Mice were trained with their cage being moved randomly between quadrant entrances from trial to trial without directly manipulating the animals. Thus, they never entered from the same start position for consecutive trials, making the wall pictures their only reliable orienting landmarks. These cues were 15x15 cm black shapes on a white background (see Materials and methods). The configuration of the 4 cues varied relative to the entrance, but the food was always placed in the same hole and nearest the X-shaped landmark (Figure 2, Materials and methods). The food-restricted mice showed persistent complex search trajectories throughout the learning period (Figure 2A and B). The latency to reach the food target was significantly negatively correlated with trial number (statistics in Figure 2D) but with a non-significant difference between the first and last trials (N=8; First trial mean ± SD = 151.34±148.91 s; Last trial mean ± SD = 80.38±43.86 s; p=0.2602). Despite the decrease in latency, mice did not appear to exhibit spatial search strategies (Figure 2C) given that the time spent in each quadrant during the probe trial is similar (target quadrant = 28.78%, other quadrants = 24.93%, 28.04%, 18.25%; one-way ANOVA: p=0.070574, F-ratio=2.71621; see Behavioral analysis in Materials and methods). This suggests latency decreased for reasons independent of spatial learning.

We found a corresponding significant positive correlation of speed and trial number (Figure 2E) with non-significant differences between first and last trials (N=8; First trial mean ± SD = 6.68±3.79 cm/s; Last trial mean ± SD = 11.40±5.07 cm/s; p=0.0704). Since the distance from start to target (d_target) varies over trials due to changing entrance location, we defined the normalized trajectory length (Figure 2F). It is the ratio between total traveled distance (d_total) and d_target; d_total/d_target = 1 is optimal. There is no significant correlation of this quantity and trial number (p=0.295). The first and last normalized trajectory lengths were not significantly different either (N=8; First trial mean ± SD = 11.79±8.25; Last trial mean ± SD = 15.51±9.92; p=0.4241).

Fitting the function d_total = B_*exp(-Trial/K) reveals the characteristic timescale of learning, K, in trial units (Figure 2F). We obtained K=26 ± 24 giving a coefficient of variation (CV) of 0.92. The mean, K=26, is therefore very uncertain and far greater than the actual number of trials. Thus, we hypothesize that the mice did not significantly reduce their distance travelled (Figure 2A, B and F) because they had not learned the food location – the decrease in latency (Figure 2D) was due to its increased running speed and familiarity with non-spatial task parameters.

We next examined hole checking as an active sensing indicator of the mouse’s expectation of the food location (see Materials and methods, Figure 4—figure supplements 1–4 and the Shortcut Video) for hole-check detection after spatial learning. Even though hole check counts increase with the total trajectory length and latency (Figure 4—figure supplement 3B and C), we expected that the mice would increase the number of hole checks near the expected food location, or along their path toward it. Thus, we split the trajectories into two halves of same duration to identify whether hole checks would increase in the second half (being closer to the target). However, the hole checking density (ratio between number of checks and distance travelled) remains constant between the first and second halves of any trial (Figure 2G), suggesting mice did not anticipate where the food is placed. Mice also did not increase their hole sampling rate as they approached the target as shown by the distance of hole checks to the target, d_{checks→food} (Figure 2H, I), again suggesting no expectation of its location. Mice do not appear to learn a landmark-based allocentric spatial map.

We tested whether the mice might acquire an allocentric spatial map if, contrary to the random entrance case, they had a single stable view of the landmarks and could associate one cue configuration with the food location (see Materials and methods). For this, each mouse entered the arena from the same entrance during the training trials (hence ‘static entrance’). Mice greatly improved their trajectory efficiency after training (Figure 3A and B), with distance and latency to food plateauing around trial 7. During early learning, mice showed random search trajectories that transformed to stereotyped trajectories towards food by late learning (Figure 3B). Typical trajectories were either (i) directly to the food, (ii) initially displaced towards the maze center then returning to a food-oriented trajectory, or (iii) initially along the wall and then turning and heading to the food (Figure 3B).

During the probe trial, mice spent the most time searching in the target quadrant (44.26%; Figure 3C), compared to the time spent in the other quadrants, respectively 19.38% (p=0.007), 16.55% (p=0.0065), 19.81% (p=0.007; p-values obtained for a pairwise t-test and adjusted for multiple comparisons via the Benjamini-Hochberg method). The latency to target strongly decreased over trials (Figure 3D) and the last trial latency was significantly reduced compared to the first (N=8; First trial mean ± SD = 255.39±153.28 s; Last trial mean ± SD = 7.38±4.03 s; p=0.004). The speed dramatically increased over trials (Figure 3E) and the ending speed was significantly greater than the initial speed (N=8, First trial mean ± SD = 6.04±2.48 cm/s; Last trial mean ± SD = 17.25±4.64 cm/s; p=0.0002). The normalized trajectory length strongly negatively correlated with trial number (Figure 3F) and the final normalized length greatly reduced (N=8; First trial mean ± SD = 18.02±11.36; Last trial mean ± SD = 1.40±0.50; p=0.0061), becoming almost optimal (Rosenberg et al., 2021).

Now the fitting of the function d_total = B exp(-Trial/K) yielded K=5.6 ± 0.5 with a CV = 0.08; the mean is therefore a reliable estimate of total distance travelled. We interpret this to indicate that it takes a minimum number of K=6 trials for learning the distance to the target (see also Figure 4—figure supplement 3D, E, F and G). Learning is still not complete because it takes 14 trials before the trajectories become near optimal.

Hole checking density varied across trajectories and differed between early and late learning trials. The density significantly decreased with trial number in the first half of trajectories (R=−0.54; p=0.0455, Figure 3G) but not the second half (R=0.17; p=0.559, Figure 3G). The hole checks also happened significantly closer to the target in the second half of the trajectory (Figure 3H; First trial mean ± SD = 48.06±11.50 cm; Last trial mean ± SD = 3.13±5.70 cm; p=3 × 10^–6). A remarkable effect of learning is that the probability of checking holes increases as the mouse approaches the food (Figure 3I). These results (Figure 3H, I) suggest that the mice may be predicting the food location and checking their prediction as they approach the estimated food location (see also Figure 4—figure supplements 3 and 4 for other quantities and scatter plots of trajectory features vs. hole checking). Hole checking near the food site also implies that the mice are not using odorant or visual cues to sense the precise food location, but instead rely on a memory-based estimate of the food-containing hole (see also Figure 4G and H). This motivated us to look at the spatial configuration of these variables to get a detailed picture of the random vs. static condition.

