Advertisement

Biological Cybernetics

, Volume 99, Issue 3, pp 197–217 | Cite as

Animal navigation: general properties of directed walks

  • Allen CheungEmail author
  • Shaowu Zhang
  • Christian Stricker
  • Mandyam V. Srinivasan
Original Paper

Abstract

The ability to locomote is a defining characteristic of all animals. Yet, all but the most trivial forms of navigation are poorly understood. Here we report and discuss the analytical results of an in-depth study of a simple navigation problem. In principle, there are two strategies for navigating a straight course. One is to use an external directional reference and to continually reorient with reference to it. The other is to monitor body rotations from internal sensory information only. We showed previously that, at least for simple representations of locomotion, the first strategy will enable an animal or mobile agent to move arbitrarily far away from its starting point, but the second strategy will not do so, even after an infinite number of steps. This paper extends and generalizes the earlier results by demonstrating that these findings are true even when a very general model of locomotion is used. In this general model, error components within individual steps are not independent, and directional errors may be biased. In the absence of a compass, the expected path of a directed walk in general approximates a logarithmic spiral. Some examples are given to illustrate potential applications of the quantitative results derived here. Motivated by the analytical results developed in this work, a nomenclature for directed walks is proposed and discussed. Issues related to path integration in mammals and robots, and measuring the curvature of a noisy path are also addressed using directed walk theory.

Keywords

Navigation Compass Cumulative errors Sensory noise Motor noise Directed walk Elementary step General Simple Bias Path integration Straightness Curvature Correlated random walk Persistent random walk Logarithmic spiral Dead reckoning Vectorial navigation Central limit theorem Positional moments Mean position Expected position Positional variance Axis of intended locomotion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

422_2008_251_MOESM1_ESM.doc (974 kb)
ESM 1 (DOC 974 kb)

