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The Journal of Neurophysiology Vol. 80 No. 2 August 1998, pp. 554-571
Copyright ©1998 by the American Physiological Society
Department of Neurology and Neuroscience, Cornell University Medical College, New York, New York 10021
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ABSTRACT |
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 Abstract
 Introduction
Methods
Results
Discussion
References
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Victor, Jonathan D. and Keith P. Purpura. Spatial phase and the temporal structure of the response to gratings in V1. J. Neurophysiol. 80: 554-571, 1998. We recorded single-unit activity of 25 units in the parafoveal representation of macaque V1 to transient appearance of sinusoidal gratings. Gratings were systematically varied in spatial phase and in one or two of the following: contrast, spatial frequency, and orientation. Individual responses were compared based on spike counts, and also according to metrics sensitive to spike timing. For each metric, the extent of stimulus-dependent clustering of individual responses was assessed via the transmitted information, H. In nearly all data sets, stimulus-dependent clustering was maximal for metrics sensitive to the temporal pattern of spikes, typically with a precision of 25-50 ms. To focus on the interaction of spatial phase with other stimulus attributes, each data set was analyzed in two ways. In the "pooled phases" approach, the phase of the stimulus was ignored in the assessment of clustering, to yield an index Hpooled. In the "individual phases" approach, clustering was calculated separately for each spatial phase and then averaged across spatial phases to yield an index Hindiv. Hpooled expresses the extent to which a spike train represents contrast, spatial frequency, or orientation in a manner which is not confounded by spatial phase (phase-independent representation), whereas Hindiv expresses the extent to which a spike train represents one of these attributes, provided spatial phase is fixed (phase-dependent representation). Here, representation means that a stimulus attribute has a reproducible and systematic influence on individual responses, not a neural mechanism for decoding this influence. During the initial 100 ms of the response, contrast was represented in a phase-dependent manner by simple cells but primarily in a phase-independent manner by complex cells. As the response evolved, simple cell responses acquired phase-independent contrast information, whereas complex cells acquired phase-dependent contrast information. Simple cells represented orientation and spatial frequency in a primarily phase-dependent manner, but also they contained some phase-independent information in their initial response segment. Complex cells showed primarily phase-independent representation of orientation but primarily phase-dependent representation of spatial frequency. Joint representation of two attributes (contrast and spatial frequency, contrast and orientation, spatial frequency and orientation) was primarily phase dependent for simple cells, and primarily phase independent for complex cells. In simple and complex cells, the variability in the number of spikes elicited on each response was substantially greater than the expectations of a Poisson process. Although some of this variation could be attributed to the dependence of the response on the spatial phase of the grating, variability was still markedly greater than Poisson when the contribution of spatial phase to response variance was removed.
Temporal coding can, in principle, provide a way that individual visual neurons can signal more than one stimulus attribute simultaneously and at least partially independently (McClurkin and Optican 1996 Physiological methods
We recorded single-unit activity in the parafoveal representation in cortical area V1 of 10 anesthetized, paralyzed macaque monkeys. Single units (25) were isolated and stable recordings maintained for sufficient time (4-6 h) for the studies reported here. All procedures involving the animals were performed in accordance with National Institutes of Health guidelines for the care and use of laboratory animals.
GENERAL PREPARATION.
Anesthesia was induced with ketamine 15 mg/kg im potentiated by xylazine 2 mg/kg im (Rompun, Haver), supplemented as needed by methohexital boluses (0.5-1 mg/kg iv) during the preparatory surgery. Pupils were dilated with atropine 1% eyedrops, and flurbiprofen 2.5% (Ocufen, Allergan) was instilled as prophylaxis against ocular inflammation. Incision sites were prepped with betadine and infiltrated with xylocaine 1%. Venous access was obtained via bilateral femoral vein cannulation. The femoral artery was catheterized for continuous blood pressure monitoring, and the trachea was cannulated for mechanical ventilation. After transfer of the animal to a stereotaxic frame, anesthesia was maintained with sufentanil (Sufenta, Janssen), 3 µg/kg iv bolus, 1-6 µg·kg SINGLE-UNIT RECORDING AND PRELIMINARY CHARACTERIZATION.
An Ainsworth tungsten-in-glass microelectrode (typical resistance, 2 M LESIONS, EUTHANASIA, AND HISTOLOGY.
At locations along the electrode track corresponding to recording sites and at an additional location at the end of the track, lesions were made by current passage (3 µA × 3 s). At the conclusion of the experiment, the animal was killed by rapid injection of a barbiturate (>15 mg/kg methohexital iv), exsanguinated via perfusion with phosphate-buffered saline, and perfused with 4% paraformaldehyde in phosphate-buffered saline. Cryostatic sections (40 µm) were stained by the Nissl method and examined under light microscopy to confirm track location in V1. Nearly all units were in granular and supragranular layers.
EXPERIMENTAL DESIGN.
Experiments to analyze the signaling of contrast, spatial frequency, and orientation were organized as diagrammed in Fig. 1. Stimuli consisted of transiently-presented full-field (4 × 4°) stationary sinusoidal luminance gratings. Stimuli were organized into runs of 16 grating presentations (237-ms duration, 710-1,026 ms between presentations), and there were 10-s gaps between runs. Between presentations of gratings within a run, and in the gaps between the runs, the display returned to a uniform field at the mean luminance. Within each run, all gratings had a fixed contrast, spatial frequency, and orientation (varied across runs as described later), and spatial phase was varied across 16 equally spaced values (in steps of 112.5 or 90°). As shown in Fig. 1, a sequence of 16 steps of 112.5° phase increments covers the same phases as would steps of 22.5° but in an order that reduces the sense of apparent motion.
