Requirement of an Allosteric Kinetics of NMDA Receptors for Spike Timing-Dependent Plasticity

Abstract

Spike timing-dependent synaptic plasticity (STDP) plays an important role in neural development and information processing in the brain; however, the mechanism by which spike timing information is encoded into STDP remains unclear. Here, we show that a novel allosteric kinetics of NMDA receptors (NMDARs) is required for STDP. We developed a detailed biophysical model of STDP and found that the model required spike timing-dependent distinct suppression of NMDARs by Ca²⁺-calmodulin. This led us to predict an allosteric kinetics of NMDARs: a slow and rapid suppression of NMDARs by Ca²⁺-calmodulin with prespiking → postspiking and postspiking → prespiking, respectively. We found that the allosteric kinetics, but not the conventional kinetics, is consistent with specific features of amplitudes and peak time of NMDAR-mediated EPSPs in experiments. We found that the allosteric kinetics of NMDARs was also valid for synaptic plasticity induced by more complex spike trains in layer II/III of visual cortex. We extracted an essential synaptic learning rule by reduction of the allosteric STDP model and found that spike timing-dependent bidirectional role of postspiking in synaptic modification, which depends on the allosteric kinetics, is the essential principle in STDP. Thus, we propose a simple hypothesis of the allosteric kinetics of NMDARs that can coherently explain critical features of spike timing-dependent NMDAR-mediated EPSPs and synaptic plasticity.

Introduction

In the mammalian brain, activity-dependent bidirectional modification of synaptic strength, including long-term potentiation (LTP) and long-term depression (LTD), has been thought to be the molecular and cellular basis of learning and memory (Bi and Poo, 2001). There are various forms of LTP and LTD, but NMDA receptor (NMDAR)-dependent LTP and LTD at excitatory synapses are the most extensively studied and the prototypic form of long-term synaptic plasticity (Malenka and Bear, 2004). The induction of LTP and LTD has been shown recently to depend on the timing between presynaptic and postsynaptic spiking (postspiking), known as spike timing-dependent synaptic plasticity (STDP) (Bi and Poo, 2001; Dan and Poo, 2004). The hallmark of STDP is a spike sequence-dependent bidirectional modification of synaptic strength within a narrow time window between prespiking and postspiking on the order of milliseconds. STDP provides significant computational advantages as a synaptic learning rule, allowing neural circuits to learn temporal sequences of inputs, to shape the temporal and spatial dynamics of neural circuits, and to code causality of external events (Gerstner et al., 1996; Song et al., 2000; Senn, 2002). Additionally, STDP reportedly contributes to the effects of sensory stimuli in the refinement of the retinotectal system in vivo (Mu and Poo, 2006; Vislay-Meltzer et al., 2006).

In STDP, prespiking → postspiking is thought to trigger high Ca²⁺ influx via NMDARs and voltage-gated Ca²⁺ channels (VGCCs), which depends on the depolarization-induced removal of the Mg²⁺ block of NMDARs and supralinear summation of EPSPs and backpropagating action potentials (bpAPs) (Hoffman et al., 1997; Stuart and Hausser, 2001; Kampa et al., 2004). The high Ca²⁺ influx is thought to induce activation of Ca²⁺-calmodulin (CaM)-dependent kinase II (CaMKII), resulting in induction of LTP (Lisman, 1989, 2001; Wang et al., 2005). In contrast, postspiking → prespiking is thought to be mediated by depolarization-dependent suppression of NMDARs (Froemke et al., 2005), which may lead to low Ca²⁺ influx via NMDARs and VGCCs. The low Ca²⁺ influx is thought to induce calcineurin (CaN) activation, resulting in induction of LTD (Lisman, 1989, 2001; Wang et al., 2005). Some forms of LTD in STDP depend on cannabinoid receptor (CB₁)-dependent suppression of transmitter release, in conjunction with activation of presynaptic NMDA autoreceptors (Sjostrom et al., 2003), whereas other forms of LTD do not require CB₁ receptor signaling (Seol et al., 2007). A new study revealed that such controversial results are attributable to different types of neurons or developmental stages (Corlew et al., 2007). Furthermore, recent studies indicate that STDP induction depends not only on the time interval between prespiking and postspiking but also on nonlinear interactions among complex timing prespiking and postspiking (Boettiger and Doupe, 2001; Sjostrom et al., 2001; Froemke and Dan, 2002; Tzounopoulos et al., 2004; Wang et al., 2005; Froemke et al., 2006). Although some phenomenological models have been proposed to describe the interactions among complex timing prespiking and postspiking (Froemke and Dan, 2002; Froemke et al., 2006; Pfister and Gerstner, 2006), the coding mechanism by which spike times are translated into STDP remains unclear because of complexity in signaling mechanisms and properties of STDP.

To address the coding mechanism of STDP, here, we focused on STDP induced in layer II/III lateral connections of visual cortex (Froemke and Dan, 2002; Froemke et al., 2005, 2006) and developed an in-depth biophysical model. We found that spike timing-dependent LTP, but not LTD, could be reproduced. The failure of induction of LTD led us to predict a requirement of slow and rapid suppression of NMDARs by Ca²⁺ · CaM with prespiking → postspiking and postspiking → prespiking, respectively. We found that the predicted allosteric kinetics of NMDARs, but not the conventional kinetics (the binding of glutamate and Ca²⁺ · CaM do not affect their binding to NMDARs each other), appears to be valid for experimental features of NMDAR-mediated EPSPs and for synaptic plasticity induced by more complex spike trains (Froemke and Dan, 2002; Froemke et al., 2006). We extracted an essential synaptic learning rule and found bidirectional regulation of NMDARs by postspiking with allosteric kinetics: the positive regulation of NMDAR activation and Ca²⁺ influx with prespiking → postspiking and the negative regulation of NMDAR activation and Ca²⁺ influx with postspiking → prespiking. We propose that the simple allosteric kinetics of NMDARs is a fundamental coding mechanism for STDP.

Materials and Methods

Development of the STDP model.

The STDP model consists of the postsynaptic membrane potential model (see Fig. 1A), the postsynaptic signaling cascades model (see Fig. 1B), and the AMPA receptors (AMPARs) trafficking model (see Fig. 1C). The GENESIS simulator (version 2.2) with a Kinetikit interface was used to numerically simulate spatiotemporal dynamics of electrophysiological and biochemical reactions in the STDP model (Bhalla and Iyengar, 1999). Detailed descriptions of these models, including block diagrams, reactions, and parameters, are described in the supplemental information (http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf), and the constructed programs are available for download (http://www.kurodalab.org/info/STDP/index.html). Here, a brief overview of the STDP model is presented as follows.

Figure 1.

Development of the STDP model. A, Electrophysiological compartment model. B, Postsynaptic signaling cascades model, which includes Ca²⁺ influx and removal and biochemical interactions. C, AMPAR trafficking model. The arrows and lines with a black circle indicate stimulatory and inhibitory interactions, respectively. Molecules required for LTP and LTD are indicated by red and blue, respectively. The detailed description of the models is provided as supplemental information (available at http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf).

Modeling of postsynaptic neuron.

