Analysis of Primate IBN Spike Trains Using System Identification Techniques. III. Relationship to Motor Error During Head-Fixed Saccades and Head-Free Gaze Shifts
Kathleen E. Cullen and
Daniel Guitton
Aerospace Medical Research Unit and the Montreal Neurological Institute, McGill University, Montreal, Quebec H3G 1Y6, Canada
 |
ABSTRACT |
Cullen, Kathleen E. and Daniel Guitton. Analysis of primate IBN spike trains using system identification techniques. III. Relationship to motor error during head-fixed saccades and head-free gaze shifts. J. Neurophysiol. 78: 3307-3322, 1997. The classic model of saccade generation assumes that the burst generator is driven by a motor-error signal, the difference between the actual eye position and the final "desired" eye position in the orbit. Here we evaluate objectively, using system identification techniques, the dynamic relationship between motor-error signals and primate inhibitory burst neuron (IBN) discharges (upstream analysis). The IBNs presented here are the same neurons whose downstream relationships were characterized during head-fixed saccades and head-free gaze shifts in our companion papers. In our analysis of head-fixed saccades we determined how well IBN discharges encode eye motor error (
e) compared with downstream saccadic eye movement dynamics and whether long-lead IBN (LLIBN) discharges encode
e better than short-lead IBNs (SLIBNs), given that it is commonly assumed that short-lead burst neurons (BNs) are closer than long-lead BNs to the motor output and thus further from the
e signal. In the
e-based models tested, IBN firing frequency B(t) was represented by one of the following: 1) model 1u, a nonlinear function of
e; 2) model 2u, a linear function of
e [B(t) = rk + a1
e(t)] where the bias term rk was estimated separately for each saccade; 3) model 3u, a version of model 2u wherein the bias term was a function of saccade amplitude; or 4) model 4u, a linear function of
e with an added pole term (the derivative of firing rate). Models based on
e consistently produced worse predictions of IBN activity than models of comparable complexity based on eye movement dynamics (e.g., eye velocity). Hence, the simple two parameter downstream model 2d [B(t) = r + b1
(t)] was much better than any upstream (
e-based) model with a comparable number of parameters. The link between B(t) and
e is due primarily to the correlation between the declining phases of B(t) and
e; motor-error models did not predict well the rising phase of the discharge. We could improve substantially the performance of upstream models by adding an
e term. Because
e = 
, this process was equivalent to incorporating
terms into upstream models thereby erasing the distinction between upstream and downstream analyses. Adding an
e term to the upstream models made them as good as downstream ones in predicting B(t). However, the
e term now became redundant because its removal did not affect model accuracy. Thus, when
is available as a parameter,
e becomes irrelevant. In the head-free monkey the ability of upstream models to predict IBN firing during head-free gaze shifts when gaze, eye, or head motor-error signals were model inputs was poor and similar to the upstream analysis of the head-fixed condition. We conclude that during saccades (head-fixed) or gaze shifts (head-free) the activity of both SLIBNs and LLIBNs is more closely linked to downstream events (i.e., the dynamics of ongoing movements) than to the coincident upstream motor-error signal. Furthermore, SLIBNs and LLIBNs do not differ in their characteristics; the latter are not, as is usually hypothesized, closer to a motor-error signal than the former.
 |
INTRODUCTION |
Saccadic eye movements in the head-fixed monkey are thought to be controlled by a feedback circuit (Fig. 1A) wherein the burst generator (B) is driven by an eye motor-error signal (
e) that is the difference between the desired angular rotation of the eye (
T) and the actual rotation of the eye that has occurred since the beginning of the saccade (
E*);
e =
T
E*. In turn, the burst generator is responsible for generating the burst in ocular motoneuron (MN) firing that drives a saccade (see review by Hepp et al. 1989
). B is thought to be composed of excitatory burst neurons (EBNs), which excite agonist MNs, and inhibitory burst neurons (IBNs), which inhibit antagonist MNs. IBNs are driven by EBNs (Sasaki and Shimazu 1981
; Strassman et al. 1986a
) and EBNs appear to be a primary target of the contralateral projections of IBNs (Strassman et al. 1986b
). It is commonly assumed that both neuron types have similar discharge characteristics (Fuchs et al. 1985
; Hepp and Henn 1983
; Luschei and Fuchs 1972
; Scudder 1988
). On the basis of their firing frequency profile, both IBNs and EBNs can be subdivided into two subclasses: long-lead bursters [long-lead IBNs (LLIBNs) and long-lead EBNs (LLEBNs)] that have a low-frequency preamble that precedes their burst and short-lead bursters [short-lead IBNs (SLIBNs) and short-lead EBNs (SLEBNs)] that have a compact burst whose duration equals saccade duration. In the classic theoretical organization of B (Fig. 1A) it was proposed that LLIBNs (or LLEBNs) project to SLIBNs (or SLEBNs) (Hepp and Henn 1983
; Luschei and Fuchs 1972
; Scudder 1988
; see reviews by Fuchs et al. 1985
; Moschovakis et al. 1996
). This structure would place LLIBNs closer to the motor-error signal than SLIBNs.

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| FIG. 1.
A: classic local feedback model for saccade generation proposed by Jürgens et al. (1981) as a modification to the original model of Robinson (1975) . This diagram was discussed in detail in the companion paper I (Cullen and Guitton 1997a ). T, desired angular rotation of eye; E*, angular rotation of eye that occurred since the onset of saccade; e(t) = T E* (eye motor error) drives B [burst neurons (BNs)]. E* is obtained by integrating BN output via an integrator that is reset to 0 after every saccade. In this schema, BN output is assumed [incorrectly, at least for inhibitory BNs (IBNs), as we have shown in companion paper I (Cullen and Guitton 1997a )] to be only proportional to eye velocity. E(t), actual eye movement generated as result of motoneuron (MN) signal passing through the plant dynamics of the eye. It is commonly assumed that e drives long-lead BNs (LLBNs) that in turn drive short-lead IBNs (SLIBNs). In this paper we investigate this hypothetical structure by studying the burst signal from the upstream perspective, i.e., the relation between e(t) and the firing frequency of B. B: local feedback model for saccade generation extended to describe control of head-free gaze shifts. H*, angular rotation of the head that occurred since onset of gaze shift and is obtained either by integrating semicircular canal signal or from neck proprioceptive feedback; g(t) = T E* H*, gaze motor error; E(t) and H(t), actual eye and head movements generated as result of eye and neck MN signals passing through the plant dynamics of the eye and head plants, respectively; OPN, omnipause neuron. In gaze control model it is often assumed that g(t) drives LLBNs that drive short-lead BNs (SLBNs). We investigate this hypothetical structure by considering the influence of eye, head, and gaze motor errors on B. See text for additional details.
|
|
In the first two companion papers of this three part series (Cullen and Guitton 1997a
,b
) we determined quantitatively the dynamic relationships between either saccadic eye motion (head-fixed) or gaze, eye, and head motion (head-free) and IBN discharge frequency. As noted in Fig. 1, we called this the "downstream" analysis. In this analysis we used system identification methods (Cullen et al. 1996
; Ljung 1987
) that permitted an objective evaluation of the lead time and the discharge frequency profile in terms of the associated saccadic eye or gaze motion.
