JouR\AL oF NEURoPHYSToLmY
Vol.48. No. l. Julv 1982. Print.d ia U.S.A
Striate Cortex of Monkev and Cat:
ContrastResponseFunction
DUANE G. ALBRECHT NNODAVID B. HAMILTON
Departmentof Psychology,IJniversity of Texas, Austin, Texas 787I2
SUMMARY
AND CONCLUSIONS
spatial frequencies and used the parameters
of the H ratio to test the predictions of two
general classesof models. If the overall gain,
compression, and saturation are set by the
absolute responselevel (response-setgain),
then the CRFs measured at different frequenciesshould shift horizontally along the
contrast axis. Results show that the measured CRFs (tested on the same cell using
different spatial frequencies) were shifted
primarily vertically, suggestingthat the gain,
compression,and saturation were set by the
absolute contrast level (contrast-set gain),
relatively independent of spatial frequency;
in terms of the H ratio, the semisaturation
contrast and the exponent were relatively
constant in comparison to the asymptotic
saturation response.Thus, the spatial frequencyresponsefunctionsare relatively constant when measured at different stimulus
contrasts.
5. There is a great deal of variation in the
location of the dynamic responserange, from
cell to cell, along the contrast axis: somecells
distribute their dynamic responserange over
the first lOVoof contrast, others the second,
etc. (relatively independentof preferred spatial frequency).One might expect this range
variation to be an important factor in behavioral contrast discrimination. To provide
an indication of the average population responseas a function of contrast,all cellswere
averaged together (percent responserelative
to each cell's maximum): the slope of the
bcst-fitting [nwer function (0.77) falls well
within the range of estimatesfound for human psychophysicalcontrast discrimination
functions.
the responses
of 247neu1. We measured
ronsrecordedfrom the striatecortexof monkeys and cats as a function of the contrast
spatialintensity of luminance-modulated
temporalsine-wavegrating patternsto provide a qualitativedescriptionand a quantitative mathematicalformulationof the contast responsefunction (CRF).
2. Qualitatively,it is possibleto provide
a generaldescriptionof the contrastresponse
function for the majority of cells as follows:
as the luminancecontrast of a pattern inin a relatively
the response
increases
creases,
linear fashion over approximately50-607o
of the responserange (generallylessthan I
log unit alongthe contrastrange),the slope
of the functionthen beginsa rapid compression to an asymptoticmaximum-saturation
responselevel. There is, however,a great
deal of variation. from cell to cell, in the
exact shapeand locationof the CRF.
3. Quantitatively,the responsesof each
cell were analyzed in terms of the leastsquares(parameteroptimized)bestfit using
four different mathematicalfunctions: linear,logarithmicpower,and hyperbolicratio.
The results of this procedureshowedthat,
( 1.4acrossthe rangeof contrastsmeasured
567o),the hypcrbolicratio (H ratio) provided the best fit for the vast majority of
striatecells:some7O9oof the cellswerebest
fitted by the H ratio and further, averaged
acrossall cells, the H ratio producedthe
leastaverageresidualvariance.
4. The contrast responsefunction is an
important factor when consideringthe spatial propertiesof corticalcells;nonlinearities
and saturation) I N T R O D U C T I O N
in the CRF (compression
The responsebehavior of sensorysystems
will necessarily
the
influence spatialtuning.
We thereforemeasuredthe CRF at different as a function of stimulus intensity has always
Society
copyright @ 1982The AmericanPhysiological
0022-307118210000-0000$01.25
211
218
D. G. ALBRECHT AND D. B. HAMILTON
been a fundamental concernof sensoryphysiology and psychophysics.The mathematical
formulations (oftentimes canonizedas laws)
used to describe the results of such experimentation have been of equal concern and
over the decadesthe sourceof long-standing
debates(33, 34). These and other concerns
have led to much researchexploring the relations between stimulus magnitude, neural
response,and behavioral response.The present study was undertaken to provide a qualitative description and a quantitative mathematical formulation of the responsesof
visual neurons in the striate cortex of monkeys and cats as a function of stimulus intensity.
Following the pioneeringinvestigationsof
Hubel and Wiesel (19, 20), striate neurons
have been the focal point of much research
and in the processmany of their important
properties have been characterized. However, one important determining factor of
the responsesof striate cells has received
very little experimental attention: namely,
the responseas a function of luminance contrast (what literature there is will be discussedbelow). This is somewhat surprising,
given that striate cells are exquisitely tuned
to respond to specific spatial-temporal vari a t i o n so f l u m i n a n c ec o n t r a s t .
The recent approachesto vision research
that usesine-wavegrating stimuli, linear syst e m s a n a l y s i s , a n d F o u r i e r m a t h em a t i c s
(and the resultant body of research-for
g e n e r a rl e v i e w ss e eR e f s . 6 , 1 0 , 3 1 , 3 2 ) p r o vide several good reasonsfor analyzing the
striate cortex contrast response function.
T h e u l t i m a t e u s e f u l n e s so f t h e l i n e a r a p proach and the frequency responsedescriptions ol the visual system rest on the degree
t o w h i c h t h e s y s t e m b e h a v e si n a l i n e a r
fashion. While striate neurons have been
shown to be linear in certain respects,nonlinearities in the contrast responsefunction
w i l l l i m i t t h e v a l i d i t y a n d u s e f u l n e s so f a n y
predictionsbased on the assumptionof linearity. In this study. striate neuronswere
e x a m i n e du n d e r c o n d i t i o n ss i m i l a r t o m a n y
other physiological and psychophysicalexperiments.The commonality of methods
acrossa variety of different experimentswill
u n d o u b t e d l y m a k e a c o m p a r a t i v ea n a l y s i s
. o the extentthat
a more likell' possibilityT
t h e s t r i a t ec o r t e x p l a y s a r o l e i n l u m i n a n c e
c o n t r a s t - d e p e n d e nvti s u a l b e h a v i o r ,t h e r e -
s u l t s o f t h i s a n a l y s i ss h o u l dc o m p l e m e n tt h e
many relevant investigations (see DISCUSstoN beiow).
M ETHODS
Preparation
The apparatus and general recording procedures are similar to those more fully described
elsewhere (1, 2, 9). Briefly, macaque monkeys
(Macaca fascicularis) and domestic cats were
preparedfor chronic experimentssome days prior
to the first neurophysiologicalrecording: under
deep barbiturate anesthesiaa rigid plastic pedestal containing a recording chamber was attached to the animal's skull.
On the day of an experiment, the animal was
anesthetizedwith a short-actingbarbiturate (thiamylal sodium) and maintained throughout the
experimenton 7 5VoNrO 125E"02 analgesia.Since
no ear, eye, or mouth bars were used.discomfort
was minimal. The animals showed no increased
aversionto the experimentersor the experimental
room as a resultof this treatment;thosepreviously
tamed remained friendly. During the recording
session,the animal rested on a foam-rubber pad
with its head held by a plate screwed into the
pedestal.It was respiredthrough an endotracheal
tube. with the respired COz being maintained at
4.5Vo.T emperaturewas maintainedwithin normal
limits by means of a thermostatically controlled
heating pad; the heart rate was monitored
throughout the experiment. The actual experim e n t s r a n f o r a b o u t 1 2 h ( l h p r e p a r a t i o n ,t h
recording,2 h recovery).
The eyeswere coveredwith contact lensesiaccommodationwas paralyzed.and the natural pupil dilated by applying cyclopentolatehydrochloride. The animal was refracted bl streak
retinoscopy,corrective lenseswere used to focus
the stimuli on the retina.and a 3-mm artificial
pupil was introduced.The eyeswere immobilized
b l c o n t i n u o u si n f u s i o no f g a l l a m i n et r i e t h i d i o d e .
Action potentialswere recordedfrom area I 7 neud l a t i n u m - i r i d i u mm i c r o r o n s u s i n g g l a s s - c o a t ep
electrodes.The action potentials were amplified
and convertedby a window discriminator to standard pulses.which were fed into and analvzedbv
an on-line computer.
Display
V i s u a l s t i m u l i w e r e g e n e r a t e dl i n e b y l i n e o n
either a) a Tektronix 654 oscilloscopeunder the
c o n t r o lo f a N o v a I 2 2 0 c o m p u t e ro r 6 ) a C o n r a c
studio monitor under the control of a PDPI l.
