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1Departments of Biomedical and 2Electrical Engineering, Vanderbilt University, Nashville, Tennessee 37235
Submitted 25 August 2003; accepted in final form 7 November 2003
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ABSTRACT |
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INTRODUCTION |
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,b). However, the unconstrained analysis (type analysis; see Johnson et al. 2001) that we used in these previous studies makes it difficult to identify definitively the specific aspects of the response that contribute to cooperative encoding of orientation information. What can be concluded is that there is a link between the temporal structure of responses and synchronized transmission of visual information. The involvement of synchrony derives from the sensitivity of type analysis results to changes in processing parameters (bin size and discharge history), as well as our analysis of synchrony [using the methods of Aertsen et al. (1989) and Gerstein et al. (1985)] together with previous studies in temporal coding (Eckhorn et al. 1988; Gray et al. 1989; Richmond and Optican 1987; Rieke et al. 1997; Snider et al. 1998; Victor and Purpura 1996).
Type analysis uses the information-theoretic measurement of KL distance (Johnson et al. 2001; Samonds et al. 2003a,2003b) to quantify the difference between two neural responses. The KL distance makes almost no assumptions about the nature of the code (e.g., rate, time, synchrony), is based on classification theory, and can be interpreted only as an indicator of the reduction in error for an optimal probabilistic classifier. Alternatively, response differences can be quantified using more constrained "distances" using cost-based metrics that do make some assumptions, e.g., whether the cortical code is based on spike counts, spike times, or spike intervals (Victor and Purpura 1996). The cost-based metrics can in turn be quantified for response discrimination by measuring the information provided by the distances (Victor and Purpura 1996).
In this study, we reanalyze single-cell responses (24 complex cells from cat area 17) from Samonds et al. (2003a) using the methods described by Victor and Purpura (1996). Our results reveal that the spike timing and spike interval information is much more precise, and that the information expressed by these properties is proportionally more substantial, for fine angle discrimination. The results of the analysis also confirm that there are clear differences in the coding of information representing fine (<10°) and coarse (>10°) differences in orientation.
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METHODS |
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The details of the physiological preparation and the recording and stimulation protocol are described in detail in Samonds et al. (2003a). Experimental procedures were performed under the guidelines established by the American Physiological Society and Vanderbilt University's Animal Care and Use Committee. Recordings in area centralis of area 17 were made in 7 adult cats anesthetized with Propofol and N20 and paralyzed with Pavulon. Fourteen cells were recorded in 6 cats with a single tungsten-in-glass microelectrode (Levick 1972). Multiunit recordings were resolved using waveform classification (Snider and Bonds 1998) to yield 7 cell pairs. Ten of the cells (20 pairs) were recorded simultaneously in one additional cat using a Bionics 5 x 5 multielectrode array. The reason there are more pairs (n = 27) in Samonds et al. (2003a) than single cells (n = 24) in this study is that we used 2 groups of 5 associated cells from the multi-electrode recordings to construct 2 groups of 10 pairs (i.e., all cells were included in this study). Visual stimuli consisted of 2-s drifting sinusoid gratings presented in a circular aperture with a 21-in. Sony Trinitron monitor (frame rate of 120 Hz, mean luminance of 73 cd/m2). The diameter of the grating ranged from 4 to 16° (mean = 9°) for the single-electrode recordings and was 10° for the multielectrode recording. The contrast was 50% for all experiments. Spatial and temporal frequencies were optimized (highest firing rate) for the single-electrode recordings and set at 0.5 cycles per degree and 2.0 Hz, respectively, for the multielectrode recording. The responses to the preferred orientation, along with fine (<10°) and coarse (>10°) variations from the preferred orientation, were collected for each cell over 200-560 stimulus repetitions. The variations for the single-electrode recordings were 3, 7, 12, 18, 25, and 33° from the preferred orientation, whereas the multielectrode orientation variations were 2° increments over a 30° range around the preferred orientation.
Single-unit metrics
We quantify the information about differences in orientation from the spike count, spike arrival times, and spike-to-spike intervals using metric-space analysis, described in detail by Victor and Purpura (1996) (see also Aronov et al. 2003; Victor and Purpura 1997, 1998). Metric-space analysis starts by defining the nature of the neural code (e.g., count, timing, intervals) and calculating a cost-based "distance" between two responses. The advantage gained from the assumption about the nature of the code is that the response is not broken down into discrete bins and is therefore not under the same sampling constraints imposed by vector-space calculations (e.g., KL distance). This allows us to examine temporal dependencies over much longer intervals (within reasonable expectations of data collection). Metric-space analysis also clarifies what aspects of the response might have contributed to the KL distance, as well as the relative amount contributed.
