Jean-Pascal Pfister
email: jean-pascal.pfister@eng.cam.ac.uk

Peter Dayan

Máté Lengyel

Synapses with short-term plasticity are optimal estimators of presynaptic membrane potentials

The trajectory of the somatic membrane potential of a cortical neuron exactly reflects the computations performed on its afferent inputs. However, the spikes of such a neuron are a very low dimensional and discrete projection of this continually evolving signal. We explore the possibility that the neuron's efferent synapses perform the critical computational step of estimating the membrane potential trajectory from the spikes. We show that shortterm changes in synaptic efficacy can be interpreted as implementing an optimal such estimator. Short-term depression arises when pre-synaptic spiking is sufficiently intense as to reduce the uncertainty associated with the estimate; short-term facilitation reflects structural features of the statistics of the pre-synaptic neuron such as up and down states. Our analysis provides a unifying account of a powerful, but puzzling form of plasticity.

 , or to a firing rate-based description of presynaptic activities 8 thereby

ignoring the fundamentally fast fluctuations in synaptic efficacies due to STP. Worse, the vast bulk of models of neural circuit information processing require synaptic efficacies to be constant over the short-term of a single computation, changing at most very slowly to average across the statistics of input, or changing only in the light of a gating mechanism [START_REF] Carpenter | Pattern recognition by self-organizing neural networks[END_REF][START_REF] Hasselmo | Acetylcholine and memory[END_REF] . These would seem to be incompatible with substantial STP. Here, we argue that, far from hindering such circuit computations, STP is in fact a near-optimal solution to a central problem neural circuits face associated with spike-based communication.

Although, as digital quantities, spikes have the mechanistic advantage of allowing regenerative error-correction, they are a significantly impoverished representation of the fast-evolving, analog, membrane potentials of the neurons concerned [START_REF] Zador | Impact of synaptic unreliability on the information transmitted by spiking neuron[END_REF][START_REF] De La Rocha | Information transmission by stochastic synapses with short-term depression: neural coding and optimization[END_REF][START_REF] Pfister | Speed versus accuracy in spiking attractor networks[END_REF] . These analog quantities are normally considered to lie at the heart of computations [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF][START_REF] Dayan | Theoretical Neuroscience[END_REF] , and so it is common to appeal to averages over space (i.e., multiple identical neurons) and/or time (i.e., slow currents) to allow them to be represented by spike trains [START_REF] Dayan | Theoretical Neuroscience[END_REF] . However, both sorts of averages are neurobiologically questionable. In many circumstances computations need to be executed in the matter of a few interspike intervals [START_REF] Thorpe | Speed of processing in the human visual system[END_REF][START_REF] Huys | Fast population coding[END_REF][START_REF] Stanford | Perceptual decision making in less than 30 milliseconds[END_REF] , precluding extensive averaging over time; and in many circuits neurons represent independent analog quantities, as in recurrent network models of autoassociative memories [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF][START_REF] Lengyel | Matching storage and recall: hippocampal spike timing-dependent plasticity and phase response curves[END_REF] , or partially independent quantities, as in surface attractor models of population codes [START_REF] Seung | Simple models for reading neuronal population codes[END_REF] . We make the alternative suggestion that the analog membrane potential of a neuron is being estimated in a statistically appropriate manner by its efferent synapses based on the spike trains that the neuron emits, and that STP is a signature of this solution.

In particular, the informativeness of an incoming spike about the membrane potential varies greatly depending on the uncertainty left by the preceding spike train. This makes the spike's impact significantly context-dependent. We demonstrate that key elements of this context-dependency are realized by synaptic depression and facilitation. Further, since incoming spikes are sparse, the behavior of the optimal estimator critically depends on prior assumptions about presynaptic membrane potential dynamics. Thus our approach allows us to make detailed predictions about how the properties of STP, implementing the optimal estimator, should be matched to the statistics of presynaptic membrane potential fluctuations.

Results

Postsynaptic potentials and the optimal estimator

We first define the optimal estimator of the continuously varying membrane potential of a presynaptic cell from its past spikes. We then show that it depends on these spikes in the same way as a particular measure of its efferent synapses' contributions to their postsynaptic membrane potentials. Since spikes are discrete, they cannot support recovery of with absolute certainty, and so the full solution to the estimation task is a posterior probability distribution [START_REF] Huys | Fast population coding[END_REF][START_REF] Anderson | Optimal filtering[END_REF][START_REF] Eden | Dynamic analysis of neural encoding by point process adaptive filtering[END_REF][START_REF] Paninski | The most likely voltage path and large deviations approximations for integrate-and-fire neurons[END_REF][START_REF] Bobrowski | Bayesian filtering in spiking neural networks: Noise, adaptation, and multisensory integration[END_REF][START_REF] Cunningham | Inferring neural firing rates from spike trains using Gaussian processes[END_REF] P u t | s 0:t

(

) over the possible values that the presynaptic membrane potential at time , , might take based on all the spikes observed so far, . The mean of this posterior is then the estimator that minimizes the squared error [START_REF] Anderson | Optimal filtering[END_REF] . We interpret the local postsynaptic potential at an excitatory synapse as representing this optimal estimate. This local potential is loosely defined as the sum of all EPSPs at this synapse (see Supplementary Note for a more precise definition); a filtered version of it is recorded in standard experiments into STP.

To be correct, the estimator must implicitly incorporate a statistically appropriate model of membrane potential fluctuations and spike generation in the presynaptic neuron [START_REF] Anderson | Optimal filtering[END_REF] . For the latter, we adopt the common characterization that a spike is created stochastically whenever exceeds a (soft) threshold [START_REF] Gerstner | Spiking Neuron Models[END_REF][START_REF] Paninski | Maximum likelihood estimate of a stochastic integrate-and-fire neural encoding model[END_REF] (Fig. 1; see also Supplementary Fig. 2). In this case, the occurrence of a spike implies that the membrane potential is likely to be high; the absence of spikes implies the membrane potential is likely to be low. Thus should increase following a spike, and decrease during interspike intervals (Fig. 1a, inset; Online Methods). The decrease should be gradual since, even at high membrane potential values, the firing rate of a neuron is limited, and so the absence of a spike is relatively weak evidence that the membrane potential is low. As required by our interpretation, the local post-synaptic potential, , at an excitatory synapse shows the same qualitative characteristics: it increases suddenly at the times of presynaptic spikes, and decays gradually towards a lower baseline between transmission events (Fig. 1a, inset).

However, this observation is only approximate. As we have mentioned, under STP, the actual size of an excitatory postsynaptic potential (EPSP) depends on the past history of spiking. Such history-dependence is also a hallmark of optimal estimation, because the evidence supplied by a spike needs to be evaluated in the context of the current state of the estimator, i.e., the current posterior [START_REF] Anderson | Optimal filtering[END_REF] . The posterior is computed on-line in a recursive manner, according to standard Bayesian filtering. In this, likelihood information from the current presence or absence of a spike, , P s t | u t (

) , is combined with the posterior computed in the previous timestep, P u t !"t | s 0:t !"t ( ) (Online Methods):

P u t | s 0:t ( ) ! P s t | u t ( ) "# # $ P u t | u t"%t ( ) P u t"%t | s 0:t"%t ( ) du t"%t (1)
Thus the impact of an incoming spike on the mean of the posterior, , will be contextdependent, varying as a function of the posterior propagated from the past. The precise nature of the context-dependence of the changes in will depend on the particular statistical model assumed for the dynamics of , P u t | u t !"t ( ) . Below, we consider two increasingly complex, models: one in which performs a random walk around a fixed `resting' (or baseline) membrane potential, ; and one in which itself also changes in time. This allows us to explore two fundamental factors contributing to the context-dependence of that correspond respectively to the effects of synaptic depression and facilitation on .

