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Previous Article | Next Article
Volume 17, Number 24, Issue of December 15, 1997
A Neural Model of Multimodal Adaptive Saccadic Eye Movement Control by Superior Colliculus
Stephen Grossberg, Karen Roberts, Mario Aguilar, and Daniel Bullock Department of Cognitive and Neural Systems and Center for Adaptive Systems, Boston University, Boston, Massachusetts 02215
ABSTRACT
How does the saccadic movement system select a target when visual, auditory, and planned movement commands differ? How do retinal, head-centered, and motor error coordinates interact during the selection process? Recent data on superior colliculus (SC) reveal a spreading wave of activation across buildup cells the peak activity of which covaries with the current gaze error. In contrast, the locus of peak activity remains constant at burst cells, whereas their activity level decays with residual gaze error. A neural model answers these questions and simulates burst and buildup responses in visual, overlap, memory, and gap tasks. The model also simulates data on multimodal enhancement and suppression of activity in the deeper SC layers and suggests a functional role for NMDA receptors in this region. In particular, the model suggests how auditory and planned saccadic target positions become aligned and compete with visually reactive target positions to select a movement command. For this to occur, a transformation between auditory and planned head-centered representations and a retinotopic target representation is learned. Burst cells in the model generate teaching signals to the spreading wave layer. Spreading waves are produced by corollary discharges that render planned and visually reactive targets dimensionally consistent and enable them to compete for attention to generate a movement command in motor error coordinates. The attentional selection process also helps to stabilize the map-learning process. The model functionally interprets cells in the superior colliculus, frontal eye field, parietal cortex, mesencephalic reticular formation, paramedian pontine reticular formation, and substantia nigra pars reticulata.
Key words: saccades; eye movements; superior colliculus; attention; learning; burst neurons; buildup neurons; parietal cortex; frontal eye fields; reticular formation; substantia nigra
INTRODUCTION
Saccades are ballistic eye movements that facilitate the survival of an animal in a rapidly changing environment. Saccadic eye movements include at least three types: visually reactive, multimodal (e.g., auditorily cued), and planned. Visually reactive saccades are reflexive movements generated by areas of rapid visual change. Auditory saccades direct the eyes toward acoustic stimuli. Planned saccades move the eye to intended targets; they "direct the eye at objects selected beforehand from the visual environment" (Becker, 1989
). Such eye movements can be made to targets that may or may not be visible when an eye movement begins.
The saccadic system can execute a movement to a planned target without being distracted by irrelevant reactive targets. However, when intense visual or auditory stimuli appear, they can take precedence over a planned movement. How and where is the shared control between visually reactive, auditory, and planned saccades achieved? In particular, visual cues are registered in retinotopic coordinates, whereas auditory cues are registered in head-centered coordinates. How are these distinct coordinate systems merged so that a particular location can be selected as a saccadic target? This transformation must be learned to align the corresponding visual and auditory representations.
The present article develops a model of how these adaptive coordinate changes take place in the deeper layers of the superior colliculus (SC) and enable a saccadic movement target to be chosen there. The neural circuitry that supports this learning process simulates neurophysiological data about the burst neurons (Waitzman et al., 1991
MATERIALS AND METHODS
To set the stage for these analyses, we summarize in this section brain regions that converge onto the SC, which in turn projects to gaze control centers. Key SC neuron types are then surveyed that influence saccadic control, and the model is introduced. In the Results, we show how the model can control multimodal, planned, and visually reactive saccades via a process of SC map learning. The model functionally interprets known anatomical connections between cells in the superior colliculus, frontal eye field, parietal cortex, mesencephalic reticular formation, paramedian pontine reticular formation, and substantia nigra pars reticulata. The model is then used to simulate physiological data of SC burst and buildup neurons during a variety of behavioral tasks and to shed light on the following types of data.
The role of the superior colliculus in the saccadic eye movement system. Saccades are mediated by pontine and mesencephalic burst circuits that are usually controlled by the superior colliculus (Goldberg et al., 1991
The deeper layers of the superior colliculus contain heterogeneous cell sizes and an intermingling of cells and axons with a large degree of overlap of dendritic fields (Grantyn, 1988
Many studies include data suggesting that the deeper layers of superior colliculus are involved in the control of visually reactive, auditory, and planned saccades (Powell and Hatton, 1969
Sensory afferents. By contrast with ascending visual input to superficial SC layers, ascending projections to deeper SC layers provide a limited distribution of contralateral visual inputs from retina, the lateral geniculate nucleus, pretectum, and superficial superior colliculus. Much heavier ascending inputs originate from contralateral auditory sources. These sources include projections from the periolivary regions of the superior olive, the nuclei of the trapezoidal body, the ventral nucleus of the lateral lemniscus, and, to a lesser extent, the external nucleus of the inferior colliculus (Edwards et al., 1979
Motor afferents. Motor afferents arise from numerous, primarily ipsilateral, sources. The posterior parietal cortex projection to the superior colliculus is primarily to the deep intermediate layers. Cells from the lateral intraparietal area of posterior parietal cortex fire before saccades and indicate the intended amplitude and direction of eye movement in motor coordinates (Gnadt and Andersen, 1988
The cortical inputs to the superior colliculus also include heavy descending projections from the frontal eye field. Anterograde tracing from the frontal eye field demonstrates a predominant projection to intermittent patches in the deep regions of the intermediate gray layer. Physiological responses of neurons in the frontal eye field generally exhibit pre- or postsaccadic activity (Goldberg and Bruce, 1990
Another important structure involved in the oculomotor system is therefore the basal ganglia that include the caudate and substantia nigra. Most substantia nigra cells need the presence of a visual target to pause before eye movements (Hikosaka and Wurtz, 1983b
