Biol. Cybern. 85, 423±436 (2001)

Conditional transitions in gaze dynamics: role of vestibular nuclei in eye-only and eye/head gaze behaviors Gin McCollum1, Richard Boyle2 1 2

Neurotology Research, Suite 303, Legacy Clinical Research and Technology Center, 1225 NE 2nd Avenue, Portland, OR 97232, USA Center for Bioinformatics, Ames Research Center, National Aeronautics and Space Administration, Mo€ett Field, CA 94035-1000, USA

Received: 17 December 1999 / Accepted in revised form: 3 May 2001

Abstract. The gaze control system governs distinct gaze behaviors, including visual ®xation and gaze reorientations. Transitions between these gaze behaviors are frequent and smooth in healthy individuals. This study models these gaze-behavior transitions for di€erent numbers of gaze degrees of freedom. Eye/head gaze behaviors have twice the number of degrees of freedom as eye-only gaze behaviors. Each gaze behavior is observable in the system dynamics and is correlated with neuronal behaviors in several, coordinated neural centers, including the vestibular nuclei. The coordination among the neural centers establishes a sensorimotor state which maintains each gaze behavior. This study develops a mathematical framework for synthesizing the coordination among neural centers in gaze sensorimotor states and focuses on the role of vestibular nuclei neurons in gaze sensorimotor state transitions.

1 Introduction Von Holst (1948) described a ®sh at ®rst tilted laterally due to a unilateral vestibular nerve lesion and then suddenly swimming upright in pursuit of a visual food source. This is a classic example of a transition between two sensorimotor states, in which dominance in spatial orientation is held by two di€erent sensory modalities ± vestibular and visual. The transition in sensory state has immediate motor consequences in (and indeed is observable by) the angle of the ®sh to gravity. Such sensorimotor state transitions are common in response to conditions in everyday life and may each be rather minor in the observable overall behavior of the individual. A typical example is the transition between maintaining visual ®xation and making a reorienting movement to a new target of interest. Visual ®xation is a behavior that can be maintained over time, localizing gaze and determining the re¯exes (such as the vestibuloCorrespondence to: G. McCollum

ocular re¯ex, VOR) and the invariants (Bloomberg et al. 1992, 1997) that maintain gaze localization. For instance, while ®xating a visual object in space, a movement of the head, either self-generated (active) or the result of an applied external perturbation (passive), is accompanied by compensatory eye movements. In this sense, visual ®xation localizes sensorimotor dynamics to a particular region of the dynamical range available to an individual (McCollum 1994, 1999b). The vestibular system allows this dynamical localization by signaling body movements that detract from visual ®xation. Gaze movements are often made using the eyes alone. Such gaze movements are typically small and con®ned to the central area of the visual ®eld. However, under certain conditions individuals restrict head movements, so that even large gaze movements are made with the eyes alone within the species' oculomotor range. For example, vestibular-de®cit patients avoid problematic head movements by using eye-only gaze movements (Pozzo et al. 1991; Curthoys and Halmagyi 1995), so that both the reorienting movement and the maintenance of visual ®xation have only the eye movement degrees of freedom. When gaze movements include a head component, the dynamical space under sensorimotor control is expanded. Boyle et al. (1996) and McCrea et al. (1999) conducted experiments in which the dynamical range of gaze movements was manipulated: monkeys made gaze movements with the head restrained or free to move. They found that vestibular nuclei neurons, and particularly the second-order vestibulospinal neurons, exhibit a switch in their ®ring rate that is contingent upon the animal's behavior: when the head is active in gaze movements, the neurons now signal the di€erence between head velocity and active head velocity ± a quantity that does not exist when the head is not actively moved. The vestibular nuclei have been recognized as a center where sensory modalities ± vestibular, visual, and proprioceptive ± converge, and from which appropriate combinations of output signals are projected to the vestibuloocular and vestibulospinal systems. These experiments indicate that the vestibular nuclei also shift

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Fig. 1. Inclusions among subsets of the gaze control system. The gaze control system includes several neural centers, which are collectives of neurons. Inclusion can be portrayed in a Venn diagram, as in the gray inset box at the upper right. Equivalently, a solid line is conventionally used in the mathematics of ordered structures to denote inclusion, as is also shown in the gray inset box. The main diagram is too complicated to show using Venn diagrams, so it uses lines to denote inclusion. For example, on the right, ``motor centers'' is shown including both the oculomotor nuclei and the spinal cord. The bold sideways labels to the left point out major distinctions between levels of analysis. The neurons are the smallest individuals considered in this study, so they are considered to be constituents. They form collectives, as cell aggregates, neural centers, and coordinated neural centers. The entire gaze control system includes all the collectives and, through them, the constituent neurons

the coordination among sensory modalities when the animal switches between sensorimotor states ± eye-only or eye/head ± for gaze movements including reorienting movements and visual ®xation (Boyle et al. 1996; Gdowski and McCrea 1999; McCrea et al. 1999). Complex systems, such as nervous systems, can often be thought of as consisting of a number of levels of analysis: constituents, collectives, and systems (Auyang 1998). A partially ordered set of components of the gaze control system is presented in Fig. 1. The vestibular nuclei are only one of several neural centers involved in gaze control and the selection and establishment of sensorimotor states (Fuchs et al. 1985; Fuller 1992c). Typically, a new gaze target is identi®ed through some sensory modality, and its location is identi®ed and put into context by the cerebral cortex. In addition, the superior colliculus, cerebellum, reticular and other areas play signi®cant roles in laying out the process of reorienting gaze. However, the anatomical and functional inclusions shown in Fig. 1 (solid lines, explained in Sect. 2.1) do not specify this process; rather, a dynamical characterization is needed, one which also speci®es the conditions under which sensorimotor state transitions occur. Whether the gaze movements are eye/head or eyeonly, both behaviors and the coordination of the neural