The local feedback loop (Fig. 1), commonly used in the oculomotor literature, assumes that the burst generator can be described equally well by both its input (motor error) and its output (movement dynamics). This assumption is not strictly true. If our downstream analyses had revealed that the firing frequency profiles of both SLIBNs and LLIBNs predict movement dynamics (e.g.,
) with very high fidelity [i.e., very high variance accounted for (VAF)] (Cullen and Guitton 1997a
,b
), then the analyses presented in this paper would have had little purpose because the discharges of all IBNs would be entirely predictable by downstream events. This consequence is because the firing frequency profile of a burst neuron (BN) cannot simultaneously predict with high VAF both eye velocity (
) and eye motor error (
e). To see this, note that by definition eye motor error
e =
T
E* = Ed
E, where Ed and E are desired and current eye positions, respectively. Hence
e = 
. It follows that if burst firing frequency B(t)
, thenB(t)
e and therefore that B(t)
f(
e).
In companion paper I (Cullen and Guitton 1997a
) we found on average no difference between SLIBNs and LLIBNs in the relationship between their firing frequency and eye movement dynamics ( i.e., neither of these neuron types was closer to the output than the other). Our most realistic model, model 8d, gave a mean population VAF of 0.34, with a range of 0.16-0.57 (SD). This result implies that other signals, notably from upstream events related to motor error, might have had an equally important influence on neuron B(t). For example, perhaps neurons that are weakly related to downstream movement dynamics are more strongly influenced by motor error. We will consider this in the present paper; we will use the same analytic approach as in companion papers I and II (Cullen and Guitton 1997a
,b
) to study "upstream" influences on B(t), notably the relationship between
e and the firing frequency profile of IBNs.
The link between
e and B firing frequency was first studied quantitatively by Van Gisbergen et al. (1981)
. They considered this problem by using phase-plane trajectories in which instantaneous firing rate was plotted against
e. The results of their analysis suggested that primate EBN discharges are better described by a nonlinear representation of
e than by simple downstream models based on eye velocity. However, their approach did not provide an evaluation and comparison of the goodness-of-fit of different models that express firing frequency in terms of different functions of
e, nor did it compare the VAF in the upstream and downstream analyses. As we have seen in companion papers I and II (Cullen and Guitton 1997a
,b
), our analytic approach permits an objective quantitative evaluation of how well a given model describes the input-output characteristics of an unknown system. In the first part of the present paper we apply these methods to the upstream analysis of IBNs in the head-fixed monkey. Our objective was twofold; to determine 1) how well IBNs code
e compared with downstream saccadic eye movement dynamics and 2) whether LLIBNs code
e better than SLIBNs given, as reviewed above, that the latter are thought to be closer to the motor output and thus further from the
e signal.
In companion paper II (Cullen and Guitton 1997b
) of this series we analyzed the discharge of IBNs with regard to their downstream effect on eye, head, and gaze motion during orienting gaze shifts made by the head-free monkey. We define gaze = eye-in-space = eye-in-head + head-in-space. Gaze shifts in the head-free monkey are, like head-fixed saccades, thought to be controlled by a feedback loop and this is shown in Fig. 1B (reviewed in companion paper II Cullen and Guitton 1997b
). In this case it is hypothesized that the burst generator is driven by a gaze motor-error signal (
g), obtained by subtracting from the desired angular rotation of gaze (
T) the angle the gaze turned since the beginning of the gaze shift,
G* (=
E* +
H*):
g =
T
G* (Fuller et al. 1983
; Guitton and Volle 1987
; Guitton et al. 1984
, 1990
; Laurutis and Robinson 1986
; Pélisson and Prablanc 1986; Pélisson et al. 1988; Roucoux et al. 1980
; Tomlinson and Bahra 1986a
,b
; Tomlinson 1990
; see review by Guitton 1992
). In the second part of the present paper we analyze, from the upstream perspective, the relationships among IBN (SLIBN and LLIBN) firing frequency profiles recorded in the head-free monkey and either eye, head, or gaze motor error or combinations thereof. Our objectives were the same as for the head-fixed analysis.
Our most important conclusion is that the signals carried by both SLIBNs and LLIBNs were similar in both head-fixed and head-free conditions and reflected better downstream dynamics than upstream motor-error signals; contrary to classical hypotheses our LLIBNS were not closer to a motor-error signal than SLIBNs and similarly SLIBNs were not closer to the dynamic output than LLIBNs. For the head-free condition the low upstream VAF made it impossible to determine whether discharges were best predicted by models using either
e,
h,
g, or some combination of these signals.
 |
METHODS |
The neurons described in this paper were obtained from the same two monkeys (Macaca fascicularis) whose IBNs were studied from the downstream perspective and described during head-fixed and head-free conditions in companion papers I and II (Cullen and Guitton 1997a
,b
), respectively. The data described herein were obtained in the same experimental sessions as for companion papers I and II. The same cells are analyzed as those described in the companion papers; these neurons were categorized as IBNs on the basis of their physiological responses during head-fixed saccades, vestibular nystagmus, smooth pursuit, and their location in the IBN area. The surgical preparation of the animals and methods used for obtaining extracellular recordings in the head-fixed and head-free conditions were identical to those described in companion papers I and II.
Models for BN firing rate
IBN discharges were analyzed in terms of the schemas of Fig. 1 in which the firing of B is related simultaneously to the upstream motor-error signal and to the downstream dynamics of the movement trajectory. In this paper we focus on upstream mechanisms. We used system identification techniques (Cullen et al. 1996
) to objectively analyze different models in which functions of either gaze, head, or eye motor error (head-free) or eye motor error (head-fixed) were used to predict IBN spike train dynamics. To obtain gaze motor error we subtracted the actual position of the visual axis from the final position of the visual axis at the end of the saccadic gaze shift. The end of the gaze shift was defined as when gaze velocity has declined to 20°/s. Similarly, for eye motor error we subtracted the actual position of the eye from the final position at the end of the saccadic eye movement. In the head-free condition this is admittably an arguable procedure because the eye at some period may be driven by the vestibular-ocular reflex away from its intended final position. However, there is no alternative procedure to estimate eye motor error, but the error should be small given that generally the head moves little before the eye saccade ends. Thus motor error was equal to movement amplitude at the beginning of a movement and zero at the end.
In companion papers I and II (Cullen and Guitton 1997a
,b
) we analyzed both the ON-direction (OND) and OFF-direction (OFFD) discharges, both of which were well correlated to movement velocity. Here we consider only the OND discharge because, as we shall see, the VAF provided by motor-error-based models was consistently lower than that provided by velocity models characterized in our previous analysis. The different models we tested are given in Table 1 and the rationale for selecting these models will be given in RESULTS and DISCUSSION. The methods for parameter estimation were the same as those described in Cullen et al. (1996)
and those used in companion papers I and II. Before considering different models we attempted to calculate the optimal delay between motor error and cell discharge. This will be considered in the next section. For each model, optimal fits were made to an ensemble of ~40 saccades or gaze shifts of different amplitudes in the range of ~5-45° in the head-fixed study and ~10-70° in the head-free study. Models were ranked according to whether a model produced an increase in the VAF as well as a simultaneous decrease in a cost index [the Bayesian information criterion (BIC)]. The BIC will decrease for an increasingly complex model only when the addition of parameters is warranted.
