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Department of Psychology and Center for Perceptual Systems, University of Texas, Austin, Texas 78712
Submitted 2 September 2003; accepted in final form 29 January 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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; Cooper and Robson 1968; De Valois et al. 1982; Movshon et al. 1978; Robson et al. 1988; for general reviews see De Valois and De Valois 1988; Geisler and Albrecht 2000; Shapley and Lennie 1985). This selectivity is very different from that of the neurons in the lateral geniculate nucleus (LGN) of the thalamus and the ganglion cells in the retina, which are not very selective for spatial frequency or orientation. In comparison, the spatial frequency bandwidth for neurons in the LGN ranges from 3 to >5 octaves (e.g., Croner and Kaplan 1995; Derrington and Lennie 1984; Hicks et al. 1983; Irvin et al. 1993; Kaplan and Shapley 1982; Rodieck and Stone 1965; So and Shapley 1981; Troy 1983a,b; Xu et al. 2002). Although it is unknown exactly how this transformation in the selectivity, from the LGN to the cortex, is produced, several different classes of models have been proposed (e.g., Anderson et al. 2000a,b; Carandini and Ringach 1997; Carandini et al. 1999; Chance et al. 1998; Douglas et al. 1995; Gilbert et al. 1990; Hubel and Wiesel 1962; Troyer et al. 1998; for recent reviews, see Albrecht et al. 2002, 2003; Ferster and Miller 2000).
Most measurements of spatial frequency selectivity have been performed using moving stimuli that were presented for relatively long durations (generally several seconds) to approximate a steady-state condition. However, natural saccadic inspection of a visual scene typically produces transient stimulation: 200- to 300-ms fixations separated by rapid eye movements. Although there are minor differences in the temporal dynamics of saccadic eye movements among humans, macaque monkeys, and cats, fixation durations of approximately 200 ms are typical across all 3 species (Evinger and Fuchs 1978; Fuchs 1967; Stryker and Blakemore 1972; for general reviews see: Carpenter 1991; Ditchburn 1973).
There is now a growing body of evidence to indicate that cortical neurons respond differently under transient, as opposed to steady-state, conditions. For example, 1) the temporal frequency tuning measured using steady-state stimuli shows relatively little low-frequency attenuation (Albrecht 1995; Hawken et al. 1996; Movshon et al. 1978), whereas the temporal response profile measured using transient stimuli decays quite rapidly, faster than would be expected based on the temporal frequency tuning (Albrecht et al. 2002; Müller et al. 1999, 2001; Tolhurst et al. 1980)1; 2) the temporal frequency tuning measured using steady-state stimuli varies as a function of contrast (Albrecht 1995; Hawken et al. 1992; Holub and Morton-Gibson 1981), whereas the temporal response profile measured using transient stimuli is relatively invariant as a function of contrast (Albrecht et al. 2002); 3) the variability of cortical neurons is approximately proportional to the mean firing rate under steady-state conditions (e.g., Geisler and Albrecht 1997; Softky and Koch 1993; Tolhurst et al. 1983), whereas under transient conditions the relationship between the mean and variance is more complex (Albrecht et al. 2002; Mechler et al. 1998; Müller et al. 1999, 2001; for recent reviews of this literature, see Albrecht et al. 2002, 2003). Given these differences between the responses of visual cortex neurons under steady-state conditions, as opposed to transient conditions, it is important to measure the spatial frequency tuning under transient conditions that are more comparable to natural fixation.
Two recent investigations of spatial frequency tuning have presented gratings for very brief temporal intervals (
20–30 ms) in rapid, temporally contiguous succession; a reverse correlation technique was then used to measure the linear component of the responses (Bredfeldt and Ringach 2002; Mazer et al. 2002).2 The goal of these studies was to measure the temporal dynamics of spatial frequency tuning over the course of the first 100 ms or so following stimulus onset. Although the results of the two studies are, on the whole, quite consistent, there are differences. First, Mazer et al. reported that the spatial frequency tuning changed very little through time (as they pointed out, the tuning was largely separable), whereas Bredfeldt and Ringach reported that the spatial frequency tuning changed substantially through time (i.e., the tuning was inseparable). Specifically, Bredfeldt and Ringach reported that during the initial time periods (at the onset of the responses) the tuning was quite broad, spanning both low frequencies and high frequencies (as they pointed out, very similar to the tuning of LGN neurons); however, through time, the responses to low frequencies were diminished, and thus the peak of the spatial frequency tuning shifted from low frequencies to high frequencies. Second, Mazer et al. reported that the latency of the response increased as spatial frequency increased (for the component of the response that they reported was not separable), whereas Bredfeldt and Ringach did not report a change in the latency as a function of spatial frequency. Specifically, they found that the onset of the responses to high frequencies and low frequencies appeared concurrently.
The steady-state method of stimulation (with durations on the order of seconds) and the brief, temporally contiguous method of stimulation (with durations on the order of 20–30 ms) have both provided important information about the properties of visual cortex neurons. However, these methods of stimulation are both different from the stimulation that occurs during normal saccadic viewing. In the present study we investigated the spatial frequency tuning of striate visual cortex neurons using stationary gratings that were presented for a temporal interval that is roughly comparable to the duration of a single fixation (i.e., 200 ms). Further, the responses were measured on a fine time scale to investigate the temporal dynamics of spatial frequency tuning during this interval.
Using this stimulus protocol, we find that for most neurons the latency of the response increases as the spatial frequency increases and that the peak of the spatial frequency tuning shifts through time from low frequencies to high frequencies. Interestingly, these dynamic changes are qualitatively consistent with what would be expected from the model proposed four decades ago by Hubel and Wiesel (1962) if one assumes, in the monkey, that magnocellular and parvocellular neurons, or in the cat, that X and Y neurons, converge on the same cortical neuron.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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The procedures for the paralyzed anesthetized preparation, the electrophysiological recording, the stimulus display, and the measurement of neural responses using systems analysis were similar to those described elsewhere (Albrecht and Geisler 1991; Albrecht et al. 2002; Geisler and Albrecht 1997; Geisler et al. 2001; Hamilton et al. 1989; Metha et al. 2001). All experimental procedures were approved by the University of Texas at Austin Institutional Animal Care and Use Committee, and conform to the National Institutes of Health guidelines. In brief, young adult cats (Felis domesticus) and monkeys (Macaca fascicularis or Macaca mulatta) were prepared for recording under deep isoflurane anesthesia. Following the surgical procedures, isoflurane anesthesia was discontinued. Anesthesia and paralysis were maintained throughout the duration of the experiment using the following pharmaceuticals. For cats, anesthesia was maintained with sodium pentothal (2–6 mg·kg-1·h-1). For monkeys, anesthesia was maintained with sufentanil citrate (2–8 µg·kg-1·h-1). For both species, paralysis was maintained with gallamine triethiodide (10 mg·kg-1·h-1) as well as pancuronium bromide (0.1 mg·kg-1·h-1). The physiological state of the animal was monitored throughout the experiment by continuous measurement of the following quantitative indices: body temperature, inhaled/exhaled respiratory gases, pressure in the airway, fluid input, urine output, urinary pH, caloric input, blood glucose level, electroencephalogram, and electrocardiogram. Microelectrodes were inserted into regions of the striate visual cortex such that the receptive fields of the neurons were located within 5° of the visual axis. Three different types of microelectrodes were used: varnish-insulated tungsten, glass pipette, or glass-coated platinumiridium. The impedances of the microelectrodes ranged from 8 to 21 M
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Action potentials were collected with a temporal accuracy of 0.1 ms and then binned to produce a poststimulus time histogram (PSTH). Note that within this report, and elsewhere (Albrecht et al. 2002), we use the terms PSTH and temporal response profile interchangeably. The bin size for the PSTH was 10 ms, and this average (centered within the 10-ms time bin) was computed every millisecond: a 10-ms running average evaluated at every millisecond throughout the entire time course of the response (cf. De Valois and Pease 1973; Enroth-Cugell and Robson 1966; Gerstein 1960; Levick and Zacks 1970). Note that for some analyses (as will be noted, when appropriate), the PSTH bin size was 1 ms.
