INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The vestibuloocular reflex (VOR) is a useful behavior with which to probe cerebellar function, primarily due to the well-defined and economical neural architecture of VOR pathways and the accessibility of these circuits for study. Modulation in the gain of this reflex (eye velocity/head velocity) represents a form of motor learning. The cerebellum is intimately involved in this process, since the cerebellar flocculus projects directly to brain stem cells that are critical participants in the VOR neuronal arcs (Langer et al. 1985), and removal or inactivation of the flocculus precludes further changes in VOR gain (Lisberger et al. 1984; Zee et al. 1981).

Ito (1972, 1982) proposed the flocculus hypothesis based on theories of cerebellar cortical function (Albus 1971; Marr 1969), wherein the flocculus is an adaptive VOR side path that inhibits the main VOR pathways. Miles and Lisberger (1981) proposed that there might be modifiable sites in both the flocculus and brain stem and that the flocculus might provide error signals to the brain stem to facilitate learning there. In support of the former proposal, Ito et al. (1974) reported changes in Purkinje cell activity of the rabbit that were coincident with changes in VOR gain, and Watanabe (1984, 1985) showed in monkey that Purkinje cells changed their response modulation during VOR following VOR gain change. Miles et al. (1980) reported that, in monkey, Purkinje cells change their firing pattern during cancellation of the VOR after learning, but argued that the direction of the change was incorrect to support the acquired VOR gain change. Lisberger et al. (1994a,b) demonstrated in monkey that both Purkinje cells and flocculus target neurons (FTNs) change their firing patterns in response to a step of head movement after learning, but that changes in the firing patterns had a latency too long to support the earliest changes in the VOR. This led them to propose the brain stem hypothesis and later the multiple site hypothesis in which the FTNs are the main adaptive site. Partsalis et al. (1995b) showed that in monkey a part of the learning remained in the FTN modulation after chemical inactivation of the flocculus, supporting the multiple site hypothesis.

Analyses of single-unit recording in relation to sensory input and/or motor output can be a powerful tool in elucidating the mechanisms of sensory-motor transformation implemented by a neuronal circuit or system. However, if the system consists of parallel and multiple stages of information processing and each stage is recursively connected by feedback loops, it is sometimes difficult to clarify the origin of the observed activity. Also when neurons receive multi-modal inputs, it may be difficult to specify which modality caused a change in neuronal activity. Floccular Purkinje cells receive vestibular and visual signals from brain stem vestibular neurons and pontine neurons, respectively, and contribute to the generation of motor commands to move the eyes. The resultant commands are fed back to flocculus as an efference copy signal. Therefore even if we see a change in the Purkinje cell firing after VOR adaptation, it may be difficult to specify the locus of the change.

One solution is to divide the neuronal circuit into subsystems that represent each stage of information processing and then identify the input-output relations of the subsystems. This system identification approach may clarify the flow of information between subsystems, and the subsystem or subsystems responsible for a behavioral change might be elucidated. Presently, we employed this approach to specify the locus of the VOR adaptation and the role of flocculus. We divided the vertical (V) VOR and optokinetic reflex (OKR) neuronal substrate into a pathway that passes through flocculus (FL pathway) and one that does not (nonFL pathway). The FL pathway was further divided into a subsystem that includes the pathway from sensory input to flocculus (preFL and FL subsystem) and a subsystem that includes the pathway from flocculus to motor output (postFL subsystem). Also, the preFL and FL subsystem and the nonFL subsystem in which multi-modal signal processing is executed were divided into subsystems, each of which represents signal processing for a single modality. Thus the information processing executed in preFL and FL vestibular, preFL and FL visual, preFL and FL efference copy, postFL, nonFL visual, and nonFL vestibular pathways could be evaluated separately.

Several similar computational studies have addressed the problem of the site of VOR motor learning. Fujita (1982) showed in his adaptive filter model that VOR adaptation could be achieved based on the framework of the flocculus hypothesis. Gomi and Kawato (1992) showed that VOR/OKR adaptation could be realized by assuming the flocculus hypothesis in their model adopting a feedback error learning scheme. Lisberger and Sejnowski (1992) and Lisberger (1994) suggested that changes in the flocculus together with those in FTNs are required to achieve stability in the VOR circuitry. The simulations of Quinn et al. (1998) suggested that both FTNs and flocculus Purkinje cells change their characteristics after VOR learning, supporting the multiple site hypothesis.

