Jennifer T Coull
email: jennifer.coull@univ-amu.fr

Julien Cotti

Franck Vidal

Differential roles for parietal and frontal cortices in fixed versus evolving temporal expectations: dissociating prior from posterior temporal probabilities with fMRI

Keywords: timing, attention, frontal, parietal, prediction, expectation, foreperiod Running

The ability to predict when an event will occur allows us to respond optimally to that event. Temporal predictability can be either fixed (prior probability) or evolving (posterior probability), in which case it is dynamically updated as a function of the elapse of time itself ("hazard function"). We used fMRI to identify the brain regions involved in either form of temporal prediction, within a single experimental paradigm. Participants performed a cued reaction time (RT) task, in which the target appeared after one of four intervals ("foreperiods") that was either predictable (temporal condition) or variable (neutral condition). As expected, RTs were faster in temporal versus neutral conditions, indicating the behavioural benefit of fixed temporal predictability. RTs also got faster as a function of foreperiod in the neutral, but not temporal, condition, reflecting the evolving temporal predictability of the hazard function. We confirmed that left inferior parietal cortex was preferentially activated by the fixed temporal predictability of temporal (versus neutral) cues. Then, by directly comparing how activity varied as a function of foreperiod in the neutral versus time conditions, we identified the neural substrates of the changes in temporal probability defined by the hazard function, whilst simultaneously controlling for changes related simply to the elapse of time itself. Whole-brain fMRI analyses (independently confirmed by anatomically guided ROI analyses) showed that activity in left inferior parietal cortex tracked the evolving temporal probabilities of the hazard function. ROI analysis further revealed a similar role for right inferior frontal cortex. Our data highlight a key role for left parietal cortex in instantiating the behavioural benefits of temporal predictability, whether predictions are fixed or dynamically evolving.

Introduction

The ability to anticipate when an event will occur allows us to optimize behaviour and, thus, conserve precious resources. For over a century now, it has been known that reaction times (RTs) are faster when the interval (or "foreperiod") between a warning cue and a target is fixed rather than variable [START_REF] Woodrow | The measurement of attention[END_REF]Klemmer, 1956).

Nevertheless, even when foreperiods are variable, RTs are significantly faster after long foreperiods than short ones [START_REF] Woodrow | The measurement of attention[END_REF][START_REF] Karlin | Reaction time as a function of foreperiod duration and variability[END_REF][START_REF] Niemi | Foreperiod and simple reaction time[END_REF].

In other words, as the foreperiod increases, so too does the speed of the eventual response. This phenomenon, known experimentally as the "variable foreperiod effect", is traditionally attributed to changes in the probability of target occurrence as a function of the elapse of time. To illustrate, imagine waiting for a bus. The probability of the bus appearing as soon as you arrive at the bus stop is low, but the probability of its appearance increases the longer you wait for it. The longer you wait, the more your expectation grows. This increasing sense of expectancy is driven by the "hazard function": the increasing conditional probability over time that an imperative event will occur, given that it has not already occurred [START_REF] Elithorn | Central inhibition: some refractory observations[END_REF][START_REF] Luce | Response Times: Their Role in Inferring Elementary Mental Organization[END_REF].

The hazard function relies on the predictive power of the unidirectional flow of time ("time's arrow"): an event that we expect to occur must do so at some time in the future, given that it has not already occurred in the past. Empirically, this means that the longer we wait for an event to occur, the more certain we are of its occurrence, and the faster we are to respond to it [START_REF] Niemi | Foreperiod and simple reaction time[END_REF] or even to perceive it [START_REF] Vangkilde | Great expectations: Temporal expectation modulates perceptual processing speed[END_REF].

Patients with lesions to the right [START_REF] Stuss | Multiple frontal systems controlling response speed[END_REF]Vallesi et al, 2007a;[START_REF] Triviño | Temporal orienting deficit after prefrontal damage[END_REF][START_REF] Triviño | Rhythms can overcome temporal orienting deficit after right frontal damage[END_REF] or left [START_REF] Triviño | Temporal orienting deficit after prefrontal damage[END_REF][START_REF] Triviño | Rhythms can overcome temporal orienting deficit after right frontal damage[END_REF] lateral prefrontal cortex (PFC) fail to show the typical RT benefits of long foreperiods. Similarly, these benefits are attenuated in healthy volunteers receiving Transcranial Magnetic Stimulation (TMS) to right, though not left, dorsolateral PFC (Vallesi et al, 2007b). fMRI data further confirm that activity in right dorsolateral PFC correlates significantly with the magnitude of the RT benefit afforded by increasing foreperiod length (Vallesi et al, 2009a,b), and peaks at the time of target onset predicted by the hazard function, whether the target actually appears then or not [START_REF] Bueti | Encoding of temporal probabilities in the human brain[END_REF]. In addition, right PFC activity is higher when targets appear after unexpectedly long, rather than unexpectedly short, foreperiods [START_REF] Coull | Orienting attention in time: behavioural and neuroanatomical distinction between exogenous and endogenous shifts[END_REF].

Collectively, these data indicate that right lateral PFC is integral in harnessing the temporally predictive information available in the simple passing of time in order to improve performance.

A priori temporal expectations, based on learned temporal associations, can also be used to optimise behaviour. For example, when a traffic light turns amber, we can predict how long it is likely to be before it turns red, allowing us to adjust driving behaviour accordingly (brake or race through). Here, colour acts as a temporally informative cue. This everyday scenario can be simulated in the laboratory by the "temporal orienting" task, an adaptation of the Posner spatial orienting of attention task.

In the spatial version of the task, cues indicate whether an impending target will appear on the left or right of the screen [START_REF] Posner | Attention and the detection of signals[END_REF]. In temporal orienting tasks by contrast, a symbolic cue informs subjects whether a target is likely to appear after a short or long delay [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF][START_REF] Nobre | Orienting attention to instants in time[END_REF]. Compared to non-informative ("neutral") or misleading ("invalid") cues, temporal cues that correctly predict the time of target onset ("valid" cues) lead to faster [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF]) and more accurate [START_REF] Correa | Attentional preparation based on temporal expectancy modulates processing at the perceptual level[END_REF][START_REF] Davranche | Orienting attention in time activates left intraparietal sulcus for perceptual and motor task goals[END_REF] responses. Neuroimaging studies have revealed a hemispheric lateralisation for temporal versus spatial orienting in left and right parietal cortices respectively [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF][START_REF] Coull | The noradrenergic alpha2 agonist clonidine modulates behavioural and neuroanatomical correlates of human attentional orienting and alerting[END_REF]. Follow-up investigations confirmed that temporal orienting selectively activated left parietal cortex (around the intraparietal sulcus) whether participants used temporal cues to enhance perceptual discriminations or speed motor responses [START_REF] Davranche | Orienting attention in time activates left intraparietal sulcus for perceptual and motor task goals[END_REF], whether responses were left-or right-sided, and whether responses were registered as manual button-presses or ocular saccades [START_REF] Cotti | Functionally dissociating temporal and motor components of response preparation in left intraparietal sulcus[END_REF]. Left parietal cortex activation has also been reported when the temporally predictable information used to optimise target detection is not provided by symbolic cues but, instead, by the temporal regularity of a rhythmic visual (Marchant and Driver, 2013) or auditory [START_REF] Bolger | Metrical rhythm implicitly orients attention in time, as indexed by improved target detection and left inferior parietal activation[END_REF] stimulus stream or a temporally predictable visual trajectory [START_REF] Assmus | Left inferior parietal cortex integrates time and space during collision judgments[END_REF][START_REF] Coull | Using time-to-contact information to assess potential collision modulates both visual and temporal prediction networks[END_REF][START_REF] Merchant | Neural responses during interception of real and apparent circularly moving stimuli in motor cortex and area 7a[END_REF].

Neuroanatomical differences in studies of the variable foreperiod effect on one hand (right lateral prefrontal cortex) and of temporal orienting on the other (left parietal cortex, mostly around the intraparietal sulcus) might reflect functional differences in the way in which temporal predictability is implemented in each paradigm. The temporally predictive information provided by temporal orienting cues (or isochronous rhythms) is fixed and constant: the cue (or rhythm) indicates a priori that the target will appear at a specific moment in time. By contrast, the temporally predictive information provided by the passage of time in variable foreperiod paradigms evolves over time and is updated dynamically. There is no specific moment in time at which the target is expected to appear; rather the probability of target appearance gradually increases as each critical moment is bypassed. In Bayesian terms [START_REF] Bernardo | Bayesian Theory[END_REF], these two forms of temporal predictability can be considered as equivalent to prior and posterior probability, respectively.

Although prior fMRI studies have investigated the neural substrates of fixed and evolving forms of temporal predictability separately, they have not yet been directly compared. In the current study, we compared the neural substrates of the fixed temporal predictability provided by temporal versus neutral cues to that of the dynamically evolving temporal predictability provided by the passage of time within a single experimental paradigm. Manipulating both forms of temporal prediction within a single paradigm allowed us to identify regions that were activated in common or differentially by these two distinct, but related, processes. In particular, we aimed to substantiate the hypothesis that left parietal cortex is important for initially implementing fixed temporal probabilites a priori, while right prefrontal cortex is involved in dynamically updating these probabilites as a function of the elapse of time [START_REF] Coull | Neural substrates of mounting temporal expectation[END_REF][START_REF] Coull | Discrete neuroanatomical substrates for generating and updating temporal expectations[END_REF][START_REF] Coull | Directing attention in time as a function of temporal expectation[END_REF].

Importantly, we used a factorial design to control for activity related simply to the passage of time itself by comparing changes over time in neutral versus temporal conditions i.e. a foreperiod x cue interaction. In the temporal condition, targets were presented after the same foreperiod durations as in the neutral condition, but the conditional probability of their appearance did not increase over time in the same way.

Rather than increasing gradually with foreperiod duration, as in the neutral condition, temporal probability in the temporal condition remains constant over time. This factorial design ensures that any activity related simply to foreperiod duration would be factored out of a between-condition comparison, pinpointing instead differences in temporal probability. Mento and colleagues recently employed a similar paradigm during EEG investigations in adults [START_REF] Mento | Spatiotemporal neurodynamics underlying internally and externally driven temporal prediction: a high spatial resolution ERP study[END_REF] or children [START_REF] Mento | Spatiotemporally dissociable neural signatures for generating and updating expectation over time in children: A high density-ERP study[END_REF]. In both studies, the authors found differential modulation of distinct EEG components as a function of temporal and neutral cues. Source localization techniques identified preferential activation of a left-lateralised premotor-parietal network following temporal cues but predominantly prefrontal activity following neutral cues. Unfortunately, the interaction between foreperiod and cue-type was not conducted.

