(1991),7,531-546.Printedin the USA.
VisualNeuroscience
$5.00+ .00
UniversityPress0952-5238/91
CopyrightO 1991Cambridge
function
Motion selectivityand the contrast-response
of simplecellsin the visualcortex
DUANE G. ALBRECHT.q.NnWILSON S. GEISLER
Departmentof Psychology,Universityof Texas,Austin
(Rrcrrvro November5, 1990;Accrprro April 8, 1991)
Abstract
The responsesof simple cells were recorded from the visual cortex of cats, as a function of the position and
contrast of counterphase and drifting grating patterns, to assesswhether direction selectivity can be
accounted for on the basis of linear summation. The expected responsesto a counterphase grating, given a
strictly linear model, would be the sum of the responsesto the two drifting components.The measured
responseswere not consistentwith the linear prediction. For example, nearly all cells showed two positions
"null phase positions"); this was true, even for the most
where the responsesapproachedzero (i.e. two
direction selectivecells. However, the measured responseswere consistent with the hypothesis that direction
selectivity is a consequenceof the linear spatiotemporal receptive-field structure, coupled with the
nonlinearities revealed by the contrast-responsefunction: contrast gain control, halfwave rectification, and
expansive exponent. When arranged in a particular sequence,each of these linear and nonlinear mechanisms
performs a useful function in a general model of simple cells. The linear spatiotemporal receptive field
initiates stimulus selectivity(for direction, orientation, spatial frequency, etc.). The expansiveresponse
exponent enhancesselectivity. The contrast-set gain control maintains selectivity (over a wide range of
contrasts, in spite of the limited dynamic responserange and steep slope of the contrast-responsefunction).
Rectification conservesmetabolic energy.
Keywords: Visual cortex, Receptive fields, Direction selectivity, Contrast gain control, Contrast response
function, Motion
Introduction
The ability to sensemotion is crucial for vision and visually
guided behavior. Neurons in the visual cortex of monkeys and
cats play a fundamental role in motion sensitivityand most are,
to some extent, selectivefor the direction of motion (Hubel &
Wiesel, 1962; 1968). The investigations of Barlow and Levick
(1965)in the rabbit retina led them to propose that the basic
"summechanism of direction selectivity could be the result of
"simple
excitatory and inhibitory connections"
mation" over
(pp. 498-500). However, other studies have approached the
problem under the general assumption that direction selectivity is inherently nonlinear: for example, that it might involve a
multiplicative or divisive interaction betweeninputs (for general
reviewsof this topic, seeNakayama, 1985;Hildreth & Koch,
1987).
Direction selectivity can be produced through strictly linear
addition and subtraction of inputs (seefor example the initial
linear stagesof the quadrature models developedby Watson &
Ahumada, 1985;Adelson & Bergen, 1985).We have testedthe
The ordering of the authors' names is alphabetical; both authors
contributed equally.
Reprint requeststo: Duane G. Albrecht, Department of Psychology,
U n i v e r s i t y o f T e x a s , A u s t i n , T X 7 8 7 1 2 ,U S A .
usefulnessand validity of a linear model (the linear quadrature
model) for describingthe propertiesof motion-sensitiveneurons
recorded from the visual cortex of monkeys and cats (Hamilton et al., 1989). The spatiotemporal transfer function of simple cells was measured using sine-wavegrating patterns of
variable spatial and temporal frequency, drifting first in one direction of motion and then in the opposite direction. The results
of theseexperiments,using drifting gratings, showed that both
the amplitude and the phase satisfied severalstrong constraints
implied by the linear quadrature model. Quantitative measurements of the spatiotemporal receptivefields of simple cells provide further experimental support for the linear mechanism
(Mclean & Palmer, 1989).
On the other hand, the responsesof direction-selectivesimple cells to stationary flickering grating patterns do not appear
to be entirely consistent with a linear mechanism. Reid et al.
(1987) developeda model of motion sensitivity based upon linear summation of lateral geniculatenucleus (LGN) inputs. The
model was used to predict direction selectivity from the responsesto stationary flickering gratings. For the 19 cells tested,
"about half of the direction selectivity is
they concluded that
due to mechanismsthat sum in a linear fashion (p. VaD."
Hamilton (1987)developedand testedsimilar predictions based
upon a linear model of direction selectivity. From a sample of
27 cells, he showed examples of some cells which did agree with
531
D.G. Albrecht ond W.S. Geisler
532
the predictionsand some which did not and then concludedthat
both linear and nonlinear mechanisms contribute to direction
selectivity.
ln many respects,simple cells are quite linear (for recent reviews, see Shapley & Lennie, 1985; De Valois & De Valois,
1988; Skottun et al., l99l). However, the contrast-response
functions of simple cells reveal nonlinear behavior' Specifically,
as the contrast increases,the responsegenerally increasesrapidly with a power-function exponent greater than 1.0 (the average for cat simple cells being approximately 2.5), and then
saturates;furthermore, when measuredat different spatial frequencies, the contrast-responsefunctions shift vertically (on
"contrast-setgain mechanism"
log-log coordinates)indicating a
(Albrecht & Hamilton, 1982).Although thesenonlinear behaviors may not be inherent to the mechanism responsible for establishing direction selectivity,they may well influence the final
mechanism,even if the directionoutput of a direction-selective
selectivemechanismis based solely upon the simple principle of
linear summation. Thus, it would seem reasonableto incorporate these contrast-responsenonlinearities into any attempt to
understandthe responsesof simple cells.
In the presentstudy, we measuredthe responsesof simple
cells to stationary gratings and drifting gratings to quantitatively
evaluate the linear and nonlinear components of direction selectivity. We also performed independent measurementsof each
cell's contrast-responsefunction. We developed and tested the
predictions of the linear model and found, in agreement with
Reid et al. (1987)and Hamilton (1987),that linear summation
alone cannot account for the responsesto stationary flickering
gratings. However, if the nonlinearitiesrevealedin the contrastresponsefunction are combined with a linear spatiotemporalreceptive field, then the predicted responsesare consistent with
the measured responses, for both drifting and counterphase
stimuli.
Methods
The proceduresfor electrophysiologicalrecording, stimulus display, and measurementof neural responsesusing linear systems
analysis have been described in detail elsewhere (Albrecht &
Hamilton, 1982;Albrecht et al., 1984;Hamilton et al., 1989).
Briefly, severaldays prior to an actual experiment, under deep
barbiturate anesthesia,a preformed rigid plastic pedestal containing a recording chamber was attached to the animal's skull.
On the day of the recording, the animal was initially anesthetized with 20 mg/kg of the short-acting barbiturate thiamylal
sodium and then maintained throughout the experiment on
75Vo NO2/25V0 02 along with 1 mg,/kglh of thiamylal sodium. The head was held rigid in stereotaxiccoordinates (without ear and eye bars) using the plastic pedestal. The eyeswere
immobilized by continuous infusion of gallamine triethiodiode
(10 mglkglh) and the animal was artificially respiratedthrough
an endotrachealthroat tube. Singleneurons were recordedusing
glass-coatedtungsten or platinum-iridium microelectrodes.
The stimuli in the present set of experiments were spatial
sine-wavegrating patterns (presentedon a Conrac studio monitor at a frame rate of 100 Hz), either drifting at a fixed temporal frequency or flickering in a stationary position at a fixed
temporal frequency. The different stimulus conditions were randomly interleaved.One presentationconsistedof a block of ten
contiguous temporal cycles.Each block was separatedby a pe-
riod of time equal to the block length; during theseseparations'
the animal viewedmean luminancewith no contrast.A minimum
of four blocks were obtained, which resultedin 40 repeatedpresentations of each stimulus condition. Action potentials were
collectedinto l-ms time bins and the resulting averageperistimulus-time histograms were Fourier analyzed. The responsemeasure was the amplitude and phase of the first harmonic
component.
Once a single neuron was isolated and classifiedas a simple
cell, its optimal orientation was determined, and held constant
throughout the experiment. The spatial-frequencytuning, temporal-frequency tuning, and contrast-responsefunction were
quantitatively measured. Direction selectivity was determined
by measuringthe responsesto the optimal grating drifting in the
preferred and nonpreferred directions. The direction-selectivity index was the ratio of the measured responsessubtracted
from one (1 - nonpreferred,/preferred).Thus, for a cell which
only respondedto movement in one direction, the index would
be 1.0; for a cell which respondedequivalently to movement in
both directions, the index would be 0.0. The responsesas a
crifunction of contrast,R(C), were fitted using least-squares
teria to the following function: R(C) = R^u"'C'/(Cn + Cio),
where R-u^ is the maximum responserate, C56the semi-saturation contrast (the contrast required to produce 5090 of the
cell's maximum response),and n the power-function exponent.
This saturating power function provides a good description of
the contrast-responsefunction of neurons in the visual cortex.
Following these preliminary experiments, the responsesto
stationary counterphaseflickering gratings were then measured
at l2 different spatial/phasepositions in 30-degsteps.The contrast, (Ln,u, - L^i)/(L^u* * l-i.), of the counterphasepattern was modulated sinusoidally. The spatial and temporal
frequencies of the gratings were held constant at the optimal
values. The full set of measurementswere performed at a contrast level near the midpoint of the dynamic responserange for
the specific cell and then, as time allowed, at other contrasts
around the midpoint. For easeof viewing, the graphs (of the responsesas a function of the position of the counterphaseflickering grating) were normalized in the following three ways: (l)
the amplitude and phase responseswere shifted horizontally so
that the amplitude functions peaked at 90 and 270 dee, Q) the
phase functions were shifted vertically so that they passed
through the origin, and (3) the amplitudes were expressedas a
percentageof the maximum response.
