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The Journal of Neuroscience, September 14, 2005, 25(37):8416-8431; doi:10.1523/JNEUROSCI.0631-05.2005
 Previous Article | Next Article 
Behavioral/Systems/Cognitive
Short-Term Synaptic Depression Causes a Non-Monotonic Response to Correlated Stimuli
Jaime de la Rocha and
Néstor Parga
Departamento de Física Teórica, Universidad Autónoma de Madrid, Canto-Blanco, 28049 Madrid, Spain
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Abstract
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Unreliability is a ubiquitous feature of synaptic transmission in the brain. The information conveyed in the discharges of an ensemble of cells (e.g., in the spike count or in the timing of synchronous events) may not be faithfully transmitted to the postsynaptic cell because a large fraction of the spikes fail to elicit a synaptic response. In addition, short-term depression increases the failure rate with the presynaptic activity. We use a simple neuron model with stochastic depressing synapses to understand the transformations undergone by the spatiotemporal patterns of incoming spikes as these are first converted into synaptic current and afterward into the cell response. We analyze the mean and SD of the current produced by different stimuli with spatiotemporal correlations. We find that the mean, which carries information only about the spike count, rapidly saturates as the input rate increases. In contrast, the current deviation carries information about the correlations. If the afferent action potentials are uncorrelated, it saturates monotonically, whereas if they are correlated it increases, reaches a maximum, and then decreases to the value produced by the uncorrelated stimulus. This means that, at high input rates, depression erases from the synaptic current any trace of the spatiotemporal structure of the input. The non-monotonic behavior of the deviation can be inherited by the response rate provided that the mean current saturates below the current threshold setting the cell in the fluctuation-driven regimen. Afferent correlations therefore enable the modulation of the response beyond the saturation of the mean current.
Key words: synaptic integration; fluctuation-driven regimen; presynaptic spike correlations; synaptic short-term depression; vesicle depletion; neural coding
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Introduction
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Since the first studies on synaptic transmission in the neuromuscular junction, Katz and Miledi (1968 ) realized that transmitter release occurs stochastically. Additional studies found that central synapses are also very unreliable (Hessler et al., 1993; Rosenmund et al., 1993) and that the transmission probability undergoes temporary changes according to the recent activity (Magleby, 1987; Zucker and Regehr, 2002). This probability decreases under depletion of the releasable transmitter vesicles (Dobrunz and Stevens, 1997), giving rise to short-term synaptic depression (STD).
Several works have explored the theoretical implications of these findings on the transmission of information in neural circuits (Abbott et al., 1997; Lisman, 1997; Tsodyks and Markram, 1997; Matveev and Wang, 2000b). One of the most relevant consequences of STD is that the synapses saturate at presynaptic rates ( ) higher than a limiting rate, that is, the mean stationary afferent current (µI) does not depend on . This saturation appears to be a major constraint on the range of within which neurons can transmit information in the stationary regimen (Abbott et al., 1997; Tsodyks and Markram, 1997). However, this argument is based on the saturation of µI and neglects that the cell response is also determined by the current fluctuations ( I). It may well be that I does not saturate as µI so that the response can be modulated beyond the limiting frequency.
The role of the current fluctuations has been attracting much attention recently (Chance et al., 2002; Kuhn et al., 2004; Moreno-Bote and Parga, 2005). They were proposed as a candidate mechanism to generate the large variability found in the activity of cortical cells (Softky and Koch, 1993), which would hypothetically operate in a fluctuation-driven regimen (FDR) (Gerstein and Mandelbrot, 1964; Shadlen and Newsome, 1994). There, the mean synaptic current generated by the network is not sufficient to drive the neurons toward threshold, and only the fluctuations may trigger discharges (van Vreeswijk and Sompolinsky, 1996). Depression transforms these fluctuations in a non-trivial manner; however, no studies have analyzed its impact on the neuron response.
Here, we use a computational model to study the role of the fluctuations on the stationary neuron response when the synapses are stochastic and show STD. We analyze the impact of STD on three factors: (1) the stochasticity of the transmission, (2) the redundant connectivity of the neurons (i.e., connections made of several contacts), and, most relevant, (3) the correlations between afferent spikes, analyzed using stimuli with different spatiotemporal structure.
It will be shown that (1) STD may lead naturally to the FDR, because it automatically sets µI below the threshold current; (2) in this regimen, fluctuations can be modulated by the input rate even if the mean current has reached saturation; (3) at high rates, STD tends to eliminate the impact of the input correlations; and (4) as a result, correlations cause I to exhibit a non-monotonic behavior as a function of , behavior that is inherited by the response rate.
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Figure 1. Illustrations of the synaptic connection characteristics in the stochastic model. A, B, Two examples with different numbers of functional contacts per presynaptic cell (M = 3 and 2, respectively). At each contact, an independent model of vesicle turnover is implemented with identical parameters. For the analysis performed in Figure 4, only the cases that, like these two examples, have the same total number of contacts N x M are compared. C, Recovery of a vacancy in the RP is stochastic and occurs independently at each vacancy. The mean recovery time  v is scaled with the size of the RP, so that cases with different N0 can be compared fairly. D, Plot of the release probability (release prob.) versus the number of releasable vesicles (Eq. 1) with U = 0.75 and N0 = 4.
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Preliminary results have appeared in abstract form (Moreno et al., 2002a).
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Materials and Methods
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The synapse model
Because connected neurons rarely are connected by a single synapse, and synapses sometimes have more than one synaptic specialization (the particular area in which vesicle exocytosis takes place) (Walmsley et al., 1998), we will consider connections made of M "functional contacts," the term used to refer to any synaptic specialization in which release takes place (Zador, 1998). Contacts sharing the same presynaptic cell will be referred to as "common" contacts. Figure 1, A and B, illustrates two examples in which each presynaptic cell establishes M = 3 and M = 2 common contacts, respectively. Cases with different values of M will be compared at a fixed total number of contacts. This is done by varying the number of presynaptic neurons as in the examples shown in Figure 1, A and B.