Stimulus presentation
The stimuli were presented on a monochrome Image Systems monitor at a frame rate of 100 Hz, with a mean luminance of 27.4 cd/m2. To overcome the nonlinearities inherent in visual displays, both hardware and software methods were used to ensure a linear relationship between the requested luminance and the measured luminance. The precision of these methods to overcome the nonlinearities in the visual display was assessed through quantitative measurements that were performed before, during, and after each experiment.
Preliminary measurements
Before the main experimental protocol, several preliminary experiments were performed, using drifting gratings, to determine the optimal orientation, spatial frequency, temporal frequency, and direction of motion. First, the optimal parameters for these 4 dimensions were determined in a qualitative fashion by adjusting the stimulus and listening to the spike trains. Second, the optimal parameters were determined quantitatively by drifting 10 contiguous cycles of each stimulus along a given dimension. The response measure was the average rate of firing. The presentation sequence of the stimulus dimensions was: temporal frequency, orientation, spatial frequency, and contrast. The responses were plotted on-line and the optimal values were determined. Third, step two was repeated with 4 randomly interleaved repetitions of 10 contiguous cycles of each stimulus along a given dimension. Finally, the contrast response function was measured using stationary gratings, as described in Albrecht et al. (2002). For the dimension of contrast, the minimum detectable contrast, half-saturation contrast, and saturation contrast (i.e., the contrast above which further increases in contrast produce little or no further increases in response) were determined (cf. Albrecht and Hamilton 1982; Bonds 1991; McLean and Palmer 1996; Sclar et al. 1990). In all of the experiments reported here, the stimuli were confined to the conventional receptive field, which was determined by expanding the size (the length and the width separately) of an optimal drifting sine wave grating (presented at a saturating contrast) until the neuron's response stopped increasing (DeAngelis et al. 1994; De Valois et al. 1985). The contrast of the grating was held constant over the conventional receptive field; the "windowing function" was one half of one cycle of a cosine (0 to 180°). The function was shifted and scaled to vary between one and zero, thus gradually modulating the contrast at the boundaries. The degree of direction selectivity was measured (Albrecht and Geisler 1991). Cells were classified as simple cells or complex cells using the criteria described by De Valois et al. (1982) as well as Skottun et al. (1991). Specifically, cells were classified as complex cells if the magnitude of the average rate of firing exceeded the magnitude of the first harmonic response. These measurements were performed in 3 monkeys and 3 cats. (Note that the same animals, and oftentimes the same cells, were also used for other stimulus protocols.) The cell types were distributed as follows: 25 monkey cells (18 complex; 7 simple) and 39 cat cells (24 complex; 15 simple).
Stimulus protocol: stationary gratings at different spatial frequencies and phases
Following the preliminary experiments, optimal stationary grating patterns were presented at a saturating contrast level (which was determined for each cell), in 8 different spatial phases, each separated by 45°. Twelve different spatial frequencies were presented; the range of these frequencies was determined based on the responses to drifting gratings such that frequencies on either side of the preferred frequency were sampled. Each of these 12 frequencies was presented at each of the 8 spatial phases, making a total of 96 unique combinations of phase and spatial frequency. Each unique combination was turned on (flashed) for 200 ms and then turned off for 300 ms, with a minimum of 40 repeated presentations (and a maximum of 80). During the 300-ms interval, when the stationary grating was turned off, the animal viewed the mean luminance. As described in detail below, this interstimulus interval (ISI) was introduced to minimize potential interactions between the sequential stimuli. The stimuli were presented in a counterbalanced fashion such that all stimulus conditions were presented an equal number of times, in a random order. With 40 repeated presentations of each condition, this stimulus protocol required about 30 min to complete.
The 300-ms ISI was introduced for several reasons. First, this interval permits recovery from any short-term pattern adaptation (Müller et al. 1999, 2001) or rapid local light adaptation (Adelson 1982; Crawford 1947; Geisler 1981; Hayhoe et al. 1987; Saito and Fukada 1975; for reviews see Hood 1998; Shapley and Enroth-Cugell 1984). Second, as will be shown in the RESULTS, for most cells, the latency of the response increases as the spatial frequency increases; thus if the stimuli are presented in close temporal proximity, then the responses to the 2 stimuli might overlap in time. For example, if a high spatial frequency is presented just prior to a low spatial frequency, with no ISI, the responses to the 2 stimuli will overlap in time (or even reverse order). Third, when a stationary stimulus is turned off, many cells produce large OFF responses that could overlap in time with the ON responses to a subsequent stimulus presented contiguously. In some cases, these OFF responses are larger than the ON responses and they can last nearly as long as the ON responses; none of the OFF responses in this sample lasted as long as 300 ms. Separating the stimuli by 300 ms precludes any of these problems.
Presenting the grating in 8 different spatial phases ensures that 1) the space average luminance remains equivalent throughout the course of the experiment over the entire receptive field, 2) both optimal and nonoptimal spatial phases are sampled, and 3) many different regions of the receptive field are stimulated with different luminance increments and decrements. Further, presenting the stationary grating in 8 different spatial phases, and then computing the average, minimizes the potential errors that are introduced by not having the origin of the stimulus located at the exact center of the receptive field. Minor offsets, or any residual eye movements (cf. Forte et al. 2002), can potentially introduce substantial errors in the estimates of the spatial frequency tuning. For example, simulations (RA Frazor, unpublished observations) show that if the origin of the stimulus (i.e., the location where all spatial frequencies are aligned in the same phase) is offset from the center of the receptive field (which is not easy to measure with a high degree accuracy; cf. Albrecht 1995; Hamilton et al. 1989) by as little as a quarter of a cycle, then the estimate of the peak spatial frequency will be off by 0.5 octaves and the estimate of the bandwidth will be off by 10%. The simulations also show that these errors are eliminated by averaging across the 8 spatial phases.
Finally, given that we are averaging the PSTHs across all of the spatial phases (for a given spatial frequency), it is important to point out that we have performed an analysis to assess the degree of any systematic variation in the overall shape of the PSTHs (for both simple and complex cells) at the different spatial phases (see Analysis of the variation in the PSTHs at different spatial phases, in the METHODS section of Albrecht et al. 2002). In general, we found that, although the amplitude of the response often varied as a function of spatial phase, the shape of the temporal response profile (i.e., the PSTH) was relatively invariant as a function of spatial phase. Approximately 95% of the variation across all 8 spatial phases could be accounted for by simply scaling the amplitude of the average of all 8 spatial phases. This observation held true for both simple and complex cells. Further, as noted in that report, there were no obvious trends in the amount of variance accounted for across animal type or cell type. Given that there is very little systematic residual variation (the median value was 4.2%) in the overall shape of the PSTHs as a function of spatial phase (compared with the average across spatial phase), and for the reasons discussed in the preceding paragraph, it therefore seemed reasonable to average the responses across spatial phase for both complex cells and simple cells.
