The Journal of Neurophysiology Vol. 78 No. 6 December 1997, pp. 3259-3282
Copyright ©1997 by the American Physiological Society
Analysis of Primate IBN Spike Trains Using System Identification Techniques. I. Relationship to Eye Movement Dynamics During
Head-Fixed Saccades
Kathleen E. Cullen and
Daniel Guitton
Aerospace Medical Research Unit and the Montreal Neurological Institute, McGill University, Montreal, Quebec H3G 1Y6, Canada
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ABSTRACT |
Cullen, Kathleen E. and Daniel Guitton. Analysis of primate IBN spike trains using system identification techniques. I. Relationship to eye movement dynamics during head-fixed saccades. J. Neurophysiol. 78: 3259-3282, 1997. The dynamic behavior of primate (Macaca fascicularis) inhibitory burst neurons (IBNs) during head-fixed saccades was analyzed by using system identification techniques. Neurons were categorized as IBNs on the basis of their anatomic location as well as by their activity during horizontal head-fixed saccadic and smooth pursuit eye movements and vestibular nystagmus. Each IBN's latency or "dynamic lead time" (td) was determined by shifting the unit discharge in time until an optimal fit to the firing rate frequency B(t) profile was obtained by using the simple model based on eye movement dynamics,B(t) = r + b1
(t); where is eye velocity. For the population of IBNs, the dynamic estimate of lead time provided a significantly lower value than a method that used the onset of the first spike. We then compared the relative abilities of different eye movement-based models to predict B(t) by using objective optimization algorithms. The most important terms for predicting B(t) were eye velocity gain (b1) and bias terms (r) mentioned above. The contributions of higher-order velocity, acceleration, and/or eye position terms to model fits were found to be negligible. The addition of a pole term [the time derivative of B(t)] in conjunction with an acceleration term significantly improved model fits to IBN spike trains, particularly when the firing rates at the beginning of each saccade [initial conditions (ICs)] were estimated as parameters. Such a model fit the data well (a fit comparable to a linear regression analysis with a R2 value of 0.5, or equivalently, a correlation coefficient of 0.74). A simplified version of this model [B(t) = rk + b1(t)], which did not contain a pole term, but in which the bias term (rk) was estimated separately for each saccade, provided nearly equivalent fits of the data. However, models in which ICs or rks were estimated separately for each saccade contained too many parameters to be considered as useful models of IBN discharges. We discovered that estimated ICs and rks were correlated with saccade amplitude for the majority of short-lead IBNs (SLIBNs; 56%) and many long-lead IBNs (LLIBNs; 42%). This observation led us to construct a more simple model that included a term that was inversely related to the amplitude of the saccade, in addition to eye velocity and constant bias terms. Such a model better described neuron discharges than more complex models based on a third-order nonlinear function of eye velocity. Given the small number of parameters required by this model (only 3) and its ability to fit the data, we suggest that it provides the most valuable description of IBN discharges. This model emphasizes that the IBN discharges are dependent on saccade amplitude and implies further that a mechanism must exist, at the motoneuron (MN) level, to offset the effect of the bias and amplitude-dependent terms. In addition, we did not find a significant difference in the variance accounted for by any of the downstream models tested for SLIBNs versus LLIBNs. Therefore we conclude that the eye movement signals encoded dynamically by SLIBNs and LLIBNs are similar in nature. Put another way, SLIBNs are not closer, dynamically, to MNs than LLIBNs.
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INTRODUCTION |
To generate a saccadic eye movement, motoneurons (MNs) are required to provide signals to the extraocular muscles that overcome the restraining forces of the orbital tissues and drive the eye quickly to its new position and hold it there. Robinson (1964)
proposed that the following fourth-order differential equation provides an accurate description of MN discharge in this process
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(1)
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where MN(t) is MN firing rate, E is the time-varying eye position, letters with an overdot are derivatives (e.g., is eye velocity), and c, r, b0, b1, b2, b3, b4 are constants. The key predictions of this formulation were as follows: 1) MNs are required to generate a brief burst (a "pulse") of action potentials to overcome the viscous drag of the orbital tissues (b1) to drive a saccade; 2) a lower frequency tonic discharge (a "step") is required to balance the elastic restoring forces of the orbital tissues to hold the eye steady in its new position (b0E); and 3) between the pulse and the step a slow decay or "slide" component exists, lasting until ~250 ms after the saccade, which just offsets the relaxation of the slow viscoelastic elements in the orbit [c
N(t)]. Subsequent extracellular recordings have confirmed these predictions and have demonstrated the pulse-slide-step nature of the MN signal (Delgado-Garcia et al. 1986; Fuchs and Luschei 1970; Fuchs et al. 1988; Goldstein and Robinson 1984; Stahl and Simpson 1995).
A number of investigators have attempted to simplify Eq. 1 by retaining only the most significant terms. For example, Keller (1973) and Van Gisbergen et al. (1981) proposed a second-order model including eye position (E), velocity (), and acceleration (Ë) terms. Goldstein and Robinson (1984) and Optican and Miles (1985) also proposed a second-order form but included as well a pole term [proportional to the derivative of the neuron's firing rate, N(t)] to explain the exponential decay of MN firing following a saccadic eye movement (the "postsaccadic slide"). More recently, Fuchs et al. (1988) demonstrated that a third-order model, with only inertia of the globe neglected in the derivation of the model Eq. 1, provides an adequate description of MN(t). However, Stahl and Simpson (1995) noted that the rejection of the second-order approximation of Eq. 1 may be premature because of questionable constraints placed on parameters by Fuchs et al. (1988).
The most prominent simplification to Eq. 1 is the first-order model (Robinson 1970)
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(2)
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In this "pulse-step" model, the burst from the MN pool (the dominant signal during saccades being proportional to eye velocity) drives the eye to the new position, whereas the tonic activity of the neuron holds the eye steady in its new position; there is no slide component [i.e., the model does not include the pole term cN(t)]. Although this model is not accurate, its wide-spread use prompted the hypothesis that the position and velocity terms are provided by separate inputs onto MNs, namely the integrator and burst generator signals, respectively. The finding that higher-order models are necessary for describing the downstream relationship between MN(t) and eye movement trajectory begs the question as to whether it is still useful to consider each term in Eq. 1 as a separate neural signal. Mounting evidence suggests that this is not the case. Indeed, neurons in the vestibular nucleus (VN) that project to MNs carry both position and velocity information and perhaps even the pole term (discussed in Fuchs et al. 1988). It is natural to ask whether other inputs to MNs also carry mixed signals. In this paper we consider the burst generator in this context.