Fig. 2A±D. Dyad connecting visual ®xation with orienting movement. This diagram is meant to be read as a mathematical statement, like an equation, rather than as an explanatory diagram. Regions (localizations of action) are in boxes. The two sensorimotor states of visual ®xation (A) and visual reorientation (B) are at the top. Solid lines denote inclusion, as is conventional in mathematics, with the included region below the one it is included in. For example, the selection of a new target (C) occurs during visual ®xation, as denoted by the solid line between them, and serves as a trigger to reorient. The coordinate systems are target centered. (Letters A±D labeling regions are irrelevant to the mathematics and are only present for purposes of indicating particular regions.) Dashed arrows denote contiguity, the mathematical relation used to express the option of leading from one region to another. A dyad is a minimal structure in conditional dynamics, in which the transitions between two states are speci®ed along with the conditions of transition

centers involved in gaze control switch abruptly between reorienting movements and visual ®xation (Yards 1967; Fuller et al. 1983; Guitton and Volle 1987; Crawford and Guitton 1997). Figure 2 shows transitions between visual ®xation and gaze reorientation, along with the conditions that lead to the transitions. (See Appendix for further explanation of notation.) During visual ®xation, a di€erent visual target may be selected (Duhamel et al. 1992), which serves as a ``trigger region'' leading to an reorienting movement (Fig. 2C). Similarly, people who wear glasses make frequent transitions between nearand far-target viewing and VOR states. Shelhamer et al. (1992) have reported experiments on conditional transitions of VOR gain, determined by behaviorally relevant cues, trigger regions in the present context. Healthy individuals perform transitions smoothly and appropriately between such dynamical regions or sensorimotor states. In contrast, vestibular patients may display ambiguity in transitions (McCollum et al. 1996) and pass inappropriately, for example, from stabile gaze to nystagmus because of head position or head shaking (Leigh and Zee 1983; Curthoys and Halmagyi 1995). The qualities of smoothness and appropriateness are formalized in conditional dynamics control systems that are complete, consistent, and unambiguous (McCollum 1999a,b); they are de®ned technically below. This paper investigates transitions between gaze sensorimotor states, focussing on the role of vestibular

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nuclei neurons in coordinating sensory modalities and motor degrees of freedom. The following section introduces the mathematical methods of conditional dynamics that will be used, along with their application at di€ering levels of analysis. Then the results of conditional dynamics analysis are presented for the eye-only and eye/head cases. 2 Conditional dynamics at the organism/systems and neuronal levels of analysis The theoretical methods used in this paper were developed for the characterization, analysis, and synthesis of vestibular-related sensorimotor states in postural control (McCollum 1994, 1999a,b; McCollum et al. 1996; McCollum 1999b), and have been applied in a range of sensorimotor systems (McCollum et al. 1995; Roberts and McCollum 1996b). Although Newtonian mechanics applies well to the movements of animals, it does not apply in the deterministic way in which it is frequently used in physics. Rather, it applies indirectly, as an explanation of physical opportunities that individuals bene®t from in selecting certain movement types (Roberts and McCollum 1996a). The processes of selecting and switching between sensorimotor states follow a regular pattern in a number of mammals, even though many numerical and proportional aspects di€er among gaze movements, for example the timing and proportion of head movement (Barnes 1979; Guitton and Volle 1987; Fuller 1992a±c). Therefore, instead of analyzing the quantitative aspects, this paper focuses on the processes of selecting and switching between sensorimotor states. 2.1 De®nitions and model criteria in conditional dynamics Modeling joins empirical observations to mathematics. An observation basic to conditional dynamics is that organisms divide behaviors into ``sensorimotor states'', such as waking versus sleeping, walking versus skipping, and visual ®xation versus gaze reorientations. Not only do these behaviors di€er, but the sensory, motor, and central neural systems are coordinated di€erently to mediate the behaviors. On the mathematical side, such behaviors are characterized by localizations in a dynamical space. For example, visual ®xation involves ®xation of gaze on a target. If the head moves up, the eyes must move down to maintain the gaze position; altogether, the head and eyes must remain within a region of dynamical space (see Fig. 3). The word region is formalized to mean a localization in dynamical space, allowing sensorimotor states to be characterized mathematically. Each region in the conditional dynamics model corresponds to a localization of action, identi®able in physiological function and, ideally, compelling as a pivotal localization. In conditions in which visual ®xation takes priority over head movements in sensorimotor coordination, both passive and active head movements are accompanied by compensatory eye movements to maintain visual

Fig. 3. Eye/head invariant that keeps the eye approximately on a visual target. Shading is used to specify the part of the eye/head plane to which action is limited. Arrows within the shaded region indicate that ¯ows within the region are in every direction. Included in the region are particular ¯ows: one head-up eye-down and the other head-down eye-up (See Appendix for further explanation of notation.)