There is no proprioceptive feedback from extraocular muscle receptors to signal directly eye position to deeper layer neurons (Stein and Meredith, 1993
Efferents. Four efferent pathways project from the superior colliculus. One pathway is ascending and reaches the thalamus, and one projects to the opposite superior colliculus. The descending efferents are involved in repositioning the eyes, head, limbs, and other peripheral sensory organs via a contralateral and an ipsilateral pathway. Although a majority of neurons without efferent projections are unimodal (Meredith and Stein, 1986
Efferent neurons of the superior colliculus convey motor commands using widespread connections with neurons in other motor areas of the brainstem and spinal cord (Moschovakis et al., 1988a
The intermediate and deep layers of the superior colliculus project to the brainstem, providing motor commands to the region that controls eye movements (Sparks and Hartwich-Young, 1989
Intrinsic neurons in the deeper layers of the superior colliculus, bursts and spreading waves. Among the efferent neurons of the superior colliculus that convey motor commands to the brainstem and spinal cord, two distinct neural activity patterns can be found, a burst and a buildup of activity. During saccades, the peak neural activity in a population of saccade-related burst neurons in the superior colliculus remains in a fixed position, and the activity level rapidly increases immediately before eye movement. Then the activity peak decays as a function of the remaining gaze error (Waitzman et al., 1991
Intracellular staining studies of alert monkeys have been used to identify collicular neurons with presaccadic activity during spontaneous eye movements with a fixed head (Moschovakis et al., 1988b 
Fig. 1. The neural activity of a burst cell (left) and a buildup cell (right) during a saccade. Each panel shows the individual rasters (top), the spike density profile (middle), and the horizontal eye position trace (bottom). Reprinted with permission from Munoz and Wurtz (1995a)
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Neurons in the colliculus that displayed activity that ceased at or near the end of a saccade were studied by Waitzman et al. (1991)
Munoz et al. (1991)
Munoz et al. (1991)
These observations suggest that the activity of tectoreticulospinal neurons reflects the change in gaze motor error (Guitton and Munoz, 1985
Munoz and Wurtz (1993)
A model of multimodal adaptive saccadic control. The neural model presented in this section, which was first reported in Roberts et al. (1994)
The model proposes the following neural mechanisms. Early in development, visual cues trigger saccades via a visually reactive saccadic system. These reactive movements are not necessarily accurate at first. The model proposes that visual error signals are caused by inaccurate foveations and trigger a learning process through which movement gains change adaptively until accurate foveations of visually reactive movements are achieved (Grossberg and Kuperstein, 1986
In particular, when auditory or planned movement vectors represent the same position as a visual target, then the former vectors learn how to map onto the SC motor error locations that represent visually reactive movement vectors. These various movement vectors can then compete when not in agreement to select winning cells the activity of which generates a focus of attention. Competition also helps to stabilize the map-learning process by suppressing all but the winning vectors, so that the learned map is not eroded by the massive flux of multiple possible target locations and the corresponding teaching signals.
Retinotopic visually reactive saccade system. Initially, saccades are executed reactively to targets defined by changing visual signals that are registered on the retina. These retinotopic signals map topographically in a natural way into a motor error map (Grossberg and Kuperstein, 1986 
Fig. 2. a, An adaptive gain (AG) stage that is used to calibrate visually reactive saccades. SG, Saccade generator. b, A corollary discharge coding current eye position that is added to the position of the target to define a head-centered spatial map. c, A head-centered representation of motor error that is produced by learning the transformation of the head-centered target representation and the final eye position. The motor error reflecting the head-centered target is then converted from a vector to a spatial map so that it can be combined with a motor error signal indicating the reactive target. d, Corticotectal map learning.
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The locus of activation in such a motor error map codes the direction and amplitude (or length) of a saccadic movement. Such an encoding is not the same thing, however, as generating an accurately calibrated saccade (e.g., see Stanford and Sparks, 1994
Table 1. Anatomical properties of cells used in the model
Cell type Property Peak decay (burst cells) Located in intermediate SC Efferent projections to: Mesencephalic reticular formation, Deeper and superficial SC neurons, Predorsal bundle Receive inhibitory input from SNr Spreading wave (build up cells) Located in intermediate layers of SC Collateral connections Efferent projections to: Mesencephalic reticular formation, Predorsal bundle Receive excitatory input from cortex Receive inhibitory input from SNr except at rostral pole Mesencephalic reticular formation Inhibitory projections to contralateral and ipsilateral SC
This spatial to temporal conversion is calibrated via a side path containing an adaptive gain stage. Early in development, if retinal error exists after a visually reactive saccade, the error is used to modify an adaptively weighted connection to the adaptive gain stage, the anatomy and neurophysiology of which model aspects of cerebellar learning (Ito, 1984
A further refinement requires an analysis of how different combinations of auditory, visually attentive, and planned eye movements, which are controlled by the parietal and prefrontal cortices, get calibrated by mechanisms downstream and parallel to the SC. Gancarz and Grossberg (1997)
Deciding among visual, auditory, and planned saccades. In addition to retinotopic coordinates, visual targets can also be stored in head-centered coordinates. For example, when a visual saccadic target must be stored in memory, intervening eye movements could render the stored target inaccurate if the location remained in retinotopic coordinates before eye movement.
Accurate visually reactive movements create a stable dynamical substrate on which a head-centered spatial map of invariant target location can form. A visual target can be converted from a retinotopic to a head-centered signal by adding the current eye position (Robinson, 1973
These planned eye movement targets share the efferent neurons of the colliculus with reactive saccade targets. However, visually reactive cells encode gaze error in a retinotopically activated motor map. Auditory and planned targets are coded in head-centered coordinates. Targets in head-centered coordinates must be adaptively mapped to a gaze motor error in retinotopic coordinates to access the correct efferent zones in the SC.