We constructed a multicompartment model of a postsynaptic neuron with Hodgkin–Huxley-like channel kinetics on the basis of the previous models (see Fig. 1A) (Migliore et al., 1999; Poirazi et al., 2003). The model has one somatic and 20 dendritic compartments (see Fig. 1A), which incorporate spike-generating Na⁺ channels, persistent Na⁺ channels, delayed rectifier K⁺ channels, A-type K⁺ channels, and L-type VGCCs with spatially inhomogeneous densities. The synaptic site is assumed to be 60 μm apart from the soma (third dendritic compartment from the soma). Presynaptic stimulation leads to probabilistic release of glutamate from presynaptic terminals, which activates postsynaptic AMPARs and NMDARs. Postsynaptic stimulation (current injection) elicits APs.

Modeling of postsynaptic signaling cascades.

We modeled Ca²⁺ influx to a spine via NMDARs and VGCCs, as well as Ca²⁺ uptake from the spine, and simulated postsynaptic Ca²⁺ concentration [Ca²⁺]_PSD in response to presynaptic and postsynaptic stimulation on the basis of the previous observations (see Fig. 1B) (Sabatini et al., 2001). We also modeled intracellular signaling cascades within the spine on the basis of the previous models (Bhalla and Iyengar, 1999; Kuroda et al., 2001; Doi et al., 2005) and observations (Winder and Sweatt, 2001; Lisman et al., 2002). The spine is divided into cytosol and postsynaptic density (PSD) compartments. We incorporated interactions among Ca²⁺, CaM, CaMKII, CaN, protein kinase A (PKA), adenylnyl cylase 1/8 (AC1/8), cAMP, phosphorylated inhibitor-I (I-1), protein phosphatase 1 (PP1), and protein phosphatase 2A (PP2A). The molecules diffuse between the cytosol and PSD compartments, where CaMKII, PP1, and PP2A interact with scaffold (cytoskeletal scaffolds). We assumed that the molecules within each compartment are well mixed, and that they deterministically interact on the basis of mass assumption.

Modeling of AMPAR trafficking.

We constructed a simplified model of phosphorylation-dependent AMPAR [glutamate receptor 1 (GluR1)] trafficking on the basis of the previous experiments (see Fig. 1C) (Malinow and Malenka, 2002; Collingridge et al., 2004; Seol et al., 2007). An AMPAR subunit, GluR1, is phosphorylated by PKA at Ser845, and PDZ (PSD-95/Discs large/zona occludens-1)-containing proteins, which associate AMPARs, are phosphorylated by CaMKII. AMPARs are involved in the following recycling processes: (1) AMPARs with phosphorylated PDZs are trapped by scaffold at PSD fraction, (2) AMPARs are diffused laterally in the membrane, and (3) dephosphorylated AMPARs (GluR1 s at Ser845) are internalized by endocytosis, and phosphorylated AMPARs are reinserted into the membrane via exocytosis. We assumed that AMPAR number trapping at scaffolds of PSD corresponds to synaptic conductance.

Electrophysiology.

Acute visual cortical slices were prepared from 2- to 4-week-old Sprague Dawley rats. Somatic whole-cell recordings were made from layer II/III pyramidal cells in current-clamp mode. EPSPs were evoked by focal extracellular stimulation, and postsynaptic APs were elicited with current injection through the recording electrode. Synaptic strength was measured as the initial 2 ms slope of the EPSP at 11–20 min after the end of induction. The detailed experimental procedures are described previously (Froemke and Dan, 2002; Froemke et al., 2005, 2006).

Results

Development of the STDP model

To understand the mechanism of spike-timing detection in STDP, we developed a realistic biophysical model of synaptic plasticity within a single postsynaptic spine, on the basis of the literature and recent experimental findings on ion channels, intracellular signaling cascades, and receptor trafficking (Fig. 1 A–C) (supplemental information, http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf). In this model, prespiking and postspiking are inputs, and the change of synaptic conductance is the output. Prespiking represents glutamate release from the presynaptic terminal, which involves short-term depression of transmitter release probability during high-frequency bursts (>2 Hz) (Tsodyks and Markram, 1997; Matveev and Wang, 2000; Froemke et al., 2006). The postspiking signal is a large increase of membrane potential via bpAPs (Hoffman et al., 1997; Stuart and Hausser, 2001), which involves frequency-dependent attenuation (Froemke et al., 2006). We used 1 Hz prespiking for 100 s over a range of constant membrane potentials, which corresponds to the postspiking signal (Artola et al., 1990; Feldman, 2000; Lisman, 2001; Stiefel et al., 2005), as test stimuli to constrain the electrophysiological and biochemical dynamics of the STDP model (Fig. 2).

Figure 2.

Voltage-dependent changes of synaptic strength. A–E, Time courses of Ca²⁺ concentration (A), CaMKII activity (B), PP1 activity (C), PKA activity (D), and CaN activity (E) at PSD in response to 100 prespikes at 1 Hz with the indicated voltage clamp of somatic membrane potential during the stimulation. Colors correspond to the voltage as indicated in E. Arrowheads indicate the onset of stimulation. The inset in A shows its time courses by a single prespiking. F, Voltage-dependent Ca²⁺ concentration at PSD. The black and gray lines indicate the mean and maximal amplitude of Ca²⁺ concentration, respectively. G, Voltage-dependent CaN, PKA, PP1, and CaMKII activities. Integrated PP1 activity (red; normalized with the maximal activity, 0.33), integrated PKA activity (dark blue; normalized by 3.68), integrated CaN activity (green; normalized by 1.75), and final CaMKII activity (light blue) against the indicated membrane potentials are plotted. H, Time courses of changes of synaptic conductance. I, Voltage-dependent changes of synaptic conductance. Data were taken at 2 and 60 min after the onset of stimulation (Obs 1 and Obs 2, respectively) as indicated in H.

At membrane potentials positive from −45 mV, the Mg²⁺ block of NMDARs was removed, and the high Ca²⁺ influx via NMDARs (Fig. 2A,F) produced sufficient Ca²⁺ · CaM. This then bound to each subunit of a holoenzyme of CaMKII and induced successive autophosphorylation of CaMKII subunits, which led to the persistent activation of CaMKII (Fig. 2B,G) (Lisman et al., 2002). The persistent CaMKII activation required the inhibition of PP1 activity (Fig. 2C,G), which dephosphorylates CaMKII. The high Ca²⁺ influx also activated PKA via AC1/8 (Fig. 2D,G) and CaN (Fig. 2E,G). PP1 was inactivated by phosphorylated I-1, the dephosphorylation and phosphorylation of which depend on CaN and PKA, respectively. The high Ca²⁺ influx inhibited PP1 via phosphorylated I-1, because its phosphorylation by PKA counteracts with dephosphorylation by CaN (Figs. 1B, 2D,E,G). Therefore, PKA activation is necessary to trigger the persistent CaMKII activation via inhibition of PP1, which is consistent with the observation that PKA acts as a gate signal for LTP induction (Blitzer et al., 1998).

In contrast, at hyperpolarized membrane potentials such as −85 mV, the low Ca²⁺ influx (Fig. 2A,F) also produced Ca²⁺ · CaM; however, this activated CaN without sufficient activation of PKA and CaMKII (Fig. 2G), and the activated CaN dephosphorylated I-1 and activated PP1 (Fig. 2C,E,G) (Winder and Sweatt, 2001). Therefore, the low Ca²⁺ influx specifically induced activation of protein phosphatases rather than of protein kinases, consistent with previous observations (Bear and Malenka, 1994). The sensitivities to Ca²⁺ influx between protein phosphatases and kinases were attributable to the distinct sensitivities of CaN and PKA to Ca²⁺ · CaM (Fig. 2G). However, in contrast to CaMKII activation, PP1 activation was transient and returned to basal levels after stimulation (Fig. 2C).