Synaptic depression and uncertainty

After eliminating the influence of spikes themselves, the simplest approximation to the statistics of the membrane potential of the presynaptic neuron is as an Ornstein-Uhlenbeck (OU) process [START_REF] Stein | A theoretical analysis of neuronal variability[END_REF][START_REF] Lansky | A review of the methods for signal estimation in stochastic di!usion leaky integrate-and-fire neuronal models[END_REF] . For this, the total input to the presynaptic cell is assumed to be Gaussian white noise [START_REF] Lansky | A review of the methods for signal estimation in stochastic di!usion leaky integrate-and-fire neuronal models[END_REF] , and is subject to leaky integration, decaying towards its resting value, (Fig. 1a; Online Methods and Supplementary Fig. 2). In this case, the posterior distribution P u t | s 0:t

(

) can be well characterized as a Gaussian, with mean , and variance , which expresses the estimator's current uncertainty about . Given the assumption about the way it is generated, observation of a spike provides evidence that is high. However, the quantitative effect on raising depends on the uncertainty . The less the uncertainty, i.e., the better known is already, the less the estimate should be influenced by a spike (Fig. 1b) and the lower the synapse's apparent efficacy. The uncertainty is determined by the evidence from past spikes; and thus we should expect the magnitude of the EPSPs to fluctuate according to this history.

Uncertainty in the optimal estimate increases during interspike intervals (Fig. 1c), because the absence of a spike is only weak evidence for a low membrane potential. Therefore, spikes that arrive after a longer period of silence should increase by more than spikes arriving in quick succession. This closely resembles, in fact all the way down to fine quantitative details, the effect of synaptic depression in a biophysically motivated canonical STP model [START_REF] Pfister | Know thy neighbour: A normative theory of synaptic depression[END_REF][START_REF] Tsodyks | Neural networks with dynamic synapses[END_REF] (Fig. 1). In that model, depression is mediated by the depletion of a synaptic resource variable which thus behaves as the biophysical analog of estimator uncertainty, .

In fact, for a paired-pulse protocol, the dynamical equations describing the time-evolution of the optimal estimator and its uncertainty, and , are formally equivalent to those of the biophysical STP model describing the time-evolution of the postsynaptic membrane potential and synaptic resource variable, and , respectively (Online Methods). As a result, higher presynaptic firing rates lead to diminishing postsynaptic responses both in the optimal estimator and in the STP model (Fig. 3a).

Synaptic facilitation and "up" state probability

The assumption that the presynaptic membrane potential follows an OU process is often too simplistic [START_REF] Shinomoto | The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex[END_REF] . Fortunately, it is straightforward in our framework to incorporate other statistical properties of membrane potential fluctuations and study their effects on the features of STP. For instance, many cortical cells show phasic activation patterns whereby their membrane potential alternates between a resting and a depolarized state, such as "up" and "down" states in the cortex [START_REF] Steriade | A novel slow (< 1 Hz) oscillation of neocortical neurons in vivo: depolarizing and hyperpolarizing components[END_REF][START_REF] Cossart | Attractor dynamics of network up states in the neocortex[END_REF] , or out-of-place-field and within-placefield activity for hippocampal place cells [START_REF] Harvey | Intracellular dynamics of hippocampal place cells during virtual navigation[END_REF] . This can be captured by extending the model for the dynamics of to allow the resting membrane potential itself to switch between two possible values, corresponding to an "up" and a "down" state (Fig. 2 ; see Supplementary Note and Supplementary Fig. 2).

In this extended model, the true current value of is unknown to the postsynaptic cell, just like . Thus, the full solution to the estimation problem is a posterior distribution expressing joint uncertainty about these two quantities (Supplementary Note). In this case, the estimated probability that is in its "up" state also reflects recent spikes, and influences ( , Fig. 2a-c) . Observing a spike when the current estimate of the membrane potential is compatible with the presynaptic neuron being in its "down" state increases this probability somewhat. Observing then a second spike in a short time window, providing substantial extra evidence for this "up" state, will thus cause a larger increment in (Fig. 2a, inset). EPSPs in the facilitating biophysical STP model and in actual facilitating synapses in a paired-pulse protocol [START_REF] Dobrunz | Very short-term plasticity in hippocampal synapses[END_REF] (Figs. 2d and3b) show the same effect.

Match between STP and the dynamics of the presynaptic cell

Our theory requires the synaptic estimator to be matched to the statistical properties of the presynaptic membrane potential fluctuations which it needs to estimate. Thus, fitting the optimal estimator to experimentally measured STP data (as shown in Fig. 3a-b) allowed us to predict, without further parameter fitting, properties of the membrane potential dynamics that the corresponding presynaptic cell type should exhibit. Testing these predictions is challenging because they are about the natural statistics of membrane potential fluctuations and thus require in vivo intracellular recordings, preferably from behaving rather than anaesthetized animals.

Nevertheless, starting from our fits to data about synaptic depression and facilitation for cerebellar climbing fibers (Fig. 3a) and hippocampal Schäffer collateral inputs (Fig. 3b), we predict membrane potential fluctuations in inferior olive neurons (Fig. 3e) and hippocampal pyramidal cells (Fig. 3f) respectively, that are in broad qualitative agreement with those found in vivo in these cell types [START_REF] Harvey | Intracellular dynamics of hippocampal place cells during virtual navigation[END_REF][START_REF] Chorev | Rhythmic episodes of subthreshold membrane potential oscillations in the rat inferior olive nuclei in vivo[END_REF] (Fig. 3c,d). Note that the absence and presence of marked "up" and "down" states in these two cell types respectively is predicted by our theory directly from the absence and presence of facilitation in their corresponding efferent synapses.

The advantage of STP

In order to quantify the advantage that STP brings to synapses for tracking the presynaptic membrane potential, we measured how well , or its biophysical analog, , performs on the estimation task in terms of the time-averaged squared error between and its estimate (Online Methods). We compared the optimal estimator of the presynaptic membrane potential (Figure 4a, red line) with the postsynaptic membrane potential occasioned by a synapse undergoing STP (blue line). These two traces are very close. Importantly, a static synapse without STP, whose fixed efficacy is still optimized for the same estimation task, performs significantly less well (green line).

Our account of synaptic dynamics assumes no transmission failures, or, equivalently, a large number of release sites between the pre-and postsynaptic cells. Therefore, we also ran simulations with stochastic synapses 4 (Online Methods) to test a more realistic regime of synaptic communication (Fig. 4b). The advantage of a dynamic over a static synapse was maintained when the number of release sites was low even in the face of transmission failures and the added variability of the dynamic synapse entailed by the stochastic restocking of vesicles.

Discussion

We showed that STP arranges for the local postsynaptic membrane potential at a synapse, , to behave as an optimal estimator of the presynaptic membrane potential, . We argued that this is central for a wide swathe of feedforward and recurrent neural circuits. In particular, it allows network computations based on analog quantities encoded in the somatic membrane potentials of neurons to be realized, even though their spikes offer only a low dimensional and discrete projection of those potentials.

As a first step, we concentrated on the interaction between a pair of cells, and on reproducing as close an estimate as possible to the exact presynaptic membrane potential on the postsynaptic side. Of course, can be changed by many factors other than synaptic currents, for example voltage signals propagating from other dendritic compartments or indeed backpropagating action potentials from the soma. However, under most experimental paradigms used to test STP [START_REF] Markram | Redistribution of synaptic e"cacy between neocortical pyramidal neurons[END_REF][START_REF] Thomson | Facilitation, augmentation and potentiation at central synapses[END_REF] , the magnitudes of these other factors are minimized, such that the somatic membrane potential recorded reflects, perhaps with some dendritic filtering, the local EPSPs at the stimulated synapse. This allows our theory to be directly applied to data obtained under such conditions (see Supplementary Note).

A significant apparent challenge to our theory, as indeed to many previous functional accounts of STP [START_REF] Abbott | Synaptic depression and cortical gain control[END_REF][START_REF] Cook | Synaptic depression in the localization of sound[END_REF][START_REF] Goldman | Redundancy reduction and sustained firing with stochastic depressing synapses[END_REF][START_REF] Mongillo | Synaptic theory of working memory[END_REF] , is that different efferent synapses of the same cell can express different forms of STP [START_REF] Markram | Di!erential signaling via the same axon of neocortical pyramidal neurons[END_REF] . Two particular classes of factors that can affect may account for this. Both factors have to do with the overall computation performed by the neuron in the network, which need not strictly factorize into individual estimation of each presynaptic membrane potential and combination of these across multiple synapses (see Supplementary Fig. S1).

First, computations might be based on estimates of different functions of the presynaptic membrane potentials than just their mean value, for instance their rates of change or higher order temporal derivatives. In this case, the different efferent synapses of a single neuron might estimate different functions, and therefore be different. It will be an interesting extension of our theory to study how synaptic estimation and single-neuron computation can be blended, rather than being performed as separate algorithmic steps.

