Sebastian Schwarz
email: sebastian.schwarz@univ-tlse3.fr

Leo Clement

Evripidis Gkanias

Antoine Wystrach

Paul Sabatier

How do backward walking ants (Cataglyphis velox) cope with navigational uncertainty?

Keywords: ants, backward movement, navigation, peeking, route following, uncertainty, view-based navigation

Current opinion in insect navigation assumes that animals need to align with the goal direction to recognise familiar views and approach it. Yet, ants sometimes drag heavy food items backward to the nest and it is still unclear to what extent they rely on visual memories while doing so. In this study displacement experiments and alterations of the visual scenery reveal that ants do indeed recognise and use the learnt visual scenery to guide their path towards the nest while walking backward. In addition, the results show that backward ants estimate their directional uncertainty by integrating multiple cues such as visual familiarity, the state of their path integrator and the time spent backward. A simple mechanical model based on repulsive and attractive visual memories captures the results and explains how visual navigation can be performed backwards.

INTRODUCTION

Central place foragerssuch as desert antsexhibit formidable navigational skills to find food and their way back home during numerous daily foraging trips [START_REF] Collett | Route learning by insects[END_REF][START_REF] Heinze | Principles of insect path integration[END_REF][START_REF] Wehner | Desert ant navigation: how miniature brains solve complex tasks[END_REF]. These ground dwellers rely on a set of navigational strategies such as path integration [START_REF] Wehner | Path integration in insects[END_REF][START_REF] Wittlinger | The ant odometer: stepping on stilts and stumps[END_REF] and visual scene navigation (Cheng, Narendra, Sommer, & Wehner, 2009;[START_REF] Zeil | Visual homing: an insect perspective[END_REF]. Path integration (PI) is the capacity to return to a starting point by constantly updating the directional and distance information (based on celestial compass cues and a stride odometer) during an outbound trip whereas visual scene navigation provides directional information by comparing memorised and currently perceived views of terrestrial cues. The literature agrees that ants continuously integrate the directional dictates of these different strategies, rather than switching between them (Collett, 2012;Hoinville & Wehner, 2018;Legge, Wystrach, Spetch, & Cheng, 2014;[START_REF] Reid | Polarised skylight and the landmark panorama provide night-active bull ants with compass information during route following[END_REF]Wystrach, Mangan, & Webb, 2015).

Current models of insect visual navigation capture well the behaviour of forward navigating ants (Baddeley, Graham, Husbands, & Philippides, 2012;Hoinville & Wehner, 2018;Wystrach, Beugnon, & Cheng, 2011;[START_REF] Wystrach | Geometry, features, and panoramic views: Ants in rectangular arenas[END_REF][START_REF] Zeil | Visual homing: an insect perspective[END_REF] however, how ants navigate while dragging a heavy food item backward remains unclear (Ardin, Mangan, & Webb, 2016;Pfeffer & Wittlinger, 2016;[START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Despite their irregular backward foot strides the ants' PI system seems as accurate as during forward movement [START_REF] Pfeffer | How to find home backwards? Locomotion and inter-leg coordination during rearward walking of Cataglyphis fortis desert ants[END_REF]Pfeffer & Wittlinger, 2016), Experiment 1 was conducted to assess the occurrence of peeking behaviour in backward walking ants across four different test conditions with one nest of Catalgyphis velox (summer 2017) located at the beginning of a straight 8.0 m  1.8 m long foraging route. The length of the route was chosen to be long enough for ants to build up proper PI information but short enough to enable the ants to drag a crumb the whole distance without being too fatigued. Three thin (0.5 cm width), large wooden boards (2.4 m  1.2 m) were connected (7.2 m  1.2 m) and placed onto the foraging route (Fig. 1a, grey rectangles). These boards enhanced the tarsi grip of the ants and provided an even substrate that minimised potential interference with small grass haulms or pebbles during tests when ants dragged their food items backward.

During training, the individually marked foragers scuttled (forward) between the nest and feeder over the connected boards and familiarised themselves with the visual surroundings. After training, individual ants were subjected to one test conditions. All tests comprised of a forager that dragged a large biscuit crumb backward. For that, trained foragers with a small food item (~ 0.2  0.2  0.2 cm) were caught and transferred into a plastic vial. The food item was then carefully and manually removed and a larger biscuit piece (~ 2.0  0.5  0.2 cm) was offered to the ant instead. The biscuit provided was large enough to force the ants to drag it backward. After the ant locked mandibles onto the large biscuit, she was transferred to the appropriate release point. Four possible test conditions were carried out with either FV-(i.e., ants with their PI vector information, captured at the feeder) or ZVants (i.e., foragers without PI information, captured just before they enter the nest; Fig. 1a). To test whether ants pay attention to the visual scenery during backward motion, FV ants were released at the feeder either with the familiar scenery (FV, condition 1) or with altered visual surroundings (FVA, condition 2): by adding large black plastic bags (~ 0.8 m  0.6 m) on one side and a large dark tarp (0.9 m  3.4 m) on the other side of the route (Fig. 1a, right panel) to avoid potential obstructions.

The objects were added a few seconds after the FVA-ant had started to home backward to ensure that the ant did not perceived any alteration before. To test the effect of route location, backward ants were released either at the beginning (ZV, condition 3) or at the middle of the route (ZVmid, condition 4).

It should be noted that upon release, the ants tested in Experiment 1 had the chance to glimpse forward while handling their large biscuit before engaging in backward movements.

Experiment 1: data and analysis

For all tests, the distance between the release point and the location at which peeking behaviours occurred was noted with the help of a measuring tape placed on the side of the board that spanned form the feeder to the nest. Tests ended as soon as the backward walking ants reached the end of the wooden boards (i.e., ~0.5 m in front of the nest entrance) or abandoned her food item for more than one minute.

Individual ants were tested only once per test but were subjected to different test conditions with at least one un-interfered training trial between tests. The sequence of tests was evenly counter-balanced across individuals (e.g., FV-FVA-ZV-ZVmid; FVA-ZV-ZVmid-FV; ZV-ZVmid-FV-FVA etc.)

Comparison were made between FV-vs. FVA-ants and ZV-vs. ZVmid-ants (Fig. 1a). Given the large inter-individual variations, paired-data was applied and thus individuals that were not tested on both FV and FVA or ZV and ZVmid conditions were excluded from the analysis. We compared both the distance at which the first peeking behaviour occurred (1 st peek distance) and the overall peek rate of individuals (i.e., number of peek/distance walked) using Wilcoxon rank sum tests, a nonparametric statistic for paired data. Given that all ants walked rather straight toward the nest along the route, distance walked could be simply approximated by the beeline distance walked along the route. Most ants walked the full route (7.2 m) except obviously ants in the ZVmid condition and some foragers that abandoned their biscuit. For the comparison of peek rate, the 7.2 m long route on the boards was divided into half (Section 1: 0 -3.4 m; Section 2: 3.4 -7.2 m; Figure 1a). Thus, during ZVmid tests ants ran only Section 2. Comparisons between ZV-vs. ZVmid-ants were conducted to separate the effect of distance walked (i.e., ZVmid vs. ZV on Section 1) from the actual location along the route (i.e., ZVmid vs. ZV on Section 2). Bayesian statistics based on the Z values from these tests were applied to evaluate which of these alternative hypotheses explain peek rate best. We used f(Z), that is, the y values on the standard normal distribution obtained from each of the Z scores, to estimate evidence in favour of each hypothesis. The ratio of the f(Z) measures of the two hypothesis gives an estimate of the 'evidence ratio'. While evidence ratios under 5 are weak, ratios over 10 are very strong (see [START_REF] Wystrach | Multiple sources of celestial compass information in the Central Australian desert ant Melophorus bagoti[END_REF] for more details).

Backward paths were recorded by using GoPro HERO3+ cameras which were manually held approx. 0.6 m above the tested ant. Therefore, a quantification of the movement speed of the ants before and after peeking could be calculated. We measured the relative distance walked by the backward ants during the five seconds preceding the onset of the first peek (i.e., before the moment when the ants released the biscuit) and five seconds after the peek (i.e., after the ant resumed backward motion).

Experiment 2

Experiment 2 was conducted to test whether ants could guide their path backwards based on terrestrial cues. To do so, we recorded the paths of backward walking ants captured near their nest and released at four different release points (summer 2018). We used two different nests of C. velox ants. For each nest, a 5.0 m  2.0 m straight foraging route was built with the nest entrance at one end and the feeder at the other end (Fig. 2a). As in Experiment 1, the route was enclosed by white plastic planks and ants were given a choice of biscuit crumbs inside the feeder to prompt foraging. However, here the ants scuttled back and forth directly on the natural ground during training. Once trained (see General methods), individual ants were captured on their way home 0.5 m before reaching their nest and subsequently released at one out of four possible locations (Fig. 2a): Feeder (F): Ants were released 0.5 m after the feeder toward the nest direction.

Before nest (BeN): Ants were released on the route, 0.5 m before their nest.

Beyond nest (ByN): Ants were released 0.5 m beyond the nest in the feeder-nest direction.

Unfamiliar (U): Ants were released ~ 30.0 m away from the nest in a visually unfamiliar location.

