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J Neurophysiol 96: 765–774, 2006;
doi:10.1152/jn.01372.2005.
Saccade-Related Spread of Activity Across Superior Colliculus May Arise
From Asymmetry of Internal Connections
Hiroyuki Nakahara,1 Kenji Morita,2 Robert H. Wurtz,3 and Lance M. Optican3
1
Laboratory for Mathematical Neuroscience and for Integrated Theoretical Neuroscience, RIKEN Brain Science Institute, Saitama;
Institute of Industrial Science, University of Tokyo, Tokyo, Japan; and 3Laboratory of Sensorimotor Research, National Eye Institute,
National Institutes of Health, Bethesda, Maryland
2
Submitted 27 December 2005; accepted in final form 24 April 2006
The superior colliculus (SC) is a layered, midbrain structure
related to rapid (saccadic) eye movements. The superficial
layers (related to vision) and the intermediate and deep layers
(related to movement) contain a nonlinearly compressed map
of retinotopic space (Fig. 1, A and B) (Cynader and Berman
1972; Robinson 1972). The central visual field is represented in
more detail than the periphery and consequently covers a larger
proportion of the SC map than does the periphery (Fig. 1B);
thus the map is compressed toward the peripheral edge along
the rostral– caudal axis. The map is approximately logarithmic
(McIlwain 1975; Ottes et al. 1986) and the maps in the
different SC layers are aligned (Mohler and Wurtz 1976;
Schiller and Stryker 1972; also see review by Sparks 1989).
Targets close to the center of the visual field are represented
rostrally in the SC (black circle in Fig. 1B) and targets in the
periphery are represented caudally (black triangle).
The deeper SC layers contain several different types of
neurons that in one classification scheme (Munoz and Wurtz
1995a) are known as buildup and burst neurons. Both types
usually have a high-frequency burst of activity (about 800
spikes/s) at the time of the saccade, but the buildup neurons
also have a slowly increasing activity (up to nearly 100
spikes/s) before the saccade. A striking but controversial aspect
of neuronal activity in the SC intermediate layers is that the
low level of activity on the buildup neurons sometimes spreads
from caudal to rostral in the SC around the time of a saccade.
It is important to emphasize that the activity that spreads is the
low buildup of activity, and not the burst (that remains centered
at the retinotopic location of the target). This intriguing observation, first made in the cat (Munoz et al. 1991a,b), was
described as a “moving hill” of activity traveling from caudal
to rostral across the map. Similar activity was later found in the
monkey SC visual motor neurons (Munoz and Wurtz 1995b),
although it was better described as a spread of activity rather
than a moving hill because there was little caudal-to-rostral
movement of the peak of activity. The spread in the monkey
has been confirmed in other experiments; a small spread in
both lateral and rostral– caudal directions was found by Anderson et al. (1998), and a spread was observed with the use of two
recording electrodes at different rostral– caudal locations (Port
et al. 2000).
The initial reports of the moving hill (in the cat) and the
spread of activity (in the monkey) led to the hypothesis that it
played a functional role in controlling the duration of eye
movements (Guitton et al. 1990; Wurtz and Optican 1994).
This hypothesis assumed, first, that the spread of activity was
driven by information about the movement feeding back to the
SC and, second, that the distribution of activity on the map
represented the distance the eye still had to go to reach the
target (i.e., the instantaneous, or dynamic, motor error). The
allure of this hypothesis was that a critical function in motor
control, calculating motor error, could be accomplished simply
by using feedback to change the distribution of activity on a
dynamic brain map.
This error computation hypothesis was controversial
(Anderson et al. 1998; Guitton et al. 1993; Sparks 1993), and
several subsequent experimental observations in the monkey
Address for reprint requests and other correspondence: H. Nakahara, Laboratory for Integrated Theoretical Neuroscience, RIKEN Brain Science Institute, Hirosawa 2-1, Wako-shi, Saitama, Japan (E-mail: [email protected]).
The costs of publication of this article were defrayed in part by the payment
of page charges. The article must therefore be hereby marked “advertisement”
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
INTRODUCTION
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Nakahara, Hiroyuki, Kenji Morita, Robert H. Wurtz, and Lance
M. Optican. Saccade-related spread of activity across superior colliculus may arise from asymmetry of internal connections. J Neurophysiol 96: 765–774, 2006; doi:10.1152/jn.01372.2005. The superior
colliculus (SC) receives a retinotopic projection of the contralateral
visual field in which the representation of the central field is expanded
with respect to the peripheral field. The visual projection forms a
nonlinear, approximately logarithmic, map on the SC. Models of the
SC commonly assume that the function defining the strength of
neuronal connections within this map (the kernel) depends only on the
distance between two neurons, and is thus isotropic and homogeneous. However, if the connection strength is based on the distance
between two stimuli in sensory space, the kernel will be asymmetric
because of the nonlinear projection onto the brain map. We show,
using a model of the SC, that one consequence of these asymmetric
intrinsic connections is that activity initiated at one point spreads
across the map. We compare this simulated spread with the spread
observed experimentally around the time of saccadic eye movements
with respect to direction of spread, differing effects of local and global
inhibition, and the consequences of localized inactivation on the SC
map. Early studies suggested that the SC spread was caused by
feedback of eye displacement during a saccade, but subsequent studies
were inconsistent with this feedback hypothesis. In our new model,
the spread is autonomous, resulting from intrinsic connections within
the SC, and thus does not depend on eye movement feedback. Other
sensory maps in the brain (e.g., visual cortex) are also nonlinear and
our analysis suggests that the consequences of asymmetric connections in those areas should be considered.
