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1 Department of Neurology and Neuroscience, Weill Medical College of Cornell University, New York City, New York 10021; 2Department of Biological Sciences, Columbia University, New York City, New York 10027; 3Johns Hopkins University, Departments of Radiology and Neurology, Johns Hopkins Hospital, Baltimore, Maryland 21287; and4 The Rockefeller University, New York City, New York 10021
Submitted 18 September 2002; accepted in final form 26 January 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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30 ms, and keeping track of which neuron fires which spike rather than just the summed local activity contribute substantially to phase coding. The contribution is both quantitative (increasing the fidelity of phase coding) and qualitative (enabling a 2-dimensional "response space" that corresponds to the spatial phase cycle). We use a novel approach, the extraction of "temporal profiles" from the metric space analysis, to interpret and compare temporal coding across neurons. Temporal profiles were remarkably consistent across a large subset of neurons. This consistency indicates that simple mechanisms (e.g., comparing the size of the transient and sustained components of the response) allow the temporal contribution to phase coding to be decoded. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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Neuronal hardware is not intrinsically noisy, however, as attested to by the machine-like precision of sensory neurons in vivo (Reich et al. 1997
) or of a cortical neuron whose inputs are carefully controlled (Mainen and Sejnowski 1995). Some have argued that the variability of cortical neurons is not a limitation of neuronal hardware in general but rather, a clue that neuronal coding makes use of rich statistical structure (Bullock 1997). Specific possibilities include the precise times of occurrence of impulses (Berry et al. 1997; Gawne 2000; Softky 1994; Théunissen et al. 1996), the pattern of intervals (Sen et al. 1996), correlations (Dan et al. 1998; Meister et al. 1995), and oscillations (Gray et al. 1989). The alternative view (Shadlen and Newsome 1998) is that variability is not a fundamental part of neural coding but rather represents a limitation that the nervous system has to deal with. In this view, reliable signals are extracted from unreliable neurons by averaging their activity across a population of similar neurons, and across appropriate intervals of time, and that detailed spatiotemporal structure simply is not relevant. Both the temporal and spatial aspects of neural coding, although often expressed as dichotomies, are best considered as continua. The timing of individual impulses must matter; the question is, over what time scale. The notion of population averages is only intended to apply to a local population; the question is, how local.
We address these issues by examining neural coding of spatial phase by neighboring pairs of neurons in primary visual cortex of the anesthetized, paralyzed macaque. Spatial phase is a fundamental aspect of spatial vision, crucial both for the extraction of local features (Burr et al. 1989) and overall scene perception (Oppenheim and Lim 1981). Certain properties of receptive fields are well known to be clustered (topographic location, size, and orientation tuning, for example). The functional organization of spatial phase tuning, however, is unknown. DeAngelis et al. (1999) study spatial phase as defined relative to the receptive field envelope and find that there is essentially no correlation among neighboring neurons. Here, we study a distinct quantity, spatial phase defined relative to a particular position in space (see DISCUSSION). We find that nearby neurons often have similar spatial phase tuning, although there is also some degree of variability. The presence of such variation does not suffice to imply that neural populations represent spatial phase on a neuron-by-neuron basis rather than via a summed population code. For coding purposes, it is useful to keep individual neurons' responses separate if their spatial phase preferences differ, but only if these differences are not overshadowed by trial-to-trial variability. Moreover, although individual neurons' responses may demonstrate a systematic dependence of their temporal characteristics on spatial phase (Victor and Purpura 1998), it is unclear how this temporal information would be useful if the temporal patterns were idiosyncratic to the individual neurons.
Our analysis addresses both of these questions. By an extension of the metric-space approach (Victor and Purpura 1997) to single-trial responses of pairs of neurons, we show that spatial phase coding is enhanced by keeping track of which neuron fires which spike. However, we also find that the extreme of a "labeled line" code is not necessary to realize these advantages. A previous shortcoming of the metric-space method is that it provided no way to compare detailed temporal characteristics across neurons. We overcome this problem here via the notion of "temporal profiles." Temporal profiles provide a quasilinear framework that accounts for the bulk of the temporal features identified by the metric-space method. Comparison of temporal profiles reveals a remarkable similarity across the population of neurons. Moreover, these temporal profiles are readily described in terms of "transient" and "sustained" components and thus provide a substrate to extract temporal information in a universal and straightforward fashion.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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Standard acute preparation techniques were used for electrophysiological recordings from single units in the primary visual cortex (V1) of the primate (cynomolgus monkeys, Macaca fascicularis). All procedures used were in accordance with institutional and National Institutes of Health guidelines for the care and experimental use of animals. Some details of the techniques used were given earlier (Mechler et al. 1998).
Experiments were performed on 14 adult animals, weighing 3–4.5 kg. Prior to surgery, animals were given atropine (0.1 mg/kg im) and then anesthetized with ketamine (10 mg/kg im). Anesthesia was maintained with sufentanil (3–6 µg · kg-1 · h-1 iv). Paralysis was induced following all surgical procedures and maintained with pancuronium (0.4 mg · kg-1 · h-1 iv). Dexamethasone (1 mg · kg-1 · day-1 im) and gentamicin (5 mg · kg-1 · day-1 im) were given to prevent the development of cerebral edema and infection, respectively. The animal was ventilated through an endotracheal tube. Heart rate, echocardiography (EKG), arterial blood pressure, and end-tidal CO2 were continuously monitored with a Model 78354A Hewlett-Packard Patient Monitor and kept in the normal physiological range. Core body temperature was maintained between 37 and 38°C using a thermostatically controlled heating pad. The EEG was obtained from frontal leads and monitored on an oscilloscope.
A limited unilateral craniotomy to expose the primary visual cortex was made overlying and posterior to the lunate sulcus (the Horsley-Clarke stereotaxic coordinates were typically 14–16 mm posterior and 14–16 mm lateral). A 1–2 mm durotomy was made for the recording electrode, which was stabilized after insertion by agarose gel.
