Ординатура / Офтальмология / Английские материалы / Eye Movements A Window on Mind and Brain_Van Gompel_2007

model based on muscle torque. In their model, neural commands do not necessarily have to be commutative to control eye movements. The authors argue that neural innervation encodes muscle torque, which is represented by a vector and, therefore, commutes. It is assumed that the direction of pull of the muscles does not depend on eye position (corresponding with inter-muscular coupling only). It is also assumed that muscle force of a muscle pair depends linearly on innervation and is not influenced by muscle length (Robinson, 1975).

Under these assumptions the model holds for relatively small eye rotations (<15 ). In addition, although the model generates realistic saccadic eye movements in 3D, it also generates post-saccadic drift. To avoid this pulse-step mismatch, Raphan (1997) has incorporated musculo-orbital coupling in the torque model. Muscle pulleys, which have previously been demonstrated histologically (Clark, Miller, & Demer, 1997, 1998, 2000), should account for this musculo-orbital coupling.

2. Muscle pulleys

Previous studies with CT and MRI scans have demonstrated that during eye rotations the bellies of the horizontal extraocular muscles (between insertion and origin) approximately stay in place with secondary positions of 30 in the vertical plane (Demer, Miller, Poukens, Vinters & Glasgow, 1995; Miller, 1989; Simonsz, Harting, De Waal, & Verbeeten, 1985). The same holds for the vertical rectus muscle bellies with secondary positions in the horizontal plane. Thus, the muscle bellies are kept in place by some structures in the surrounding tissue, which implies that the effective pulling direction of the rectus muscle with respect to the globe remains the same also for eye rotations larger than 15 .

Demer et al. (1995), Demer, Miller, & Poukens (1996) and Clark et al. (1997, 1998, 2000) have argued that previously identified anatomical structures, called muscle pulleys (Simonsz, 2003), act like sleeves around the extraocular muscles and serve as their functional origins. The position of the pulleys with respect to the eye globe determines their effect on the direction of pull of the muscle. In the most extreme situation, this direction of pull does not change at all regardless of eye position (Figure 1). It is suggested that the surrounding tissue contains elastin, collagen and smooth muscle, especially in the area where the pulleys should stabilize the muscle bellies. The smooth muscle tissue receives rich innervation. The neurotransmitters involved suggest both excitatory and inhibitory control, which indicates that the pulleys actively influence the paths of the extraocular muscles. The location of these pulleys and their displacement also were determined by high-resolution MRI. The results suggest that the pulleys are musculoorbital tissue connections. On the other hand, the paths of the extraocular muscles posterior to the pulleys undergo small displacements during changes of gaze. These displacements are more consistent with inter-muscular mechanical coupling. Histological research results are not conclusive in this matter. There are arguments both for musculo-orbital and inter-muscular coupling.

a dynamical model of the ocular motor plant that a correct position of orbital pulleys is in fact sufficient to obtain a perfect pulse-step match for any saccade in 3D.

3. Commutative eye movements and ocular motor instabilities

In patients with ocular motor instabilities and/or misalignments, including patients with strabismus or CN, 3D recordings demonstrate that rotation axes describing the orientation of the eye orientation typically are not restricted to a single horizontal plane (Dell’Osso and Daroff, 1975), but also have components in the vertical and torsional planes (Apkarian, Bour, van der Steen, & Faber, 1996 and Apkarian, Bour, Van der Steen, & Collewijn, 1999; Averbuch-Heller, Dell’Osso, Leigh, Jacobs, & Stahl, 2002; Korff et al., 2003; Ukwade, Bedell, & White, 2002). This may imply that in CN patients eye movements do not obey Listing’s law. The question arises whether these two extra components of CN have their origin in the central nervous system or whether these components are the result of the anatomical structure of the eye-globe with its surrounding tissue (Averbuch-Heller et al., 2002). In the latter case, it is of interest whether aberrant anatomical structures of the eye-globe play a role in the generation of the three components of CN and/or the occurrence of misalignments.

A way to investigate this issue is to measure in 3D the amplitude and direction of particularly the fast phase of CN during various viewing conditions, including different gaze directions, binocular and monocular viewing. Eye-movement recordings are then compared with computer simulations of the dynamical model (Fetter, Haslwanter, Misslisch, & Tweed, 1997; Raphan, 1998; Schnabolk and Raphan, 1994), which predicts that dynamic variations in the amplitude and direction of CN are related to gaze in a systematic way.

