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The Journal of Neuroscience, January 1, 2000, 20(1):470-484
Membrane Potential and Firing Rate in Cat Primary Visual Cortex
Matteo Carandini1, 2, 3 and David Ferster3 1 Institute for Neuroinformatics, Swiss Federal Institute of Technology and University of Zurich, CH-8057 Zurich, Switzerland, 2 Howard Hughes Medical Institute and Center for Neural Science, New York University, New York 10003, and 3 Department of Neurobiology and Physiology, Northwestern University, Evanston, Illinois 60208
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
We have investigated the relationship between membrane potential and firing rate in cat visual cortex and found that the spike threshold contributes substantially to the sharpness of orientation tuning. The half-width at half-height of the tuning of the spike responses was 23 ± 8°, compared with 38 ± 15° for the membrane potential responses. Direction selectivity was also greater in spike responses (direction index, 0.61 ± 0.35) than in membrane potential responses (0.28 ± 0.21).
Threshold also increased the distinction between simple and complex cells, which is commonly based on the linearity of the spike responses to drifting sinusoidal gratings. In many simple cells, such stimuli evoked substantial elevations in the mean potential, which are nonlinear. Being subthreshold, these elevations would be hard to detect in the firing rate responses. Moreover, just as simple cells displayed various degrees of nonlinearity, complex cells displayed various degrees of linearity.
We fitted the firing rates with a classic rectification model in which firing rate is zero at potentials below a threshold and grows linearly with the potential above threshold. When the model was applied to a low-pass-filtered version of the membrane potential (with spikes removed), the estimated values of threshold (
54.4 ± 1.4 mV) and linear gain (7.2 ± 0.6 spikes · sec
1 · mV
As it was for stimulus orientation, threshold was also independent of stimulus contrast. The rectification model accounted for the dependence of spike responses on contrast and, because of a stimulus-induced tonic hyperpolarization, for the response adaptation induced by prolonged stimulation. Because gain and threshold are unaffected by visual stimulation and by adaptation, we suggest that they are constant under all conditions.
Key words: threshold; summation; iceberg; tuning; linearity; orientation; contrast; adaptation; simple cells; complex cells
INTRODUCTION
A mechanism that contributes to the remarkable selectivity of cells in the visual cortex is the action potential threshold. Because neurons in the visual cortex are mostly quiet in the absence of visual stimulation, their average membrane potential at rest must lie somewhat below firing threshold. In principle, therefore, the tuning of the firing responses of visual cortical neurons could represent the tip of an iceberg: just as icebergs are wider below the surface of the water than above it, the tuning of the synaptic inputs to a cell could be broader below threshold than above it. In the domain of orientation selectivity, a comparison of tuning curves measured from the membrane potential (Nelson et al., 1994
; Pei et al., 1994
Another major theme in the current research on the primary visual cortex centers on simple cells and regards the degree to which the responses of these cells are linear. Linearity was implicit in the original descriptions of simple cells (Hubel and Wiesel, 1962
In the experiments presented in this paper, we have examined the relationship between membrane potential and firing rate in neurons of the cat visual cortex. We have found that the iceberg effect does contribute significantly to orientation and direction selectivity: the orientation tuning of cortical cells as measured from their action potentials is considerably sharper than the orientation tuning measured directly from the membrane potential.
We have also investigated the linearity of the membrane potential responses and found that threshold also increased the distinction between simple and complex cells. This distinction is commonly based on the linearity of the spike responses to drifting sinusoidal gratings. In many simple cells, such stimuli evoked substantial elevations in the mean potential, which are nonlinear. Being subthreshold, these elevations would be hard to detect in the firing rate responses. Moreover, just as simple cells displayed various degrees of nonlinearity, complex cells displayed various degrees of linearity.
A final issue that we have addressed concerns the relationship between membrane potential and firing rate. We have tested what is perhaps the simplest model for this relationship: the rectification model (Granit et al., 1963
A preliminary version of the results has been presented in abstract form (Carandini and Ferster, 1998
MATERIALS AND METHODS
Details of most procedures have been described previously (Ferster and Jagadeesh, 1992
Experimental preparation. Young adult cats were anesthetized with intravenous sodium thiopental and placed in a stereotaxic headholder. Paralytic agents (gallamine or pancuronium) were administered to minimize motion of the eyes, and the animals were artificially respirated. Phenylephrine hydrochloride and atropine sulfate were applied to the eyes to retract the nictitating membranes, dilate the pupils, and paralyze accommodation. Contact lenses with artificial pupils (4-mm-diameter) were inserted.
