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1Department of Mathematics, 2Center for the Neural Basis of Cognition, 3Center for Neuroscience at University of Pittsburgh, and 4Department of Neurobiology, University of Pittsburgh, Pittsburgh, Pennsylvania; and 5Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland
Submitted 5 August 2004; accepted in final form 6 December 2004
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ABSTRACT |
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INTRODUCTION |
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; Bi and Poo 1998; Debanne et al. 1998; Magee and Johnston 1997; Markram et al. 1997; Sjöström et al. 2001; Yao and Dan 2001; Zhang et al. 1998). In this spike-timing–dependent plasticity (STDP), if a presynaptic spike precedes a postsynaptic spike within a window of tens of milliseconds, the corresponding synapse potentiates. If a presynaptic spike arrives after a postsynaptic spike within a similar window, the synapse depresses. The details of the signaling mechanism that the synapse uses to detect the timing between pre- and postsynaptic spikes are not fully known (Bi 2002; Bi and Poo 2001; Senn 2002; Sjöström and Nelson 2002).
Calcium has long been suggested as a signaling agent for long-term potentiation (LTP) and depression (LTD) (Lisman 1989; Malenka et al. 1988; Yang et al. 1999) and more recently STDP (Bi and Poo 2001; Bi and Wang 2002; Sjöström and Nelson 2002). Calcium can enter dendrites through N-methyl-D-aspartate (NMDA) channels or local voltage-gated calcium channels (VGCCs). Although there is experimental and modeling support for postsynaptic calcium as a critical signal to initiate molecular mechanisms leading to synaptic plasticity, open questions remain. It is not clear whether a calcium-based mechanism is sufficient to explain all of the experimental data for STDP. It is also uncertain if such a mechanism at the same time accounts for the induction of classical LTP and LTD. Additionally, it is unknown what outcomes such a system would yield for multiple pre- and postsynaptic spikes. We use a combination of computational modeling and experiments to address these questions.
Some recent modeling efforts have considered postsynaptic calcium concentration levels as the signal to differentiate pre- and postsynaptic spike orderings and timings (Abarbanel et al. 2003; Karmarkar and Buonomano 2002; Shouval et al. 2002). In these models, high postsynaptic calcium levels lead to LTP and low calcium levels lead to LTD. Models using this hypothesis can capture certain of the phenomena of STDP and/or classical LTP/LTD. They predict, however, that when a presynaptic spike precedes a postsynaptic spike by a sufficiently long time interval, calcium levels drop low enough so that the synapse depresses. While one publication reports such a result (Nishiyama et al. 2000), other experiments do not see the effect. Additionally, there is evidence that this second LTD regimen at positive (pre-before-post) timing may be due to feedforward inhibition in the native circuitry (Aihara et al. 2003; Togashi et al. 2003).
In this paper, we show that, for any deterministic signaling system based only on postsynaptic calcium levels, this pre-before-post depression is unavoidable. Furthermore, a level-based system conflicts fundamentally with our experimental results using triplets of pre- and postsynaptic spikes, which indicate that when a presynaptic spike is followed by a postsynaptic spike that is followed by another presynaptic spike (pre-post-pre), there is no change in synaptic strength, whereas the complementary arrangement (post-pre-post) leads to LTP.
As an alternative, we present a model based on the detection of the postsynaptic calcium time course (Holthoff et al. 2002; Ismailov et al. 2004; Sabatini et al. 2002; Yang et al. 1999; Yuste et al. 1999) that can reproduce STDP outcomes for spike pairs, triplets, and quadruplets, as well as classical LTD and LTP results induced by repeated presynaptic stimulation (Cho et al. 2001; Dudek and Bear 1992; Wang and Wagner 1999). In particular, LTD depends on the width of the calcium signal, namely the durations and relative timings of periods that the calcium signal spends above certain thresholds, interacting nonlinearly. Our model postsynaptic cell is based on the experimentally derived CA1 pyramidal cell model of Poirazi et al. (2003), with synaptic receptor kinetics based on time courses found experimentally (Andrasfalvy and Magee 2001; Perouansky and Yaari 1993), and calibrated to match dendritic calcium profiles reported in the literature (Koester and Sakmann 1998; Murthy et al. 2000; Sabatini et al. 2002). Plasticity in the model is produced by distinct calcium-driven modules for LTD and LTP that interact to determine the final plasticity outcome (Wang et al. 2005). Furthermore, the LTD module includes a veto, through which appropriately timed periods of moderate calcium concentration can block LTD components. Plasticity outcomes are determined dynamically, through the evolution of differential equations that are driven by changes in calcium concentration. To our knowledge, this is the first computational model for STDP to fully use the calcium time course, although Abarbanel et al. (2003) have harnessed certain aspects of the calcium time course for reproducing experimental STDP results.
The specific model that we present here represents an example of a general set of components that seem to be necessary to reproduce experimental spike pair and triplet STDP results from hippocampal culture, using postsynaptic calcium dynamics. This is consistent with experimental findings that have uncovered postsynaptic components of STDP in hippocampal culture (Li et al. 2004; Wang et al. 2005); since specific presynaptic components (e.g., Sjöström et al. 2003) have not been implicated in this preparation, these are not included in the model. Our computational results indicate that any model that aims to reconcile the pair and triplet data using the postsynaptic calcium signal alone will require some amount of fine-tuning. Consistency with classical LTP/LTD further constrains any model. The inherent sensitivity of a calcium-based detection system may contribute both to the variability in plasticity outcomes induced by certain experimental protocols and to the disparity in results of triplet experiments in different experimental settings (Froemke and Dan 2002; Sjöström et al. 2001; Wang et al. 2005).
