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The Journal of Neuroscience, July 15, 2002, 22(14):6129-6157
Detection and Discrimination of Relative Spatial Phase by V1
Neurons
Ferenc
Mechler,
Daniel S.
Reich, and
Jonathan D.
Victor
Department of Neurology and Neuroscience, Weill Medical College of
Cornell University, New York, New York 10021
 |
ABSTRACT |
Edge-like and line-like features result from spatial phase
congruence, the local phase agreement between harmonic components of a
spatial waveform. Psychophysical observations and models of early
visual processing suggest that human visual feature detectors are
specialized for edge-like and line-like phase congruence. To test
whether primary visual cortex (V1) neurons account for such
specificity, we made tetrode recordings in anesthetized macaque monkeys. Stimuli were drifting equal-energy compound gratings composed
of four sinusoidal components. Eight congruence phases (one-dimensional features) were tested, including line-like and edge-like waveforms. Many of the 137 single V1 neurons (recorded at 45 sites) could reliably signal phase congruence by any of several
response measures. Across neurons, the preferred spatial feature had
only a modest bias for line-like waveforms. Information-theoretic analysis showed that congruence phase was temporally encoded in the
frequency band present in the stimuli. The most sensitive neurons had
feature discrimination thresholds that approached psychophysical
levels, but typical neurons were substantially less sensitive. In
single V1 neurons, feature discrimination exhibited various
dependences on the congruence phase of the reference waveform. Simple
cells were over-represented among the most sensitive neurons and on
average carried twice as much feature information as complex cells.
However, the distribution of the indices of optimal tuning and
discrimination of relative phase was indistinguishable in simple and
complex cells. Our results suggest that phase-sensitive pooling of
responses is required to account for human psychophysical performance,
although variation in feature selectivity among nearby neurons is considerable.
Key words:
spatial feature detection; feature discrimination; phase-selective nonlinearity; congruence phase; edge; line; macaque; primary visual cortex; transinformation; simple and complex cells
 |
INTRODUCTION |
Psychophysical studies of spatial
vision have demonstrated the importance of spatial phase information in
shape perception (Burton and Moorhead, 1981
;
Oppenheim and Lim, 1981
), texture discrimination
(Klein and Tyler, 1986
; Rentschler et al.,
1988
), and contour integration (Field et al.,
1993
; Kovacs and Julesz, 1993
; Dakin and
Hess, 1999
). Edge-like and line-like features are examples of
salient spatial cues defined by phase. Detection thresholds for
compound gratings (Tolhurst, 1972
; Shapley and Tolhurst, 1973
; Tolhurst and Dealy, 1975
), and
the discrimination sensitivity for the relative spatial phase of
harmonic components of compound gratings (Burr, 1980
;
Badcock, 1984a
, b
; Burr et al., 1989
) as well as the
phase dependence in monocular rivalry (Atkinson and Campbell,
1974
) and afterimages (Georgeson and Turner,
1985
), are all consistent with the existence of two classes of
feature detectors, one tuned to edge-like and the other to line-like
waveforms. Human discrimination of relative phase requires contrasts
markedly above detection threshold, (Nachmias and Weber,
1975
), indicating that the mechanism underlying discrimination
is nonlinear.
The prevailing view of early vision posits localized and spectrally
band-limited image analysis at multiple spatial scales. The privileged
role of lines and edges as features in human vision is posited to
derive from phase congruence (Morrone and Burr, 1988
).
This is illustrated in Figure 1. Phase congruence denotes a local
phenomenon whereby harmonic components across spatial scales share a
common phase and, consequently, reinforce that phase by summation.
Edges and lines are examples of salient phase congruence across spatial
scales. Sensitivity to phase congruence requires the existence of local
mechanisms that compare relative phase information across multiple scales.
Theoretical work also motivates these experiments. The nonlinear
feature detector model developed by Burr and Morrone
(1992)
derives an edge versus line feature dichotomy from the
orthogonal odd versus even symmetry of the spatial function of these
features' cross-section. The first stage of their model consists of
even/odd symmetry-sensitive linear spatial filters, idealized cortical simple cells. The second stage, intended to represent complex cells,
implements a local energy operator: squared filter outputs are summed
within a single orientation band in a phase-specific manner. At the
final stage, features are identified by a winner-take-all localization
of maxima in the map of feature energy. The model of Burr and
Morrone (1992)
makes successful qualitative predictions of
illusions, quantitative predictions of thresholds, and testable predictions for the roles of simple and complex cells in feature detection and discrimination.
Our paper expands on earlier studies that assayed with spatial compound
gratings the feature (relative phase) selectivity of single neurons in
the primary visual cortex (V1) of cat (De Valois and Tootell,
1983
; Levitt et al., 1990
) and monkey
(Pollen et al., 1988
). We found that nonlinearities
contributed to feature coding in the entire frequency band of the
stimulus. Most response harmonics, but not the DC, were tuned to
features. Preferred features were rather evenly distributed in V1
(edges or lines were not overtly over-represented) and also varied
within local clusters. Feature discrimination threshold in the most
sensitive V1 neurons approached human psychophysical thresholds. These
statements held for both simple and complex cells. The pattern of
feature tuning and discrimination observed in V1 neurons puts new
constraints on our models of cortical circuits.
Parts of this paper have been published previously at the 1998 and 1999 Annual Meeting of The Society for Neuroscience (Mechler et al.,
1998a
, 1999
).
 |
MATERIALS AND METHODS |
Physiological preparation. Standard acute preparation
techniques were used for electrophysiological recordings from single units in the V1 of the primate (cynomolgus monkeys, Macaca
fascicularis). All procedures were in accordance with
institutional and National Institutes of Health guidelines for the care
and experimental use of animals. Some details of the techniques have
been given earlier (Mechler et al., 1998b
).
Experiments were performed on 14 adult animals, weighing 3-4.5 kg.
Before surgery, animals were given atropine (0.1 mg/kg, i.m.) and then
anesthetized with ketamine (10 mg/kg, i.m.; Ketaset, Fort Dodge, IA).