The nature of the neural code is determined by the particular cost-based metric selected for calculation of the distance between two responses. The metric can be the number of spikes (Dcount), the arrival time of spikes (Dspike), or the interval time between two spikes (Dinterval). The distance is determined by finding the minimum total cost to transform one spike train into another spike train following a path that is established by a set of elementary steps (Victor and Purpura 1996). The cost between two responses is determined by first the difference in the number of spikes (deleting or inserting a spike) and then by either the shift in spike times, or a change in the interval. The cost calculated by a shift in time or change in an interval is scaled by a "cost per unit time" variable q. The total cost of translating one spike train into another is the total deletions and insertions plus the total time shifted (or interval time lengthened/shortened) multiplied by q. Conceptually, 1/q represents the temporal precision of the particular metric and a value of q = 0 for Dspike[q] or Dinterval[q] is equivalent to Dcount. We tested distances for q values of 0, 1, 2, 4, 8, 16, 32, 48, 64, 96, 128, 256, 512, 768, and 1024 s-1.
A stimulus-dependent clustering method (Victor and Purpura 1996) is then used between stimulus trials and across a stimulus set to create a confusion matrix and calculate the transmitted information. The information signifies how reliable the metric distances are with respect to distinguishing stimuli. We use a stimulus set of 2 (the preferred orientation and a small or large perturbation of orientation) to calculate the information H(q) about fine and coarse orientation discrimination to make a comparison with the results from KL distance measures from type analysis (Samonds et al. 2003). Because the metric-space analysis requires much less data, we only used 20 stimulus repetitions from the available data set. We use the method of random reassignment of stimulus classes to estimate bias (mean of 200 samples) (Panzeri and Treves 1996).
We produce 95% confidence intervals for each information estimate using the variance of the estimated bias. However, these intervals should not actually be interpreted as confidence limits of the information estimates. The intervals are actually limits on the consistency of the random data and are likely larger than the estimation confidence limits, especially when considering estimates that approach the 1-bit ceiling (maximum possible information for 2 stimuli).
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RESULTS |
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) using metric-space analysis (Victor and Purpura 1996) to define information differences between responses to two orientations. We first determined the optimal q value (temporal resolution with the maximum information) for the spike time metric, followed by the optimal q value for the spike interval metric. We then made comparisons of the amount of information provided by the spike time and interval metric (at the optimal q) to the information provided by only the number of spikes. Finally, we explored how the information varies across time (using the first 500 ms, 1 s, and the entire 2 s of the responses) to determine whether the information provided by the temporal structure of the responses is a transient or sustained property. Spike time metric
Figure 1, A and B are representative examples of the function of information versus the spike timing scaling parameter q. The information for q = 0 is the amount of information available from the total spike count of the responses. For values of q > 0, the timing of the spikes influences the distance metric relative to orientation, so the information represents what is provided by the temporal structure of the responses. As q becomes larger, the temporal resolution (1/q) of the metric becomes finer, reflecting the change in available information as the temporal code becomes more precise. Each data point in Fig. 1, A and B represents an information measurement using a particular cost-scaling value (q) when calculating the spike timing distance. Lines are cubic splines fitted to the estimates and error bars are 95% confidence intervals for information estimates.
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For coarse orientation discriminations (see gray data points), the information is effectively represented in the number of spikes (or rate) with only minor enhancement from the temporal structure that arises from very coarse periodic modulations in the rate. We reason that the temporal coding is from rate modulations because the optimal q value for Fig. 1A is 4 s-1 and for Fig. 1B is 3 s-1, which both fall into the range of temporal frequencies (1-8 Hz) used for the drifting gratings.
To provide a clearer picture of the temporal structure that produces the information in Fig. 1, we produced a raster plot (Fig. 2A) and poststimulus time (PST) histogram (Fig. 2B) of the data we used to calculate the information in Fig. 1A for a small orientation difference. The raster plot and PST histogram reemphasize the impossibility (i.e., zero information) of distinguishing the responses solely on the basis of the number of spikes. We are nevertheless able to observe a consistent difference between the responses to the preferred orientation and a 4° displacement from the preferred orientation. The spikes from the response to the preferred orientation appear to arrive slightly (about 35 ms) later than the spikes from the response of the nonoptimal orientation. Even casual observation of the raster plot (supporting the quantitative measurement of 0.15 bits in Fig. 1A) demonstrates that temporally coordinated changes in firing make it possible to distinguish the responses for this particular cell to a very small difference in orientation.