At each of these contacts, a simple stochastic model of vesicle turnover is implemented (Vere-Jones, 1966; Maass and Zador, 1999; Wang, 1999). The model assumes a vesicle "releasable pool" (RP) of size N0, that is, it can hold up to N0 vesicles. Vesicles within this pool are thought to be "docked" and will also be referred to as "readily releasable" vesicles. We assume the "univesicular release hypothesis," which states that, independently of the number of docked vesicles, an action potential (AP) can at most trigger the fusion of one vesicle per functional contact (Edwards et al., 1976). The release of a docked vesicle is modeled stochastically, that is, on arrival of a spike, a release is triggered with a certain probability. This probability is a function of the current number of releasable vesicles, n, as follows:
 | (1) |
as proposed in a study of the hippocampal CA3–CA1 synapse (Dobrunz and Stevens, 1997; Dobrunz, 2002). This dependence on n is illustrated in Figure 1 D, where one can observe that the parameter U represents the release probability when there is only one vesicle ready. The recovery of vesicles when the RP is not fully replenished is also modeled randomly: given that the number of vesicles ready to become docked is much larger than N0 (De Camilli et al., 2001), the replenishment of a vacancy in the RP can be taken as the first event from a Poisson process with homogeneous mean recovery time v (Fig. 1C). This modeling is compatible with the experimental observation that the recovery of the release probability from depletion can be fitted with an exponential (Dobrunz and Stevens, 1997) (data not shown). When comparing instances with RPs of different sizes, the recovery time v will be normalized so that the ratio v /N0 is the same in all of the examples. This normalization is sketched in the two examples shown in Figure 1C, in which the case with N0 = 4 has a recovery time twice as large as that with N0 = 2. This scaling makes the comparison between synapses with different N0 values easier, because (as will be shown later) the current statistics in the limit of large depends only on the ratio v /N0.
The average state of the synapses will be described by the probability that an afferent spike reaching a functional contact triggers the release of a vesicle. This "transmission probability" (Pt) will be obtained by computing over a long stimulus the fraction of functional contacts that, being hit by an AP, elicit a synaptic response.
Presynaptic activity: description of the stimuli
We will consider four different types of presynaptic activity: (1) "uncorrelated" stimuli, (2) "synchronous" stimuli, (3) "autocorrelated" stimuli, and (4) "phase-locked periodic" stimuli.
Figure 2 shows a spike rastergram for each stimulus type, which serves to explain graphically the reason of their choice. Because spikes are independent, the raster obtained with the uncorrelated stimulus looks very homogeneous in comparison with the rest, because it lacks any kind of spatial or temporal structure (Fig. 2 A). In contrast, the other three stimuli were chosen as simple examples with only spatial (Fig. 2 B, synchronous), only temporal (Fig. 2C, autocorrelated), and both spatial and temporal structure (Fig. 2 D, phase-locked). In each of these rasters, the corresponding structure is easily caught by the eye. Because of the presence of correlations, the global instantaneous activity of each stimulus (drawn below each raster), presents a different profile: despite its stochastic nature, the uncorrelated stimulus displays a much smaller temporal variability than the other three.
The stimuli are defined in terms of the "statistics" of the spike trains. Therefore, different trials do not reproduce the position of the spikes precisely, only the statistics of the presynaptic trains is the same. During a trial, each of the N presynaptic neurons elicits a spike train composed of a sequence of APs at the times  , which we write as (i = 1,2,... N):
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Its mean firing rate () is defined as the mean number of spikes per unit time. Cross-correlations are quantified in terms of the correlation between two spikes of different spike trains (i  j),
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the brackets indicating average over trials, whereas a similar expression is used to define the autocorrelation, which measures the degree of correlation between two spikes of the same spike train,
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Uncorrelated stimuli. This stimulus is built under the assumption of complete independence between any two spikes (regardless of whether they are fired by the same or different cells). In practice, the stimulus consists of a population of presynaptic neurons firing independent Poisson trains with the same constant rate, . We will use these stimuli to probe the synaptic model and the effects of varying different synaptic parameters; the elicited neuron response will serve as a control condition with which to compare the responses to correlated stimuli.
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Figure 2. Spike rastergrams of the presynaptic population for the four stimuli studied. A, The uncorrelated stimulus lacks any kind of spatiotemporal structure and produces a very homogeneous rastergram. B, The synchronous stimulus has a clear spatial structure, because some spikes from different cells fall aligned in time (inset), but it lacks any temporal structure. C, The autocorrelated stimulus presents a temporal structure appreciable as horizontal filaments. The inset illustrates three rasters from a single neuron firing with different autocorrelation magnitude: as  increases, the spikes tend to cluster in bursts. D, The phase-locked periodic stimulus displays both temporal and spatial correlations. The inset shows rasters from three cells. The common probability distribution function of spike occurrence is shown as dashed lines. The global instantaneous activity (below each raster) was obtained from a single trial using a bin size dt = 10 ms. Thus, it exhibits the variability perceived by the postsynaptic neuron in the spike count in a time window of the size of its membrane time constant (i.e., m = 10 ms). These four examples show that spike correlations increase this variability enormously. Parameters are as follows: N = 10,000, = 5 Hz,  = 0.002 (B), = 1.5 (C), f = 40 Hz, and  = 4ms(D).
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Synchronous stimuli. The synchronous stimulus represents the simplest example of instantaneous cooperative behavior between cells. To isolate the effect of spatial correlations from temporal correlations, we consider individual trains without autocorrelations (i.e., they all follow a Poisson statistics with the same rate). The different trains, however, are not generated independently, but they have a certain degree of synchrony, which means that each cell, beyond discharges occurring synchronously by chance, emits a fraction of its spikes at the same time as some other cells (Fig. 2 B, inset). This is quantified in terms of the cross-correlation between the spike trains of two presynaptic cells, which takes the following form:
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The correlation coefficient () gives the probability above chance that, given that fiber i produced a spike at time t, fiber j produces another spike at the same time. It quantifies with values between 0 and 1 the strength of the cross-correlations. It is 0 when all of the presynaptic neurons fire independently, and it is 1 when they fire replicas of essentially the same spike train.
