The Journal of Neurophysiology Vol. 80 No. 2 August 1998, pp. 832-848
Copyright ©1998 by the American Physiological Society

Temporal Firing Patterns of Purkinje Cells in the Cerebellar Ventral Paraflocculus During Ocular Following Responses in Monkeys II. Complex Spikes

Yasushi Kobayashi^{1, 2}, Kenji Kawano^{3, 4}, Aya Takemura³, Yuka Inoue³, Toshihiro Kitama³, Hiroaki Gomi^{4, 5}, and Mitsuo Kawato¹

¹ ATR Human Information Processing Research Laboratories, Kyoto 619-0288; ² JST-CREST, Aichi 444-8585; ³ Neuroscience Section, Electrotechnical Laboratory, Ibaraki 305-8568; ⁴ JST-CREST, Ibaraki 305-8568; and ⁵ NTT Basic Research Laboratories, Nippon Telegraph and Telephone Corporation, Kanagawa 243-0198, Japan

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Kobayashi, Yasushi, Kenji Kawano, Aya Takemura, Yuka Inoue, Toshihiro Kitama, Hiroaki Gomi, and Mitsuo Kawato. Temporal firing patterns of Purkinje cells in cerebellar ventral paraflocculus during ocular following responses in monkeys. II. Complex spikes. J. Neurophysiol. 80: 832-848, 1998. Many theories of cerebellar motor learning propose that complex spikes (CS) provide essential error signals for learning and modulate parallel fiber inputs that generate simple spikes (SS). These theories, however, do not satisfactorily specify what modality is represented by CS or how information is conveyed by the ultra-low CS firing rate (1 Hz). To further examine the function of CS and the relationship between CS and SS in the cerebellum, CS and SS were recorded in the ventral paraflocculus (VPFL) of awake monkeys during ocular following responses (OFR). In addition, a new statistical method using a generalized linear model of firing probability based on a binomial distribution of the spike count was developed for analysis of the ultra-low CS firing rate. The results of the present study showed that the spatial coordinates of CS were aligned with those of SS and the speed-tuning properties of CS and SS were more linear for eye movement than retinal slip velocity, indicating that CS contain a motor component in addition to the sensory component identified in previous studies. The generalized linear model to reproduce firing probability confirmed these results, demonstrating that CS conveyed high-frequency information with its ultra-low firing frequency and conveyed both sensory and motor information. Although the temporal patterns of the CS were similar to those of the SS when the sign was reversed and magnitude was amplified ~50 times, the velocity/acceleration coefficient ratio of the eye movement model, an aspect of the CS temporal firing profile, was less than that of the SS, suggesting that CS were more sensory in nature than SS. A cross-correlation analysis of SS that are triggered by CS revealed that short-term modulation, that is, the brief pause in SS caused by CS, does not account for the reciprocal modulation of SS and CS. The results also showed that three major aspects of the CS and SS individual cell firing characteristics were negatively correlated on a cell-to-cell basis: the preferred direction of stimulus motion, the mean percent change in firing rate induced by upward stimulus motion, and patterns of temporal firing probability. These results suggest that CS may contribute to long-term interactions between parallel and climbing fiber inputs, such as long-term depression and/or potentiation.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

A remarkable feature of the Purkinje cells in the cerebellum is that each receives two major afferents that differ dramatically in their firing dynamics: multiple parallel fiber inputs that generate simple spikes (SS) at rates up to several hundred discharges per second and a single climbing fiber input that generates complex spikes (CS) at rates that do not exceed more than a few discharges per second (Thach 1968). The type of information transmitted by the ultra-low CS firing rate and the effect of a signal with such low temporal resolution on the cerebellum are still not completely resolved.

The present study quantitatively examines CS function by examining CS and SS responses in the ventral paraflocculus (VPFL) during ocular following responses (OFR) in awake monkeys (Miles et al. 1986) and quantifying the relation between the two discharges and the retinal slip and the eye movement. Recent modeling studies of the temporal firing profiles of SS during OFR using an inverse-dynamics model (a linear combination of eye acceleration, velocity, and position) demonstrated that SS in the VPFL encode dynamic motor commands (Gomi et al. 1998; Kawano et al. 1996; Shidara et al. 1993). In the present study, in addition to the electrophysiological experiments, this model has been extended to a more sophisticated generalized linear model (Kawato 1995) to analyze the correlation between the ultra-low CS firing rate and the motor commands or retinal slip.

OFR are tracking movements of the eyes evoked by movements of a visual scene and are thought to be important for the visual stabilization of gaze. It was advantageous for several reasons to study CS function by recording them in the VPFL during OFR. The OFR are primarily under negative feedback control because this behavior is primarily in response to retinal slip, which is the difference between the image motion and the eye movement. The early phase of the OFR, however, is controlled in an open-loop manner, and this early phase has been shown to be subject to long-term adaptive modification by visual error signals (Miles and Kawano 1986). OFR are reflexes induced by the retinal slip, so it is technically easy to obtain a large number of trials, thus increasing the signal-to-noise ratio. It is possible to quantify the correlation between the sensory error signal (retinal slip) and CS firing because it is possible to accurately control the parameters of the visual stimulus. SS evoked during OFR have been recorded and characterized in Purkinje cells in the VPFL (Gomi et al. 1998; Kawano et al. 1996; Shidara and Kawano 1993; Shidara et al. 1993).

The climbing fiber projections from the inferior olive (IO) to the VPFL have been well characterized (Gerrits and Voogd 1982, 1989; Langer et al. 1985). In the rabbit flocculus, a large number of CS are evoked by movement of a large visual stimulus (Graf et al. 1988; Simpson and Alley 1974). CS also were recorded in the VPFL of monkeys during tracking of a small target (Stone and Lisberger 1990b). Thus a considerable number of CS are expected to be evoked in the VPFL during OFR.