In contrast to these computational approaches in which model parameters were determined heuristically, there has been no study employing system identification techniques that used real experimental data to determine characteristics of the subsystems that compose the VOR. Our method can identify the transfer function of each of the subsystems described using experimental data. No presumption for adaptability of any of the subsystems was made, and thus all subsystems were free to change their characteristics in parallel with VOR gain adaptation.

The results illustrate that characteristics of the preFL and FL subsystems related to head movement change with VVOR adaptation while those of other preFL and FL subsystems remain stable. In keeping with our previous work, the characteristics of the nonFL subsystem related to head movement that includes the Y-group and FTNs also change with the learning. With the use of this information, a role for the cerebellar flocculus in VOR adaptation is discussed. Preliminary reports have appeared (Hirata et al. 1999). A glossary of terms used for experimental paradigms is provided in Table 1.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Description of the model

Figure 1A is a diagram of the known synaptic linkage within the VVOR/VOKR pathways, and Fig. 1B is the model. In Fig. 1A, input to the circuit arises from motion of the head, the visual surround, or both. The output is an eye movement. Eye movements are fed back via an external physical feedback loop to a summing junction where they are combined with visual movements and head movements computed as retinal inputs. Head velocity and acceleration are decoded by the vestibular organ and act as inputs to superior vestibular nucleus (SVN) and position vestibular pause (PVP) neurons in medial vestibular nucleus (MVN) (McCrea et al. 1987; Mitsacos et al. 1983). The flocculus receives head motion information via the floccular projecting neurons (FPNs) (Zhang et al. 1993) that modulate exclusively to head motion, not eye motion at the 0.5-Hz sinusoidal stimuli applied. Signals similar to those on FPNs reflecting head motion have also been recorded as input elements within the primate flocculus (Lisberger and Fuchs 1978; Miles et al. 1980). Retinal slip information arrives as mossy fiber inputs at the flocculus via the middle temporal area (MT)/medial superior temporal area (MST)/dorsolateral pontine nucleus (DLPN) or nucleus reticularis tegmenti pontis (NRTP) circuit (Kawano et al. 1992, 1994). An efference copy of ascending eye movement commands is relayed, e.g., via axon collaterals of the ascending vestibular nucleus neurons terminating in the midline paramedian tract (PMT) cells that are prefloccular nuclei (Blanks et al. 1983; Buttner-Ennever et al. 1989; McCrea et al. 1987). The output of these midline PMT nuclei provides a mossy fiber input that reflects an efference copy of ascending oculomotor commands (Buttner-Ennever et al. 1989; McCrea et al. 1987). Signals reflecting a copy of the ascending oculomotor commands present on vestibular nuclear neurons have also been recorded in primate flocculus as input elements (Lisberger and Fuchs 1978; Miles et al. 1980). Another candidate source for an efference copy signal might be the prepositus hypoglossi. The Y group receives head motion information via SVN interneurons reflecting both anterior and posterior canal input (Blazquez et al. 1994, 2000) and Y-group and SVN-FTNs receive the terminals of floccular Purkinje cells (Langer et al. 1985; Partsalis et al. 1995a; Zhang et al. 1995). FTN and Y group outputs project to the extraocular motor neurons that send the final command to move the eyes. This diagram does not include the OKN indirect pathway because the stimulus currently employed was outside of the frequency range in which the OKN indirect pathway is activated. Further, signals carried on climbing fibers to the flocculus have not been addressed in the present experiments. VOR gain change or motor learning within the context of the VOR implies that the brain's assessment of the magnitude of head motion is the variable that is reevaluated or learned. This head motion signal is carried by mossy fibers. Climbing fiber signals might be more related to underlying mechanisms that achieve VOR learning and are not addressed here. This diagram does not show visual pathways that do not pass through the flocculus to avoid complexity of the diagram, but these are included in the model in Fig. 1B.