T a b l e so f l u m i n a n c e st o s p ec t f y 'e a c hp a t t e r n( s e l f a d d r e s s i n ga r r a y s ) w e r e s t o r e d i n t h e c o m p u t e r
a n d s e n t t o t h e D / A c o n t r o l l i n gs c o p el u m i n a n c e
o n e l i n e a t a t i m e . s y n c h r o n i z e tdo t h e r a s t e rs c a n
STRIATE CORTEX: CONTRAST RESPONSE
of the monitor. Calibration ensuredthat the grating contrastsusedwere within the display'slinear
range (the linear range exceeded 6070 contrast,
the maximum used was 56Vo). Patterns were
drifted across the scope by changing the starting
position in the stimulus array on each successive
frame. To rotate the patterns, we placed the scope
in a large wheel that rotated the whole unit. The
scopeface was viewed through a circular aperture
in a large white screenmaintained at roughly the
same mean luminance level (27.4 cd/m2). The
aperture subtendedl8o at the 57-cm viewing distance used for cats and 6o for monkeys at a viewing distanceof 172 cm.
rABLE L
Linear
Log
Power
H ratio
Mathematical formulations
R(C)=A+B.C
R(C)=A+B.logro(C)
R(C) = A.CB
R(C) = R*",.(C"/(C'+ C'o'))
responsesof neurons in the visual cortex of
monkeys and cats as a function of the contrast intensity of visual stimuli. To this end,
we measuredthe responsesof 247 cells ( I l0
cells from monkey, 137 from cat) to optimal
spatial-temporal frequency sine-wave gratE xperimental procedure
ing patterns presented at different contrasts.
Once the responseof a single cell was clearly
In order to characterize the resulting conisolated, its receptive field was located and cen- trast responsefunctions (CRFs) quantitatered on the display scope. Its preferred orientatively, we performed a least-squaresfit of the
tion, direction of movement, spatial frequency,
responsesof each cell, using several different
and temporal frequency were approximately demathematical formulations. Thus, for extermined by listening to the spike trains while
we asked whether the responsesof
varying these parameters.Bar stimuli were then ample,
particular
cell were best fitted by a linear
a
used to classify the cell as simple or complex acor perhaps a logarithmic function, the cricording to the criteria of Hubel and Wiesel (19).
On the basis of these preliminary measurements, terion for best fit being determined by which
the responsesof the cell to various spatial and function accounted for the largest portion
temporal frequencieswere quantitatively assessed of the variance in responseacross contrasts
with the orientation and direction of motion held (that is, which function produced the least
constant at the optimum values. (For cells that
residual variance). Four different functions,
did not show length inhibition, the grating was
shown in Table l. were used to analyze the
kept elongated;for thosecellsthat did show length
responsesof all 24'7 neurons: linear, logainhibition, the grating length was decreasedto the
rithmic, power, and hyperbolic ratio.
optimum.)
We then proceededto measure the contrast responsefunction (for all 247 cells) while all other
factors were held constant. Eight different cont r a s t s( 1 . 4 . 2 . 4 , 4 . 0 , 6 . 6 , I 1 . 5 , 1 9 . 0 ,3 3 . 0 , 5 6 . 0 )
were presentedin a randomly interleavedfashion.
Each presentation consistedof 20 cycles followed
by l5 s of no-pattern luminance; cumulative responsesat a given contrast consistedof a minimum of 40 repetitions and a maximum of 100.
For 22 cells this procedure was repeated using
several different test spatial frequencies. The averaged peristimulus time histograms (PSTHs)
were collected in 5-ms time bins; from these
PSTHs an on-line Fourier harmonic analysiswas
computed. For complex cells, the averageresponsc
rate (minus the spontaneousactivity), the DC
component, was used as the responsemeasure; for
simple cells, amplitude of modulation (minus the
spontaneousactivity), the first harmonic component, was used as the responsemeasure.
R E S UL T S
The primary goal of this study was to investigate and quantitatively characterize the
Contrast responsefunction
A QUALITATIVEDESCRIPTION.To begin, it
is important to emphasize that there is a
great deal of variation, from cell to cell, with
respect to the exact form of the CRF: some
cells are, with little doubt, best fitted by a
linear function while others are best fitted
by a log contrast function, still others by a
power function. Furthermore, and perhaps
of greater significance,there is a great deal
of variation in the dynamic range of contrasts covered by a given cell: some cells
distributing their response range from I to
l07o contrast, others from l0 to 20Vo, etc.
There is also a great deal of variation in the
slopesof the CRFs (on log-log coordinates:
from lessthan I to greater than 5). The variationsnoted above,and others,will be quantified in the following sections;however,the
variation can be qualitatively seen in Fig. I
where a variety of typical CRFs are shown
plotted on log-log coordinates.
220
D. G. ALBRECHT AND D. B. HAMILTON
30
o
z
o
o.
o to
U
G
*
a - RAIrO
n,2.7
n ' 5.5
Cs.
6.E
e ' 0.61
% coilTRAST
rtc. l. Contrast response functions for nine representative striate neurons; percent response (relative to the
maximum response)is plotted on log-log coordinates as a function of the luminance contrast of spatial-temporal
sine-wave grating patterns. The smooth curve drawn through responsesof each cell is the best-fitting function of
four candidates (H ratio, log, linear, power). As can be sccn, there is a grcat deal of variation from cell to cell
with respect to the exact shapc and relative position of each ccll's contrast response.Some cells are best fitted by
a log function, others by a power function; howcver, most are bcst fitted by the hyperbolic ratio. Note the variation
in the position (along the contrast axis) where the dynamic responserangc is distributed. Animal type (monkey
or cat) and ccll typc (simple or complex) are specified in the upper left corner of each graph (animal type-cell
type); variations in this rcgard will bc prcsentcd below (the few cells shown hcre should not be taken as indicative
of any ccll-rype lrcnds).
The responsesof each of the nine cells
shown in Fig. I were analyzed for a leastsquares fit using the four functions shown
in Table l; the line drawn through the measured rcsponsesof each cell represents the
best fit of the four (the function and the parameters are as specified). The six cells
shown in A-F are typical examplesof striate
CRFs best fitted by the hyperbolic ratio; as
will be shown below, this function proved to
be the best descriptor of the CRF for the
overwhelmingmajority of striate cells. The
cells shown in G and H were best fitted by
a log contrast relationship; the responses
221
STRIATE CORTEX: CONTRAST RESPONSE
shown in 1 were best fitted by a power function. Quantitative normative statistics will
be presentedbelow concerning how the entire population of celis and the various
subgroups (cat, monkey, simple, complex)
were distributed among the four functions.
Again, the point we wish to emphasize(and
illustrate in Fig. I ) is the variation from cell
to cell with respectto the form of the CRF.
Nevertheless,it is possible to provide a
general qualitative description that applies
to the majority (some 80-907o)of the striate
contrast responsefunctions. In general then,
as the contrastof a grating pattern increases.
the responseof a striate cell increasesin a
relatively linear (possibly logarithmic, see
DISCUSSIoN
below) fashion; the slopeof this
linear increaseis steepand thus coversa restricted contrast range (generally less than
I log unit of contrast). At approximately 50609oof the maximum responseof a cell, the
slope of the function begins a rapid decline;
that is, the function begins an accelerating
compression. Ultimately, the response totally saturates(the slope hyperbolically approacheszero) and remains at the saturated
level or, in some cases,actually decreasesto
some extent.
Take for example one of the cells shown
in Fig. I (say Fig. I C). Over a contrast range
of l-8Va, the responses
of this cell cover some
60Voof the cell's total responserange in a
relatively linear fashion. However, beyond
this linear range the cell distributes the remaining 40Voof its dynamic responserange
over a contrast range from 8 to 309o;beyond
3070contrast the responseis saturated and
virtually static. Thus, for this cell, over a
little less than I log unit of contrast the responsewas essentiallylinear, and then over
the next 0.5 log unit of contrast the response
was compressingto a saturated maximum
responsethereafter.While this generalcharacterization is not applicable to all striate
cells, as will be shown below, it is a good
general descriptor for some 8O-907oof the
total population.
The cells shown in Fig. I have been labeled as cat, monkey, simple, or complex;
however. these should not be taken as nece s s a r i l ye x e m p l a r vo f a n y v a r i a t i o n a m o n g
the cell classes.As will be seen,the similarities among these different cell groups far
exceed the differences. All the data pre-
sentedin the followingtableswill be broken
down in termsof thesegroups.A discussion
of the generaltrendsfor all cellswill precede
of group differences.
a final discussion
CLASSICAL
FUNCTIONS.
ThC
SCATCh fOT A
general function to describe the intensity
responsebehavior of sensory systems adequately has a long history. The two functions
that seem to have received the greatest
amount of attention are the log function, or
F e c h n e r ' sl a w ( 1 4 ) , a n d t h e p o w e r f u n c t i o n ,
or Stevens'law (l{). We felt it was important to analyze the contrast-intensity responsefunction of striate neurons from the
perspectiveof thesetwo functions in addition
to a strict linear function (the relative fit of
the hyperbolic ratio will be analyzed below).