Anesthesia was maintained with sufentanil citrate (3-6
µg · kg
1 · hr
1,
i.v.; Sufenta, Janssen, Titusville, NJ), and muscle paralysis was
induced (after all surgical procedures) and maintained with pancuronium
bromide (0.1 mg · kg
1 · hr
1,
i.v.). Dexamethasone (1 mg/kg, i.m.) and gentamicin (5 mg/kg, i.m.)
were given to help prevent the development of cerebral edema and
infection, respectively. The animal was ventilated through an
endotracheal tube. Heart rate, EKG, arterial blood pressure, and
end-tidal CO2 were continuously monitored with a Model
78354A Hewlett-Packard Patient Monitor and kept in the normal
physiological range. Core body temperature was maintained between 37 and 38°C using a thermostatically controlled heating pad. The EEG was
obtained from frontal leads and monitored on an oscilloscope.
A limited unilateral craniotomy to expose the primary visual cortex was
made overlying and posterior to the lunate sulcus (the Horsley-Clarke
stereotaxic coordinates were typically 14-16 mm posterior and 14-16
mm lateral). A 1-2 mm durotomy was made for the recording electrode,
which was stabilized after insertion by agarose gel.
Extracellular recording. Spike responses of single units
were recorded extracellularly. We used either traditional glass-coated tungsten microelectrodes (single tip; typical resistance 2 M
) (Merrill and Ainsworth, 1972
; Ainsworth et al.,
1977
), or quartz-coated platinum-tungsten fibers tetrodes
(Thomas Recording, Giessen, Germany). Tetrodes had a conical tip, with
four contacts of ~1 M
each, ~25 µm apart: one at the apex and
three arranged in radial symmetry on the conical surface. A stepper
motor advanced either type of electrode in 1 µm steps.
The signals from the electrode or tetrode channels were passed through
a unity gain (for the tetrode, multi-channel) differential head-stage
amplifier (NB Labs, Denison, TX, or NeuraLynx, Tucson, AZ), and then
further amplified and filtered (0.3-6 kHz pass-band, NeuraLynx
eight-channel differential amplifier). Analog candidate spike
waveforms, as detected by a threshold criterion, were digitized at 25 kHz within a short (~1.2 msec) temporal window containing the peak
amplitude, and then recorded on computer disk (Discovery software,
DataWave Technologies, Longmont, CO). Multiple single units were
isolated by cluster analysis of spike waveforms initially performed
on-line (Autocut, DataWave Technologies), then off-line [custom
software (Reich, 2001
)]. Isolation criteria included
stability of principal components of spike waveforms and a 1.2 msec
minimum interspike interval consistent with a physiologic refractory
period. Spike times for further data analysis were identified off-line to 0.1 msec, the accuracy to which the clocks of the recording computer
and the stimulus generator were synchronized.
Histology and laminar assignment of recording sites.
Experiments lasted for 4-5 d, at the end of which the animal was
killed by infusion of a lethal dose of methohexital (Brevital;
Eli Lilly & Co., Indianapolis, IN). After transcardiac perfusion with
4% paraformaldehyde in PBS, a block of the occipital cortex containing the penetration was saved for histological reconstruction of the electrode track. The block was cut in 40-µm-thick parasagittal sections, approximately parallel with the plane of the electrode penetration. Lesioned landmarks and fluorescent tracing aided track
reconstruction. Electrolytic lesions (5 µA × 5 sec, electrode positive) were made, on withdrawal after recording was completed, at
two or more points along all the tracks made with an Ainsworth single
electrode, and on some tracks made with tetrodes. Fluorescent full-track tracing was made with the lipophilic dye Dil (D-282; Molecular Probes, Eugene, OR). The dye, applied in a thin coat on the
tetrode tip before penetration, left a ~40- to 200-µm-wide trace
from entry to the point of deepest penetration. These traces were
easily identified in fluorescent micrographs prepared from sections
before Nissl staining. In the same sections, the laminar boundaries
were identified from the overlaid light micrographs of the Nissl
density taken after Nissl staining. Lesions were also best identified
on the Nissl-stained sections. Laminar positions of the recording sites
were estimated relative to the pattern of Nissl density along the
reconstructed electrode track after correction for tissue shrinkage.
With this method we successfully identified the laminar position of
two-thirds of the recording sites. Sites near a laminar boundary within
the precision of reconstruction were classified as located in either
lamina across the boundary. However, even with good histology,
occasionally landmark positions could not be found or remained
ambiguous, and laminar positions were either not assigned to recording
sites or could only be classified in one of three gross divisions
(granular, supragranular, or infragranular layers).
Optics. The eyes were treated with anti-inflammatory
(Ocufen) and anti-bacterial (neomycin) ophthalmic solutions. Pupils
were dilated with topical application of 1% atropine sulfate
(Atrosulf-1; Optics Laboratories Co., Fairton, NJ) and covered with
gas-permeable contact lenses (Metro Optics Inc., Houston, TX) under
eyelids retracted with 6-0 chromic gut sutures. Artificial pupils (2 mm) and corrective lenses were used to focus the stimulus on the
retina. Optical correction was estimated by retinoscopy and then
refined by optimizing responses of isolated single units to high
spatial frequency visual stimuli.
Visual stimulation. Foveae were mapped on a tangent board by
back-projection with an ophthalmoscope. The receptive fields of
isolated neurons were mapped on the same board with a laser. The
standard simple/complex classification, based on the modulation ratio,
was used: if the fundamental of the response to a drifting grating of
near optimal spatial parameters was larger than the DC component (after
subtraction of the maintained rate of firing), then the cell was cast
as simple, and complex otherwise (Movshon et al., 1978b
;
De Valois et al., 1982
; Skottun et al.,
1991
).
Visual stimuli were generated by a special purpose stimulus generator
(Milkman et al., 1978
, 1980
) under the control of a PDP-11/93 computer and
displayed on a Tektronix 608 monochrome oscilloscope (green phosphor;
150 cd/m2 mean luminance; 270.32 Hz frame refresh).
The luminance of the display was linearized with lookup tables in the
range of 0-300 cd/m2. At the 114 cm viewing
distance of the animal, the stimuli appeared in a 4° circular
aperture on dark background.
After isolation of single units, their receptive fields were
characterized in a standard way using drifting sine gratings: tuning
was measured first for orientation, then for spatial frequency, and
finally for temporal frequency, each parameter optimized for subsequent
tuning measurements. The contrast response function was measured using
the optimal sine grating. When multiple single units were
simultaneously isolated with tetrodes, receptive-field characterization
was always done for the most responsive unit, and often for a second
unit. For many neurons, the receptive field was also characterized with
pseudorandom black-and-white checkerboards modulated by long
(212-1 frames) binary m-sequences at 67.58 Hz. Our
implementation of m-sequence stimuli and associated analysis procedures
have been described in detail previously (Victor, 1992
;
Reid et al., 1997
; Reich et al.,
2000
).