Extracellular recording
Spike responses of single units were recorded extracellularly. We used single-tip (Ainsworth et al. 1977; Merrill and Ainsworth 1972) electrodes, typical resistance 2 M
, or quartz-coated platinum-tungsten fibers tetrodes (Thomas Recording, Giessen, Germany). Tetrodes had a conical tip, with four contacts of 1 M each, 25 µm apart: one at the apex and three arranged in radial symmetry on the walls of the conical surface. A stepper motor advanced either type of electrode in 1-µm steps.
The signals from the electrode or tetrode channels were passed through a multi-channel differential head-stage amplifier (NB Labs, Denison TX, or Neuralynx, Tucson AZ) and then further amplified and filtered (0.3- to 6-kHz pass-band, Lynx-8 multichannel differential amplifier, Neuralynx). Candidate spike waveforms, detected by a threshold criterion, were digitized at 25 kHz within a short (1.2 ms) temporal window containing the peak amplitude and digitally recorded (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). Isolation criteria included stability of principal components of spike waveforms and a 1.2-ms minimum interspike interval consistent with a physiologic refractory period. Spike times for further data analysis were identified off-line to 0.1-ms precision, the accuracy to which the clocks of the recording computer and the stimulus generator were synchronized.
Histology and laminar assignment of recording sites
Experiments lasted for 4–5 days, at the end of which the animal was sacrificed by infusion of a lethal dose of methohexital. After transcardiac perfusion with 4% paraformaldehyde in phosphate-buffered saline, a block of the occipital cortex was processed for histological reconstruction of the electrode track in Nissl-stained sections, aided by electrolytic lesions (5 µA x 5 s, electrode positive) on some tracks and deposition of the lipophilic fluorescent dye DiI (D-282; Molecular Probes, Eugene, OR) on others. This confirmed that all recordings were in V1. The precise laminar position was identified for two-thirds of the recording sites, and the relationship with respect to the granular layer was identified in the remainder.
Optics
The eyes were treated with anti-inflammatory (fluribuprofen) and anti-bacterial (neomycin/polymyxin/dexamethasone) ophthalmic solutions at least daily. Pupils were dilated (1% atropine) and covered with gas-permeable contact lenses. 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.
Cell classification
We used the F1/F0 modulation ratio to classify cells as simple or complex: 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 considered simple and complex otherwise (De Valois et al. 1982; Movshon et al. 1978a; Skottun et al. 1991). We recognize that the dichotomous nature of this classification has recently been called into question (Mechler and Ringach 2002), but we used it because it is quantitative, widely accepted, and likely related to how responses depend on spatial phase.
Visual stimulation
Visual stimuli were generated by a special purpose stimulus generator (Milkman et al. 1978, 1980) under the control of a PDP-11/93 computer and displayed on a Tektronix 608 monochrome oscilloscope (green phosphor, 150 cd/m2 mean luminance, 270.32-Hz frame refresh). The luminance of the display was linearized with lookup tables in the range of 0–300 cd/m2. At the 114 cm viewing distance of the animal, the stimuli appeared in a 4° circular aperture on dark background.
Neuronal 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. Receptive field characterization was always done for the most responsive unit and often for a second unit.
Responses as a function of spatial phase
To assess responses as a function of spatial phase, we presented full-field (4 x 4°) sinusoidal luminance gratings at or near the optimal orientation and spatial frequency at each of 16 equally spaced spatial phases. Stimuli were organized into runs of eight repetitions (236 ms each) of a single spatial phase, separated by 710 ms presentations of a uniform field at the mean luminance. After each of these sets, the spatial phase was changed randomly. Typically four to eight repetitions of the block of unique runs were obtained, with the order of runs within each block randomized, resulting in 32–64 presentations of each of the 16 spatial phases. Runs were aborted if spike discrimination became unreliable or if there was a major change in responsiveness.
Data analysis
Off-line data analysis was performed with custom software written in C and in the Matlab (Mathworks, Natick MA) programming environment. Our goal was to determine the extent to which spatial phase is represented in single-trial responses to static gratings and to characterize the features of spike trains that provide this representation. To this end, we extended the metric-space approach (Victor and Purpura 1996, 1997) from single-cell recordings to responses of simultaneously recorded cells. As described in the following text, this allows us to characterize the role played by spike counts, spike times, and the distribution of spikes across pairs of simultaneously recorded neurons. This approach separates these roles by constructing notions of dissimilarity ("metrics" or rules for calculating distances) between neural responses. Characterization of neural coding via metrics avoids making strong assumptions (such as approximate linearity of the stimulus-response relationship or approximate Poisson-ness of the spike trains) about the neural response.
Single-unit metrics
This basic family of metrics—the "spike time" metrics of Victor and Purpura (1996, 1997), determines whether spike timing carries information about the stimulus in addition to the information carried by spike count. If the timing of individual spikes does carry additional information, the metrics determine (via a parameter q, defined in the following text) the temporal precision at which spike timing is relevant. In a "spike time" metric, the distance between two spike trains is defined as as the minimal total "cost" of a sequence of elementary steps that transforms one spike train into the other. The following elementary transformation steps are allowed: insertion of a spike (for a cost of 1), deletion of a spike (for a cost of 1), and shifting a spike by an amount of time 
t (for a cost of qt). Because a spike in one train can always be transformed into a spike in another train for a cost of 2 (via its deletion from one train and insertion into the other), spikes are seen as similar only if they occur within 2/q of each other. When q = 0 s-1, spike timing is irrelevant to the distance calculation, and the metric reduces to a comparison of responses based solely on the number of spikes. To use the metrics to identify features of neural coding, we determine the extent to which the metrics induce a "stimulus-dependent clustering" of the responses (see following text). That is, we calculate a quantity H (formally, an information) that indicates the extent to which responses to the same stimulus are close (i.e., similar), while responses to different stimuli are distant (i.e., dissimilar). The calculation of H depends on the pairwise distances between responses to identical stimuli and to distinct stimuli and thus on the parameters of the metric (here, q). If spike timing carries no phase information, then H will be maximal at q = 0 s-1. This is because alternative metrics with q > 0 are influenced by the timing of spikes, which is hypothesized to be irrelevant to the identity of the stimulus. The converse situation is that the timing of spikes, and not just the number of spikes, systematically depends on spatial phase, and this dependence provides information about spatial phase that is not provided by the spike counts. In this case, H will be maximal for some q > 0. This formalizes the notion that a metric that considers spike timing at the appropriate temporal scale will be able to identify the stimulus that elicited a response with greater certainty (i.e., higher H) than a metric that ignored spike timing. In this case, the value of q at which H is maximal determines the precision of spike timing (as 1/q, in units of seconds) that is relevant to coding. For further details and discussion, see Victor and Purpura (1996, 1997).