View larger version (43K): [in this window] [in a new window]   Fig. 1. Vertical vestibuloocular reflex (VOR) and optokinetic reflex (OKR) neuronal circuit (A) and model (B). In A, FPN and FTN are the floccular projecting neurons and the floccular target neurons in superior vestibular nuclei (SVN), respectively, Y is dorsal Y group, Int. is the inter-neurons in SVN projecting to Y group. P is a floccular Purkinje cell, and g is a granule cell. PVP is position vestibular pause neurons in medial vestibular nuclei (MVN). MT, MST and DLPN are the middle temporal visual area, the medial superior temporal area, and the dorsolateral pontine nuclei, respectively. MN, extraocular motor neurons; LTN, lateral terminal nucleus of the accessory optic system. Colors in the background in A correspond to colors of the blocks in B. In B, G(s), G(s), and G(s) are prefloccular/floccular subsystems each of which represents a transfer function of prefloccular/floccular efference copy pathway, vestibular pathway, and visual pathway, respectively. The 3 components are added in the flocculus and form the Purkinje cell simple spike (SS) output. GpostFL(s) represents a transfer function of postfloccular pathway, which transfers the Purkinje cell SS activity to a part of the motor command. G(s) and G(s) represent transfer functions of nonfloccular visual and vestibular pathways, respectively. Corresponding neuronal circuit to G(s) is not shown in A. The block on the left hand side of G(s) represents a nonlinear transformation of retinal slip to linearize its relationship to Purkinje cell activity. h(t), d(t), r(t), f(t), ecopy(t), and x(t) are head movement, optokinetic stimulus movement, retinal slip movement, floccular Purkinje cell SS activity, efference copy signal, and eye movement, respectively.

Figure 1B is a block diagram of the VVOR/VOKR system configured around the flocculus reflecting the pathways in Fig. 1A. This model consists of three components, namely 1) prefloccular/floccular (preFL and FL), 2) postfloccular (postFL), and 3) nonfloccular (nonFL) systems. The preFL and FL system includes signal processing executed in prefloccular pathways and in the flocculus itself, and its output is indicated as floccular Purkinje cell simple spike (SS) activity. In the current study, only SS firing was analyzed as mentioned above. For this reason, Fig. 1B does not include a pathway from the inferior olive to flocculus (in this report the term flocculus should be understood to mean both the true flocculus and the ventral paraflocculus). The three kinds of mossy fiber input to the flocculus are explicitly described as separate pathways in the preFL and FL system. In each pathway, the sensory or original efference copy signal is processed by a subsystem G(s), G(s), or G(s) for vestibular, visual, or efference copy signals, respectively, and is converted into floccular Purkinje cell SS activity. The postFL system G_postFL(s) and subsequent oculomotor muscle plant transfer Purkinje cell activity into eye movements. The nonFL system consists of two pathways, each processing vestibular or visual sensory signals and converting them into eye movements. Signal processing executed in the nonFL visual and vestibular pathways are described by the transfer functions G(s) and G(s), respectively.

The following assumptions were made for the model structure based on previous physiological evidence. 1) The interaction of vestibular, visual, and efference copy signals in the flocculus is linear in the stimulus range employed (Lisberger and Fuchs 1978; Miles and Fuller 1975). 2) Interaction between signals from floccular and nonfloccular pathways is linear in the stimulus range employed (Robinson 1977). The plausibility of these assumptions was tested by predicting system outputs in response to unknown inputs (cf. Evaluation of model adequacy and estimated parameters).

The following system identification technique was developed to identify characteristics of each subsystem by using sensory input, motor output signals, and floccular Purkinje cell activity.

System identification technique

One direct method to identify a system's characteristics is to measure the input and output of a system simultaneously. In biological systems, however, this direct approach is not always possible due to technical difficulties and to the structural complexity of the system. Thus we embarked on the following approach to identify each subsystem in the model. The method consists of four steps.

Presently only slow phase eye movements and corresponding Purkinje cell SS activity were studied, assuming that slow phase and quick phase eye movements are generated by independent mechanisms. Also it was assumed that each subsystem is time invariant during one recording set that consists of about 5 min of several visual-vestibular mismatch paradigms.

IDENTIFICATION OF THE PRE-FLOCCULAR/FLOCCULAR SUBSYSTEMS: G(s), G(s), G(s). The system equation for the preFL and FL system in the Laplace transform domain is expressed as follows
where s denotes the Laplace operator while F(s), H(s), R(s), and E(s) denote Laplace transforms of floccular Purkinje cell SS firing rate f(t)¹, vertical head movement h(t) (head position), retinal slip r(t) (retinal slip position), and efference copy ecopy(t), respectively. Position, velocity, and acceleration of the eye, head, and retinal slip movements are vertical unless otherwise specified. G(s) is a transfer function of the semicircular canals, and since it can be assumed that its characteristics are not affected by the gain of VOR (Miles et al. 1980), it is incorporated into G(s) hereafter. G_r(s) is a transfer function of the retina. We also assume that its characteristics are not affected by the gain of the VOR and incorporate it into G(s). Therefore the above equation becomes

(1)

If it is assumed that 1) head velocity and acceleration signals, 2) retinal slip velocity and acceleration signals contribute to Purkinje cell firing modulation, and that 3) efference copy signals convey eye velocity information, the system equation in the time domain can be expressed as the following multiple linear regression model