The CS in the VPFL have previously been well characterized during smooth pursuit eye movement by Stone and Lisberger (1990b), who concluded that CS were driven by contralaterally or upward directed image motion. CS were modulated out-of-phase with SS. By spike-triggered averaging analysis, they concluded that CS during steady-state pursuit were driven by the retinal slip associated with imperfect pursuit.

In this study, we have advanced the quantitative understanding of CS during OFR in regard to the following four points and have provided critical data to examine the major theories of CS function. First, we developed a new statistical method for quantitatively analyzing what information is encoded in the temporal patterns of CS firing rate. With this new technique, we have demonstrated that the CS firing probability carries very high-frequency temporal information that matches that of the SS. Second, cell-to-cell negative correlations between the firing characteristics of SS and CS for individual cells were revealed. Third, although in previous studies the sensory aspects of CS were well studied, we have added new evidence for a motor-related nature of CS. Fourth, we examined the relationship between the velocity of eye movement or retinal slip and the firing rates of CS over a wide range of step ramp speeds.

	METHODS

Abstract Introduction Methods Results Discussion References

Our methods for preparing the monkeys, for presenting visual stimuli, and recording the simple spike of Purkinje cells (P cells) are presented in detail in the preceding paper (Gomi et al. 1998). Here, we describe only those additional techniques that were used in the analysis of simple- and complex-spike responses.

Visual stimuli

Visual stimuli were designed to study the directional selectivity of neural firings (experiment 1), effects of changes in stimulus velocity on neural firings (experiment 2), and temporal patterns of firing rate and firing probability (experiment 3).

EXPERIMENT 1. The directional firing characteristics of 13 cells were examined by moving the stimuli in eight directions ( = 0, 45, 90, 135, 180, 225, 270, and 315°) at a constant speed of 80°/s. The stimulus was presented and moved 40 times (40-77 trials, mean = 57 trials) in each direction while recording from each cell (320-616 trials, mean = 456 trials altogether). Because the latency of the change in SS firing rate during the OFR is ~40 ms from the onset of stimulus motion (Shidara and Kawano 1993), the spike modulation for each stimulus direction was calculated as the mean firing rate minus the spontaneous firing rate over an interval extending from 40 to 220 ms after the onset of stimulus motion. Spontaneous rate was calculated as the mean firing rate over an interval from 100 to 40 ms after the onset of stimulus motion. The preferred direction of SS or CS of each cell was calculated as the direction of the average vector of modulation vectors for each direction, defined as a two-dimensional vector with the same direction as the stimulus motion () and a length equal to the spike modulation defined above. Because we did not use a three-dimensional (3-D) planetarium projector system (Graf et al. 1988), we cannot directly argue about the preferred axis of rotation in 3-D space from our experimental data.

EXPERIMENT 2. Stimuli moving at six or eight different velocities (+80, +40, +20, 20, 40, and 80°/s or + 80, +40, +20, +10, 10, 20, 40, and 80°/s) were presented while recording from 12 cells. For each cell, the stimulus was moved either vertically or horizontally, so that it would overlap with the preferred and anti-preferred directions of SS. Upward and contralateral motion were assigned positive polarity. At least 70 trials (76-217 trials, mean = 134 trials) were performed at each stimulus velocity for each cell (608-1,592 trials, mean = 948 trials).

EXPERIMENT 3. Stimuli moving directly upward at 80°/s were presented to nine vertical axis cells (V cells). For V cells, directly upward is close to the preferred direction of CS and the antipreferred direction of SS. Upward moving stimuli were presented >300 times (312-901 trials, mean = 579 trials) while recording from each cell.

For improving data reliability in the analysis of the firing characteristics of CS for each cell, we focused on obtaining a large number of trials rather than on increasing the number of recorded cells.

Recording technique

Purkinje cells were identified by the presence of SS and CS (Thach 1968). Before trial sessions, we carefully discriminated each single unit using a time-amplitude window discriminator, and we checked that CS and SS were derived from the same cell by confirming the occurrence of a brief pause of the SS after the CS (Sato et al. 1992). After the sessions, SS and CS were discriminated with a time resolution of 1 or 2 ms off-line using custom software, running on a SUN SPARK station, which clusters groups of spikes by amplitude, duration and wave form by principal component analysis.

Generalized linear model

In our previous studies, the SS firing rate was reproduced directly using an inverse-dynamics representation model, which is a linear weighted summation of the eye acceleration, velocity, and position (Gomi et al. 1998; Kawano et al. 1996; Shidara et al. 1993). The very low CS firing rate precludes the direct use of this method for the analysis of the CS temporal firing profile, but the firing probability rather than the firing rate itself can be modeled. The low CS firing rate highlighted the binomial nature of the spike count. The variance was not constant, invalidating the minimum-square-error method for parameter estimation, and the standard deviation had the same magnitude as the mean, rendering the correlation coefficient rather insensitive to the goodness of fit. The fluctuations in the CS firing rate were largely due to the variance of the binomial distribution. Theoretically, the firing rate (spikes/s) multiplied by the time bin (s) converges to the firing probability p as the trial number n goes to infinity with a standard deviation of the binomial distribution. The following is a typical example of a firing probability with the same order of magnitude as its standard deviation. The standard deviation 0.003 is close to the signal itself for p = 0.005 (2.5 spikes/s multiplied by a 2-ms bin) and n = 500. Thus for CS, the low value of the actual correlation coefficient and the predicted value does not necessarily mean a poor fit by the model. On the other hand, for SS, firing at 100 spikes/s, the signal (i.e., probability p = 0.2) is much larger than the noise (i.e., the standard deviation is 0.02), and thus use of the correlation coefficient is valid.

We confirmed that the number of CS and SS, X_i, which accumulated within the time t =i bin for n trials, obeyed the following binomial distribution (Kobayashi et al. 1995)

(1)

where y_i denotes the realized value of the stochastic variable X_i, that is, the observed spike number. p_i is the spike occurrence probability within the time t = i bin. The firing probability p(t) as a function of time t was modeled by the following generalized linear summation of acceleration, velocity, and position of eye movement, which is a smooth function of time

(2)

(3)

where , , and  denote the acceleration, velocity, and position of the eye, and M, B, and K denote their coefficients, respectively, while  denotes the time delay between spike discharge and eye movement, and C is a constant. The sigmoid function S constrains p(t) to values between 0 and 1. This is a specific example of a generalized linear model (McCullagh and Nelder 1989).