The specific activation of signaling molecules regulated trafficking of AMPARs and thus regulated AMPAR synaptic conductance (Figs. 1C, 2H) (Malinow and Malenka, 2002; Collingridge et al., 2004; Seol et al., 2007). Persistent activation of CaMKII phosphorylated PDZs, including synapse-associated protein 97 and stargazin. The phosphorylation facilitated AMPAR trapping by cytoskeletal scaffolds (scaffolds) at PSD (Fig. 1C) and resulted in a persistent increase of AMPARs at PSD and of conductance, resulting in expression of LTP (Fig. 2H). Note that the persistent CaMKII activation was the only stabilizing mechanism for LTP and that no other stabilizing mechanisms, such as protein synthesis and AMPAR clustering, were modeled in this study. In contrast, both PP1 and CaN dephosphorylated PDZs and GluR1 at Ser845 (a subunit of AMPARs). Dephosphorylation of PDZs and GluR1 at Ser845 resulted in the dissociation of AMPARs from PSD and facilitated the internalization of AMPARs into the endosome, respectively (Fig. 1C). The internalized AMPARs were maintained in the dephosphorylated state by the action of PP1 and CaN, which prevented the AMPARs from being reinserted, and resulted in a decrease of conductance through AMPAR channels (Fig. 2H). However, because the activation of CaN and PP1 was transient, the number of AMPARs within the PSD and, correspondingly, the total prespike-evoked synaptic conductance eventually returned to the basal level. This result indicates that the stabilization of LTD cannot be explained with the experimentally observed time constants of AMPAR trafficking and suggests the presence of an unidentified pathway or pathways that are responsible for the maintenance of LTD. The mechanisms of persistency for LTD remain unknown; however, the saturation-dependent trafficking of AMPARs (Hayer and Bhalla, 2005) or the clustering of interacting receptors (Shouval, 2005) may be possible mechanisms. Although depression was not persistent, short-term, voltage-dependent changes of synaptic strength in our model were highly correlated with depolarization-dependent LTP and hyperpolarization-dependent LTD observed experimentally (Fig. 2I) (Artola et al., 1990; Stiefel et al., 2005). Similarly, we experimentally found that, by use of pharmacological manipulation, high, moderate, and low NMDAR activities are correlated with potentiation, no change (neither potentiation nor depression), and depression, respectively (supplemental Fig. 1, available at www.jneurosci.org as supplemental material). Because the NMDAR activity depends on the membrane potential, this result is consistent with synaptic plasticity induced by direct depolarization (Malenka and Bear, 2004). Therefore, we confirmed that our STDP model is, in principle, consistent with membrane potential-dependent synaptic plasticity and with related molecular events.

Prediction of the allosteric kinetics of NMDARs

We next explored whether the experimentally determined dynamics of STDP can be reproduced in our biophysical model. We used a brief current injection to the somatic compartment to trigger postspiking and bpAPs within the spine. Neither prespiking nor postspiking with 1 Hz for 100 s alone induced synaptic plasticity (see Fig. 4E) (Johnston et al., 2003). Induction of LTP by prespiking → postspiking is thought to be triggered by accelerated Ca²⁺ influx via NMDARs and VGCCs, which depends on the depolarization-induced removal of the Mg²⁺ block of NMDARs and supralinear summation of EPSPs and bpAPs (Hoffman et al., 1997; Stuart and Hausser, 2001; Kampa et al., 2004). Induction of LTD by postspiking → prespiking has been shown recently to be mediated by depolarization-dependent suppression of NMDARs (Froemke et al., 2005), which may lead to reduced Ca²⁺ influx through NMDARs and VGCCs. Moreover, the direct binding of Ca²⁺ · CaM to NMDARs reduces mean open time and leads to an overall reduction of open probability, resulting in suppression of NMDARs (Ehlers et al., 1996; Rycroft and Gibb, 2002, 2004b). Therefore, we first examined whether accelerated or suppressed Ca²⁺ influx can be induced by prespiking → postspiking or postspiking → prespiking via NMDARs, respectively (Fig. 3). Prespiking → postspiking led to accelerated Ca²⁺ influx via bpAP-dependent removal of the Mg²⁺ block of NMDARs (Fig. 3A,B), which led to induction of LTP (supplemental Fig. 2, available at www.jneurosci.org as supplemental material). However, postspiking → prespiking did not suppress Ca²⁺ influx (Fig. 3A,B), in contrast to the experimental evidence (Froemke et al., 2005), and did not induce LTD (supplemental Fig. 2, available at www.jneurosci.org as supplemental material). This result indicates the existence of an unknown mechanism for suppression of Ca²⁺ influx.

Figure 4.

Prediction of the allosteric kinetics of NMDARs. A, Schematic (top) and detailed mechanism (bottom) of the allosteric kinetics of NMDARs. τ₁ and τ₂ were the time constants of Ca²⁺ · CaM binding to glutamate (Glu)-unbound and Glu-bound NMDARs, respectively, where Ca²⁺ · CaM concentration is 1 μm. Ca²⁺ · CaM bound and Ca²⁺ · CaM unbound NMDARs, which bind to Glu, show the low and high open probability, respectively (Ehlers et al., 1996; Rycroft and Gibb, 2002, 2004b). C₀, C₁, C₂, C₃, C₀ · CaM, C₁ · CaM, C₂ · CaM, and C₃ · CaM are the concentrations of closed-state NMDARs, and O and O · CaM are those of open-state NMDARs (bottom). CaM represents 2 Ca²⁺ · CaM or 3 Ca²⁺ · CaM, and the numerals on arrows are rate constants in μm⁻¹s⁻¹ or s⁻¹, which were set on the basis of the experimental observation (bottom). The complete description of the allosteric kinetics of NMDARs is provided on-line (available at http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf). B, Time courses of spike timing-dependent [Ca²⁺]_PSD by a single pairing of prespiking and postspiking in the allosteric model with the indicated spike timing in C. C, Spike timing-dependent [Ca²⁺]_PSD in the allosteric model with the indicated τ₁ and τ₂. D, Time courses of spike timing-dependent synaptic conductance in the allosteric model. One hundred pairings of the prespiking and postspiking with 1 Hz were given. E, Spike timing-dependent synaptic conductance in the allosteric model at 2 min (Obs 1) and 60 min (Obs 2) after the onset of stimulation (left). The apparent time constants for LTP and LTD at 2 min were 36.1 and 53.0 ms, respectively. The synaptic conductance with 100 prespikes and postspikes alone at 2 min (black) and 60 min (gray) after the onset of stimulation is also plotted (right, bars). F, Integrated PKA activity (normalized by 3.06) and final CaMKII activity. G, Integrated PP1 activity (normalized by 0.34) and integrated CaN activity (normalized by 1.53).

Figure 3.