For horizontal saccades, excitatory burst neurons (EBNs) project to the ipsilateral abducens nucleus (ABD) to activate it during saccades and act together with a group of inhibitory burst neurons (IBNs) that silence antagonist abducens MNs (Hikosaka and Kawakami 1977; Hikosaka et al. 1978; Igusa et al. 1980; Sasaki and Shimazu 1981; Scudder et al. 1988; Strassman et al. 1986a,b; Yoshida et al. 1982). IBNs and EBNs discharge vigorously or "burst" during ipsilateral saccadic eye movements and are thought to carry similar signals, because 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).
In recent models of the head-fixed saccadic system (Fuchs et al. 1985; Jürgens et al. 1981; Robinson 1975; Scudder 1988), a saccade is controlled by a feedback loop wherein an eye motor-error signal (desired change in eye position minus actual change in eye position) is the "upstream" signal that activates burst neurons (BNs) whereas the "downstream" output of these neurons drives MNs (Fig. 2B). From the downstream perspective and within the context of the simple model Eq. 2, BNs provide the eye velocity signal (for review see Fuchs et al. 1985). Accordingly, most prior analyses of BN activity have focused largely on either the relationship between peak eye velocity and peak discharge frequency or global characteristics of the discharge such as the relationship between number of spikes (NOS) and horizontal component of saccade amplitude (Kaneko and Fuchs 1981; Scudder et al. 1988; Strassman et al. 1986a,b; Yoshida et al. 1982). These are static relationships that do not provide information concerning the time-varying relationship between BN spike trains and resultant eye movement trajectories. Only the study by Van Gisbergen et al. (1981) considered time-varying or dynamic relationships. They asked whether the difference in the ON-direction (OND) and OFF-direction (OFFD) responses of BN signals could provide some of the terms in Eq. 1 other than velocity and examined, with phase-plane trajectories, the influence of acceleration. Using a method that required removing the acceleration term by trial and error, they suggested that the burst discharge is dynamically related to both eye velocity and acceleration.
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| FIG. 2.
A: location of recorded IBNs was verified with the aid of a marking lesion in monkey L. Lesion was made in area located just dorsal to the region where putative IBN neurons were recorded. Dotted vertical line, path of electrode; filled circles, locations of cells previously recorded at the same location. B: classic local feedback model for saccade generation proposed by Jürgens et al. (1981) as a modification to Robinson's (1975) model.  E*, eye motor efference signal giving angular rotation of the eye that has occurred since onset of saccade is obtained by integrating burst neuron (BN; B) output that encodes eye velocity. T, desired angular rotation of the eye. Difference between T and E* provides an estimate of the eye motor error [ (t)] relative to desired change in eye rotation. Saccade is triggered when omnipause neurons (OPNs) stop firing, so their inhibitory influence on the BN pool is removed. Activity of B can be studied from two perspectives: influence of B on the eye movement trajectory (downstream analysis) and the response of B to eyemotor-error signal driving it (upstream analysis). , summing junction; E(t), resultant eye position.
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This paper is the first in a series of three companion papers wherein we analyze in detail the discharges of one class of BNs, IBNs. We have focused on IBNs because 1) their anatomic location is easily defined, which facilitates microelectrode access; 2) they show an excellent static relationship to saccade metrics; and 3) they are driven by EBNs and EBNs appear to be a primary target of the contralateral projections of IBNs (Sasaki and Shimazu 1981; Strassman et al. 1986a,b). In this paper we employ objective computer-based optimization algorithms to characterize the discharges of BNs in the context of Eq. 1. Our objective is to demonstrate which of the terms in Eq. 1 are important for describing the discharge of this class of premotor neurons. Using algorithms founded in system identification theory (Ljung 1987)we sought to construct the mathematical model that best describes the input-output relationship between an IBN's firing rate and properties of the saccadic eye movement trajectory. In the second companion paper of this three paper series (Cullen and Guitton 1997a) we apply these techniques to the analysis of IBN discharges in the head-free monkey to discover whether burst discharges are best correlated to eye, head, or gaze trajectories. In the third companion paper of this series (Cullen and Guitton 1997b) we model IBN firing rate as a function of the upstream motor-error signal (Fig. 2B) and compare upstream and downstream models in terms of how well they account for the neural discharge pattern.
Our analysis method accomplished the following: 1) provided an objective method for calculating lead time and for comparing the goodness-of-fit of different models; 2) provided an indication of whether it is warranted to increase the goodness-of-fit by using more complex models that have more free parameters; and 3) permitted a quantitative evaluation of the relationship between estimated initial firing rates and/or biases and movement characteristics such as initial and final eye position, saccade amplitude, and peak eye velocity. The results of our analysis indicate that 1) contrary to the current assumption, primate short-lead IBNs (SLIBNs) and long-lead IBNs (LLIBNs) encode eye movement dynamics equally well, and 2) eye velocity-based models that included a bias term (r), which is inversely related to the amplitude of the saccade, provide a good description of BN spike trains.
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METHODS |
Two monkeys (Macaca fascicularis) were prepared for chronic extracellular recording. The surgical preparation of the animals and the methods used for obtaining the extracellular recordings from brain stem neurons were similar to those previously described by Cullen and McCrea (1993). The IBNs described in this study were identified on the basis of their characteristic burst discharge during rapid orienting horizontal eye movements in the ipsilateral direction and their location in the IBN area caudal and ventral to the ABD (see descriptions for cat in Hikosaka and Kawakami 1977; Kaneko and Fuchs 1981; Yoshida et al. 1982; and monkey in Scudder et al. 1988). Figure 1 illustrates the anatomic distribution of the entire population of IBNs recorded in this study. The location of these neurons was determined by using reconstructions of electrode tracts, stereotaxic locations relative to the ABD, and later histological confirmation. Figure 2A illustrates the marking lesion made in monkey L that was made just dorsal to the location where two IBNs were recorded. Cells were also characterized as BNs on the basis of their burst responses during quick phases of vestibular nystagmus evoked during horizontal rotation of the vestibular turntable and also by their lack of response during slow phases of nystagmus and smooth pursuit eye movements.
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| FIG. 1.
Reconstructed locations of the entire population of inhibitory burst neurons (IBNs) analyzed in this study are illustrated on stereotaxic cross sections of the macaque brain stem. Sections were drawn at 400-µm intervals. Filled circles, short-lead IBNs (SLIBNs); open circles, long-lead IBNs (LLIBNs); ABN, abducens nucleus; PHN, nucleus prepositus hypoglossi; nVII, 7th nerve.