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NAKAHARA, MORITA, WURTZ, AND OPTICAN
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did not support it. The spread moves laterally on the SC map
as well as caudal to rostral (Anderson et al. 1998) and is
difficult to see near the rostral pole (where small errors would
have to be prominently represented) (Anderson et al. 1998;
Port et al. 2000), both of which are hard to reconcile with the
error model. [The spread of activity is so slight that it has not
been detected by functional imaging that is sensitive enough to
detect the presaccadic burst of activity (Moschovakis et al.
2001).] The motor error idea is also inconsistent with the
observed timing of the spread, which reaches the rostral pole
either too early (Quaia et al. 1999) or too late (Soetedjo et al.
2002) to indicate that motor error is zero and that the saccade
should be ended. Probably the most compelling reason for
rejecting the error computation hypothesis is the result of
muscimol inactivation of the SC (Aizawa and Wurtz 1998;
Quaia et al. 1998). One of the strongest predictions of the error
computation model is that interrupting the caudal-to-rostral
spread with muscimol should delay its arrival at the rostral SC,
thereby producing a longer saccade. However, in those lesion
experiments saccade length did not increase. Taken together,
these experiments show two things: 1) sometimes a low level
of activity spreads through the SC buildup neurons at around
J Neurophysiol • VOL
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SC Distance (mm)
the time of a saccade, but 2) that spread is not well correlated
with the motor control of the saccade. These results cast
considerable doubt on the hypothesis that the spread in the SC
results from a feedback signal and is computing the remaining
error of the ongoing saccade. However, it is still possible that
the SC receives input signals that are saccade related, but are
not used to compute saccadic error. For example, the end of the
SC burst is tightly correlated with the end of the saccade
(Waitzman et al. 1991) and interruption of a saccade by
stimulation of the brain stem omnipause neurons leads to a
pause in SC burst neurons (Keller and Edelman 1994); both of
these SC activity changes would require feedback about when
the saccade ends.
If the spread across the SC buildup neurons is not caused by
an external, feedback signal, then what is its origin? Here we
consider the possibility that the spread of activity arises as a
consequence of the connections within the SC. We first assume, as have other models (Arai and Keller 2005; Arai et al.
1994; Droulez and Berthoz 1991; Lefèvre et al. 1998; Optican
1995; Ottes et al. 1986; Trappenberg et al. 2001), that neurons
within the SC are uniformly distributed, i.e., the neural density,
or the number of cells per square millimeter, is the same
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D
C
FIG. 1. Asymmetries produced by mapping between
the visual field and superior colliculus (SC). A: logarithmic transform between horizontal meridian of the visual
field (in degrees of visual angle) and SC (in mm) (Ottes et
al. 1986; Robinson 1972). B: topographic mapping between visual hemifield and contralateral SC. Thick red
and thin green lines show horizontal and vertical meridians, respectively, and lighter red curves show increased
eccentricity. C: visual-asymmetric and SC-symmetric
connection field. Symmetry of yellow spokes on SC map
indicates the connection field is isotopic, and asymmetry
of the spokes on the visual map emphasizes anisotropy of
the visual field. Density of red shading indicates strength
of connections. D: visual-symmetric and SC-asymmetric
connection fields. In both C and D, the center of the
connection fields is the same, (R, ␪) ⫽ (20°, 30°). E and
F: shape of the visual and SC connection fields for the
maps in C and D just above. Abscissa is the distance along
the ␪ ⫽ 30° line used for the example field shown in C and
D with zero as the center of the field. Because of the
nonlinear projection of the visual field onto the SC map,
the fields can be symmetric on one map or the other, but
not both.
MODEL OF SPREADING ACTIVITY IN THE SC
METHODS
Connection symmetry
First, we assumed that neurons within the SC are uniformly distributed (same number of cells per square millimeter everywhere on
the map), as was assumed in previous studies. Second, we assumed
that connection strength was determined by the distance measured on
either the SC map (the SC-symmetric kernel) or in the visual field (the
visually symmetric, also called the SC-asymmetric kernel; see following text). Here the symmetry, either SC or visual space, refers to being
isotropic (the same in all directions) as well as homogeneous (the
same for neurons everywhere) in the space (see RESULTS for details).
The present study notes that the connection kernel cannot be
symmetric in both visual space and SC space (Fig. 1) because of the
logarithmic mapping. We therefore must decide how to define the
“distance” between any two neurons, i.e., how to define the kernel.
Simply defining distance as the physical separation between the two
cells on the collicular map results in the SC-symmetric kernel (Fig.
1C), which was assumed in most previous SC models (Arai et al.
1994; Droulez and Berthoz 1991; Gancarz and Grossberg 1999;
Lefèvre and Galiana 1992; Optican 1995; Short and Enderle 2001;
Trappenberg et al. 2001). In this kernel, equally distant neurons have
the same connection strength.