In the current study, temporal cues predicted that the target would appear after one of four different foreperiods whereas neutral cues provided no temporally precise information (Figure 1). We used four foreperiods, rather than only two, so as to track the way in which temporal predictability evolved over time more dynamically than was possible in our previous studies [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF][START_REF] Cotti | Functionally dissociating temporal and motor components of response preparation in left intraparietal sulcus[END_REF][START_REF] Davranche | Orienting attention in time activates left intraparietal sulcus for perceptual and motor task goals[END_REF][START_REF] Coull | Functional anatomy of timing differs for production versus prediction of time intervals[END_REF]. Sensorimotor factors and foreperiod durations were precisely matched across temporal and neutral cue conditions. In the temporal condition, the probability of target onset time was fixed and constant (100% certain) across the four foreperiods. In the neutral condition, it changed as a function of foreperiod. We therefore hypothesized that differences in neural activity between the temporal and neutral conditions, averaged across the four foreperiods, would identify regions associated with the behavioural benefits of fixed a priori temporal predictions. On the other hand, we hypothesised that during the neutral condition, in the absence of any other temporal information, participants would rely upon temporally predictive information inherent in the passage of time itself (i.e. the hazard function) to optimise performance. Any change in neural activity as a function of foreperiod in the neutral condition, as compared to the temporal condition (i.e. the cue x foreperiod interaction), would therefore identify regions selectively involved in the dynamic updating of temporal predictability as a function of time.

Methods

Participants

Sixteen healthy right-handed volunteers (mean age=26.9 years; SD = 5.9 years), with no record of neurological or psychiatric disorders and normal or corrected-to-normal vision, participated in the fMRI experiment. All volunteers gave written informed consent to the study protocol, which had been approved by the local research ethics committee. The study was performed in accordance with the ethical standards laid down in the Declaration of Helsinki.

Task

Participants performed a cued visual reaction time (RT) task. The visual cue either predicted the time at which an imminent target would be presented (temporal condition) or acted simply as a trial onset signal, providing no temporally precise information about the time of target presentation (neutral condition). Each trial began with either the temporal or neutral cue, followed by a variable foreperiod (i.e. the interval between the cue and the target) and then brief presentation of an imperative visual target (Figure 1).

Participants were informed that they could use the temporal cue to predict the time of target onset, so as to respond to its appearance as quickly as possible. They were told that the neutral cue would not help them to predict target onset but that they should still respond to the target as quickly as possible. Therefore, upon presentation of the cue participants could either predict with 100% certainty when the target would appear (temporal cue condition) or else could not predict its appearance in advance but could update the likelihood of its appearance as a function of elapsing foreperiod (neutral cue condition). In our experiment, the neutral cue condition is equivalent to the classic "variable foreperiod" paradigm [START_REF] Woodrow | The measurement of attention[END_REF][START_REF] Niemi | Foreperiod and simple reaction time[END_REF].

Participants were instructed to avoid responding before the target had appeared (i.e. anticipatory responses). Reaction times were calculated as the time between visual presentation of the target and the button-press response.

All stimuli were white and presented in the centre of a black background. A background stimulus, consisting of four white concentric circles, was constantly present throughout the task. The temporal cue comprised a brief (117ms) brightening of one of the four circles. Brightening of the smallest (innermost) circle indicated that the target would appear after the shortest (533ms) foreperiod, brightening of the next largest circle indicated that it would appear after a 950ms foreperiod, brightening of the next largest circle that it would appear after a 1467ms foreperiod, and brightening of the largest (outermost) circle indicated that the target would appear after the longest (2083ms) foreperiod. Note that these rather unusual foreperiod durations (533/950/1467/2083ms) were constrained by the screen refresh rate of the monitor: they approximate to our intended foreperiod durations of 500/1000/1500/2000ms. The neutral cue comprised a brief (117ms) brightening of all four circles indicating that the target could appear after any one of the four foreperiods (533/950/1467/2083ms) with equal likelihood. The target comprised a cross (+), which was briefly (117ms) superimposed over the four circles.

Speeded responses to the target were registered with a brief right finger button press.

Average trial length was approximately 1.5s. Inter-trial intervals varied between 2-3s (in steps of ~130ms). Temporal and neutral trials were intermixed and the foreperiod varied on a trial-by-trial basis. The combination of the variable inter-trial interval with the variable foreperiod meant that the effective interval between consecutive trials varied between 3.3s-4.9s.

The two cues (temporal/neutral) and the four foreperiods (533/950/1467/2083ms) constituted a 2 x 4 factorial design. During each of the scanning runs, participants performed a total of 112 trials, comprising 56 temporal cue trials and 56 neutral cue trials (14 trials per foreperiod per cue condition), presented in random order. Across the four scanning runs, this gave a total of 56 trials per foreperiod per condition. Prior to scanning, participants performed a training session to learn the association between the four temporal cues and the four foreperiods. This session began with blocked presentation (6 trials) of each of the 4 cue-foreperiod temporal condition combinations. This was followed by a block of 24 temporal cue trials in which the 4 foreperiods were cyclically presented in ascending order of foreperiod duration (from shortest to longest), then a block of 24 temporal cue trials in which the 4 foreperiods were presented in completely randomised order. Finally, participants performed a practice block in which temporal and neutral cue trials (24 trials each) were randomly intermixed. Scans were acquired at the Marseille fMRI centre (http://irmfmrs.free.fr) using a 3-T (3T-MAGNEX) Bruker Medspec 30 / 80 Advance whole-body MRI system (Ettlingen, Germany), equipped with a head coil. EPI was used to obtain T2*-weighted fMRI images in the axial plane, using an interleaved slice acquisition sequence. The acquired image volume consisted of 30 × 4 mm transverse slices, with an inter scan interval (TR) of 2.0s.

The size of this image volume allowed us to scan the entire cerebral cortex and most of the cerebellum.

Four functional scanning runs, each comprising 234 scans (approximately 8 minutes per run), were acquired for each participant. Run order was counterbalanced across subjects. Within each functional run, the eight trial-types (temporal/neutral cue x 4 foreperiods) were presented in randomised order to ensure optimisation of event-related signal strength [START_REF] Josephs | Event-related functional magnetic resonance imaging: modelling, inference and optimization[END_REF]. For 3 of the participants, a failure in stimulus synchronization software meant that data from only three of the four functional runs were usable. A structural MRI was acquired (using a standard T1-weighted scanning sequence, 1 mm 3 resolution) to allow anatomically specific localization of significant areas of brain activation.

Data Analysis

Behavioural Data

Reaction times (RTs) to detect the target were recorded during the fMRI session and analysed with a repeated measures ANOVA, incorporating cue (time/neutral) and foreperiod (533/950/1467/2083ms) as within-subjects factors. Trials with anticipatory responses and RTs faster than 100ms were excluded from the analysis (9.9% of all trials).

Post-hoc pairwise comparisons were Bonferroni corrected. We also examined the relationship between the RT benefit of time versus neutral cues (RTs in the neutral condition minus RTs in the time condition) to the RT benefit of foreperiod duration (linear regression of RT as a function of foreperiod in the neutral condition minus linear regression of RT as a function of foreperiod in the time condition) using Pearsons's correlation.

fMRI Data

Image processing and statistical analysis of fMRI data were conducted with SPM8 (http://www.fil.ion.ucl.ac.uk/spm/software/spm8). The first 5 images of each run were included to allow for magnetic field saturation and were discarded prior to image processing. All remaining functional images were slice-time corrected using the middle slice in time as reference (slice 27; interleaved acquisition). These images were then realigned to correct for head movement between scans. All functional images were then spatially normalised by matching each image to the standard SPM8 EPI template, resampled to 3-mm isotropic voxel size, and were spatially smoothed using isotropic Gaussian kernels of 8 mm full-width half-maximum (FWHM).

We then conducted event-related analysis of the preprocessed fMRI data. For each individual participant, stimulus-evoked neural responses were modelled as events with zero duration (delta functions), time-locked to the onset of the cue, and convolved with the canonical hemodynamic response function. We modeled eight separate cue regressors, which represented the factorial combination of cue-type (time/neutral) and foreperiod (533/950/1467/2083ms). We also modelled one regressor of no interest that was time-locked to the onset of the target during both temporal and neutral trials, which accounted for any activity related to execution of a motor response, and six regressors of no interest that modelled the movement parameters derived from the realignment procedure. Images were adjusted for low-frequency physiological drifts, using a highpass filter of 128 seconds, and global changes were removed by proportional scaling.

The general linear model was used to generate statistical parametric maps of the tstatistic, transformed to maps of corresponding Z values, which provided parameter estimates of event-related activity at each voxel for each experimental condition. At the first level of analysis, each of the 16 participants was analysed separately to define patterns of activation for each comparison of interest for each participant. We defined three main contrasts. First, to identify regions associated with the fixed temporal predictions induced by temporal cueing, we compared activity during the temporal versus neutral trials averaged across all foreperiods (i.e. the main effect of cue). This contrast [1 1 1 1 -1 -1 -1 -1] identified regions whose activity was significantly greater in the temporal condition than the neutral condition. Second, to identify regions whose activity increased as a function of foreperiod duration, we measured how cue-related activity was parametrically modulated by foreperiod duration averaged across both cue conditions (i.e. the main effect of foreperiod). This contrast [-2 -1 1 2 -2 -1 1 2] identified regions whose activity significantly increased as foreperiod duration increased in both temporal and neutral trials. Third, to identify regions associated with the dynamic updating of temporal predictions as a function of the passage of time (i.e. the hazard function), we directly compared the parametric modulation of activity as a function of foreperiod in neutral versus temporal trials (i.e. the cue x foreperiod interaction). This contrast [2 1 -1 -2 -2 -1 1 2] identified regions whose activity increased as a function of foreperiod duration in the neutral condition, but not in the temporal condition. By using the interaction term to index the hazard function, we controlled both for foreperiod duration (the same foreperiods were used in both temporal and neutral conditions) and temporal predictability generally (both temporal and neutral conditions induce temporal predictions but temporal predictions evolve dynamically as a function of foreperiod duration in the neutral condition, but remain fixed across foreperiod duration in the temporal condition). The interaction term therefore indexes those regions associated specifically with the dynamic evolution of temporal predictions (i.e. the hazard function) rather than temporal prediction generally.

