Journal of Vestibular Research, Vol. 7, No.4, pp. 347-367,1997 Copyright © 1997 Elsevier Science Inc. Printed in the USA. All rights reserved 0957-4271197 $17.00 + .00 ELSEVIER PIT S0957-4271(96)00176-0
Review
VESTIBULAR-NECK INTERACTION AND TRANSFORMATION OF SENSORY COORDINATES
T. Mergner,* W. Huber,* and W. Beckert
*Neurological University Clinic, Freiburg; tSection Neurophysiology, University of Ulm, Germany Reprint address: Dr. T. Mergner, Neurologische Klinik, Neurozentrum, Universitatsklinikum, Breisacher Str. 64, 0-79106 Freiburg, Germany. Tel: +49-761-270-5313 (5306); Fax: +49-761-270-5310; E-mail: [email protected]
o Abstract - The article considers findings and Introduction concepts on vestibular-proprioceptive interaction for self-motion perception and postural control If we want to use the sensory information pro under the form of simple describing models. It vided by the sensory systems located in our points out that vestibular-neck interaction is only heads (vestibular, visual, auditory) for postural a small fraction of an extended mechanism of co ordinate transformations. This links together 'the or perceptual control of upright stance, our different parts of our bodies, so that sensory infor brain has to take into account that the head is mation arising in one part of the body can be used movable with respect to the trunk and, hence, for perceptual or motor tasks in other parts. Par also relative to the support surface we are stand ticular emphasis is put on the problems that arise ing on. Consider, for example, the semicircular from imperfect signal transduction in the vestibu canals of the vestibular system and one of their lar semicircular canal systems at low stimulus fre targets during postural control, the leg muscles. quencies/velocities. Also, a "down-and-up-chan Having a fixed orientation in the head, each ca neling" principle is suggested, by which the body nal will indicate head velocity in specific planes support is linked via coordinate transformations defined by head anatomy. It is therefore conve to the internal notion of physical space provided nient, and concurs with intuition, to treat these by the vestibular system. Furthermore, the follow ing question is addressed: how does the brain use signals as being "coded in head coordinates," visual input to overcome the vestibular deficien although the cause of these signals is an acceler cies, at the risk of visual self-motion illusions? Fi ation with respect to a fixed inertial coordinate nally, a conceptual mode! of postural controi is system. On the other hand, the leg muscles used presented in which a proprioceptive feedback tn body sway, act about axes de that links the body to its support surface is merged fined by the position of the corresponding foot with a loop for postural stabilization in space. on its support. Therefore, their innervation can © 1997 Elsevier Science Inc. be viewed in terms of the intended rotation (or torque) about these axes and, hence, as being o Keywords - vestibular-neck interaction; "coded in foot coordinates." proprioception; visual-vestibular interaction; The way a given canal signal acts upon a postural control; coordinate transformation; human. given muscle in the act of stabilization obvi ously depends on canal orientation with respect to the foot and, hence, on head-on-trunk-to-foot
347 348 T. Mergner et al position. 1 A human investigator who wants to stitute an important organizational principle of understand the mechanisms of postural control, the eNS, for both perception and motor control. may interpret this dependence as the result of a The need for a coordinate transformation of coordinate transformation. This interpretation vestibular signals was anticipated by von Holst invites further abstraction: In dealing with pos and Mittelstaedt (1) in their classical paper on tural reflexes one may postulate (i) signals that the "Reafferenzprinzip", in which they state directly indicate position, velocity or accelera that, due to a subtractive interaction of vestibu tion with respect to an inertial frame of refer lar and neck afferents, control of trunk posture ence (which can only be arrived at by appropri by the vestibular system proceeds as if the vesti ate coordinate ~ransformations of primary sensory bular organs were actually located in the trunk signals), (ii) a computation of corrective signals (" .. derOrganismus sich so verhiilt, als ob der (for example, torques) therefrom, and finally "Lagesinn" normaliter im Leib siiBe ... "; p (iii) a transformation of these desired torques 475).2 Studies in decerebrate cat support the no into innervations coded in coordinates of the tion of the trunk posture being controlled by a participating muscle. subtractive interaction of vestibular and neck Whether functional entities corresponding afferents (2,3). Also, recordings of vestibular nu to these constructs exist in the real central ner clear neurons during lateral head and/or trunk vous system (CNS) is, as yet, an open question. tilt in the decerebrate cat (4) and during horizon We cannot exclude that the three separate steps tal rotations in the intact cat (5) revealed sub of the human mind outlined above are lumped tractive (and additive) interactions between ves into a dedicated network of neurons providing tibular and neck inputs. Moreover, subtractive multiple (polysynaptic) connections of every and additive interactions of vestibular and neck canal with every muscle, as would be the case afferents have also been observed in the cere with a technical learning network trained to re bral cortex of the intact cat (6,7). produce human postural behavior. However, we In the intact cat, certain head and body pos- feel that the CNS is much more structured than 'tures during natural behavior are reminiscent of technical networks, and that its hierarchical ar what one might expect from a subtractive vesti chitecture could well accommodate abstract lev bular-neck interaction (8), whereas experimen els of processing between the primary sensory tal stimulation of these inputs do not readily and motor levels, which undoubtedly would con reveal the expected interactions. In humans, re tribute to its flexibility. On the motor side, a cer flexes reminiscent of the cervico-spinal ones in tain type of abstraction can be recognized in the the decerebrate cat are seen only in the first tight intraspinal coupling of synergistic muscle month after birth (9,10) or following extended groups. Therefore, we conceive of the coordi brain lesions (11,12). In the healthy adult, by nate transformation as being performed in a glo contrast, for a long time only sparse evidence bal way, before a compound motor plan is for for a vestibular-neck interaction in postural con mulated to concert muscle actions. Moreover, trol was obtained (13). we know from psychophysical studies that on More recent studies on human posture con the perceptual leveL humans can indeed trans trol directly support the notion of the above out form their primary sensory signals into such ab lined coordinate transformation. With natural stract notions as 'head velocity in space' or 'trunk body sway during tilt of a platform, compensa velocity in space', to name only two. Accord tory postural reactions are essentially indepen ingly, in the present article, we proceed from the dent of the head-on-trunk position. In contrast, notion that coordinate transformations may con- the body sway resulting from galvanic vestibu lar stimulation changes direction in direct rela tion to the horizontal head-on-trunk and trunk-
I W e here proceed from the assumption that the main source of inertial stabilization of posture is the vestibular system, although somatic graviceptors may also play a role (see 2This idea goes back to Mittelstaedt's PhD thesis of 1949 Mittelstaedt, this volume). (see Mittelstaedt, this volume). Sensory Coordinate Transformations 349 on-foot excursion (14-16). Galvanic stimulation oratories. In doing so, we will interpret a previ with the head in the primary position leads to a ously suggested model of this interaction from body sway towards the ear where the anode is the point of view of coordinate transformations fixed, whereas the same stimulus with the head between a hierarchy of superimposed coordi horizontally rotated 30° to the right side, for in nate systems, and we will also discuss problems stance, evokes a body sway that also is shifted arising from imperfect signal transduction by 30° towards the right. Apparently, the galvanic receptors. Moreover, in view of their impor vestibular stimulus, which is fixed in head coor tance for ego-motion perception, we will touch dinates' is transformed into space coordinates on certain aspects of the interaction of visual by means of neck afferents-much as a physi cues with vestibular and proprioceptive ones. cal, and hence head position dependent acceler Finally, we will try to link these considerations ation would be. Whereas with the physical (tilt) to the control of body posture and present a hy stimulus, this transformation compensates for pothesis on how to merge a proprioceptive feed head-on-trunk rotation and keeps the postural re back loop stabilizing the body relative to a piv action constant, it makes the reaction directly oting support with a vestibular one controlling dependent on the neck input with galvanic stim body