Multi-unit metrics
To extend the analysis to multiple neurons, two changes are required. First, we represent a simultaneously recorded response of multiple neurons as a single sequence of labeled events. That is, a multi-unit response is considered as a "labeled spike train," where a label assigned to each spike indicates its neuron of origin. For a population of L neurons, we use the integers 1, 2,..., L as labels. These labels are merely abstract tags—no serial ordering is implied. Second, we introduce a new parameter k into the metric. The new parameter k describes the importance of the neuron of origin of a spike, much as q describes the importance of the time of a spike. Metrics used to characterize coding in such labeled spike trains allow all of the elementary transformations used by the single neuron metric (insertion of a spike, deletion of a spike, and moving a spike). In addition, these metrics include an elementary step in which a spike's label (i.e., neuron of origin) is changed. This transformation is assigned a cost of k.
The dimensionless parameter k thus indicates the importance of distinguishing spikes that are fired by different neurons. When k = 0, the multi-unit metric reduces to the single-unit metric described earlier because there is no cost for reassigning the neuron of origin of a spike. This amounts to considering the neuron of origin to be irrelevant to coding, a situation we designate as a "summed population" code. When k 
2, spikes from different neurons are never considered to be similar because the cost of matching them (by changing the label of either 1) is no less than the cost of deleting a spike from one neuron's response and inserting it into another neuron's response. Thus for k 2, the distance between two multi-unit responses becomes equal to the sum of the distances between individual cell responses calculated by the single-unit metric. Because each neuron's activity is considered independently and noninter-changeably labeled, we designate this situation a "labeled line" code. The range of values of k from 0 to 2 provides a continuum between the extremes of the "summed population code" (k = 0) and the "labeled line code" (k = 2). For intermediate values, spikes in two multi-unit responses that occur sufficiently close to each other in time can be seen as similar even if they are fired by different neurons.
An efficient algorithm for the calculation of single-unit metric distances has been described before (Victor and Purpura 1996). Its running time is proportional to N2, where N is the typical number of spikes fired by a neuron. For this work, we used a straightforward extension of this algorithm to multi-unit metrics (Victor and Purpura 1997). For a population of L neurons, the running time of this extension is on the order of N2L. A more efficient version of the algorithm has been recently developed (Aronov 2003) in which the running time is on the order of NL+1.
Stimulus-dependent clustering
The next stage of our analysis is to determine to what extent our candidate notions of distance provide for discrimination of responses to different spatial phases. That is, for each candidate metric, we determined the extent to which repeated responses to the same spatial phase lie at shorter distances from each other than responses to different spatial phases (i.e., whether the responses to each spatial phase form separate clusters). To quantify such clustering, we utilized the approach of Victor and Purpura (1996, 1997), which calculates the amount of "transmitted information," Hdata, in bits. This information-theoretic quantity is upwardly biased for finite data samples (Carlton 1969; Panzeri and Treves 1996; Treves and Panzeri 1995). We estimated the upward bias by repeating the calculation of transmitted information for data sets that consisted of a random reassignment of responses to the stimulus classes (Panzeri and Treves 1996). We corrected the information estimate by subtracting the mean of 10 such calculations from Hdata, and we denote the difference Hdata – Hresampled by H.
The net result of the preceding calculations is thus a scalar measure H of the strength of stimulus-dependent clustering derived from each member of a family of notions of similarity. The family of notions of similarity is parametrized by the extent to which spike timing (q) and neuron of origin (k) are significant. This analysis (calculation of the distances between the responses, calculation of Hdata and Hresampled from the pairwise distances, and calculation of the final measure of clustering H = Hdata – Hresampled) was performed for all (q,k) pairs in which q = 0, 1, 2, 4,..., 512 s-1, and k = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.25, 1.5, 1.75, and 2.
Multidimensional scaling
The preceding analysis indicates the extent to which the responses to each of the spatial phases are discriminable and whether this discrimination is best accomplished by consideration of the timing of responses (q > 0), the neuron of origin of each response (k > 0), or both. However, it ignores the size of the difference between pairs of stimuli. Among our stimuli, some pairs were quite similar to each other (e.g., 22.5° apart in phase), whereas others were diametrically opposite (180° apart in phase). Features of spike trains that convey information about spatial phase should not only support discrimination of stimuli that are of different spatial phases but, also, should provide a "representation" of spatial phase. That is, responses to similar spatial phases should be more nearly alike than responses to very different spatial phases. Moreover, spatial phase is intrinsically a cyclic quantity. It is therefore natural to ask whether the neural responses, when compared by a candidate metric, have a corresponding behavior.
To determine whether response features indeed provide for representation (and not merely discrimination) of spatial phase, we need to go beyond the information-theoretic quantity H, which indicates discriminability but not relationships. That is, we need to consider the geometry of the response clusters elicited by gratings at each spatial phase and not only whether the clusters are distinct.