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METHODS |
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Low-density cultures of dissociated embryonic rat hippocampal neurons were prepared according to a previously described protocol with minor modifications (Bi and Poo 1998). Hippocampi were removed from E17–E19 embryonic rats and treated with trypsin for 20 min at 37°C, followed by washing and gentle trituration. The dissociated cells were plated on poly-L-lysine–coated glass coverslips in 35-mm petri dishes with 30,000–90,000 cells per dish. The culture medium was Dulbecco's modified Eagle’s medium (DMEM; BioWhittaker) supplemented with 10% heat-inactivated bovine calf serum (Hyclone), 10% Ham's F12 with glutamine (BioWhittaker), and 50 U/ml penicillin-streptomycin (Sigma). Twenty-four hours after plating, one-third of the culture medium was replaced by the same medium supplemented with 20 mM KCl. Both glial cells and neurons grow well under these culture conditions. The optimal period for these cultures to be used is 8–15 div (days in vitro), during which connections of 50–500 pA were commonly found. Very strong synapses (>500 pA) in these cultures express little LTP (Bi and Poo 1998) and were excluded from the study.
Simultaneous whole cell perforated-patch recordings from two glutamatergic neurons were carried out with patch-clamp amplifiers (Axopatch 200B, Multiclamp 700A, Axon Instruments) at room temperature. The pipette solution contained the following (in mM): 136.5 potassium gluconate, 17.5 KCl, 9 NaCl, 1 MgCl2, 10 HEPES, 0.2 EGTA, and 200 µg/ml amphotericin B (pH 7.3). The external bath solution was a HEPES-buffered saline (HBS) containing the following (in mM): 150 NaCl, 3 KCl, 3 CaCl2, 2 MgCl2, 10 HEPES, and 5 glucose (pH 7.3). Throughout the recording, the culture was perfused with fresh bath medium at a rate of 
2 ml/min, using an automated perfusion system (VC64, Warner Instrument) that allows for rapid change of solutions. The neurons were visualized with a phase-contrast inverted microscope (Leica DMIRB). Signals (filtered at 5 kHz) were acquired at a sampling rate of 10 kHz using a 16-bit digitizing board (DigiData1320, Axon Instruments or E6035, National Instruments) interfaced with either pClamp8 software (Axon Instruments) or a LabView-based customized program recently developed in our laboratory. Series resistances and input impedance were monitored by a test hyperpolarizing pulse (5 mV, 10 ms). In general there was no significant change in series resistance (10–30 M
) and input impedance (300–500 M) following repetitive pairing protocols. Data were accepted for analysis only in the cases where series resistance and input impedance did not vary beyond 10% throughout the experiment. Trials showing significant run-up or run-down during the control period (>5% over 10 min) were also excluded from further analysis. Recorded excitatory postsynaptic current (EPSC) traces were analyzed using the pClamp program (Axon Instruments). STDP ratio was calculated from the averaged EPSC amplitude within 10 min before and that between 15 and 30 min after the stimulation paradigm.
Two-compartment CA1 neuron model
We used a two-compartment CA1 model consisting of a soma and a localized dendritic region, which allowed for consideration of back-propagation of action potentials initiated experimentally in the soma. The starting point for our model was the experimentally calibrated multi-compartment CA1 pyramidal cell model developed by Poirazi et al. (2003). We reduced the model to two compartments in a way such that the voltage dynamics in the single dendritic region qualitatively matched those in a typical dendritic compartment of Poirazi et al. (2003). In the rest of the paper, we refer to the dendritic compartment in our model as a "spine" to represent the fact that we assume that there is minimal calcium diffusion out of this compartment (Goldberg et al. 2003; Sabatini et al. 2002) and that calcium is well-mixed throughout the compartment (Kovalchuk et al. 2000; Murthy et al. 2000; Sabatini et al. 2002). However, it should be noted that true spines are not well-developed in rat hippocampal cultures. We further reduced the model by systematically removing conductances that did not significantly affect voltage dynamics in response to current injections or to synaptic inputs; the reduced and full two-compartment models yield similar calcium time courses. We also replaced the calcium dynamics of the Poirazi et al. (2003) model with a modified version of those of Traub et al. (1994) to replicate the calcium profiles reported in Koester and Sakmann (1998), Yuste et al. (1999), Murthy et al. (2000), and Sabatini et al. (2002). Furthermore, we used the more natural form of kinetic equations for current activations and inactivations given in Traub et al. (1994)
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z(v),
z(v) in Eq. 1, are still from Poirazi et al. (2003). Note that in Eq. 1 and all subsequent equations, differentiation is with respect to time.
The ordinary differential equations for the voltage potentials in the compartments in the reduced model take the form
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), a delayed rectifier potassium current (IKdr in Poirazi et al. 2003), and a calcium-activated potassium afterhyperpolarization current (ImAHP in Poirazi et al. 2003); ICa is a high-threshold calcium current (ICaL in Poirazi et al. 2003); Icoup denotes the electrical coupling between the compartments in µA/cm2, given by Icoup = 1.125(vj –vi) in the equation for compartment i; and Iin corresponds to current injections (in the soma) or to AMPA and NMDA synaptic currents, Isyn,AMPA and Isyn,NMDA (in the spine). The forms of these currents are specified in Poirazi et al. We normalized the membrane capacitance Cm to 1 µF/cm2 and we modified the conductances of these currents to maintain appropriate voltage dynamics in our compartmental reduction; the values used (in mS/cm2) appear in Table 1.