For each of these three contrasts, the 16 relevant activation maps (one per subject) were entered into a second level of analysis, and statistical inferences for each contrast were derived using one-sample t-tests in SPM. The resulting group-level maps were characterised in terms of both peak amplitude and spatial extent. Statistical maps of significant change in the amplitude of response were initially thresholded at p<0.001 (uncorrected for multiple comparisons) at the voxel-level, with the threshold for significance being set at p<0.05, corrected for multiple comparisons (FWE) at the voxel or cluster level. Activations that passed the threshold for significance were either comprised of a significantly large number of voxels (cluster level) or were of a significantly high amplitude (voxel level). In this way, we did not fail to report activations that were small in size (and so might fail to pass the cluster level threshold) but strongly activated (and so would pass the voxel level threshold). We report all areas surviving this threshold, which is corrected for interrogation of the entire brain volume.

Parameter estimates (beta values) in significant clusters were extracted using the Marsbar region of interest toolbox [START_REF] Brett | Region of interest analysis using an SPM toolbox. 8th International Conference on Functional Mapping of the Human Brain[END_REF] and plotted to aid data interpretation.

To complement the whole-brain analyses described above, and given the strong anatomical hypotheses outlined in the Introduction, we conducted independent ROIbased analyses of activity in left parietal cortex and right prefrontal cortex. We explored how activity changed as a function of experimental condition in AAL-defined left inferior and superior parietal cortices and right inferior (pars triangularis and pars opercularis) and middle frontal gyri. Beta values for each of the eight experimental conditions (2 cuetypes x 4 FPs) were extracted from each of the four ROIs using MarsBar [START_REF] Brett | Region of interest analysis using an SPM toolbox. 8th International Conference on Functional Mapping of the Human Brain[END_REF]. We conducted repeated measures ANOVAs for each ROI separately, with cue (time/neutral) and foreperiod (533/950/1467/2083ms) as within-subjects factors. We also used Pearson's correlations to explore the relationship between performance and regional activity by correlating the difference in the slope of the RT data in neutral versus temporal trials with the difference in activity in neutral versus temporal trials. We correlated RT slope with activity measured at the longest (2083ms) foreperiod, since this is the time at which the hazard function should exert its greatest influence. ROIs were defined anatomically and independently from the whole-brain analysis, thereby avoiding the error of "double-dipping".

Results

Behavioural data

There were highly significant main effects of both cue condition and foreperiod on RTs.

Responses were significantly faster (F(1,15)=104.79, p<0.0001) when the target was preceded by a temporal cue (244ms ±11.9) rather than a neutral cue (307.46ms±11.53ms) and got significantly faster (F(3.45)=18.28, p<0.0001) as foreperiod duration increased.

Importantly, the main effect of foreperiod was qualified by a significant cue condition x foreperiod interaction (F(3,45)=100.35, p<0.0001), such that RTs got faster as a function of foreperiod following the neutral cue (F(3,45)=85.17, p<0.0001) but not following the temporal cue (Figure 2a). Instead, RTs in the temporal condition were actually fastest at the shortest foreperiod (F(3,45)=7.99, p<0.001). Pairwise comparisons at each foreperiod also confirmed results of prior studies showing that the speeding of RTs induced by temporal (versus neutral) cues was evident at the short foreperiods (all p< 0.0001 at the 533ms/950ms/1467ms foreperiods) but not at the longest one (p>0.1) [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF][START_REF] Coull | Orienting attention in time: behavioural and neuroanatomical distinction between exogenous and endogenous shifts[END_REF][START_REF] Griffin | Orienting attention in time[END_REF][START_REF] Correa | The attentional mechanism of temporal orienting: determinants and attributes[END_REF].

The RT benefit induced by temporal versus neutral cues was significantly correlated to the RT benefit of foreperiod duration in neutral versus temporal trials (r= -0.65, p<0.005). Specifically, the more subjects benefitted from the temporal cue, the more they also benefitted from increasingly long foreperiods in the neutral condition (Figure 2b). We used two complementary approaches to analyse fMRI data. First, we interrogated the entire brain volume for regions of activity identified by functionally defined contrasts.

Second, we examined patterns of functional change in anatomically defined regions of interest. While the first approach is functionally driven but anatomically impartial, the second approach is anatomically driven and functionally impartial.

Functionally guided whole brain analyses

Temporal cueing: fixed temporal predictions

The main effect of cue condition [temporal-neutral] confirmed prior reports that left parietal cortex is activated significantly more by temporal, than neutral, cueing [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF][START_REF] Cotti | Functionally dissociating temporal and motor components of response preparation in left intraparietal sulcus[END_REF][START_REF] Davranche | Orienting attention in time activates left intraparietal sulcus for perceptual and motor task goals[END_REF][START_REF] Coull | Functional anatomy of timing differs for production versus prediction of time intervals[END_REF]. We found two distinct clusters of activity in left parietal cortex (Figure 3; Table 1), one in inferior parietal cortex (BA40) and the other located more caudally in superior parietal cortex (BA7). Interestingly, the right-sided homologue of the superior parietal cluster was also significantly more activated by temporal, than neutral, cueing (Figure 3), as was right posterior occipital cortex, left basal ganglia (putamen) and thalamus (pulvinar) (Table 1).

Figure 3 about here please Table 1 about here please

Foreperiod duration

The main effect of foreperiod [parametrically increasing foreperiod duration] showed that activity in SMA, dorsal premotor cortex bilaterally, left frontal operculum, left putamen and right lingual gyrus increased with increasing foreperiod (Table 2a).

Hazard function: evolving temporal predictions

The interaction between cue and foreperiod identified a single cluster of activation in left inferior parietal cortex, located in the intraparietal sulcus (Table 2b), indicating that activity in this region increased as a function of foreperiod more in the neutral condition than in the temporal condition (Figure 4, top panel).

Figure 4 about here please

There were no other clusters of activation that survived the correction for multiple comparisons at the whole brain level. However, given the strong a priori hypothesis that right prefrontal cortex would be activated by the hazard function [START_REF] Coull | Orienting attention in time: behavioural and neuroanatomical distinction between exogenous and endogenous shifts[END_REF]Vallessi et al, 2007a,b;2009a,b;[START_REF] Bueti | Encoding of temporal probabilities in the human brain[END_REF][START_REF] Triviño | Temporal orienting deficit after prefrontal damage[END_REF] we conducted a small volume correction (SVC) of this contrast, using a mask comprising AAL-defined right inferior frontal (pars triangularis and pars opercularis) and middle frontal gyri. This SVC analysis revealed a significant cluster of activation in right inferior frontal cortex (Figure 4, bottom panel; Table 2b). We also observed a third cluster of activity in left dorsolateral prefrontal cortex (x,y,z co-ordinates= -36, 14, 25; 23 voxels; Z=3.97) that did not survive the correction for multiple comparisons but which, given prior reports of bilateral prefrontal involvement in the hazard function [START_REF] Triviño | Temporal orienting deficit after prefrontal damage[END_REF][START_REF] Triviño | Rhythms can overcome temporal orienting deficit after right frontal damage[END_REF], we report here for completeness.

We also examined the cue x foreperiod interaction in the opposite direction and found that activity in medial occipital cortex (x,y,z co-ordinates= 15, -82, -8; 71 voxels; Z=4.18) increased as a function of foreperiod in the temporal condition but not in the neutral condition

Table 2 about here please

Anatomically guided ROI analyses

Left inferior parietal cortex

A significant main effect of cue (F(1,15)=17.60, p=0.001) indicated that left inferior parietal cortex was significantly more activated during the time condition than the neutral condition. There was no main effect of foreperiod (F(3,45)=1.25, ns) though a significant cue x foreperiod interaction (F(3,45)=3.88, p=0.015) revealed that activity in this region increased as a function of foreperiod duration during the neutral trials (F(3,45)=4.55, p<0.01), but remained constant across foreperiod duration during temporal trials (F(3,45)=0.14, ns) (Figure 5). Pairwise comparisons at each foreperiod (Bonferroni corrected for multiple comparisons) showed significantly higher activity in the temporal than neutral condition at the 533ms, 950ms and 1467ms foreperiods (all p<0.05).

However, there was no difference in activity between temporal and neutral conditions at the 2083ms foreperiod. Pearson's correlations revealed a strong relationship between the RT benefits of the hazard function and parietal activity at the crucial longest (2083ms) foreperiod (r = -0.673, p<0.005). Specifically, the more negative the slope of the RT data in the neutral, versus temporal, condition, the more parietal activity increased in the neutral, relative to temporal, condition (Figure 6a).

Left superior parietal cortex

A significant main effect of cue F(1,15)=34.00, p=0.0001) indicated that left superior parietal cortex was significantly more activated during the time condition than the neutral condition (Figure 5). There was no main effect of foreperiod (F(3,45)=0.36, ns), nor cue x foreperiod interaction (F(3,45)=0.80, ns).

Right inferior frontal gyrus

Although there were no significant main effects of cue (F(1,15)=0.29, ns) or foreperiod (F(3,45)=1.96, ns), a significant cue x foreperiod interaction (F(3,45)=4.65, p<0.01) revealed that activity in right inferior frontal gyrus increased as a function of foreperiod duration during the neutral trials (F(3,45)=5.55, p<0.005), but remained constant across foreperiod duration during temporal trials (F(3,45)=0.07, ns) (Figure 5). Pairwise comparisons at each foreperiod (Bonferroni corrected for multiple comparisons) showed that the significantly higher activity in the temporal versus neutral condition at 950ms (p<0.05) was reversed at 2083ms, by which point there was significantly higher activity in the neutral than temporal condition (p<0.05). Pearson's correlations revealed a significant relationship between the RT benefits of the hazard function and frontal activity at the longest (2083ms) foreperiod (r = -0.538, p<0.02). Specifically, the more negative the slope of the RT data in the neutral, versus temporal, condition, the more frontal activity increased in the neutral, relative to temporal, condition (Figure 6b).

Right middle frontal gyrus

There were no main effects of cue (F(1,15)=3.38, ns), foreperiod (F(3,45)=1.17, ns), nor cue x foreperiod interaction (F(3,45)=1.53, ns).