Although quantitative, the analysis so far only showed a broad comparison between random versus static entrance experimental conditions. We developed a method to map out the directions that the mice stepped towards more frequently from each particular location in the arena, yielding a displacement map (see Statistical analysis of trajectories in Materials and methods). This map can generate a precise estimate of the mean directionality of the trajectories for each experimental condition (Figure 4A, B, E, F and p-values in Figure 6—figure supplement 1). Additionally, we also made a frequency plot of the checks for each particular hole in the arena, yielding the spatial distribution of hole checks (Figure 4C, D, G and H). These two maps are combined to give the Target Estimation Vector (TEV). The TEV is interpreted as the mouse’s estimate of the target position (pink vectors in Figure 4A, B, E and F).

In random entrance training, no significant directionality was found in the arena (Figure 4A and B), again consistent with a random search. The spatial distribution of hole checks migrated from near the walls towards the center of the arena but remained random and displaced from the target (Figure 4C and D, and also Figure 2—figure supplement 1), consistent with the broad results of Figure 2. Conversely, there was a clear and significant flow of trajectories for mice trained with static entrance (Figure 4E and F; p<0.0001; see Materials and methods for the definition of the p-values assigned to directions). The hole checks reflected this and became solely concentrated along the route from start to target (Figure 4G and H). In the static entrance condition, the entropy (uncertainty) of the number of hole checks near (<20 cm) the target was lower than that for the far (>20 cm) condition (Figure 6—figure supplement 1). In other words, mice were more consistent in the number of their hole checks near the target compared to far from the target, suggesting that they had an internal representation of their proximity to the food site.

We compared the deviation between the TEV and the true target vector (that points from start directly to the food hole; Figure 4I). While the random entrance mice had a persistent deviation between TEV and target of more than 70^o, the static entrance mice were able to learn the direction of the target almost perfectly by trial 6 (TEV-target deviation in first trial mean ± SD = 57.27^o ± 41.61^o; last trial mean ± SD = 5.16^o ± 0.20^o; p=0.0166). A minimum of 6 trials is sufficient for learning both the direction and distance to food (Figure 4I) (Figure 3F) (see Discussion). The kinetics of learning direction to food are clearly different from learning distance to food since the direction to food remains stable after Trial 6 while the distance to food continues to approach the optimal value.

Under the path integration hypothesis, it is expected that error accumulates as the mouse walks (McNaughton et al., 2006). This motivated us to investigate the frequency of hole checks per area near the target (within 20 cm) vs. far (further from 20 cm away), Figure 4J and K. The density of hole checks in both conditions remained constant for random entrance mice, regardless of training duration. However, static entrance mice had significantly higher density of hole checks near the target (Figure 4K; p-values: random-far vs. static-far=0.005; static-far vs. random-near=0.03; random-near vs. static-near=0.01). The mean position of hole checks near (≤20 cm) the target is interpreted as the mouse estimated target (Figure 4C, D, G and H; green + sign = mean position; green ellipses = covariance of spatial hole check distribution restricted to 20 cm near the target). This finding together with the displacement and spatial hole check maps (Figure 4F and H, respectively) corroborates the heatmap of time spent in the target quadrant (Figure 3C), suggesting a positive correlation between hole checks and time searching (see also Figure 4—figure supplement 3C). Thus, we use the distance from this point to the entrance as the magnitude of the TEV. This definition reveals that the TEV is nearly coincident with the direct route between start location and the food-containing hole (pink vectors in Figure 4F), consistent with the near optimal normalized trajectory length previously discussed (Figure 3F).

These results demonstrated that mice can learn the spatial location of the food target from a static entrance. We next investigated whether the performance gain can be attributed to the cues by manipulating the mouse’s entrance. Well-trained static entrance mice were subject to rotation of their home cage by 180°,+90°, or –90° for probe trials in order to start from a different location. We used a different cohort of N=8 mice for each rotation. The mice were not directly handled during the rotation procedure. If the mice were using the distal landmarks, they would still be expected to find the correct food location. If they relied on self-motion cues, they should travel to the REL, since this is the spot where the target would have been if the mouse had been trained from that entrance. In other words, going to the REL means that the trajectory is anchored to the mouse’s start point and not to the wall cues. As illustrated in Figure 1B, the displacement between each start-REL pair is exactly equal from the mouse’s point of view (e.g. going 70 cm forward then 30 cm to the left reaches the respective REL of the target).

Following all rotations, mice searched at the REL instead of the correct location for the food reward regardless of the arena having wall cues (Figure 5A, B and C), meaning that they were not using landmarks to compensate for rotation. In fact, the trajectory kinetics of the 180^o-rotated probe for reaching the REL target (mean ± SD latency = 4.80 ± 3.06 s; speed = 14.60 ± 5.21 cm/s; normalized distance to REL = 1.96 ± 0.90) is indistinguishable (within one SD or less) from the kinetics of the last trial of static entrance for reaching the true target. The criterion for reaching the REL target was getting within 5 cm of its position (the average inter-hole distance is 10 cm).

Figure 5

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We also show the displacement maps (mean trajectory directionality), hole check spatial profile and TEV for the 180^o-rotated probe trial (Figure 5C). The TEV now points to the REL, and the hole checks accumulate along the route to the REL. These maps are also very similar to the static entrance maps in Figure 4. Thus, the mice failed to use allothetic cues for navigation in the novel start location test.

It is possible that mice, like gerbils (Collett et al., 1986), choose one landmark and learn the food location relative to that landmark (e.g. the ‘X’ in Figure 3B); in this case, the mice would only have to associate their initial orientation to that landmark without identifying the image on it. We used two additional controls to assess this possibility. Mice were trained without any cues on the wall, eliminating the possibility that they provided orienting guides. The learning curve of mice with or without the 2D wall cues were not significantly different (Figure 4—figure supplement 5).

Mice that were well trained in the lighted maze were given trials in complete darkness. These mice still showed the same learning curve compared to untrained mice or mice following rotation (Figure 4—figure supplement 5).