Alternatively, we could define distance as the separation in visual
space between the two points that map to those neurons, i.e., neurons
receiving equally separated inputs in visual space have the same
connection strength. The separation in space can lead to a visually
symmetric kernel simply by Hebb’s postulate (Hebb 1949) that “cells
that fire together, wire together.” Two separate stimuli will be unlikely
to appear together consistently because natural visual stimuli appear
everywhere relative to the fovea with equal probability. Thus the
probability that two neurons fire together will depend on whether a
single stimulus excites them both, which will be more likely to happen
if the receptive fields of those neurons are close together in visual
space. This postulate leads to a visually symmetric kernel. In fact,
symmetry in the visual space, but not on the brain map, has been
demonstrated in V1 (Angelucci et al. 2002). As shown below, the
visually symmetric kernel, once projected back to the SC, becomes
asymmetric (Fig. 1D).
The shape of both kernels can be appreciated by looking at a slice
along the horizontal meridian. The asymmetry of a kernel in a space
shows up as a skewed curve in that space (Fig. 1, E and F; Fig. 2, C
and D).
Intracollicular connections
We used computer simulations of an SC model with either a
symmetric or asymmetric kernel (see following text). The kernel
defines the strength and distribution of the connections among SC
neurons and thereby governs the behavior of the SC model. Here we
explain qualitatively the background and methods used in the present
study. Quantitative details can be found in the APPENDIX.
Logarithmic mapping from visual field to SC
The nonlinear mapping between the visual hemifield and the
contralateral SC found in monkeys by Robinson (1972) was fit by
Ottes et al. (1986) with the equations
x ⫽ Bx log
冋冑
册
R2 ⫹ 2AR cos 共␪兲 ⫹ A2
A
冋
R sin 共␪兲
y ⫽ By arctan
R cos 共␪兲 ⫹ A
册
(1)
RESULTS
(2)
where (x, y) represents Cartesian coordinates on the SC topographic
map (in mm) and (R, ␪) represents polar coordinates in degrees in the
visual field (A ⫽ 3.0°, Bx ⫽ 1.4 mm, and By ⫽ 1.8 mm).
J Neurophysiol • VOL
The kernel used in these simulations was defined as the difference
between an excitatory function (E) and an inhibitory function (I). For
the function E, we considered a Gaussian function either on SC (an
SC-symmetric kernel) or in the visual field (an SC-asymmetric kernel). For the function I, a Gaussian function defined “local” inhibition,
and a saturating function of the sum of all SC activities defined
“global” inhibition (see the APPENDIX). The two different forms of
inhibition represent the short-distance, local effects of inhibitory
connections within the SC and the long-distance, more global effects
of inhibition from outside the SC [e.g., from the substantia nigra
(Hikosaka and Wurtz 1983)]. Using either kernel (SC-symmetric or
-asymmetric) and either inhibition (local or global), the temporal
evolution of the SC neural population activity was simulated by
solving a partial differential equation (Eq. A5 in the APPENDIX).
Overview of SC-symmetric and -asymmetric kernels
The SC-symmetric and -asymmetric kernels define two different models. Because of the nonlinear map from the visual to
the SC, the kernel can be symmetric in only one space or the
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everywhere on the map. Second, the spatial distribution and
strength of the connections of one SC neuron with its neighbors
defines a function, or kernel. All saccade models necessarily
contain a kernel, but our model differs in our assumptions
about the shape of this kernel.
Many previous SC models (e.g., Arai et al. 1994; Droulez
and Berthoz 1991; Gancarz and Grossberg 1999; Lefèvre and
Galiana 1992; Optican 1995; Short and Enderle 2001; Trappenberg et al. 2001) assumed that the kernel is determined by
the distance between two neurons; i.e., if the distance is the
same, the connection strength is the same, between any pair of
neurons. Such a kernel is circularly symmetric on the SC (e.g.,
a Gaussian function). Figure 1C illustrates this symmetry by
showing one sample neuron (in yellow; not to scale) on the SC.
It also shows that the projection of this kernel back onto the
visual field is not symmetric because of the logarithmic projection of the visual field onto the SC (Fig. 1A).
There is no evidence, however, that SC kernels are a symmetric function of the distance between two neurons. We
therefore explore an alternative possibility: that the kernel is
defined by the separation in visual space between two stimuli.
The logarithmic projection (see METHODS and RESULTS) of the
visual space onto the SC makes this kernel asymmetric (Fig.
1D). Here, we study the consequences of having either symmetric or asymmetric kernels. We show that asymmetric connections within the SC can cause activity to spread across the
SC during the movement without any exogenous inputs (if
inhibition is low enough). Furthermore, we show that the
spread caused by an asymmetric kernel is consistent with
recent experimental findings, including those inconsistent with
the spread acting as a motor error signal. Thus we propose that
an asymmetric connection kernel causes the spread of activity
on the SC, and thus external feedback is not a prerequisite for
the spread. This raises the possibility that symmetry in connection kernels may be an important factor in the distribution
of brain activity in other areas that have nonlinear brain maps.
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NAKAHARA, MORITA, WURTZ, AND OPTICAN
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other, but not both. (Of course, the kernel could be asymmetric
in both spaces, but those cases are outside the scope of this
paper.)
Figure 1 shows the correspondence between the kernels for
the visual field and for the SC map. When the connection
kernel in the model is determined by the distance on the SC
between two neurons, the connection kernel is SC-symmetric
(Fig. 1C, right). Here, symmetry means that the kernel is both
isotropic (the same in all directions) and homogeneous (the
same for neurons everywhere) in the space. The kernel is
asymmetric in visual space when it is projected back through
the nonlinear map from the SC into visual space (Fig. 1C, left),
severely distorting the distances. The fact that the kernel is
anisotropic in visual space is illustrated by cross sections
through them in Fig. 1E. It is also inhomogeneous on visual
space (Fig. 2A). The distortion in the visual field is greater for
sites near the center of the field (Fig. 2, A and C) as a result of
the logarithmic transformation between visual field and SC
map (Fig. 1A).