The transformation takes place in the model in three steps. First, the transformation between a head-centered target position and a motor error vector (viz., the direction and amplitude of the desired eye movement) is learned. This transformation is learned by computing the difference between the head-centered target position and the final eye position after a reactive movement terminates (see Fig. 2c). This computed difference is a motor error vector. Because reactive movements are rendered accurate by cerebellar learning, the final eye position is the target position after such a movement. In other words, the motor error vector between the stored head-centered target position and the final eye position should equal zero. Learning of the transformation is thus accomplished by a process that reduces the error vector to zero (Grossberg and Kuperstein, 1986
Such a transformation can be learned in response to any number of head-centered maps, including auditory, visually attentive, and planned movement maps. In each case, when a new target is instated and read out at the motor error vector cells of its map, the present eye position is subtracted from it. This difference codes the desired movement to the new target. Thus the motor error vector cells not only control a learned coordinate change but also compute movement vectors.
Groh and Sparks (1992)
Further experiments are needed to determine whether these representations take the form of motor vectors and/or the motor error maps that the model also invokes; namely, the second step of the model converts these motor vectors into locations on a topographic map, which is called the motor error map (Fig. 2c). This step transforms large activity levels in the motor vector code to caudal positions in the topographic map and small activity levels to rostral positions (Grossberg and Kuperstein, 1986
The third step is a learned transformation from the maps of the auditory, visually attentive, and planned motor errors to the map of visually reactive motor errors at the buildup cell or SW layer (Fig. 2d). This transformation renders the initially visually reactive map also sensitive to multimodal and planned targets. For example, it is proposed to be the means whereby frontal eye field (Schlag-Rey et al., 1992
Retinotopic and head-centered coordinate system alignment. The various maps from head-centered to motor error coordinates are learned by associating two representations of the same target position in space. For example, after the visually activated head-centered parietal map forms, a visual target can activate both the peak decay layer in the visually reactive pathway and the head-centered maps in the parietal and prefrontal cortices (Fig. 2d). These pathways are associated by transforming the targets in head-centered coordinates into a gaze motor error map the output signals of which converge on the spreading wave layer, in which they use associative learning to become adaptively aligned with the visually reactive map. From this perspective, visually driven output signals from the peak decay, or burst cell, layer define teaching signals to the spreading wave, or buildup cell, layer at which unimodal inputs converge from multiple cortical loci (Fig. 2d). This is the central reason in the model why both cell types exist. Learning enables the motor error vectors from these unimodal inputs to map onto the correct locations within the spreading wave layer using the teaching signals from the burst cell layer as aligning cues.
In response to each active burst cell location, a Gaussian teaching function across position is sent to the buildup cell layer (Fig. 2d). This teaching signal enables maximal learning to occur at the peak of the Gaussian, whereas less learning occurs farther away. Each error vector is hereby associated with a population of SC cells, with the most active cell situated at a map location that best codes the correct saccadic direction and amplitude. Such distributed population learning has several functional roles. First and foremost, it enables new target locations that have not been practiced during development to generate accurate saccades by using the Gaussian to interpolate locations that have been practiced. Secondary consequences are that an SC population code determines saccadic movements (Sparks and Nelson, 1987
Map learning takes place when a visual cue onset is coded by both the head-centered and visually reactive pathways. Consistent simultaneous activity in both pathways allows the location in the cortical error map that is activated by the head-centered representation to sample the position that correctly codes the desired eye movement in the reactive pathway on a number of learning trials. Activation of mismatched locations by discordant cues are statistically uncorrelated and get washed out by competitive interactions across the layers. In this way, the planned target is adaptively transformed from a head-centered representation to a gaze motor error vector and then to a gaze motor error map that is adaptively aligned with the map of the gaze motor error in the visually reactive pathway. The error map in this layer resembles the directional maps described for deep layers of the superior colliculus (Sparks and Mays, 1980
Spreading waves, peak decay, and map learning. Why does map learning produce a system characterized by a spread of activity across the buildup cell layer during saccades? When a gaze motor error signal is sent to the saccade generator (Fig. 3), an eye movement begins. As the movement progresses, the motor error vector decreases due to the negative feedback from the eye position corollary discharge. The declining error vector excites a series of loci on the cortical error map. As a result, the commanded location at the buildup cell layer shifts across the map. The buildup cell layer thus exhibits a spreading wave as an emergent property of the circuit that makes auditory, planned, and visually reactive commands dimensionally consistent so that they can be adaptively mapped into one another. The spreading wave results from continuous updating of the adaptive motor error map as the movement progresses. 
Fig. 3. Emergence of the spreading wave from the learned map using corollary discharges.
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This hypothesis helps to explain why the buildup of activity across the SC is so broad. Each new error vector maps into a new Gaussianly distributed location at the SW layer, and the cells of this layer take a while for their activity to decay. In addition, signals from the decaying activity of the burst cells cause a residual secondary peak of buildup cell activity to gradually decay. These properties are simulated below to fit SC data.
Our present interpretation of the process by which error vectors are updated as the saccade unfolds uses feedback loops that include the parietal and frontal cortices. There is, however, no logical requirement that would make it impossible for such a feedback loop to also be closed using noncortical sites. If the cortical loops are the only ones that exist, then lesions of all the appropriate parietal and prefrontal representations should eliminate the spreading wave but not the ability to generate saccades directly via burst and buildup activities that would not spread toward the fixation cells during a saccade. Moreover, note that the cortical input to the colliculus is excitatory, not inhibitory. Thus cutting this input would not have the same effect as cutting inhibitory feedback in a classical negative feedback loop.
Topographically organized excitatory feedback from the spreading wave to the peak decay layer allows a resonant activation to occur between corresponding locations in the two layers. Because resonant activation drives the map-learning process, it is critical to restrict its spatial locus (Fig. 4a). A nonspecific inhibitory signal from the spreading wave layer, proposed to be mediated by the mesencephalic reticular formation (Edwards and de Olmos, 1976 
Fig. 4. a, Resonance of visually reactive position during a saccade. b, Blocking of irrelevant targets during a planned saccade. Open circles are sources of inhibition. For simplicity, the motor error vector and map are combined.