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Experimental paradigms and data collection
During the experiments the head-fixed monkey was seated in a primate chair that was fixed to the suprastructure of a vestibular turntable. For a juice reward the monkey was trained to orient to a target light that was projected onto a cylindrical screen surrounding the turntable. The spot of light was generated by a HeNe laser that was projected onto the screen via a system of two mirrors, mounted above the monkey's head, whose positions were controlled by galvanometers (General Scanning). The screen onto which the visual target was projected was monochromatic and evenly illuminated. The target was stepped between positions ±5, 10, 20, and 35° horizontal relative to the straight ahead position. The monkeys were also trained to orient to a food target that appeared unexpectedly on any side of an opaque screen facing the monkey (the "barrier paradigm," see Guitton et al. 1984 for details). This procedure provided us with a rich variety of saccade vectors, starting positions, and, for saccades of the same vector, a variety of velocity profiles.
On average, the saccades in this study were slower than those reported previously in the monkey (e.g., Fuchs 1967). Figure 3A illustrates the relationship between saccade duration and saccade amplitude for the data set obtained from one animal (monkey H). As expected, saccade duration was related to saccade amplitude; however, there was considerable variation across the data set. For example, the duration of a 30° saccade ranged from 60 to 160 ms. This variability was a result of the wide range of velocities that the monkey utilized to generate a saccade of a given amplitude (Fig. 3B). The peak eye velocity generated during a 30° saccade ranged from 250 to 800°/s. In general, the saccadic eye velocities in our study were highly variable and often similar to those that were reported in the literature for humans (Robinson 1964). The difference between the metrics of the saccades in our study and other primate studies may have arisen as a result of two factors: 1) the more natural behavioral paradigms employed in our study resulted in monkeys producing less "stereotyped" saccade trajectories to orient to targets (
70% of the saccades in each neuron's data set) and 2) M. fascicularis were used in this study, whereas most previous studies have used M. mulatta. Although these two types of monkeys have a similar oculomotor range, it is more difficult to obtain consistent behavioral performance in M. fascicularis monkeys (Cullen, unpublished observation).
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| FIG. 3.
Relationships between (A) saccade duration and saccade amplitude and (B) peak saccade velocity and amplitude. Data points represent all the saccades that were analyzed for monkey H.
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For this paper BN activity was recorded during rapid voluntary orienting eye movements (saccades) made with the animal's head held fixed. Data for the head-free condition are reported in Cullen and Guiton 1997a. A 386 microcomputer equipped with a data translation DT2825 data acquisition board, a DATAQ waveform scroller board, and a CED1401 peripheral device (Cambridge Electronic Design) was used for controlling visual and vestibular stimuli, for acquiring data, and for on-line display and analysis of data. Eye movements were recorded by using the magnetic search coil technique. The recorded position signals were low-pass filtered (250 Hz, 8 pole Bessel) and sampled at 1,000 Hz by the computer. The low-pass filtering shifted the recorded position signals backward in time by 2 ms; the resultant delay was taken into account in our subsequent analysis.
Extracellular recordings were obtained with enamel insulated tungsten microelectrodes (7-10 M
impedance, Frederick Haer), which were inserted into the cerebellum through a 22-gauge guide tube. The guide tube was positioned with an X-Y stage (Narishige) that was attached to the monkey's recording well. The microelectrode was advanced along the Z axis into the brain stem with a miniature hydraulic drive (Narishige). Spike potentials were amplified and filtered (band-pass 400 Hz to 10 kHz). A unit's isolation was continuously monitored, so that the level of a windowing circuit was set at an appropriate level to generate a pulse coincident with the rising phase of a spike. This pulse was sent to the event channel of the intelligent peripheral (CED1401) device, logging the event times of the neuronal action potentials (see Cullen and McCrea 1993).
Position signals and the single unit activity were simultaneously recorded on Digital Audio Tape (DAT) tape (20 kHz maximum sampling frequency). The DAT recorded data were often replayed off-line, so that the unit data could be carefully retriggered [this option was particularly important for the head-free recordings presented in the companion papers (Cullen and Guitton 1997a,b)].
Analysis of BN discharges
During off-line analysis sampled eye movement traces were digitally filtered (Matlab) at 125 Hz, because Fourier analysis of ocular saccades has revealed little power above 50 Hz (for review, see Cullen et al. 1994, 1996). Eye velocity was derived digitally from position data. A nonlinear function relating the NOS in the unit discharge to polar angle of the saccade was employed to determine the preferred direction of each IBN (Scudder et al. 1988). The onset and offset of a saccade were defined using a 20°/s eye velocity criterion. Burst duration was defined as the time between the onset and offset of the burst. Burst onset was defined as when the first spike of the burst occurred. Because these neurons frequently generated a few additional spikes following a saccade or gaze shift, burst offset was defined as when 95% of the spikes in the burst had occurred. A spike density function, in which a Gaussian function [5 (SD) ms] was convolved with the spike train, was used to represent the neural discharge. Choosing a Gaussian of this width effectively low-pass filtered the neural discharge so that its frequency content was similar to that of the coincident eye movements (see Cullen et al. 1996 for details). In a previous study (Cullen et al. 1996) we had determined that decreasing the width of the Gaussian by as much as 50% had little effect on the results of our dynamic analysis (latencies and parameters were not markedly changed). Only those saccades in which the amplitude of the vertical position component was less than one-third of the horizontal component were studied; i.e., the amplitude of the horizontal component was between 95 and 100% of the overall movement amplitude. We considered these to be horizontal saccades.
Dynamic models of BN firing rate
We used system identification techniques (Ljung 1987) for the analysis of IBN spike train dynamics. The use of these algorithms permitted optimization across multiple eye trajectories simultaneously and provided a method to objectively determine 1) how well a given model predicts a neuron's discharge pattern; 2) whether increasing the complexity of a model is permissible; 3) what a BN's dynamic latency is; and 4) the relationship between initial conditions (ICs) and quantities describes saccade trajectories. Details of these techniques can be found in Cullen et al. (1996) wherein we describe system identification techniques as they relate to BN analysis. IBN discharges were evaluated in terms of the currently favored schema (Fig. 2B) (Jürgens et al. 1981) of the saccade control system that states that burst firing rate is related simultaneously to the upstream motor-error signal (Cullen and Guitton 1997b) and to the downstream dynamics of the saccade trajectory (this paper; Cullen and Guitton 1997a). In RESULTS of this paper we compare different linear and nonlinear dynamic models that predict IBN firing rate on the basis of downstream signals during head-fixed saccades.
METHODS FOR PARAMETER ESTIMATION.