®xation. Such sensorimotor priority or ``governance'' is expressed mathematically as inclusion. An example of the inclusion of dynamical ¯ows is shown in Fig. 3. The maintenance of eye/head movements within the region shown at the top of the ®gure, for any trajectory, includes a coordinated movement head-up and eyes-down (Fig. 3, lower left) and a coordinated movement headdown and eyes-up (Fig. 3, lower right). Figure 3 uses the mathematical convention that a solid line denotes inclusion, with the included region lower on the page; Figs. 1±4 and 6 contain inclusions. Thus, Fig. 3 is to be read as a mathematical statement, like an equation, rather than as an explanatory diagram. The formal use of the relation of inclusion is a strong constraint on the model. Each region must be modeled so that the speci®ed inclusions hold rigorously. Maintaining the inclusion relation strongly distinguishes diagrams such as in Fig. 3 from, for example, ¯ow charts and linear systems diagrams. In conditional dynamics, only those degrees of freedom that are limited are speci®ed in the diagram; the others continue to exist and have values. For example, Fig. 3 does not specify whether the eyes are open or closed, nor the positions of the arms or legs, even though they could a€ect gaze. When a restriction of those other variables is intended, it must be speci®ed. However, inclusion is not enough to specify the transitions between regions in which quite di€erent dynamics may apply. A major example in this study is the di€erence between the behavioral and neural dynamics maintaining visual ®xation and the behavioral and

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Fig. 4A±Q. Eye-only control system for visual ®xation, gaze reorientation, and transitions between. A±D reproduces the system-level dyad (Fig. 2), with eye movements reduced to one degree of freedom. Regions G and N±P are at a collective level of analysis beyond the range of this study. They are only included to clarify the relations between the system level and the regions characterizing neuronallevel mechanisms, H±M, that are central to this study. In H and M, the directions of the eye movements are in the directions of sensitivity of the PVP neurons. The variable g is de®ned in Fig. 5. See text (Sect. 3.1) for further information

neural dynamics producing a gaze reorientation. Certain included regions lead to a change in sensorimotor state. Although many regions may be speci®ed as included in visual ®xation, it is the change in visual target that leads to a reorienting movement. The relation between a change in visual target and a reorienting movement is formalized as contiguity (represented by dashed arrows in Fig. 2) (McCollum 1994, 1999b).

Whenever contiguity is speci®ed between two regions, it must be supported by plausible physical or neural dynamics. A strength of conditional dynamics is that it is not necessary to specify the dynamics in every case, because there are a range of dynamical ¯ows that may join two regions (Roberts and McCollum 1996a). This allows analysis of systems in which the dynamics are complex or poorly understood.

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Conditional dynamical modeling begins with an evaluation of the data, to judge whether there is enough experimental evidence of localizations of action and conditional transitions. Not every localization or transition need have been investigated experimentally; often, evidence at a di€erent level of analysis can attest to overall properties of the conditional dynamical space (such as those discussed in Sect. 2.2), which then allow predictions of regions and contiguities. It is essential to specify mathematically the spaces in which localizations of action occur, so that inclusions hold rigorously and can be veri®ed by inspection. (These spaces are typically subspaces of a much higher-dimensional space; see Appendix.) It is also essential, in specifying inclusions, to be clear and explicit about which unspeci®ed degrees of freedom have ®xed ± as opposed to varying ± values. Experimentally investigated conditional transitions are then speci®ed as contiguity, resulting in a descriptive conditional space for the behavior. Some form of mathematical completeness (such as those discussed in Sect. 2.2) allows a mathematical analysis of the conditional dynamical space, so that the model can be used to identify absences and inconsistencies in the experimental record and to make predictions. 2.2 Conditional dynamics models that join di€erent levels of analysis Figure 1 presents the gaze control system, consisting of constituent neurons, collectives of neurons and neural centers, and the system. At each level of analysis, di€erent functional concepts and dynamical spaces apply. Conditional dynamics joins dynamical spaces at di€erent levels of analysis, along with the conditional relations between them. We will focus on the constituent, neuronal level in the vestibular nuclei, and its relationship to the system level. Conditional dynamics has been used at the system level of analysis to investigate transitions in the postural system. Shifts in the sensory and sensorimotor states governing postural adjustments are frequent and unproblematic in healthy individuals (McCollum et al. 1996; McCollum 1999b); so are eye and head movements. At the system level, a control system must be distributed and autonomous. For these reasons, control systems for postural adjustments have been required to be complete, consistent, and unambiguous, in the sense: (1) that any region can be entered and exited, (2) that any dynamics speci®ed is applicable to allow a contiguity, (3) that all speci®ed contiguities are usable, and (4) that two contradictory (and nonsummable) processes are not required at the same time (McCollum 1999a,b). The control system shown in Fig. 2 is called a ``dyad'', and is complete, consistent, and unambiguous based on the above criteria. It consists of two sensorimotor states ± visual ®xation and visual reorientation (Fig. 2A,B) ± along with the conditions under which transitions occur, given by the two trigger regions (Fig. 2C,D). If the two trigger regions are nonintersecting, then the control system is stabile in the sense