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Rostral migration of activity in the spreading wave layer from its original location erodes feedback excitation to the burst cell map at which visually reactive targets are stored. The eroding excitatory input thus leads to decay of the fixed peak of activity because the error in foveating the target decreases. That is why this spatial map is referred to as the PD layer.
Reconciling auditory, planned, and reactive saccade targets. Auditory, planned, and visual targets compete for attention (Kowler et al., 1995
Two sources of inhibition suppress irrelevant visually activated target locations (Fig. 4b). One source is the nonspecific inhibition discussed in the previous section. The second inhibitory source is interrupted during an eye movement at both the spreading wave and peak decay layers. At the spreading wave layer, this latter source of inhibition is eliminated when the target is presented to allow activity to build at this layer. At the peak decay layer, it is released at the location of the chosen target gaze error. Inhibition remains to other cells in the peak decay layer, therefore preventing their activation when the spreading wave activity shifts across the map.
Anatomical and neurophysiological SC correlates and sites of attentional target selection. The connections of cells in the model, summarized in Figure 5a, closely correlate with known anatomy and neurophysiology of the superior colliculus. Figure 5b depicts this correspondence to anatomical data in the superior colliculus. These connections are also summarized in Table 1, which relates anatomical evidence to different cell types in the model (Hikosaka and Wurtz, 1983a 
Fig. 5. a, Schematic representation of the model of local interactions in the superior colliculus. b, Anatomical correlates of the model depicted in a. SNr, Substantia nigra; PPRF, paramedian pontine reticular formation; MRF, mesencephalic reticular formation.
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Attentional selection of a saccadic target may be progressively elaborated in several brain regions. For example, it is known that movement commands in the parietal cortex are attentively modulated (Mountcastle et al., 1981
It is also known, however, that planned saccades can be made when the SC is lesioned (Schiller and Sandell, 1983 
Fig. 8. The pattern of recorded activity across buildup cells in monkey superior colliculus (left) (reprinted with permission from Munoz and Wurtz, 1995b
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Fig. 10. (a) Burst and (b) buildup cell responses (reprinted with permission from Munoz and Wurtz, 1995a
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Realistic simulations of the physiological response properties of both burst and buildup cells in five different saccade paradigms are simulated in the Results. The mathematical equations and parameters of the model are summarized in Appendix.
RESULTS
Map-learning simulations
All simulations used a fourth order Runge-Kutta algorithm with a fixed step size of 0.0025. The first simulations demonstrate how the multimodal map is learned. The adaptive weight zij from the ith cell in the spatial error map to the jth cell in the PD layer grew if their activities Xi and Pj were simultaneously large, where j is the cell corresponding to the initial gaze motor error. For a saccade encoded by an initial gaze motor error at cell 12, the weights z12j from cell 12 in the planned motor error map to cells near a j of 12 in the SW layer also grow because of the Gaussian spread of the PD activity that is input to the SW layer. This spread was initially very broad and covered over half of the SW layer (Fig. 6a). PkGk-j in Figure 6a shows the spatial width of the teaching signal from the kth PD cell to the jth SW cell, where Pk is the activity of the kth PD cell and Gk-j is the strength of the Gaussian filter connection to the jth SW cell.
Fig. 6. a, The activity across the spatial error map (Xi) and the Gaussian spread of PD activity to the SW layer. b, The resulting adaptive weights (zij) after 1000 randomly distributed target presentations.
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Learning was performed during 1000 randomly generated saccades. All of the adaptive weights zij were initialized to 1.0, and the reactive input Rk to the PD layer was set equal to 200. All of the weights zij that resulted from training during these 1000 saccades are shown in Figure 6b. These weights were used to generate activity at the SW layer during saccades, with each saccade made to a target at a different gaze motor error. The activity profile at the SW layer for each different saccade is shown in Figure 7. The SW activity is shown at a specific time after each target presentation but before eye movement has started. The vertical lines correspond to the location in which the peak decay activity was found. In this figure, the maximally active cell at the SW layer corresponds to the location of the peak decay activity. This correspondence indicates that the learned weights provide an accurate mapping from the spatial error map to the SW layer. 
Fig. 7. The weights zij from Figure 6 used to show the activity across the SW layer at a time after the target is presented at different gaze errors but before eye movement begins. The maximum activity in the SW layer is at the same location as the PD layer activity (shown by the vertical lines), implying that an accurate mapping was learned.
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Burst and buildup cell simulations
The next simulations are of the time course of activation of burst and buildup cells during a saccade, using the adaptive weights learned above (Munoz and Wurtz, 1995b
Several factors complicate the distribution of activity at the SW layer. One factor is that the input to the SW layer is spatially distributed even in response to a fixed motor error (Figs. 2d, 3). A second factor is that activity builds up and decays in response to the input to SW cells at a finite rate, even as the dynamic motor error that causes it is changing. Finally, as the activity moves across the SW layer, excitatory input to the active location at the PD layer erodes. This decrease in excitatory input causes the activity at the PD layer to decrease (Fig. 9). A secondary peak of activity in the SW layer near the location of the initial motor error is visible, even as the saccade ends, because of residual input from the PD layer (Fig. 8). 
Fig. 9. The pattern of recorded activity across burst cells in monkey superior colliculus (left) (reprinted with permission from Munoz and Wurtz, 1995b
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The release from inhibition by the substantia nigra at the PD layer, together with the increasing SW activity input, causes a rapid increase in the activity at the PD layer before eye movement, followed by activity the declining amplitude of which, at the same location, covaries with residual motor error (Fig. 9).
Next we simulate the dynamics of burst and buildup cell activities during visual, overlap, memory, and gap tasks (Munoz and Wurtz, 1995a
The simulation results for each of the four saccade paradigms are displayed below (see Figs. 10, 11, 12, 13). In each of the figures, there are data comparing the physiological responses of a burst cell (top left) and a buildup cell (bottom left) with the simulated responses of a PD cell (top right) and a SW cell (bottom right). Above both the biological and simulated cell responses are two time lines that indicate the status (on or off) of the fixation point and the external target stimulus. Below each graphed cell response is a line indicating the current eye position throughout the simulation. 