To numerically generate the synchronous stimulus, we first generated a Poisson "mother" train with rate /. Next, each individual train was built as a thinned copy of the mother train from which some spikes, randomly picked, were eliminated with a probability of 1 – per spike.
Autocorrelated stimulus. This stimulus is a simple example of cooperative behavior between spikes from the same cell. We chose a stimulus with positive temporal correlations between pairs of spikes from the same cell but not between spikes from different presynaptic trains, which has a simple numerical implementation. More specifically, we generated independent renewal trains with exponential autocorrelations [for details, see de la Rocha et al. (2002)]:
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and without cross-correlations: Cij(t,t') = 0(i j). The magnitude of the autocorrelations is measured by the dimensionless parameter , which intuitively sets the excess of probability of, given one spike, finding another one within a time range c. Notice that can be varied keeping the firing rate fixed. The inset in Figure 2C shows three examples of spike trains generated with the same rate = 10 Hz, the same correlation range c = 2 ms, but different values: 0 (which is a Poisson process), 0.6, and 1.5. It is clear that, for larger values of , the APs cluster in bursts made of more spikes (although their exact number varies from burst to burst). The interval between consecutive spikes within a burst is of the order of c.
Phase-locked periodic activity. As a last example, we combined a simple periodic temporal structure with a spatial organization in which the cells coordinate their firing by phase-locking their spikes to an external oscillation of frequency f. Individual spike trains are constructed in the following way: each cell emits one AP at the same temporal phase of the oscillation, but with a certain jitter. The value of the jitter of every spike is drawn from a Gaussian distribution of deviation , truncated on the sides at a half-period distance (to prevent the tails of consecutive Gaussians from overlapping). The parameter sets the precision of the phase-locking of each neuron, and its value is kept fixed as the oscillation frequency is varied. This pattern of spikes, which represents the output activity of a network that oscillates rhythmically (Buzsaki and Draguhn, 2004), is illustrated in Figure 2 D.
Synaptic current
The activity of the N neurons of the presynaptic population gives rise to the stimulus excitatory current Istim(t). Each AP can produce at most the release of M vesicles. The release from a single vesicle is modeled as an instantaneous pulse of current of size J. Thus, the transformations of the train of APs into a sequence of instant current pulses is formulated as followsa:
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The sums on the right run over presynaptic neurons (i = 1... N), over common contacts for each presynaptic neuron (n = 1... M), and over the releases produced at each particular contact (k = 1... rel). We ignored the variability in the number of common contacts found in experiments (Gil et al., 1999; Silver et al., 2003), so that all connections are made of the same number of them. On the contrary, we captured the variability of quantum response across contacts by taking the efficacies Ji,n randomly distributed with a Gaussian of mean J and coefficient of variation  = 0.2–0.4 (Gil et al., 1999). In the simulations, we will specify the value of the ratio of the efficacy and the membrane capacitance, J/Cm, which measures the quantal amplitude (i.e., the mean amplitude of the EPSP produced by the release of a single vesicle) in voltage units.
The background current Ibg(t), which represents the afferent activity from other cells of the network or from different brain areas, is generated by Poisson events activating one excitatory and one inhibitory synapse at constant rates E and I, respectively. These rates are of the order of several spikes per millisecond, because they represent the superposition of thousands of presynaptic trains at low cortical spontaneous frequencies (e.g.,  1–2 Hz).
The total instantaneous synaptic current reaching the target neuron is the sum I(t) = Istim(t) + Ibg(t). We will normally be interested in its mean and SD. To evaluate them, we will divide the duration T of a long trial into small bins (of size dt), build the histogram of the current, and compute its mean µI and deviation I as follows:
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where data are taken starting from a time when the response of the target neuron has reached its stationary state (depending on the input rate the stationary regimen has reached after 0.1–1 s).
Without background component, the mean current µI has a simple analytical expression, because it is equivalent to the mean total charge entering the cell per unit time: because each of the NM contacts is hit by APs per unit time, and only a fraction Pt of them trigger the influx of the quantum charge J, we have the following:
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The deviation I measures the magnitude of the "fluctuations" produced by the stochasticity of the incoming spikes and of the synaptic transmission.b
The model target neuron and the analysis of the response
We model the target neuron as a simple leaky integrate-and-fire (LIF) unit (Ricciardi, 1977). This model describes the dynamics of the membrane potential V(t) when it is below threshold by the following equation:
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Here, Cm is the total membrane capacitance, gL is the leak conductance, and EL is the leak potential. The membrane time constant is m = Cm/gL. The terms Istim(t) and Ibg(t) represent the stimulus and background current, respectively (see above). Eq. 11 is used until V(t) reaches the spike generation threshold  . At that point, an AP is discharged, and the potential V(t) is reset to H, where it is held during a refractory time ref. The "current threshold" is defined as the value of the current that depolarizes the neuron exactly at its firing threshold; for the integrate-and-fire neuron, it reads I = gL.
In this work, we will deal only with the "stationary" regimen, that is, the response of the neuron will be computed after the transmission probability of its synapses has reached its steady state. We will characterize the response mainly by means of the output firing rate (out), defined as the mean number of output APs per unit time. The output firing rate as a function of the afferent rate will be referred to as the "response function."
We will also compute the coefficient of variation of the response interspike intervals, CVout, defined as the ratio between the SD and the mean of the output interspike interval. This is a measure of the variability of the output trains, which is 0 for purely periodic trains, equals 1 for Poisson inputs, and is usually >1 for bursty spike trains. For the case of a phase-locked stimulus, we will also compute the response vector strength, VSout, which measures the degree of phase-locking of spike responses varying between 0, for no temporal alignment, and 1, for perfect phase-locking (Goldberg and Brown, 1969).