Results
The initial goal of this study was to assesswhether the motionsensitivepropertiesof simple cells could be accountedfor on the
basis of a linear mechanism for direction selectivity. The basic
strategy was to measure the direction selectivity of each cell
using drifting gratings and then assessthe degreeto which the
linear model could predict the responsesto stationary counterphase flickering gratings.
Counterphasegratings are composed of two gratings of
equal contrast moving in opposite directions. Given a linear
mechanism for direction selectivity, and the principle of additivity in linear systems,the responseto a counterphasegrating
would be a weighted sum of the responsesto the individual
components present in the stimulus. Although the weight of
each component in the stimulus is equal, the weight of each
533
Visuol cortex motion and contrast
component in the responsewill depend upon the degreeof selectivity. For example, given a linear cell that is completely direction selective(i.e. no responsein the nonpreferred direction),
the weight given to the component drifting in the nonpreferred
direction would be zero, and the weight given to the component
drifting in the preferred direction would be one. As the direction selectivity changes from cell to cell, the weight given to
each component in the counterphase stimulus would vary accordingly.
Linear summation
This section describesthe expected responsesto counterphase
gratings presentedin different spatial positions, given a strictly
linear model, where the direction selectivityis simply due to linear summation of inputs (seeeqns. (Al-A5) in the Appendix).
These expectationsare then compared to the measured responsesof 4l simple cells. (Note that adding halfwave rectification does not affect the predictions describedin this section.)
The solid lines in Fig. lA plot the predictedamplitude and
phase of responsefor a linear nondirection selectivecell, as a
function of the position of a counterphase flickering grating.
From the work of Enroth-Cugell and Robson (1966;seealso'
Hochstein & Shapley, 1976; Movshon et al., 1978;De Valois
et al., 1982),we know that the amplitude of the responseshould
"null phase
be a sinusoidalfunction of spatial position with two
(We
the
have
adopted
zero.
positions," where the responseis
positive,
with
the
plotting
as
amplitudes
all
of
convention
phasesranging from 0-360 deg.) The temporal phase of the response should consist of two values separated by 180 deg: 90
over half of the range of spatial position (0-180 deg) and 270
over the other half (180-360deg). The filled circlesin Fig. 1A
plot the measured amplitude and phase responsesfor a representative nondirection-selectivesimple cell with a direction selectivity index of 0.08. As can be seen,the expectationsfrom a
linear model provide an adequate fit for this cell. The measured
amplitude varies as a function of position in a fashion which is
similar to the absolute value of a sine function, with two positions where the responseis near zero; the phase responsescluster around the two expected values, 90 and 2'70.
The solid lines in Fie. lB plot the predicted amplitude and
phase for a totally direction-selectivecell. Given strict linearity,
B. DIRECTIONSELECTIVE
SELECTIVE
A, NONDIRECTION
lrJ
roo
roo
a
a
o
f
F
qo
( L 5 0
=
0
o
r80
o
360
o
r80
360
JbU
UJ
@
r80
r80
6 .
A
0
o
t8O
360
O
r80
360
POSITION
cell (B) as a function of the position
Fig. l. Measuredresponsesof a nondirection selectivecell (A) and a highly direction-selective
of a counterephaseflickering stationary grating pattern. The amplitude of the responseis plotted in the upper panels and the
phase in the lower panels (note that 0 and 360 are the same point). The solid lines are the predictions from the model which
assumessrrict linearity, including the mechanismsresponsiblefor direction selectivity. The strictly linear model provided a reasonably good fit to the measuredresponsesof the nondirection-selectivecell (direction index 0.08). The predicted and measured
amplitudes varied much like the absolute value of a sine function. The predicted phasesare 90 deg for half the positions (0-180)
and270 for the other half (180-360); the measuredphasesclusteredaround thesetwo values with a slope near zero. On the other
hand, the strictly linear model provided a very poor fit ro the measured responsesof the direction-selectivecell (direction index 0.95). The predicted amplitudes are constant, whereas the measured amplitudes varied in a fashion similar to the absolute
value ofa sine function. The predicted phaseschanged continuously with a slope of 1.0, whereasthe measuredphasesclustered
into two groups separatedby 180 deg, and the slope within each group was approximately 0.6. (Stimulating contrast/peak res p o n s ef o r ( A ) i s 0 . 1 / 5 5 . 6 , a n d f o r ( B ) i s 0 . 0 8 / 4 7 ' 5 . )
D.G. Albrecht and W.S.Geisler
534
the amplitude would not change as a function of position, and
the phasewould changecontinuously,with a slope of one. As
noted above, these linear predictions are a simple consequence
of the fact that while the stimulus is actually composed of two
gratings drifting in opposite directions, a linear direction-selective cell will only be affected by the component moving in the
cell's preferred direction. If the cell is only responding to one
of the drifting components, changing the position of the counterphase grating would be equivalent to changing the starting
position of that component. Hence, changingposition would
not affect the responseamplitude (only the responsephase).
The filled circlesin Fig. 18 plot the measuredamplitude and
phase responsesfor a representativedirection-selectivesimple
cell with a direction index of 0.95. As can be seen,the expectations from a strictly linear model do not provide an adequate
fit for this cell. Like the nondirection-selectivecell, the measured amplitude varies as a function of position in a fashion
which approximates the absolute value of a sine function, and
there are two locations where the responseis near zero. The
phase responsesdo change continuously, as expectedfrom the
linear model; however, they clusterinto two distinct groups separated by 180deg, and the slopewithin eachgroup is closerto
0.5 than 1.0.
In order to quantify these response patterns for the total
population of cells, we indexed the size of the changein amplitude as a function of position by taking the maximum and minimum valuesof the responseand computing the following ratio:
(max - min)/(max * min). The value of this amplitude index
was then compared to the direction-selectivityindex. In general'
given a linear model, the amplitude index should be equal to
one minus the direction-selectivityindex.
Consider the expected changes in the amplitude index for
cells with different degrees of direction selectivity. Given a
completely nondirection-selectivelinear cell, there will be two
positions of the grating where the inputs sum to a maximum
value and two positions where the inputs sum to a minimum
"null phase positions"). Thus, for a cell with a direcvalue (the
tion index near 0.0, the amplitude changeswould be large: the
amplitude index would approach 1.0. Given a totally directionselectivelinear cell, the responseamplitude will be constant independent of position. Thus, for a cell with a direction index
near 1.0, the amplitude changeswould be small: the amplitude
index would approach 0.0. Given a direction-selectivelinear cell
betweenthesetwo extremes,the amplitude will changewith position: however, the differencesbetweenthe maximum and minimum values will decreaseas the direction selectivityincreases.
"null phase position"
(Note that, strictly speaking, the term
would be inappropriate for a linear cell with any degree of direction selectivity becausethe inputs do not perfectly cancel at
"
any position of the counterphasegrating; minimum phase position" would be perhaps a more accurate term.)
Figure 2A shows the distribution of the amplitude index for
the total sampleof cells. The amplitude for most cellsshows sizable changesas a function of the position of the counterphase
grating (only four cells had an index less than 0.5). The mean
of the distribution was 0.79. Figure 28 shows the scatter plot
for the amplitude index vs. the direction index for all cells. The
solid line, with a slope of -1.0, is the prediction of the linear
model. As can be seen, nearly all of the cells in this sample
showed larger changesin amplitude than was expected from a
linear filter. The slope of the regressionline for the data points
was -0.21: the coefficientof correlation was -0.32. Even the
a
-J
J
lrJ
o
l)
l!
o
E
tiJ
.D
R
l
z
0.s
A M P L I T U D CHANGE
E
A
r.o
1."
a
a
lrJ
(9
a
o
o
o
a
z
:
o . .
t
a
a
a
I
o
lrJ n
" .q"
o
:)
tr
J
(L
=
0
o
0.5
l.o
D I R E C T I OS
NE L E C T I V I T Y
B
Fig. 2. A: Distribution of the amplitude change index for the entire
population of cells. To provide a quantitative indication of the change
in amplitude as a function of the position of the counterphasegrating,
the following ratio was computed from the maximum and minimum
responses: (max-min)/(max+min). For linear nondirection-selective
cells, the amplitude index should be close to one; for linear directionselectivecells, the amplitude index should be close to zero. As can be
seen, all cells showed large changesin amplitude as a function of grating
position. The mean amplitude index was 0.79' B: Scatterplot of the amplitude index vs. the direction-selectivity index' In general, given a
strictly linear model for simple cells, the amplitude index should be
equal to one minus the direction-selectivity index' This expected relationship is indicated on the scatterplot by the straight line, with a slope
o f - 1 . 0 . A s c a n b e s e e n ,t h e c h a n g e si n a m p l i t u d e w e r e m u c h l a r g e r
than expectedfrom the strictly linear model and the degreeof direction
selectivity was only weakly correlated with the amplitude index. The
s l o p e o f t h e r e g r e s s i o nl i n e f o r t h e d a t a p o i n t s w a s - 0 . 2 1 ; t h e c o e f f i c i e n t o f c o r r e l a t i o nw a s - 0 . 3 2 .
most direction-selective cells showed sizable changes in amplitude as a function of position. For example, the directionselective cell illustrated in Fig. 1B had an amplitude index of
0.76, yet its direction-selectivity index was 0.95.
The slope of the phase function was measured for the total
population of cells and then compared to the direction-selectivity
index. (The slope was determined by fitting separate straight lines
to the datapointswhichfell within thexaxis rangesof 20-160deg
and 200-340deg.)As noted above,for a linear nondirectioncell,the phasesshouldclusterinto two distinctgroups
selective
by 180degand the slopewithin eachgroupshouldbe
separated
\{\
Visual cortex motion and contrast
0.0. For a totally direction-selectivecell, the phases should
changecontinuously from 0-360 with a slope of 1.0. ln general,
given a strictly linear filter, the slope of the phase function
would be equal to the direction-selectivity index.