Data analysis
Our goal is to determine the extent to which spike trains elicited by transient presentations of static gratings can represent the contrast, spatial frequency, or orientation of the grating and to what extent this representation depends on spatial phase. We wanted to minimize assumptions about how stimulus attributes might be represented, both in terms of the relevant features of a spike train, and the mapping between these features and the parameterization of the stimulus. For example, the attribute of contrast naturally runs monotonically from 0 to 1, but one cannot assume that spike counts represent this attribute in a linear fashion, and there may be a contribution to the representation of contrast from bursts, latency changes, or other temporal features. Representation of spatial attributes raises additional issues. For example, the attribute of spatial frequency runs monotonically from low to high, but a typical neuron produces the largest response at an intermediate spatial frequency, and responses to spatial frequencies significantly below or above this optimum might have far fewer spikes. The time courses of these off-peak responses might (or might not) have consistent differences. Thus although in a formal sense spatial frequency is a monotonic variable, there is certainly no reason to assume that this attribute is represented in a "linear" fashion, and it is even unclear whether it is represented in a monotonic fashion.
METRICS.
The metrics we consider include comparisons based solely on the number of spikes (called the "spike count" metric, Dcount), as well as a family of distances (parameterized by a quantity q) that are sensitive to the temporal pattern of the spikes ("spike time" metrics, Dspike[q]). The parameter q (s STIMULUS-DEPENDENT CLUSTERING.
The second step of the analysis is a determination of the extent to which each metric induces stimulus-dependent clustering. The experimental set-up defines a set of stimulus classes s1, s2, . . . , sC, (e.g., one for each spatial frequency). Based solely on the calculated pairwise distances, one can cluster the recorded spike trains into response classes r1, r2, . . . , rC, (Victor and Purpura 1996a
BIASES DUE TO FINITE SAMPLES.
For finite data samples, the information estimate H given by Eq. 1 is upwardly biased (Carlton 1969 RELEVANCE OF H.
The clustering measure H specifies the extent to which stimuli that differ in one or more attributes lead to responses that have systematic differences. That is, statistically significant values of H imply a systematic representation of a stimulus attribute in the spike train (i.e., that it has been encoded) but do not necessarily imply that the visual system has the capacity to decode it.
COMPARISON WITH ASSESSMENT OF TUNING.
To assess tuning, one chooses a measure of response size, such as the spike count or a particular Fourier component, and asks how this measure depends on one or more stimulus parameters. Here we examine a set of measures of the dissimilarity (or difference) between two responses and ask how these measures depend on the choice of stimuli. This is a more general strategy. A measure of response size always can be turned into a measure of dissimilarity Interaction of spatial phase with contrast, spatial frequency, and orientation
We will begin by presenting an analysis of several data sets in detail and then will present a summary of our findings across the recorded units. The detailed presentation also will help clarify our approach to the analysis of how the encoding of contrast, spatial frequency, and orientation interacted with spatial phase. We will develop two clustering measures: Hpooled, in which responses from all spatial phases are pooled, and Hindiv, in which each spatial phase is treated individually. These information-theoretic measures indicate the extent to which the spike discharge represents (i.e., has the potential to signal) a particular spatial attribute in a context in which spatial phase is allowed to vary (Hpooled) or held fixed (Hindiv). As we have pointed out above, these quantities are not used as absolute measures of information, and we will focus on comparing them, comparing their evolution over time, and comparing their dependence on the stimulus parameter of interest.
SAMPLE DATA SET 1: SPATIAL FREQUENCY ENCODING IN A SIMPLE CELL.
Responses of a simple cell to gratings at three spatial frequencies (0.5, 2, and 4 cycles/deg) are shown in Fig. 2. This simple cell was directional and orientationally tuned and had a spatial frequency cutoff of 6 cycles/deg, and the illustrated responses were collected at its optimal orientation. At each spatial frequency, systematic phase dependence of the response is evident. For example, at 0.5 cycles/deg, the largest responses occur for spatial phases in the range 157.5-270°, and responses to gratings with spatial phases near 0° are minimal. At 2 cycles/deg, the largest responses occur for spatial phases 247.5-337.5°, and responses to gratings with spatial phases in the range 112.5-180° are small. At this spatial frequency, a prominent off-discharge also is present when the on-response is large. At 4 cycles/deg, responses are smaller, and the dependence of response on spatial phase is less marked, but there is still a response maximum for phases in the range 45-135°. Because of the joint dependence of response size on spatial frequency and spatial phase, the size of the response does not necessarily indicate the spatial frequency of the grating. For example, a moderate response could either indicate the presence of an 0.5 cycles/deg grating near a null spatial phase or a 4 cycles/deg stimulus near the peak spatial phase. This intuitive analysis, namely that spatial frequency and spatial phase are jointly encoded, is supported by the quantitative analysis we now describe.
SAMPLE DATA SET 2: ORIENTATION ENCODING IN A COMPLEX CELL.
Figure 4 shows the responses of a complex cell to gratings that varied in orientation and spatial phase. As seen in the figure, the complex cell was tuned orientationally in response to static presentations of gratings, as was demonstrated for neurons in both V1 and V2 in the study by K. P. Purpura and L. M. Optican (unpublished results). Responses to drifting gratings (not shown) were tuned to the same orientation and were direction selective as well. This unit had a spatial frequency cutoff of 2 cycles/deg, and the illustrated responses were collected at 1 cycles/deg near its optimum. As is seen from the response histograms, there is a maximal response at an orientation of 90°, with smaller responses often present at the neighboring orientation of 112.5°, and smaller still at 67.5°. Although some dependence on spatial phase is present, these three orientations contained the largest responses at each spatial phase. At most spatial phases, the largest response was at 90°. Although some spikes are present during presentations of orientations that are removed from this peak, these spikes typically do not occur during the onset of the grating, and the rasters (not shown) suggest that they are not systematically present (i.e., noise).
SAMPLE DATA SETS 3 AND 4: CONTRAST ENCODING.
Figure 6 shows the responses of a complex cell to gratings that varied in contrast. The unit was orientationally tuned and direction selective. The illustrated responses were collected at 1.0 cycles/deg, near its spatial cutoff and at the preferred orientation of 135°. Responses are somewhat noisy, and there is little dependence of response size on spatial phase.
SUMMARY ACROSS DATA SETS.
