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Department of Neurobiology and Anatomy, University of Texas Medical School at Houston, Houston, Texas
Submitted 14 November 2007; accepted in final form 23 May 2008
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ABSTRACT |
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INTRODUCTION |
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The postulate promulgated by Hebb (1949)
that the connection from neuron A to neuron B would be strengthened if neuron A "repeatedly or persistently takes part in firing" neuron B implies a temporal-order rule for spike firing. The synapse from neuron A to neuron B should strengthen only if the firing of neuron A precedes the firing of neuron B; if neurons A and B fired simultaneously or if neuron A fired after neuron B, then neuron A could not have caused neuron B to fire. Hebb further theorized that this postulate would cause groups of neurons to organize into cell assemblies, with recurrent excitatory connections causing the assembly to fire in a coordinated fashion. Finally, when one cell assembly formed and consistently preceded the firing of another cell assembly, the connections from assembly A to assembly B would strengthen selectively, such that the activation of assembly A (by sensory input, for example) could cause the subsequent activation of assembly B (which could activate assembly C, and so on) with no further external input necessary to drive the system. Hebb called this sequential activation of cell assemblies a "phase sequence."
Subsequent discovery of the phenomenon of long-term potentiation (LTP) has led support to these notions (Bliss and Lømo 1973). In particular, the discovery that activity of the presynaptic cell must precede the activity of the postsynaptic cell to produce LTP demonstrated the causal nature of the synaptic potentiation (Levy and Steward 1983). More recently, the phenomenon of spike-timing–dependent plasticity (STDP), in which pairs of pre- and postsynaptic spikes can cause LTP or LTD (long-term depression), depending on whether the presynaptic or postsynaptic spike occurred first, respectively (Bi and Poo 1998; Markram et al. 1997), further strengthened the notion that the sequential order of neuronal firing can be reflected in the pattern of weights stored in the synaptic matrix of the network.
Numerous models have utilized the temporally asymmetric nature of LTP induction to create representations of neuronal sequences to solve various temporal-sequence–dependent tasks. Levy (1989, 1996) used such rules to solve hippocampus-related tasks such as Pavlovian trace conditioning and spatial path disambiguation. Blum and Abbott (1996) used temporally asymmetric LTP in a model of Morris water-maze learning, in which the learning rule created a representation in which learned paths toward the escape platform were encoded in the synaptic weight matrix of a network of hippocampal place cells. The model of Blum and Abbott made the strong prediction that, if an animal made repeated trajectories along a stereotyped path, the place fields of the neurons would shift backward (in the direction opposite to the animal's motion) as the result of the asymmetric LTP induction. This prediction was confirmed experimentally by Mehta and colleagues (1997, 2000), who showed that the center of mass (COM) of place fields shifted backward and the shape of the place field could become negatively skewed, under some conditions, as a rat ran repeated laps on a narrow track. A subsequent study demonstrated that the backward shift depended on N-methyl-D-aspartate receptors (NMDARs) (Ekstrom et al. 2001). A firing-rate model of STDP was able to capture both the asymmetric changes of the weight distribution onto the postsynaptic cell and the asymmetric expansion of the output (i.e., the place field) of the cell (Mehta et al. 2000). Yu et al. (2006) subsequently demonstrated that an integrate-and-fire network model could also show the backward-shift phenomenon.
STDP is one of many induction protocols that cause NMDAR-dependent changes in synaptic strength, and the above-discussed models used a learning rule that explicitly simulated variants of this one induction protocol. Different learning rules describe other forms of LTP/LTD induction (e.g., 100-Hz stimulation, 1-Hz stimulation, and theta-burst stimulation; Bliss and Lømo 1973; Dudek and Bear 1992; Larson et al. 1986), and a complete account of plasticity at these synapses requires a rule (or set of rules) that incorporates all these known forms of plasticity induction. Shouval and colleagues (2002) proposed a biophysical model that uses postsynaptic Ca2+ concentration as a signal that can account for all of these different forms of plasticity. The calcium-dependent plasticity (CaDP) model replicated the effects of tetanic stimulation, low-frequency stimulation, theta-burst stimulation, and STDP. Moreover, it made a novel prediction that there were actually two regions of LTD in the STDP curve, not just one, because postsynaptic spikes that occurred at a long interval after a presynaptic spike actually caused LTD. This prediction has been confirmed experimentally in at least some synapses (Nishiyama et al. 2000; Wittenberg and Wang 2006). An additional consequence of the CaDP model is that STDP plasticity curves are frequency dependent (Shouval et al. 2002), as observed experimentally (Markram et al. 1997; Sjostrom et al. 2001).