Finally, to completely rule out both landmark and possible olfactory cues, we trained and tested mice in total darkness. Head direction tuning may be impaired in sighted mice navigating in darkness (Asumbisa et al., 2022), but spatial learning was not affected in our experiments. Mice can also use localized odor sources as landmarks for spatial learning (Fischler-Ruiz et al., 2021) and we therefore attempted to eliminate such cues (see Discussion and Materials and methods). The mice were still able to learn the food location in the absence of visual and olfactory cues (Figure 4, Figure 4—figure supplement 5). These experiments demonstrate that the mice can learn to efficiently navigate from a fixed start location to the food location using path integration of self-motion cues.

Furthermore, the location of the targets in all instances of the experiment was carefully chosen to avoid cues from the arena’s circular geometry. If the target was placed near the center of the arena, the learning task would be trivial and would involve little spatial inference. Conversely, if the target was placed too close to the wall, thigmotaxis would be sufficient to get near the food. Also, the line going through the center of the arena that is perpendicular to the line connecting start to center has to be avoided (see Figure 1B for reference), otherwise the system would be symmetric to rotation or could be simplified to a left-or-right choice that would involve little spatial learning. Finally, the target cannot be placed near the start positions. Thus, in the static entrance experiments, the chosen target position is always more than 35 cm away from the wall, and more than 60 cm away from the start. In the random entrance case, the target is placed more than 30 cm away from the wall, and more than 40 cm away from the closest starting location.

Our results established that mice learn the spatial location of a food reward using path integration of self-motion cues. Have the mice learned a flexible cognitive map? A test of cognitive mapping would be if mice could take a short cut and navigate from one remembered location to another along a novel, unreinforced path. The ‘two food location experiment’ set out to test this possibility by training mice to travel to food from their home cage to two very differently located sites (chosen according to the rules mentioned in the previous section). The two locations were trained sequentially so that one site was trained first (target A) followed by training on the second site (target B). A probe trial without food after training target A was designed to check if mice had learned it successfully. Only one hole was filled with food in any given training trial. In a second probe trial after training on B, mice were tested to see whether they can travel between the two remembered locations (food is absent; probe B-A trial) despite no prior training nor reinforced experience on the short cut route (Figure 1E).

Mice successfully learned the location of target A, evidenced by their direct search trajectories and significant decreases in distance traveled (Figure 6A), with the same performance as in the static entrance case. During the learning of target A, the distance of the mouse to the future location of target B was always larger than that expected by chance, that is to a randomly chosen location in the maze (see Materials and methods, Figure 6—figure supplement 2). In fact, the trajectories stayed, on average, approximately 40 cm away from the forthcoming target B position. In other words, the mice did not learn, by chance, direct unreinforced routes from ‘near B to A’. In Figure 6A, for example, the mouse passed within 16 cm of the future target B site in an early learning trial, but this close approach does not appear to constitute a B->A trajectory. The first probe trial (without food) confirmed that mice had learned the location of target A. After not finding food in A, the mice re-initiated a random search (Figure 6B).

Figure 6 with 2 supplements see all

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After the probe trial, mice were trained to learn target B. During this training, three mice (Mouse 33, 35, 36) first learned a route to target B but, on subsequent searches, they would sometimes first check the site of the previous target A and then travel to the correct target B along a direct short cut route (see Discussion). Nevertheless, on average the mice kept farther from the previous site A than the expected by chance (Figure 6—figure supplement 2). A first visit to B then to A was never observed during training because mice go back home after getting the food in B.

After learning the location of target B (Figure 6C), food was also omitted from this target for Probe B-A. As illustrated in Figure 6D and the Shortcut video (Video 1) (all mice shown in Figure 6—figure supplement 2), five of eight mice were observed to go from B to A via varying trajectories (namely, mice numbered 33, 35, 36, 57, and 59). Two (number 58 and 60) of the remaining three mice went first to A and subsequently to B via direct or indirect routes; these mice had previously taken direct routes to B, suggesting that going to A first is not due to a learning deficiency. The remaining one, mouse 34, went from B to the start location and then, to A. This mouse had previously taken the B-A route during training. In all cases, the mice clearly remembered the location of both targets, even though target A had not been presented or rewarded for 4 days.

Video 1

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The geometric and kinetic features of these experiments for early and late learning of each target, and for Probe B-A, are presented in Figure 6E–H. For defining the trajectory quantities in the Probe B-A trial, the ‘start’ position is taken as target B, and the ‘target’ is the A site (e.g. the normalized trajectory length is the ratio between total traveled and direct B-A distances). The quantities in probe B-A are statistically indistinguishable from late learning of targets (i.e. trials 14 A and 8-B). Conversely, all the Probe B-A values are significantly different both from trial 1-B and from randomized B-A trajectories (see Materials and methods; p<0.05; except for the density of hole checks which is statistically equal for all considered trials, Figure 6H). Therefore, the behavior of the mice when going from B to A in the probe is compatible with the behavior of an animal that acquired spatial memory about the trajectory, even though this trajectory was never reinforced. This suggests that the mouse computed the shortcuts without prior experience (see Shortcut video Video 1) for an example of shortcutting and associated hole checks.

We calculated the displacement and hole check maps and the TEV for the whole training protocol, including the probe B-A (Figure 7). Again, the data show that, after training for A, the TEV pointed directly to A and hole checks accumulated along its path (Figure 7A and B). The beginning of training for B (trial 1-B) generated random search patterns, while the TEV and hole checks tended towards A. After learning B, we see that the TEV and hole checks completely shifted to it (Figure 7C, D, E and F). Remarkably, the probe B-A trajectories revealed a strong directed flow with a TEV pointing from B to A, whilst hole checks visibly accumulated along this route (Figure 7G and H). The TEV-target deviation remained close to zero in late learning and probe B-A (Figure 7I), whereas the area density of hole checks is increased near the target compared to far only after learning and in probe B-A (Figure 7J). These maps suggest the emergence of a cognitive map guiding the mice when taking the B to A unrehearsed shortcut routes.

Figure 7

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As a stringent control, we performed a two-food-locations experiment with a 180^o-rotated probe B-A trial (Figure 8; N=8). Landmarks were never present (neither for training nor for probe). In the rotated probe B-A, the mice went to the REL of B and then to the REL of A. The displacement map, hole check spatial distribution and TEV now all pointed from REL B to REL A (instead of B to A). This is consistent with both the independent experiments of the rotated probe with static entrance training and of the two-food location. Mice can thus take a novel short cut and have therefore computed a cognitive map based on a fixed starting point and self-motion cues alone.

