The paddle controller
4.3. A N OSCILLATOR FOR THE PADDLE CONTROLLER
v E 1m | ¤ ! if I otherwise Upward . A E 1mK¤1 | ! if I otherwise Downward . 48!
The gainfactor of the ipsilateral side is unaltered.
This small addition to the gain control system does not only prevent unnecessary energy expenditure. It also increases the speed of correction of erroneous outputs by reducing the counter-action from the muscle working "in the wrong direction". Too small an output in an upward stroke is corrected faster when the gain to the depressor muscle is reduced.
4.3.2 d Formal description of the oscillator The complete oscillator described above can be specified by the following set of coupled equations ("
8
is the weight of the mutual inhibition betweenI]
andI ; it is normally set to
1.0): I] E /^~ 4"Ó= T ] " 8 I (4.9) I E / B ~ 4" T " 8 I ] (4.10) / B ~ 0 2 /^~M0 h 2 (4.11) T] E 8 [z + K ] (4.12) E I] (4.13) I2] E : (4.14) v E 1 I2 ] (4.15) L E K ¥ (4.16) E 8 °³ + 1°D 1iD ~ B ¶ (4.17) D ~ B L £ L (4.18)
These equations are for the lefthand, or upstroke half-centre of the network; an identical set can be written down (exchangingufordandLforR) for the downstroke half-centre.
4.3.2 e Misbehaviour of the oscillator In this form the oscillator meets the requirements for a paddle controller: it generates a smooth oscillation, the amplitude of which is shown to be proportional to the input (equation
94 CHAPTER4. THE PADDLE CONTROLLER
4.5). The incorporated gain control ensures a sufficiently fast reaction to changing paddling commands (/^~
), and compensates for small disturbances encountered by the paddles. Large transients in the input cause problems, however; especially sharp increases imme- diately followed by a drop to a lower input level. In this case an identical component turns
up in both K and such that
L Ó L L B i1! ì L B iA!B¯ D 419! but max0 D ~ 2 max0 L L 2 max *( L B 1!=I L B ) 3 max0 L B L B 2 ë D 420!
This identical component must be a result of the capacitances of the two leaky integrators (which are the only elements in the oscillating network that have memory).
In (other) words, the gain control loopmeasuresa value for the maximum amplitude that does not match the actual maximum amplitude5
. The identical component decays in time as the leaky integrators discharge, and begin to represent the new average level of their input. During this time period a severe problem thus arises.
4.3.3 An oscillator with shunting delayed self-inhibition
The identicalcomponentseems to be caused by the subtractive nature of the inhibition of theTleaky integrators on theIinterneurones. It was therefore decided to investigate a network where this inhibition is of the shunting type. Ideally these two types of inhibition should be interchangeable. The shunting inhibition, however, due to the fact that it divides by the inhibiting signal, was thought to result in an identical factor in the output. An identical
factorwould be easy to account for by the gain control loop. From this point onwards, the networks with subtracting versus shunting self-inhibition will be referred to astype-1 and
type-2(oscillators, paddle controllers) respectively.
It turns out that a shunting self-inhibition introduces its own problems. As can easily be seen from figures 4.4 and 4.5 the basic network now becomes a pair of mutually inhibiting
de Vries-Rose machines(see chapter 3). De Vries-Rose machines are not inherently oscillatory, but stable. Two mutually inhibiting de Vries-Rose machines produce, when the same input is presented to both machines with a small delay between the two presentations, a short oscillation with a duration dependant on " , z and ? . Then they converge to a situation
where the sum of the two outputs equals the response of a single de Vries-Rose machine with the same parameters to the same input. Thus, for a constant input/\~
andz
?
10, and will each converge to approximately ¦
/~ º\"Õ!º 2 ¦ /^~ º 4"Õ! , since a single de
Vries-Rose machine converges to approximately6 ¦
/~ º\"Õ! .
5Formally speaking: the conditions for equation 4.3 are no longer met 6This can easily be seen: let
T denote the input to a de Vries-Rose machine, letQ denote the steady state output of the machine and let
denote the synaptic weight of the inhibitory connection. Then we have, approximating the shunting inhibition (T%§äQ ) with a regular division (T©¨;Q ):
QªóÆT%§ (
LQ )«
T
4.3. AN OSCILLATOR FOR THE PADDLE CONTROLLER 95