(2)

where _h [spk/s per deg/s²], _h [spk/s per deg/s], _r [spk/s per deg/s²], _r [spk/s per deg/s], _e [spk/s per deg/s], and  [spk/s] are regression coefficients and denote sensitivities of Purkinje cell firing to head acceleration acc_h(t), velocity vel_h(t), retinal slip acceleration acc_r(t), velocity vel_r(t), efference copy signals ecopy(t), and the dc component of Purkinje cell firing, respectively. _h [s], _r [s], and _e [s] denote delays between head movement and Purkinje cell activity, retinal slip and Purkinje cell activity, and efference copy signals and Purkinje cell activity, respectively. (t) is the error term whose mean is 0. The assumptions made above are plausible if we consider that 1) the vestibular nerve signal conveys mainly head velocity and acceleration (Goldberg and Fernandez 1971), 2) the contribution of retinal slip position was significant only in the very low-frequency range (Kobayashi et al. 1998) that was not presently used, and 3) mossy fibers convey an efference copy of oculomotor motor commands to the flocculus predominantly as an eye velocity signal (Miles et al. 1980; Stone and Lisberger 1990).

Step 1: identification of G(s), G(s).

(3)

Step 2: identification of G(s).

IDENTIFICATION OF POST-FLOCCULAR AND NON-FLOCCULAR SUBSYSTEMS: G_postFL(s), G(s), G(s). The system equation for the postFL and nonFL systems in the Laplace domain is expressed as follows
where G_m(s) and X(s) denote a transfer function of the ocular muscle plant and a Laplace transform of eye movement (eye position), respectively. Since it is assumed that the characteristics of G_m(s) as well as G(s) and G_r(s) do not change with VOR gain change, they were incorporated into G(s), G(s), or G_postFL(s). Thus the above equation becomes

(5)

This can be rewritten as
It has been shown that Purkinje cell firing encodes a part of eye movement inverse-dynamics in terms of a linear combination of eye acceleration, velocity, and position (Shidara et al. 1993). This means that the postFL pathway can be described by a second-order linear system. If we assume that nonFL subsystems can also be characterized by a second-order linear system within the input range currently employed, the above equation can be expressed in the time domain as follows

(6)

where a_x [spk/s per deg/s²], b_x [spk/s per deg/s], c_x [spk/s per deg] denote sensitivities of Purkinje cell firing rate to eye acceleration acc_x(t), velocity vel_x(t), and position pos_x(t), respectively while _x [s] denotes a delay time between Purkinje cell activity and eye movement.  is the dc term and (t) is the error term whose mean is 0. a_r, b_r, a_h, _r, and _h together with a_x, b_x, c_x, and _x determine the second-order transfer functions of nonFL subsystems and postFL subsystem as follows


Thus identifying each nonFL subsystem and postFL system results in determining the regression coefficients in Eq. 6.

Step 3: identification of G_postFL(s) and G(s).

(7)

Step 4: identification of G(s).

(8)

Evaluation of model adequacy and estimated parameters

The model adequacy was checked by examining the residual of Purkinje cell firing following the regressions executed in steps 1, 2, 3, and 4 (residual check), and the predictions of Purkinje cell firing in response to visual-vestibular mismatch stimuli (prediction check) such as VOR_e, VOR_r, VOR_s (see Experimental paradigms below for stimulus condition), which were not used to determine the model parameters. The amplitude distribution and autocorrelation function of the residual were compared with those of Purkinje cell firing rate during no external stimulation. The idea is that if the amplitude distribution and autocorrelation functions of the residual match those of the Purkinje cell firing during the no stimulus condition⁴ (no stimulus in dark for steps 1 and 3, in light for steps 2 and 4; see Experimental paradigms for details), the information encoded in the Purkinje cell firing by the applied external stimuli has been successfully extracted or predicted. The Ansari-Bradley test was applied to test any statistical difference in the amplitude distributions of residual and Purkinje cell firing during no external stimulus. The autocorrelation function of the residual was considered to be the same as that of Purkinje cell firing during no external stimulus if the former resides in a ±0.1 range of the latter. Cells that passed this residual check following each of the four steps of regression and the prediction check were subjects for further analysis.