that it is nonreverberating. For example, settling is completed and gaze is on target before a new trigger to reorient occurs. (Settling may require a corrective saccade or a slide, see Goldstein and Robinson (1986); for simplicity these are omitted, because including them would obscure the main point here.) Reverberating control systems are suspected in the case of some vestibular patients (McCollum et al. 1996; McCollum 1999a). Spontaneous nystagmus, as in positional or head-shaking nystagmus (Leigh and Zee 1983; Curthoys and Halmagyi 1995), is evidence of a control system that has been destabilized and has become reverberating. The dyad shown in Fig. 2 gives the foundation to the investigation of neuronal gaze mechanisms in a logically sound control system of a type already used to model vestibular-related postural control systems. The sensorimotor states shown in Fig. 2 are at the system level of Fig. 1. However, this study relates the system level to the constituent neuronal level of Fig. 1. At the constituent or collective levels of analysis, not all control systems are distributed or autonomous. Rather, control systems involving the constituent and collective levels of analysis display a range of logical structures. If a system and a mechanism always occur together ± for example, remembering Grandmother and the activation of the grandmother cell ± then the two regions would ideally be denoted as the same, and the control system is the same. The Mauthner cell of teleosts and amphibians may be the closest familiar example in which the behavior (escape) and activity in the neuron are typically coincident (Faber and Korn 1978). In mammalian nervous systems, there is rarely such a coincidence. Instead, sensorimotor control is complex, involving constituent neurons in a range of collectives within the system. Transfer function studies (e.g., Robinson 1977) typically address collectives of neural centers that cooperatively behave linearly, under conditions of sinusoidal stimulation. In contrast, the present approach is designed to address the dynamics of individual neural types under various conditions. Once all the relevant types are investigated, the model should reduce to give the same results as transfer function approaches do ± when the same collectives are considered ± under the same conditions. (Here, the term ``reduce'' is used in the sense that quantum mechanics reduces to Newtonian mechanics in the appropriate limit.) The present study investigates the logical structure, in terms of inclusion and contiguity, between states of vestibular nuclei neurons and two sensorimotor states of visual ®xation and reorienting at the system level. The method is to join the two levels in a conditional dynamics analysis, showing how vestibular nuclei neurons mediate gaze behaviors. The resulting analysis is not complete (criteria 2±4 above are followed, but not criterion 1), but is a beginning with one neural center to build a synthesis of the collective control of oculomotor behavior by several neural centers. Besides the systemlevel dyad, the regions included will address several of the principal and known vestibular nuclei cell types and their relationships to the system-level sensorimotor

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states of visual ®xation and reorientation movement. Besides the autonomous, distributed style of control found at the system level, the control systems include triggered sequences in which a subset of one region acts as a trigger region leading to another region, as in a dyad, and behaviors that are governed from outside, especially the manipulation of neuronal behavior by a€erent input. 3 Eye-only and eye/head gaze behaviors By specifying regions rather than points, and families of trajectories rather than single trajectories, conditional dynamics allows the synthesis of a range of data into a control system that speci®es opportunities for a variety of gaze movements. For example, gaze reorientations between two speci®ed points can be performed in a variety of ways, with di€ering velocities and ratios of head and eye involvement (Phillips et al. 1995). The present study is limited to visual ®xation and reorienting movements, and omits other gaze behaviors, such as smooth pursuit and head tracking. The relevant issues here are how the phases of the gaze reorientation are sequenced and how the sensory signals are distributed. Vestibular and proprioceptive sensations, generated either through rea€erence or external perturbations, along with the motor command and e€erence copy signals, play changing roles in gaze behavior. The following presentation begins with visual ®xation, because it is simpler than reorientation and serves as an introduction to the notation and mode of argument of conditional dynamics. 3.1 Eye-only The dyad discussed above (Fig. 2) as a systems-level control system for transitions between visual ®xation and reorientation is reproduced in Fig. 4A±D as a systems-level context for the functions of vestibular neurons. In Fig. 4A±D it is drawn with only one degree of freedom of eye movements. This dyad is the same for eye-only and eye/head gaze movements, but the sensory and motor mechanisms responsible for the movements di€er. For eye-only gaze movements, all head movements are assumed to result from external perturbations, with the head unmoving with respect to the trunk. A primary role of the vestibular nuclei in eye-only gaze control is to regulate the signals from the vestibular nerve a€erents to the oculomotor neurons. Included in visual ®xation (Fig. 4A) is visual ®xation with trunk movement (Fig. 4E). The highly correlated relationship between eye and trunk movement (Fig. 4Q) is normally associated with visual ®xation (Fig. 4A,E) and implemented by the vestibular signal on oculomotor neurons (Fig. 4F,G). (Note that Fig. 4F is not included in Fig. 4E because Fig. 4F also includes the VOR in the dark, which is not characterized here, although it is also included in Fig. 4Q.) Among secondary vestibular neurons, the vestibular signal is largely

imparted to the oculomotor neurons by a cell type termed the position-vestibular-pause (PVP) neuron (Fig. 4H), each of which has particular pause characteristics (McCrea et al. 1980, 1987a,b; Scudder and Fuchs 1992); additional drive is also supplied by burst tonic neurons (Cullen et al. 1993) and perhaps the eye-head-velocity neurons (McConville et al. 1996; however, cf. Scudder and Fuchs 1992). Because of the inhibition that silences the PVP cell during saccades, even when the head is restrained from moving, the relationship of its ®ring rate to eye movement has a variable e€ectiveness g (Fig. 5). The vestibular signal is variably e€ective at the beginning and end of a gaze saccade (Dichgans et al. 1974; Fuller et al. 1983; Guitton and Volle 1987; Tabak et al. 1996). The discrete structure shown here allows that variation. The variable e€ectiveness g is introduced both as a notational convenience (an abbreviation allowing the graphs of Fig. 5 not to appear within Fig. 4) and as a recognition of the importance of the variable e€ectiveness g as a physiological variable. It is g that is modulated by extra-vestibular (EV) inhibition. We use the term ``EV inhibition'' in a broad sense, without specifying identi®ed cell types. The exact nature of the EV inhibition (Fig. 4I±L) to the PVP or other vestibular nuclei cells participating in di€erent vestibular re¯exes is not pertinent to the present analysis. The physiological dynamics on the space of EV inhibition and g (Fig. 4I,J) connect the extreme regions of the EV inhibition/g space (Fig. 4K,L) and therefore include them. Each extreme region includes a particular region in PVP ®ring rate/ VOR eye movement space (in Fig. 4, K includes H and L includes M). Because PVP ®ring rate has static eye (Scudder and Fuchs 1992) and head (McCrea et al. 1996) position sensitivity, the range of PVP ®ring rate is labelled ``low'' to ``high''; the particular dependence on static eye position is omitted for simplicity of presentation and is straightforward to introduce. Each PVP neuron has a particular direction of saccade sensitivity, determined by the direction of saccades for which EV signals inhibit the individual neuron. The inhibitory input is formalized as a dynamical ¯ow, increasing the EV inhibition at the same time as it decreases the e€ectiveness g. The ¯ow (Fig. 4I) governs movement within inhibition/g space (Fig. 4K to Fig. 4L) and the cessation of delivery of the vestibular signal to the oculomotor neurons by the PVPs, each neuron in its particular direction or range of directions (Fig. 4H±M). These mechanisms, including the EV inhibition and PVP neurons, are not autonomous but are governed by conditions imposed upon them. Thus, they are not shown as forming a dyad or as involving trigger regions, but rather as simple results of dynamical ¯ows. The in¯uence of the EV inhibition on the PVPs (Fig. 4H±M) forms a module within the conditional dynamical synthesis of eye-only gaze control (Fig. 4). Even though the module is not an autonomous control, it serves as a prototype for similar neuronal control mechanisms, such as in Sect. 3.2. Between the system-level dyad (Fig. 4A±D) and the neuronal module (Fig. 4H±M) are presumed collective