Parameter values
Unless specified otherwise, we will use the following parameter values: integrate-and-fire model, EL = 0 mV, Cm = 300 pF, m = Cm/gL = 10 ms, ref = 2 ms, = 15–9 mV, H = 10–6 mV; synaptic model, v /N0 = 600 ms, U = 0.75, J/Cm = 0.25 mV, N0 = 1–8; background, E = 3.7 spikes/ms, I = 1.2 spikes/ms, JE/Cm = 0.25 mV, JI/Cm =–0.35 mV. The total number of synaptic functional contacts N x M equals 2000.
Simulations
Because the synaptic current is made of instantaneous pulses of current formally modeled as Dirac  functions (Eq. 7), the evolution of the potential (Eq. 11) can be solved in the simulations exactly [i.e., the numerical method used was not approximative (e.g., Runge-Kutta), but it was an exact integration]. The response of the neuron was simulated from three to five trials of 40–100 s, obtaining statistical errors for the output variables of the size of the symbols used in the plots. All simulations were performed on a PC running under SUSE Linux. All graphs were made using the plotting tools Grace and Xfig.
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Results
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We will start by introducing the "causal" relationship that, in our model, exists between depression and the FDR. It will be shown that, in this regimen, capturing the synaptic stochasticity in the model of STD has an enormous impact on the response behavior of the target cell. This will be illustrated by probing two different models of STD with uncorrelated poissonian stimuli. The response here will serve as a control condition to compare the response to correlated stimuli analyzed later. The number of functional contacts will also be proved to be very important in shaping the response function.
The fluctuation-driven regimen
One of the most outstanding effects of STD is that the mean stationary EPSP amplitude decreases with the afferent spike rate , behaving as 1/ for large enough presynaptic rate (Abbott et al., 1997; Tsodyks and Markram, 1997) (Fig. 3A). The mean current µI is proportional to the product of the mean EPSP and (Eq. 10) and therefore saturates as increases (Fig. 3B). The average membrane potential, which can be approximated by  V  µI/gL, is also upper bounded and its maximum value is given by the following:
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where µmax is simply obtained by substituting in Eq. 10 the rate of releases Pt by its maximum value, given by the rate at which vesicles recover, 1/v. The question now is whether this upper bound can constrain the cell to be, regardless of the presynaptic rate, in the so-called "fluctuation-driven regimen" (Gerstein and Mandelbrot, 1964; Calvin and Stevens, 1968; Shadlen and Newsome, 1994). In the FDR, the mean depolarization must be lower than the threshold, a condition that is formulated as follows:
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which after substituting Vmax by the expression above can be reformulated as follows:
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Assigning some realistic values to the parameters on the right-hand side allows us to estimate the number of presynaptic neurons required to violate this inequality. Taking = 15 mV, MJ/Cm = 1 mV (which is a representative value of the nondepressed EPSP size), v = 500 ms, and m = 10 ms, we obtain that, if N < 750, the condition given in Eq. 13 is satisfied. We confirmed this result by going through the same analysis with a conductance-based LIF model obtaining a boundary of N < 600 (see Appendix). Although the total number of afferent connections received by a cortical neuron exceeds these figures (Braitenberg and Schüz, 1991), estimations of the size of what can be considered a population of neurons encoding the same information fall below this number [e.g., around a hundred (Shadlen and Newsome, 1998)].c
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Figure 3. Comparison between the stochastic and the deterministic models of STD. The two models (stochastic, squares; deterministic, circles) are compared in the case of asynchronous Poisson input spike trains. A, The transmission probability, which represents the mean normalized EPSP size in the deterministic model, decreases for high enough as 1/. B, The mean current, which coincides for both depression models, saturates below I (dotted line), implying an FDR. C, The current deviation behaves differently in each model: in the deterministic model, it shows a non-monotonic behavior and takes substantially lower values than in the stochastic model in which it is approximately monotonic. D, This affects significantly the value of out, which is much smaller for the deterministic model and shows a non-monotonic behavior inherited from I. D, Inset, out in the suprathreshold regimen (the threshold was reduced to = 7.4 mV), where I does not determine the response behavior and the two models give very similar output rates. E, F, Current and the potential traces for the stochastic (E) and the deterministic (F) models when = 60 Hz. The current traces at this high input rate reveal that, whereas the large current fluctuations produced by the stochastic model enable V(t) to reach threshold, the deterministic model does not generate large enough fluctuations, and the response is smaller. The current histograms are plotted on the right. The value of µI is superimposed on the current traces (solid line) together with the values µI ± I (dashed lines). is represented by the dotted line, whereas I is indicated in the histograms by arrowheads. Parameters are as follows: stochastic model, M = 5, N0 = 1; deterministic model, J/Cm = 1.25 mV, M = 1; all models, = 0.4, = 9.3 mV (for plots A–D), 7 mV (D, inset), and 10.5 mV (for the traces), and H = 6 mV. The background activity was not included. The rest of the parameters are as specified in Materials and Methods. spk/sec, Spikes/second.
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Depression therefore implies that a neuron receiving an input signal from a few hundred cells will work in the FDR. This is true in the stationary state, that is, after the short transient interval required to depress the synapses. In the FDR, current fluctuations play a fundamental role in driving the response of the neuron, because the membrane potential reaches only on the arrival of positive fluctuations of I(t). That is why their correct modeling takes special importance in this regimen.