Requirement of the distinct suppression of NMDARs. A, Time courses of spike timing-dependent [Ca²⁺]_PSD by a single pairing of prespiking and postspiking in the STDP model (the no-allosteric model) with the indicated spike timing in B; 1, prespiking → postspiking (T_post − T_pre = +128 ms; black); 2, prespiking → postspiking (+16 ms; red); 3, postspiking → prespiking (−32 ms; blue). B, Spike timing-dependent Ca²⁺ increase in the no-allosteric model. The mean and maximum [Ca²⁺]_PSD are indicated by black and gray, respectively. The mean denotes the integration of Ca²⁺ induced by a single pairing of prespiking and postspiking divided by 1 s. T_post was defined as the time of 5 ms after the onset of current injection represented by an α-function (500 pA, 0.5 ms), and T_pre was defined as the time of the onset of glutamate release. The positive (red) and negative (blue) intervals correspond to the prespike → postspike and postspike → prespike timing, respectively. C, Spike timing-dependent suppression of NMDARs under the blockage of Ca²⁺ influx via NMDARs. Expected Ca²⁺ influx, if the NMDARs open, are plotted with the indicated time constant of Ca²⁺ · CaM binding to NMDARs, τ. Ca²⁺ influx via NMDARs was normalized by unsuppressed Ca²⁺ influx (C–E). D, Ca²⁺ influx via NMDARs induced by a prespiking alone with the indicated τ. The inset shows the time courses of the total NMDAR-mediated Ca²⁺ influx. E, Spike timing-dependent Ca²⁺ influx via NMDARs.

The failure to suppress Ca²⁺ influx by postspiking → prespiking led us to examine the mechanism of suppression of NMDARs by Ca²⁺ · CaM. We changed the time constant of Ca²⁺ · CaM binding to NMDARs under the blockage of NMDAR-dependent Ca²⁺ influx and examined the suppression of NMDARs solely by VGCC-dependent Ca²⁺ influx, which is triggered by postspiking (Fig. 3C). NMDARs were suppressed by VGCC-dependent Ca²⁺ influx with the rapid time constant (2.3 ms) but not with the slow time constant (230 ms). This indicates that suppression of NMDARs by VGCC-dependent Ca²⁺ influx, which may reflect the time window for LTD induction with postspike → prespike timing (Froemke et al., 2005), requires a rapid binding of Ca²⁺ · CaM to NMDARs. We also examined suppression of NMDARs by NMDAR-dependent Ca²⁺ influx induced by prespiking alone (Fig. 3D). Ca²⁺ influx was gradually reduced as the time constant of Ca²⁺ · CaM binding to NMDARs became rapid because of the self-suppression of NMDARs by Ca²⁺ influx through themselves. This indicates that the slow binding of Ca²⁺ · CaM to NMDARs is required to avoid self-suppression of NMDARs. We examined the suppression of NMDARs by both NMDAR- and VGCC-dependent Ca²⁺ influx and found that suppression of NMDARs did not change regardless of spike timing with any time constant and resulted in a failure of spike-timing detection (Fig. 3E). This result indicates that, with postspike → prespike timing, a rapid binding of Ca²⁺ · CaM to NMDARs is required to suppress NMDARs by VGCC-dependent Ca²⁺ influx, whereas with prespike → postspike timing, a slow binding of Ca²⁺ · CaM to NMDARs is required to avoid the rapid self-suppression of NMDARs.

This requirement of distinct suppression of NMDARs in a spike timing-dependent manner led us to predict an allosteric kinetics of NMDARs: rapid binding of Ca²⁺ · CaM to glutamate-unbound NMDARs and slow binding of Ca²⁺ · CaM to glutamate-bound NMDARs (Fig. 4A) (supplemental infor-mation, available at http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf). For simplicity, we changed the association and dissociation time constants without changing the affinity of Ca²⁺ · CaM to NMDARs. Hereafter, the STDP model (with or without the allosteric kinetics of NMDARs) denotes the allosteric or no-allosteric model, respectively. The allosteric model led to suppression of Ca²⁺ influx with postspike → prespike timing and accelerated Ca²⁺ influx with prespike → postspike timing (Fig. 4B,C), resulting in STDP (Fig. 4D,E), although any no-allosteric model with rapid or slow binding of Ca²⁺ · CaM to NMDARs did not produce the appropriate Ca²⁺ influx and subsequent STDP (supplemental Fig. 2, available at www.jneurosci.org as supplemental material). Note that the curve in LTP at 2 min was graded, but it became bistable at 60 min. Such slow increase is one of the unusual properties of spike timing-dependent LTP observed in experiments (Bi and Rubin, 2005).

In the allosteric model, accelerated Ca²⁺ influx with prespike → postspike timing specifically induced CaMKII and PKA activation and PP1 suppression (Fig. 4F,G), which induced LTP (Fig. 4D,E). Persistent CaMKII activation requires a suprathreshold Ca²⁺ influx because of the nature of its autophosphorylation loop, so the high Ca²⁺ influx, rather than the low and sustained Ca²⁺ influx, preferentially triggers CaMKII activation and subsequent LTP induction (Fig. 4C,E,F) (supplemental Fig. 2, available at www.jneurosci.org as supplemental material). In contrast, suppressed Ca²⁺ influx with postspike → prespike timing specifically induced PP1 activation without PKA activation (Fig. 4C,F,G), which induced LTD (Fig. 4D,E). Because PKA was not fully activated by the suppressed Ca²⁺ influx (Fig. 2G), PP1 activity was not suppressed and exhibited the highest activity at the low Ca²⁺ level (Fig. 2G). Also, because the PP1 activation has several steps from Ca²⁺, the PP1 activation requires enough duration of Ca²⁺. Therefore, the low and sustained Ca²⁺ influx, rather than the high and transient Ca²⁺ influx, preferentially induces LTD (Fig. 4C,E,G; supplemental Fig. 2, available at www.jneurosci.org as supplemental material). These results indicate that the allosteric kinetics of NMDARs can reliably detect prespike and postspike timing, which lead to STDP.

Features of the allosteric kinetics of NMDARs

To test the prediction of the allosteric kinetics of NMDARs, we measured the NMDAR-mediated EPSPs (NMDAR-EPSPs), which should directly show the effect of the allosteric kinetics, under the conditions where any component of AMPAR-EPSPs was not involved (Fig. 5A). We compared the amplitudes and time of NMDAR-EPSPs between the models and the experiments (Fig. 5B). In the allosteric model, the peak amplitudes of the NMDAR-EPSPs decreased with postspike → prespike timing (Fig. 5C, top). The peak amplitudes of the NMDAR-EPSPs in the no-allosteric model with rapid suppression (τ = 2.3 ms) also slightly decreased; however, those with slower suppression (τ = 230 ms) did not decrease with postspike → prespike timing (Fig. 5C, top). Experimentally, the peak amplitudes of the NMDAR-EPSPs also decreased (Fig. 5C, bottom). This experimental result is consistent with the results of both the allosteric model and the no-allosteric model with rapid suppression. We then compared the peak time of NMDAR-EPSPs (Fig. 5D). In the allosteric model and the no-allosteric model with slow suppression, the peak time of the NMDAR-EPSPs remained constant regardless of the spike timing, whereas the peak time of the NMDAR-EPSPs in the no-allosteric model with rapid suppression was delayed with postspike → prespike timing, because the decay of NMDAR-EPSPs is suppressed by Ca²⁺ influx via NMDARs themselves (Fig. 5D, top). In the experiments, the peak time of the NMDAR-EPSPs was not delayed and remained constant (Fig. 5D, bottom). This experimental result is consistent with the results in the allosteric model and in the no-allosteric model with slow suppression. We further examined the amplitude at the decay phase (decay amplitude), because the effect of the allosteric kinetics can appear more clearly during the decay phase. The decay amplitudes of the NMDAR-EPSPs in the allosteric model decreased; however, those in the no-allosteric models did not decrease with postspike → prespike timing (Fig. 5E, top). Experimentally, the decay amplitudes of the NMDAR-EPSPs also decreased (Fig. 5E, bottom). This experimental result is consistent with the result of the allosteric model but not with that of the no-allosteric models. We confirmed that the allosteric model gives a better fit to the experimental data than the no-allosteric models by measuring S(θ, Σ), a sum of variance and covariance of mean squared errors between experiment and simulation in peak amplitude, peak time, and decay amplitude, normalized by the best-fit simulation (Fig. 5F). As S(θ, Σ) becomes closer to 1, the model fit improves. S(θ, Σ) of the allosteric model was closer to 1 than that of the no-allosteric models, regardless of the time constant. This indicates that the allosteric model gives a much better fit to the experimental data than the no-allosteric models. Therefore, only the allosteric model appears to be consistent with the experimental properties of NMDAR-EPSPs.