For downstream models, BN discharge was the output and a function of parameters describing saccade trajectory. Therefore these were inverse models because burst frequency was the output of the system, rather than the input as it is in Fig. 2B. This analysis procedure was necessitated by the estimation techniques that required that the noise be in the output of the model formulation; this is conveniently placed in the BN discharge (Cullen et al. 1996). By using the IBN firing rate as the output, we were able to make comparisons between the fits computed for different plausible models (Tables 2 and 3). For each model optimal fits were made to an ensemble of ~40 saccades of different amplitudes between 5 and 45° to either the left or right, depending on the neuron's OND. We chose to characterize the discharges of IBNs in terms of horizontal eye movements, because IBNs project to the MNs that innervate the muscles that are responsible for producing horizontal eye movements (i.e., the lateral and medial recti). Note, in a preliminary analysis we analyzed the responses of several cells along their optimal response direction (see RESULTS) and found that goodness-of-fit, relative importance of estimated parameters, and the optimal latencies estimated in our models were not markedly changed. Generally, the starting positions of the saccades in each data set were well distributed over a range of ±20° on either side of the center position. Optimal fits to the neuron's discharge during OFFD saccades were carried out separately.
ESTIMATION OF DYNAMIC LATENCY.
We shifted the burst in time relative to the saccade by the latency (the optimal dynamic delay td) that provided the best fit of the BN firing rate when a simple but accurate downstream model was used (see section on lead time in RESULTS). It is important to note that the entire portion of the temporally shifted spike train that was coextensive with the saccade duration was used to fit the model. Thus every spike within this interval contributed to the latency calculation. When the BN was shifted by the appropriate td, the main portion of the burst was typically aligned with the duration of the saccade. Consequently, the action potentials discharged by the neuron before the main portion of the burst (the burst "prelude") were typically rejected by the optimization algorithms and were not included in the fit. The estimated delay (td) was then applied to the other more complex models (i.e., it was a fixed parameter) (see Cullen et al. 1996 for details).
COMPARISON OF MODEL FITS.
The goodness-of-fit of each model was determined by considering the root mean square (RMS) of the fit errors and the variance accounted for (VAF). The VAF was defined as [1 
(var/SD)] where var is the variance of the fit's error and SD is the standard deviation of the data set about its mean. Using VAF allowed us to normalize the goodness of a model's fit across neurons. It should be noted that the VAF for a linear model is equivalent to the square of the correlation coefficient (R; i.e., a VAF of 0.5 indicates that 50% of the variability in the unit's firing rate is explained by the model and would correspond to R = 0.71). We tried models of different complexities. A more complex model has more adjustable parameters and hence has the potential for providing a better fit to the data. The question then arises as to whether increasing model complexity is warranted. As described in Cullen et al. (1996) we answered this question objectively by using the Bayesian information criterion (BIC) (Schwarz 1978); a "cost index" that indicates whether increasing the complexity of a model is justified when the accompanying decrease in the error of the fit is taken into account. We computed the BIC for each model estimation; a decrease in the BIC value indicated that an increase in model complexity was warranted. If this index did not decrease when a more complex model was used, then the model was considered to be no better at describing the data than the more simple model.
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RESULTS |
We present data from twenty-eight IBNs that were recorded during saccadic eye movements in two head-fixed monkeys. A characterization of the activity of the cells in terms of the trajectory of head-fixed saccades is presented in this paper. In our companion papers we 1) analyze the activity of these neurons in the head-free condition and compare the results with the head-fixed data (Cullen and Guitton 1997a) and 2) consider IBN discharges in terms of their response to a motor-error signal (Cullen and Guitton 1997b).
General discharge characteristics
Neurons were identified as IBNs on the basis of their anatomic location (see METHODS) and activity during saccadic and horizontal smooth pursuit eye movements and horizontal vestibular nystagmus. The neurons were identified on the basis of their characteristic burst discharge during horizontal saccadic eye movements in the ipsilateral direction. We determined that the cells were not the "burst-driver" neurons that were reported in a similar region in the cat (Ohki et al. 1988) by verifying that they were inactive during slow phases of vestibular nystagmus in both directions. It was also verified that these neurons were inactive during smooth pursuit eye movements and therefore that none could be categorized as the reticular formation smooth pursuit cells previously described by Tomlinson and Brance (1991). The neurons in this study were categorized as either SLIBNs (n = 16) or LLIBNs (n = 12), depending on whether the mean period between the onset of the first spike and the onset of eye velocity was 
15 ms or >15 ms, respectively; we used this arbitrary definition to be compatible with the prior IBN study of Scudder et al. (1988).
The discharge of a typical SLIBN (H0415) is illustrated in Fig. 4, A-C. This neuron generated a strong burst of firing in the OND (ipsilateral) and weak burst in the OFFD (contralateral), with the first spike leading ipsilateral saccadic eye movements on average by 11 ± 2.8 (SD) ms (Fig. 4A). In Figs. 4 and 5 and all subsequent figures showing spike frequency, the spike discharge was shifted to the right by a time equal to the calculated dynamic lead time of each neuron (see Estimation of lead times). The duration of this neuron's burst was tightly correlated with the duration of the saccade (slope = 0.9; R = 0.91) and the total NOS during a burst was proportional to the horizontal component of the amplitude of the saccadic eye movement (slope = 1.5; R = 0.92). During smooth pursuit eye movements this cell only discharged during saccades and did not discharge during slow-phase smooth pursuit movements in either the ipsilateral or contralateral directions (Fig. 4B). During vestibular stimulation this cell discharged only during quick phases and did not discharge during ipsilateral or contralateral slow phases (Fig. 4C).
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| FIG. 4.
Behavior of typical SLIBN (H0415; A-C) and LLIBN (H0421, D-F) during saccadic eye movements (A and D), smooth pursuit (B and E), and vestibularly induced eye movements (C and F). A strong burst occurred whenever saccade, or vestibular quick phase, was directed toward the ipsilateral side. Duration of each neuron's burst was correlated with the duration of the saccade (R = 0.91 and R = 0.74, respectively) and total number of spikes (NOS) generated during a burst was proportional to horizontal component of the amplitude of saccadic eye movement (R = 0.92 and R = 0.62, respectively). Note that the IBNs did not discharge in response to slow-phase smooth pursuit or vestibularly induced eye movements in either the ipsilateral or contralateral direction. Top trace in each panel, output of spike threshold discriminator. SD, spike density function showing firing frequency distribution; h, horizontal eye velocity; Eh, horizontal eye position; Ev vertical eye position; Hh, horizontal head position; Th, horizontal target position.
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| FIG. 5.