To determine this geometry, we used a standard technique, multidimensional scaling (Kruskal and Wish 1978). In general, multidimensional scaling embeds a set of point in a Euclidean space so that the distances between the points correspond to prespecified numbers. In the present application, the "points" correspond to the individual responses. The "distances" correspond to the distances provided by a metric with specified values of q and k. Thus multidimensional scaling assigns coordinates to each spike train so that the standard Euclidean distances between spike trains are the best possible approximations of the distances given by a metric. According to the (standard) procedure of Kruskal and Wish (1978), this best fitting Euclidean embedding is found by determining the eigenvectors of a matrix Ajk whose entries are given by
 | (1) |
Fitting ellipses
Each of the 16 stimuli elicits a subset (of size M/16) of the M responses and thus corresponds to a subset of the M embedded points found by the preceding multidimensional scaling procedure. For each of the 16 spatial phases, the location of this corresponding subset of embedded points can be summarized by its centroid. (In contrast, H summarizes the extent to which these subsets overlap but ignores their relative locations.) The coordinates of each centroid are simply the average of the coordinates of the corresponding set of spike trains as provided by multidimensional scaling. One would expect the 16 centroids of these clusters in our experiment to lie along a closed curve due to the cyclic nature of spatial phase. Because linear mechanisms are a fundamental ingredient of receptive field models and because a linear response must fall on or near an ellipse (see APPENDIX) (see also Movshon et al. 1978a), we sought to characterize the positions of these centroids in terms of bestitting ellipses.
We therefore fitted ellipses to the 16 centroids of responses to each of the spatial phases. We found best-fit ellipses by minimizing the mean squared distance between the centroids and 16 points located at constant phase intervals around an ellipse whose shape, position, and orientation were allowed to vary freely. (This step can be carried out as a linear regression.) To characterize the arrangement of response clusters, we quantified the shapes of the best-fit ellipses by their axis ratios. The shape of the ellipse can vary from a doubly covered line segment (an axis ratio of 0) to a circle (an axis ratio of 1). The former extreme indicates that only one mechanism effectively contributes to the response. The latter extreme corresponds to two spatiotemporal mechanisms in quadrature (Emerson 1997; Emerson and Huang 1997; Heeger 1992; Marcelja 1980; Pollen et al. 1985), a situation in which the cycle of spatial phase is faithfully represented by the circular trajectory of responses.
We quantified the goodness of fit of these ellipses by the variance in the layout of the centroids left unexplained by the 16 corresponding points on the best-fit ellipse.
Temporal profiles
Multidimensional scaling of the pairwise metric distances creates an arrangement of responses in an abstract space that depends on their temporal structure, but it does not identify which aspects of the neural responses contribute to this arrangement. The analysis described in the following text seeks a simple temporal interpretation for the coordinates identified by multidimensional scaling. In particular, if the coordinates of the embedded responses could be derived by linear operations applied to the responses (see APPENDIX), then this procedure will identify them. If the embedded responses do not form an ellipse but nevertheless may be construed as a combination of a small number of factors (that do not vary sinusoidally with spatial phase but combine linearly), this procedure will also identify these factors and their time courses.
Let us assume that multidimensional scaling has embedded the M responses into a D-dimensional response space. We seek up to D distinct factors ("temporal profiles") whose linear superposition accounts for the observed embedding coordinates. We choose a desired temporal resolution for the calculation of the temporal profiles and divide the response period into K time bins, corresponding to the desired resolution. Because we postulate that these D factors operate linearly on the response trains, our analysis seeks a K x D matrix P whose entries are the weight of each of the D mechanisms in each of the K bins.
We initially consider single-unit responses. The coordinates of the M responses (as provided by multidimensional scaling) constitute a coordinate matrix C, of size M x D. Corresponding to the desired number of bins K in the temporal profiles, we create a second representation of the M single-unit responses by an M x K matrix R. A row of R contains the number of spikes of a particular response in each of the K bins. To find a correspondence between linear operations on the spike trains and the embedded coordinates C, we seek a KxD matrix P such that RP best approximates C in the least-squares sense, other than an arbitrary translation. Allowance for an arbitrary translation is necessary since the origin of the embedding identified by multidimensional scaling does not necessarily correspond to the null spike train, and the addition of an arbitrary translation to all embedded coordinates does not change their relative distances. To include the effects of an arbitrary translation, we append a column of 1's to R to form an M x (K + 1) matrix R'. We then solve for the (K + 1) x D matrix P' such that R'P' best approximates C in the least squares sense (a standard linear regression). The last row of P' corresponds to the arbitrary translation, i.e., the coordinates of the null response. With the last row deleted, the nth column of P' is the temporal profile Pn(t) of the nth dimension in the response space, with the understanding that time t is discretized into K bins. That is, Pn(t) indicates the weight with which the response at time t contributes to the nth coordinate in the multidimensional response space. Explicitly, it provides an approximation for the nth coordinate of the mth spike train
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n is the translation constant for the nth dimension, and the response is considered on the time interval a 
t b.
The notion of temporal profiles is readily extended to multi-unit responses, as is this method of calculating them. For a response containing spikes from L neurons, we construct an M x KL matrix R by horizontally concatenating L matrices (each M x K) constructed by the preceding method from the individual units' responses. As in the preceding text, we augment the matrix R by appending a column of 1's, to form an M x (KL + 1) matrix R'. We then solve for a (KL + 1) x D matrix P' such that R'P' best approximates C in the least squares sense. The nth column of this matrix P' contains L segments of length K. The lth segment of the nth column of P' is the temporal profile of the lth neuron along the nth dimension. We denote this segment, a list of K numbers, by Pn,l (t), with the understanding that time t is discretized into K bins. The nth coordinate of the mth multi-unit response can then be approximated by
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n is the translation constant for the nth dimension, obtained from the final column of P'. To the extent that Eqs. 2 and 3 provide a good match for the embedding identified by multidimensional scaling, they imply that the pairwise distances between spike trains can be recovered from linear operations on the responses. Because the metric distances are not created from linear operations on the spike responses, there is no a priori guarantee that such approximations will be accurate—so the very existence of temporal profiles makes a nontrivial statement about the temporal representation of phase. Moreover, because the temporal profiles indicate how the coordinates are derived from the temporal structure of individual responses, they provide an interpretation of the coordinates and a way to compare coding across populations of neurons.