Figure 3A shows the distribution of the slope for the total
sampleof cells. For 7090 of the population, the slope was less
than 0.5; only two cells had a slope greaterthan 0.7. The averageslope for the distribution was 0.35. Figure 3B shows the
scatter plot for the slope vs. the direction selectivity. As can be
seen,the majority of cells fall below the value expectedfrom a
strictly linear mechanism(indicatedby the solid line with a slope
of 1.0). The slopeof the regressionline for the data points was
0.52;the coefficientof correlationwas 0.70. Even the most di-
o
J
J
r!
C)
lr
o
E
lrj
d)
)
z
0.5
o
l.o
SLOPE
A
t.o
UJ
d
q o.5
)
o
o.5
o
r.o
DIRECTION
SELECTIVITY
B
F i g . 3 . A : D i s t r i b u t i o n o f t h e p h a s ei n d e x . T o p r o v i d e a q u a n t i t a t i v e
index of the change in the phase responses,as a function of the position of the counterphasegratings, the averageslope was computed. For
linear nondirection-selectivecells, the slope should be close to zero; for
linear direction-selectivecells, the slope should be close to one. As can
be seen, for most cells the slope was well below 0.5; the largest value
w a s 0 . 7 5 . T h e m e a n s l o p e w a s 0 . 3 5 . B : S c a t t e rp l o t o f t h e s l o p eo f t h e
phase responsesvs. the direction-selectivity index. ln general, given a
strictly linear model of simple cells, the slope should be equal to the
direction-selectivity index. This expected relationship is indicated in
the scatter plot by the straight line with a slope of 1.0. As can be seen,
the slopeswere lessthan expectedfrom the strictly linear model and the
degreeof direction selectivitywas moderately correlated with the slope.
T h e s l o p e o f t h e r e g r e s s i o nl i n e f o r t h e d a t a p o i n t s w a s 0 . 5 2 ; t h e c o e f ficient of correlation was 0.70.
cellsdid not have a slope of 1.0; for example,
rection-selective
cell shown in Fig. 28 was
the slope of the direction-selective
0.61.
Linear summation, rectification, and responseexponent
The results presentedabove clearly indicate that simple cells
cannot be describedas strictly linear filters. One possibleexplanation is that the direction-selectivemechanism itself is nonlinear. Another explanation,one that we considerin the Discussion,
is a phase-specificdirection-selectiveinhibition: a nonlinear silent inhibition from the nonpreferred component. However,
thesetypes of explanationsrequire postulatingrelativelycomplicated and undocumented(although see Dean et al., 1980)
A more parsimoniousapproachis to first
nonlinear processes.
considerother well-known nonlinearities.
One obvious nonlinearity of simple cells is seenin the responsesto sine-wavestimuli: becausesimple cells have little or
no maintained discharge,and they cannot produce negativere"halfwave rectified." Any responses,the responsewaveform is
alistic model of simple cells should include this nonlinearity and
thus in the sectionswhich follow, the models incorporate halfwave rectification. However, halfwave rectification following
linear summation of inputs cannot account for the results presentedin the section above becausethe responsesto both drifting and counterphasegratings are merely diminished by a fixed
amount (the amplitude of the fundamental harmonic is diminished by 5090).
Another prominent nonlinear property of simple cells is the
expansivepower-function growth in responseto contrast, before the onset of compressionand saturation (Albrecht & Hamilton, 1982; Sclar et al., 1990).The power-function exponent
varies from cell to cell; it is generallygreaterthan 1.0; the averagevalue is 2.5. If the exponent reflects an expansivenonlinearity occurring after linear summation, it could potentially
explain the discrepancybetweenthe responsesto drifting vs.
counterphasegratings. In this section,we developand test the
hypothesisthat simple cell direction selectivityis basedupon linear
summation of inputs and that this linear summation is then followed by halfwave rectification and an expansivenonlinearitythe exponentmodel (cf . Heeger,in press,for a similar proposal).
The exponent model is presentedformally in the Appendix (see
e q n s .( . 4 6 - , 4 1 l ) ) .
Consider the potential effectsof an expansivenonlinearity
on the responsesto counterphase flickering gratings presented
in different positions. Recall that given a strictly linear filter,
the magnitudeof variation in the counterphaseamplitudes(as
a function of position) is inverselyrelated to the degreeof direction selectivity(the amplitude index is 1.0 minus the direction
mechanism
index). If the output of a linear direction-selective
passesthrough an expansivenonlinearity, the variation in the
counterphaseamplitudeswould be enhanced-the strongerresponsesat the peaks (90 and 2'70ded would be disproportionately increasedrelative to the weaker responsesat the troughs
(0 and 180 deg). Similarly, the direction selectivityof the cell
would be enhanced; the stronger responsesin the preferred direction wouid be disproportionately increased relative to the
weaker responsesin the nonpreferred direction' The phases
would be unaffected by the expansivenonlinearity and thus the
slopesof the phasefunctions would directly reflect the linear dimechanism.
r e c ti o n - \ e l e c t i v e
To illustrate the basic effect of an expansivenonlinearity on
D.G. Albrecht and W.S. Geisler
536
A
roo
/",
UJ
l
o
J
o=
a
.i-\
f
F
cu
a
o
a
o
r80
360
g
r8O
360
360
lrJ
a
r80
I
(L
n
P0slTloN
Fig. 4. The effect of applying an erpansiveresponseexponent after linear summation is illustrated for a highly direction-selectivecell (direction index 0.95). The solid line in (A) shows the amplitudes expected
from a strictly linear direction-selectivecell with a direction index of
0.'72. A cell with this degree of direction selectivity shows some variation in amplitude (amplitude index 0.28) and the phasescluster into two
groups with a slope equal to the direction index. The dashed lines in (A)
show the effect of applying a power-function exponent of 4.2 (taken
from the contrast-responsefunction for this cell) to the expected responses.As can be seen,the exponent considerably enhancesthe variation in the amplitude and increasesthe peakedness'Furthermore, the
exponentenhancesthe expecteddirection selectivityfrom 0.72-0.99' The
solid line in (B) shows the expected phases before and after applying
the exponent: the exponent has no effect on the phases. (Stimulating
c o n t r a s t / p e a k r e s p o n s e: O . l / 6 5 . 9 . )
the responses to counterphase flickering stimuli, Fig. 4 plots the
measured amplitudes and phases of a direction-selective cell (direction index of 0.95, amplitude index of 0.87, and slope index
of 0.65). The solid curve in Fig. 4.A shows the predicted amplitudes of a strictly linear model with a direction index of 0.72;
the dashed curve shows what happens to the predicted amplitudes if they are passedthrough an exponentiationstage(the exponent for this particular cell, taken from its contrast-response
function, was 4.18). Note that the expansivenonlinearity results
in a nonsinusoidalrelationship betweenresponseamplitude and
position (an exponentiatedsinusoidalrelationship); specifically,
the exponent increasesthe difference between the maximum and
minimum values, narrows the width of each half-cycle, and increasesthe peakedness.For this cell, the exponent enhancedthe
predicted variation in the amplitude from 0.28-0.83, which
more closelymatchesthe measuredvariation of 0.87. The exponent also enhanced the predicted direction selectivity from
0.'72-0.99, which more closely matches the measureddirection
selectivityof 0.95. Becausethe exponent has no effect on the
phases,the solid line in Fig. 48 shows the predictedphaseswith
and without the exponent;the predictedslope, 0.72, is similar
to the measuredslope, 0.65.
ln the exponent model, the predictions for normalized counterphase responsesare completely determined by two parameters:(1) the direction selectivityof the linear mechanism(which
determinesthe weight given to each of the drifting components)
and (2) the nonlinear exponent (which determinesthe degreeof
responseexpansion).Both of theseparameterscan be estimated
from the responsesto drifting gratings. Specifically,by measuring the exponent n of the contrast-responsefunction and by
measuring the ratio of the responsesin the two directions, it is
possibleto make a parameter-freeprediction of the normalized
counterphaseresponsesas a function of position for both amplitude and phase. (The direction selectivityof the linear mechanism is given by the rth root of the ratio of the responsesin
the preferred and nonpreferred directions; see eqns. (A9) and
(A10) in the Appendix.)
Figure 5 replots the measured responsesto counterphase
cell is shown
gratings shown in Fig. 1; the nondirection-selective
cell is shown in Fig. 58.
in Fig. 5.A and the direction-selective
The expansiveexponent (taken from the contrast responsefunccell and 3 .22 f or the
tion) was LL86 for the nondirection-selective
direction-selectivecell; the measured direction-selectivity indices were 0.08 and 0.95. The smooth curves through the data
points represent the parameter-free predictions expected from
these measuredexponents and direction selectivities.As can be
seen,the fit to both the amplitude and the phasedata is quite
good.
Figure 6 shows the results of this analysis for nine typical
cells. For each cell, the exponent and the direction selectivityof
the linear mechanism were predetermined as described above
and thus there were no free parameters.The measureddirection
selectivityd, along with the exponent r and the direction selectivity of the linear mechanism D, are shown in the lower right
corner of each plot. These nine cells were chosen to illustrate a
range of direction selectivitiesand a range of contrast-response
exponents. From left to right, the measured direction selectivity decreases;this is reflected as a reduction in the slope of the
phase functions. From top to bottom the exponent decreases;
this is reflected as a reduction in the amplitude variation and,
as expected, the effect of the exponent is more evident for the
cells with a greater degreeof direction selectivity. The fit to all
41 cellswas quite good: for over half of the cellsthe predictions
accounted for more than 9090 of the variance (the averagevalue
was 8390).
