An internal model of the physics of moving objects
In this section, I suggest that the eye velocity sensitivity of PCs results from a positive feedback circuit through the floccular complex that can be viewed as a model of the inertia of smoothly moving objects. presents the evidence that the eye velocity component of PC firing is due to a corollary discharge input (from Stone and Lisberger, 1990
). Here, the target started with a step-ramp motion. After 400 ms, a signal specifying the monkey’s eye position was fed back to the experimental-control computer and used to drive target position. As a result, the target was stabilized with respect to the moving eye, and moved wherever the eye moved for an interval of 300 ms. During the interval when the target was stabilized (between the vertical dashed lines), the eye continued to move without loss of speed, and the simple spike firing rate of the PC under study remained high without a noticeable decrease. Thus, both smooth eye motion and the elevated simple spike firing of PCs are maintained even when retinal image motion is removed by image stabilization. In contrast, the discharge of MT neurons shows a clear decrease when an analogous experiment is performed (Newsome et al., 1988
), implying that MT responses during pursuit are driven by retinal signals while floccular responses are driven by extra-retinal signals.
Figure 6 Response of a representative PC during pursuit with a stabilized target. From top to bottom the traces are average simple spike firing rate, superimposed target and eye position, and eye velocity. The two vertical dashed lines indicate the interval when (more ...)
An interpretation in terms of an internal model is outlined by . The interpretation starts from two facts: i) the disynaptic connections to motoneurons imply that floccular output drives eye velocity; and ii) floccular inputs reflect the ongoing eye velocity. Thus, floccular PCs control eye velocity and receive feedback about eye velocity, implying that the eye velocity component of PC firing is configured as a positive feedback loop. In the model represented by , eye velocity at time t
]) is driven by a combination of two signals: eye velocity at time t
] ) and image velocity from time t
ms ( İ
− 100]). If Δt
is small, then the positive feedback loop acts as an integrator and will maintain automatically both eye velocity (Morris and Lisberger, 1987
) and the discharge of PCs (Stone and Lisberger, 1990
) during steady-state pursuit, even in the absence of image motion during perfect tracking or stabilized vision. Thus, a positive feedback loop through the floccular complex predicts the two features of the data in , that both eye velocity and simple spike firing persist when image motion from the target is eliminated. Absent positive feedback, the velocity of the eye would decay quickly to zero during stabilized vision because of the mechanics of the oculomotor plant.
Figure 7 Configuration and implications of the postulated internal model of the inertia of moving objects. A: Proposed positive feedback of a corollary discharge of eye velocity through the floccular complex. Eye velocity at time t is driven by the sum of eye (more ...)
Perhaps the positive feedback configuration of the eye velocity component of PC firing provides neural inertia, ensuring that an eyeball in smooth pursuit motion remains in motion. Similarly, physical inertia will cause real objects in motion to remain in motion. Therefore, we can view the positive feedback of eye velocity through the floccular complex as an internal model of the physics of objects in our world. In a similar vein, Cerminara et al. (2009)
showed that the activity of PCs in the lateral cerebellum models the velocity of a moving object even when the target disappears briefly. Further, Angelaki et al. (2004)
and Yakusheva et al. (2008)
came to a similar conclusion, that the representation of tilt and translation in the brainstem, the cerebellar cortex of the nodulus, and the fastigial nucleus represented an internal model of the physics of the world. Perhaps internal models of the physics of the world are a general operating principle in many areas of the cerebellum.
Another way to think about the function of (and necessity for) an eye velocity positive feedback pathway through the floccular cortex is outlined by . Consider the pursuit of step-ramp target motion. The onset of target motion creates retinal image motion that drives pursuit. However, as eye velocity accelerates and tracks the target almost perfectly, retinal image motion disappears. Thus, a transient retinal image velocity (solid trace in the middle graph of ) leads to a sustained eye velocity (solid trace in top graph of ). An eye velocity positive feedback circuit would act as an integrator and convert a transient input into a sustained output. It also would allow image motion to serve as a command for changes in eye velocity, which are eye accelerations. shows the plausibility of these ideas: the eye acceleration at the initiation of pursuit (solid trace in bottom graph of ) follows the image motion by 100 ms, but shows a remarkable resemblance to the image motion when shifted forward in time and scaled to superimpose on eye acceleration (dashed versus solid trace in middle panel of ).