Second, the long-run efficacies of synapses in a neural circuit also play a key part in the computations it performs [START_REF] Dayan | Theoretical Neuroscience[END_REF][START_REF] Martin | Synaptic plasticity and memory: an evaluation of the hypothesis[END_REF] , suggesting that presynaptic membrane potentials scaled by these computational factors should be reproduced instead. In this case, the optimal estimator just scales too. However, if the synapse suffers from saturation or other similar nonlinear constraints, then the form of STP that minimizes the error in the estimate of the scaled presynaptic potential would have to reflect these non-linearities, and change as the long-run weight alters. These changes could take potentially quite complex forms. Since different efferent synapses have different long-term weights, they could exhibit different forms of STP. Indeed, metaplasticity of STP due to long-term potentiation has been observed experimentally [START_REF] Markram | Redistribution of synaptic e"cacy between neocortical pyramidal neurons[END_REF][START_REF] Manabe | Modulation of synaptic transmission and long-term potentiation: e!ects on paired pulse facilitation and EPSC variance in the CA1 region of the hippocampus[END_REF] .

We should also note that differences in STP across efferent synapses have been predominantly shown for inhibitory interneurons [START_REF] Markram | Di!erential signaling via the same axon of neocortical pyramidal neurons[END_REF][START_REF] Reyes | Target-cell-specific facilitation and depression in neocortical circuits[END_REF][START_REF] Koester | Target cell-dependent normalization of transmitter release at neocortical synapses[END_REF] . In our theory, estimation of the membrane potential comes in the service of network computations, which are typically posed in terms of the excitatory principal cells rather than inhibitory interneurons. Therefore, the theory does not fully extend to cover inhibitory neurons.

Our theory employs a standard account of the relationship between and the presynaptic spikes, and so is dual to the suggestion 47 that generation is more complex so that estimation can be straightforward even with a static synapse. The latter account is not easy to reconcile with the fluctuations evident in STP. Others have pursued ideas more similar to ours about adaptive gain control mechanisms in neurons generally acting as optimal filters [START_REF] Wark | Timescales of inference in visual adaptation[END_REF] , and dynamic synapses specifically acting as estimators of presynaptic firing rates [START_REF] De La Rocha | Information transmission by stochastic synapses with short-term depression: neural coding and optimization[END_REF] (see also B. Cronin, M. Sur & K. Koerding. Soc. Neurosci. Abstr., 663. 6, 2007) or interspike intervals [START_REF] De La Rocha | Information transmission by stochastic synapses with short-term depression: neural coding and optimization[END_REF] . Some of these studies do not encompass STP; others address depression but not facilitation. Finally, those studies aim mainly at predicting a general advantage (if one exists at all 7 ) of dynamic synapses over static ones.

Our approach is unique in suggesting that synaptic dynamics are matched to the statistics of presynaptic membrane potential fluctuations which we were able to demonstrate at least qualitatively (Fig. 3c-f). Even such a qualitative match is noteworthy, given that the STP data we fitted were not extensive, were recorded in vitro under potentially very different stimulation regimes and neuromodulatory milieux for synaptic dynamics than those relevant in vivo, and given our highly simplified statistical model of presynaptic membrane potential dynamics. For example, a common dynamical motif shared by many neurons, including olivary neurons and hippocampal pyramids, and ignored by our model, is the presence of subthreshold membrane potential oscillations [START_REF] Harvey | Intracellular dynamics of hippocampal place cells during virtual navigation[END_REF][START_REF] Chorev | Rhythmic episodes of subthreshold membrane potential oscillations in the rat inferior olive nuclei in vivo[END_REF] .

The simplicity of our model of presynaptic membrane potential dynamics makes it hard to provide a direct biological interpretation of the optimal parameter values (Online Methods) from the fits to data. Further theoretical work would be necessary to incorporate higher-order statistical regularities of these dynamics into the model. Further empirical studies recording STP under more naturalistic conditions and in vivo membrane potential recordings from the same identified pair of neurons, or at least the same cell type, will also be required. These would jointly license more direct comparisons and interpretations. Such experiments would of couse be technically challenging. However, the link to optimal estimation offered by our theory provides them with the potential to test directly an important facet of neural circuit computations.

Figures and legends

Figure 1. Estimating the presynaptic membrane potential from spiking information. (a) Sample trace (black) of the presynaptic membrane potential generated from an Ornstein-Uhlenbeck process. When the membrane potential exceeds a soft threshold, action potentials (vertical black lines) are generated. The optimal estimator of the presynaptic membrane potential (red line: mean estimate , red shading: one standard deviation ) closely matches an optimally tuned canonical model of short-term plasticity 11 (blue). Inset shows a magnified section. (b) EPSP amplitude of the optimal estimator (red: mean s.t.d.) and of the canonical model of short-term plasticity (blue: mean s.t.d.) as a function of the estimator uncertainty . Note that EPSP amlitudes in the biophysical model tend to be smaller than those in the optimal estimator which is compensating for a somewhat slower decay in the biophysical model (see (a) inset). (c) The dynamics of the scaled uncertainty (red) closely match the resource variable of the canonical model of STP (blue), .

Figure 2. Estimating the presynaptic membrane potential when the resting membrane potential randomly switches between two different values. (a) Presynaptic subthreshold membrane potential with action potentials (black), its optimal mean estimate ( , red line) with the associated standard deviation ( , red shading), and the postsynaptic membrane potential in a model synapse [START_REF] Mongillo | Synaptic theory of working memory[END_REF] with optimally tuned short-term plasticity (blue line). Inset: the (scaled) optimal estimator (red solid line) strongly depends on the estimated probability of being in the "up" state (red dot-dashed line). (b) EPSP amplitude in the optimal estimator depends on its uncertainty (x-axis, ), and the change in the estimated probability that the presynaptic cell is in its "up" state (color code,

). (c) The estimated probability that the presynaptic cell is in its "up" state (red) tracks the state of the presynaptic neuron (black) as it randomly switches between its "up" and "down" states. (d) EPSP magnitudes in the optimal estimator against EPSP magnitudes in the model synapse. (Online Methods), in the optimal estimator (red), an optimally fitted dynamic synapse (blue), and an optimally fitted static synapse, without short-term plasticity (green). When (entirely stochastic spiking), spikes are generated independently of the membrane potential, and as a consequence, all models fail to track the membrane potential. As becomes larger (more deterministic spiking), the dynamic synapse model matches the optimal estimator in performance and significantly outperforms the static synapse. Realistic values for (here mV) have been found to be between 2 and 3 in L5 pyramidal cells of somatosensory cortex [START_REF] Jolivet | Predicting spike timing of neocortical pyramidal neurons by simple threshold models[END_REF] . (b) Estimation performance with stochastic vesicle release as a function of the number of synaptic release sites, . The dynamic synapse (blue) tracks the performance of the optimal estimator (red) well and outperforms the static synapse (green) at all values of . The performance of all estimators decreases only when the number of independent release sites becomes very low ( or ). When is large ( ), synaptic transmission becomes deterministic, even though spike generation itself remains stochastic (with parameter shown by the arrow in a).

# $ % & ' 2 , t ()t 2 - . / / 0 1 2 2 ! N u t ; ût ,, t 2 -. 0 1 (9)
Note that the smaller the bin size is, the closer this posterior distribution is to a Gaussian. The expected firing rate of the presynaptic cell at time is obtained from the normalization condition P u t | s 0:t ( ) u t |s 0:t = 1,

! t = g u t ( ) u t |s 0:t = g 0 exp " ˆ u t + " 2 # t 2 2 $ % & ' ( ) (10)
The mean and variance of the posterior in Eq. 9 evolve (in continuous time, by taking the limit !t " 0) as 5 :

ˆ ˙ u t = 1 ! u rest " ˆ u t [ ] + #$ t"% 2 S t " & t [ ] (11)
˙ ! t 2 = 2 " ! OU 2 # ! t 2 [ ] # $ t % 2 ! t 4 (12)
where denotes the spike train of the presynaptic cell represented as a sum of Dirac delta functions, and is an arbitrary small positive constant that ensures that at the time of a spike , the update of is based on the variance just before the spike . (A similar, but not identical, derivation can be found in Ref. 26.) Equation 11indicates that each time a spike is observed, the estimated membrane potential should increase proportionally to the uncertainty (variance) about the current estimate. In turn, this estimation uncertainty then decreases each time a spike is observed (Eqs. 10 and 12). Conversely, in the absence of spikes, the estimated membrane potential decreases while the variance increases back to its asymptotic value. It can be shown [START_REF] Pfister | Know thy neighbour: A normative theory of synaptic depression[END_REF] that the representation of uncertainty about the membrane potential by is self-consistent because it is predictive of the error of the mean estimator, .