Figure 8

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Although the mice can take varying trajectories, the TEV suggests that they continuously update their knowledge of the direction from start to food throughout their trajectories. This further suggests that the neurons coding for head direction are essential for spatial learning in our maze. Heading direction can be derived from optic flow signals (Burlingham and Heeger, 2020; Horrocks et al., 2023). In the static entrance experiments, the heading direction is constantly updated throughout the entire run (Figure 4—figure supplement 3G), having the mouse veer toward the food only in the last segment of the trajectory (Figure 4—figure supplement 3Giii). Hippocampal neurons of rats randomly foraging in a real world or virtual reality environment can develop directional responses based on rich and stable optic flow signals and without requiring vestibular input (Acharya et al., 2016). In our maze, landmarks and home cage openings may drive optic flow signaling irrespective of whether they act as local visual cues. This is consistent with the hypothesis that visual input can guide navigation via a central retinal stream for landmarks and a peripheral retinal stream for optic flow input (Saleem, 2020). Our in the dark experiments suggest that other sensory input might also drive the head direction network.

The vestibular system encodes natural motion (Cullen, 2019; Mohammadi et al., 2024) and contributes to the generation of head direction responses (Cullen and Taube, 2017; Taube, 2007). We hypothesize that, in the ‘dark’ experiments, vestibular cues are a major signal supporting spatial learning. In agreement with VR studies (Mao et al., 2020; Yang et al., 2024), we hypothesize that contextual visual cues of the starting point and arena boundary anchor directional information derived from optic flow, vestibular and perhaps proprioceptive signals to fixed environmental features.

Five days before training, mice are transferred from standard animal facility cages to the experimental home cages for habituation. On the first day, mice are allowed to habituate to their new home cages. On the second day, the door dividing the home cage and the arena is removed and each mouse is given free access to explore the empty arena for 10 min. There are no extra-maze landmarks present during pretraining. Animals are food restricted, with 10% of their body weight in food given back each day which they could consume ad libitum. On day 3, randomly chosen 50% of the holes in the arena are filled with food treats (a piece of Cheerio), and each mouse is given 10 min to forage for food. This is repeated on day 4, where 25% of the arena’s holes are filled with food treats. On day 5, only four holes placed at the maze center contained treats. Mice pass the pre-training stage when they successfully found treats at all locations within 20 min. Mice mostly confined their search to circular trajectories near the wall of the maze for Days 3 and 4 but, after training with food near the maze center (Day 5), they checked holes throughout the maze (Figure 1—figure supplement 1A). Training in the spatial learning task then commenced.

The mice are trained to locate a food reward that has a fixed relationship with four visual cues on the walls. The protocol mimics classic Morris Water Maze setups to test allocentric landmark-based learning. One of the holes in the area is capped from the bottom with a food-containing insert that is not visible from the surface. Extra-maze landmarks (described in the Visual Cues section) are placed on the walls of the maze to serve as location cues. At the start of the trial, the door is slid upwards so the mice can enter the maze, and the mice are given a maximum of 20 min to find the target hole. The home door is kept open to permit the mice free access to return to their home cage during the trial; preliminary experiments indicated that closing the door perturbed the mice. The door is re-inserted after the mouse has found the food reward and returned home. If the mouse has not returned home by itself after 1 min of finishing the food reward, it is gently guided back by the experimenter. There is an inter-trial interval of approximately 20 min.

Initial experiments used two trials per day but this was increased to speed up learning; our analyses and graphs are truncated at 14 trials to permit averaging over all the mice. In most experiments, three trials a day were given, over 6 days; at the end of each trial the mouse’s home is moved to a different entrance. The home location was changed for each trial; the floor was washed and rotated by 90°, 180°, or 270° between trials, independent on whether the home entrance was the same (static trials) or rotated (random trials). The location of the entrance is randomized with the following constraints: no two trials in a row have the same entrance, and all entrances are selected the same number of times so there is no bias. On the 7th day, a probe trial is given where the mouse is allowed to search for food in the absence of any food. Learning continues over 1 more day and, on the 9th day, a reversal trial is given where the visual landmarks are rotated by 180°. The location of the home cage is randomized before every trial. In this task, the mouse would have to learn the invariant relation between the food hole and up to four visual cues in order to locate the food.

Four black symbols on white backgrounds: a square, a cross, vertical bars, and horizontal bars. The cues are taped 5 cm above the floor. The square was 15 by 15 cm. The cross consisted of two 2.5 cm wide and 14 cm long bars. The four vertical and four horizontal bars were 2.5 cm wide and 14 cm long. Previous behavioral studies have shown that mice can discriminate the visual stimuli we use (Jacobs et al., 2009; Prusky and Douglas, 2004; Prusky et al., 2000). Recording of neurons in mouse V1 have additionally shown that orientation selective cells in mice visual cortex can discriminate our visual stimuli (Long et al., 2015).

This protocol allows mice to navigate by potentially using visual cues in cooperation with path integration or by path integration alone. The setup is the same as visual cues training, except mice enter from the same entrance each trial instead of a randomized entrance. Mice are considered well trained once their latency learning curve has plateaued for three successive trials. The mice’s latencies had plateaued by 14 trials but training continued till 18 trials.

After 18 trials, reversal trials were done with no food present and the mouse’s home cage rotated 180°. The mouse was allowed to search for food for 10 min. If the mice were able to learn the invariant landmark/food spatial relationship, we would expect them to search at the learned food site. If they had used path integration of self-motion cues to navigate from the fixed entrance to the food, we would expect them to search at the rotationally equivalent location (REL).

The four quadrants of the arena’s floor are identical. This means that the position of the holes in the second quadrant (Q2, Figure 1B; also Figure 4—figure supplement 3A) is equal to the position of the holes in the first quadrant (Q1) rotated by 90°Counter clockwise (CCW) about the center of the arena. Q3 is Q1 rotated by 180^o (CCW) and Q4 is Q1 rotated by –90^o (i.e. 90°CW). In other words, every hole in Q1 has an REL in each quadrant achieved by the respective rotation. The REL target works in the same way: it is the position (relative to the entrance used in the trial) where the food would have been had the mouse been trained from the entrance it used in the trial. To illustrate this, consider Figure 1D: we put the food (target) in a particular hole in Q1 for a mouse that is trained to enter from Q2 (such as the example in Figure 1B) in the static entrance protocol. The vector that points from the mouse’s start point (in Q2) to the target in Q1 (i.e. the target vector) is ‘anchored’ to the start position at Q2. For example, in the mouse’s perspective, the target vector is equivalent to going forward for 70 cm, and then turn left and follow for another 30 cm. If in a probe trial, we now let the mouse enter from Q4 instead, there are two options: either the mouse uses the landmarks and searches for the food in Q1 (where the food actually used to be), or it follows the target vector (70 cm forward +30 cm left) and goes to the REL location of the target in Q3. Following the target vector is a sign of path integration using self-motion signals (idiothetic cues).