Figure 5 about here please

Functionally guided ROI analysis

As a supplementary post-hoc analysis, we defined a smaller (3mm radius) functional ROI in middle frontal gyrus, centred on the co-ordinates (x,y,z,= 52, 40, 26 mm) provided by Vallesi et al (2009a). Beta values in each experimental condition were extracted from this ROI using MarsBar [START_REF] Brett | Region of interest analysis using an SPM toolbox. 8th International Conference on Functional Mapping of the Human Brain[END_REF] and the pattern of activity across conditions was analysed with a repeated measures ANOVA that included cue (time/neutral) and foreperiod (533/950/1467/2083ms) as within-subjects factors. There were no significant main effects of cue (F(1,15)=0.42, ns) nor foreperiod (F(3,45)=0.86, ns). However, the significant cue x foreperiod interaction (F(3,45)=3.64, p<0.05) indicated that activity in right middle frontal gyrus increased as a function of foreperiod duration during the neutral trials (F(3,45)=2.98, p<0.05), but remained constant across foreperiod duration during temporal trials (F(3,45)=0.48, ns).

Discussion

Being able to predict when an event is going to occur improves sensorimotor processing of that event. Using a cued reaction time (RT) task, we confirmed that both the fixed temporal predictability of temporal cues, and the evolving temporal predictability of the hazard function, significantly speeded RTs. To identify areas linked to different forms of temporal expectation, we interrogated the entire brain volume for regions whose activity significantly changed as a function of cue (temporal/neutral) or the interaction between cue and foreperiod duration. fMRI analyses showed that left inferior parietal cortex, particularly around the intraparietal sulcus, was engaged both by the fixed temporal predictability of temporal cues and by the dynamic updating of temporal probabilities in the neutral cue condition (the hazard function). Right inferior frontal cortex, on the other hand, was engaged by dynamically evolving, but not fixed, prediction mechanisms. The results of these whole-brain analyses were confirmed by independent anatomically and functionally guided ROI analyses.

Behavioural benefits of fixed and evolving temporal predictability

As predicted, RTs to detect the target were faster after temporal cues than neutral cues, reflecting the behavioural benefits of temporal cueing. RTs were also faster for longer foreperiods, reflecting the behavioural benefits of the hazard function. Importantly, a significant interaction between cue and foreperiod indicated that although RTs got faster as a function of foreperiod in the neutral condition, they remained relatively constant in the temporal condition. These data clearly reflect the way in which the probability distribution evolves over time in the neutral condition: the probability of the target appearing at the shortest foreperiod after a neutral cue was 25%, rising to 33% at the second foreperiod, then to 50% at the third, until reaching 100% probability only at the fourth and longest foreperiod. On the other hand, the probability distribution in the temporal condition was fixed and constant: the probability of the target appearing at the cued foreperiod after a temporal cue was 100% for all four foreperiods. Consequently, RTs were significantly faster after temporal cues than neutral ones at the three shorter foreperiods, but no different at the longest foreperiod when the probability of target appearance was 100% in both conditions. It could be argued that the speeding of RTs as a function of foreperiod in the neutral condition does not reflect the effects of temporal predictability but, instead, simply reflects the longer period of time available in which to prepare and hone the motor response. However, RTs didn't decrease as a function of foreperiod in the temporal condition, even though foreperiods in this condition afforded the same amount of time to prepare the response. The differential pattern of RTs in the temporal versus neutral condition is therefore more likely to reflect a process that is differentially engaged by the two cue conditions (e.g. fixed versus evolving temporal predictability) than a more fundamental one that is common to both (e.g. time to prepare a motor response).

Incidentally, RTs in the temporal condition were not just invariant across the three longest foreperiods but, collectively, were actually slower than RTs at the shortest foreperiod. This is reminiscent of the pattern observed in fixed foreperiod paradigms in which, by contrast to variable foreperiod paradigms, RTs are slower at longer foreperiods (Klemmer, 1956;Niemi and Naataanen, 1981). Timing estimates (Gibbon, 1977) are more variable for longer durations. Therefore, even when the foreperiod is 100% predictable (in fixed foreperiod paradigms or the temporal cueing condition of the current experiment) predictions of target onset time will be less accurate at long foreperiods, resulting in slower and more variable RTs (see also [START_REF] Piras | Implicit, predictive timing draws upon the same scalar representation of time as explicit timing[END_REF].

Left inferior parietal cortex activity indexes both fixed and evolving temporal predictions

Confirming prior fMRI [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF][START_REF] Cotti | Functionally dissociating temporal and motor components of response preparation in left intraparietal sulcus[END_REF][START_REF] Davranche | Orienting attention in time activates left intraparietal sulcus for perceptual and motor task goals[END_REF][START_REF] Coull | Functional anatomy of timing differs for production versus prediction of time intervals[END_REF] and source-localised EEG [START_REF] Mento | Spatiotemporal neurodynamics underlying internally and externally driven temporal prediction: a high spatial resolution ERP study[END_REF] results, we found that activity in left lateral inferior parietal cortex was significantly higher in temporal, compared to neutral, cue trials, implicating this region in the performance benefits of fixed a priori temporal predictions. We further demonstrated that activity located slightly more medially, in left intraparietal sulcus, increased as a function of foreperiod in the neutral condition but not in the temporal condition. Since the objective probability of target appearance increases over time in the neutral condition but not in the time condition, we argue that this specific pattern of activity reflects the neural signature of the hazard function. By employing exactly the same foreperiods in both time and neutral conditions we controlled for incidental dynamic processes that also vary over time, such as motor preparation, response inhibition or sustained attention. Such processes would be engaged by the time condition as well as the neutral condition, and so are effectively cancelled out within the factorial design. Instead, and as confirmed by the RT data, any changes in neural activity we observed as a function of foreperiod in the neutral condition, but not in the time condition, are likely to reflect temporal expectancies that evolve over time i.e. the hazard function. Providing further support for our findings, the same pattern of activity was found in the left inferior parietal cortex ROI defined by the AAL database (Figure 5) as in the left intraparietal sulcus cluster identified by wholebrain analyses (Figure 4). In the left inferior parietal cortex ROI, activity was significantly higher in the time condition than the neutral condition at the shortest foreperiods. However, as target probability increased over time in the neutral condition, the patterns of activity in time and neutral conditions gradually converged, until meeting at the longest foreperiod at which point the probability of target occurrence was 100% in both time and neutral conditions. These patterns of brain activity neatly mirror the behavioural data. Indeed, activity in left inferior parietal cortex strongly correlated with the RT benefits of the hazard function.

Whole-brain analysis further revealed that the fixed temporal expectation provided by a temporal cue activated left and right superior parietal cortices, as well as left inferior parietal cortex. In contrast to the pattern of activity in left intraparietal sulcus however, activity in superior parietal cortices did not vary as a function of foreperiod duration. Indeed, ROI analyses allowed us to functionally dissociate distinct forms of temporal expectation in discrete regions of parietal cortex: left inferior parietal cortex, including intraparietal sulcus, was implicated in both fixed and evolving temporal expectations whereas left (and right) superior parietal cortex was engaged by fixed temporal expectations only. This dorsal-ventral distinction for distinct aspects of temporal expectation is reminiscent of [START_REF] Corbetta | Control of goal-directed and stimulus-driven attention in the brain[END_REF] scheme for the role of parietal cortex in attention: in this model, superior parietal cortex directs attention to relevant locations or objects in a goal-directed way, whereas more inferior regions of parietal cortex (temporoparietal junction) redirect attention to unexpected events in a more stimulus-driven way. Analogously, our data indicate that superior parietal cortex is implicated in the a priori deployment of attention towards a fixed moment in time (goaldirected), whereas inferior parietal cortex and intraparietal sulcus are involved in dynamically reorienting attention to the next possible foreperiod if the target has not yet appeared (stimulus-driven). Alternatively, we may conceptualize these distinct processes in Bayesian terms: they reflect the orientation of attention in time as a function of prior and posterior probabilities respectively.

Although inferior parietal cortex and intraparietal sulcus activations were leftlateralised, superior parietal cortex activations were observed in both left and right hemispheres. Bilateral activation of parietal cortex has previously been reported for the performance benefits of temporal versus neutral cues [START_REF] Coull | Functional anatomy of timing differs for production versus prediction of time intervals[END_REF] or spatial versus neutral cues [START_REF] Gitelman | A large-scale distributed network for covert spatial attention: further anatomical delineation based on stringent behavioural and cognitive controls[END_REF]. However, when attentional cueing in the temporal and spatial domains were compared directly, [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF] found hemispheric lateralization for temporal versus spatial cueing in left and right parietal cortices respectively. These data suggest that both left and right parietal cortices are engaged whenever a selective attentional cue (either temporal or spatial) is compared to a neutral, non-informative cue, but that left and right parietal cortices are preferentially engaged by temporal and spatial attentional cueing respectively. Incidentally, the preferential activation of left inferior parietal cortex by temporal versus spatial cues in this earlier study [START_REF] Coull | Where and When to Pay Attention: The Neural Systems for Directing Attention to Spatial Locations and to Time Intervals as Revealed by Both PET and fMRI[END_REF]) also indicates that this region is implicated in temporal orienting over and above any spatial component of the temporal cue (small circle=short/large circle=long) that might be thought to have led to the parietal activation. Indeed, fixed temporal probabilities activate left inferior parietal cortex even in an entirely non-spatial paradigm employing isochronous auditory rhythms [START_REF] Bolger | Metrical rhythm implicitly orients attention in time, as indexed by improved target detection and left inferior parietal activation[END_REF].

Overall, our fMRI data indicate a common anatomical substrate for temporal prediction in left inferior parietal cortex, whether the probabilities underlying that prediction are fixed (temporal cues) or evolving (hazard function). Moreover, the RT benefits of temporal cues correlated significantly with the RT benefits of the hazard function, suggestive of a common functional mechanism: the more participants benefitted from the temporal information conveyed by the hazard function the more the benefitted from the temporal information conveyed by the temporal cues. Collectively, these data suggest that left inferior parietal cortex is responsible for implementing the response benefits of temporal prediction, however that prediction has been generated. Prior electrophysiological studies in monkeys have shown that parietal activity reflects the accumulation of sensory evidence over time leading to a particular response [START_REF] Roitman | Response of neurons in the lateral intraparietal area during a combined visual discrimination reaction time task[END_REF][START_REF] Kiani R Hanks | Bounded integration in parietal cortex underlies decisions even when viewing duration is dictated by the environment[END_REF] and, more specifically, that neural firing in parietal cortex mirrors the hazard function (Janssen and Shadlen, 2005). Our whole brain fMRI approach not only confirms that parietal cortex is important in making use of the hazard function to optimize responding but, in addition, reveals that the changing temporal probabilities of the hazard function are also tracked by activity in inferior frontal cortex.