The stimulus protocol, saccades, and saccadic suppression
Although the 200-ms presentation interval provides a good approximation to the average duration of a single fixation in humans, macaque monkeys, and cats (Evinger and Fuchs 1978; Fuchs 1967; Stryker and Blakemore 1972; for general reviews see Carpenter 1991; Ditchburn 1973), the 300-ms ISI does not provide a good approximation to the average duration of a single saccade. In comparison to humans, macaque monkeys make faster saccades (25–55 ms; Fuchs 1967) and cats make slower saccades (50–150 ms; Evinger and Fuchs 1978). Note, however, that because visual cortex neurons are so selective along many different stimulus dimensions, it is highly unlikely that any single neuron will be stimulated during every fixation while viewing a complex natural image (see Geisler and Albrecht 1997; their Fig. 13 and related text). In fact, it is highly likely that there will be many fixations in between. We chose 300 ms to minimize interactions between stimuli and to maximize data collection.
Another way in which this experimental protocol does not approximate normal saccadic inspection of a visual scene stems from the fact that the eyes are paralyzed and thus there is a lack of coordination between the time course of the stimulus presentation protocol and any potential saccadic suppression signals, which might be generated by a corollary discharge from an oculomotor mechanism that produces saccadic eye movements. However, although saccadic suppression has been demonstrated psychophysically in the visual system as a whole (e.g., Campbell and Wurtz 1978; Riggs et al. 1974; Volkmann et al. 1968 1978; however, see Carpenter 1991; Greenhouse and Cohn 1991), physiological experiments in awake behaving monkeys suggest that saccadic suppression does not appear to be a factor at the level of the striate cortex (Bair and O'Keefe 1998; Gur and Snodderly 1997; Gur et al. 1997; Judge et al. 1980).
Quantitative index of the spatial frequency shift
As will be shown in the RESULTS, for most cells the latency of the response increases as a function of spatial frequency. This fact, coupled with the transient nature of the PSTH, produces a shift in the peak of the spatial frequency tuning through time from low frequencies to high frequencies. In general, this pattern of results is systematic and robust within the large-amplitude initial transient component of the temporal response profile for all of the cells that show a frequency shift, and thus the pattern of results can be described during this interval with a reliable, quantitative index: the peak of the spatial frequency response function during this interval. However, because there is a great deal of heterogeneity in the shapes of the temporal response profiles from cell to cell (see Albrecht et al. 2002; Figs. 4 and 5), the pattern of results following the initial transient varies from cell to cell and becomes more difficult to describe in a systematic, unified fashion. To be more specific, as the amplitude of the response decreases through time following the initial transient, the trends in the frequency shifts become more complex from cell to cell. Further, the responses are more variable within each cell because the amplitude is reduced and thus the signal-to-noise ratio is also reduced, as a consequence of the mean–variance relationship (for a review, see Geisler and Albrecht 1997). For example, for those cells that show a trough after the initial transient, the same trends that are observed during the initial transient are generally repeated in the rise to the sustained component. For those cells that do not show a prominent trough, and for those cells that show a small-amplitude sustained component, no systematic trends can be observed in the spatial frequency tuning following the initial transient, although presumably, the frequency and latency shifts are present during this interval as well. For these reasons, the quantitative index of the shift in the peak of the spatial frequency response function (i.e., the spatial frequency that produced the largest response) was restricted to the large-amplitude initial transient component of the temporal response profile, and the extent of this interval was determined in a qualitative fashion for each individual cell. Specifically, the interval began when the responses to the frequency with the shortest latency exceeded the spontaneous activity and the interval ended when the responses to the frequency with the longest latency reached a local minimum. For this sample of cells, the mean value of the duration of the initial transient component is 41.3 ms (SD = 17.7). It should be noted that the determination of the extent of the initial transient component, in addition to the specific range of spatial frequencies that was sampled, could affect the estimates of the magnitude of the frequency and latency shifts, given in Fig. 7 and Table 1, for this sample of neurons.
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Given the inherent variability within the responses of visual cortex neurons, and the finite number of repeated presentations of any given stimulus, it is possible that observed patterns in the responses are a consequence of random variation as opposed to systematic variation. To determine the likelihood that the observed variation in the responses is a consequence of chance alone we assess the sampling distribution that would be expected by chance alone (see, e.g., Albrecht et al. 2002; Edgington and Bland 1993; Gallant et al. 1996; Müller et al. 2001; for a general discussion see Edgington 1995).
In this report, we use this statistical analysis to assess the degree to which PSTHs for a given cell are invariant across spatial frequency. Specifically, we compute the average PSTH and then scale and shift this average to fit the PSTH for each spatial frequency. To determine the amount of residual variation (following scaling and shifting the average PSTH) that would not be expected from chance alone, we performed the following analysis. First, the PSTH was averaged across all spatial frequencies, without any scaling. Second, the average PSTH was scaled and shifted to fit the measured responses at each spatial frequency. Third, the relationship between the mean rate of firing and the variance was measured. With these properties measured (the average PSTH, the optimized values for the scaling and shifting, and the mean–variance relationship), the following statistical analysis was performed. 1) Random variation was introduced around the scaled and shifted average PSTH for each spatial frequency (given the mean–variance relationship and the exact number of repeated presentations). 2) These "new-randomized" PSTHs were then averaged across spatial frequency. 3) The "new-randomized" average PSTH was scaled and shifted to fit the "new-randomized" PSTH for each spatial frequency. 4) The percentage of variation accounted for was computed. 5) Steps 1–4 were repeated on 10,000 occasions to obtain the sampling distribution for the estimate of the residual variation that would be expected based on the specific properties of each cell under the null hypothesis that there was no systematic variation in the shape of the PSTH across spatial frequency.
We also use this statistical analysis to assess whether the observed latency and frequency shifts are greater than what would be expected by chance alone. Specifically, for each individual neuron in the sample, the average PSTH was scaled and shifted to fit the measured responses at each spatial frequency; in so doing, the latency and frequency shifts were removed. Then, random variation was introduced around the scaled and shifted average PSTH for each spatial frequency (given the mean–variance relationship and the exact number of repeated presentations). Finally, the latency and frequency shifts were measured. This procedure was repeated on 10,000 occasions to obtain the sampling distributions for the latency and frequency shifts that would be expected based on the specific properties of each cell under the null hypothesis that there was no latency or frequency shift.
Detection performance as a function of the integration interval
Detection performance was measured using methods that are similar to those described in detail elsewhere (Albrecht et al. 2002; Geisler and Albrecht 1995, 1997; see also Müller et al. 2001; Tolhurst et al. 1983; for a general discussion within this context see Barlow 2003). Specifically, the signal-to-noise ratio was measured as a function of the interval of integration following response onset. The signal-to-noise ratio d' is equal to the absolute value of the difference in the means divided by the square root of the average variance. Detection accuracy is related to d' by the standard normal integral; when d' is equal to 1.0, this is equivalent to 75% correct detection in a 2 alternative forced choices task (see Green and Swets 1966). To measure this detection performance, the following analysis was performed, using PSTHs where the bin size was 1 ms (not 10 ms). First, the PSTH for the optimal spatial frequency was averaged across all spatial phases. Second, this average PSTH was scaled to match the response at the optimal spatial phase. (The average PSTH was used because it is the most reliable measure of the PSTH shape; see RESULTS.) Third, we measured the relationship between the response mean and the response variance for each integration interval; that is, the variance proportionality constant (see Geisler and Albrecht 1997). Fourth, using this scaled PSTH and the variance proportionality constant, we determined the mean and the SD of the responses within the scaled PSTH for temporal intervals that increased in 1-ms steps, starting at the point in time when the responses to the preferred spatial frequency began (i.e., when we determined that the response appeared to be greater than the spontaneous rate of firing). Finally, the signal-to-noise ratio was determined for each interval relative to the mean and SD of the spontaneous rate of firing (i.e., the base rate), measured over the same time interval. In so doing, this procedure assesses the ability of the cell to detect its optimal spatial frequency and phase in comparison to no stimulus.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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PSTH as a function of spatial frequency
Figure 1 shows the responses of 3 neurons recorded from within the monkey visual cortex (A, C, and E), and 3 neurons recorded from within the cat visual cortex (B, D, and F), measured over the course of a 200-ms interval, for different spatial frequencies. Each curve shows the PSTH for a particular spatial frequency, evaluated at 1-ms intervals using a 10-ms running average (see METHODS). The systematic trends in the initial transient component are more easily seen in Fig. 2 responses of the same cells plotted on a restricted time scale.