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Figure 3 is the population histogram of the optimal q values when measuring the information using the spike time metric distance. Twenty-three out of the 24 cells had information in the timing of the spikes in addition to information from the spike count. The temporal code for fine discrimination (black bars) of orientation is much more precise, with an average optimal q = 109.0 s-1 or 9.2 ± 7.2 ms temporal resolution, than for coarse discrimination (white bars) of orientation, with an average optimal q = 14.5 s-1 or 68.9 ± 43.5 ms temporal resolution.
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Figure 4, A and B are representative examples of the function of information versus q when testing the spike interval metric. Again, the information for q = 0 is the amount of information available from the spike count of the responses. The spike interval metric is similar to the spike time metric in that 1/q conceptually represents the temporal resolution of the metric. The difference between the metrics is that the interval metric compares the time between sequential pairs of spikes (a relative measurement of time), whereas the spike time metric measures the absolute spike time. As in the previous section, each point in Fig. 4, A and B is the information estimate (debiased with 95% confidence error bars) for a particular q, and the lines are cubic splines fitted for smoothing.
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One form of interval-based coding that has been linked to orientation is bursting (Cattaneo et al. 1981a,1981b; DeBusk et al. 1997). Bursts are defined as groups of 2 or more spikes with intervals 
8 ms (DeBusk et al. 1997). We compared the orientation tuning of bursts against that for all spikes (i.e., firing rate) to demonstrate how bursting might influence the information we measure with the interval metric. Figure 5A is an example of how the tuning seen with bursts only is refined over that from firing rate, as described previously by Cattaneo et al. (1981a). On average, the half-height bandwidth of those spikes in bursts (25.0 ± 7.2°) is 28.1 ± 10.3% narrower than the bandwidth when all spikes are included (35.1 ± 10.6°). The peak firing rates of our sample of cells ranged from 9 to 74 sps (mean = 32.7 ± 20.8 sps). In Fig. 5B, the interspike interval histogram illustrates the predominance of bursts in the responses (on average, 24.9 ± 13.2% of spikes were found in bursts for n = 24 cells) and how the magnitude of the "bursts peak" (centered at 3 ms) varies with a 6° difference in orientation, similar to the results demonstrated by Cattaneo et al. (1981a,b) and DeBusk et al. (1997). Orientation has been found to modulate the number of spikes contained in a burst, as opposed to the number of bursts (DeBusk et al. 1997).
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In the previous two sections, we examined how the q parameter (or temporal resolution) varied the amount of information that was available from spike time and spike interval metric-space analysis. This reveals the precision of temporal coding (timing or intervals) for discrimination of fine and coarse angular differences. Here we examine exactly how much information is provided by the temporal coding and compare these measurements to the quantity of information provided by the spike count.
Figure 7 shows the average amount of information provided by the 2 temporally based metrics, along with the rate information (count), for small (left) and large (right) differences in orientation. The temporal structure (based on either temporal metric) provides more information in both cases, with the spike time metric providing slightly more information than the spike interval metric. The timing information becomes more critical in the case of fine discrimination of orientation, where on average almost no information is provided by the spike count. For discrimination of large angular differences, an average of 0.24 bits of information is provided by the spike count. The small amount of information from the spike timing (0.24 bits) and intervals (0.09 bits) for fine discrimination should still be considered substantial when considering that we are only comparing 2 stimuli (maximum possible information = 1 bit).
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In the Spike time metric section, we illustrated that the timing information was a result of orientation-dependent phase shifts in the responses. However, only in 5/24 cases were we clearly able to demonstrate that the phase shift occurred throughout the response and was not simply a result of changes in the initial response latency with respect to onset of the stimulus. Heller et al. (1995) found that the first spike time (i.e., transient latency) accounted for 35% of the total information in V1 cells. Examination of the PST histograms for 20/24 of our cells (mentioned above) suggests that the information gained from spike timing and phase shifts can occur throughout the response and is not simply a consequence of the variation of initial response latency.
