Mathematical and Computational Theory of Animal Navigation

Allen Cheung

Summary: Navigation is an umbrella term which encompasses a wide range of sensorimotor and information processing tasks, many of which are still poorly defined or understood. A primary goal of theme 2 is to develop a theoretical understanding of the underlying principles common to all navigating agents, animal or robot. To achieve this goal it has been necessary to dissect navigational tasks into their most basic components, to understand the information required to solve those tasks, to use those mechanistic building blocks to generate complex biomimetic behaviours, and to see whether sophisticated methods can pick the correct building blocks from observed behaviour.

Schematic representation of information flow

A. Schematic representation of information flow during social insect foraging. In this scheme, navigation (medium range) directly controls locomotion. In turn, higher level commands and strategies direct the navigation system.

B. Graphical example of an unknown mixture of simulated (unbiased) idiothetic and allothetic directed walks (IDWs and ADWs) with varying magnitudes of random angular displacement errors, and unknown axes of intended locomotion. In this example, the decision algorithm was correct 28 out of 30 times (ADWs = solid lines, IDWs = dotted lines), being unable to decide in the remaining two (dashed lines).

C. Computer simulated PI (path integration) homing and searching trajectory expressed in egocentric coordinates, showing damped oscillatory behaviour and chaotic point attractors.

D. A schematic illustration of the mapping of a straight trajectory in real space to a neural representational space. This is an example of an allocentric static vectorial PI system, the only noise-tolerant class of PI systems.


  • Cheung, A. (2010) The fourth moment of the radial displacement of a discrete correlated/persistent random walk, Journal of Theoretical Biology. 264: 641-644. (PDF File 408 KB)
  • Cheung, A., Vickerstaff, R. (2010) Finding the way with a noisy brain, PLoS Computational Biology (in press). (PDF File 1,592 KB)
  • Cheung, A. (2009) Mathematical and neural network models of medium range navigation during social insect foraging, In: Jarau and Hrncir (eds.) Food Exploitation by Social Insects: Ecological, Behavioral, and Theoretical Approaches. Taylor & Francis Group LLC.
  • Cheung, A., Stürzl, W., Zeil, J., Cheng, K. (2008) The information content of panoramic images II: View-based navigation in nonrectangular experimental arenas, Journal of Experimental Psychology: Animal Behaviour Processes. 34(1): 15-30. (PDF File 1,362 KB)
  • Cheung, A. (2008) From behaviour to brain dynamics, In: Marinaro M., Scarpetta S. and Yamaguchi Y. (Eds) Dynamic Brain – from Neural Spikes to Behaviors, Lecture Notes in Computer Science, vol 5286, pp91-95. Springer Berlin / Heidelberg. (PDF File 341 KB)
  • Cheung, A., Zhang, S.W., Stricker, C., Srinivasan, M.V. (2008) Animal navigation: General characteristics of directed walks, Biological Cybernetics. 99: 197-217. (PDF File 610 KB)
  • Garratt, M., Cheung, A. (2009) Obstacle avoidance in cluttered environments using optic flow, Australasian Conference on Robotics and Automation (ACRA).  (PDF File 1,719 KB)
  • Luu, T.L., Cheung, A., Ball, D., Srinivasan, M.V. (2011) Honeybee flight: A novel ‘streamlining’ response, Journal of Experimental Biology (Accepted). 
  • Stürzl, W., Cheung, A., Cheng, K., Zeil, J. (2008) The information content of panoramic images I: The rotational errors and the similarity of views in rectangular experimental arenas, Journal of Experimental Psychology: Animal Behaviour Processes. 34(1): 1-14. (PDF file 2,596 KB)
  • Vickerstaff, R., Cheung, A. (2010) Which coordinate system for modelling path integration, Journal of Theoretical Biology. 263: 242-261.   (PDF File 619 KB)

Faculty of 1000 post-publication peer reviews by: Dyer A, Rosa M (27 Jun 2011), Giurfa M (28 Jul 2011), Avargues-Weber A, Chittka L (29 Jul 2011): 2011.

This work was also highlighted by The Scientist as the Editor’s Choice in Neuroscience (see media interest).

Ongoing Collaborative Research

Luu T, Cheung A, Ball D, Srinivasan MV. Using the virtual reality setup of Luu et al (2011), the abdominal pitch angle of tethered flying honeybees is measured under different conditions to probe visually-driven flight behaviour. (Manuscript in preparation. Provisional title: “Factors affecting the visually driven abdominal pitch response in tethered flying honeybees”).