Portions of this material were presented at the annual meetings of the Society for Neuroscience (2000) and The Association for Research in Vision and Ophthalmology (2001).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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Data from all cells that met these criteria (38 simple cells, 32 complex cells) were submitted to metric space clustering analysis, ellipse fitting, and the derivation of temporal profiles as single units (see METHODS). We used all pairs of simultaneously recorded units obtained from these 70 neurons for analysis of ellipse fitting and temporal profiles by neuron pairs. These paired recordings consisted of 29 simple-simple pairs, 11 simple-complex pairs, and 17 complex-complex pairs, many of which were overlapping. Because of the computational burden required by existing algorithms, not all of these paired data sets were subjected to the metric space clustering analysis. Instead, this analysis was restricted to pairs of simultaneously recorded cells of the same class (either simple or complex). At sites at which three or more units of the same class were recorded, cells were randomly paired for analysis (and, if the number of cells was odd, a randomly chosen unpaired cell was not analyzed). This resulted in a total of 22 simple cells and 18 complex cells (Table 1) analyzed in disjoint simultaneously recorded pairs (11 simple-simple pairs, 9 complex-complex pairs). For each analysis that follows, we examine representative neuron pairs before presenting the results of the population.
Stimulus-dependent clustering
The goal of this analysis was to determine how spatial phase information is coded across a local population of neurons by comparing joint responses of pairs of neurons and responses of individual neurons considered in isolation. We compared transmitted information in single-unit and pair responses and explored their dependence on the precision of spike timing (measured by the parameter q) and the importance of the neuron of origin of each spike (measured by the parameter k).
Sample data set 1: pair of simple cells with similar phase preferences
Analysis of responses of two simultaneously recorded simple cells to gratings at 16 spatial phases are shown in Fig. 2A (left). The two cells clearly have phase-dependent responses and similar spatial phase preferences. Responses of both cells are the largest at spatial phases 135–225° and are small outside of this range. In the range of the preferred phases, both cells respond with a transient peak at the onset of the stimulus followed by a sustained ON response, both of which are phase dependent. Both cells show negligible dependence of the OFF response on spatial phase.
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The two cells were first analyzed independently with the single-unit metric (curves in Fig. 2A, middle). Maximal clustering for these cells is achieved for q in the range of 16 to 32 s-1. This indicates that stimulus-dependent clustering is stronger when the temporal structure of responses is taken into account (q > 0 s-1) than when only the number of spikes is considered (q = 0 s-1). As q increases beyond the optimal values, H decreases, eventually to chance level (H 
0). This decline in H for values of q > 32 s-1 indicates that comparing responses with a metric that is sensitive to very small temporal shifts of spikes degrades the relationship between spatial phase and response cluster. In other words, taking into account the timing of spikes improves the sharpness of the dependence of responses on spatial phase but only down to a particular timing resolution. This timing resolution is measured by 1/q. In typical data sets, such as this one, the temporal precision 1/q is in the range 30–60 ms. For many cells, such as the one denoted by squares in Fig. 2A, H shows a substantial rise even for small values of q. That is, analysis of the responses with only a coarse sensitivity to spike times leads to a substantial improvement of stimulus-dependent clustering.
The two cells were then analyzed jointly with the multi-unit metric, for a mesh of ordered pairs (q,k), with the same values of q as in the preceding text and k ranging from 0 to 2, sampled more finely at lower values. The index of joint response clustering (denoted by Hjoint to distinguish it from calculations based on single units) is plotted as a function of these parameters as the surface in Fig. 2A. For constant values of k, Hjoint depends on the timing parameter q in a way similar to H for the single-unit responses, achieving a maximum for q in the range 16–32 s-1. In contrast, Hjoint has very little dependence on k for any value of q. This indicates that, in this data set, distinguishing spikes that are fired by different neurons has no effect on the strength of stimulus-dependent clustering. That is, for the purpose of determining spatial phase from these neurons' responses, it suffices to consider them indistinguishable members of a population.
The preceding observation does not mean, however, that the two neurons are redundant—merely that they are no less informative if one ignores which neuron fired which spike. Indeed, if the contributions of cells to stimulus-dependent clustering were completely redundant, the values of Hjoint would be no greater than the single-unit values of H. Figure 2A thus shows that the responses are not completely redundant: at constant q, Hjoint is greater than the either of the single-unit values of H across the entire range of values for k. On the other hand, if there were no redundancy, the values of Hjoint would be equal to the sum of the single-unit values of H. It does not achieve this value (curve indicated by +'s) at any value of q and thus the contributions of the two cells to stimulus-dependent clustering are partially redundant.
Redundancy index
To quantify the degree of independence of the single neuron responses, we used a redundancy index (Reich 2001b). The redundancy index is given by
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The redundancy index for this data set is plotted as a function of k in Fig. 2A (right). When analysis is restricted to k = 0, the redundancy index is 0.60. At the optimal value of Hjoint, the redundancy index is the same (0.60 at k = 0.2). Thus for this pair of cells, the two neurons together provide more information about which spatial phase was present than either neuron alone, but there is no increase in the fidelity of the phase representation associated with keeping track of which cell fired which spike.
Sample data set 2: pair of simple cells with dissimilar phase preferences
Responses of two simple cells with dissimilar phase preferences are shown in Fig. 2B (left). Each cell discharges a transient ON response to gratings in a range of spatial phases and a transient OFF response to gratings at the opposite phases. The cell denoted by circles has the largest ON responses at spatial phases 135–225° and the largest OFF responses at 315–45°. The cell denoted by squares has an ON and OFF response at spatial phases 225–315° and 45–135°, respectively, which are nearly orthogonal to the preferred phases of the first neuron.
Independent analyses of the two cells with the single-unit metric yield results (curves in Fig. 2B, middle) similar to those described for the data set in Fig. 2A. Stimulus-dependent clustering of both cells' responses is greatly improved when their temporal structure is taken into consideration. Maximal clustering is achieved for q in the region of 16–64 s-1. The maximal values of H are 1.32 bits (the cell denoted by squares) and 1.46 bits (the cell denoted by circles). These values are atypically high, being more than three times greater than the average across simple cells. Nevertheless, they are only 1/3 the value expected of perfect clustering (log216 = 4).