Current fluctuations in the deterministic and stochastic models of STD
To show the relevance of the stochastic nature of the synaptic transmission, we compare the current deviation and the neuron response produced by two models of STD, namely, the stochastic model of vesicle turnover described in Materials and Methods and the widely used deterministic model of "averaged" synaptic responses (Abbott et al., 1997; Tsodyks and Markram, 1997). When N0 = 1, both models give, by construction, the same mean current (Fig. 3B), but they lead to current fluctuations of different magnitude and different behavior as a function of the rate (Fig. 3C). Because the deterministic model was proposed to fit synaptic responses averaged over trials (Tsodyks and Markram, 1997; Varela et al., 1997), it lacks the trial-to-trial variability observed in synaptic transmission (Stevens and Wang, 1994), and therefore the fluctuations it generates are smaller than those produced by the stochastic model, particularly at high input rates (Fig. 3C). Specifically, in the stochastic model, a smaller probability of transmission, Pt, increases the fraction of failure spikes. In contrast, Pt equals the normalized mean EPSC in the deterministic model, and its decrease produces just smaller PSC amplitudes (something that contradicts the quantal hypothesis) without filtering any of the incoming spikes. At very large rates, the current in the deterministic model is composed of a series of PSCs of negligible size, closely spaced in time. This synaptic current has almost no fluctuations and resembles, in the limit of large input rate, an injection of constant current. This subtle difference makes I grow monotonically with toward its saturation value in the stochastic model, whereas in the deterministic model I shows a non-monotonic behavior tending to 0 at high input rates (Fig. 3C).
Because the target cell is in the FDR, the amplitude of the afferent fluctuations mainly determines its response function. Thus, out is much larger and saturates monotonically in the stochastic model, whereas in the deterministic model it inherits the non-monotonic behavior of I and vanishes at high rates (Fig. 3D). Figure 3, E and F, illustrates traces generated by both models at = 60 Hz. Both produce the same mean depolarization, but the smaller magnitude of the current fluctuations in the deterministic model generates smaller fluctuations of the potential, making the neuron fire at a lower rate.
If the mean current saturates above threshold (i.e., the suprathreshold regimen), the two models give similar rates, because in this case the response is basically determined by the mean current drive and the fluctuations play a secondary role (Fig. 3D, inset).
The stochastic model seems, in summary, a more appropriate model to investigate the response in the FDR, given that the deterministic model leads to the artifact of vanishing fluctuations at high , an effect that would contaminate the analysis performed in what follows.
Effect of the synaptic parameters M and N0 on the neuron response
We turn now to investigate the relevance of the size of the RP (N0) and the number of functional contacts per presynaptic neuron (M) in the response to uncorrelated stimuli. Studying the response produced by synapses with several functional contacts will serve us to understand the impact of spike synchrony analyzed later. This happens because multiple functional contacts trivially give rise to the synchronous release of vesicles, which will produce the same effect qualitatively as a synchronous stimulus.
The size of the RP is a controversial issue. Different experiments in hippocampal slices have led to different values: whereas Stevens and collaborators report values between 2 and 20 with mean N05 (Dobrunz and Stevens, 1997; Murthy et al., 2001), other groups have reported that the immediately releasable pool, or "primed" pool, has a size close to 1 (Hanse and Gustafsson, 2001). Theoretical analysis of paired-pulse depression in cortical neurons has reached the same conclusion, namely, that in some synapses there must be a last "bottleneck" pool that only holds around one vesicle (Matveev and Wang, 2000b).d For a fair comparison of synapses with different N0, the recovery time was adjusted to keep the ratio v/N0 fixed (see Materials and Methods) (Fig. 1C).
Regarding the number of common contacts, it is a widely variable quantity across brain areas. An average number from the somatosensory cortex lies around 5 for pyramidal cells (Markram et al., 1997; Silver et al., 2003) and 17 for GABAergic interneurons (Gupta et al., 2000). Among noncortical synapses, one can find examples in which M rises up to 15–20 in the spinal cord (Walmsley, 1991), 20 in the cerebellum (Pedroarena and Schwarz, 2003), and >1000 in the famous end-bulbs of Held (Held, 1893; Schneggenburger et al., 2002). For a fair comparison of cases with different M values in the following analysis, the total number of contacts, N x M, is kept fixed (Fig. 1A,B).
The comparison of the transmission probability (Fig. 4A) and the mean number of readily releasable vesicles (Fig. 4B) for two different RP sizes (N0 = 1 and N0 = 4) reveals that these two synaptic types differ only at low frequencies ( < 15 Hz). In this range, synapses with larger N0 have a larger Pt and, on average, have more vesicles ready for release. However, as the input rate increases, more spikes reach the synapses per unit time and the RP becomes more depleted. Both the transmission probability and the mean number of releasable vesicles tend to 0 as 1/ (dashed lines in Fig. 4A,B represent the fit 1/). At these high rates, synapses with different N0 are indistinguishable. In other words, a synapse with N0 > 1 behaves as a synapse with a single vesicle RP, but with a recovery time constant v/N0 (Matveev and Wang, 2000a; de la Rocha, 2003). This can be understood by noting that, at high input rates, the RP is normally empty and sometimes has at most one docked vesicle. Thus, if for instance N0 = 2, after complete depletion of the RP, the replenishment of one vesicle can come from the recovery of any of the two vacancies, so that it can be viewed as a single vesicle process with a twice as large recovery rate.
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Figure 4. Impact of the RP size N0 and the number of common contacts M on the response. A, B, The transmission probability (A) and the mean number of ready-releasable vesicles (B) of synapses with N0 = 1 (open) and 4 (solid) is plotted against . Regardless of the value of N0, both quantities follow the same 1/ dependence at high (dashed lines are the fit 1/). C, The mean current, which is independent of M, saturates to the same value for both N0 values. Before saturation, it is larger in the case N0 = 4 because of the larger value of Pt. The dotted line represents I and indicates that the cell is in the FDR. D, E, The current deviation is plotted for several values of M in the two cases N0 = 1 (D) and N0 = 4 (E). In both cases, synapses with more common contacts generate a larger I, but at high rates the deviation converges to a common value regardless of the number M. F, G, The response firing rate acquires a resonant behavior when the number of functional contacts is large enough (for M  5if N0 = 4 and for M 10 if N0 = 1). The asymptotic value ofout does not vary with M, meaning that at a large input rate, different M values give rise to the same response. F, inset, shows the maximumout versus M. The dashed line indicates the asymptotic value of out (i.e., the out obtained when   ). Parameters are as in Figure 3A–D except that = 10.6 mV and H = 9 mV. Background is not included. Notice the log scale used in the horizontal axes. spk/sec, Spikes/second.