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; Müller et al. 1999, 2001; Tolhurst et al. 1980). Second, for many of the neurons, there is a brief trough in the firing rate between the transient and the sustained component, as described in previous reports (Albrecht et al. 2002; Müller et al. 2001; Tolhurst et al. 1980). Approximately half of the cells in this sample show a trough; for some cells the trough is very small and for other cells it is quite pronounced. Third, the shapes of the temporal response profiles for the different spatial frequencies appear to be qualitatively similar: they appear to differ by a vertical scaling of the magnitude and a horizontal shift in time. Fourth, as will be shown, the horizontal shift is systematic: for most neurons, as the spatial frequency increases, the latency increases. Spatial frequency tuning through time
Figure 3 shows 3-dimensional plots ("ribbon-plots") of the measured responses illustrated in Figs. 1 and 2. Spatial frequency is plotted along the x-axis, time is plotted along the y-axis, and response magnitude is plotted along the z-axis. Note that there is no smoothing of the 10-ms time bins in these plots. Thus for example, in Fig. 3A, there are 432 discrete measured average responses; that is, the average responses to 12 spatial frequencies measured over 36 time bins, with each bin being the average response to 320 repetitions of the same spatial frequency at that point in time. These plots illustrate the changes in spatial frequency tuning that occur over time. As can be seen, these frequency–time plots are tilted diagonally to the right, indicating that spatial frequency tuning shifts from low frequencies to high frequencies through time. Early in time, tuning is restricted to low frequencies and there are no responses to high frequencies. Later in time, tuning is primarily restricted to high frequencies because the responses to low frequencies are greatly diminished. This pattern of results corresponds to 1) the fact that the latency of the response to low spatial frequencies is shorter than the latency of the response to high frequencies and 2) the fact that the responses to all spatial frequencies are transient (cf. the transient responses illustrated in Figs. 1 and 2). For example, in Fig. 3A, at the early time intervals, the tuning is restricted to low frequencies because the responses to high frequencies are delayed and thus those responses are not yet contributing (cf. Fig. 2A); conversely, later in time, the tuning is primarily restricted to high frequencies because the responses to low frequencies have decayed and thus those responses are no longer contributing as strongly.
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Some cells show little or no latency shift
It is important to point out that the pattern of results illustrated in Figs. 1, 2, 3, 4, 5 is not representative of all of the neurons in the sample. Specifically, some of the neurons do not show any shift in the latency as a function of spatial frequency, whereas other neurons show only a minor shift. None of the neurons in this sample showed a statistically significant negative shift (see analysis given below). Figure 6 plots the responses of a neuron for which the latency of the response is essentially equivalent for the 12 different spatial frequencies (which in this case spanned a relatively large range: 2.6 octaves). As can be seen, for this neuron there is little, if any, latency or frequency shift. This observation holds for other cells as well.
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The spatial frequency shift through time can be quantified for the entire sample of neurons by measuring the difference between the peak spatial frequency at the beginning and at the end of the large-amplitude initial transient component (see METHODS). Figure 7A plots this quantitative index of the frequency shift (expressed in terms of octaves) for the entire sample of neurons. As can be seen, the frequency shift is small for some cells and large for others, with a virtual continuum in between. Across the sample as a whole, the frequency shift is approximately one octave (mean = 0.91; median = 0.99). The breakdown of the frequency shift across animal type and cell type is given in Table 1; the asterisks indicate pairwise comparisons that are statistically significant at the 0.05 level of confidence (using a t-test). Using a randomization test (see METHODS), we determined that the frequency shifts for 20 of the 64 neurons fall within the 95% confidence interval expected from chance alone (i.e., for these 20 neurons the shifts are not statistically significant). Of these 20 neurons, 3 are monkey simple cells, 4 are monkey complex cells, 7 are cat simple cells, and 6 are cat complex cells. The median frequency shift for these cells is 0.09 octaves.
The latency shift can be quantified for the entire sample of neurons by measuring the difference between the latency for the lowest and highest spatial frequencies. Figure 7B plots this quantitative index of the latency shift for the entire sample of neurons. As can be seen, the latency shift is small for some cells and large for others, with a virtual continuum in between. Across the sample as a whole, the latency shift is approximately 30 ms (mean = 32.8 ms; median = 31.8 ms). The breakdown of the latency shift across animal type and cell type is given in Table 1. Using a randomization test (see METHODS), we determined that the latency shifts for 5 of the 64 neurons are not statistically significant. Of these 5 neurons, 1 is a monkey complex cell, 2 are cat simple cells, and 2 are cat complex cells. The median latency shift for these cells is 5.3 ms.
Average PSTH as a function of spatial frequency
In a previous investigation (Albrecht et al. 2002), we measured the responses of visual cortex neurons as a function of contrast using a stimulus protocol that was similar to the protocol used in this investigation (i.e., the stimuli were stationary gratings that were flashed on for 200 ms and off for 300 ms). In that study, we found that the shape of the temporal response profile (the PSTH) was relatively invariant as a function of contrast. Specifically, we found that, across the sample as a whole, approximately 95% of the variation in the responses as a function of contrast could be accounted for by simply shifting and scaling the average PSTH.
Visual inspection of the responses shown in Fig. 1 within this report suggests that the temporal response profiles for these cells are qualitatively similar across spatial frequency. To assess this observation quantitatively for all of the neurons, we calculated the percentage of variation that could be accounted for by simply scaling and shifting the average PSTH. Across the sample as a whole, approximately 90% of the variation can be accounted for by shifting and scaling the average temporal response profile (median value = 92%; mean value = 89%). To assess whether the residual variation is random, as opposed to systematic, we performed a statistical analysis (see METHODS). The results of this analysis reveal that for 7 of the 64 neurons the residual variation is indeed statistically significant at the 99% confidence level. In other words, for these 7 neurons (all monkey complex cells), there are systematic changes in the shapes of the temporal response profiles as a function of spatial frequency. However, it is important to note that the magnitude of this systematic residual variation is relatively small: both the mean and the median value are approximately 3%. At the 95% confidence level, 14 of the 64 neurons (9 monkey complex cells and 5 cat complex cells) show systematic deviations from the average temporal response profile; the magnitude of the systematic deviation for these 14 neurons is about 2%.
These results suggest that the spatial frequency shifts that can be seen in Fig. 3 are a consequence of 1) the latency shift as a function of spatial frequency, coupled with 2) the transient nature of the invariant temporal response profile. Further, these results have an important implication: If the latency shift is taken into consideration and the PSTHs are aligned, then the shape of the spatial frequency tuning function is relatively invariant (cf. Figs. 2, 3, 6, 9, and 10 in Albrecht et al. 2002).