posture in space. In the appendix, we give ulation. a brief outline of the mathematics describing the In everyday life we perform most of our pos 3D-coordinate transformations of vestibular in tural reactions more or less automatically, re put by proprioceptive input, which we think serving our limited capacity of attention for other may be applicable to both perception and pos tasks we are performing while standing or walk tural control. ing. However, we can focus our attention on, and consciously perceive, our bodies' motion' and the evoked postural reactions. Therefore, it ap Transformation of Vestibular Signals pears possible to learn about the central pro cessing of the involved sensory cues by study Figure 1A illustrates the need for a transfor ing self-motion perception, using classical tools mation of the vestibular signals involved in the of vestibular psychophysics and developing control, or the perception, of trunk posture. A new ones. The psychophysical approach may be subject who intends to maintain an upright pos advantageous in several respects. Experiments ture is tilted forward while his head is in a ret can be designed to involve primarily passive roflexed position. His otolith signals will indi stimulations, to be open loop (no feedback by cate head-in-space excursion (angle HS) which, postural reaction), to be independent of the in the simple situation of Figure 1A (axes of body's biomechanical properties and constraints, trunk and head rotation collinear), equals the and to consider simultaneously or successively sum of the trunk-in-space excursion (angle TS) several individual aspects of a complex stimulus and of the head-on-trunk excursion (angle HT; condition. Also, the psychophysical approach HS = TS + HTL This relation can be inter- may reveal global aspects of the central process pre tee m, the case of 21. coordinate t~'ans- ing that are similar to those that possibly underlie formation that describes how the physical TS the fOffi1ulation of compound motor reactions. signal acts upon the vestibular system (Figure In the following, we will sketch a picture of IB). Under more general conditions (TS-axis and how we see the role of coordinate transforma HT-axis not colinear), the transformation corre tion of vestibular signals for motion perception sponds to a matrix operation (compare Appendix). and postural control. First, we will consider the Let us briefly state some conventions used notion that coordinate transformation basically throughout this article: Rotations about positive serves the purpose of reconstructing, in our angels are counterclockwise (TS is negative in brains, the physics of our bodies and the outside the example of Figure 1A), and coordinate sys world. Second, we shall reconsider findings on tems are right-handed. Uppercase pairs of letters vestibular-proprioceptive interaction for human represent acutal physical quantities, lowercase ego-motion perception from this and other lab- pairs of letters stand for an internal representa- 350 T. Mergner et al
A
PHYSICS NERVOUS SYSTEM
B -- ..... XT I Head I NECK on PROP. I ht Trunk HT I SYSTEM (angle) I I I I I r (-) ~ (+) I I ESTIMATION a= Trunk PHYSICAL MOTICN I TRUNK-in-SPACE ..... in .. HEAD-in-SPACE VEST. .... 'PTS I (3D-Back - Space TS (3 D-Coordinat e HS SYSTEM hs ts I Transformation) (angle ) Transformation) I
Figure 1. Schematic illustration of the need for 3D-coordinate transformation of vestibular signals used for pos tural control. (A) Conventions used. Head coordinates (XH,YH,ZH), trunk coordinates (XT,YT,ZT), and space coordi nates (Xs,Ys,Zs), of a subject whose trunk is tilted forward while his head is in retroflexed position (right-handed coordinate systems). For F, "foot point", and H, "head point", see Appendix. (8) Simplified model showing how the nervous system reconstructs physical trunk-in-space (TS) motion for self-motion perception (ns). The ves tibular stimulus of head-in-space (HS) motion results from a coordinate transformation of TS by head-on-trunk motion (HT) corresponding to the kinematic relation between head and trunk. A back transformation of the in ternal representation of HS (vestibular signal hs) by the neck proprioceptive signal ht yields an estimate of TS (ts). If TS and HT are colinear, these transformations correspond to simple summations (signs in parentheses).
tion of those quantities, and the '-/i-prefix stands known inclination) also be derived by proprio for perception, or more precisely, the value re ceptively monitoring the foot-to-Ieg angle, back ported by the subject. transformation is the only means when the sup HS, the transform of TS, causes an otolith re port surface is changing its inclination (for ex sponse (hs) reflecting the magnitude of HS. In ample balance platform). order to learn about the underlying trunk-in Back transformation of vestibular signals space excursion, this physiological signal must would be a perfect means to reconstruct the be be submitted to an inverse transformation using havior of body parts other than the head if the the proprioceptive signal ht as an indicator of head-based sensory mechanisms were flawless. head-on-trunk excursion. In our simple exam However, these transducers do not function like ple, the inverse transformation simply consists ideal physical measuring instruments. In partic of subtracting ht from hs (ts = hs - ht). Again, ular, the well known high pass characteristics of in the general 3D case, the mathematical de the semicircular canals strongly affect the mea scription involves matrices (compare Figure IB surement of low frequency rotations about earth and Appendix). Although information on trunk vertical axes (rotations about horizontal axes position could in certain cases (fixed support of are registered by both the canals and the Sensory Coordinate Transformations 351 otoliths, with the latter detecting the low fre tion was created by having the subjects sit on a quency rotations that are missed by the canals). Baniny chair equipped with a head rotation de There.is evidence for a partial correction of this vice that, if the chair rotated by, say, 12° to the canal deficiency by central mechanisms; pri right, counter-rotated the head by 12° to the left. mary vestibular canal afferents of squirrel mon By this manipulation, only the subject's trunk key have a time constant of about 5 s (17), will rotate with respect to space, whereas his whereas secondary vestibular neurons of alert head remains stationary. Thus, only the neck macaques exhibit time constants of up to 20 s, proprioceptive system is stimulated (HT = 12° which reduces again to abQut 5 s under general to the left), but not the vestibular system. On de anesthesia (18). Moreover, whereas the primary mand, subjects would report and quantitatively human time constant is estimated at 8 s [Iga estimate each of the three aspects of this stimu rashi, cited in (17)], perceptual studies suggest lus condition, namely HT, HS, TS (19,20). values of more than 20 s [see (19)]. However, a full compensation is not possible as this would require the neural implementation of mathemat Perception of Head-on-Trunk ical integration; such a function would be unsta Rotation (Ij;HT) ble because of both low frequency sensory noise and inherent fluctuations. It is conceiv U sing sinusoidal rotations in the dark at fre able, though, that the imperfections of other quencies of 0.025 to 0.4 Hz, we found that the transducers (otolith afferents, neck propriocep perception of neck rotation was almost ideal in tion) are, indeed, by and large corrected by cen terms of gain G and phase (G = 1; phase differ tral mechanisms since, unlike for the canals, the ence between tlJHT and HT = 0°). Gain is de required transfer function would not be plagued fined here as the ratio of estimated or indicated by a denominator of zero. head-on-trunk angular displacement to actual The low frequency deficiency of the canals displacement (0 = tlJHTIHT). Detection thresh poses a problem in the scheme of Figure lB. old in terms of angular velocity showed values Consider a head erect, upright standing subject of about 0.3°/s. Gain, phase and velocity thresh whose head is slowly rotated in the horizontal old were essentially independent of stimulus plane about the stationary trunk. Given the in frequency. In the present context we therefore complete central compensation for the high pass will always proceed from the assumption that characteristics of the canals, the internal repre the compound neck proprioceptive system (which sentation of this rotation (hs) will be too small may include a central corrective mechanism; com at low frequency. In contrast, that of head-on pare the Introduction) can be characterized by trunk rotation (ht) provided by neck propriocep a transfer function of unity over the frequency tion is, according to our assumptions, essen range tested. tially ideal. As a result, the back-transformation hs - ht would yield a non-zero signal. Hence, the subject would experience an illuso:-y trunk rotation and possibly a perceptual conflict with Perception of Head-in-Space his notion of standing on a firm ground. Experi Rotation (Ij;HS) mental evidence supporting the reality of this problem, and our hypothetical solution to it, will In contrast to tlJHT, the perception of head-in be considered below (Vestibular-Neck Interac space rotation during pure neck stimulation is tion and Appendix). subjected to an illusion when vision is ex cluded; subjects feel their heads rotated in space, with the direction being determined by Vestibular-Neck Interaction the head-on-trunk excursion. The illusion is small at high frequencies (>0.1 Hz), but reaches We have studied vestibular-neck interaction a considerable "gain" at low frequencies (gain in young healthy volunteers. Pure neck stimula- here denotes perceived head-in-space displace- 352 T. Mergner et a/