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x denotes calcium concentration in micromolar, 0,x denotes the baseline resting calcium level, d relates to the calcium diffusion rate from the soma to the spine, and nbuff controls the strength of nonlinear calcium buffering, whereas buff scales a linear buffering term. The currents ICa,soma and ICa,spine in Eq. 3 denote VGCC terms (as in Eq. 2), with 
denoting a factor to convert from units of current to units of concentration, itself depending on buffering. We take to have magnitude 0.01 for both compartments, which reflects similar volumes of active zones, independent of overall compartmental volumes. The other parameters here are soma = spine = 0.083/ms, 0,soma = 0.05 µM, 0,spine = 0.07 µM, d = 1,000 ms, nbuff = 6, and buff = 0. These parameters were selected to match experimental data on dendritic calcium dynamics (Koester and Sakmann 1998; Murthy et al. 2000; Sabatini et al. 2002; Yuste et al. 1999), with resting calcium in particular constrained by specific experimental results (Maravall et al. 2000; Pozzo-Miller et al. 1999; Yuste et al. 1999).
The current flow through NMDA channels includes several different ion types. For the calcium current through the NMDA channels in the spine, denoted ICa,NMDA in Eq. 3, and the total current through these NMDA channels, which we denote Isyn,NMDA, we used equations of the form
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, 1993), we use
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The AMPA synaptic current is given by
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; Perouansky and Yaari 1993). These take the form
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]NMDA = 20/ms, 
NMDA,rise =2 ms, NMDA,fast = 10 ms, NMDA,slow = 45 ms, []AMPA = 20/ms, AMPA,rise = 0.58 ms, AMPA,fast = 7.6 ms, and AMPA,fast = 25.69 ms. Note that fpre(t) represents a step pulse that is turned on to initiate the postsynaptic effect when a presynaptic spike occurs. Calcium detectors
The mechanism for plasticity in our model involves a biophysically plausible calcium detection system that responds to calcium and then changes the strength of the synapse accordingly. In the model, three detector agents (P, A, V) respond to the instantaneous calcium level in the spine compartment. The interactions of these three agents, together with two others (D, B) that they influence, act to track the calcium time course in the spine (see Fig. 3A). More specifically, different calcium time courses lead to different time courses of P and D, which compete to influence a plasticity variable W. This variable W is used as a measure of the sign and magnitude of synaptic strength changes from baseline. Note that this scheme is significantly different from a detection of peak calcium levels, in which the change in W would be determined by how large spine calcium becomes during an appropriate set of spikes. The interactions between agents within our detector system qualitatively resemble the pathways influencing the regulation of Ca2+/calmodulin-dependent protein kinase II (CaMKII) (Bhalla and Iyengar 1999; Lisman 2001).
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; Lisman and Zhabotinsky 2001). We set
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suggest that the presence of protein phosphatase 1 (PP1) can influence the steady-state calcium sensitivity of CaMKII. We included an analogous effect through the weak competition term cpAP term in the P equation, although one could alternatively choose to alter the Hill equation for P to obtain similar results. A detects calcium levels that exceed the lower threshold of 0.6 µM, and its fast time constant ensures that the time course of A follows that of calcium rather closely, but with some smoothing out of peaks and valleys. If a postsynaptic spike precedes a presynaptic input by a small 
t, then A accumulates over both, reaching a threshold for the activation of B. This double-filter acts as a width detector for the calcium concentration. If enough of B accumulates, then D is activated. This double-filter system also eliminates depression from post-pre pairings with t outside of the appropriate depression window. There may be other ways to achieve these effects but our analysis suggests that the elements of width detection and modular competition are essential (see RESULTS).
Sufficiently large P values increase W, while sufficiently large D decreases W, setting up a potential second, more significant, competition between potentiation and depression, beyond the cpAP term in the P equation, which ensues if the mechanisms for both are activated. Like P and A, the variable V responds to a calcium threshold, namely 2 µM, and its fast time constant gives close tracking of calcium level. V acts as a "veto" on the accumulation of B, through the competitive term cdBV in the B equation. This veto prevents calcium levels that are sustained near or above 2 µM from activating the LTD side of the W equation. The existence of such an interaction is motivated by the inhibitory effect of protein kinase A (PKA) on PP1, but it represents the integrated effect of many pathways, with an effective threshold set by our choice of 
(spine). The LTD block that the veto provides is relevant for pre-post spike pairings, as well as for post-pre-post triplet experiments and for classical LTP/LTD.
Simulations
All simulations were performed using the software XPPAUT (Ermentrout 2002). We used the fourth-order Runge-Kutta method for numerical integration in XPPAUT, with a step size of 
t = 0.025 ms.
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RESULTS |
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In the model, the calcium concentrations in the spine resulting from pre- or postsynaptic (somatic) suprathreshold stimulation alone do not elicit plasticity. After presynaptic stimulation, a small amount of calcium enters through the NMDA channels. The AMPA channels induce a limited depolarization, which provides a small relief of magnesium block. With postsynaptic stimulation alone, calcium enters through VGCCs only.