The CaDP model, in combination with a model of metaplasticity, is able to reproduce the experience-dependent formation of selective receptive fields, as found experimentally in visual cortex (Yeung et al. 2004). The purpose of the present study was to determine whether this model could also reproduce the backward-shift effect of hippocampal place cells—a phenomenon and system it was not designed to address. We found that the CaDP model, with no substantial changes from previous work on visual cortex, could indeed reproduce the backward-shift effect of place fields. Moreover, the model solved a problem of previous place-field shift models, in that the dynamics of the model significantly slowed down the backward shift after a number of laps, in agreement with experimental data, rather than allowing the shift to continue indefinitely.
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METHODS |
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A feedforward model was constructed with 1,000 input cells (the input layer) and one output cell receiving weighted inputs from the input layer (Fig. 1A ). All of the input cells had identical, nonplastic, Gaussian place fields and the centers of these place fields were evenly distributed along a circular track with a circumference of 2 m. During all the simulations presented here, the rat ran around the track at a constant speed of 0.5 m/s. We have varied this speed within the range of 0.25–1 m/s and found no qualitative change in the shifting of place fields. The width (at half-maximum) of the input place fields was set at 0.3 m, which was derived from empirical data (Lee et al. 2004; Mehta et al. 1997, 2000) and had been used in our previous model (Yu et al. 2006). The place field's magnitude at a given location determined the probability that the respective input cell would generate a spike
 | (1) |
), x0 is the center of the place field, xi is the current position of the rat, and 
determines the width of the place field (full width at half-height = 2.355). Total synaptic input to the postsynaptic cell was generated randomly by
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is the Heaviside function, which has the value 0 if its argument is <0 and 1 if its argument is >0. The current is given in milliamperes and the synaptic weights (wi) are given directly in units of current.
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 | (3) |
l is the time constant for this leak current (25 ms; Song et al. 2000); Is is the synaptic input current; and β, in units of 
/ms, is a scaling factor that controls the total synaptic input level to ensure that the peak firing rate of the output cell is in the frequency range of 10–100 Hz found in the experiments (Lee et al. 2004; Mehta et al. 2000). If V reaches the firing threshold of –50 mV, a postsynaptic spike is generated and a back-propagating action potential BPAP(t) is inserted with a total amplitude of 100 mV; 75% of this value decays rapidly (fB = 3 ms) and the remainder decays slowly (sB = 25 ms). The BPAP is an important component of this model, in that it controls the amount of calcium that flows into the spine (Shouval et al. 2002). CaDP learning rule
The synaptic weights are plastic and subject to the CaDP learning rule (Shouval et al. 2002; Yeung et al. 2004), in which postsynaptic [Ca2+] determines synaptic plasticity. Synaptic weights are updated as
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 | (5) |
 | (6) |
 | (7) |
([Ca2+]) is a calcium-dependent learning rate (Fig. 1B) and ([Ca2+]) is a U-shaped function of calcium that determines the direction and amplitude of plasticity (Fig. 1C). Equations 5–7 define the functional form of and (Fig. 1, B and C), with the parameters (p1, p2, p3, p4) = (2, 0.5, 3, 0.00001) and (
1, β1, 2, β2) = (0.3, 40, 0.5, 40).
The STDP curve generated by the CaDP model at 1-Hz pairing is shown in Fig. 1D. Different from the "standard" STDP curve described in previous experimental and modeling papers (Bi and Poo 1998; Song et al. 2000; Yu et al. 2006), this STDP curve has a second LTD window in the pre-post region (
t >0) that peaks around +40 ms. The existence of this second LTD window is supported by recent reports in hippocampal slice studies (Nishiyama et al. 2000; Wittenberg and Wang 2006) and has an important impact on the present results (see following text).