The dynamics of the membrane potential estimator in equations 11 and 12 is closely related to the dynamics of short-term depression. This can be shown formally by taking the limit when presynaptic spikes are rare. In this case, equations 11-12 can be rewritten 5 as:

! u t " v0 ! ût " m + Ĵ Ŷ # t !$ 2 S t ( 13
)
! ! t 2 " 1 " ! t 2 # D " Ŷ ! t "$ 2 S t ( 14
)
where is the normalized variance of the estimator. The other constants involved are , , [START_REF] Loebel | Multiquantal release underlies the distribution of synaptic e"cacies in the neocortex[END_REF] , and

ˆ J = 1/ ! " # $ 3 % # 2 ( ) , ˆ Y = ! " # $ 4 % #
ˆ ! D = 1/ 2/! + " # 2 % # 2
(
) . , and are the stationary posterior mean, variance, and expected firing rate in the optimal estimator (Eqs. 9-10) in the absence of presynaptic spikes. More precisely, from Eqs. 11 and 12, we have

0 = 1 ! u rest " ˆ u # [ ] " $% # 2 & # (15)
0 = 2 ! " OU 2 # " $ 2 [ ] # % $ & 2 " $ 4 (16)
where the expected firing rate is ! " = g 0 exp(# ˆ u " + # 2 $ " 2 /2). Although it is difficult to get an explicit expression for as a function of the model parameters only, from Eqs. 15 and 16 we can still express it as a function of the stationary variance

ˆ u ! = u rest " 2 # $ OU 2 $ 2 ! "1 % & ' ( ) * (17)
Equations 13-14 directly map the posterior mean, and (normalized) variance, onto the postsynaptic membrane potential, and the synaptic resource variable in a canonical, biophysically-motivated model of a synapse undergoing synaptic depression (c.f. Equations 20-21).

The switching Ornstein-Uhlenbeck process

In the previous section, we modeled presynaptic membrane potential fluctuations with an Ornstein-Uhlenbeck process around a constant resting membrane potential. Here we generalize this process by letting the resting potential itself change. In this switching OU process, the resting membrane potential is not fixed but randomly switches between two levels, and , corresponding to "up" and "down" states (Supplementary Fig. 1b)

P u t rest | u t!"t rest ( ) = 1! # ! "t if u t rest = u + and u t!"t rest = u + # ! "t if u t rest = u ! and u t!"t rest = u + 1! # + "t if u t rest = u ! and u t!"t rest = u ! # + "t if u t rest = u + and u t!"t rest = u ! $ % & & ' & & (18)
where and are the rates of switching to the "down" and "up" states, respectively. Similarly to equation 4, the presynaptic membrane potential evolves as an Ornstein-Uhlenbeck (OU) process around the resting potential which is now time-dependent:

P u t | u t !"t ,u t rest ( ) = N u t ; u t !"t + 1 # u t rest ! u t !"t ( ) "t, $ W 2 "t % & ' ( ) *
Spike generation is described by the same rule as before (Eqs. 6-7). Although we were able to develop some analytical insight into the behavior of the optimal estimator in the case of a switching OU process (Supplementary Note), a full analytical treatment remains a challenging task. Thus, the results displayed in Fig. 2 have been obtained by using standard particle filtering techniques [START_REF] Doucet | Sequential Monte Carlo methods in practice[END_REF] (see below).

Biophysically motivated STP model

The model we used was taken directly from Ref. 11 as a canonical model of a synapse undergoing STP. It describes how the postsynaptic potential , the synaptic resource (responsible for depression) and the utilization factor (responsible for facilitation) covary in time:

˙ v t = v 0 ! v t " m + J y t!# x t!# S t ( 20
) ˙ x t = 1! x t " D ! y t!# x t!# S t ( 21
) ˙ y t = Y ! y t " F + Y (1! y t!# )S t ( 22
)
where is the postsynaptic resting membrane potential, is the postsynaptic membrane constant, is (the static part of) synaptic efficacy, is the maximal synaptic utilization (and the rate of increase in in response to a spike), is the time constant of synaptic depression, and is the facilitation time constant. Note that if the facilitation time constant is very short ( ), then can be replaced by in Eqs. 20 and 21, resulting in pure depression. Also note that this standard model ignores the finite rise time of EPSPs. However, since rise times are usually about an order of magnitude faster than decay time constants, the effects of this approximation on the estimation performance of a synapse (as shown in Fig. 4) are expected to be negligible and in any case affect the static and the dynamic versions of the model equally.

Measuring the performance of estimators

The performance of an estimator, , was measured as its rescaled root mean squared error (Fig. 4):

P = 1 ! 1 " OU 1 T 0 T # ût ! u t ( ) 2 dt $ % & ' ( ) 1 2 (23)
where can be substituted with to measure the performance of the biophysical models. Note that this provides a suitably normalized measure of performance as perfect estimation results in , and an estimator that outputs the expected mean presynaptic membrane potential, thereby completely ignoring presynaptic spikes, has .

Stochastic release

Fig. 4 shows the performance of static and depressing synapses and the optimal estimator for the case of stochastic vesicle release. Here, we provide the details of the calculations involved following Ref. 4.

Depressing synapses. Let denote the total number of independent release sites. Each site can release at most one vesicle each time presynaptic spike occurs. Each released vesicle gives rise to a quantum postsynaptic response, where is the maximal EPSP amplitude. The postsynaptic membrane potential evolves as (24) where is the membrane time constant, is the postsynaptic resting potential, and is the presynaptic (delta) spike train. The total number of vesicles released at time in response to a presynaptic spike, , depends on the number of vesicles that are ready to fuse, , and the utilisation fraction . More precisely, at the time of a spike, is drawn from a binomial distribution:

The number of ready-to-fuse vesicles decreases by each time there is a spike, and increases stochastically back to with a time constant in the absence of spikes. Formally, the dynamics of is given by (26) where , and denote the stochastic restocking times produced by an inhomogeneous Poisson process with intensity . It is easy to show that taking the expectation of Eqs. 24 and 26 over the stochastic release and restocking events, and setting , we get back the standard model of shortterm depression (see Eqs. 20 and 21).

Static synapses. If we take the limit of a short time constant for depression, i.e. , the restocking of the vesicle described by Eq. 26 becomes instantaneous and therefore we have . As a consequence, the number of released vesicles at the time of a spike is given by (27) Optimal estimator. In the case of stochastic vesicle release, the variables of the optimal estimator evolve as

! u t = 1 ! u rest " ût # $ % & + '( t 2 n t r NY S t " ) t # $ * % & + ( 28
) ! ! t 2 = 2 " ! OU 2 # ! t 2 $ % & ' # ( t ) 2 ! t 4 (29)
where the number of released vesicles is given by Eq. 27. Note that if we replace by its expectation , we get back the deterministic optimal estimator derived in Eqs. 11-12.

The "utilisation" parameter describes the probability of release for a presynaptic spike in all three models. For a fair comparison, it was chosen to be for all models, which optimized the performance of the dynamic synapse in the deterministic case for (the value we used for the stochastic simulations), and which is consistent with experimentally reported values for the probability of in cortical pyramidal-topyramidal cell connections [START_REF] Thomson | Facilitation, augmentation and potentiation at central synapses[END_REF] .

Numerical simulations for the optimal estimator

We evaluated the mean posterior , the conditional means , and the conditional variances , numerically using a standard particle filtering technique [START_REF] Doucet | Sequential Monte Carlo methods in practice[END_REF] . In practice, we used particles, each of which was two-dimensional (u t (i ) ,u t rest,(i ) ), i = 1,…, N part . They evolved according to:

u t rest,(i ) ! P u t rest,(i ) | u t !"t rest,(i ) ( ) (30)
u t (i ) ! P u t (i ) | u t !"t (i ) ,u t rest,(i ) ( ) (31)
where

P u t rest | u t !"t rest (
) is given by Eq. 18 and P u t | u t !"t ,u t rest ( ) by Eq. 19.