Compound gratings. In our experiments, 1D gratings
were drifting at or near the optimal orientation and direction for the V1 neurons. With the spatial origin centered on the display, the spatiotemporal light variation
I(x,t) around a spatiotemporal mean
intensity I0 in a single drifting sine grating
is described, in cosine formulation for convenience, by:
|
(1)
|
where C is the Michelson contrast (defined as
C = [max(
I)
min(
I)]/I0),
is spatial frequency (c/°), f is temporal
frequency (in Hz), and
is relative phase (in radians). At time
zero, the intensity peak is at position 
/2k
(so, if
= 0, it is at the origin). The drift velocity of the grating
is v = f/
. Compound gratings
are linear combinations (spatiotemporal superpositions) of these single
sine gratings.
Each of our compound-grating stimuli is constructed from four of
these single-grating harmonic components. We use a superposition of odd
harmonics. That is, the mth component grating is chosen to
have a frequency equal to 2m-1 times the fundamental.
Consequently, the light variation around the mean intensity in the
mth component, Sm(x,t), is given by:
|
(2)
|
Thus, the four gratings included a fundamental and its third,
fifth and seventh harmonic (see Fig. 2a, boxed
area), each with a contrast inversely proportional to the harmonic
number, and at the same drift velocity v = Fk/
k = f/
. For the fundamental, we used a
low-frequency sine grating (typically,
= 0.25 c/°, and
f = 0.78 Hz; v = 3.1 °/sec). These
fundamentals were selected so that the higher harmonics up to the
seventh fell within the pass-band of most cells. Across the set of
compound gratings, the spatial and temporal frequencies and the
contrasts of the four components were unchanged, but the phases were
varied systematically to specify the shape of the compound waveform.
With the above notation, the light variation (around the mean
intensity) in the compound grating stimuli that we used is given
by:
|
(3)
|
Thus,
plays the role of the
congruence phase, i.e., the phase shared by all components at
x = 0 and t = 0 (Fig. 1). As seen in
Figure 2b, we sampled the
congruence phase in eight equal steps on the [0,
) phase interval to
construct eight different compound waveforms. The amplitudes of the
four component gratings were chosen so that, when combined with phase
=
/2, these components constitute the first four non-zero
Fourier components of a square wave (or edge; see Fig. 2a).
Because the amplitudes of the components were the same for each
stimulus, all the compound gratings thus constructed had equal energy.
For a comprehensive discussion of the mathematical properties of our
compound gratings, see Appendix.

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Figure 1.
The definition of congruence phase, . At the
location of phase congruence, components reinforce the local spatial
feature that dominates the compound waveform. Depending on their
congruence phase, , the sum of the same four component gratings can
give rise to very different spatial compound waveforms. On the
left, the components are combined in cosine phase ( = 0). The harmonic components coincide at their peaks, leading to a
waveform of alternating bright and dark lines. On the right,
components are combined in sine phase ( = /2). The harmonic
components coincide at their position of maximal slopes, leading to a
periodic sequence of on- and off-edges approximating a square
wave.
|
|

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Figure 2.
Construction of our compound grating
stimuli. a, A square wave (edge) is a linear combination of
an infinite series of spatial sine waves. This Fourier decomposition of
the edge contains only the odd harmonics of the fundamental spatial
frequency f, each with amplitude inversely proportional to
its harmonic index. Note that the components have the same relative
phase ( = /2 at the location of the spatial feature, the
edge. This identical relative phase of the components at the location
of the spatial feature is called the congruence phase. b,
The eight equal-energy compound luminance gratings used in our
experiments (thick lines) were built of four sinusoidal
components (thin lines), the first four non-zero components
of an edge (f through 7f shown
boxed in a). The congruence phase, , is varied
in eight equal steps counterclockwise around the phase circle
[0, ). The spatial waveform of the compound gratings varies
smoothly with , from line-like ( = 0) through edge-like
( = /2) back to line-like ( = ) through
intermediate transient waveforms. Notice that the line-like waveform
obtained with = is a half-cycle shifted version of the
waveform with = 0. Correspondingly, the variation in waveform
observed throughout the [0, ) phase interval is repeated on the
[ ,2 ) phase interval, with a half-cycle shift in the compound
waveforms. Because all stimuli were presented as drifting
waveforms, this spatial shift is equivalent to a half-period temporal
delay. Therefore, stimuli on the [ ,2 ) phase interval duplicate
those in the [0, ) phase interval.
|
|
Note that the phase parameter
specifies the shape of each compound
grating. As the phase parameter increases from 0 to
, the compound
waveform smoothly varies, from line-like (at
= 0), to
edge-like (at
=
), and then back to line-like (via a different sequence of waveforms). This sequence of waveforms is then
repeated as
varies along the [
,2
) interval. Note that a
waveform constructed with a particular value of
is shifted by half
a period (either in time or in space) when
is replaced by
+
, and thus does not produce new stimuli. In summary, by varying a
single phase-parameter on just half the circle, we create a "feature
space" of one-dimensional (1D) equal energy compound gratings. We
call the corresponding parameter space the "phase circle," keeping
in mind that it comprises the periodic continuation of the [0,
)
interval. In Figure 2b, this feature space is illustrated with the eight equally spaced samples around the phase circle that we
used in these experiments.
Note that although the edge-like combination of an infinite number of
sine components is convergent (because it is the Fourier series of an
edge; see Fig. 2a) the infinite series does not converge for
any other phase congruence. Consequently, with the exception of the
edge-like stimulus, the peak (Michelson) contrast of each compound
waveform would grow without limit, albeit slowly, as additional
odd-harmonic components were added. However, this does not lead to any
practical difficulties, because we use only a finite set of gratings
for all phase combinations. For a fixed set of components, the
Michelson contrast in our feature space decreases monotonically (as a
cosine function of congruence phase) from line to edge in either
direction on the phase circle. The Michelson contrast is largest for
the line-like waveform (congruence phase
= 0), the contrast of
which at peak is
, and smallest for the edge-like waveform (congruence phase
=
/2), the contrast of which at peak corresponds to
. We set the contrast of the fundamental component C to 0.5 so
that the modulation of the four-component line-like waveform had a Michelson contrast of 0.84. The root-mean-square contrast was 0.38 for
each compound grating.