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The mean current produced by synapses with N0 = 1 and N0 = 4 is plotted in Figure 4C. In both cases, vesicle depletion produces the saturation of µI at high input rates. Because, as it was just shown, at high rates these synaptic types are indistinguishable, it is expected that they produce the same current statistics. Before the mean saturates, synapses with larger N0 generate a larger µI because the transmission probability is larger. With the parameters chosen here, µI saturates below current threshold (Fig. 4C, dotted line), which implies that the target cell works in the FDR at all input rates.
The current SD is non-monotonic
As opposed to the mean current, which depends only on the total number of functional contacts, N x M, current fluctuations are very sensitive to the particular value of M (Fig. 4D,E): when the synaptic connections have several common contacts, the variability of the current is larger because the PSCs are made of 1, or 2,..., or M quantum events. In contrast, if M = 1, the PSCs are always unitary and only the random trigger of two simultaneous releases by different presynaptic cells may occasionally generate larger PSCs. This can be seen in Figure 4, D and E, where I is plotted for several M values (for N0 = 1 and N0 = 4, respectively): at low and moderate input rates, M sets the gain of the fluctuations so that larger M values produce larger I values. However, as increases, I converges to a value independent of M. This asymptotic convergence, in combination with the increase at low rates, endows I with a non-monotonic behavior when M > 1, regardless of the value of N0.
Why does the effect of having M > 1 vanish at high input rates? Figure 5 illustrates the explanation: top and bottom diagrams illustrate the sequence of releases (bars) and recovery intervals (horizontal bands) produced at five common contacts by a sequence of afferent spikes (top bars) reaching the presynaptic terminals at = 2 Hz (diagram A) and = 40 Hz (diagram B). Because N0 = 1, when a spike arrives during the recovery interval of a contact, it produces a failure (dots) and otherwise it triggers a release (i.e., U = 1). At the arrival of the first spike, the five contacts are fully recovered, and the first AP triggers the synchronous release of five vesicles. In the case in which = 2 Hz, each contact has time to recover before the next spike arrives, and thus the synchrony in the releases across the M common contacts is maintained. The PSCs generated at this input rate are large, because they are normally composed of three or four quantum events, and therefore they generate large current fluctuations. In contrast, when = 40 Hz, the second AP arriving at the presynaptic terminal finds all contacts depleted and the next "unitary" release does not occur until one contact is eventually replenished. Because the replenishment of vesicles occurs independently at each contact, releases occur asynchronously producing mostly unitary PSCs that give rise to smaller current fluctuations. In conclusion, at high rates the releases produced at common contacts become effectively asynchronous, because the transmission probability becomes so small that, on arrival of an AP, at most one of the contacts succeeds in releasing transmitter.
Non-monotonic behavior of the neuron response
Because the target cell has been set in the FDR, this non-monotonic modulation of the current fluctuations has a great impact on the output firing rate (Fig. 4F,G): the response function inherits the non-monotonic behavior of I and converges at high input rates to a value independent of the number of functional contacts. Because M sets the gain of the fluctuations for low and moderate values of , it consequently also sets the gain of the response rate.
The non-monotonicity is more prominent for synapses with N0 = 4 than for those with N0 = 1. This occurs mainly as a consequence of the larger mean current for N0 = 4 (Fig. 4C) but also because the magnitude of the current fluctuations is a little larger in that case (Fig. 4, compare D, E). Figure 4F, inset, shows the maximum output rate plotted versus M for those two cases: one sees that N0 sets the sensitivity of the response to variations of the gain parameter M. For this reason, the non-monotonic shape of out appears clearly for all M > 4 when N0 = 4, whereas a much larger value of M (M > 9) is required when N0 = 1.
A graphical explanation of why the response function acquires a resonant shape can be viewed in Figure 6. The figure shows the afferent spikes (top rasters), transmitter releases (bottom rasters), synaptic current (top traces), and membrane potential (bottom traces) produced by a presynaptic population with a rather large M = 16, which serves to illustrate the effect more clearly. These variables were monitored at low ( = 5 Hz) and high ( = 100 Hz) input rate. At = 5 Hz, the fewer APs per time unit trigger synchronous releases at common contacts (Fig. 6A, bottom raster); because we set N0 = 4, these synchronous releases were approximately preserved even in the case that incoming spikes came closely spaced in time (Fig. 6C). Because of the large PSCs produced, the potential could reach threshold very frequently, giving rise to a large spiking response. At = 100 Hz, the structure of common contacts is not perceptible in the release raster (Fig. 6B, bottom raster). The PSCs generated, which are now more, are mostly unitary and the fluctuations of the current are severely reduced. The potential, which has a mean that now lies closer to the threshold, does not fluctuate in the strong manner that it did at low input rate, and therefore it reaches fewer times. When the input rate is very low ( < 2 Hz), although the magnitude of the PSCs can be very large, they are so few that the output rate is also low (traces not shown).
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Figure 5. Desynchronization of releases produced at M common functional contacts. A, B, In each diagram, top bars represent the input spike train for = 2 and 40 Hz, respectively. Below, each row represents the releases (bars) produced by those spikes in each of the M = 5 synaptic contacts established by a presynaptic cell. The releases are followed by the vesicle recovery intervals (horizontal boxes), something that occurs randomly and independently across contacts. During these intervals, spikes fail to triggeranyresponse(solid dots); otherwise they produce are lease (i.e., we set N0 = 1 and U = 1). It is clear that, at low rates ( = 2in A), the releases occurring at the common contacts are well synchronized, whereas, at higher rates ( = 40 Hz in B), they occur almost independently. Note the different time scale of each diagram.