In contrast, the visually symmetric kernel (Fig. 1D, left) is
asymmetric when it is projected onto the SC (Fig. 1D, right).
Straight lines near the center of the visual field become more
curved on the SC map than lines in the periphery. This kernel
is anisotropic on the SC (e.g., Fig. 1F). The kernel is also
inhomogeneous on the SC (Fig. 2, B and D). The distortion in
the SC is greater for sites more rostral on the SC map, which
correspond to sites nearer the center of the visual field. Thus
the kernel is called SC-asymmetric.
In simulation studies below, we focused on the effects of
different kernels on neurons that show a buildup or small
prelude of activity before the saccadic burst, such as buildup
neurons in the SC. Because burst neurons have no such
activity, they cannot participate in the spread of the prelude.
We assume that the main difference between buildup and burst
neurons is the amount of inhibition they receive, with burst
neurons receiving so much inhibition before saccades that they
show no buildup (Quaia et al. 1999).
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Difference in spread in SC-symmetric and
-asymmetric kernels
We first determined the consequences of providing a localized input at one point on the SC map with the two kernels.
With an SC-symmetric kernel, activity evoked around the
location of the initial input waxes and wanes, but remains
centered at the initial site, with no spread being evident (Fig.
3A). In contrast, with an SC-asymmetric kernel (Fig. 3B), the
activity does not remain at the input site but spreads toward the
rostral SC. This autonomous spread occurs because of the
asymmetry of the kernel, i.e., the rostral– caudal difference in
strength (cf. Droulez 1991). Figure 4 shows the difference
more clearly by showing a section through both asymmetric
A
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B
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Early
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Late
3. In the model simulation the spatiotemporal evolution of neuronal
activity depends on the symmetry of SC connections. A: SC-symmetric
connection field gives stationary locus of activity after persistent input at one
point [black cross at (R, ␪) ⫽ (15°, 30°)]. B: SC-asymmetric connection field
causes initial locus of activity to expand and move rostrally. SC maps show
frames of spatiotemporal activity moving from left to right (early to late) after
the start of input. Yellow asterisk indicates the shift of the center of gravity of
simulated activity. Model produces a spread of activity only on the map with
asymmetric connections on the SC.
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C
FIG. 2. Homogeneity of connection fields. A: SC-symmetric connection field is homogeneous (i.e., the same for
neurons at all locations) in the SC space, but not in the
visual space. B: visual-symmetric connection field is homogeneous in the visual space, but not in the SC space. In
A and B, 2 connection fields are shown, having their
centers at (R, ␪) ⫽ (10°, 30°) (orange) and (40°, 30°)
(green). C and D: shape of each of the 2 connection fields
for the 4 maps in A and B. Same formats as Fig. 1, E and
F. Two connection fields for each map are superimposed
by aligning their centers and indicated by their corresponding colors (orange and green). In C the shapes of the 2
connection fields are the same at different locations on the
SC map but different on the visual field map, whereas in D
they are different on SC but the same on the visual field.
MODEL OF SPREADING ACTIVITY IN THE SC
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FIG. 4. SC-asymmetric connection shifts the activity
toward the rostral SC, leading to the caudal-to-rostral
spread. A: initial neural activity distribution on SC (slice
through the midline). Dotted line indicates the center of
mass of the activity. Symbols a, b, and c indicate the
locations of the 3 representative neurons used in B and
D. B: SC-asymmetric connections, derived from the
visually symmetric kernel (of Mexican-hat type; see
METHODS), are shown for the 3 neurons at a, b, and c
(orange, red, and magenta lines, respectively). Magnitude of each kernel is scaled by the magnitude of the
activity shown in A. C: total influence (normalized) of
the neural activity in A, through the SC-asymmetric
kernel, on other neurons, called the input field. Center of
mass of the activity kernel is indicated by green dotted
line, which is to the left of the center of mass of the
initial neural activity in A (black dotted line). Input field
is asymmetric so that it drives activity to spread to the
left. D and E: SC-symmetric connections are shown in
the same format as B and C. Center of the mass of the
input field and the initial neural activity now overlap
each other. Input field is symmetric so that activity does
not spread.
1
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(Fig. 4B) and symmetric (Fig. 4D) kernels. The net effect of the
kernel is to create an input for all the SC neurons. The
asymmetric kernel gives rise to an input field that is skewed
rostrally (Fig. 4C), which causes the neural activity to spread.
The symmetric kernel creates an input field that is trapped
locally by balanced excitation and inhibition (Fig. 4E), so there
is no spread. In Fig. 3, the center of gravity of the activity
(COA) is shown to indicate the spread. The COA shown here
moves more than that reported in the monkey (Anderson et al.
1998; Munoz and Wurtz 1995b) because the COAs reported by
those authors include both the spread and the stationary burst.
The burst is an order of magnitude greater than the spread,
whereas the COA here is computed using only the prelude
activity on buildup neurons.
Model parameters that alter the spread
Permutations of these simulations show that the autonomous
spread is robust; the spread occurs with a wide range of
parameters, as long as the connection kernel is SC-asymmetric.