References

  1. Atkinson RPD, Rhodes CJ, MacDonald DW, Anderson RM (2002) Scale-free dynamics in the movement patterns of jackals. Oikos 98: 134–140CrossRefGoogle Scholar
  2. Barlow HB, Levick WR (1965) The mechanism of directionally selective units in rabbit’s retina. J Physiol 178: 477–504PubMedGoogle Scholar
  3. Barry C, Hayman R, Burgess N, Jeffery KJ (2007) Experience-dependent rescaling of entorhinal grids. Nat Neurosci 10: 682–684PubMedCrossRefGoogle Scholar
  4. Bartumeus F, Peters F, Pueyo S, Marrasé C, Catalan J (2003) Helical Lévy walks: adjusting searching statistics to resource availability in microzooplankton. Proc Nat Acad Sci USA 100: 12771–12775PubMedCrossRefGoogle Scholar
  5. Batschelet E (1981) Circular statistics in biology. Academic Press, LondonGoogle Scholar
  6. Berg HC (1983) Random walks in biology. Academic Press, New YorkGoogle Scholar
  7. Bovet P, Benhamou S (1988) Spatial analysis of animals’ movements using a correlated random walk model. J Theor Biol 131: 419–433CrossRefGoogle Scholar
  8. Bracher C (2004) Eigenfunction approach to the persistent random walk in two dimensions. Phys A 331: 448–466CrossRefGoogle Scholar
  9. Brown CT, Liebovitch LS, Glendon R (2007) Lévy flights in Dobe Júhoanso foraging patterns. Hum Ecol 35: 129–138CrossRefGoogle Scholar
  10. Brown R (1828) A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Edinburgh New Philos J, pp 358–371Google Scholar
  11. Byers J (2001) Correlated random walk equations of animal dispersal resolved by simulation. Ecology 82(6): 1680–1690CrossRefGoogle Scholar
  12. Cheung A, Zhang SW, Stricker C, Srinivasan MV (2007) Animal navigation: the difficulty of moving in a straight line. Biol Cybern 97: 47–61PubMedCrossRefGoogle Scholar
  13. Claes I, Van den Broeck (1987) Random walk with persistence. J Stat Phys 49: 383–393CrossRefGoogle Scholar
  14. Collett TS, Rees JA (1997) View-based navigation in hymenoptera: multiple strategies of landmark guidance in the approach to a feeder. J Comp Physiol A 181: 47–58CrossRefGoogle Scholar
  15. Dacke M, Nilsson D-E, Scholtz C, Byrne M, Warrant EJ (2003) Animal behaviour: insect orientation to polarized moonlight. Nature 424: 33PubMedCrossRefGoogle Scholar
  16. Dudchenko PA, Bruce C (2005) Navigation without landmarks: can rats use a sense of direction to return to a home site?. Connect Sci 17: 107–125CrossRefGoogle Scholar
  17. Einstein A (1905) On the motion, required by the molecular kinetic theory of heat, of small particles suspended in stationary liquid. Ann Physik 17: 549–560CrossRefGoogle Scholar
  18. Etienne AS, Maurer R, Berlie J, Reverdin B, Rowe T, Georgakopoulos J, Séguinot V (1998) Navigation through vector addition. Nature 396: 161–164PubMedCrossRefGoogle Scholar
  19. Etienne AS, Maurer R, Séguinot V (1996) Path integration in mammals and its interaction with visual landmarks. J Exp Biol 199: 201–209PubMedGoogle Scholar
  20. Grey DR (1989) Persistent random walks may have arbitrarily large tails. Adv Appl Prob 21: 229–230CrossRefGoogle Scholar
  21. Gothard KM, Skaggs WE, McNaughton BL (1996) Dynamics of mismatch correction in the hippocampal ensemble code for space: Interaction between path integration and environmental cues. J Neurosci 16: 8027–8040PubMedGoogle Scholar
  22. Hafting T, Fyhn M, Molden S, Moser M-B, Moser EI (2005) Microstructure of a spatial map in the entorhinal cortex. Nature 436: 801–806PubMedCrossRefGoogle Scholar
  23. Horn BKP, Schunck B (1981) Determining optical flow. Artif Intell 17: 185–203CrossRefGoogle Scholar
  24. Ingemar JC (1991) Blanche—an experiment in guidance and navigation of an autonomous robot vehicle. IEEE Trans Robot Autom 7: 193–204CrossRefGoogle Scholar
  25. Jeffery KJ (2007) Integration of the sensory inputs to place cells: what, where, why, and how?. Hippocampus 17: 775–785PubMedCrossRefGoogle Scholar
  26. Jeffery KJ, Anand RL, Anderson MI (2006) A role for terrain slope in orienting hippocampal place fields. Exp Brain Res 169: 218–225PubMedCrossRefGoogle Scholar
  27. Kareiva PM, Shigesada N (1983) Analyzing insect movement as a correlated random walk. Oecologia 56: 234–238CrossRefGoogle Scholar
  28. Kreyszig E (1993) Advanced engineering mathematics. Wiley, SingaporeGoogle Scholar
  29. Larralde H (1997) Transport properties of a two-dimensional “chiral” persistent random walk. Phys Rev E 56: 5004–5008CrossRefGoogle Scholar
  30. McCulloch CE, Cain ML (1989) Analyzing discrete movement data as a correlated random walk. Ecology 70(2): 383–388CrossRefGoogle Scholar
  31. Malkiel B (1973) A random walk down Wall Street. W W Norton & Company Inc., New YorkGoogle Scholar
  32. Mantegna RN, Stanley HE (1994) Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys Rev Lett 73: 2946–2949PubMedCrossRefGoogle Scholar
  33. Marsh LM, Jones RE (1988) The form and consequences of random walk movement models. J Theor Biol 133: 113–131CrossRefGoogle Scholar