An alternate approach to evaluating this model (similar to
the approach taken by Reid et al.,1987) is to compare the direction selectivitymeasuredfrom the responsesto drifting gratings with the direction selectivity expected from the responses
to stationary gratings. This analysiswas also performed on the
entire sample of cells. Specifically, for each cell, the amplitude
and phase responses(as a function of the position of the counterphaseflickering grating) were fitted using the linear-summation model, followed by rectification and the exponent of the
contrast-response function. The exponent for each cell was
fixed (from the independent set of measurementsof the contrast-responsefunction); however, the direction selectivityof the
linear mechanism was free to vary. The optimized value of the
linear direction selectivity, along with the measured exponent
of the contrast-responsefunction, were then used to predict the
537
Visual cortex motion and contrast
B . D I R E C T I OSNE L E C T I V E
A. NONDIRECTIO
SE
NL E C T I V E
roo
r00
3u
50
IJ
o
f
F
=
o=
0
0
0
r80
JbU
o
r80
360
r80
560
0
r80
360
LrJ
a
r80
TL
o
POSITION
Fig. 5. Measured responsesof a nondirection-selectivecell (A) and a direction-selectivecell (B) as a function of the position
o f a c o u n t e r p h a s e g r a t i n g . T h e s e a r e t h e s a m e d a t a p o i n t s s h o w n i n F i g1. ; h o w e v e r , t h e s m o o t h c u r v e s a r e t h e e x p e c t e d r e s p o n s e s
from a model which assumesa linear-summation mechanism for direction selectivity followed by an expansive nonlinear exponent (the exponent model). The exponent for each cell was determined from an independent measurementof the contrastresponsefunction. For the cell shown in (A) the exponent was 1.86 and for the cell shown in (B) the exponent was 3.22. As can
be seen, the responsesexpectedfrom linear summation followed by the nonlinear exponent provide a good fit to the measured
counterphase responsesof both the nondirection-selectivecell and the direction-selectivecell.
measureddirection selectivityof each cell (seeeqn. (Al0) in the
Appendix).
The scatter plot shown in Fig. 7 illustrates the results of this
analysis for all 4l cells; the direction selectivity predicted from
the responsesto stationary flickering gratings is plotted as a
function of the direction selectivity measured from the responsesto drifting gratings. As can be seen,the predicted and
measured values were, on average, quite similar. The straight
line, with a slope of 1.0, illustratesperfect correspondence.The
slope of the regressionline for the data points was 0.86; the coefficient of correlation was 0.89.
Contrast goin, linear summation, rectification,
and response exponent
The model described above (linear summation followed by an
expansive exponent) provides a good description of the responsesto counterphase gratings presentedat a fixed contrast
(within the dynamic range of a cell). However, it does not take
into account responsecompression and saturation. A more accurate model will have to include saturating nonlinearities.
The simplest hypothesis is that the expansive behavior at
lower contrasts and the compressive behavior at higher contrasts are due to the same responsenonlinearity - the response
nonlinearity model. However, this hypothesis seemsunlikely,
based upon the results of previous studies. Albrecht and Hamilton (1982) measuredthe contrast-responsefunction of striate
cells at different spatial frequencies. As they argue, if the saturation were due to a final responsenonlinearity, then saturation would occur at higher contrasts for nonoptimal spatial
frequencies,the magnitude of the saturated responsewould be
the same for all frequencies, and the spatial frequency tuning
would change with contrast. Instead, they found that saturation
occurred at approximately the same contrast for all spatial frequencies, the magnitude of the saturated responsewas different for each frequency, and the spatial-frequency tuning
remained relatively invariant. They point out that this behavior
"contrast-set gain" mechawould be produced if a saturating
nism preceded responsegeneration. This earlier work, in conjunction with the results presented above, lead us to propose
that there are two separatenonlinearities, a contrast-gain control nonlinearity that is responsible for saturation and a final
response-exponentnonlinearity, that is responsible for expansion-the contrast-gain/exponent model.
These two alternative hypotheses(the responsenonlinearity
model andthe contrast-gain/exponent model) can be discriminated by measuring contrast-responsefunctions for counterphase gratings presentedat different spatial positions. On the
one hand, if the saturation and expansion are part of the same
final responsenonlinearity, then the contrast-responsefunctions
UJ
o
f
F
J
c
=
rr n
vn
v
<
J
n
v
a
r80
360
360
360
360
LrJ
a
t/
I
(L
a
D =O . 5 3
t
n =4.tB
r80
360
r80
D=O.O2
n =2.39
d =o.o7
r80
D= O . 3 4
n=2.31
d =0 . 6 2
360
r80
360
r80
360
POSTT|ON
lrJ
o
l
lJ
o-
roo
roo
a
a
a
50
50
6ar
360
360
360
360
ld
D=O.5l
n =2 . t l
d =o.78
!
(L
r80
D =O.OB
n = 2.03
d =o . l 4
D =O . 4 8
n=?.25
d=o.77
r80
r80
360
t80
560
360
POSITION
lrl
o
f
rI nn
vv
a
roo
a
a
a
a
F
J
(L
roo
a
oo-
a
a
a
a
3U
a
=
r80
r80
360
360
360
lrJ
a
2 r8o
I
(L
o --.
D= o . s o
/
r80
r80
r80
D =O . 3 8
n= 1.47
d =o . 5 l
n =t . 3 4
d =0.67
n=0.96
d =0 . 8 9
r80
360
360
r80
360
PosrTloN
Fig. 6. Measured responsesof nine different cells as a function of the position of a counterphasegrating pattern. The smooth
curves through the data points are the predictions from the model which assumesa linear mechanism for direction selectivity
followed by a nonlinear responseexponent (rhe exponent model). The exponent for each cell was determined from an independent measurementof the contrast-responsefunction. As can be seen,the linear-summation model for direction selectivity,
followed by the contrast-responseexponent, provides a good fit to the measured responsesfor these nine cells. The nine cells
were organized in this figure such that from left to right direction selectivity decreasesand from top to bottom the exponent
decreases.As would be expected, the slopes of the phasestend to decreasefrom left to right, and the amplitude change tends
to decreasefrom top to bottom. In the lower right corner of each graph, the following three values are listed: n = measured
exponent of the contrast response function, d = measured direction selectivity, and D = the direction selectivity of the linear
mechanism. The direction selectivity of the linear mechanism (D) was determined by removing the effect of the exponent on
the measureddirection selectivity (d). (Stimulating contrast/peak firing rate for each cell: from left to right, top row, 0.1/65.9,
0 . 1 5 / 7 7. 5 , 0 . 0 5/ 5 7. 0 . m i d d l e r o w , 0 . 1 5 / 4 9 . 5 , 0 . 0 4 /4 7 . 5 , 0 . 1 3 / 1 4 . 8 , a n d b o t t o m r o w , 0 . 2 / 2 4 . 5 , 0 . 1 2 / 3 0 . 4 , 0 . 0 6 / 2 7. 3 . )
538
539
Visualcortexmotion and contrast
control occurs prior to the expansiveexponent (the model is formally defined in the Appendix, seeeqns. (Al2-A16)). Figures8C
and 8D plot the samedata points shown in Figs. 8A and 88; the
smooth curves are the best fit of the contrast-7oin/exponent
model. As can be seen,the predictions are quite accurate. The
model correctly predicts that saturation occurs at a fixed contrast, that the magnitude of the saturated responsevaries with
position, that the shape of the curve which describesamplitude
as a function of position remains invariant with contrast (i.e.
the curves at different contrasts only differ by a scale factor),
and that the responsesat nonoptimal positions remain nonop-
D I R E C T I OS
NE L E C T I V I T Y
r.o
o
lrj
F
o
o
I t t
lrj
E
(L
n
o
0.5
1.0
MEASURED
F i g . 7 . C o m p a r i s o n o f t h e d i r e c t i o n s e l e c t i v i t yp r e d i c t e d f r o m t h e
responsesto counterphase flickering gratings with the direction selectivity measuredfrom the responsesto drifting gratings for the total population of cells, assuming linear summation followed by the measured
exponent of the contrast-responsefunction. In this analysis, which differs from the analysis illustrated in Figs. 5 and 6, the exponent model
was fitted to each cell (using least-squarescriteria) with the direction
selectivityof the linear mechanismfree to vary. The exponent was taken
from the measured contrast-responsefunction. As can be seen in this
scatter plot, the correspondencebetween the measured and predicted
d i r e c t i o n s e l e c t i v i t yw a s q u i t e g o o d . T h e s t r a i g h t l i n e , w i t h a s l o p e o f
1.0, illustrates perfect correspondence.The slope of the regressionline
f o r t h e d a t a p o i n t s w a s 0 . 8 6 ; t h e c o e f f i c i e n to f c o r r e l a t i o nw a s 0 . 8 9 .
measured at different spatial positions would reach saturation
at different contrasts and the final saturated response levels
would be identical (for both optimal and nonoptimal positions).
On the other hand, if the contrast-goin/exponent model is correct, then the contrast-responsefunctions would reach saturation at the same contrastsand the final saturatedresponselevels
would vary with the spatial position of the counterphasegrating.
Figure 8 plots the responses of a typical cell to counterphase gratings presented at different positions and contrasts.
In Fig. 8A, responseis plotted as a function of position at five
different contrast levels(0.02, 0.04, 0.08, 0.16, and 0.32). In
Fig. 88, responseis plotted as a function of contrast at four different phase positions (52, 82, 142, and 172). The smooth
curves through the data points are what would be expected if
both the saturation and the expansive exponent were due to a
final responsenonlinearity (in this case, a saturating power
function). As can be seen, the measured responsesdo not follow the predicted trend; instead, the responsestend to saturate
at the same contrast and the magnitudes of the saturated responsesare different for each position (responsesat nonoptimal
positions remain nonoptimal, independentof contrast).
