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  Visual fixation and tracking by flies: Mathematical properties of simple control systems

Poggio, T., & Reichardt, W. (1981). Visual fixation and tracking by flies: Mathematical properties of simple control systems. Biological Cybernetics, 40(2), 101-112. doi:10.1007/BF00344289.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-F0C9-6 Version Permalink: http://hdl.handle.net/21.11116/0000-0006-6C60-5
Genre: Journal Article

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Poggio, T1, 2, Author              
Reichardt, W1, 2, Author              
Affiliations:
1Former Department Information Processing in Insects, Max Planck Institute for Biological Cybernetics, Max Planck Society, ou_1497801              
2Max Planck Institute for Biological Cybernetics, Max Planck Society, ou_1497794              

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 Abstract: The study of the orientation behavior of flies requires the consideration of a few simple control systems for fixation and tracking. In this paper two such control systems are analyzed, in terms of the corresponding difference and differential equations. The first control system corrects with a delay ɛ the angular trajectory proportionally to the error angle ψ; the second control system also corrects proportionally to the error angle ψ but only when the absolute value of ψ is increasing. The differential equations are ψ∘(t)=−α′ψ(t−ε)+A′ ((0)) and ψo(t)=−α′ψ(t−ε)⋅u[ψ(t−ε)ψo(t−ε)]+A′ ((*)) u[ ] being the step function (u[x]=1 if x>0, otherwise u[x]=0). Under suitable restrictions on the parameters it is proved (a) that the difference equations xn+1=xn−αxn+A ((0')) and xn+2=xn+1−αxn+1⋅u[xn+1⋅(xn+1−xn)]+A, ((*')) which can be associated to Eqs. (0) and (*), are “asymptotically equivalent” (for large n) if the time scale is “smoothed” over two time units and (b) that the second equation, with 0<α<2, always converge to a set of oscillating solutions of period 2 for arbitrary initial conditions. Numerical simulations show that the delay-differential equations behave in a similar way. We have also demonstrated with computer simulations that both control systems can satisfactorily predict the 3-D trajectory of a fly chasing another fly. The main biological implications of the analysis are: (1) The two control systems are practically equivalent descriptions of the fly's control of flight on a “coarse” time scale (2 times the fly's delay), consistently with an earlier more general derivation of Eq. (0) (Poggio and Reichardt, 1973). (2) On a fine time scale the second control system is characterized by an asymptotic oscillation with period twice the fly's delay. It is conjectured that a wide range of control systems of the same general type must have a similar oscillatory behavior. Finally, we predict the existence of asymptotic oscillations in the angular trajectory and the torque of tracking flies (if a control system of the second type is involved to a significant extent). Such oscillations should have a basic period of twice the effective reaction delay, and should be best detectable outside the binocular region. Closed loop experiments and the analysis of free flight trajectories may provide critical tests of this prediction.

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 Dates: 1981-03
 Publication Status: Published in print
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 Identifiers: DOI: 10.1007/BF00344289
BibTex Citekey: 1680
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Title: Biological Cybernetics
  Other : Biol. Cybern.
Source Genre: Journal
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Publ. Info: Berlin : Springer
Pages: - Volume / Issue: 40 (2) Sequence Number: - Start / End Page: 101 - 112 Identifier: ISSN: 0340-1200
CoNE: https://pure.mpg.de/cone/journals/resource/954927549307