For all tests, ants were captured in a plastic vial, offered a large biscuit crumb to incite backward walking and, once the ant had grabbed the cookie, released within a lampshade at the middle of a large wooden board (2.4 m  1.2 m). Individual ants were tested only once per test, but could be subjected to different tests conditions, with at least one un-interfered training trial between tests. The wooden board was set in place just before and removed just after each test. The board was centred on the current release location with the long side along the feeder-nest direction (Fig. 2a) as it represents the expected homing direction. The board provided an even substrate Experiment 2: data and analysis.

The backward paths and locations of peeking behaviour were noted. For each peek, the duration (i.e., the time the ant was not dragging the biscuit) was recorded.

The actual trajectory of the path during forward motion was not recorded. Recording continued until the ants either reached the edge of the board or abandoned their large crumb for more than 1 min.

The recorded paths were digitised as (x, y) coordinates using GraphClick (Arizona Software). Path characteristics such as direction, meander and peek location were computed and analysed with Matlab (Mathworks, Matick, MA, USA). The meander index measures how much the path changes direction from segment to segment and was calculated as the mean angular deviation in direction between successive segments, each 0.3 m [START_REF] Schwarz | Ocelli contribute to the encoding of celestial compass information in the Australian desert ant Melophorus bagoti[END_REF]. Differences between test locations were determined using a generalised linear mixed effect model with repeated ants as random effect and Tukey's post hoc corrections. For peek durations, a model for proportional (binomial) data was applied with 0 to 60 s (the duration at which recording was stopped) reported between 0 and 1.

RESULTS

Experiment 1

Experiment 1 was designed to investigate whether the occurrence of peeking behaviour in backward walking ants depends on factors such as: path integration, terrestrial cues, the location along the route or the distance walked.

FV vs. ZV: effect of path integration

In both FV and ZV conditions, ants started to pull their biscuit toward the nest and mostly maintained that direction. However, the PI state had a visible effect on the occurrence of peeking behaviours: ZV-ants peeked earlier (first peek distance ZVants: mean ± SD = 3.28 ± 2.19 m; FV-ants: mean ± SD = 5.90 ± 1.93 m; Wilcoxon rank sum test: P < 0.001, Z = -3.458) and thrice as much (overall peek rate ZV-ants: mean ± SD = 0.63 ± 0.63 peek/m; FV-ants: mean ± SD = 0.19 ± 0.29 peek/m; Wilcoxon rank sum test: P < 0.001, Z = -3.751) than FV-ants (Fig. 1c, blue vs. green). Also, ZV-ants occasionally abandoned their large food item and did not resume backward movements (6 out of 33), whereas FV-ants never abandoned theirs (0 out of 32). A Fisher's exact test verified a significant difference (P = 0.032). It seems clear that the lack (or conflicting) PI information decreases the ant's directional certainty.

FV vs. FVA: effect of altered visual familiarity

To test the potential effect of the level of an altered visual scenery on backward walking ants, two conditions were conducted: (1) FV-ants homing backward on the unaltered, usual route, and (2) FVA-ants, homing backward on the same route but with altered visual surroundings (Fig. 1a). The alternation of the visual surrounding was effected only after the ants had started their journey backward, so that differences indicate that ants paid attention to terrestrial cues while walking backward. Results showed that FVA-ants peeked more often than FV-ants, however, this effect was weak, and reaches significance only in Section 2 (Wilcoxon rank sum test: P = 0.027, Z = 3.751) but not in Section 1 of the route (Wilcoxon rank sum test: P = 0.688, Z ~ 0; Fig. 1d), due to a statistical floor effect. Indeed, a low rate of peeking in the first section of the route was expected, given that the path integration vector is longer and thus stronger at the beginning of the route home (Wystrach et al., 2015).

Second and most importantly, FVA-ants travelling in the altered visual environment displayed their first peek earlier along the route as compared to FV-ants on the familiar route (Wilcoxon rank sum test: P = 0.044, Z = 2.016; Fig. 1d).

Together, this suggests that backward walking ants paid attention to terrestrial cues.

We also noted that, as for ZV-ants (see above), FVA-ants tested in the altered familiar condition abandoned their biscuits significantly more often than FV-ants (FVA: 6 out of 31 vs. FV: 0 out of 32; Fisher's exact test: P = 0.022). Here again, it seems that the altered visual scenery decreases the directional certainty of backward walking ants.

ZV vs. ZVmid: effect of location

We investigated the potential effect of the location along the route by releasing zero vector ants either at the beginning (ZV) or in the middle of the familiar route (ZVmid; Fig. 1a,c). Consequently, ZVmid-ants walked only Section 2, while ZV-ants moved along both sections. Ants displayed their first peek on average slightly earlier when released at the middle of the route (ZVmid) than when released at the beginning of the route (ZV; Wilcoxon rank sum test: P = 0.040, Z = 2.062; Fig. 1e). Also, the peek rate displayed by ZVmid-ants along Section 2 (the only section they walked) was higher than ZV-ants along Section 1 (Wilcoxon rank sum test: P = 0.005, Z = -2.814) but similar to the peek rate displayed by these ZV-ants along Section 2 of the route (Wilcoxon rank sum test: P = 0.796, Z = -0.2585; Fig. 1e). A Bayesian evidence ratio (see Methods) was computed to estimate whether Section 1 (i.e., different location, same distance walked) or Section 2 (i.e., same actual location, different distance walked) of ZV-ants' peek rate resembles most ZVmidants' peek rate. The obtained evidence ratio was 50.74 in favour of Section 2, which equals 'overwhelming evidence' for an effect on peek rate of the actual location along the route rather than the distance walked.

Peeking and walking speed

Interestingly, in all conditions, and for the vast majority of the individuals, ants walked backward on average twice as quickly after peeking than before peeking (Fig. 1b).

This supports the idea that a peeking event increases the ant's directional certainty for some time.

Experiment 2

Experiment 2 was conducted to test whether ants could guide their path backwards using terrestrial cues. To do so, we recorded the paths of backward walking ants captured near their nest and released at four different release points.

Unlike as in Experiment 1 all tested ants in Experiment 2 were prevented from monitoring the visual surrounding before dragging their food item backward as a lampshade was blocking the whole panoramic view upon release (see Methods).

Hence, any difference in the taken direction must result from visual information perceived while the ants were dragging their crumb backwardat least until they peeked for the first time.

Ants can guide their backward path

To see whether ant paths were directed in different direction across release locations, the Y-valuesthe end position along the feeder-to-nest line relative to the release pointwere examined. The Y-values at the end of the foragers' path varied across released location (ANOVA: F = 21.96, P < 0.001; Fig. 2c). Ants released at the feeder were directed toward the nest and hence obtained the highest Y-values (Tukey's post-hoc test F vs. BeN, ByN and U: Zs > 3.75, Ps < 0.001; Fig. 2c). Ants in unfamiliar tests (U) showed no directional preference along the Y-axis (Fig. 2c), as expected given the lack of familiar visual information at this location. Interestingly, ants from BeN and ByN conditions differed significantly in their final Y-values on the board (Tukey's post-hoc test BeN vs. ByN: Z = 3.47, P = 0.003). The medians of both of these groups are close to the actual nest location, showing that they used familiar visual cues to search at the nest (Fig. 2c). Differences between conditions could also be observed in path meander (ANOVA: F = 9.07 P < 0.001). Ants from the feeder test showed straighter paths than ants from all other conditions (Tukey's post-hoc test F vs. BeN, ByN and U: Zs > 3.68 Ps < 0.002; Fig. 2c). No difference in meander among the remaining test conditions could be determined. Indeed, BeN-, ByN-and U-ants were expected to search on the board: BeN-and ByN-ants due to the proximity of the nest and U-ants due to the of the lack of familiar visual information.

Overall, these data show that ants could use familiar visual cues to adequately direct their backward paths.

Remarkably, analysis of the paths displayed purely backwards, before the first peek (or until the ant left the board if she did not peek), showed a similar pattern of results for both the Y-values reached (ANOVA: F = 11.37 P < 0.001) and path meander (ANOVA: F = 3.52 P = 0.024) (Fig. 2D). Despite purely backward movements, ants released at the feeder travelled significantly longer distances along the feeder-nest direction (Tukey's post-hoc test: F vs. BeN, ByN and U: Zs > 3.29 Ps < 0.006; Fig. 2d) and displayed straighter paths (Tukey's post-hoc F vs. ByN and U: Zs > 2.65 Ps < 0.03; F vs. BeN: Z = 1.88 P = 0.235; Fig. 2d) before peeking. The three other groups (BeN, ByN, U) were indeed expected to search on the board and to perform a similar level of path meander, as observed. Differences in the Y-values reached when released on each side of the nest did not reach significance using Tukey's post-hoc test (BeN vs. ByN: Z = 2.084, P = 0.158), however, the pattern of results followed what was expected if ants were using views to direct their path toward the nest. Ants released before (BeN) and beyond (ByN) their nest both moved on average toward the nest location, that is, in opposite direction from their release points; and ants released at the unfamiliar test site (U) showed less directed paths (Fig. 2d). The differences in paths characteristics is reflected if one considers the probability of obtaining the expected order of path endpoint across the four test conditions (Y-value: F > BeN > U > ByN) is 1/4! = 0.042.