  34. McNaughton BL, Battaglia FP, Jensen O, Moser EI, Moser M-B (2006) Path integration and the neural basis of the “cognitive map”. Nat Rev Neurosci 7: 663–678PubMedCrossRefGoogle Scholar
  35. Milford MJ (2008) Robot navigation from nature. Springer Tracts in Advanced Robotics, vol 41Google Scholar
  36. Milford MJ, Wyeth G, Prasser D (2006) RatSLAM on the edge: revealing a coherent representation from an overloaded rat brain. IEEE/RSJ international conference on intelligent robots and systems, IEEE, pp 4060–4065Google Scholar
  37. Mittelstaedt H, Mittelstaedt ML (1982) Homing by path integration. In: Papi F, Wallraff HG (eds) Avian navigation. Springer, Berlin, pp 290–297Google Scholar
  38. Nossal R, Weiss G (1974) A descriptive theory of cell migration on surfaces. J Theor Biol 47: 103–113PubMedCrossRefGoogle Scholar
  39. O’Keefe J, Burgess N (1996) Geometric determinants of the place fields of hippocampal neurons. Nature 381: 425–428PubMedCrossRefGoogle Scholar
  40. O’Keefe J, Nadel L (1978) The hippocampus as a cognitive map. Clarendon Press, OxfordGoogle Scholar
  41. Quirk GJ, Muller RU, Kubie JL (1990) The firing of hippocampal place cells in the dark depends on the rat’s recent experience. J Neurosci 10: 2008–2017PubMedGoogle Scholar
  42. Pearson K (1905a) The problem of the random walk. Nature 72: 294CrossRefGoogle Scholar
  43. Pearson K (1905b) The problem of the random walk. Nature 72: 342CrossRefGoogle Scholar
  44. Ramos-Fernández G, Mateos JL, Miramontes O, Cocho G, Larralde H, Ayala-Orozco B (2004) Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi). Behav Ecol Sociobiol 55: 223–230CrossRefGoogle Scholar
  45. Rayleigh (1905) The problem of the random walk. Nature 72: 318CrossRefGoogle Scholar
  46. Reichardt W (1961) Autocorrelation, a principle for the evaluation of sensory information by the central nervous system. In: Rosenblith W (eds) Sensory communication. MIT Press/Wiley, Cambridge/New York, pp 465–493Google Scholar
  47. Reynolds AM, Smith AD, Menzel R, Greggers U, Reynolds DR, Riley JR (2007) Displaced honey bees perform optimal scale-free search flights. Ecology 88: 1955–1961PubMedCrossRefGoogle Scholar
  48. Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27: 832–835CrossRefGoogle Scholar
  49. Save E, Nerad L, Poucet B (2000) Contribution of multiple sensory information to place field stability in hippocampal place cells. Hippocampus 10: 64–76PubMedCrossRefGoogle Scholar
  50. Shlesinger MF (1995) Comment on “Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight”. Phys Rev Lett 74: 4959PubMedCrossRefGoogle Scholar
  51. Srinivasan MV (1990) Generalized gradient schemes for the measurement of two-dimensional image motion. Biol Cybern 63: 421–431PubMedCrossRefGoogle Scholar
  52. Srinivasan MV (1994) An image-interpolation technique for the computation of optic flow and egomotion. Biol Cybern 71: 401–415CrossRefGoogle Scholar
  53. Tchen CM (1952) Random flight with multiple partial correlations. J Chem Phys 20: 214–217CrossRefGoogle Scholar
  54. Thrun S (1997) To know or not to know: on the utility of models in mobile robots. AI Mag 18: 47–54Google Scholar
  55. Thrun S (2003) Robotic mapping: a survey. In: Lakemeyer G, Nebel B (eds) Exploring artifical intelligence in the new millenium. Morgan Kaufmann, San Francisco, pp 1–35Google Scholar
  56. Viswanathan GM, Afanasyev V, Buldyrev SV, Murphy EJ, Prince PA, Stanley HE (1996) Lévy flight search patterns of wandering albatrosses. Nature 381: 413–415CrossRefGoogle Scholar
  57. Viswanathan GM, Buldyrev SV, Havlin S, da Luz MGE, Raposo EP, Stanley HE (1999) Optimizing the success of random searches. Nature 401: 911–914PubMedCrossRefGoogle Scholar
  58. Wehner R (1992) Arthropods. In: Papi F (eds) Animal homing. Chapman and Hall, London, pp 45–144Google Scholar
  59. Wehner R (1994) The polarization-vision project: championing organismic biology. Fortschr Zool 39: 103–143Google Scholar
  60. Wiltschko W, Wiltschko R (2005) Magnetic orientation and magnetoreception in birds and other animals. J Comp Physiol A 191(8): 675–693CrossRefGoogle Scholar
  61. Wu H, Li BL, Springer TA, Neill WH (2000) Modelling animal movement as a persistent random walk in two dimensions: expected magnitude of net displacement. Ecol Model 132: 115–124CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Allen Cheung
    • 1
    • 2
    • 3
    Email author
  • Shaowu Zhang
    • 3
  • Christian Stricker
    • 4
  • Mandyam V. Srinivasan
    • 1
    • 2
    • 3
  1. 1.Thinking Systems, Queensland Brain InstituteUniversity of QueenslandBrisbaneAustralia
  2. 2.School of Information Technology and Electrical EngineeringUniversity of QueenslandBrisbaneAustralia
  3. 3.Centre of Excellence in Vision Science, Research School of Biological SciencesAustralian National UniversityCanberraAustralia
  4. 4.The John Curtin School of Medical ResearchAustralian National UniversityCanberraAustralia

Personalised recommendations