Data analysis. Off-line data analysis was performed in the
Matlab programming environment using custom software. In general, fast
Fourier transforms were used whenever Fourier analysis is mentioned.
The details of the information analysis based on Fourier metrics have
been given previously (Mechler et al., 1998b
). Matlab toolbox functions, as well as custom programs, were used to perform tests of statistical significance. Specifics of each data analysis will
accompany the description of the corresponding results.
 |
RESULTS |
Data were obtained from V1 neurons with parafoveal receptive
fields (centered at 2-5° eccentricity). Following convention, we
used the modulation ratio (see Materials and Methods) for the classification of V1 neurons: if the modulation ratio exceeded 1.0, neurons were classified as simple cells, and complex cells otherwise. A
total of 226 data sets were collected from 137 neurons (88 complex and
49 simple) from 45 recording sites. Criteria for quantitative analysis
were (1) good isolation was maintained throughout the experiments
described below, and (2) responses to at least one of the compound
gratings were reliable (d' > 1.0 for the amplitude of any
of the first six Fourier components of the response in comparison to
the blank condition, or
across these first six components). Slightly more than half of the data
sets met these criteria. These 121 data sets from 32 recording sites
included 78 data sets from 46 complex cells and 43 data sets from 31 simple cells. (Some cells yielded two data sets from compound gratings
of different drift velocity). Note that in each recorded cluster the
fundamental frequency and orientation of the compound gratings were
optimized for one cell only (usually the most robustly responding one).
Because grating parameters were not necessarily optimal for each cell
in the cluster, the fraction of cells that could yield responses that
met analysis criteria (had they been stimulated with gratings of
optimal orientation) may be higher than 77/137. Cells that did not meet
the above selection criteria for analysis typically also responded
poorly to the component gratings presented alone at the selected
frequencies and orientation.
Feature tuning in V1 neurons
Our aim in this study was to gain insight into how V1 neurons
signal and discriminate spatial waveforms, including those that resemble salient spatial features such as edges and lines. These features are presumed salient because of spatial phase congruence. We
know that although appropriate symmetry-selective filtering is
necessary, linear filtering alone cannot explain the underlying feature-extraction mechanism. Subcortical visual processing involves nonlinear transformations, but these transformations are primarily related to adjustment of overall gain and dynamics, and are not orientation or feature specific. Thus, the neuronal circuitry that
performs feature extraction in primates is almost certainly at a
cortical level.
The neuronal implementation of feature extraction, however, is as yet
unknown. Natural candidates for the pre-filters are V1 simple cells the
receptive field profiles of which have the appropriate even or odd
symmetry as required by a local energy model. Although the analysis of
phase selectivity to spatial compound gratings is a necessary step in
understanding the relationship of these neurons to feature extraction,
only a few studies of single neurons evaluated this directly: De
Valois and Tootell (1983)
and Levitt et al.
(1990)
in the cat, and Pollen et al. (1988)
in
the monkey. Our study extends these earlier works by examining
responses to more complex (f + 3f + 5f + 7f)
compound gratings at a closely spaced set of relative phases, and also responses to the components themselves. To obtain good statistical confidence, we typically recorded responses for 100 repeats of each
stimulus. With tetrodes, we simultaneously probed multiple nearby
neurons, thus examining the local variation of phase selectivity of V1
neurons. These measures allowed us to address questions about spatial
feature extraction in V1 that have both neurophysiological and
psychophysical implications.
The defining feature of simple cells is the simple, approximately
linear fashion in which they appear to sum spatial stimuli within their
classical receptive fields (Hubel and Wiesel, 1962
), but
it is well recognized that this approximate spatial linearity is
typically compounded with various types of nonlinearity
(Movshon et al., 1978a
; Albrecht and
Geisler, 1991
; Carandini et al., 1997a
). Strict
linearity mandates that a response contain only components at those
temporal frequencies that are present in the stimulus. If simple cells
were strictly linear, the amplitude and phase of each harmonic
component of their response to the compound grating would depend only
on the corresponding component grating in the stimulus. The presence of
other stimulus components, or the phase in which they are combined,
should be irrelevant. Consequently, if we were to restrict the response
measure to a single harmonic present in the stimulus, the magnitude and
phase of this response harmonic would be identical for all of the
compound gratings, up to a phase offset corresponding to the phase
offset in the stimulus. Moreover, responses at even harmonics should be
absent, because the stimulus components are restricted to the first
four odd harmonics. However, nonlinearities are expected in the
response to compound gratings even in simple cells. The
most obvious nonlinearity in all V1 neurons is a spike threshold. Other
nonlinearities expected in all V1 neurons include contrast gain control
(Albrecht and Hamilton, 1982
; Bonds,
1989
; Heeger, 1992
), which is thought to be
phase-insensitive, and pattern adaptation (Maffei et al.,
1973
; Carandini et al., 1997b
,
1998
), which may be
phase-sensitive. The aim of the initial analysis was to identify the
effects of these nonlinearities in the responses of simple cells to
compound gratings. We also asked whether nonlinear responses are tuned to spatial waveforms, and if so, how the tuning is distributed in the
population of V1 simple cells.
Responses of a paradigmatic simple cell are shown in Figure
3. This layer 4C
simple cell had
little spontaneous activity in the absence of visual stimuli (shown as
the blank condition, i.e. a uniform screen of luminance set at the mean
of the grating stimuli in Fig. 3a). Single drifting gratings
(those in Fig. 3a, as well as other sine gratings used for
characterizing the neuron; data not shown) elicited responses that
seemed close approximations to half-wave rectified sinusoids the
modulation frequency of which was that of the first harmonic component
of the stimuli. This behavior is characteristic of typical simple
cells, both in our data and as previously reported (Movshon et
al., 1978a
; Skottun et al., 1991
). Responses
elicited by the set of eight compound gratings are shown in Figure
3b, organized according to the position of the compound
gratings in the feature space. This simple cell responded with a robust
burst of spikes to the passage of an OFF-transient (luminance
decrement), present to variable extent in each of the eight waveforms.