Experimental setup and surgical procedure

Two adult male squirrel monkeys weighing between 800 and 900 g were utilized for these experiments. Animals were placed in a primate chair daily for several weeks before any surgery was performed to acclimatize them to the experimental setup. All surgery was performed in a sterile operating suite using induction by Ketamine and inhalation anesthesia using Isofluorane. For head fixation, a stainless steel bolt was secured to the occiput using small, stainless steel screws and dental cement. This bolt fit into a receptacle on the monkey chair to fix the head in the center of a magnetic field generated by two sets of field coils driven in quadrature. For eye movement recording, a prefabricated eye coil constructed of Teflon-insulated stainless steel wire (Cooner) was implanted under the conjunctiva at the limbus of either eye, sutured to the sclera, and the twisted ends of the coil wire led to an occipital plug. A stainless steel recording chamber aimed at the flocculus was implanted and secured with dental cement over a fenestra in the skull. All wounds were treated daily with antibiotics. The Animal Welfare and Use Committee of Washington University approved all procedures and experiments.

Animals were seated in a primate chair with their heads fixed. The chair was placed in the center of a white cylindrical screen 1 m diam (extending 36 cm above and 50 cm below the animal's head) on which black random dots were projected. This is the optokinetic stimulus (OKS). The right or left side of the animal was placed down so that rotation of the OKS about an earth vertical axis produced upward or downward nystagmus. The eye coil output was led to a phase-locked detector whose output gave signals proportional to horizontal and vertical eye position. Vertical and horizontal eye velocities were calibrated. Horizontal and vertical eye position, OKS velocity, and chair velocity were continuously digitized at a sampling frequency of 200, 100, and 100 Hz, respectively, with the use of a CED 1401 interface (Cambridge Electronic Design) for display and storage using the Spike-2 program. The output of a threshold detector, signaling the occurrence of action potentials, was sampled with a resolution of 0.01 ms and stored in the same way. Raw data were also stored on VCR tape (Neurodata PCM).

Experimental paradigms

Stimuli used for all paradigms in the present experiment were 0.5-Hz sinusoids. Although the system identification technique mentioned above is valid for inputs with a wider frequency range, we used this stimulus to directly compare present results with previous related works using the same type of stimulus (Miles and Braitman 1980; Miles et al. 1980; Partsalis et al. 1995a,b; Watanabe 1984, 1985; Zhang et al. 1993, 1995). Therefore only gains and phases at 0.5 Hz were evaluated in the transfer functions estimated in steps 1, 2, 3, and 4 above. To change the animals' VOR gain, suppression of the VOR (VOR_s) or reversal of the VOR (VOR_r) and enhancement of the VOR (VOR_e) paradigms were utilized for low gain and high gain training, respectively.⁵ In the VOR_s paradigm, the chair moves in phase with and at the same speed as the OKS (peak speed: ±40 or ±80 deg/s). In VOR_r, the chair moves in phase with the OKS but with a velocity amplitude twofold that of the chair (chair peak speed: ±40 deg/s). In VOR_e, the chair moves out of phase with the OKS at twice the speed (chair peak speed: ±40 deg/s). Animals were trained toward low or high VVOR gains for 4-7 h a day. During the training, Purkinje cells were isolated at various stages of VVOR gain. After the isolation VF, VOR_d, VOR_r, VOR_s, VOR_e, nostimL, and nostimD were performed for about 60 s each. VOR_r, VOR_s, and VOR_e were recorded to check the performance of the model that was identified by using VF and VOR_d paradigms, while no stimulus paradigms were used as references to evaluate the residual following the model fit (see System identification technique). If the cell was still isolated after this recording set, the training was continued and the same recording set was repeated every hour until we lost the cell.

Unit recording, identification of Purkinje cells

Floccular and ventral parafloccular V zone Purkinje cells were identified by the occurrence of complex spikes (CS) and by their characteristic discharge patterns during vertical VOR_r, VOR_s, or VOR_e. CS could not always be recorded simultaneously during SS recording but were usually seen in close physical proximity to the sites of SS recording.

Data handling

Data analysis and model simulation were performed in Matlab (Mathworks) running on a Pentium III based PC. Vertical eye velocity, eye acceleration, head acceleration, and OKS acceleration were calculated by using a three-point low-pass differential digital filter (Usui and Amidror 1982) from vertical eye position, eye velocity, head velocity, and OKS velocity, respectively. Low-pass filters (combination of 5, 7, and 9 point moving average FIR filters: cutoff frequency 10 Hz) were applied to reduce high-frequency noise in the eye acceleration exaggerated by the differential filtering. Retinal slip velocity was calculated as eye velocity-OKS velocity-head velocity after interpolating (cubic spline) OKS and head velocity traces so that their sampling frequency and time stamp match those of eye velocity. Retinal slip acceleration was then calculated from slip velocity by using the same digital filters used to obtain eye acceleration. Purkinje cell instantaneous firing rate (FR) was calculated as the reciprocal of each inter-spike interval (Partsalis et al. 1995a,b; Zhang et al. 1993, 1995).