There was an (importance) weight, , assigned to each particle, which was updated according to

w t (i ) ! w t "#t (i ) P s t | u t (i ) ( ) , i = 1,…, N part (32)
In each step of the simulation, all weights were renormalised such that . The particles were resampled when the weights became strongly uneven. Formally, if the number of effective particles at time , defined as

N t eff = i =1 N ! w t (i ) ( ) 2 " # $ % & ' (1 (33)
fell below a given threshold then all particles were resampled and the weights were all set back to . The empirical mean and variance of the posterior membrane potential distribution were determined as

ût ! i =1 N ! u t (i ) w t (i ) (34) ! tot ,t 2 ! i =1 N " u t (i ) ( ) 2 w t (i ) # ût 2 (35)
The numerical evaluation of , ! t + ( ) [START_REF] Zucker | Short-term synaptic plasticity[END_REF] , and ! t " ( )

2 followed the same procedure except that the summation was restricted to the particles that were in the "up" (resp. "down") state and the weights were renormalized accordingly.

Model parameters for simulations

Unless otherwise noted, the presynaptic membrane time constant was set to ms. The spiking determinism parameter was mV, and was set such that Hz. Fig. 3a : The fitted parameters were , ms and Hz. Each one of those 3 parameters were then sequentially perturbed by (resp. ). The area between the minimum and the maximum of those 6 conditions is represented by the dark (resp. light) gray shading. mV, ms, mV, Hz, Hz, mV, and was set such that Hz. Each one of those 7 parameters were sequentially perturbed by (dark gray shading) and by (light gray shading). Note that in this figure, we do not display the experimental data point for the shortest inter-spike interval (ISI = 5 ms) because our current spiking model does not include the effects of refractoriness and burstiness which may dominate estimation at such short intervals.

Figure 3 .

 3 Figure 3. The optimal estimator reproduces experimentally observed patterns of synaptic depression and facilitation. (a) Synaptic depression in cerebellar climbing fibers (circles: mean s.e.m.; redrawn from Ref. 6) and in the model (solid line), measured as the ratio of the amplitude of the eighth and first EPSP as a function of the stimulation rate during a train of eight presynaptic spikes. (b) Synaptic facilitation in hippocampal Schäffer collaterals (circles: mean s.e.m.; redrawn from Ref. 39) and in the model (solid line), measured as the ratio of the amplitude of the second and first EPSP as a function of the interval between a pair of presynaptic spikes. Shading in (a) and (b) shows the robustness of the fits (Online Methods): model predictions when best-fit parameters are perturbed by (dark gray) or (light gray). (c) -(f) Comparison between in-vivo data and samples traces from the model. (c) In vivo intracellular recordings from the (anaesthetised) rat inferior olive nucleus (reproduced from Ref. 40). (d) In vivo

Figure 4 .

 4 Figure 4. Estimation performance of the presynaptic membrane potential. (a) Peformance as a function of the determinism of presynaptic spiking,(Online Methods), in the optimal estimator (red), an optimally fitted dynamic synapse (blue), and an optimally fitted static synapse, without short-term plasticity (green). When (entirely stochastic spiking), spikes are generated independently of the membrane potential, and as a consequence, all models fail to track the membrane potential. As becomes larger (more deterministic spiking), the dynamic synapse model matches the optimal estimator in performance and significantly outperforms the static synapse. Realistic values for (here mV) have been found to be between 2 and 3 in L5 pyramidal cells of somatosensory cortex[START_REF] Jolivet | Predicting spike timing of neocortical pyramidal neurons by simple threshold models[END_REF] . (b) Estimation performance with stochastic vesicle release as a function of the number of synaptic release sites, . The dynamic synapse (blue) tracks the performance of the optimal estimator (red) well and outperforms the static synapse (green) at all values of . The performance of all estimators decreases only when the number of independent release sites becomes very low ( or ). When is large ( ), synaptic transmission becomes deterministic, even though spike generation itself remains stochastic (with parameter shown by the arrow in a).

Fig. 1 :Fig. 2 :

 12 Fig. 1 : The standard deviation of the presynaptic membrane potential was mV, the resting membrane potential was

Fig. 3b :

 3b Fig. 3b : The lower baseline potential was fixed at mV. The fitted parameters were: mV, ms, mV, Hz, Hz,

Fig. 4 :

 4 Fig. 4 : We set mV, mV, and mV.

N

Acknowledgements

We thank M. Häusser and R. Brown for useful references, and L. Abbott, Sz. Káli and members of the Budapest Computational Neuroscience Forum for valuable discussions. This work was supported by the Wellcome Trust (J-P.P., M.L., P.D.) and the Gatsby Charitable Foundation (P.D.).

Author contributions

J.P.P. and M.L. developed the mathematical framework. J.P.P performed the numerical simulations. All authors wrote the manuscript.

Online Methods

The optimal estimator

Our goal is to understand the factors contributing to the mean of the posterior, or the estimate

corresponding to the postsynaptic potential under our interpretation, and in particular to the size of change in this estimate in response to an incoming spike, the analogue of the size of an EPSP.

The Ornstein-Uhlenbeck process

The generative model involves two simplifying assumptions. First we assume that presynaptic membrane potential dynamics are discrete time and Markovian (Supplementary Fig. 1a)

In particular, we assume that the presynaptic membrane potential evolves as an Ornstein-Uhlenbeck (OU) process, given (in discrete time steps of size and thus as an AR(1) process ) by

where is the resting membrane potential of the presynaptic cell (assumed to be constant here), is its membrane time constant, and is the step size for the random walk behavior of its membrane potential. Because both and are assumed to be constant, the marginal variance of the presynaptic membrane potential, ! OU 2 = ! W 2 "/2, is stationary.

The second assumption is that spiking activity at any time only depends on the membrane potential at that time:

In particular, we assume that the spike generating mechanism is an inhomogeneous Poisson process. Thus, at time , the neuron emits a spike ( ) with probability , and therefore the spiking probability given the membrane potential can be written as:

We further assume that the transfer function is exponential (Supplementary Fig. 1c): (7) where and are the base rate and determinism of the spike generation process, respectively.

Since equations 3 and 5 define a hidden Markov model, the posterior distribution over can be written in a recursive form:

That is, the posterior at time , P u t | s 0:t ( ) , can be computed by combining information from the current time step with the posterior obtained in the previous time step, P u t!"t | s 0:t!"t ( ) . Note that even though inference can be performed recursively, and the hidden dynamics is linear-Gaussian (Eq. 4), the standard (extended) Kalman filter cannot be used for inference because the measurement does not involve additive Gaussian noise, but rather comes from the stochasticity of the spiking process (Eqs. 6-7).

Performing recursive inference (filtering), as described by Eq.


Monique M Maat
email: mdemaat@alysis.nl

Theodorus A Tijssen

Roger J Brüggemann

Huibert H Ponssen

Monique M De Maat

PHARMACOKINETICS AND DISPOSITION Paracetamol for intravenous use in medium-and intensive care patients: pharmacokinetics and tolerance

Keywords: Paracetamol, Pharmacokinetics, Safety, Critically ill patients

Paracetamol for intravenous use in medium-and intensive care patients: pharmacokinetics and tolerance

Introduction

Paracetamol is a frequently used drug on the Intensive Care (ICU) and Medium-Care (MCU) units. It is used both as an analgesic and an antipyretic to reduce oxygen consumption. Paracetamol is, when used in therapeutic dosages, generally well tolerated. However, in overdosage, it can cause severe hepatic necrosis [START_REF] Forrest | Clinical pharmacokinetics of paracetamol[END_REF][START_REF] Gelotte | Disposition of acetaminophen at 4, 6, and 8 g/day for 3 days in healthy young adults[END_REF]. In the ICU/MCU, paracetamol is mostly administered as a rectal formulation. As the patient has to be turned to administer paracetamol rectally, for the nursing staff at these wards, it is not the most practical way of drug administration. In addition, diarrhea or rectal surgery limits the use of suppositories. Another issue in this patient category is the unpredictable enteric absorption, which also argues for the use of an i.v. formulation of paracetamol.

