HIGH/

BANDWIDTH

. " ^.s

F1EIECY CCPS) ^l.

~~

- I I

2. 3.

"·

^{5. 10.}

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noted that, for any given frequency, the phase lag increases as the bandwidth of the Gaussian noise stimulus increases {low, medium, and high). This is particularly noticeable for subject GSC. It is interest- ing to compare this behavior with the corresponding increase in phase lag as the number of frequencies is increased in the "sum of sinusoids"

stimulus (Figure 7. 8). In section 1. 4, we suggested that the increase in phase lag could be due to the increase in spectral content of the stimulus, i.e., more sinusoids. However, it is now apparent that this is not a satisfactory explanation. First, all three Gaussian wave- forms contain all frequencies from DC to about 3 cps; they differ only in the high-frequency cutoff (corner frequencies). Second, comparison with Figure 7. 8 shows that the phase lag for the low-bandwidth Gaussi- an stimulus is somewhat less than that for any of the "sum of sinu- soids" motions, but that the phase lag for medium- and high-bandwidth Gaussian motion is greater. Thus, it appears as if phase lag is a function of the bandwidth of the target motion spectrum. Strangely enough, increasing the number of sinusoids from 4 to 7 or 13 causes the phase curve to become more erratic, but if the number of frequen- cies is increased until the spectrum is continuous, the resulting phase curve becomes quite smooth and well-behaved. One possible explana- tion is that the 7- and 13-frequencies stimuli were transmitted so that all frequency components were approximately equal in magnitude. In contrast, the Gaussian noise gradually tapers off at the higher fre- quencies.

3. Discussion

We have seen that the oculo1noto1· tracking system is highly

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non-linear. If the number of frequencies in the target motion is small enough, in general the tracking eye movements are composed of a sum of those sinusoids present in the stimulus. However, as the number of frequencies increases, or as the bandwidth of the spectrum of target motion gets wider, the phase lag at a given frequency increases. Any discrepancy between the phase lag expected by the authors and the measured values has, in the past, been attributed to the action of a predictor. Michael and Jones (4) measured the phase lag of tracking eye movements when target motion consisted of a single sine wave on which had been superimposed Gaussian noise of varying bandwidth. At a given frequency, as the noise bandwidth increased, the phase lag likewise became larger. This was interpreted as progressive failure of the predictor as the target motion became more "unpredictable. ¹¹ However, in our experiments with target motion consisting of nothing but Gaussian noise, we have found that phase lag became larger as the total bandwidth of the target spectrum increased. Clearly, it does not seem reasonable to describe "low-bandwidth Gaussian noise" motion as more predictable than "high-bandwidth" motion. It appears then that phase lag at a given frequency is more dependent on the overall bandwidth of target motion than on "predictability" of that motion. We shall return to this point later.

In essence, the existence of a predictor is a direct consequence of describing the tracking oculomotor system by linear transfer func- tions. To see how this happens, let us examine the work of Dallas and

Jones (1). Target motion consisted of band-limited Gaussian noise with a half-power cutoff at 1. 25 cps. The anthors plotted the gain and

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phase of the tracking eye movements as functions of frequency. (Only horizontal eye motions and monocular viewing were considered. )

From the gain curve, appropriate corner frequencies were chosen, and the closed-loop transfer function could then be written directly:

G( . ) _ 44 ( 5

+

jw)

JW - 2

(2. 2

+

jw)(-w

+

12jw

+

100)

(1)

However, this function G(jw) dictates phase lags cj>'(jw) which are smaller than the experimentally determined values <j>(jw). The differ- ence is then attributed to a constant delay T of about 70 ms ; this is not an unreasonable value for the transit time it would take nerve im- pulses to travel from the retina to the lateral geniculate, to the cere- bral cortex, then to the oculomotor nuclei, and finally to the extra-

ocular muscles. As we shall see below, the net transit times for the subjects used in this thesis are very close to this value. A constant delay T simply adds the term

-jwT

e (2)

to G(jw) of Eq. (4), and this does not affect the gain characteristics.

Assuming unity feedback, the open-loop transfer function g(jw) can be obtained directly from the closed-loop function G(jw):

=

^G(jw)

g(jw) 1 - G(jw) ⁽³⁾

This procedure was then repeated for pure sine wave stimuli, and the resulting open-loop transfer function g'(jw) was computed.

Since g(jw) and g'(jw) are different, it is necessary to postulate a predictor P(jw) which can be "switched in" series with the forward loop whenever it is decided that the target is predictable. Then

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P(jw)

= g'Pw»

_{g JW} ⁽⁴⁾

This is derived graphically: the gain of the predictor is first com- puted, and this function in turn dictates the phase characteristics.

This phase function does not agree with the experimental values, and this difference is finally attributed to pure prediction. The "pre- dieter" consists of a low-pass minimum phase network whose gain is larger than unity at all frequencies within its bandwidth, and of a pure

"phase-advance" which increases linearly with frequency. However, for a constant delay T, phase

cp

is linearly related to frequency f:

cp =

^{360fT , (5)}

and the "phase-advance" can be translated as a constant "negative de- lay" of about 214 ms, as measured from the authors ¹ curve. It is ex- tremely difficult to visualize such a cortical predictor which manages to anticipate target motion by 214 ms regardless of the frequency.

However, the nature of the predictor is a direct consequence of the description of the oculomotor system by means of linear transfer functions.

Since we have found that the system is highly non-linear, we shall abandon all attempts to describe it by means of linear transfer functions. Consequently, we rid ourselves of the gain-phase restric- tions imposed by functions such as G(jw) of Eq. ( 1). In fact, the very concept of phase is misleading, since it really has obvious mean- ing only for linear systems. We will then translate all phase values to constant delay times by means of Eq. (5). Figure 7.12 shows the phase curve experimentally determined for single sine waves by Dal-

c

--~~~~---~~---~----,i---~~~~--~~--~---~---

I . -

I ,.

, 1

#2 Phase lag due to constant delay

Figure

?.121

Phase lag of the oculomotor system for tracking targets whose motion consists of single sinusoids

.2

Curve 1: Values measured by Dallos & Jones (1) Curve 2: Least-means-squared approximation to

Curve 1 assuming that phase lag is result of constant delay

.3 ^.4

⁰

5

1 2

3