Although the transient of the opposite polarity, an ON-transient
(luminance increment), is also present in each stimulus waveform, this
cell fired only minimally during its passage in most conditions. This
sensitivity to spatial contrast polarity is characteristic of a linear
spatial integrator followed by a threshold. Because of the threshold,
an elevation in firing rate in a linear response to one polarity is not
matched by a drop in firing rate to the opposite polarity.

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Figure 3.
Typical responses to compound gratings
and their components recorded from a V1 simple cell, a layer 4C
neuron (L400306as). For each condition, thick lines
(bottom) represent the time course of luminance variation
across one repeat of the stimulus near the center of the receptive
field. (A repeat is one period at the fundamental temporal frequency of
F1 = 0.78 Hz.) Note that the temporal
waveforms are not the same as the spatial waveforms depicted in Figure
2, but are related by mirror symmetry and translation because the
stimuli depend on time and space through the combination
vx ft (see Eq. 3). Raster plots
(middle) show the spike responses recorded for 100 repeats.
Poststimulus time histograms (top) show the average firing
rate variation in 20 msec bins. a, Responses to the
component sinusoids presented individually
{F1, F3,
F5, F7}. The
blank condition is also included (top). b,
Responses to compound waveform stimuli. Stimuli and responses are
arranged around the circle of the feature space (as in Fig. 2) and
labeled by their congruence phase, . Also included is the response
to the true edge: it is directly left of the response to the
compound grating with the edge-like congruence phase ( = /2).
|
|
Note the similarity between the response to the full edge (Fig.
3b, true edge) and the response to the stimulus
that approximates an edge via its first four components (Fig.
3b, "edge"). For this cell, the response to
the full edge is slightly narrower in time. This indicates that the
pass-band of the linear receptive field of the cell was broad enough so
that one or more stimulus components of the edge above the seventh
harmonic affected the response of the cell. In most neurons, however,
responses to the full edge and its truncated approximation were
indistinguishable. Thus, the pass-bands of most neurons were
sufficiently narrow so as to exclude the details present in those
higher harmonics. This is expected given the average 2-2.5 octave
spatial frequency bandwidth (full width at half-height) of macaque V1
neurons (De Valois et al., 1982
).
The above observations were quantified by Fourier analysis. There is a
more general reason for doing the Fourier analysis: we have no a priori
knowledge of which response component carries feature dependent
signals. Although nonlinear interactions may act to enhance selectivity
toward a particular spatial feature, this need not be consistent across
all response components. First, we consider conventional scalar
response measures defined on Fourier amplitudes alone and in
combination, the analysis of which is relatively straightforward. Next,
we present an analysis of the Fourier amplitudes and phases jointly (as
vectors in the complex plane), which is perhaps more demanding, but
also more interesting, because the complex measures have larger
signaling capacity attributable to the extra degree of freedom in the phases.
Feature tuning in scalar response measures
Figure 4a shows the
analysis of Fourier amplitudes of the responses of the simple cell from
Figure 3 to the sine gratings presented alone. Selective tuning to
gratings of various spatial and temporal frequencies, drifting at a
constant speed, is indicated by the response amplitudes measured at the
fundamental frequency of each grating (amplitudes marked with
thick bars). Note that the grating contrast was scaled as in
the components of an edge: the contrast of first component was three,
five, and seven times larger than the contrast of the second, third,
and fourth components, respectively. This means that the simple cell
was even more sensitive to gratings of high frequencies than this plot
indicates, i.e., the high-frequency cut-off in the pass-band of this
cell fell beyond the seventh harmonic, because its response to this
stimulus was unequivocal (m = 4 in Fig. 4a).
Nonlinear responses to single gratings are indicated by non-zero
components at multiples of the fundamental frequency for each grating.
The approximately
/2 ratio of the response fundamental over the DC
component of the response is consistent with these components
originating from half-wave rectification. (An exact
/2 modulation
ratio is expected for a perfect half-wave rectifier).

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Figure 4.
Mean Fourier amplitudes of the responses shown in
Figure 3. Error bars indicate 95% confidence limits on the mean.
a, Fourier components (DC and
F1 through F8) of
responses to simple grating stimuli. From front to
back: blank screen (at the mean luminance of the edge), the
first four non-zero drifting sine components of an edge (Eq. 2), and
the full edge. b, Fourier components of responses to
drifting compound gratings. For this simple cell and most other V1
neurons, nearly all response energy is contained at these eight
frequencies. The maximum amplitude (across congruence phase) of most
response harmonics predicts similar optimal waveforms for this simple
cell, ( /2 opt 3 /4, i.e., between
90 and 135°). For clarity, error bars are shown only for the
line-like waveform. Insets at the bottom show a
snap shot of the `edge' and `line' stimuli.
The second copy of the `line' ( = ) is a
half-cycle shifted version of the first ( = 0).
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Nonlinearities are also seen in the response to the full edge
(Fig. 4a, true edge). One manifestation of
nonlinearity is the presence of responses at even harmonics, as
described above. A second manifestation is that the responses measured
at the odd harmonics to either of the compound gratings (Fig.
4b) or the full edge (Fig. 4a) is not equal to
the responses to the corresponding gratings presented alone. For this
cell, the individual grating responses would predict that the peak
component of the response to each compound gratings or the full edge
occurs at the third harmonic frequency
(F3), but in fact it occurs at
F1 or F2. Although some
Fourier components above the eighth harmonic temporal frequency (F8) are still significant, the
overwhelming part of the response energy is contained in the DC and the
first eight components.
For this and other simple cells, examination of the Fourier amplitudes
of the responses to compound gratings (Fig. 4b) reveals that
F1 has both the largest response amplitude and
the largest variation of amplitude across the stimulus set. At each
frequency, linearity predicts identical Fourier amplitudes for all
compound gratings. Note that although the approximate constancy of the DC component is consistent with the linear prediction in this simple
cell (cell of Fig. 3), which thereby gives the DC component the poorest
feature tuning, most other Fourier amplitudes show systematic variation
(i.e., tuning) with stimulus congruence phase. Moreover, this tuning
seems similar across components. Judging by the maximum amplitude of
most components, the optimal waveform for this simple cell has a
congruence phase
/2
opt
3
/4 (between 90 and 135°). By any one of these response measures, therefore, this cell is tuned neither for edges nor lines but for an
intermediate waveform.
