Mathematical analysis of the navigational process in homing pigeons

Animal navigation: a noisy magnetic sense?

Diverse organisms use Earth's magnetic field as a cue in orientation and navigation. Nevertheless, eliciting magnetic orientation responses reliably, either in laboratory or natural settings, is often difficult. Many species appear to preferentially exploit non-magnetic cues if they are available, suggesting that the magnetic sense often serves as a redundant or 'backup' source of information. This raises an interesting paradox: Earth's magnetic field appears to be more pervasive and reliable than almost any other navigational cue. Why then do animals not rely almost exclusively on the geomagnetic field, while ignoring or downplaying other cues? Here, we explore a possible explanation: that the magnetic sense of animals is 'noisy', in that the magnetic signal is small relative to thermal and receptor noise. Magnetic receptors are thus unable to instantaneously acquire magnetic information that is highly precise or accurate. We speculate that extensive time-averaging and/or other higher-order neural processing of magnetic information is required, rendering the magnetic sense inefficient relative to alternative cues that can be detected faster and with less effort. This interpretation is consistent with experimental results suggesting a long time course for magnetic compass and map responses in some animals. Despite possible limitations, magnetoreception may be maintained by natural selection because the geomagnetic field is sometimes the only source of directional and/or positional information available.

Cognitive map-based navigation in wild bats revealed by a new high-throughput tracking system

Seven decades of research on the "cognitive map," the allocentric representation of space, have yielded key neurobiological insights, yet field evidence from free-ranging wild animals is still lacking. Using a system capable of tracking dozens of animals simultaneously at high accuracy and resolution, we assembled a large dataset of 172 foraging Egyptian fruit bats comprising >18 million localizations collected over 3449 bat-nights across 4 years. Detailed track analysis, combined with translocation experiments and exhaustive mapping of fruit trees, revealed that wild bats seldom exhibit random search but instead repeatedly forage in goal-directed, long, and straight flights that include frequent shortcuts. Alternative, non-map-based strategies were ruled out by simulations, time-lag embedding, and other trajectory analyses. Our results are consistent with expectations from cognitive map-like navigation and support previous neurobiological evidence from captive bats.

Behavioural traits of individual homing pigeons, Columba livia f. domestica, in their homing flights

Homing tracks of two groups of pigeons, Columba livia f. domestica, were analyzed in view of difference between individual birds and correlations between characteristic variables, looking at the initial phase while the pigeons were still at the release site, and the homing phase separately. Individual birds differed significantly in their flying speed during the initial phase, and one pigeon tended to stay longer at the release site than the others. There were no significant differences in steadiness and efficiency, indicating that all pigeons homed equally well. Differences in correlation dimension, a variable reflecting the complexity of the navigational process, reflect differences in the use of navigational information, with one bird apparently using less complex information than others. The flying speed during the initial phase was positively correlated with the flying speed during the homing phase. During the homing phase, the steadiness of flight and the efficiency of homing were closely correlated, and both tended to be positively correlated with the correlation dimension, suggesting that birds that use more complex navigational information home more efficiently.

Mathematical analysis of the homing flights of pigeons based on GPS tracks

To analyse the effect of magnetic and olfactory deprivation on the homing flight of pigeons, we released birds from a familiar site with either their upper beak or their nostrils anaesthetized. The tracks were analysed by time lag embedding to calculate the short-term correlation dimension, a variable that reflects the degrees of freedom and thus the number of factors involved in a system. We found that higher natural fluctuations in the earth's magnetic field characterized by A _P-indices of 8 and above caused a reduction of the correlation dimension of the control birds. We thus separated the data into two groups according to whether they were recorded on magnetically quiet days or on days with higher magnetic fluctuations. Anaesthetizing the upper beak had no significant effect. Making pigeons anosmic reduced the correlation dimension on magnetically quiet days, but did not cause any reduction on days with higher fluctuations. Altogether, our data suggest an involvement of magnetic cues and olfactory factors during the homing flight and point to a robust, multi-factorial map.

Novel Insights into the Map Stage of True Navigation in Nonmigratory Wild Birds (Stone Curlews, Burhinus oedicnemus)

In the map-and-compass model of true navigation, animals at unfamiliar sites determine their position relative to a destination site (the map stage) before progressing toward it (the compass stage). A major challenge in animal navigation research is to understand the still cryptic map stage in general and the map stage for free-ranging wild animals in particular. To address this challenge, we experimentally translocated wild, nonmigratory birds (stone curlews [Burhinus oedicnemus]) far from their nests and GPS-tracked their subsequent movements at high resolution and for long durations. Homing success was high and cannot be explained by random chance or landmark navigation, implying true navigation. Although highly motivated to return home, the homing trajectories of translocated birds exhibited a distinct, two-phase pattern resembling the map and compass stages: a long, tortuous wandering phase without consistent approach home, followed by a short and direct return phase. Birds retranslocated to the same site initially repeated the original wandering path but switched to the return phase earlier and after covering a smaller area; they returned home via a different path but with similar movement properties. We thus propose the map learning hypothesis, asserting that birds resolve the map by acquiring, potentially through learning, the relevant navigation cues during the wandering phase.