We performed additional tests for the use of visual landmarks in the Visual Cues + Static Entrance protocol. Well-trained mice had their home cages rotated to another quadrant and tested on how well they found the food from a new starting location. Mice were rotated 180°, 90°, and then –90° with respect to their original location (Figure 5A, B and C). Mice were given six consecutive trials at each new location. One control group of mice had the visual cues on the arena wall removed during rotation training. If mice were able to use visual cues, we would expect them to improve their search efficiency when visual cues were available. Lack of improvement or similar performance compared to the control group without visual cues suggests that the mice do not rely on the type of visual cues we provided for spatial learning and navigation.

To control for the effects of all visual information, one cohort (four mice) was trained and tested without any cues (Figure 4—figure supplement 5A and B). A second cohort (four mice) was trained in the light according to the Visual Cues +Static Entrance protocol, then the lights are turned off and the mice are given a trial in darkness (Figure 4—figure supplement 5C and D). The lights were opened in between trials so the mice did not acclimatize to the darkness. Mice were tracked using infrared LEDs with emission spectra detected by our camera. The following conditions were tested: darkness during a regular trial with food present in the arena and during a probe trial where there is no food present in the arena; both conditions gave the same results.

We additionally both trained and tested a cohort (four mice) in darkness and with control of potential odors. A recent study has shown that placing sighted mice in darkness impairs entorhinal cortex head direction cell tuning (Asumbisa et al., 2022), but here the mice successfully learned to find food and with the same time course of learning (Figure 4—figure supplement 5E and F).

This protocol aims to test flexibility of spatial learning using only path integration (N=8 mice). No visual cues are placed on the arena walls. Mice entered the arena from the same entrance each trial. Food was placed in a target well in ‘Target A’ and mice were trained to find this location for 18 trials, 3 trials a day. A probe trial (‘Probe A’) was done after trial 18. For trials 19–26, the food was moved to a different location, ‘Target B’, and mice are trained to find the new location. We were careful to choose Target B so that Target A and B were not symmetric with respect to the mouse’s entrance. A second probe trial (‘Probe B-A’) was done after trial 26. The purpose of the ‘Probe B-A’ trial is to check whether mice take a shortcut between B (latest learned target position) and A (first learned target position). For some cohorts, the training continued until trial 34 and a third probe was given.

The mice (N=8) were trained exactly as in the ‘Two food location’ experiment described above. However, the mice start location was rotated by 180^o prior to the second probe trial. This protocol joins the ‘Static entrances with rotated probe’ protocol with the ‘Two food location protocol’, and is designed to provide further support for our path integration hypothesis. Each of the trained targets have their REL counterparts (180^o rotated around the center of the arena). Now, the purpose of the rotated ‘Probe B-A’ trial is to check which of the two options will happen: either mice take shortcuts between B (latest learned target position) and A (first learned target position); or they take shortcuts between REL B and REL A, supporting path integration via self-motion cues.

The procedure described in this section is applied to calculate trajectory directionality (and hence the target estimation vector, TEV) and the hole checking spatial distributions. Each mouse in the Random Entrance experiment started from a different quadrant of the arena, for each consecutive trial, and the target was fixed relative to the arena (global reference frame). However, in order to increase sampling, we need to coherently align the mice entrances, generating the mouse’s perspective reference frame (see Figure 1—figure supplement 1B–F). Each experiment batch was performed with four mice, such that at any given trial, a given mouse entered from quadrant 1 (Figure 1—figure supplement 1C), another entered from quadrant 2 (Figure 1—figure supplement 1D), the next entered from quadrant 3 (Figure 1—figure supplement 1E), and the last entered the arena from quadrant 4 (Figure 1—figure supplement 1F). Most of the experiments, then, consisted of two different batches of four mice that had to be conveniently aligned to increase sampling. In this figure, we aligned all the entrances to the Start point at the top of the arena (trajectories were rotated accordingly), emphasizing the four possible positions of the target from the mouse’s perspective.

We had to find a way to align all the targets in a given trial to one of the four possible positions, such that the experiment stays coherent over trials (i.e. the target randomly switches in between these four positions from trial to trial). For example, in Figure 1—figure supplement 1B–F, we show trial 14. The target position for each starting quadrant is labeled with a red letter A and a red circle, whereas the target for the previous trial is labeled with a green circle and a green letter A subscripted with ‘trial 13’. These two positions (trial and previous trial) must always be different to keep the random characteristic of the experiment.

Notice that the target positions in each of the panels D, E, and F in Figure 1—figure supplement 1 are simply rotated relative to the target positions in panel C. The target configuration in panel D is 90°Clockwise rotated relative to the target configuration in panel C. The configuration in panel E is 180°Clockwise rotated, and panel F, 270°Clockwise rotated; both angles are relative to the configuration in panel C. Thus, in order to sample all the mice together, we simply rotate the trajectories from the mice that started in quadrant 2 [panel D] counter-clockwise by 90^o; the trajectories from the mice that started in quadrant 3 [panel E] are rotated by 180°Counter-clockwise; and the trajectories starting from quadrant 4 [panel F] are rotated by 270°Counter-clockwise. This reduces all the experiments to the first quadrant, allowing us to sample all the trajectories of all the mice together in each individual trial.

We developed an algorithm to detect the mouse’s behavior of sniffing holes to detect food as a measure of active sensing. The hole checking events are used to infer the mouse’s memory and uncertainty about its environment. The procedure described here is applied independently to each trial. We compute the spatial distribution $P\left(X_i,Y_i\right)$ for the count of hole checks for each hole $i$ positioned in $\left(X_i,Y_i\right)$ in the arena, accumulated across all mice in a given experiment (Figure 4C, D, G and H; Figure 5C; Figure 7B, E, F and H; Figure 8C; and Figure 1—figure supplement 1B–F; usually N=8 mice for each experiment), and normalized by the total count. The frequency of checks in each hole is coded both in the color and size of the filled circles: larger and darker shaded circles correspond to larger number of checks in that particular hole (lighter shaded pink small circles are a small number of checks). The black ellipsis marks the covariance of the $P\left(X_i,Y_i\right)$ spatial distribution (‘x’ in the figures marking the mean position of hole checks). It is constructed from the eigenvalues and eigenvectors of the covariance matrix $C$ of the hole check coordinates $\left(X_i,Y_i\right)$ weighted by $P\left(X_i,Y_i\right)$: the eigenvectors give the directions of the ellipsis semi-axes, whereas the eigenvalues give the width of each semi-axes. The center of the ellipsis (mean of the ellipsis’ foci) is aligned with the mean of the spatial distribution.