Because nest-directed path sections were achieved before the ants triggered their first peek and the visual panorama was revealed to them only after they had started backward motion, the differences across locations show that ants can recognise and use the familiar visual cues to guide their path while moving backward and without the need of peeking.

Interestingly, several ants released at the feeder (4 out of 16 ants) displayed nestdirected backward paths across the whole recording board without performing a single peek and by keeping their body orientation away from the feeder-nest direction by at least 90 (bold paths in Fig. 2d).

Peek duration and past information

We also tested whether peek duration was influenced by the test condition and the number of previously displayed peeks (Fig. 2b). The feeder condition was excluded from this analysis as these ants were expected to move in a straight line and exit the board so that the actual peek number of a given ant may correlate with the location where the ant peeks. The three other groups (BeN, ByN, U) on the other hand, are expected to search on the board so any effect of the peek number is unlikely to be attributed to a specific location on the board. Interestingly, peek duration, which was recorded up to 60 s, was strongly influenced by the number of peeks previously displayed by the ant (GLM peek number: F = 17.09, P < 0.001; Fig. 2b) and not the actual test condition (GLM condition: F = 0.17, P = 0.841; Fig. 2b). The more peeks an ant had previously displayed the longer its current peeking lasts. This shows that the ant's peeking behaviour is also modulated by past information. Whether it is the time passed or the number of peek previously displayed cannot be disentangled here.

DISCUSSION

Ants dragging a heavy food item backward occasionally trigger a so-called 'peeking behaviour' or 'peek': ants drop their food and turn around to look forward.

Aligning their body in a familiar direction enables them to recognise the learnt visual panorama and store this direction to follow it during their subsequent backward path [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. It is clear that ants gain directional information from learnt terrestrial cues when peeking forward. However, whether or not they use terrestrial cues while dragging their food item backward is less clear. Several of the current results demonstrate that ants are indeed able to do so, raising question about the underlying mechanisms.

Ants still recognise terrestrial cues while walking backward

Experiment 1 shows that the visual scenery experienced while walking backward influences the occurrence of peeking behaviour. First, zero vector (ZV) ants (i.e., ants captured at the nest and thus deprived of useful path integration (PI) information), displayed their first peek earlier when starting their backward journey halfway along the route than when starting at the beginning of the route (Fig. 1e). Second, full vector (FV) ants (i.e., captured at the feeder and thus having a PI vector pointing towards the nest) displayed their first peek earlier along the route if the surrounding scenery was artificially altered (FVA, Fig. 1d). This was true even though the scene was manipulated only after the ants had started dragging their biscuit backward indicating that ants paid attention to the familiar scenery while walking backward. It should be noted that this effect was weak (Fig. 1d), possibly because the alteration of the scene was not obvious enough [START_REF] Schwarz | Desert ants (Melophorus bagoti) navigating with robustness to distortions of the natural panorama[END_REF].

In Experiment 2, ants could guide their trajectories based on terrestrial cues while walking backward. Ants were released on a board (ruling out the use of olfactory cues) and within a lampshade at different location relative to the nest (Fig. 2). The visual world was revealed to them only after they had started their backward journey.

Nonetheless and despite the lack of PI homing vector, their paths were oriented in the expected direction (i.e., the nest) resulting in differences between test conditions. Importantly, this was also true for the portion of path displayed before their first peek, that is, displayed purely backward (Fig. 2d).

In sum, ants can use learnt terrestrial visual cues while walking backward to guide their path as well as decide whether and when to peek forward. The next section discusses potential explanations.

Mental rotation or combining attractive and repulsive views?

How can ants recognise views backward? This is a puzzling question given that the assumption of current models of visual homing states that views must be retinotopically aligned to provide directional information (Ardin, Peng, Mangan, Lagogiannis, & Webb, 2016;Baddeley et al., 2012;Collett, Graham, & Collett, 2017;[START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF][START_REF] Möller | A model of ant navigation based on visual prediction[END_REF][START_REF] Murray | The role of attractive and repellent scene memories in ant homing (Myrmecia croslandi)[END_REF][START_REF] Narendra | Mapping the navigational knowledge of individually foraging ants, Myrmecia croslandi[END_REF]Wystrach, Cheng, et al., 2011;[START_REF] Wystrach | Snapshots in ants? New interpretations of paradigmatic experiments[END_REF][START_REF] Zeil | Catchment areas of panoramic snapshots in outdoor scenes[END_REF] although some other processes may be also at work (Wystrach et al., 2012). Recently, it has been suggested that ants may perform some sort of mental rotation to compare misaligned views (Ardin, Mangan, Wystrach, & Webb, 2015;Ardin et al., 2016), which may be achieved if views are encoded in the frequency domain [START_REF] Stone | An anatomically constrained model for path integration in the bee brain[END_REF]. However, one can wonder why, if they can achieve mental rotation, would the ants peek at all? A capacity for mental rotation is hard to reconcile with the result of previous experiments where ants would not adjust their backward trajectory at all unless they peeked to align their body in the correct direction [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF].

An alternative hypothesis would be that backward dragging ants may assess the maximum, rather the minimum of the rotational image difference function [START_REF] Zeil | Catchment areas of panoramic snapshots in outdoor scenes[END_REF]. However, this fails to explain why ants succeed to use terrestrial cues to guide their path backward on a two-way route but not on a one-way route [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Also, to assess the maximum (as for the minimum [START_REF] Wystrach | Optimal cue integration in ants[END_REF]) for homing requires ants to scan the world to evaluate the relative mismatch across the different rotations; here however, the ants could maintain a straight backwards path without scanning (Fig. 2d).

Here we suggest instead a third hypothesis: ants may still need to align their body in the familiar direction to recognise memorised views but possess a memory bank of views learnt while facing in multiple directions and not only toward the nestas suggested in a related manner concerning learning-and orientation walks [START_REF] Zeil | The learning walks of ants (Hymenoptera: Formicidae)[END_REF]. Notably, views learnt while facing in the anti-nest direction could be treated as repulsive when homing (Fig. 3a). The familiarities resulting from the comparison of the currently perceived view with both attractive (nest facing) and aversive (feeder facing) visual memories could simply be compared in a way somewhat analogous to an opponent process, as recently modelled in the light of the insect's neural circuitry (Le [START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF]. The signal resulting from this comparison informs the ant about whether to move toward or away from the currently faced direction, without the need for scanning (Le [START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF]. The only additional assumption needed for this model to explain backward guidance is that, if the ant is holding a large food item, 'moving away' means going backward because the ant cannot lift the cookie to turn its body. Fig. 3a illustrates the functionality of this simple model.

In the light of this model, we can wonder whether ants on the way home can also use the visual memories stored during their outbound trips (i.e., when they went from the nest to the feeder) as repulsive. This idea challenges the opinion that ants treat in-and outbound trip visually separately depending on the motivational state [START_REF] Harris | Priming of visual route memories[END_REF][START_REF] Wehner | Ant navigation: one-way routes rather than maps[END_REF]. Instead, ants may always recall both their memorised inbound and outbound facing views but treat them as repulsive or attractive depending on their current motivational state.

This attraction/repulsion model explains several observed phenomena of the current and past studies. (1) In a former experiment [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF] backward ants were not able to correct their path at all while walking backward because, in this particular set-up, in-and out-bound routes were spatially separated (as a one-way circuit) so that no outbound views where available to potentially help out backward homing ants. (2) In the current Experiment 2, backward ants released at the feeder (F) carried on in the correct nest direction (Fig. 2d) because they recognised outbound views oriented toward the feeder, driving them away from (or opposite to) this direction (Fig. 3a). (3) Alteration of the visual surrounding would trigger earlier peeking behaviours because the familiarity of the feeder facing (outbound) views would be equally altered, disrupting the repulsive effect and thus reducing the overall directional drive (Fig. 1d). (4) Assuming that outbound views near the feeder are more familiar than in the middle of the route (ants perform learning walks at the feeder: [START_REF] Nicholson | Learning walks and landmark guidance in wood ants (Formica rufa)[END_REF], the repulsive effect would be stronger for ants released at the feeder than in the middle of the route, yielding to a stronger directional drive and hence less peeking near the feeder (Fig. 1d). ( 5) Further, it was surprising in Experiment 2 that ants released close to the nest could direct their backward paths toward the nest (BeN, ByN; Figure 2c,d). Although this verifies that they recognised familiar views they nonetheless tended to peek often and even abandoned their cookie close to the nest (9 out of 28 in BeN-and 4 out of integration between 'attractive and 'repulsive view'. During learning walks around the nest, ants appear to store indeed both nest and anti-nest oriented views [START_REF] Jayatilaka | The choreography of learning walks in the Australian jack jumper ant Myrmecia croslandi[END_REF]. Even if these nest views may be all extremely familiar, the integration of attractive (nest-oriented) and repulsive (anti-nest oriented) views would result in a low overall directional drive, and thus lead to high peek rates (and a high probability for abandoning the crumb) but nonetheless guide the ants toward the nest. This very same idea explained the recent puzzling observation that ants behaved similarly at the nest and in unfamiliar settings, but not on the foraging route [START_REF] Murray | The role of attractive and repellent scene memories in ant homing (Myrmecia croslandi)[END_REF]. ( 6) Finally, two recent studies revealed the importance of outbound trips for homing ants. Ants with outbound views during training were more efficient in homing than ants with only inbound views during training [START_REF] Freas | Landmark learning, cue conflict, and outbound view sequence in navigating desert ants[END_REF][START_REF] Freas | Terrestrial cue learning and retention during the outbound and inbound foraging trip in the desert ant, Cataglyphis velox[END_REF]. But whether homing ants actually used their outbound view as suggested here, or simply learnt homing views by turning back during their outbound trips remains to be seen.