To collate the observations in individual data sets across the population of units recorded (Table 1), we proceeded as follows. For each data set, we identified peak values of Hpooled and Hindiv (as a function of q) for each analysis period (the first 100, 256, and 473 ms). These maximum values of H were averaged separately for simple cells and complex cells and for each attribute (contrast, spatial frequency, or orientation) that was studied. Averaged values were normalized by the maximal average value achieved for that attribute (for Hpooled and Hindiv, any of the 3 analysis periods, and either cell type). The rationale for this normalization is to compare encoding with spatial phase held fixed to encoding with spatial phase allowed to vary, in simple and complex cells, and to examine how encoding evolves over time. Averages for each attribute were separately normalized because our experimental protocol (different numbers and ranges of contrasts, spatial frequencies, and orientations) would confound a comparison of absolute values across modalities. The averaged, normalized values of Hpooled and Hindiv are presented in Fig. 10.
Joint encoding of two stimulus parameters
The preceding analysis investigated the extent to which the output of a neuron in primary visual cortex can represent a single stimulus attribute. However, contrast, spatial frequency, and orientation interact to determine a neuron's response. Next, we examine data sets in which two parameters were varied in addition to spatial phase to determine to what extent joint representation of multiple attributes is affected by variations in spatial phase.
DATA SET IN DETAIL.
Figure 11 shows responses of an oriented, directionally selective V1 simple cell to gratings at three contrasts and two orientations, the preferred orientation (Fig. 11A), and an off-peak orientation (Fig. 11B). This unit had a spatial frequency cutoff of 6 cycles/deg, and the responses illustrated were recorded with 2 cycles/deg gratings. This is one of the most clear-cut "simple" cells we encountered: responses are strongly dependent on spatial phase. There was a sufficiently high maintained discharge so that one could see a reduction in firing accompanying a stimulus 180° away from the peak spatial phase. Orientation and phase interact (cf. the 2 orientations and the spatial phases of 90 and 270°) but (within each phase) orientation tuning did not depend on contrast. The clustering analysis (Fig. 12) reflects this interaction of spatial phase with orientation in that Hindiv exceeds Hpooled for each of the analysis intervals.
SUMMARY ACROSS DATA SETS.
Observations from individual data sets were collated as described above for the single-parameter experiments. However, because there were relatively few experiments with each pair of parameters (Table 1), all two-parameter experiments were pooled (Fig. 10D). Although the degree of clustering in simple cells is higher, simple cells show a confounding effect of spatial phase (Hindiv greater than Hpooled for all analysis intervals), whereas complex cells do not (Hpooled greater than Hindiv for all analysis intervals).
Contribution of spatial phase to response variability
Several authors have reported that the variability in a single neuron's response is greater than that expected from a Poisson process (Holt et al. 1996
Summary of results
Our major aim was to analyze how spatial phase interacted with contrast, spatial frequency, and orientation to produce changes in the temporal pattern of spike discharges in the transient response. We use the term "phase independent" to refer to the representation of a particular attribute if, in a statistical sense, the representation of the attribute of interest in the temporal pattern of the response is not degraded by including the responses to gratings with different spatial phases. That is, with phase-independent representation, the value of the attribute can be determined from the neural response even if spatial phase is not fixed. (A phase-independent representation may be the result of phase-invariant responses. But phase-independent representations also can arise if the stimulus attribute of interest and spatial phase both influence the response but in a manner in which the effects of spatial phase do not constitute a confound.) Conversely, we use the term phase dependent to refer to the representation of a particular attribute if the spatial phase must be fixed to determine the value of the attribute from the neural response. For contrast (Fig. 10A), representation was strongly phase dependent in simple cells. Complex cells represented contrast in a phase-independent manner in the initial response segment (the first 100 ms), but the full on response (256 ms) depended jointly on contrast and spatial phase. For spatial frequency (Fig. 10B), representation was phase dependent in simple and complex cells. For orientation (Fig. 10C), representation was phase dependent in simple cells but phase independent in complex cells. Finally, although changes in spatial phase influence grating responses in most neurons, it is not the source of the supra-Poisson variability across trials reported by us and by others (Tolhurst et al. 1981 Implications for receptive field organization
The classification (Skottun et al. 1991 Spatial phase, spatial frequency, and location
Early cortical circuitry must process visual information for a variety of purposes. For some purposes (e.g., resolution), point-like receptive fields represent information in the most immediately useful form, while for other purposes (e.g., texture analysis), receptive fields that are tuned to specific spatial frequencies represent information in the most immediately useful form. However, a cortex that contained, at every spatial location, a full complement of cortical neurons that behaved as local Fourier analyzers subserving every orientation, spatial frequency, spatial phase, and bandwidth would be highly redundant. The Gabor-like spatial structure of simple cortical cell receptive field profiles is well recognized (Kulikowski and Vidyasagar 1986
INTRODUCTION
Abstract
Introduction
Methods
Results
Discussion
References
; McClurkin et al. 1991a,b; Purpura et al. 1993). We previously have shown (Purpura et al. 1993; Victor and Purpura 1996a) that in both simple and complex cells, temporal coding contributes significantly to the representation of contrast, spatial frequency, and orientation, but the relationship of spatial phase to temporal coding remains largely unexplored.
; Spitzer and Hochstein 1985a,b). In qualitative terms, a simple cell's characteristic modulated response to a moving grating is consistent with linear combination of signals from subregions of the receptive field, leading to a marked dependence of responses on spatial phase. In contrast, a complex cell's characteristic steady elevation of firing rates in response to a moving grating is typically considered to be indicative of additive combination of signals across an array of rectifying subunits (Spitzer and Hochstein 1985b) and would lead to responses that are independent of spatial phase.
i.e., a rate, or spike count, code. It is clear that this assumption is not justified (McClurkin and Optican 1996; McClurkin et al. 1991a,b; Purpura et al. 1993; Victor and Purpura 1996a). Conceivably, simple cells might be able to exploit temporal coding to signal stimulus attributes in a manner that is not confounded by spatial phase even though the firing rate envelope might be strongly dependent on spatial phase. Conversely, temporal coding of multiple stimulus attributes in the discharge of a complex cell (Victor and Purpura 1996a) might be confounded by changes in spatial phase unless the putative subunits that make up a complex cell's receptive field (Spitzer and Hochstein 1985b) combine in a temporally coherent fashion.