We therefore analyzed all of the 24'l cells
with respect to the least-squaresbest fit of
a linear, log, and power relation (refer to
Table I ). A breakdown of how many cells
were best fitted by each of the three candidate functions is shown in Table 2. It should
be clear from this data that across all
subgroups a log function provides the best
fit (compared to a linear or power function) for the vast majority of striate cells
( s o m e8 0 % ) .
If we look at the data from the total population in a slightly different way', by analyzing the residualvariance unaccountedfor
after finding the best-fit parametersfor each
function, we find that the average residual
per point is 271 + 14 (SE) for linear, 204
t l 1 ( S E ) f o r p o w e r ,a n d 1 2 0 + 8 ( S E ) f o r
log. We can thereforeconcludethat over the
c o n t r a s tr a n g e t e s t e d ( l - 5 6 E o ) , a l o g f u n c tion in comparisonto a linear or power function providesa much better fit to the contrast
responsefunction of striate cells.
Given that the responsesof most striate
cellstend to compressand saturate at higher
responserates, as describedabove, it is not
too surprising that a log function should fit
better than a linear function. If we were to
restrict the analysisto the beginning portion
of the CRF (say the first log unit of cont r a s t ) , t h e a n a l y s i sc o u l d p o t e n t i a l l yp r o d u c e
rather different results (see ntscusstoN).
H o w e v e r ,d e m o n s t r a t i n gt h a t a l o g o r l i n e a r
function provides a better fit over a (judiciously selected) restricted range becomes
s o m e w h a tu n t e s t a b l e( p a r t i c u l a r l l ' s i n c e ,a s
222
D. G. ALBRECHT AND D. B. HAMILTON
TABLE 2.
Percentage distributions of best fts
All Cells
Linear
Log
Power
Cat Cells
Monkey Cells
Total
(247)
Simple
(ls5)
Complex
(e2\
Total
(137)
Simple
(83)
Complex
(54)
Total
( l l0)
Simple
('72)
6
8l
I3
7
'19
3
86
ll
4
86
l0
6
84
l0
2
87
1
78
l5
74
l8
14
ll
Complex
(38)
R
5
84
ll
The best-fitting (parameter optimized) linear, log, and power functions were derived for the responsesof each
cell individually and then the residual variance was compared. Shown are percentage of cells best fitted (least
residual variance) by linear, log, and power functions, broken down across the various subgroups. Thus the first
three columns provide the p€rcentages for all cells, the second three columns show all cat cells. the last three
columns show all monkey cells. As can be seen, given these three functions and responsesover a range of l-56Vo,
some 8070 of the population is best fitted by a log function. Values in parenthesesare numbers of cells.
one restrictsthe range,the actual differences
expected are not very large-given small
perturbations, many nonlinear intensity responsefunctions are well approximated by
a linear function).
HypERBoLtc RATIo. As discussed above,
the contrast responsedata from striate cells
can be qualitatively describedas linear over
a restricted range, then showing gradual
compression, and finally total saturation.
This type of behavior is quite adequately
characterized by a hyperbolic function (H
ratio) of the form
(C) = R..,'(C ltC" + Cro"))
response
where R-"* refers to the maximum response
rate, C5o refers to the semisaturation contrast (the contrast required to produce 507o
of the cell's maximum response),and n, the
exponent, relers to the rate at which the
changesoccur. This function was first used
TABLE 3.
Percentage distributions of best fits
All Cells
Total
(247)
Linear
Log
Power
H ratio
by Naka and Rushton (30) to fit voltageintensity data from retinal S potentials. It
has since been used to describethe intensity
responsefunctions of retinal neurons in a
wide variety of vertebrate species(4, 5, ll,
I 3, 16, 22, 37 ) in addition to luminance sensitivity measured in human psychophysical
s t u d i e s( 3 , 1 5 , 1 7 , l 8 ) .
We wished to determine the validity of
using this relationship as a general descriptor of the contrast response behavior of
striate neurons.We therefore began by asking two experimental questions:a) which of
the four functions, linear / logI power/hyperbolic, provided the best fit for the largest
proportion of neurons; and b) which of the
four accounted for the largest proportion of
the variancein responseacrosscontrast (that
is, which producedthe least averageresidual
variance acrossall cells).
The answer to the first question is shown
in Table 3. Of the four functions. the hv-
4
19
7
70
Simple
(155)
5
19
8
68
Cat Cells
Complex
Total
Simple
Monkey Cells
Complex
Total
Simple
(e2)
(r37)
(83)
(s4)
(rr0)
(72)
3
19
5
73
3
2t
6
70
4
20
5
7t
2
22
7
69
5
15
9
7t
5
t7
ll
65
Complex
(38)
5
I
J
3
19
T h e b e s t - f i t t i n g( p a r a m e t e r o p t i m i z e d ) l i n e a r , l o g , p o w e r , a n d H r a t i o f u n c t i o n s w e r e d c r i v e d f o r t h e c o n t r a s t
r e s p o n s e so f e a c h c e l l i n d i v i d u a l l y a n d t h e n t h e r e s i d u a l v a r i a n c e w a s c o m p a r e d . S h o w n a r e t h e p e r c e n t a g ed i s t r i b u t i o n s o f t h e c e l l s b e s t f i t t e d b y e a c h o f t h e f o u r f u n c t i o n s ( b r o k e n d o w n a c r o s sa n i m a l t y p e a n d c e l l t y p e a s
i n T a b l e 2 ) . V a l u e s i n p a r e n t h e s c sa r e n u m b c r s o f c e l l s .
223
STRIATE CORTEX: CONTRAST RESPONSE
perbolic ratio provides the best fit for some
7lVo of the cells and a log function for some
197o.This general trend is seen across all
groups: monkey, cat, simple, complex. The
answer to the second question is shown in
Fig. 2, where the distributions of residual
variance are shown for each function type;
the means and standard errors associated
with these distributions are shown in Table
4 broken down for the various subgroups.
From these results it becomesclear that the
hyperbolic ratio is by far the best general
descriptor for the striate cell contrast responsefunctions. The averageresidual varia n c ep e r d a t a p o i n t w a s 3 8 . 4 + 3 . 5 ( S E ) : t h e
log function is not really a very close second
(l2o + 7.8(SE)).
To help illustrate the type of fit these four
functions are providing for a typical striate
cell, the data points for a particular cell are
shown in Fig. 3 plotted on linear-linear coordinateswith the best fitting a) linear function, b) log function, c) power function, and
d) H ratio. As can be seen,a strictly linear
function over the entire range providesa very
poor fit; as will be shown below, when the
analysis is restricted to the first I log unit
of contrast,both a linear relationshipas well
as a log relationship provide reasonablefits.
Over a larger range of contrasts,a log function provides a much better fit than a strict
linear function, since it not only characterizes the first few responsesquite adequately,
but then turns (concave downward) to accommdate the compressionand saturation
of the cell's response.The log compression,
however, is not nearly rapid enough to accommodate the cell's acceleratingcompression. The values of the parameters of the
power function that best fit thesedata points
result in a function that compressesmuch
like a log function (concavedownward). Of
the four, the H ratio providesthe best total
description of the linear dynamic range in
addition to the nonlinear compressionand
saturation that this cell shows.
The values of the parameters of the hyperbolic function vary considerablyfrom cell
to cell, as would be expectedgiven the qualitative differencesthat can be seen in the
CRFs of different cells. These valuescan be
used to quantify some of this variation in a
u s e f u l w a y . T h e s e m i s a t u r a t i o nc o n s t a n t i s
an excellent indication of the overall sensi-
80
70
50
40
30
to
U'O
J
J
lrJ "^
o'w
G e-- o
frl
@
-?o
z a
bQo
i:t
o
roo
200
300
400
500
)550
LINEAR
i .272
S . E. 1 3 . 7
20r
:l
o
too
200
300
400
500
>550
RESIDUAL VARIANCE
F t c . 2 . D i s t r i b u t i o n ss h o w i n g t h e a v e r a g ev a r i a t i o n
between the measured responsesand the predicted responses(following parameter optimization ). Responses
of each cell were fitted by the four functions (linear.
l o g , p o w e r , h y p e r b o l i cr a t i o ) a n d t h e n t h e s q u a r e dd e v i a t i o n p e r p o i n t w a s i n d e x e d .W h i l e s o m e c e l l s a r e b e s t
fitted by a linear relationship, others by a power function
or log function, averaged across all cells, the H ratio
provides a better description of striate cell contrast responsefunctions.
tivity of each cell, sinceit tells us the contrast
requiredto reach a fixed criterion of response
(namely 507oof the maximum response);
f u r t h e r m o r e , t h e s e m i s a t u r a t i o nc o n s t a n t
falls on the linear portion of the dynamic
range. The distributions of the values of all
three parametersare shown in Fig. 4 for all
cells, and the means and standard errors
across all subgroupsare shown in Table 5;
this analysis includes all cells except some
99o of the cells that produced values of C5s
that exceeded1007ocontrast. An analysisof
c o n s t a n ti n
t h e s ec e l l sw i t h a s e m i s a t u r a t i o n
excessof 1007ocontrast showed that they
were best fitted by either a strict linear func-
224
D. G. ALBRECHTAND D. B. HAMILTON
TABLE 4.