The information surface from joint analysis of the same cells (Fig. 2B, middle) shows a similar dependence of response clustering on q as the surface from the data set in Fig. 2A. However, in contrast to measurements from pairs of cells with similar phase preferences, the values of Hjoint in the present data set are strongly dependent on k. At k = 0, the values of Hjoint are not significantly higher than the values of H derived from single-unit responses, whereas at near-optimal values of q, Hjoint increases greatly with k. The maximum value of Hjoint (as a function of q) levels off at approximately k = 0.5 but continues to rise slightly until k = 2. When the joint analysis is limited to k = 0, the redundancy index for this pair of cells is 0.94 (Fig. 2B, right). For k > 0, however, it decreases to a much lower value of 0.44 (Fig. 2B, right), indicating that distinguishing spikes fired by different neurons improves stimulus-dependent clustering in the present data set. If neurons are not distinguished (k = 0), the addition of the second cell does not significantly increase the amount of transmitted information, corresponding to a redundancy index of 1. That is, if responses are simply pooled, the benefits that might result from reduced noise (due to independent contributions from each neuron) are offset by the penalty of combining responses with distinguishable phase preferences.
Sample data set 3: pair of complex cells
Responses of two simultaneously recorded complex cells are shown in Fig. 2C (left). Each cell has prominent ON and OFF discharges with both transient and sustained components at all spatial phases. The average firing rates of both cells do not vary significantly with phase.
As seen from the values of H in Fig. 2C (middle), stimulus-dependent clustering of single-unit responses is considerably weaker for the present data set than for simple cells, as would be expected from the qualitative phase-insensitivity seen in the response histograms (Fig. 2C, left). For both neurons, the amount of transmitted information is near chance (0 bits) at q = 0 s-1, corresponding to the observation that the number of spikes in the cells' responses did not vary systematically with spatial phase. At positive values of q, however, H becomes significantly higher than chance and reaches a maximum at q in the region of 32 s-1. Thus a modest amount of information about the spatial phase of the stimulus is encoded in the temporal structure of responses, even though the total number of spikes carries no information. As in the previous examples, the peak value of q indicates that the spike times are informative on a time scale of 1/q= 30 ms.
In contrast to the previous data sets, analysis of joint responses reveals a strong dependence of clustering on k (Fig. 2C, middle). At k = 0, the values of Hjoint are lower than the values of H derived from individual responses. Thus simple addition of the two responses confounds the coding of spatial phase. This corresponds to a redundancy index >1 (Fig. 2C, right). At near-optimal values of q, however, Hjoint increases with k and reaches values that are even higher than the sum of the individual measurements. This corresponds to synergistic coding of spatial phase, and a redundancy index of <0 (Fig. 2C, right). The generally low values of H, especially at q = 0 s-1, and the increase in Hjoint for k > 0 were typical of the recordings of pairs of complex cells. However, synergistic coding, clearly demonstrated by these two cells, was not typical in our data sets.
Summary across data sets
Figure 3 summarizes the clustering analysis across the data sets. For individual cells that constituted the analyzed pairs (22 simple and 18 complex), the average behavior of H is shown in Fig. 3A. The dependence of the peak value of H on the F1/F0 modulation ratio for these individual neurons is shown in Fig. 3B, and characteristics of the stimulus-dependent clustering analysis are summarized in the first two columns of Table 2. At all values of the metric parameter q, individual responses of simple cells exhibited stronger stimulus-dependent clustering than did those of complex cells (Fig. 3A). At q = 0 s-1 (the spike count metric), the values of H for complex cells were usually not above chance level, while for simple cells they were usually highly significant (Table 2). This is consistent with the notion that response magnitude is phase dependent in simple cells and phase independent in complex cells. However, most simple and complex cells yielded values of H that were above chance level when spike timing was taken into consideration, consistent with our previous findings. Maximal clustering occurred at q > 0 s-1for 20 of the simple cells (91%) and 16 of the complex cells (89%). The average value of H at optimal q was 2.2 times higher than the value at q = 0 s-1 for simple cells and 2.9 times higher for complex cells. Thus temporal coding can allow both simple and complex cells to transmit more than twice the amount of information about spatial phase than is contained in spike counts. Qualitatively, H depended on q in a similar way for the two classes of cells. Maximal clustering was achieved, on average (geometric mean), at q = 24 s-1 for simple and 26 s-1 for complex cells, corresponding to a temporal precision of 40 ms.
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It might appear surprising that modulation ratio and H were not tightly linked. In principle, eye movements might artifactually reduce the measured value of H. However, the rasters of Fig. 1 show that phase tuning was stable over the course of an experiment. This rules out the possibility that eye movements affect our measurements. (This was a data selection criterion.) More likely, modulation ratio (or phase tuning) and H are not closely linked because H depends critically on the signal-to-noise ratio, whereas modulation ratio and phase tuning depend primarily on signal. There are additional reasons that information values can be low even for a narrowly tuned cell with high a signal-to-noise ratio. If a threshold limits responses to only one portion of the spatial phase gamut, it will necessarily reduce the number of spatial phases that can be distinguished on the basis of the neuronal response. Thus paradoxically, responses of a more narrowly tuned neuron can contain less information about spatial phase. Comparison of Fig. 2, A and B, show an example of this. The neurons of Fig. 2A are narrowly tuned, responding well to only five or six of the stimuli, and have values of H of 0.4 bits. The neurons of Fig. 2B respond to a broader range of phases and have values of H of >1 bit.
Although there was a dramatic difference between the average values of H across simple and complex cells (Fig. 3A and Table 2), this difference does not necessarily mean that simple cells, as a class, signal spatial phase, while complex cells do not. As seen in Fig. 3B, the range of peak values of H for simple and complex cells was overlapping. Two other aspects of this figure indicate that neurons cannot be classified as simple and complex based on the amount of transmitted information about spatial phase: some simple cells have low values of H and the maximal value of H covaries with the modulation ratio, not the classification (simple vs. complex) of the neuron per se. That is, within the cells of either category, higher modulation ratios are associated with larger values of H; the classification cutoff at a modulation ratio of 1 plays no special role These observations are in keeping with findings concerning the selectivity of responses of simple and complex cells to the relative spatial phases of compound gratings.