ment referred to the actual head excursion rela A variety of different vestibular-neck stimu tive to the trunk). lus combinations was used. If the head is ro tated about the stationary trunk (stimulating both the vestibular and the neck proprioceptive Perception of Trunk-in-Space systems; HS = HT), ~TS is zero at all frequen Rotation (ljJTs) cies, that is, subjects veridically perceive the trunk as stationary. This observation led us to The perception of trunk-in-space rotation hypothesize that tVTS can be explained by a (lin during neck stimulation presents a mirror image ear) subtraction of the underlying vestibular and of ljiHS: its gain is large at high frequencies and neck signals (which, for the case of head-only clearly decreases at low frequencies. rotation, would indeed yield WTS = t~(HS - HT) = Relnarl(ably, for b()th and gain not 0), This hypothesis was confirmed by the re only varied with frequency, but depended also sponses to other stimulus combinations; if these on stimulus magnitude, being less (WHS) or combinations involve trunk rotation, ljiTS reflects larger (tVTS) when stimuli of 24°/s instead of the deficiencies and illusory components of its 12°/s peak velocity were used. Conceivably, vestibular and neck constituents, respectively this non-linear behavior is due to a detection (which cancel only for HS = HT). threshold of trunk-in-space velocity during pure During head rotations about the stationary neck stimulation, which we have estimated to trunk, the perception of head-in-space motion be of the order of 1.2°/s (19). (~HS) is also veridical. It exhibits a gain close to During pure vestibular stimulation (head and unity and almost zero phase, in sharp contrast to trunk rotated as a whole), our subjects per the behavior during pure vestibular stimulation ceived no head-on-trunk rotation, in accordance described above. with physical reality. Furthermore, they exhib Figure 2 shows a signal flow diagram de ited very similar ~HS and ~TS. Remarkably, scribing the relationships between vestibular gain, phase, and threshold of these vestibular and neck proprioceptive stimuli and the result perceptions resemble quite closely those of ing perceptions \jJHT, t\JHS, and \jJTS. For the sake neck ljiTS. of simplicity and symmetry, we here consider
NECK 0.3°/s . . ht HT J¥ \fIHT
+
+ 'PHS VEST. ~ hs ts' H. S 0----; I; I---~.)-----+ '¥TS
RECOV,
Figure 2. Dynamic model of vestibular-neck interaction for the perception of horizontal head-to-trunk, head-in space, and trunk-in-space angular displacement ('I'HT, 'l'HS, and 'lfTs, respectively). In this and the following three figures, symbols with dot represent physical velocities (for example, H'S, H'T) or internal representations thereof (h's, h·t, t's). Dashes discriminate different internal representations of same physical parameter (for ex ample, t's and t's' both represent trunk velocity in space, but differ in their lower time constant). Interaction takes place at the level of the internal velocity representations, which then are integrated (Jdt) to yield per ceived angular displacements on the output side. The velocity thresholds of 0.3°/s and 1.2°/s are those obtained during pure vestibular and pure neck stimulation (19). A, transfer function of the vestibular system. RECOV., partial recovery of low frequency content, increasing time constant of fs' by comparison to t's. For further de tails see text. [Simplified from (19).] (J) PHYSICAL SrrMULf
I i t,i FOOT ! FS in c> T FOOT SPACE FS SLED
Figure 3. Model of the perception of foot, trunk, and head rotation in space (WFS, WTS, and WHS, respectively). Extension of model in Figure 2. The vestibular canal sys tem reacts to angular acceleration (second derivative, i)2, of HS) and produces a velocity signal with high pass characteristics, by way of a "leaky" integration (f and high pass). The vestibular signal is "down-channeled" to yield a representation of foot-in-space (f's; WFS), and then "up-channeled" to yield 1JFTS and 'l'HS. Note that the proprioceptive signals intervening in down-channeling are given the high pass characteristics (A) of the vestibular channel. [Slightly modified from (21 ).J
(J.) 01 (J.)
£ ac, ",,1 4* X&;~''iffi¥¥f 4Jj~'"'R~~;~~""--""""""----~ -7··"~.~«=,.~ 'f$ * ,) ¥_"A 4 ,·!"9U"$.4@if ___ i"r' '<---~~~~~-'- ;i'~Y'1'Afi'!A:;:4¥!!~g~_ a& 354 T. Mergner et al head-in-space velocity (H"S) and head-to-trunk the actual TS are congruent. In behavioral velocity (H'T) as the specific physical stimuli terms, this would mean: When we move in an acting upon the system, although in reality the environment that does not provide us with a re primary stimulus for the vestibular channel is liable stationary external reference and we head acceleration (H"S) and may be head-to therefore lose orientation, stopping trunk mo trunk displacement (RT) for the neck proprio tion and, possibly, firmly anchoring our trunks ceptive channel. However, assuming linearity at (for example, by ground contact) may help us to the input side, the scheme in Figure 2 can easily reorient. Proceeding from such a stabilized plat be transformed to ret1ect these facts, without af form, head position and movement could then fecting the resulting internal velocity represen be determined using only the neck propriocep tations /z and /z" see tive system with its close-to-unity transfer func The scheme proceeds from the basic idea tion. The head, in turn, then would represent a that our brain creates an internal representation reliable platform for visual orientation in space. of trunk behavior, to begin with, because the trunk represents the natural platform for the head. The primordial role of the trunk is under Stack of Superimposed lined by the observation that naive subjects gen Coordinate Systems erally relate to their trunks when asked to de scribe their orientation and movement in space The model in Figure 2, which provides an in in the absence of vision. Also, as a rule, patients ternal representation of the trunk as a basis for with bilateral vestibular loss refer to their trunks the head, may be extended both further upward in every psychophysical tasks requiring a space and further downward. By downward extension reference in the absence of vision. Trunk repre we can take into account rotations other than sentation is obtained by summing the vestibular that of the neck, for example. rotations between signal with a modified version of the proprio shoulder girdle and pelvic girdle, between the ceptive neck signal ('Ah Of). The modification be latter and the legs, and between the legs and the stows on the neck signal the transfer character feet, which contribute to the head-in-space and istics ('A) of the vestibular signal; in other head-to-foot excursion in an upright standing words, the compound neck signal with unity subject. To the extent that reliable propriocep transfer function mentioned above is passed tive information on these rotations is available, through an internal model of the vestibular chan the brain can use the vestibular information to nel. As a consequence, tjJTS becomes zero during infer the kinematic state of the feet and, hence, head rotation on the stationary trunk (HS = RT), of the body support surface. irrespective of stimulus frequency, which is Evidence for such a "downward channeling" "correct". With all other stimulus combinations of the vestibular signal has been obtained in a (HS=I= HT), however, \'~TS deviates from reality psychophysical study similar to the one de at low frequencies by (1 - 'A)(HS - HT). scribed above, in which vestibular stimulation As mentioned above, the internal "computa was obtained by rotating head and trunk en bloc tions" of t s = h 's- 'A h't (or ts = hs- 'Aht, if we and proprioceptive stimulation by rotating sub argue in the position domain; compare Figure jects' feet relative to their seated trunks (21). 1). This can be viewed as a back transformation The results obtained were, indeed, analogous to from a head-based into a trunk-based system, the ones obtained with neck stimulation. Fur which, in the more general case of rotations thermore, it could be shown that leg and neck about different axes, must be described by ma proprioceptive effects sum up linearly as long trix operations. In view of the illusions that can as the rotation axes are colinear. arise from the unavoidable imperfections of the Consequently, the model shown in Figure 2 vestibular channel (compare above), there may was extended to include leg proprioception be a strategy in the freely behaving subject to (Figure 3). In the extended model the common zero t s because this is the only state where his base for the perception of trunk and head mo internal image of trunk-in-space behavior and tion in space is an internal representation of foot Sensory Coordinate Transformations 355
