Neural network models for the gaze shift system in the superior colliculus and cerebellum

Neural Networks 15 (2002) 811–832 www.elsevier.com/locate/neunet

Neural network models for the gaze shift system in the superior colliculus and cerebellum Xiaoxing Wanga,b,*, Jesse Jinb, Marwan Jabria a

School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW 2006, Australia b School of Information Technologies, The University of Sydney, Sydney, NSW 2006, Australia Received 21 September 2001; revised 10 May 2002; accepted 10 May 2002

Abstract We investigate the role that the superior colliculus (SC) and the cerebellum might play in generating gaze shifts. The discharge of cells in the intermediate layers of the SC is tightly linked to the occurrence of saccades. Many studies have demonstrated that the cerebellum is involved in both eye and head movements. When the head is unrestrained, large amplitude gaze shifts are composed of coordinated eye and head movements. In this study, we propose that the gaze saccades system is controlled by a feedback loop between the SC and the cerebellum. The SC only encodes retinal coordinates and controls the eye displacement (to move the fovea to the target), while the cerebellum deals with the gaze programming and controls the head displacement. When a target appears in space, the buildup cells within the SC decode the target signal in the retina before the saccade onset, and input the signal of the gaze displacement to the cerebellum. The cells in the cerebellum vermis encode the initial position of the eye in the orbit. The gaze displacement is decomposed into the head amplitude and the eye amplitude within the cerebellum. There are two output signals from the cerebellum. One signal controls the head movement. The other is projected back to the SC, and forms a component of the saccade vector to control the eye movement. The sum of the vectors provided by the cerebellum and the vector provided by the burst cells in the SC indicates the direction and the amplitude of the desired movement of the eye during the saccade. We propose a cerebellum model to predict the displacements of the eye and head under the condition that the position of the target signal in the retina and the initial position of the eye in the orbit are known. The results from the model are close to that observed physiologically. We conclude that before gaze shift onset, the cerebellum may play an important role in decomposing the gaze displacement into an eye amplitude and head amplitude signal. q 2002 Published by Elsevier Science Ltd. Keywords: Neural networks; Eye movement; Superior colliculus; Cerebellum; Gaze shifts; Modeling

1. Introduction Behavioral studies using monkeys and humans as subjects show that gaze shifts are accomplished by moving both the eyes and the head in the same direction toward the target when the head is unrestrained (Bizzi, Kalil, & Morasso, 1972; Misslisch, Tweed, & Vilis, 1998; Munoz, Guitton, & Pelisson, 1991; Sparks & Groh, 1995). It is well know that gaze shift is the sum of the eye-in-head movement and head-in-space movement, gaze ¼ eye/ space ¼ eye/head þ head/space. Because of the complex, fast and accurate coordination involved, the mechanisms for eye/head gaze shifts have led to a lot of questions. For example, where is the gaze displacement signal decomposed into separate eye and head * Corresponding author. Tel.: þ 612-9351-4494; fax: þ 612-9789-2208. E-mail address: [email protected] (X. Wang). 0893-6080/02/$ - see front matter q 2002 Published by Elsevier Science Ltd. PII: S 0 8 9 3 - 6 0 8 0 ( 0 2 ) 0 0 0 6 5 - 5

displacement signals? How does the position of the eyes in the orbit influence the contribution of the eye movement and the contribution of the head movement? What areas in the brain encode the retinal coordinate frames, and what areas in the brain encode the head-centred coordinate frames? How does the coordinate transformation take place? Many hypotheses attempt to account for the mechanisms of eye/head gaze shifts. For example, the ‘separate channel hypothesis’ (May & Porter, 1992), the ‘gaze displacement hypothesis’ (Freedman & Sparks, 1997a; Freedman, Stanford, & Sparks, 1996), the ‘superposition hypothesis’ (Grantyn, Dalezios, Kitama, & Moschovakis, 1996; Moschovakis, Dalezios, Petit, & Grantyn, 1998), and the ‘cerebellum hypothesis’ (Lefevre, Quaia, & Optican, 1998; Quaia, Lefevre, & Optican, 1999). Separate channel hypothesis states that both the displacement command of the eye movement and the head movement are generated by the superior colliculus (SC;

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May & Porter, 1992). The pathway from the upper SGI projects to pontine regions implicated in the control of eye movements, and the pathway originating in the lower SGI projects to pontine regions implicated in head movement control (Freedman & Sparks, 1997a,b). Gaze hypothesis assumes that the SC encodes the sum of eye and head displacements, i.e. gaze displacements (Freedman et al., 1996; Guitton, Crommelinck, & Roucoux, 1980). Such commands are decomposed downstream from the SC into separate commands to eye and head controllers in such a way that the contribution of the eye movement decreases while the contribution of the head movement increases when initial eye position shifts in the direction of the desired movement (Moschovakis et al., 1998). Superposition hypothesis rests on the assumption that position-sensitive slow drifts temporally overlap positioninvariant saccades that are electrically evoked from the SC (Grantyn et al., 1996). It further rests on the assumption that slow drifts evoked from the SC are due to anatomic projections that bypass the burst generator (Grantyn et al., 1996; Moschovakis et al., 1998). Cerebellum hypothesis (Lefevre et al., 1998; Quaia et al., 1999) assumes that the saccades are controlled by two parallel pathways, the SC and the cerebellum. The Optican group (Lefevre et al., 1998; Quaia et al., 1999) assigned the cerebellum a more important role in saccades because the lesions of the SC do not result in large and enduring deficits. They devised a model of the saccade eye movement in which the characteristics of the saccades are determined by the cooperation of two parallel pathways, one through the SC and the other through the cerebellum (Lefevre et al., 1998; Quaia et al., 1999). However, they did not consider the gaze saccade and how the gaze displacement signal is decomposed into separated eye and head displacement signals. In the present study, we propose that the cerebellum integrates both the position of the target signal in the retina and the position of the eye in the orbit, and decomposes the gaze displacement signal into separate eye and head displacement signals. The buildup cells in the SC provide a gaze displacement signal to the cerebellum. The burst cells in the SC form a saccadic vector which is composed of two components: the first one is a position-invariant vector depending on the target signal within the SC. The second one comes from the cerebellum and encodes the desired eye displacement information which is the result of gaze programming. Freedman and Sparks (1997a) reported their experimental data for gaze shifts, and concluded that it is possible to accurately predict the amplitudes of the eye and head components of a gaze shift if the displacement of the target and the initial position of the eyes in the orbits are known. In the present study, we generated a mathematical model to describe the prediction. The

results from the model are close to those observed physiologically.

2. Background 2.1. Which areas in the brain encode the head-centred coordinate frames? It is clear that the SC encodes the retinal coordinate frames. There is a motor map in the intermediate and deep layer of the SC with the fovea represented rostrally (Cynader & Berman, 1972; Robinson, 1972; Schiller & Koerner, 1971; Sparks & Groh, 1995). The motor map topographically point-to-point matches the visual receptive field on the retina (Ottes, Van Gisbergen, & Eggermont, 1986; Schiller & Koerner, 1971; Schiller & Stryker, 1972; Van Gisbergen, Van Opstal, & Tax, 1987). The position of the visual target signal within the motor map specifies the gaze error (Freedman & Sparks, 1997a; Freedman et al., 1996; Tweed, 1997). However, it is not clear which areas in the brain are responsible for encoding the head-centred coordinate frames (Cullen & Guitton, 1997). It follows that the areas in the brain which encode the position of the eye in its orbit are also not known as the eye position is partially determined by the head-centred coordinate. The eye position signal has been widely recorded in the nucleus reticularis tegmenti pontis, the cerebellum (McElligott & Keller, 1984; Ron & Robinson, 1973; Seung, 1996), the central thalamic nuclei, the frontal eye field and the supplementary eye field (Russo & Bruce, 1993). May and Porter (1992) suggest that the head-centred coordinate should be within the SC. They proposed that the SC generates two separate commands: a command originating in the upper SGI to move the eyes and a command originating in the lower SGI to move the head. However, other investigators argued against their hypothesis (Freedman et al., 1996). Zipser and Andersen (1988) suggested that the posterior parietal cortex (area 7a) encodes head-centred coordinates, and proposed a back-propagation neural network model for the retinal-to-head coordinate transformations (Groh & Sparks, 1992). Ron and Robinson (1973) pointed out a property of saccades evoked by cerebellar stimulation, that the direction and amplitude of the saccade varies with initial eye position. This is in contrast to the SC or the frontal eye field which indicates that the cerebellar vermis encodes the position of the eye in its orbit (Davson, 1990; Schall, 1991). In our model, we assume that there is a head-centred coordinate in the cerebellum (McElligott & Keller, 1984; Hirai, 1988; Ron & Robinson, 1973), and that the cerebellum vermis encodes the initial position of the eyes relative to the orbits before any eye movement (Davson, 1990; Schall, 1991; Seung, 1996).

