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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
TOP
ABSTRACT
INTRODUCTION
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
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
Theoretical work also motivates these experiments. The nonlinear feature detector model developed by Burr and Morrone (1992)
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
Parts of this paper have been published previously at the 1998 and 1999 Annual Meeting of The Society for Neuroscience (Mechler et al., 1998a
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
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
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
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
Visual stimuli were generated by a special purpose stimulus generator (Milkman et al., 1978
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
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:
where C is the Michelson contrast (defined as C = [max(
(1) min(
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
(so, if
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/
Thus,
(3)
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Figure 1. The definition of congruence phase,
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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 (
Note that the phase parameter
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
, and smallest for the edge-like waveform (congruence phase . 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
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)
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
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
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Figure 3. Typical responses to compound gratings and their components recorded from a V1 simple cell, a layer 4C
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
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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, ( 
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
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.
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
7
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 (
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,
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,
(4) 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
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
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
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 (
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 (
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
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
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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 ( , are often <0.05. The optimal stimulus (
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
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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.
The other kind of third-order nonlinear interaction that leads to responses at the fundamental frequency consists of contributions such as F3