Fig. 11. (a) Burst and (b) buildup cell responses (reprinted with permission from Munoz and Wurtz, 1995a
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Fig. 12. (a) Burst and (b) buildup cell responses (reprinted with permission from Munoz and Wurtz, 1995a
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Fig. 13. (a) Burst and (b) buildup cell responses (reprinted with permission from Munoz and Wurtz, 1995a
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Visually guided paradigm simulation
In the visually guided saccade simulation, shown in Figure 10, fixation point offset coincides with target presentation, at which point eye movement begins. At the SW layer, only rostral pole fixation cells (not plotted) are active while the fixation point remains on. At fixation point offset and target onset, the fixation cell activity decays, whereas a hill pattern builds up across the orientation buildup cells. The peak of the hill corresponds to the initial gaze motor error. When the fixation cell activity ceases, the hill travels from the caudal to the rostral end of the map. The activity eventually reaches the fixation zone and stops moving. The SW cell activity rises and falls gradually as does buildup cell activity. The fall of the activity at the locus of the initial peak of the SW layer extends beyond the end of the eye movement.
When the target is presented, activity at the PD layer is produced at cell 20 corresponding to the initial gaze motor error. The peak decay cell activity begins later than does SW cell activity and coincides closely with the beginning of eye movement. This relationship is similar to that of burst cell activity in comparison with buildup cell activity. PD cell activity rises and falls abruptly as does burst cell activity, and the fall of the PD cell activity coincides with the end of the eye movement. Overlap paradigm simulation
In the overlap saccade, shown in Figure 11, target presentation precedes fixation point offset. Activity at the PD layer grows and generates a burst after the fixation point is turned off. An eye movement then begins. Initially at the SW layer, while only the fixation point is on, maximum activity is produced only at the fixation cells. When both the fixation point and the target are on, the fixation cell becomes active, and a hill of activity builds up across the buildup cells. The SW cell response subsequently decays because of habituation (see Zij in Equation 9 of Appendix) until the fixation light is turned off. When the fixation point is turned off and only the target remains on, the hill of activity builds up to a higher level and travels across the map as a spread of buildup cell activity.Memory-guided paradigm simulation
In the memory-guided saccade, shown in Figure 12, both target onset and offset precede fixation point offset. Activity at the PD layer grows into a burst after the fixation light is turned off. When only the fixation point is on, only the SW fixation cells are active at the SW layer. When the target is flashed, the orientation buildup cells exhibit activity along with the fixation cells. The level of orientation cell activity remains constant while the fixation point remains on but increases and then migrates once the fixation point is turned off.Gap paradigm simulation
In the gap saccade, shown in Figure 13, fixation point offset precedes target onset. Activity at the PD layer coincides with target onset and then decays. When only the fixation point is on, the fixation cell at the SW layer is the only SW cell active. When neither the fixation point nor the target is on, the fixation cell activity quickly decays, and there is no activity at the SW layer. At target onset, an activity hill builds up across orientation buildup cells. Note that this buildup occurs at a quicker rate than in the visually guided saccade simulation of Figure 10. This shorter latency can be compared with the production of an express saccade. Express saccades are often elicited during the gap saccade task (Fischer and Weber, 1993
These simulations demonstrate how the PD cells in the model can be identified with burst cells and how the SW cells in the model can be identified with buildup cells. Table 2 summarizes these comparisons.
Table 2. Comparisons between cell types
Cell type Property Reference Peak decay (burst cells) Decaying, fixed peak of activity Waitzman et al. (1991) Movement response in both regular and express saccades Munoz and Wurtz (1995a) Spreading wave (buildup cells) Spreading wave of activity Munoz et al. (1992) Sustained response from visual target onset through eye movement in both regular and express saccades Munoz and Wurtz (1995a) Multimodal response enhancement and depression Stein and Meredith (1993) Fixation traveling wave Sustained response while eye remains fixated Munoz and Wurtz (1995a) Multimodal enhancement and depression simulations
Auditory inputs to the model SC can be transformed from a head-centered into a motor error map by using the same type of circuit that planned and attended visual targets use (see Fig. 5a). The same model mechanisms transform all head-centered signals into a motor error map. This coordinate transformation is thus a general engine for linking head-centered to motor error commands. Because each corticocollicular pathway is unimodal, each such pathway needs to compute a motor error vector (Fig. 5a), but all of these error vectors can then use the same PD layer teaching signals and SW layer cells to determine the winning-target locations.
We now show that the SW membrane equations that combine visual and auditory input produce multiplicative response enhancement in cells at the spreading wave layer for coincidentally located visual and auditory targets and response depression in cells when the targets are not at the same location, as also occurs in vivo (Stein and Meredith, 1993
The activity across the SW layer (Fig. 14) at the end of this sensory response period is shown for all three target presentations. The SW layer activity is shown during presentation of the single multimodal target (a), the unimodal target (b), and the separate auditory and visual targets (c). In each graph, the activities of all buildup cells at the SW layer are shown, and the diameters of the circles are directly proportional to the activity of the SW cell at each corresponding location. The value of the slider bars at the top of each graph reflects the gaze motor error of the visual or auditory target when the target is presented. The location of the filled circles also indicates this gaze motor error. 
Fig. 14. The simulated activity of all buildup cells at the SW layer at the end of the sensory response period for each of three target presentations. The SW layer activity is shown during presentation of (a) a single multimodal target, (b) a unimodal target, and (c) separate auditory and visual targets. In each graph, the diameters of the circles are directly proportional to the activity of the SW cell at each corresponding location. The value of the slider bars at the top of each graph and the location of the filled circles reflect the gaze motor error of the visual (Av) or auditory (Aa) target.