Saccades and postsaccadic drifts, if any, were eliminated from eye movement traces by using an automated desaccading algorithm that was visually checked on the computer screen. Data periods corresponding to saccades and the postsaccadic slide were eliminated from both head and OKS movements as well as from Purkinje cell FR.

Gain of the VOR was measured as vertical eye velocity divided by head velocity. More precisely, the following regression was executed and the coefficient c was referred to as the gain of the VOR
where vel_x(t) and vel_h(t) are desaccaded, vertical eye velocity and head velocity traces during VOR_d, respectively, d denotes the dc term that was close to 0 in most cases, and (t) is the error term.  [s] is a delay between eye velocity and head velocity from which the phase of the VOR at 0.5 Hz was calculated as 180°. The coefficients c and d were determined by solving the normal equation derived from the above equation to minimize the square sum of (t) with a given . The value of  that gives the minimum square sum of (t) was globally searched.

After this preprocessing, all signal traces were resampled at the sampling points of desaccaded Purkinje cell FR data to allow signal processing in the analysis mentioned in the System identification technique section.

We also analyzed the cells with a conventional method (Watanabe 1984, 1985) that averages Purkinje cell firing over cycles and fits sinusoidal wave to it to compare results with the new method. BFGS algorithm (Polak 1971) was used for this nonlinear parameter estimation of the amplitude and phase parameters of the sinusoid.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Fifty-three floccular V zone Purkinje cells that showed downward eye movement sensitivity were well isolated and recorded for extended periods at 78 VVOR gains between 0.4 and 1.5 from 2 monkeys. Cells with clear upward eye movement sensitivity were very rare and were not evaluated in this report. The 53 cells were isolated while animals were being adapted by visual-vestibular interaction paradigms, thus those that did not show firing modulation during these paradigms were not recorded in the current experiment. The quantitative analysis of these 53 cells is the subject of the remainder of this report. Thirteen cells of the 53 could be continuously recorded for up to 7 h and thus provided data at multiple VOR gains.

Figure 2 illustrates the behavioral paradigms presently employed and the accompanying Purkinje cell responses in a recording ensemble of a typical cell at a normal VVOR gain of 0.98. Only 10 s of over 60 s of recorded traces are shown. In each eye velocity trace, the sharp deflections are fast phases of nystagmus or saccadic eye movements and will not be considered further.

View larger version (57K): [in this window] [in a new window]   Fig. 2. Eye movements and corresponding Purkinje cell SS activities during experimental paradigms employed. VF, VORd, VORs, VORe and VORr are visual following, VOR in dark, suppression of VOR, enhancement of VOR, and reversal of VOR paradigms, and NostimL and NostimD are no stimulus in light and dark condition, respectively. E is vertical eye velocity in deg/s, IFR is instantaneous Purkinje cell SS FR in spk/s, and D and H are optokinetic stimulus (OKS) and vestibular chair velocity in deg/s. The OKS and chair traces are shown in the same panels in gray and black lines, respectively. Ten seconds of about 60 s of data for each paradigm are shown.

During VOR_d modulation amplitude calculated by averaging over 30 stimulus cycles in this Purkinje cell is very small (±4.8 spk/s). Other Purkinje cells also showed small modulation during VOR_d no matter what the VOR gain (mean modulation amplitude of entire sample to 40 deg/s head velocity is 12.3 ± 8.0 spk/s). However, this small modulation amplitude clearly correlates with VVOR gain state (Fig. 3). During VF this Purkinje cell FR increased during downward eye movement. Modulation in each stimulus cycle varied substantially. In VOR_e Purkinje cell FR increases during downward eye velocity or upward head velocity. During VOR_s the eye is relatively stable within the orbit, and Purkinje cell FR increases during downward head movement or small upward eye movement (eye velocity not completely suppressed). During VOR_r, Purkinje cell FR increases during downward eye movement or downward head movement. Among these visual-vestibular mismatch paradigms, the largest modulation was usually observed during VOR_r, suggesting that all inputs to Purkinje cells evoked by this stimulus are additive (Lisberger and Fuchs 1978) (see Fig. 11). The bottom two panels in Fig. 2, nostimL and nostimD, illustrate the baseline firing herein defined as the intrinsic noise underlying Purkinje cell FR modulation. The extinction of the light (nostimD) results in a decrease of the spontaneous background discharge (75 of 78 samples, 96.2%).