Ants combine multiple cues to estimate directional uncertainty and trigger peeks.

It is known that ants combine the directional dictates of the current visual familiarity with their PI in a weighted fashion [START_REF] Wehner | Steering intermediate courses: desert ants combine information from various navigational routines[END_REF]. Notably, the direction indicated by the current view is more weighted as the current view is familiar (Legge et al., 2014) and the direction indicated by the PI is more weighted as the PI vector length increases [START_REF] Wystrach | Optimal cue integration in ants[END_REF]. Backward walking ants appear to weight these cues in the same fashion. Fig. 3b shows how such an integration of cues captures the peek rate observed across our conditions and distance walked along the route. Notably, this explains why ZV-ants peeked earlier and more often than FV-ants (Fig. 1b), as observed in North African Cataglyphis fortis ants (Pfeffer & Wittlinger, 2016) and why peek rate increases as the distance walked along the route increases (Fig. 1d,e), as observed in Myrmecia croslandi ants (see Fig. S2 in [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF].

Interestingly, such an estimate of directional certainty seems not only to influence the occurrence of peeking but also whether the peeking ants decided to return to their biscuit or abandon it. FV-ants in familiar visual surroundings and therefore with the highest directional certainty, dragged their biscuit all along the 7.20 m route without exception (32 out of 32). In contrast, some ants abandoned their biscuits in both FVA (6 out of 31) and ZV (6 out of 33) conditions. Finally, it is worth mentioning that ants clearly increased the speed of their backward motion after peeking (Fig. 1c). The increase in speed is likely not only a mere consequence of a short recovery from the dragging activity but also a gain in navigational certainty as this sudden speed increase is also apparent when displaced foragers reach their familiar route and recognise the familiar scenery (pers. observ.). Furthermore, a decrease in speed when ants run off their PI has been described (Buehlmann, Fernandes, & Graham, 2018). Hence it seems that the speed of movement reflects the strength of the directional drive and therefore directional certainty. If peeking behaviour seems to increase directional certainty, the later seem to decrease with time spent backwards (Fig. 3c), which also make functional sense.

Ants gather information about the time spent backward.

Recently, it has been shown that directional information based on terrestrial cues is obtained when ants face forward during peeks and must therefore be stored in a short-term memory while the ant is subsequently walking backward [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Our results suggest that short-term memory also influences the ants' navigational certainty. The time spent peeking forward varied (as already noted by Pfeffer and Wittlinger (2016)) with some ants 'peeking' for less than a second while other spent more than 60s (after which recording was stopped) without returning to their cookie. Interestingly, the more an ant had peeked before during a test run, the longer her current peek lasted (Fig. 2b). This shows that ants somehow gather information across time to adjust their current certainty: either information about the overall time spent backward or information about the number of peek previously displayed (see also Fig. 3c).

Former experiments have already shown that the behaviour of ants can be modulated by recent experience in the order of seconds to minutesa form of hysteresis [START_REF] Graham | Insect navigation: do ants live in the now[END_REF]. For instance, ants display a so-called backtracking behaviour only if they have recently perceived the nest surrounding [START_REF] Wystrach | Backtracking behaviour in lost ants: an additional strategy in their navigational toolkit[END_REF]. Also, homing ants display higher meander in their paths when recapitulating a well-known route for the second time in a row (Collett, 2014;[START_REF] Wystrach | Running paths to nowhere: repetition of routes shows how navigating ants modulate online the weights accorded to cues[END_REF]. Finally, ants can remember the compass direction of a wind gust after being blown (Wystrach & Schwarz, 2013). In the present case, ants seem to also build up information about the recent past. Even though it makes functional sense it remains to be seen what neural or physiological mechanisms underlie this phenomenon.

Conclusion

This study confirmed that ants walking backward are not just paying attention to celestial cues but combine multiple information from their PI, the recognition of terrestrial cues and temporal information such as the time they spent backward. All this information seems to be integrated in an overall directional drive. This drive, which reflects the current directional certainty, seems to (1) guide the ants backward path, (2) triggers peeking behaviour if too low and (3) finally dictates whether or not to return to their food item during peeks. Importantly, this study shows that ants uses familiar terrestrial cues while navigating backward. In addition to the attractive memories facing toward the nest, the hypothesis that homing ants use a collection of repulsive visual memories facing away from the nest and possibly stored during their outbound trip was put forward (Le [START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF]. In the light of this hypothesis, visual navigation forward or backward can then simply be achieved by using the relative familiarity between both sets of opposite valence memories. As often with research on insect navigation, remarkably flexible behaviours incite researchers to endorse the simplest explanations [START_REF] Wystrach | What can we learn from studies of insect navigation[END_REF]. repulsive (bold red arrow on right left scheme). Given that the neighbouring green arrow is not aligned, the closest aligned attractive view is further away, beyond the nest (bold green arrow on left scheme). Overall, the ants at this position (still facing downward) will be more repulsed than attracted by the current facing direction and thus walk backward toward the nest. In contrast, when beyond the nest ByN (still facing downward), the attractive views will match better than the repulsive views and the ants will thus turn around and walk backward toward the nest. Whatever the position and orientation of the ants around the nest, the agent will be drawn towards the nest (b) Directional drive across test conditions of Experiment 1 in Section 1 and 2. Ants are tested in FV (full-vector), FVA (full-vector altered familiarity) and ZV (zerovector) conditions. Overall directional drive (dark blue vectors) results from the integration of path integration (cyan vectors, the longer the path integration vector the stronger its directional drive) and learnt view (yellow vectors, the more familiar the view the stronger its directional drive). (c) Directional drive resulting from the recognition of a learnt view (yellow vectors) decrease with time spent facing in a different direction. Low directional drive results in lower speed (dashed arrow) and eventually peeking. Here memorised views are assumed to be facing upward and are thus recognised only when facing upward during peeking (small grey arrow) and not while facing downward during backward motion. Note that the second peek triggers a lower directional drive than the first (see also Fig. 1b).

INTRODUCTION

Central place foragerssuch as desert antsexhibit formidable navigational skills to find food and their way back home during numerous daily foraging trips [START_REF] Collett | Route learning by insects[END_REF][START_REF] Heinze | Principles of insect path integration[END_REF][START_REF] Wehner | Desert ant navigation: how miniature brains solve complex tasks[END_REF]. These ground dwellers rely on a set of navigational strategies such as path integration [START_REF] Wehner | Path integration in insects[END_REF][START_REF] Wittlinger | The ant odometer: stepping on stilts and stumps[END_REF] and visual scene navigation (Cheng, Narendra, Sommer, & Wehner, 2009;[START_REF] Zeil | Visual homing: an insect perspective[END_REF]. Path integration (PI) is the capacity to return to a starting point by constantly updating the directional and distance information (based on celestial compass cues and a stride odometer) during an outbound trip whereas visual scene navigation provides directional information by comparing memorised and currently perceived views of terrestrial cues. The literature agrees that ants continuously integrate the directional dictates of these different strategies, rather than switching between them (Collett, 2012;Hoinville & Wehner, 2018;Legge, Wystrach, Spetch, & Cheng, 2014;[START_REF] Reid | Polarised skylight and the landmark panorama provide night-active bull ants with compass information during route following[END_REF]Wystrach, Mangan, & Webb, 2015).

Current models of insect visual navigation capture well the behaviour of forward navigating ants (Baddeley, Graham, Husbands, & Philippides, 2012;Hoinville & Wehner, 2018;Wystrach, Beugnon, & Cheng, 2011;[START_REF] Wystrach | Geometry, features, and panoramic views: Ants in rectangular arenas[END_REF][START_REF] Zeil | Visual homing: an insect perspective[END_REF] however, how ants navigate while dragging a heavy food item backward remains unclear (Ardin, Mangan, & Webb, 2016;Pfeffer & Wittlinger, 2016;[START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Despite their irregular backward foot strides the ants' PI system seems as accurate as during forward movement [START_REF] Pfeffer | How to find home backwards? Locomotion and inter-leg coordination during rearward walking of Cataglyphis fortis desert ants[END_REF]Pfeffer & Wittlinger, 2016), however, guidance based on terrestrial visual cues seems disrupted [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Indeed, evidence suggests that to recognise the familiar terrestrial scenery ants need to align their body in the familiar forward direction [START_REF] Narendra | Mapping the navigational knowledge of individually foraging ants, Myrmecia croslandi[END_REF]Wystrach, Cheng, et al., 2011;[START_REF] Zeil | Visual homing: an insect perspective[END_REF]. This is probably why ants dragging a food item backward occasionally display a so-called 'peeking' behaviour: the ant stops pulling, drops its food item and turns around to look forward. If the scenery is familiar, the ant quickly returns to her food item and adjusts her backward path in the newly corrected homing direction. It seems clear that during these few moments facing forward in a familiar direction, ants recover and store the correct direction; and subsequently rely on celestial cues to maintain this new bearing when traveling backward [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. In this case navigation is discretised into different sources of information being used sequentially rather than simultaneously. Also, 'peeking' involves the decision to trigger a distinct and observable behaviour when navigational information is needed. This behaviour therefore provides a good opportunity to investigate how ants estimate their navigational uncertainty and as a corollary, which navigational information they have access to.