In companion papers I and II (Cullen and Guitton 1997a
,b
) we determined the preferred direction of each cell and found that, on average for the population, this was nearly parallel to the horizontal axis. We also noted that restricting our analysis to the horizontal component of a movement did not change the model fits compared with using the actual movement vector. In the present analysis we did not repeat these calculations for motor error but assumed, as before, that analyzing the horizontal component was adequate. However, in two cells we did test whether the neglect of the small vertical component of movement had a significant effect on the results of the motor-error model fits. To do this we compared the results obtained by using the full vector to those using only the horizontal component and we found no significant change in the model fits.
Estimation of lead time
Each BN's lead time was calculated by the two methods employed in companion paper I (Cullen and Guitton 1997a
). In the first method, for each cell the average lead time was determined by calculating the average period between the onset of the first spike and the onset of eye velocity during head-fixed saccades. In companion papers I and II (Cullen and Guitton 1997a
,b
) this method was used to classify cells as SLIBNs and LLIBNs. In this paper we have kept the same classification of SLIBNs (n = 16) and LLIBNs (n = 12) as used in those papers. Recall that the mean period between the onset of the first spike and the onset of eye saccades during head-fixed saccades was taken as
15 ms for SLIBNs and >15 ms for LLIBNs.
In the second method the unit discharge was shifted by the time (td) required to obtain an optimal fit for a given dynamic model. In our analysis of downstream models (Cullen and Guitton 1997a
,b
) we used the simple downstream model 2d
|
(1)
|
to estimate each cell's td, where r is a bias term, b1 is a gain term, and
is the input of either eye or gaze velocity. We attempted to use a similar approach to determine the optimal dynamic lead time for the head-fixed upstream model 1u
|
(2)
|
where r is a bias term; a1, a2, a3, and a4 are gain terms; and
e is eye motor error. This is model 1u of our upstream analysis (Table 1). Figure 2A compares the results of applying this procedure to the discharges of SLIBN L0702 during head-fixed saccades using models 2d (
) and 1u (
). The abscissa gives the time the burst was shifted, which is plotted relative to the onset of the burst. The ordinate gives the VAF provided by each equation. On average, this neuron burst 14 ms before saccade onset and the value 0 on the abscissa (vertical line) gives the VAF when the burst is not shifted in time relative to the onset of the movement; the dashed vertical line on the right indicates when the movement started. (Note that, with any time shift, only that portion of the neural discharge that overlapped with the saccade's duration was used in the optimization algorithm.) Figure 2A indicates that as a unit's discharge was shifted in time, the VAF provided by model 2d changed and the optimal dynamic lead time was defined as the latency for which the maximal VAF was obtained in the model fit. In companion paper I (Cullen and Guitton 1997a
) the value estimated for td for neuron SLIBN L0702 was 14 ms; i.e., in the downstream analysis, the burst needed to be shifted on average 14 ms toward the onset of the movement. (Exceptionally, for this neuron the dynamic lead time equaled that based on the first spike.)

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| FIG. 2.
A: relation between variance accounted for (VAF) andtime by which burst is shifted, either toward or away from the saccade. This was determined, for cell L0702, during head-fixed saccades using downstream (companion paper I, Cullen and Guitton 1997a ) model 2d ( ) and upstream model 1u ( ). Optimal dynamic lead time td was defined as latency for which maximal VAF was obtained. Using model 2d, the burst had to be shifted 14 ms toward saccade; i.e., td = 14 ms. By comparison model 1u gave td = 3 ms; the burst needed tobe shifted 3 ms further away from the saccade than its initial positionat t = 0. Thus the upstream and downstream analyses are in conflict. Dashed vertical line on right indicates the onset of the saccade, in this case 14 ms after the onset of the first spike. B: schematic representation of the results described in (A) above using a typical firing frequency profile, B(t), and the movement trajectory. e profile is mirror image of eye position profile E. , eye velocity. In this example the first spike in burst precedes saccade by 14 ms and model 2d required that burst be shifted 14 ms toward saccade as shown in the 3rd panel from bottom. Internal neural e signal, e.g., from the superior colliculus (SC), precedes the saccade by 15 ms (2nd panel from bottom). Model 1u requires that burst precedes SC signal by 17 ms, or equivalently, the saccade by 32 ms (bottom panel). Figure demonstrates that a realistic estimate of dynamic lead time can be identified by using eye velocity-based model 2d but not eye motor-error-based model 1u.
|
|
Figure 2A also shows the optimal lead time identified by using the upstream eye motor-error-based model of model 1u. In contrast to the downstream estimate of td, model 1u gave
3 ms; that is the maximal VAF was obtained in the model fit by shifting the unit discharge an additional 3 ms forward in time from its mean position of 14 ms before the movement. Put another way, for an optimal motor error fit the firing frequency profile needed to be positioned 17 ms before the onset of the eye motor-error profile. These calculations were made with the use of eye motor-error profiles obtained from the actual eye trajectories. However, in reality the motor-error signal is thought to be generated in the brain stem, a likely structure being the superior colliculus (SC) (Munoz et al. 1991
; Munoz and Wurtz 1995
; Waitzman et al. 1991
).
Figure 2B demonstrates that a realistic estimate of dynamic lead time can be identified with the use of the eye velocity-based model 2d but not the eye motor-error-based model 1u, if neurophysiological constraints are considered. With regard to this signal processing in the brain stem our optimal lead-time calculations imply that the burst of neuron L0702 should precede by 17 ms the internally generated motor-error signal emanating from the SC. This requirement generates the conflicting situation illustrated in Fig. 2B. Note first that SC burst discharges precede ongoing eye movements by ~20 ms and this corresponds to the latency of an electrically evoked saccade (Sparks 1978
; Sparks and Mays 1980
; see reviews by Guitton 1991
; Sparks and Mays 1990
). By comparison, electrical stimulation of the SC during an ongoing saccade perturbs the movement at a latency of ~10 ms (Miyashita and Hikosaka 1996
; Munoz et al. 1991
). Thus the functional latency would appear to be somewhere in between these values (~15 ms). Thus, as illustrated in Fig. 2B, if the IBN burst should precede the SC signal by 17 ms (as required by the upstream analysis) then it must precede the eye movement by 32 ms. This contradicts the downstream td. Conversely, if the downstream optimization calculation requires that the IBN burst precedes the movement by 14 ms then this burst should lag the SC signal by 1 ms, which contradicts the upstream analysis. The following two questions arise: 1) What is the source of this ambiguous result? and 2) What is the realistic timing? The answer to the first question lies in the fact that eye motor error is a monotonically declining function, whereas both IBN discharges and eye velocity profiles are monotonically declining only after peak firing rate and eye velocity, respectively, have been attained. As we will consider further in the DISCUSSION, in relation to Fig. 10, the schematic of Fig. 2B illustrates that the optimization algorithm minimizes the influence of the initial rising portion of the unit's discharge by requiring that the cell discharge be shifted in time such that less of the rising phase in firing frequency coincides with the motor-error trajectory. However, in doing so the algorithm ignores a substantial portion of a neuron's initial discharge. In addition, shifting the discharge to exclude this portion of the discharge produces latencies with little physiological relevance. In answer to the second question, a more realistic lead time would be to have the IBN burst discharge lag the motor-error signal by ~2 ms; the sum of, for example, SC brain stem conduction time (Guitton and Munoz 1991
) and one synaptic delay, given that BNs are driven monosynaptically by the SC (Chimoto et al. 1996
). This would place burst lead time relative to movement at almost 13 ms, compatible with the results of our downstream analysis using model 2d (companion paper I, Cullen and Guitton 1997a
).

