The paddle controller
4.3. A N OSCILLATOR FOR THE PADDLE CONTROLLER 91 Alternatively D
~
B
might be measured by spindle-like proprioceptors in the paddling muscles (cf. thec -system in vertebrate muscles). The input would then represent a measure
of the force excerted on the paddle. Although some processing is now required to map the sensory information toD
~
B
, this option has the advantage that it can detect and compensate for disturbances of the paddle movement. Another advantage is the fact that the same system can take care of the gain regulation (cf. the -system in vertebrate muscles).
4.3.2 b Propriocept of the expected oscillation amplitude The expected maximum oscillation amplitude is defined as the maximum oscillation ampli- tude that the oscillator will attain in the limit: D
lim o D ~ B 3
. The mapping of input/ ~
to expected amplitude D
, which must be represented in the gain control loop, can then be
determined by measuring the averageD
~
B
(D
~
B
) for a range of combinations of inputs (/\~
) and relevant oscillator parameters. It has been shown above that the time constants of the self-inhibition affect only the period of the oscillation. This leaves/~
and"Õ as independant
variables. A curve-fit can then be done to find a function that performs the correct mapping
/^~ + "Ó! $% D .
Simulations showed that a practical observable is the fraction /\~ º D
~
B
. This fraction is independant of /^~
, which proves that the oscillator is indeed truly resonant (at least over the ranges of /^~
and "Õ that were used
4 ). Statistical analysis of a (" , /~ º D ~ B ) dataset for z ? 10 indicates that D ~ B
(and thus D ) can be approximated by:
D D ~ B ¯ /~ 169 10 l 4 " 2 0485" 1 0963 45!
It can be seen that for fixed" a neural network that determines D
from /~
can consist of just one neurone receiving a copy of/\~
; training it is as simple as setting its gain. Summarising, we assume the following set of relations:
@ L L D @
D Elementary oscillation times a gain factor
D ~ L Ü L instantaneous amplitude D ~ B L £ L £¯ max 0 D ~ 2
instantaneous maximum amplitude
D 0/^~
2
å
/~
expected maximum amplitude
46!
A feedback control loop that continuously regulates the gain of the generated oscillation so that its maximum amplitude approximates the expected value can be determined by using a signal of the form:
E 8 ³y + D 1 D ~ B 1 ¶ 47! could for example be a signal that modulates the response of a muscle to excitation,
or sets the gain of a motor neurone. Both nominator and denominator have an additional component of 1 to prevent divisions by zero as well as zero values. Zero values arising
3So:
v is themeasuredmaximum oscillation amplitude; is the value it is expected to have
4
92 CHAPTER4. THE PADDLE CONTROLLER
when D is zero cause the leaky integrator to discharge: this results in erroneous (too low) values afterD
rebound.
An advantage of this type of gain control, which essentially compares the envelope of the output (i.e. the extremes between which it oscillates) with the expected envelope is that it does not depend on signals that should follow the oscillation. It can therefore be used for a wide range of frequencies of oscillation. Systems that do depend on information that follows the output of the CPG (e.g. leg position) can prove to be too slow to effectively cope with high frequencies. This is known to be the case in the feedback control of the cockroach locomotion controller: at high walking speeds the feedback lags behind too far to be useful. Some additional mechanism for use at high walking speeds might be present to perform some form of control over several cycles (Zill, 1985).
4.3.2 c A more elegant gain control
The control described by neurone definition 4.7 is a simple, efficient solution to the problem of controlling the oscillating network’s gain. It is not very elegant in that it operates in an identical way on both output channels. When for instance the output needs to be amplified while in a downward stroke, both o and are amplified, even though only needs to
be amplified. There is no need to amplify o ; it should rather be damped.
Fu % Σ I2R X Ou Od G Gu Figure 4.7
A more sensible gain control loop.
In order for the paddler to save a little energy and reduce wear and tear on its muscles and tendons, some logic can be added to ensure that only the dominant output channel receives the full gain; the other channel receives a smaller gain. In figure 4.7 this is shown for the case of a downward stroke; a mirror structure performs the same function for an upward stroke. Two interneurones, I2]
and I2 , measure which channel has the largest output by
subtracting them. Since would-be negative outputs are clipped to zero (as described by formula 1.3) only one ofI2]
andI2 will have an output larger than 0.
The difference signalsI2]
andI2 are used todecreasethe gain factor of the contralateral
side (i.e. the channel with the lowest output) by a shunting inhibition. This process is given by the following equations: