isoforms have been measured to be 3–4 days (Cohen et al., 2013), and a more complex model would be needed to simulate CaMKII turnover. Such a model would need to represent CaMKII subunit turnover and replacement within a CaMKII holoenzyme, and also account for the likelihood that a persistently active population of postsynaptic CaMKII should be modeled as a separate pool. Singh and Bhalla (2018) have recently modeled turnover and subunit exchange among CaMKII holoenzymes and simulated bistability, with persistence of enhanced CaMKII activity for at least 60 days. As regards maintenance of L-LTP by recurrent synaptic reactivation (model variant III), this maintenance mechanism is relatively insensitive to protein turnover rates. Kinases, and the synaptic tag, can deactivate or decay relatively quickly, as long as reactivation events occur frequently enough so that reactivation of kinases, and possibly increased translation of postsynaptic proteins, induces periodic elevations of TAG and synaptic weight to counteract ongoing decay.

As regards maintenance of L-LTP by bistable PKMζ activity (model variant I) our model assumes a standard value of 0.02 min⁻¹ for the PKMζ deactivation rate constant. If deactivation corresponds to degradation, then the degradation time constant is 50 min. Cohen et al. (2013) did not measure the turnover time of synaptic PKMζ, and as far as we know it has not yet been determined. Eden et al. (2011) measured half-lives of ~100 proteins from a human cancer cell line and found values ranging from 45 min to 23 h. These data suggest that if PKMζ deactivation corresponds to degradation, our assumed rate constant is at the upper end of plausible values. With the current model, we are not able to substantially decrease this rate constant without impairing model dynamics. If the PKMζ degradation time is assumed to be substantially longer, e.g., several days as found for the majority of synaptic proteins (Cohen et al., 2013), a more complex model would be needed. Such a model might assume a relatively rapid sequestration of constitutively active PKMζ, or an inactivating modification of PKMζ followed by slow degradation.

A basic assumption of our illustrated simulations is that processes necessary for the induction of L-LTP are also necessary for maintenance of L-LTP of engram synapses. This assumption will need to be empirically tested. We assumed the synaptic tag necessary for “capture” of plasticity-related proteins (PRPs) during induction of L-LTP remains essential during maintenance of L-LTP to allow reinforcing PRP capture. Indeed, Shema et al. (2011) previously suggested that persistent, selective upregulation of tagging at strong synapses may capture and localize PKMζ at those synapses, helping to maintain synaptic strength. If this assumption is correct, the tag is predicted to be persistently set to some extent, or periodically reactivated, at strong synapses. During maintenance, the activities of CaMKII, ERK, and PKMζ are all predicted to be necessary insofar as these kinases participate in tag setting, or enhance synaptic AMPAR incorporation. But autonomous bistability in these activities may (model variants I, II, IV) or may not (variant III) be necessary.

The tag identity has yet to be established, but several molecular species have been suggested to participate including the BDNF receptor TrkB (Lu et al., 2011) and polymerized actin (Ramachandran and Frey, 2009). One or more of these species is predicted to be persistently, or recurrently, upregulated at strong synapses. But the extent of upregulation during L-LTP maintenance, required to counter passive synaptic weight decay, may be substantially less than the increase during induction of L-LTP (Figures 2C2, 3A2,B2). Thus, it may be difficult to empirically demonstrate tag persistence or recurrence. Nonetheless, a promising strategy to further delineate the feedback mechanism(s) responsible for maintaining L-LTP and LTM is to investigate whether, and which, kinase activities or suggested tagging molecules are upregulated selectively at strengthened engram synapses.

Methods

Equations Prior to Positive Feedback

In the model, LTP is induced by three tetanic stimuli with a standard inter-stimulus interval (ISI) of 5 min. As in our previous study (Smolen et al., 2006), each individual tetanus is modeled as inducing: (1) A square-wave increase in synaptic Ca²⁺ concentration, [Ca_syn], to a peak value A_Casyn. The brief increase has a duration Ca_dur, with a standard value of 3 s that is similar to data (Malenka et al., 1992, Pologruto et al., 2004). Otherwise, Ca²⁺ remains at a basal value Ca_bas; (2) An increase in cAMP concentration [cAMP] to a value A_cAMP, with duration d_cAMP. d_cAMP has a standard value of 1 min, similar to data (Bacskai et al., 1993; Vincent and Brusciano, 2001). Otherwise cAMP is at cAMP_bas; (3) An increase in the zeroth-order rate constant for activation of Raf kinase by its upstream effector Ras. This rate constant is denoted k_fRaf and has a basal value of k_fbasRaf. Upon stimulus it increases to (A_STIM + k_fbasRaf). The duration of increase does not appear constrained by data. We assumed a plausible duration of 1 min.