; Morgan et al. 1991; Oppenheim and Lim 1981; Shapley et al. 1990; Tadmor and Tolhurst 1992; Victor and Conte 1996). It is natural to assume that spatial phase, because of its close relationship to position, is encoded by the locus of activity across a population of neurons. However, to the extent that neurons may be regarded as local Fourier analyzers (DeValois and DeValois 1988; De Valois et al. 1985), spatial phase and spatial position are distinct entities (Ohzawa et al. 1996). From the point of view of local Fourier analyzers, features such as lines, edges, and smooth gradations are superpositions of grating patches in specific relative phases. Changing the phase but not the position of the patches changes the nature of the feature, whereas changing their position (but not their relative phases) translates the feature. Thus neural representation of the nature of a feature and its location might require more than a simple spatial code.
, 1997a) in V1, and in the work investigating the temporal encoding of gratings in V1 and V2 (K. P. Purpura and L. M. Optican, unpublished results), spatial phase was not explicitly examined in part because of the limited control of eye position available in awake behaving animals (Creutzfeldt et al. 1987). In this paper, we analyze the temporal structure of responses of single neurons in V1 in the anesthetized, paralyzed animals (under conditions in which spatial phase is controlled precisely), to address the issues raised above.
).
METHODS
Abstract
Introduction
Methods
Results
Discussion
References
1·h1 iv. A few animals were refractory to sufentanil at 6 µg·kg1·h1, and for these animals, urethan (400-500 mg/kg iv loading, 200 mg/kg iv every 12 h) was substituted for sufentanil. Dexamethasone 1 mg/kg iv was administered at the start of the experiment and daily thereafter to reduce cerebral edema. Procaine penicillin G 25,000 U/kg im and benzathine penicillin G 25,000 U/kg im (Pen BP-48, Pfizer, New York, NY) were administered as prophylaxis against surgical infection. Gentamicin (5 mg/kg im daily) was given if fever, hypoxia, increased tracheal secretions, or chest auscultation suggested the development of infection. Every 12-24 h, the corneas were irrigated with Ringer and flurbiprofen was instilled. Local antibiotic (bacitracin, neomycin, and polymyxin B ointment) was applied to the conjunctivae if a discharge was present.
1·h1 iv, and anesthesia with sufentanil or urethan was maintained. Core temperature, monitored with a rectal thermistor, was maintained at 37°C with a thermostatically controlled heating blanket. Ventilator settings were adjusted to maintain an end-expiratory CO2 at 30-35 mmHg. Supplemental oxygen was administered every 6 h, and electrocardiograms and oxygenation were monitored continuously. Hydration (lactated Ringer solution with 5% glucose, 2-3 ml·kg1·h1) was maintained throughout the experiment.
) was advanced through a small durotomy until the action potential of a single neuron was discriminated reliably by a window discriminator (Winston Electronics, Millbrae, CA) either alone or augmented by one or more analog "hoops" (Tucker-Davis Technology, Gainesville, FL) that placed amplitude and latency criteria on later phases of the spike waveform. The receptive field was mapped onto a tangent screen and ocular dominance was determined by auditory criteria. In all subsequent recording, the nondominant eye was occluded. A first-surface mirror was adjusted to align the receptive field with the center of a computer-driven CRT display (mean luminance 150 cd/m2 with a green phosphor, subtending 4 × 4° at the viewing distance of 114 cm). This display system, a modification of the system described by Milkman et al. (1980), provides for a 256 × 256-pixel raster at 270 Hz with look-up table correction of intensity-voltage nonlinearities. Although every attempt was made to align the center of the receptive field with the center of the display (spatial phase 0), it is recognized that this alignment is to some extent arbitrary, and our analysis strategy does not depend on absolute knowledge of spatial phase.
Lmin)/(Lmax + Lmin)] of 0.5-1.0 the spatial frequency and temporal frequency of which were determined by the auditory assessment. Spatial frequency tuning was determined by responses to gratings at each of eight spatial frequencies (typically 0.25, 0.5, 1.0, 2.0, 3.0, 4.0, 6.0, and 8.0 cycles/deg) at a contrast 0.5-1.0, the orientation of which was determined by the quantitative orientation tuning run, and the temporal frequency of which was determined by the auditory assessment. In most units, a contrast response function also was determined by responses to drifting gratings at contrasts of 0.0625, 0.125, 0.25, 0.5, and 1.0, (optimal orientation and spatial frequency, temporal frequency determined by the auditory assessment), and temporal tuning was assessed by responses to 1-, 2-, 4-, and 8-Hz drifting gratings at the optimal orientation, spatial frequency, and contrast. In all of these tuning runs, stimuli were presented in randomized order in four to eight blocks. Each stimulus was presented continuously for 11 s, the last 10 s of which were Fourier analyzed at the stimulus frequency and its second harmonic to quantify responses. In a few cases, the quantitative characterization led to tuning functions for spatial or temporal frequency that differed substantially from the auditory assessment. In these cases, the quantitative characterization was repeated with these modified values.
) on the basis of whether their response to a drifting grating of high spatial frequency was predominantly a modulated response at the fundamental frequency (simple cells) or elevation of the mean (complex cells). Confidence limits for Fourier coefficients were determined by the T2circ statistic (Victor and Mast 1991).
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FIG. 1.
Layout of a spatial frequency experiment. Stimuli were presented transiently for 237 ms and grouped into runs of 16 presentations that included all spatial phases once. From run to run, spatial frequency and the initial spatial phase were varied in a balanced fashion. In other experiments, contrast, orientation, or two of the parameters (contrast, spatial frequency, orientation) were varied from run to run, along with initial spatial phase.