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The average sensory response of a SW cell was used to determine the effects of multimodal response enhancement or depression compared with unimodal response. This measure was considered analogous to the mean number of impulses produced by a neuron during presentation of a sensory cue that was used by Stein and Meredith (1993)
According to the formula used by Stein and Meredith (1993)
where CM is the average number of impulses evoked by the combined-modality stimulus and SMmax is the average number of impulses evoked by the most effective single-modality stimulus. When the average activities at cell 10 in the unimodal target simulation and the single multimodal target are compared, the response enhancement is:
The increase in SW cell activity in the multimodal case occurs because there is excitatory input not only from the visual spatial error map but also from the auditory spatial error map. This increase in excitatory input pushes the SW cell activity into the linear region of the signal function c(t) in Equation 9 of the Appendix, thereby producing additional excitatory feedback to this cell and enhancing its activity. In the case in which a unimodal target is presented, the excitatory input from the visual spatial error map to the SW cell is insufficient to produce SW cell activity in the linear region of the signal function; thus the SW cell activity remains in its slower-than-linear region, and the excitatory feedback to this cell is negligible.
Increasing the spatial separation of a visual and an auditory stimulus produces an activity pattern across the SW layer that reflects a gaze change biased toward the closer target. If both targets are spatially coincident, the combination of the two excitatory bell-shaped inputs corresponding to the visual and auditory targets in the planned pathway produces a bell-shaped activity profile in the SW layer the peak of which corresponds to the motor error of the target location. If the disparity between the two targets is increased, the combination of the two activity profiles produces a bell-shaped activity pattern with an initial peak location representing the gaze motor error between the two targets. If the spatial separation is increased further, the activity pattern that is produced has a peak closer to the gaze error representing the medial target.
This bias in the peak location is produced by the combination of the two activity profiles from the auditory and visual spatial error maps. Because of the distribution of weights from the error vector map to the spatial error map, the bell-shaped activity profile corresponding to a target with a small gaze error is narrower than is the activity profile corresponding to a target with a larger gaze error. As a result, the map locations where the two activity profiles overlap are closer to the target with the smaller gaze error. When the two activity profiles are combined, this results in a profile with a peak closer to the location coding the smaller gaze error.
When the average activities at cell 10 in the unimodal target simulation and the simulation of multimodal targets at separate locations are compared, the response depression is:
The depression in SW activity at cell 10 in the multimodal case can be compared with the SW activity in the unimodal case. This depression results because the excitatory input from the visual spatial and the auditory spatial error maps do not significantly overlap. Therefore, in both the unimodal and multimodal cases, the excitatory input to the SW cell is virtually the same. This amount of excitatory input allows the SW cell activity to remain in the slower-than-linear region of the feedback function, and the excitatory feedback to this cell is negligible. The response depression increases in the multimodal case because the growth of activity at a gaze error location between the visual and auditory targets pushes the surround activity at cell 10 into the linear region of the signal function c, producing additional inhibitory feedback to this cell.
DISCUSSION
Recent data on the superior colliculus reveal a spreading wave of activation the peak of which codes the current gaze error (Munoz et al., 1991
The more recent model of Optican (1995)
The Optican (1995)
The present model is consistent with the main data about burst and buildup cells and with many aspects of previous models. However, this model also analyzes how the coordinate systems that the superior colliculus uses to control movement arise and how multiple types of movement commands are rendered dimensionally consistent via learning and compete to select and attend to a final movement command. Whereas Optican's model suggests that both burst and buildup signals are needed to produce output signals, it does not analyze the role that we propose they play in adaptively aligning and coordinating multiple types of movement signal. The present model suggests that these issues are central ones for understanding the functional organization of the saccadic system in general and the superior colliculus in particular.
For example, reflexive and volitional commands both share control of saccadic eye movements. Visually reactive cells encode gaze error in a retinotopically activated motor map, whereas auditory targets are obviously registered in head-centered coordinates, and it has been proposed that visually attended target representations in parietal cortex are also computed in head-centered coordinates (Andersen et al., 1985
This transformation is functionally rationalized by the fact that calibration between visual inputs and eye movement commands is learned early in development within a visually reactive saccade system (Grossberg and Kuperstein, 1986
The SACCART neural network model developed here shows that the map-learning process that combines visually attentive, auditory, and planned saccade commands provides a functional rationale for both burst and buildup cell activity patterns and provides fits to data from these cells in visual, overlap, memory, and gap behavioral paradigms (Munoz and Wurtz, 1995a
Rucci et al. (1997)
Although the SACCART model presented here assumes that the planned input is stored in head-centered coordinates before being transformed into motor error coordinates, data on parietal cortex support at least two different viewpoints. The data of Andersen et al. (1990)
Finally, it is important to note the effect predicted by the current model in the event of a parietal lesion that interrupts the transcortical feedback to the colliculus. The primary role of the cortical projection to the SC is to allow attentive cortical selection of the target, not to provide an input corresponding to residual motor error. In fact, one effect of the model cortical input that creates the spreading wave is to cooperate with the signals that could control the saccade in the absence of cortical input. Thus according to the model, if cortical input is interrupted, visually guided saccades can still occur, but they would be expected to be hypometric. This outcome has been reported for patients with posterior parietal cortical lesions (Duhamel et al., 1992
FOOTNOTES
Received May 27, 1997; revised Sept. 23, 1997; accepted Sept. 25, 1997.
This work was supported in part by Air Force Office of Scientific Research Grant F49620-92-J-0499 (S.G. and K.R.), Defense Advanced Research Projects Agency Grant N00014-92-J-4015 (S.G. and K.R.), and Office of Naval Research Grants N00014-92-J-1309 (S.G., K.R., M.A., and D.B.) and N00014-95-1-0409 (S.G., K.R, and D.B.). We thank Diana Meyers for her valuable assistance in the preparation of this manuscript.
Correspondence should be addressed to Dr. Stephen Grossberg, Department of Cognitive and Neural Systems and Center for Adaptive Systems, Boston University, 677 Beacon Street, Boston, MA 02215.