It is possible to think of the eye velocity output from the floccular complex as a representation of target velocity in world coordinates. Such a signal could come from cortical areas, where similar extraretinal responses have been reported (e.g. Newsome et al., 1988
; Fukushima et al., 2000
), or it could be constructed in and brainstem-cerebellum positive feedback circuit, as I have proposed here. There are theoretical advantages to positive feedback with the short latencies provided by a brainstem-cerebellar loop: longer latencies would have a propensity to oscillate. Further, a plausible neural substrate exists in the relationship between the floccular complex and the nucleus prepositus. Thus, while I cannot exclude a cortical origin for eye velocity output from the floccular complex, I favor the alternative that it arises in subcortical circuits.
An internal model of downstream processing
It is traditional to think of inverse dynamic models in terms of compensation for the physical plant of the effector organ. In the oculomotor system, the brainstem velocity-to-position integrator proposed by Skavenski and Robinson (1973)
is widely viewed as “the” inverse dynamic model that compensates for plant dynamics. To think about the dynamics of floccular output, I will extend the concept of the “plant” to comprise both the eyeball and the brainstem circuits that are downstream from floccular output. My thinking is premised partly on the fact that smooth pursuit is an evolutionarily recent specialization compared to the vestibulo-ocular reflex and saccades: pursuit is most capable of driving highly effective smooth tracking of small objects in primates (although not all species have been studied).
Pursuit signals emanate from the floccular complex and then share the brainstem circuits that must already have been specialized to transform vestibular and saccadic commands into the correct eye movements. Therefore, floccular output may have had to adapt to provide dynamics that are suitable given the pre-existing brainstem circuits: from the perspective of the floccular output, the brainstem and eyeball together may function as the “plant”. To drive eye movement effectively, the output of the floccular complex should have dynamics that complement the filter properties of downstream processing, including the final oculomotor circuits in the brainstem and the mechanics of the eyeball. Here I suggest that the mossy fiber inputs to PCs create the output that would be expected if an inverse model of those dynamics existed in the floccular cortex.
To understand the dynamics of floccular output and its possible function as an inverse model of downstream dynamics, I ask how the pooled floccular output must be transformed to generate the smooth pursuit eye movements recorded at the same time. To do so, we went one step further than did Shidara et al. (1993)
. Krauzlis and Lisberger (1994)
started by averaging the simple spike responses of a sample of PCs to obtain a pooled floccular output during pursuit of step-ramp target motion in the on-direction () and off-direction (). Comparison of the average firing rates (solid traces) and eye velocities (dashed traces) reveals two problems that make it impossible for a single set of downstream circuits, with a single set of dynamics, to convert both firing rates into eye movement. First, the amplitudes do not match: downstream circuits would need to amplify the responses during off-direction pursuit selectively. Second, the dynamics of eye velocity (Luebke and Robinson, 1988
) and firing rate (Krauzlis and Lisberger, 1994
) are different at the onset and offset of pursuit: for on-direction pursuit, there is a large overshoot in firing rate at the onset but not the offset of pursuit; for off-direction pursuit, the situation is the opposite. To resolve these problems, Krauzlis and Lisberger (1994)
assumed that the output from the floccular complexes on the two sides of the brain might be combined before being transformed by a single brainstem circuit. They created an opponent floccular output (), computed as the difference between the averaged outputs from the floccular complexes on the two sides of the brain, and then processed the opponent output through a model of the brainstem and plant dynamics (). The output of the model provided a good account of the eye movements recorded simultaneously with the PC firing rates at the onset and
offset of pursuit for step-ramp target motions ().
Figure 8 Population average of the firing rate of 20 PCs and the effects of transforming it with a model of the brainstem and oculomotor plant. A, B: Solid and dashed traces show the population averages of simple spike firing rate and eye velocity for pursuit (more ...)