Under the path integration hypothesis, it is expected that error accumulates as the mouse walks (McNaughton et al., 2006). This is consistent with the increase in number of hole checks per unit area near the target (Figure 4J and K; ‘near’ = less than 20 cm away from the target). We consider that the change in number of hole checks measures the variability of the mouse’s estimate of the target position.

We therefore calculate the mean and covariance of the distribution of hole check events restricted to within 20 cm of the target (again, the green ‘x’=mean). The distance from the start to this restricted mean is used to scale the TEV vector (see Target Estimation Vector).

The arena design forces the mouse to put its nose very close to the hole to be able to see the food or detect any food odour. With that in mind, we defined two methods for detecting a hole check (see Figure 4—figure supplement 1A and B, and the Shortcut video Video 1). The two methods are complementary, and the second can find hole-check events missed by the first method: we apply the first method, then perform a visual check on the data to see if there were potential hole-checks that were missed. Then, we apply the second method to capture any remaining events.

The first method is the minimum velocity criterion and provides a high threshold for hole check identification. Four conditions must be satisfied simultaneously to detect an event: (i) the nose of the mouse must be within 3 cm of an arena hole; (ii) the velocity has to be less than 20% its maximum value; (iii) the velocity must be at a minimum; and (iv) the velocity has to have dropped by at least 5 cm/s to reach that minimum.

The second method is more inclusive and defines a hole-checking event by a simple slowing down event, provided that: the nose is within 3 cm of an arena hole, and the slowing down is enough for the velocity to drop past the 20% threshold of its maximum.

The detected events by the application of these two methods in sequence are marked for all trajectory samples of the Probe B-A in as examples. A particular case is shown in more details in Figure 4—figure supplement 1, and in the Shortcut video (Video 1) with the recording of a mouse’s performance. It is worth mentioning, there is a clear hole-check missed by our algorithm in the 25–27 s range. This is because the velocity of the mouse was already below the 20% maximum before and after the hole check, hence the two sets of criteria defined above were not met for this particular event and it was counted by the manual check. Otherwise, we clearly see that all the other events are captured by sequential use our two methods.

This analysis is a way to visualize mouse trajectories and directionality across all the mice for each trial. It is related to a velocity map of the arena, except that here, the arrows point in the direction of most probable movement instead of the direction of the velocity. Figure 4—figure supplement 1B shows a scheme for how we compute this map, and we detail it below. We call this quantity as the ‘displacement map’, since it gives the displacement of the mouse for each position in the arena.

In this section, we explain how to calculate the displacement map for a single sample of $N$ mice. The procedure here is applied to each individual trial independently (either during learning, or for a probe trial). This map is used for inferring the learned directionality of the target (relative to entrance). The error, average and significance of this quantity is estimated from first principles by a jackknife procedure explained in the next section. In the figures we show the average displacement map vectors that were found to be significant. These maps are then used to compute the target estimation vector (TEV; detailed in the last section).

We start by overlaying a lattice of size $L$ on top of the arena recording. This means that there are $L$ boxes on the $x$ (horizontal) direction, and $L$ boxes on the $y$ (vertical) direction, making a total of $L^2$ boxes in the lattice. Each box defines a lattice site with coordinates $\left(x,y\right)$, such that $x$ and $y$ are integers between 1 and $L$. Each mouse corresponds to an independent observation of a trajectory, so we overlay all mice trajectories for each trial separately in the calculations below. Next, we map the coordinates of the trajectories into the lattice coordinates in order to obtain a temporal sequence of visited lattice sites in each trial. We are interested in counting the number of times that each subsequent pair of adjacent lattice sites appears in this sequence, regardless of what mouse it came from. This will be used to define the displacement map, detailed in what follows.

The displacement map $\overrightarrow{M}\left(x,y\right)$ is a vector field defined on the lattice that assigns to each site the preferential direction of movement. This direction is estimated from the trajectories data. Mathematically, it can be written as $\overrightarrow{M}\left(x,y\right)=\left[M_h\left(x,y\right),M_v\left(x,y\right)\right]$, where $M_h$ is the horizontal component and $M_v$ is the vertical component. The horizontal component expresses the trend to move to the left or right (along the lattice $x$-axis), and the vertical component in the perpendicular direction, that is ‘up’ or ‘down’ in the top view of the lattice of Figure 4—figure supplement 1D (along the lattice $y$-axis). Each component is a function of the lattice coordinates, $\left(x,y\right)$. The vector $\overrightarrow{M}$ is defined as the spatial gradient of the probabilities to move out of ${\displaystyle \left(x,y\right)}$ towards one of its adjacent sites. Thus, its components are given by

where ${\displaystyle P_{\to }(x,y)}$ is the probability of stepping right, that is from $\left(x,y\right)$ to $\left(x+1,y\right)$; $P_{\leftarrow }\left(x,y\right)$ is the probability of stepping left [from $\left(x,y\right)$ to $\left(x-1,y\right)$]; $P_{\uparrow }\left(x,y\right)$ is the probability of stepping up [from $\left(x,y\right)$ to $\left(x,y+1\right)$]; and $P_{\downarrow }\left(x,y\right)$ is the probability of stepping down [from $\left(x,y\right)$ to $\left(x,y-1\right)$]. The probabilities of stepping out of $\left(x,y\right)$ must be normalized, therefore $P_{\to }\left(x,y\right)+P_{\leftarrow }\left(x,y\right)+P_{\uparrow }\left(x,y\right)+P_{\downarrow }\left(x,y\right)=1$. If there is no trajectory going through a particular site, the probability of going from that site into each of the directions is equal to the ‘null’ probability, $P_0=1/4$. Time sequences where the mouse does not move are ignored, since we are only interested in the movement between adjacent sites.