To quantitatively test the hypothesis that there are separate
contrast-gain and expansiveexponent nonlinearities, we evaluated a specific version of the contrast-gain/exponent model.
This version consisted of a multiplicative contrast-gain mechanism, a linear-summation mechanism, halfwave rectification,
and a final expansiveexponent. The rectification and expansive
exponent occur after linear summation and the contrast-gain
timal.
The predictionsare determinedby four parameters:the direction selectivityof the linear mechanism(D), the expansiveexponent of the final nonlinearity n, and two associatedwith the
contrast-gain mechanism-a semi-saturationconstant (450)
and an exponent (ln). The optimized values for these parame t e r sw e r e D = 0 . 5 8 , n = 2 . 4 1. C s o = 0 . 0 2 5 . a n d m : 1 . 6 3 .
(When the contrast-responsefunctions were fitted using a saturating power function, the exponent was found to be 2.19,
slightly lessthan 2.47,because,in the contrast-gain/exponent
model, the slope of the contrast-responsefunction is determined
by both the contrast-set gain nonlinearity and final response
nonlinearity.) If contrast is held constant,the two parameters
of the contrast-gain mechanism are fixed and the model becomes equivalent to the simpler model describedin the section
above (the exponent model). Thus, the predictions of the contrast-gain/ exponent model for all of the data measured at a
fixed contrast (e.g. Figs. 4-7) are identical to the predictions of
a model consisting of linear summation followed by rectification and an expansive exponent.
The contrast-gain/ exponent model is consistent with other
known characteristicsof simple cells. The model correctly predicts the contrast-responsefunctions measuredat different spatial frequenciesand orientations. Furthermore, it predicts the
fact that spatial-frequency selectivity, orientation selectivity, direction selectivity, and degree of ocular dominance are relatively
invariant with contrast. It can also predict the phase and amplitude responses,measured as a function of spatial and temporal frequency.
Discussion
A linear mechanism for direction selectivity
The first goal of this investigation was to test the hypothesis
that the direction selectivityof simple cells is based upon linear
summation. To do this, we presentedstationary flickering gratings at different positions and contrasts. The linear model predicts that the responsesto a counterphasegrating should be the
sum of the responsesto the two drifting components. For example, the responsesof a highly direction-selective cell to a
counterphasegrating should be no different than the responses
to a drifting grating moving in the preferred direction. The results clearly showed that a model of simple cellswhich is strictly
linear (i.e. not only the direction-selectivemechanism is linear,
but all other aspectsas well) does not produce accurate predictions. However, when the nonlinearitiesevident in the contrastresponsefunction are taken into account (contrast-gaincontrol
and expansive responseexponent), a linear direction-selective
mechanism does produce accurate predictions.
D.G. Albrecht and W.S.Geisler
540
R E S P O N S EN O N L I N E A R I T Y
r oo
lrl
o
o
350
l
F
J
(L
=
- GAIN/ EXPONENT
CONTRAST
roo
too
A
50
0
tBO
360
POSITION
O
20
40
CONTRAST
Fig. 8. Responsesto a counterphase grating at different positions and contrasts. In (A), the amplitude of responseis plotted
as a function of position at five different contrasts(0.02, 0.04, 0.08, 0.16, and 0.32). In (B), the responsesare plotted as a function
of contrast for four different positions of the grating (52, 82, 142, and 172).The smooth curves through the measuredresponses
show the predictions assuming a linear direction-selectivemechanism followed by a saturating responsenonlinearity that is expansive at low contrasts and compressiveat high contrasts. As can be seen,this model, responsenonlinearity model, provides
a very poor fit to the measured responses.In (C) and (D), the same measured responsesare plotted as described above. The
smooth curves show the predictions assuming a linear direction-selectivemechanism coupled with a saturating contrast-setgain
control mechanism which scalesthe value of the sum dependent upon the overall level of contrast; this scaledvalue is then followed by an expansiveresponseexponent. As can be seen,this model, contrast-goin/exponent model, provides a very good fit
to the measured responses.
C o nt rast- gain / expo nent mode I
We are led to suggestthat the responsesof simple cells might
be usefully modeled, as illustrated in Fig. 9, with four basic
stages(cf. Heeger, in press, for a similar proposal). The fundamental stimulus selectivities(for direction, orientation, spatial
frequency, etc.) are establishedthrough a linear spatiotemporal filter (i.e. linear summation over spaceand time). Halfwave
rectification and an expansive exponent are applied after linear
summation; contrast-gain control is applied before the expansive exponent. This model is an extension of the strictly linear
model presentedin Hamilton et al. (1989)and is consistentwith
the results presented there.
At this point in time, we cannot be certain about the specific
physiological and anatomical mechanismswhich constitute
these four stages.The contrast-setgain control observedin cortical cells may be the result of mechanisms in the retina, the
LGN, the cortex, or all three. Some of this contrast-gain control is likely to be occurring in the retina (for a review see,
Shapley & Enroth-Cugell, 1984); some may be feedforward
gain control from LGN to cortical cells; and some may be due
to a cooperative network in the cortex (seeBonds, 1990, for a
541
Visual cortex motion and contrast
R E C T I F I C A T I O NE X P A N S I V E
LINEAR
EXPONENT
SUMMATION
OUTPUT
I NPUT
CONTRAST
GAI'i CONTROL
t'ig. 9. The contrast-gain/exponent model for simple cells. The model
consistsof four components. The linear component (a linear spatiotemporal filter) performs summation over spaceand time; it establishesthe
fundamental stimulus selectivities(such as direction, orientation, spat i a l f r e q u e n c y ) .T h e c o n t r a s t - s egt a i n c o n t r o l c o m p o n e n t ' a s a t u r a t i n g
multiplicative mechanism, scalesthe responsesas a function of contrast;
it maintains the stimulus selectivities independent ol contrast. The
rectification component placesa threshold at zero response;it conserves
metabolic energy. The expansiveresponseexponent enhancesstimulus
selectivity. The wide arrows indicate global inputs dependent upon
responsespooled across multiple cells. The thin arrows indicate local
inputs dependent upon the responseswithin a single cell (or sequence
of cells).The contrast-gaincontrol is drawn below the other stageswith
ivide arrows on both sides,becauseit may be applied at any stageprior
to the responseexponent and it may receiveinput from both higher and
lower levels. This model predicts the responsesof simple cells to drifting and counterphase gratings as a function of contrast.
view of Hildreth & Koch, 1987). In responseto drifting and
counterphase gratings, this multiplicative direction-selective
model does not produce modulations at the fundamental frequency; rather, it predicts sine-squared responses(frequency
doubling) prior to the integration stage, and increased mean
rate after the integration stage. Thus, the predictions of this
model are inconsistent with the measured responsesof simple
cells (whether or not the integration stage is included).
On the other hand, the results do not rule out all alternate
hypotheses.The results could potentially be due to some other
combination of linear and nonlinear operations. For example,
consider a model in which the outputs of two overlapping linear mechanisms,each selectiveto the opposite directions of motion, are halfwave rectified, differenced, and then halfwave
rectified again. Becauseof the rectification, one of the two direction-selectivemechanisms is silent in the sensethat it produces no overt response.This model predicts that the responses
to drifting and counterphase gratings are halfwave-rectified sine
waves which modulate at the frequency of the stimulus. In responseto counterphasegratings presentedin different positions,
the model predicts responsessomewhat similar to the responses
illustrated in Fig. 6. However, the exponent model and the contrast-gain/exponent model fit the data more accurately and they
are more parsimonious (in that they are based upon simple linear summation coupled with the well-known nonlinearitiesof
t h e c o n t r a s t - r e s p o n sf u
en c t i o n ) .
Contr(tst-setgain control maintqins stimulus selectivities
discussion of related issues). The predictions of the controst'
gain/exponent modelare compatible with any of these possibilities (because the contrast-gain control can occur at any point
prior to the expansivenonlinearity; seeAppendix). The linear
mechanism responsible for stimulus selectivity (i.e. the linear
spatiotemporal receptive field) could be due to summation of
LGN inputs, summation of inputs from other cortical cells, or
both. Again, the predictions of the contrast-gain/exponent
model are not inconsistent with these possibilities, becausethe
ordering of linear operations (within the linear-summation stage)
does not affect the final output.
The expansive exponent may be the result of a variety of
physiological mechanisms within the simple cell; it could be a
simple static nonlinearity (as described in the Appendix), or it
might be a multiplicative gain mechanism set by a time-averaged
responsewithin the cell. The predictions of these two alternatives are very similar; the only major difference is that the static
nonlinearity will changethe shapeof the poststimulus-time histograms. For the present experiments,the predictions are identical (except for a scaling lactor). Rectification may be the result
of a responsethreshold at zero. Interestingly,rectification could
contribute to the exponent: a responsethreshold greater than
zero would increasethe slope of the contrast-responsefunction
(plotted in log-log coordinates), and thus a threshold could be
one component of the expansivenonlinearity' These two nonlinearities (rectification and the expansiveexponent) could occur at any point after the linear summation.
The results of the present experiments may rule out some alternate hypotheses.For example, consider a nonlinear Reichardttype model in which the outputs of two nondirection-selective
mechanisms,with different spatial positions and time constants,
are multiplied and then integrated (similar to Fig. 1A in the re-
Cortical neurons have a limited dynamic response range. As
stimulus strength increases,responsemagnitude generally increasesrapidly as a power function with an exponent greater
than one and then saturates. While the expansiverelationship
between stimulus and responsecan enhance stimulus selectivity (seediscussionbelow), the saturation introduces a problem
which could potentially diminish stimulus selectivity.