Average residual variance
All Cells
Linear
Simple
(l5s)
Complex
(e2\
Total
037)
Simple
(83)
Complex
(54)
Total
(ll0)
Simple
(72)
Complex
(38)
2'7|
251
+16
305
+25
279
+18
267
+21
299
+33
262
+21
235
+25
315
+38
Il3
+ 9
t32
t04
+ 9
96
+19
n7
+ 1 4
+ 18
l4l
+ 13
134
+ 1 6
154
+21
189
+ 13
186
+ 13
+ 1 4
204
+24
228
+ l8
207
+ 1 9
! t J
268
+ 3l
35
+'7
43
+ 6
120
+ 8
Power
204
+ ll
H ratio
Monkev Cclls
Total
(247)
+ 1 4
Log
Cat Cells
38
+ 4
230
4l
) t
1 1 4
J+
J{
+ 5
+ 5
+ 6
4l
+ 7
48
+10
After finding the best-fitting (parameter optimized) linear. log, power, and H ratio functions for each cell, the
residual variance was computed. Shown is the residual variance + SE per point, averaged across all cells (as well
as broken down into the subgroups) for each function. Values in parenthesesare number of cells.
tion (6122)or a powerfunctionwith param- howeveronly half of the function (primarily
etersthat forcea C5egreaterthan 100.The the linear portion) appears in the range of
hyperbolicfunctionwill fit thesedata points; measuredcontrasts.
LINEAR
A point worth developing here concerns
the search for a general mathematical formulation (general, accurate, and simple).
C
L O G A R I T H IM
roo
tof
EXPONENT
t=?.e
s . E' o . r
'of
50
,of
lr,
an
=
o
.L
at,
EI
E
oL
o
qn
?\
POWER
4
2
5
5
0
HYPERBOLICRATIO
te
roo
u) ta
J-J
I
3ro I
G r olrJ
E!
:Eo
50
aQoo
20
0
2
5
5
0
0
2
% CONTRAST
5
5
0
lo
,.MAX
o
Y- tl5
s.E ' 2.7
80
r t c . 3 . R e s p o n s e so f a t y p i c a l s t r i a t e c e l l a s a f u n c tion of contrast are plotted on linear-linear coordinates
showing the best-fitting(parameter optimized) Iinear,
l o g , p o w e r ,a n d h y p e r b o l i cr e l a t i o n s h i p sA
. s can be seen,
t h e H r a t i o p r o v i d e st h e b e s t d e s c r i p t i o no f t h e t y p i c a l
C R F : a l i n e a r i n c r c a s eo v e r a r e s t r i c t e dr a n g e f o l l o w e d
b y a c c el e r a t i n g c o m p r e s s i o n t o u l t i m a t e s a t u r a t i o n .
While a log function and power function do show res p o n s ec o m f r r e s s i o nt .h e c o m p r e s s i o nd o c s n o t a c c e l e r a t e
o r s a t u r a t e .A l i n e a r f u n c t i o n , o v e r t h e e n t i r e r a n g e o f
c o n t r a s t sm e a s u r e d ,i s c l e a r l y i n a p p r o p r i a t e( s e c F i g . I 4
f o r a n a n a l v s i so f l i n e a r i t y o v e r a r e s t r i c t c d r a n g e ) .
90
roo rio 120 130 t40 t50 ) r 5 0
F I c . 4 . D i s t r i b u t i o n o f t h e o p t i m i z e d p a r a m e t e r so f
the hyperbolic relationship for each cell (save 22, or 9Io.
w i t h C r o i n e x c e s so f 1 0 0 % ; s e e t e x t ) . E a c h c e l l h a s a
slightly different set of values. The exponent specifies
t h e r a t e o f c h a n g e ,o r s l o p eo f t h e f u n c t i o n : t h e C 5 6 .o r
s e m l s a t u r a t r o nc o n s t a n t , m o v e s t h e c u r v e h o r i z o n t a l l y
a n d p r o v i d e sa g o o d i n d e x o f t h e c o n t r a s t s e n s i t i v i t ya t
h a l f t h e m a x i m u m r e s p o n s eR
: . . , s p e c i f i e st h e s a t u r a t i o n p o i n t ( v a l u e sg r e a t e r t h a n 1 0 0 i n d i c a t e t h a t s o m e
c e l l s h a d n o t v e t r e a c h e ds a t u r a t i o n a t t h e h i g h e s tc o n t r a s t m e a s u r e d) .
STRIATE CORTEX: CONTRAST RESPONSE
TABLE5.
H ratio constants
All Cells
Total
(22s)
Simple
(l 39)
Exponent
2.9
+ 0.1
2.8
+ 0. I
Semisaturation
19.3
+ 0.9
a
Maximum
225
I 15.0
!
z-l
19.8
l-l
l 17.0
+ 4.0
Cat Cells
Complex
(86)
Total
(t27)
Simple
(' l-t)
Monkey Cells
Complex
(50)
3.0
2.5
+ 0.16 + 0.12
2.5
+ 0.15
2.6
+ 0.2
18.6
+ 1.5
t5.2
+ 1.25
15.5
+ 1.06
lll.7
nl.0
il1.0
t
t:.2
i
5.2
t.l
Total
(e8)
3.4
+ 0.13
Simple
(62)
Complex
(36)
3.3
+ 0 .1 7
3.5
+ 0.2
16.0 24.0
+ 1.9 + 1.5
25.0
+ 2.0
22.r
+ 2.5
I I1.0
+ 3.8
t24-O
+ 8.4
1t2.0
+ 5.5
120.0
+ 5.7
Values are averages of means + SE of the optimized parameters of the best-fitting hyperbolic relationship
computed for cach cell (save 22,or99o ofthe population, with Ce in excessof 1007o;see text). Values in parcntheses
are numbersof cells.
While the H ratio may be well suited to accommodate some 707oof the total population
of cells. its value would be diminished if it
was grossly inaccurate in describingthe rest
of the population.Fortunately, this is not the
case. Those CRFs that are best fitted by a
log function or a power function with an
exponent less than 1.0 are quite adequately
describedby the H ratio as well; the ultimate
saturation is simply moved to higher contrasts. The point is that, with the goal of
parsimony in mind, it is possibleto describe
some 907, of all striate cells using the H
ratio; the increase in residual variance (as
indicatedin Fig. 2 and Table 4) is reasonably
small.
DYNAMIC RANGEVARIATION. TheTe is a
great deal of variation from cell to cell in
the absolute location of the dynamic responserange along the contrast axis; different cellsrespondover different rangesofcontrast. This variation is illustrated in Fig. 5
where the contrast responsefunctions for
four different cells are shown; these cells all
had very similar spatial frequency tuning
and were all encountered during the same
electrode penetration. As can be seen, the
dynamic responserange of each cell covers
a slightly different range of contrasts.At a
given low contrast, one cell may already be
saturated, another silent, and a third may
be optimally positioned such that the contrast falls within the dynamic linear range.
Such range variation could be an important
factor in behavioralcontrast discrimination;
in general. when considering the activation
of a large population of cells, increasingthe
contrast of a grating produces an increase
in both a) the overall number of action potentials produced as well as b) the overall
number of cells responding.
To providea quantitative indication of this
range variation for the total population of
cells we computed (from the best-fitting
function for each individual cell) the contrast required to reach 50Voof each cell's
maximum respons€(that is, we computed
the semisaturationcontrast):the valueswere
80
l!
vr Tct
z o
460
u)
l!
850
aaqo
30
o
t
o
?
o 3 0 4 A
% CONTRAST
5
0
6
0
rtc. 5. Contrast response functions of four striate
cells plotted on lincar-linear coordinates to illustrate the
variation that occurs, from cell to cell, in the location
of the dynamic portion of the responserange. along the
contrast axis. Thcse four cells had similar spatial tuning
and wcrc cncountcrcd during a l-mm penetration pcrpcndicular to laminae. One cell begins responding at l%
contrast and saturates by l0%t another cell does not
b c g i n r c s p o n d i n gu n t i l l 0 % c o n t r a s t .
226
D. G. ALBRECHTAND D. B, HAMILTON
4a
30
F
2 Zla
,-,ra
4
a!
to
-
.21
.14
.46
.6-3
.86
.2=
._ ,.0
.5
2.2
3.?