Recently, Perfalgan TM , a parenteral formulation of paracetamol, became available in The Netherlands [START_REF]Perfalgan 10 mg/ml, solution for infusion[END_REF]. Only a limited amount of pharmacokinetic and pharmacodynamic studies have been performed with this formulation [START_REF] Flouvat | Bioequivalence study comparing a new paracetamol solution for injection and propacetamol after single intravenous infusion in healthy subjects[END_REF][START_REF] Würthwein | Pharmacokinetics of intravenous paracetamol in children and adolescents under major surgery[END_REF][START_REF] Holmer Pettersson | Plasma concentrations following repeated rectal or intravenous administration of paracetamol after heart surgery[END_REF]. A study in a small group of healthy male volunteers showed a mean peak serum concentration (C max ), observed at the end of a 15-min i.v. infusion of 1 g paracetamol (Perfalgan), of 29.9 mg/L (range 17.1-49.3 mg/L) and a mean area under the plasma concentration-time curve (AUC inf ) from time zero to infinity of 57.6 mg*h/L [START_REF] Flouvat | Bioequivalence study comparing a new paracetamol solution for injection and propacetamol after single intravenous infusion in healthy subjects[END_REF]. Peak serum concentrations of 10-20 mg/L are considered to be the therapeutic antipyretic range [START_REF] Korpela | Paracetamol-misused good old drug?[END_REF]. Serum paracetamol concentrations providing analgesia are unknown but are assumed to be higher than the antipyretic serum concentrations [START_REF] Korpela | Paracetamol-misused good old drug?[END_REF]. In children undergoing a tonsillectomy, paracetamol was administered in a dose of 40 mg/kg both orally and rectally [START_REF] Anderson | Analgesic efficacy of paracetamol in children using tonsillectomy as a pain model[END_REF]. Paracetamol serum concentrations > 20 mg/L yielded insufficient painkilling effects in 15% of children [START_REF] Anderson | Analgesic efficacy of paracetamol in children using tonsillectomy as a pain model[END_REF]. In contrast, Hahn showed that an i.v. dose of only 5 mg/kg paracetamol (administered as propacetamol), by which an initial serum concentration of 14 mg/L was reached, yielded sufficient analgesia [START_REF] Hahn | Analgesic effect of i.v. paracetamol: possible ceiling effect of paracetamol in postoperative pain[END_REF]. A study by Beck et al. demonstrated a C max of only 10.4 mg/L and 17.2 mg/L after rectal administration of paracetamol at 20 mg/kg and 40 mg/kg, respectively [START_REF] Beck | The pharmacokinetics and analgesic effects of larger dose rectal acetaminophen (40 mg/kg) in adults: A double-blinded, randomized study[END_REF]. Furthermore, another study compared the bioavailability of paracetamol after oral and i.v. (propacetamol) administration [START_REF] Holmér Pettersson | Early bioavailability of paracetamol after oral or intravenous administration[END_REF]. Oral administration of paracetamol as part of multimodal pain management immediately postoperatively resulted in a huge and unpredictable variation in plasma concentration compared with the i.v. administration [START_REF] Holmér Pettersson | Early bioavailability of paracetamol after oral or intravenous administration[END_REF]. Predictability of serum concentrations, quick achievement of C max after administration, and a higher C max (with probably larger therapeutic effect [START_REF] Piquet | Lack of acetaminophen ceiling effect on RIII nociceptive flexion reflex[END_REF][START_REF] Piletta | Central analgesic effect of paracetamol but not aspirin[END_REF]) compared with oral [START_REF] Hahn | Analgesic effect of i.v. paracetamol: possible ceiling effect of paracetamol in postoperative pain[END_REF] or rectal [START_REF] Holmer Pettersson | Plasma concentrations following repeated rectal or intravenous administration of paracetamol after heart surgery[END_REF] administration) could argue for use of the parenteral formulation of paracetamol. In addition, ease of administration of drugs in the ICU/MCU, particularly with drugs that need to be administered several times a day, is a factor worth considering.

Perfalgan was introduced in the ICU/MCU of the Albert Schweitzer Hospital (Dordrecht, The Netherlands) mid-2005. In our clinical experience, we noted that approximately 25% of patients showed an episode of hypotension shortly after administration of paracetamol i.v. and that an estimated 8% of the patients needed an intervention to regain an acceptable blood pressure level. This adverse event is described in the product information of Perfalgan, but only with an incidence of 0.01-0.1% [START_REF]Perfalgan 10 mg/ml, solution for infusion[END_REF]. At this time, pharmacokinetic data of Perfalgan in clinical patients, particularly ICU and MCU patients, is limited. We therefore wanted to study the pharmacokinetics of paracetamol after i.v. administration in ICU/MCU patients. In addition, we wanted to determine the incidence of infusion-related hypotension in this patient population.

Methods

Patients and data collection

All adult patients admitted to the ICU or MCU were eligible provided they had an arterial catheter and were to start paracetamol 1 g i.v. four times daily. Only patients or their representatives who gave written informed consent were included in this study. Approval was obtained from the Institutional Review Board. Contraindications for paracetamol administration [START_REF]Perfalgan 10 mg/ml, solution for infusion[END_REF] were considered for every patient and were as follows:

& Hypersensitive for paracetamol, propacetamol hydrochloride, or one of the additives of the infusion solution and/or & Severe hepatocellular insufficiency (which is left to the discretion of the responsible intensivist)

The following patient characteristics were collected at baseline: age, gender, height/weight, type of admission on ICU or MCU, clinical chemistry parameters (liver, kidney, lactate), activated partial thromboplastin time (APTT), prothrombin time (PT), indication for paracetamol, Acute Physiology And Chronic Health Evaluation (APACHE) II score on the day paracetamol was studied, and body temperature. SBP and diastolic blood pressure (DBP) were recorded at time point (T)=0 min (just before start of paracetamol infusion), T=15 min (end of paracetamol infusion), and T=30 min. SBP and DBP were measured using an arterial line. A reduction of at least 10 mmHg in SBP at T=15 min and T=30 min was considered clinically relevant. Interventions to correct hypotension were recorded until 60 min after the end of paracetamol infusion (T=75 min). During this 60 min period, syringe changes, fluid boluses, administration of other drugs, change of infusion rate of current drugs, and changes in ventilator settings were reduced to a minimum. If, in case of urgent handling, one of the above-mentioned actions had to be executed (for instance, due to paracetamol-induced hypotension), it was written down.

Study drug and blood samples

Perfalgan 10 mg/ml, infusion solution (Bristol-Myers Squibb B.V., Woerden, The Netherlands) was administered in a dose of 1 g four times daily by means of a 15-min infusion of 100 ml=1,000 mg ready-to-use solution [START_REF]Perfalgan 10 mg/ml, solution for infusion[END_REF].

Start of the infusion was T=0. Blood samples were drawn at the following time points: T=0 (just before paracetamol infusion), 15, 30, 45, 60, 120, 180, 240, and 359 min (= just before next paracetamol administration). The exact time of sampling was noted down.

Bioanalysis

Blood samples were obtained at specified time points, centrifuged (4,000 rpm, 5 min), and serum was separated. Serum samples were stored at -20°C until analysis. Concentration of paracetamol in the serum sample was proven to be stable during at least 2 months after withdrawal (data on file). Concentrations of paracetamol in serum were quantitatively determined with fluorescent polarization immunoassay (Cobas Integra 400, Roche Diagnostics, West Sussex, UK). Within-run variation and total variation for low as well as high concentrations (9.9 mg/L, 32.9 mg/L, and 97.4 mg/L) were within a range of 0.7-5.8% and 4.4-7.5%, respectively. Lower limit of detection of the analysis was 0.2 mg/L.

Pharmacokinetic analysis

Pharmacokinetic parameters were derived by noncompartmental analysis with WinNonlin, version 4.1 (WinNonlin 4.1, Pharsight, USA). The highest observed serum concentration was defined as C max . The terminal log-linear period (log C versus t) was defined by the last data points (n≥3). The absolute value of the slope (β/ln10) was calculated by least squares linear regression analysis. The elimination half-life (t 1/2 ) was calculated using the equation t 1/2 =ln2/β. The area under the serum concentration time curve (AUC) was obtained using the log-linear trapezoidal rule from 0 to the last data point and then extrapolated to infinity (AUC inf ). Clearance (CL) was calculated by dividing the dose by the corresponding AUC inf . The volume of distribution (V d ) was calculated by dividing CL by β.

Statistics

SPSS for Windows, version 16.0 (SPSS Inc., Chicago, IL, USA) was used to perform statistical analyses. Results of pharmacokinetic parameters are expressed as median and interquartile range (IQR). Pharmacokinetic parameters (C max , AUC inf , CL, V d , and t 1/2 ) were compared using the Mann-Whitney U test. SBP at T=15 and T=30 min was compared with that at T=0 min with the Wilcoxon signedrank test. Relationships between the occurrence of a relevant reduction in SBP at T=30 min and possible risk factors were tested by univariate analysis with logistic regression. Factors considered were baseline characteristics including gender, age, APACHE II, admittance to ICU/ MCU, lactate, and SBP at baseline. For these analyses, a p value of ≤0.05 was considered statistically significant.