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| FIG. 10.
Head-fixed analysis. Eye motor-error-based fits (VAF = 0.27), using model 1u, to eye velocity trajectories generated during head-fixed saccades, for the same 3 example saccades that were shown in Figs. 5, 6, and 8 for neuron L0702. These results were surprising, because they indicate eye motor-error-based model 1u actually fits the eye velocity trajectories better than the unit's firing frequency profile during the same saccade (compare to Fig. 3A, top panel; VAF = 0.13). These results suggest that the correlation between motor error and firing frequency may in large measure be a result of inherent correlation between motor error and velocity.
|
|
 |
RESULTS |
In this paper we study the same population of 28 IBNs whose downstream effects during head-fixed saccades and head-free gaze shifts were presented in companion papers I and II (Cullen and Guitton 1997a
,b
), respectively.
Head-fixed saccades: dynamic models linking IBN discharges to eye motor error
Table 1 shows the four upstream models we tested in this paper (models 1u-4u) including mean population values of the parameters that were estimated. The mean population VAF and BIC values for these models are shown in Table 2. The first model we tested, number 1u, investigated the hypothesis of Van Gisbergen et al. (1981)
that cell firing is represented by a nonlinear function of
e. In our formalization of this proposal we used a fourth-order nonlinear function of
e. The VAF produced by this model was low (mean VAF = 0.15; Table 2) for all neurons including SLIBNs and LLIBNs, except SLIBN H1015 for which the VAF = 0.43.
In the downstream analysis of companion paper I (Cullen and Guitton 1997a
) we found that a model's fit could be considerably improved by incorporating a bias term that varied from saccade to saccade. We tried this approach in model 2u (Table 1). Overall, the mean VAF (0.25) increased dramatically relative to model 1u, with a corresponding decrease in the BIC value indicating that the use of 43 parameters was warranted.
Figure 3 illustrates for our example SLIBN L0702 (A) and LLIBN H0409 (B) the model fits to firing rate for the same saccades considered in companion paper I (Cullen and Guitton 1997a
) for model 1u (top panel) and model 2u (middle panel). The VAFs for units L0702 and H0409 were 0.13 and 0.15 (model 1u) and 0.27 and 0.38 (model 2u), respectively.

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| FIG. 3.
Head-fixed analysis. Examples of upstream model fits of IBN firing rate, B(t), profiles during saccades for example SLIBN L0702 (A) and LLIBN H0409 (B). Top row: fit (heavy solid line) to the firing frequency profile (light line around shaded area) of model 1u, a 4th-order nonlinear function of eye motor error. Second row: fit of model 2u that contained a motor error term and a variable bias term rk that was estimated as a parameter for every saccade. Third row: fit of model 4u that contained a pole term that gave a low-passed filtered version of e. In this model an initial firing rate, B(t0), was estimated separately for every saccade. Values B(t0) and rk are listed separately, above each row, for each of the 3 example saccades. Parameter values estimated using >40 saccades given below each model fit. Bottom 2 traces: accompanying eye velocity and eye motor-error trajectories temporally shifted by estimated dynamic lead time td.
|
|
The estimated biases in model 2u were correlated with the amplitude of the saccade for the majority of SLIBNs (10/16) and for one-half of LLIBNs (6/12); i.e., 57% of the population. [This is similar to the downstream analysis, models 6d-8d of companion paper I (Cullen and Guitton 1997a
).] In fact, nearly every IBN that demonstrated such correlations for downstream models 6d and 7d also demonstrated correlations for eye motor-error-based model 2u. For these cells the values estimated for initial conditions and biases were inversely correlated with saccade amplitude (mean SLIBNs, R =
0.66; LLIBNs, R =
0.65). Figure 4 illustrates this property for our example cell H0925. Significant correlations between biases and peak saccade velocity were observed for only two SLIBNs, but this could have arisen because amplitude and peak velocity are correlated. As in the downstream models, a relationship between estimated biases and the eye position before saccades was investigated and found to be significant for only two SLIBNs. No relationship was observed between estimated biases and the final position of the eye for any of our neurons. In contrast, note that the initial firing rate taken directly from the data was not correlated with any of the measured saccade parameters.

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| FIG. 4.
Head-fixed analysis. Negative correlation among estimated biases rk in model 2u and saccade amplitude for LLIBN H0925. Correlation coefficient R for this cell = 0.73. Comparable relationships were seen in 50% of IBNs.
|
|
The amplitude dependence of the majority of neurons suggested the use of model 3u that brought together the dependence on motor error with that on amplitude using only three parameters. This model was not so good as model 2u and gave a mean VAF of 0.15, similar to that given by model 1u but with two less parameters. In the DISCUSSION we will consider the limitations of motor-error models 1u-3u, the significance of their parameters, and how a model fit can be improved by introducing a pole term [
(t)] to produce model 4u.
Head-free gaze shifts: dynamic models linking IBN discharges to gaze and eye motor error
Table 3 shows the four downstream models we tested for the head-free condition. They are the same as those tested for the head-fixed condition except that now we used either
g or
e as the input. Model 1u tests the hypothesis that unit firing can be represented by a nonlinear function of either gaze or eye motor error. This is the head-free extension of the Van Gisbergen et al. (1981)
hypothesis. The top two panels in Fig. 5 illustrate the predicted discharge rate of our example SLIBN L0702 for the same three gaze shifts that were illustrated for this cell in companion paper II (Cullen and Guitton 1997b
). The model, with either gaze or eye motor error as the input, appears in these examples to give a reasonable fit through this neuron's average firing rate; however, the overall VAFs provided by both the gaze- and eye-based models were low at 0.14 and 0.13, respectively. For all the neurons taken together the mean VAF given by model 1u was poor (Table 4): 0.12 and 0.13 in the gaze- and eye-based models, respectively.

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| FIG. 5.
Head-free analysis. Example of model fit to SLBN L0702 activity; based on 4th-order nonlinear function of motor error (model 1u, Table 3). Comparison of top and 2nd panels illustrates that models based on gaze and eye motor error, respectively, provided comparable fits (heavy solid line) of the neuronal discharge (shaded areas), albeit using different parameters; VAFs were 0.14 and 0.13 in top and 2nd panels, respectively. Values of parameters, estimated using 40 gaze shifts accompanying this neuron's discharge, provided below each model fit. Two bottom traces: gaze and eye motor-error trajectories that have been shifted in time relative to the burst by estimated dynamic lead time td for this neuron.
|
|
Model 2u, in which individual bias values were fit to each discharge, provided a much improved fit and the population VAFs were 0.25 and 0.22 in the gaze- and eye-based models, respectively. The first and second panels from the top in Fig. 6 show the fits to our example data; model 2u gave VAFs of 0.19 and 0.20 for cell L0702 in the eye- and gaze-based models, respectively.