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The complex changes in the spatial frequency response function that occur through time could potentially provide information to subsequent neurons beyond what is contained in the total spike count, summed over the duration of a single fixation. Whether this additional information is extracted by subsequent neurons depends on how the responses are integrated. Therefore it is worth considering the consequences of several different temporal integration strategies that could, in principle, be used by subsequent neurons.
One strategy would be to integrate the responses of visual cortex neurons over relatively brief temporal intervals. Consider, for example, the effect of integrating over specific 10-ms time intervals for the cell illustrated in Fig. 3A. When the running 10-ms integration interval is centered at 40 ms, as opposed to 60 ms, the spatial frequency tuning is considerably different; the ranges of spatial frequencies covered are almost nonoverlapping. Further, the bandwidths of these tuning functions for these brief integration intervals would be only a fraction of the bandwidth for a long integration interval. In general, as the integration intervals become longer, these changes in spatial frequency tuning become smaller (although the responses become more reliable; see detection analysis, below).
Next, consider the strategy of integrating (summing action potentials) over intervals that begin at a fixed point in time: at stimulus onset, or at some point after stimulus onset. Here we consider the effects of increasing the duration of the summation interval on spatial frequency tuning. We take the beginning of the intervals to be the time at which the shortest latency responses begin. Figure 8 plots the spatial frequency tuning for progressively longer integration intervals, increasing in 1-ms steps, for the 6 neurons illustrated in Fig. 1. (Note that in this analysis the bin size for the PSTHs is 1 ms, not 10 ms.) The smooth curve through the data points, at each sequential integration interval, shows the best fit of a function that provides a good description of the spatial frequency tuning of visual cortex neurons (an asymmetrical Gabor function; see Geisler and Albrecht 1995, 1997). For example, to be even more specific, consider the cell shown in Fig. 8A. The first set of data points and the fitted curve at the bottom of the panel, show the spatial frequency tuning when the responses are integrated over only 1 ms, starting at 40 ms (see figure caption); the second set of data points and fitted curve, just above the lowermost one, show the spatial tuning when the responses are integrated over a total of 2 ms (i.e., 40 and 41 ms); the integration interval for the next set includes 40, 41, and 42 ms, for a total of 3 ms, and so forth. As the integration interval increases, the spatial frequency tuning is initially dynamic for very short integration intervals and then becomes relatively stable for longer, but still brief, integration intervals. For the first integration interval, with a duration of 1 ms, the peak spatial frequency is 0.89 cycles/deg, whereas for the integration interval with a duration of 20 ms, the peak frequency is 2.11 cycles/deg. After this point in time, the peak frequency changes very little; for example, when the integration process is extended to 200 ms (not shown in the figure), the peak frequency is 2.26 cycles/deg.
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Effect of the integration interval on detection performance
For most visual cortex neurons, a large fraction of the action potentials to stationary stimuli occurs during the initial transient component of the PSTH (cf. Fig. 1; see also Albrecht et al. 2002; Müller et al. 1999, 2001; Tolhurst et al. 1980). Given this observation, it is possible that reliable information could be transmitted to subsequent neurons early within a single 200-ms fixation interval. However, the reliability of the transmitted information cannot be determined from the PSTH alone because both the mean and the variance of the response change throughout the time course of the 200-ms interval (Albrecht et al. 2002; Frazor 2002; Müller et al. 1999, 2001). To assess the reliability of the transmitted information, we measured detection performance as a function of the duration of the integration interval using a signal detection analysis (see METHODS, as well as Albrecht et al. 2002; Geisler and Albrecht 1995, 1997; see also Müller et al. 2001; Tolhurst et al. 1983; for a general discussion within this context see Barlow 2003).
Figure 9 plots the signal-to-noise ratio (d') as a function of the duration of the integration interval, for the cells shown in Figs. 1, 2, 3, 4, 5. As can be seen, the detection performance improves quite rapidly as the integration interval increases and becomes quite reliable in a brief period of time. The results of this analysis for the sample as a whole are summarized in Table 2. As can be seen, the 75% correct detection threshold (d'= 1.0) is achieved within a relatively brief period of time after response onset, approximately 25 ms (mean = 24.7, median = 18.7, SE = 2.9). Further, 90% of the maximum performance is achieved approximately 54 ms after response onset for the sample of monkey neurons and 109 ms after response onset for the sample of cat neurons. This difference between cats and monkeys could potentially be related to the fact that the duration of saccades is shorter in monkeys (25–55 ms; Fuchs 1967) than that in cats (50–150 ms; Evinger and Fuchs 1978). Because of the longer duration of the saccades in cats, they are afforded a longer interval in which to assess the information contained within the previous fixation.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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), that incorporates known differences in the properties of LGN cells. Motion selectivity in depth
When an object approaches, the spatial frequency content of its retinal image will shift over time from higher frequencies toward lower frequencies. Thus if the latency of the response of a given neuron increases as a function of spatial frequency, then the neuron would produce a larger response to approaching stimuli than to either stationary or receding stimuli because the excitation from the higher frequencies and lower frequencies would overlap in time, for the approaching object. Given that the latency of the response for most visual cortex neurons does, in fact, increase as a function of spatial frequency (see Fig. 7B), one hypothesis to consider is that cortical neurons are selective for motion in depth (i.e., behave like looming detectors or approach detectors; Akase et al. 1998; Cynader and Regan 1978; Poggio and Fischer 1977; Toyama and Kozasa 1982; Wang and Frost 1992; for a review of the related literature see Sherk and Fowler 2001). For this hypothesis to be viable, the rates of frequency shift through time would have to match the rates of motion in depth that would be useful within a behavioral context. However, the mean rate of change for this sample of neurons is approximately 0.05 octaves/ms. This rate of change is too fast to afford an adaptive response because it is not reached until an object is so close to the eye that there would be <50 ms until impact. Nonetheless, several cells in the sample have rates of frequency shift that are substantially less than the mean. The cell with the slowest rate of change could respond to an approaching object when the time to impact is about 140 ms. This cell could possibly contribute to some behaviors such as the menace reflex, which occurs on the order of 80–100 ms (LeGrand 1957; Westheimer 1968). However, for the majority of cells in this sample, the spatial frequency shift is too rapid.
Coarse-to-fine processing
It has been suggested that for certain computational problems it may be valuable to use low-frequency information to constrain the analysis of high-frequency information (e.g., Anderson and Van Essen 1987; Marr and Poggio 1979; Menz and Freeman 2003; Nishihara 1984; Parker et al. 1997; however, see Smallman 1995). Given that the low spatial frequency information is available in most visual cortex neurons before the high-frequency information, one might entertain the hypothesis that the shift in spatial frequency selectivity from low to high could provide the input to a coarse-to-fine analysis.
Consider one potential class of mechanism that could provide input to a coarse-to-fine analysis. This class of mechanism would consist of separate populations of subsequent neurons that integrate the responses of striate neurons over brief sequential time intervals following stimulus onset, where each of the separate populations is responsible for coding a different time interval and hence a different spatial frequency band. A necessary component of such a mechanism would be some form of gating process that synchronizes the integration interval for each population with the onset of the stimulus. During saccadic inspection, this gating process could conceivably be based on a corollary discharge from the oculomotor mechanism that produces a saccadic eye movement (cf. Reich et al. 2001), or on the first wave of activity (e.g., the first population to respond sets the gating for the remaining populations).