By means of simulation of eye movements, the current study shows that fast components in the vertical and/or torsional plane may be explained by a displacement of musculoorbital coupling or inter-muscular mechanical coupling with respect to the optimal position (no pulse-step mismatch). It is concluded that the observed torsional/vertical components of nystagmus in CN can be considered as the result of deviant plant characteristics only in the horizontal plane and their origin not necessarily has to be found in the central nervous system. In conclusion, non-commutative eye movements in CN patients with ocular motor abnormalities could be attributed to aberrant anatomical structures including a displacement of musculo-orbital and/or intermuscular coupling.

4. Methods

4.1. Subject

In the horizontal planes the patient displayed classic congenital nystagmus (Abadi and Dickinson, 1986; Apkarian et al., 1996; Dell’Osso and Daroff, 1975) and interocular

amplitude disconjugacy. A variable esotropia and hypermetropia was present. The left eye was generally the preferred eye of fixation particularly in primary position. Ophthalmic evaluation revealed albinism with foveal and macular hypoplasia, peripheral fundus hypopigmentation and iris diaphany. Being an albino, the patient had blue irises, light brown hair colour and does not tan when exposed to sunlight. Optical refraction yielded spherical corrections of OD (right eye): S = +10; OS (left eye): S = +11 25. Corrected Snellen acuity at 6 m viewing distance was 0.1 (20/200) OD viewing, 0.15 (20/300) OS viewing. At near viewing (20 cm) OU (both eyes) acuity was 0.4 (20/50). The albino patient showed no evidence of stereopsis; color vision was normal. Visual fields as tested with static perimetry were normal.

4.2. Three dimensional recordings

Measurement of ocular rotation in 3D for the patient and five healthy control subjects was accomplished with the dual scleral induction coil technique (Bartl et al., 1995; Ferman et al., 1987a; Jansen, Ferman, Collewijn, & Van den Berg, 1987a,b). Raw data were digitized (12 bit) with a sample frequency of 500 Hz (Bour, 1984). Data were collected from ten channels; two channels for horizontal and vertical target position, four channels for OD and four channels for OS. For each eye, two channels are derived from the direction coil that picks up the eye rotation signals in the horizontal and vertical plane. The other two channels are derived from the torsion coil that yields the signals of clockwise and anticlockwise rotation of the eye in the torsional plane. In all cases, the eye orientations are calculated as rotation vectors from the signals, using Mathematica 5.0. (Wolfram, 1999) The various offsets were taken into account (Hess, Van Opstal, Straumann, & Hepp, 1992). Calibration consisted of an in vitro and an in vivo protocol.

For the in vitro calibration, the following procedure was implemented. The straightahead position (0 ) was used as the reference target. The three gain components of the direction and the torsion coil (coil vectors) were obtained by directing a gimbal system to symmetrically positioned targets with respect to the reference target. Via a rotation matrix (Collewijn, 1975; Robinson, 1963), sine and cosine components of direction and torsion coils were converted to 3D Fick coordinates. After Haustein (1989), Fick coordinates were converted to rotation vectors (Apkarian et al., 1999).

For in vivo calibration, cooperation of the patient was required. The subject fixated a reference target positioned at straight ahead. To obtain the corresponding ‘zero fixation’ position for each eye, monocular measurements were implemented. Following the experimental protocol, all successive and successful “zero position fixations” within a given test session were marked for further offline calibration and analysis. To compensate for long-term drift and/or sudden shifts of eye-movement signals (particularly those from the torsion coil) repeated fixation of the zero fixation reference target was required. A third-order polynomial was fit through successive “zero-fixations” that extended over a given fixation measure. In addition, zero-fixations were parsed into foveation periods (Dell’Osso and Daroff, 1975).

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Subjects had to look at a translucent screen at a distance of 114 cm which subtended a viewing angle of approximately 24 by 24 . The target consisted of a laser spot (0 1 diameter) that was back-projected on the screen and the position of the spot was changed by a servo-controlled mirror system. The laser spot jumped from the reference position (0, 0) to an eccentric position and back to the reference position. For the normal control subjects, 48 positions were located on 6 meridians at 0, 30, 60, 90, 120 and 150 with an eccentricity of 5, 10, 15 and 20 . The duration of each fixation position was 1 s. For the CN patients the stimulus grid consisted of 28 targets presented in horizontal, vertical and oblique meridians from primary position to ±30 eccentricity.