4.8
7.2
C.r -
CYCLES / OEGREE
F l c . 6 . T o q u a n t i f y t h e r a n g e v a r i a t i o n , t h e a v e r a g e s e m i s a l u r a t i o nc o n t r a s t ( t h e c o n t r a s t r e q u i r e d t o r e a c h
509oof a cell's maximum response)is plotted conjointly with (as a function of) preferred spatial frequency (cells
recorded from cat cortex are averaged together in the left panel, monkey cells in the right panel). One standard
deviation is plotted on each side of the mean to indicate the variation in the location of the dynamic response
range that occurs, from cell to cell, across the range of preferred frequencies. Such range variation indicates that
cells distribute their limited dynamic response ranges over different restricted contrast ranges: multiple channels
along the contrast axis.
then averagedover sequentialfrequency in- sponsesmight one expect when the contrasts
tervals.The resultsof this analysisare shown of a given spatial pattern exceed the linear
in Fig. 6, where the means and standard range of a given cell. To test severaldifferent
deviations of the semisaturation contrasts possibilities,we measured the CRFs of 22
are plotted as a function of spatial frequency. cells at several different spatial frequencies
As can be seen,the standard deviations in- and then analyzed the data using the hydicate the variation that occurs, from cell to perbolic equation, which is capable of discell, in the location of the dynamic range; tinguishingdifferent models (cf. Refs. 17, l8
further, the average values indicate that a n d R e f . l 5 ) .
One simple model, response-set gain,
thesedistributionsare very similar acrossthe
would place the burden of the nonlinear
range of preferred spatial frequencies.
compressionand saturationon the final comContrast response and spatial tuning
mon response-generating
mechanism of the
While someof the CRFs of neuronsin the individual striate neuron: the gain is set by
striate cortex are in fact strictly linear (some t h e r e s p o n s el e v e l ( R o b s o n ( 3 1 ) d i s c u s s e s
5Vaof the total population across the range similar theoretical issues and some of the
of such a model; seealso Evans
of contrasts measured), the vast majority consequences
have a dynamic linear responserange only ( 1 2 ) f o r a s i m i l a r d i s c u s s i o ni n r e f e r e n c et o
over a limited contrast range (generally less the auditory system).This compressionwould
than I log unit of contrast,dependingon the be applied after the spatial summation had
cell's particular exponent and semisatura- occurred. Thus, for example, a given cell
tion contrast). Outside of the linear range, may have a restricted responserange and a
the cells are a) essentially silent, b) com- fixed maximum response (R,"^) beyond
pletely saturated at the maximum, or c) in which it cannot be expected to operate; it
the processof nonlinear compressionto the therefore compressesany and all excitation,
asymptotic level. In other words, beyond independentof other stimulus variables(e.g.,
their respectivedynamic linear range, the spatial frequency, temporal frequency, oriCRFs of most striate cells are quite nonlin- entation, etc.) solely on the basisof how far
ear (the range of the latter generally ex- up the responseaxis a certain responsehad
moved. Such a mechanism can potentially
ceeding the former).
Given compressionand saturation of the result in the cell producing equivalent recontrast responsefunction. it is appropriate sponsesto optimal stimuli (presentedat or
and important to ask what the consequences beyond the saturation level) and nonoptimal
of such nonlinearitiesmight be on the spatial s t i m u l i ( p r e s e n t e da t h i g h c o n t r a s t ) ;o n c ea n
processingof striate cells. What sort of re- o p t i m a l s t i m u l u sp r o d u c e st h e m a x i m u m r e -
STRIATE CORTEX: CONTRAST RESPONSE
sponse,nonoptimal stimuli can catch up if
their contrast is raised.
Given such a mechanism,which has been
demonstratedfor other visual neurons (25),
one would expect the CRFs measured at
different spatial frequenciesto shift to the
right horizontally as the spatial frequency
is varied from the optimum value. Or, in
terms of the hyperbolic relationship, the
maximum rate of firing (R-",) and the exponent (n) would be expectedto remain constant and only the semisaturation constant
(Cso) should increase(directly proportional
to the frequency attenuation factor). Such
a scheme would have the desirable quality
of producing (from a nonlinear CRF) identical spatial-frequencysensitivity curves independentof the responsecriterion used (the
FREo
z
o
o
U
o
"
' 50
227
sensitivity ratio between different frequencies would remain constant, at least up to
saturation). Since the nonlinear gain is determined by the response,if we hold the responselevel constant, the effect of the nonlinearity should be nullified (cf. Ref. 3l).
A secondmodel, contrast-setgain, would
place the burden of the nonlinear compression and saturation on a mechanism that
precedesthe striate responsegeneration and
possibly even the spatial summation of the
light distribution: the gain is set by the contrast level. It is possiblethat some mechanism prior to the striate neuron (or at least
prior to the neuron's response)clips or compressesthe potential excitation more as a
function of the luminance contrast and not
the actual responselevel. If the input to the
'MAX
2t
)o2
365
1 8
r 6
?2
27
2 7 9
209
2 6
1 2 7
8 7
2.9
16 5
5 3
' 5C
E' M A t
22
65
524
23
64
3C
2
26
80
222
c5
27
66
39
o
,
'
0
u,^^
Yivv
d
0
FR € o
n
C:c
Ruax
a
r 5
22
24
40
46
471
327
33
48
262
31
49
92
30
,,
"5o
323
4,o
32
43
35
43
o5
4 5
roo
rc0
% CONTRAST
FIc. 7. Contrast responsefunctions for four cells measured using several different test spatial frequencies.The
best-fitting H ratio was found for each test frequency; values of parameters, along with the test frequencies, are
shown to the right of each curve. Qualitatively, curves appear to shift vertically downward along the responseaxrs
a s t h e f r e q u e n c y i s v a r i e d f r o m t h e o p t i m u m v a l u e . Q u a n t i t a t i v e l l , t h i s e f f e c t i s s e e n i n v a l u e s o f p a r a m e t e r so f
t h e b e s t - f i t t i n gH r a t i o s : w i t h i n a g i v e n c e l l , t h e e x p o n e n t a n d t h e s c m i s a t u r a t i o nc o n t r a s t a r e r e l a t i v e l y c o n s t a n t
a c r o s ss p a t i a l f r e q u e n c yi n c o m p a r i s o nt o R . . , ( w h i c h v a r i e sc o n s i d e r a b l y ) S
. u c h v e r t i c a l s h i f t s p r e s e r v et h e r e l a t r v e
f r e q u e n c y r e s p o n s ef u n c t i o n i n d e p e n d e n to f c o n t r a s t . T h e s e r e s u l t s a r e m o r e c o n s i s t e n tw i t h a c o n t r a s l - s e tg a i n
mechanism as opposed to a response-setgain.
D. G. ALBRECHT AND D. B. HAMILTON
sp
I
I
I
oL
3
J
UJ
C)
l!
l'f
lrj
@
- l
I
l
3oL
roo
Jr
I
I
6L
0
50
too
% OEVIATION
FROMMEAN
r t c . 8 . D i s r r i b u t i o n so f d e v i a t i o n si n t h e H r a t i o p a rameter values for the contrast responsefunctions of ihe
2 l c e l l s m e a s u r e da t d i f f e r e n t t e s i s p a t i a l f r e q u e n c i e s .
The percent deviation from the mean within a given cell
was computed for each parameter. These population
s t a t i s t i c ss u b s t a n t i a t et h e b e h a v i o r i l l u s t r a t e d i n F i e . 7 :
the primary shift of a givencell'sCRF. when measired
at different spatial frequencies, is vertical and not horrzontal. As the test frequency is varied from the optim u m v a l u e , t h e e x p o n e n t a n d s e m i s a t u r a t i o nc o n t i a s t
r e m a i n r e l a t i v e l y f i x e d w h i l e t h e R . " , d e c r e a s e si n versely proportional to the attenuation factor. The attenuation factor thus remains relativelyconstantindependent of contrast.
striate cell already carried the compressivesaturation function, then the striate cell
would reflectthe same function, independent
of spatial frequency. Compressionand saturation would occur at the same contrasts
(not response)for all spatial frequencies.
Given this mechanism,one would exDect
t h e C R F t o s h i f t v e r t i c a l l yd o w n w a r d a s t h e
spatial frequency was varied from the optimal value. In terms of the hyperbolic relationship, we would expect the exponent (n)
and the semisaturationconstant (Cso) to remain fixed at different spatial frequencies
a n d o n l y t h e m a x i m u m r e s p o n s er a t e ( R - " , )
to vary inversely with the frequency attenuation factor. This second scheme would
have the desirable quality of producing
(from nonlinear CRFs) identical spatial frequency responsecurves, independentof the
contrast used to measure the responses(the
responseratio between different frequencies
would remain constant acrosscontrist).