It made no difference whether the input was sustained as in the
example in Fig. 3B or transient as in Fig. 5A. The relative
strength of excitation and inhibition could be varied over
substantial ranges without disrupting the spread; halving the
strength of the excitatory connections (Fig. 5B) or doubling the
strength of the inhibitory connections (Fig. 5C) did not substantially change the magnitude of the spread. [Of course, the
inhibition must be less than the excitation, e.g., quadrupling the
strength of inhibition eliminated the spread (Fig. 5D).]
We next varied the nature of the inhibition. The commonly
used symmetric kernel has a center-surround shape, with a
small excitatory center superimposed on a larger inhibitory
surround (a “Mexican hat” function). It is well known, however, that there is another major source of inhibition in the SC
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Caudal
other than that resulting from local connections; projections
from the substantia nigra pars reticulata of the basal ganglia
(Hikosaka and Wurtz 1985) provide a more global inhibition
throughout the SC map. We therefore compared the effects of
the two types of inhibition tested separately: local inhibition as
used in the previous simulations (Fig. 6A) and global inhibition
in the absence of any local inhibition (Fig. 6B). There was a
spread of activity with global inhibition just as there was with
local inhibition, but the characteristics of the spreads were
different; the COA stays near the location of the initial input
longer in the case of global inhibition.
Simulations of abnormal conditions
We also tested the model under two abnormal conditions to
show the robustness of the spread in our model. The first shows
the effect of inactivating a part of the SC, a common experimental procedure (e.g., Aizawa and Wurtz 1998). With the
SC-asymmetric model, inactivation at a spot on the map does
not halt the spread of activity because the intrinsic connections
provide many paths around the lesion (Fig. 7). Under the old
hypothesis—that the spread is indicating motor error—it would
represent a vector (with both magnitude and direction) and
always point along a streamline (lines of isoelevation in Fig.
1B) toward zero, which is in the rostral SC (fixation zone)
(Optican 1995). Thus if the spread is motor error, purely driven
by feedback signal, and if there is a lesion on the streamline
between the initial site of activation and the fixation zone, the
spread would “fall into the lesion” and disappear. Feedback
could not generate a perpendicular component that would drive
the activity around the lesion and then toward the rostral pole.
Thus the results of the inactivation experiment disprove the
error computation hypothesis (Aizawa and Wurtz 1998; Quaia
et al. 1998). In contrast, we have now demonstrated by the
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Kernel on SC
Resulting SC Activity
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SC-asymmetric model that the spread can continue by getting
around the lesion because of the asymmetry of the intrinsic
connections.
In the second experiment two stimuli were presented at
once, another common experimental procedure (e.g., Edelman
and Keller 1998; Robinson 1972). In these experiments, monkeys usually made a saccade toward one stimulus or the other,
but sometimes made a saccade to a point between the two. The
observed SC activity stayed at both locations throughout the
movements (Edelman and Keller 1998). For the SC-asymmetric model, two initial inputs were provided and the spread from
each of the activated SC sites was followed (Fig. 8). In the case
of local inhibition, one spot always prevailed and spread
toward the rostral SC whereas the other died out (Fig. 8A).
With global inhibition each point led to a spread of activity that
coalesced into one stream of activity (Fig. 8B). Thus the global
A
B
inhibition model is more consistent with experimental results
that show an “averaged” saccade.
DISCUSSION
Consequences of symmetric and asymmetric SC connections
We explored a consequence of the nonlinear mapping of the
visual field onto the SC, a two-dimensional structure. The
logarithmic projection from visual space to SC means that
there are two logically consistent ways to interconnect SC
neurons, either with a kernel that is symmetric in SC coordinates (i.e., SC-symmetric) or with a kernel whose image,
projected back into the visual field, is symmetric (i.e., necessarily SC-asymmetric). Our model shows a spread of activity
only when there is an asymmetry in the connections within the
Local Inhibition
FIG. 6. Differences in spread with local or global
inhibition. Dynamics of simulated SC activity are
shown with local lateral inhibition as in previous
examples (A) and with global inhibition (B). Location
of persistent input is (R, ␪) ⫽ (40°, 30°). Figure format
as in Fig. 3. Simulated spread with only local inhibition is reminiscent of the “moving hill” in the cat,
whereas with global inhibition the spread is more
similar to the spread of activity in the monkey.
Global Inhibition
Early
Late
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0
FIG. 5. Effect of the shape of SC-asymmetric connection field
on simulated spread of activity. Left column: relative sizes of the
excitatory (red) and inhibitory (blue) connections in SC. Right
column: sequence of frames of SC activity induced by a transient
input. In A, the connection shape is the same as in Fig. 3B. In B
the amplitude of excitation is halved, and in C the inhibition is
doubled with no change in the spread. In D the inhibition is
quadrupled, and the spread is eliminated. Dotted curves in B–D
indicate the connection shape in A for comparison. Simulated
spread is relatively unaffected by a 2-fold change in the relative
sizes of excitation or inhibition, but the spread is eliminated when
the inhibition is much larger than the excitation.
MODEL OF SPREADING ACTIVITY IN THE SC
A
B
Inactivation with Local Inhibition
FIG. 7. Effects of a small lesion in the model SC.
Spread goes past the lesion (indicated by a dashed
circle) with both local (A) and global (B) inhibition.
Format is the same as in Fig. 3.