CaMKII is assumed to be activated by a fourth-power Hill function of cytoplasmic Ca²⁺ concentration. This Hill functions constitutes a minimal representation of the activation of CaM kinases by calmodulin (CaM), because four Ca²⁺ ions bind cooperatively to CaM and CaM-Ca₄ activates CaM kinases. Data show a steep [Ca²⁺] dependence that can be characterized by a Hill coefficient ≥4 (Bradshaw et al., 2003).

The following ordinary differential equation (ODE) results for activation of CaMKII.

For cAMP to activate PKA, two cAMP molecules must bind cooperatively to the regulatory subunit of a PKA holoenzyme (Herberg et al., 1996). Thus, our qualitative representation of PKA activation assumes the activation rate is a Hill function of the second power of [cAMP].

In Equation 4, the time constant τ_PKA is given a value of 15 min, resulting in substantial accumulation of PKA activity over three tetani with interstimulus intervals of 5 min.

At the synapse, Raf kinase is phosphorylated and activated by stimuli, and by a small basal rate constant.

MAP kinase kinase, or MEK, is phosphorylated and activated by Raf. In turn, MEK activates ERK. Saturable Michaelis-Menten terms have been commonly used to describe phosphorylations and dephosphorylations of the kinases in the ERK cascade (e.g., Pettigrew et al., 2005), and are used here. Phosphatase dynamics are not explicitly modeled. Only doubly phosphorylated MEK and ERK are considered active. Concentrations of singly phosphorylated forms are given by conservation conditions.

As discussed previously (Smolen et al., 2006, 2014), data suggest that CaMKII, PKA, and ERK all play roles in setting the synaptic tag required for induction of late LTP. The molecular nature of the tag has not been characterized. We assume three phosphorylation sites, one for each kinase, and represent the amount of active synaptic tag as the product of the fractional phosphorylations of each site. The differential equations for the dynamics of these fractional phosphorylations, and their multiplication to give the level of active tag, are as in Smolen et al. (2014).

PKMζ is translated after stimulus and is constitutively active, with activity denoted PKM_act. Its translation is increased by activation of CaMKII and ERK. As a minimal representation of these dynamics, CaMKII and ERK are each assumed to phosphorylate sites denoted P_CK2 and P_ERK. The rate of synthesis of active PKMζ is assumed proportional to the product of P_CK2 and P_ERK, along with a small basal rate of synthesis, k_transbaspkm.

As in Smolen et al. (2006), the model hypothesizes a limiting factor, a protein P_lim, to prevent the synaptic weight W from increasing indefinitely with repeated simulation. As W increases, P_lim is posited to decrease via incorporation into the synapse. The rate of increase of W is given as a product of the amount of TAG, the concentration of the plasticity-related protein PRP, the activity of PKMζ, and the concentration of the limiting protein P_lim. There is also a small basal rate of increase to balance ongoing decay.

The rate of P_lim consumption is assumed to increase as the rate of W increases, yielding a similar ODE. However, because PKMζ is assumed to increase W by enhancing AMPAR incorporation into the synapse (Migues et al., 2010), PKMζ is not included in the ODE for P_lim (i.e., P_lim is not taken to be AMPAR).

Simulation of Positive Feedback

Equations 2–22 as written describe the induction of late LTP (L-LTP) that decays over several hours, without any positive feedback to sustain L-LTP indefinitely. In simulations of persistent L-LTP, three types of feedback were implemented: (1) Sustained autoactivation of CaMKII, (2) Sustained autoactivation of PKMζ, or (3) Ongoing reactivation of a potentiated synapse, resulting in recurrent reinforcement of L-LTP. Only one feedback type was implemented at a time. These feedback types were, respectively, implemented as follows, by modifying the above equations:

Type 1: The right-hand side of Equation 3, describing the rate of activation of CaMKII, was increased by adding a quadratic Hill function describing autoactivation,

Type 2: The right-hand side of Equation 20, describing the rate of activation of PKMζ, was increased by adding a quadratic Hill function describing autoactivation,

Type 3: Equations 2–22 were unaltered. Instead, ongoing brief, spontaneous elevations of [Ca_syn], [cAMP], and the rate constant k_fRaf for Raf activation were added. Similarly to our previous study (Smolen, 2007), positive feedback was implemented by making the amplitudes of these elevations proportional to an increasing Hill function of synaptic weight. For this qualitative model, the Hill function was chosen to be fairly steep (power of 5) so that only potentiated synapses undergo significant reactivation. During each brief reactivation event, the elevated levels of [Ca_syn], [cAMP], and k_fRaf are,

The elevation duration was 3 s for Ca_syn and 1 min for cAMP and k_fRaf. Outside of elevations [Ca_syn] = Ca_bas, [cAMP] = cAMP_bas, and k_fRaf = k_fbasRaf.