In general, the nonlinear signature of complex cell responses to
the compound gratings is that even-order Fourier harmonics dominate the
response. In the typical complex cell, unlike the typical simple cell,
the largest response component as well as the response component with
the largest phase-dependent modulation is the DC or the second harmonic
component, F2. Figure
5 shows the responses of six more V1
neurons (mostly complex cells). As a group, these give a sense of the
variety of phase-selective responses encountered in V1; individually,
each is selected to emphasize a distinct point. Figure 5a
shows the responses of a typical complex cell. For this cell, unlike
for the typical simple cell, the poststimulus time histograms for
drifting gratings, especially at high frequencies, are unmodulated. For
compound gratings, the response histograms for this cell are
characteristically bimodal, with a response transient corresponding to
the passage of the stimulus transient of both contrast polarities. This
contrasts with the unimodal histograms seen for the paradigmatic simple cell (Fig. 3). For each drifting waveform, there are two response peaks
approximately half a period apart (in terms of the fundamental), but
their size and ratio vary systematically with the congruence phase.
Thus, the typical complex cell shows a strong nonlinearity (domination
of the response energy by even-order harmonics), but the
phase-dependent variation manifest in the size and ratio of the peaks
diverges from what is expected of a phase-insensitive energy
operator.

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Figure 5.
Response histograms for six V1 cells that exhibit
the variety of response patterns observed in our sample. Responses to
compound gratings are shown ordered around the phase circle, as in
Figure 3b, and the responses to the blank as well as the
four component gratings (equivalent to Fig. 3a), in columnar
arrangement inside the phase circle (blank on
top). Vertical scale bars indicate size of the peak
response. a, Typical complex cell (450213.u); vertical scale
60 spikes/sec; fundamental period 315.7 msec. b, The complex
cell that was most sensitive and had highest signal-to-noise in our
sample (431115.s); vertical scale 350 spikes/sec; 315.7 msec.
c, The simple cell that was the most sensitive and had the
highest signal-to-noise in our sample (440909.t); vertical scale 350 spikes/sec; 1263 msec. d, A complex cell that approximates a
broadly tuned edge detector (490707.s); vertical scale 30 spikes/sec; 1263 msec. e, A complex cell that responds only
to the full edge (shown above the response to the four-component
approximation of the edge) but not to the four-component compound
gratings (470320.t); vertical scale 20 spikes/sec; 1263 msec.
f, A borderline simple/complex cell that approximates a
broadly tuned line detector (440813.s); vertical scale 100 spikes/sec;
1263 msec.
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Figure 5, b and c, respectively, shows the
responses of the complex and simple cell that had the highest gain and
the least noisy responses in our sample. Both follow with high fidelity the higher harmonic modulations present in the stimulus. The simple cell responses exhibit a tendency of firing to be restricted to one-half of the stimulus period, indicative of dominant odd-harmonic Fourier components in the response. The response histograms of the
complex cell exhibit the opposite tendency, toward a firing pattern
that is replicated in each half of the stimulus period, indicative of
dominant even-harmonic Fourier components in its response. However,
these descriptions are caricatures, and most cells within our sample of
>100 V1 neurons showed intermediate behavior. (The ability of the even
and odd response harmonics to signal congruence phase is given in a
systematic population analysis below.)
Each neuron discussed so far was typical in that it had a more or less
vigorous response to each congruence phase, but with a variable
response waveform. On the basis of the response histograms alone,
therefore, it is difficult to tell by eye for most neurons whether they
are selective to one or the other spatial waveform to any significant
degree, and a quantitative analysis of the responses is necessary.
However, a minority of the neurons were quite selective to certain
waveforms to a degree that was obvious even from a cursory examination
of their response histograms. Figure 5d-f
presents examples of such phase-selective neurons. Figure 5d
shows a complex cell that was broadly tuned to edges. Figure
5e shows another edge-selective complex cell that was quite responsive to the full edge but barely to the four-component edge-like compound grating. For this cell, most grating components probably fell
below its pass-band, but it fulfilled the criteria for analysis based
on d' (see above). This behavior was rare (only 2 of 137 cells in our sample). The final example, a borderline simple/complex cell shown in Figure 5f, can be described as a (broadly
tuned) line detector. This cell preferred an approximately line-like waveform (for the congruence phases tested, the largest peak of the
response histogram occurs at
opt
7
/8). In
general, only a few neurons in the entire sample of 77 V1 neurons that
were analyzed exhibited such obvious phase preference.
Some V1 cells (such as the simple cell in Fig. 3) signal variation of
congruence phase predominantly in their odd response harmonics, and
other cells (such as most complex cells in Fig. 5) signal congruence
phase predominantly in their even response harmonics. Therefore, scalar
measures of the even and odd response energy are also obvious
candidates for further analysis. For the simple cell of Figures 3 and
4, some of these measures are examined in Figure
6.

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Figure 6.
The dependence of various scalar response measures
on the congruence phase, for the simple cell of Figures 3 and 4. To
describe the feature tuning of the cell in each measure, the data
(open symbols) were fit (thick lines) with a
five-parameter second-order harmonic function (Eq. 4) independently for
each response measure. The optimal congruence phase
( opt; arrows), and the selectivity
measure based on circular variance (1 CV),
were extracted from the fits. Error bars represent the 95% confidence
intervals around the mean. a, Mean firing rate
(DC), opt = 0.75 rads (136°);
1 CV = 0.03; b, response energy in
first four even harmonics: opt = 0.63 rads
(114°); 1 CV = 0.18; c, response
energy in first four odd harmonics: opt = 0.67
rads (120°); 1 CV = 0.19; d, total
response energy: opt = 0.59 rads (106°);
1 CV = 0.18.
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The four response measures shown here are the mean firing rate (Fig.
6a, DC), the even-harmonic energy (defined as the summed squared amplitudes of the DC and harmonics 2, 4, 6, and 8). (Fig. 6b), the odd-harmonic energy (summed squared amplitudes of
harmonics 1, 3, 5, and 7) (Fig. 6c), and the total response
energy (summed squared amplitudes of the DC and the first eight
harmonic components of the response) (Fig. 6d). The
linear prediction that the response is independent of congruence
phase fails. Each of these response measures systematically depends on
the stimulus phase, and, for the three energy measures, this dependence
is substantial.