View larger version (52K): [in this window] [in a new window]   Fig. 3. Three-dimensional polar diagram of Purkinje cell firing pattern during VORd recorded at different VOR gains. Each black dot represents a single Purkinje cell. The x-y plane represents amplitude (radius) and phase (angle) of the Purkinje cell firing modulation in a polar coordination. The z-axes represents VOR gain at which the cells were recorded. Note that a clear correlation between VOR gain and the amplitude of Purkinje cell modulation exists.

Of these paradigms, the VF was used to identify the preFL and FL visual and efference copy subsystems G(s), G(s), nonFL visual subsystem G(s), and postFL subsystem G_postFL(s). VOR_d is used to identify preFL and FL vestibular subsystem G(s) and nonFL vestibular subsystem G(s). VOR_e, VOR_s, VOR_r are used to check the adequacy of the model. If the model can predict Purkinje cell firing during the paradigms that were not used to identify the subsystems, the model is probably useful in interpreting the VVOR/VOKR system. NostimL and nostimD are used to evaluate the residual after identification of the subsystems and predictions of Purkinje cell firing.

Figure 3 illustrates a three-dimensional (3D) polar diagram of Purkinje cell modulation recorded at different VOR gains, representing VOR gain versus modulation amplitude and phase calculated by averaging over more than 28 cycles during VOR_d paradigm. Note that the modulation amplitude to 40 deg/s of head velocity is very small ranging from 0.4 to 37 spk/s (mean 12.8 ± 8.5 spk/s). However, this small modulation amplitude changes in parallel with VOR gain along the phase angle of around 20 deg. The modulation amplitude of the cell population increases in phase with head velocity as VOR gain increases, and it decreases in phase with head velocity or increases in an out-of-phase manner with head velocity. This result is comparable with Watanabe's result (Watanabe 1984, 1985). The observed increase in Purkinje cell modulation in-phase with head velocity after high gain adaptation may be causal in the increase in FTN and Y group modulation out-of-phase with head velocity. The FTN and Y group modulation causes larger eye velocity (Partsalis et al. 1995a; Zhang et al. 1995). That is, the observed change in the Purkinje cell modulation is in the correct direction to induce the adaptation. However, we cannot simply conclude from this result that the adaptation occurred in flocculus. To pinpoint the site or sites responsible for the observed change in Purkinje cell firing pattern and VVOR gain change, we utilized the system identification technique as follows.

Figure 4 illustrates a summary of the signal processing of a typical cell executed in step 1 to determine the characteristics of the preFL and FL subsystems G(s) and G(s) using the VF paradigm. In the retinal slip velocity and acceleration panels, black dots are the desaccaded, resampled and nonlinearly transformed signals, and the faint trace is the original slip signal. It has been shown in similar multiple linear regression analyses that coefficients for retinal slip signals are very small (Hirata et al. 1998; Suh et al. 1999), even though flocculus receives retinal slip information via DLPN as illustrated in Fig. 1A. A possible reason for the small coefficients in the linear regression analysis is a nonlinear relationship between retinal slip and Purkinje cell firing as seen in firing properties of lateral terminal nucleus cells in relation to slip velocity (Mustari and Fuchs 1989). If such a nonlinear signal transformation also exists in the preFL and FL visual pathway, linear regression analyses cannot properly evaluate the contribution of slip signals.

View larger version (61K): [in this window] [in a new window]   Fig. 4. Reconstruction of Purkinje cell SS FR during VF paradigm (bottom) by a multiple regression model consisting of retinal slip velocity (top), retinal slip acceleration (2nd), efference copy (3rd), and a dc term (not shown) for a typical cell. In the top 3 panels, gray lines are original traces including saccadic periods and black lines are after desaccade. In the bottom panel, the gray trace is the original Purkinje cell SS FR, and the black dots are the reconstruction by the model. Ten seconds of about 60 s of data are shown.