This class of model is strongly constrained by the fact that the spatial frequency shifts of striate neurons are so rapid (30 ms; see Fig. 7). The greater the number of spatial frequency bands that are extracted, the smaller the required integration intervals (e.g., 2 spatial frequency bands would require integration intervals of 15 ms, 4 spatial frequency bands would require 7.5 ms); but, as the integration interval decreases, the signal-to-noise ratio of the information also decreases. Thus to obtain reliable information, larger populations of neurons would be required to code each spatial frequency band. Further, as the integration interval decreases, the temporal accuracy of the gating signal must increase. Given these constraints, it is uncertain whether this class of model is plausible. Finally, although not inconsistent with this class of model, it is worth noting that the well-established spatial frequency selectivity of individual striate neurons could provide robust input to a coarse-to-fine analysis.
Relationship to direction selectivity
Measurements of the spatiotemporal receptive fields of direction-selective cortical cells show that there is a systematic change in the preferred spatial phase through time (DeAngelis et al. 1993; Emerson et al. 1987; Hamilton et al. 1989; Palmer et al. 1991). The measurements in this report show that, for many cells, there is a systematic change in the preferred spatial frequency through time. It is worth considering whether these two phenomena are related.
One way to produce a direction-selective neuron is to combine neurons with different response latencies and spatial positions (Adelson and Bergen 1985; Barlow and Levick 1965; Reichardt 1961; Watson and Ahumada 1985). Thus the spatial frequency shifts reported here might be an epiphenomenon of the mechanisms that produce direction selectivity, if, in direction-selective cells, there is a trend for the spatially displaced inputs with longer latencies to be tuned to higher spatial frequencies. To assess this potential relationship within this sample of neurons, we measured the correlation between the magnitude of the latency shift and the magnitude of the direction selectivity. (Note that the direction selectivity was determined during the preliminary measurements; see METHODS.) We found a small positive correlation that was not significant. We also note that the dynamic changes in phase selectivity that determine the direction selectivity of cortical neurons extend over a time interval of 100–200 ms (DeAngelis et al. 1993; Hamilton et al. 1989; Palmer et al. 1991), which is considerably longer than the dynamic changes in spatial frequency selectivity reported here (see Fig. 7). Therefore it seems unlikely that the dynamic changes in the spatial frequency response function result from the mechanisms that produce direction selectivity.
Latency of the response as a code for spatial frequency
We have known for many decades that the latency of the response of visual cortex neurons decreases as the contrast increases (e.g., Albrecht 1978, 1995; Carandini and Heeger 1994; Carandini et al. 1997; Dean and Tolhurst 1986; Reid et al. 1992). Further, it has been suggested that this systematic relationship between latency and contrast could potentially be used to signal the magnitude of the contrast (Gawne 1999; Gawne et al. 1996; Reich et al. 2001; Richmond et al. 1997). Comparable logic could be applied to the variation in the latency of the response as a function of spatial frequency that is described within this report. Note, however, that contrast and spatial frequency information would be confounded. Further, it is important to keep in mind that response latency is not only influenced by contrast and spatial frequency (see Figs. 1, 2, 3, 5, and 7B). Response latency is also influenced by many other stimulus attributes: for example, color (Cottaris and De Valois 1998), motion (Hamilton et al. 1989), luminance (Petersen et al. 2001), and texture (Rossi et al. 2001). Presumably, the greater the number of stimulus attributes that affect response latency, the more sophisticated the neural processing would have to be to interpret the stimulus attributes that are producing the latency differences. This reduces the likelihood that the systematic relationship between the latency of the response and the spatial frequency plays an important role in coding spatial frequency.
Latency of the response as a function of spatial frequency and behavioral response times
We have known for several decades that behavioral response times in humans increase as a function of spatial frequency (e.g., Breitmeyer 1975; Breitmeyer et al. 1981; Harwerth and Levi 1978; Tolhurst 1975). In this study we found that, for most cells, response latency increases as a function of spatial frequency. In a separate study (DG Albrecht, unpublished observations) we found that there is a significant positive correlation across cells between response latency and peak spatial frequency. (In that study, the latency to the peak of the response was measured using a stationary flashed grating and the peak spatial frequency was measured using drifting gratings.) For the monkey sample, the correlation was 0.35 (P < 0.01; n = 49); for the cat sample, the correlation was 0.41 (P < 0.01; n = 65). It is possible that both of these properties of cortical neurons combine to produce the observed behavioral effects; however, without the variation across cells, the variation within cells would largely cancel.3 For the sample of neurons recorded from within the monkey visual cortex, the magnitude of the variation across cells is approximately twice the magnitude of the human behavioral effects (Rudd 1988). Therefore it is possible that the responses of striate visual cortex neurons contribute to the systematic relationship between human behavioral response times and spatial frequency.
Other studies of the temporal dynamics of spatial frequency tuning
Bredfeldt and Ringach (2002) measured the development of spatial frequency tuning during the initial onset of the response, on a fine time scale, using a reverse correlation procedure in the frequency domain. Similar to the results reported here, they found that the spatial frequency tuning changed over the time course of the response and that the preferred spatial frequency shifted from lower frequencies to higher frequencies. In contrast to the results reported here, they found that the onset latency of the response was the same across the dimension of spatial frequency. Thus at the onset of the responses, the tuning was quite broad, spanning both low frequencies and high frequencies: As they pointed out, the tuning was very similar to that of LGN neurons.
Mazer et al. (2002) also measured the development of spatial frequency tuning using a reverse-correlation procedure. Similar to the results reported here, they found that the latency of the response increased as a function of spatial frequency. In contrast to the results reported here, and the results reported by Bredfeldt and Ringach (2002), they found that the spatial frequency tuning was largely separable through time (i.e., they found that the peak spatial frequency was relatively invariant throughout the time course of the response). However, as they pointed out, to the extent that the latency of the response changes as a function of spatial frequency, there must be some component of the tuning that is changing through time.
There are many possible factors that might help account for the differences in the results reported here and the results reported in the previous studies. Within this section, we consider four factors (see also Simoncelli et al. 2004).
First, the previous studies used a reverse-correlation procedure to measure the spatial frequency dynamics. They used this procedure to measure the linear component of the responses (see footnote 2). It is well documented that striate cortex neurons possess many pronounced nonlinearities (e.g., response rectification, response expansion, contrast gain control, contrast adaptation, response refractory period, response saturation, response supersaturation, and so forth; for a current general review of this literature see Albrecht et al. 2003). To the extent that one or more of the factors that produce the temporal dynamics of the spatial frequency response function are a consequence of these, or other, nonlinearities, it is difficult to directly compare the measurements of the linear component with the measurements in this report, which contain not only the linear component, but also the nonlinear components.
Second, in the reverse-correlation studies, the duration for which any single spatial frequency was static on the retina was relatively short (20–30 ms), whereas in the present study, the duration is relatively long (200 ms). It is possible that there could be differences in rapid pattern adaptation (Müller et al. 1999, 2001) as well as rapid local light adaptation (Crawford 1947; Saito and Fukada 1975; for a review see Shapley and Enroth-Cugell 1984). For example, using the psychophysical probe-flash paradigm to investigate light adaptation mechanisms in human subjects, it has been shown that much of the multiplicative gain change occurs within 50 ms after the onset of the adapting field, and although it decays more slowly at offset, it is well under way within 200 ms (Adelson 1982; Geisler 1981; Hayhoe et al. 1987; for a general review see Hood 1998).
Third, in the reverse-correlation studies, the spatial frequencies were presented contiguously, whereas in the present study, there is a 300-ms ISI. The logic for the 300-ms ISI is described at length in the METHODS section; in brief, the ISI is designed to minimize potential interactions (as a consequence of latency differences and OFF responses) between the sequential stimuli. To the extent that there are spatial and temporal nonlinearities involved in these responses, even the estimate of linear component measured under these two stimulus conditions might not be the same.