Here two experiments with backward walking ants were carried out to investigate the following questions: (1) Do ants still pay attention to visual terrestrial cues when walking backward? (2) Does this visual information enable ants to guide their backward path? (3) Which information is used by ants to estimate uncertainty and trigger a peeking behaviour?

METHODS

Study animal and site

The experiments were carried out with Spanish desert ants Cataglyphis velox on a field site with diverse grass and bush vegetation at the outskirts of Seville (37°19'55"N 5°59'20"W) during June 2017 and 18. Cataglyphis velox show typical characteristics of a desert ant such as diurnality, thermophily and solitary foraging (Cerda, 2001). As in other ant species, navigation and orientation in C. velox is predominantly based on vision derived from terrestrial and celestial cues [START_REF] Mangan | Spontaneous formation of multiple routes in individual desert ants (Cataglyphis velox)[END_REF][START_REF] Wystrach | Optimal cue integration in ants[END_REF].

General methods

Two experiments were conducted: Experiment 1 in 2017 and Experiment 2 in 2018. Both set-ups shared the following methods. Ants were restricted to forage on a straight route (for specific dimensions see below) between their nests and a feeder.

The routes were cleared from all vegetation that could interfere with ant observations and enclosed by thin white plastic planks (10 cm high) that were dug halfway into the ground. The slippery surface of the planks prevented ants foraging elsewhere while minimising the obstruction of surrounding views [START_REF] Wystrach | Ants might use different viewmatching strategies on and off the route[END_REF]. Ants could freely travel between nest and feeder, which was a ~ 15  15  15 cm plastic bowl sunk into the ground that contained several kinds of sweet buttery biscuit crumbs. The walls of the bowl were covered with a thin layer of Fluon and prevented ants from climbing. Ants that dropped into the feeder and picked up a crumb were marked individually with coloured acrylic or enamel paint (Tamiya).

During training, ants could leave the feeder via a small wooden ramp. Ants were considered trained and ready for testing once they had performed at least five foraging runs and were able to reach the feeder from the nest in a straight line (without colliding into any barriers). During tests (see below) the feeder ramp was removed to prevent other homing ants from interfering.

Experiment 1

Experiment 1 was conducted to assess the occurrence of peeking behaviour in backward walking ants across four different test conditions with one nest of Catalgyphis velox (summer 2017) located at the beginning of a straight 8.0 m  1.8 m long foraging route. The length of the route was chosen to be long enough for ants to build up proper PI information but short enough to enable the ants to drag a crumb the whole distance without being too fatigued. Three thin (0.5 cm width), large wooden boards (2.4 m  1.2 m) were connected (7.2 m  1.2 m) and placed onto the foraging route (Fig. 1a, grey rectangles). These boards enhanced the tarsi grip of the ants and provided an even substrate that minimised potential interference with small grass haulms or pebbles during tests when ants dragged their food items backward.

During training, the individually marked foragers scuttled (forward) between the nest and feeder over the connected boards and familiarised themselves with the visual surroundings. After training, individual ants were subjected to one test conditions. All tests comprised of a forager that dragged a large biscuit crumb backward. For that, trained foragers with a small food item (~ 0.2  0.2  0.2 cm) were caught and transferred into a plastic vial. The food item was then carefully and manually removed and a larger biscuit piece (~ 2.0  0.5  0.2 cm) was offered to the ant instead. The biscuit provided was large enough to force the ants to drag it backward. After the ant locked mandibles onto the large biscuit, she was transferred to the appropriate release point. Four possible test conditions were carried out with either FV-(i.e., ants with their PI vector information, captured at the feeder) or ZVants (i.e., foragers without PI information, captured just before they enter the nest; Fig. 1a). To test whether ants pay attention to the visual scenery during backward motion, FV ants were released at the feeder either with the familiar scenery (FV, condition 1) or with altered visual surroundings (FVA, condition 2): by adding large black plastic bags (~ 0.8 m  0.6 m) on one side and a large dark tarp (0.9 m  3.4 m) on the other side of the route (Fig. 1a, right panel) to avoid potential obstructions.

The objects were added a few seconds after the FVA-ant had started to home backward to ensure that the ant did not perceived any alteration before. To test the effect of route location, backward ants were released either at the beginning (ZV, condition 3) or at the middle of the route (ZVmid, condition 4).

It should be noted that upon release, the ants tested in Experiment 1 had the chance to glimpse forward while handling their large biscuit before engaging in backward movements.

Experiment 1: data and analysis

For all tests, the distance between the release point and the location at which peeking behaviours occurred was noted with the help of a measuring tape placed on the side of the board that spanned form the feeder to the nest. Tests ended as soon as the backward walking ants reached the end of the wooden boards (i.e., ~0.5 m in front of the nest entrance) or abandoned her food item for more than one minute.

Individual ants were tested only once per test but were subjected to different test conditions with at least one un-interfered training trial between tests. The sequence of tests was evenly counter-balanced across individuals (e.g., FV-FVA-ZV-ZVmid; FVA-ZV-ZVmid-FV; ZV-ZVmid-FV-FVA etc.)

Comparison were made between FV-vs. FVA-ants and ZV-vs. ZVmid-ants (Fig. 1a). Given the large inter-individual variations, paired-data was applied and thus individuals that were not tested on both FV and FVA or ZV and ZVmid conditions were excluded from the analysis. We compared both the distance at which the first peeking behaviour occurred (1 st peek distance) and the overall peek rate of individuals (i.e., number of peek/distance walked) using Wilcoxon rank sum tests, a nonparametric statistic for paired data. Given that all ants walked rather straight toward the nest along the route, distance walked could be simply approximated by the beeline distance walked along the route. Most ants walked the full route (7.2 m) except obviously ants in the ZVmid condition and some foragers that abandoned their biscuit. For the comparison of peek rate, the 7.2 m long route on the boards was divided into half (Section 1: 0 -3.4 m; Section 2: 3.4 -7.2 m; Figure 1a). Thus, during ZVmid tests ants ran only Section 2. Comparisons between ZV-vs. ZVmid-ants were conducted to separate the effect of distance walked (i.e., ZVmid vs. ZV on Section 1) from the actual location along the route (i.e., ZVmid vs. ZV on Section 2). Bayesian statistics based on the Z values from these tests were applied to evaluate which of these alternative hypotheses explain peek rate best. We used f(Z), that is, the y values on the standard normal distribution obtained from each of the Z scores, to estimate evidence in favour of each hypothesis. The ratio of the f(Z) measures of the two hypothesis gives an estimate of the 'evidence ratio'. While evidence ratios under 5 are weak, ratios over 10 are very strong (see [START_REF] Wystrach | Multiple sources of celestial compass information in the Central Australian desert ant Melophorus bagoti[END_REF] for more details).

Backward paths were recorded by using GoPro HERO3+ cameras which were manually held approx. 0.6 m above the tested ant. Therefore, a quantification of the movement speed of the ants before and after peeking could be calculated. We measured the relative distance walked by the backward ants during the five seconds preceding the onset of the first peek (i.e., before the moment when the ants released the biscuit) and five seconds after the peek (i.e., after the ant resumed backward motion).

Experiment 2

Experiment 2 was conducted to test whether ants could guide their path backwards based on terrestrial cues. To do so, we recorded the paths of backward walking ants captured near their nest and released at four different release points (summer 2018). We used two different nests of C. velox ants. For each nest, a 5.0 m  2.0 m straight foraging route was built with the nest entrance at one end and the feeder at the other end (Fig. 2a). As in Experiment 1, the route was enclosed by white plastic planks and ants were given a choice of biscuit crumbs inside the feeder to prompt foraging. However, here the ants scuttled back and forth directly on the natural ground during training. Once trained (see General methods), individual ants were captured on their way home 0.5 m before reaching their nest and subsequently released at one out of four possible locations (Fig. 2a): Feeder (F): Ants were released 0.5 m after the feeder toward the nest direction. Experiment 2: data and analysis.

The backward paths and locations of peeking behaviour were noted. For each peek, the duration (i.e., the time the ant was not dragging the biscuit) was recorded.

The actual trajectory of the path during forward motion was not recorded. Recording continued until the ants either reached the edge of the board or abandoned their large crumb for more than 1 min.

The recorded paths were digitised as (x, y) coordinates using GraphClick (Arizona Software). Path characteristics such as direction, meander and peek location were computed and analysed with Matlab (Mathworks, Matick, MA, USA). The meander index measures how much the path changes direction from segment to segment and was calculated as the mean angular deviation in direction between successive segments, each 0.3 m [START_REF] Schwarz | Ocelli contribute to the encoding of celestial compass information in the Australian desert ant Melophorus bagoti[END_REF]. Differences between test locations were determined using a generalised linear mixed effect model with repeated ants as random effect and Tukey's post hoc corrections. For peek durations, a model for proportional (binomial) data was applied with 0 to 60 s (the duration at which recording was stopped) reported between 0 and 1.