The opponent population response in shows transient overshoots and undershoots at the onset and offset of pursuit that are symmetrical. Because of the excellent prediction obtained with this approach, Krauzlis and Lisberger (1994)
concluded that the “transient overshoots exhibited in the firing rate of PCs can provide appropriate compensation for the lagging dynamics of the oculomotor plant”. In an earlier study, Stone and Lisberger (1990)
had provided evidence that the transient overshoots, while correlated with eye acceleration, are actually driven by visual motion inputs to the floccular complex. Thus, the transients provided by the visual mossy fiber inputs to floccular PCs contribute a critical component that allows the pooled floccular output to compensate for dynamics in the downstream processing. Without the extra transient provided by the visual input, the eye movement driven by floccular output would show lower eye acceleration than it does. The logic outlined here does not prove that the floccular complex contains an inverse model of downstream processing. But, it does provide a valid way to interpret the data: the output from the floccular complex appears to be customized to compensate for the dynamics of downstream processing, a signature of inverse dynamics models.
The floccular complex appears to perform one additional transformation of its inputs to configure its outputs in a way that matches the dynamics of downstream processing. Recall that the vestibular and saccadic commands for eye movements comprise command signals for eye velocity. To share brainstem circuits with these other kinds of eye movements, floccular output also should be a command for smooth eye velocity. Yet, the eye movement inputs to the floccular complex are mainly mossy fibers that arise in the nucleus prepositus. They transmit signals that are dominated by eye position with a significant but smaller component related to eye velocity (McFarland and Fuchs, 1992
; McCrea and Baker, 1985
; Langer et al., 1985
). In contrast, the simple-spike outputs from the floccular complex are dominated by eye velocity (Miles and Fuller, 1976
; Lisberger and Fuchs, 1978a
), with a significant but smaller component related to eye position in some neurons (Noda and Warabi, 1982
). Somehow, the circuits of the floccular cortex must reduce the eye position input and emphasize the eye velocity input. Only by providing an output related to eye velocity can the floccular complex co-opt the brainstem VOR pathways and drive smooth pursuit eye movements with the correct dynamics.
An internal model of the kinematics of the eye and head
Here, I suggest that the combination of head and eye movement inputs to floccular PCs acts as an internal model of the kinematics of smooth gaze control during smooth eye and head movement. As illustrated in , it is possible to move gaze through space either by rotating the eyes with a stationary head (A), or by rotating the head and keeping the eyes stationary in the orbit (B). Either strategy has the same effect of changing the angle of the eyes with respect to the stationary world, even though the motor signals and strategies are quite different. Further, the action caused by the vestibulo-ocular reflex (VOR), which reacts to head rotation with an equal and opposite eye rotation, means that the eyes continue to point at the same location in space even though both the head moves in space and the eyes move in the orbit (C). The ultimate goal of smooth eye movement is to point the eyes at the desired stationary or moving object, using visual tracking systems to rotate gaze smoothly and vestibular mechanisms to compensate for head turns and prevent gaze from being destabilized by our own motions.
Figure 9 Cartoon and schematic diagrams showing parallels of gaze velocity in orbit and its internal model in PCs. A–C: Eye and head movements achieved under different tracking conditions. A: Tracking of smooth target motion through smooth eye motion in (more ...)
The situation cartooned in also is defined by an equation that describes how the combination of eye motion in the orbit ( Ė
) and head motion in space (
) determines gaze motion (Ġ
), defined as the velocity of the eyes in space:
The same relationship and terminology could be used for eye/head/gaze position or acceleration. Equation (3)
defines one aspect of the kinematics of the orbit, namely the combination of rotation of the eyes and head to control where the eyes are pointed in the visual world. As I discussed in relation in , the discharge of many floccular PCs combines signals related to eye and head velocity to achieve an output signal. Because the two sets of signals are equally weighted, at least on average, they produce an output that is related to gaze velocity and were named the “gaze velocity Purkinje cells” or “GVP” cells by Robinson (1981)
. Because they mimic the combination of eye and head velocity that occurs physically in the orbit (), the GVP cells () can be regarded as the output of an internal model of eye-head kinematics. The head and eye movement signals appear to enter the floccular complex independently, leading me to assume that the internal model resides in the floccular cortex.