Stratton P, Cheung A, Wiles J, Kiyatkin E, Sah P, Windels F. In vivo extracellular action potential waveforms from brains of awake and anaesthetized rats are analysed and compared. Waveform variability is quantified to determine the likely error rate when standard spike classification techniques are used. (Manuscript in preparation. Provisional title: “Action potential waveform variability limits multi-unit separation in freely behaving rats”).

Cheung A, Ball D, Milford M, Wyeth G, Wiles J. Mathematical derivations, computer simulations and robot experiments are used to examine the neurocomputational implications of a stable representation of place but an unstable head direction system, in rodents navigating in darkness. (Manuscript in preparation. Provisional title: “Maintaining a cognitive map in darkness: the need to fuse boundary knowledge with path integration”).

Related Activities: Conferences Attended, Workshops Attended, Invited speaker

Invited Presentations

Cheung, A. (2011) Animal spatial navigation: need for a “cognitive map”? The 1st QBI-MCN Symposium on Systems Neuroscience. Brisbane, Australia (29-30 Sep 2011).
Cheung, A. (2011) Uncertainty of places and paths during navigation. Workshop on Insect Homing: Mechanisms and Models. Bielefeld, Germany (31 Aug – 2 Sep 2011), funded by the Centre for Interdisciplinary Research (ZiF) in Bielefeld, Germany.
Cheung, A. (2010) Limits of animal navigation: a theoretical perspective. The 9th International Congress of Neuroethology, Salamanca, Spain (2-7 Aug 2010), travel grant funded by the German Research Foundation.

Oral presentations

Cheung, A. (2011) Knowing thy place. Thinking Systems Frontiers 2011: Intelligent Machines, Robots, Human-Computer Interaction & the Science-Arts Nexus, Sydney, Australia (8-9 Dec 2011).
Cheung, A. (2010) Mathematical and computational theory of animal navigation. Thinking Systems Symposium, Brisbane, Australia (22 Nov 2010).
Cheung, A. (2009) Expected radial displacement of random and directed walks. ANZIAM: 45th Applied Mathematics Conference, Caloundra, Australia (1-5 Feb 2009).
Cheung, A. (2007) From behaviour to brain dynamics. The 12th International Summer School on Neural Networks, Erice, Italy.

Poster presentations

Luu, T., Cheung, A., Ball, D. and Srinivasan, M.V. (2011) Flying is believing: Honeybee flight in a virtual arena. Thinking Systems Frontiers 2011: Intelligent Machines, Robots, Human-Computer Interaction & the Science-Arts Nexus, Sydney, Australia (8-9 Dec 2011).
Luu, T., Cheung, A., Ball, D. and Srinivasan, M.V. (2010) Honeybee flight: A novel ‘streamlining’ response. The 9th International Congress of Neuroethology, Salamanca, Spain.
Cheung, A. et al. (2009) Learning to navigate. Thinking Systems Joint Symposium, Brisbane, Australia (17-18 Nov 2009).


Invited speaker at the 7th ACEVS-CVS Summer School on Animal Navigation (1st-5th Dec 2008).
Main organizer of the 8th ACEVS-CVS Summer School on Animal Navigation (23rd-27th Nov 2009), held for the first time at UQ.

List of Media Interest

Review of Luu et al. (2011) by: Grant (2011) Flight of the honeybee. The Scientist (December 1, 2011). Editor’s choice in neuroscience.

Comment in ABC Science about new article:
Lent, D., Graham, P. and Collett, T. (2010) Image-matching during ant navigation occurs through saccade-like body turns controlled by learned visual features. Proceedings of the National Academy of Science USA. 107(37): 16348-16353.

Comment in New Scientist about new article:
Souman, J., Frissen, I., Sreenivasa, M. and Ernst, M. (2009) Walking straight into circles. Current Biology. 19: 1-5.

Supervision of undergraduate, summer or honours students related to TS

Benjamin Sinclair, TS Summer Scholar: “Deconstructing a squiggle”.
Kieran McLean, TS Winter Scholar: “Boxed in”.
Nathan Hearn, TS Summer Scholar: “Using a boundary neural code for navigation in the dark”.

Where to Next?

I have taken up a junior faculty position at the Queensland Brain Institute (from 2011 to 2016), to continue and build on my research into the neurocomputational theory of animal spatial navigation. The aim of this research program is to extend the foundation theory on path integration and spatial representations, in order to develop a theoretically rigorous understanding of multimodal interactions, information acquisition and processing, by neurons and neural circuits during spatial navigation.