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lective and system levels of analysis (Fig. 4P,D). Like arriving at erect stance, settling on a new gaze position involves a complex of neural mechanisms to maintain that position: it is not only a position but also a sensorimotor state. The principal mechanism relevant to the vestibular nuclei in the eye-only condition is the reassertion of vestibular control over eye movements, formalized as the inclusion of J in P and of K and H in D in Fig. 4. 3.2 Eye/head

Fig. 5. De®nition of the variable g: e€ectiveness of PVP neurons in mediating vestibular input to VOR eye movements. The variable g, shown along the large vertical axis to the left, is 1 for a particular PVP neuron when it is active in its direction of sensitivity to ocular saccades, and thus most e€ective in VOR eye movements. When EV inhibition reaches the PVP, it reduces the activity of the cell, reducing its e€ectiveness in VOR eye movements in its on-direction and reducing g to 0

motor control processes (Fig. 4N±P) that are beyond the scope of this study. The motor dynamics processes for both horizontal and non-horizontal eye movements are subsumed here (Gaymard and Pierrot-Deseilligny 1999). The responsible motor centers are collectives of many neural types, especially including the nucleus prepositus hypoglossi (McCrea 1988). Because conditional dynamics formalizes relationships and the conditional transitions between them, neural centers and neural populations may appear in multiple regions. Although the relaxation of the reorienting movement and settling on the new visual target is clear in system-level recordings (Dichgans et al. 1974; Boyle et al. 1996; McCrea et al. 1999), it is a complex neural transition at the col-

The degree of freedom that is added when gaze movements include both eye and head ®ts into the organization of gaze reorientation in a way that is familiar from eye-only gaze reorientation (Fig. 4), adding complication modularly. The one degree of freedom that is added is active head movement with respect to the trunk. (Active trunk movements are not considered here.) In the system-level dyad (Fig. 6A±D), each region is augmented by one degree of freedom, the head angle. Thus, for example, there is a range of eye angles for which gaze can be on target (black region in Fig. 6A), because gaze direction depends on both eye and head angle. Similarly, a wider range of gaze reorientation movements is possible. In gaze reorientation, eye and head movements are held in relationship (Fig. 6N). As part of this eye/head coordination, the vestibular signal that normally maintains eye position is suppressed by the EV inhibition on the PVP (Fig. 6H±M). The action of the EV inhibition is expressed by a logical control module of the same organization for the eye-only and eye/head cases (Figs. 4H±M, 6H±M). (The only di€erence is that the labels of the coordinate axes in Fig. 6H± M have been abbreviated to squeeze them into a smaller space.) However, subtraction of active head movement signals (Fig. 6R±W) allows the eyes to move relative to the trunk rather than to the head. (The variable k is de®ned in Fig. 7.) In both eye-only and eye/head gaze orientation, innervation of vestibular neurons relaxes vestibular control and visual ®xation (Figs. 4A, 6A) so that gaze reorientation may take place (Figs. 4B, 6B). In the eye-only case, this innervation is the EV inhibition that relaxes the e€ectiveness of vestibular control over VOR eye movements (Fig. 4I±M). In the eye/head case, the motor control of the gaze reorientation movement (Fig. 6N) includes both the increase of EV inhibition to lower g (Fig. 6I) and also the innervation that subtracts active head movement from the vestibular signal (Fig. 6R). As in the eye-only case, these dynamics are reversed as the movement comes to an end (Fig. 4O,J; Fig. 6O,J,S). As the reorienting movement relaxes (Figs. 4P and 6P), the invariants are reinstated: the eye-target invariant (Figs. 4E and 6E) over eye movements (Figs. 4J,H, 6J, H) and the normal eye/head position invariant (Fig. 6Z) over head movements (Fig. 6V). This manipulation of the vestibular projection is expressed in a logical control module of the same discrete structure (Fig. 6R±W)