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Fig. 1. Activities of fixation cells, burst cells, buildup cells. PABEF, activity of the fixation cell; QGBCDEN, activity of the burst cell; QGHMN, activity of the buildup cell.

2.2. Where is the gaze displacement signal decomposed into separate eye and head displacement signals? Three hypotheses have been proposed to interpret gaze programming. SC hypothesis. The gaze displacement signal is decomposed within the SC (May & Porter, 1992). However, this hypothesis may have some fundamental problems (Freedman et al., 1996). Downstream hypothesis. The gaze displacement signal is decomposed downstream from the SC (Freedman & Sparks, 1997a,b; Phillips, Fuch, Ling, Iwamoto, & Votaw, 1997). This hypothesis is based on the ‘gaze displacement hypothesis’, which states that because a gaze displacement signal is derived from the locus of activity within the motor map of the SC, and the gaze decomposition does not take place within the SC, then logically, the gaze programming must occur downstream from the SC. Upstream hypothesis. The gaze displacement signal is decomposed upstream from the SC. It is well known that the activities of the collicular neurons encode the amplitude and direction of the saccade (Ju¨rgens, Becker, & Kornhuber, 1981; Robinson, 1972). The vector of the saccade indicates the displacement of the eye movement, not the position of the eye at the end of the saccade. This displacement of the eye movement is the result of gaze programming, which includes the information of the initial position of the eye in its orbit. Therefore, the gaze displacement signal is decomposed upstream from the SC. A number of studies support the downstream hypothesis and a number of other studies support the upstream hypothesis. Which hypothesis is correct? We consider both hypotheses reasonable, even though they look contradictory. We think that the desired gaze displacement (not separate eye and head displacement commands) and the eye displacement (after gaze is decomposed) are different events within the SC. They occur at different times and are performed by the different collicular neurons. Before answering the question we recall the properties of the collicular neurons. There is a general agreement that there are three types of cells that discharge in relation to

saccades within the SC: the buildup cells, the burst cells and the fixation cells (Munoz & Wurtz, 1995a,b). See Fig. 1. The burst cells have a high-frequency burst that occurs just before the saccade onset. The buildup cells discharge at a low frequency beginning before a saccade onset. The fixation cells are located in the rostral pole of the SC. During active visual fixation, the fixation cells show an increased discharge rate and suppress saccade via inhibitory connections onto the burst cells and buildup cells in the caudal end of the SC (Munoz & Wurtz, 1995a,b). The buildup cells, with a low frequency, encode the visual target signal in the motor map (Mays & Sparks, 1980; Munoz & Wurtz, 1995a,b; Sparks & Groh, 1995). We assume that the buildup cells form a gaze displacement that is topographically place-coded in the motor map within the SC, and project the gaze displacement signals into the cerebellum. For the sake of simplicity, the terms ‘visual signal’, ‘target signal’, ‘target displacement’, ‘retinal error’, ‘gaze displacement’, or ‘gaze error’ that are used in the literature are called ‘target signal’ or ‘gaze displacement’ in this paper, even though they have slightly different meaning (Freedman & Sparks, 1997a). The latency from the discharge of the buildup cells to the saccade onset is quite long, more than 100 ms (Munoz & Wurtz, 1995a,b). Some investigators thought that the latency is too long to explain this delay by proposing a simple loop through any single region of the visual cortex (Stein & Meredith, 1991). We think it is necessary to spend some time to decompose the gaze displacement before a saccade onset. During this latency the gaze-displacement signal is decomposed into separate eye and head displacement signals within the cerebellum. From the above analysis, the ‘downstream hypothesis’ occurs at the target signal onset, and the gaze displacement is formed by the buildup cells. The buildup cells project the gaze displacement signal to the downstream from the SC. The population of burst cells with a high frequency is generally thought to generate a saccadic vector in the motor map (Sparks & Groh, 1995). Earlier studies believed that each neuron discharged maximally prior to a saccade onset which had a particular direction and amplitude, regardless of

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the initial position of the eye in its orbit (Sparks, Holland, & Guthrie, 1976; Sparks & Mays, 1980; Wurtz & Goldberg, 1972). Recent experiments have shown that the activities of these neurons are influenced by the initial position of the eye in its orbit (Freedman & Sparks, 1997a; Moschovakis et al., 1998; Opstal, Hepp, Suzuki, & Henn, 1995; Pare´ & Munoz, 2001). In the present study, we assume that the saccadic vector is composed of two components: the first one is a position-invariant vector that depends on the target signal within the SC (Moschovakis et al., 1998). The second one comes from the cerebellum and encodes the desired eye displacement information that is the result of the programming. Since gaze programming is based on both the target signal and the initial position of the eye in its orbit, the second component contains the information of the initial position of the eye in its orbit. A simulation reported in this paper demonstrate that the activities of a population of burst cells is influenced by the initial position of the eye in its orbit. From the above analysis, the ‘upstream hypothesis’ occurs near saccade onset, and the desired eye displacement is formed by the burst cells. The burst cells receive the eye displacement signal upstream of the SC. 2.3. What conditions should be satisfied for the area in which the gaze displacement is decomposed? If an area in the brain performs gaze programs, it should have the following properties: 1. Pathways between the SC and this area. 2. The ability to encode the initial position of the eye in its orbit. 3. The ability to project a head displacement signal to the area in which head movement is controlled after gaze programming has been performed. 4. There are cells whose activities beginning before a saccade onset, so that they receive the signals of the gaze displacement from the buildup cells in the SC. The cerebellum satisfies above requirements, as explained below: 1. Experiments show that there are several pathways between the SC and the cerebellum (Dean, 1995; Gonzalo-Ruiz, Leichnets, & Smith, 1988; Huerta & Harting, 1984; Lefevre et al., 1998; May, Hartwich-Young, Nelson, Sparks, & Porter, 1990). The dorsal part of the intermediate and deep layers of the SC are the major source of descending pathways. It terminates in two important precerebellar nuclei: the inferior olive (IO) and the nucleus reticularis tegmenti pontis. Both the nuclei project into the fastigial nucleus. The pathway from the cerebellum to the SC is derived from the cells in the caudal fastigial nucleus and projects to both the caudal end and the rostral end of the intermediate layer of the SC (May et al., 1990).

2. The results of lesion, stimulation and recording studies in the cerebellum converge on the interpretation that the cerebellum serves to calibrate the saccade generation system by signaling the position of the eye in its orbit (Schall, 1991; Seung, 1996). The eye position signal in the cerebellum may be derived from the eye muscle proprioceptive afferent (Davson, 1990; Fuchs & Kornhuber, 1969; Maekawa & Kimura, 1980; Schwartz & Tomlinson, 1977; Steinbach, 1987). 3. The classical experiments (Mussen, 1927; Pollock & Davis, 1927) demonstrated that the cerebellum is involved in the control of the head movement. When the middle lobe of the cerebellum was stimulated, the head rotated towards the side stimulated, either the anterior or posterior direction, depending on the electrode location; when the anterolateral (or posterolateral) part of the middle lobe was stimulated, the head flexed backward (or forward) and to the side stimulated. Hirai discussed the cerebellar pathways contributing to head movement (Hirai, 1988). The reciprocal connections between the cerebellum and the vestibular nuclei comprise the mechanisms for the control of head movement (Keller, 1991) 4. There is abundant evidence for saccade related neurons in the vermis of the cerebellum (Kase, Miller, & Noda, 1980; Llinas & Wolfe, 1977; McElligott & Keller, 1984). The activities of the long-lead burst mossy fibers begin, on average, 160 ms before the saccade onset (Kase et al., 1980; Schall, 1991). They may receive the gaze displacement signal, which comes from the SC. 2.4. Overview of the model structure We outline the structure of the model for the gaze generation system to provide a general idea of the mechanism used. This is shown in Fig. 2. The buildup cells decode the target signal in the retina before a saccade onset, which forms the gaze displacement signal to the cerebellum. The cells in the cerebellum vermis encode the initial position of the eye in its orbit (Davson, 1990; Schall, 1991). The gaze displacement is decomposed into the head and eye contribution within the cerebellum. The output signal for the head movement goes to the brainstem and is carried by reticulospinal and vestibulospinal neurons. These neurons have extensive interconnections with other oculomotor structures, suggesting that the signal they carry may be both eye and head related. The output signal for the eye movement is projected back to the SC. The sum of the vectors provided by the cerebellum and the vector provided by the burst cells in the SC indicates the direction and amplitude of the desired movement of the ensuing eye movement. The signal coming from the cerebellum carries information of the initial position of the eye in its orbit. This is the reason why the SC activity is influenced by eye position (Opstal et al., 1995).