Calcium time courses, but not necessarily levels, differ across paired stimulation protocols
In a paired protocol of a presynaptic stimulation followed by a postsynaptic stimulation 10 ms later (pre-10-post), we see a large influx of calcium through the NMDA channel due to the removal of the magnesium block by the BPAP (Fig. 1A). In the pre-40-post scenario, the influx of calcium is reduced because the arrival of the BPAP at the dendrite comes later. While the slow closing of the NMDA channels still allows calcium influx to be enhanced by this unblocking, the peak calcium level is lower in the pre-40-post scenario than in pre-10-post (Fig. 1A) because NMDA channels are more inactivated by the time the magnesium block is removed. The peak calcium level will continue to decrease as the pre-post interstimulus interval is lengthened. This inevitable reduction in calcium level will necessarily lead to a zone of pre-post depression in any model that attributes potentiation to high calcium levels and depression to lower calcium levels, independent of where the levels are set.
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Note that when paired stimulation is applied repeatedly at 1 Hz, calcium decays to very low levels between pairs. Thus each pairing induces a postsynaptic calcium response that is quite similar to the ones shown for single pairings. The same applies with triplet combinations of pre- and postsynaptic stimulation (Fig. 1, C and D).
Hippocampal culture triplet experiments do not support calcium level detection
Previous studies showed that in hippocampal cultures, LTP could be induced when a post-pre spike pair was followed by a third, postsynaptic spike resulting from a suprathreshold EPSP (Bi and Poo 1998). To examine further the plasticity outcomes of prescribed spike triplets, we performed experiments in hippocampal cultures. These experiments revealed an interesting asymmetry in the resulting STDP (Fig. 2). Consistent with our previous results, post-pre-post triplets, with 10-ms intervals between all successive stimuli, result in potentiation. On the other hand, pre-post-pre triplets, again with 10-ms intervals between stimuli, result in no consistent change. This is surprising because the calcium influxes resulting from the two triplet patterns are similar. Figure 1, C and D, shows the calcium time courses for pre-post-pre and post-pre-post stimulation triplets with 10-ms intervals between all successive stimuli. These are compared with the calcium time course for the pre-10-post paradigm. Note that the peak calcium levels shown in Fig. 1, C and D, in both triplet cases are approximately the same as that for the pre-10-post case, with the peak level in the post-pre-post scenario actually slightly below that for pre-10-post, due to the potassium A current. Since calcium levels do not build up over subsequent triplets, if calcium level alone were used as a signal to select a direction of synaptic plasticity, the prediction would be that potentiation would arise in both cases, with greater potentiation in pre-post-pre than in post-pre-post. This is in complete contrast to what we observe in experiments (Fig. 2), although such behavior has been reported in a different experimental system (Sjöström et al. 2001) and is discussed further below.
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The similarity of the calcium profiles across pre-post, post-pre, and triplet scenarios (Fig. 1) implies that, to reproduce our experimental plasticity results (Fig. 2, Bi and Poo 1998), any detector based exclusively on calcium must use the entire calcium time course. From an examination of the calcium time courses in Fig. 1 and the experimental results in Fig. 2, we surmise that a detection system should incorporate three features: 1) calcium levels above a high-threshold (e.g., 4 µM; see Fig. 1A) trigger potentiation; 2) levels that exceed a low threshold (e.g., 0.6 µM) and remain above for a minimum continuous time (e.g., 45 ms) trigger depression (i.e., the width of the calcium signal must be accounted for; see Fig. 1B); and 3) levels exceeding a mid-level threshold (e.g., 2 µM) trigger a veto of the depression components of the model. In particular, with such a veto in place, the relative timings and durations of periods of low and moderate calcium become relevant to the plasticity outcome, enhancing the dependence of plasticity on aspects of the calcium time course other than levels.
As a specific example of a model showing that these features are sufficient for distinguishing STDP scenarios, we constructed a calcium time course detector consisting of five interacting elements together with a readout variable, all described by ordinary differential equations. We do not imply that this implementation is the only or even the optimal way to codify our rules. This merely represents a straightforward possibility using simple chemical kinetics. The interactions of the model's components are shown in Fig. 3A, whereas the actual model equations, with a description of their actions, are given in METHODS. The equations can be viewed as incorporating relatively independent early signaling modules for LTD and for LTP, as recently suggested on the basis of experimental results (Wang et al. 2005). For simplicity, we assume that changes in synaptic strengths are proportional to changes in the readout variable (W) and that the effects of the potentiation (P) and depression (D) components on W sum linearly. Thus if both are activated at appropriate levels, they can cancel each other. Temporal details of the calcium time course determine the relative strengths of the potentiation and depression components, and hence the state attained by the readout. Figure 3B summarizes the saturation levels, denoted by W
, of W for various fixed levels of P and D. Note that the evolution of W is sensitive to P and D on much different scales. These scales are completely arbitrary and could be changed by a rescaling of D; we left the scales disparate for ease of visualization in Fig. 7D.