Results

Patient characteristics

Patients were included from December 2006 until April 2007. During this period, data of 38 patients was collected. Table 1 presents the baseline characteristics of these patients.

The patient population was predominantly male (71%) and had a median age of 68 years. Most patients were hemodynamically stable postoperative patients and stayed on the MCU for observation after surgery (68%). The indication of paracetamol in the included patients was, consequently, predominantly pain (Table 1).

Pharmacokinetic analysis

Figure 1 shows the serum paracetamol concentration versus time data for individual study participants. The data set consisted of 28 full pharmacokinetic curves (nine samples per patient). From the ten other patients, five to eight samples were available. Table 2 presents the pharmacokinetic data of all included patients and for the different groups, i.e., patients admitted to the MCU and ICU. The median peak serum concentration (C max ) was 25.92 mg/L (IQR 21.05-33.61 mg/L) and ranged from 8.32 to 59.92 mg/L (Table 2). The other pharmacokinetic parameters (AUC inf , CL, V d , and t 1/2 ) showed, in accordance with the C max , a large variation (Table 2).

Statistical analysis showed a trend to a larger C max (p=0.052) and a significantly smaller V d (p=0.033) in MCU patients when compared with ICU patients. Although not statistically different (p>0.05), CL and AUC inf of paracetamol in MCU patients were 47% lower and 100% higher, respectively, when compared with ICU patients. Hypotension Table 3 presents data on observed blood pressure (diastolic and systolic) at the different time points. Blood pressure could be assessed from 36 of the 38 patients at both time points. At T=15 and 30 min, eight (22%) and 12 (33%) of the 36 patients had a reduction in SBP of at least 10 mmHg when compared with baseline (T=0), respectively. Six of eight patients that showed a relevant reduction in SBP at T=15 min had a persistently low SBP at T=30 min (both compared with T=0). In three of the eight patients, there was a further decay in SBP at T=30 min compared with T=15 min. In patients with a clinical relevant reduction in SBP, SBP dropped with 6.5-31% (n=8) and 6.5-44% (n=12) at T=15 and T=30 min, respectively, when compared with baseline. Two patients with a clinically relevant reduction in SBP were admitted to the ICU; all other patients were admitted to the MCU. Overall, SBP was significantly reduced (p<0.05) at T=15 min and 30 min when compared with SBP at the start of the paracetamol infusion (Table 3). The logistic regression analysis did not identify any possible risk factors for a relevant reduction in SBP. Figure 2 shows the APACHE II scores for patients with and without a clinically relevant reduction in SBP.

Six patients (16%) needed an intervention to regain blood pressure control. Three had a clinically relevant reduction in SBP at T= 15 or 30 min. Increase of noradrenaline infusion occurred in four patients (11%) and a fluid bolus (gelatin) was given in five patients (13%). Two patients were extremely restless and/or experienced diaphoresis at time point 15 min; both had a clinical reduction in SBP at the defined time points (one at T=15 and one at T=30 min), but no intervention was performed during the following time period (until 60 min after end of paracetamol infusion).

Discussion

Here we describe the pharmacokinetics of paracetamol after i.v. administration in a population of patients admitted to the MCU and ICU. In this observational study, paracetamol was administered as a ready-to-use solution (Perfalgan). In a study by Flouvat et al., the pharmacokinetics of paracetamol was compared with propacetamol, an injectable prodrug of paracetamol requiring reconstitution [START_REF] Flouvat | Bioequivalence study comparing a new paracetamol solution for injection and propacetamol after single intravenous infusion in healthy subjects[END_REF]. The pharmacokinetic parameters in that study were roughly comparable with the parameters of the patients in our study. A difference, however, was observed between the t 1/2 found in healthy volunteers compared with patients in our study: 2.72 h versus 1.63 h, respectively. A smaller V d in our population probably accounts for this difference. In addition, AUC inf and CL were lower and higher, respectively, in our study when compared to the data of Flouvat et al. [START_REF] Flouvat | Bioequivalence study comparing a new paracetamol solution for injection and propacetamol after single intravenous infusion in healthy subjects[END_REF].

We also studied whether there was a difference in the pharmacokinetic parameters of patients admitted to the ICU and MCU. ICU patients showed a trend to a smaller C max and a significantly higher V d when compared with MCU patients. Our hypothesis is that ICU 1).

Our ICU patients had a 32% lower C max and a 122% higher CL (and consequently a lower AUC inf ) than the healthy volunteers in the study of Flouvat et al. [4]. This was, however, an observational study. Thus, these results must be confirmed in a controlled study with a calculated sample size. Also, whether the observed differences in pharmacokinetics resulted in differences in pharmacodynamic effect cannot be said. Neither pain nor body temperature was monitored in the context of the study.

Furthermore, we found a significant reduction in SBP 15 and 30 min after start of paracetamol infusion when compared with SBP at baseline. In addition, a clinical relevant reduction in SBP (at least 10 mmHg) was seen in 22% and 33% of included patients at 15 and 30 min, respectively. The incidence of hypotension is thus higher than described in the product information for Perfalgan [START_REF]Perfalgan 10 mg/ml, solution for infusion[END_REF]. Moreover, 16% of the included patients needed an increase in noradrenaline infusion or a fluid bolus to regain an acceptable blood pressure. A prospective study in critically ill patients showed 27 events of hypotension in 14 patients after administration of propacetamol. Mean arterial pressure (MAP) dropped significantly in these patients, requiring active intervention (initiation or increase of noradrenaline or a fluid bolus) in 33% of the events [START_REF] Hersch | Effect of intravenous propacetamol on blood pressure in febrile critically ill patients[END_REF]. Three other reports of paracetamol-induced hypotension in ICU patients were found in the literature [START_REF] Brown | Acetaminophen-induced hypotension[END_REF][START_REF] Boyle | Paracetamol administration is associated with hypotension in the critically ill[END_REF][START_REF] Mackenzie | Effects of acetaminophen administration to patients in intensive care[END_REF]. Boyle and colleagues [START_REF] Boyle | Paracetamol administration is associated with hypotension in the critically ill[END_REF] found a significant reduction in MAP within 15 min of oral paracetamol administration, which persisted for 90 min. Based on the published results so far, the occurrence of paracetamol-induced hypotension seems thus independent of route of administration or formulation. Whether the high incidence of hypotension is dependent on the specific population (ICU or MCU patients) in which it is administered is not clear. Mackenzie found in their study that MAP was lower in male than in female patients after administration of paracetamol and was correlated with age and baseline MAP [START_REF] Mackenzie | Effects of acetaminophen administration to patients in intensive care[END_REF]. We, however, did not identify any specific risk factors for the occurrence of hypotension. In particular, APACHE II score was not significantly related with an episode of hypotension (see Fig. 2).

In conclusion, we describe the pharmacokinetics of paracetamol after i.v. administration in a large population of patients admitted to the ICU or MCU. Overall, data were roughly comparable with earlier publications, but differences were noted in the ICU population. Further research on the importance of this finding in relation to paracetamol's analgesic or antipyretic effect might be needed. In addition, our data on paracetamol-induced hypotension confirms the subjective observations noted by our nursing staff. As its incidence is high, hypotension caused by paracetamol, a drug frequently used in the ICU/MCU population, must not be overlooked.