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Fig. 2. The model of the gaze generation system showing the flow of information through time. FEF, frontal eye field; LIP, lateral intraparietal area. The buildup cells discharge before saccade onset. They encored the visual target, and form the signal of the gaze displacement. The burst cells discharge immediately prior to the saccade onset. The saccadic vector is composed of two components: the first one is a position-invariant vector that depends on the target signal within the SC. The second one comes from the cerebellum and encodes the desired eye displacement information that is the result of the gaze programming.

2.5. How is the gaze displacement signal decomposed into separate eye and head displacement signals? It is possible to predict the displacement of eye and head components of gaze shift if the position of the target signal in the retina and the initial position of the eye in its orbit is known (Freedman & Sparks, 1997a). We propose a cerebellum model to predict the displacements of the eye and head, and derive several equations from the data reported by Freedman and Sparks (1997a) to describe the gaze programming. For example, Eq. (1) describes the slope of the linear relationship between the gaze displacement and the eye amplitude, and Eq. (2) describes the slope of the linear relationship between the gaze displacement and the head contribution. ( SlopeEye ¼

1

if gaze # 25 þ x

10=ð55 2 xÞ if gaze . 25 þ x

( SlopeHead ¼

0

if gaze # 25 þ x

ð45 2 xÞ=ð55 2 xÞ

if gaze . 25 þ x

ð2Þ

where ‘gaze’ is the gaze displacement; x, the initial eye position, for example, x ¼ 20 means the eyes begin deviated in their orbit by 208 contralateral to the direction of the gaze shift; SlopeEye, the slope of the linear relationship between the gaze displacement and the eye amplitude; SlopeHead, the slope of the linear relationship between the gaze displacement and the head contribution. The comparison between the results of the equations and the data cited from Freedman and Sparks (1997a) are shown in Figs. 3 and 4. It is clear that the data from the equations is close to data observed physiologically (Freedman & Sparks, 1997a). 2.6. How does the coordinate transformation take place?

ð1Þ

Fig. 3. The position of the eyes in their orbits can alter the eye contribution to the movement. The dark line and blue line are the data reported by Freedman and Sparks (1997a). The red line is the result of Eq. (1). (Cent) the eyes are initially centered in their orbits; (Con) the eyes begin deviated in their orbits contralateral to the direction of the gaze shift.

The visual stimuli are imaged on the retinas and project to the SC in retinal coordinates. The eye position signals can be added to form the representations in head-centered coordinates, and space-centered coordinates can be formed by adding head position information. Most available data on the coordination of eyes and head were obtained from movements directed along the horizontal meridian (Barnes, 1979; Bizzi et al., 1972; Guitton & Volle, 1987; Phillips, Ling, Fuch, Siebold, & Plorde, 1995; Zangemeister & Stark, 1982). These one-dimensional studies revealed several features of eye/head coordination (Bizzi et al., 1972; Zangemeister & Stark, 1982). Two- or three-dimensional models have been proposed, too (Tweed, 1997). However, the effects of the initial position of the eye in its orbit have not been considered by the models. In this study, we propose a one-dimensional coordinate transformation model concerned with both the initial position of the eye

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Fig. 4. The position of the eyes in their orbits can alter the head contribution to the movement. The dark line and blue line are the data reported by Freedman and Sparks (1997a). The red line is the result of Eq. (2). (Cent) the eyes are initially centered in their orbits; (Con) the eyes begin deviated in their orbits contralateral to the direction of the gaze shift.

in its orbit and the final position of the eye in its orbit after a saccade.

3. Models 3.1. The structure of the SC computational model There are three layers in the SC, shown in Fig. 5. The first layer is the superficial layer. The cells in this layer receive visual target information from two visual pathways, one is a direct pathway from the retina, and the other is an indirect pathway from the visual cortex. Visual stimuli are imaged on the superficial layer and may be projected to the intermediate and deep layers (Lee, Helms, Augustine, & Hell, 1997). The second layer is the burst layer, and the third is the buildup layer. Fixation cells, which are another type of collicular cell, exist in the rostral end of buildup layer (Munoz & Wurtz, 1995a,b). Oculo motoneurons control eye movements when they are excited by the medium lead burst neurons (MLBN) in

the pons. MLBN are strongly inhibited by omnipause neurons (OPN), which fire tonically between saccades and pause during saccades. Fixation cells also fire tonically between saccades and pause during saccades, and project an excitatory input to the OPN (Quaia, Aizawa, Optican, & Wurtz, 1998a; Quaia, Optican, & Goldberg, 1998b). When a target appears, the frontal eye fields (FEF) and the lateral intraparietal area (LIP) supply an excitatory input to the buildup cells and the burst cells in the SC. We assume that the buildup cells have a low threshold by construction, and fire immediately. As their discharge activity increases, so does their inhibition of the fixation cells. However, the burst cells have a high threshold. Even though the burst cells and the buildup cells receive input at the same time, the burst cells do not fire until the level of inhibitory input from the fixation cells is sufficiently low. After the eye and the head movements are decomposed by the cerebellum, an inhibitory signal is projected to the rostral part of the SC from the cerebellum. It is the inhibition of the cerebellum that withdraws the suppression of the fixation cells to the burst cells and removes the excitatory input of the fixation cells to the OPN. The OPN withdraws the inhibition to the MLBN. Therefore the MLBN excites the oculo motoneurons (Quaia et al., 1998a,b). The oculo motoneurons drive the movement of the eye. To illustrate how the model works, we divide the saccade gaze system into several parts and separate into several subfigures as follows. We define below the symbols and variables used in the following sections. MðF; LÞ is the excitatory input representing the map that includes the target stimulus subject of the saccade. This map includes the signals that came from the FEF and LIP. OBd ; OBr ; and OFx are, respectively, the output of the buildup, burst and fixation cells. a Bd, a Br, and a Fx are, respectively, the activity of the buildup, burst and fixation cells. u Bd, u Br,

Fig. 5. The structure of the computational model of the SC. MLBN, medium lead burst neurons; OPN, omnipause neurons; VC, visual cortex; FEF, frontal eye fields; LIP, lateral intraparietal area, BR, burst cells; BD, buildup cells; FX, fixation cells. Solid line, excitatory; Dashed line, inhibitory.

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Fig. 6. Fixation periods.

and u Fx are, respectively, the thresholds of the buildup, burst and fixation cells. Other symbols are introduced in the body of the description below. Fixation. During periods of visual fixation both of the buildup cells and the burst cells are inhibited by the fixation cells (Fig. 6). The output spiking density of a fixation cell is

Fig. 8. Composition of the saccade vector.

 X



Br Br OFx ¼ f aFx 2 WiBd ·OBd ·O 2 W i i

¼

8 > > > > > <



2 aFx 2

P

1þe > > > > > :0

1



2W Br ·OBr WiBd ·OBd i



if aFx 2

X

WiBd ·OBd 2 W Br ·OBr þ Q . uFx i

i

ð3Þ

i

otherwise

where WiBd and OBd i are, respectively, the weight and output of the ith buildup cell. Q is a constant to keep uFx . 0: During fixation, the activity a Fx suppresses the buildup and burst Pcells. When the buildup cells discharge, their activities WiBd ·OBd inhibit the fixation cells. At saccade i termination, the signal of the current eye position (as the target is foveated) inhibits the excitatory burst cells. So the fixation cells regain their high activity, and in doing so, inhibit both the buildup and burst cells until the next saccade. Target appears. Before the initiation of a saccade, the FEF and LIP project the signal of target position into the buildup cells at the caudal part of the SC. Buildup cells fire and suppress fixation cells (Fig. 7). The equation   Fx Fx SBd ¼ f MðF; LÞ 2 WBd ·O 2 uBd ¼

8 <1

  Fx Fx ·O 2 uBd . 0 if MðF; LÞ 2 WBd

:

otherwise

0

ð4Þ

is the ‘switch’ representing a buildup cell when it is Fx ready to discharge. WBd represents the weight of a

Fig. 7. Target appears.