; Treves and Panzeri 1995). A typical experiment might thus consist of varying the contrast between two values (e.g., 0.5 and 1.0) and varying orientation across three values (e.g., peak, peak + 22.5°, peak + 45°). For each of these six contrast × orientation pairs, there were 16 runs (to provide each spatial phase with the opportunity to be presented first). This block of 6 × 16 = 96 unique runs, presented in randomized order, thus contained 16 presentations of each of the six grating stimuli at each of 16 spatial phases. For each run, the initial spatial phase and sequence of spatial phases was chosen in a pseudorandom fashion, so that for each contrast, spatial frequency, and orientation, each spatial phase was presented exactly the same number of times and presented in each serial position within runs exactly the same number of times. This arrangement was designed to counterbalance any effects of contrast adaptation. Several (typically 2-4) repetitions of the block of unique runs were obtained, with the order of runs within each block randomized. Runs were aborted if spike discrimination became unreliable or if there was a major change in responsiveness. Spike times were recorded with a resolution of 1.2 ms (1/3 of the frame time) by the DEC 11/93 computer that sequenced the runs and controlled the visual stimulator.
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TABLE 1.
Summary of experiments performed
, 1997a) and describe briefly here. We consider a series of candidates for the notion of a "metric" (distance or dissimilarity) between spike trains. Our primary assumption is that if a candidate metric reflects the manner in which stimuli are represented by neural discharges, then distances between individual responses to the same stimulus will be small, whereas distances between individual responses to distinct stimuli will be large. Thus for each candidate metric, the analysis breaks into two stages: a calculation of distances between response pairs and an assessment of the extent to which these distances indicate stimulus-dependent clustering. Each metric is a way of comparing individual responses with each other; the clustering calculation examines the relationship between stimuli and these individual responses.
1) indicates the sensitivity of the distance to the precise timing of spikes. For spike trains to be seen as similar in the sense of Dspike[q], they must have a similar number of spikes and the times of these spikes must agree to within 1/q. More precisely, the distance between two spike trains Sa and Sb in the sense of Dspike[q] is defined as the minimal cost required to transform Sa into Sb via a sequence of any of the following transformations: insertion of a spike, which entails a cost of 1; deletion of a spike, which entails a cost of 1; and shifting a spike by an amount of time 
t, which entails a cost of qt. For q = 0, the metric Dspike[q] collapses into the spike count metric Dcount. The first step in data analysis thus consists of calculation of the distances between all pairs of individual responses, for each of the candidate metrics (Dcount and Dspike[q], for q ranging from 1 to 512 s1 in octave steps).
, 1997a). In essence, the clustering algorithm puts each spike train into the class that corresponds to the stimulus that elicited the closest set of observed responses in the other trials. Application of the clustering algorithm to each response yields a partition of the Ntot observed spike trains into an array N(s
, r
) that tallies the number of times that a response in class r was elicited by a stimulus in class s.
, r) that is nonzero only on the diagonal-that is, no stimuli are misclassified. The other extreme, an absence of stimulus-dependent clustering, corresponds to an array N(s, r) that is randomly filled. Between these extremes, the array N(s, r) is larger on the diagonal than offcorresponding to a situation in which individual responses to each stimulus form partially overlapping clouds (as illustrated in Victor and Purpura 1997a, Figs. 9 and 13). A dimensionless quantity to quantify clustering is the "transmitted information" H (Abramson 1963) of the matrix N(s, r). H is given by
where logarithms are taken to the base 2. Perfect clustering of C equally probable stimulus classes corresponds to H = log C, and random clustering corresponds to H = 0. We stress that we interpret the transmitted information H merely as an index of stimulus-dependent clustering. The calculation of H is in no way intended to be optimal (for this would entail additional assumptions concerning the nature of stimulus encoding and response variability) nor to reflect how biological processes extract information from spike trains.
![− log <LIM><OP>∑</OP><LL><SUB><IT>b</IT></SUB></LL></LIM><IT>N</IT>(<IT>s</IT><SUB>α</SUB>, <IT>r</IT><SUB><IT>b</IT></SUB>) + log <IT>N</IT><SUB>tot</SUB>]](/content/vol80/issue2/fulltext/554/img002.gif)
(1)
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FIG. 9.
Analysis of encoding of contrast for the data set of Fig. 8. 
, Hpooled; 
, Hindiv, plotted as in Fig. 3. Analyses are performed on the responses restricted to the first 100 ms (A), the first 256 ms (B), and the first 473 ms (C).
; Panzeri and Treves 1996; Treves and Panzeri 1995). Thus we estimated conservatively (Panzeri and Treves 1996) the upward bias of the estimate (Eq. 1) for H by repeating the above calculations for synthetic data sets that consisted of a random reassignment of the observed responses to the stimulus categories. The mean of 10 such calculations will be denoted by H0, and all comparisons that we present will be based on the empirically corrected value H H0, rather than H. A formula that asymptotically estimates the upward bias has recently been developed (Panzeri and Treves 1996; Treves and Panzeri 1995). Strictly speaking, this formula is not applicable here, because our clustering procedure violates the "independent-binning" assumption required for its derivation. Nevertheless for a few example data sets, comparison of our empiric estimate of the bias by resampling and the analytic formula were similar. Finally, the similarity of results across 16- and 4-phase data sets was further evidence that our results were not merely due to sample-size biases in the estimates of H.
). Because the overall aim of the study is to determine whether or not representation of stimulus parameters is disrupted by spatial phase, we chose a wide, but sparsely sampled, range of stimulus parameters (as described earlier) to make H large but to keep its bias small.
), but direct electrical stimulation necessarily induces a change in the temporal pattern of spikes as well. Conversely (Roelfsema et al. 1994), there are functional correlates associated with changes in temporal pattern of spikes, even when there is no change in overall firing rate.