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Fig. 6A±Z. Eye/head control system for visual ®xation, gaze reorientation, and transitions between. A±D reproduces the system-level dyad (Fig. 2), showing eye and head movements on separate axes. Because the gaze direction is the sum of the eye-in-head and head-on-trunk angles, gaze direction is shown as a black, diagonal region in A±D. As in Fig. 4, regions G and N±P are at a collective level of analysis beyond the range of this study. Regions H±M form a module characterizing the elimination of PVP ®ring from the VOR during eye movements. Regions R±W form a similar module characterizing the elimination of active head movement from the signals of vestibular neurons. Note that the invariant Z bears a relationship to V similar to that of F to H. The variable k is de®ned in Fig. 7. See text (sect. 3.2) for further information

(Boyle et al. 1996; Gdowski and McCrea 1999; McCrea et al. 1999). For eye-only gaze movements, visual ®xation requires only an invariant relating eye movement to head/trunk movement, to keep the eye on target (Figs. 4E, 6E). When gaze movements include both eye and head, visual ®xation requires an extra invariant, to govern the relationship between the head and the eye (Fig. 6Z). The normal eye/head position is variable and can be voluntarily altered. Nevertheless, for each species and individual there is a typical relationship between the eyes and the head in each gaze position given by eye position relative to the trunk. The data showing that active head movements are subtracted out of vestibular signals are recent (Boyle et al. 1996; Gdowski and McCrea 1999; McCrea et al. 1999); the exact mechanisms are as yet unknown. However, the functional e€ect and the basic roles of vestibular nuclei neurons are clear: head position becomes a free variable, inessential to the gaze movement, because the vestibular neurons are able to subtract out the position of the carrier of the vestibular endorgans ± the head ± during active head movements. The normal eye/head position invariant (Fig. 6Z), along with the coordinated eye/head movement (Fig. 6N), is a vehicle for reducing an eye/head gaze movement to an eye-only movement. If the normal position requires no correlation between eye and head position (analogous to the relaxation of correlation in Fig. 6M, W), then the eyes are free to move regardless of head position. If, in addition, no attempt is made to move the head along with the eyes in the coordinated eye/head movement (Fig. 6N), then Fig. 6 is reduced functionally to Fig. 4. This is a possible mechanism for voluntarily producing eye-only gaze movements, both by healthy individuals and by vestibular patients (Pozzo et al. 1991; Curthoys and Halmagyi 1995). 4 Discussion For visual ®xation, gaze reorientation, and the transitions between the two sensorimotor states, this paper has addressed: (1) mediation by ®ring rate dynamics in the vestibular nuclei and (2) the di€erence between eye-only and eye/head control. The ®ring rate dynamics were found to express the di€erence in degrees of freedom by recombining sensory information as appropriate for

Fig. 7. De®nition of the variable k: e€ectiveness of head movement in VOR eye movements. The variable k is de®ned in terms of the correlation between head and eye movements, because the particular neuronal mechanism is unknown, although it is plausible that it is e€erence copy (McCrea et al. 1999). The variable k, shown along the large vertical axis to the left, is 1 when head and eye movements are negatively correlated and 0 when they are uncorrelated

di€erent movement phases in eye-only and eye/head gaze control. The pause in PVP ®ring rate during ocular saccades suppresses information that would distract the gaze system from its movement purpose. In addition, during active head movements, vestibular nuclei neurons are observed to signal the di€erence between head velocity and active head velocity. From the nervous system's point of view, it reorganizes to move the head independent of the trunk, bringing into play a new

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motor control degree of freedom, active head velocity. When head motion is a motor-control degree of freedom, the vestibular system must distinguish active from passive head motions, or it fails to balance and coordinate gaze and posture sensory modalities. By distinguishing active and passive head movements, it speci®es head movement as a motor-control degree of freedom. Thus, even though the role of the vestibular nuclei is primarily sensory, by providing information appropriate for motor purposes, the vestibular nuclei also express the motor degrees of freedom. The role of specifying motor control variables is of a complexity worthy of the vestibular nuclei, because motor control variables are of di€erent types and maintain a variety of relationships. When the center of reference of eye movements is shifted from the head to the trunk (McCrea et al. 1999), the head is released to be a free variable, irrelevant to eye movements. To the extent that inhibition of the PVP signal to oculomotor neurons releases eye position from the VOR, eye position becomes a control variable. However, unless either the head-on-trunk angle or the eye-in-orbit angle is the only gaze variable, both are free variables with respect to gaze direction, the essential control variable. Adding to the complexity of specifying motor control variables, head-on-trunk and eye-in-orbit are bound by a range of relationship mechanisms such as the VOR, to facilitate gaze control. This study has developed a modular format for characterizing actions of vestibular neurons and their a€erents. The module ®ts in a control structure including the system-level dyad governing visual ®xation, gaze reorientation, and the transitions between the two sensorimotor states. The module is suitable for expressing further neuronal mechanisms, such as the action of the ®xation cells of the superior colliculus on saccade generation (Munoz and Guitton 1991; Munoz and Wurtz 1993). Thus, this study provides a mathematical framework for synthesizing the neuronal and collective mechanisms making up the gaze control system. The control systems developed in this study (Figs. 4, 6) include the system-level along with neuronal-level mechanisms in the vestibular nuclei. Appropriate concepts and variables di€er between the two levels of analysis (Auyang 1998). At the system level, the primary variables measured are eye, head, and trunk movements. At the neuronal level, the primary variables measured characterize ®ring rate patterns. Each of these has its own characteristic dynamics, occurring in appropriate spaces of variables. By including both in the same control system, conditional dynamics allows a synthesis of disparate dynamics and time courses (McCollum 2001). Rather than one system-level set of dynamical variables, the conditional dynamics synthesis portrays a multilevel, multiply-connected dynamics, formed of the dynamical spaces of physical and physiological processes, joined in variable hierarchies and chains. By selecting one particular sensorimotor state and one level of analysis, the conditional dynamics can be reduced to a simpler dynamical description, from which