IBN firing rate behavior during ON-direction (OND) and OFF-direction (OFFD) saccades for 3 example IBNs; SLIBN L0702 and LLIBNs H0409 and H0925. A, B: units L0702 and H0409 were typical of 1/2 the cells in our population. For these IBNs there was a striking similarity in the shape of the neural firing frequency and eye velocity trajectories during horizontal saccades directed ipsilateral to the recording site (cell's OND), whereas the cells discharged few, if any, spikes during contralateral saccadic eye movements in the cell's OFFD. C: in contrast, the other 1/2 of cells in our population demonstrated a substantial OFFD response. OFFD response of unit H0925 was particularly robust and resembled an attenuated version of the OND response. Vertical line, 400 spikes/s, 400°/s, and 40°; SD, spike density function; , horizontal eye velocity; E, horizontal eye position.
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Figure 4, D-F, illustrates the discharge of a typical LLIBN (H0421). This neuron generated a preamble of lower frequency discharge followed by a burst of firing, with the first spike in the preamble leading ipsilateral saccadic eye movements by 21 ± 7.9 ms (Fig. 4D). The duration ofthis neuron's spike train was correlated with the duration of the saccade (slope = 0.9; R = 0.74) and the total NOS generated during the train was proportional to the horizontal component of the amplitude of the saccadic eye movement (slope = 0.9; R = 0.62). As was the case for SLIBNs, this neuron did not discharge in response to slow-phase smooth pursuit or vestibularly induced eye movements in either the ipsilateral or contralateral directions (Fig. 4, E and F).
IBN preferred direction
The IBNs in this study burst vigorously for ipsilaterally directed saccades, and they typically discharged far fewer spikes during contralateral (OFFD) and pure vertical saccades. Figure 5 shows examples of discharges of three neurons we selected to illustrate our data throughout this paper and the companion papers (Cullen and Guitton 1997a,b): SLIBN L0702, LLIBN H0409, and LLIBN H0925. The latter neuron had the best OFFD discharge in our population. Note the clear correlation between burst frequency and saccade velocity for discharges in the ONDs and OFFDs for H0925 and in the OND for cells L0702 and H0409. A precise measure of a neuron's optimal saccade direction was calculated by fitting a nonlinear function to relate the NOS in a unit's discharge to the polar angle of the saccade (see METHODS). The saccade amplitudes used to determine the preferred direction of each cell in the study ranged from 15 to 25°. The preferred directions in our population (Fig. 6) varied from +29° (upward) to 30° (downward). The mean preferred direction was 1.0 ± 15.2° closely aligned with the horizontal stereotaxic plane (~15° upward from the plane of the horizontal canals). Our analysis of eye movements that were horizontal, or nearly so, is consistent with the mean preferred horizontal direction of IBNs (see METHODS).
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| FIG. 6.
Preferred direction of each of the IBNs in our population is illustrated in polar coordinates; length of vector indicates sensitivity of each IBN measured in spikes/deg, according to the scale on left. Mean preferred direction of the entire sample of IBNs was 1° (range +29 to 30°) suggesting that the direction of the population response of these cells was very closely aligned with ipsilateral horizontal. Average population vector is represented by the heavy arrow. Saccades included in this analysis varied in amplitude over a range of 15-25°. Each unit's optimal direction was determined by fitting the data with the sum of 2 sinusoids by using a nonlinear least squares algorithm. Inset: direction tuning of an example IBN (SLIBN H0409) is illustrated by a plot of the NOS vs. saccade direction.
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Estimation of lead times
The time by which a BN's discharge led eye saccades was calculated by two methods. Histograms illustrating the results of applying each method are shown in Fig. 7. In Fig. 7A the lead time was determined by calculating the period between the onset of the first spike and the onset of eye velocity. This method was used in many previous studies of IBNs and as a criterion to categorize the cells as either SLIBNs or LLIBNs (e.g., Scudder et al. 1988). In the second method (Fig. 7B), by using our optimization algorithms described in detail in Cullen et al. (1996), a neuron's dynamic lead time (td) was left as an unknown and determined by shifting progressively the unit discharge in time relative to the saccade until a best fit was obtained for models 2d and 6d (Table 2)
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and
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where r, the bias term; b1, the velocity gain; b2, the acceleration gain; c, the coefficient of the pole; and td are all varied until RMS error is minimized for the whole data set. These equations are accurate dynamic models of firing rate, model 2d being the simplest and 6d the most complex to analyze. Model 2d has three parameters to optimize, whereas 6d, having the pole term 
(t), has 44 (when 40 saccades are analyzed). For two neurons in which we studied lead time in detail, Eq. 3 provided a clear minimum error for a particular value of td. With Eq. 4 it was more difficult to optimize the value of td, because the influence of this quantity in the optimization process was diluted by the larger number of parameters (Cullen et al. 1996). Model 6d was less sharply "latency tuned" than model 2d. Nevertheless, we found no difference between the lead time estimates given by Eqs. 3 and 4. Accordingly, for all neurons we will present data obtained from Eq. 3. Figure 7B shows that the dynamic method provides latency estimates that have far less scatter than those obtained by using the onset of the first spike(Fig. 7A).
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| FIG. 7.
Histograms illustrating lead time by which IBN discharges led head-fixed saccades. Lead time was obtained by 2 methods: (top) time between onset of 1st spike and onset of eye velocity; (bottom) dynamic lead time td obtained by shifting unit discharge in time until optimal fit was obtained for the model, B(t) = r + b1(t td). On average, the burst of monkey IBNs led ipsilateral head-fixed saccades by 17.0 ms (SD = 9.3) and 13.4 ms (SD = 3.1) by using onset of 1st-spike and dynamic estimation methods, respectively.
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The relationship between the lead times calculated by using each method is plotted in Fig. 8. The values obtained from the two methods were not well correlated for the 28 primate neurons we analyzed (R = 0.51). However this relationship improved once a single outlier (the value within parentheses) was removed from the analysis (R = 0.67). A continuity can be seen between the dynamic lead times estimated for SLIBNs (
) and LLIBNs (
). The time-of-first-spike method yielded a mean lead time for SLIBNs that was statistically similar to the one using dynamic analysis with Eq. 3 (11.6 ± 3.8 ms vs. 11.8 ± 2.7 ms, respectively). However, for LLIBNs the first-spike method yielded significantly longer lead times (24.2 ± 9.7 ms vs. 15.5 ± 2.2 ms, respectively). For the entire population of neurons, including SLIBNs and LLIBNs, the mean lead times were 17.0 ± 9.3 ms and 13.4 ± 3.1 ms for the first-spike and dynamic methods, respectively. The difference between these values was significantly different (Student's t-test,P < 0.01). It is noteworthy that the mean lead time estimated dynamically for neurons classified as SLIBNs was significantly less (P < 0.001) than that estimated, using the same method, for neurons classified as LLIBNs [11.8 ± 2.7 ms vs. 15.5 ± 2.2 ms, respectively, a difference of 3.7 ms to be compared with the 8 ms estimated by Scudder et al. (1988)]. Only five neurons (4 LLIBNs and 1 SLIBN) had dynamic lead times longer than 15 ms.