The stepping-out probabilities are computed from the mouse trajectory in the following way. This procedure is applied to all mice overlayed together in each individual trial. First, a step is defined as moving from a box at $\left(x,y\right)$ to one of its four adjacent boxes, say the one on the right $\left(x+1,y\right)$. Now, if we wanted to calculate the probability of going right from a position $\left(x,y\right)$, we need to go through the sequence of visited lattice sites looking for $\left(x,y\right)$ in an instant followed by $\left(x+1,y\right)$ in the immediately next instant. We count the number of times $S_{\to }\left(x,y\right)$ that this pair appears. The count of the transitions of $\left(x,y\right)$ to $\left(x+1,y\right)$, in this example, is made regardless of when these transitions happened during the trajectory. The only condition is that the position $\left(x,y\right)$ must be immediately followed by $\left(x+1,y\right)$. The same counting is made for the transitions from $\left(x,y\right)$ to $\left(x-1,y\right)$ and stored in $S_{\leftarrow }\left(x,y\right)$, from $\left(x,y\right)$ to $\left(x,y+1\right)$ in $S_{\uparrow }\left(x,y\right)$, and from $\left(x,y\right)$ to $\left(x,y-1\right)$ in $S_{\downarrow }\left(x,y\right)$.

With these sums of steps performed, we can calculate the probability of stepping right, left, up, or down. First, note that initially the chance of going to any direction is $P_0$, assuming we know nothing about the trajectories. Now, given that we observed $S_{\to }\left(x,y\right)$ steps going to the right, we need to update the chance of going to the right using the union (i.e. sum) of the observed chance $S_{\to }\left(x,y\right)/n_s$ with $P_0$, where $n_s$ is the total number of steps counted toward any direction in all lattice sites in a given trial:

where $W_{\to }\left(x,y\right)$ is the direction weight of the action ‘step to the right from $\left(x,y\right)$’. Alternatively, this can be written as a weighted sum of memory-driven stepping with probability 1 and random stepping with probability $P_0$:

We employ Equation 2 because the mouse could, in principle, have chosen any of the other three directions. This ensures that every time the memory is increased (adding $S_{\to }/n_s$ to the weight), the random contribution for that step decreases; this justifies the last term in Equation 2, where the intersection between the memory term and the ‘null’ probability, $S_{\to }{P}_0/n_s$, is subtracted from the aforementioned union. One can easily see that the formula guarantees that the null probability is automatically recovered, that is $W_{\to }\left(x,y\right)=P_o$ , when there are zero observed steps in a given direction at position $\left(x,y\right)$, and that $W_{\to }\left(x,y\right)=1$ if all steps are to the right at position $\left(x,y\right)$.

Finally, we use these weights to calculate the probabilities, first defining the total weight of stepping out of $\left(x,y\right)$,

and then calculating the probabilities by

The same is made for the other directions (left, up and down). Equation 4 ensures that the probability of stepping out of $\left(x,y\right)$ is always normalized. This is then applied to Equation 1 to obtain the displacement map.

Let us go through the simple example shown in Figure 4—figure supplement 1C–F. In this case, the accumulated trajectories in the center box located at $\left(x,y\right)=\left(\mathrm{3,3}\right)$ are such that there are two passes going up and one pass going down. Thus, each direction has the following weight [given by Equation 2]: ${\displaystyle W_{\uparrow }\left(3,3\right)=\frac{2}{3}+P_0-\frac{2}{3}{P}_0=\frac{3}{4}}$, $W_{\downarrow }\left(\mathrm{3,3}\right)=\frac{1}{3}+P_0-\frac{1}{3}{P}_0=\frac{1}{2}$, and $W_{\to }\left(\mathrm{3,3}\right)=W_{\leftarrow }\left(\mathrm{3,3}\right)=P_0=\frac{1}{4}$. Applying Equation 4, we obtain the probabilities $P_{\uparrow }\left(\mathrm{3,3}\right)\approx 0.43$, $P_{\downarrow }\left(\mathrm{3,3}\right)\approx 0.29$, and $P_{\to }\left(\mathrm{3,3}\right)=P_{\leftarrow }\left(\mathrm{3,3}\right)\approx 0.14$. This results in a step map vector $\overrightarrow{M}\left(\mathrm{3,3}\right)=\left[\mathrm{0,0.14}\right]$. This vector represents the preferred direction of movement out of site $\left(\mathrm{3,3}\right)$, meaning the mice are likely to pass in the vertical direction, going from bottom to top (hence a positive vertical quantity); and are not picky regarding left or right (hence a null horizontal quantity).

The trajectories of the mice obey box-scaling independently of trial or experimental condition. Box- scaling is a standard method to measure the dimensionality of curves, and is described in Falconer, 2004. In other words, the number of boxes needed to cover any trajectory scales linearly with the lattice dimension $L$; see Figure 4—figure supplement 1F. This means that the choice of $L$ is somewhat arbitrary, and we chose $L=11$ to make each box the size of a few centimeters as expected for the size of a place field from a DG cell (GoodSmith et al., 2017).

The method above gives a single displacement map $\overrightarrow{M}\left(x,y\right)$ and enables the calculation of the TEV $\overrightarrow{D}$ (see below) for each trial of an experiment containing $N$ mice. We have to generate multiple samples for the same experiment using different mice to estimate the significance of the mean displacement direction calculation. We employ a jackknife procedure to achieve this goal. It consists of the ‘leave one out’ rule: this means that a sample of $N$ mice give $N$ unique jackknife samples, each of which containing $N-1$ unique mice. Then, we get $N$ estimates ${\overrightarrow{M}}_k\left(x,y\right)$, with $k$ from 1 to $N$, by applying the previously described procedure to each jackknife sample of $N-1$ mice (instead of the whole sample). Finally, we calculate the average, $\overrightarrow{M}\left(x,y\right)=\frac{1}{N}{\sum }_{k=1}^N{\overrightarrow{M}}_k\left(x,y\right)$ for the displacement map. This is the vector map that we show as small arrows colored from blue to red (Figure 4A, B, E and F; Figure 5C; Figure 7A, C, D and G; Figure 8C; and ; we only show statistically significant averages).