RESULTS

Experiment 1

Experiment 1 was designed to investigate whether the occurrence of peeking behaviour in backward walking ants depends on factors such as: path integration, terrestrial cues, the location along the route or the distance walked.

FV vs. ZV: effect of path integration

In both FV and ZV conditions, ants started to pull their biscuit toward the nest and mostly maintained that direction. However, the PI state had a visible effect on the occurrence of peeking behaviours: ZV-ants peeked earlier (first peek distance ZVants: mean ± SD = 3.28 ± 2.19 m; FV-ants: mean ± SD = 5.90 ± 1.93 m; Wilcoxon rank sum test: P < 0.001, Z = -3.458) and thrice as much (overall peek rate ZV-ants: mean ± SD = 0.63 ± 0.63 peek/m; FV-ants: mean ± SD = 0.19 ± 0.29 peek/m; Wilcoxon rank sum test: P < 0.001, Z = -3.751) than FV-ants (Fig. 1c, blue vs. green). Also, ZV-ants occasionally abandoned their large food item and did not resume backward movements (6 out of 33), whereas FV-ants never abandoned theirs (0 out of 32). A Fisher's exact test verified a significant difference (P = 0.032). It seems clear that the lack (or conflicting) PI information decreases the ant's directional certainty.

FV vs. FVA: effect of altered visual familiarity

To test the potential effect of the level of an altered visual scenery on backward walking ants, two conditions were conducted: (1) FV-ants homing backward on the unaltered, usual route, and (2) FVA-ants, homing backward on the same route but with altered visual surroundings (Fig. 1a). The alternation of the visual surrounding was effected only after the ants had started their journey backward, so that differences indicate that ants paid attention to terrestrial cues while walking backward. Results showed that FVA-ants peeked more often than FV-ants, however, this effect was weak, and reaches significance only in Section 2 (Wilcoxon rank sum test: P = 0.027, Z = 3.751) but not in Section 1 of the route (Wilcoxon rank sum test: P = 0.688, Z ~ 0; Fig. 1d), due to a statistical floor effect. Indeed, a low rate of peeking in the first section of the route was expected, given that the path integration vector is longer and thus stronger at the beginning of the route home (Wystrach et al., 2015).

Second and most importantly, FVA-ants travelling in the altered visual environment displayed their first peek earlier along the route as compared to FV-ants on the familiar route (Wilcoxon rank sum test: P = 0.044, Z = 2.016; Fig. 1d).

Together, this suggests that backward walking ants paid attention to terrestrial cues.

We also noted that, as for ZV-ants (see above), FVA-ants tested in the altered familiar condition abandoned their biscuits significantly more often than FV-ants (FVA: 6 out of 31 vs. FV: 0 out of 32; Fisher's exact test: P = 0.022). Here again, it seems that the altered visual scenery decreases the directional certainty of backward walking ants.

ZV vs. ZVmid: effect of location

We investigated the potential effect of the location along the route by releasing zero vector ants either at the beginning (ZV) or in the middle of the familiar route (ZVmid; Fig. 1a,c). Consequently, ZVmid-ants walked only Section 2, while ZV-ants moved along both sections. Ants displayed their first peek on average slightly earlier when released at the middle of the route (ZVmid) than when released at the beginning of the route (ZV; Wilcoxon rank sum test: P = 0.040, Z = 2.062; Fig. 1e). Also, the peek rate displayed by ZVmid-ants along Section 2 (the only section they walked) was higher than ZV-ants along Section 1 (Wilcoxon rank sum test: P = 0.005, Z = -2.814) but similar to the peek rate displayed by these ZV-ants along Section 2 of the route (Wilcoxon rank sum test: P = 0.796, Z = -0.2585; Fig. 1e). A Bayesian evidence ratio (see Methods) was computed to estimate whether Section 1 (i.e., different location, same distance walked) or Section 2 (i.e., same actual location, different distance walked) of ZV-ants' peek rate resembles most ZVmidants' peek rate. The obtained evidence ratio was 50.74 in favour of Section 2, which equals 'overwhelming evidence' for an effect on peek rate of the actual location along the route rather than the distance walked.

Peeking and walking speed

Interestingly, in all conditions, and for the vast majority of the individuals, ants walked backward on average twice as quickly after peeking than before peeking (Fig. 1b).

This supports the idea that a peeking event increases the ant's directional certainty for some time.

Experiment 2

Experiment 2 was conducted to test whether ants could guide their path backwards using terrestrial cues. To do so, we recorded the paths of backward walking ants captured near their nest and released at four different release points.

Unlike as in Experiment 1 all tested ants in Experiment 2 were prevented from monitoring the visual surrounding before dragging their food item backward as a lampshade was blocking the whole panoramic view upon release (see Methods).

Hence, any difference in the taken direction must result from visual information perceived while the ants were dragging their crumb backwardat least until they peeked for the first time.

Ants can guide their backward path

To see whether ant paths were directed in different direction across release locations, the Y-valuesthe end position along the feeder-to-nest line relative to the release pointwere examined. The Y-values at the end of the foragers' path varied across released location (ANOVA: F = 21.96, P < 0.001; Fig. 2c). Ants released at the feeder were directed toward the nest and hence obtained the highest Y-values (Tukey's post-hoc test F vs. BeN, ByN and U: Zs > 3.75, Ps < 0.001; Fig. 2c). Ants in unfamiliar tests (U) showed no directional preference along the Y-axis (Fig. 2c), as expected given the lack of familiar visual information at this location. Interestingly, ants from BeN and ByN conditions differed significantly in their final Y-values on the board (Tukey's post-hoc test BeN vs. ByN: Z = 3.47, P = 0.003). The medians of both of these groups are close to the actual nest location, showing that they used familiar visual cues to search at the nest (Fig. 2c). Differences between conditions could also be observed in path meander (ANOVA: F = 9.07 P < 0.001). Ants from the feeder test showed straighter paths than ants from all other conditions (Tukey's post-hoc test F vs. BeN, ByN and U: Zs > 3.68 Ps < 0.002; Fig. 2c). No difference in meander among the remaining test conditions could be determined. Indeed, BeN-, ByN-and U-ants were expected to search on the board: BeN-and ByN-ants due to the proximity of the nest and U-ants due to the of the lack of familiar visual information.

Overall, these data show that ants could use familiar visual cues to adequately direct their backward paths.

Remarkably, analysis of the paths displayed purely backwards, before the first peek (or until the ant left the board if she did not peek), showed a similar pattern of results for both the Y-values reached (ANOVA: F = 11.37 P < 0.001) and path meander (ANOVA: F = 3.52 P = 0.024) (Fig. 2D). Despite purely backward movements, ants released at the feeder travelled significantly longer distances along the feeder-nest direction (Tukey's post-hoc test: F vs. BeN, ByN and U: Zs > 3.29 Ps < 0.006; Fig. 2d) and displayed straighter paths (Tukey's post-hoc F vs. ByN and U: Zs > 2.65 Ps < 0.03; F vs. BeN: Z = 1.88 P = 0.235; Fig. 2d) before peeking. The three other groups (BeN, ByN, U) were indeed expected to search on the board and to perform a similar level of path meander, as observed. Differences in the Y-values reached when released on each side of the nest did not reach significance using Tukey's post-hoc test (BeN vs. ByN: Z = 2.084, P = 0.158), however, the pattern of results followed what was expected if ants were using views to direct their path toward the nest. Ants released before (BeN) and beyond (ByN) their nest both moved on average toward the nest location, that is, in opposite direction from their release points; and ants released at the unfamiliar test site (U) showed less directed paths (Fig. 2d). The differences in paths characteristics is reflected if one considers the probability of obtaining the expected order of path endpoint across the four test conditions (Y-value: F > BeN > U > ByN) is 1/4! = 0.042.

Because nest-directed path sections were achieved before the ants triggered their first peek and the visual panorama was revealed to them only after they had started backward motion, the differences across locations show that ants can recognise and use the familiar visual cues to guide their path while moving backward and without the need of peeking.

Interestingly, several ants released at the feeder (4 out of 16 ants) displayed nestdirected backward paths across the whole recording board without performing a single peek and by keeping their body orientation away from the feeder-nest direction by at least 90 (bold paths in Fig. 2d).

Peek duration and past information

We also tested whether peek duration was influenced by the test condition and the number of previously displayed peeks (Fig. 2b). The feeder condition was excluded from this analysis as these ants were expected to move in a straight line and exit the board so that the actual peek number of a given ant may correlate with the location where the ant peeks. The three other groups (BeN, ByN, U) on the other hand, are expected to search on the board so any effect of the peek number is unlikely to be attributed to a specific location on the board. Interestingly, peek duration, which was recorded up to 60 s, was strongly influenced by the number of peeks previously displayed by the ant (GLM peek number: F = 17.09, P < 0.001; Fig. 2b) and not the actual test condition (GLM condition: F = 0.17, P = 0.841; Fig. 2b). The more peeks an ant had previously displayed the longer its current peeking lasts. This shows that the ant's peeking behaviour is also modulated by past information. Whether it is the time passed or the number of peek previously displayed cannot be disentangled here.

DISCUSSION

Ants dragging a heavy food item backward occasionally trigger a so-called 'peeking behaviour' or 'peek': ants drop their food and turn around to look forward.