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; Holmes 2000; Lisman 1994; Lisman and Zhabotinsky 2001; Lisman et al. 2002; Malenka et al. 1989; Malinow et al. 1989; Mulkey et al. 1994; Zhabotinsky 2000). The potentiation detector is motivated by the phosphorylation cascade of CaMKII. The depression detector is a hypothesized abstraction of the kinetics of dephosphorylation agents such as PP1. While it is not our intention to model particular detector variables as direct representatives of particular molecules, experimentally deduced Hill coefficients (with respect to Ca2+) relevant for the phosphorylation and dephosphorylation of CaMKII were used (Holmes 2000; Lisman and Zhabotinsky 2001), supplying each differential equation with a biophysically plausible calcium sensitivity. The veto system is motivated by the competition of kinases and phosphotases, specifically the action of PP1 inhibitory agents such as PKA (e.g., Bhalla and Iyengar 1999; Lee et al. 2000). We stress that the precise form of the detector is unimportant as long as it incorporates the three proposed features. The model developed here is given to demonstrate that these features are sufficient to reproduce plasticity outcomes in a variety of experimental paradigms. Plasticity outcomes match experiments and derive from detector responses to calcium time course
To compare the plasticity outcomes from our calcium time course detection system to experiments on stimulus pairs and triplets, we examine the saturation levels (W) of the readout variable W in response to simulations of experimental stimulus paradigms. The detector is able to reproduce the STDP window of Bi and Poo (1998) as well as our triplet results (Fig. 2), while giving no plasticity for isolated pre- and postsynaptic spikes (data on isolated spikes not shown). Figure 4 displays the cumulative plasticity results for spike pairing simulations of our full model system, together with previously published data from spike pairing experiments (Bi and Poo 1998). In these simulations, the postsynaptic cell was induced to fire by current injection at a time t after the presynaptic cell was stimulated. Note that negative t values correspond to post before pre, whereas positive t values correspond to pre before post. For each t, the experiment was repeated at 1 Hz until saturation, typically requiring about 20 pairings. Even in the saturated state, W may oscillate slowly about a fixed level, and accordingly, the W levels shown in Fig. 4 were generated by averaging W over a 1-s period. We stress that this model does not yield depression in pre-post scenarios with large interstimulus intervals, as occurs in previously published models based on calcium levels (Abarbanel et al. 2003; Karmarkar and Buonomano 2002; Shouval et al. 2002).
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The pre-post-pre case, as seen in Figs. 1C and 7A1, has a similar calcium time course to pre-10-post and to post-pre-post but leads to no consistent synaptic plasticity (Figs. 7C, blue curve, and 2). As mentioned above, the distinctive feature of the pre-post-pre case is a slightly higher shoulder in the decaying phase of the calcium profile, relative to pre-10-post (Fig. 1C, arrow). This shoulder is enough to counteract the effects of the veto because the calcium level spends a sufficiently long time below the level needed for veto activation but still above the level needed to trigger depression, as seen in Fig. 7A2. The resulting activation of D offsets the simultaneously induced activation of P (Fig. 7D). This leads to a lack of preferred direction of change of the synapse (Fig. 7C). This clearly shows how finely tuned the detector must be so that the triplet interactions can be made consistent with the experiments. In DISCUSSION, we consider possibilities that may make the system more robust.
In addition to spike triplets, we compared the predictions of our model with recent experimental data on spike quadruplets (Wang et al. 2005) and found qualitative agreement for the same parameter values used in the pair and triplet simulations. As in the triplet case, different arrangements of spikes produced different plasticity results, with no obvious rule based on the spike times.
Effects of active conductances in dendrites
It has been suggested that the outward potassium A-current plays a crucial role in STDP. Specifically, an EPSP may quickly and transiently inactivate the A-current, boosting the amplitude of a back-propagating AP that occurs just afterward (Magee and Johnston 1997; Migliore et al. 1999; Watanabe et al. 2002). The postsynaptic depolarization, and hence the extent of A-channel inactivation, depends on calcium influx, and the state of the A-channels in turn feeds back on calcium influx; although the A-channels do not themselves allow direct calcium influx, they are relevant to the postsynaptic calcium time course. Using published parameter values for the A-current, we find that blocking the A-current in the dendritic compartment has little qualitative effect on the STDP results produced by pre-post and post-pre stimulus pairs in our model.
A-current blockade does alter triplet results, however. Specifically, if the A-current is blocked in pre-post-pre simulations, then potentiation results. This potentiation occurs because the calcium influx via both L-type and NMDA calcium channels is increased. Higher calcium levels lead to stronger activation of the potentiation component of the model, and they also activate the veto more strongly. By the time the veto wears off, the calcium influx has returned to baseline levels. Thus the depression component of the model experiences only the usual activation, and the balance of LTP/LTD is tipped to favor potentiation. Post-pre-post simulations under A-current blockade still give potentation. In post-pre-post simulations, the Mg2+ block of NMDA channels is fully removed by the second postsynaptic stimulation, even with the A-current intact, and thus A-current blockage has little effect on plasticity. In summary, A-current blockade leads to potentiation in both pre-post-pre and post-pre-post experiments, which matches the findings obtained in triplet experiments by Sjöström et al. (2001). This suggests that differences in densities of dendritic potassium channels may contribute to differences in plasticity outcomes of triplet experiments observed in different preparations.
Another experimental manipulation that has been performed in the context of STDP is the blockade of L-type calcium channels. Blocking L-type calcium channels in our model eliminates depression, but not potentiation, in the paired STDP protocol. This also agrees with the results seen in experiments (Bi and Poo 1998).
Modular signaling for LTP and LTD
In recent experimental work on STDP in hippocampal culture, Wang et al. (2005) found that pharmacological block of CaMKII eliminates pre-post plasticity and leads to LTD in triplet paradigms. Complementing this result, blockade of calcineurin leads to no plasticity from post-pre pairings and LTD in triplet experiments (Wang et al. 2005). Corresponding alterations may be introduced in our model, based on its modular structure, and yield similar plasticity outcomes. In particular, blocking the growth of the LTP component P in our model eliminates pre-post plasticity, while blocking the rise of the initiator element A on the LTD side of our model eliminates post-pre plasticity and yields LTP from both pre-post-pre and post-pre-post triplet experiments (data not shown). However, in the case when P is blocked, the model does not reproduce the LTD observed experimentally in post-pre-post experiments, perhaps relating to an overly strong veto effect in the model.