Right inferior frontal cortex is engaged by dynamic updating of temporal predictions over time

Activity in right inferior frontal cortex increased as a function of foreperiod in neutral, but not temporal, trials, reflecting the performance benefits of the hazard function.

These data support previous findings that right lateral frontal cortex is implicated in the dynamic updating of temporal probabilities over time [START_REF] Stuss | Multiple frontal systems controlling response speed[END_REF]Vallesi et al, 2007bVallesi et al, , 2009a;;[START_REF] Bueti | Encoding of temporal probabilities in the human brain[END_REF]. Interestingly, the contrast between neutral and temporal trials as a function of foreperiod identified an additional sub-threshold cluster of activity in left prefrontal cortex, lending some support to prior neuropsychological reports that RT benefits of the hazard function depend upon the integrity of both left and right prefrontal cortices (Vallesi et al, 2009a;[START_REF] Triviño | Temporal orienting deficit after prefrontal damage[END_REF][START_REF] Coull | Discrete neuroanatomical substrates for generating and updating temporal expectations[END_REF].

Results of our whole-brain analysis were confirmed and supported by independent anatomically defined ROI analyses. Activity in the right inferior frontal gyrus ROI (comprising pars opercularis and pars triangularis) tracked the increasing temporal predictability of target appearance during neutral trials but did not change as a function of foreperiod duration during temporal trials, suggesting that it is implicated specifically in temporal predictability that evolves as a result of the hazard function. Prior neuropsychological investigations of patients with prefrontal lesions, however, indicate that right prefrontal cortex is critical not only for implementing the hazard function but also for the fixed temporal predictions carried by temporal cues [START_REF] Triviño | Temporal orienting deficit after prefrontal damage[END_REF][START_REF] Coull | Discrete neuroanatomical substrates for generating and updating temporal expectations[END_REF]. Although we found that activity in right inferior frontal cortex was significantly higher in the temporal than neutral condition at the 950ms foreperiod (though not at the 533ms or 1467ms foreperiods) this pattern was reversed at the longest 2083ms foreperiod. Therefore although right inferior frontal cortex may be engaged more by temporal than neutral cues at short foreperiods, this pattern is reversed at longer ones, thereby effectively cancelling out any differential activity between cue conditions when data are averaged across foreperiod duration. In other words, the role of right inferior frontal cortex in temporal predictability depends critically upon foreperiod duration, strongly suggesting that it reflects the evolving temporal probabilities of the hazard function. Indeed, activity in right inferior frontal cortex significantly correlated with the RT benefits of the hazard function.

The apparently conflicting neuropsychological and fMRI results might be reconciled by distinguishing the ways in which temporal cueing was measured in these studies. In the current fMRI investigation, we compared temporal cues to neutral cues. In the neuropsychological studies of Triviño and colleagues, they compared temporally valid cues to temporally invalid ones (i.e. the target appeared either when expected (valid) or earlier/later than expected (invalid)). While the comparison of temporal to neutral cues indexes the RT benefits of temporal cues, the comparison of valid to invalid cues indexes both the RT benefits of valid cues and the RT costs of invalid cues. Prior fMRI investigation has shown that a direct comparison of temporally invalid to temporally valid cues activates not only left inferior parietal cortex but also a bilateral network of frontal regions, including right prefrontal cortex [START_REF] Coull | Orienting attention in time: behavioural and neuroanatomical distinction between exogenous and endogenous shifts[END_REF]. It is therefore conceivable that performance of right prefrontal patients in the neuropsychological studies was underpinned by an effect on the attentional shifting mechanism required by temporally invalid trials, rather than an effect on the performance benefits of temporally valid cues. A neuropsychological study comparing temporal to neutral cues would help resolve this issue.

ROI analyses further revealed distinct patterns of functional activation in discrete regions of right prefrontal cortex. While activity in right inferior frontal gyrus increased as a function of foreperiod duration in neutral but not temporal trials, reflecting the hazard function, there was no significant change in activity of middle frontal gyrus either as a function of foreperiod duration nor as a function of temporal versus neutral cues. These data contrast with previous fMRI findings, which localize the neural mechanisms underlying the hazard function to more dorsal regions of prefrontal cortex (Vallesi et al, 2009a;[START_REF] Bueti | Encoding of temporal probabilities in the human brain[END_REF] than found here. Experimental differences across studies1 make it premature to draw definitive conclusions about the specific region of right prefrontal cortex mediating the behavioural benefits of the hazard function. In addition, our ROI analysis was anatomically coarse, with the middle frontal ROI encompassing a large, functionally heterogeneous area of cortex (Figure 5), which may have diluted any activity related specifically to the hazard function. We therefore conducted a supplementary post-hoc ROI analysis of our data, in which the ROI was functionally (rather than anatomically) defined using the co-ordinates provided by Vallesi et al (2009a). Activity in this more limited region of middle frontal gyrus significantly increased as a function of foreperiod duration in neutral but not temporal trials, mimicking the pattern we found in inferior frontal gyrus, and reflecting the evolving temporal probabilities of the hazard function over time.

SMA activity increases as a function of foreperiod duration independently of temporal predictability

Activity in SMA, premotor cortex bilaterally and left frontal operculum increased as a function of foreperiod duration in neutral and temporal conditions equally. Activity in left putamen also increased as a function of foreperiod duration in both neutral and temporal conditions, though was consistently higher in the temporal condition. These parametric increases in activity during both temporal and neutral conditions indicate that these regions are not specifically implicated in the hazard function: their activity increased as a function of foreperiod duration in the temporal condition, in which predictability remains fixed across foreperiods, as much as in the neutral condition.

These findings are at odds with prior fMRI data suggesting that SMA activity specifically tracks the hazard function [START_REF] Cui | Go: Amplitude of the fMRI signal encodes expectation of cue arrival time[END_REF][START_REF] Bueti | Encoding of temporal probabilities in the human brain[END_REF]. In a variable foreperiod task with neutral cues, [START_REF] Cui | Go: Amplitude of the fMRI signal encodes expectation of cue arrival time[END_REF] found phasic increases in SMA activity at the moment of target appearance, with SMA amplitude varying as a function of foreperiod duration (i.e. in line with the hazard function). By contrast, when the time of target appearance was explicitly provided by a visual countdown, SMA activity no longer varied as a function of foreperiod. The authors therefore concluded that SMA activity reflected the active updating of temporal probabilities (i.e the hazard function) rather than the simple passage of time. However, by including both neutral and temporal cues in the current study, we have shown that SMA activity varies with foreperiod duration both when participants have prior temporal expectations about target onset time (temporal condition) as well as when they update temporal expectations as a function of time (neutral condition). The main difference between our temporal cue condition and Cui et al's (2009) countdown condition, both of which fully predict when the target will appear, is the need to actively time the foreperiod following a temporal cue but not during a countdown. Taken as a whole, these data suggest that SMA might be indexing the active timing of foreperiod duration -either in order to update probabilities in the neutral cue condition or to predict target onset time in the temporal cue condition. By contrast, in [START_REF] Cui | Go: Amplitude of the fMRI signal encodes expectation of cue arrival time[END_REF] countdown condition, no active timing was required: participants had simply to wait until the countdown reached zero. This interpretation is consistent with prior fMRI [START_REF] Wencil | Carving the clock at its component joints: neural bases for interval timing[END_REF][START_REF] Coull | SMA selectively codes the active accumulation of temporal, not spatial, magnitude[END_REF] and single cell electrophysiological [START_REF] Mita | Interval time coding by neurons in the presupplementary and supplementary motor areas[END_REF][START_REF] Merchant | Measuring time with different neural chronometers during a synchronization-continuation task[END_REF][START_REF] Merchant | Interval tuning in the primate medial premotor cortex as a general timing mechanism[END_REF][START_REF] Crowe | Dynamic representation of the temporal and sequential structure of rhythmic movements in the primate medial premotor cortex[END_REF] studies showing that SMA activity increases with active or voluntary timing of stimulus duration.

Foreperiod-related activity increases in both temporal and neutral conditions might instead (or also) reflect motor preparation or response inhibition. Indeed, SMA has previously been implicated in motor preparation [START_REF] Deiber | Mesial motor areas in selfinitiated versus externally triggered movements examined with fMRI: effect of movement type and rate[END_REF][START_REF] Cunnington | The preparation and execution of self-initiated and externally-triggered movement: a study of event-related fMRI[END_REF]Hoffstaedter et al, 2013;[START_REF] Rae | Selection and stopping in voluntary action: A meta-analysis and combined fMRI study[END_REF] and response inhibition [START_REF] Mostofsky | Response inhibition and response selection: two sides of the same coin[END_REF][START_REF] Rae | Selection and stopping in voluntary action: A meta-analysis and combined fMRI study[END_REF], both of which are processes that would be increasingly engaged over the foreperiod as the participant waits to make their response.

Alternatively, activity in SMA might simply reflect the time needed to make the motor response: [START_REF] Grinband | The dorsal medial frontal cortex is sensitive to time on task, not response conflict or error likelihood[END_REF] found that preSMA activity was higher for slow RTs than fast ones. However, our own data argue against this interpretation. SMA activity increased as a function of foreperiod in both neutral and temporal conditions equally, despite the fact that RTs got faster with foreperiod duration in the neutral condition but remained constant over foreperiod in the temporal condition. Moreover, SMA became increasingly active as foreperiod duration lengthened, which, in the neutral condition, was actually associated with increasingly shorter RTs. However, our cluster of activation was located in SMA proper whereas that of [START_REF] Grinband | The dorsal medial frontal cortex is sensitive to time on task, not response conflict or error likelihood[END_REF] was located more rostrally within preSMA, which might explain at least some of the differences in our findings.