To describe the dependence of each of these response measures on
spatial phase, we used the method of least squares to fit a harmonic
function of the congruence phase,
to the response measure,
R:
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(4)
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This five-parameter fitting function is a natural choice for the
following reason. The complex amplitudes of the response harmonics are
well approximated by an ellipse parametric in twice the congruence
phase, as demonstrated empirically in Figure 9 and analytically
(considering contributions up to and including fourth-order nonlinear
contributions) in the Appendix. Given such an elliptical dependence of
the complex amplitudes of the individual harmonics on congruence phase,
one can show that the dependence of an energy measure on congruence
phase will be a function of the form of Equation 4. For each response
measure considered, we defined the optimal congruence phase,
opt, as the phase at which the curve fitted by
Equation 4 takes its maximum.
In the circular feature space used here, the sharpness of the tuning to
features of a response measure (i.e., its feature selectivity) is
naturally measured by the circular variance (CV) of the
response measure (Mardia, 1972
). The CV is
defined as:
i.e., 1 minus the length of the vector-averaged response measures.
To apply this measure, we take the response amplitudes Rk from the fitted curve and
k to
be the congruence phase. The length of the vector-averaged value (the
measure 1
CV) approaches 1 in the limit of
narrow tuning, and 0 for a response measure that is independent of
congruence phase. The measure (1
CV) is a
global measure of the selectivity of tuning, and, for simple unimodal
tuning functions, it is monotonically related to the conventional local
measures of selectivity such as bandwidth or modulation depth.
For the simple cell in Figure 6, the four response measures, although
not equally sharply tuned, yield very similar optimal phases
(arrows). This is remarkable because one might expect that they reflect the effects of different nonlinearities. For this cell,
the optimal compound waveforms had a congruence phase
opt
/3 (120°). The DC was least tuned to
congruence phase (any tuning in the DC is attributable to
nonlinearities of at least fourth order; see Appendix), and the three
energy response measures were about equally selective when measured by
circular variance (1
CV was 0.03 for DC, ~0.18 for
each energy measure).
The analysis shown for the simple cell in Figure 6 was also carried out
for the examples of Figure 5 (mostly complex). Figure 7 summarizes quite similar results for
the DC and the three energy measures. The DC (open circles)
usually predicted the same optimal congruence phase, but in most cases
was a less selective measure than the energy measures, as quantified by
the CV. Although a greater selectivity is expected for the
energy measures than for the DC merely because the energy (impulses
squared/seconds squared) but not the DC (impulses/second) is a
squared quantity, the full extent of the observed selectivity
difference is not explained by units of measurement. In the case of the
typical complex cell in Figure 7a, the even energy
(squares) and odd energy (triangles) are
similarly tuned, but the even energy dominates. The dominance of the
response by even energy is even more pronounced in the case of the
complex cell in Figure 7b. In this case, and in the case of
the "edge-detector" (Figure 7d), the even and odd energy are also differently tuned. (Note that although the odd energy is very
small, the measured values are highly reliable, as determined by the
illustrated bootstrap confidence limits.) However, in most cases when
the even and odd response energies were both substantial, such as in
the cases of the simple cell (Fig. 7c) and the line detector
(Fig. 7f), the two scalar measures tended to be
similarly tuned. Note that Figure 7, b and c
shows the cells with the highest signal-to-noise ratios in our sample
of V1 neurons; the error bars of the other cells are more typical.

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Figure 7.
The dependence of the same scalar response
measures as in Figure 4, the DC (open circles), odd energy
(open triangles), even energy (open squares), and
total response energy (filled circles), on congruence
phase for the six examples of Figure 5. Panels correspond to those in
Figure 5. Note that the vertical scale for the energy measures
(left) and the DC (right) differ. For each cell,
the optimal phase ( opt), and the phase
selectivity based on circular variance (1 CV)
given below are estimated from the total response energy.
Vertical dotted lines and arrowheads indicate the
optimal congruence phase. Error bars indicate 95% confidence limits.
The continuous lines are the best fitting second-order
harmonic functions (Eq. 4). a, Cell 450213.u,
opt = 0.97 rads (=174°); 1 CV = 0.153; b, cell 431115.s,
opt = 0.63 rads (=117°); 1 CV = 0.126; c, cell 440909.t,
opt = 0.56 rads (=100°); 1 CV = 0.140; d, cell 490707.s,
opt = 0.56 rads (=101°); 1 CV = 0.428; e, cell 470320.t,
opt = 0.99 rads (=179°); 1 CV = 0.169; f, cell 440813.s,
opt = 0.91 rads (=163°); 1 CV = 0.492.
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Figure 8 shows an example of how phase
tuning varies locally in V1. These four complex cells, recorded
simultaneously by a tetrode, exhibit considerable difference in phase
sensitivity (gain), selectivity, and preference. This is representative
of the variation of these parameters in local V1 ensembles. Cell 1, the
cell with the highest gain in this local cluster, and cell 2 are least
selective: their tuning curves (Fig. 8c, left)
approximate what would be expected from a strict (phase-insensitive)
energy calculation. In comparison, cell 3 (the least sensitive in this cluster) and cell 4 (the cell comparable in sensitivity to cell 2) are
both well tuned but tuned to different preferred phases (Fig.
8c, right). Cell 3 is tuned to a waveform the
congruence phase of which is intermediate between that of a line and an
edge. (Judged from its responses shown in Fig. 8a, cell 3 seems simple but it was classified as a complex cell on the basis of
its response to the optimal single grating.) Cell 4 is tuned to a
line-like waveform.

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Figure 8.
Four complex cells simultaneously recorded by
tetrode (infragranular layers). a, b, Response
histograms. Vertical scale bar indicates 150 spikes/sec for Cell 1, and
50 spikes/sec for Cells 2-4. Horizontal scale bar indicates the 1263 msec fundamental (F1) stimulus period.
a, Responses to single sine components presented alone.
b, Responses to compound gratings with eight different
congruence phases. Data sets corresponding to different cells are in
concentric arrangement. c, The dependence of three energy
measures on congruence phase plotted for the four cells as in Figure 7
(odd energy, triangles; even energy, squares;
total energy, filled circles). Optimal congruence phase
( opt, arrowheads) and the phase
selectivity based on circular variance (1 CV)
are estimated from the total response energy: Cell 1 (450509.s), opt = 0.02 rads (=4.2°); 1 CV = 0.066;
Cell 2 (450509.t), opt = 0.91 rads
(=163°); 1 CV = 0.058; Cell 3 (450509.u), opt = 0.79 rads (=142°); 1 CV = 0.277; Cell 4 (450509.v),
opt = 0.09 rads (=15.5°); 1 CV = 0.271.