(support) motion in space (f"s), which is ob of his or her body support, but would not have a tained from the vestibular signal h 's by subtrac perceptually derived confirmation of this as tion of neck and leg proprioceptive signals sumption. which have been given the transfer characteris The principle of coordinate transformation tics of the vestibular signal (Ah't, At f). Repre may as well be extended upward, so as to in sentations of trunk and head motion in space are clude the perception of object motion in space. then obtained by adding the original tf and h't Consider a subject who is rotated on a Barany signals with approximately unity transfer char chair while viewing a light spot that is being acteristics. kept in fixed alignment with his head. The ob We speculate that similar principles could ject is perceived as moving in space. The gain govern the organization of postural reflexes, and phase characteristics of this perception as with the interaction of vestibular and proprio well as its threshold are very similar to those of ceptive inputs for legs and arms taking place at the self-motion perception in this condition, that different levels of the spinal cord. This, possi is, they reflect the characteristics of the vestibu bly, could explain the seemingly puzzling find lar signal (24). When the object is rotated rela ing that electromyographic responses evoked by tive to the head during the chair rotation, the galvanic vestibular stimulation have shorter la resultant object motion perception can be de tencies in the soleus muscles, when subjects scribed as the sum of the vestibular signal and a maintain their balance by the feet muscles, than vi suo-oculomotor signal; the latter is a com in the triceps brachii muscles, when they main pound of (i) an efference copy of the eye move tain their balance by holding with their arms ment by which the object is being tracked, and (22). Moreover, there is direct electrophysiologi (ii) a residual retinal slip signal. cal evidence for a "downward and upward" Interestingly, the just mentioned experiments channeling of vestibular information in the spinal on object motion perception (24) also support cord; there are ascending spinal pathways that the above notion of a "down channeling". Pure carry, in addition to muscle afferent input from neck stimulation evoked an object motion illu limbs and neck, also vestibular information (23). sion that could be related to the illusory percep An experimenter who finds that the percep tion of head-in-space rotation resulting from the tion of head-in-space motion during head-on same stimulation~ This result could have been trunk or trunk-on-foot rotation has a similar anticipated on the assumption that the various "gain" and threshold as the perception of rela internal perceptions form a consistent picture of tive motion between body parts (based on pro the outside world (an assumption which is not prioception) may conclude that the head-in-space always borne out!). Indeed, if neck stimulation perception is primarily determined by proprio induces a non-zero ~HS and if, on the other ception. According to our model, he is correct if hand, the object does not move with respect to subjects are standing on firm ground. In other the head, consistency requires that the object be words, as long as subjects feel their support is perceived as moving along with the head. stationary, they may use only proprioceptive Leg proprioceptive stimulation iL these ex cues for their head and trunk motion perception. periments (24) proved to be effective, as well. Only when there is an indication of support in When vestibular and neck stimulation were be stability do they incorporate the motion of the low threshold for self-motion in space, trunk support inferred from down-channeling the ves on-leg rotation (feet touching firm ground while tibular information into their perception. Possi legs sway back and forth as chair oscillates) re bly, it is because of this mechanism that we so stored an almost veridical object motion percep 'smoothly' can use vehicles, which at times tion. Thus, under the experimental conditions provide a stable platform in inertial terms (dur tested, it appears that the perception of object ing constant velocity), but may accelerate or de motion can be incorporated into the model celerate at other times. A patient with complete shown in Figure 3. This is achieved by adding a vestibular loss with eyes closed, in contrast, further coordinate transformation that trans could only make assumptions about the stability forms the object(eye)-versus-head rotation (that 356 T. Mergner et al is, vi suo-oculomotor signal) into an object-in Let us consider for a moment the drum as just space signal by means of the head-in-space sig another movable part of the subject's body, and nal h 's '. The last transformation can be viewed the visuo-oculomotor system as an equivalent as a back-transformation of the visuo-oculomo of a proprioceptive system signaling the relative tor signal by means of vestibular signals to re motion of this artificial body part. To start with, cover object motion in space after its distortion let both foot support and drum be stationary. by a forward transformation resulting from the When we then extend our principle to the visual movements of various platforms (trunk, head) scene (Figure 4) by adding an essentially ideal on top of each other. vi suo-oculomotor signal v h to the internal rep Also, "artificial" neck :Jroprioceptive stimu resentation of head in space (h's '), an internal lation by muscle vibration affects perceived lo representation of scene motion in space is cre cation of a visual target (25-27). Furthermore, ated. In this situation, there is high congruency the role of proprioception for perceived motion between signals related to the foot support and of a visual object in haptic contact with the those related to the visual scene (for example body has repeatedly been investigated in a t's' = t'v in Figure 4). A subject estimating 'l"TS, meanwhile classical paradigm called the "ocu for instance, would deliver a veridical judgment, lobrachial illusion"; in this paradigm, a light and the experimenter would be unable to infer emitting diode (LED) attached to the stationary from this judgment whether the subject derived finger in the otherwise dark room is seen as it from the visual or the vestibular-propriocep moving if the illusion of an arm movement is tive reference. elicited during vibratory excitation of muscle The situation is clearly different when the spindles of an arm muscle (for literature and foot support is moving. The internal representa model, see 28). tion of head in space (h 's') then is affected by the non-ideal vestibular transfer characteristics (h,'s' = h's + (1-A)hf;h/= h·t+ ... + 1'1; A, Visual-Vestibular and vestibular transfer function) and the scene would Visual-Proprioceptive Interaction appear to move by (A -1)P·S. Moreover, even if the scene were perceptually stabilized by some The "down- and up-channeling" principle unknown mechanism (for example, by cogni outlined in Figure 3 suggests that, perceptually, tion, suggesting that a particular scene, as it al the body support is linked by coordinate trans ways has been, is stationary), there would be a formations to the internal notion of physical problem with taking it, in tum, as a reference space provided by the vestibular system. If the for head or trunk rotation in space, since visual support is experienced as stationary with re contingent and vestibular-contingent signals spect to space, it is taken as a reference for would be no longer congruent (for example, t".5' -=1= trunk and head motion (two-way coordinate t 'v; Figure 4), at least at those stimulus frequen transformations, according to the down- and up cies where /",«1. channeling in Figure 3). This principle has been We see no silnple solution to this problem formulated, in the first place, for situations in and think that subjects in such situations have to which vision of the surroundings is excluded. resort to a perceptual measure of visual-vestib We now turn to situations where subjects are ular conflict (see below), cognition, experience, surrounded by, and are viewing, an optokinetic and so on in order to decide for one or the other drum or, in a more general sense, a visual scene reference. This notion is supported by previous which may be either stationary or moving both studies from this laboratory on human self-mo in space and relative to the head of the subject. tion perception during various visual-vestibular We shall see that matters become complicated stimulus combinations [sinusoidal rotations (29)], when we include in the coordinate transforma in which subjects tended to base their perception tions an external object which perceptually is on either the vestibular-contingent or the visual interpreted as a spatial reference for some rea contingent reference, depending on internal or ex sons (for example, cognition, experience). ternal biasing factors (for example, instruction). Sensory Coordinate Transformations 357
VESTIB.r------, SYS...... VISUAL,..------1 SYS. : SPACE : _ i..-t--.: VISUAL : I REFERENCE I VS-O ~ REFERENCE I I I Lo ______.J •.••.••••• : ~____ I
visual scene to head, vh
+ Aht-- +
CONGRUENCY?
fs fs ..... - - - - -CONGRUENCY? ------~ fv foot support :- - SP~6E- -: foot support: -VtSUAL- -: re ,~e:.~~E~~":! re I~~~~E~~E..'