The average behaviors of Hjoint for the 11 pairs of simple cells and 9 pairs of complex cells are shown in Fig. 3, C and D, respectively. For the joint responses, the dependence of clustering on temporal resolution was similar to that for the individual responses. That is, the qualitative dependence of Hjoint on q was the same at all values of k, indicating that the temporal structure of responses contributes to stimulus-dependent clustering independently of the neuron of origin. For a pair of simple cells, the information transmitted when decoded as a summed population (0.56, "q optimal, k = 0" in Table 2) was less than when decoded in a manner that was sensitive to which neuron fired which spike (0.69, "q optimal, k optimal"). This improvement corresponds to a drop in the average redundancy index (Fig. 3E) from 0.8 at k = 0 to 0.6 at optimal k.
For complex cells, the improvement of stimulus-dependent clustering for optimal k (typically near 1) was greater than for simple cells (Fig. 3D). While clearly less than the amount of information transmitted by pairs of simple cells about spatial phase, the amount of information (0.13, "q optimal, k optimal" in Table 2) transmitted by a pair of complex cells is not negligible. To put this quantity in perspective, this amount of information supports a performance level of 70% correct on a two-alternative forced choice task.
The behavior of the redundancy index in complex cells indicates diversity across this subpopulation. On average (Fig. 3E), the redundancy index declines from 1.70 at k = 0 to just under 1 at optimal k. This reflects an admixture of two kinds of behavior: pairs such as the one in Fig. 2C, in which there is very little redundancy at sufficiently high k, and other pairs, for which the redundancy index remains high even for large k. The latter group includes pairs in which the maximal value of H for one of the neurons is near zero. This leads to high values of the redundancy index as long as Hjoint < H1 + H2.
One would expect that the importance of the neuron of origin of individual spikes would be larger for pairs of cells with dissimilar phase preferences than for pairs of cells with similar phase preferences. For simple cells, this intuition is confirmed (Fig. 4A). We quantified the importance of neuron of origin (i.e., of which neuron fired which spike) by the drop in the redundancy index, calculated between k = 0 and optimal k. We correlated this value (on the ordinate) with a single parameter rtuning (on the abscissa) that quantifies the similarity of phase preferences. rtuning was obtained as follows. We took the average firing rate over the entire 473-ms interval as the response measure at each spatial phase and set rtuning equal to the standard (Pearson) correlation coefficient of the cells' responses across the 16 spatial phases. Three pairs of simple cells had negative values of rtuning, indicating largely opposite phase preferences. These pairs were also the ones with the largest differences (>0.5) between the two redundancy indices (e.g., data set in Fig. 2B). Conversely, for the four pairs of simple cells with the most similar phase preferences (rtuning > 0.97), the difference between the two redundancy indices was <0.005 (e.g., data set in Fig. 2A). For complex cells, there was a greater range of the redundancy index, but no clear correlation with phase tuning similarity—most likely because spike counts poorly reflect the phase tuning of complex cells, and because measurement of the redundancy index is less reliable when values of H are small.
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Figure 4A suggests that simultaneously recorded neurons tend to have similar phase preferences (i.e., rtuning tends to be close to 1 rather than randomly distributed between -1 and 1) but is limited to the randomly selected 20 pairs of cells (11 simple-simple, 9 complex-complex) for which we performed the stimulus-dependent clustering analysis parametric in q and k. Figure 4B shows rtuning for all pairs of simultaneously recorded neurons that met the selection criteria (29 simple-simple pairs, 11 simple-complex pairs, and 17 complex-complex pairs; see Table 1). The tendency for the phase tuning of neighboring neurons to be similar (rtuning > 0) is more clearly evident, especially for simple-simple pairs (mean rtuning = 0.68) but also for simple-complex (mean rtuning = 0.29) and complex-complex (mean rtuning = 0.30) pairs. Also, there is no evident tendency for a quadrature relationship between neighboring neurons, which would correspond to rtuning = 0.
Geometry of response clusters
The analysis so far focused on the extent to which the responses to the 16 different spatial phases fell into distinguishable clusters but provided no insight into the relationship between these clusters. For example, this analysis could yield identical information (H) values for a neuron that primarily confounded each spatial phase with its opposite (an idealized ON-OFF neuron with little noise) and for a neuron that confounded each spatial phase with its neighbor (an idealized linear neuron with some noise). Yet these distinctions are important because they can suggest the kinds of receptive-field mechanisms that distinguish spatial phases. Additionally, even a high degree of clustering would be useless in representing spatial phase unless the responses varied with spatial phase in some systematic manner.
The first step in characterizing the geometry of the response clusters was multidimensional scaling of metric distances between single-trial responses. For single neurons, we used the single-unit metric with q = 32 s-1; for joint responses of simultaneously recorded neurons, we used the multi-unit metric with q = 32 s-1 and k = 1. These choices were close to maximal across the entire population of analyzed neurons. We chose a uniform value of q and k for this analysis so that we would not confound differences between recordings with the dependence of the analysis of individual recordings on q and k. The details of the results of multidimensional scaling do change with the metric parameters, but this change is gradual, and our main conclusions do not depend on the specific choices of q and k within the range q = 8–128 s-1 and all k > 0.5.
After embedding of the responses in a 10-dimensional Euclidean space by multidimensional scaling, we characterized the geometry of the response clustering by fitting ellipses to the centroids of responses to each spatial phase. Separable receptive fields predict a response trajectory that doubly covers a line segment (moving sinusoidally as a function of spatial phase), while linear but inseparable receptive fields predict an elliptical trajectory with a nonzero minor axis (see METHODS and APPENDIX). Therefore it is important to determine whether the fitting of a response trajectory by an ellipse is significantly better than a fit with a doubly covered line segment, i.e., an ellipse with an axis ratio of zero.