The parameters other than the time delay were estimated using Fisher's scoring method by maximizing the following likelihood function, L or its logarithm l (the maximum likelihood method)

(4)

m is the bin number included in one experiment. In experiment 3, m = 126, because all of the spike counts within a 2-ms bin were collected to calculate y_i from 0 to 250 ms after the onset of stimulus motion; m = 756 (6 velocities) or m = 1,008 (8 velocities) in experiment 2 because stimuli with different velocities produced different sets of data. The time delay leading to the maximum likelihood was globally searched at every 2-ms step from 20 to 20 ms.

Let L₁ denote the likelihood evaluated by the maximum likelihood estimator in p_i (_i = y_i/n), which is the best possible model but with a large degree of freedom m, while L₀ denotes the maximum likelihood of the current model. If the model is good, the likelihood ratio ( = L₀/L₁) is close to 1, but if the model is poor, the ratio approaches 0. The deviance D, 2log expressed in the following equation is always positive and approaches zero as the fit becomes better and becomes large as the fit becomes poorer

(5)

Generally, a smaller deviance indicates a better fit. Because the deviance increases in proportion to m, the deviances in experiment 2 were divided by 6 or 8 (the number of different stimulus velocities) for comparison with that of experiment 3 in Table 1.

To further examine the sensory and motor characteristics of CS and SS, we compared the ability of a model based on the sensory error and the eye movement to reproduce the firing probability. Thus the model was based on a generalized linear combination of the acceleration, velocity, and the position of the retinal slip as well (the difference between stimulus position and eye position)

(6)

where , , r, C_r, and  denote the acceleration, velocity, and position of the retinal slip, a constant, and the delay between the onset of stimulus motion and spike discharge, respectively. The delay between the retinal slip and the spikes was searched globally from 30 to 70 ms.

	RESULTS

Abstract Introduction Methods Results Discussion References

CS and SS during OFR

We recorded SS and CS during OFR from 34 Purkinje cells. Figure 1 shows stimulus and eye movement (velocity, acceleration, and position) and examples of the responses to upward and downward stimulus motion at 80°/s. The characteristic short latency (~50 ms from stimulus to movement) and the complex acceleration of OFR were evident (Miles et al. 1986).

View larger version (42K): [in this window] [in a new window]   FIG. 1. Eye movements and complex spikes (CS) and simple spikes (SS) during ocular following responses (OFR). Left: responses to 80°/s upward stimulus motion. Right: responses to 80°/s downward stimulus motion. Stimulus velocity (thin line in A and B) and eye velocity (thick line in A and B), eye acceleration (C and D), and eye position (E and F) as functions of time after the onset of stimulus motion during OFR are shown. G and H: examples of rastergrams of the SS and CS responses in a V cell during OFR to 20 presentations of upward (G) and downward (H) test ramps. Bars represent SS and the circles represent CS.

The Purkinje cells were categorized into two groups. One population (23/34 cells) exhibited an increase in CS and a decrease in SS firing rate in response to upward moving stimuli (Fig. 1G), and an increase in SS and a decrease in CS firing rate in response to downward moving stimuli (Fig. 1H). These cells were termed V cells. The other population (11/34 cells) exhibited an increase in SS and a decrease in CS firing rate in response to ipsilateral stimulus motion, and an increase in CS and a decrease in SS firing rate in response to contralateral stimulus motion. These cells were termed horizontal cells (H cells).

Directional tuning of CS (experiment 1)

To quantify the spatial tuning characteristics of SS and CS, the responses to moving the stimulus in eight different directions at 80°/s were recorded (Fig. 2). The aggregated activities of eight V cells are shown. Both SS and CS were modulated by vertically moving stimuli, but they were not modulated by horizontal stimuli. Both SS and CS had some degree of spontaneous firing. Downward moving stimuli elicited increases in the SS firing rate and decreases in CS rate, both beginning ~40 ms after the onset of stimulus motion. Upward moving stimuli elicited decreases in the SS firing rate and increases in CS rate with similar (40 ms) latencies. The latencies of the changes in SS and CS are more quantitatively examined later by analysis of their temporal profiles.

View larger version (43K): [in this window] [in a new window]   FIG. 2. SS and CS peristimulus histograms in response to moving the stimulus in 8 different directions. Aggregated responses from 8 V cells are shown. Position of each histogram corresponds to the direction of the stimulus motion. Ip, ipsilateral; Up, upward; Co, contralateral; Dw, downward.

The preferred directions of SS and CS for individual cells are shown in Fig. 3, A and B. Preferred directions of the SS and CS for each cell were calculated as described in METHODS. The mean of the preferred directions of SS for V cells was 273.7 ± 27.4° (mean ± SD), that of SS for H cells was 1.8 ± 6.7°, that of CS for V cells was 84.7 ± 10.7°, CS for H cells was 189.4 ± 5.6°. The mean difference between the preferred directions of SS and CS was 173 ± 16°, which is close to 180° (Fig. 3, A and B). Thus the reciprocity between the preferred direction of SS and CS was shown.

View larger version (31K): [in this window] [in a new window]   FIG. 3. Directional tuning properties of SS and CS. Preferred directions of SS (thin lines) and CS (thick lines) for 8 V cells (A) and 5 H cells (B) are shown. C-F: mean ± SD modulation of SS and CS in V cells (C and E) and H cells (D and F) as a function of the direction of stimulus motion. Abscissa (C-F) is the direction of stimulus motion in degrees measured counterclockwise from the ipsilateral direction (ipsilateral = 0°, upward = 90°, contralateral = 180°, and downward = 270°). Means were fitted by a cosine tuning function (dotted line).