) among their inputs. The relevant time scale of these coincidences (and thus of the temporal structure of spike trains) may range from submillisecond to many milliseconds, depending on the biophysical mechanisms involved (Bourne and Nicoll 1993; Softky 1994). Our approach explicitly recognizes these possibilities and uses appropriate mathematical methods, including nonparametric elements, to address them. In view of the ongoing debate concerning the relevance of detailed firing patterns (Shadlen and Newsome 1995; Softky 1995), we consider this to be an appropriately conservative strategy.
for example, Dcount is the difference in the number of spikes contained in the two trains that it compares. However, measures of dissimilarity need not correspond to measures of response size, as in the case of Dspike[q] for q > 0.
; Victor et al. 1997).
RESULTS
Abstract
Introduction
Methods
Results
Discussion
References
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FIG. 2.
Responses of a V1 simple cell to gratings that vary in spatial frequency (rows: 0.5, 2, and 4 cycles/deg) and spatial phase (columns: steps of 22.5°). Stimulus onset is at time 0. Vertical line at 237 ms marks the disappearance of the stimulus. Contrast: 0.5. Orientation: 90° (preferred). Unit 21/2.
H0. Hpooled was calculated for the spike count metric Dcount and each of the spike time metrics Dspike[q] (for q ranging from 1 to 512 s1 in octave steps). Second, the data set was partitioned into 16 subsets, 1 for each spatial phase. Within each of these 16 subsets, stimulus-dependent clustering was assessed by a calculation of H H0. For this calculation, each stimulus class consisted of a single spatial frequency at a single spatial phase, and the only responses used to calculate H or H0 were responses obtained at that spatial phase. The resulting 16 values of H H0, one for each spatial phase, were averaged, to obtain Hindiv. In essence, Hindiv indicates the extent to which the spike trains represent spatial frequency in a context in which spatial phase is held fixed, whereas Hpooled indicates the extent to which the spike trains represent spatial frequency in a context in which spatial phase is allowed to vary.
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FIG. 3.
Analysis of encoding of spatial frequency for the data set of Fig. 2. , Hpooled (a measure of the representation of spatial frequency, in the context that spatial phase is allowed to vary). , Hindiv (a measure of the representation of spatial frequency, in the context that spatial phase is held fixed). A-C: contrast 0.5, with data analysis restricted to the first 100 ms (A), the first 256 ms (B), and the first 473 ms (C). D-F: contrast 1.0, with data analysis restricted to the first 100 ms (D), the first 256 ms (E), and the first 473 ms (F). Hpooled and Hindiv have been corrected for estimated bias via a resampling procedure and have been calculated for a range of spike time metrics Dspike[q], as well as the spike count metric Dcount, plotted at q = 0. Missing symbols indicate that the calculated values of H did not exceed the value expected by chance.
1, corresponding to a temporal precision (1/q) of ~15-60 ms. As q increases beyond this point, Hindiv decreases, eventually to chance levels. This indicates that the pattern of spikes at a higher temporal resolution (<15 ms) does not appear to depend in a systematic way on the stimulus. Other data sets show a drop in values of Hindiv and Hpooled at lower values of q, indicating a proportionately more coarse temporal resolution. This timescale for the "informative" precision of a spike agrees with our previous findings in recordings in the awake macaque (Victor and Purpura 1996a) and results of others (Heller et al. 1995) using a different analytic technique. It shows that there is a substantial difference between the informative precision of a spike and the intrinsic precision of the neural spike-generating mechanism (Mainen and Sejnowski 1995; Reich et al. 1997).
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FIG. 4.
Responses of a V1 supragranular complex cell to gratings that vary in orientation (rows: steps of 22.5°) and spatial phase (columns: steps of 22.5°). Responses to orientations of 0, 22.5, and 157.5° contained very few spikes and are not shown. Stimulus onset is at time 0. Vertical line at 237 ms marks the disappearance of the stimulus. Contrast: 1.0. Spatial frequency: 1 cycles/deg. Unit 20/1.
1. But in this case, the increase of Hpooled for q > 0 over Hpooled for a spike count code (q = 0) is small, indicating that there is only a minimal contribution of the temporal structure of the response to the representation of orientation. Additionally, this increment is only seen for the higher of the information curves (Hpooled) and not for Hindiv. That is, for this unit, a spike count code is superior for representing orientation provided that each orientation is presented at a single spatial phase. The major finding, that Hpooled exceeds Hindiv, confirms the intuition that for this unit, the representation of orientation is phase independent. Hindiv is clearly not zero, but the fact that it is smaller than Hpooled indicates that there is no confounding of spatial phase and orientation, in contrast to the interaction between spatial phase and spatial frequency analyzed in Figs. 2 and 3.
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FIG. 5.
Analysis of encoding of orientation for the dataset of Fig. 4. , Hpooled; , Hindiv, plotted as in Fig. 3. Analyses are performed on the responses restricted to the first 100 ms (A), the first 256 ms (B), and the first 473 ms (C).
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FIG. 6.
Responses of a V1 supragranular complex cell to gratings that vary in contrast (rows: 0.125, 0.25, 0.5, and 1.0) and spatial phase (columns: steps of 22.5°). Stimulus onset is at time 0. Vertical line at 237 ms marks the disappearance of the stimulus. Spatial frequency: 1.0 cycles/deg. Orientation: 135° (preferred). Unit 19/2.
consistent with the notion that complex cells respond in a phase-invariant manner (Skottun et al. 1991). However, as opposed to the previous data sets, the effect of temporal coding is dramatic: Hpooled and Hindiv are near zero for the spike count code (q = 0) and only become substantial for distances that are sensitive to the temporal pattern of spikes (Dspike[q], for q in the range 8-32 s1).
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FIG. 7.
Analysis of encoding of contrast for the data set of Fig. 6. , Hpooled; , Hindiv, plotted as in Fig. 3. Analyses are performed on the responses restricted to the first 100 ms (A), the first 256 ms (B), and the first 473 ms (C).
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FIG. 8.