Presynaptic stimulation experiments yield a "No Man's Land"
In addition to matching STDP experiments, our model calcium detector yields results that are compatible with LTP/LTD induced by presynaptic stimulation under postsynaptic dendritic voltage clamp (Cho et al. 2001) (Fig. 8). In these experiments, 200 presynaptic stimuli are applied at 1 Hz. As the postsynaptic cell is held at progressively higher voltage levels, depression emerges, a "No Man's Land" (Lisman 2001) where there is no plasticity occurs at moderate voltages, and finally potentiation results at the most depolarized levels. Figure 8A shows this trend, as produced by our model. Furthermore, in experiments done with a voltage clamp to –10 mV, increasing the concentration of a calcium buffer by a moderate amount switches potentiation to depression, while sufficiently large additional increases in buffer concentration eliminate plasticity (Cho et al. 2001). Our model also reproduces this result at any level of voltage clamping at which potentiation occurs at default buffer concentrations (Fig. 8B).
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; Wang and Wagner 1999). While the result shown in Fig. 8C was obtained under postsynaptic voltage clamp, an analogous crossover occurs without the clamp. Without voltage clamp, we find that the crossover frequency decreases as we increase the AMPA conductance (data not shown), which must be considered as an unknown parameter for these particular simulations, since switchover experiments in hippocampal slices were performed using extracellular field EPSPs to drive the postsynaptic cell (Wang and Wagner 1999). Similar crossover results, which have been likened to the BCM sliding threshold rule (Bienenstock et al. 1982), have been shown in several earlier STDP models (Abarbanel et al. 2002; Izhikevich and Desai 2003; Senn et al. 2001; Shouval et al. 2002).
Finally, to show further the role of postsynaptic calcium in LTP/LTD, we examined the consequences of depolarization of the model postsynaptic cell to 0 mV, in the absence of presynaptic stimulation. Our model reproduces the experimental finding (Conti and Lisman 2002; Malenka et al. 1988) that this paradigm leads to no plasticity. In the model, postsynaptic depolarization to 0 mV inactivates VGCCs, preventing sufficient calcium influx for the induction of plasticity.
The similarity between the results produced by our calcium detector model (e.g., Fig. 8) and those seen experimentally is consistent with the idea that STDP and classical LTP/LTD share similar intracellular mechanisms.
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DISCUSSION |
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Examination of the calcium time course across different scenarios indicates that a calcium time course detector system should incorporate three rules: 1) a high threshold for potentiation, 2) a combined low-amplitude, long-duration signal detector for depression, and 3) a mid-level veto of depression. While calcium levels still play a role in this system, the latter two ingredients in particular yield a strong dependence on temporal dynamics of the calcium signal. Furthermore, we have shown that for a detector that is sensitive only to calcium levels, depression in pre-post scenarios with large interstimulus intervals is unavoidable. This phenomenon, which is not supported by experiments, does not arise in our model. While the calcium time course detector in this work uses a small number of elements to convert the above rules into accurate reproductions of plasticity outcomes across experimental scenarios, this success requires some degree of fine-tuning. The similarity of calcium time courses seen in our simulations, across scenarios that yield different plasticity outcomes in experiments, suggests that variability in plasticity outcomes from the repeated application of certain experimental protocols will be inevitable in a calcium-based system.
Veto mechanism eliminates pre-post depression with calcium time course detection but not with level detection
Note that the ability of our detector to eliminate the depression seen in other models when the presynaptic stimulation precedes the postsynaptic stimulation by a long time interval follows because the depression trigger is a time course detector composed of three elements, rather than a simple level detector. In our model, in addition to contributing to the absence of pre-post depression, the veto prevents depression in the post-pre-post case and is responsible for the "No Man's Land" for intermediate levels of calcium in the classical LTP/LTD protocol of Cho et al. (2001) (Fig. 8). In the latter case, the veto works to limit the response of the depressive elements in the model, such that a cancellation between depression and potentiation occurs.
It is important to note that, in a pure calcium level detection model for STDP, the presence of a veto is still not sufficient to eliminate pre-post depression with large interstimulus intervals. This is because as the time interval between pre- and postsynaptic events increases, the calcium level must necessarily fall (Fig. 1A) (Karmarkar and Buonomano 2002). Eventually, the calcium triggered by the pre-post stimuli will fall to the level found in the post-10-pre case, which causes depression. Thus at this level, the veto will not be activated, and depression will occur. The calcium time course detector avoids this outcome because when the calcium level falls to such a low level, its time course is no longer broad enough to trigger depression.
In theory, the lack of consistent synaptic conductance changes observed experimentally in the pre-post-pre case could result from unreliable synaptic transmission. Specifically, release failure in pre-post-pre experiments could effectively yield a sequence of pre-post and post-pre events leading to a cancellation between potentiation and depression signals. As the modulation of release probability is responsible for synaptic depression and facilitation, this idea could explain some of the variability in outcomes of pre-post-pre experiments in terms of differing biases toward paired-pulse depression or facilitation within different presynaptic cells. However, although the presynaptic release probability in these cultures has not been measured directly, most synapses studied here display weak synaptic depression and facilitation, and preliminary results suggest that outcomes of pre-post-pre experiments in culture are independent of these short-term plasticity features of the presynaptic cell (Wang et al. 2005). This leads us to favor the idea that pre-post-pre variability derives at least in part from heterogeneity in parameters associated with postsynaptic calcium time-course detection rather than presynaptic effects alone.