numerical conclusions can be drawn. That is, by selecting particular conditions, the conditionality characteristic of living systems can be removed. Thus, conditional dynamics sets up a conditional framework, within which the partial dynamics of one ®xed circuit, with ®xed dynamics, has a context. The major strength of circuit analysis techniques, such as transfer function techniques of linear systems analysis, is in monostate, sinusoidal behaviors. Adding sensorimotor state transitions to such analyses removes them from their area of greatest mathematical power and requires a complicating layer of formalism. Otherwise, simplifying assumptions are required, such as the reduction of nystagmus to the slow phase or the con¯ating of the systems and neuronal levels of analysis. By providing a context for numerical circuit analyses, conditional dynamics is complementary to them. In order to model a physiologically autonomous gaze control system ± to the extent that is biologically appropriate ± dynamical ¯ows will eventually be required to arise from dynamics within the system in question. For example, the dynamics of EV inhibition will be required to be included in the model, rather than simply their e€ects (Figs. 4I,J 6I,J). A number of mechanisms might be responsible for the inhibition; for example, a pathway involving the reticular inhibitory burst neurons (Strassman et al. 1986), intrinsic vestibular nuclei circuits (Roy and Cullen 1998), or an e€erence copy of the motor command projected onto the neurons (McCrea et al. 1999). If the EV inhibition includes multiple mechanisms, as is likely, then it is at the collective level. Similarly, the apparent e€erence copy shifting the movement center of reference from the head to the trunk (McCrea et al. 1999) should eventually be within the model, rather than simply within its e€ects (Fig. 6R,S). By providing a mathematical framework for addressing the synthesis of the gaze control system, the results of this study open a range of scienti®c questions. A key question is the organization of the gaze control system at various collective levels. Fuller (1992c) introduces the collective-level functional concepts of strategic, integral, and biomechanical control, labeled in Fig. 1 as identifying goals, laying out processes, and accomplishing behaviors. Until the control system includes empirically convincing collective units, we do not know to what extent Fuller's concepts are re¯ected in gaze control systems. Collective units that are ripe for synthesis include premotor and motor control (Figs. 4N, 6N) and settling to a new gaze position (Figs. 4P, 6P). In ®lling in the constituent and collective levels, biological insight will be gained into the distribution of tasks that allows completeness, consistency, and lack of ambiguity at the system level. For example, is it necessary that the vestibular nuclei collective unit be responsible for assigning the roles of sensory modalities? Or, on the contrary, is a role such as that necessarily distributed? There is no reason to expect rigidity in collective mechanisms of gaze control. More likely, gaze control components combine in a ¯exible number of collectives.

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Within the vestibular nuclei themselves, there is a large variety of ®ring rate patterns, with signals correlated with many di€erent relationships between motor control variables (Gdowski et al. 2000). A plausible picture of gaze control is that each vestibular neuron plays multiple roles in a range of gaze control collectives. Acknowledgements. We thank Werner Graf for pointing out von Holst's paper, Stefan Glasauer for helping to ®nd it, F. Owen Black for discussions of his clinical observations of vestibular patients, and Patrick D. Roberts and Gerhard Magnus for helpful comments on the manuscript. This research was supported in part by NIH/NINDS NS27050 (RB).

Appendix: Notation of high-dimensionality sets Sets of high dimensionality may contain subsets of lower dimensionality in which there is more detail to specify. It is possible to denote sets of high dimensionality in such a way that one may focus on dimensions of particular interest. The following examples display low-dimensional cases in which there is an overlap between the usefulness of set notation and the notation of separate dimensions. To begin, consider two-dimensional sets. Examples: 1. A closed disc D in a plane P, P  D. 2. A line L in a plane P, P  L. 3. One edge E of a square S, S  E. Alternatively, the square could be denoted as the product of two unit intervals ‰0; 1Šx  ‰0; 1Šy and the edge as ‰0; 0Šx  ‰0; 1Šy  ‰0; 1Šx  ‰0; 1Šy . 4. A moving line within a square: ‰f…t†Šx  ‰0; 1Šy  ‰0; 1Šx  ‰0; 1Šy , where f(t) is some function of time t  < with range [0, 1]. 5. A moving interval within a square: ‰g…t† d; g…t†‡ dŠx  ‰0; 1Šy  ‰0; 1Šx  ‰0; 1Šy , where d  0:5 and g(t) is some function of time t  < with range ‰d; 1 dŠ. 6. A moving square within a square, where the motions in the x and y directions follow di€ering functions of time: ‰r…t† d; r…t† ‡ dŠx ‰s…t† d; s…t† ‡ dŠy  ‰0; 1Šx ‰0; 1Šy , where d  0:5 and r(t) and s(t) are functions of time t  < with range ‰d; 1 dŠ. The inner square is the intersection of two perpendicular moving intervals (as in example 5): ‰r…t† d; r…t† ‡ dŠx  ‰0; 1Šy  ‰0; 1Šx  ‰0; 1Šy and ‰0;1Šx  ‰s…t† d;s…t† ‡ dŠy  ‰0; 1Šx  ‰0; 1Šy (Fig. A1). 7. A moving square within a square, where the motions in the x and y directions form a two-dimensional ®gure such as a circle, so that it is more illuminating to draw them as a two-dimensional ®gure. Now consider examples of three-dimensional sets consisting of a cube within a cube: 8. A moving cube that is the intersection of three perpendicular moving intervals: ‰a…t† d; a…t† ‡ dŠx ‰b…t† d; b…t† ‡ dŠy  ‰c…t† d; c…t† ‡ dŠz  ‰0; 1Šx  ‰0; 1Šy  ‰0; 1Šz , where d  0:5 and a(t), b(t), and c(t) are functions of time t  < with range ‰d; 1 dŠ (Fig. A2). This is the three-dimensional analogue of the two-dimensional example 6.