Inactivation with Global Inhibition
Early
Late
Role of inhibition in the SC spread of activity
One of the implications of our model simulations is that the
amount and distribution of inhibition are critical in determining
B
network behavior. The level of inhibition set in the model
determines whether there is any spread (Fig. 5, C and D). Even
when inhibition was set low enough to allow the spread, the
nature of the spread differed, depending on whether the inhibition was local or global. With local inhibition, the caudal part
of the activity decayed as the rostral part spread, creating a
moving hill of activity similar to what Munoz et al. (1991b)
found in the cat (cf. Fig. 6A). In contrast, with global inhibition, the caudal part of the activity remained at the initial site,
whereas the activity spread rostrally, creating a spread of
activity similar to what Munoz and Wurtz (1995b) found in the
monkey (cf. Fig. 6B). We do not know whether this similarity
is simply coincidental, or indicates a species difference, because the cat and the monkey are the only species studied
sufficiently to evaluate the nature of the spread of activity (but
see also below).
Is there any evidence at the level of single neurons in the SC
for the characteristics assumed in our model? On the key point
of inhibition, there is abundant evidence for extensive inhibitory connections within the intermediate layers of the SC
where the buildup neurons are located. At an anatomical level,
axon terminals that contain the precursor of ␥-aminobutyric
acid (GABA) are extensive in the cat (Lu et al. 1985) and
across species roughly a third of the neurons are GABAergic
(Mize 1992). Electrical stimulation of SC intermediate layers
in awake monkeys (Munoz and Istvan 1998) and in ferret brain
slices (Meredith and Ramoa 1998) produces inhibition both
locally and at substantial distances across the intermediate
layers. Thus there is ample evidence that there are substantial
numbers of neurons intrinsic to the SC that can provide
Two Stimuli with Local Inhibition
FIG. 8. Temporal evolution of activity when 2
stimuli are presented simultaneously to the model SC.
A: local lateral inhibition leads to a single site “winning” over the other, and then spreading rostrally. B:
global inhibition does not lead to a winner-take-all
competition, so both sites remain active. Activity at
these sites coalesces and spreads rostrally.
Two Stimuli with Global Inhibition
Early
Late
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SC. The logarithmic projection map defines the SC-asymmetric connection (Figs. 1 and 2), although it is the nature of the
intrinsic connections—not the nature of the projection map—
that gives rise to the spreading of activity.
Simulations with our model demonstrated that an SC map
with asymmetric connections produces a spread of activity on
the map from caudal to rostral (Fig. 3). This is an autonomous
spread in the absence of any external feedback signals. The
spread was ubiquitous over a wide range of parameters (Fig. 5)
and for both local and global inhibition (Fig. 6). Thus SCasymmetric connections could underlie the observed perisaccadic spread of activity on the SC, without any saccadic error
feedback to the SC. In contrast, simulation of the SC-symmetric connections never led to any such spread of activity (Fig. 3).
Therefore a model of SC-symmetric connections requires extrinsic input to produce the spread. Computationally, this is a
general phenomenon not only on the SC but in any brain area
with symmetric connections (Amari 1977); i.e., symmetric
connections lead to so-called neutrally stable attractors (Amari
1977; Seung 1996; Zhang 1996). Indeed, previous SC models
with symmetric connections had to add further directional
(anisotropic) intrinsic connections or specific extrinsic drives
to induce the spreading activity. The same approach is also
used in other areas, e.g., to move activity across head-direction
cells in hippocampus (Zhang 1996).
A
771
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NAKAHARA, MORITA, WURTZ, AND OPTICAN
Comparison of spread in the model to spread in the SC
The model can be critical in providing clues as to the nature
of the spread in the SC of the low-level buildup or prelude
activity, but only if the characteristics of the simulated spread
and the experimentally observed spread are similar. We think
there are several such similarities. First, the spread of activity
in the monkey SC has been shown to extend laterally as well
as rostrally (Anderson et al. 1998), which is the case with the
simulated spread (Fig. 3B). Second, another characteristic of
J Neurophysiol • VOL
the spread in the monkey is that it can be variable from trial to
trial and from neuron to neuron (Port et al. 2000). In the model
such variability is easily produced by varying the strength of
inhibition (Fig. 5, C and D). Although it is not obvious that the
strength of local inhibitory connections would vary in the
brain, variation in global inhibition from the substantia nigra
would be reasonable. Third, the activity in the cat is best
described as a moving hill of activity (Munoz et al. 1991b),
whereas that in the monkey is better described as a spread of
activity (Munoz and Wurtz 1995b). As described above for the
model, local inhibition produced a moving peak of activity
similar to that in the cat, whereas global inhibition produced a
spread of activity similar to that in the monkey (Fig. 6). Fourth,
simulated inactivation did not stop the spread of activity on the
SC map (Fig. 7). Finally, we have shown that if the inhibition
is too great, activity cannot spread; thus more inhibition on
burst than buildup neurons may explain why burst neurons in
the SC do not show a buildup of activity.
Implications of the SC-asymmetric connection model
A few generalizations can be made with respect to the SC.
First, the spread of activity can occur in the absence of any
feedback signal and thus need not be involved in saccadic
control. Second, the spread may require a reduction of inhibition from outside (e.g., the substantia nigra pars reticulata). If
that reduction occurred around the time of a saccade, it would
explain why the spread is always correlated with a saccade, and
it predicts that a reduction at other times should also reveal a
spread of activity. Third, if the spread is caused by asymmetric
intrinsic connections, it may be just an epiphenomenon, with
no contribution to movement control. This is not surprising,
given its small size relative to the burst of activity that drives
the saccade. Fourth, it is possible that the spread may have a
function we do not recognize, i.e., in establishing connections
during development or as part of a feedforward controller in
orienting behavior.