Responses of a V1 simple cell to gratings that vary in contrast (rows: 0.25, 0.5, and 1.0) and spatial phase (columns: steps of 22.5°). Stimulus onset is at time 0. Vertical line at 237 ms marks the disappearance of the stimulus. Spatial frequency: 0.5 cycles/deg. Orientation: 135° (preferred). Unit 26/2.
1). The dependence on q is relatively small and of unclear significance.
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FIG. 10.
Summary across data sets. Normalized values of Hpooled ( and 
) and Hindiv (
and ) are plotted as a function of the length of the analysis interval, for simple cells ( and ) and complex cells ( and ). Averaging is carried out for contrast experiments (A), spatial frequency experiments (B), orientation experiments (C), and experiments in which 2 parameters were varied (D). D pools results from contrast × spatial frequency experiments, contrast × orientation experiments, and spatial frequency × orientation experiments.
that is, Hindiv is greater than Hpooled for an analysis of the first 256 ms. Finally, when the off response is included (
473 ms), Hindiv is comparable with Hpooled for simple and complex cells, indicating phase-independent representation in both cell populations. This is primarily a result of an increase in Hpooled, indicating that the off response is relatively phase independent. Additionally, Hindiv decreases somewhat for both cell types. The significance of this decrease is unclear, but it may indicate that the contrast dependence of the off response is distinct from that of the on response and hence confounds the representation of contrast when it is included in the response.
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FIG. 11.
Responses of a V1 simple cell to gratings that vary in contrast (rows: 0.125, 0.25, and 0.5), orientation (A: 67.5°, the preferred orientation; B: 112.5°) and spatial phase (columns: steps of 90°). Stimulus onset is at time 0. Vertical line at 237 ms marks the disappearance of the stimulus. Spatial frequency: 2 cycles/deg. Unit 28/5.
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FIG. 12.
Analysis of encoding of contrast for the dataset of Fig. 11. , Hpooled; , Hindiv, plotted as in Fig. 3. Analyses are performed on the responses restricted to the first 100 ms (A), the first 256 ms (B), and the first 473 ms (C).
; Softky and Koch 1993; Tolhurst et al. 1981, 1983; Victor and Purpura 1996a). One possible contributing factor to this (especially in studies in awake animals) is that small fluctuations in eye position effectively lead to changes in spatial phase, and hence, greater variability (Gur and Snodderly 1987). Thus reliable but phase-dependent responses might be mistaken for responses with intrinsically high variability. We investigated this possibility directly by comparing response statistics with and without explicit variation of spatial phase.
View this table:
TABLE 2.
Fraction of analyses inconsistent with Poisson statistics
View this table:
TABLE 3.
Comparison of descriptive statistics with Poisson expectations
N, where N is the mean number of spikes per trial. For each data set, this distribution was compared with the observed fraction of trials with n spikes via the 
2 test (Table 2). When responses to all spatial phases were pooled, all data sets deviated in a highly significant manner (P < 0.001) from the expectations of a Poisson process. When responses to each spatial phase were considered individually, 73% of the data sets (85% of those derived from simple cells, 44% of those derived from complex cells) had a response distribution that deviated in a highly significant manner (P < 0.001) from Poisson expectations. To ensure that these findings were not the result of inclusion of a small number of outliers, this analysis was repeated after exclusion of the responses that were in the upper quartile of the spike count distribution. This truncated distribution was compared with a similarly truncated Poisson distribution. Again, all pooled data sets were inconsistent (P < 0.001) with Poisson expectations, as were most phase-specific data sets (78% at P < 0.05, 60% at P < 0.001).
from 3.05 to 2.63 (P < 0.01 by paired t-test) for simple cells and from 3.04 to 2.31 (P < 0.05 by paired t-test) for complex cells. However, after the removal of the variance due to spatial phase, the variance/mean ratio was still >1 for all cells examined (minimum 1.58, maximum 4.80, geometric mean 2.53), and again, there was no significant difference (P > 0.05 by t-test) between simple cells (geometric mean 2.63) and complex cells (geometric mean 2.31). (The lack of a difference between simple and complex cells does not imply that there is no difference in the dependence of responses on spatial phase between simple and complex cells in generalas noted earlier, this analysis only included data from either cell class recorded under conditions in which a phase dependence was apparent.) Across all data sets, variation in spike count due to variation in spatial phase accounted for an average of 14% of the variance (range: 2-30 ± 8%, mean ± SD), but this source of variance was not nearly enough to account for the excess variance compared with the expected variance of a Poisson process.
, 1983), further indicating that changes in spatial phase are at most a minor contributor to response variability in cortical neurons.
DISCUSSION
Abstract
Introduction
Methods
Results
Discussion
References
, 1983).
including a reduction or elimination of temporal coding (Mechler et al. 1997). However, transient presentation more closely mimics the time course of retinal stimulation that occurs during a sequence of physiological visual fixations (Viviani 1990). Moreover, the use of drifting gratings in these experiments necessarily would have induced a technical confound between spatial phase and time lags in neural circuits, because changing the initial spatial phase of a drifting grating is the same as shifting it in time.
) of simple and complex cells on the basis of whether their responses were phase dependent or not might lead to the expectation that in simple cells all attributes are represented in a phase-dependent manner, whereas in complex cells, all attributes are represented in a phase-independent manner. As described above, our data do not conform to this expectation. There are several factors that likely underlie this departure. First, the classification of simple and complex cells is not dichotomous. Many cells show both phase-dependent and phase-independent behavior (Pollen et al. 1988; Spitzer and Hochstein 1985a): a cell may be classified as complex because of its phase-invariant responses at high spatial frequencies yet may display prominent phase-dependent responses at low spatial frequencies. Thus especially in experiments that compare responses across a range of spatial frequencies, complex cells may display hallmarks of phase dependence.
Hpooled (Fig. 7B) for the optimal spike time metric Dspike[q]. That is, that number of spikes in the response to a grating may be relatively independent of spatial phase, even though their timing is strongly dependent on phase.