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Another notable point is that responses of cell 4 to compound gratings
have a single mode (Fig. 8b, innermost histograms), much
like those of simple cells, but its responses to single sine gratings,
except at the lowest spatial frequencies (Fig. 8a,
histograms in rightmost column), consist mostly of spike
rate elevation and only weak modulation, the defining characteristic of
complex cells. Such apparently mixed behavior was observed in many
cells of both classes (as defined by their responses to single
gratings) in our sample: simple cells could have strong even harmonic
components in response to compound gratings (as in Fig. 7c),
whereas complex cells could have strong odd harmonics in response to
compound gratings. Mixed behavior, intermediate behavior between what
is expected for an "ideal" simple and ideal complex cell, was
reported earlier in cat area 17 neurons studied with contrast-reversed single gratings (Spitzer and Hochstein, 1985
). However,
the mixed behavior observed by those authors was based on absolute
phase (position) sensitivity, not on the sensitivity to relative phase (or feature) as observed in this study.
Feature tuning in vector response measures
The energy measures considered above are sensitive to response
size but not timing. This extra degree of freedom present in the phases
may also make it possible for the responses to encode the stimulus
space (a circle), which is of genuinely two-dimensional (2D) topology
and which the scalar measures are incapable of encoding. To determine
whether this is indeed the case, we next consider a joint analysis of
the amplitude and phase of response components. We begin this analysis
on the simple cell of Figures 3, 4, and 6. Figure
9a shows the dominant response
component, F1, plotted as a vector on the
complex plane for each of the eight compound gratings.
F1 is referenced to the phase of the fundamental
stimulus component by subtracting the congruence phase
(Eq. 2) from
the measured phase of F1. (This plotting
convention corresponds to h = 1 in the Appendix.) With
this phase reference, a linear response would be represented by the
same complex number for each stimulus: the eight plotted responses
would all coincide at a single point. The expected position of the
linear response is the center of the dark disk
(m = 1 alone) in Figure 9a, which represents
the response to the fundamental grating component presented alone. Deviation from this, as indicated by the lawful arrangement of responses on a loop, indicates the effects of phase-sensitive nonlinear
interactions between the different harmonic components of the stimulus.
Because our stimuli, by design, contained only odd harmonics of the
fundamental frequency, nonlinear contributions at the fundamental can
be attributable only to odd-order nonlinearities. (For details on how
our stimulus design determines the frequency- and phase-signature of
nonlinearities, see the Appendix.) Third-order interactions, the
odd-order nonlinearities with the lowest order, are likely the largest
contributors to F1. As detailed in the Appendix,
third-order nonlinearities are of two kinds, with different implications for how their phase dependence affects the shape of the
locus plotted in Figure 9.

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Figure 9.
Amplitude and phase of the first four
Fourier harmonics in the response, represented by a vector quantity in
the complex plane, for the simple cell shown in Figures 2-4. The
center of each shaded circle represents the mean
response to a compound grating. Circles indicate 95%
confidence of the mean. The distance of a point from the origin
indicates the magnitude of the response, and the direction represents
its phase plotted with the phase correction indicated by h
(see Appendix). Progression of congruence phase ( ) on the
phase-circle (i.e., on the fitted ellipse) is indicated by
circular arrow in separate insets at the
bottom right of each panel. The linear prediction
(dark circle) is indicated only for the odd harmonics
present in the compound gratings (it is zero at other frequencies), and
is estimated by the response to the component alone (i.e.,
m = 1 for the F1 plot, and
m = 2 for the F3 plot). The
response to the full edge is similarly indicated (light
circle), except the F3 plot where it fully
overlapped the response to the four-component approximation of the
edge. Deviation from linearity, as indicated by the lawful arrangement
of responses on a closed loop, is caused by interaction between the
different harmonic components of the stimulus. The ellipse, fitted as
described in Results, is a good descriptor of the trajectories,
although goodness of fit, as assessed by the p values of the
 , are often <0.05. The optimal stimulus
( opt) predicted by the most distant point on the
ellipse from the origin and found by interpolation on the ellipse
(arrowhead) is similar in the four response harmonics and
comparable to the values obtained from scalar response measures in
Figure 6. a, Fundamental (F1)
response; opt = 0.68 rads (122°);
p = 0.013; b, second harmonic
(F2) response; opt = 0.72 rads (130°); p < 0.001; c, third
harmonic (F3) response;
opt = 0.69 rads (124°); p > 0.130; d, fourth harmonic (F4)
response; opt = 0.65 rads (117°);
p = 0.095.
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To get a better view of the details of the F1
responses in Figure 9a, we present an expanded version in
Figure 10. One kind of third-order
nonlinearity that can contribute to F1 is
represented by the combination F1 + Fk
Fk (see
n = 3; p = 1 in Appendix and Table A1). The phase
of this nonlinear contribution covaries with that of the fundamental
because the phases of Fk and
Fk in the stimulus cancel each other. For
these interactions, the convention used for plotting phases in Figure
9, namely, offsetting by the phase of the fundamental grating
component, will lead to a plotted response vector that is independent
of congruence phase. (This is because the congruence phase
is
identical to the phase of the fundamental grating.) That is, these
components can contribute to a difference between the average response
to the compound gratings and the response to F1
alone, but they cannot contribute to differences among the responses to
the eight compound grating stimuli. Their contribution is represented
graphically in Figure 10 as the displacement between the center of the
ellipse (blue star) and the response to
F1 alone (red disk).

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Figure 10.
An expanded view of a portion of Figure
9a. Red circle is linear prediction, and
blue circle is full edge. See Results for details. A
snapshot of each compound grating is shown next to the corresponding
responses.
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The other kind of third-order nonlinear interaction that leads to
responses at the fundamental frequency consists of contributions such as F3
F1
F1,
F5
F3
F1, (n = 3;
p =
1 in Appendix and Table A1). The raw phase of
these responses varies as 
not
. Thus, after subtraction of the
phase of the fundamental (i.e., the congruence phase
), their
contribution rotates as
2
. Each of these third-order
nonlinearities, if present