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Figure 6. Current and potential traces produced by synapses with many common contacts. In A and B, the rasters represent spikes from the presynaptic population (top) and releases triggered by six presynaptic cells (bottom). The grid dashed lines group together releases from common contacts. The bottom traces show the synaptic current (top) and the membrane potential (bottom) of the postsynaptic cell. The dotted line represents the threshold. A and B represent low and high input rates, respectively. A, Because M = 16, the releases triggered by common contacts look like vertical bars when they occur synchronously at low input rate. Releases inside the circle are redrawn at a larger scale in C. Synchronous releases generate very large fluctuations of the synaptic current, which make the membrane potential fluctuate strongly, giving rise to a large output rate. B, When the input rate is = 100 Hz, releases from common contacts do not show any synchronous structure and produce current with a smaller variability. The membrane potential shows smaller fluctuations, and because the target cell operates in the FDR, they give rise to a lower output rate. Parameters are M = 16 and N0 = 4, and the rest are as in Figure 3A–D.
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Synchronous stimulus
We will now discuss a correlated stimulus in which the presynaptic neurons fire synchronous Poisson trains with the same constant rate and a correlation coefficient (see Materials and Methods). We will show that the impact of synchrony across afferent APs resembles very much that of synchronous releases produced by common synaptic contacts, examined above.
As a general rule, modifying the correlations between presynaptic trains, while keeping their rate fixed, does not alter the "mean" synaptic current but may change the current deviation substantially. As a consequence, any information conveyed by the spatiotemporal correlations of the input is transmitted by the fluctuations of the current but not by the mean drive. For reliable static synapses, synchronizing the afferent spikes increases the probability of synchronous releases, which give rise to PSPs composed of several unitary events. This increases the magnitude of the fluctuations (Shadlen and Newsome, 1998; Salinas and Sejnowski, 2000; Moreno et al., 2002b), something that generally increases the output rate, particularly in the FDR (Salinas and Sejnowski, 2000; Moreno et al., 2002b).e For synapses showing STD, synchrony increases I too, although in a nonuniform way with (Fig. 7A): for low rates, the increase is larger than for high rates generating a non-monotonic behavior of I as a function of (Moreno et al., 2002a). This occurs because the impact of synchrony is weighted by the transmission probability Pt, which decreases with (Fig. 3A) (J. de la Rocha, R. Moreno-Bote, and N. Parga, unpublished observations). In other words, at high rates, the input synchrony does not affect the response, because it becomes rather unlikely that the simultaneous arrival of APs triggers the corresponding synchronous releases.
The non-monotonic behavior of I is again inherited by out, which displays a tuned dependence on with a "preferred" presynaptic rate, p. The preferred rate depends on the synaptic parameters v, U, and N0 (data not shown). However, it remains almost unaltered to variations of within a wide range of values of the correlation coefficient ( 0.01–0.1 for N0 = 1). Furthermore, sets the amplitude of the tuning curve acting as a "gain control parameter," for low and moderate input rates, within the range 0.01–0.06 (for N0 = 1, see Fig. 7B). The maximum response initially follows an approximately linear relationship with and later saturates (Fig. 7C).
In Figure 7D, we lowered the threshold while keeping the rest of the parameters fixed, so that the cell sits in the suprathreshold regimen. As expected, the response no longer has a resonant shape but increases monotonically for all correlation values. Increasing the degree of synchrony increases the rate, especially at low at which µI < I (i.e., the cell is still in the FDR), and depression is not very prominent. This change, however, does not represent an increase of the gain, as it approximately does in the subthreshold regimen (Fig. 7B). Hereafter, we will therefore restrict the analysis to the FDR and will come back to the general case in Discussion.
The gain modulation observed in the FDR and attributable to the afferent spike correlations has important consequences in the transmission of information carried by the rate to the postsynaptic cell. In the absence of synchrony, I is barely modulated by the input rate (Fig. 7A, full circles), so that the information can be transmitted by µI only in the range of low and moderate values, because at high presynaptic rates, µI undergoes very little modulation (Fig. 7A, solid line). Synchronizing the afferent APs acts as a gain for I (Fig. 7A), which because of depression may acquire a prominent non-monotonic behavior. Thus, when the mean current is saturated, the current fluctuations become the main carrier of information, enabling the modulation of the response (1) within a larger dynamical range of the response and (2) for a wider range of (Fig. 7A,B). For instance, for = 0.05, the response undergoes a 10 Hz variation in the range = 20–100 Hz (Fig. 7B), whereas µI barely varies in this range (Fig. 7A, solid line).
Figure 7E shows traces of the current and the membrane potential at three presynaptic rates, namely, at 1 Hz, at the preferred frequency p = 9 Hz, and at 80 Hz, for = 0.04. Apart from the synchronized stimulus, the target cell receives a constant background activity (see Materials and Methods). As the rate increases, the positive fluctuations in the current (produced by synchronous releases) occur more often but are smaller, and the mean current increases, taking the mean depolarization closer to the threshold. Therefore, the neuron responds maximally at p = 9 Hz, because the current reaches a compromise between large fluctuations and large mean (note that the preferred frequency in Fig. 7B does not produce the maximal fluctuations in Fig. 7A). The histograms of the current at these three rate values are shown in Figure 7, F–H, respectively (thick lines), along with the asynchronous case, = 0 (thin lines), drawn for comparison. At each frequency, both histograms have the same mean (dotted lines), which always falls below I (arrowheads in the figure), but at = 1 and 9 Hz they differ substantially (Fig. 7F,G). In these two cases, synchronous releases skew the distribution and increase its deviation, something that increases the output rate significantly. At higher rate ( = 80 Hz), the skewness, which is the signature of synchronous releases, disappears and the two distributions become similar.