Figure 5.

Features of the allosteric kinetics in NMDAR-EPSP. A, Derivation of NMDAR-EPSPs by postspiking → prespiking. NMDAR-EPSPs (right) was derived by the subtraction of AP alone from the NMDAR-EPSP plus AP in the absence of AMPAR activation (left, solid line) in presence of CNQX, a specific AMPAR antagonist (Froemke et al., 2005) (left, dashed line). B, Time courses of NMDAR-EPSPs by postspiking → prespiking in experiment (T_post − T_pre = −10 ms, red; prespike-only as control, gray), in the no-allosteric model (−1, −2, −4,…, −256 ms) with rapid (τ₁ = τ₂ = 2.3 ms) and slow (τ₁ = τ₂ = 230 ms) suppression, and in the allosteric model (−1, −2, −4,…, −256 ms; τ₁ = 2.3 ms and τ₂ = 230 ms), as indicated. C–E, Peak amplitude (C), peak time (D), and decay amplitude (E) of NMDAR-EPSPs in the models (top) and the experiments (bottom). The responses were normalized by the control responses. The decay amplitude was taken at the time when three times longer than the largest peak time in each set. The responses of the no-allosteric model with rapid and slow suppression (blue and green, respectively) and the allosteric model (red) are plotted (top). The timing-dependent NMDAR-EPSPs in the experiments were fitted by a single exponential curve using the least square method (bottom). The apparent time constants of the responses of NMDAR-EPSPs versus the spike timing, τ_app, are indicated. F, Evaluation of goodness of fit of the no-allosteric and allosteric models with the experimental data. The time constant in the no-allosteric model was changed as indicated, and the measure S(θ, Σ) was plotted. S(θ, Σ) is a sum of variance and covariance of errors between experiment and simulation in peak amplitude, peak time, and decay amplitude, normalized by variance and covariance of residuals in the best-fit model. S(θ, Σ) is given by , where f (x_i, θ) is the fitting function, and θ is the parameter vector, Σ is the variance–covariance matrix of residuals of the best-fit model (τ₁ = 2.3 ms and τ₂ = 2.3 × 10⁴ ms), Y_i is a column vector of the peak amplitude, peak time, and decay amplitude of ith NMDAR-EPSP, and x_i is T_post − T_pre of the EPSP (Gallant, 1987). Note that the closer to 1 the S(θ, Σ) becomes, the better the model fits the experimental results. Arrowheads denote the time constants used in the no-allosteric and allosteric models. G, Goodness of fit against various time constant of Ca²⁺ · CaM binding to NMDARs, τ₂, in the allosteric model. The τ₂ was changed as indicated, and the S(θ, Σ) was measured. The arrowhead indicates τ₂ (230 ms) used in the allosteric model.

We also examined the robustness of the allosteric kinetics of NMDARs by measuring of goodness of fit against various time constants of Ca²⁺ · CaM binding to NMDARs (Fig. 5G). We fixed τ₁ as 2.3 ms from the experimental NMDAR suppression (Fig. 3C) (Froemke et al., 2005) and changed τ₂ as indicated. We found that the allosteric kinetics can fit the NMDAR-EPSPs well in a wide range over 230 ms (100-fold difference between the time constants τ₁ and τ₂). In general, allosteric kinetics often accompanies a huge difference in time constants. For well-known allosteric kinetics of Ca²⁺ · CaM binding to nonphosphorylated and phosphorylated CaMKII, the difference in time con-stants is ∼1000-fold (supplemental information, available at http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf) (Meyer et al., 1992). Compared with such allosteric kinetics, a difference in time constants in the allosteric model of NMDARs (100-fold) is likely to be reasonable and physiological.

Complex spike train coding by the allosteric kinetics

We further tested the allosteric model with more complex spike patterns, shown previously to also induce STDP (Froemke and Dan, 2002; Wang et al., 2005; Froemke et al., 2006). We found that the allosteric model correctly predicted the direction of synaptic plasticity in spike triplet experiments (Fig. 6A,B) (supplemental Fig. 3, available at www.jneurosci.org as supplemental material) (Froemke and Dan, 2002), which indicated that the allosteric model can be extrapolated over STDP induced with more complex spike trains. We also validated the allosteric model with spike quadruplet experiments (Fig. 6C) (Froemke and Dan, 2002). We found a high correlation of the direction of synaptic plasticity between the allosteric model and the experiments (Fig. 6C) (Froemke and Dan, 2002). Finally, we validated the allosteric model using paired spike bursts instead of paired prespikes and postspikes (Fig. 7) (Froemke et al., 2006). The presynaptic burst after a postspike induced LTD at a similar extent, regardless of numbers of prespikes both in simulation and experiment (Fig. 7A). This is because of decreased probability of glutamate release during high-frequency stimulation (Tsodyks and Markram, 1997; Matveev and Wang, 2000) and of saturation of glutamate binding to NMDARs by a single prespike. The presynaptic burst before a postspike induced LTP, but the amplitude of LTP decreased as the number of prespikes increased (Fig. 7B). This is because a burst of five presynaptic spikes induced glutamate release mostly during the first or second spiking in the burst resulting from short-term depression; thereby the effective interval between prespiking and postspiking was prolonged, resulting in a reduction of the amplitude of LTP. The amplitudes of LTP with one or two prespikes in experiment are higher than those in the allosteric model because of the slower increase of LTP in the allosteric model; however, the amplitudes of LTP in experiment and the allosteric model became similar at longer periods (>60 min) (data not shown), which is in principle not inconsistent with the experimental data (Froemke et al., 2006). The postsynaptic burst with a single prespike induced a gradual shift from LTD to LTP as the number of postspikes increased both in experiment and in the allosteric model (Fig. 7C). The increase of the number of postspikes induced repetitive removal of the Mg²⁺ block from NMDARs, and this action overcame the suppression of NMDARs by Ca²⁺ · CaM, resulting in a gradual shift from LTD to LTP (Fig. 7C). The burst frequency-dependent synaptic plasticity was also similar between the experiments and the allosteric model (Fig. 7D) (Froemke et al., 2006). This result is consistent with the frequency-dependent synaptic plasticity at other synapses (Sjostrom et al., 2001). Therefore, the allosteric model appeared to be consistent with a wide range of complex spike train-dependent synaptic plasticity in layers II/III of visual cortex.

Figure 6.