Finally, in the present set of measurements, the length of the integration intervals used in the analysis of the measured responses is shorter. As the integration interval is increased, any spatial frequency shifts will be diminished (cf. Fig. 8, as well as the discussion in Mazer et al. 2002).
Possible mechanisms for spatial frequency dynamics
It has been known for many decades that, in comparison to the LGN cell inputs, neurons in the visual cortex can be quite selective for orientation, spatial frequency, and other stimulus qualities (see references in the INTRODUCTION). Hubel and Wiesel originally proposed that the receptive field properties of striate cortex neurons could be produced by simply summing the responses of a row of LGN cells whose receptive fields are aligned (see Fig. 19 in Hubel and Wiesel 1962). Since then, many different structural, biophysical mechanisms have been proposed to account for the increased stimulus selectivity that takes place from the LGN to the cortex. Some of the structural, biophysical mechanisms that have been proposed are: expansive voltage-spike transduction, noisy membrane potential, recurrent excitation, intracortical inhibition, correlation-based inhibition, synaptic depression, nonspecific suppression, shunting inhibition, tonic hyperpolarization, strong push–pull inhibition, and changes in membrane conductance. Some of these mechanisms rely on feedforward inputs, some rely on feedback inputs, and others rely on local, as well as far-reaching, lateral interconnectivity. For recent discussions and reviews of this substantive and growing body of literature, see Abbott et al. (1997), Adorjan et al. (1999), Albrecht et al. (2003), Anderson et al. (2000a,b), Ben-Yishai et al. (1995), Carandini and Ringach (1997), Carandini et al. (1999), Chance et al. (1998), Douglas et al. (1995), Ferster and Miller (2000), Gilbert et al. (1990), Hirsch et al. (1998), Hupe et al. (2001), Kayser et al. (2001), Lamme et al. (1998), Miller and Troyer (2002), Murthy and Humphrey (1999), Nelson et al. (1994), Pugh et al. (2000), Shu et al. (2003), Somers et al. (1995), Stetter et al. (2000), Troyer et al. (1998), and Worgotter and Koch (1991).
The dynamic properties of the spatial frequency response function could be the result of some or all of these different types of structural, biophysical mechanisms. Here we describe a simple feedforward structural model, similar to the one proposed by Hubel and Wiesel, that incorporates some of the known differences between the magnocellular (M) and parvocellular (P) neurons in the LGN. Specifically, M cells respond about 10–15 ms faster than P cells and have receptive fields that are about twice the size of the receptive fields of P cells (see references given in the APPENDIX). The model relies on convergent input from the M and P cells, which was first proposed by Mazer et al. (2002) to potentially account for the latency differences that they reported as a function of spatial frequency in striate neurons.
Although it is not yet known whether all V1 neurons receive input from both M cells and P cells, it is becoming increasingly clear that many V1 neurons do receive contributions from both M cells and P cells (e.g., Allison et al. 2000; Nealey and Maunsell 1994; Vidyasagar et al. 2002; for a discussion and review of this developing line of research, see Maunsell 1992; Merigan and Maunsell 1993). Further, it is important to point out that this class of model is not limited to mixing M and P inputs. The fundamental requirement is that the inputs to a cortical cell must have different response latencies that covary with spatial frequency. Nealey and Maunsell (1994) report that there are latency variations within each class of LGN cell. Finally, note that the properties of X and Y cells within the cat LGN are somewhat similar, in the context of this class of model, to those of M and P cells in the monkey LGN. Specifically, Y cell receptive fields are approximately twice the size of X cell receptive fields (Stone et al. 1979) and the response latencies of Y cells are approximately 10–15 ms shorter (Sestokas and Lehmkuhle 1986).
In this simple demonstration, a cortical cell receptive field is constructed by summing ON-center and OFF-center M and P cells in the fashion shown in the top portion of Fig. 10. The fundamental parameters of this model that are responsible for producing the frequency and latency shifts are constrained (i.e., not allowed to vary), using physiological measurements taken from previous reports. (See the APPENDIX for details.) The bottom portion of Fig. 10 shows the predictions (solid curves) of this model for the responses (dots) of the monkey cell shown in Fig. 2C; the PSTHs for 6 different spatial frequencies are illustrated. The arrow in each panel indicates the same point in time; specifically, the latency to the peak of the response for the lowest spatial frequency shown. As can be seen, the model qualitatively captures the important aspects of the data: 1) as the spatial frequency increases, the latency of the response increases, and 2) the shape of the temporal response profile is relatively invariant. Note that this cell is representative of the sample as a whole because the magnitudes of the spatial frequency shift and latency shift are both approximately equal to the median values for this sample (cf. Figs. 4, 5, and 7). It remains unknown whether the measured variation within and between the M and P cells, and the X and Y cells, could account for the measured variation in the spatial frequency selectivity through time for all of the cells.
It is interesting to note that these predictions are robust to variations in many of the properties of this class of model (e.g., the ratio of center to surround radius, the relative strength of center and surround, and so forth). On the one hand, this adds to the plausibility of this class of model. On the other hand, it demonstrates that different versions of the model could produce comparable results. Indeed, there are undoubtedly many different types of structural models that could also produce comparable results. With these caveats in mind, we note that the goal of this exercise was not to develop a detailed structural model, but rather to explore whether the temporal dynamics of the spatial frequency response function could potentially be produced in a parsimonious fashion using realistic combinations of LGN inputs.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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Here we describe a simple feedforward structural model, similar to the one proposed by Hubel and Wiesel (1962; their Fig. 19), that incorporates some of the known differences between the M cells and P cells in the LGN. The model is illustrated pictorially in the top portion of Fig. 10. The spatial parameters of the M and P cells are required to be consistent with the measurements reported by Croner and Kaplan (1995), which are similar to (but more extensive than) earlier measurements (e.g., Crook et al. 1988; DeMonasterio and Gouras 1975; Derrington and Lennie 1984; Hubel and Wiesel 1960; Irvin et al. 1993). The spatial properties of the M and P receptive fields are represented by a difference of Gaussians. The temporal properties are represented by 2 half-Gaussians with different SDs before and after the peak, along with an asymptote after the peak. We assume that the ON and OFF inputs are equal in strength, and that the M and P inputs are also equal in strength.
As noted in the text, the fundamental parameters of the model that are responsible for producing the frequency and latency shifts are constrained, and not free to vary, using physiological measurements taken from previous reports from other laboratories. Specifically, the ratio of the diameters of the M and P centers is set to 2.5, the ratio of the center to surround strength is set to 2.0, the SD of the M center size is set to 0.1°, and the ratio of the diameters of the center and surround is set to 7.2 (Croner and Kaplan 1995). The difference in the M and P cell latencies is set to 11.5 ms (Maunsell et al. 1999). The 4 free parameters in the model that describe the temporal response profile (along with the optimized values in parentheses) are: the latency to the peak of the response for the M cells (56.7 ms), the rise time SD of the PSTH (7.6 ms), the decay time SD of the PSTH (6.4 ms), and the fraction of the maximum response at which the PSTH asymptotes after the peak (0.41). Note that the shape of the temporal response profile is the same for both the M and P cells; the only difference between the profiles was that, for the P cells, the response was delayed using the measured latency difference reported by Maunsell et al. (1999). The free parameter that describes the peak spatial frequency is the center-to-center separation between cells of a given type (5.5 SDs).
