The paddle controller
4.3. A N OSCILLATOR FOR THE PADDLE CONTROLLER 89 When referring to the "oscillation generated by the network" I will usually mean D
~
; the"output oscillations of the network"refer to K and .
Oscillators can be classified with respect to the type of oscillation they produce. For many oscillators this classification can be done along an axis of increasing variability of amplitude and decreasing variability of frequency and increasing sluggishness. The relaxation oscillator is found at one end of this scale, and the resonant oscillator on the other end (Wilson and Waldron 1968). Most known biological oscillators are of an intermediate type.
When a simple subtraction is used for the inhibitory synapses, the network is a resonant oscillator because both the elementary oscillators that form the two half-centres of the net- work are resonant. This is easy to prove. A leaky integrator as used in the self-inhibition can be characterised by the fact that a given activity change takes a fixed amount of time. After all, one of the definitions of the time constant of a leaky integrator is the time necessary for a 3 dB (approximately 67%) change in output of the filter. This means that the time for the self-inhibition to reach a certain percentage of the input to the oscillator is independant of the level of input. Therefore the moment when the I interneurone becomes self-inhibited or self-dis-inhibited is determined not by the input, but solely by the parameters of the network. The same applies to the moment when one half-centre yields to the other. In other words the period of the oscillation generated by such a resonant network is constant; only its amplitude varies with the input.
The frequency of the oscillation depends on both
+
??! and the timestep h
of the simulation. The latter dependancy is a result of the mutual exchange of inhibition between
I]
andI , which, being aflip-flop, takes place at a rate proportional to h
.
4.3.2 Feedback control of the dynamics of the oscillator
When the network of figure 4.5 is simulated, it turns out that in order to ensure a stable, non-decaying oscillation rather large time constants are necessary in theTneurones. These large values fory and? result in the low responsiveness typical of resonant oscillators. Thiscan pose a severe problem when fine control of locomotion is required; therefore some form of feedback is necessary to control the dynamics of the pacemaker.
The most straight forward solution is shown in figure 4.6: a gain control mechanism that, based on a copy of/^~
, determines a gainfactor necessary to generate the desired oscillation.
In other words, applying the gainfactor to the generated oscillation should result in an
oscillation with the maximum amplitudeD
~
B
that corresponds to/ ~
.
Such a control mechanism would require a small network that can be tuned (or trained) to represent the expected output amplitude as a function of the input. Also a reading of the currently generated oscillation will be needed. The latter could be accomplished by measuring the position of the paddles, with a receptor like e.g. the wing-hinge stretch receptor possessed by certain moths (Yack and Fullard, 1993). This solution is very easy with respect to measuring the amplitude. However, it requires anexactinternal representation of the desired paddle or wing beat pattern in order to compute an online gainfactor.
It would be much easier just to match themaximum amplitudeof the currently generated oscillation to the maximum amplitude of the desired oscillation. In other words: only the bounds between which the oscillation should take place are important, and not the exact form of the oscillation. Therefore a phase-independant sensory determination of the maximum amplitude is needed, i.e. of the current extreme values of
L
and
L
between which the
90 CHAPTER4. THE PADDLE CONTROLLER Measured amplitude Oscillator Ao* Io Fu Fd Ou Od +1 G τ % amplitude X X Expected Ae Figure 4.6
Simple gain control loop. The output oscillations and£ are amplified with a gainfactor such that the resulting oscillationQ
}
Q
has the correct maximum amplitude ( ~ B
Å ). The%
U
neurone divides the measured maximum amplitude (
~ B
) by the expected amplitude (u ) through shunting inhibition (indicated as}H}k ).W
U
acts as a multiplier.
4.3.2 a Propriocept of the generated oscillation amplitude How can the maximum amplitude be measured in a phase-independant way, while the oscillation (D
~
) is represented explicitly in the network only as its components and
(respectively
L
,
L
)? How can one determine the maximum amplitude an oscillation will
attain from just one sample?
For an oscillation that results from the counteraction (difference) of two antagonistic oscillations L and L the average ©D « of the sum D L t L
equals the maximum
amplitude when both
L
and
L
are identical, in counterphase, and symmetric around their
median (0.5 A). In other words:
© D « 1 { 0 L Ó L 42! when 1 { 0 L 1 { 0 L 05D 43! with L and L
oscillating between 0 andD , so that
L L Ö 0 D + D 2 , and the period of the oscillations.
In all (except perfectly sawtoothed) oscillationsD will fluctuate around the average value ©JD
«
. In oscillations generated by the pacemaker networks described in this chapter these fluctuations are small, and the gain control loop serves to damp them even further. Also
L
and
L
are at least identical and in counterphase. Therefore one can safely take
D ~ B L £ L 44!
Summarising, the input to the gain control system (representing the instantaneous output amplitude of the oscillator) can be taken directly from the two output channels (
L
and
L
)
of the oscillator. The simplest way to do this would be to use an efferent copy of the signals that are sent to the muscles. This would not require any preprocessing to map the signal to
D
~
B
4.3. AN OSCILLATOR FOR THE PADDLE CONTROLLER 91