Our approach to estimating significance of the minor axis was the following. The null hypothesis is that the apparent minor axis is no longer than what might be expected due to chance. We generated surrogate trajectories by reflecting randomly chosen subsets of the 16 centroids over the major axis of the ellipse—i.e., negating their coordinate along the direction of the minor axis but leaving their coordinate along the direction of the major axis unchanged. (All transformations were made parallel to the plane of the best-fit ellipse and only involved coordinate changes along the minor axis.) Under the null hypothesis that the centroids were positioned at random along the minor axis but perhaps systematically along the major axis, these surrogates would be just as probable as the observed data. The best-fit ellipse was then recalculated for each of 1,000 such sets of surrogate centroids, and the in-plane variance explained by the best-fit ellipse was tabulated. If the original centroids were positioned at random along the minor axis, then random reflection would be just as likely to improve the goodness of the elliptical fit as to worsen it. If, on the other hand, centroids were positioned systematically in two dimensions around an ellipse, reflections would typically decrease the goodness of fit. We defined plineseg as the fraction of the surrogate sets for which ellipses explained more of the variance than for the original trajectory. Data sets with plineseg < 0.05 were thus fit by ellipses significantly better than by a doubly covered line segment. For some data sets, ellipses accounted for the centroid positions significantly better than doubly covered line segments (by this measure), yet both of these fits accounted for only a small fraction of the variance. Because of this possibility, we considered an elliptical fit to be significant only if it passed the above test for systematic arrangement in two dimensions and it explained >50% of the total variance.
Sample data set 1
In Fig. 5, we present this analysis of the same three data sets shown in Fig. 2. Figure 5A, top and middle, shows the first two dimensions of the embedding provided by multidimensional scaling of the individual responses illustrated in Fig. 2A (simple cells with similar phase tuning). Figure 5A, bottom, shows the first two dimensions of the embedding of the joint responses. In each case, along the first dimension, the projections appear to be similar to doubly covered line segments. Correspondingly, all best-fit ellipses are highly eccentric, with axis ratios equal to 0.036 for the cell denoted by a circle, 0.049 for the cell denoted by a solid square, and 0.026 for the pair of cells analyzed jointly. These ellipses account for 81, 79, and 83% of the variance, respectively. All three ellipses yield high values of the preceding statistic plineseg, indicating that along the second dimension, the trajectories are no more consistent with ellipses than would be expected by chance variation from a doubly covered line segment.
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Sample data set 2
Figure 5B shows the same analysis for the cells of Fig. 2B (simple cells with distinct phase preferences). Unlike the single-unit response trajectories in Fig. 5A, the trajectories in B appear to vary systematically along the axis of the second dimension. This is even more evident for the joint response trajectory, which is nearly an ellipse. In the plane of the first two dimensions, single-unit trajectories appear to be ellipses that have been twisted around the major axis. Best-fit ellipses to these trajectories have axis ratios equal to 0.075 for the cell denoted by a circle and 0.127 for the cell denoted by a solid square. These ellipses explain 92 and 85% of the variance and are both significant, with plineseg < 0.01 and plineseg < 0.001, respectively. The joint response trajectory has a much higher axis ratio of 0.425 and explains 88% of the variance with plineseg < 0.001.
As we will see in the following text, the first dimension in the multidimensional space indicates the difference between the ON and OFF responses (see Temporal profiles). Centroids move in one direction along the first dimension as the ON response increases and in the other direction as the OFF response increases. If the number of spikes in the ON and OFF discharges were the only attribute of the response that varied systematically with spatial phase, one would expect significant systematic variation of the responses only along this dimension, resulting in doubly covered line segments as response trajectories. The coding analysis (Figs. 2 and 3) demonstrates that temporal structure, and not just response magnitude, varies systematically with spatial phase. In multidimensional scaling of the second data set, centroids of responses with equal magnitudes of the ON and OFF components are separated along the second dimension. This must be due to differences in the temporal structure of these responses and not just changes in response magnitude. That is, temporal coding must underlie a systematic arrangement of the responses in a second dimension of the response space.
Sample data set 3
Figure 5C shows multidimensional scaling of the cells of Fig. 2C (2 complex cells). Although responses to certain spatial phases seem to lie apart from the bulk of the responses, the trajectories of both single- and multi-unit responses are not similar to ellipses. Indeed, the best-fit ellipses to these trajectories explain 49 and 50% of the variance for the cells denoted by a circle and a solid square, respectively, and 48% of the variance for the joint responses.
Summary across data sets
Results of elliptical fitting are summarized in Table 3. The first two columns indicate that there were significant quantitative differences between simple and complex cells. For complex cells, ellipses explained a smaller fraction of the variance than for simple cells, but their axis ratios tended to be larger. The former difference is expected from the fact that responses of complex cells vary less systematically with spatial phase than responses of simple cells. The latter difference does not have a clear explanation on this basis. As pointed out in the preceding text, any two-dimensional trajectory requires two (or more) spatial mechanisms, each coupled to distinct temporal responses. The greater axis ratio in complex cells suggests that the spatial and temporal distinctions between these mechanisms were greater in complex than in simple cells.
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The last three columns focus on the joint responses of pairs of cells. The fraction of data sets with response trajectories that were significantly elliptical was larger for pairs of cells analyzed jointly than for individual cells. The axis ratios of the ellipses were also higher for the joint responses. That is, their trajectories were more nearly circular. This difference was substantial for data sets such as the one in Fig. 5B in which the phase tunings of the individual neurons were very different. However, for most data sets, the average axis ratios were only slightly higher for pairs of cells than for individual neurons.
The goodness of the elliptical fit was similar for single neurons' responses and for joint responses. As seen in Fig. 6, in both cases the goodness of fit was strongly correlated with the amount of spatial phase information in the responses as measured by stimulus-dependent clustering. Response trajectories that strongly deviated from ellipses were produced primarily by those data sets that exhibited weak stimulus-dependent clustering. As H and Hjoint increased, the fraction of response variance explained by ellipses increased almost to 1. This indicates that, for the neurons that produced the most discriminable responses to spatial phases, the ellipse is indeed the geometry that expresses how these responses depend on spatial phase.