Calcium influx
The local NMDAR-mediated calcium concentration follows a first-order linear differential equation
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= 20 ms is the calcium passive decay time constant (Markram et al. 1995; Sabatini et al. 2002). The current I depends on the association between presynaptic spike times and the postsynaptic depolarization level as I = gmf(t, tpre)H(V), where f describes the dynamics of the glutamate–receptor interaction, H describes the voltage-dependent magnesium block of the NMDAR, and gm determines the NMDAR conductance. If we measure calcium concentration in micromoles (µM), then the units of gm are in µM/(ms mV). However, it is best to view the calcium concentration as a unitless, phenomenological variable. On a presynaptic spike, f reaches its peak value of 1, and then 70% of this value decays with a fast time constant (fN = 50 ms), and the remainder decays with a slower time constant (sN = 200 ms). H describes the voltage dependence of the NMDAR (Jahr and Stevens, 1990).
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Metaplasticity
Like other Hebbian-type plasticity rules, the CaDP rule by itself is not stable. Metaplasticity is a homeostatic mechanism that has been used with the CaDP model to regulate membrane activity (Yeung et al. 2004). Metaplasticity controls postsynaptic calcium, and thereby the activity level of the postsynaptic neuron, through a slow, activity-dependent regulation of NMDAR conductance (gm). Metaplasticity is implemented through a voltage-dependent kinetic model of NMDAR insertion and removal
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). Units for k+ are [1/ms] and for k– are [1/ms mV2]. There are significant experimental indications that, in cortex, reduced activity increases the time constant of NMDARs and increased activity reduces this time constant (Carmignoto and Vicini 1992; Philpot et al. 2001; Quinlan et al. 1999; Rao and Craig 1997). This type of change is qualitatively similar to the rule assumed here, in that there is a negative feedback of the total charge influx through NMDARs. There are also indications that changing the activity levels in culture can cause a change in the total number of NMDARs (Watt et al. 2000). Initial conditions
Initially the output neuron is connected to the input neurons by a Gaussian function such that the output cell initially forms a place field. Each input place cell is defined by its own central location, x0, which is different for each input cell. For the output cell, we define its center as location j in the input space and, using this notation, the initial weights wi are connected according to the rule
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RESULTS |
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; Lee et al. 2004; Mehta et al. 1997, 2000; Yu et al. 2006). On average, the COM of the distribution of synaptic weights onto the output cell began to shift backward from the very start of the simulation (between laps 1 and 2) before apparently reaching asymptote around laps 7 and 8 (Fig. 2B). Similarly, the COM of the place fields, on average, also shifted backward from the beginning of the simulation (Fig. 2C).
Concomitant with the asymmetric strengthening of the synaptic weights onto the output cell was a decrease in the NMDAR conductance (Fig. 2D), as the result of the increased firing of the postsynaptic cell. This homeostatic mechanism, which was previously incorporated into the CaDP model to rein in runaway plasticity (Yeung et al. 2004), was also the reason that the backward shift slowed down significantly. The CaDP model at low frequencies acts like a causal STDP rule and causes the backward shift of place fields, as described earlier. However, as observed experimentally, causality matters less at high frequencies and plasticity is controlled primarily by the pre- and postsynaptic firing rates (Shouval et al. 2002; Sjostrom et al. 2001). This loss of causality is demonstrated in Fig. 3, A–C, which shows the STDP curves produced by the model at a constant NMDAR conductance level of gm = –0.0025. Without metaplasticity, increases in postsynaptic firing rate (Fig. 3C) would have eliminated the backward shift. That is, all spike-time pairings would have induced LTP and no LTD, resulting in a symmetric expansion of the place field in both directions. However, due to the metaplasticity mechanism in the model (Fig. 2D), the NMDAR conductance is decreased as the firing rate increases (to gm 
–0.00125). In Fig. 3, D–F, we show the resulting STDP curves at this new conductance level. At low frequencies no LTP is observed and only a slight LTD remains, whereas at 20 Hz significant bidirectional plasticity remains, but without the sharp transition at t = 0. The reduced plasticity at low frequencies and the reduced asymmetry at higher frequencies eliminate the place-field expansion as the firing rates increase and critically slow down the backward shift.
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; see Fig. 2B of that paper) demonstrated such a transient, forward shift in the COM in the later laps of the 17 laps recorded in that experiment. The reason for the reversals in direction of change of the weight distribution was the different time courses of the STDP (wij) and the homeostatic plasticity (gm). Initially, the STDP caused the weights to change in a way that produced the backward shift. As the firing rates increased, the magnitude of gm decreased until it eventually put a brake on the increase in firing (Fig. 4D). At this point, gm began to increase and, since the firing rate was already high, the STDP curve favored a forward shift of the weights. Eventually, gm reached a stable level around lap 35 and the model parameters were such that the synaptic weight changes reached a steady state with few overall changes to the distribution thereafter.