Sensitivity of calcium time course detection implies variability in plasticity outcomes
Our model detector system distinguishes between the various calcium time courses of pre- and postsynaptic spike pairs and triplets, eliminates depression in pre-post scenarios with large interstimulus intervals, and matches classical LTP/LTD experiments. However, it also requires fine-tuning. To ensure that pre-post-pre produces no preferred direction of synaptic change, the balance between the veto level and the depression trigger must be precisely set. Small changes in neuronal excitability, such as altering the strength of the A-current or the AHP current, can alter the results. These seem to be unavoidable aspects of any detector system that distinguishes between calcium time courses induced in the pre-post, post-pre, and triplet scenarios, as evidenced by the similarity of the calcium time courses appearing in these scenarios, seen in Fig. 1. The sensitivity of the detector leads us to conclude that responses to multi-spike patterns cannot be universal and consistent across all synapses if changes in synaptic strengths are based exclusively on a calcium signal coming from NMDA channels and VGCCs (see also Ismailov et al. 2004; Saudargiene et al. 2004). Future experiments that examine the dependence of STDP outcomes on synaptic location within the dendritic tree would provide a further test of the nature of this variability. This conclusion is also in keeping with the variability seen across systems in which spike pairings (reviewed in Abbott and Nelson 2000) or multiple spike interactions (Froemke and Dan 2002; Sjöström et al. 2001; Wang et al. 2005) have been explored; however, differences in plasticity outcomes across preparations likely also depend on differences in features, such as short-term plasticity and regulation of BPAPs, across these preparations.
Possible mechanisms for added robustness
The sensitivity of our model to changes in certain parameters may relate to the experimentally observed variability in STDP outcomes for each fixed protocol (e.g., Bi and Poo 1998; Wang et al. 2005) (Fig. 2). On the other hand, there may be ways to augment the calcium signal in the model to add robustness to a system that converts calcium time courses into synaptic plasticity outcomes. For example, calcium is heavily buffered in the dendrite, and it has been proposed that spatial separation of calcium entering through NMDA channels versus the VGCCs could simplify the spike-time detector (Karmarkar and Buonomano 2002). While there is experimental evidence that calcium is spatially isolated within a spine (Nimchinsky et al. 2002; Sabatini et al. 2002), the evidence for further localization to areas inside the spine is inconclusive. Furthermore, our computational analysis indicates that models based on spatial separation of calcium still require a significant amount of fine-tuning to account for all the STDP data sets. The separation hypothesis suggests that synaptic NMDA calcium (ICaS) induces potentiation, whereas depression stems from the interaction of VGCC calcium (ICaL) with ICaS or some other agent. This second component is needed to account for the fact that ICaL levels are similar in depressing post-pre and nondepressing post-alone scenarios. Without a veto, models based on separation, with interaction of ICaL and ICaS for depression, lead to depression in pre-post scenarios with long interstimulus intervals (data not shown), exactly as seen in our model when the veto is omitted and in other calcium-based models (Abarbanel et al. 2003; Shouval et al. 2002). Karmarkar and Buonomano (2002) circumvent this issue by attributing depression to the interaction of ICaL with metabotropic glutamate receptors, which they postulate are inactive in pre-post scenarios; this amounts to imposing a veto on depression in all pre-post interactions. However, this implies that there is an additional mechanism that is able to distinguish between pre-post and post-pre pairings. Even with a veto, spatial separation models still require some tuning. In particular, when the veto activation threshold is lowered sufficiently to prevent depression in the pre-40-post case, fine-tuning is required to allow enough depression to occur to cancel potentiation in the pre-post-pre case. Nonetheless, because of the myriad of possible ways to implement calcium detection with spatially separated calcium sources, their role in shaping plasticity merits attention in future work.
An important calcium source that we have not explicitly modeled is calcium release by intracellular calcium stores, which has been implicated as possibly contributing to depression. The decision not to treat this factor directly was motivated by the fact that the release from these stores has been found to be slow compared with the time scales required to resolve STDP (Kovalcuk et al. 2000; Sabatini et al. 2002), and blockade of calcium-induced calcium release does not significantly affect calcium influx in postsynaptic spines (Kovalchuk et al. 2000). Furthermore, since we selected parameter values relevant to calcium dynamics to match experimental observations of calcium signals (Koester and Sakmann 1998; Murthy et al. 2000; Sabatini et al. 2002), any short-term effects of calcium stores are implicitly encoded in the dynamics of our model. Longer-term effects of stores, which may be relevant for consolidating the effects of potentiation and depression, have been neglected.
Finally, it is also possible that stochasticity in the calcium dynamics due to small numbers of calcium ions in the spine contributes significantly to STDP in some systems. It has recently been shown (Shouval and Kalantzis 2005) that stochasticity may lessen depression at long pre-post intervals in a model based on calcium levels alone. However, it is not yet known how this model could be adjusted to reproduce the experimental results with triplet stimulation (Fig. 2), given the extreme similarity of peak calcium levels in pre-post, pre-post-pre, and post-pre-post scenarios (Fig. 1). Furthermore, the inclusion of stochasticity in the dynamics of calcium in our model has little impact on plasticity outcomes.