The statistical significance of the average $\overrightarrow{M}\left(x,y\right)$ map in each lattice position $\left(x,y\right)$ is built from first principles as follows (see Figure 4—figure supplement 2). The main feature of each of the vectors in this map is its direction (i.e. the angle it makes with the positive x-axis). Strong directionality means that the vectors ${\overrightarrow{M}}_k\left(x,y\right)$ all pointed roughly in the same direction (i.e. roughly same angles), whereas weak directionality means uniformly distributed angles around the circle (Figure 4—figure supplement 2A and B). The stronger the directionality of the sample ${\overrightarrow{M}}_k\left(x,y\right)$ the more significant the average $\overrightarrow{M}\left(x,y\right)$. A simple measure of the spread of the angles (i.e. the directionality strength) is the standard deviation (S.D.) of the sample angles that ${\overrightarrow{M}}_k\left(x,y\right)$ make with the positive x-axis: stronger directionality implies a smaller S.D. (Figure 4—figure supplement 2C). Thus, the significance ($p$) of directionality is how likely it is for a sample of $N$ uniformly distributed angles to display a given S.D. value collectively.

The S.D. of $N$ angles (with zero average) is $\sigma =\sqrt{\frac{1}{N-1}{\sum }_{k=1}^N{\theta }_k^2}$, where the ${\theta }_k$ are uniformly distributed from –180^o to 180^o. If we could determine the probability density function $\rho \left(\sigma \right)$, then the significance would be given by Hair et al., 2013 $p\left(S.D.\right)={\int }_0^{S.D.}\rho \left(\sigma \right)d\sigma$, which is the probability of observing a given S.D. from the sample. In other words, $p\left(S.D.\right)$ is the probability that an observed S.D. came from a set of $N$ uniformly distributed angles with zero mean. Thus, given an experimental observation S.D. from the jackknife sample, we will know what is the probability $p$ that the angles from the sample were uniformly distributed in the circle (i.e. have weak directionality). A small $p$ value, therefore, indicates strong directionality (since having small S.D. is very unlikely for uniformly distributed angles – Figure 4—figure supplement 2D).

We need to estimate $\rho \left(\sigma \right)$ to be able to calculate $p$ for any S.D. Although it has an integral representation similar to the Chi distribution (Tomé and Oliveira, 2015), it is easier to make a numerical estimation: we fix, for instance, $N=8$ (since we have 8 jackknife samples) and generate 10,000 independent $\sigma$ values applying the S.D. formula above to $N$ independent uniform angles ${\theta }_k$ from –180^o to 180^o. The normalized histogram of all the observed $\sigma$ is a good estimate of $\rho \left(\sigma \right)$ (Figure 4—figure supplement 2D). The $p$ value is obtained by numerically integrating $\rho \left(\sigma \right)$ from 0 to the observed S.D. value (Figure 4—figure supplement 2E).

Now, it suffices to calculate the S.D. of the angles of the vectors ${\overrightarrow{M}}_k\left(x,y\right)$ in the jackknife sample (unbiased to make the average vector $\overrightarrow{M}\left(x,y\right)$ have $\theta =0^{\circ }$). An observed S.D.=30°Corresponds to p=10^–4 (i.e. the sample has very strong directionality). We only show in the figures the displacement vectors that have S.D.≤5^o, yielding vanishing p<10^–7 (all vectors regardless of significance are shown in for comparison for random and static entrance experiments). Comparing to Figure 4, we notice that, as expected, only vectors from trials that are supposed to have random directionality vanish (i.e. trials for random entrance experiments, and trial 1 for static entrance). The vectors in strongly directed flow trials (such as trial 14 in static entrance experiments) remain. The same happens for the two-food location experiment.

The target estimation vector (TEV) is an estimate of the position of the target based only on the observed trajectories and hole checks of the mice. We write the average TEV as $\overrightarrow{D}=\frac{1}{N}{\sum }_{k=1}^N{\overrightarrow{D}}_k$, and ${\overrightarrow{D}}_k=D{\widehat{u}}_k$ is the TEV for a particular jackknife sample $k$. The magnitude $D$ is inferred from the hole checks and the direction ${\widehat{u}}_k$ is calculated from the displacement map ${\overrightarrow{M}}_k\left(x,y\right)$ as follows. The total sum of the displacement map ${\overrightarrow{M}}_k\left(x,y\right)$ over all arena sites $\left(x,y\right)$ gives the estimate of the learned direction ${\widehat{u}}_k={\overrightarrow{u}}_k/\left|{\overrightarrow{u}}_k\right|$, with ${\overrightarrow{u}}_k={\sum }_{\left(x,y\right)}{\overrightarrow{M}}_k\left(x,y\right)$. The magnitude $D$ is the distance from start to the average of the hole check distribution restricted to 20 cm around the target (i.e. $D$ is the distance from the start to the ‘x’ in the green ellipsis in Figure 4G and H; Figure 5C; Figure 7B, F and H; Figure 8C; and Figure 1—figure supplement 1B–F). In the case of the Probe B-A trial, we use $D$ as the distance from B (instead of ‘start’) to the restricted average of the hole check distribution around target A. In trials where no preferred direction is detected (such as in the random entrances experiment, or in the first trial after switching from target A to B), the magnitude $\left|{\overrightarrow{u}}_k\right|$ is already roughly the distance from start to the arena’s center (expressing the randomness in the displacement map), and hence we take the TEV to be just ${\overrightarrow{D}}_k={\overrightarrow{u}}_k$ instead.

The error in the magnitude $D$ of the TEV is obtained from the covariance matrix $C$ of the spatial distribution of hole checks restricted to less than 20 cm of the target. It is given by ${\sigma }_D=\sqrt{{\lambda }_1+{\lambda }_2}$, where ${\lambda }_{\mathrm{1,2}}$ are the eigenvalues of $C$. In the plots [Figures 4, 5, 7 and 8], we simply represent the covariance matrix by the green ellipses in the same way we do for the uncensored distribution (see Active sensing and hole-check detection). The error in the direction of $\overrightarrow{D}$ is simply the S.D. of the angles of ${\overrightarrow{D}}_k$ with respect to the positive x-axis (calculated by shifting $\overrightarrow{D}$ to 0^o, and making the angles of ${\overrightarrow{D}}_k$ between –180^o and 180^o, similarly to what is done for the displacement map). This gives the pink shaded circular sector accompanying the TEV in the figures of the displacement maps and the error bars in the TEV-target deviation plot (Figures 4I and 7I).

• Leonard Maler

• André Longtin
• Leonard Maler
• Jean-Claude Beique

• André Longtin
• Leonard Maler

• André Longtin

• Leonard Maler

• André Longtin
• Leonard Maler
• Jean-Claude Beique