connection between a fixation cell and a buildup cell. Eq. (4) states that when the difference in activity between the excitatory input MðF; LÞ and the inhibitory Fx Fx ·O of the fixation cells exceeds the threshold input WBd Bd u , the buildup cells will discharge (their switch is on). It is easy to see that the threshold of the buildup Fx cells can be selected such that, for small WBd values, the buildup cells will fire as soon as the input MðF; LÞ is non-zero. Decomposition of the gaze displacement signal. The buildup cells project the signal of gaze displacement into the cerebellum. The cerebellum decomposes the gaze displacement signal into the desired eye amplitude signal and desired head amplitude signal. Composition of the saccade vector. The saccadic vector is composed of two components: the first one is a positioninvariant vector that depends on the target signal within the SC. The second one comes from the cerebellum and encodes the desired eye displacement information that is the result of programming. Because gaze programming is based on both the target signal and the initial position of the eye in its orbit, the second component contains the information of the initial position of the eye in its orbit. A saccade vector is formed in the motor map of the burst layer (Fig. 8). Burst cells fire and eye movement start. The cerebellum projects an inhibitory input to the fixation cells in the rostral end of the SC. The decrease in activity of the fixation cells causes them to release their inhibition on the burst cells, and

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Fig. 9. Saccade termination.

the burst cells fire. That is, 2

2

CB " ) FX # ) BR ") burst cells fire

Fig. 10. The equations in the saccade system.

then 2

þ

þ

2

BR " ) FX # ) OPN # ) MLBN " ) OM " ) eyes start moving where " means activity increases and # means activity decreases. ) þ means excitatory and ) 2 means inhibitory. The equation ! ! ! X Bd Bd Br Fx Fx Br S ¼ f MðF; LÞ þ Wi ·Oi 2 WBr ·O 2 u ðAÞ i

¼

8 > > <1 > > :0

if MðF; LÞ þ

X

! WiBd ·OBd i

Fx Fx ·O . uBr ðAÞ 2 WBr

i

otherwise

ð5Þ Fx is the switch for a burst cell to fire. WBr is a weight from a fixation cell to a burst cell. The threshold u Br(A) of the switch is, by construction, an increasing function of amplitude A from the target to the rostral pole. The values Fx of the threshold u Br and the weights WBr are such that the burst cells cannot discharge at the presentation of the input MðF; LÞ: At saccade onset, the interaction of increased excitatory input to the buildup cells and decreased inhibitory input to the fixation cells allows the burst cells to discharge. The switching off of S Br causes a clipping of activities in the buildup cells. The output spiking density of a burst cell is defined as 8 ðt > < aBr ðuBr Þ 2 gðOBr ðtÞÞdt if SBr ¼ 1 0 OBr ðtÞ ¼ ð6Þ > : Br 0 if S ¼ 0

where aBr ðuBr Þ is the activity of the cell when S Br is turned on. g(O Br(t)) is the inhibitory input representing the current eye position error signal Ðfrom the brainstem reticular formation feedback loop. t0 gðOBr ðtÞÞdt is the integrated input of the feedback loop. When a burst cell discharges ðt ¼ 0Þ; its output O Br reaches a maximal value a Br(u Br).

The output of the cell decreases with time. The decrease in activity of the burst cells causes them to release their inhibition on the fixation cells which then reinstate their high discharge rate and disable the burst cells (see Eq. (5)). During saccade. The burst cells will discharge until the motor signal is reduced to zero. The current eye position is derived by the tonic oculo motoneurons (OM) that integrate input from the burst cells. This current eye position is fed back to the burst cells as one component of the motor error. The excitatory burst cells are inhibited by the signal of the current eye position. Saccade termination. When the fovea reaches the target, and the dynamic motor error is reduced to zero, the activity of the burst cells decays rapidly. This causes the burst cells to withdraw their inhibition of the fixation cells. The fixation cells activate OPN. This leads to inhibition of MLBN and cessation of saccade signal to the oculo motoneurons, and the eyes stop moving (Fig. 9). 2

þ

2

þ

BR # ) FX " ) OPN " ) MLBN # ) OM # ) eyes stop moving Fig. 10 shows the relationship among the equations. 3.2. Coordinate transformations among the retina coordinate, head coordinate and space coordinate For simplicity, we only consider the horizontal gaze shift, and assume that the center of the head coordinate is the same as the center of the space coordinate when a target appears in the retina. During a large gaze saccade, the eye moves toward the target limited by eccentricities of less than 408 horizontally (Misslisch et al., 1998). The symbols used in this section are defined below (Fig. 11). TR1 is the position of a target in the retina when a target appeared, EH1, the position of the eye in its orbit when a target appeared. The position of the eye means the position of the center of the retina coordinate in the head coordinate, TS1, the position of a target in space before a saccade. That

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Fig. 11. Coordinate transformation. Top panel, the coordinate before a saccade. TR1, the position of a target in the retina when a target appeared; EH1, the position of the eye in its orbit when the target appeared in the retinal receptive field; TS1, the position of the target in space before a saccade. Bottom panel, the coordinate after a saccade. HS2, the distance of the head movement in space during a gaze saccade; EH2, the position of the eye in its orbit when the fovea fixes on the target after the saccade; TR2, the amplitude of the eye movement in the retinal coordinate; TS2, the position of the target in space at the end of saccade.

is TS1 ¼ EH1 þ TR1

With Eqs. (7) and (11), ð7Þ

At saccade end, the fovea fixes on the target. The target should be located at the center of the retina coordinate. HS2 is the amplitude of head movement in the space coordinate during the gaze saccade, EH2, the position of its eye in the orbit when the fovea fixes on the target, TR2, the amplitude of eye movement in the retinal coordinate. That is, the distance that the fovea moves, TS2, the position of the target in space at saccade end. Within a certain range u, the eyes move freely without head movements. Two cases have been considered. If lTS1 l # u; the eyes move freely without head movements. So, TR2 ¼ TR1 ; if lTS1 l . u; then the head has to move the eye. We have TR1 ¼ TR2 þ HS2

ð8Þ

EH2 ¼ EH1 þ TR2

ð9Þ

Eq. (9) means that the position of the eye at the saccade end is equal to the sum of the position of eye in the head before the saccade and the amplitude of the eye movement. TS2 ¼ EH2 þ HS2

ð10Þ

Eq. (10) means that after a saccade the position of the target in space is equal to the sum of the position of the eye in the head and the position of the head in space. From Eqs. (9) and (10), TS2 ¼ EH1 þ TR2 þ HS2 As the target is fixed in the space, that is TS1 ¼ TS2

ð11Þ

TS1 ¼ EH1 þ TR1 ¼ EH1 þ TR2 þ HS2 ¼ TS2 and TR1 ¼ TR2 þ HS2 This is Eq. (8) (shown as Fig. 11). 3.3. The structure of the neural network model in the cerebellum The function of the cerebellum neural network model is to predict eye and head movements in a gaze shift. There are four layers in the cerebellum model: the input layer, the granule layer, the purkinje layer and the fastigial layer, shown in Fig. 12. The fastigial layer lays in the deep cerebellar nuclei (DCN). The input layer has two parts. One encodes position of the target in the retina (Fig. 13) and the other encodes the position of the eye relative to the head. The mossy fibers send the inputs to the granule layer where the granule cells make a coordinate transformation from the retinal coordinate and the head coordinate to the space coordinate, and calculate the position of the target in space. According to this position the signals for eye and head movement are separated into the purkinje layer by the parallel fibers. The fastigial layer is the output layer. The outputs of the fastigial layer have two pathways. One pathway returns to the SC and encodes the predictions of eye movement. The other pathway projects to the neck motoneurons that control the head movement. The IO provides the control rules. The climbing fibers (Cfs) project the information of the rules to the purkinje layer and the fastigial layer. The inhibitory input from the fastigial layer

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Fig. 12. The function of the cerebellum neural network model is to predict the eye and head movements in a gaze shift. There are four layers in the cerebellum model: the input layer, the granule layer, the purkinje layer and the fastigial layer. The input layer has two parts. One encodes the position of the target in the retina and the other encodes the position of the eye relative to the head. The fastigial layer lays in the deep cerebellar nuclei (DCN). The outputs of the fastigial layer have two pathways. One pathway returns to the SC that encodes the predictions of eye movement. The other pathway projects a signal to control the head movement. The inferior olive (IO) provides the control rules. (Le) eyes move to left; (Re) eyes move to right; (Lh) head moves to left; (Rh) head moves to right.