In conclusion, our complementary whole-brain and ROI approach has allowed us to differentiate discrete patterns of functional response in distinct brain regions. Fixed temporal expectations established a priori by temporal cues activated left inferior and superior parietal cortices whereas temporal expectations that evolve over time in line with the hazard function activated left intraparietal sulcus and right inferior frontal cortex. This pattern of results supports and extends prior EEG observations of predominantly left-lateralised premotor-parietal activity for temporal cues and predominantly rightlateralised prefrontal activity for neutral cues [START_REF] Mento | Spatiotemporal neurodynamics underlying internally and externally driven temporal prediction: a high spatial resolution ERP study[END_REF][START_REF] Mento | Spatiotemporally dissociable neural signatures for generating and updating expectation over time in children: A high density-ERP study[END_REF]. By manipulating both temporal cueing and foreperiod duration within a factorial design, we found that left inferior parietal cortex plays a key role in mediating the RT benefits of temporal predictability, whether temporal probabilities are fixed or evolve dynamically as a function of time. -7, 64 -51, -10, 52 45, -10, 46 -54, 11, 4 -30, -10, 7 15, -70, -2 -27, -58, 46 54, 8 2 -Brain regions whose activity increased parametrically as a function of foreperiod duration. The table shows areas whose activity increased with foreperiod duration either (a) in both temporal and neutral cue conditions or (b) in neutral but not temporal cue conditions. Activations are significant at p<0.05 corrected for multiple comparisons at the whole brain-level or in an anatomically defined region of interest (*).

Figure legends

Figure 1 Each trial began with presentation of a visual cue, comprising four concentric circles. In the time condition, the size of the highlighted circle predicted the duration of the interval ("foreperiod") between the cue and the target. For example, the smallest circle indicated that the target would be presented after a 533ms foreperiod, whereas the largest indicated a 2083ms foreperiod. In the neutral condition, all four circles were highlighted, indicating that the target would be presented after a random foreperiod. The target consisted of an upright cross, superimposed upon the background image of the four concentric circles. The participants' task was to detect the appearance of the target as quickly as possible. They were encouraged to use the temporally predictive information conveyed by the temporal cue to response speed.

Figure

  Figure 1 about here please

Figure

  Figure 2 about here please

Figure 2 (

 2 Figure 2 (a) On average, mean reaction times (RT) were faster following temporal versus neutral cues, reflecting the behavioural benefit of fixed temporal predictions. RTs were faster as a function of increasing foreperiod in the neutral condition, but not in the temporal condition, reflecting the behavioural benefit of the hazard function. Error bars reflect standard errors. (b) Across participants, the RT benefits of the hazard function (slope of RT data across foreperiod in neutral trials -slope of RT data across foreperiod in temporal trials) correlated significantly with the RT benefit of temporal cues (RT in neutral trials -RT in temporal trials). The more negative the RT slope in neutral trials, the greater the speeding effect of temporal cues. Each point represents an individual participant.

Figure 3

 3 Figure 3Compared to neutral cues, temporal cues selectively activated left inferior parietal lobe (IPL) and superior parietal lobes (SPL) bilaterally, across foreperiod (FP) duration. Significant clusters of activation are thresholded at p<0.05 (corrected for multiple comparisons across the entire brain volume) and displayed on a transverse slice of the template brain provided by SPM software. L=left, R=right. Accompanying plots illustrate the mean level of activity (beta values) in significant clusters during temporal (purple) or neutral (blue) cue trials.

Figure 4

 4 Figure 4Activity in left intraparietal sulcus (IPS) and right inferior frontal gyrus (IFG) increased as a function of foreperiod (FP) duration during the neutral, but not temporal, condition. Significant clusters of activation are thresholded at p<0.05 (corrected for multiple comparisons across the entire brain volume) and displayed on sagittal and coronal slices of the template brain provided by SPM software. L=left, R=right. The accompanying

Figure 5

 5 Figure 5Four distinct Regions of Interest (ROIs) were defined by AAL and are displayed on a template brain from MRIcron (www.mricro.com/mricron). Activity in left inferior parietal lobe (IPL) and right inferior frontal gyrus (IFG) (ROIs) increased as a function of foreperiod (FP) duration during the neutral, but not temporal, condition. Activity in superior parietal lobe (SPL) was higher in the temporal, versus neutral, condition, independent of FP duration. Activity in middle frontal gyrus (MFG) did not change as a function of either cue or FP. The accompanying plots illustrate the mean level of activity (beta values) in ROIs during temporal (purple) or neutral (blue) cue trials. L=left, R=right. *=significant (p<0.05) effect. ns=non-significant.

Figure 6

 6 Figure 6Across participants, the RT benefits of the hazard function (RT slope in neutral trials -RT slope in temporal trials) correlated significantly with increases in activity during neutral versus temporal trials in (a) inferior parietal cortex and (b) inferior frontal cortex. Activity was measured in within the AAL-defined ROIs illustrated in Figure4. Each point represents an individual participant. ROIs are displayed on an axial slice of the template brain from MRIcron (www.mricro.com/mricron) to complement the views shown in Figure4.

Table 1 -

 1 Brain regions preferentially activated by temporal versus neutral cueing, averaged across foreperiod. Activations are significant at p<0.05 corrected for multiple comparisons at the whole brain-level. BA=Brodmann's Area.

	Anatomical structure	x,y,z co-ordinates	cluster size	Z score
		(mm)	(voxels)
	temporal -neutral
	left inferior parietal cortex (BA40)	-48, -34, 37	43	4.80
	left superior parietal cortex (BA7)	-18, -76, 43	171	4.65
	right superior parietal cortex (BA7)	18, -79, 55	240	4.65
	right posterior occipital cortex	36, -91, 10	74	4.15
	left basal ganglia (putamen)	-27, 11, 10	84	4.37
	left thalamus (pulvinar)	-9, -10, 7	45	3.97
	Anatomical structure	x,y,z co-ordinates	cluster size	Z score
		(mm)	(voxels)
	Increasing foreperiod duration
	(a) temporal and neutral
	SMA	6,
	left dorsal premotor cortex
	right dorsal premotor cortex
	left frontal operculum
	left basal ganglia (putamen)
	right lingual gyrus
	(b) neutral not temporal
	left inferior parietal cortex
	right inferior frontal cortex

In Vallesi et al (2009), fixed versus evolving forms of temporal predictability were presented in separate blocks, whereas in our study fixed and evolving predictabilities were intermixed within a block. In[START_REF] Bueti | Encoding of temporal probabilities in the human brain[END_REF], analyses measured the temporal profile of activity, identifying areas whose activity showed a phasic peak of activity at predictable times. By contrast, in our study we identified more tonic changes in the amplitude of regional activity as a function of increasing predictability.

Acknowledgements

This work was supported by European Collaborative Research Projects in the Social Sciences (06_ECRP_FP009) and the Agence Nationale de la Recherche (ANR-12-BSH2-0005-05). We would also like to thank Bruno Nazarian for practical assistance. These data appeared in abstract form in Coull JT, Cotti J, Vidal F (2014) Increasing activity in left inferior parietal cortex and right prefrontal cortex with increasing temporal predictability: An fMRI study of the hazard function.


Oussama Mortada

Abedel Halim Zahr

Jean-Christophe Orlianges

Aurelian Crunteanu

Matthieu Chatras

Pierre Blondy

J-C Orlianges

Analysis and optimization of acoustic wave micro-resonators integrating piezoelectric zinc oxide layers

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Acoustic filters are among the most compact devices that can be used for RF communication systems [START_REF] Aigner | SAW and BAW technologies for RF filter applications: A review of the relative strengths and weaknesses[END_REF][START_REF] Reinhardt | Acoustic technologies for advanced RF architectures[END_REF][START_REF] Ruby | Acoustic FBAR for filters, duplexers and front end modules[END_REF], and recent improvements of composite MEMS resonators enable to achieve very high quality factors Q, allowing for low-imprint devices and operating frequencies as high as 8 GHz [START_REF] Rinaldi | 5-10 GHz AlN Contour-Mode Nanoelectromechanical Resonators[END_REF] [START_REF] Lanz | Solidly mounted BAW filters for 8 GHz based on AlN thin films[END_REF]. A lot of research efforts were focused on bulk acoustic wave (BAWs) [START_REF] Flament | A complete UMTS transmitter using BAW filters and duplexer, a 90nm CMOS digital RF signal generator and a 0.25µm BiCMOS power amplifier[END_REF][START_REF] Catherinot | A general procedure for the desing of BAW filters[END_REF][START_REF] Giraud | Bulk acoustic wave filters synthesis and optimization for multi-standard communication terminals[END_REF] devices which offered many potentialities for smart RF components or systems, especially as an alternative to surface acoustic waves (SAWs) filters [START_REF] Liu | Switchable low loss SAW filter bank with SAW notch filters[END_REF] [START_REF] Lu | Reconfigurable multiband SAW filters for LTE applications[END_REF].

Here we present a novel piezo-acoustic ZnO-based contour-mode device fabricated on suspended silicon membrane. The general structure of this type of resonator consists of metallic interdigitated (IDTs) electrodes acoustically coupled within a piezoelectric zinc oxide (ZnO) film suspended on monocrystalline (100) silicon membranes. The design optimization of such devices allows reaching high quality factor Q and high electromechanical coupling coefficient k t 2 [START_REF] Ossama | A zinc dioxide-on-silicon MEMS resonator for narrowband filtering[END_REF]. These two parameters are important factors for the integration of such piezoelectric micro-resonators for filter applications since they are key elements to achieve a high figure of merit which is directly associated with the quality factor and the electromechanical coupling coefficient (FoM = k t 2 .Q ) [START_REF] Shi | Lithium Niobate on Silicon Dioxide Suspended Membranes: A Technology Platform for Engineering the Temperature Coefficient of Frequency of High Electromechanical Coupling Resonators[END_REF].

In this paper, a detailed study has been achieved on micro-resonators topology, in order to identify the optimum geometrical parameters (especially the length, the number of IDTs electrodes and the thickness of ZnO layer) affecting the micro-resonators performances. In order to study the effect of these parameters and experimentally checking their influence, we have designed, fabricated, and tested micro-resonators devices operating around 700 MHz. This frequency corresponds to the first excited acoustic mode displayed by a device incorporating a 500 nm thick ZnO layer and using IDTs with w pitch = 4-µm electrode pitch (see Figure 1).

We investigated different IDTs lengths, (L=12λ, 17λ, 21λ and 25λ corresponding to 100/130/170/200 µm), different numbers of IDTs electrodes (n= 80/ 50/ 40/ 25) on ZnO layers with thicknesses between 200 and 800 nm. We studied the influence of all these parameters on the extracted quality factor Q and the electromechanical coupling coefficient k t 2 while comparing the measured responses of the micro-resonators to the simulated results obtained using the MBVD electrical model (Modified Butterworth Van Dyke).