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| FIG. 6.
Head-free analysis. Fits using model 2u (Table 3) to SLBN L0702 activity for 3 example gaze shifts illustrated in Fig. 5. Model incorporated motor error and variable bias terms. Models based on gaze (top panel) and eye (2nd panel) motor error provided comparable fits (heavy solid line) of the neuron's discharge (shaded curves), albeit using different parameters. (VAF = 0.21 and 0.20 for gaze and eye motor-error-based predictions, respectively.) Bias values rk estimated for separate gaze shift listed above each row. Values of parameters, estimated using 40 gaze shifts accompanying this neuron's discharge, provided below each model fit. Accompanying gaze and eye motor-error traces (bottom 2 panels) have been shifted in time by the estimated dynamic lead time td for this cell.
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In the head-fixed analysis of upstream models the estimated biases in model 2u were correlated with saccade amplitude in 57% of the cells. In contrast, for head-free upstream models, the estimated biases in
g-based model 2u were correlated to gaze shift amplitude in only 25% of all the neurons and in
e-based model 2u were correlated to eye movement amplitude in only 18% of all the neurons. Consequently, for upstream models the dependence of biases on both eye and gaze shift metrics was less robust head-free than head-fixed.
Some of these relationships are illustrated in Fig. 7 for the same example IBN (H0925) whose data were shown in Fig. 4 and in the companion papers (Cullen and Guitton 1997a
,b
). For this neuron, the bias term in
g-based model 2u was not correlated with eye or gaze amplitude (Fig. 7, A and B; R =
0.19 and
0.11, respectively). In the preceding upstream head-fixed analysis we found that incorporating a bias that depended on saccade amplitude (model 3u) lowered considerably the VAF relative to model 2u and did not improve the fit over model 1u. A similar result was found in the head-free analysis, as shown in Table 4.

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| FIG. 7.
Head-free analysis. Relationships, obtained from model 2u, between estimated bias rk and amplitude of entire gaze shift (A) and ocular component (B) of gaze shift for LLIBN H0925. Only a minority of IBNs that demonstrated head-fixed correlations between saccade metrics and variable biases in upstream model 2u also demonstrated head-free relationships in corresponding gaze or eye velocity-based models. For this neuron, a relationship was observed between estimated rk and the amplitude of head-fixed saccades (Cullen and Guitton 1997a ).
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Head-free gaze shifts: dynamic models linking IBN discharges to head motor error
Figure 8 demonstrates the fits of dynamic models based on head motor error to the discharge of neuron L0702 during three example gaze shifts. The top panel illustrates the model fit obtained when head motor error was used as an input to model 1u (VAF = 0.11). In the second panel head motor error was used as an input to model 2u (variable bias) and this slightly improved the fit (VAF = 0.20). For the entire population of cells the VAF of 0.25 given by model 2u (Table 5) was equal to that obtained with the use of either gaze or eye motor-error-based fits (Table 4). We did not try model 3u (with a head amplitude dependent bias) because of the poor performance of this model for eye in the head-fixed condition, and for both eye and gaze in the head-free condition. In summary, for head motor-error-based models 1u and 2u, the VAFs were generally equivalent to those of the eye- or gaze-based versions of these models.

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| FIG. 8.
Head-free analysis. Head motor-error-based model fits to SLIBN L0702 activity for the 3 example gaze shifts illustrated in Figs. 5 and 6. A 4th order, nonlinear function of head motor error (model 1u, Table 5; top panel) fits the data with a VAF of 0.11 for this cell. When we used a model with a variable bias term, estimated as free parameter for every movement (model 2u), the fit improved (2nd panel, VAF = 0.21). This cell was atypical in that fit was similar to those obtained when gaze or eye motor error were inputs to this model (see Fig. 6). The 3 values listed above the 2nd panel represent estimated biases rk. Values of parameters, estimated using 40 gaze shifts, provided below each model fit for this neuron. Accompanying gaze, eye, and head motor-error traces (bottom 3 panels) have been shifted in time by estimated dynamic lead time td for this neuron.
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 |
DISCUSSION |
We have applied the system identification methods described in Cullen et al. (1996)
to analyze the discharge of both SLIBNs and LLIBNs in terms of motor-error signals in the head-fixed and head-free monkey. Our approach permits an objective comparison, to be made in the following sections, among downstream models that predict the discharges of these neurons by using quantities that describe movement trajectories (e.g., eye or gaze velocity, acceleration, etc.) (see Cullen and Guitton 1997a
,b
) and upstream models that relate the discharges to motor error. Such a comparison is important to evaluate current models of the oculomotor and gaze-motor circuitry (Fig. 1). Our most important conclusion is that the signals carried by both SLIBNs and LLIBNs are similar and reflect better downstream dynamics than upstream motor-error signals; LLIBNS are not closer to a motor-error signal than SLIBNs and similarly SLIBNs are not closer to the dynamic output than LLIBNs.
Head-fixed saccades: significance of parameters estimated in eye motor-error-based models
The static nonlinear model 1u gave a VAF of only 0.15 in spite of containing up to fourth-order motor-error terms. Removing the third- and the fourth-order terms decreased the VAF even further (not shown). This model failed to predict the initial slope of the discharge rate for saccades <20° (Fig. 3A). However, for 30° >
e > 20° the initial rise and peak discharge were better predicted as shown in Fig. 3B (top left panel). This was because as saccade size increased, the fourth-order term became rapidly dominant so for ~
e > 30°, the predicted initial discharge was negative (not shown). Because of the low VAF and initially negative discharge for
e > 30°, we do not retain model 1u as a plausible upstream model.
The estimated biases in model 2u were correlated with the amplitude of the saccade in 57% of the population. Recall that a similar amplitude-dependent effect was noted in our downstream analysis of model 7d in companion papers I and II (Cullen and Guitton 1997a
,b
). Given that the upstream and downstream models were fit to the same data set, we can compare directly the biases estimated in the two calculations. The biases estimated for eye motor-error-based models were significantly larger than the biases estimated for eye velocity-based models. For example, for cell H0925 we find rk2u = 0.75 rk7d + 263 (R = 0.83). Model 3u, with three parameters, incorporated the amplitude dependence of the bias term. However, the VAF (0.15) given by this model decreased substantially from that given by model 2u because of a drastic reduction in free parameters from 41 to 3.
The amplitude-dependent effects revealed in the downstream analysis should reflect a similar amplitude-dependent effect of the upstream signal that drives IBNs. In fact, studies in the SC have suggested that saccade-related BNs of the rostral colliculus, which discharge during small amplitude saccades, burst more strongly than those of the caudal colliculus, which discharge during larger amplitude saccades (D. Munoz, personal communication). Because cells near the rostral fixation zone of the SC appear to carry both fixation and movement command signals (Munoz and Wurtz 1993
), it is possible that to generate small saccades this additional burst modulation is required to overcome an inappropriate fixation signal.