). K. P. Purpura and L. M. Optican (unpublished results) found that the initial 50 ms after stimulus onset carried measurable amounts of information about the orientation and spatial frequency of transiently presented sinewave gratings. There was a rapid rise in information between 50 and 100 ms followed by a slower rise during the following 100 ms. This suggested that the rise in information between 100 and 200 ms was due to local recurrent and feedback circuits and that the prolonged tonic activity in feedforward pathways may contribute to temporal encoding through the activation of and interaction with membrane components in cortical neurons that produce bursts and other temporal patterns. In sum, whereas the initial visual response reflects the geniculocalcarine connections in a straightforward way, the remainder of the response is influenced heavily by inputs from other cortical neurons. We showed (Fig. 10A) that for complex cells, the initial 100 ms represents contrast in a phase-independent manner, but the full on response shows substantial phase dependence. The initial component may well be explained by a superposition of feedforward nonlinear receptive field subunits (Spitzer and Hochstein 1985b); we hypothesize that the later, phase-dependent components represent the influence of intrinsic cortical activity.
in this case, the absence of a maintained discharge.
) are summed before rectification (as originally proposed by Spitzer and Hochstein 1985b) or if there is a local rectification within these elongated regions as well (Purpura et al. 1994; Victor and Conte 1991). Although there is some evidence that the orientation specificity of cortical cells in Layer IV is determined by their subcortical inputs (Ferster 1987; Reid and Alonso 1995), there is also evidence that intracortical processing plays a substantial role in orientation tuning (Bonds 1989; Morrone et al. 1982; Sillito 1975; Sillito and Jones 1996), particularly as the response evolves in time (Ringach et al. 1997; Volgushev et al. 1995; K. P. Purpura and L. M. Optican, unpublished data). Orientation-specific inputs from other cortical neurons (either excitatory or inhibitory) can lead to the phase-independent representation of orientation that we observe, provided that these inputs act as nonlinear subunits or have distinctive time courses. (Otherwise, their impact would merely be to change the effective sensitivity profile of the receptive field.) To remain consistent with our findings that spatial-frequency representation is phase dependent, we postulate that these subunit inputs span a broad range of spatial scales and thus do not contribute to spatial frequency tuning. In this way, intracortical connectivity among cells that share a common orientation could provide a mechanism for phase-independent representation of orientation but not spatial frequency.
). Interactions of oriented subunits that span a range of spatial scales was shown to be the crucial computational element required to account for isodipole texture selectivity (Purpura et al. 1994).
; Kulikowski et al. 1982; Ohzawa et al. 1996), and theorists have advanced arguments for a variety of evolutionary and developmental pressures that favor this kind of structure (Atick 1992; Daugman 1990; Field 1987; Olshausen and Field 1996) as a compromise between the demands of analyses localized in space and analyses localized in the Fourier domain.
). For other purposes, orientation and spatial position are key but spatial phase can be ignored. For example, an object's boundary can be demarcated by a thin line or an edge (luminance step). From the point of view of a local analyzer centered at this boundary, a thin line would appear to have cosine phase (or antiphase, depending on polarity), whereas the edge would appear to have rising or falling sine phase (depending on the direction of the luminance gradient). Each of these local features has distinct phase characteristics, but once they have been extracted for image segmentation, only their orientations and positions are important.
i.e., even- and odd-symmetric receptive fields or quadrature pairs (Emerson 1997; Emerson and Huang 1997; Field and Tolhurst 1986; Liu et al. 1992; Rentschler and Treutwein 1985). Our data support another strategy to resolve the conflicting demands of phase-independent and phase-dependent representation. As summarized in Fig. 10, cortical neurons can represent contrast, orientation, and, to a limited extent, spatial frequency in a phase-independent manner. Optimal clustering is achieved for values of q in the range 16-64 s1, corresponding to a temporal precision (1/q) of ~15-60 ms. On the other hand, when spike trains are viewed with a somewhat higher temporal resolution (up to q = 128 s1) then spatial phase itself is represented in a systematic manner (Victor et al. 1997). This finding was present in both simple and complex cells and thus appears to be more general than the well-known spatiotemporal progression within the receptive field thought to underlie direction selectivity in simple cells (McLean and Palmer 1989; McLean et al. 1994; Movshon et al. 1978; Reid et al. 1987, 1991; Saul and Humphrey 1992b). We point out that although spatiotemporal quadrature has theoretical advantages for motion analysis (Adelson and Bergen 1985), mere spatial and temporal offsets of separable receptive field elements suffice to represent spatial phase via the temporal structure of the response. The temporal asynchrony across phases could be generated by lagged geniculate cells (Saul and Humphrey 1992a), but much smaller temporal offsets (of only 20 ms) also would suffice to account for our observations. Offsets in this range could be generated by the intrinsic dynamics of signal integration in cortical neurons (Frégnac et al. 1996). A receptive field the spatial components of which are offset but not orthogonal in time plays the dual role of representing a range of spatial phases (when its discharge is viewed as a high-resolution spike time code) and the effective contrast at a single spatial phase (when its discharge is viewed as an overall rate code). When the discharge is viewed with an intermediate temporal resolution, our data show that certain stimulus attributes may be represented in a phase-independent manner.
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ACKNOWLEDGEMENTS |
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The authors acknowledge the comments and insights of Prof. Bruce Knight and Dr. Ken Miller, the assistance of D. Reich, N. Schiff, F. Mechler, and M. Conte with the experiments, and the assistance of A. Canel and A. Hoffman with software development and data analysis.
This work was supported by National Institutes of Health Grants EY-9314 (J. D. Victor) and NS-01677 (K. P. Purpura), The McDonnell-Pew Foundation (K. P. Purpura), and The Hirschl Trust (J. D. Victor).
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FOOTNOTES |
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Address for reprint requests: J. D. Victor, Dept. of Neurology and Neuroscience, Cornell University Medical College, 1300 York Ave., New York NY 10021.
Received 24 October 1997; accepted in final form 24 April 1998.
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REFERENCES |
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Abstract
Introduction
Methods
Results
Discussion
References
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ELJA, S.,
BISHOP, P. O.
Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex.
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