Visual stimulation. Visual stimuli consisted of monocularly presented drifting sine-wave gratings displayed on a Tektronix (Wilsonville, OR) 608 oscilloscope screen using a Picasso stimulus generator (Innisfree, Cambridge, MA). The peak contrast used was 64%, and the mean luminance (kept constant throughout the experiments) was 20 cd/m2. Optimal spatial frequency was determined from computer-generated spatial frequency tuning curves. Grating size, position, and temporal frequency were adjusted to be optimal, usually by hand.
To generate orientation tuning curves, stimuli of 12 different orientations (0-330°) were presented in random order, 4 sec for each orientation. The contrast of the gratings was usually 47%, and the block of stimuli included an additional 4 sec blank screen presentation. This block of 13 stimuli was repeated two to five times for each cell, with a different randomized order each time.
To generate contrast-response curves, stimulus blocks consisted of seven optimally oriented stimuli with contrasts logarithmically spaced between 1 and 64%, which were randomly presented. Test stimuli were 4 sec long and were preceded by 4 sec adaptation stimuli (20 sec before the first test stimulus), as previously described (Carandini and Ferster, 1997
Intracellular recording. Whole-cell patch recordings in the current-clamp mode were obtained from neurons of area 17 of the visual cortex using the technique developed for brain slices by Blanton et al. (1989)
. Membrane potentials were low-pass-filtered and digitized at 4 kHz, and the timing of spikes was logged with 250 µsec accuracy.
Response measures. To obtain tuning curves for the membrane potential and spike train responses we considered two response measures, the mean and the modulation. The mean response was the average over the 4 sec stimulus presentation, whereas the modulation was the peak-to-peak amplitude of the best-fitting sinusoid at the stimulus frequency (obtained by fast Fourier transform). For this analysis, individual spikes were treated as Dirac
functions.
Tuning curves. The orientation tuning of the responses was fitted with a descriptive function. This function is the sum of two Gaussians and is defined on the circle. The two Gaussians are forced to peak 180° apart and to have the same width
:
In the above expression, O is the stimulus orientation (between 0 and 360°), and the angle brackets indicate angular values expressed between
(1) To allow us to report a single preferred orientation and tuning width for each signal, membrane potential and firing rate, the mean and the modulation for each signal were fitted together. In particular, although the base response and the heights of the two Gaussians were allowed to differ for mean and modulation, an additional constraint was applied such that the fits to these measures had the same preferred orientation, Op, and tuning width,
Measures of response tuning. From the parameters of Equation 1, it is easy to obtain some widely used measures of response tuning, namely the direction index and the orientation tuning half-width.
The direction index is a common measure of direction selectivity (Schiller et al., 1976
The tuning half-width is a common measure of the narrowness of orientation tuning (Campbell et al., 1968
Coarse potentials and firing rates. To test the rectification model of firing rate encoding, we obtained coarse membrane potential traces and firing rates. The coarse membrane potential traces, V(t), were obtained as follows. First, we identified the time of occurrence of spikes by searching for maxima in the derivative of the membrane potential. We then identified the starting and ending times of the typical spike for each cell, including afterhyperpolarizations. Spikes typically began at t0 =
RESULTS
We recorded intracellularly from 41 cells in the cat primary visual cortex and measured their orientation tuning with drifting sinusoidal gratings. Twenty-nine of these cells responded with at least one spike/sec to stimuli of the preferred orientation. Twenty-eight of these 29 cells had a clear preference for orientation and are the object of this study.
The mean resting potential of the cells was
Membrane potential responses to different orientations
From extracellular recordings, it is known that in response to drifting gratings simple and complex cells exhibit rather different spike trains: those of simple cells are strongly modulated at the stimulus frequency, whereas those of complex cells consist principally of an elevation in the mean firing rate (Movshon et al., 1978c
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Figure 1. Membrane potential responses of two cells to stimuli of preferred orientation drifting in the preferred direction (left) and in the nonpreferred (opposite) direction (right). A, Responses of a simple cell (cell 61). The grating stimulus drifted at 4 Hz. Each bar of the grating elicited a strong modulation in the membrane potential response. B, Responses of a complex cell (cell 24). The grating stimulus drifted at 2 Hz. The responses it elicited contained only a mild component at the stimulus frequency. The dotted horizontal lines indicate the resting potential.