We have shown that models 1u-3u, representing possible formulations of the hypothesis that an eye motor-error signal drives BNs, have low VAF. Indeed, these observations raise the question as to whether there does exist a motor-error model formulation that can improve on models 1u-3u. We will consider this in a subsequent section.
Comparison between SLIBNs and LLIBNs
In the classical view of the oculomotor circuitry (see INTRODUCTION) LLIBNs encode
e better than SLIBNs. We tested this hypothesis by examining the relationship between discharge lead time (based on the 1st spike to distinguish SLIBNs from LLIBNs) and VAF estimated by models 1u-3u. Figure 9A shows this result for our population of IBNs by using model 3u, each point representing one cell. The mean VAF is not systematically related to mean lead time and is not lower for LLIBNs (
) compared with SLIBNs (
). The results using model 3u were typical of those also obtained for models 1u and 2u. This general lack of a relationship between VAF and cell lead time is consistent with our finding that there was no significant difference in the VAF provided by any of the upstream models for SLIBNs versus LLIBNs (Table 2). Taken together, the results indicate that SLIBNs and LLIBNs encode
e equally well. Put another way, LLIBNs are not closer to the
e signal than SLIBNs. An analogous finding showing no difference between SLIBNs and LLIBNs was made for the downstream analysis in companion paper I (Cullen and Guitton 1997a
).

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| FIG. 9.
Head-fixed analysis. A: relationship between lead time (time between onset of 1st spike and the saccade) and VAF by model 3u. Open triangles, LLIBNs; filled circles, SLIBNs. Neurons with short-lead times do not have higher VAF (i.e., LLIBNs are not "closer" to the motor error signal than SLIBNs. B: relationship between VAF given for any cell by upstream model 3u and VAF, for the same cell, given by downstream model 8d. Each point represents one cell: open triangles, LLIBNs; filled circles, SLIBNs. Neurons with high downstream VAFs generally do not have low upstream VAFs, and vice versa. Only 1 neuron had a higher upstream VAF than downstream VAF, but the difference between the 2 VAFs was small.
|
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Although no difference existed between SLIBNs and LLIBNs in terms of the mean signals carried by each population, it is possible that among the entire population some neurons may be closer to
e, whereas others are closer to the final motor output. We tested this hypothesis by plotting, in Fig. 9B, the upstream VAF by model 3u versus the downstream VAF by model 8d (from companion paper I, Cullen and Guitton 1997a
). Model 8d was our best downstream model and 3u was chosen for the comparison because it is of the same form and has the same number of parameters. The plot shows that all neurons except SLIBN H0922 had a higher downstream VAF than upstream VAF. Even neuron H1015, which had a high upstream VAF (0.57), had an even higher downstream VAF (0.68). Thus in 27 of 28 IBNs the discharge frequency profile was best linked to movement dynamics; but in the one exception the upstream VAF was still quite low (VAF = 0.25).
Our conclusion that there is no serial link from LLIBNs to SLIBNs has other experimental support. Although Raybourn and Keller (1977)
suggested that long-lead BNs (LLBNs) preferentially receive inputs from the SC, a more recent study in the cat demonstrated that the majority of both SLEBNs and SLIBNs receive clear monosynaptic projections from SC output cells (Chimoto et al. 1996
). Furthermore, Strassman et al. (1986a
,b
) demonstrated, by using intracellular staining techniques, that at least some LLIBNs and LLEBNs project directly to the abducens motor nucleus in the squirrel monkey. Hence, taken together all the results presented argue against subdividing either IBNs or EBNs into two distinct functional populations on the basis of lead times.
On the other hand, an argument borrowed from Scudder et al. (1988)
against a functional LLIBN projection to abducens MNs is that LLIBN dynamic lead time begins l5.5 ms before a saccade; i.e., ~6.5 ms before the pause in the antagonist MNs. This interval is very long for a monosynaptic connection. However, LLIBNs could begin the process of MN hyperpolarization that could be secured by the SLIBN burst. However, Scudder et al. (1988)
also pointed out the lack of a decline in MN tonic discharge beginning during the LLIBN prelude. Clearly, the functional significance of the LLIBN discharge prelude still needs to be resolved as well as a better understanding of the specific morphophysiological circuitry underlying the brain stem burst generator.
Is there a good head-fixed upstream model?
The choice of an optimal upstream model based on motor error is difficult because no model stood out clearly as having a far superior VAF. It is possible to generate an upstream model formulation that improves on models 1u-3u. We tested such a model that included a first-order
e signal, a derivative of firing rate (pole term), and a bias term (Table 1, model 4u). In this formulation the pole term [
(t)] provides a mechanism for transforming the step-ramp shape of
e into a realistic firing frequency profile. The pole term permits this by essentially combining the decaying
e and initial condition terms with a low-pass filtered step input approximately proportional to r + a1
e. This combination can generate a firing frequency profile that rises to a peak and then decays as can be seen in Fig. 3, third panel from top. In model 4u the initial firing rate [B(t0)] was estimated as a separate parameter for the 40 saccades in the data set and the number of parameters to be estimated was 43. This led, not surprisingly, to the highest VAF of our upstream models (Table 2, mean VAF = 0.33). The BIC value was lower for model 4u than for the other upstream models, indicating that the increased complexity of this model was justified.
In model 4u the values estimated for the bias terms were highly variable, frequently taking on large negative values. The standard deviations of the estimates of parameters in this model were very large when compared with those of parameter estimates generated in our other upstream models (Table 1). We obtained a similar result in our downstream analysis in companion paper I (Cullen and Guitton 1997a
) when we used model 6d, the downstream model with a pole term in which the initial conditions were fit as parameters. For these reasons we rejected 6d as a viable model. Model 4u is a complex equation to optimize (Cullen et al. 1996
). It was our best upstream predictor, but only marginally compared with model 2u, which had 41 parameters and gave a VAF of 0.25. This, together with the high variance in the parameter estimates and the need to fit a parameter to each saccade, indicates, as for model 6d, that model 4u does not give us useful insights into how IBNs respond to upstream signals during saccadic eye movements.
To gain a better understanding of what the VAFs in models 1u-4u signify it is important to point out that an eye motor-error-based model predicts eye velocity as well as it predicts firing frequency. An example of this is illustrated in Fig. 10 for the same three example saccades for which the fits to SLIBN L0702 were shown in Fig. 3A. When we used our approximation (model 1u) to the Van Gisbergen et al. (1981)
hypothesis to estimate the actual
(rather than firing frequency) profiles of these saccades, we obtained mean VAF = 0.27. However, when we used model 1u to estimate the firing frequency of L0702 we obtained, averaged over all saccades, mean VAF = 0.18 (Table 2). Thus model 1u predicted
better than B(t), because (as noted earlier in relation to Fig. 2B) both velocity and motor error decay with time, and after peak
, was attained are correlated, albeit to a limited degree. Hence, the best population upstream VAF = 0.33 we obtained with model 4u undoubtedly reflects the inherent correlation between
e and
and in turn between
and B(t).
