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

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

$$\mathop \psi \limits^ \circ (t) = - \alpha '\psi (t - \varepsilon ) + A'$$

((0))

and

$$\mathop \psi \limits^o (t) = - \alpha '\psi (t - \varepsilon ) \cdot u[\psi (t - \varepsilon )\mathop \psi \limits^o (t - \varepsilon )] + A'$$

((*))

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

$$x_{n + 1} = x_n - \alpha x_n + A$$

((0'))

and

$$x_{n + 2} = x_{n + 1} - \alpha x_{n + 1} \cdot u[x_{n + 1} \cdot (x_{n + 1} - x_n )] + A,$$

((*'))

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.