The membrane potential of the simple cell (Fig. 1A) was strongly modulated at the temporal frequency of the stimulus (4 Hz). This modulation was stronger for the stimulus drifting in the preferred direction (left) than in the opposite direction (right). In this cell as well as in all other simple cells in our sample, the modulation in membrane potential was seldom symmetrical around the resting potential of the cell: the membrane potential spent more time above rest than below it. Thus, together with a strong modulation, the membrane potential responses of simple cells exhibited a noticeable increase in their mean.
By contrast, the membrane potential response of the complex cell (Fig. 1B) consisted mainly of an elevation in the mean. This elevation was accompanied by a gradual hyperpolarization and reduction in spike frequency during the course of the stimulus presentation, which is most likely a consequence of pattern adaptation (Carandini and Ferster, 1997
The effects of changing stimulus orientation on the responses of the simple cell are illustrated in Figure 2. Here the responses were averaged over each stimulus cycle, so each trace represents the average response of the cell to the passage of one bar of the grating over the receptive field. The firing rate responses (Fig. 2A) are typical of many simple cells: the cell is strongly tuned for orientation, gives no response to stimuli of nonpreferred orientations, and displays a marked preference for one direction of motion (270°) over the opposite (90°).
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Figure 2. Cycle averages and spike histograms, as a function of stimulus orientation, for the simple cell in Figure 1A. The first column refers to a blank stimulus, and the subsequent columns refer to 12 stimulus orientations, spanning the range between 0 and 360° in 30° steps. Responses are averaged over one stimulus cycle (0.25 sec). A, Firing rate. B, Membrane potential. Cell 61.
The strength of the tuning of the firing rate responses is only partly inherited from the underlying membrane potential responses (Fig. 2B) and appears to receive a substantial contribution from the spike threshold. In particular, the tuning of the membrane potential responses appears to be broader than that of the firing rate responses in at least three ways. First, although stimuli at orientations flanking the preferred orientation did modulate the membrane potential and increase the mean membrane potential, they did not elicit firing. Second, the mean membrane potential at all orientations was more positive than in the absence of visual stimulation, an effect that is not visible in the firing rate responses (which were zero in both cases). Third, at the preferred orientation the difference between the firing rate response in the two opposite directions of motion (90 and 270°) was far greater than the differences between the corresponding membrane potential responses.
These same observations can be made for the complex cell shown in Figure 1 (Fig. 3). As in the simple cell, stimuli at all orientations evoked a depolarization relative to rest. Moreover, visual stimuli at orientations surrounding the preferred orientation (30 and 210°) increased the mean membrane potential of the cell but did not elicit substantial firing responses, creating a substantial difference in the orientation tuning width measured from the two types of responses. Finally, the two opposite directions of motion at the preferred orientation (60 and 240°) elicited firing rate responses that were far more dissimilar than the corresponding membrane potential responses.
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Figure 3. Cycle averages and spike histograms, as a function of stimulus orientation, for the complex cell in Figure 1B. Format as in Figure 2. Responses are averaged over one stimulus cycle (0.5 sec). Cell 24.
Orientation selectivity of firing rate and membrane potential responses
To quantify the responses (membrane potential or firing rate) to drifting gratings, we used two measures: mean and modulation. The first is simply the average response measured over the stimulus duration. The second is the peak-to-peak amplitude of the sinusoid at the grating frequency that best fits the response (i.e., two times the amplitude of the first harmonic of the response). If the response were a perfect sinusoid, its modulation would be its peak-to-peak amplitude.
The orientation tuning of the mean and modulation of both the membrane potential and the firing rate responses of the simple cell is illustrated in Figure 4. The modulation component was large and well tuned both in the firing rate (Fig. 4B) and in the membrane potential (Fig. 4D). The mean component was much smaller but similarly tuned.
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Figure 4. Orientation tuning of the simple cell in Figures 1A and 2. Top, Firing rate. Bottom, Membrane potential. Left, Mean responses. Right, Response modulation. Gray areas indicate confidence intervals for the responses to a blank stimulus. Their width and the length of the error bars on the data points are twice the SE of the measurements. In the top panels the confidence intervals are infinitesimal: the response to the blank was always 0 spikes/sec. The thin curves indicate the fits of a descriptive tuning curve (Eq. 1). The thick lines in the top panels indicate the predictions of the rectification model of firing rate, obtained from the membrane potential responses. Cell 61.