Implications of intrinsic connections
The spread of activity we have found in the model with
asymmetric kernels may be applicable to other maps in the
brain because all sensory maps in the brain are nonlinear. Are
there, then, signs that activity can spread across such maps
when inhibition decreases? One example may be the wellknown visual, aural, and tactile hallucinations that can accompany migraine headaches (for review, see Dahlem and Chronicle 2004). Lashley (1941) proposed that because the aura
appeared to move, it might be attributed to a spread of activity
across the cortex. The mechanism for this spread is not well
understood, but has been associated with a wave of depression
that spreads across the cortex (spreading depression of Leão
1986) or with changes in network properties (Dahlem et al.
2000; Reggia and Montgomery 1996; Schwartz 1980). Our
theoretical study suggests that activity might spread across
brain areas that have asymmetric intrinsic connections, simply
because inhibition is sufficiently reduced. This idea is consistent with the finding that prophylactic drugs for migraine
augment cortical inhibition and is more specific than a mechanism of general “hyperexcitability” (Dahlem and Chronicle
2004). Thus although phenomena (e.g., pathological and ex-
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intrinsic and extrinsic inhibition. There are also many areas
that provide extrinsic GABAergic input to SC, including such
oculomotor-relevant areas as substantia nigra pars reticulata,
zona incerta, and mesencephalic reticular formation (Appell
and Behan 1990). These inputs can cover extensive areas of the
SC and could easily be the source of the global inhibition in our
model. In addition, it might well be that studies using electrical
stimulation within the SC antidromically activated these external inputs, rather than just intrinsic neurons within the SC; this
remote effect would then have been included in the assessment
of long-range connections within the SC. In any case, it is clear
that there is a substrate of neurons supporting both local and
global inhibition within the SC.
There is currently no evidence for or against the other key
element in the model, the rostral– caudal asymmetry in the
strength of the local connections. There is no evidence of
asymmetry in long-distance connections in the intermediate
layer in either the cat (Behan and Kime 1996) or the ferret
(Meredith and Ramoa 1998). Local connections can be inferred
from studies of identified neurons in the cat (for summary see
Grantyn and Moschovakis 2004), but the functional extent of
connections is most clearly shown by the use of tissue slices
that allow precise measures of distances between sites of
stimulation and affected neurons. The technique revealed the
interaction between excitation and inhibition in SC by showing
that a reduction of GABAergic inhibition in rat intermediate
layers produces synchronized bursts of activity similar to the
burst before saccades (Isa et al. 2004; Saito and Isa 2003). The
extent of this excitatory and inhibitory action is shown most
exactly by SC slice experiments using photoactivation of caged
glutamate (Pettit et al. 1999; also see McIlwain 1982). Such
experiments show that strong excitatory responses are elicited
by photostimulation ⱕ300 ␮m from a recorded neuron’s soma
with only weaker responses out to 500 ␮m, whereas inhibitory
responses were restricted to photostimulation within 150 ␮m
(Özen et al. 2004). The picture that emerges is one in which the
distances for monosynaptic connections are short and the reach
of local inhibition is shorter than that for excitation. The short
distances produce no problem for the model, but the apparently
shorter range of inhibition than excitation does; the model
assumes a more extensive reach of inhibition than excitation.
However, the slice results are based on monosynaptic connections and our model does not require these local connections to
be monosynaptic. On another key point of rostral– caudal
asymmetry of these local connections, we are unaware of any
evidence for such asymmetry in the length of connections or
their strength. The slice technique, however, should be able to
contribute to the answer to this question as well as to the
relative reach of excitation and inhibition, once polysynaptic
connections are considered.
MODEL OF SPREADING ACTIVITY IN THE SC
perimental hallucinations) that arise in other brain maps may
have characteristics quite different from the spread of activity
we have considered in SC, we suggest that what they may have
in common is the consequence of reduced inhibition on maps
with asymmetric connections.
APPENDIX
Simulation details
The excitatory kernel E(x, x⬘) and the inhibitory kernel I(x, x⬘) were
represented by a two-dimensional Gaussian function of distance
d(x, x⬘)
a SC
冑2␲sSC
冋
dSC共x, x⬘兲2
2
2sSC
册
(A1)
冋
dvis共x, x⬘兲2
2
2svis
册
(A2)
or
a vis
冑2␲svis
exp ⫺
for the SC-symmetrical or visually symmetrical kernels, respectively.
Thus the shape of the kernel is like a “Mexican hat” [i.e., the
difference of the two Gaussian-shaped functions E(x, x⬘) and I(x, x⬘);
Fig. 4]. Here dSC(x, x⬘) indicates the distance between x and x⬘
measured in the SC space (in millimeters), whereas dvis(x, x⬘) indicates the distance measured in the visual space (in degrees) between
two points in the visual hemifield corresponding to x and x⬘; aSC and
avis scale the magnitude of the connection strength, respectively; sSC
and svis are the respective SDs. The parameters were set to (aSC, sSC)
values of (8, 0.1) and (8, 2) for the SC-symmetrical lateral excitation
and inhibition, respectively, or (avis, svis) values of (8, 1.5) and (18,
18) for the visually symmetrical lateral excitation and inhibition,
respectively, throughout the paper (except for Fig. 5).