In the absence of detailed data, the frequency of these reactivations was set low enough so that the synaptic tag variable TAG would relax to near its basal value between reactivations, to accord with current data that the tag decays within ~90 min after LTP induction (Frey and Morris, 1997, 1998). The frequency was also set high enough to allow W to be maintained in an upper state for a reasonable value of the synaptic weight decay time constant τ_ltp. To implement this, we used a Gaussian random variable for the intervals between successive reactivations. This variable was updated for each new reactivation, using the Box-Mueller algorithm. The mean interval was 500 min and the standard deviation was 150 min. A minimum interval of 10 min was imposed.

To simulate only induction, not maintenance, of L-LTP (Figures 2A1,A2), Equations 23–27 were omitted.

Parameter Values and Numerical Methods

Standard parameter values are used in all simulations except as noted otherwise. Without positive feedback, these are as follows:

Ca_bas = 0.04 μM, cAMP_bas = 0.06 μM, Ca_dur = 3 s, d_cAMP = 1 min, A_Casyn = 0.8 μM, A_cAMP = 0.25 μM, A_STIM = 0.13 min⁻¹.

K_casyn = 0.7 μM, k_fck2 = 180 μM min⁻¹, τ_ck2 = 1 min, K_camp = 1.0 μM, τ_PKA = 15 min.

RAF_TOT = MEK_TOT = ERK_TOT = 0.25 μM, k_fbasRaf = 0.0075 min⁻¹, k_bRaf = 0.12 min⁻¹, k_fMEK = 0.6 min⁻¹, k_bMEK = 0.025 μM min⁻¹, K_MEK = 0.25 μM, k_fERK = 0.52 min⁻¹, k_bERK = 0.025 μM min⁻¹, K_ERK = 0.25 μM.

k_transpkm = 0.2 μM min⁻¹, k_transbaspkm = 0.0015 μM min⁻¹, k_dpkm = 0.02 min⁻¹.

k_phos1 = 0.15 μM⁻¹ min⁻¹, k_deph1 = 0.008 min⁻¹, k_phos2 = 0.8 μM⁻¹ min⁻¹, k_deph2 = 0.2 min⁻¹, k_phos3 = 0.06 μM⁻¹ min⁻¹, k_deph3 = 0.05 min⁻¹, k_phos4 = 0.1 μM⁻¹ min⁻¹, k_deph4 = 0.1 min⁻¹, k_phos5 = 2.0 μM⁻¹ min⁻¹, k_deph5 = 0.1 min⁻¹.

[PRP] = 1.0 μM, k_ltp = 500 μM⁻² min⁻¹, k_ltpbas = 0.01 min⁻¹, τ_ltp = 300 min, K_lim = 0.2 μM, k_Pl = 6.0 min⁻¹, k_Plbas = 0.0035 μM min⁻¹, τ_Pl = 100 min.

Positive feedback introduces new parameters (Equations 23–27). For positive feedback via CaMKII autoactivation, or via PKMζ autoactivation, new standard parameter values are, k_CaMKII = 4.0 μM min⁻¹, K_CaMKII = 1.0 μM, k_PKM = 0.028 μM min⁻¹, K_PKM = 0.75 μM. Also, k_ltp = 70 μM⁻² min⁻¹ for CaMKII autoactivation, and 300 μM⁻² min⁻¹ for the simulation of Figure 2B with PKMζ autoactivation. For the simulation of Figure 2C with PKMζ increasing the synaptic tag, k_ltp = 240 μM⁻² min⁻¹, and the coupling coefficient k_TPKM = 0.0001 μM⁻¹.

For feedback via synaptic reactivation k_ltp = 480 μM⁻² min⁻¹, K_W = 4.0. τ_ltp was increased to 3200 min and k_ltpbas was correspondingly decreased, to 0.00086 min⁻¹.

Model variant IV (Figure 5) used the parameter values of the synaptic reactivation simulation (Figure 3C) with the following exceptions. Reactivation events were weakened by reducing A_cAMP, A_Casyn, and A_STIM (Equations 25–27) to 80% of their standard values. PKMζ positive feedback was included (Equation 24), but the coupling constant k_PKM was reduced to 0.0078 μM min⁻¹. These changes ensured neither form of positive feedback could sustain bistability on its own. Other changes were: k_transbaspkm = 0.00045 μM min⁻¹, k_dpkm = 0.006 min⁻¹, k_ltp = 180 μM⁻² min⁻¹, k_ltpbas = 0.003 min⁻¹, τ_ltp = 800 min. The mean and standard deviation of reactivation frequency were (200 min, 50 min).