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| FIG. 8.
Relationship between lead time values obtained by 2 separate methods that were used in this study: the onset of 1st-spike and dynamic estimation method. The 2 lead time measures were not well correlated for total sample of IBNs (n = 28; R = 0.51). However, this relationship improved considerably (R = 0.67) when a single outlier () was removed from the analysis. This analysis shows that SLIBNs () tended to have shorter dynamic lead times than LLIBNs (). Distinction between SLIBNs and LLIBNs was made according to relative timing of 1st spike to onset of saccade.
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Relationships between IBN activity and saccade metrics
Linear relationships between the duration of IBN discharges and ipsilaterally directed saccadic eye movements were reported in a number of previous studies and our data are comparable. Figure 9, A and B, illustrates the relationship between burst duration and saccade duration for neurons classified as SLIBNs and LLIBNs, respectively. The insets show data obtained from neurons that we selected as examples and whose data are illustrated throughout the paper. The heavy lines show the population means whose slopes, intercepts, and correlation coefficients are given in Table 1. The slopes for SLIBNs and LLIBNs were not significantly different (P = 0.11), but the intercepts were (P < 0.05). In Table 1 we also show data from Scudder et al. (1988). Both data sets concur; the average burst duration of SLIBNs closely approximates the duration of the associated saccade, whereas for LLIBNs the discharge preamble produces spike trains that are longer than saccades. The present analysis suggests that the duration of this preamble, for a given cell, is constant (because the slope is ~1) and independent of saccade duration (and amplitude).
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| FIG. 9.
A: relationship between burst duration and saccade duration in the head-fixed condition for population of SLIBNs. Heavy line, average regression fit of this relationship (slope = 0.87, intercept = 10.6). Inset: data for our example SLIBN (unit L0702). B: relationship between burst duration and saccade duration in the head-fixed condition for population of LLIBNs. Heavy line, average regression fit of this relationship for LLIBNs (slope = 0.97, intercept = 26.5). Inset: data for our example LLIBN (unit H0409).
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A number of studies have reported a strong relationship between the NOS in an IBN's discharge and the horizontal amplitude of ipsilaterally directed saccades. To obtain the NOS in the portion of the burst that directly drives saccades, we shifted the spike train in time by the dynamic lead time (td, calculated by using model 2d of Table 2) and counted the NOS during the discharge period that was aligned with each saccade (e.g., between the vertical dashed lines in Fig. 4A). The NOS was proportional to the amplitude of the horizontal component of the saccadic eye movement for each neuron. (Because the saccades we analyzed were nearly horizontal, using the overall amplitude did not influence this conclusion.) This relationship is illustrated for SLIBNs and LLIBNs in Fig. 10, A and B, respectively. The heavy lines show the population means whose slopes, intercepts, and correlation coefficients are shown in Table 1, as well as the equations given by Scudder et al. (1988). The slope and intercept of the mean lines were not significantly different for SLIBNs and LLIBNs (P = 0.41 and 0.11, respectively) and are also similar to Scudder et al. (1988). We conclude that our method of defining the interval over which NOS were counted, by using an objective estimate of lead time, preserves the strong relationship that was observed in previous studies between total NOS in a burst and the horizontal amplitude of a saccade.
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| FIG. 10.
A: relationship between total NOS in a discharge and saccade amplitude in the head-fixed condition for population of SLIBNs. Heavy line, average regression fit of this relationship (slope = 1.1, intercept = 9.3). Inset: data for our example SLIBN (unit L0702). B: relationship between total NOS and saccade amplitude in the head-fixed condition for population of LLIBNs. Heavy line, average regression fit of this relationship for LLIBNs (slope = 0.95, intercept = 11.9). Inset: data for example LLIBN (unit H0409).
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Previous studies also reported a relationship between IBN peak firing frequency and peak saccade velocity (Scudder et al. 1988; Yoshida et al. 1982). We did not find this relationship for SLIBNs and LLIBNs (Table 1; mean R = 0.39) to be as robust as that in Scudder et al. (1988); it was significant for 38 and 8% of our SLIBNs and LLIBNs, respectively. This difference may be due to the less-stereotyped saccade profiles that were elicited during some of our protocols (see METHODS and DISCUSSION).
Dynamic models of IBN discharges during head-fixed saccades in OND
The foregoing analysis of IBN lead times and relationships with saccade metrics has 1) characterized our cells as typical IBNs by using criteria similar to those of other studies and 2) introduced and employed a dynamic estimate of discharge frequency lead time, which is further utilized in the following text to evaluate different models that relate IBN spike train dynamics to eye movement parameters during OND saccades. Note that the action potentials discharged by the neuron before the main portion of the burst (the burst prelude) and after (the tail) were typically excluded by the optimization algorithms (see METHODS). The resulting fits are illustrated throughout the paper for the example SLIBN (L0702) and LLIBN (H0409).
Table 2 shows the eight downstream models we investigated. It also provides the means and SDs of the parameters estimated for each model for SLIBNs, LLIBNs, and the entire population of IBNs. Table 3 gives the VAF and BIC produced by each of these models for each of the 28 neurons in this study as well as the means and SDs for SLIBNs, LLIBNs, and for the entire population.
We will now consider each model in turn beginning with the simplest.
BURST FREQUENCY PROPORTIONAL TO EYE VELOCITY.
Model 1d in Table 2 is the simplest. This often used model addresses the question of whether instantaneous burst frequency is proportional to eye velocity. It is based on the well-known relationship (discussed previously) between peak discharge frequency and peak eye velocity (Kaneko and Fuchs 1981; Scudder et al. 1988; Strassman et al. 1986a,b; Yoshida et al. 1982).
The VAF and BIC values for model 1d are shown in Table 3. This model was a poor predictor of the data as indicated by the low VAFs. Indeed, a negative VAF in 12 of 28 of the neurons implies that the variance of this model's fit error was actually larger than the variance of the data about a constant mean level.
ADDITION OF BIAS TERM.
Model 2d is an extended form of 1d and can be viewed as a first-order linear approximation to a nonlinear relationship between B(t) and where r is the bias term.
When this bias term was added to the model (Table 2), the VAF for the entire population increased very significantly from 0.04 to 0.30 (Table 3). Note that this model actually provided a fit to the data that was on average as good as that described by a correlation coefficient of 0.55 in a linear regression analysis. The accompanying decrease in the average BIC value (from 10.23 to 9.60) indicates that the addition of a bias term was warranted.