Fig. 1

 1 Fig. 1 Concentration-time data of paracetamol. ♦ Individual paracetamol serum concentration. Insert: Solid line represents the median value of the paracetamol serum concentration at the different time points in the study population. Dashed lines represent first and third quartiles of paracetamol serum concentration

Fig. 2

 2 Fig. 2 Acute Physiology And Chronic Health Evaluation (APACHE) II scores in patients with and without a clinically relevant reduction in systolic blood pressure (SBP). *Median value

Table 2

 2 Pharmacokinetic data AUC inf area under the serum concentration versus time curve extrapolated to infinity, CL clearance, C max maximum serum concentration, ICU intensive care unit, IQR interquartile range, MCU medium care unit, t 1/2 elimination half-life, t max time to C max , V d volume of distribution *Mann-Whitney U test

	Median [IQR] (min-max)
		All patients	MCU	ICU
		N=38	N=26	N=12	p value*
	C max (mg/L)	25.92 [21.05-33.61] (8.32-59.92)	26.97 [23.51-37.21]	20.35 [16.13-28.69]	0.052
	AUC inf (mg/L*h)	42.28 [21.36-49.93] (13.19-123.64)	47.99 [31.60-69.89]	24.64 [18.56-54.54]	0.129
	CL (L/h)	23.65 [15.71-40.54] (7.49-67.04)	20.84 [14.34-31.65]	39.78 [18.43-53.87]	0.137
	V d (L)	52.01 [44.41-74.36] (15.44-152.11)	50.88 [42.11-61.61]	71.09 [54.20-86.32]	0.033
	T 1/2 (h)	1.63 [1.07-2.30] (0.35-4.73)	1.63 [1.13-2.29]	1.76 [0.92-2.32]	0.963

Eur J Clin Pharmacol (2010) 66:713-719

Acknowledgments No sources of funding were used to assist in the preparation of the study. The authors thank the individuals who participated in the study. Michell Fritse and Dick van Steenis are kindly acknowledged for analysis of the serum samples and the nurses from the ICU and MCU for the collection of the samples. Ton Post is acknowledged for his help with the inclusion of patients in this study.

Conflicts of interest

The authors declare that they have no conflict of interest.


Luigi Calvo

Giovanni Pistone

Sabrina Arnone

Daniela Colomba

Salvatore Amico

Antonella Giacalone

Pietra Vitale

Calogero Nicosia

Eliana Barone

Rosario Scaglione

Giuseppe Licata

Salvatore Corrao
email: s.corrao@tiscali.it

Polymyalgia rheumatica and vertebral fractures: a 1-year pilot controlled study

Keywords: Polymyalgia rheumatica, Vertebral fractures, Osteoporosis, Controlled clinical trial

come

Introduction

Polymyalgia Rheumatica (PMR) is an inXammatory condition of unknown cause that is characterized by an aching and morning stiVness in the shoulder and pelvic girdles other than the same kind of symptoms in the cervical region.

Well-documented data are available from literature about PMR and osteoporosis [START_REF] Homik | Calcium and vitamin D for corticosteroid-induced osteoporosis[END_REF][START_REF] Homik | Bisphosphonates for steroid induced osteoporosis[END_REF][START_REF] Weyand | Polymyalgia rheumatica and giant cell arteritis[END_REF]. In particular, the bene-Wcial eVects of corticosteroid therapy in the treatment of PMR may be oVset by the occurrence of corticosteroidrelated osteoporosis. So, several studies have addressed the question of what dose of corticosteroids does not determine deleterious eVect on bone density: the results of these studies prove that this dose is uncertain (4). On the other hand, in our knowledge, there are no data about both the frequency of vertebral fractures and related clinical features in recently diagnosed PMR patients. Moreover, no data exist about the possibility that vertebral fracture in PMR patients could be independent of steroid therapy.

For this reason, we aimed to investigate this last topic by a case cohort study with a 1-year follow-up for each polymyalgia rheumatica patient who had experienced a vertebral fracture within the Wrst month after the diagnosis.

Methods

We recruited ten consecutive patients who experienced vertebral fractures (VF-group) between during a previous 2-year period (2006-2007 years) aVected by Wrst diagnosed polymyalgia rheumatica, followed-up for 1 year, and without any other signiWcant associated condition. All the fractures were found out in acute symptomatic patients and had occurred during the Wrst month after the diagnosis. As a control group, we studied ten control patients, without vertebral fractures and with a follow-up of 1 year, randomly selected among a larger group of patients aVected by PMR. Randomization was performed by links in order to obtain a matched control group from some relevant variables (age, BMI, gender, comorbidities and drug therapy).

All the patients were referred to the Rheumatologic outpatient clinic of our Hospital trust, and all met the ACR criteria for diagnosis of PMR [START_REF] Chuang | Polymyalgia rheumatica: a 10-year epidemiologic and clinical study[END_REF][START_REF] Healey | Long-term follow-up of polymyalgia rheumatica: evidence for synovitis[END_REF]. Moreover, in each patient of both groups PMR was diagnosed within 1 month from the onset of symptoms.

Each patient underwent an X-ray at baseline, after 12 months, and when experienced an acute pain along spine. The vertebral fractures were Wrst evaluated by X-ray after an acute aching episode and conWrmed by axial computerized tomography. We analysed the following data: eritrosedimention rate (ESR), visual analogical scale score (VAS) for pain assessment, methyprednisolone daily dosage. Each patient had been monthly evaluated by the aforementioned clinical and laboratoristic parameters during the 1-year follow-up period. The two groups did not meaningfully diVer for gender and age, neither there were meaningful diVerences for steroidal dosing, ESR, VAS at baseline. Moreover, the two groups did not diVer for percentage of patients overweight, hypertensive and diabetic (Table 1). There were no signiWcant diVerences about drug therapy (antidiabetic and antihypertensive) between the two groups.

Statistical analysis

The results are presented as mean value § standard deviation, median (interquartile range, as lower and upper quartile) and percentages when appropriated. Unpaired t test was used for comparison of means, Mann-Whitney U test for comparison of medians, and Fisher's exact test for comparison of proportions. Area under curve (AUC) of ESR and VAS, from baseline to 12 month control, was computed for comparison between groups. Predictive values (post-test likelihood) with change, prevalence (pretest likelihood), sensitivity, speciWcity and likelihood ratios with robust conWdence intervals were computed from ESR and VAS values for prediction of vertebral fractures. ROC plots and a ROC analysis were performed to individuate best sensitivity and speciWcity cut-oV points. A two-tailed alpha value of <0.05 was considered to be signiWcant.

Results

Study population variables at baseline are shown in Table 1.

Figures 1 and2 show ESR and VAS trends of the VFgroup compared to the control one during 1-year follow-up.

Although the total amount of corticosteroids during the Wrst 3 months of observation was not signiWcantly diVerent between the two groups, the VF-group resulted with a higher and statistically signiWcant (P = 0.0135) median corticosteroid 12-month total dosage [3,480 mg (2,805-3,030) versus 2,760 mg (2,666.25-3,247.5)]. The VF-group had statistically signiWcant higher ESR and VAS AUC when compared to control group (median ESR AUC, 484.75 vs. ESR at the baseline (cut-oV >80 mm) showed a speciWcity of 90% (95%CI 56-100) and sensitivity of 70% (95%CI 35-93). VAS diVerence from Wrst to second month (cut-oV •3) showed a speciWcity of 90% (95%CI 56-100) and sensitivity of 80% (95% CI 44-97).

Discussion

To our knowledge, this study is the Wrst that appraises the incidence of vertebral fractures after the clinical onset of the PMR. Our study deals with a case cohort study about 10 patients aVected by PMR with recent vertebral fractures (VF-group) compared to 10 patients aVected by PMR without vertebral fractures (NF-group). This is a pilot-study considered both the design and the sample size. However, this kind of design has allowed a strict selection of controls closely matched to the cases for meaningful variables.

So, conclusions of our study and related hypothesis could be suYciently strong to both elicit interest in scien-tiWc community on this topic and consequently draw sound clinical trials. Our results show that ESR was higher in VF-group in comparison with controls (P = 0.0403) at baseline. This Wnding suggests that subjects who will develop vertebral fractures precociously after the onset of PMR could belong to a subgroup of PMR subjects with a higher probability of vertebral fractures independent from steroid osteoporotic bone eVects.

In conclusion, our preliminary results point out that vertebral fractures both represent a problem for patients aVected by PMR, during the Wrst months after the onset of symptoms, and could be predicted from commonly used laboratory (ESR) and clinical (VAS) Wndings. Apart from the importance to investigate pathophysiological mechanisms, an eVort to understand the importance of our data and the needs for a precocious instrumental evaluation for osteoporosis, in this kind of patients, might be investigated in future studies for better utilizing economic resources.

Fig. 1 Fig. 2

 12 Fig. 1 Box-wisker plot of ESR value year trend. Comparison between VF (vertebral fracture) and control groups (median value: thick line, 1st and 3rd quartiles values, lower and upper box limits, min and max values, lower and upper wisker limits)

Table 1

 1

	Patient characteristics
	and variables at baseline: control
	group versus VF-group

123 288.25; P = 0.0001; median VAS AUC, 70.75 vs. 43.5 P < 0.0001);