to the IO carry a feedback of the predictions being established in the granule layer. 3.3.1. Methods In the cerebellum model, there are 26 mossy fibers between the input layer and the granule layer for training the neural network. Sixteen mossy fibers encode the position of the target in the retina and are organized in two sides, R and L. The numbers of R and L indicates the eccentric position of the target in the retina. There are eight values R1 ¼ 0:03; R2 ¼ 0:05; R3 ¼ 0:1; R4 ¼ 0:2; R5 ¼ 0:35; R6 ¼ 0:55; R7 ¼ 0:65; R8 ¼ 0:8 for R. R1 ¼ 0:03 means the target is near the center of the retina, R4 ¼ 0:2 means the target is 208 eccentric from the center of the retina towards the right and R8 ¼ 0:8 means the target is 808 eccentric from the center of the retina towards the right. Similarly, there are eight values L1 ¼ 20:03; L2 ¼ 20:05; L3 ¼ 20:1; L4 ¼ 20:2; L5 ¼ 20:35; L6 ¼ 20:55; L7 ¼ 20:65; L8 ¼ 20:8 for L. L8 ¼ 20:8 means the target is located at 808 on the left side of the

Fig. 13. The buildup cells in the SC code the location of targets relative to the fovea in the retina, then pass the signals to the cerebellum.

center of the retina. The other 10 mossy fibers encode the position of the eyes in their orbit. Letters ‘Ra’, ‘Rb’, ‘Rc’, ‘Rd’, and ‘Re’ express eye locations at right 308 (Ra), right 208 (Rb), right 158 (Rc), right 108 (Rd), and middle (Re). Letters ‘La’, ‘Lb’, ‘Lc’, ‘Ld’, and ‘Le’ express eye locations at left 308 (La), left 208 (Lb), left 158 (Lc), left 108 (Ld), and middle (Le). A combination of both the number and the letter indicates the initial state of the target and the eye when the target appears. For example, L6Rb means the eyes are located right 208 in their orbits, and the target is located left 558 on the retina. We always assume the center of the head coordinate is the same as the center of the space coordinate when the target appears in the retina. There are 10 granule cells in the granule layer, denoted as Gk ; k ¼ 1; 2; …; 10: The mossy fibers contact the granule cells at the glomeruli which are the sites for the input signal distribution to the granule cells. On the one hand L connects with G1, G3, G5, G7, G9, and R connects with G2, G4, G6, G8, and G10. On the other hand a connects with G1, G2, b connects with G3, G4, etc. e connects with G9, G10. The mossy fibers excite the granule cells, and the granule cell axons form the parallel fibers that contact the dendrites of the purkinje cells. The parallel fibers carry the integrated mossy fiber signals to the purkinje cells. The DCN neurons in the fastigial layer receive the signals from the purkinje cells. The purkinje cells and DCN neurons are organized in microcomplexes. Each microzone consists of a parasagital strip of the purkinje neurons that project to the same DCN neurons and a Cf that contacts cells in the complex. The purkinje cells are arranged in a 4 £ 10 matrix, thus defining

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Fig. 14. The synapse weight between the granule layer and the purkinje layer. The numbers on the x-axis indicate the granule cells. The area of the squares indicates the weight between the granule cells and Lh, Rh, Le, Re on the purkinje layer, respectively.

four microcomplexes each having 10 purkinje cells. The model has one DCN neuron per microzone that receives the projections from the purkinje cells in the same complex and all 26 mossy fibers. There are four DCN neurons in the fastigial layer. The outputs of two DCN neurons encode the predictions of amplitude of the eye movements and return to the SC. The outputs of the other two DCN neurons predict the amplitude of head movements in space, and project to the neck motoneurons that control head movement. The Cf originates from the IO and encodes the sensory prediction error which is distributed to the cerebellum, thus it serves as a ‘teacher’ in the supervised learning environment. The learning is postulated to occur at synapses of neurons that contact Cfs using long-term potentiation (LTP) and long-term depression (LTD). In the model, a Cf projects to all purkinje cells and DCN neurons in a microzone. The learning in the cerebellum is postulated to have a close relationship with the firing of the Cfs. When the activity of the IO is below the base firing rate, opposite learning rules are used at the synapses. The normalization of the weights is then achieved by the combination of LTP and LTD rules. The weight update rule is defined as: Dw ¼ 2aInðtÞCfðtÞ ¼ 2aInðtÞðexðtÞ 2 OðtÞÞ where w is the weight between the granule layer and the purkinje layer, In(t) is the input from the mossy fibers, Cf(t) is the activity of the Cf, ex(t) is the expected output, O(t) is the actual output. The basic firing rate of the IO is modeled to be 0. When Cf(t) is negative, it is actually representing the activity below the base firing rate. a is the Purkinje cell learning rate which is selected as 0.04. The training of the model is to learn the pattern to separate the eye movement and the head movement. The memory of such a pattern is stored at the parallel fiber – purkinje synapses during the LTP – LTD training, which is reflected in Fig. 14. The numbers on the x-axis indicate the granule cells. The area of the squares indicates the weights between the granule cells and Lh, Rh, Le, Re on the purkinje layer, respectively. We start training with a preset weight vector of all weights set as 1. Fig. 15 shows the mean square error versus the training cycles. The errors drop quickly at the beginning of training, and gradually approaches 0 after around 1400 training cycles. When the synapse weights between the granule layer and

Fig. 15. The training error curve versus the training cycles. The x-axis is the number of training cycles. The y-axis is the error after each training cycle.

purkinje were fixed, we simulated the head contribution to the gaze shift and the eye amplitude using 80 mossy fibers that encode the input signals of the gaze amplitude for the gaze shifts initiated when the eyes were centered in their orbits (Fig. 16(A) and (B)), deviated contralateral to movement direction by 108 (Fig. 16(C) and (D)), 208 and 308. The results of the simulation of the cerebellum model are close to data obtained physiologically (Freedman & Sparks, 1997a), which is shown in Fig. 17. 3.4. A mathematical model for gaze programming When the head is free to move, the eyes and head can move together to accomplish large redirections of the line of sight (Barnes, 1979). The head and eye movement amplitudes are influenced by the position of the eyes in their orbits (Russo & Bruce, 1993; Van Opstal, Hepp, Suzuki, & Henn, 1995). Freedman and Sparks (1997a) reported their experimental data of gaze shifts, and concluded that it is possible to accurately predict the amplitudes of the eye and head components of gaze shifts if the displacement of the target and the initial position of the eyes in their orbits are known. In the present study, we generated several equations to describe the prediction according to the experimental data reported by Freedman and Sparks (1997a). For simplicity, the following discussion is restricted to horizontal gaze shift only, and neglects the other two rotational degrees of freedom of the eye. It is well know that gaze shift is the sum of the eye-in-head movement and headin-space movement. Considering the effect of the initial position of the eye in its orbit, we have G ¼ EðpÞ þ HðpÞ

ð12Þ

where G is the gaze displacement; p, the initial position of the eye in its orbit; E, eye displacement; H, head contribution. The head contribution to the gaze shift is defined as the amplitude of the head movement that occurs between head movement onset and gaze movement end. The horizontal eye position is chiefly controlled by two extra ocular muscles, the lateral and medial recti. When the eye is still, it rests at the mechanical equilibrium point at

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Fig. 17. The data reported by Freedman and Sparks (1997a). The results of the simulation of the cerebellum model (Fig. 16) close to the data observed physiologically.

Fig. 16. The results of the simulation of the cerebellum model. The head contribution to the gaze shift and the eye amplitude plotted as a function of gaze amplitude for the gaze shifts directed along the horizontal meridian initiated when the eyes were centered in the orbits (A and B), and deviated contralateral to movement direction by 108 (C and D).

which the net torque due to the muscles and passive orbital tissues vanishes. For horizontal gaze shifts with eyes centered and gaze shifts smaller than about 258, experimental data indicate that the head movement does not contribute to the change in gaze position. The 25 mark is called maximum eyemovement-only (MXEO), because within ^ 258 the eyes move freely without head contributions. For a gaze shift larger than about 358, the eye position at the end of the gaze shift does not exceed 358, and the maximal contribution of the eyes is below 408 (Guitton & Volle, 1987). The 35 mark is called the minimum head-movement-only (MIHO) because the head moves only if the eyes initial position in their orbits are at 358 ipsilateral to the direction of the gaze shift. This range of allowed eye-in-head positions is also called the effective oculomotor range (EOMR; Misslisch et al., 1998). The properties of the saturation of the eye position (EONR or MIHO) have been well studied (Guitton & Volle, 1987; Misslisch et al., 1998). However, studies about the properties of MXEO are not abundant in the literature. The MXEO and the MIHO are fixed for a subject even though they may have ^ 58 errors. However, different subjects may have different MXEO and MIHO, depending on their habits, the size of their eye-orbits, and their vision capabilities. Eyes in center. When the eyes are initially centered in