The results of measuring the contrast responsefunction at different spatial frequencies are shown in Fig. 7 for four different
striate cells. The measured responseswere
fitted by the hyperbolic relationship and the
values of the parameters that produced the
least-squaresfit are shown to the right of
each individual curve. As can be seen, the
predictions of the response-setgain model
were not fulfilled; the CRF of a given cell
does not shift to the right (as the spatial
frequency is varied from the optimum) but
rather shifts downward. This qualitative effect can be seenquantitatively by examining
the values of the hyperbolic parameters,for
each cell individually. In general, the exponent (n) and the semisaturation contrast
(C5s) are relatively constant while the maximum responserate is quite variable.
To provide a quantitative indication of
how a larger population of cells behavesunder similar circumstances.we computed the
averagedeviation from the mean value of
each parameter, within a given cell, as frequency was varied from the optimum. The
distributions of these deviations are shown
in Fig. 8 for a sample of 22 cells. The distributions of n and C56are clearly shifted to
the left of the R."* distributiou the medians
demonstratethat most of the variation across
frequency is occurring in the value of the
R-"* parameter: the curves shift primarilv
vertically.
The cell shown in Fig. 78 is replotted
again in Fig. 9 with the specificpredictions
(elaborated above) from the two different
models(response-set
gain versuscontrast-set
gain) superimposed
on the actual data points.
The curves in Fig. 9A were generated by
holding n and C5econstant,leavingonly R_".
free to vary with frequency (to produce the
best fit). The curves in Fig. 9^Bwere generated by holding R.", and n constant and
only allowing Cr6 to vary. As can be seen,
the contrast-setgain model providesa much
better fit to the actual data than does the
response-set
gain model.
The type of behavior illustrated in Figs.
7-9 has several interesting implications for
STRIATE CORTEX C O N T R A S T R E S P O N S L
A
CONTRAST- SET
B R E S P O N S.ES E . r G A I N
a , /
o
a
a
o
z
o
o
U
o.^
I
a
a
\ lU
o
U
Y
I
d
ah
ro
0
r00
% @ilTRAST
rlc. 9. Predicted contrast functions (solid lines) and measured contrast responsefunctions for a representative
c e l l . I n , 4 , l h e r a t e o f c o m p r e s s i o na n d s a t u r a t i o n i s s e t b y t h e o v e r a l l l e v e l o f c o n t r a s t , i n d e p e n d e n to f s p a t r a l
f r e q u e n c y ;t h e t e s t f r e q u e n c yp r i m a r i l y i n f f u e n c e st h e m a x i m u m r a t e . I n 8 , t h e r a t e o f c o m p r e s s i o na n d s a t u r a t r o n
i s d e t e r m i n e d b y t h e o v e r a l l r e s p o n s el e v e l : t h e t e s t f r e q u e n c y p r i m a r i l f i n f l u e n c e st h e s e m i s a t u r a t i o nc o n r r a s t .A s
can be seen, predictions from the contrast-set gain model provide a more accurate description of the measured
resDonses.
the overall spatial tuning characteristicsof
striate cells. First of all, as noted above,the
ratio of the responsesproduced by different
spatial frequencies(the frequency attenuation factor) remains fixed, independent of
the contrast of the patterns. Thus, for example, examining the responsesof the cell
shown in Fig. 9,4 shows that frequency 0.5
cycle/deg produces 7 or 8Voof the response
generated by the optimum (1.0 cycle/deg),
relatively independentof the test contrast of
the patterns.
However, the same is not the case for the
spatial frequency sensitivity curves generated at different response criteria. As the
fixed criterion of responseincreases,the sensitivity ratio between a nonoptimal frequency and an optimal frequency would not
remain constant. If a very high criterion of
responseis chosenfor the cell shown in Fig.
9A (say beyond 40 spikes/s), only a very
small band of frequencieswill be capable of
reaching that criterion. at any contrast.
To illustrate these observations,we have
replotted the data shown in Fig. 9 in terms
of the spatial-frequencyresponse(Fig. l0l)
a n d s e n s i t i v i t y( l 0 B ) f u n c t i o n s .A s c a n b e
seenthe frequencyresponsecurvesvary little
with the fixed contrast measurements.
whereas the sensitivity curves vary considerably with the fixed criterion of response
measurements.The bandwidths besideeach
curve help quantify the observations.It is
worth noting that the bandwidth of the spatial-frequency contrast sensitivity curve becomesvery narrow given a high criterion of
response; if a subsequent mechanism required a similar responselevel for transmission,the spatial frequencyinformation would
be quite specific.
Dffirences and similarities among
cell types
Analysis of the data in terms of the various
cell types-simple, complex, cat, monkeyrevealed only relatively small variations in
the general trends outlined above. The statistics shown in Tables 2 through 5 are broken down in terms of these subgroupsto facilitate suchcomparisons.For example,Table
3 shows that some 707o of the total population is best fitted by the H ratio: this per-
230
D . G ALBRECHT AND D. B. HAMILTON
A C C N S T A N TC J \ T R A S T
B C O N S T A N ' IR
' ESPONSE
o0
CONTRAS:
x :
339.
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a:
EANDWrDrts
069
66?.
072
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rr!
iii
z
o
LJ
6 t n
o
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o
ll
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8 A N0 W r 0 r a
=
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032
a 17
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2
o
ae
{\
r.0
0.1
t.o
o
S P A T I A L F R E Q U € I { C(Yc v u o e c )
rlc. 10. Spatial tuning functions of the cell shown in Fig. 9. In ,4, the spatial frequency response functrons
measured using three different contrasts are plotted on log-log coordinates. In I, the spatial irequency sensitivrry
functions measured using three different response critcrion livels are plotted. As can be seen, the overall shape
and bandwidth change little as the contrast of the test is varied; however. the tuning changes considerably as the
responsecrlterion is varied (the bandwidth being inversely related to the response criterion). It is worth noting
t h a t i f a s u b s e q u e n ct e l l u l a r m e c h a n i s m( r e c e i v i n gi n p u t f r o m a s t r i a t e n e u r o n s u c h a s t h e o n e i l l u s t r a t e d i n F r g s .
9 and l0) required a relatively high level of response(say above 809oof a cell's maximum). the resulting sensitivity
would be rather low: however, the spatial frequency information transmitted would be quite specific.
centagedid not changeacrosssimple and
complexcells in cat, but did show a small
(but significant)variationacrossthe mean
vafuesof monkeysimple(65Eo)versuscomlex (79Vo\cells.The relativemaenitudeof
A CAT AVERAGERESPONSE
this differenceis typical of the variarion
found in all the tablesacrosscell types:the
generaltrend shownby the total population
of cells is valid acrossall subgroups;however,there are often small (but, given the
8 MONKEYAVERAGERESPOilISE
to0
5U
lrJ
U'
z
o
(L,^
U' IU
UJ
E.
lQ
x: SIMPLE
^ ' COMPLEX
ro
30
root
3
% CONTRAST
x . SIMPLE
^ ' COMPLEX
to
30
too
F I G . I l . A v e r a g e p o p u l a t i o n r e s p o n s eb, r o k e n d o w n i n t e r m s o f a n i m a l t y p e a n d c e l l t y p e . i s s h o w n t o p r o v i d e
s o m e i n d i c a t i o n o f c e l l t y p e v a r i a t i o n . T h e p er c e n t r e s p o n s eo f e a c h c e l l ( r e l a t i v e t o t h e m a x i m u m ) w a s a v e r a g e d
a c r o s sa l l c e l l s o f a g i v e n t y p e ( C R F s o f a l l c e l l s w e r e t h u s w e i g h t e d e q u a l l y ) a n d p l o r t e d o n l o g - l o g c o o r d i n a r e s .
For cat. the H ratio providedthe best 6t to data (n = 1.4, Cr = 12.5):for monkel, a power function provided
t h e b e s t f i t ( s l o p e = 0 . 7 8 ) . f ) i f f e r e n c e sb e t w e e ns i m p l e a n d c o m p l e x c e l l s a r e d i f f i c u l r r o d r s c e r n .
STRIATE CORTEX: CONTRAST RESPONSE
E-RATIO
B M O N K E YA V E R A G H
A CA]' AVERAGEH-RATIO
l!
o
z
o
o( , t olrj
E
*
% C0{TRAST
ntc. 12. Toprovideacharacterizationof theaverageCRFforcomparisonpurposes.theupperandlowerbounds
(2 SE above and below the parameter means) of the average H ratio are shown for the population of cat cells and
monkey cells separately. As can be seen, the mean exponent and semisaturation contrast are greater for monkey
i n c o m D a r i s o nt o c a t .
sample size, statistically significant) variations.
One way of illustrating the similarities
and differences among cell types is to generate average contrast response functions.
To provide such characterizations,we developed two separate methods, each isolating different types of information. The first
method providesan indication of the average
percent excitation level (an average population response)of all cells within a given
class(say, all monkey simple cells) as a function of stimulus contrast. The results should
teli us what percentageof the total excitation
possible,for a given classof cell, is achieved
at any given contrast.