Averaged measures of goodness-of-fit were also calculated separately for SLIBNs and LLIBNs for the models 1d [Table 3: VAF (short-lead) = 0.05; VAF (long-lead) = 0.03] and 2d [Table 3: VAF (short-lead) = 0.32; VAF (long-lead) = 0.28]. There was no significant difference in the VAF by each of these models nor by any other model we tested for the two subpopulations of cells; accordingly in Table 2 we pooled the results for both SLIBNs and LLIBNS and display the average value for each parameter in each model. The considerable improvement in the fit of IBN firing rate that occurred with the addition of the bias term is illustrated for three example saccades for SLIBN L0702 (Fig. 11A) and LLIBN H0409 (Fig. 11B). The goodness-of-fit to the SLIBN is particularly striking.
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| FIG. 11.
Examples of model fits to IBN firing frequency profiles during head-fixed saccades for our 2 example IBNs illustrated in Figs. 5, 9, and 10, SLIBN L0702 (A) and LLIBN H0409 (B). The most simple downstream model tested [top row: B(t) = b1(t td)] generally provided a poor fit of IBN discharge during saccades. However, addition of bias term to the model greatly improved our ability to estimate IBN discharges (2nd row). In contrast, addition of acceleration term to this eye velocity-based model (3rd row) was not very effective in further decreasing model error of the fits to either short-lead BNs (SLBNs) or long-lead BNs (LLBNs) discharges. Values of parameters, estimated by using >40 saccades, are provided below each model fit. Three example saccades are illustrated for each IBN where shaded trajectories represent unit activity and superimposed curves represent the fits produced by each model. Bottom 2 traces: accompanying eye velocity and position trajectories that were shifted in time by estimated optimal dynamic latency td.
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IMPORTANCE OF ACCELERATION TERM.
The next downstream model we investigated utilized a simple approximation to the hypothesis of Van Gisbergen et al. (1981), that unit firing can be represented by a nonlinear function of eye velocity plus an eye acceleration term. As a first approximation to this proposal, the nonlinearity in velocity was (as in Eq. 3) simply represented by the bias term (r; Table 2: model 3d). For all our neurons the addition of the acceleration term (b2) had little effect on increasing the VAF (or decreasing the BIC); the addition of an acceleration term increased the mean VAF from 0.30 to 0.31 (Table 3). The same was true when the SLIBNs and LLIBNs were considered separately. Examples of the marginal improvement in the fit of the IBN firing rate that accompanied the addition of an acceleration term are illustrated in Fig. 11.
NONLINEAR VELOCITY TERMS.
A more complex nonlinear function of eye velocity was tested that included a third-order nonlinearity in addition to an eye acceleration (Table 2: model 4d). The addition of the higher-order eye velocity terms (d1 and d2) decreased the error of the model slightly (Table 3: mean population VAF = 0.33). Taken together the results so far show that in downstream models a bias term is much more significant than either acceleration or nonlinear velocity terms for representing IBN discharges.
ESTIMATION OF POLE TERM.
A model was also investigated which in addition to eye velocity, acceleration and bias terms allowed for the estimation of a pole term in the system (the derivative of the IBN firing rate; Table 2: models 5d and 6d). This corresponds to the slide component of the MN's response described in Eq. 1 by the term cN(t). The addition of this term provided a slightly better model fit than model 3d when the ICs in firing rate (necessitated by the pole term) were taken from the data (Table 3: VAF = 0.31 in model 3d vs. 0.37 in model 5d). However, when the ICs were estimated as parameters (Table 2: model 6d), the fit improved considerably (Table 3: VAF = 0.37 in 5d vs. 0.54 in 6d). For model 6d the fits to the data were as good, on average, as those described in a linear regression analysis by R values of 0.74 and 0.73 for SLIBNs (VAF = 0.55) and LLIBNs (VAF = 0.53), respectively. Examples of fits to firing rate are shown in Fig. 12 for the same example neurons and saccades as in Fig. 11. The improvement in fit when ICs were estimated as parameters (2nd panel from top) rather than taken from the data set (top panel) can be appreciated by visual inspection. In Fig. 12 the numbers above each firing frequency profile give the initial frequency B(t0).
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| FIG. 12.
Model fits are illustrated for example SLIBN L0702 (A) and LLIBN H0409 (B) for the same saccades illustrated in Fig. 11. A model, which included eye velocity, eye acceleration, bias, and pole terms, produced particularly good fits of IBN discharges when firing rate at beginning of each saccade was estimated as a parameter (2nd row) rather than taken from data (top row). A model, without a pole term, for which bias term was estimated separately for each saccade (3rd row), provided fits that were nearly comparable to those of row 2. Parameters estimated from >40 saccades are given below each model fit. Values B(t0), listed separately for each of 3 saccades, represent actual values of firing rate at beginning of saccade (top row) and the values estimated by algorithm for initial firing rate when B(t0) was included as parameter in the model (2nd row). Values of rk listed separately for each of 3 saccades in the 3rd panel represent bias estimated for each saccade. Top 3 rows: shaded area shows actual firing frequency profiles; heavy solid line, estimated firing. Bottom 2 traces: accompanying eye velocity and position trajectories that were temporally shifted by estimated optimal dynamic latency td.
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A model of the form 6d requires the estimation of many more parameters (equivalent to the number of saccades included in the analysis) than one in which the ICs were taken directly from the data (model 5d). To determine whether increasing the model complexity to this extent (from 4 to 44 parameters) was warranted, it was necessary to consider the BIC. Comparison of the BIC values for models 3d, 5d, and 6d (Table 3) shows that the addition of ICs as parameters to be evaluated was warranted (mean BIC = 9.57, 9.41, and 8.84, respectively). Model 6d, however, suffered from unrealistic values of the bias term and this will be considered in the DISCUSSION.
ESTIMATION OF VARIABLE BIAS.
In model 6d, the average time constant of the decaying ICs is long relative to saccade duration and given by the value c = 2.4 s (see Table 2; DISCUSSION). Thus the ICs were effectively constants where the model estimated a different constant value for each saccade that was the sum of the estimated value of the bias and the constant IC. This observation led us to specifically test a model in which only one bias term was estimated separately for each saccade (Table 2: model 7d). This variable bias model had nearly as many parameters as did model 6d (42 vs. 44, respectively); however, the method of estimation was simpler for a model of this form than for models that include a pole term (see Cullen et al. 1996). Example fits obtained using model 7d are illustrated in Fig. 12 (3rd panel from top). This model provided fits that were nearly as good as those given by model 6d (Fig. 12; Table 3: VAF = 0.54 for model 6d vs. 0.52 for model 7d). The possible physiological implications of the variable bias are considered in the following section and in DISCUSSION.