Dr. Roberts's present address: Cognex, One Vision Drive, Natick, MA 01760.
Dr. Aguilar's present address: Machine Intelligence Group, Massachusetts Institute of Technology Lincoln Labs, 244 Wood Street, Lexington, MA 02173.
APPENDIX
This section lists the mathematical equations and parameters of the model.
Reactive pathway
The visually reactive pathway in the model learns to accurately foveate visual targets. These visual targets are registered in a retinotopic map, the peak decay (PD) layer. Reactive signals that activate a region within this map topographically excite the corresponding region in the spreading wave (SW) layer. The locus of activation at the PD layer defines the direction and amplitude of a saccadic movement.
Two sources of inhibition suppress irrelevant visually activated target locations in this map. A nonspecific signal from the SW layer provides a constant level of inhibition to the peak decay layer. In addition, the model substantia nigra inhibits this layer but is released at the topographic location K encoding the saccadic eye movement. Excitatory feedback from the spreading wave to the PD layer, along with the release of substantia nigra inhibition, overcomes the nonspecific inhibition at the location encoding the target. Because all other locations are inhibited, the selected location can remain active throughout the saccade without interference from competing reactive targets. This resonant activation between layers allows a target choice to be made at the PD layer while interfering targets are suppressed. By gaining access to the SW layer, planned saccade targets can suppress saccades to reactive targets and instantiate the desired command. Peak decay
This resonant circuitry at the peak decay layer is implemented by defining the change in the activity level at each cell. The peak decay cells (Pk) combine excitatory input (Rk) reflecting the reactive target input and excitatory input (Sk) from the spreading wave layer. In addition, the peak decay cells are inhibited by model mesencephalic reticular formation (MRF) input [m( Sj)], by fixation cell input from the rostral pole of the SW layer (W1Fk), and by model substantia nigra input (NPk). The following membrane, or shunting, equation defines the change in activity at the peak decay layer.
(1) Reactive input
where the following functions obtain:
SW signal function
where K identifies the cell location encoding initial gaze error. MRF input
Spread of fixation cell inhibition
SNr input
When the target is turned on, the reactive input Rk excites the location at the peak decay layer corresponding to the initial gaze motor error with a maximum reactive input value. After the target is turned off or the eye begins to move, the reactive input is shut off. Activity of each spreading wave cell excites the corresponding peak decay cell via a sigmoid function f(Sk). All of the input to this function is in the slower-than-linear region of the sigmoid. Migration of the spreading wave away from the initial location diminishes the strength of the mutual excitation established with the PD layer, causing the activity peak in this layer to decay.
Inhibitory input to this layer is provided from the MRF, the fixation cell of the spreading wave layer, and the substantia nigra. The MRF inhibits the entire peak decay layer if there is any activity at the buildup cells of the spreading wave layer. If there is activity, the MRF input to the peak decay layer is established at a constant level of inhibition.
The fixation cell of the spreading wave layer inhibits cells at the peak decay layer based on the distance between the fixation cell and the peak decay cell. The fixation cell inhibits peak decay cells that encode smaller initial gaze motor errors more strongly than peak decay cells that encode larger errors. This stronger inhibition helps to produce quicker decay in cells that encode smaller initial gaze motor errors than in those that encode larger errors so that the complete decay of activity coincides to eye fixation in both cases. When the initial gaze motor error is small, eye fixation occurs quickly, and the peak decay activity correspondingly returns to zero at this time. Thus several factors work together to shut off the peak decay cells.
Inhibition from the substantia nigra decreases to the peak decay cell encoding the initial gaze motor error once the fixation point is turned off. After the first 150 time steps subsequent to fixation point removal, the SNr inhibition begins to gradually decrease. After 375 time steps have passed, the SNr inhibition decreases more rapidly. The effect of this SNr activity pattern produces an initial slow rise of peak decay activity followed by a strong burst. Head-centered pathway: auditory, visually attentive, or planned saccades
The second pathway in the model provides the ability to perform saccades to targets coded in head-centered coordinates. The case of planned saccades will be discussed for definiteness. First, a planned target signal is combined with the current eye position signal to produce the gaze motor error vector. Next, the current gaze motor error (Vh) is converted to a two-dimensional representation [X(r,q)]. Finally, this planned target input is combined with the reactive target input at the spreading wave layer.From head-centered targets to motor difference vectors
At this stage, the combination of head-centered target coordinates and eye position coordinates consists of a set of paired cell populations coding the error between the two in vector coordinates. Each cell population is defined as part of an agonist-antagonist pair coding a given degree of freedom of the oculomotor plant. For example, a pair may code the horizontal degree of freedom; one member of the pair codes activity of the muscle that pulls the eye to the right, and the other codes activity of the muscle that pulls the eye to the left. For simplicity, the oculomotor plant can be approximated by two degrees of freedom, horizontal and vertical, so there are four cell populations that define the muscle lengths of four muscles each of which control up, down, left, or right movements in a two-dimensional plane in head-centered coordinates. The current gaze motor error Vh, where h = 0, 1, 2, or 3, corresponds to the vector difference between the desired muscle length (xh*) and the current muscle length (xh) (Grossberg and Kuperstein, 1986
![V<SUB>h</SUB>=0.075[x<SUP>*</SUP><SUB>h</SUB>−x<SUB>h</SUB>]<SUP><UP>+</UP></SUP>](/content/vol17/issue24/fulltext/9706/img014.gif)
(2)
Parameter A defines the amplitude of the desired eye movement, and D is the direction of the movement in radians. The average direction (Dh) coded by each eye position vector element is:
(3)
The current muscle length is updated when the eye moves by invoking the simplest possible movement rule, because events downstream from the SC are not the focus of the present work. A wide variety of similar rules generate analogous results. It is assumed that the current muscle length xh integrates activity from the spreading wave layer:
(4)
if SJ exceeds the threshold activation of 0.01. The value of J reflects the retinotopic position of SJ, the maximum activity in the spreading wave layer.