If the responsesaturation found in simple cells were determined solely by the overall magnitude of the response,it would
causenonpreferred stimuli (presentedat high contrasts)to produce responsesas large as preferredstimuli (seeRobson 1975,
for a discussionof this issue). However, previous work has
shown that for most simple cells, spatial-frequencytuning, orientation tuning, eye preference,and direction selectivity are to
a large extent unaffected by increasesin contrast (Albrecht'
1978,Figs. l7-19; Albrecht & Hamilton, 1982,Figs. 7-10; Sclar
& Freeman, 1982;Li & Creutzfeldt, 1984).Taken together with
the resultsof the presentexperiments,thesefacts argue for contrast-set gain control which attenuates responses dependent
upon the overall level of contrast.
In order for a multiplicative gain control to produce responsesaturation as a function of steady-statecontrast, the gain
factor must decreaseas contrast increasesso as to exactly cancel
the increased stimulation. Becausethis saturation is achieved
within a few temporal cycles, the gain control would have to occur quite rapidly (within a few hundred milliseconds) and thus
it should perhaps be distinguished from the rather slow type of
cortical cell adaptation which occurs following prolonged exposure to high contrasts(see,for example,Albrecht et al., 1984;
Ohzawa et al., 1985).This rapid contrast-gaincontrol is perhaps
"global inhibition" documented by Bonds
more similar to the
(1990),the "rapid suppression"documentedby Nelson (1991)'
"contrast normalization" proposed by Heeger (in press).
and the
542
The net result of the contrast-setgain control mechanism is
that the fundamental stimulus selectivitiesof a particular cell
are maintained independent of the overall level of contrastin spite of the limited responserange and the steepslopesof the
contrast-responsefunction.
Another consequenceof contrast-setgain control is that,
once above the saturation contrast (which is often quite low),
the stimulus-response relationship is highly constrained: each
stimulus can only evoke one particular responselevel and thus,
in comparison to a linear system, the responserate is more
uniquely tied to the stimulus. For example, the maximum responselevel signalsa unique stimulus (the stimulus that is simultaneouslyoptimized on all relevant dimensions).Other response
levels signal particular setsof nonoptimal stimuli which are determined by the tuning function of the cell along the relevant
dimensions.Consider a polar plot of the two dimensionsof spatial frequency and orientation. Each response level defines a
concentric ring obtained by slicing the tuning function at that
responselevel; only those stimuli on this ring could have generated that particular response level. The size of the ring decreasesas the responselevel increases,collapsingto a point at
the maximum response,thus illustrating that the higher the responserate the more unique the relationship between stimulus
and response.
Rect ification conserves energy
Simple cells have little or no maintained discharge and, when
stimulated with a sine wave, only half of the modulation appearsin the response(see,for example,Movshon & Tolhurst,
1975;Albrecht & De Valois, l98l). Thus, the responsesappear
to be halfwave rectified. If simple cells did have maintained discharges,at half their maximum responserates, then both halves
of the modulation could potentially be represented. Because
"reversed-polaritypairs") are rethey do not, two cells (i.e.
quired for transmissionof the information-one for the top
half of the signal, and one for the bottom half. Although doubling the number of neurons might seem inefficient, it greatly
reducesthe number of action potentials produced, the amount
of synaptic transmitter substancereleased,and hencemetabolic
effort.
To compare the efficiency of thesetwo alternatives(one cell
on at half-max vs. a pair of rectifying cells), consider the metabolic effort associatedwith encoding the luminance variations
of sine-wave stimuli. For a cell to maintain a continuous responseat half the maximum rate, the metabolic activity would
obviously be quite high. Furthermore, since the average response of the cell would not vary with stimulus contrast, the
metabolic rate would remain high whether the cell was responding to an actual visual contrast or simply waiting for a potential visual contrast.
This would not be true for pairs of rectifying cells. When the
contrast is zero, the averageresponserate will be zero-a clear
savings. However, even if the contrast of the stimulus is high
and the two rectifying cells are responding at their maximum
levels, the total responsegeneratedby both cells would be approximately half that of the one cell. Furthermore, whenever
the responsesof the two cells are below saturation, then even
smaller responseswill be generatedby the rectifying cells.
In general, the metabolic savings factor s for sine-wave stimuli is inversely related to the fraction / of the saturated response
evoked by an optimal sine wave (s = r/4f). Given that simple
D.G. Albrecht and W.S. Geisler
cells are so selective(for position, orientation, spatial frequency,
direction of motion, etc.), their preferred stimulus is not always
present and thus most cells will be far below the maximum saturated responseand the savingsin metabolic effort will be enormous. Supposethat the visual cortex as a whole is responding
on averageat l9o of its maximum level (the actual number is
undoubtedly much smaller), then using rectified pairs would reduce the metabolic activity by a factor of 78.
Exponent enhoncesstimulus selectivities
The responsesof cortical cells in both cat and monkey increase
rapidly as a function of the strength of stimulation-the responseincreaseswith a power-function exponentgreaterthan
one. The value of the exponent in this relationship varies from
cell to cell; the mean for cat simple cells is 2.5 and thus the
slopes are very steep (in log-log coordinates). The steep slopes
of the contrast-responsefunction could potentially serveto enhance any and all previously establishedstimulus selectivities.
By itself, an exponentiation stage (with an exponent greater
than one) could not produce spatial-frequencytuning or orientation tuning; nor could it produce direction selectivityor ocular dominance. Furthermore, it may or may not enhance
discriminability, dependingupon where noise is introduced (see
Geisleret al., 1991,for a detailedanalysisof discriminationin
single neurons). However, any stimulus preferencesinherent in
the input to the exponentiation stagewill be greatly magnified.
Consider what happens to the responsesto two separate
stimuli, a grating drifting to the left and a grating drifting to the
right, as these responsespass through the exponentiation stage
of a particular direction-selectivecell with an exponent of, say,
2.5. For one of thesestimuli, the grating drifting to the right,
linear summation of inputs resultsin a net sum of, say, 2.0; for
the other stimulus, a grating drifting to the left, linear summation of inputs results in a net sum of, say, 6.0. When thesetwo
linear sums are passedthrough the exponentiation stage their
relative magnitude is dramatically altered: the grating drifting
to the right produces a responseof 5.6 and the grating drifting
to the left produces a responseof 88.2. Thus, before the exponentiation stage, the preferred stimulus produced a net linear
sum which was only three times greater than the nonpreferred
stimulus, whereasafter the exponentiation stage, the preferred
stimulus produced a responsewhich was nearly l6 times greater
than the nonpreferredstimulus.
The enhancementof stimulus selectivityintroduced by an expansiveexponenteasesthe structural requirementsfor establishing highly selectivereceptive fields. Given a strictly linear
relationship between input strength and responsemagnitude,
the structural requirementsare more demanding. For example,
consider what would be required to produce a cell with a direction selectivity index of 0.95 - a very direction-selectivecell. If
the contrast-responserelationship was strictly linear, then the
spatiotemporal receptive field would have to be very oriented
in space-time.Although not impossible,this linear receptive
field is actually quite difficult to realize physically (seeWatson
& Ahumada, 1985; Adelson & Bergen, 1985). On the other
hand, given an exponentiation stage, the demands on the spatiotemporal receptivefield are considerablyless,and much easier to realize. This point is clearly illustrated in Fig. 10 where
direction selectivity after expansion is plotted as a function of
direction selectivity before expansion, given different exponents.
Similar logic applies to other stimulus selectivities. ln the
Visual cortex motion and contrasl
543
t00
0.8
3
o
F
f
z- ^U
U.b
o .
5
uo
o-
z
x
U
t :
UJ
i\\
:r \
:\ \
it\
: l
=
oa
\
" t \
: l \
a a
a\
UJ
F
r
"l
..\
o
-'t/.''"'
0.1
A R
t.0
Ar\
SPATIALFREOUENCY
a
o.2
0.4
0.6
0.8
r.0
B E F O R EE X P O N E N T ( D )
l'ig. 10. Effect of expansive exponents on direction selectivity. The
horizontal axis is the direction-selectiveindex of a linear filter before
an erpansivenonlinearity; the vertical a..risis the direction-selective
index
after an expansive nonlinearity. The smooth lines show the effects of
e\ponents ranging from 1.0-6.0, in unit steps.The curves illustrate that
an erpansiveexponent can substantiallyincreasethe direction selectivity.
For erample, if the index is 0.3 before expansion, it is increasedto approrimately 0.65 by an exponent of 3.0 and to approximately 0.9 by an
e r p o n e n to f 6 . 0 .
caseof orientation tuning, a greater degreeof selectivity could
be attained with a shorter receptivefield. In the caseof spatialfrequencl' tuning, a greater degree of selectivity could be attained uith fewer numbers of excitatory and inhibitory regions.
If the rather broad spatial-frequencytuning of the center/surround inputs to the cortex are passedthrough a cascadeof neural elements(each with exponentsgreater than one), the tuning
at the end of the serieswill be very narrow.
Figure 1I illustratesthe potential effect of an expansiveexponent on spatial-frequency selectivity. The circles connected
b1' solid lines plot the measuredresponsesof a center/surround
receptivefield, recordedfrom the LGN of a cat, as a function
oi spatial frequency. Although this cell does show a certain degree of spatial-frequency tuning, it is not very selective: the
bandwidth at half-height is approximately 2.5 octaves.The circles connectedby dashed lines illustrate what would happen to
the responsesif they were passedthrough a cell with a contrastresponseexponent of 2.5; the circles connected by dotted lines
illustratethe consequences
of cascadingthrough a secondcell
u'ith an exponent of 2.5 (or equivalently, instead of two cells in
series,one cell with an exponent of 5.0). As can be seen,the
mere addition of an exponentiationstageproducesconsiderable
narrouing of the spatial-frequency
tuning. An exponentof 2.5
reducedthe bandwidth to approximately 1.5 octaves,which is
closeto the averagevalue for cat simple cells.A secondexponent of 2.5 reducedthe bandwidth still further to aooroximatelv
1. 0 o c t a v e s .