Aligning their body in a familiar direction enables them to recognise the learnt visual panorama and store this direction to follow it during their subsequent backward path [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. It is clear that ants gain directional information from learnt terrestrial cues when peeking forward. However, whether or not they use terrestrial cues while dragging their food item backward is less clear. Several of the current results demonstrate that ants are indeed able to do so, raising question about the underlying mechanisms.

Ants still recognise terrestrial cues while walking backward

Experiment 1 shows that the visual scenery experienced while walking backward influences the occurrence of peeking behaviour. First, zero vector (ZV) ants (i.e., ants captured at the nest and thus deprived of useful path integration (PI) information), displayed their first peek earlier when starting their backward journey halfway along the route than when starting at the beginning of the route (Fig. 1e). Second, full vector (FV) ants (i.e., captured at the feeder and thus having a PI vector pointing towards the nest) displayed their first peek earlier along the route if the surrounding scenery was artificially altered (FVA, Fig. 1d). This was true even though the scene was manipulated only after the ants had started dragging their biscuit backward indicating that ants paid attention to the familiar scenery while walking backward. It should be noted that this effect was weak (Fig. 1d), possibly because the alteration of the scene was not obvious enough [START_REF] Schwarz | Desert ants (Melophorus bagoti) navigating with robustness to distortions of the natural panorama[END_REF].

In Experiment 2, ants could guide their trajectories based on terrestrial cues while walking backward. Ants were released on a board (ruling out the use of olfactory cues) and within a lampshade at different location relative to the nest (Fig. 2). The visual world was revealed to them only after they had started their backward journey.

Nonetheless and despite the lack of PI homing vector, their paths were oriented in the expected direction (i.e., the nest) resulting in differences between test conditions. Importantly, this was also true for the portion of path displayed before their first peek, that is, displayed purely backward (Fig. 2d).

In sum, ants can use learnt terrestrial visual cues while walking backward to guide their path as well as decide whether and when to peek forward. The next section discusses potential explanations.

Mental rotation or combining attractive and repulsive views?

How can ants recognise views backward? This is a puzzling question given that the assumption of current models of visual homing states that views must be retinotopically aligned to provide directional information (Ardin, Peng, Mangan, Lagogiannis, & Webb, 2016;Baddeley et al., 2012;Collett, Graham, & Collett, 2017;[START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF][START_REF] Möller | A model of ant navigation based on visual prediction[END_REF][START_REF] Murray | The role of attractive and repellent scene memories in ant homing (Myrmecia croslandi)[END_REF][START_REF] Narendra | Mapping the navigational knowledge of individually foraging ants, Myrmecia croslandi[END_REF]Wystrach, Cheng, et al., 2011;[START_REF] Wystrach | Snapshots in ants? New interpretations of paradigmatic experiments[END_REF][START_REF] Zeil | Catchment areas of panoramic snapshots in outdoor scenes[END_REF] although some other processes may be also at work (Wystrach et al., 2012). Recently, it has been suggested that ants may perform some sort of mental rotation to compare misaligned views (Ardin, Mangan, Wystrach, & Webb, 2015;Ardin et al., 2016), which may be achieved if views are encoded in the frequency domain [START_REF] Stone | An anatomically constrained model for path integration in the bee brain[END_REF]. However, one can wonder why, if they can achieve mental rotation, would the ants peek at all? A capacity for mental rotation is hard to reconcile with the result of previous experiments where ants would not adjust their backward trajectory at all unless they peeked to align their body in the correct direction [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF].

An alternative hypothesis would be that backward dragging ants may assess the maximum, rather the minimum of the rotational image difference function [START_REF] Zeil | Catchment areas of panoramic snapshots in outdoor scenes[END_REF]. However, this fails to explain why ants succeed to use terrestrial cues to guide their path backward on a two-way route but not on a one-way route [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Also, to assess the maximum (as for the minimum [START_REF] Wystrach | Optimal cue integration in ants[END_REF]) for homing requires ants to scan the world to evaluate the relative mismatch across the different rotations; here however, the ants could maintain a straight backwards path without scanning (Fig. 2d).

Here we suggest instead a third hypothesis: ants may still need to align their body in the familiar direction to recognise memorised views but possess a memory bank of views learnt while facing in multiple directions and not only toward the nestas suggested in a related manner concerning learning-and orientation walks [START_REF] Zeil | The learning walks of ants (Hymenoptera: Formicidae)[END_REF]. Notably, views learnt while facing in the anti-nest direction could be treated as repulsive when homing (Fig. 3a). The familiarities resulting from the comparison of the currently perceived view with both attractive (nest facing) and aversive (feeder facing) visual memories could simply be compared in a way somewhat analogous to an opponent process, as recently modelled in the light of the insect's neural circuitry (Le [START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF]. The signal resulting from this comparison informs the ant about whether to move toward or away from the currently faced direction, without the need for scanning (Le [START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF]. The only additional assumption needed for this model to explain backward guidance is that, if the ant is holding a large food item, 'moving away' means going backward because the ant cannot lift the cookie to turn its body. Fig. 3a illustrates the functionality of this simple model.

In the light of this model, we can wonder whether ants on the way home can also use the visual memories stored during their outbound trips (i.e., when they went from the nest to the feeder) as repulsive. This idea challenges the opinion that ants treat in-and outbound trip visually separately depending on the motivational state [START_REF] Harris | Priming of visual route memories[END_REF][START_REF] Wehner | Ant navigation: one-way routes rather than maps[END_REF]. Instead, ants may always recall both their memorised inbound and outbound facing views but treat them as repulsive or attractive depending on their current motivational state.

This attraction/repulsion model explains several observed phenomena of the current and past studies. (1) In a former experiment [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF] backward ants were not able to correct their path at all while walking backward because, in this particular set-up, in-and out-bound routes were spatially separated (as a one-way circuit) so that no outbound views where available to potentially help out backward homing ants. (2) In the current Experiment 2, backward ants released at the feeder (F) carried on in the correct nest direction (Fig. 2d) because they recognised outbound views oriented toward the feeder, driving them away from (or opposite to) this direction (Fig. 3a). (3) Alteration of the visual surrounding would trigger earlier peeking behaviours because the familiarity of the feeder facing (outbound) views would be equally altered, disrupting the repulsive effect and thus reducing the overall directional drive (Fig. 1d). (4) Assuming that outbound views near the feeder are more familiar than in the middle of the route (ants perform learning walks at the feeder: [START_REF] Nicholson | Learning walks and landmark guidance in wood ants (Formica rufa)[END_REF], the repulsive effect would be stronger for ants released at the feeder than in the middle of the route, yielding to a stronger directional drive and hence less peeking near the feeder (Fig. 1d). ( 5) Further, it was surprising in Experiment 2 that ants released close to the nest could direct their backward paths toward the nest (BeN, ByN; Figure 2c,d). Although this verifies that they recognised familiar views they nonetheless tended to peek often and even abandoned their cookie close to the nest (9 out of 28 in BeN-and 4 out of integration between 'attractive and 'repulsive view'. During learning walks around the nest, ants appear to store indeed both nest and anti-nest oriented views [START_REF] Jayatilaka | The choreography of learning walks in the Australian jack jumper ant Myrmecia croslandi[END_REF]. Even if these nest views may be all extremely familiar, the integration of attractive (nest-oriented) and repulsive (anti-nest oriented) views would result in a low overall directional drive, and thus lead to high peek rates (and a high probability for abandoning the crumb) but nonetheless guide the ants toward the nest. This very same idea explained the recent puzzling observation that ants behaved similarly at the nest and in unfamiliar settings, but not on the foraging route [START_REF] Murray | The role of attractive and repellent scene memories in ant homing (Myrmecia croslandi)[END_REF]. ( 6) Finally, two recent studies revealed the importance of outbound trips for homing ants. Ants with outbound views during training were more efficient in homing than ants with only inbound views during training [START_REF] Freas | Landmark learning, cue conflict, and outbound view sequence in navigating desert ants[END_REF][START_REF] Freas | Terrestrial cue learning and retention during the outbound and inbound foraging trip in the desert ant, Cataglyphis velox[END_REF]. But whether homing ants actually used their outbound view as suggested here, or simply learnt homing views by turning back during their outbound trips remains to be seen.

Ants combine multiple cues to estimate directional uncertainty and trigger peeks.

It is known that ants combine the directional dictates of the current visual familiarity with their PI in a weighted fashion [START_REF] Wehner | Steering intermediate courses: desert ants combine information from various navigational routines[END_REF]. Notably, the direction indicated by the current view is more weighted as the current view is familiar (Legge et al., 2014) and the direction indicated by the PI is more weighted as the PI vector length increases [START_REF] Wystrach | Optimal cue integration in ants[END_REF]. Backward walking ants appear to weight these cues in the same fashion. Fig. 3b shows how such an integration of cues captures the peek rate observed across our conditions and distance walked along the route. Notably, this explains why ZV-ants peeked earlier and more often than FV-ants (Fig. 1b), as observed in North African Cataglyphis fortis ants (Pfeffer & Wittlinger, 2016) and why peek rate increases as the distance walked along the route increases (Fig. 1d,e), as observed in Myrmecia croslandi ants (see Fig. S2 in [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF].