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their orbits, the relationship between the eye amplitude and the gaze amplitude is characterized by two linear functions. When the gaze amplitude G # 258, the eyes move without any head contribution. The eye amplitude is equal to the gaze amplitude, and the slope of the linear relationship between the eye amplitude and the gaze amplitude is equal to 1. E ¼ G;

and

E kE ¼ ¼ 1 when G # 25 G

ð13Þ

where E is the eye amplitude; G, the gaze amplitude; kE ; the slope of the linear relationship between the eye amplitude and the gaze amplitude. Head movement does not contribute to a change in gaze position, and the slope of the linear relationship (ratio) between the head contribution and the gaze amplitude is equal to 0. When the gaze amplitude G . 258, the head begins to contribute to the gaze shift. Assuming a subject has a maximal visual target of 808 (maximal initial retinal error), the head contribution increases linearly with increasing gaze amplitude for movements between 258 and 808. Because the horizontal gaze amplitude is approximately equal to the target displacement amplitude (Freedman & Sparks, 1997b), we describe here the maximal gaze amplitude using the maximal visual target. The maximal gaze amplitude here involves only eyes and head movements, not shoulder or body movements. When the gaze amplitude reaches its maximum of 808, the eye movement amplitude saturates at an amplitude of 358. As the range of the gaze amplitude is from 258 to 808, the slope of the linear relationship between the eye amplitude and the gaze amplitude is kE ¼

eye amplitude 35 2 25 10 ¼ ¼ < 0:18 gaze amplitude 80 2 25 55

ð14Þ

When the gaze amplitude . 358, the head begins to contribute to the gaze shift. When the gaze amplitude ¼ 808, the head reaches its maximum position. The maximum contribution of the head is the difference between the maximum gaze amplitude and the maximum eye amplitude, equal to 80 2 35 ¼ 45: The slope of the linear relationship between head contribution and gaze amplitude is kH ¼

H 80 2 35 45 ¼ ¼ < 0:82 when 25 , G , 80 G 80 2 25 55 ð15Þ

It is clear that kE þ kH ¼

E H G þ ¼ ¼1 G G G

ð16Þ

When the eyes are initially centered in their orbits, the gaze

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shift can be described as G ¼ Eð0Þ þ Hð0Þ ( E when G # 25 ¼ 25 þ kE ðG 2 25Þ þ kH ðG 2 25Þ when 25 , G , 80 ð17Þ The eye amplitude and the head contribution can be described as ( G when G # 25 Eð0Þ ¼ 25 þ kE ð0Þ·ðG 2 25Þ when 25 , G , 80 ð18Þ ( 0 when G # 25 Hð0Þ ¼ kH ð0Þ·ðG 2 25Þ when 25 , G , 80 Here we present the head contribution equation H(0). As a matter of fact, head contribution is only an epiphenomenon of eye amplitude, total head amplitude, and the relative timing of the two platforms. The neural system drives the total head movement, and is careless with the head contribution (Freedman & Sparks, 1997a,b). Contralateral position of the eye. When the eyes begin to deviate in their orbits by 108 contralateral in the direction of the gaze shift, the relationship between the eye amplitude and the gaze amplitude is characterized by two linear functions. When the gaze amplitude G # 358, the eyes move without head contribution. The eye amplitude is equal to gaze amplitude, and the slope of the linear relationship between eye amplitude and gaze amplitude is equal to 1. E ¼ G;

and

kE ¼

E ¼ 1 when G # 35 G

ð19Þ

The head contribution is null, and the slope of the linear relationship between head contribution and gaze amplitude is equal to 0. When the gaze amplitude . 358, the head begins to contribute to the gaze shift. Assume a subject has a maximum gaze displacement of 808. When the gaze amplitude ¼ 808, the eye movement amplitude saturates at 458. Because the range of gaze amplitude is from 358 to 808, the slope of the linear relationship between eye amplitude and gaze amplitude is kE ð210Þ ¼

E 45 2 35 10 ¼ ¼ < 0:22 G 80 2 35 45

ð20Þ

when 35 , G , 80 When the gaze amplitude . 458, the head begins to contribute to the gaze shift. When the gaze amplitude is 808, the head reaches its maximum position. The maximum contribution of the head is the difference between the maximum of gaze amplitude and the maximum of eye amplitude, equal to 80 2 45 ¼ 35: The slope of the linear relationship between head contribution and gaze amplitude

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Table 1 The comparison between the model results and the physiological data Eye Amplitude

T Eq. S

Head Contribution

0

210

220

230

0

210

220

230

0.24 0.18 0.12

0.27 0.22 0.18

0.34 0.29 0.25

0.42 0.40 0.44

0.76 0.82 0.88

0.73 0.78 0.82

0.66 0.71 0.75

0.58 0.60 0.56

is kH ð210Þ ¼

H 80 2 45 35 ¼ ¼ < 0:78 G 80 2 25 55

ð21Þ

when 35 , G , 80 When the eyes begin to deviate in their orbits by 108 contralateral in the direction of the gaze shift, the gaze shift can be described as G ¼ Eð210Þ þ Hð210Þ ( E when G # 35 ¼ 35 þ kE ðG 2 35Þ þ kH ðG 2 35Þ when 35 , G , 80 ð22Þ The eye amplitude and head contribution can be described as ( G when G # 35 Eð210Þ ¼ 35 þ kE ð210Þ·ðG 2 35Þ when 35 , G , 80 ( 0 when G # 35 Hð210Þ ¼ kH ð210Þ·ðG 2 35Þ when 35 , G , 80 ð23Þ Similarly, Eqs. (24) and (25) describe the slope kE and kH when the eyes begin to deviate in their orbits by 208 and 308 contralateral to the direction of the gaze shift, respectively. kE ð220Þ ¼

ð35 þ 20Þ 2 45 10 ¼ ¼ 0:29; 80 2 45 35

ð80 2 45Þ 2 10 25 kH ð220Þ ¼ ¼ ¼ 0:71 80 2 45 35 ð35 þ 30Þ 2 55 10 ¼ ¼ 0:4; kE ð230Þ ¼ 80 2 55 25

ð24Þ

ð25Þ

55 2 30 2 10 15 kH ð230Þ ¼ ¼ ¼ 0:6 55 2 30 25 The results of these equations are close to the observed physiological data (Freedman & Sparks, 1997a), shown in Table 1. In this study, we provide a quantitative analysis for decomposing the gaze displacement signal and propose a mathematical model to predict the displacement of the eye and head components of the gaze shift. We conclude that the displacements of the eye and the head components are

Fig. 18. Weighted averaging of activity at points B and C yields the same movement as activity at the center of the active population A.

determined by five factors: the current gaze displacement, the initial eye position, the upper limit of the range in which the eyes move freely without head contribution, the lower limit of the range in which only the head moves, and the maximal displacement of the visual target (the maximal initial retinal error). The model has the potential for dealing with the gaze decomposition under other conditions, for example, when the initial eye position is ipsilateral to the direction of gaze shift, or when the direction of gaze shift is extended to the vertical or oblique direction. The results of the quantitative analysis may benefit the study of eye velocities and head movement components, for example, the velocity of the eye may be considered as the function of the slope of the linear relationship between eye amplitude and gaze amplitude. 3.5. The formation of the saccade vector It has been assumed that the exact trajectory of a saccade is determined by the activity of the entire population and that the information is not extracted from only the most active cells in the population at a subsequent stage of neural processing (Lee, Rohrer, & Sparks, 1988; McIlmain, 1976; Sparks et al., 1976; Van Gisbergen et al., 1987; Van Opstal & Van Gisbergen, 1989; Vogels, 1990). The trajectory of a saccade could be based on the weighted sum of the simultaneous movement tendencies produced by the activity of a large population of less finely tuned neurons. The contribution of each neuron to the direction and amplitude of the movement is relatively small. The population-averaging model (Fig. 18) assumes that the region of the collicular neurons active before a given saccade occupies a symmetrical area within the motor map in the SC. Only the neurons in the center of the active population discharge maximally before the programmed movement. For each subset of the active neurons (Fig. 18(B)) producing a movement tendency with a direction and amplitude other than the programmed movement, there will be a second subset of active neurons (Fig. 18(C)) producing an opposing movement tendency such that the resultant of the two movements will have the programmed direction and amplitude (Lee et al., 1988; McIlmain, 1976; Sparks et al., 1976; Van Gisbergen et al., 1987). However, the neurons, which are not in the center of the active population, such as subset neurons B and C, have a spike activity less