Next, the vector difference (Vh) is converted from a one-dimensional to a two-dimensional map representation [X(r, )], which can subsequently be input to the map in the spreading wave layer. To transform a spatial vector into the corresponding location in a spatial map, both weight and threshold gradients are used to map incoming signals (Grossberg and Kuperstein, 1986
(5)
where the two-dimensional representation is defined in coordinates of the amplitude r and direction of eye movement. The function [x]+ indicates that the value of M(r,
Functions Wh(r, (r,
![W<SUB>h(r,&thgr;)</SUB>=0.64r[<UP>cos</UP>(&thgr;−D<SUB>h</SUB>)]<SUP><UP>+</UP></SUP>.](/content/vol17/issue24/fulltext/9706/img019.gif)
(6)
The thresholds determine the level above which a signal is permitted to generate activity and are a function of the amplitude (r) of the eye movement:
(7)
Evidence of similar gradients to the superior colliculus have been reported by several authors (Jayaraman et al., 1977
The two-dimensional activity is normalized to produce a single peak of activity:
(8)
Equation 8 approximates the action of a recurrent on-center, off-surround network (Grossberg, 1973 Spreading wave layer
Because the spatial representation of the planned target is aligned with that of the reactive target, the two signals can be combined at the jth cell in the spreading wave layer to produce a change in the activity of a cell via a membrane, or shunting, equation (Grossberg, 1973Spreading wave (buildup cells)
The buildup cells (Sj where 1 < j < N) combine excitatory input (Xi) reflecting the planned target input and excitatory input Pk from the peak decay cell layer. The inputs (Xi) are multiplied, or gated, by transmitter weights (Zij) that habituate in an activity-dependent manner. In addition, the buildup cells are inhibited by MRF input [m(
(9)
Habituative intermodal path strength
where: PD signal function
Spread of input from PD to SW
MRF input
SNr input
Feedback signal function
Surround weight
![c(x)=[x−0.035]<SUP><UP>+</UP></SUP>.](/content/vol17/issue24/fulltext/9706/img030.gif)
In greater detail, Equation 9 says that excitatory input to this layer is provided from the planned target spatial map 2 kg(PkGk-j), and recurrent feedback 40c(Sj). When the target is turned on, the planned target input excites a region at the spreading wave layer with maximum activity corresponding to the initial gaze motor error. The planned input to this layer remains even after the target is turned off or the eye begins to move. The peak decay layer is designed so that only one cell in the sum 4
Inhibitory input 40m( Spreading wave (fixation cells)
The fixation cells (Sj, where j = 1) combine excitatory input (F) reflecting fixation input and weighted excitatory input (2
(10) Fixation input
where SW buildup input kernel
Intermodal map learning
For the multimodal and planned pathways to gain access to the visually reactive pathway, the different coordinate systems of these pathways are adaptively aligned via associative learning. This learning process needs to be stable through time, despite the possible interference from multiple sensory cues. In addition, the eye movement calibration learned by the visually reactive pathway at the cerebellar adaptive gain stage needs to be maintained after multimodal learning.
Multimodal and planned signals are relayed to the SW layer where they compete for attention with visually reactive saccades via the recurrent on-center off-surround interactions in Equation 9. Learning occurs between the motor error map of each multimodal and planned input source and the SW layer. After the learning process begins, it is stabilized by the congruence of saccade commands from multimodal and visually reactive pathways. If multimodal and visually reactive positions agree, then learning reinforces the current map alignment. If the commands disagree, then both learning and the distracting saccade target are suppressed by their mismatch.
The equations for one such learned map are shown below. Each map is determined by the same mechanisms but can use its own motor difference vector to map into the SW layer. Learning proceeds according to the following associative learning rule, which is often called the outstar learning rule (Grossberg, 1968
![<FR><NU>dz<SUB>ij</SUB></NU><DE>dt</DE></FR>=&agr;n(X<SUB>i</SUB>)[p(P<SUB>k</SUB>G<SUB>k<UP>−</UP>j</SUB>)−z<SUB>ij</SUB>],](/content/vol17/issue24/fulltext/9706/img035.gif)
(11)
Variable zij in Equation 11 defines the strength of the learned connection between the ith cell in the multimodal map, the activity of which is Xi, and the jth cell of the SW layer. The kth cell in the PD layer (Pk) acts as a teaching signal and is input to the SW layer using a Gaussian spread (Gk-j). The adaptive weights zij all start out with the same value. When Xi is sufficiently active, n(Xi) > 0 in Equation 11, so learning commences. The term n(Xi) > 0 hereby gates learning on and off. When n(Xi) > 0, the adaptive weight tracks the teaching signal p(PkGk-j). Such tracking behavior is called steepest descent. Thus the outstar learning rule is a gated steepest descent law. In particular, whenever the intermodal and visually reactive targets agree, both n(Xi) and p(PkGk-j) are positive, and the weight zij converges toward the PD activity Pk that is projected to site j in the SW layer. The weights zij are bounded because the PD layer input (Pk) is bounded by 1.2, the upper bound of the excitatory term in Equation 1. Learning continues until the intermodal target input becomes aligned with the visually reactive PD that projects to the SW. The learning rate is = 0.5 in Equation 11.
Learning is sped up and rendered more robust when the intermodal spatial map can sample neighboring locations in the SW layer. As described above, the excitatory input from the PD layer to the SW layer shows a Gaussian fall-off. In addition, the gating input n(Xi) is active over a broad region. The spatial distribution of the predictive and PD inputs enables learning to interpolate across several map locations. As a result, distributed learning of the entire map can occur using only a small set of sampled positions.
Three conditions cooperate to achieve stable learning. First, only one visually reactive target is active at the PD layer during learning. Second, the gaze motor error vector remains fixed during the beginning of the target presentation. This period is when most of the learning occurs. Third, the learning rate parameter
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