Do the GVP cells represent the output from an internal model of the mechanics of the orbit, or of the interaction between the VOR and visually-guided smooth tracking? The latter idea, expressed in , is based on the concept that a command for eye motion in the orbit is generated by adding floccular output related to gaze velocity and vestibulo-ocular signals calibrated to compensate for head turns and stabilize gaze. During head motion in light or dark, the VOR causes eye motion that is equal in size and opposite in direction to head motion so that gaze motion is minimized. The same is true of GVP cells: during the VOR in light or dark the head and eye velocity components cancel each other so that simple-spike firing rate is, on average, unmodulated. Because the VOR results from vestibular inputs to brainstem pathways, there is no need for the floccular complex to intervene. In fact, the eye velocity positive feedback through the floccular complex would be detrimental during the VOR. If active, the eye velocity positive feedback would integrate a vestibular input that signals head velocity, creating a command for a constant eye acceleration. By nulling the eye velocity positive feedback with a head velocity input, the positive feedback can be suspended during the VOR, but enabled during visual tracking.
Figure 10 Schematic diagram showing the coordination of smooth eye movements evoked by visual and vestibular mechanisms. Visual tracking drive is assembled in the floccular complex by the weighted sum of signals proportional to eye and head velocity. The visual (more ...)
In normal monkeys, we cannot distinguish the two alternatives that GVP cells form internal models of the physical kinematics of the orbit versus the interaction between the VOR and visually-guided tracking. Because zero gaze velocity is the baseline status of the VOR, the two alternatives predict the same data for PCs. However, the degeneracy of the two alternatives is broken after learning in the VOR because the physical gaze velocity driven by the adapted VOR is non-zero. If a monkey wears magnifying spectacles for several days to persistently double the size of the visual image, then the amplitude of the VOR increases so that the images seen through the spectacles are stabilized during head turns (Miles and Fuller, 1974
). Subsequent vestibular rotation in darkness reveals that the gain of the VOR has increased dramatically. Because eye rotation is now much larger than head rotation, the VOR in the dark produces a large physical gaze motion in space.
Recordings from GVP cells after motor learning in the VOR support the alternative that the floccular complex may be acting as an internal model of the interaction of the VOR and visually-guided smooth tracking. Increases in the gain of the VOR are associated with increases in the strength of the vestibular input to GVP cells (Miles et al., 1980b
; Lisberger et al., 1994b
; Hirata and Highstein, 2001
), modeled by the parameter d
in Equation (1)
and . As a result, the null point for cancellation of the eye and head velocity inputs to GVP cells moves: from the control situation where eye and head velocity are equal to the adapted situation where a larger eye movement is needed to null the vestibular input. After increases in the gain of the VOR, the null point for head and eye velocity inputs occurs when eye movements are larger than head movements and approximately equal to those induced by the adapted VOR in darkness. It appears that learning in the VOR adjusts the signal processing in the floccular complex in a way that maintains a good model of the interaction of the VOR and visually-guided smooth tracking, rather than a good model of the kinematics of the orbit.
In thinking about the seemingly complex situation that occurs when spectacles are used to adjust the gain of the VOR, it is important to remember that motor learning in the VOR is intended, in real life, to respond to deficits in vestibular inputs or the strength of extraocular muscle action. Thus, motor learning is an adjustment that is intended to restore the situation where eye velocity is equal in amplitude and opposite in direction to head velocity. The parallel adjustment in the vestibular inputs to GVP cells in association with motor learning in the VOR would maintain the situation where floccular simple-spike output is unmodulated when physical gaze velocity is zero.
I suggest that the floccular cortex is the site of the internal model of the kinematics of the orbit, and I propose that the learned adjustment of the internal model also occurs in the cerebellar cortex. It is tempting to think of learning in the cerebellar cortex in terms of the original cerebellar learning theory (Marr, 1969
; Albus, 1971
; Ito, 1972
), but there is now evidence for multiple sites of plasticity in the cerebellar cortex (Hansel et al., 2000; Mittman and Hausser, 2007
; Jorntell and Hansel, 2006
). There also is some evidence for fast learning in the cerebellar cortex (Medina and Lisberger, 2008
) and slower learning in the deep cerebellar nuclei (Kassarkjian et al., 2005; Shutoh et al., 2006
). Future research will have to localize the sites of internal model learning more precisely in the cerebellar cortex and determine the relative importance of learning in the deep nucleus and the cerebellar cortex at different phases of motor learning.