To quantify the directional dependency of the SS and CS modulations, the directional tuning data of SS or CS averaged over the V cell or the H cell population were fitted by a cosine function of the direction of stimulus motion

(7)

where f and  denote the mean firing rate and the direction of stimulus movement ( = 0, 45, 90, 135, 180, 225, 270, and 315°), respectively. The preferred directions of the averaged data (_p) were computed by averaging the preferred directions across the population. a and b denote the regression coefficient and the intercept of the regression equation, respectively, which were determined by the least-square method. a and b indicate the magnitude of direction-dependent modulation of the firing rate and the spontaneous firing rate, respectively. For the SS data, a and b were 29.2 and 14.8 in V cells and 38.2 and 0.7 in H cells. For the CS data, a and b were 0.78 and 0.06 in V cells and 0.86 and 0.3 in H cells. The averaged data and the fitted curves are shown in Fig. 3, C-F. The data and the fitted curves were well correlated (r = 0.99 for SS in V cells, r = 0.99 for SS in H cells, r = 0.96 for CS in V cells, and r = 0.90 for CS in H cells), indicating that the CS and SS directional tuning characteristics were well modeled by the cosine function. The cosine directional tuning curves of SS and CS were 180° out of phase. Because the mean direction-dependent modulation of the CS firing rate (a = 0.82) was 0.024 of that of SS (a = 33.7), the mean change in the CS firing rate depending on stimulus directions was only 2.4% of that of SS.

When the preferred direction of SS was plotted against that of CS as in Fig. 4, the slope of the regression line was close to 1.0 (0.82) and its intercept was close to 180° (155°). These results provide quantitative evidence that the spatial tuning properties (including the preferred direction) of CS are opposite to those of SS. In 13 cells examined, the preferred direction of SS recorded from each cell correlated with the preferred direction of CS recorded from the same cell with a coefficient of 0.90 (P = 0.001). The data were separated into two clusters (i.e., H and V cells). Although no significant correlation existed when each cluster of data were regressed separately, the above significant correlation for all the data at least indicates global reciprocity of the preferred directions of the SS and CS.

View larger version (20K): [in this window] [in a new window]   FIG. 4. Preferred direction of SS for each cell is plotted against that of CS. Line represents the linear regression of the data.

Effects of stimulus velocity on CS and SS (experiment 2)

Twelve cells were studied and findings for one example are shown Fig. 5. The SS firing rate increased and the CS rate decreased with increased downward stimulus velocity (retinal slip velocity) and the resulting downward eye movement. Moreover, the SS firing rate decreased and the CS rate increased with increased upward stimulus velocity and the resulting upward eye movement.

View larger version (40K): [in this window] [in a new window]   FIG. 5. SS and CS of an example V cell in response to a wide range of stimulus velocities. Stimulus was moved vertically at +80, +40, +20, +10, 10, 20, 40, and 80°/s. Upward motion was assigned positive polarity. Mean stimulus (thin line) and eye (thick line) velocities (left) and peristimulus time histograms with CS (middle) and SS (right) binned in 1-ms intervals are shown for each stimulus velocity for which 95 trials were obtained.

To quantify the correlation between the SS and CS firing rates and retinal slip or eye velocities, the mean SS and CS firing rates were plotted against mean retinal slip and mean eye velocity. Considering its time delay, eye movement was averaged over the time interval from 50 to 300 ms after the onset of stimulus motion. SS and CS were averaged from 40 to 290 ms after the onset of stimulus motion. Retinal slip was averaged from 0 to 250 ms after the onset of stimulus motion. The mean SS firing rate was a monotonically decreasing function of the slip velocity. The curve from each individual cell was a sigmoid function, that is, a decreasing, monotonic, saturating function (Fig. 6A). In contrast, the mean SS firing rate was approximately a linear function of the eye velocity (Fig. 6B). The correlation coefficient between the mean SS firing rate and eye velocity for each of 12 cells was calculated and then averaged (mean = 0.99). The absolute value of the mean correlation coefficient (0.91) between the SS firing rates and the slip velocities was statistically significantly smaller (P = 0.0001) than that between SS and eye velocity. This observation provides quantitative evidence that the relationship between the SS firing rate and eye velocity is more linear than the relationship between the SS firing rate and slip velocity.

View larger version (37K): [in this window] [in a new window]   FIG. 6. Velocity tuning curves of SS and CS. Following relations are shown: SS firing rate and retinal slip velocity (A), SS firing rate and eye velocity (B), CS firing rate and retinal slip velocity (C), CS firing rate and eye velocity (D), SS and CS firing rate (E). Bold solid line shows the regression line of all the data. F: retinal slip velocity (circle) as a function of the stimulus velocity, and the eye velocity (cross) as a function of the stimulus velocity. Dotted line shows the line with a slope of 1.0 and passing through the origin. Retinal slip was calculated by subtraction of eye movement from stimulus movement. Upward or contralateral motion was assigned positive polarity. Each set of data points connected by a line represents data from an individual cell (n = 12).

The mean CS firing rate of each cell was approximately an increasing, saturating function of the slip velocity, although some exceptions can be seen especially at large slip velocities (Fig. 6C). At stimulus velocities between 40 and 80°/s, some cells exhibited an increase, some a decrease, and some no change in CS firing rate. As found for SS, the relationship between the mean CS firing rate and the eye velocity was more linear (Fig. 6D). The mean correlation coefficient between CS firing rate and eye velocity (0.93) was statistically larger than (P = 0.008) that between CS firing rate and slip velocity (0.89), again providing quantitative evidence that the relationship between CS firing rate and eye velocity is more linear than the relationship between CS firing rate and slip velocity.

The correlation coefficient between the mean SS and CS firing rate at each stimulus velocity was calculated for each of 12 cells (Fig. 6E) and then averaged (mean = 0.91 ± 0.06; P = 0.05). This result demonstrates that there is a reciprocal relationship between the mean SS firing rate and the mean CS firing rate with respect to their stimulus and eye velocity dependence. The very large negative slope (48.2) of the regression line in Fig. 6E for the average data of 12 cells indicates that modulation of CS firing rate depending on different stimulus speeds was opposite in sign and only 2.1% of that of SS.