A striking result of our analysis is that a simple downstream model (companion paper I, Cullen and Guitton 1997a
, model 8d) that included only three parameters
a bias, a saccade amplitude gain, and a velocity gain
provided a much better fit of neuron discharge than our upstream models 1u and 3u. Furthermore, model 8d provided a fit about equal to that by models 2u and 4u for which every saccade had a free parameter. [Compare model 8d in Table 3 of Cullen and Guitton (1997a)
, VAF = 0.34; with model 4u of Table 2, VAF = 0.33.] Furthermore, the even simpler downstream model 2d with two free parameters (a fixed bias term and a velocity term) provided almost as good of a fit as model 4u, and the best downstream model 7d containing a variable bias and a velocity term did much better.
One way to generate a better upstream model fit is to include in the equation a term proportional to
e. As noted in the INTRODUCTION,
e = 
; this procedure is equivalent to removing the classic distinction shown in Fig. 1 between upstream and downstream views of the burst generator. Thus the model
brings together upstream model 3u and downstream model 8d. The mean VAF by this model for our population of IBNs was 0.34, the same as that given by model 8d and a little better than model 4u. Thus adding the
e term to model 3u increased the VAF from 0.15 to 0.34, whereas there was no effect of adding the
e term to model 8d. Put another way, when eye velocity (or equivalently the derivative of eye motor error) is available as a parameter, eye motor error is not needed by the optimization algorithm. These results state unequivocally that, during saccades, the activity of nearly all IBNs is linked entirely to downstream events (i.e., the dynamics of ongoing eye movement) and not to upstream motor-error signals, as described in the classic local feedback model of saccade generation (Fig. 1).
Head-free gaze shifts: significance of parameters estimated in gaze and eye motor-error-based models
The VAFs provided by models 1u and 3u were very low (0.1) in both the eye-based and gaze-based models, even lower than in the head-fixed condition. Because of these poor fits it was not possible to determine, using these models, whether the cells were driven by either eye or gaze motor error. Hence, models 1u and 3u will not be considered further. Model 2u more than doubled the VAF: using gaze motor error, VAF = 0.25; using eye motor error, VAF = 0.22. However, this improved performance was at the expense of having one free parameter (the bias) per saccade.
We have focused so far on models that are based on either eye or gaze motor error. However, the possibility remains that a model incorporating simultaneously both eye and head motor error might fare better. In companion paper II (Cullen and Guitton 1997b
) we found that this approach did not increase the VAF significantly. Accordingly, we tried a modification to 2u of the form
When this model is applied to our example neurons L0702 and H0409, the VAF (0.21 and 0.28, respectively) was similar to that provided by eye- and gaze-based models. Furthermore, as we saw in Table 5, if only the dependence on head motor error was considered, the VAF remained at about the same low value (0.25). Indeed, the mean VAF provided by model 2u was ~0.25 independent of whether eye, head, gaze, or combinations thereof are used. Thus model 2u cannot be used in any form that permits a choice of the relevant upstream signals. This is perhaps because the VAF by model 2u is still too low and specifically, as we saw in Fig. 10, because the VAF may be caused by a correlation between the simultaneously declining slopes of the motor error and firing frequency profiles.
As in our analysis of head-fixed saccades, we found that our head-free upstream motor-error models generally fit the velocities of movements as well as, or even better than, they fit the unit discharge. For example, when we used model 1u based on gaze motor error to estimate the discharge of cell L0702 we obtained a mean VAF = 0.14. When we used model 1u to estimate the actual gaze, eye, and head velocity trajectories generated during the gaze shifts associated with the discharge of L0702 we obtained VAFs of 0.33, 0.35, and 0.10, respectively. Hence, any VAF < 0.35 between motor error and firing frequency profile could be because of the correlation between motor error and velocity, the latter as we have amply shown in companion paper II (Cullen and Guitton 1997b
) being correlated to firing frequency.
In our head-fixed study in the first part of this paper we asked whether a model of the form 4u could provide a useful model of IBN discharges. The filtering characteristics of this model, combined with the initial conditions fit as parameters, provided in principle the means for obtaining the most accurate fit to the neuron firing frequency profile. In our head-free analysis using model 4u, the average population VAF increased and BIC decreased considerably compared with model 2u (Table 4): VAF = 0.29 and 0.32 for the population in gaze- and eye-based models, respectively. Although the eye-based model gives a slightly higher VAF, the difference between the two was not significant. Figure 11 shows the fits of model 4u for our example cell L0702: the gaze- and eye-based inputs seemed to give excellent average fits through the "noise" in each example profile, but gave mean VAFs averaged over 40 gaze shifts of only 0.22 and 0.21, respectively. As was the case for our head-fixed analysis, the estimated biases were negative and the standard deviations of the parameters in this model were very large compared with those estimated in the other models. This implies that model 4u does not provide a useful description of IBN discharges during head-free gaze shifts.

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| FIG. 11.
Head-free analysis. Fits to the activity of SLIBN L0702, using model 4u (Table 3), for 3 example gaze shifts illustrated in Figs. 5, 6, 8, and 10. This model provided the best mean population VAF and incorporated a pole term for which initial firing rate B(t0) was estimated as a free parameter for every saccade. Models based on gaze (top panel) and eye (2nd panel) motor errors provided comparable fits ((heavy solid line)) of neuronal discharge (shaded curves), albeit with different parameter sets. With gaze motor error as input, VAF = 0.22; with eye motor error as input, VAF = 0.21. Values of parameters, estimated using 40 gaze shifts, provided below each model fit for this neuron. Three values listed above each fit represent the values B(t0). Accompanying gaze and eye motor-error traces (bottom 2 panels) have been shifted in time by estimated dynamic lead time td.
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An important finding of our analysis of head-fixed saccades was that the activity of primate SLIBNs and LLIBNs was better described by downstream models than by upstream models. These results were also true in the more general case of head-free gaze shifts. An upstream model with five parameters based on a nonlinear function of gaze or eye motor error (model 1u, VAF = 0.13) was not as good as a simple two-parameter downstream model 2d(VAF = 0.21 in Table 4 of Cullen and Guitton 1997b
) that included only velocity (gaze or eye) and fixed bias terms. This observation held, regardless of whether a neuron was a SLIBN or a LLIBN. When initial conditions were estimated for each saccade, the best upstream model 4u (43 parameter, VAF = 0.32) was still not as good as the comparable downstream model 6d and, furthermore, was comparable to the downstream model 8d (4 parameters). Hence, for head-free gaze shifts, just as for head-fixed saccades, IBN discharges were consistently better described by downstream rather than upstream models.
It should be noted that in the above head-fixed and head-free analyses of motor-error-based models, we focused on predicting the activity of IBNs during OND saccades and gaze shifts. Our downstream analyses found that OFFD discharges, when they were significant, were similar to OND discharges in that they encoded eye velocity, head-fixed (Cullen and Guitton 1997a
), and eye and head velocity, head-free (Cullen and Guitton 1997b
). Consequently, we expect that applying the upstream analyses detailed previously to the OFFD response of IBNs would likewise reveal that motor-error-based models are not as good at predicting IBN OFFD responses as eye velocity-based models containing a comparable number of terms.

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| FIG. 12.