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curve and the curve (Fig. 1) did not qualitatively alter the results (data not shown). The speed of metaplasticity was characterized by the parameter d, which multiplies both k+ and k– and therefore changes the convergence rate of metaplasticity (normally d = 1). We found that d can range from 0.25 to 4 without qualitatively changing the simulation results.
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All the simulations described to this point were started from an initially well structured place field of the output neuron (Fig. 1A). However, if simulations are started from a nonstructured place field, this model often generates well-structured place fields. Figure 6 shows the results of 4 different simulation runs in which the initial connections between the input cells and the output cell were random. In three of these simulations, well-structured place fields developed over the course of the simulations and, in one simulation, the development of the place field failed. Overall, structured place fields developed in 16/20 simulation runs. Although in experimental data place fields do not form from such random, nonspecific firing patterns (Frank et al. 2004), the ability to develop structured place fields from a random initial state demonstrates that the model is robust in terms of its relative insensitivity to starting conditions.
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DISCUSSION |
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; Mehta et al. 2000; Yu et al. 2006). However, because the causal nature of STDP is preserved throughout these simulations, there is no reason that these backward shifts would terminate or even slow down, in the absence of additional mechanisms. The backward shift observed experimentally, however, does slow down and seems to terminate or asymptote (Ekstrom et al. 2001; Lee et al. 2004; Mehta et al. 1997, 2000; Yu et al. 2006). Furthermore, in previous models of the backward shift, plasticity was based on an abstract and frequency-independent model of STDP. The assumption of frequency independence is not consistent with experiments that have shown that the STDP curves are frequency dependent (Markram et al. 1997; Sjostrom et al. 2001) and that at high frequencies LTP dominates. Previous papers have described calcium-dependent plasticity rules that can account for various induction protocols, including the frequency dependence of STDP (Abarbanel et al. 2003; Rubin et al. 2005; Shouval et al. 2002). When such plasticity rules are coupled with metaplasticity, they can produce selective receptive fields (Yeung et al. 2004).
In this study we use the CaDP model of synaptic plasticity in combination with a model of metaplasticity, a combination that was previously used to simulate receptive field plasticity in visual cortex (Yeung et al. 2004). Here we applied this combination to simulations of place field plasticity in the hippocampus. The results showed a robust backward shift in the place field COM, accompanied by an increase in firing rates, as observed experimentally (Lee et al. 2004; Mehta et al. 1997; Yu et al. 2006). We also find that this backward shift of the place field COM critically slows down as a result of metaplasticity. We have obtained these results without assuming any additional stopping mechanisms, by simply applying a model previously developed to account for visual cortex plasticity to simulations of place field plasticity. The general trends in these simulations—such as backward shifts, increase in firing rates, sharpening of place fields, and termination of shift—are robust to significant parameter changes. For some parameters, the results exhibit damped oscillations in the place field locations that are coupled to oscillations in the value of the NMDAR conductance. Such oscillations may also be present in some experimental results (see Fig. 2B of Mehta et al. 1997), but they are not a robust prediction of our model, since they are parameter dependent. Nonetheless, damped oscillations are characteristic of many coupled systems of dynamical equations; for example, they are exhibited in the BCM (Bienenstock–Cooper–Munro; Bienenstock et al. 1982) rule when the time constant of the sliding modification threshold is significantly slower than the plasticity time constants (Cooper et al. 2004).