To address more definitively whether the information we measured resulted from latency differences (i.e., a transient effect) or reflected a sustained temporal code in all 24 cells, we measured the information with the 3 metrics using only the first 500 ms, only the first 1 s, and the entire 2 s of the response (which in effect represents a function of the accumulated information). We were able to make reliable information estimates for all 3 durations for 21/24 cells. For 16/21 cells, the information increased for each successively longer duration. The 5 cells that appeared to contain all of the information in the initial 500 ms included responses that showed no modulation or only moderate modulation. However, 11/16 complex cells with moderate or no noticeable modulation still showed persistent temporal coding. The average results for all 3 metric information measurements are plotted in Fig. 9.
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DISCUSSION |
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) suggested that the information available for discrimination of fine differences in orientation was not a result of the spike rate, but some form of temporal and cooperative coding. In the case of larger orientation differences, the KL distance analysis did suggest that the responses could be discriminated on the basis of the spike rate (e.g., a random Poisson distribution of spikes). By varying parameters of the KL distance (bin width and discharge history), we determined how temporal and spatial dependencies contributed to fine orientation discrimination and suggested that the temporal structure of responses also played a significant role (Samonds et al. 2003a,b). The bin width and incorporation of discharge history (temporal dependencies) varied the KL distance (Samonds et al. 2003a), and the cooperation increased with increasing population size (spatial dependencies) (Samonds et al. 2003b). However, we did not separate and determine the extent to which features such as the rate and temporal structure of individual responses supported response discrimination. The metric-space analysis of the data presented here confirms that very little information is provided by the spike count for fine discrimination of orientation, but that coarse discriminations can be accomplished using only the spike count. The metric-space analysis more clearly confirmed that temporal structure, in the form of precise spike timing or spike-to-spike intervals, provides substantial information that can be used for discriminating small angle differences. The temporal structure contributes proportionally very little to discriminating larger angle differences, with the exception of very slow rate modulations. There is also agreement between the KL distance analysis and the metric-space analysis with respect to the temporal scale (about 4-12 ms and about 2-30 ms, respectively) of the coordination of responses for fine angle discrimination.
Although both methods define a "distance" between responses and produce results based on information theory, the measurements start from nearly opposite ends of the spectrum. KL distance makes almost no assumptions on the nature of the code, is a vector-space measurement (i.e., based on bins), and is calculated between 2 estimated probability distributions (Johnson et al. 2001). Metric distances assume the underlying code, do not need to overcome high-dimensional uncertainties, and are calculated between single responses (Victor and Purpura 1996). These very considerable differences make it all the more reassuring that both analytical methods yield qualitatively and, in some cases, quantitatively similar results.
Orientation and phase discrimination
Victor and Purpura (1996) also explored the role of spike timing and intervals for encoding orientation using metric-space analysis. Although they used a larger stimulus set (8 orientations) and flashed gratings, our results compare favorably with their orientation analysis of V1 responses. They similarly found a large percentage of V1 cells with information provided by temporal structure on the basis of either the spike timing (80%) or spike interval (80%) metric. The reason our percentage (23/24 or 96%) is higher could be from the particular set of cells examined. Our cells were biased for supragranular layers and complex cells. Victor and Purpura (1996) did find higher percentages of temporal coding for V2 cells (92% and 94%). Another difference could be simply that we explored a more detailed set of stimuli, as well as the more obvious differences between flashed and drifting gratings. We tested approximately 5° (fine) or approximately 15° (coarse) differences around the preferred orientation, whereas Victor and Purpura (1996) tested 22.5° increments across the entire range of tuning for each cell.
Because Victor and Purpura (1996) did not test smaller differences in orientation, we can only directly compare their optimal q results with our coarse discrimination of orientation results, especially because we have shown q depends so highly on the difficulty of the orientation task. Their average optimal q was approximately 10 s-1 for the spike timing metric and the spike interval metric. For our data, the average q was 14.5 and 33.6 s-1, respectively, which we would expect to be slightly higher, given that for even coarse orientation discrimination our increments were smaller than 22.5°.
Finally, the amount of information gained from the temporal structure between the two studies is nearly the same. We showed that in the case of the amount of information, the results were similar between fine and coarse discrimination of orientation (Fig. 8). Across the entire data set, Victor and Purpura (1996) measured nearly the same amount of information for the spike timing (0.171 bits) and interval (0.107 bits) metric (normalized for experimental design so that 1 bit is the maximum possible) that we measured in the present study (approximately 0.2 and approximately 0.1 bits, respectively).