Complex spike timing-dependent synaptic plasticity. A, B, Changes of synaptic strength by spike triplets in the allosteric model (A) and in the experiments (B) (Froemke and Dan, 2002) (reproduced with permission from Nature). One prespiking and two postspiking (left), where t₁ indicates the relative timing of the first postspiking to prespiking (T_{post 1} − T_pre), and t₂ indicates the relative timing of the second postspiking to prespiking (T_{post 2} − T_pre). Two prespiking and one postspiking (right), where t₁ = T_post − T_{pre 1} (T_{pre 1}, first prespiking) and t₂ = T_post − T_{pre 2} (T_{pre 2}, second prespiking). Data in A were taken at 2 min after onsets of stimulation and averaged across three trials with different profiles of glutamate release based on presynaptic short-term depression (SEM <0.26%). C, Changes of synaptic strength by spike quadruplets. Synaptic conductance in the allosteric model versus correspondence in the experiments was plotted (Froemke et al., 2005). Open symbols, Prespiking → postspiking → postspiking → prespiking; filled symbols, postspiking → prespiking → prespiking → postspiking. Simulation data were averaged across three trials (SEM <0.38%).

Figure 7.

Burst-dependent synaptic plasticity. A, B, Changes of synaptic strength by presynaptic bursts with a single postsynaptic spike. The presynaptic spikes (with the indicated numbers) with 100 Hz after (A) or before (B) a postsynaptic spike was given at 1 Hz for 100 times. The interval between the postspiking and the nearest prespiking was 6 ms. Closed circles indicate synaptic conductance in the allosteric model (average of three trials; SEM <0.21%), and open circles indicate the correspondence in experiment (mean ± SEM) (Froemke et al., 2006) (A–D). C, Changes of synaptic strength by postsynaptic bursts with a single presynaptic spike. The postspikes (with the indicated numbers) with 100 Hz together with a prespike were given at 1 Hz for 100 times, and synaptic conductance was plotted. The interval between the prespiking and the first postspiking was 6 ms. D, Burst frequency-dependent synaptic plasticity. Five pairings of prespiking → postspiking (circles) and postspiking → prespiking (squares) with the indicated frequencies were given at 1 Hz for 100 times, and synaptic conductance was plotted (average of three trials; SEM <0.26%).

Requirement of postsynaptic NMDARs for the induction of LTD

We further compared the role of NMDARs in the induction of LTD between the allosteric model and the experimental results. Based on experiments, two NMDAR-dependent scenarios have been proposed for the induction of LTD by spike pairing. These two scenarios differ on the presumed synaptic location of NMDARs: one depends on activation of presynaptic NMDA autoreceptors (Sjostrom et al., 2003; Bender et al., 2006), and the other depends on postsynaptic NMDARs (Froemke et al., 2005; Seol et al., 2007). Therefore, we experimentally examined the requirement of postsynaptic NMDARs for the induction of LTD.

Through the recording electrodes, we loaded postsynaptic neurons with (+)-5-methyl-10,11-dihydro-5H-dibenzo [a,d] cyclohepten-5,10-imine maleate (MK-801) (1 mm), an NMDAR antagonist, and examined whether LTD could be induced (Fig. 8). Postspike → prespike pairing did not induce LTD in cells filled with MK-801 (change in EPSP slope, 5.6 ± 9.5%; n = 3; p < 0.03) but induced LTD in nearby cells (<30 μm; change in EPSP slope; −35.9 ± 9.0%), which were recorded at the same time, indicating that injection of MK-801 fully blocked the induction of LTD. We confirmed that intracellular MK-801 (1 mm) blocked NMDARs to the same degree as bath-applied AP-5 (50 μm); intracellular MK-801 reduced NMDAR-EPSP amplitude by −76.9 ± 5.5% (n = 5), whereas extracellular APV reduced NMDAR-EPSP amplitude by −80.2 ± 4.8% (n = 4; p > 0.6, compared with intracellular MK-801) in the presence of CNQX (20 μm) to block AMPARs and picrotoxin (10 μm) to block GABA_A receptors. These results indicate that postsynaptic NMDARs are required for the induction of LTD and support our hypothesis that postsynaptic NMDARs are responsible for the induction of LTD at this synapse.

Figure 8.

Requirement of postsynaptic NMDARs for the induction of LTD in STDP. Intracellular MK-801 (1 mm) prevented pairing-induced LTD. Examples of data from two simultaneously recorded neurons in the same slice are shown. Top, Recording with MK-801 in the internal solution. Bottom, Recording from a nearby neuron (∼25 μm away). At the time indicated by the arrow, postspike → prespike pairing [T_post − T_pre = −9.6 ms (top), −10.3 ms (bottom)] was repeated. Red and blue lines indicate increase (11.8%; p > 0.2) and decrease (−44.8%; p < 10⁻¹⁰) in synaptic strength. EPSP slope is shown in the top panel, and input resistance (R_i) is shown in the bottom panel.

Spike timing-dependent Ca²⁺ influx

Recent Ca²⁺ imaging experiments at a different synapse revealed that prespiking → postspiking triggered accelerated Ca²⁺ influx, whereas postspiking → prespiking did not show significant suppressed Ca²⁺ influx (Nevian and Sakmann, 2004). To investigate whether our model could reproduce this experimental result, we included in our simulation the presence of a Ca²⁺ indicator in the allosteric model (Fig. 9A). We found that suppression of Ca²⁺ influx (which was seen in the absence of the Ca²⁺ indicator) (Fig. 4B,C) was not triggered by postspiking → prespiking in the presence of the Ca²⁺ indicator. This is because rapid absorption of free Ca²⁺ by the Ca²⁺ indicator results in the reduction of Ca²⁺ binding to CaM, and the suppression of NMDARs by Ca²⁺ · CaM was not consequently triggered, resulting in disappearance of suppressed Ca²⁺ influx by postspiking → prespiking.

Figure 9.

Spike timing-dependent Ca²⁺ influx. A, Time courses of Ca²⁺-bound Ca²⁺ indicator in cytosol (Conc, 100 μm; Kd, 580 nm) by prespiking → postspiking (T_post − T_pre = 4 ms; left) and postspiking → prespiking (−4 ms; center). Time courses of Ca²⁺-bound Ca²⁺ indicator by a prespike alone were overlaid. Maximum of Ca²⁺-bound Ca²⁺ indicator by prespiking, prespiking → postspiking and postspiking → prespiking are shown (right). B, Time courses of [Ca²⁺]_PSD and [Ca²⁺]_cytosol by prespiking → postspiking with the indicated timing.

In addition, Ca²⁺ influx via NMDARs in the allosteric model seems to be far higher than Ca²⁺ influx via VGCCs, although some Ca²⁺-imaging experiments have shown that Ca²⁺ influx via NMDARs is comparable with Ca²⁺ influx via VGCCs (Koester and Sakmann, 1998; Nevian and Sakmann, 2004) (Figs. 3A, 4B). This is simply because the observation site in the allosteric model was a PSD compartment, where NMDARs were highly concentrated, and the amplitudes of Ca²⁺ influx from NMDARs and VGCCs became comparable in the cytosol (Fig. 9B). Therefore, the ratio of Ca²⁺ influx via NMDARs and VGCCs is also consistent with the experimental results.

