(wetware) limitations that demand nonlinear constituents of our model studies. V. The components of the complete locomotion system (i.e. the CPG and the propulsion
system) should be tuned to each other in such a way that the system is able to respond adequately to any changes in the motor commands that can normally occur. In other words, supposing the appendage that generates the thrust is adequate for its task, the CPG should be tuneable to drive the appendage so as to ensure that the generated thrust follows changes in the motor command as well as possible. This is where sensory feedback providing information about the state of the appendage comes into play. Using this information the behaviour of the CPG can be modulated to ensure that its output is appropriate to achieve the current goal as efficiently as possible, given the inputandthe present state (position, load and other pecularities) of the appendage.
4.1.4 Overview of this chapter
After the above introduction to the subject (4.1.2) and inventory of requirements and bound- ary conditions for the paddle controller (4.1.3) it is useful to briefly review some of the findings on central pattern generators and the oscillators that underly these CPGs. This is the topic of section 4.2.A possible model for a paddle controller will be introduced in section 4.3. It generates two alternating outputs that innervate the paddle levator and depressor: one antagonistic pair of muscles. The controller to be presented in this section is based on a description (Wilson and Waldron 1968) of the locust flight system. The latter is responsible for wing control during flight, and it is well studied, both empirically and theoretically. The choice is motivated by the strong similarity between paddling and flapping wings, and by the fact that the system can generate smooth oscillations. The output of the paddle controller is then used as innervation to the levator and depressor muscles of the paddles. The details of the relation between muscle innervation to muscle force will not be included in the model: they are not of prime interest at this stage. Instead a linear relation will be assumed to be an acceptable first-order approximation.
It turns out that it is relatively simple to construct a working, oscillating, network. This network consists of two identical half-centres as described by Wilson and Waldron (1968) that are oscillatory due to a self-inhibitory delayed feedback. A reciprocal inhibition between the two half-centres ensures that they generate identical oscillations that are in counterphase (section 4.3). Therefore the difference between the oscillations, a measure for the net force excerted by the antagonistic muscle-pair, is a smooth oscillation of twice the amplitude and half the frequency of the oscillations. Its amplitude turns out to vary linearly with the input, and its frequency depends only on the parameters of the network. The network is a resonant oscillator (section 4.3.1).
Although the oscillator generates a smooth oscillation with a simple correspondance between input and amplitude, it cannot be used in this form as a paddle controller. The output of the controller becomes a stable, non-decaying, oscillation only for large time constants in the delayed feedback. This also makes the controller slow in its response to changes in input. For this reason a control circuit is added in section 4.3.2. This circuit measures the current net force, compares it to the expected net force, and sets the gain of the muscle innervation (the output of the controller) accordingly. Both perception and gain
82 CHAPTER4. THE PADDLE CONTROLLER
regulation of this gain control loop might be accomplished through spindles or analogous structures in the muscles.
When the gain in both innervations is scaled by the same amount, excessive forces might be excerted on the paddle by the two antagonistic muscles, and this might still only yield a slightly higher net force. To prevent this situation, the control circuit reduces the gain in the muscle that excerts the least force (section 4.3.2 c).
The gain control circuit works well for a large range of input changes, but fails when there are large transients or steps in the input (section 4.3.2 e). When these arise, the network enters a state of identical tonic activity in the two half-centres, with only small oscillations superimposed. Still, the net force resulting from this output pattern has the expected size. Although it is modulated to a far too small extent, it leaves the gain control circuit inactive.
This unwanted phenomenon is known to arise in natural CPGs too (see section 4.3.5). A first attempt (section 4.3.3) to tackle it was based on the assumption that the tonic activity results from memory in the delayed self-inhibition. In that case the identical toniccomponent
in the two motor neurone outputs might be changed into an identical tonicfactorby changing the nature of the self-inhibition. In the so-called type-1 oscillator a subtractive self-inhibition was used: the type-2 oscillator will use a shunting self-inhibition. Since a shunting inhibition performs what is essentially a division by the inhibiting signal, an identical component could become an identical factor, which can then be corrected for by the gain control system.
Use of a shunting self-inhibition necessitates an additional mutual inhibition between the two half-centres in order to have non-decaying oscillations. Due to these changes the simple relation between input and output becomes considerably more complex; both amplitude and frequency of the oscillation now vary with varying input. This is in better accordance with biological CPGs (section 4.3.4).
Unfortunately these changes did not prove to be sufficient to prevent tonic output pat- terns. Closer scrutiny of the problems showed that tonic activity results mainly when the phase of the oscillation is reset by a step or transient in the input. Also, the gain control system registers a net force that is temporarily too large. Therefore an additional inhibitory connection from the gain control system to the delayed feedback is tested in section 4.3.5. Such an inhibitory connection performs both a reset of the memory in delayed self-inhibition, as well as a temporary frequency shift.
With this connection, tonic activity no longer impairs the functioning of the CPG. Also a step or transient in the input no longer results in a phase reset; instead the output is smoothly corrected to its new desired value. In this form the CPG will be used as a paddle controller. To be of any use for locomotion with paddles, an additional model circuit is needed that specifies the amount of thrust generated by a certain amount of paddling. A solution to the problem of converting an oscillation into a corresponding thrust will be presented in section 4.4.1. The resulting mapping from motor command to thrust generated by a paddle driven by the paddle controller turns out to agree surprisingly well with the initial model outlined above (section 4.1.1). Transient dynamics of the system which cause temporary deviations from the steady state thrust are actively supressed by the feedback gain control loop. Although the system itself consists of non-linear components, the mapping from motor command to generated thrust is almost linear. As was assumed in the initial model, this linearity arises from an appropriate co-variation of paddle frequency and amplitude with the motor command.
The similarity in the mapping from motor command to thrust between the linear model and the type-2 paddle controller leads to the prediction that paddlers equipped with the two models will show similar behaviour. In vivo experiments show that this is indeed the case; section 4.4.2.