In Fig. 2, we suggested that one of the reasons the spike timing contributes to orientation discrimination is that the cells are encoding phase differences in the responses attributed to phase differences in stimulation. A change in orientation will result in a phase change in portions of the receptive field for individual cells. For example, with a left-to-right drifting grating, upper portions of the receptive field (RF) of a cell will be activated earlier with clockwise rotation of the drifting grating (Fig. 10A). Also, a grating much larger than the RF of the cells would result in similar phase changes depending on the RF location within the overall grating dimensions (Fig. 10B).
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) using flashed gratings suggest that temporal coding occurs even without the coherent motion, although possibly with less precision (the protocol differences do not allow direct comparisons, as described above). Victor and Purpura (1998) and Aronov et al. (2003) also measured temporal coding for spatial phase with stationary gratings. Both studies demonstrated that the temporal structure played a crucial role in encoding spatial phase (enhancing rate information) and might even play a crucial role in extracting one-dimensional features from multidimensional stimulation (Victor and Purpura 1998). The temporal coding of phase with stationary gratings is, again, less precise (25-50 ms; Victor and Purpura 1998) than what we measure with fine discrimination of orientation for drifting gratings (9.2 ms), but does exceed or match the precision we measure for coarse discrimination of orientation (68.9 ms). One would expect changes in phase and orientation to have similar influences on response properties with respect to spatial locations in receptive fields. Because the receptive fields of cells also have temporal properties in addition to the spatial characteristics, even stationary stimulation may lead to similar mechanisms underlying temporal coding as those described in Fig. 10. Primary visual cortical cells might even multiplex angle and direction information contained in the orientation of a drifting grating through temporal coding at different times (transient vs. sustained portions) that mimic the different temporal scales found in adaptation (Muller et al. 1999; van Wezel and Britten 2002). Timing, coordination, and cooperation
Resolving the differences in precision (with presumably similar spiking mechanisms) can be accomplished, in part, with the main finding reported here—the dependency of coding mechanisms on the stimulus (i.e., smaller changes in orientation are represented more robustly in the precise timing of spikes, whereas larger changes are represented in the spike count). The dependency and precision of temporal coding on the form of stimulation has also been shown for contrast (Reich et al. 1997), motion (de Ruyter van Steveninck et al. 1997), and transience (Mechler et al. 1998). The dynamic nature of neural coding might also be an indication of the dynamic nature of the spiking mechanism itself. As stimulation changes, the integrative properties of cortical cells change (Azouz and Gray 2003; Koch et al. 1996), leading to a continuum between rate and temporal coding (Reike et al. 1997) or a continuum between integration and synchronization (Rudolph and Diexthle 2003). The dynamics of conductance, spiking, and integrative mechanisms have been documented throughout the brain (Gray and McCormick 1996; Traub and Miles 1991; Traub et al. 1999).
The patterns of spikes (e.g., bursts and oscillations) that arise from the dynamic mechanisms described above form a likely substrate for the interval information that we measure for fine discrimination of orientation. There are clearly links between these interval-based patterns and orientation (Cattaneo et al. 1981a,b; DeBusk et al. 1997; Eckhorn et al. 1988; Gray et al. 1989; see also Fig. 5). The information we measure from intervals is not mutually exclusive from the information we measure from timing and is likely why the results from the metrics typically parallel each other on a qualitative level. The bursts and oscillations play a role in the plasticity of the coordination and synchronization among a network of cells (Eckhorn et al. 1988; Gray et al. 1989; Lisman 1997; Snider et al. 1998).
In Fig. 9, we showed that our timing information was not simply a transient effect that reflected changes in the initial response latency. Gawne et al. (1996) and Reich et al. (2001) have both shown latency to encode for contrast, but not orientation. The orientation-dependent phase shift in responses that we find (Fig. 2) would predict a link between latency and orientation, but examination of latency across the entire stimulus set does not necessarily show any predictable organization between latency and orientation, which would agree with past conclusions (Gawne et al. 1996; Reich et al. 2001). This is probably because the phase change with orientation depends on the spatial relationship between the stimulus and the RF (e.g., Fig. 10A). What the phase shifting might indicate is that the information we measure about spike timing is a result of computationally fast and efficient cortical processing mechanisms based on time shifts (Hopfield 1995; Milton and Mackey 2000; Wyss et al. 2003).