To generate these population response
functions, we summed together the normalized responses(percent relative to maximum) of each and every cell within a given
group acrossall contrasts,and then divided
by the number of cells in that group. The
resulting data points (responsemeans) for
all cell typesare shown in Fig. I l. Each point
representsthe averagepercent excitation of
all the cells within a particular group produced by the given contrast. As can be seen,
the differencesbetweensimple and complex
c e l l sa r e q u i t e s m a l l .
There is no compelling a priori reason to
expect thesedata points to fit any particular
mathematical function. Each cell has a siven
dynamic range distributed over a limited
contrast range. Thus, at say ll%o contrast,
some cells will have already begun to saturate while others have not yet begun to respond at all. As the contrast increases,more
and more cells are activated; whether this
increase in the number of cells responding
(in addition to the increasein the response
level of each cell) should be linear or otherwise is an empirical question. The data
points for cat cortical cells (both simple and
complex), shown in Fig. llA, were best fitted by the hyperbolic relationship drawn
t h r o u g h t h e p o i n t s ( n : 1 . 4 , C s o= 1 2 . 5 ) .
The data points for monkey cortical cells
(both simple and complex), shown in Fig.
I18, were best fitted by the power function
drawn through the points (slope = 0.78).
The secondmethod for characterizing the
general CRFs of different cell types was to
averageacrossthe values of the parameters
of the best-fitting hyperbolic relationships
(across all but the 9Vo with semisaturation
constantsbeyond 100). We took the average
values and then added or subtracted two
standarderrors from each to producethe two
most distant functions,as shown in Fig. l2A
for cat and l2B for monkey. There does
appear to be a distinct difference between
the average cat and monkey contrast responsefunctions.The 95Volimits (of the est i m a t e s o f t h e m e a n s )d o n o t o v e r l a p u n t i l
D. G. ALBRECHT AND D. B. HAMILTON
EXPONENT
SEMI.SATURATION
CEL
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oof
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- - |I
ozo I
d'['
I
rof
I
o: o Lf
H
O
54o z l
,.- |
oL
5
7
8
o 5
[iONKEY CELL
oof
6
.of
,or
I
20r
,or
I
rof
I
o["
o
rO
1
2
3
4
5
6
oLf
7
8
o
5
lo
t5
20 25 30 35 40)40
Flc' 13. Distribution of the H ratio constants broken down in terms
of animal type (monkey or cat).
the contrast reachessome20Vo.Thesedifferencescan be seenin the distributionsof
the slopesand semisaturation
constantsfor
cat and monkeycellsshownin Fig. 13.The
average exponent is certainly greater for
monkeycellsand the averagesemisaturation
contrastis alsogreater.Simpleand complex
cell averageH ratio functionswere not different.Referringback to Table 5, it is apparentthat the averagevaluesof the populations were virtually unchanged airois
simpleand complexcellswithin a givenanimal (well within +2 SE).
DISCUSSION
The major goal of this investigation was
to describe and characterize, both qualitatively and quantitatively, the responsesol
neuronsin the visual cortex of monkeys and
cats as a function of the contrast or iniensitv
of spatial stimuli. To this end we *"urur"i
the responsesof 24'7 striate neurons as a
function of the contrast of optimal spatialtemporal sine-wave grating patterns, and
then analyzed the fit of the responses(using
least-squarescriteria) with respect to foui
potentially useful mathematical relationships:linear, logarithmic, power, and hyperbolic ratio.
Qualitatively, the results of these experiments show that in general the responie of
most striate neuronsincreasesin a relativelv
linear fashion over a restrictedranse of con-
trasts (the first 0.5-1.0 log unit of contrast);
as the contrast increasesbeyondthe dynamic
linear range,the responsebeginsto compress
(over an equivalent range of contrast, 0.51.0 log unit) and then ultimately, the responsesaturates at a maximum rate of fir_
ing. Quantitatively, the results of the analysis indicate that, of the four mathematical
formulationstested,the hyperbolicratio provided the most accurateand generaldesCription of the contrast response function of
striate neurons. The H ratio numericallv
specifiesall three salient characteristicsof
the responsefunction: the linear increasegradual compression,and saturation.
Linear versus log functions
The previousbrief reports concerningthe
relationship of striate cell responsesto the
contrast of sine-wavegrating patterns have
been somewhat contradictory: some suggesting a log function, others a linear funition. Maffei and Fiorentini (26) provide six
examples of striate cell CRFs that appear
qualitativcly to be besr fitted by a log contrast function over some range of conirasts,
often followed by saturarion. Ikeda and
W r i g h t ( 2 1 ) s h o w o n e e x a m p l eo f a c o r t i c a l
CRF and state that, in general,the responses
of striate cells increaseas the log of contrast.
O t h e r s t u d i e s( l , 2 8 , 3 6 ) r e p o r t t h a t t h e r e lationship between responseand contrast is
approximately linear over restricted ranges:
however, beyond that limit, the respon"ses
STRIATE CORTEX: CONTRAST RESPONSE
l))
tend to compressandfor saturate.It should linearfunctionprovidesthe bestdescription
be notedthat noneof the abovereportswere of the response
overrestrictedrangesof conspecificallydesignedto providea systematic trast. Even if we set asidethe problemof
quantitativeevaluationof the contrastre- judiciouslyselectingthe limits of the response function from a large sample of strictedrange,it is nevertheless
the casethat
the
striatecells;the data consistof qualitative asthe contrastrangetestedis decreased,
comparisons
of small samplesof individual actualdifferencesbetweenthe two functions
Measuringsuch small differcellswith no quantitativeindexof how can- is decreased.
didatefunctionsmight fit (or residualvari- enceswould not only be difficult, but in a
This point is ilance)acrossa largesampleof neurons.The practicalsenseunnecessary.
of
resultsreportedhere may help clarify some lustratedin Fig. 14 wherethe responses
a typicalcell areplottedacrossa 1.6log unit
of the existingcontradictions.
The first point worth raisingconcernsthe range of contrasts(1.4-56V"):the leastgreat variety of CRF typesthat commonly squaresanalysis,however,was restrictedto
occurin a randomsampleof striatecells;as the five responsesthat occur over the first
illustratedin Fig. I and numericallyspeci- 0.9 log unit of contrast( 1.4-I l.5Vo).As can
fied in Tables 2 through 4, over a contrast be seen,the fit providedby eachof thesetwo
over
rangeof 1.4-56Vo
somecellsare best fltted functionsis virtually indistinguishable
by a strict linear function,othersby a log the first 0.9 log unit of contrast.In favor of
function.It wouldthereforenot be surprising the linear functionthe sum of squareswas
to find examplesof eachin a sampleof cells. slightlyless;however,notethat the log funcA secondpoint concernsthe criteriausedto tion fit on the basisof the first five points is
determinewhich relationshipbestdescribes certainlyin a better positionto capturethe
with the exception sixth point, whereasthe linear function is
the measuredresponses;
of the one cell shownby Ikeda and Wright quite distant.In the end, as the analysisis
(21), thereare no reportsof a numericalin- restrictedto smallrangesof contrasts(small
the numberof functionsthat
dex used to ascertainquantitativelywhich perturbations),
functionprovidedthe bestfit. Giventhe ex- can providea goodfit increases.
Somefinalconsiderations
concernthegoal
tant variation,objectivenumericalcriteria
seemrequired.
of accuratesimplicity.Sinceeitherfunction
A third pointconcerns
the methodological may well fit the first 0.5- 1.0log unit of confeasibility(and practicalnecessityor desir- trast, parsimonywould moveus in favor of
ability) of distinguishingwhethera log or the linear assumption.However,the ulti-
L I N E A R( R e s r n r c r e o )
LOG (RESrRrcrEo)
roo
trJ
@
z
o
(L
U)50
lrJ
E,
;q
-o
25
25
50
0
% CONTRAST
50
n l c . 1 4 . C o n t r a s t r e s p o n s ef u n c t i o n f o r a t y p i c a l s t r i a t e c e l l p l o t t e d o n l i n e a r - l i n e a rc o o r d i n a t e s t; h e s o l i d c u r v e s
s h o w t h e b e s t - f i t t i n gl i n e a r a n d l o g f u n c t i o n w h e n t h e l e a s t - s q u a r eas n a l y s i si s r e s t r i c t e dt o t h e f i r s t f i v e d a t a p o i n t s
( 0 . 9 l o g u n i t o f c o n t r a s t ) .A s c a n b e s e e n ,t h e r e i s v e r l ' I i t t l e d i f f e r e n c ei n t h e g o o d n e s so f f i t b e t w e e nl o g a n d l i n e a r
w h e n t h e a n a l y s i si s r e s t r i c t e d ib o t h f u n c t i o n s p r o v i d e a n a d e q u a t ef i t .