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then those of the neurons in the center A. We assume the spike activity of the neurons in the active population have a Gaussian distribution. The sum of the spike activity of the active population is: ~ a; rÞÞ þ Oð~cða þ 1808; rÞÞ O ¼ Oð~að0; 0ÞÞ þ W½Oðbð

vectors O and O0 : S ¼ O þ O0 ¼ OðaÞ 2 O0 ðaÞ þ

O ¼ OðaÞ þ

n X

Wj ½Oðaj ; rj Þ þ Oðaj þ 1808; rj Þ

ð28Þ

j¼1

where Oða; rÞ represents the discharge activity of the neurons at point ða; rÞ: The activities of the population of the burst cells determine the direction and amplitude of the eye displacement. The activities of the population depend on the initial eye position in their orbits. We assume that the eye displacement signals that come from the cerebellum affect the whole population of burst cells, not only one cell, and form a vector in quite a different direction to the direction of the position-invariant vector. Eq. (29) describes the vector that comes from the cerebellum: 8 9 n < = X W 0j ½O0 ðaj ; rj Þ þ O0 ðaj þ 1808; rj Þ O0 ¼ 2 O0 ðaÞ þ : ; j¼1 ð29Þ The saccade vector S, which determines the eye displacement of the ensuing saccade, is the sum of

½Wj Oðaj ; rj Þ

2 W 0j O0 ðaj ; rj Þ þ ½Wj Oðaj þ 1808 ; rj Þ 2 W 0j O0 ðaj þ 1808; rj Þ

2 2

where r is the distance between B, C and A. a is the angle of BC and RA. On the left of the Eq. (26), O represents the sum of output spiking activity of the three subsets of neurons A, ~ a; rÞÞ and B and C. On the right of Eq. (26), Oð~að0; 0ÞÞ Oðbð Oð~cða þ 1808; rÞÞ represent the output spiking activity of the subset neurons A, B, and C, respectively. Eq. (27) indicates the Gaussian distribution of the spike activity of the neurons. P is the peak value of the discharge activity of the neurons, and h is a constant, which determines the shape of the Gaussian surface. a~ ð0; 0Þ is the desired vector produced by neurons at point A in the center of the active region where output spiking ~ a; rÞ and ~cða þ 1808; rÞ activity reaches the peak value P. bð are the vectors produced by the subsets of active neurons at points B and C. Point C has an equal distance r from the point A in anatomic space and is symmetrical to point A with point B. The relationship among a~; b~ and ~c in polar coordinates is shown in Fig. 18. Considering more subsets of active neurons, then Eq. (26) can be rewritten as:

n X j¼1

ð26Þ

~ a; rÞÞ ¼ Oð~cða þ 1808; rÞ ¼ Pe2h r Oð~að0; 0ÞÞ ¼ P; Oðbð ð27Þ

825

ð30Þ

We simulated the formation of the saccade vector using MATLAB. The data of the eye displacement signal comes from the cerebellum model. The results suggest that as the eyes begin in an increasingly contralateral position, the activities of the population of the burst cells increase, and the amplitude of the eye component increase, as shown in Fig. 19.

4. Discussion 4.1. Open or closed loop The models of the SC saccade generation system can be essentially grouped into two major classes (Guitton, 1991): 1. Those where the SC signal functions in open-loop and specifies only the initial vector of eye motor error. 2. Those where the SC lies within a feedback loop and specifies the instantaneous vector of eye-motor error. Some people believe that saccadic eye movements are too fast to be visually guided, for example, a saccade of 208 amplitude typically has a duration of 70 ms. Saccades, therefore, are said to be ballistic movements; much as the flight of a golf ball toward its goal which, once started, can no longer be influenced (Becker, 1991). There are some open loop models, such as Scudder –Van Gisbergen model (Scudder, 1988; Van Gisbergen, Van Opstal, & Schoemakers, 1985), the Tweed and Vilis (1987) model. Recently, new models have emerged which assumes that the SC provides instantaneous motor error rather than only initial motor error as assumed by the open-loop models. For example, the Munoz model (Munoz, 1988; Munoz & Wurtz, 1995a,b), the Galiana model (Galiana & Outerbridge, 1984) and the Optican model (Lefevre et al., 1998; Quaia et al., 1998b, 1999) are closed-loop models. Our SC –cerebellum model is a closed-loop model. The feedback loop includes the SC, cerebellum, and brainstem. The population of burst cells provide the desired direction and amplitude of the impending saccade. The stronger the burst cell’s output, the faster the saccade, but the correct amplitude is assured by feedback control. The saccadic vector in the motor map is the sum of a collicular vector and a vector that comes from the cerebellum. The size of the collicular vector depends on the position of the target in the retina, while the vector coming from the cerebellum involves the initial position of the eye relative to head.

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Fig. 19. Simulation for the gaze saccade system. The height of the hill in the upper graph of A, B, C, D indicates the saccadic vectors in the SC. The position of the collicular cells at the caudal side of the SC corresponds to the target stimuli position in the retina. The lower graph indicates the initial position of the eye in its orbit. These results suggest that when the eyes begin in an increasingly contralateral position the activities of the population of the burst cells increase, and the amplitude of the eye component increases.

The size of the collicular vector tends to be maximum for large eccentricity of the target. The function of the vector coming from the cerebellum is to reduce the size of the collicular vector. Therefore, a lesion of the cerebellum causes overshooting of the saccade, which also occurred in the simulation that we did. 4.2. SC or cerebellum ablation does not abolish eye saccades Cerebellum lesions induce permanent deficits, affecting dramatically the accuracy and consistency of saccades (Optican & Robinson, 1980; Zee et al., 1981).

The fact that saccades can still be generated following SC ablation indicates that at least one other area of the brain is responsible for generating visually guided saccades. Some work has revealed one type of saccade that the SC is uniquely responsible for (Schiller et al., 1987); ablation of the SC permanently prevents express saccades, which are short latency (70 ms) saccades to predict visual targets (Fischer & Boch, 1983; Boch et al., 1984; Boch & Fischer, 1986; Mayfrank et al., 1986). Some results show that two parallel pathways mediate visually guided eye movements, one depending on the SC and the other on the FEF (Keating et al., 1983; Schiller, 1985; Keating & Gooley, 1988). The FEF pathway depends

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on input from the striate cortex (Mohler & Wurts, 1977). The results of ablation studies indicate that subjects can still generate saccades following FEF lesions. Combined, however, bilateral lesions of the FEF and SC result in a permanent loss of saccades (Schiller et al., 1979, 1980). A reasonable working hypothesis is that the SC pathway is responsible for reflexive, orienting saccades, and the FEF pathway is responsible for directing visual attention, the more voluntary saccades (Schiller et al., 1987). Recently Hanes and Wurtz (2001) studied the interaction of the FEF and SC for saccade generation. Their research lead the conclusion that in the non-ablated monkey the direct FEF – brain stem pathway is not functionally sufficient to generate accurate saccades in the absence of the indirect pathway that courses from the FEF through the SC to the brain stem circuitry. They suggested that the recovery of function following SC ablation that has been seen in previous studies must result not from the use of an already functioning parallel pathway but from neural plasticity within the saccadic system. In our model, the cerebellum performs the functions for gaze saccade generation. The cerebellum provides a headcentered coordinate, encodes the initial position of the eye relative to the head, and receives a signal that comes from the SC and encodes the position of the target in the retina. The cerebellum predicts the eye and head movements in a gaze saccade, and sends both the eye and head movement signals to the SC and neck motoneurons (as the cerebellum connects with neck motoneurons; De Zeeuw & Koekkoek, 1997), respectively. 4.3. Interpretations for the end of a saccade There are several interpretations for the end of a saccade by the physiological data. For example, one is the SC pathway hypothesis (Munoz & Wurtz, 1995a,b; Scudder, 1988). When the fovea reaches the target, and the dynamic motor error is reduced to zero, the activity of the burst cells then decays rapidly. Burst cells withdraw their inhibition from the fixation cells. The fixation cells activate OPN. This leads to inhibition of MLBN and cessation of saccade signal to the oculo motoneurons, and the eyes stop moving. Another is the cerebellum pathway hypothesis from Optican group (Lefevre et al., 1998; Quaia et al., 1999). They assume the fastigial neurons burst twice: an early burst and a late burst. The early burst of the fastigial neuron inputs to the MLBN, combines with the input of SC neurons, to determine the initial direction and speed of the saccade. In contrast to the early burst observed for saccades in the preferred direction, a late burst is produced in correspondence with saccades in the opposite direction. The Optican group proposed that this late burst is generated by the cerebellum to actually end the saccade when the eyes are approaching the target (Lefevre et al., 1998; Quaia et al., 1999). Our model considers both the SC pathway and the