SS and CS dependencies on either the slip velocity or the eye velocity were examined in Fig. 6, A-D. Because OFR is essentially induced by the retinal slip, one may wonder that the slip velocity is very highly correlated with the resultant eye velocity, thus the above analyses might not be sensible. Plotting the averaged slip velocity (cross) and the averaged eye velocity (circle) as functions of the stimulus velocity resolved this issue (Fig. 6F). Even in the small stimulus velocity range (from 40 to 40°/s), the two curves had opposite curvatures. Furthermore, for the largest stimulus speeds (80 and 80°/s), the eye velocity showed clear signs of saturating, whereas the retinal slip kept increasing. Thus even for the averaged behavior, the retinal slip and the eye movement were considerably different. Marked differences in their transient behaviors will be given in the following text.

Temporal patterns of CS firing rates (experiment 3)

In V cells (n = 9), the SS firing rate decreased and the CS firing rate increased in response to upward 80°/s stimulus motion (Fig. 7A). Moreover, CS and SS of an individual cell appeared to be affected to similar extents by the same stimulus motion (e.g., cells 1 and 2 exhibited relatively small changes in both CS and SS firing rate, whereas cell 3 exhibited relatively large changes in both CS and SS firing rate). Furthermore, although the percent change in the SS firing rate was much larger than that of CS (note the 10 times difference in ordinate scales between the left and right columns), the SS temporal firing profiles were similar to those of CS if the sign was reversed and the magnitude scaled.

View larger version (32K): [in this window] [in a new window]   FIG. 7. Reciprocal relationships between the CS and SS firing rates. A: SS and CS firing rates evoked by upward stimulus motion at 80°/s (1-ms bins from 0 to 300 ms after the onset of stimulus motion) for 3 example cells. B: mean percent change in SS firing rate for each cell is plotted against that of CS. Solid line represents the linear regression of the data. C: instantaneous SS firing rates (binned in 2-ms intervals from 0 to 300 ms after the onset of stimulus motion) of cell 1 shown in A were plotted as a function of those for CS in the same time bins. Solid line represents the linear regression of the data (n = 150; r = 0.38; P = 0.0001).

There was a significant cell-to-cell negative correlation (0.76, P = 0.01) between the mean magnitude of change from the spontaneous activity in the SS and CS firing rates in response to upward 80°/s stimulus motion (Fig. 7B). This result statistically supports the above qualitative observation suggesting that the magnitude of change in the CS firing rate parallels the magnitude of change in the SS firing rate. Because the slope of the regression line is 17.9, the population mean change in CS firing rate is 5.6% of that of SS, for the preferred direction of CS and the anti-preferred direction of SS. The y-axis intercept of the regression line (Fig. 7B) was not significantly different from zero (P = 0.23), which suggests that the SS firing rate is unmodulated if the CS firing rate is unmodulated and vice versa.

There was a statistically significant negative correlation (P = 0.0001) between the instantaneous CS and SS firing rates within the same 2-ms time bin (Fig. 7C) for cell 1 shown in Fig. 7A. The instantaneous SS and CS firing rates were negatively correlated for all nine cells examined in experiment 3. For seven of the nine cells, the negative correlation was statistically significant (P < 0.01). This analysis provides statistical evidence that the SS temporal firing profile is similar but of opposite sign to the CS temporal firing profile.

Reproduction of firing probability of CS

The result of using a generalized linear model of eye movement to reproduce the SS and CS firing probability in experiment 3 is shown in Fig. 8 for five individual cells. The mean correlation coefficient between the observed firing rate and the estimated firing probability was 0.84 ± 0.12 for SS and 0.48 ± 0.10 for CS for the nine cells in experiment 3. The mean deviance of the CS firing probability for the eye movement model (equation 2) was smaller than that of the SS (Table 1), indicating that the CS firing probability was more accurately reproduced than that of the SS. Furthermore, the deviance of the CS firing probability was smaller than that of the SS firing probability on an individual cell basis for seven of the nine Purkinje cells analyzed. As the number of trials performed while recording from a single cell approaches infinity, the firing rate (spikes/s) multiplied by the time bin (s) should approach the firing probability p. On the other hand, if the trial number n is finite, there are large fluctuation in firing rate with a standard deviation due to the binomial distribution (see METHODS). Thus the rapid fluctuation of the firing rate from the predicted firing probability observed in Fig. 8 is not mainly an error due to the model but rather sampling noise inherent to the stochastic spike-count data itself.

View larger version (45K): [in this window] [in a new window]   FIG. 8. Reproduction of the SS (left) and CS (right) firing probability from the eye movement for 5 example V cells in response to upward stimulus motion at 80°/s using a generalized linear model. Thin curves show the observed firing rate in 2-ms bins, and the thick curves show the estimated firing probability within the corresponding time bin. Note that the ordinate scales for the firing rate (right) and the firing probability (left) were matched such that the asymptote of the former overlaid that of the latter. Accumulated trial numbers for the 5 cells were 901, 899, 396, 327, and 487 from top to bottom. Data are shown from top to bottom in order of increasing mean change in firing rate in response to stimulus motion.

A generalized linear model of eye movement also was used to reproduce the SS and CS firing probability from experiment 2. A single set of parameters was estimated for each cell to reproduce the firing patterns for all stimulus velocities. As in experiment 3, the mean deviance of the CS firing rate was smaller than that of the SS (Table 1) and the deviance of the CS firing rate was smaller than that of the SS firing rate for all 12 individual cells recorded in experiment 2. Thus these data also indicate that the CS firing probability was better reproduced from eye movement than was the SS firing probability.