Figure 4. Extended version of the "down-and-up-channeling" principle (Figure 3), which includes the visual scene (by adding a signal of visual scene relative to head, v·h). The scene has the potential to become a spatial reference (vs = 0; cognition and so on). Head, trunk, and foot motion can also be referred to this visual refer ence, a fact that allows to evaluate congruency between visual-contingent and vestibular-contingent self-mo- tion perceptions. Symbols as in Figure 3. A.h·f ...... A.rf denote representations of H'T ...... L'F, shaped by a model of the vestibular channel, and h's' and t's' are representations of H'S and H'T obtained by internal calcu lation.
In these experiments, perceptual rivalry be pattern (in order to discourage them from consid tween the vestibular-contingent and visual-con ering it a visual object instead of a background), tingent reference for self-motion perception was The subject's body and the optokinetic pattern sharpened by having subjects, who were sitting were sinusoidally rotated, independently of each on a stationary turning chair, fixate at a small other, in order to create various visuo-vestibular visual object, behind which the optokinetic stimulus combinations (see Figure SA). As shown background was moving. When subjects at in the figure, if pattern rotation was in phase tended to this object, it appeared to move in with chair rotation (c,d), perceived body displace space (Dunker's induced motion). The occur ment was more and more underestimated when rence of this illusion facilitated circular vection stimulus frequency was decreased, and eventu (CV), that is, a state in which perceived self-ro ally became reversed in direction (d). In contrast, tation is solely determined by the visual channel perceived displacement was essentially indepen (scene actll1g: as seemingly stationary spatial dent of frequency when the pattern was station reference). We attribute this rapid switching to ary (b) or was rotated counter to the chair (a). CV to a mechanism that tries to minimize con Interestingly, the results could be described flicts between object and self-motion perception. by a model (Figure SB) which, without the con However, we feel that rivalry between purely flict box, is analogous to that in Figure 2, in that vestibular-contingent and visual-contingent ref from the vestibular signal 11 's, a version of the erences is not the only possible mode of visual head-to-visual scene signal h'v is subtracted, vestibular interaction for human self-motion which has received the frequency characteris perception. Visual-vestibular interaction proved tics of the vestibular signal. This yields an inter to be quite different when we changed the visual nal representation of visual scene motion in stimulation and the instruction (30). Specifically, space (v ·s). Adding to v's the original signal h'v subjects were presented with only an optokinetic results in an improved signal of head rotation stimulus and were instructed to stare through the (h 's') that is superior to the vestibular h s signal 358 T. Mergner et al in conditions where the scene is stationary. Note before, but nevertheless attended to it, whereas that an implementation of this visual improve during chair rotation with the pattern stationary, ment of the vestibular h 's' signal in Figure 4 the same instruction left the visual effect essen would be helpful in conditions in which body tially unchanged (unpublished observations). This and support are moved en bloc (for example, led us to add a further mechanism to the model during riding in a vehicle; with h f = 0 in h 's' = in Figure 5B (thin lines); it represents a visual h's + (1 - i\.) hf, h 's' = h :S), but could not rem vestibular conflict mechanism [compare (31,32)] edy the problems outlined above for the general in which an "expected visual signal" is derived case (body and support moving independently from the visual signal h 'v under the assumption of each other, h:S' = h's + (1-'A)h f; behavior that h 'v is caused by self-motion. This expected ally, this would correspond to a condition in visual signal is obtained by endowing h'v with which we move within a moving vehicle and try the frequency characteristics of the vestibular to estimate \vhether the visual scene IS station channel ('Ah 'v; in the model it is actually the ary). Thus, although the scheme in Figures 4 and same as the one used to constnlct v ·s). Subtraction 5 provides solutions for special cases (foot sup of this signal from the vestibular signal yields a port/visual scene stationary), there is no general conflict signal that, when high (as during pure solution by linear interaction for the problem of visual stimulation), suppresses the visual contri estimating self-motion in those cases where the butions to 'AHS in the model. Interestingly, with support and/or the scene are moving. This calls a pure visual stimulus under the form of veloc for a non-linear interaction. ity step, the model predicts an illusory "l'Hs with In the experiments just mentioned, we noted delayed onset (and gradual build up), similar to that during counter-phase stimulus combina that of CV occurring with optokinetic stimula tions, the visual effect on \If HS was considerably tion at constant speed [see (33)]. less when subjects looked through the pattern as We have mentioned these aspects of visual-
B CONFLICT
HS= ao= CONST. VS= -aO(a) ------400(b)~ .. ' :>~.;>;;;;:/' VH aO(c) .•..•.. >. o .... ,/' ......