234
D. C. ALBRECHTAND D. B. HAMILTON
roo
SINGLE- CHANNEL
M U L T-I C H A N N E L
lrl
@
z
o
(Lqn
auu
Lr,l
tts
POWER
be
n
o
30
600
30
% CONTRAST
60
Flc l5' When considering the overall behavioral effect of the striate cell CRFs on the contrast-incremenr
t h r e s h o l d sa n d m a g n i t u d e e s t i m a l i o n s ,o n e c a n t h i n k o f a ) t h e o v e r a l l c o m p r e s s i v en a t u r e o f e a c h c e l l ' s C R F ( a
single-channel process), or b) the variation in the location of each cell's dynamic response range (a multichannel
process). The function shown in the single-channel model is the best-fitting power function (iveraged across all
cells); the function shown in the multichannel model is the average H ratio shifted to cover diffJrent contrasr
ranges.
mate nonlinear compression that striate
CRFs show over an equally extensiverange
of contrasts argues for the accuracy of the
log function. The main flaw in the accuracv
of the log function is that it does not compress as rapidly as striate cells and it does
not show saturation.
Fortunately our options are not restricted.
As we have shown in the RESULTSsection
above. the H ratio provides a mathematical
relationship that is very similar to what is
actually measuredfor striate cells: linear increaseover a restricted range, gradually accelerating compression,to an asymptotic final saturation. All three components of
striate cell contrast responsebehavior are
summarized in this one accurate and simple
formulation.
lowing inaccuracy will result: optimal stimuli will produce the largest responsesat low
contrastsbut as the contrast increases.nonoptimal stimuli will generate responsesas
large as optimal stimuli. This state of affairs
is illustrated in Fig. 98 where, given the response-setgain curves, a contrast of 607owill
produce essentiallyequivalent responsesto
a set of spatial frequencieswhich, when presentedat lower contrasts,producequite nonequivalent responses. Such unreliability
would seem undesirable.
However, as shown in Figs. 7-10, the
compression and saturation of the striate
cells' contrast responsefunction do not seem
to be a simple function of the absolute response magnitude; the point at which the
responseto a given frequency will begin to
compress and saturate is set more by the
absolute value of the contrast (and not the
response). The response to each test frequency will begin to compressand saturate
at the same fixed contrast. The net result of
this contrast-setgain mechanismis to maintain the spatial tuning characteristicsindependent of stimulus contrast.
Contrast response and spatial variable
The nature of the contrast responsefunclion will necessarilyinfluencethe spatial-analytic capabilities of cortical cells, particularly as the function deviatesfrom linearity.
The nonlinear aspectsof the striate cell contrast response function (compression and
saturation) can potentially have adverseeffects on the ability of a cortical neuron to Psychophysical invest i gations
signal the presenceof a specificspatial-tem- of contrast
poral luminance contrast reliably and acThere are now several reports of human
curately. If the compressionand saturation psychophysicalsuprathresholdcontrast disare set by the absolute responsesize (that
crimination functions(7, 24, 29, 35) and
is, if the gain is responseset), then the fol- m a g n i t u d e e s t i m a t i o n f u n c t i o n s ( 8 , 2 3 ) .
235
STRIATE CORTEX: CONTRAST RESPONSE
These studies were all performed using similar stimuli: luminance-modulatedsine-wave
grating patterns. While the relations between stimulus strength, neural activity,
behavioral discrimination. and sensation
magnitude are certainly not obvious or experimentally demonstrated (cf. Ref. 38), it
is neverthelessworth examining the behavior
of neuronsunder conditions similar to those
used in psychophysical experiments and
comparing the results from the two analogous experiments.
The psychophysicalmeasurementsof the
contrast discrimination function (CDF) all
agree that the just noticeable difference in
contrast of a test grating increaseswith the
contrast of the background grating, qualitatively in accord with Weber's law. Quantitative estimates of the slope of the CDF
plotted on log-log coordinates range from 0.5
to l.0, dependingon the exact methods(generally falling short of Weber's law). Several
possible neurophysiological correlates at the
level of the striate cortex can be suggested.
It is possiblethat the increasein the contrast just noticeable difference simply reflects the type of compression shown by single striate cells. It is certainly the case for
most cellsthat over a range of contrastsfrom
I to 56Eo,the increment in contrast required
to producea given increment in responsewill
generally increase along the contrast axis
(the slope of the function relating response
to contrast generally decreases).If we use
a power function to characterize the responsesof all of the cells in our sample, we
find that the average slope of the function
(after least-squaresparameter optimization
for each cell) was 0.59 + 0.002 (SE) for all
cat cells and 0.85 t 0.004 (SE) for all monkey cells; a comparisonof the means across
simple and complex cells revealedno significant differences.Thus, the compressivecontrast responsefunctions shown by individual
striate neuronsmay be a contributing factor
in the compressive behavioral contrast discrimination function.
A quire differcnt considerationstemsfrom
the fact that diffcrent striate neurons distribute their respective dynamic response
range over different portions of the contrast
range. Some cells begin responding at l%
contrast and are completely saturated by
l0% contrast;others begin respondingat 57o
too
A=7.4
I
e = O . 7, \
lrj
U'
z
o
&
@ t' -o
uJ
G
\ro=o''
ae
t 8. o.a:
t
3
t
o
3
0
CoNTRAST
%
t
o
F I G . 1 6 . D a t a p o i n t s s h o w t h e a v e r a g e r e s p o n s eo f
t h e e n t i r e p o p u l a t i o no f 2 4 7 c o r t i c a l c e l l s ; t h e n o r m a l ized contrast response functions from each and every
cell were averaged together across contrast to provide
some indication of how a large populalion of visual cortical cells might respond during the course of a typical
behavioral contrast-discrimination task. To derive the
two power functions shown, 2 SE of each population
mean were added and subtractedi these upper and lower
b o u n d s w e r e t h e n a n a l y z e d f o r a l e a s t - s q u a r e sf i t ( t h e
saturated responseat 5670,primarily from the cat cells,
was excludedfrom this analysis-compare Fig. I l: if
5 6 9 oi s i n c l u d e d ,v a l u e so f t h e h i g h a n d l o w p a r a m e t e r s
b e c o m eA
: = 9 . 7 . 6 . 2 . B : 0 . 6 . 0 . 7 ) . T h e s ee s l i m a t e so f
the slope of the population contrast response function
( t h e a v e r a g eb e i n g 0 . 7 7 ) a r e a l l w e l l w i t h i n t h e r a n g e
of estimates found for behavioral contrast discrimina-
contrast,and still others at l0 or 207o.Thus,
as a background contrast increases,the responselevel of some cells would increaseto
saturation while others would begin to respond.Given this, one might expectthat over
some range of background contrasts (perhaps l-307o) there would always be some
group of cells respondingover their respective dynamic linear ranges.
We might term the first suggestiona nonlinear single-channelmodel, since the behavioral CDF might reflectthe averagecompressive physiological CRF. The second
model might best be termed a linear mult i p l e - c h a n n em
l o d e l , s i n c et h e C D F w i l l r e flect the summed activity of multiply' posit i o n e d( a l o n gt h e c o n t r a s ta x i s ) l i n e a rC R F s .
T h e s e t w o m o d e l sa r e i l l u s t r a t e di n F i g . l 5
where the average power function for all
o
lJo
D. G. ALBRECHT AND D. B. HAMILTON
cells is plotted (a single channel) and the
average H ratio has been shifted left and
right about the mean semisaturation(a multichannel model).
Regardlessof whether these models will
help describe the physiology of behavioral
contrast discrimination, it must surely be the
case that many cells (a population of cells
and not just an individual cell) will be activated by the inducing background contrast
and the test contrast. It is therefore useful
to have some indication of how the activity
of a large population of cells (preferably the
entire ensemble of cells in the cortex) increaseswith contrast. To this end, we averaged together (across all cell types: cat,
monkey, simple, complex) the normalized
responses (percent relative to maximum,
thus giving each cell equal weight) ol each
and every cell from our sample of 247 cells.
While there are a varietv of different
methods (and basic assumptions)one could
use to derive a population response,the average population contrast responsefunction
shown in Fig. 16 provides us with the sum
total responsefor a sample of 247 visual cortical neurons; this summed population responseis presumably somewhat comparable
to the average cortical response that is
evoked during the course of a typical behavioral contrast discrimination task. The
slopeof the power function that best fits the
average responseof all the neurons added
together (over the l-33Vo contrast range) is
0.77, a value well within the range of the
psychophysicalestimates.
ACKNOWLEDGMENTS
W e t h a n k W . S . G e i s l e r .R . L . D e V a l o i s . L . G . T h o r e l l . a n d t h e U n r v e r s i t yo f T e x a s .
Received 3l July l98l; accepted in final form 22
Februarv 1982.
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