How exactly would phase shifts encode orientation information? The simplest explanation would be that the response timing relative to stimulus onset in cortex results in response changes further down the line of visual processing. In this case, the cortex would depend on stimulus onset reference signals such as transient responses (Bair 1999; Victor and Purpura 1996) or in more natural viewing conditions, from saccade-dependent responses (Park and Lee 2000).
Although the results presented here represent only single-cell analysis, there is certainly a link between the temporal structure seen in single cells and how the structure results in the temporal coordination and greater information among a population of cells (or vice versa) (Aronov et al. 2003; Eckhorn et al. 1988; Gray et al. 1989; Reich et al. 2001; Samonds et al. 2003a,2003b). The link between timing and coordination leads us to another explanation of how phase shifts might encode orientation with filterlike assemblies (Samonds et al. 2003b). A filterlike assembly takes advantage of the sensitivity of coincidence detection (Abeles 1982) for reliable transmission of a signal for the "preferred" orientation of an assembly of cells. The assembly "rejects" signals from orientations away from the preferred orientation because the synchrony is weaker and therefore has smaller chance of reaching the next layer of visual processing.
Figure 10B provides us with a situation in which the synchrony between two cells would change as a result of phase shifts in the individual responses. Figure 11, A and B illustrate a very simple example of how this could occur. At the preferred orientation (Fig. 11A), the spikes for both cells are synchronized and increase the chances of firing a third cell (Abeles 1991) that receives the signal from 2 cells that would have receptive fields organized in the manner shown in Fig. 10B. At nonoptimal orientations (Fig. 11B) the responses shift earlier (cell 1) and later (cell 2), resulting in asynchrony and less chance of cell 3 firing (i.e., rejected in a filterlike process). Because the above situation depends on the relative timing between cells, the trial-by-trial correlation of response latency between cells (Fries et al. 2001) makes the case more plausible. The admittedly simplistic example of Fig. 11 resembles the classic orientation encoding model described by Hubel and Wiesel (1962), with the exception of being based on timing rather than rates. Because of this similarity, we should also note that our example would also be dependent on many of the complexities of network and synaptic interactions that have evolved from Hubel and Wiesel's description (Shapley et al. 2003).
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; Grothe and Klump 2000; Lestienne 2001). Precise spike timing and coordinated responses start in the retina (Berry et al. 1997; Meister 1996; Meister et al. 1995) and are preserved in the lateral geniculate nucleus (Alonso et al. 1996; Dan et al. 1998; Reinagel and Reid 2000). Cortical neurons have the capabilities for maintaining very precise spike timing (Mainen and Sejnowski 1995) and the results reported in this article, as well as previous results, suggest preservation of the coordinated activity in visual cortex (Alonso and Martinez 1998; Samonds et al. 2003a,2003b).
In conclusion, a popular view of orientation coding in cortex is that whether as a population of cells (Dayan and Abbott 2001) or as even single cells (Bradley et al. 1987), detailed orientation information is most effectively encoded on the slopes of the tuning curve, away from the preferred (or "peak") orientation. This presumes that information is contained in the average firing rate, which undergoes the most reliable and largest changes in this range of activation for the cell. This position ignores the most active portion of the cell's response range. Although there is admittedly very little change in the average firing rate around the preferred orientation of the cell, the firing pattern undergoes many other changes that not only may provide detailed orientation information, but may more reliably pass on this information (Reyes 2003). At the preferred orientation of the cell, the cell operates more as a coincidence detector (Azouz and Gray 2003) with more reliable and precise transmission of spiking and timing information (Lisman 1997; Snider et al. 1998). Coarser orientation information may be represented in the form of the spike rate where groups are essentially active or inactive depending on the orientation. In the region of the preferred orientation, a cooperative and coordinated code based on timing reliably transmits more detailed orientation information.
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ACKNOWLEDGMENTS |
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GRANTS
This work was supported by National Eye Institute Grant RO1EY-03778-20.
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FOOTNOTES |
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Address for reprint requests and other correspondence: A. B. Bonds, Department of Electrical Engineering, Vanderbilt University, 255 Featheringill Hall, 400 24th Ave. South, Nashville, TN 37235 (E-mail: ab{at}vuse.vanderbilt.edu).
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