We examined whether temporal patterns of firing probability in SS and CS encode significantly sensory or motor information. The firing probabilities of CS and SS were reconstructed via the generalized linear model of eye movement (Eq. 2) or retinal slip (Eq. 6), then the deviances were compared between the eye movement model and the retinal slip model. To improve the data reliability, we averaged the data from nine cells recorded in experiment 3 as shown in Fig. 9. Because of this population averaging, the stochastic noise in the firing rate of CS was reduced. The temporal patterns of retinal slip position, velocity, and acceleration and eye position, velocity, and acceleration and SS and CS elicited by upward 80°/s stimuli are shown in Fig. 9. Thick lines in Fig. 9, G and I, show the firing probability reconstructed for SS and CS respectively by the generalized linear model of the retinal slip. Thick lines in Fig. 9, H and J, show the firing probability reconstructed by the generalized linear model of eye movement for SS and CS, respectively.

View larger version (33K): [in this window] [in a new window]   FIG. 9. Population average of the retinal slip, eye movement, CS and SS of 9 cells in experiment 3 (5,214 trials). Left: retinal slip position (A), velocity (C), and acceleration (E) in responses to 80°/s upward stimulus motion. Right: eye position (B), velocity (D), and acceleration (F) responses to the same stimulus motion. G and H: thin traces show the population average of firing probability of SS. Thick traces in G and H show the estimated firing probability of SS from retinal slip and eye movement, respectively. I and J: thin traces show the population average of firing probability of CS. Thick traces in I and J show the estimated firing probability of CS from retinal slip and eye movement, respectively.

It may appear superficially that the temporal patterns of the retinal slip and the eye movement are similar and thus that it is statistically difficult to discriminate which signal better reconstructs the firing frequency patterns. But actually even the position temporal patterns are quite different between the two signals unless an appropriate time shift is introduced, whereas the velocity and acceleration are entirely different with negative correlations. The firing data were best modeled by retinal slip ~40 ms time delayed, and it was modeled best by the eye movement ~10 ms time advanced. Thus for estimating statistically the extent of similarity of the two signals, we first time delayed the retinal slip by 40 ms and time advanced the eye movement by 10 ms and then calculated the correlation of the two signals. The correlation coefficient between retinal slip acceleration 0-200 ms after onset of the stimulus motion and eye acceleration 50-250 ms from onset of the stimulus motion was 0.012. Thus the acceleration patterns were little correlated. The correlation coefficient between velocity of retinal slip and velocity of eye movement was 0.60. Thus the velocity patterns were negatively correlated. The coefficient for position was 0.97. Thus in summary, the position patterns were highly positively correlated, but the dynamic components (acceleration and velocity) of the two temporal patterns were entirely different.

Comparison of the deviances for the eye movement model and the retinal slip model, as shown in the left two columns of the Table 1 indicates that the SS and CS firing probabilities were reproduced as well or better from the retinal slip than from eye movement for the upward 80°/s stimulus in experiment 3. But we do not think this is a statistically important observation. To reliably and more rigorously compare the two statistical models in reproducing the experimental data, the generalization capability of the models should be tested using the data from experiment 2 because a large variety of stimuli and responses are essential to test the goodness of the models. The mean deviances in experiment 2 are shown in the right two columns of Table 1. Here, both the SS and CS were better reproduced from the eye movement than the retinal slip. Instead of directly comparing the deviance values themselves, an index of the sensory-motor nature of the signals was calculated as the deviance for the eye movement divided by that for the retinal slip. If the index is smaller than 1, the signal will be more motor in nature, whereas if it is larger than 1, the signal will be more sensory. First, because the mean of the index for CS (0.95) was close to 1, CS were equally well reproduced from either the retinal slip or the eye movement. Second, the mean of the index for SS (0.85) was smaller than that for CS, suggesting that SS are more motor in nature than CS (and conversely, that CS are more sensory in nature than SS).

The generalized linear model of the eye movement shown in Eq. 2 nonlinearly transforms the linear weighted summation of the eye acceleration, eye velocity, eye position, and the constant term, M·(t + ) + B·(t + ) + K·(t + ) + C by the sigmoid function S defined in Eq. 3. The bold solid curves in Fig. 10, A and B, denote this summation, that is, the argument of S or the contents of the square bracket in Eq. 2 for the SS and CS, respectively. Here, we use the same population data from the 9 cells in experiment 3, which were already shown in Fig. 9, B, D, F, H, and J. The noisy curves denote the inverse of the sigmoid function of the actual firing data: S¹(y_i/m) = log[(y_i/m)/(1  y_i/m)]. Because the bold solid curve S¹(p) well approximates this noisy curve, the good fit of the generalized linear model was reconfirmed. The four thin solid curves shown in Fig. 10 indicate the four terms, i.e., M·(t + ), B·(t + ), K·(t + ), and C for the SS and CS, in A and B, respectively. C of the CS was smaller than that of the SS by ~4, indicating that the spontaneous firing rate of the CS was about exp(4) = 0.02 times of that of the SS. The acceleration, velocity and position curves of the CS shown in Fig. 10B had the opposite polarity but similar magnitudes to those of SS in Fig 10A. This reconfirmed that the temporal patterns of firing frequency for the CS are similar to those for the SS if the sign is reversed and the magnitude is scaled by dividing by exp(4) = 0.02. 

View larger version (26K): [in this window] [in a new window]   FIG. 10. Bold solid curves in A and B indicate the linear weighted summation of the eye acceleration, eye velocity, eye position and the constant term, M·(t + ) + B··(t + ) + K·(t + ) + C for the SS and CS, respectively. Four thin solid curves shown in A for the SS and in B for the CS indicate the 4 terms in these summations. Noisy curves indicate the inverse of the sigmoid function of the actual firing rates. We use the same aggregated data from the 9 cells in experiment 3, which were already shown in Fig. 9, B, D, F, H, and J.

Next, we examined the relative contributions of the three factors (acceleration, velocity, and position) in reconstructing the SS and CS temporal profiles by calculating the following variance accounted for (VAF) of the eye acceleration, velocity, and position, respectively

(8)

(9)

(10)

The VAF indicates what proportion of the total modulation in the S¹ transformed firing frequency could be accounted for by each of the three terms. V[x] denotes the variance of x. A larger VAF indicates a larger contribution of that component. VAF_a, VAF_v, and VAF_p for the SS were 0.05, 0.51, and 0.55, respectively. VAF_a, VAF_v, and VAF_p for the CS were 0.34, 0.24, and 0.29, respectively. These results indicate that the eye velocity component was the most dominant in the SS and eye acceleration component was the most dominant in the CS.