The paper is organized as follows. In part II we present the design and the fabrication process followed by modeling techniques for this type of micro-resonators in part III. In part IV, the performance of these devices are experimentally investigated and compared to the theoretical analysis (electrical model) and to finite-element -modeling (FEM). We report and discuss (in parts V and VI) three key device performance parameters: power handling, nonlinear electrical characterization and temperature coefficient of frequency (TCF). Part VII concludes the paper by summarizing the contributions of our study and setting future perspectives for device implementation For a given composite ZnO/Si resonator thickness, the mechanical resonance of such structure can be written [START_REF] Zuo | Very high frequency channel-select MEMS filters based on self-coupled piezoelectric AlN contour-mode resonators[END_REF]:

II. ZINC OXIDE MICRO-RESONATORS: DESIGN AND FABRICATION

.

where w p is the electrode pitch, E eq and φ eq are the equivalent modulus of elasticity and mass density of the composite ZnO/Si layers.

The proposed micro-resonators are fabricated following the procedure shown in Figure 2. The three-mask fabrication process is conducted on low resistivity (30 Ω.cm) 2 μm thick SOI wafers. A 1 µm thick thermal oxide layer is grown on both sides of the SOI wafer, to serve as hard mask for further wafer bulk micromachining [START_REF] Chatras | A surface-mountable membrane supported filter[END_REF], but also for electrical isolation.

SOI wafers were used because of the high intrinsic Si material acoustic quality, near-zero built-in stress and predictable mechanical properties of the top single-crystal silicon layer, which is used as the suspended membrane [START_REF] Petersen | Silicon as a mechanical material[END_REF]. The first processing step is back-side etching. A photolithography is carried to expose wafer back-side areas and SiO 2 is etched (Figure 2 Hydroxide) at 85°C. Etching is completed when the 2-µm thick Si membranes are released and suspended on the SOI substrate.

Next, a 200 nm thick molybdenum (Mo) layer is DC sputtered on the wafer at 300°C (Figure 2 (d)). The bottom Mo electrode is then patterned, followed by the deposition of ZnO film layer by Pulsed Laser Deposition (Figure 2 (e)). The ZnO thin film show a pure (002) texture and the full width half maximum (FWHM) of the (002) diffraction peak is equal to 0.07˚. More information and detailed description about the quality of deposited ZnO layer can be found in our previous work [START_REF] Ossama | A zinc dioxide-on-silicon MEMS resonator for narrowband filtering[END_REF]. Lastly, inter-digital fingers (200 nm thick of Au) are deposited using the lift-off procedure on the ZnO surface (Figure 2 (f)). A close-up scanning electron microscope (SEM) images of the fabricated micro-resonators with different IDTs lengths and numbers can be seen in Figure 3.

III. MODELING AND SIMULATION

The measured responses of the fabricated devices were fitted to the Modified Butterworth van Dyke (MBVD) model shown in Figure 4, which expresses the electrical behavior of these piezoelectric micro-resonators and then equivalent electrical parameters were extracted. The motional resistance R m comes from the mechanical damping transformed to the electrical domain; C m and L m are motional capacitance and impedance respectively and used to represent, with R m , the mechanical motion of the device. The capacitor C f is added to account for the feedthrough between IDTs electrodes; it is a function of finger pitch (w p ), length (L) and spacing (w s ). C p and R p are respectively shunt capacitance and shunt resistance. C p is determined by the static capacitance of the electrode areas and R p represents loss from the leakage current and dielectric loss from the piezoelectric material. R f expresses losses induced by the silicon membrane, this membrane is very low resistivity (30 Ω.cm). Lastly, R s come from transmission line losses.

L n

The quality factor Q was computed using the following equation [START_REF] Wanling Pan | A low-loss 1.8GHz monolithic thin-film piezoelectric-on-substrate filter[END_REF]: Following the FEM simulations it was deduced that the resonance appearing at 1.2 GHz belong to a silicon membrane bulk resonant mode while the modes between 0.8 and 1 GHz are spurious flexural modes coming from reflexions into the input/ output electrodes. These modes can be easily suppressed by modifying the micro-resonator geometry.

Q (2)

IV. DEVICE MEASUREMENTS

The fabricated prototypes have been tested using a ROHDE & SCHWARZ-ZVA24 Vector Network Analyzer and a Cascade probe station using GSG probes. Tested devices were directly probed and connected to the measurement instrumentation without the use of any external electronic interface. A short load open through (SLOT) calibration was performed.

A. Q Evolution for different lengths and numbers of IDTs

A theoretical study of the scattering [S] parameters was performed using the equivalent electrical model MBVD to extract the values of the electrical model components describing the micro-resonators. These values enables to calculate the quality factor Q, which will facilitate the interpretation of this analysis. Second, for a fixed IDTs length L=12λ, the quality factor increases from 730 (for n = 25 IDTs) up to 1180 (for n=40 IDTs) then down to 612 and to 590 for n=50 and n=80 respectively. We can notice a high peak quality factor when n=40 IDTs. This behavior is repeated for the other three lengths L = 17λ, 21λ and 25λ (Figure 6). The presented experimental results show that Q improves drastically when n=40 and the length L=12λ.

Looking at [S] parameters of the measured circuit presented in Figure 7, we noted the presence of spurious modes in close proximity of the main mechanical resonance of certain micro-resonators: micro-resonators having a large number of IDTs electrodes have a plurality of resonance modes which are mixed together at the main mechanical resonance so that the resonance peak becomes less selective and less "clean" which corresponds to a low Q.

2D FEM analysis was performed using COMSOL software in order to evaluate the experimental performances of these micro-resonators. Figure 7 shows a comparison between these simulations and measured micro-resonators. It shows how spurious modes appear and multiple when the number of IDTs is increasing. It is due to the reflected acoustic waves in the system, especially when the suspended Si membrane is large and the resonator includes a high number of IDTs (Figure 8).

By reducing the number of IDTs electrodes, resonance peaks are very selective because spurious modes are strongly attenuated especially for IDTs having n=40 which corresponds to the highest Q, and disappear completely when n=25 IDTs. On the other hand, when n=25, the equivalent motional impedance R m increases dramatically and this is why the quality factor go down again.

B. Evolution of k t 2 for different lengths and numbers of IDTs

The electromechanical coupling coefficient k t 2 expresses the degree of energy interaction between the mechanical and electrical domains. According to (3), it depends on resonance and anti-resonance values:

(3) k t 2 measurement results and their evolution with the length and number of IDTs electrodes are shown in Figure 9. For a given length, the k t 2 values decreased with increasing the number of IDTs electrodes. Thus, for a length L=12λ, the k t 2 takes its largest value of 8.84% when the number of IDTs n is equal to 25 and this value decreases to 5.67% when n increases to 80. The same evolution is observed with L= 21λ, where k t 2 = 7% in the case of 25-n IDTs electrodes and decreases to 4.3%

for 80-n IDTs electrodes.

This behavior does not only depend on the number of IDTs electrodes but also on their length. From the data represented on figure 8, we can also see that the k t 2 varies also with the variation of IDTs electrode length: smaller lengths result in a greater coupling coefficient. For example, for 40-n IDTs, the k t 2 decreases from 7.4% when L=12λ to 5.34% when L=25λ.

FIG. 9. Variation of k t 2 from a sample of three equivalent circuits at 700MHz with four different IDTs electrodes number and four different IDTs lengths.

The figure of merit (FOM) can be calculated and is presented in Figure 10.

C. Evolution of device's quality factor with different ZnO thicknesses

A group of three resonator devices with three different thicknesses of ZnO (200 nm, 600 nm and 800 nm) were fabricated and tested.

The three devices were designed with the same number of IDTs (n=80) and length (L=25λ). Measurement results are

shown in Figure 11. The resonance frequency response of the micro-resonators decreases from 765 MHz down to 672 MHz with increasing the ZnO thickness from 200 nm to 800 nm. These results were somehow expected since the composite ZnO/Si composite layer thickness is increasing and the resonant frequency value of the device is inversely proportional to its thickness [START_REF] Pan | Multiple-frequency thickness-mode thin-film piezoelectric-on-substrate filter array[END_REF]. It can be seen that an excellent agreement is obtained between the MBVD model and the measured results, clearly showing the validity of Q extraction using the equivalent electrical circuit.

The FEM simulations were employed for predicting the vibration frequencies of the ZnO micro-resonators. The measured and simulated results of resonant frequency and Q for resonators incorporating ZnO layers with thicknesses from 200 nm to 1000 nm are represented in Table II. A discrepancy of less than 4% was found between the FEM analysis and experimental data, further proving that the actual behavior of these micro-resonators can be described by conventional models. Figure 12 present a comparison of the calculated f.Q product between measured and simulated results. From the results presented in table 2 we can notice that decreasing ZnO thickness increases the quality factor from 430 (for the 800 nm thick ZnO layer) to 1600 (for the 200 nm thick ZnO). This behavior can be explained by the fact that thin ZnO layers allow to take full benefit of the very high quality factor of the single crystal silicon membrane. However, reducing the ZnO thickness to less than 200 nm would results in poor-crystalline ZnO layers with modest piezoelectric properties. The results presented so far clearly show that thin ZnO layers on Si membranes are well suited for the fabrication of high Q resonators.

V. POWER HANDLING AND NONLINEAR CHARACTERIZATION OF SINGLE MICRO-RESONATORS

The power-handling capability and nonlinear behavior of the ZnO micro-resonators were experimentally investigated by measuring the transmission response at various input power levels. The input power was swept from 5 to 22 dBm. We visually noticed that by increasing the input power, the suspend resonators were subject to mechanical vibrations. These vibrations were amplified by applying large electric fields to the piezoelectric layer especially at the resonance frequency, so the input power level was limited to 22 dBm in order to protect the probing high-frequency pads.

A. Set-up

Figure 13 presents the set-up used for power characterization of the micro-resonators [START_REF] Sahyoun | Nonlinear behavior of AlN in BAW and CRF devices for high RF electric field[END_REF]. We use the ZVA 24 GHz analyzer to generate a poser RF signal at port 1 and we monitor the transmitted power signal at port 2. FIG. 13. Set-up characterization of micro-resonators for high RF power level.

A power amplifier (PA) EM Power between 10 MHz and 1 GHz is used for amplifying the input RF signal by 43 dBm.