The longstanding discrepanciesbetween the degreeof selectivitl' and the exact shape of the receptive field (for recent revieri of this issue,seeTadmor & Tolhurst, 1989)could perhaps
be accountedfor by the exponentof the contrast-responsefunction. The basic discrepancyis that the number of parallel excitatorv and inhibitorv resions in the measuredreceotivefield does
Fig. 11. Effect of expansiveexponentson spatial-frequencyselectivity.
The filled circles connected by solid lines plot the responsesof a cell
recorded from the lateral geniculate nucleus of a cat, as a function of
the spatial frequency of a drifting sine-wavegrating pattern. Although
this cell shows some degree of selectivity for spatial frequency, as expected from the center/surround receptive field, it is rather broadly
tuned; the bandwidth of the cell at half-height is 2.5 octaves.The filled
circles connected by dashed lines plot the effect of applying an expans i v e e x p o n e n to f 2 . 5 ( t h e a v e r a g ev a l u e o f c a t s i m p l e c e l l s )t o t h e m e a sured responsesofthe cell. The filled circles connected by dotted lines
plot the effect of applying a second exponent of 2.5 (or, equivalently,
applying an exponent of 5.0 directly to the LGN responses).As can be
seen, the net effect of an exponent greater than l 0 is to enhance the
selectivity.After applying an exponent of 2.5, the bandwidth decreased
to 1.5 octaves(which is close to the averagevalue for cat simple cells);
after cascading through a second exponenr of 2.5, the bandwidth is
decreasedstill further to 1.0 octaves. As this figure illustrates, the expansive contrast-responseexponentsof cortical cells can potentially enhance stimulus selectivity.
not always correspond to the number expected from the measured spatial frequency responsefunction. An expansiveexponent could help explain this discrepancyin two ways. First, it
would narrow the bandwidth of the measuredspatial-frequency
responsefunction (as describedabove). Second,it would diminish the responsesof the weaker peripheral regions of the receptive field and enhance the responsesof the dominant central
regions. Both of these effects would contribute to the discrepancy between measurementsin the space domain and spatialfrequency domain.
Barlow (1972) and others (for a review, see Maunsell &
Newsome, 1987) have suggestedthat as sensory information
flows from the peripheral receptors through the thalamus and
then on through the various regions of the cerebral cortex, the
stimulus selectivity of the sensory neurons tends to increase:
the receptorsrespond to a very broad range of stimuli, whereas
the higher level neurons respondto a very narrow range of stimuli. As a consequence,the responsesof the higher level neurons
could potentially signal the presenceof complex stimulus features. The exponent of the contrast-responsefunction can help
achievethis type of stimulus selectivity:if each stageof processing along a sensorypathway simply exponentiatesthe final output (using exponents greater than one), then the stimulus
selectivity inherent in the summation of inputs would be enhanced. The recent report from Sclar et al. (1990) is consistent
with this basic notion. They demonstrated that the average
value of the exponent of the contrast-response
function for neu-
D.G. Albrecht and W.S.Geisler
544
rons in the visual pathway of monkeys increasesfrom 1.6 in the
parvocellular layer of the LGN to 2.4 in the striate cortex to 3.0
in the middle temporal visual area. Although one benefit of
thesesteepcontrast-responsefunctions might be enhancedcontrast sensitivity (as they point out), another benefit might be
generalizedenhancementof stimulus selectivityfor higher level
neurons.
Acknowledgments
We thank the Universityof Texas.We alsothank Larry Sternfor assistingwith dataanalysisand Gail Geislerfor draftingthe figures.This
wassupportedin part by NIH GrantEY02688to W.S.Geisler.
research
References
MAUNsELL,J.H.R. & Nrwsor,,m,W.T. (1987).Visual processingin monkey extrastriate cortex. Annual Review of Neuroscience 10,363-401.
M c L r a N , J . & P e r u r n , L . ( 1 9 8 9 ) .C o n t r i b u t i o n o f l i n e a r s p a t i o t e m p o ral receptive field structure to velocity selectivity of simple cells in
area l7 of caI. Vision Research 29,675-6'79.
MovsnoN, J.A., Tnoupsorq, I.D. & Torrrursr, D.J' (1978).Spatial summation in the receptivefields of simple cells in the cat's striate correx. Journal of Physiology 283, 53-77.
MovsnoN. J.A. & Tornunsr, D.J. (1975). On the responselinearity of
neurones in cat visual cortex. Journal of Physiology 249,56-57P.
Nerevarrre, K. (1985). Biological image motion processing:A review.
Vision Research 5, 625-660.
NarsoN, S.B. (1991). Temporal interactions in the cat visual system. I.
Orientation-selective suppression in the visual cortex' Journal of
Neuroscience ll, 344-356.
Onzrwe, 1., Scren, G. & FTIEMAN,R.D. (1985). Contrast gain control
in the cat's visual system. Journal of Neurophysiology 54, 651-667 .
RErr, R.C., Soooar, R.E. & SsaprEv, R.M. (1987)' Linear mechanisms of directional selectivity in simple cells of cat striate cortex.
Proceedings of the Nntional Academy of Sciencesof the U.S.A.84'
8740-8744.
R o B S o N ,J . G . ( 1 9 7 5 ) .R e c e p t i v ef i e l d s : S p a t i a l a n d i n t e n s i v er e p r e s e n tation of the visual image. In Handbook of Perception, Vol. 5: Vis i o n , e d . C e n r r n r r r r , E . C . & F n m o r u . q . NM, . P . , p p . 8 l - 1 1 2 . N e w
York: Academic Press.
Scren, G. & FrsrltlN, R.D. (1982). Orientation selectivityin the cat's
striate cortex is invariant with stimulus contrast' Experimental Brain
Reseorch 46,457-461.
S c r . q . RG. .. . M . l u N s n r r , J . H . R . & L E N N I E ,P . ( 1 9 9 0 ) .C o d i n g o l i m a g e
contrast in central visual pathways of macaque monkey. Vision Re-
energymodels
AorrsoN,E.H. & BrnceN,J.R.(1985).Spatiotemporal
for the perceptionof motion. Journal of the Optical Societyof
America2,284-299.
Arnrncsr, D.G. (1973).Analysisof visualform. DoctoralDissertation'
Universityof California,Berkeley.
to
ArrnrcHr, D.G. & DE Verots,R.L. (1981).Striatecortexresponses
periodicpatternswith and without the fundamentalharmonics.
J ournal o-fPhysiology319, 497-514.
Arnrrcrrt. D.G., Falnen, S.B.& HavnroN, D.B' (1984).Spatialconrecordedin the cat'sviof neurones
trastadaptationcharacteristics
sualcortex.Journal of Physiology347,'713-739.
search 30, l-lo.
D.B. (1982).Striatecortexof monkey
ArnnEcnr, D.G. & HarrarrroN,
Snanny, R. & ENnorn-CucErr, C. (1984). Visual adaptation and retof
Neurophysiology
Journal
and cat: Contrastresponsefunction.
inal gain controls. Progress in Retinsl Research 3' 263-346.
48,21' 7-237.
.lpSneruv, R. & LrNNrtr, P. (19S5). Spatial frequency analysis in the viA neurondoctrinefor
Bmlow, H.B. (1972).Singleunitsand sensation:
t
sual system.Annual Review of Neuroscience 8, 547-583.
Perceptionl, 3'11-394.
perceptualpsychology?
S r o r r u N . B . C . . D r V a r o I s ,R . L . , G n o s o r , D . H . , M o v s s o r ' J . A . , A r of directionBlnrow. H.B. & LrvIcr, W.R. (1965).The mechanism
B R E c H r ,D . G . & B o N o s , A . B . ( 1 9 9 1 ) .C l a s s i f y i n gs i m p l e a n d c o m ally selectiveunits in rabbit'sretina. Journal of Physiology178'
plex cells on the basis of responsemodulation. Vision Reseorch31,
477-504.
I 079-I 086.
BoNos,A.B. (1990).Roleof inhibitionin the specificationof orientaTeouon, Y. & Tourunsr, D.J. (1989).The effect ofthreshold on the retion selectivityof cellsin the cat striate cortex. VisualNeuroscience
lationship between the receptive-field profile and the spatial-frea
/l
<<
D r e N . A . F . . H r s s , R . F . & T o r H u n s r , D . J ' ( 1 9 8 0 ) .D i v i s i v e i n h i b i t i o n
involved in direction selectivity. J ournal of Physiology 308, 84-85P.
Dr Verots, R.L., Arnrucsr, D.G. & Tuornr, L.G. (1982).Spatial frequency selectivity of cells in macaque visual cortex. Vision Research
22,545-559.
Dn VeroIs, R.L. & Dr V.r.roIs,K.K. (1988). Spatial Zrslon' New York:
Oxford University Press.
ENnorn-CucErr., C. & Rossow, J.G. (1966). The contrast sensitivityof
retinal ganglion cells of the cat. Journal of Physiology 181' 511-552.
Gnrsrrn, W.S., ArsnrcHr, D.G., ServI, R.J. & SeuNorns, S'S. (1991).
Discrimination performance of single neurons: rate and temporal
pattern informarion. Journal of Neurophysiology 66' 334-362.