The ratio of the velocity and acceleration coefficients (B/M) in the generalized linear model of eye movement (Eq. 2) for the population data shown in Figs. 9 and 10 was computed. (B/M) for SS was 55 and for CS was 26. This confirms that the SS contained the larger velocity component or the CS contained the larger acceleration component.

In both experiment 3 and experiment 2, the mean acceleration, velocity, and position coefficients of eye movement (M, B, and K) for CS and SS generally had opposite signs but were of the same order of magnitude (Table 2). The mean C of CS was smaller than that of SS by ~4. This also indicates a lower firing probability of the CS than the SS. Taken together, this indicates that the percent change in the CS firing rate is approximately exp(4) = 0.02 that of the SS firing rate. The onset of SS and CS modulation preceded the onset of eye movement by a similar amount: the average  (Eq. 2) from 21 cells from experiment 2 and experiment 3 was 5.1 ± 10.3 ms for CS and 10.8 ± 6.0 ms for SS (means ± SD; not significantly different P > 0.05; see also Table 2). Because M, B, K, C, and  reflect the temporal firing probability profile, these results indicate that the CS temporal firing probability profile was similar to that of SS if the sign was reversed and the modulation amplitude scaled. Thus these data provide additional quantitative evidence with high temporal resolution (2-ms bin) that the SS and CS temporal profiles were similar but opposite in sign.

We have demonstrated previously that SS recorded in the VPFL exhibit temporal firing profiles that closely follow an inverse dynamics representation of eye movements and that the ratio between the acceleration and velocity coefficients is close to that of motor neurons, thus indicating their role as the dynamic motor commands (Gomi et al. 1998; Kawano et al. 1996; Shidara et al. 1993). In the present study, we modeled firing probability by generalized linear models instead of using linear models for firing frequency. Because the sigmoid (logarithmic) function in the generalized linear model can be approximated by an exponential function if its argument is negative and its absolute value is large, the acceleration, velocity, and position coefficients of the linear model can be approximated from the corresponding coefficients of the generalized linear model by multiplying by exp(C). Thus the ratio of the coefficients can be compared directly between the linear model and the generalized linear model used in the present study.

The mean velocity and acceleration coefficient ratio of eye movement model (B/M) for 21 cells in experiments 2 and 3 for SS was 56, which is close to that for motor neurons (67) (Keller 1973); thus these results are consistent with the hypothesis that SS provide dynamic motor commands (Gomi et al. 1998; Kawano et al. 1996; Shidara et al. 1993). The mean ratio (B/M) for 21 cells in experiments 2 and 3 for CS was 28. This indicated the temporal profile of SS contained the larger velocity component or that of CS contained the larger acceleration component. The same conclusion was drawn already from application of the same model to the accumulated data from nine cells in experiment 3 (Figs. 9 and 10). The VAF analysis of the same data also reconfirmed this.

Plots of the best-fit parameters of the coefficients M, B, K, and C for the SS data against the best-fit parameters for the CS data for each of the 21 Purkinje cells from experiments 2 and 3 indicate that the acceleration, velocity, and position coefficients for the SS and CS data generally had opposite signs (note the quadrant) but were of the same order of magnitude, even on a cell-to-cell basis (Fig. 11). The regression lines in Fig. 11, A and B, were constrained to pass through the origins for the following two reasons. First, the results shown in Fig. 7B indicate that SS were unmodulated if CS were unmodulated. This suggests that the inverse dynamics coefficients were zero for both SS and CS, so that the origin was a default data point. Second, the results of experiment 1 indicate that both CS and SS were unmodulated by the stimulus direction perpendicular to their optimal and antioptimal directions. Thus a large number of data points concentrate on the origin. The slopes of the regression lines for M (P = 0.02) and B (P = 0.00001) were significantly more negative than zero based on a Student's t-test. This result indicates that there are cell-to-cell negative correlations between the SS and CS coefficients and consequently cell-to-cell negative correlations between the SS and CS temporal firing patterns. M and B are functionally more important than K because SS provide only the dynamic part of the motor commands (Gomi et al. 1998; Kawano et al. 1996; Shidara et al. 1993).

View larger version (32K): [in this window] [in a new window]   FIG. 11. Estimated parameters M, B, K, and C, and their confidence intervals are shown in A-D, respectively. Estimated coefficient for SS of each cell is plotted against that of CS using the same scale for the ordinate and the abscissa. Center of each cross indicates the maximum likelihood estimation, whereas the lengths of the vertical and horizontal bars indicate the square roots of the asymptotic variance for SS and CS, respectively. Crosses represent data from 9 cells responding to upward stimulus motion at 80°/s and data from 12 cells in which 6 or 8 different velocities were used. Estimated coefficients from the averaged data to upward stimulus motion at 80°/s (5,214 trials) are plotted as double circles.