Fig. A1. Example 6. Solid lines indicate inclusion of moving rectangles within a square. The moving rectangles intersect to form a smaller, moving square

9. A moving cube that is the intersection of a moving square (as in example 7) with a moving interval (as in example 5): ‰jx …t; x; y† d; jx …t; x; y† ‡ dŠx ‰jy …t; x; y† d; jy …t; x; y† ‡ dŠy ‰k…t† d; k…t† ‡ dŠ  ‰0; 1Šx  ‰0; 1Šy ‰0; 1Šz , where d  0:5 and j : …t; x; y† ! …x; y† and k : …t† ! …z† are functions of time t  < with range ‰d; 1 dŠ in the spatial directions (Fig. A3). Three lessons become clear from these examples. First, the relationships between time-dependent motions in di€erent dimensions are clari®ed by using the set inclusion notation of the ®gures (Hasse diagrams). (See also Fig. 1.) Second, when specifying functional relationships in a subset of dimensions, such as x and y in example 9, it is convenient to group those dimensions together. For example, a slice of a cube might be denoted simply as ‰m…t† d; m…t† ‡ dŠx and drawn as an interval of a line segment, when the entire space is understood to be three-dimensional. As a high-dimensional example, consider the moving cube of example 9 embedded in a two-hundred-dimensional cube. The essence of the subset is still expressed by Fig. A3, given that there is no functional relationship or constraint that a€ects the cube and depends on the other 197 dimensions. Despite the diculty of depiction, by using set inclusion and separating dimensions in a way that is natural to the set in question, sets of high dimension and with elaborate functional relationships can be clearly denoted. The notation accommodates many possibilities that have not been illustrated. For example, the func-

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Fig. A2. Example 8. Moving intervals within a cube intersect to form a smaller, moving cube

Fig. A3. Example 9. Moving intervals of di€erent dimensionality intersect to form a moving cube

tional relationships can be conditional and discrete, and the directions of the coordinates can be speci®ed to change as a function of time and other coordinates. Even in the above examples, the time dimension has been suppressed. Expressing the time dimension includes explicit dynamics. For example, consider a slice of a cube moving from low to high along a sigmoid curve (Fig. A4). The cube is now explicitly included in a four-dimensional space, including time. The cube itself is a subset, with time suppressed ± that is, left unspeci®ed. The graph of

k(t), specifying the dynamics of the moving slice, is drawn in z  t space, with the x and y dimensions suppressed. The moving slice within the cube is a subset of both of these spaces. Further detail may be speci®ed by depicting regions along the path of the moving slice (Fig. A5). Within the time sequence of moving slices is included the slice at the initial position (bottom left). The slice at the ®nal position is also included (bottom right). The relation of contiguity (McCollum 1994, 1999a) connects the two positions, as denoted by a dashed arrow. Contiguity

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Fig. A4. Dynamics represented using the explicit time dependence of an interval moving within a cube

depends on the dynamics speci®ed above by the graph of k(t) in z ´ t space. The contiguity relation is used to specify regions that occur in the course of dynamics. Dynamics may be conditionally applicable to the system. The use of inclusion and contiguity allows the speci®cation of the regions in state space in which dynamics are applicable, along with the results. References Auyang SY (1998) Foundations of complex-system theories. Cambridge University Press, Cambridge Barnes GR (1979) Vestibulo-ocular function during co-ordinated head and eye movements to acquire visual targets. J Physiol (Lond) 287: 127±147 Bloomberg JJ, Reschke MF, Huebner WP, Peters BT (1992) The e€ects of target distance on eye and head movement during locomotion. Ann NY Acad Sci 656: 699±707 Bloomberg JJ, Peters BT, Smith SL, Huebner WP, Reschke MF (1997) Locomotor head-trunk coordination strategies following space ¯ight. J Vestib Res 7: 161±177 Boyle R, Belton T, McCrea RA (1996) Responses of identi®ed vestibulospinal neurons to voluntary and re¯ex eye and head movements in the alert squirrel monkey. Ann N Y Acad Sci 781: 244±263 Crawford JD, Guitton D (1997) Primate head-free saccade generator implements a desired (Post-VOR) eye position command by anticipating intended head motion. J Neurophysiol 78: 2811±2816 Cullen KE, Chen-Huang C, McCrea RA (1993) Firing behavior of brain stem neurons during voluntary cancellation of the horizontal vestibuloocular re¯ex, II. Eye movement related neurons. J Neurophysiol 70: 844±856 Curthoys IS, Halmagyi GM (1995) Vestibular compensation: a review of the oculomotor, neural, and clinical consequences of unilateral vestibular loss. J Vestib Res 5: 67±107 Dichgans J, Bizzi E, Morasso P, Tagliasco V (1974) The role of vestibular and neck a€erents during eye-head coordination in the monkey. Brain Res 71: 225±232

Fig. A5. Contiguity. Dynamics provides the opportunity to move from one region to another. The moving interval from Fig. A4 moves from the bottom to the top of the cube. These two positions thus become contiguous under the given dynamics, as denoted by the dashed arrow. The contiguity is directional, because no dynamical process is given that moves the interval from top to bottom

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