4.3. Analysis

Listing plane is determined by fitting a plane through the data points using a least-square fitting procedure. The equation of this plane is given by

where nlp is the “normal” vector of Listing’s plane, and d is the distance of the plane to origin (0, 0, 0). With nlp and d, the orientation of this Listing’s plane is determined. The deviation from Listing’s plane is determined by calculating the standard deviation (SD) of the distance of the data points with respect to the plane. The x, y and z directions of the eye do not necessarily coincide with the x, y and z directions defined by the experimental setup. This has the effect that when Listing’s plane is measured, it usually is skewed by some angle with respect to the fronto-parallel plane defined by y and z directions of the magnetic field setup. The rotation vector data points can be adjusted such that Listing’s plane is equivalent to the fronto parallel plane. With the adjustment, the deviation from Listing’s plane at each eye orientation is given by only the component rx. The SD of the deviation from Listing’s plane for all eye orientations N is

The ‘reference orientation’ is the orientation when the eye is looking straight-ahead (along the x axis, defined by the magnetic field setup). The ‘primary orientation’ is the orientation when the eye is looking along the ‘normal vector’ of Listing’s plane, given by n. If the eye is rotating from this primary orientation to any other orientation, the component rx (torsion) equals 0.

4.4. Three dimensional model simulations

The most important aspect of the dynamical eye-movement model of Schnabolk and Raphan (1994) is that the eye muscles generate torque to rotate the eye. When the restoring torque of the orbital tissues counterbalances the torque applied by the extra-ocular muscles,

a unique equilibrium point is reached. The relationship between restoring torque and eye orientation yields the unique torque-orientation relationship. With this approach the signals from the central nervous system can be treated as vectors. These signals are converted to muscle torques, which are also vectors and thus commute. Therefore, the ocular motor plant must convert torque (vector) into orientation (non-vector). According to this model the problem of non-commutativity of eye orientation is then solved by the ocular motor plant and not by the central nervous system. The central nervous system only has to generate a two dimensional (2D) (horizontally and vertically) ocular motor command.

The diagram shows that the neural pulse-step generator, instead of the one-dimensional (1D) operator in the Robinson model (Robinson, 1975), uses three operators to transform the neural input signal r into a motoneuron signal mn. Hp is the system matrix for the velocity-to-position integrator, Gp is the coupling from the neural signal r to the integrator and Cp is a matrix that transforms the output of the integrator xp to a motoneuron step signal xp. For the direct premotor-to-motor coupling and the generation of the pulse from the velocity command there is a matrix D. The sum of pulse D r and step Cpxp commands (Robinson, 1975) yields the total motoneuron signal mn, which is transformed by the muscle-matrix M into a torque-related signal m. Thus, the transformation of neural input signal r into a torque-related signal m can be described by the following equations:

A schematic diagram of the 3D model of eye-movement control is shown in Figure 2. The final position of the eye is determined by the balance between the muscle force and

the force exerted by the connective tissue. To obtain an accurate model, the description of the torque-orientation transformation has to account for both of these factors. The dynamical system (see Equation 5) describes the balance between the active torque m exerted by the extraocular muscles and the passive torque, where k is the coefficient of elasticity, J is the moment of inertia tensor, B is the matrix representing viscous damping, and and h are the angular velocity vectors of the eye relative to space and the head relative to space, respectively. The resulting eye position in rotation vectors is defined by the angle and the vector n. The actual derivation of the equations of this dynamical system that describes eye orientation as a function of the applied torque is too elaborate to discuss in this context (but see Schnabolk and Raphan, 1994).

Figure 2. Schematic diagram of 3D model of eye-movement control. According to the Robinson model the internal saccade velocity command is split up in a direct pulse and a velocity-to-position integrated step. The pulse step signal of the motoneuron innervates the muscle. The 3D coupling from neural innervation to muscle force is given by matrix M. Finally, the active muscle torque counterbalances the passive visco-elastic forces of the eye globe with surrounding tissue and fat.