Short-term modulation of SS by CS

The cross-correlation analysis was applied to the SS (Fig. 12A) and the CS (Fig. 12B) firing for nine cells presented with upward 80°/s stimulus motion (experiment 3). Figure 12 shows the results for one example cell. The apparent cross-correlation R_app was calculated directly by CS spike trigger averaging of the SS (Fig. 12C). The stimulus-dependent cross-correlation R_stm was similarly calculated after shuffling the impulse trains (Perkel et al. 1967; Toyama et al. 1981) (Fig. 12D). The true interaction (net cross-correlation R_net = R_app  R_stm) between SS and CS (Fig. 12E) was determined by subtracting the stimulus-dependent cross-correlation from the apparent cross-correlation. The proportion of the SS discharge modulation SS_CS that is a direct consequence of short-term effect of CS can be evaluated by the convolution integral of the CS firing pattern with the net cross-correlation obtained above (Fig. 12F)

(11)

Comparison of the convolution integral of CS (Fig. 12F) and the SS peristimulus time histogram (Fig. 12A) indicates that the estimated CS-induced SS modulation was negligible compared with the actual SS modulation; the ratio of the estimated CS-induced SS modulation (averaged over the interval from 0 to 250 ms from the onset of the stimulus motion after subtraction of the prestimulus firing rate) and the actual SS modulation after subtraction of the prestimulus firing rate was very small (0.006) (for the 9 cells, mean ± SD = 1.2 × 10² ± 1.4 × 10²). It is important not to overestimate the stimulus-dependent pseudo correlation and, consequently, to underestimate the net correlation and the CS-induced SS modulation. The magnitude of the net cross-correlation between SS and CS observed in the present study was approximately the same order of magnitude as in previous observations (30-50 spikes/s firing rate during the pause and 10-30 ms pause duration) (Sato et al. 1992; Stone and Lisberger 1990b). Consequently, the reciprocal relationship between SS and CS cannot be explained by a short-term CS-induced SS modulation. A similar conclusion was derived from studies of other species (Simpson et al. 1996).

View larger version (41K): [in this window] [in a new window]   FIG. 12. Cross-correlation analysis of SS and CS. A: SS peristimulus time histogram. B: CS peristimulus time histogram. C: apparent cross-correlation between SS and CS, Rapp, which was directly calculated by CS spike trigger averaging of SS. D: stimulus-dependent pseudo cross-correlation between SS and CS, Rstm, which was similarly calculated after shuffling the impulse trains (Perkel et al. 1967; Toyama et al. 1981). E: net cross-correlation between SS and CS, Rnet, determined by subtracting the stimulus-dependent cross-correlation in D from the apparent cross-correlation in C. F: SS modulation, SSCS, which is accounted for by the short-term effect on SS by CS. Data shown in A-F were obtained from responses of 1 cell to upward stimulus motion presented 487 times at 80°/s.

	DISCUSSION

Abstract Introduction Methods Results Discussion References

Theories of CS function

The view that climbing-fiber inputs function as detectors of control errors is accepted rather widely but there is still much argument as to whether they act in real-time via the direct influence of CS on targets of the Purkinje cells and/or via a short-term modulatory action on SS patterns or in longer term by inducing long-lasting changes in the potency of parallel fiber-Purkinje cell synapses. More specifically, the primary theories regarding the function of CS are summarized as follows. Some previous researchers have not excluded the possibility that multiple theories may be correct so that the CS have multiple functions.

1)CS have been shown to be elicited by unexpected perturbations during wrist movements in awake monkeys (Gilbert and Thach 1977), during skilled locomotion (Andersson and Armstrong 1987) and a step-like movement (Gellman et al. 1985) in awake cats, and during walking in decerebrate ferrets (Lou and Bloedel 1986, 1992). Because the cerebellum is involved in controlling both posture and movement, unexpected perturbations might be considered as errors (Oscarsson 1980) in postural performance and movement, because they imply a mismatch between desired and actual movement. In this connection it has been suggested that the mean firing rate of CS during several hundred milliseconds represents a sensory error signal (e.g., retinal slip). This hypothesis was derived from experimental data obtained in the rabbit flocculus during eye movements induced by movement of a large visual field (Graf et al. 1988; Simpson and Alley 1974). Furthermore, in the monkey VPFL during smooth pursuit eye movement induced by small target motion, transient retinal slip was shown to correlate with the occurrence of a single CS during steady-state pursuit (Stone and Lisberger 1990b). In Ojakangas and Ebner's study, CS was shown to be coupled to a velocity-related error signal during a voluntary arm movement (Ojakangas and Ebner 1994).

2)The CS have been suggested to be real-time motor commands that modulate SS (Mano et al. 1986) because for tens of milliseconds after a CS, there is a pause in SS firing (Bell and Grimm 1969) and/or there is a short-term modulation of SS discharges for several hundred milliseconds after CS firing (Ebner and Bloedel 1981).

3)The electrical coupling between IO neurons (Llinás et al. 1974; Sotelo et al. 1974) has been shown to cause a degree of CS synchrony among groups of Purkinje cells (Sugihara et al. 1993; Wylie et al. 1995). In the vestibulocerebellum in alert rabbit, CS synchrony was demonstrated during eye movement (De Zeeuw et al. 1997a). This characteristic, and the observation that CS have relatively rhythmic firing patterns (Welsh et al. 1995), has led to the suggestion that CS are phasic motor commands involved in controlling the timing of movement execution.

4)Results of physiological, anatomic, and behavioral studies, as well as the existence of long-term depression at parallel-fiber/Purkinje-cell synapses, support the proposal that the climbing fibers are involved in motor learning. In summary, SS provide motor commands that are regulated by CS via modulation of the efficacy of the parallel fiber inputs (Albus 1971; Ito 1984; Marr 1969).

Kawato and colleagues (Kawato and Gomi 1992a,b; Kawato et al. 1987) extended earlier learning models by formulating a computationally explicit feedback-error-learning model. In this model, CS are assumed to be copies of feedback motor commands generated by a crude feedback control circuit, and thus CS are suggested to be sensory error signals in motor coordinates. The model predicts that the cerebellar cortex acquires an inverse dynamics model of a controlled object as a result of this learning. There is an ongoing debate as to whether the plasticity at the parallel-fiber/Purkinje-cell synapse is the elementary process underlying motor adaptation and motor learning and what role climbing fibers may have in this process.

These hypotheses variously predict the information conveyed by CS, the relationship between SS and CS, and the function of CS. The information conveyed by CS is suggested to be either sensory error (hypotheses 1 and 4), motor commands (hypothesis 2), or timing of movement (hypothesis 3). The feedback-error-learning model (Kawato and Gomi 1992b) predicts intermediate properties of CS, that is, that CS are derived from sensory error signals but already are represented temporally and spatially as feedback motor commands. CS and SS are suggested to be either independe