Although the CaDP model produces a robust backward shift in the place field COM, we do not find the emergence of a significant negative skew in the shape of the place fields. Indeed, the development of a negative skew is experimentally variable as well, seemingly dependent on a number of parameters such as the brain region and perhaps the nature of the experimental task (Huxter et al. 2003; Lee et al. 2004; Mehta et al. 1997, 2000; Yu et al. 2006), as well as on the precise definitions of the place-field boundaries (Mehta et al. 2000). For example, Lee et al. (2004) reported the development of negative skewness in CA3 place fields but not in simultaneously recorded CA1 place fields. The second LTD region present in the deterministic CaDP model at positive t (Fig. 1D) likely results in a trimming of the left-hand flank of the place field as the shift develops. Therefore the second LTD window is a likely factor preventing the emergence of a negative skew in the model. If the second LTD window were not present, or if it were significantly smaller, a greater negative skew might develop, as shown in previous models using a standard STDP curve with a single LTD window (Mehta et al. 2000; Yu et al. 2006). It has been previously shown that stochastic synaptic transmission (Shouval and Kalantzis 2005) or an additional veto mechanism (Rubin et al. 2005) can reduce or eliminate the second LTD window in a calcium-dependent model. Including stochastic synaptic transmission or a veto mechanism is therefore likely to result in the emergence of a negative skew, but stochasticity or additional veto mechanisms are beyond the scope of this paper. Nonetheless, these considerations suggest that the development of negative skewness in place fields may be related to the conditions under which a second region of LTD is present physiologically in the STDP curve.
Although the general trends in this model are robust, the precise dynamics of individual simulations are very different. The variability between the dynamics of individual simulations depends on the precise system parameters, the initial conditions, and the randomness of specific simulations. Such variability of the dynamics of place-field shifts is also observed experimentally. For example, Lee and Knierim (2007) showed that, although at a population level there may be a consistent backward trend in COM shifts in CA3 or CA1, there is a large amount of variability at the single-neuron level. Some place fields can shift forward, others can shift backward, and others remain stable. Interestingly, there appears to be a greater coherence of shifting in the place fields of CA3 cells and in the tuning curves of head direction cells compared with the place fields of CA1 cells, which may reflect the influence of network dynamics of the two former regions forcing a more coherent population response to the variable synaptic changes occurring at the single-cell level (Lee et al. 2004; Yu et al. 2006).
Metaplasticity is an experience-dependent change in synaptic plasticity (Abraham and Bear 1996) and various forms of metaplasticity have been observed experimentally. An appropriate choice of a metaplasticity model (Yeung et al. 2004) can serve as a negative feedback mechanism that is similar to the sliding threshold of the BCM rule (Bienenstock et al. 1982). Such a metaplasticity mechanism is also able to account for some forms of synaptic scaling (Turrigiano 1999; Yeung et al. 2004). The form of metaplasticity used here, adopted from Yeung et al. (2004), was implemented as a voltage-dependent change in the NMDAR conductance. There is evidence that properties of NMDAR conductance are activity dependent (Bellone and Nicoll 2007; Carmignoto and Vicini 1992; Philpot et al. 2001) and these changes may contribute to metaplasticity. However, a similar negative feedback could be implemented by other specific mechanisms as well, such as activity-dependent changes in calcium buffer concentration or spine shape, and these different mechanisms are likely to produce qualitatively similar results.
In summary, the present study adopted a model of synaptic plasticity that was originally derived and tested against visual cortex data (Yeung et al. 2004) and applied the model to the phenomenon of the asymmetric expansion of hippocampal place fields. With no significant changes in parameters, the model was able to replicate the backward shift of place fields. Moreover, the combination of CaDP and metaplasticity automatically solved a problem with previous models (Mehta et al. 2000; Yu et al. 2006) by causing the backward shifting to slow down after a number of laps, consistent with the experimental data. Similar receptive field shifts occur in visual cortex (Fu et al. 2002; Yao and Dan 2001) and hippocampus (Ekstrom et al. 2001; Lee et al. 2004; Mehta et al. 1997, 2000; Yu et al. 2006) as the result of appropriate spike-timing manipulations. Because the CaDP model can account for physiological data from both in vitro and in vivo preparations, it may capture the essence of a general cortical mechanism of synaptic plasticity (Shouval et al. 2002).
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GRANTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: J. J. Knierim, Department of Neurobiology and Anatomy, University of Texas Medical School at Houston, P.O. Box 20708, Houston, TX 77225 (E-mail: james.j.knierim{at}uth.tmc.edu)
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ABSTRACT
INTRODUCTION
0.2–0.5 mM), the synaptic weight is weakened (long-term depression [LTD]). When postsynaptic [Ca2+] is high (>0.5 mM), the synaptic weight is strengthened (long-term potentiation [LTP]). D: the spike-timing–dependent plasticity (STDP) curve generated by the CaDP learning with 1-Hz pairing. A presynaptic spike train and a postsynaptic spike train, each at 1 Hz and with a total of 50 spikes, were paired at the time difference specified in the x-axis (