For some parameters, empirical data were used to determine standard values. As described in Smolen et al. (2006), the model's total concentrations of Raf, MEK, and ERK, and the peak concentration of active PKA as determined by stimulus and kinetic parameters, are similar to experimental values. The peak concentration of active CaMKII due to a tetanus is approximately 10% of the estimated total CaMKII concentration (Bhalla and Iyengar, 1999). Data were also used to constrain the amplitude and duration of Ca²⁺ and cAMP elevations (Smolen et al., 2006).

However, for many parameters, such as biochemical rate constants in vivo, data are lacking to determine values. Therefore, these parameters were chosen by trial and error to yield dynamics similar to empirical time courses, when the latter are available. In simulations that do not assume bistable CaMKII activity, CaMKII decays with a time constant of ~1 min. This brief lifetime is consistent with data describing CaMKII activity in dendritic spines (Murakoshi et al., 2017). PKA activity increases by ~150% during L-LTP induction, and decays post-tetanus with a time constant of 15 min. This amplitude, and decay time constant, are consistent with data (Roberson and Sweatt, 1996). Simulated postsynaptic ERK activation lasts ~90 min post-tetanus. This duration is plausible but is bracketed by empirical data. One study suggests ERK remains phosphorylated for as long as 8 h after tetanus (Ahmed and Frey, 2005). However, other studies (English and Sweatt, 1997; Liu et al., 1999) suggest briefer activation of ~30 min. Because long-lasting ERK activity could regulate transcription important for L-LTP, we suggest further study of ERK kinetics is warranted. After tetanus the synaptic tag (variable TAG) returns to baseline, or decays to a persistent but low activity to maintain L-LTP, within ~3 h. This decay time is similar to data (Frey and Morris, 1997, 1998). L-LTP induction (the increase of W) is complete in ~2 h, similar to the time required for empirical induction of L-LTP with BDNF (bypassing early LTP) (Ying et al., 2002). Parameters in the phenomological representations of positive feedback loops (CaMKII autoactivation, persistent PKMζ activity, and synaptic reactivation) in Equations 23–27 are not determined by empirical data. Rather, values for these parameters were chosen by trial and error to yield bistable switches of synaptic weight that were induced by simulated tetani, but with both stable states robust to perturbations (weak stimuli). The resulting standard parameter value set cannot be claimed to be unique, and the simulated time courses of some variables, such as the individual TAG substrates Tag-1–Tag-3, are not constrained by data.

In Figures 2–5, some variables are vertically scaled, scale factors are as follows. Figures 2A1,A2: 0.12 for [CaMKII_act], 7.0 for [ERKPP], 50.0 for [PKA_act], 0.1 for W, 700.0 for TAG, 2.0 for [PKM_act]. Figures 2B1,B2, same as Figures 2A1,A2 except 0.6 for [PKM_act]. Figures 2C1,C2, same as Figures 2B1,B2. Figures 3A1,3A2, same as Figures 2A1,A2 except 0.3 for [PKM_act]. Figures 3B1,B2: 0.12 for [CaMKII_act], 11.0 for [ERKPP], 50.0 for [PKA_act], 0.1 for W, 1000.0 for TAG, 1.5 for [PKM_act]. Figure 4A, same as Figures 3A1,3A2. Figure 4B, same as Figures 3B1,3B2. Figure 5, same as Figure 3B except 0.75 for [PKM_act].

The forward Euler method was used to integrate ODEs and, for the simulations of Figures 2A,B, we verified that using the fourth-order Runge-Kutta method did not yield significant differences. The time step was 10 ms. Model variables were initialized to 0.0001 (μM or arbitrary units) excepting variables given by conservation conditions. Prior to stimulus, model variables were equilibrated for at least two simulated days. The slowest variables, W and [PKM_act], were set to equilibrium basal values determined by other variables, except for Figures 4A,B for which W and [PKM_act], were initialized in their upper state. Models were programmed in Java.

Java programs to reproduce the simulations in Figures 2A,B, 3A,B have been deposited in the Github online database (https://github.com/psmolen0/Smolen_Byrne_LTP_Model), as files fig2ab3a.java and fig3b.java. These programs use model variants I-III, and will also be deposited in ModelDB. These programs are also included as supplementary manuscript files.

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

Author Contributions

PS and JB conceived and directed the project. PS developed the model and carried out simulations. PS, DB, and JB participated in analyzing results and writing the manuscript. All authors contributed to the article and approved the submitted version.

Funding

This study was supported by National Institutes of Health grant NS102490. Other than general grant support, the funder had no role in directing the research or preparing the manuscript.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We thank H. Shouval and W. Sossin for comments on an earlier version of the manuscript.

References