ICs AND BIASES DEPEND ON SACCADE PARAMETERS.
For 14 cells
the majority of SLIBNs (9/16) and many LLIBNs (5/12)the values estimated for ICs in model 6d were inversely and significantly correlated with the metrics of saccades. This was not the case for the example neurons shown in Figs. 11 and 12, and so these relationships are illustrated for another LLIBN (H0925) in Fig. 13, A and B. The estimated ICs for this cell were well correlated with saccade amplitude (R = 0.86) and less strongly related to peak eye velocity (R = 0.62). For those neurons with significant correlations, ICs were better correlated with saccade amplitude (mean SLIBNs, R = 0.70; LLIBNs, R = 0.67) than with peak saccade velocity (mean SLIBNs,R = 0.63; LLIBNs, R = 0.59). Note that in all these neurons the value of the ICs varied inversely with amplitude and peak velocity (as in Fig. 13).
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| FIG. 13.
Analysis of relationship between estimated initial conditions (ICs) and biases for LLIBN H0925. A and C: for this cell both ICs [IC = B(t0) + r; model 6d] and estimated biases (rk; model 7d) were negatively correlated with amplitude of the saccade (R = 0.86 and 0.89, respectively). In addition, estimated ICs and biases were also correlated with peak velocity of saccades for this neuron (B, D). However, this relationship was less robust. Insets: example of difference in the firing rate profiles generated by LLIBN H0925 during small vs. large saccades. Shaded area, actual firing frequency; superimposed solid line, estimated firing frequency using model 7d. Note that peak heights of burst firing frequency profiles are comparable in spite of the nearly 10-fold difference in amplitude between the 2 saccades. Best fit, by using model 7d, estimates a bias of 300 spikes/s for smaller saccade and 25 spikes/s for larger saccade, thereby producing a good fit of the unit discharge during wide range of saccade amplitudes.
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For 9 of the 14 cells whose ICs were correlated to metrics, the bias term estimated for model 7d (Fig. 13, C and D) was also linearly correlated with the amplitude and/or peak velocity of the saccade. The estimated biases for cell H0925 were well correlated with saccade amplitude (R = 0.89) and less strongly related to peak eye velocity (R = 0.56). The bias term was largest for small saccades. This implies that model 2d, which has a fixed bias, averaged across all saccade amplitudes should be best at predicting medium-amplitude saccades. Furthermore, model 2d should underevaluate the discharge frequency of small saccades and overevaluate that of large ones. This is precisely what Fig. 14 shows for neuron H0925.
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| FIG. 14.
Downstream model containing eye velocity and bias term (model 2d) was estimated to predict firing behavior of LLIBN H0925 during 40 saccades ranging in amplitude from 5 to 45° [variance accounted for (VAF) = 0.32]. Globally estimated model was then applied to 3 distinct subsets of original data set, each of which included only saccades within a specific amplitude range. In general, the model underestimated discharge of small saccades ranging from 5 to 10° (A) and overestimated discharge of large saccades ranging from 25 to 30° (C). The model fit was best for subset of saccades whose amplitudes were in the middle range (15-20°) of saccades contained in original data set (B). Accordingly, VAF when globally estimated model 2d was applied to subset of mid-range saccades was 0.49% (B), but only 0.15 and 0.25% when applied to the subsets of small and large saccades (A and C, respectively). Results of this analysis of LLIBN H0925 and that illustrated in Fig. 13 indicate that the downstream model in which bias term could be estimated separately for each saccade (model 7d) would better describe discharge of this neuron. For A-C, shaded area shows actual firing frequency, superimposed solid line represents estimated frequency; 2nd and 3rd traces, accompanying eye velocity and position trajectories temporally shifted by estimated dynamic latency td.
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We also investigated whether a relationship existed among either the estimated biases and/or ICs and the position of the eye before and/or following a saccade. For two SLIBNs there was a significant relationship among the estimated ICs, as well as the estimated biases, and the position of the eye before the onset of the saccade; however, no relationship was observed with final position for any IBN in this study. No relationships were ever observed among the ICs when they were taken directly from the data (model 5d) and any of the measured saccade parameters (amplitude, peak velocity, or initial or final eye position).
EFFECT OF SACCADE AMPLITUDE ON IBN SPIKE TRAIN DYNAMICS.
Our finding that estimated ICs and biases were often correlated with saccade amplitude led us to construct a more simple model that included an amplitude-dependent term. This model (Table 2: model 8d) is a simple extension of model 2d, to which a term that is proportional to saccade amplitude (r1E) was added. The addition of the eye amplitude term to model 2d resulted in comparable improvements in VAFs of LLIBNs (Table 3: VAF = 0.28 in model 2d vs. 0.32 in model 8d) and SLIBNs (Table 3: VAF = 0.32 in 2d vs. 0.36 in 8d). On average, model 8d was slightly better at describing IBN discharges than the nonlinear function of eye velocity in model 4d (Table 3: VAF = 0.34 in 8d vs. 0.33 in 4d). However, model 8d did not fare as well as 7d, but it had 38 less parameters.
EFFECT OF EYE POSITION.
The final two models that were tested addressed the issue of whether eye position was relevant in estimating IBN discharges. The simplest model included two terms, an eye position term and a bias. For each neuron this model was far less effective at describing the discharges of cells than the simple model of Eq. 3 that included an eye velocity term and a bias (VAF = 0.05 vs. 0.30 in model 2d). Furthermore, the addition of a position term to model 2d resulted in only a negligible increase in the VAF (VAF = 0.31).
Downstream model errors for SLIBNs vs. LLIBNs
We have shown that, as a population, SLIBNs have a shorter dynamic lead time than LLIBNs. This is compatible with a linear signal processing stream wherein it is possible that LLIBNs project to SLIBNs that in turn project to MNs (Fuchs et al. 1985). However, we argue here and in the companion paper (Cullen and Guitton 1997b) that this is not the case. To determine whether SLIBNs encode downstream-eye movement dynamics better than LLIBNs, we examined the relationship between lead time and the VAF values estimated for each downstream model. The VAF was not lower for those neurons that had shorter lead times; hence, SLIBNs do not appear closer than LLIBNs to the motor output for any of the eight models that we tested. The relationship between lead time and VAF for model 8d is illustrated in Fig. 15.
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| FIG. 15.
Relationship between IBN lead times and VAF provided by downstream model 8d (Table 2). There was no clear relationship among lead times calculated by using either onset of 1st spike ( ) or the estimated dynamic lead time ( ) and VAF provided by this model or any other downstream model that was tested. This shows that there is no distinction between SLIBNs and LLIBNs regarding their link to motor output.
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OFFD discharges
The
Abstract
Introduction
Abstract