16° (d) /' ;' H8 -4
I j I I I 0.025 0.05 0.1 0.2 0.4 8TIM. FREQUENCY [Hz]
Figure 5. Self-motion perception in a special condition of visual-vestibular interaction not covered by the scheme in Figure 4. (A) Experimental findings. Sinusoidal whole-body rotation on a turning chair (HS: peak dis placement, ±8°) was used, combined with four different optokinetic stimuli: a, rotation of visual scene with re spect to space, VS, in counter-phase with, and at same amplitude as HS; b, scene stationary in space; c, scene in phase and of same amplitude as HS; d, scene in phase with, and with double amplitude of HS. The fairly low gain at high frequency (0.5 to 0.7) may be due to an adaptational phenomenon linked to frequent repetitions of the stimuli [see (19)]. Gain of perceived self-motion (PHS) exhibited little dependence on optokinetic stimulus at high frequency, but was increasingly mOdified by optokinetic stimulation when stimulus frequency decreased, with response phase even being reversed in combination d. (8) Model used to describe the results. Note anal ogy to model in Figure 2. V'H, velocity of visual scene motion relative to head. A measure of visual-vestibular conflict is added, which transiently (vestibular time constant) suppresses the visual contribution to 'l"vs and "l"HS by means of a "choke control" (double black triangles). [From (30), slightly mOdified.] Sensory Coordinate Transformations 359 vestibular interaction here to point out the fact (about point F; see inset). Only displacements that the coordinate transformations from vestib in vertical rotation planes are considered. ular to visual space suggested in Figure 4 may In the proprioceptive feedback loop (solid often be obscured by other mechanisms, which lines), it is the angle between body and foot try to either suppress (conflict) or enhance CV (BF) which is being sensed and controlled. For (Dunker's induced motion) on the one hand, the sake of simplicity, we let BF = 0° if the lon and to make use of the visual input to overcome gitudinal body axis is perpendicular to the foot the deficiency of the vestibular (canal) system support. As long as this support is in a horizon for low stimulus frequencies/velocities, on the tal and stable position, bi = 0 constitutes the de other hand. sired state of the proprioceptive loop (bf; inter Little is known, so far, of coordinate trans nal representation of BF). Any deviation from formations in terms of visual-proprioceptive in this state (caused, for example, by an externally teraction for self-motion perception (right part applied torque during a push against the body) of Figure 4, downward channeling from visual evokes a compensatory muscle torque acting on reference). In previous pilot experiments, we the ankles to reduce this deviation. have investigated the effect of trunk rotation un A way to activate solely the proprioceptive der the stationary head (neck stimulation) over a loop would be to stimulate selectively muscle frequency range of 0.025 to 0.8 Hz in a percep proprioceptors by vibration (tonic vibration re tual state of a stable CV (29). Gains of visually flex; box VIBR. STIM. in figure 6). For a some and neck induced "¥TS were similar, being es what more natural example, consider a subject sentially independent of stimulus frequency. who stands on an unstable platform, holding his Thus, it came as no surprise when we found that body in place by seizing a ground-based post, 'l"TS during combined stimulations can ~ be de and tries to keep the platform in its initial hori scribed by a simple summation of neck and vi zontal position. There is evidence that in such a sually induced effects. situation, where body equilibrium is safe, the vestibular loop considered below no longer con tributes to the "postural" reactions of the legs Postural Control (34). Note that we assume that the direct loop has properties of an automatic load compensa If vestibular input is to be used for postural tion [similar to equilibrium point models (35, control of upright stance, this necessarily in 36)], receiving a single input (set point signal) volves coordinate transformations. As mentioned from a supervisor (here the vestibular loop) that in the Introduction, there is some direct experi considers more global aspects of the behavioral mental evidence for the occurrence of such situation. transformations. In the following, we shall dis The vestibular loop (dashed) basically infers cuss a conceptual model of postural control whether the support is actually stable and hori that, at least qualita6vely, takes this evidence zontal, or not. To this end, it creates an internal into account, and that incorporates a very simi representation.. of the orientation of the foot. lar notion of coordinate transformation as the and hence of the suppon, with respect to space one considered in the context of ego-motion (FS). Signal f" then is used to change the set perception. The model has been developed as a point of the proprioceptive loop to a value of hypothetical answer to the question of how a -ff); the proprioceptive loop responds to this proprioceptive feedback loop for body stabiliza change by a corresponding adjustment of BF tion relative to the support surface could coop which, assuming perfect control, results in BF = erate with one for body stabilization in space. - FS and therefore keeps the body upright in The model (Figure 6) resembles in its basic as space (BS = BF + FS = 0). pects those shown in Figures 1 through 5. It How does the vestibular system create the considers a person standing on a platform, with foot representation fs? On the physical side of the platform being pivotable in space and the the model, the position of the foot relative to body being tilted en bloc relative to the platform space (FS) is kinematically transformed by the 360 T. Mergner et al
PHYSICS & BIOMECHANICS NERVOUS SYSTEM
MUSCLE TORQUE NEURAL CONTROLLER
DIRECT PROP. LOOP bs'
(J) bf + ~ E¢6~~~~L~=-! M~~~A I._- Figure 6. Conceptual model of vestibular-proprioceptive interaction for postural stabilization of upright stance. Note that, as in Figure 1, interaction here is shown in terms of angular displacement, not velocity, for the sake of simplicity. BF, bOdy-an-foot angle; BS, body-in-space angle. For details, see text. body-to-foot orientation (BF) to yield BS, nal path from galvanic vestibular stimulation which acts on the vestibular system. On the side (box GALV. STIM.) traverses only the central of the central nervous system, the resulting ves back transformation (in contrast to external tibular representation of body-in-space behavior physical stimuli, which act through both a forth (bs) then is submitted to a neural back transfor and back transformation) and is therefore well mation (BF = FB-I) in which the leg proprio suited to study the characteristics of this neural ceptive signal bf intervenes. The result is fs, the stage. Corresponding experimental results have "vestibular image" of FS, which, in the case of been mentioned in the Introduction. co-planar rotations, simply corresponds to bs - bf Note that after injection of fs into the proprio ceptive loop, a "vestibulo-proprioceptive image" of body orientation in space (bs') arises, which Conclusion will be different from the vestibular image bs if the BF transformation introduces errors or if the We have tried to point out that the classical intervening hf signal is distorted to endow it with concept of vestibular-neck interaction actually the characteristics of the vestibular channel (note, is only a small fraction of an extended mecha however, that in the vertical planes low fre nism that links together the different parts of quency input is provided by the otolith systems). our bodies and that allows sensory information By way of this mechanism, it should be no arising in one body part to be used for percep problem to balance the body on a rigid tilted tual or motor tasks in other body parts. The platform with the help of the direct propriocep mechanism also provides linkages between the tive loop, since the vestibular-proprioceptive body and the outside world, for instance, with a loop always adds what is needed for a compen visually or haptically perceived object or refer sation of the tilted support. ence. It would be a simplistic view, however, to Finally, it is clear from Figure 6 that the sig- conceive of perceptions and motor reactions as 362 T. Mergner et al 5. Anastasopoulus D, Mergner T. Canal-neck interaction 25. Biguer B, Donaldson IM, Rein A, Jeannerod M. Neck in vestibular nuclear neurons of the cat. Exp Brain Res muscle vibration modifies the representation of visual 1982;46:269-80. motion and direction in man. Brain 1988; 111: 1405-24. 6. Becker W, Deecke L, Mergner T. Neuronal responses 26. Roll R, Velay JL, Roll JP. Eye and neck proprioceptive to natural vestibular and neck stimulation in the ante messages contribute to the spatial coding of retinal rior suprasylvian gyrus of the cat. Brain Res 1979; 165: input in visually oriented activities. Exp Brain Res 139-43. 1991 ;85 :423-31. 7. Mergner T, Becker W, Deecke L. Canal-neck interac 27. Taylor JL, McCloskey Dr. Illusions of head and visual tion in vestibular neurons of the cat's cerebral cortex. target displacement induced by vibration of neck mus Exp Brain Res 1985;61:94-108. cles. Brain 1991 ;114:755-9. 