Figure 5 plots retinal slip versus Purkinje cell FR during VF paradigm of the same cell as in Fig. 4. In the top panel of Fig. 5, desaccaded Purkinje cell FR during VF paradigm is plotted against desaccaded retinal slip velocity as dots. There is a clear nonlinear relationship between the slip velocity and Purkinje cell FR that shows saturation at slip velocities beyond a ±10 deg/s range. A sigmoidal function a/{1  e^{b(rslp_vel+c)}} that is commonly used to express this kind of saturation property was fit to the plots and superimposed on them as a black line. For curve fitting, parameters a, b, and c were estimated by using a nonlinear optimization method, BFGS algorithm (Polak 1971), which is categorized in quasi-Newton methods. Due to such nonlinearity, a linear regression analysis may result in a small coefficient for retinal slip velocity. In other words, a multiple linear regression model that just includes a linear term of retinal slip velocity is not adequate to evaluate the contribution of slip velocity to Purkinje cell firing, because it can only evaluate its "linear contribution." Therefore a nonlinear transformation using the fitted sigmoidal function was included in the model as shown in Fig. 1B to evaluate the slip contribution more properly (see APPENDIX for more details of this nonlinear transformation). The middle panel in Fig. 5 shows the same figure for slip acceleration. In contrast to slip velocity, slip acceleration is almost linearly related to Purkinje cell firing. Different sigmoids were used for different cells. Estimated sigmoids for all the cells examined are superimposed in the bottom panel of Fig. 5. Thick traces are their averages.

View larger version (64K): [in this window] [in a new window]   Fig. 5. Linearization of retinal slip velocity and acceleration to Purkinje cell FR. The top panel illustrates retinal slip velocity vs. Purkinje cell FR during VF (dots) and a fitted sigmoidal function. Coefficients of the sigmoidal function were estimated by using a nonlinear optimization method under the least-squares criteria. Only desaccaded parts of the data were used. Note the saturation in Purkinje cell FR outside a ±10 deg/s range of retinal slip velocity. The 2nd panel shows the same for retinal slip acceleration. Note that the relatively linear relation between the retinal slip acceleration and Purkinje cell FR. The bottom 2 panels are superposition of the fitted sigmoidal functions of all cells examined (dashed lines). The left panel is for retinal slip velocity, and the right panel is for retinal slip acceleration. Thick solid traces are averages.

The efference copy trace in Fig. 4 is shown in a similar format to retinal slip. Eye velocity was used for the efference copy of eye movement. Dots are the desaccaded and resampled traces, and the faint trace is the original eye velocity. The P cell firing panel shows Purkinje cell FR (faint) and the reconstructed firing rate (dots) by a multiple linear regression model with the above three explanatory variables and the dc term (Eq. 3). The regression was performed by using the entire 60 s of data. As can be seen, the dots roughly overlie the original Purkinje cell FR trace; however, there is still considerable variability in the firing evident both above and below the dots. This variability can be accounted for by an evaluation of the residual noise in the firing pattern following the extraction of the signal components by the regression and comparison of this noise to that of the spontaneous Purkinje cell discharge in the nostimL (Fig. 6). The estimated coefficients are evaluated in Fig. 12. The delay terms _x, _r estimated for the entire 78 samples are 0.0176 ± 0.0078 and 0.054 ± 0.011 s, respectively.

View larger version (55K): [in this window] [in a new window]   Fig. 6. Residual check following the regression executed in Fig. 4. Top left: the residual FR for a typical cell shown in Figs. 4 and 5. Top right: the histogram of the residual (black bars) and that of Purkinje cell FR during the nostimL condition (spontaneous firing rate; gray line). Note that there is a slight deviation of the 2 histograms from a Gaussian distribution and that the 2 histograms are almost identical. Bottom: autocorrelation functions of the residual (black), the original FR before the regression (light gray), and the spontaneous firing rate (dark gray). Note that shape of the autocorrelation function of the residual is almost a delta function, and identical to that of the spontaneous firing rate.

Figure 6 illustrates the evaluation of the residual component of Purkinje cell FR following the regression illustrated in Fig. 4. The top right panel compares the amplitude distribution of the residual (black bars) that deviates slightly from a Gaussian distribution, and the spontaneous firing rate during nostimL (gray line). The difference between these two non-Gausian distributions was not statistically significant (P > 0.397, Z = 0.2635, Ansari-Bradley test). The bottom panel compares the autocorrelation functions of the spontaneous firing rate during nostimL (dark gray) and the residual after the regression (black). The autocorrelation functions were calculated from each data set with the same data length so that the variance in the estimation for the functions are comparable. The oscillation seen in the autocorrelation function of the original firing (light gray) is not seen in that of the residual, and its delta function like shape is almost identical to that of the spontaneous firing rate. The similarity in autocorrelation functions assures that the frequency content of the two signals are identical and that of the amplitude distribution assures that the two signals have the same probability density function. These similarities in frequency content and probability density function indicate that the two signals are statistically identical if we focus only on the linear