Extraction of an essential rule of STDP

We extracted a simple and essential learning rule by reduction of the allosteric model. The principle of Ca²⁺ influx in the allosteric model is that, with prespike → postspike timing, spiking prevents the suppression of NMDARs by Ca²⁺ · CaM via the allosteric kinetics of NMDARs, and postspiking triggers the removal of the Mg²⁺ block of NMDARs by depolarization of the postsynaptic membrane potential, resulting in accelerated Ca²⁺ influx and induction of LTP (Fig. 10A, left). In contrast, with postspike → prespike timing, postspiking triggers the suppression of NMDARs by Ca²⁺ · CaM via the allosteric kinetics of NMDARs, resulting in suppressed Ca²⁺ influx and induction of LTD (Fig. 10A, right). Note that increase of membrane potential by postspiking does not affect NMDARs with postspike → prespike timing, because membrane potential returns to the resting potential when prespiking arrives. Thus, postspiking positively and negatively regulates NMDAR activation via increase of membrane potential and suppression by Ca²⁺ · CaM, respectively. Based on this principle of the allosteric model, we write the simple STDP model as follows:

where NMDAR(t), V(t), and Ca(t) represent NMDAR activity, membrane potential, and Ca²⁺ · CaM concentration, respectively; τ_NMDAR, τ_V, and τ_Ca are the decay time constants of NMDAR(t), V(t), and Ca(t), respectively; V_rest is the resting membrane potential; AP is the amplitude of the backpropagating action potential; Ca_VGCC is the Ca²⁺ influx via VGCC; K_Ca is the effective Ca that gives the 50% inhibition of the maximal activity of NMDAR; t_pre and t_post are the timings of the prespiking and postspiking, respectively; (Ca)^max is the maximal value of Ca by a single pairing of prespiking and postspiking; θ_LTP and θ_LTD denote the thresholds of Ca for LTP and LTD, respectively; and A_LTP, A_LTD, and A_LTP/LTD correspond to the amplitudes of LTP, LTD, and consequent synaptic strength, respectively. We set τ_NMDAR = 40 ms, τ_V = 6 ms, τ_Ca = 20 ms, V_rest = −65 mV, AP = 40 mV, Ca_VGCC = 1.3, K_Ca = 0.3, θ_LTP = 6.2, θ_LTD = 4, A_LTP = 40, and A_LTD = 20.

Figure 10.

Simple STDP model. A, The principle of STDP. Postspiking positively regulates NMDARs (NMDAR) via increase of membrane potential (V) with prespiking → postspike timing (left, solid line), resulting in induction of LTP (left). With this timing, Ca²⁺ · CaM (Ca) does not affect NMDAR via the allosteric kinetics (left, dashed line). Postspiking negatively regulates NMDAR via Ca with postspike → prespike timing (right, solid line), resulting in induction of LTD (right). With this timing, V does not effectively affect NMDAR, because V returns to the basal level when a prespike arrives (right, dashed line). B, NMDAR(t), V(t), and Ca(t) by a single pairing of prespiking and postspiking with the indicated timing (Eqs. 1–3). The dashed line in NMDAR (top, right) indicates NMDAR with prespiking alone, and the dashed lines in Ca (bottom) indicate a linear sum of Ca induced by a prespike alone and a postspike alone with the indicated timing. C, Spike timing-dependent Ca (left) and synaptic strength (right). The high and low Ca induced by a single pairing of prespiking and postspiking with the indicated timing were converted into LTP (above the threshold, θ_LTP) and LTD (below the threshold, θ_LTD), respectively (Eqs. 4, 5). D, Synaptic plasticity induced with spike triplets. The relative spike timing of triplets, t₁ and t₂, is the same as in Figure 6.

Equations 1 and 2 represent the decays of NMDAR activity and membrane potential, respectively. Equation 3 represents that Ca²⁺ increase depends on the voltage-dependent activation of NMDARs with an exponential decay, where (0.0223[V(t) − V_rest] + 0.5) denotes the voltage dependence of NMDARs, which is obtained by the linearization of the voltage dependence of NMDARs in the allosteric model around the resting membrane potential (supplemental information, available at http://www.kurodalab.org/info/STDP/Urakubo2008SI.pdf). In Equation 4, at t = t_post, postspiking triggers the increase of voltage via bpAP and Ca²⁺ increase via VGCC, whereas at t = t_pre, NMDAR is activated in a reciprocal manner of Ca. Equations 3 and 4 indicate that Ca acts on NMDAR only at t = t_pre but not later, which means that the suppression of NMDARs reflects Ca²⁺ level at prespiking but not after prespiking. Therefore, Equations 3 and 4 represent the allosteric kinetics of NMDARs. The difference in Equation 4 used when t = t_pre and t = t_post enables LTP induction with the prespike → postspike timing and LTD induction with the postspike → prespike timing. Equation 5 represents that the high Ca²⁺ influx results in LTP via CaMKII, and the low Ca²⁺ influx results in LTD via PP1.

The high and low Ca were induced with the prespike → postspike and the postspike → prespike timing, respectively (Fig. 10B). With the prespike → postspike timing, NMDAR is fully activated because of the absence of Ca at t = t_pre, whereas with the postspike → prespike timing, NMDAR is weakly activated because of the presence of Ca at t = t_pre (Fig. 10B) (Eq. 3). This dependence of NMDAR on Ca at t = t_pre clearly indicates that the intracellular Ca²⁺ concentration, when NMDARs are activated, carries the previous prespike and postspike-timing information. The simple STDP model was capable of reproducing the spike timing-dependent Ca (Fig. 10C, left) and consequent STDP (Fig. 10C, right). Note that the high and low Ca, but not moderate one, triggered LTP and LTD, respectively. The time window for LTP depends on the decay time constant of NMDARs, and that for LTD on the decay time constant of Ca²⁺ increase by VGCC, which are consistent with the allosteric model. Synaptic plasticity induced by spike triplets was coherently reproduced (Fig. 10D). These results were similar to those in the allosteric model (Figs. 4B–E, 6A,B), indicating that the simple STDP model retains the essential properties of the allosteric model. Thus, the bidirectional role of postspiking via the allosteric kinetics of NMDARs in STDP, rather than the detailed parameters and complicated molecular interactions, is the essential principle of STDP.

Discussion

We show that the simple allosteric kinetics of NMDARs codes complex spike-timing information into STDP. The allosteric kinetics of NMDARs depends on the different time constants of Ca²⁺ · CaM to glutamate-bound and glutamate-unbound NMDARs. In general, the allosteric kinetics accompanies the changes of affinity, and it is possible that the allosteric kinetics of NMDARs also accompanies the changes of Ca²⁺ · CaM affinity for glutamate-bound and glutamate-unbound NMDARs. Similar results were obtained if the affinity of glutamate for Ca²⁺ · CaM-unbound NMDARs were over four times higher than that for Ca²⁺ · CaM-bound NMDARs (data not shown), although the affinity of glutamate to NMDARs is not significantly affected by binding Ca²⁺ · CaM to NMDARs (Zhang et al., 1998). The principle of our prediction is that any similar allosteric kinetics of NMDARs that depends on the timing between prespiking and postspiking can account for STDP, even if STDP is Ca²⁺ · CaM independent. Taking into consideration that STDP has been experimentally observed across a wide range of preparations, a single molecular property, such as an allosteric kinetics of NMDARs, seems more reasonable to account for STDP, rather than other complex molecular events, such as phosphorylation/dephosphorylation, because the former is more robust against variable expression levels of molecules.