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Figure 7. Effect of STD using a synchronous stimulus. The synaptic current and the neuron response are analyzed when presynaptic Poisson trains are synchronized with correlation coefficient (Fig. 2 B). A, B, Current deviation I and output versus for several degrees of synchrony. Synchrony endows the current fluctuations with a resonant behavior, which is inherited by the response rate. The degree of synchrony has almost no effect on the position of the maximum, but it finely sets the gain the response function. The mean current µI is also shown in A (solid line). C, Maximum response versus . Dashed line indicates a symptotic value of out. D, The threshold was lowered to = 11 mV to analyze the response in the suprathreshold regimen. E Current (top) and potential (bottom) traces for three input frequencies with = 0.04. Horizontal dotted line represents threshold. F–H, Current histograms computed from the traces shown in E (thick lines), along with histograms for the = 0 case (thin lines F, Inset, A magnification in the y-axis of the same plot in which the tail cannot be perceived. The vertical dotted line represents µI and the arrowheads represent the position of I. Parameters are as follows: = 15 mV, H = 10 mV, M = 5, and N0 = 1. White and black arrowheads in A signalI for the subthreshold and suprathreshold regimens. Background is included. The rest are as Figure 3A–D. spk/sec, Spikes/second.
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Figure 8. Autocorrelated spike trains. Simulation examples of fluctuations produced in the spike count (A, B) and release count (C, D, N0 = 4; E, F, N0 = 1) by a single spike train with and without autocorrelations (left and right panels, respectively) with a fixed rate of = 5 Hz. The spike count (or release count) per bin is plotted versus time on the left axes. Monotonically increasing lines represent the cumulative spike count (or release count) on the right axes. A, B, The Poisson input normally generates a single spike per bin and occasionally two. In contrast, the case of = 1.5 produces large positive fluctuations in the spike count because of the presence of bursts. C–F, These large spike count fluctuations are transmitted to the release count only if N0 = 4 (D), and they are essentially filtered if N0 = 1(F). When M = 5, the release count fluctuations are amplified. Concerning the total release count produced, it can be observed that the train with autocorrelations elicits fewer releases than the Poisson train regardless of N0 (compare D with C, and F with E). Bin size dt = 10 ms.
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To better understand the consequences that synchrony and STD may have in the transmission of information, we went beyond the analysis of the output rate and studied higher-order statistics of the response. In this direction, we performed a simulation of two target neurons receiving APs from nonoverlapping subpopulations, which are part of the same population of synchronously firing cells. We then computed the output correlation coefficient of the target pair, out, which because of the afferent correlations is different from 0. The observed general trend is that, as synapses depress with an increasing afferent rate, the correlation coefficient out drops off to 0 (data not shown). This is completely consistent with what we just saw, namely, that as synapses become more unreliable the impact of input synchrony vanishes, affecting the output firing rate but also weakening the ability of the presynaptic population to generate additional synchrony in the output ensemble of cells. We will go over the implications of this result in Discussion.
To summarize, the impact of synchronized incoming spikes is very similar to that of synaptic connections with several common contacts, but the functional implications are very different, because can be a dynamical variable, whereas M was a fixed parameter. Synchronized stimuli increase the current fluctuation gain at low and moderate input rates, something that enhances the gain of the non-monotonic response function.
Autocorrelated stimulus
Neurons in cortical and subcortical areas often display positive autocorrelations in their spike times (Bair et al., 1994; Dan et al., 1996; Baddeley et al., 1997; Goldman et al., 2002), meaning that the emission of a spike by a presynaptic neuron increases the probability of observing a new discharge of the same neuron over a short time interval (e.g., 10–20 ms). In other words, spikes are not homogeneously spread in time, but they tend to come in clusters. A typical case of short-range positive autocorrelations is the emission of bursts observed in many different areas (Lisman, 1997; Sherman, 2001; Krahe and Gabbiani, 2004). We used a particular instance of input statistics that produces spike trains with short-range positive exponential autocorrelations (see Materials and Methods). The magnitude of the autocorrelations is set up by a parameter , which yields a Poisson process for = 0 and trains with a bursty temporal structure for 1.5 ("bursts" composed of a variable number of spikes with mean 4–5). Examples of these spike trains for several values of can be seen in Figure 2C, inset.
For nondepressing synapses, positive autocorrelations have the same qualitative effect as spatial cross-correlations (Moreno et al., 2002b): the impact produced by incoming clusters of APs does not depend on whether the spikes within the cluster come from the same or from different presynaptic cells.f This simple principle seems not to be applicable when synaptic transmission depends on the presynaptic activity. With facilitating synapses, for instance, a cluster of spikes coming from the same or from different cells, would not produce the same depolarization: only if they are emitted by the same presynaptic neuron, the synapse facilitates and gives rise to a larger depolarization. However, we will see that, under the choice of certain parameter values of the stochastic model, bursts of spikes may produce qualitatively the same effect as the synchronous activity considered before, although with a quantitatively smaller overall impact.
Clustering the incoming spikes can increase the fluctuations of the synaptic current, because it may induce the temporal summation of consecutive PSCs elicited by the spikes within the burst. This increase of the fluctuations is analyzed at the level of the release statistics in Figure 8, where we compare two inputs with the same rate but different correlation magnitudes: = 0 (Poisson) and = 1.5 (bursts). Figure 8, A and B, shows the number of spikes falling in bins of size m versus time.g As expected, the correlated case produces large positive fluctuations because of the existence of bursts and the uncorrelated case looks very homogeneous, producing one or no spikes per bin and only occasionally two. Figure 8, C and D, depicts the number of "releases" produced by those spike trains when the RP size is N0 = 4 and M = 5. The bursty input still produces larger fluctuations than the Poisson stimulus. Notice that, because every AP can produce from zero up to five releases, the variability of the releases seems amplified with respect to that of the spike trains.
Figure 8, E and F, also shows a sequence of releases but for synapses with N0 = 1. In contrast to the case N0 = 4, the large fluctuations of the bursty input have been severely filtered out so that now both bursty and Poisson inputs produce fluctuations of similar magnitude. This difference can be easily understood: bursts produced after a long silent interval might find the RP fully replenished. Then, if N0 = 1, the burst can release at most one vesicle per contact, whereas if N0 > 1, the same burst can release up to N0 vesicles. This explains that, only in the case N0 > 1 and when preceded by a long enough silent interval, a cluster of APs can be faithfully translated into a significant fluctuation in the number of releases. At high rates, the required long silent periods never occur. The model with N0 |
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Abstract
Introduction