In this system Listing’s law is automatically obeyed as long as the velocity commands that drive the pulse-step generator only have pitch and yaw components and no roll components. The model is physiologically quite accurate and generates realistic saccades and smooth pursuit eye movements. Raphan (1997) made adjustment to the original model of Schnabolk and Raphan (1994) and introduced a modifiable muscle-matrix M in such a way that it was dependent of eye orientation Mp . The original constant diagonal matrix used by Schnabolk and Raphan (1994) was multiplied by a rotation matrix RM, to incorporate the pulley-effect (Figure 3).

= k

where MP is the muscle matrix with pulleys, n is the axis of eye and muscle torque rotation, is the rotation angle of the muscle torque, is the rotation angle of the eye and k is a constant ratio between the latter two that indicates the level of pulley-effect.

Vertical pulley

Horizontal pulley

Vertical pulley

Vertical pulley

B

A

Horizontal pulley

Vertical pulley

Figure 3. Instead of one pulley constant for all three muscle pairs, three pulley constants for each muscle pair can be used. The ky and kz value for the horizontal and vertical muscle pair are defined by the quotient of the angles: A/B.

4.5. Different pulleys for each muscle pair

For k = 0, there is no pulley-effect whatsoever. The direction of muscle pull remains constant in the orbital frame and RM reduces to the identity matrix that was used in the original model. For k = 1 the pulley effect is such that the direction of pull of the muscles always remains constant with respect to the eye regardless of the eye position. Simulations of Raphan (Fetter, 1997; Raphan, 1997) show that with k = 0 48 the smallest transient torsional components are found during a saccade. This corresponds with the value k ≈ 0 5 that Quaia & Optican (1998) found for a negligible pulse-step mismatch.

A restriction of this approach is that there is no separate k for the horizontal, vertical and torsional plane. Therefore, RM was replaced by the product of the pulley effect in three planes of rotation:

It must be noted that when the neural innervation is 2D, there is no “torsional” innervation, only horizontal and vertical innervation. The effect is that since r x = 0 mm x = 0, the value of kx has no significance. The matrix R becomes essentially a 3 by 2 matrix, equal

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to Equation 6 with the left column removed. The pulley matrix converts neural input, which is two-dimensional, to muscle torque, which is three-dimensional:

Thus now the pull-direction of the extra ocular muscles can be adjusted for horizontal (kz) and vertical (ky) planes separately. Since the velocity command that drives the pulse-step generator only has pitch and yaw components the value of kx is irrelevant. The dynamical model of Raphan with the addition of a 2D to 3D pulley-matrix R was implemented in a computer-program that solved the differential equations numerically (Mathematica 5.0; Wolfram, 1999). The constants were the same as in the original model:

H = −0 03333 G = 0.3333

C = 2.944

D = 0.1389

J = 5 10−7 kg m2

m = 2 493 10−7 kg m2/s2 spikes/s) B = 7 476 10−5 kg m2/s/rad

k = 4 762 10−4 kg m/s 2/rad kz = between 0 and 1

ky = between 0 and 1

The angular head velocity h was taken as zero.

5. Results

5.1. Normal controls

To determine Listing’s plane, and the deviation from this plane in two normal control subjects, a ‘calibration circle’ paradigm has been used. With a least square error method Listing’s plane was fitted through the 3D eye positions recordings during saccades and fixations. Subsequently, all data were rotated into the frontal parallel plane. The SD of the rotation vector data from this plane is then determined using Equation 2. The SD from Listing’s plane for all normal control subjects did not exceed 1 3 .

5.2. Patient recordings

Eye-movement recordings of a patient with CN showed a component in the torsional plane that exceeded those of healthy normal controls. Figure 4 shows a recording obtained from a patient with CN. The left column shows the 3D recording when the patient makes

10

0

–10

torsional

200

0

–200

100

0

–100

torsional

Time (s)

Figure 4. Three dimensional search-coil recording of a patient with congenital nystagmus. From top to bottom the position in the horizontal, vertical and torsional planes (rotation vectors) and the velocity in the horizontal vertical and torsional planes is shown. OD refers to the right eye and OS refers to the left eye. The horizontal CN clearly shows components in both the vertical and torsional planes. Note that a downward saccade for OD is associated with a clockwise torsion and a decrease in jerk amplitude in the torsional plane, whereas an upward saccade for OD is associated with a counterclockwise torsion and an increase of the jerk amplitude in the torsional plane.