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cerebellum pathway. The model of the termination of a gaze shift is shown in Fig. 20. We assume the cerebellum predicts the desired eye amplitude E p and the desired head amplitude H p for a large saccadic gaze shift. The signal E p goes back to the burst layer in the SC and forms the saccade vectors in the motor map. The desired head amplitude signal H p goes to the brainstem and is carried by reticulospinal and vestibulospinal neurons. Vestibular nuclei (VN) may connect the neck motoneurons which controls the head movement. We assume there are two feedback loops. One is between burst cell and oculo motoneurons. The burst cells will discharge until the motor signal is reduced to zero. The current eye position E0 is derived by the tonic oculo motoneurons that integrate input from the burst cells. This current eye position E0 is fed back to the burst cells as one component of the motor error. The excitatory burst cells are inhibited by the signal of the current eye position. We assume that the vestibulo-ocular reflex (VOR) is inhibited by the difference between current eye position and desired eye amplitude, Ep 2 E0 : The VOR is switched off when the burst cell bursts. However, when the eye is on target, Ep 2 E0 ¼ 0; the VOR is switched on, and any additional head motion is compensated for by VOR. Another feedback loop may be between VN and neck motoneurons. The distance of current head movement (or the angle of the neck rotation) H0 is derived by the neck motoneurons, and is fed back to VN. VOR may be suppressed or even completely inactivated until the eye is on target. The active head movement H0 is compared with the desired head amplitude H p that is predicted by the cerebellum. There is a tendency for head movement to stop when the difference of H p and H0 is close to zero. However, the final stop of the head movement depends on the compensated function of the VOR. 4.4. The effects of a lesion of the cerebellum Quaia et al. (1998a,b) showed that muscimol injections into the intermediate layers of the SC alter the trajectory of movement and confirmed previously reported effects on latency, amplitude, and speed of saccades. They assume that another system, acting in parallel with the SC, contributes to the determination of saccadic trajectory. Our model provided this parallel system that encoded the signal from the cerebellum. Lesions of the cerebellum results in deficits in accuracy of saccades (Lefevre et al., 1998; Stein & Meredith, 1991). The reason may be interpreted as the cerebellum not providing the vector of eye movement to compose the saccade vector in the motor map of the SC. In the normal case, a saccadic vector OA is the sum of a collicular vector OB and a vector OC from the cerebellum, as shown in Fig. 21(a). The vector OC may have a quite different direction to OB (Opstal et al., 1995). O is the starting point for the saccade; the square T is the final eye position. After a cerebellum lesion, however, the vector OC disappears. The

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Fig. 20. The cerebellum predicts the desired eye amplitude E p and the desired head amplitude H p. The signal E p goes back to the burst layer in the SC, and forms the saccade vectors in the motor map. The desired head amplitude signal H p goes to the brainstem. Vestibular nuclei (VN) may connect the neck motoneurons which controls the head movement. There are two feedback loops. One is between burst cell and oculo motoneurons. The other may be between VN and neck motoneurons. We assume that the vestibulo-ocular reflex (VOR) is inhibited by the difference between current eye position E0 and desired eye amplitude, E p 2 E0 . The VOR is switched off when the burst cell bursts. VOR is suppressed or even completely inactivated until the eye is on target. When Ep 2 E0 ¼ 0; the eye is on target, the VOR is switched on. The final stop of the head movement depends upon the compensated function of the VOR. EP, initial eye position; GD, gaze displacement; T/R, target-re-retina; E/H, eye-re-head; H/S, head-re-space; solid line, excitatory; dashed line, inhibitory.

retinal vector OB makes the saccade overshoot, and the eye reaches the position T0 that is different from the correct position T. The results of a simulation are shown in Fig. 21(b) and (c).

5. Conclusion In this paper, we proposed a SC model and a cerebellum model. Both the models compose a gaze generation system. To confirm the mechanism of the model, we made three

simulations: gaze displacement decomposed within the cerebellum, formation of a saccade vector within the SC, and the affects of lesions on the cerebellum. The results of the simulations show that the behavior of the model is similar to behaviors observed physiologically. The present study suggests that the buildup cells project the gaze displacement signal to the cerebellum and the burst cells form a saccade vector with the eye displacement signal coming from the cerebellum while the gaze displacement signal is decomposed into eye and head contribution within the cerebellum. We discussed the coordinate transformation

Fig. 21. A saccadic vector OA is the sum of a collicular vector OB and a vector OC from the cerebellum. O is the starting point for the saccade. The square T is the final eye position in the normal case. The square T0 is the eye position after a cerebellum lesion. In this case, vector OC disappears, the retinal vector OB makes the saccade overshoot. (b) Before the cerebellum lesion, the activity of the burst cell in the caudal part of SC is normal. (c) After the cerebellum lesion, the activity of the burst cell increases.

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concerned with the eye position in its orbit, and proposed several equations to describe the gaze displacement decomposition.

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% feed through network granule ¼ zeros(1,10);

This research has been supported in part by a grant from the Australian Research Council. We are grateful to E.G. Freedman for helpful comments in part on drafts of the manuscript.

for i ¼ 1:2; for k ¼ 1:5; if ((inputs(i) . 0)&(inputs(k þ 2) . 0)) granule(k þ ((i 2 1) p 5)) ¼ tanh((inputs(i) þ 0.0 p inputs(k þ 2))); end end end

Appendix A. The programming used for the simulation of the cerebellum model using the simulator MATLAB

mse ¼ 0; EXP_DCN ¼ fscanf(input_file,‘%f’,[1 4]); fprintf(output_file,‘%f’,inputs); fprintf(output_file,‘%f’, EXP_DCN);

Acknowledgments

function [ ] ¼ cerebellum( ) % initialization inputs ¼ zeros(1,7); granule ¼ zeros(1,10); PF ¼ ones(1,10,4); Purkinje ¼ zeros(1,4); CF ¼ zeros(1,4); DCN ¼ zeros(1,4); EXP_DCN ¼ zeros(1,4); DCN_Weights ¼ ones(1,8,4) p 0.4; for i ¼ 1:4 DCN_Weights(:,:,i) ¼ DCN_Weights(:,:,i) þ [0.6 0 0 0 0 0 0 0]; end PF_lrate ¼ 0.2; DCN_lrate ¼ 0.05; mse_array ¼ 0; % file setup input_file ¼ fopen(‘input_data1.txt’,‘r’); output_file ¼ fopen(‘output_data.txt’,‘w’); mse_file ¼ fopen(‘mse.txt’,‘w’); mse ¼ 1; % read input iteration ¼ 0 while iteration , 3000 iteration ¼ iteration þ 1; if feof(input_file) frewind(input_file); end inputs ¼ fscanf(input_file,‘%f’,[1 7]);

for i ¼ 1:4; temp ¼ PF(:,:,i); Purkinje(i) ¼ tanh(sum(temp. p granule,2)); temp_k ¼ zeros(1,8); temp_k(1) ¼ 2 Purkinje(i); for k ¼ 1:7; temp_k(k þ 1) ¼ inputs(k); end DCN(i) ¼ tanh(sum(temp_k. p DCN_Weights(:,:,i),2)); if DCN(i) , 0 DCN(i) ¼ 0; end % fprintf(output_file,‘%f’,DCN); % fprintf(output_file,‘%f’,EXP_DCN); % fprintf(output_file,‘\n’); CF(i) ¼ EXP_DCN(i) 2 DCN(i); PF(:,:,i) ¼ PF(:,:,i) 2 granule p PF_lrate p CF(i); T ¼ PF(:,:,i); for k ¼ 1:10 if T(k) , 0; T(k) ¼ 0; end end PF(:,:,i) ¼ T; % weight update DCN_Weights(:,:,i) ¼ DCN_Weights(:,:,i) þ temp_k p DCN_lrate p CF(i); mse ¼ mse þ CF(i)^2; end %mse ¼ mse/i; mse_array ¼ [mse_array; mse]; fprintf(mse_file,‘%f \n’,mse); fprintf(output_file,‘%f’,DCN); fprintf(output_file,‘\n’);

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% DCN % EXP_DCN % mse end weight_file ¼ fopen(‘weight.txt’,‘w’); for i ¼ 1:4 w ¼ PF(:,:,i) fprintf(weight_file,‘%f’,w); fprintf(weight_file,‘\n’); end w1 ¼ [PF(:,:,1);PF(:,:,2);PF(:,:,3);PF(:,:,4)]; subplot(2,1,1); hintonw(w1) subplot(2,1,2); plot(mse_array); % finishup fclose(weight_file) fclose(input_file) fclose(output_file) fclose(mse_file)

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