8. Roberts TDM. Neurophysiology of postural mecha 28. DiZio P, Lathan CE, Lackner JR. The role of brachial nism. 2nd ed. London: Butterworths; 1978. muscle spindle signals in assignment of visual direc 9. Peiper A, Isbeli H. Uber die Kbperstellung des Sauglings. tion. J Neurophysiol 1993;70: 1578-84. Jahrbuch Kinderheilkunde 1927; 115: 142-76. 29. Mergner T, Becker W. Perception of horizontal self i O. Gesell A, Ames ;B. Tonic neCK refiex dmi symmetro rotation: Multisensory dnd cognitive aspects. In: War tonic behavior. J Pediat 1950;36:165-76. ren R, Wertheim AH, eds. Perception and control of 1 I. Simons A. Kopfhaltung und Ruhetonus. Z Neurol Psy self-motion. Hillsdale, NJ, London: Lawrence Erlbaum; chiatr 1923;80:499-549. 1990:219-63. 12. Walshe FMR. On certain tonic or postural reflexes in 30. Mergner T, Schweigart G, Kolev 0, Hlavacka F, hemiplegia, with special reference to the so-called Becker W. Visual-vestibular interaction for human 'associated movements.' Brain 1923 ;46: 1-37. ego-motion perception. In: Mergner T, Hlavacka F, eds. 13. Nashner LM, Wolfson P. Influence of head position Multisensory control of posture. New York: Plenum and proprioceptive cues on short latency postural Press; 1995:157-68. reflexes evoked by galvanic stimulation of the human 31. Zacharias GL, Young LR. Influence of combined visual labyrinth. Brain Res 1974;67:86-8. and vestibular cues on human perception and control of 14. Lund S, Broberg C. Effects of different head positions horizontal rotation. Exp Brain Res 1981 ;41: 159-71. on postural sway in man induced by reproducible vesti 32. Kolev 0, Mergner T, Kimmig H, Becker W. Detection bular error signal. Acta Physiol Scand 1983;117:307-9. thresholds for object motion and self-motion during 15. Tokita T, Ito Y, Takagi K. Modulation by head and vestibular and visuo-oculomotor stimulation. Brain Res trunk positions of the vestibulo-spinal reflexes evoked Bull 1996;40:451-58. by galvanic stimulation of the labyrinth. Observations by 33. Dichgans J, Brandt T. Visual-vestibular interaction: labyrinthine evoked EMG. Acta Otololaryngol (Stockh) effects on self-motion perception and postural control. 1989;107:327-32. In: Held R, Leibowitz H, Teuber HL, eds. Handbook of 16. Hlavacka F, Krizkova M, Bodor 0, Mergner T. Adjust sensory physiology, vol 8. New York: Springer; 1978: ment of equilibrium point in human stance following 7.55-804. galvanic vestibular stimulation: influence of trunk and 34. Fitzpatrick R, Burke D, Gandevia S. Task-dependent head position. In: Stuart DG, ed. Motor control VII. reflex responses and movement illusions evoked by Motor Control Press; Tucson (Az): 289-93. galvanic vestibular stimulation in standing humans. J 17. Fernandez C, Goldberg JM. Physiology of peripheral Physiology 1994;478:363-72. neurons innervating semicircular canals of the squirrel 35. Bizzi E, Hogan N, Mussa-Ivaldi FA, Giszter S. Does monkey. 2: Response to sinusoidal stimulation and the nervous system use equilibrium point control to dynamics of peripheral vestibular system. J Neurophys guide single and multiple joint movements? Behav ioI1971;34:661-75. Brain Sci 1992;15:603-13. 18. Buettner DW, Buttner D, Henn V. Transfer characteris 36. Feldman AG. Once more for the equilibrium point tics of neurons in vestibular nuclei of the alert monkey. hypothesis (lambda model). J Mot Behav 1986;18:17-54. J NeurophysioI1978;41:1614-28. 37. Andersen RA, Snyder LH, Li CS, Stricanne B. Coordi 19. Mergner T, Siebold C, Schweigart G, Becker W. nate transformations in the representation of spatial Human perception of horizontal head and trunk rotation information. CUff Opin NeurobioI1993;3:171-6. in space during vestibular and neck stimulation. Exp 38. Soechting JF, Flanders M. Moving in three-dimen Brain Res 1991;85:389-404. sional space: frames of reference, vectors, and coordi 20. Mergner T, Nardi GL, Becker W, Deecke L. The role of nate systems. Annu Rev Neurosci. 1992; 15: 167-91. canal-neck interaction for the perception of horizontal 39. Hepp K. Oculomotor control: Listing's law and all that. trunk and head rotation. Exp Brain Res 1983;49:198-208. Curr Opin Neurobiol 1994;4:862-8. 21. Mergner T, Hlavacka F, Schweigart G. Interaction of 40. Schweigart G, Heimbrand S, Mergner T, Becker W. vestibular and proprioceptive inputs. J Vestib Res 1993; Role of neck input for the perception of horizontal head 3:41-57. and trunk rotation in patients with loss of vestibular 22. Britton TC, Day BL, Brown P, Rothwell JC, Thompson function. Exp Brain Res 1993;95:533-46. PD, Marsden CD. Postural electromyographic responses 41. Cappa S, Sterzi R, Vallar G, Bisiach E. Remission of in the arm and leg following galvanic vestibular stimu hemineglect and anosognosia during vestibular stimula lation in man. Exp Brain Res 1993 ;94: 143-51. tion. Neuropsychologia 1987;25:775-82. 23. Coulter JD, Mergner T, Pompeiano O. Effects of static 42. Vallar G, Bottini G, Rusconi ML, Sterzi R. Exploring tilt on cervical spinoreticular tract neurons. J Neuro somatosensory hemineglect by vestibular stimulation. physiol 1976;39:45-62. Brain 1993;116:71-86. 24. Mergner T, Rottler G, Kimmig H, Becker W. Role of 43. Karnath HO. Subjective body orientation in neglect and vestibular and neck inputs for the perception of object the interactive contribution of neck muscle propriocep motion in space. Exp Brain Res 1992;89:655-68. tion and vestibular stimulation Brain 1994;117: 1001-12. Sensory Coordinate Transformations 363 Appendix Cartesian, space-fixed coordinate system. The description of rotations is slightly more compli Thre e-Dimensional cated. The most intuitive way is to indicate (i) Coordinate Transformation the axis about which the rotation is performed and (ii) the angle. However, this description be In order to achieve conceptual simplicity, in comes unwieldy when one is interested in more the previous sections of the article, only rota complex rotational motions in three-dimen tions within one plane, or alternatively, about sional space, with rotation axes whose orienta one axis or about several parallel axes were con tion and position are varying with time. sidered. Obviously. a fully three-dimensional de Therefore, other parameterizations rely on scription is necessary for more general situations the fact that rotations form a mathematical (for example, when the axis of the head-on-trunk group and can be represented by 3 X 3-matri rotation is not parallel to the axis of the trunk ces. To explain this in more detail, we introduce to-space rotation). It is the aim of this appendix the following three coordinate systems: to develop the mathematics which can be used to describe such transformations and which are (S): The immobile, space-fixed system. The needed for computer simulations of the ideas z-axis is vertical, the x- and y-axes are developed above. horizontal. Let us consider a simple "two-joint-model," similar to the one sketched in Figure 1, in which (T): The mobile, trunk-fixed system. The the trunk as a whole, from feet to neck, is con z-axis is along the trunk, the y-axis is sidered as a rigid rod. The head is allowed to ro parallel to the line between the shoul tate about any axis through a point H,$ which ders, and the x-axis is perpendicular. roughly corresponds to the head joints (crossing (H): The mobile, head-fixed system. The point of interaural and naso-occipitallines). The z-axis is along the vertical head axis, trunk in turn can rotate with respect to a space the y-axis is parallel to the interaural fixed coordinate system about any axis through line, and the x -axis is perpendicular. a "foot point" F. In addition, translations of the foot point F are possible. Depending on the sit On upright standing, the axes of all three coor uation under study, it would be straightforward dinate systems are not rotated with respect to to generalize this model by introducing addi one another. The following relations hold for tional rotational degrees of freedom, for exam any quantity a/vector character in these coordi ple, through the hip-joints or through the knees, nate systems: as is discussed for example in reference (1). Mathematically, the position of the head Xr = TS· Xs with respect to the trunk is described by 3 rota tional degrees of freedom, for example, by 3 an Xli = HT· xr gles. Analogously, the position of the trunk with respect to fixed space is described by 3 rota tional and 3 translational degrees of freedom. Here, XH, XT, Xs stand for one and the SCllne Consequently, a complete description of the physical quantity of vector character, for exam static posture of the model body is given by 9 ple, the gravitational force or an angular veloc numbers. The dynamic description accordingly ity, as measured in the three different coordi requires 9 time-dependent functions. The math nate systems. In the following, vector quantities ematical expressions for velocities and acceler are typeset as bold face lower case letters, for ations of any point in the model body will in example, x, and matrices as pairs of bold face volve the first and second time-derivatives of capitals, for example, TS. TS is an orthogonal 3 X these 9 functions. 