On the basis of our results we propose a modified form of classic (Fig. 1A) local feedback model for head-fixed saccade generation. This schema gives mathematical steps necessary to convert a motor-error signal into the signal given by our downstream model 8d, which was our most plausible description of IBN discharges. A pure motor-error signal is first inverted and differentiated by box labeled d/dt; this converts e to . Bias term(r0 + r1 E) is then added to this signal to provide model 8d (companion paper I Cullen and Guitton 1997a ). This is the signal carried by BN represented by box labeled B. Possible sources of eye motor-error signal and (r0 + r1 E) signal are considered in text. In addition, the bias term (r0 + r1 E) is offset at each of the neural integrators. Removal of variable bias term is also required at the level of motor nucleus if transfer function between MNs and plant dynamics is linear. Inset: we found no differences between SLIBNs and LLIBNs with respect to the signals they carry; thus contrary to the schema illustrated in Fig. 1A, LLIBNs and SLIBNs are equally close to the e signal.
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Models of control of head-free gaze shifts and their predictions of burst characteristics
A number of models have been proposed to explain the generation of head-free gaze shifts. The earliest models emanated from the work of Bizzi and colleagues (Bizzi et al. 1971
; Morasso et al. 1973
; Whittington et al. 1984
) and proposed that BNs can be classified into two distinct groups: gaze-related and saccade-related BNs, depending on whether the number of spikes in their burst is proportional to gaze or eye amplitude, respectively. In companion papers I and II (Cullen and Guitton 1997a
,b
) we discussed extensively our results in this context and demonstrated that our IBNs cannot be classified according to this scheme, even though they carry both an eye and a head velocity signal in the head-free condition.
Tomlinson (1990)
proposed a modification to the Whittington et al. (1984)
model that used a feedback arrangement to generate a gaze motor-error signal that, in turn, drove BNs. As we have shown, our IBNs
both SLIBNs and LLIBNs
do not carry an eye or gaze motor-error signal; the VAF between firing frequency and motor error is best explained by a correlation between frequency and velocity and the fact that eye velocity is, in turn, correlated with eye motor error. In a more recent model, Philips et al. (1995) proposed four types of BNs: command, eye, head, and gaze cells. Of these cell types, the first codes eye motor error, head-fixed and head-free; the second codes eye movement dynamics, head-fixed and head-free; the third codes head dynamics; and the fourth codes gaze motor error. Taken together the results of companion papers I-III (this paper; Cullen and Guitton 1997a
,b
) indicate that our IBNs do not fall into any of these proposed categories.
The model proposed by Galiana (Galiana and Guitton 1992
; Guitton et al. 1990
) successfully predicts the discharge properties of our BNs. This model uses only one type of BN and a gaze feedback system in which the burst generator is driven by both gaze motor error (e.g., from the SC) and phasic signals from other sources also under the control of the error signal. In particular, this model predicts that the relationship between number of spikes (NOS) and gaze amplitude should be consistently better than the relationship between NOS versus eye or head amplitude during head-free gaze shifts. This result was the case in the present analysis (Cullen and Guitton 1997b
) as well as in our analysis of cat IBNs (Cullen et al. 1993
). We will describe in more detail the predictions of this model in a subsequent paper (unpublished manuscript).
Conclusions
In this series of papers we have used the system identification methods described in Cullen et al. (1996)
to analyze and compare the discharges of both SLIBNs and LLIBNs in terms of downstream models (companion papers I and II Cullen and Guitton 1997a
,b
) and upstream (or motor-error-based) models in the head-fixed and head-free monkey. The approach taken in these studies permitted an objective evaluation of the discharges of IBNs by using these two general classes of models.
For the head-fixed condition, on the basis of the results of both the present paper and companion paper I (Cullen and Guitton 1997a
), we conclude that both SLIBN and LLIBN discharges encode saccade dynamics best. For our population this was expressed, on average, by model 8d
Thus if we conserve the feedback loop arrangement of Fig. 1A (i.e., the hypothesis that a signal proportional to eye motor error lies upstream of the burst generator) it is necessary to postulate some signal processing between
e and B. The mathematical steps necessary to convert
e to model 8d are expressed in Fig. 12. Recall from the INTRODUCTION that
e = 
e. Assuming that there exists a pure motor-error signal, then mathematically it is necessary to subject it to a sign inversion and differential in the box labeled "
d/dt," after which the bias and amplitude dependent terms r0 + r1
E are added. The output of B is described by model 8d. The manner in which the signal is treated at the level of the MNs is discussed in companion paper I (Cullen and Guitton 1997a
).
For the head-free condition, the processing is identical to that shown in Fig. 12 except that the motor-error signal is now gaze motor error rather than eye motor error and the output of B is B(t) = r0 + r1
E + b1
+ g1
. The manner in which this complex signal is generated upstream of B is highly speculative and will not be considered here.
The question arises as to where and how the generation of signals proposed by Fig. 12 occurs in the brain stem oculomotor circuitry. The SC is an important element in this signal processing stream because it is known to generate, in the head-fixed animal, either the initial eye motor error (Guitton 1991
, 1992
; Sparks and Mays 1990
) or instantaneous eye motor error (Munoz and Wurtz 1995
; Waitzman et al. 1991
). Similarly, the SC may generate either an initial gaze motor-error signal (Freedman and Sparks 1997
) or an instantaneous gaze motor-error signal (Munoz et al. 1991
) in the head-free animal. It has long been accepted that the SC of the monkey projects to LLBNs (Raybourn and Keller 1977
). As mentioned in an earlier section, it has been demonstrated that cat SLIBNs receive monosynaptic projections from SC output cells (Chimoto et al. 1996
). This finding provides anatomic evidence that LLIBNs are not closer to the SC than SLIBNs.
There is considerable recent evidence that the output neurons of the SC in the cat and monkey also encode eye or gaze velocity (Berthoz et al. 1986
; Lee et al. 1988
; Munoz et al. 1991
; Stanford et al. 1996
). Furthermore, as we have pointed out in the first section of the DISCUSSION, it is possible that SC output cells also carry a bias signal whose value varies inversely with saccade amplitude. Such an observation suggests that the collicular output already, at least in part, carries a signal of the form: r0 + r1
E + b1
during head-fixed saccades. The instantaneous eye motor-error signal that may also emanate from the SC may therefore be more implicated in stopping the saccade via, e.g., a SC fixation cell-omnipause neuron connection (Paré and Guitton 1994
; Paré et al. 1994
) and not in modulating BN firing frequency.
 |
ACKNOWLEDGEMENTS |
We are grateful to H. L. Galiana for many helpful discussions and to C. G. Rey for contribution to the development of analysis methods and software. We thank Dr. J.A.M. Van Gisbergen for a number of useful suggestions that helped improve this manuscript greatly. We also thank Dr. W. M. King for critically reading this manuscript.
This study was supported by the Medical Research Council of Canada, the National Institutes of Health, and the Human Frontiers Science Organization.
 |
FOOTNOTES |
Address for reprint requests: K. E. Cullen, Aerospace Medical Research Unit, 3655 Drummond St., Montreal, Quebec H3G 1Y6, Canada.
Received 13 August 1996; accepted in final form 2 July 1997.
 |
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