Interestingly, such an estimate of directional certainty seems not only to influence the occurrence of peeking but also whether the peeking ants decided to return to their biscuit or abandon it. FV-ants in familiar visual surroundings and therefore with the highest directional certainty, dragged their biscuit all along the 7.20 m route without exception (32 out of 32). In contrast, some ants abandoned their biscuits in both FVA (6 out of 31) and ZV (6 out of 33) conditions. Finally, it is worth mentioning that ants clearly increased the speed of their backward motion after peeking (Fig. 1c). The increase in speed is likely not only a mere consequence of a short recovery from the dragging activity but also a gain in navigational certainty as this sudden speed increase is also apparent when displaced foragers reach their familiar route and recognise the familiar scenery (pers. observ.). Furthermore, a decrease in speed when ants run off their PI has been described (Buehlmann, Fernandes, & Graham, 2018). Hence it seems that the speed of movement reflects the strength of the directional drive and therefore directional certainty. If peeking behaviour seems to increase directional certainty, the later seem to decrease with time spent backwards (Fig. 3c), which also make functional sense.

Ants gather information about the time spent backward.

Recently, it has been shown that directional information based on terrestrial cues is obtained when ants face forward during peeks and must therefore be stored in a short-term memory while the ant is subsequently walking backward [START_REF] Schwarz | How Ants Use Vision When Homing Backward[END_REF]. Our results suggest that short-term memory also influences the ants' navigational certainty. The time spent peeking forward varied (as already noted by Pfeffer and Wittlinger (2016)) with some ants 'peeking' for less than a second while other spent more than 60s (after which recording was stopped) without returning to their cookie. Interestingly, the more an ant had peeked before during a test run, the longer her current peek lasted (Fig. 2b). This shows that ants somehow gather information across time to adjust their current certainty: either information about the overall time spent backward or information about the number of peek previously displayed (see also Fig. 3c).

Former experiments have already shown that the behaviour of ants can be modulated by recent experience in the order of seconds to minutesa form of hysteresis [START_REF] Graham | Insect navigation: do ants live in the now[END_REF]. For instance, ants display a so-called backtracking behaviour only if they have recently perceived the nest surrounding [START_REF] Wystrach | Backtracking behaviour in lost ants: an additional strategy in their navigational toolkit[END_REF]. Also, homing ants display higher meander in their paths when recapitulating a well-known route for the second time in a row (Collett, 2014;[START_REF] Wystrach | Running paths to nowhere: repetition of routes shows how navigating ants modulate online the weights accorded to cues[END_REF]. Finally, ants can remember the compass direction of a wind gust after being blown (Wystrach & Schwarz, 2013). In the present case, ants seem to also build up information about the recent past. Even though it makes functional sense it remains to be seen what neural or physiological mechanisms underlie this phenomenon.

Conclusion

This study confirmed that ants walking backward are not just paying attention to celestial cues but combine multiple information from their PI, the recognition of terrestrial cues and temporal information such as the time they spent backward. All this information seems to be integrated in an overall directional drive. This drive, which reflects the current directional certainty, seems to (1) guide the ants backward path, (2) triggers peeking behaviour if too low and (3) finally dictates whether or not to return to their food item during peeks. Importantly, this study shows that ants uses familiar terrestrial cues while navigating backward. In addition to the attractive memories facing toward the nest, the hypothesis that homing ants use a collection of repulsive visual memories facing away from the nest and possibly stored during their outbound trip was put forward (Le [START_REF] Möel | Opponent processes in visual memories: a model of attraction and repulsion in navigating insects' mushroom bodies[END_REF]. In the light of this hypothesis, visual navigation forward or backward can then simply be achieved by using the relative familiarity between both sets of opposite valence memories. As often with research on insect navigation, remarkably flexible behaviours incite researchers to endorse the simplest explanations [START_REF] Wystrach | What can we learn from studies of insect navigation[END_REF].

  during tests and prevented ants to use potential familiar olfactory cues from the ground or the nest (BeN and ByN tests). A grid pattern (0.25 m  0.25 m) drawn on the board enabled paths to be transcribed onto gridded paper. The lampshade (beige opaque fabric, 0.5 m diameter; 0.4 m height) surrounded the ant upon release and obstructed any familiar terrestrial views; the top of the lampshade was open providing the ant with celestial compass cues. Once the ant had pulled the large crumb backward for 0.1 m, the lampshade was lifted and the visual surrounding was revealed. This ensured that the ants could not utilise any familiar scenes before starting their backward path. This sets Experiment 2 apart from Experiment 1 where tested ants had the chance to glimpse forward before dragging the biscuit backward.

Figure1.

  Figure1. Dynamics of peeking behaviours in terms of path integration and visual

Figure 2 .

 2 Figure 2. Backward path characteristics and peeking behaviour at different release

Figure 3 .

 3 Figure 3. Models on how ants may combine navigational information. The longer the

  Before nest (BeN): Ants were released on the route, 0.5 m before their nest.Beyond nest (ByN): Ants were released 0.5 m beyond the nest in the feeder-nest direction.Unfamiliar (U): Ants were released ~ 30.0 m away from the nest in a visually unfamiliar location.For all tests, ants were captured in a plastic vial, offered a large biscuit crumb to incite backward walking and, once the ant had grabbed the cookie, released within a lampshade at the middle of a large wooden board (2.4 m  1.2 m). Individual ants were tested only once per test, but could be subjected to different tests conditions, with at least one un-interfered training trial between tests. The wooden board was set in place just before and removed just after each test. The board was centred on the current release location with the long side along the feeder-nest direction (Fig.2a) as it represents the expected homing direction. The board provided an even substrate during tests and prevented ants to use potential familiar olfactory cues from the ground or the nest (BeN and ByN tests). A grid pattern (0.25 m  0.25 m) drawn on the board enabled paths to be transcribed onto gridded paper. The lampshade (beige opaque fabric, 0.5 m diameter; 0.4 m height) surrounded the ant upon release and obstructed any familiar terrestrial views; the top of the lampshade was open providing the ant with celestial compass cues. Once the ant had pulled the large crumb backward for 0.1 m, the lampshade was lifted and the visual surrounding was revealed. This ensured that the ants could not utilise any familiar scenes before starting their backward path. This sets Experiment 2 apart from Experiment 1 where tested ants had the chance to glimpse forward before dragging the biscuit backward.

Iberian

  Figure1

Acknowledgements

We are grateful for Xim Cerda and his helpful team at CSIC Seville for permanent assistance in logistics and administration during field work. We also thank Cornelia Buehlmann, Scarlett Dell-Cronin, Cody Freas and Michael Mangan for manual and moral support during field preparation and data collection. Finally, we are grateful for the constructive feedback of Paul Graham on the manuscript and the comments of two anonymous reviewers. The study was partly financed by The European Research Council, 759817-EMERG-ANT ERC-2017-STG.

Acknowledgements

We are grateful for Xim Cerda and his helpful team at CSIC Seville for permanent assistance in logistics and administration during field work. We also thank Cornelia Buehlmann, Scarlett Dell-Cronin, Cody Freas and Michael Mangan for manual and moral support during field preparation and data collection. Finally, we are grateful for the constructive feedback of Paul Graham on the manuscript and the comments of two anonymous reviewers. The study was partly financed by The European Research Council, 759817-EMERG-ANT ERC-2017-STG.

Acknowledgements

We are grateful for Xim Cerda and his helpful team at CSIC Seville for permanent assistance in logistics and administration during field work. We also thank Cornelia Buehlmann, Scarlett Dell-Cronin, Cody Freas and Michael Mangan for manual and moral support during field preparation and data collection. Finally, we are grateful for the constructive feedback of Paul Graham on the manuscript and the comments of two anonymous reviewers. The study was partly financed by The European Research Council, 759817-EMERG-ANT ERC-2017-STG.

Acknowledgments

Author contributions

Conceptualisation: S.S., A.W.; Methodology: S.S., A.W.; Formal analysis: S.S., A. W. for different locations (grey dashed arrows) with the ant always facing downward. For instance, when facing downward at the BeN location, the closest aligned view is repulsive (bold red arrow on right left scheme). Given that the neighbouring green arrow is not aligned, the closest aligned attractive view is further away, beyond the nest (bold green arrow on left scheme). Overall, the ants at this position (still facing downward) will be more repulsed than attracted by the current facing direction and thus walk backward toward the nest. In contrast, when beyond the nest ByN (still facing downward), the attractive views will match better than the repulsive views and the ants will thus turn around and walk backward toward the nest. Whatever the position and orientation of the ants around the nest, the agent will be drawn towards the nest (b) Directional drive across test conditions of Experiment 1 in Section 1 and 2. Ants are tested in FV (full-vector), FVA (full-vector altered familiarity) and ZV (zerovector) conditions. Overall directional drive (dark blue vectors) results from the integration of path integration (cyan vectors, the longer the path integration vector the stronger its directional drive) and learnt view (yellow vectors, the more familiar the view the stronger its directional drive). (c) Directional drive resulting from the recognition of a learnt view (yellow vectors) decrease with time spent facing in a different direction. Low directional drive results in lower speed (dashed arrow) and eventually peeking. Here memorised views are assumed to be facing upward and are thus recognised only when facing upward during peeking (small grey arrow) and not while facing downward during backward motion. Note that the second peek triggers a lower directional drive than the first (see also Fig. 1b).