3-matrix, which maps (S)-coordinates into (T) In the following, translations are described coordinates, and analogously, HT is an orthog by specifying the coordinates of the point F in a onal3 X 3-matrix that maps (T)- into (H)-coor- 364 T. Mergner et al dinates. The third of these equations follows by tion of the angles HS = HT + TS and, likewise, inserting the first into the second. The orthogonal TS = HS - HT is equivalent to TS = HT-l . (or synonymously, rotation) matrices HT and TS HS = TH . HS. These correspondences served as are characterized by the following properties: a recipe for generalizing the uni-planar models of the previous sections to full 3D models. HTf. HT = 1 = TSf . TS detHT = 1 = detTS. Dynamical Input Thus, the inverse matrix HT- l is identical with the transpose BTt, and we shall occasionally de How can dynamic processes be described? note it by TH. The above properties imply that First we consider translational stimuli. If the only 3 out of 9 matrix elements are independent. initial position of the foot point F is iel and F is Hence, TS and HT contain the 6 rotational de moved with a velocity vet), the following sim grees of freedom of the model body (2,3). ple relation holds: There are several conventions for parameter izing rotation matrices by three numbers. For f ( t) = f 0 + f~ v (t') d t ' example, the three Euler angles [see, for exam ple, (2) or (4)] specify the magnitudes of three or in differential form: successive rotations about three predefined d axes. However, for numerical purposes (that is, d/(t) = vet). [2] computer simulations), it is more convenient to employ the full 3 X 3 matrices, so that we do The situation is more complicated for rota not need any explicit parameterization of the ro tions. Let the trunk rotate with respect to fixed tation matrices. space with an angular velocity vector WTS(t), Special consideration has to be taken to de with the foot point F as the fix point of the rota scribe position vectors (mathematically: affine tion. Remember that any rotational motion is vectors), since the three different coordinate uniquely described by such an (possibly time systems have their origins in different places. dependent) angular velocity vector, whose di Let f be the position vector of the foot point F rection specifies the instantaneous axis of the - which is the origin of the trunk -system in rotational motion and whose length measures Figure 1- as measured in (S )-coordinates. f the instantaneous rotation velocity. Then be contains the three translational degrees of free tween the (S)-to-(T) transformation matrix dom of the model. Furthermore, let h be the po TS(t) and CUTS(t) the following equation holds, sition vector of the point H - which is the origin in analogy to equation [2]: of the head-system in Figure 1-as measured in o wTS,3(t) -wTS,2(t) (T)-coordinates. By definition, h is a constant. d TtTS(t) = TS(t) . - WTS, 3 (t) 0 WTS, I (t) [3J Then, for any position vector the following rela W . (t) -W , l(t) 0 tions hold: TS 2 TS The mathematically interested reader might fr = TS(rs - f) note that each row in the resulting matrix on the fH = HT(fr-h) right-hand side is equivalent to the vector prod uct of WTS with the corresponding row of TS. fH = HT(TS(fs - f) - h). [lJ The rows of d/dtTS can also be viewed as the In the special case of rotations in a single plane instantaneous velocities relative to fixed space (colinear axes), we denote the angle of rotation of the tips of the unit vectors ex, ey, ez of the (ro with the same letters as the corresponding ma tating) trunk system. A similar matrix equation trix; for example, TS would be the angle corre governs the time evolution of the matrix HT, sponding to rotation TS. The combination of which describes the position of the head with two successive rotations in the same plane, for respect to the trunk. The corresponding angular example, HS= HT . TS is equivalent to the addi- velocity is called WHT. 366 T. Mergner et al verse of the third equation in [1]. The anatomi tegration similar to the one in equation [4], us cal parameter rH specifies the location of the ing the information from WTS(T)*. In Figure 1, otolith organs and is given in (H)-coordinates. these operations are implemented in the box The position vector h of the point H (heads 3D-Back-Transformation. joints) is in (T)-coordinates, and f is the position Similarly, an internal representation of the vector of the foot point F. As result we obtain position of the feet (f*) can be obtained by two r s, the otolith position in (S)-coordinates. integrations (that is, by inversion of equations In the second equation, the acceleration as of [6J, calculating fk from aH* under consideration the otoliths is obtained by twice differentiating of WHS*, HT*, and TS* and the knowledge i\ with respect to time and subtracting the about the initial orientation (and magnitude) of space-fixed vector of gravitation g. Finally. in the gravitational field. Thereby. in the mathe order to determine the projection of this acceler matical sense, the orientation and position of ation upon the head coordinate system (which the body can be fully reconstructed. determines the direction of the acceleration-re In the previous sections we have pointed out lated forces with respect to utriculus and saccu that the canal signal WHS* is plagued by high lus) the back transformation from (S) to (H) in pass transfer characteristics (see Figure 2), the third equation of [6] is required. which appear to be more or less isotropic in all directions. In contrast the low frequency char acteristics of WHT* are practically ideal. As a re Back Transformation sult, if the reconstruction of WTS * is performed directly as suggested by equation [7], WTS* has As already noticed, the posture of the model the same undesirable high-pass characteristics. body is completely characterized by the two However, the consideration made in section matrices TS and HT, and the foot point vector f. Vestibular-Neck Interaction can be applied: Let In this section, we discuss how an internal rep us denote by A the canal transfer function (that resentation of TS and its time derivative (essen is, WHS* = AWHS)' Bestowing the same transfer tially WTS) can be reconstructed by the brain characteristics on WHT*, thus replacing WHT* in from the available afferent information in order equation [7] with the modified proprioceptive to obtain an internal representation of body pos signal AWHT*, a modified version of WTS*' is ture and movement in space [compare (4)]. obtained. This improved signal is, for example, For this we consider the angular velocity of zero (veridical), with head rotation on the sta the head WHS as signaled by the canal system tionary trunk. As mentioned in the same sec (WHS *) and the head-to-trunk orientation HT tion, an improved WHS*' can be obtained by a sensed by the neck proprioceptive system (HT*; further coordinate transformation, using asterisk denoting brain signals such as afferent wHS*' HT*(wTS*' + wHT*)' activity and internal representations of physical = quantities). Then, the angular velocity of the We simulated the coaxial conditions like "head trunk relative to fixed space, measured in trunk rotation on the stationary trunk" or "trunk for coordinates, can be obtained by inversion of the ward tilt with the head in retroflexed position", equation [5]: as mentioned in the sections Transformation of Vestibular Signals as well as Vestibular-Neck W (T)TS * = (T H * w HS * ) - w HT * . [7] Interaction. The three-dimensional simulation model was implemented using, for simplicity, For colinear rotations this means that the head the experimentally observed vestibular transfer on-trunk angular velocity is subtracted from the characteristics for all three canal planes. Our total (head-in-space) angUlar velocity to yield experimental findings were well described by the trunk-in-space angular velocity. Note that at the model, as was the case with our earlier 2D this stage of the reconstruction, WTS(T)* is repre models (19,21). For the vertical planes, we im sented in (T)-coordinates. The trunk-to-space plemented an independent channel for otolith orientation TS* can then be estimated by an in- signals, using at present ideal transfer character- Sensory Coordinate Transformations 367 istics, apart from a detection threshold (also, no lateral direction with a concurrent horizontal attempt was made to merge canal and otolith head-on-trunk rotation mentioned in the Intro signals, since little is known to date about ca duction. The appropriate functioning of the nal-otolith interaction for self-motion percep model was verified by comparing its reconstruc tion and postural control). For both the canal tion of trunk and head orientations with the "ac and the otolith channels, we applied various tual" orientations. These simulations confirmed head and trunk rotations abut non-colinear axes, that the models presented in the previous sections like the condition of a body-in-space sway in are not restricted to uniplanar situations. REFERENCES 1. Johansson R, Magnusson M. 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