Generality of results
The arguments given here support the idea that calcium levels alone cannot be used to distinguish experimentally observed spike pair and triplet results, independent of modeling issues. Using a biophysical CA1 cell representation (Poirazi et al. 2003) to simulate calcium time courses, we examined whether there is enough information in the calcium time course in itself to distinguish various STDP protocols. We found that this should be the case provided three basic rules are observed. We then constructed a postsynaptic calcium time course detector that implements these rules. This detector system is based on the pathways influencing CaMKII regulation (Bhalla and Iyengar 1999; Lisman 2001). There may be alternative means of constructing such a detector but our analysis suggests that all of its qualitative features are necessary in the absence of additional assumptions. To keep our model as simple as possible, we did not attempt to derive a detailed biophysical model of the molecular pathways that respond to calcium signals. Our intention was to show feasibility and plausibility of a calcium time course detector. However, employing more biological realism may be worthwhile in the future.
We adapted an experimentally calibrated CA1 model for this work. It is possible that some differences may arise between the precise dynamics of cultured CA1 neurons and the dynamics of neurons in the in vitro preparations used in the calibration of the CA1 model. However, the details of the parameter choices and the number of compartments in our model are actually not important for our overall conclusions, namely that there is enough information in the postsynaptic calcium time course to distinguish the various scenarios of STDP, but any mechanism of synaptic plasticity based on this signal is necessarily sensitive to parameter changes. The essential feature of any calcium detection system is the interaction of NMDA channels with back-propagating action potentials. The magnesium block is the key timing element for the system. As a result, as long as this feature is captured qualitatively, inaccuracies in the parameters of our model can be tolerated without significantly altering our findings. Moreover, as long as we choose a reasonable coupling parameter between the somatic and dendritic compartments, the amplitude of the BPAP arriving in the model dendrite should be compatible with that of some point on the dendritic tree of a CA1 pyramidal cell (Holthoff et al. 2002; Saudargiene et al. 2004). We also note that changes in the buffering rates of calcium in the model, to first approximation, yield a compression or expansion of our calcium time courses. This scaling affects the tuning required of the detector, but the qualitative plasticity outcomes remain unchanged (Fig. 9A). Finally, it is important to note that the sensitivity in our model is not a sensitivity to noise in spike timing. Indeed, our plasticity outcomes are robust to spike timing jitter (Fig. 9B).
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). However, at low frequencies of paired stimulation, including 0.1 and 5 Hz, this cannabinoid mechanism alone is insufficient to account for timing-dependent LTD. Meanwhile, the importance of postsynaptic mechanisms has been stressed by numerous results on classical LTP and LTD, as well as experimental findings that postsynaptic calcium elevation is necessary for hippocampal STDP (Li et al. 2004). It will be interesting to see whether future experiments will uncover a presynaptic component of STDP in hippocampal preparations, given that plasticity outcomes of spike triplet experiments differ across different preparations (Froemke and Dan 2002; Sjöström et al. 2001; Wang et al. 2005) (Fig. 2). If so, presynaptic components of STDP may also be incorporated into the framework of our model in future work, perhaps with modifications to take into account additional time scales of signaling. Predictions and consequences for synaptic computations
The results of Sjöström et al. (2001), Froemke and Dan (2002), and our work suggest that a universal plasticity rule for trains of multiple spikes based exclusively on spike times may not exist. In particular, the experiments do not support a rule based on a linear combination of pairs of spikes, as suggested by several elegant STDP models (Abarbanel et al. 2002, 2003; Karbowski and Ermentrout 2002; Karmarkar and Buonomano 2002). While these models could be adjusted to account for the multi-spike experiments of Sjöström et al. (2001), they would be more hard-pressed to account for the findings of Froemke and Dan (2002) or for our spike triplet results. Our triplet results are also not captured by computational studies in which pre-post-pre and post-pre-post outcomes are symmetric (Abarbanel et al. 2003).
In our model, A-current blockade can lead to potentiation in both pre-post-pre and post-pre-post experiments (Sjöström et al. 2001). Furthermore, our model supports the idea that LTP or LTD can be unmasked in pre-post-pre experiments (Wang et al. 2005) and that they can cancel in experiments invoking plasticity using presynaptic stimulation with postsynaptic voltage clamp (Cho et al. 2001). Conversely, we predict that if precisely timed caged calcium release could be used to bolster calcium levels during the decay phase of a pre-post or post-pre-post experiment, so that the calcium time course was converted to that of the pre-post-pre scenario, LTP could be canceled. Our analysis also predicts the existence of a molecular veto mechanism that is harnessed by calcium signals above a moderate threshold to suppress LTD. Finally, our model suggests that slowing the decay rate of the L-type VGCCs, while simultaneously adjusting parameters to maintain calcium influx rate and peak calcium levels, would lead to pre-post-pre LTP, because the resulting broadening of the calcium signal above the LTP threshold would dominate over that below the LTP threshold (data not shown). This slowing would also broaden the LTD window in post-pre spike pair experiments and introduce LTD under pre-post with long interspike intervals, where the LTP module is not activated, due to a prolonged calcium time course (Fig. 9B).
The dependence of synaptic plasticity on calcium time course would lead to two additional computational consequences. First, plasticity outcomes induced by similar stimuli would not be consistent across all synapses and neural systems, but would depend on the specific properties of the local calcium, synaptic, and membrane kinetics. For example, local increases in dendritic excitability tend to promote LTP in pre-post-pre scenarios in our model while broadening the LTD window in spike pair experiments (data not shown). Second, synaptic plasticity contingent on calcium time course would display an enhanced sensitivity to the precise temporal details of spike trains, relative to plasticity based on levels alone. This added acuity could enable synaptically connected circuits to learn to differentiate more intricate sequences of spike times.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: J. Rubin, University of Pittsburgh, Dept. of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260 (E-mail: rubin{at}math.pitt.edu)
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INTRODUCTION