For the firing rate, that the mean (Fig. 4A) was tuned similarly to the modulation (Fig. 4B) is simply a consequence of the lack of firing rate responses at rest. Because the resting firing rate was zero, the effect on the mean of an increase in rate in one phase of the responses could not be compensated by a decrease at another phase. For the membrane potential, by contrast, there is no corresponding constraint. Indeed, the lower limit for the membrane potential (the reversal potential of potassium ions) was well below the resting potential of the cells. The similarity in tuning between the mean (Fig. 4C) and the modulation component (Fig. 4D) is caused by the tendency pointed out in the description of Figure 1: the modulation in the membrane potential was larger above the resting potential than below it.
A comparison of the orientation tuning curves for firing rate and membrane potential in Figure 4 confirms that the firing rate responses are more sharply tuned than the membrane potential responses. For example, stimuli flanking the preferred orientation (240 and 300°) gave membrane potential responses that were ~20% as large as the response at the preferred orientation (270°). Yet the firing rate responses to these stimuli were zero, indicating that the tuning width of the firing rate was smaller than the spacing between orientations (30°). In addition to the width of the tuning, the difference in tuning between membrane potential responses and firing rate responses applies most notably to the relative sizes of the responses to the two opposite directions of motion, 90 and 270°. Although the membrane potential responses to a grating drifting in the 270° direction are only marginally larger than those to a grating drifting in the opposite direction, the difference in firing rate responses in the two conditions is substantial.
A similar analysis in terms of mean and modulation can be performed for the complex cell of Figures 1B and 3. The results of such an analysis are illustrated in Figure 5. Consistent with the observations made on the traces, the mean component (Fig. 5C) of the membrane potential response is substantially larger than the modulation component (Fig. 5D). Similarly, the mean component (Fig. 5A) of the firing rate response is larger than the modulated component (Fig. 5B). In a complex cell, then, the stimulus tuning is mostly expressed in the left panels, which report the response means.
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Figure 5. Orientation tuning of the complex cell of Figures 1B and 3. Format as in Figure 4. Cell 24.
To compare the tuning of the different response measures, mean and modulation of firing rate and membrane potential, we fitted the responses with the descriptive function in Equation 1 and obtained estimates of the direction index and tuning half-width (see Materials and Methods). Even in the face of the restrictions that we imposed to limit the number of free parameters, the fits were generally good. They are illustrated by the thin curves in Figures 4 and 5 and in many subsequent figures.
The values for the direction index confirm that for the cells in Figures 4 and 5, the encoding of subthreshold events into firing rates substantially increased the selectivity for direction of motion. Indeed, for the simple cell in Figure 4 the direction index was 0.25 for the potential modulation (Fig. 4D) and 0.79 for the modulation of the firing rate (Fig. 4B). For the complex cell in Figure 5 the direction index was only 0.09 for the mean membrane potential (Fig. 5C) and 0.82 for the mean firing rate (Fig. 5A).
On the other hand, the values for the tuning half-width of these two cells do not suggest a substantial difference between firing rate and membrane potential in terms of orientation selectivity. Indeed, for the cell in Figure 4 the tuning half-width was 18° for the firing rate and 21° for the membrane potential, and for the cell in Figure 5 the tuning half-width was similarly narrow in the two signals (17°).
Before concluding that in these cells the encoding of membrane potential into firing rates did not sharpen the tuning, however, one should consider that half-widths of 17-18° are the lowest that can be measured from our data. This limit arises from the 30° spacing of our stimuli on the orientation axis. Because in these two cells the firing rate responses were zero at all orientations except the preferred, it is likely that the true tuning half-width for the firing rate was actually <17°. For the membrane potential, on the other hand, the presence of data points on the slopes of the tuning curves (Figs. 4D, 5C) indicates that the data would not be fitted by narrower tuning curves.
The difference in tuning sharpness between the membrane potential and the firing rate is most evident in cells that are more broadly tuned, where our sampling limitations do not play a role. This is illustrated in Figure 6, which contains the tuning curves for four additional cells, two complex and two simple. For three of these cells, the orientation tuning of the firing rate responses was significantly sharper than that of the membrane potential responses. These are the first complex cell (Fig. 6A-D, half-widths of 23° for the firing rate and 40° for the membrane potential), the first simple cell (Fig. 6I-L, half-widths of 20° for the firing rate and 39° for the membrane potential), and to some degree the second simple cell (Fig.6M-P, half-widths of 28° for the firing rate and 38° for the membrane potential). For the second complex cell (Fig. 6E-H), instead, the tuning half-widths of the firing rate and of the membrane potential were similar (28 and 30°).