Using an even-symmetric receptive field, the model was fitted to the measured responses for the cell shown in Fig. 2C (using a leastsquares procedure). The results are illustrated in the bottom portion of Fig. 10.
A descriptive model
Descriptive function models of the responses of visual cortex neurons can be useful for a variety of different applications, including 1) testing various competing hypotheses; 2) assessing the performance of neurons in tasks such as discrimination or identification; and 3) developing functional models, in concert with descriptive functions for other dimensions, to predict the responses of single neurons or populations of neurons under a wide variety of stimulus situations (for further discussion of this and related issues, see Albrecht et al. 2002, 2003). Here we present a simple, atheoretical, mathematical model, which provides a reasonable description of the responses reported here. [To help visualize this model, see Fig. 17 in Albrecht et al. (2002).] Although the model does not capture all of the variation in the responses as a function of spatial frequency and time, it does capture more than 80% of the variation (the median value is 85%).
First, consider quantifying the shape of the PSTH. It has been demonstrated that a simple Gaussian function provides a good description of the initial transient rise and fall of the PSTH (Müller et al. 2001). However, many cells have a sustained plateau that is not captured by a simple Gaussian. We have shown that this sustained portion can be quantified by incorporating an additional "
Gaussian" (Albrecht et al. 2002). The second Gaussian is added to the whole Gaussian after the peak of the temporal response
 | (A1) |
Equation A1 describes the relative response as a function of time rt, where ba is the Gaussian half-bandwidth, bb is the " Gaussian" half-bandwidth, 
gives the relative magnitude of the Gaussian to " Gaussian," t is time, and 
u(u) is the latency to the peak of the response (which may depend on the spatial frequency u). Although this descriptive function (which we term a "1 Gaussian") will not incorporate all of the diversity from cell to cell (e.g., those cells with secondary oscillations, and so forth), it provides a good description for most cells. It would, of course, be possible to incorporate additional components to describe the more unusual characteristics of some cells; however, this would increase the number of free parameters and the complexity of the overall resulting descriptive function.
As in the main body of the report, we define latency to be the latency to the peak of the response and not to the initial onset of the response. Note that because the initial rise time in the responses to the stationary gratings is steep for most cells, time 0 (at the onset of the stimulus) is several SDs removed from the peak of the response, and thus the fitted value of the Gaussian at rt(0) is essentially zero.
Second, consider quantifying the shape of the spatial frequency response function. For most cells, the relative response as a function of spatial frequency ru(u) can be described using an asymmetrical Gabor equation (Geisler and Albrecht 1997)
 | (A2) |
), one could use the term asymmetrical Gaussian as well; we felt the term Gabor was more appropriate because we were characterizing responses in the Fourier domain.
Third, consider quantifying the latency to the peak of the response as a function of spatial frequency u(u). For most cells the latency can be described using a power function plus a vertical offset (for the shortest latency)
 | (A3) |
 | (A4) |
Most of the residual variation that is not captured by the descriptive function model is most probably a consequence of one, or more, of the following factors: 1) None of the component descriptive functions captures all of the variation in the responses along a given dimension; for example, the 1 Gaussian does not capture the secondary (and tertiary) oscillations in the temporal response profile (cf. Figs. 4, 5, 6, and 18 in Albrecht et al. 2002). 2) The responses across dimensions are not completely independent; for example, there are some variations in the shape of the temporal response profile across spatial frequency (see the section in RESULTS, Average PSTH as a function of spatial frequency, and METHODS). 3) Because the variability of cortical cells is systematically related to the mean rate of firing, the overall maximum firing rate of a cell can become an important factor (e.g., Geisler and Albrecht 1997; Geisler et al. 1991). 4) The number of repeated presentations of each stimulus configuration is limited. 5) There is random/stochastic variability that stems from the inherent response variability of the particular cell. To determine (with a high degree of confidence) whether the residual variation is systematic, as opposed to stochastic, it is necessary to perform the type of statistical analysis described in METHODS (see also footnote 1 in Albrecht et al. 2002).
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ACKNOWLEDGMENTS |
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This research was supported by the National Eye Institute Grant EY-02688 and by the University of Texas.
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FOOTNOTES |
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1 If there is very little low-frequency attenuation, then the response to the 200-ms step function should be considerably more sustained than what is observed for the majority of neurons. 
2 Over the past several decades, reverse-correlation methodology has been used to measure the linear component of the responses of sensory neurons (Anzai et al. 1999; Chichilnisky 2001; DeAngelis et al. 1993; de Boer and Kuyper 1968; Eggermont et al. 1983; Emerson et al. 1989; Jones and Palmer 1987; Marmarelis and Marmarelis 1978; Mazer et al. 2002; McLean et al. 1994; Palmer et al. 1991; Reid et al. 1997; Ringach et al. 1997; for recent reviews, see Nykamp and Ringach 2002; Ringach et al. 1997; Simoncelli et al. 2004).
3 Consider 2 cells with different preferred spatial frequencies, yet overlapping spatial tuning. These two cells would respond to the same spatial frequency at different times. This is because the given spatial frequency would be high for the cell that prefers lower spatial frequencies, and low for the cell that prefers higher spatial frequencies; the latency differences would cancel. Thus without the variation in latency between cells, the average response time would not vary as a function of spatial frequency, except perhaps at the boundaries.
Address reprint requests and other correspondence to D. G. Albrecht (E-mail: albrecht{at}psy.utexas.edu).
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40 separate occasions, in a random fashion. Responses at a given spatial frequency were averaged across the repetitions and the spatial phases. A: responses measured for a neuron recorded from within the monkey visual cortex: poststimulus time histograms (PSTHs) are plotted for 12 different spatial frequencies: 1.17, 1.24, 1.4, 1.56, 1.87, 2.33, 2.8, 3.11, 3.5, 3.73, 4.67, and 5.6 cycles/deg. Note that for each PSTH there is a large magnitude initial transient component followed by a brief trough and a subsequent return to a smaller-magnitude sustained component. Further, note that the shapes of the PSTHs are qualitatively similar across the different spatial frequencies; however, as spatial frequency changes, the PSTHs appear to scale vertically and shift horizontally. C and E: responses of 2 additional monkey neurons. The same basic pattern of results is evident. The spatial frequencies presented to the cell in C were: 0.67, 0.8, 1.0, 1.2, 1.33, 1.5, 1.6, 2.0, 2.4, 2.67, 3.0, and 4.0 cycles/deg. Spatial frequencies presented to the cell in E were: 0.6, 0.67, 0.8, 1.0, 1.07, 1.20, 1.33, 1.6, 2.0, 2.4, 2.67, and 3.0 cycles/deg. B, D, and F: responses for 3 neurons recorded from within the cat visual cortex. Once again, the shapes of the PSTHs for each cell appear to be relatively invariant and, although it is somewhat difficult to visualize, as spatial frequency changes, the PSTHs appear to scale vertically and shift horizontally. Spatial frequencies presented to the cell in B were: 0.14, 0.18, 0.2, 0.22, 0.26, 0.34, 0.36, 0.4, 0.44, 0.54, 0.66, and 0.80 cycles/deg. Spatial frequencies presented to the cell in D were: 0.34, 0.36, 0.4, 0.44, 0.54, 0.66, 0.8, 0.88, 1.0, 1.06, 1.34, and 1.6 cycles/deg. Spatial frequencies presented to the cell in F were: 0.66, 0.8, 0.88, 1.0, 1.06, 1.34, 1.6, 1.78, 2.0, 2.66, 3.2, and 4.0 cycles/deg.