HanrrtoN. D.B. (1987). The Phase Tronsfer Function of Visual Cor'
ticol Neurons. Doctoral Dissertation, University of Texas, Austin.
HeunroN, D.B., ALBRrcHr, D.G. & Gnrsrrr., W.S' (1989). Visual cortical receptivefields in monkey and cat: spatial and temporal phase
transfer function. Vision Research 29, 1285-1308'
HEncrn, D.J. (in press).Computational model of cat striate physiology.
To appear in Computational Models of Visual Perceplion, ed' MovsuoN, J.A. & L.c.Nov,M.
H n o n r t u , E . C . & K o c n , C . ( 1 9 8 7 ) .T h e a n a l y s i so f v i s u a l m o t i o n :
From computational theory to neuronal mechanisms. Annual Review of Neuroscience 10,4'71-533.
HocnsrrrN, S. & SHenrrv, R.M. (1976). Quantitative analysis of retinal ganglion cell classifications. Journal of Physiology 262,237-264.
Husn, D.H. & WrrsEr, T.N. (1962). Receptivefields' binocular interaction, and functional architecture in the cat's visual cortex. Jozrnol of PhysioloqY 160, 106-154.
Hunrr, D.H. & WrEsrr, T.N. (1968).Receptivefields and functional architecture of monkey striate cortex. Journal of Physiology 195,
215-243.
O. (1984). The representation of contrast and
Lr, C. & Cnrutzruor,
other stimulus parameters by single neurons in area l7 of the cat'
Pflugers Archiv 401, 304-314.
quency tuning curve in simple cells of the cat's striate corlex. Visual
Neuroscience3, 445-454.
W e r s o N ,A . B . & A n u N r . l o e ,A . J . , J n . ( 1 9 8 5 ) 'M o d e l o f h u m a n v i s u a l motion sensing. Journal oJ the Optical Society of America 2,
322-341.
Appendix
This Appendix
describes the quantitative models that were con-
sidered in trying to account for the responsesof simple cells to
drifting and counterphase gratings.
Linear model
To begin, let L(x,/) representthe spatiotemporal stimulus,
where x is space and I is time. The linear mechanism integrates/filters the input stimulus by a spatiotemporal impulseresponsefunction, h(x,t). Without loss of generality,we can
assumethat the receptive field is positioned at zero. Thus, the
predictedresponseover time, R(l), is
R(l) : L(x,t) x* ft(x,r)l'=6
(Al)
where x* representsthe operation of two-dimensional convolution. The predicted responsesto gratings drifting in opposite
directions are
R,(r) = lH(u,-w)lC cos(-2rwt + 6@,-w\)
(42)
545
Visual cortex motion and contrast
and
R2Q) :
l H ( u , w ) l C c o s ( 2 r w t+ Q ( u , w ) )
(A3)
whereCis the contrastof the gratings,lH (u,-w)l and lH(u,w\
are the valuesof the amplitude spectrumof h(x, /) at the spatial (u) and temporal (w) frequenciesof the drifting gratings,
and 6@,-w\ and 6(u, w) are the values of the phase spectrum
at these frequencies. Note that changing the sign of w reverses
direction of motion; without loss of generality, we can assume
that -rr is associatedwith motion in the nonpreferred direction.
The response to the mean luminance does not appear in the
above equations becausewe are only consideringthe amplitude
and phase of the fundamental.
Comparing eqns. (A2) and (A3) shows that the ratio of the
r e s p o n s ea m p l i t u d e s i n t h e t w o d i r e c t i o n s i s l H ( u , - w \ l /
lH(u,w)1. Thus, the ratio of the responseamplitudes equals
the ratio of the amplitude spectrumvaluesof the linear summation mechanism. The direction-selectivity index used in the
presentpaper is defined to be one minus the ratio of the responsein the nonpreferred direction to that in the preferred direction. Therefore, the predicted direction-selectivityindex, is
trum values of the linear mechanism raised to the exponent of
the responsenonlinearity. Therefore, the direction selectivityindex d is
d:r
- lH(u,-w)l'/lH(u,w)l'.
(A9)
Combining eqns. (A4) and (A9) shows that the relationship between the direction-selectivityindex D of the linear mechanism
and the direction-selectivityindex d of the final responseis
given by
(A10)
D:t-(l-d)1/'
This relationship was used to estimateD from the measuredresponsesto drifting gratings.
The predictedresponseto counterphasegratings is
R 3( l ) : m a x l j . 5 l H ( u , - w ) 1 C c o s ( - 2 r ( u x o + w t )
+ 6(u,-w))
+ 0 . 5 1 H ( u , w ) l C c o s -( 2 r ( u x s - w t \
+ 6@,11,)),01'. (A11)
D : | - lH(u.-wll/ H(u,wll.
(.A4)
Becausea counterphasegrating can be written as the sum of
two gratings drifting in opposite directions (scaledby 0.5 to preservethe mean luminance and contrast), the predicted response
is given by
R r ( / ) : 0 . 5 1 H ( u , - w )l C c o s (- 2 r ( u x s +
trt\ + 6(u,-w)\
+ 0 . 5 1 H ( u , w ) l C c o s- (2 r ( u x o - w t ) + 6 ( u , w \ )
(A5)
where x6 is the position of the counterphasegrating. Once normalized to the peak responseand to a responsephaseof zero,
the predicted responses(as a function of x6) depend only on
the direction-selectivityindex D of the linear mechanism.
Exponent model
This model is identical to the linear-summation model except
that the output is rectified and then an exponent n is applied.
Thus, the predictedresponseis given by
R ( l ) : m a x l L ( x , t )x x h ( x , t ) , 0 1 '
(A6)
wheretaking the mox with respectto 0 performsthe halfwave
to drifting gratingsare
The predictedresponses
rectification.
R r ( / ) : m a x [ l H ( u , - w ) l C c o s-(2 r w t + 6 ( u , - w ) ) ' 0 ] '
(A7)
and
R 2 ( l ) : m a x [ l H ( u , w ) l Cc o s ( 2 r w+t 6 ( u ,] t ) ) ' 0 1 ' . ( A 8 )
For this model, the ratio of the responseamplitudesin the
t w o d i r e c t i o niss l . I 1 ( 4 , - w ) l ' / l H ( u , w ) 1 " .T h u s ,t h e r a t i oo f
amplitudesequalsthe ratio of the amplitudespecthe response
Once normalized to the peak responseand to a responsephase
of zero, the predicted responses(at a fixed spatial and temporal frequency), as a function of x6, depend only on the direction-selectivity index D of the linear mechanism and the
exponentn of the final responsenonlinearity. Thesetwo parameters can be estimated from the responsesto drifting gratings,
thus making it possible to generateparameter-free predictions
for responsesto counterphase gratings (seeFigs. 5 and 6). Alternatively, n can be estimatedfrom the contrast-responsefunction for drifting gratings and D can be estimated from the
responsesto counterphasegratings in order to make parameterfree predictionsof the measureddirection selectivity(seeFig. 7).
Cont rast -gain / expo nen t mode I
This model is identical to the above model exceptthat contrastgain control is applied before the responseexponent. The contrast-gain control mechanismscalesthe input stimulus by a gain
factor g( e ) that is a function of the average (rms) spatiotemporal contrast C over some region of spaceand time. Thus, the
predictedresponseis given by
R ( l ) : m a x l e ( e ) L G , t )* * h ( x , t \ , 0 ) " .
(A12)
The contrast-gainfactor is givenby the following equation:
c ( c ) : e n \ / ( e n+ e { u l
(A13)
where m is an exponent greater than zero, and C56is the halfsaturation constant.
There are several points to make about this model before
considering the predictions for drifting and counterphasegratings. (1) Becausethe gain-control factor is determinedby the
averagecontrast, it acts like a constantin eqn. (A12) and can
be factored out. Thus, it is apparent that the predictions of the
model are the same whether the contrast-gain control is before
the linear summation, after the linear summation, after the half-
D.G. Albrecht and W.S.Geisler
546
wave rectification, or distributed acrossall three levels. (2) On
the other hand, the responseexponent is dependent on the response magnitude and hence occurs last. (3) There are several
ways that the averagecontrast might have been computed. For
simplicity, we used the rms calculation over one temporal and
spatial cycle, but the predictions for the presentexperimentsare
not strongly dependent on the type of averaging.
The predicted responsesto drifting gratings are
R r ( / ) = m a x f l H ( u , - w ) l e ( f l C c o s( - 2 r w r + Q Q , - w ) ) , 0 1 '
(Ar4)
and
R 2( / ) : m a x l l H ( u , w ) l g ( d ) C c o s ( 2 r w t + 6 @ , ) t ) ) , 0 1 '
(Al5)
Combiningeqns.(A14-Al5) showsthat onceagainthe ratio of
in the two directionsis lH(u,-w)l'/
the response
amplitudes
l H ( u , w ) l ^, a n d t h u se q n .( A l 0 ) s t i l lh o l d s .
gratingsis
The predictedresponseto counterphase
R 3( / ) : m a x l \ . 5 lH ( u . - w ) lg ( C ) C c o s (- 2 z r( u x o* w t )
+ O(u,-w))
1 ( u , w ) lg ( C ) C c o s (- 2 r ( u x o- w t )
+ 0 . 5H
+ 6(u,)/)),01".
(Al6)
phase
and to a response
Oncenormalizedto the peakresponse
gratingsas a
to counterphase
of zero,the predictedresponse
indexD
functionofx6, dependonly on the direction-selectivity
exponentn. The
and the final response
of the linearmechanism
normalizedpredictionsare identicalto thoseof the exponent
model at a fixed contrast,but differ if contrastis varied.