Two power circulators with third port connected to a load of 50 ohms are introduced before the PA and after the ZVA analyzer output port 1 to protect them from reflected power from the device. An attenuator is placed before the ZVA input port 2 to protect it from the high power circulating in the characterization chain. A "transmission response" calibration is realized at a power level of 18 dBm before measurement to correct the errors of the transmission response.

B. Power handling in micro-resonators

Two examples of power sweep measurement are shown in Figure 14. This test is realized to find the bifurcation point which can be considered as the highest level of power that the device can handle before large nonlinearity is recorded. It is also defined as the input power at which magnitude shows a vertical slope in amplitude and displays the verge of instability [START_REF] Shahmohammadi | Nonlinearity reduction in silicon resonators by doping and reorientation[END_REF]. Looking at Figure 14, we mention that the nonlinear changes are confined around this frequency where we may notice a severe frequency shift for f r of 5 MHz with increasing the input power. In the frequency regions out-side the resonant frequency area, the piezoelectric layer is not in the resonance mode and behaves as a static capacitance depending on the permittivity of the material (ZnO): we do not noticed any dielectric change with S-parameters measurements so the ZnO's permittivity ε 33 is constant.

The power handling test was realized for three micro-resonators with the same number of IDTs n=40 (this number is chosen because of the very high quality factor, as demonstrated in part 3) but different lengths (12λ, 17λ and 25 λ). As shown in Figure 14, for the presented two micro-resonators, the resonance peaks are slightly bend at the maximum power (22 dBm) but no bifurcation point is detected. This is a very unique situation where a suspended micro-resonator can support 22 dBm without breaking and also without any bifurcation point. The low resistivity of SOI wafers employed for device fabrication may explain these results. It was demonstrated in [START_REF] Shahmohammadi | Nonlinearity reduction in silicon resonators by doping and reorientation[END_REF] that for similar micro-resonators fabricated on high doping level of silicon substrate (which result in a low resistivity) are less susceptible to be driven to nonlinearity compared to higher doped wafers. Insertion losses remain stable until 14 dBm and increase after this power level.

In order to analyze the nonlinearity degree of these devices, we define a normalized frequency shift S N which is calculated as the difference between the shifted peak frequency (f s ) and the center frequency (f r ) divided by f s [START_REF] Shahmohammadi | Concurrent enhancement of Q and power handling in multitether high-order extensional resonators[END_REF]:

S N = (4)
The nonlinearity behavior can be detected when S N takes a value different of zero. Figure 15 shows that the nonlinearity begins at lower power level when micro-resonators have small IDTs. The power level dependence of S N can be separated into two zones. In the first one between 5 dBm and 20 dBm, the micro-resonator with higher IDTs lengths has a smaller S N than Our results show that ZnO-based micro-resonators can handle high power levels. This is a clear evidence of the fact that the piezoelectric micro-resonators technology can be advantageously used for high power applications.

VI. TEMPERATURE STABILITY

The frequency-temperature characteristic (TCF) of ZnO micro-resonators was also measured in the temperature range from 293 to 363 K. The temperature was varied in 10 K increments and the S-parameters measurements were performed after a 5 minute period of temperature stabilization. This variation is mainly caused by the dependence of the acoustic velocity on temperature. The acoustic velocity is determined by (E/ρ) 1/2 , where E and ρ are the Young's modulus and mass density, respectively. For ZnO, Young's modulus and the mass density decreases as temperature increases due to the material softening. The dominant effect on the Young's modulus result in decreasing of the acoustic velocity as temperature increases, resulting in a negative TCF (resonant frequency decreasing with increasing temperature).

To extract the device TCF, the resonant frequency shift with temperature is shown in Figure 17 for both micro-resonators and fitted with a linear equation. Both micro-resonators show approximately the same TCF (64.72 ppm/K and 65.7 ppm/K).

These values are near to the TCF of a conventional ZnO FBAR device (-70 ppm/°C) demonstrated in [START_REF] Yu | Ultra Temperature-Stable Bulk-Acoustic-Wave Resonators with SiO2 Compensation Layer[END_REF]. Future work will focus on increasing the operating frequency of the micro-resonators towards GHz frequencies domain.

Also, the demonstration of complex filtering functions using this technology will be very interesting. Finally, these devices can be used to fabricate sensors because of their high piezoelectric sensibility especially in term of temperature as we saw.

SUPLEMENTARY MATERIAL

See supplementary material for more information concerning: X-ray diffraction curve of ZnO thin films deposited on molybdenum film (presented in part II), the used MBVD model (presented in part ΙΙΙ) and values of extracted electrical components of MBVD model (presented in part IV).

Figure 1

 1 Figure1shows a cross-section of the proposed ZnO-based contour-mode micro-resonator. It is made with metal IDTs electrodes acoustically coupled to a suspended composite ZnO/Si structure. When a RF signal is applied across the IDTs, the piezoelectric layer moves perpendicularly to the substrate plane, leading to successive displacements and charge generation by acoustic pressure from the input to the output electrodes. In practice, the zinc oxide piezoelectric film is sandwiched between patterned gold top electrodes and a molybdenum (Mo) floating bottom electrode. By looking at Figure1, two types of acoustic waves can be detected in these micro-resonators: some waves are propagating in the ZnO film; others are propagating in the silicon membrane. First ones are propagating in all the thickness of the ZnO layer, but the second ones are guided elastic waves, referred as two-dimensional Lamb waves or plate modes. They are highly dispersive, and have very high propagation speeds. It is clear that the excitation field induced by the combined electrode system makes it possible to optimize the selective electric transduction of these guided modes.

FIG. 1 .FIG. 2 .

 12 FIG. 1. Finite element method (FEM) simulation showing the membrane displacement during propagation of an acoustic wave in ZnO and Si membrane layers. The figure shows only few of the IDTs finger electrodes as representatives of the overall device behavior.

  (b)) using buffered hydrofluoric acid (BHF), followed by 450 μm deep silicon etching (Figure 2 (c)) using TMAH (Tetra Methylamine

FIG. 3 .

 3 FIG. 3. SEM picture of the fabricated devices: eight micro-resonators having different lengths L of IDTs electrodes and different numbers n of IDTs electrodes.

FIG. 5 .Figure 5

 55 FIG. 5. Comparison between the measured S 21 parameters and the corresponding MBVD model with parameters extraction for fabricated micro-resonators (n=25 & L=21λ)

Figure 6

 6 Figure 6 reports the average quality factor Q recorded from a collection of 12 combinations of n and L. Q is compared among the same IDTs lengths of micro-resonators having different number of IDTs. From the data represented on Figure 6 a clear trend relating the measured Q vs electrode number and length of IDTs is observed.

  comparing micro-resonators having the same IDTs number in Figure6(n = 25 IDTs for example), we can notice that decreasing IDTs length increases the quality factor, from 413 (for the 25λ IDTs length) to 730 (for the 12λ IDTs length).

FIG. 7 .FIG. 8 .

 78 FIG. 7. Plot of measured and FEM simulated admittance response of four different micro-resonators: (a) with 25IDTs, (b) with 40IDTs, (c) with 50IDTs, (d) with 80IDTs. Spurious modes appear when the number of IDTs is increasing.

FIG. 10 . 2 .

 102 FIG. 10. Variation of FOM of contour-mode micro-resonators with four different IDTs electrodes number and four different IDTs lengths. The optimization of n and L enabled a record-high electromechanical coupling k t 2 of 7.4% with a high quality factor Q equal to 1180. This have been achieved by mechanically coupling 40-n IDTs of 100µm (L=12λ) in width, resulting in a figure of merit FoM (FOM= k t 2.Q) of 87. This performance surpasses the current existing technologies (TableI) designed to operate in the same frequency range.

FIG. 11 .

 11 FIG. 11. Measured and simulated transmission response of fabricated ZnO resonators with different ZnO thicknesses

13 FIG. 14 .

 1314 FIG. 14. Frequency response around 700 MHz of two micro-resonators for different injected different power levels: (a) n=40 & L=12λ, (b) n=40 & L=17λ.

FIG. 15 .

 15 FIG.[START_REF] Petersen | Silicon as a mechanical material[END_REF]. The normalized frequency shift (S N ) as a function of delivered power for two different combinations of micro-resonators operating around 700MHz.

  Two micro-resonators with different IDTs number and lengths were tested: first one with n=25 & L=21λ and the second one with n=40 & L=12λ. The resonant frequency variation with temperature of one of a measured micro-resonator response is presented in Figure 16.

FIG. 16 .

 16 FIG.[START_REF] Wanling Pan | A low-loss 1.8GHz monolithic thin-film piezoelectric-on-substrate filter[END_REF]. Transmission response around the resonant frequency of a fabricated micro-resonator measured in the temperature range from 293 to 363K.

FIG. 17 .

 17 FIG. 17. The frequency shift with temperature measured and fitted with linear equation for two micro-resonators: (a) n=25 & L=21λ, (b) n=40 & L=12λ VII. CONCLUSION

L=50λ, n=80IDTs L=50λ, n=50IDTs L=50λ, n=40IDTs L=50λ, n=25IDTs L=25λ, n=25IDTs L=21λ, n=25IDTs L=17λ, n=25IDTs L=12λ, n=25IDTs FIG

  .[START_REF] Rinaldi | 5-10 GHz AlN Contour-Mode Nanoelectromechanical Resonators[END_REF]. The equivalent MBVD model considering all losses present in the structure.

TABLE I .

 I Performance comparison between state-of-art piezoelectric micro-resonators.

	State of art	f r (GHz)	k t 2	Q	FOM	ref
	AlN-LOBAR	1	1%	3891	39	[17]
	AlN-LFE	2,8	2,5%	1855	45	[18]
	LiNbO 3 -LFE	0,5	11,5%	362	41	[19]
	This work	0,7	7,4%	1180	87
	f r is the resonance frequency 2 is the electromechanical coupling coefficient k t Q is the quality factor
	FOM is the figure of merit
	LOBAR: lateral overtone bulk acoustic resonator
	LFE: Lateral field excitation

TABLE II .

 II Comparison between simulated and measured values of the resonance frequency f r and the quality factor Q.

		Resonance Frequency f r (MHz)	Quality factor Q
	ZnO thickness (nm)	Measurement	FEM Simulation	Measurement	FEM Simulation
	200	765	767	1600	1675
	400	-	727	-	955
	600	703	705	618	635
	800	672	679	430	450
	1000	-	657	-	400