Pigeon navigation: different routes lead to Frankfurt

Background

Tracks of pigeons homing to the Frankfurt loft revealed an odd phenomenon: whereas birds returning from the North approach their loft more or less directly in a broad front, pigeons returning from the South choose, from 25 km from home onward, either of two corridors, a direct one and one with a considerable detour to the West. This implies differences in the navigational process.

Methodology/Principle Findings

Pigeons released at sites at the beginning of the westerly corridor and in this corridor behave just like pigeons returning from farther south, deviating to the west before turning towards their loft. Birds released at sites within the straight corridors, in contrast, take more or less straight routes. The analysis of the short-term correlation dimension, a quantity reflecting the complexity of the system and with it, the number of factors involved in the navigational process, reveals that it is significantly larger in pigeons choosing the westerly corridor than in the birds flying straight - 3.03 vs. 2.85. The difference is small, however, suggesting a different interpretation of the same factors, with some birds apparently preferring particular factors over others.

Conclusions

The specific regional distribution of the factors which pigeons use to determine their home course seems to provide ambiguous information in the area 25 km south of the loft, resulting in the two corridors. Pigeons appear to navigate by deriving their routes directly from the locally available navigational factors which they interpret in an individual way. The fractal nature of the correlation dimensions indicates that the navigation process of pigeons is chaotic-deterministic; published tracks of migratory birds suggest that this may apply to avian navigation in general.

A model for navigational errors in complex environmental fields

Many animals are believed to navigate using environmental signals such as light, sound, odours and magnetic fields. However, animals rarely navigate directly to their target location, but instead make a series of navigational errors which are corrected during transit. In previous work, we introduced a model showing that differences between an animal׳s 'cognitive map' of the environmental signals used for navigation and the true nature of these signals caused a systematic pattern in orientation errors when navigation begins. The model successfully predicted the pattern of errors seen in previously collected data from homing pigeons, but underestimated the amplitude of the errors. In this paper, we extend our previous model to include more complicated distortions of the contour lines of the environmental signals. Specifically, we consider the occurrence of critical points in the fields describing the signals. We consider three scenarios and compute orientation errors as parameters are varied in each case. We show that the occurrence of critical points can be associated with large variations in initial orientation errors over a small geographic area. We discuss the implications that these results have on predicting how animals will behave when encountering complex distortions in any environmental signals they use to navigate.

Following the sun: a mathematical analysis of the tracks of clock-shifted homing pigeons

We analysed the tracks of clock-shifted pigeons from six releases to determine how they cope with the conflict between their sun compass and other navigational cues. Time-lag embedding was used to calculate the short-term correlation dimension, a parameter that reflects the complexity of the navigational system, and with it, the number of factors involved. Initially, while pigeons were still at the release site, the short-term correlation dimension was low; it increased as the birds left the site, indicating that the birds were now actively navigating. Clock-shifted pigeons showed more scatter than the control birds, and their short-term correlation dimension became significantly smaller than that of the controls, remaining lower until the experimental birds reached their loft. This difference was small, but consistent, and suggests a different rating and ranking of the navigational cues. Clock-shifted pigeons do not seem to simply ignore the information from their manipulated sun compass altogether, but appear to merely downgrade it in favour of other cues, like their magnetic compass. This is supported by the observation that the final part of the tracks still showed a small deviation in the expected direction, indicating an effect of clock-shifting until the end of the homing flight.

Development of the navigational system in homing pigeons: increase in complexity of the navigational map

In the present study we analysed GPS-recorded tracks from pigeons of different ages from 11 sites between 3.6 and 22.1 km from the home loft, which revealed changes in the navigational system as the birds grew older and became more experienced. The efficiency of juveniles in their first year of life, at only 0.27, was rather low, indicating that the young birds covered more than three times the direct distance home. In the second year, after a standard training programme, the efficiency of the same birds increased to 0.80 and was no longer different from that of older pigeons. The short-term correlation dimension, a variable that reflects the number of factors involved in the navigational process, also increased with age. In juveniles, it was markedly lower than in the other two groups, but even in yearlings it was still significantly lower than that of old pigeons, indicating that the navigational map of yearlings is still developing. Our results indicate that the map system, although functional in the first year of life, continues to become more complex - older pigeons seem to either consider more navigational factors than younger ones or at least weigh the same factors differently.