# Sensory Systems/Insects/Halteres
## Halteres (Gyroscopes for Flies)
### Introduction

Halteres are sensory organs present in many flying insects. Widely thought to be an evolutionary modifcation of the rear pair of wings on such insects, halteres provide gyroscopic sensory data, vitally important for flight. Although the fly has other relevant systems to aid in flight, the visual system of the fly is too slow to allow for rapid maneuvers. Additionally, to be able to fly adeptly in low light conditions, a requirement to avoid predation, such a sensory system is necessary. Indeed, without halteres, flies are incapable of sustained, controlled flight. Since the 18th century, scientists have been aware of the role halteres play in flight, but it was only recently that the mechanisms by which they operate have been better explored.

### Anatomy

The haltere evolved from the rearmost of two pairs of wings. While the first has maintained its usage for flight, the posterior pair has lost its flight functions and has adopted a slightly different shape. The haltere is visually comprised of three structural components: a knob-shaped end, a thin shaft, and a slightly wider base. The knob contains approximately 13 innervated hairs, while the base contains two chordotonal organs, each innervated by about 20-30 nerves. Chordotonal organs are sense organs thought to be solely responsive to extension, though they remain relatively unknown. The base is also covered by around 340 campaniform sensilla, which are small fibers which respond preferentially to compression in the direction in which they are elongated. Each of these fibers is also innervated. Relative to the stalk of the haltere, both the chordotonal organs and the campaniform sensilla have an orientation of approximately 45 degrees, which is optimal for measuring bending forces on the haltere. The halteres move contrary (anti-phase) to the wings during flight. The sensory components can be categorized into three groups): those sensitive to vertical oscillations of the haltere, including the dorsal and ventral scapal plates, dorsal and ventral Hicks papillae (both the plates and papillae are subcategories of the aforementioned campaniform sensilla), and the small chordotonal organ. The basal plate (another manifestation of the sensilla) and the large chordotonal organ are sensitive to gyroscopic torque acting on the haltere, and there is also a population of undifferentiated papillae which are responsive to all strains acting on the base of the haltere. This provides an additional method for flies to distinguish between the direction of force being applied to the haltere.

#### Genetics

As Homeobox genes were being discovered and explored for the first time, it was found that the deletion or inactivation of the Hox gene Ultrabithorax (Ubx) causes the halteres to develop into a normal pair of wings. This was a very compelling early result as to the nature of Hox genes. Manipulations to the Antennapedia gene can similarly cause legs to become severely deformed, or can cause a set of legs to develop instead of antennae on the head.

### Function

The halteres function by detecting Coriolis forces, sensing the movement of air across the potentially rotating fly body. Studies have indicated that the angular velocity of the body is encoded by the Coriolis forces measured by the halteres. Active halteres can recruit any neighboring units, influencing nearby muscles and causing dramatic changes in the flight dynamics. Halteres have been shown to have extremely fast response times, allowing these flight changes to be performed much more quickly than if the fly were to rely on its visual system. In order to distinguish between different rotational components, such as pitch and roll, the fly must be able to combine signals from the two halteres, which must not be coincident (coincident signals would diminish the ability of the fly to differentiate the rotational axes). The halteres are capable of contributing to image stabilization, as well as in-flight attitude control, which was established by numerous authors noting a reaction from the head and wings to inputs from the components of the rotation rate vector. contributions from halteres to head and neck movements have been noted, explaining their role in gaze stabilization. The fly therefore uses input from the halteres to establish where to fixate its gaze, an interesting integration of the two senses.

### Mathematics

Recordings have indicated that halteres are capable of responding to stimuli at the same (double-wingbeat) frequency as Coriolis forces, the proof of concept that allows further mathematical analysis of how these measurements can occur. The vector cross-product of the halteres' angular velocity and the rotation of the body provide the Coriolis force vector to the fly. This force is at the same frequency as the wingbeat in both the pitch and roll planes, and is doubly fast in the yaw plane. Halteres are capable of providing a rate damping signal to affect rotations. This is because the Coriolis force is proportional to the fly's own rotation rate. By measuring the Coriolis force, the halteres can send an appropriate signal to their affiliated muscles, allowing the fly to properly control its flight. The large amplitude of haltere motion allows for the calculation of the vertical and horizontal rates of rotation. Because of the large disparity in haltere movement between vertical and horizontal movement, Ω$_1$, the vertical component of the rotation rate, generates a force of double the frequency of the horizontal component. It is widely thought that this twofold frequency difference is what allows the fly to distinguish between the vertical and horizontal components. If we assume that the haltere moves sinusoidally, a reasonably accurate approximation of its real-world behavior, the angular position γ can be modeled as: $\gamma ={\frac {\pi }{2}}\sin(\omega t)$ where ω is the haltere beat frequency, and the amplitude is 180, a close approximation to the real life range of motion. The body rotational velocities can be computed, given the known rates (the roll, pitch, and yaw components are labeled below with 1, 2, and 3, respectively) from the two halteres' (Ω$_b$ being the left and Ω$_c$ being the right haltere) reference frames, respective to the body of the fly with the following calculations:
$W_{1}=-{\frac {\Omega _{b3}+\Omega _{c3}}{2\sin(\alpha )}}$
$W_{2}={\frac {\Omega _{b3}-\Omega _{c3}}{2\cos(\alpha )}}$
$W_{3}=-{\frac {\Omega _{b1}+\Omega _{c1}}{2}}$
α represents the haltere angle of rotation from the body plane, and the Ω terms are, as mentioned, the angular velocity of the haltere with respect to the body. Knowing this, one could roughly simulate input to the halteres using the equation for forces on the end knob of a haltere:
$F=mg-ma_{i}-ma_{F}-m{\dot {\Omega _{i}}}\times r_{i}-m\Omega _{i}\times (\Omega _{i}\times r_{i})-2m\Omega _{i}\times v_{i}$
m is the mass of the knob of the haltere, g is the acceleration due to gravity, r$_i$, v$_i$,} and a$_i$ are the position, velocity, and acceleration of the knob relative to the body of the fly in the i direction, a$_F$ is the fly's linear acceleration, and Ω$_i$ and Ώ$_i$ are the angular velocity and acceleration components for the direction i, respectively, of the fly in space. The Coriolis force is simulated by the 2mΩ × v$_i$ term. Because the sensory signal generated is proportional to the forces exerted on the halteres, this would allow the haltere signal to be simulated. If attempting to reconcile the force equation with the rotational component equations, it is worthwhile to remember that the force equation must be calculated separately for both halteres.

### References

* ↑ J. L. Fox and T. L. Daniel (2008), "A neural basis for gyroscopic force measurement in the halteres of Holorusia.", J Comp Physiol, 194: 887–897
* 1 2 Rhoe A. Thompson (2009), "Haltere Mediated Flight Stabilization in Diptera: Rate Decoupling, Sensory Encoding, and Control Realization.", PhD thesis, University of Florida
* 1 2 J. W. S. Pringle (1948), "The gyroscopic mechanism of the halteres of diptera.", Phil Trans R Soc Lond B, 233 (602): 347–384