CaN is required to suppress NMDARs with postspiking → prespiking (Froemke et al., 2005). Phosphatase activity of CaN is not likely to detect such short timing simply because Ca²⁺-dependent activation of phosphatase activity of CaN requires a longer time constant (Klee et al., 1998). In addition, dephosphorylation of NMDARs by CaN or other related phosphatases was required to associate Ca²⁺ · CaM to the NMDARs (Rycroft and Gibb, 2004a). Therefore, CaN activity may be a prerequisite condition for detecting the postspiking → prespiking, such as dephosphorylation of NMDARs, and this is implicitly incorporated into NMDAR kinetics in the allosteric model.

Although particular detailed parameters of our model may be unsupported by experimental observations, our main conclusion in this study is the bidirectional role of postspiking based on the allosteric kinetics of NMDARs in STDP as shown by the model reduction, and this does not critically depend on the exact details of most of the parameters. We emphasize that the signaling cascades in the allosteric model should be regarded as a tentative description, rather than a complete description. For example, some signaling cascades in the allosteric model, such as PP1, are based on the previous reports in other types of synaptic plasticity such as LTD induced by prolonged low-frequency presynaptic stimulation (LFS) (Mulkey et al., 1993), but they have not been shown to be directly involved in STDP. Also, in the allosteric model, LTP induced by high-frequency presynaptic stimulation, but not LTD induced by prolonged LFS (typically 900 times at 1 Hz), could be reproduced (data not shown) (Bear and Malenka, 1994). This implies the possibility that a missing pathway, requiring a long duration of the stimulus (>10 min), may be responsible for LTD induced by prolonged LFS. Note that 100 prespikes at 1 Hz alone did not induce LTD both in simulation and experiment (Fig. 4E) (Mizuno et al., 2001; Johnston et al., 2003). Additionally, another class of STDP has been shown to involve a distinct form of LTD that is independent of postsynaptic NMDARs and may be presynaptically expressed (Sjostrom et al., 2003; Bender et al., 2006). The STDP in layer II/III of visual cortex is not consistent with doublet-dependent STDP (Wittenberg and Wang, 2006) and triplet-dependent STDP (Wang et al., 2005) in hippocampal CA1 neurons. This suggests the possibility of requirement of an additional pathway(s) for the STDP in hippocampal CA1 neurons. The extension of the model by incorporating the future progress will shed light on the general principles of distinct classes of STDP.

A recent study reported that NMDAR subtypes are selective for the direction of synaptic modification in STDP (Gerkin et al., 2007). Our results indicate that the kinetics of NMDARs should detect the interaction between prespiking and postspiking. Together, the selectivity of NMDAR subtypes for the direction of synaptic modification may be attributable to the subtype-dependent different kinetics of NMDARs in Ca²⁺ · CaM binding or in the voltage dependence, both of which can be considered part of the postspiking signal.

Several biophysical models of STDP have been proposed in which postsynaptic Ca²⁺ influx also serves as a spike-timing detector (Karmarkar and Buonomano, 2002; Shouval et al., 2002; Rubin et al., 2005). These models assume that the basal Ca²⁺ influx induced by uncorrelated prespiking and postspiking (less than −200 or >100 ms) is far lower than that for LTD and LTP, and the postspiking → prespiking induces moderate Ca²⁺ influx for LTD, whereas the prespiking → postspiking induces high Ca²⁺ influx for LTP. However, with this assumption, it is difficult to avoid unnecessary LTD induction by the prespiking → postspiking with a long interval, which has not usually been observed in experiments (Sjostrom and Nelson, 2002), although some modifications have been proposed to avoid this (Rubin et al., 2005; Shouval and Kalantzis, 2005). Also, a biophysical model has been proposed to explain synaptic plasticity induced by spike triplets (Shah et al., 2006). However, this model uses the unnecessary LTD induction by the prespiking → postspiking to explain synaptic plasticity induced by spike triplets. In contrast, in the allosteric model, we set that the prespiking → postspiking, the uncorrelated spiking, and the postspiking → prespiking induced the high, moderate, and low Ca²⁺ influx, which led to potentiation, no change (neither potentiation nor depression), and depression, respectively (Koester and Sakmann, 1998; Cho et al., 2001; Lisman, 2001; Froemke et al., 2005) (Fig. 10C) (supplemental Fig. 4, available at www.jneurosci.org as supplemental material). This makes the allosteric model free from the unnecessary induction of LTD by the prespiking → postspiking, regardless of timing interval, and the allosteric model can coherently explain synaptic plasticity induced by a variety of complex spike trains within a single framework. Additionally, we experimentally found that the partial blockage of NMDAR activity induces depression, and the facilitation of NMDAR activity with removal of Mg²⁺ induces potentiation (supplemental Fig. 1, available at www.jneurosci.org as supplemental material). The biphasic synaptic modification against NMDAR activity is consistent with the hypothesis of the three Ca²⁺ levels for synaptic plasticity, because Ca²⁺ influx dominantly depends on NMDARs. Together, the three Ca²⁺ levels for synaptic plasticity is substantially evident from both phenomenological experiments (synaptic plasticity induced by spike doublets, spike triplets, and more complex spike trains) and pharmacological experiments (supplemental Fig. 1, available at www.jneurosci.org as supplemental material).

Our allosteric model belongs to the one-coincidence detector model. A model of two-coincidence detectors for STDP has also been proposed (Karmarkar and Buonomano, 2002). In the model, voltage-dependent Mg²⁺ removal from NMDARs detects prespiking → postspiking for LTP, whereas metabotropic GluRs (mGluRs) and Ca²⁺ influx via VGCCs detects postspiking → prespiking for LTD. Recently, some experiments have revealed that such secondary coincidence detectors are involved in LTD of STDP at other synapses: mGluRs at synapses between layer IV and layer II/III neurons in the somatosensory cortex (Bender et al., 2006) and CB₁ receptors at synaptic connections between layer V pyramidal cells (Sjostrom et al., 2003). Taking our findings of requirement of rapid suppression by postspiking into consideration, these molecules should interact with a postspiking signal with the time constant on an order of a few ten milliseconds. Such rapid time constant may depend on the ion flux or changes of voltage, rather than phosphorylation/dephosphorylation processes or catalytic activities of downstream of mGluR1 and CB₁ receptors.

We analyzed the sensitivity of the amplitude of STDP against the conductance of AMPARs, because the amplitude of LTP in STDP has been reported to be dependent on the size of the initial EPSPs (Bi and Poo, 1998; Sjostrom et al., 2001), which depends on the AMPAR activity, and this issue is theoretically related to the boundary condition on synaptic plasticity, whether the changes of synaptic strength depend on the current synaptic weight or not (Song et al., 2000; van Rossum et al., 2000; Cateau and Fukai, 2003). The relative amplitudes of LTP induced by prespiking → postspiking remained the same against the changes of conductance of AMPARs (supplemental Fig. 5, available at www.jneurosci.org as supplemental material). This means that the amplitude of LTP is positively correlated with the size of the initial EPSPs, which is similar to the theoretical study of the multiplicative learning rule (van Rossum et al., 2000; Cateau and Fukai, 2003). However, additional experimental studies are necessary to conclude this issue.

In conclusion, the allosteric model appears to be consistent with specific features of STDP, such as spike timing-dependent NMDAR-EPSPs and spike doublet-, triplet-, quadruplet-, and burst-dependent synaptic plasticity within a single framework. This study provides a novel concept of the bidirectional role of postspiking in induction of synaptic plasticity by the allosteric kinetics of NMDARs.