When simulating global inhibition, we omitted the term I(x, x⬘) but
added the term G(t), given by
G共t兲 ⫽ G0 ⫹ ␺
冋冕
册
r共x, t兲␳dx
x僆SC
(A3)
where G0 represents a constant and ␺⵺ is a sigmoidal function given
by (Naka and Rushton 1966)
␺ 共z兲 ⫽
azn
z ⫹ zn
x⬘僆SC
Inputs representing visual target signals were simulated either by
adding positive constants to the equations of ⭸r(x, t)/⭸t (persistent
input) or by setting the initial conditions r(x, 0) to some positive
values (transient input) for all the discretized compartments included
in circles around the input locations. Specifically, we assumed H(x,
t) ⫽ 0.1 for all t (all the figures except for Fig. 5) or set r(x, 0) ⫽ 1
(Fig. 3). The radius of the circle around the input locations was set to
0.1 mm (all the figures except for Fig. 5) or 0.2 mm (Fig. 5). To
simulate partial inactivation of SC in Fig. 7, we set H(x, t) ⫽ ⫺1,000
for all t for all the compartments included in a 0.3-mm-radius circle
centered at (R, ␪) ⫽ (7.5°, 30°). In the simulation with double stimuli
in Fig. 8, we applied two persistent inputs at (R, ␪) ⫽ (20°, 45°) and
(R, ␪) ⫽ (20°, ⫺45°), whose strengths differ moderately: H(x, t)
values of 0.1 and 0.12, respectively.
ACKNOWLEDGMENTS
H. Nakahara thanks S. Amari and O. Hikosaka for support.
Present address of K. Morita: Amari Research Unit, RIKEN Brain Science
Institute, Hirosawa 2–1, Wako-shi, Saitama, Japan.
GRANTS
⭸r共x, t兲
⫽ ⫺r共x, t兲
␶
⭸t
再冕
Simulating inputs and inactivation
(A4)
n
0
where a is the saturation value of ␺(z); z0 is a value that gives half the
maximum; and n is a constant that determines the maximum slope.
Because the shape of the function ␺⵺ is sigmoidal, the overall form
of G(t) is also sigmoidal and its value is bounded between G0 and
G0 ⫹ a. All neural activities are summed together by 兰x僆SC r(x, t)␳dx
and fed into ␺⵺, thereby implementing the global feedback inhibition. The parameters were set as follows: G0 ⫽ 0.05; a ⫽ 300; z0 ⫽
600; and n ⫽ 8.
To run the simulation of the dynamics, we adopted the following
neural field equation (Amari 1977; Wilson and Cowan 1973)
⫹␴
respectively [where I(x, x⬘) is zero in cases using global inhibition]; ␳
is the density of the neurons (assumed to be uniform in SC space);
H(x, t) is the external input to the neuron at locus x at time t; G(t) is
the strength of the global inhibition (which is zero when using lateral
inhibition); ␴⵺ is the piecewise-linear activation function consisting
of a threshold and a saturation such that ␴(z) ⫽ 0 (if z ⱕ 0), z (if 0 ⬍
z ⱕ ␩), and ␩ (if ␩ ⬍ z). This integral–partial differential equation
(IPDE) was solved (MATLAB, The MathWorks) using discretization
methods. Specifically, the two-dimensional plane of the SC was
subdivided into small compartments. To facilitate computer simulations, we chose compartments distributed uniformly with respect to
visual space (each is 1 ⫻ 1°) by translating the uniform neural density
on the SC to the density in visual space. Thus the IPDE was
approximated by a set of ordinary differential equations describing the
activities of the representative neuron in each compartment. The
parameters were set to: ␶ ⫽ 1; ␳ ⫽ 200; and ␩ ⫽ 1. Because we are
interested in the qualitative rather than quantitative properties of the
dynamics, we simply set the time constant ␶ to be unity; thus the
variable t represents only relative time under this parameter set.
Specifically, it would be inappropriate to compare the value of t
between the lateral inhibition case (Figs. 3, 5, and 6A) and the global
inhibition case (Fig. 6B).
关E共x, x⬘兲 ⫺ I共x, x⬘兲兴r共x⬘, t兲␳dx⬘ ⫹ H共x, t兲 ⫺ G共t兲
冎
(A5)
where x denotes the location in the SC; t is time; r(x, t) is the neuronal
activity of the neuron at x; ␶ is the simulation time constant (see
following text); E(x, x⬘) and I(x, x⬘) represent the strength of the lateral
excitation and inhibition from the neuron at x⬘ to the neuron at x,
J Neurophysiol • VOL
This work was supported by Grant-in-Aid on Priority Areas (C) 17022049
(System study on higher-order brain functions) from the Ministry of Education,
Culture, Sports, Science and Technology (MEXT) of Japan and Grant-in-Aid
for Young Scientists (A) 17680029 from the Japan Society for the Promotion
of Science (JSPS) to H. Nakahara; the Fund from the US–Japan Brain
Research Cooperative Program of MEXT, Japan; JSPS Research Fellowships
for Young Scientists Grant 10834 to K. Morita; and Intramural Research
Program, National Eye Institute, National Institutes of Health, Department of
Health and Human Services grants to L. M. Optican and R. H. Wurtz.
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