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Figure 6. Orientation tunings of two complex cells and two simple cells. The format of each group of four panels is as in Figure 4. A-D, E-G, Complex cells (cells 86 and 28). I-L, M-P, Simple cells (cells 68 and 71). These cells are arranged in order of spike modulation index: 0.88, 0.92, 1.43, and 1.54. The corresponding potential modulation indices are 0.41, 0.37, 0.56, and 1.84.
The iceberg effect and orientation selectivity
The narrower tuning of the firing rate responses with respect to the membrane potential responses is a general property of cat V1 cells. The extent of this phenomenon is illustrated in Figure 7A, where we compared the tuning width of the two responses as estimated for each cell in our population. The abscissa marks the tuning half-width for the firing rate responses, and the ordinate marks the tuning half-width for the membrane potential. For the overwhelming majority of cells, the points lie to the left of the identity line, indicating that the firing rate was more sharply tuned than the membrane potential. Indeed, the mean tuning width of the firing rate responses was 23 ± 8°, whereas the mean tuning width for the membrane potential was 38 ± 15°. The median difference in tuning width between firing rate and membrane potential responses was 10°. This difference was <2° in one-fourth of the cells and >25° in another fourth of the cells.
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Figure 7. Comparison of orientation tuning in the membrane potential responses and in the firing rate responses. A, Orientation tuning width at half-height, obtained from fits such as those in Figure 6. B, Direction index, computed from the sum of the mean and modulation components. Open symbols, Simple cells; filled symbols, complex cells. Lines mark the identity between abscissa and ordinate.
The degree to which firing rate is more narrowly tuned than membrane potential is even more striking if one considers that we are most likely overestimating the tuning width of the firing rate. As mentioned above, because we sampled the orientation axis in rather coarse, 30° intervals, we cannot resolve orientation tuning curves with half-widths <17°. For half of the cells, this measurement limit was reached by the tuning of the firing rate responses. The half-width of the tuning of these responses was then conservatively estimated to be 17°, resulting in the vertical streak of points in Figure 7A. The true half-width for these firing rate responses, however, would lie somewhere to the left of that streak. In most of those cells the half-width of the membrane potential responses was large, often >30°, and if we knew the true half-width of the firing rate, the difference between the two would be even larger than what appears in the graph.
The methods of recording with the whole-cell patch technique are more invasive than those used by most studies of visual responses in cat V1, which were conducted extracellularly. Consequently, it is possible that the firing rate responses that we observed could be unnaturally sharp. For example, the perfusion of the cell with the electrode solution could have altered the resting potential or the amplitude of synaptic potentials of the cells. Some of these possible changes could have generated an artifactual difference in tuning between the membrane potential and firing rate responses.
This concern is soon dispelled when the tuning width of the firing rate responses in our sample is compared with the results of previous extracellular studies (Fig. 8). This comparison suggests that the tuning widths that we report for the firing rate are, if anything, broader than those reported in the literature. An early study of orientation tuning (Campbell et al., 1968
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Figure 8. Orientation tuning of the firing rate responses as measured in published extracellular studies and in our intracellular recordings. The measure of tuning width in the abscissa is the half-width at half-height. A, Replotted from Campbell et al. (1968)
The iceberg effect and direction selectivity
In addition to being more sharply tuned than the membrane potential responses for stimulus orientation, the firing rate responses tended to be more selective for stimulus direction. This phenomenon has already been reported for simple cells (Jagadeesh et al., 1993
Comparably dramatic increases in direction selectivity were observed in a number of cells. This property is summarized for all of our cells in Figure 7B, where the direction index (calculated from the sum of the mean and modulation components) is plotted for the membrane potential versus the firing rate responses. Essentially all the points lie on the right side of the identity line, indicating that in the greatest majority of the cases the direction index was higher when measured from the firing rate than when measured from the membrane potential. In only three cells was the direction index of the firing rate smaller than that of the membrane potential (the tuning of one of these cells is in Fig. 6E-H), but these cells were only mildly direction-selective. The average direction index for spikes in our population was 0.61 ± 0.35, whereas the average index for the membrane potential was only 0.28 ± 0.21. This confirms that threshold substantially accentuates direction selectivity in the same way that it does orientation selectivity.
As with the measurements of tuning width, it is of importance to know whether the sampling bias of our technique led us to record from an unnaturally direction-selective set of cells. Again, this concern is dispelled by a comparison with previously published data obtained extracellularly (Gizzi et al., 1990
Nonlinearity of the responses of simple cells
One of the properties implicit in Hubel and Wiesel's (1962)
In particular, a sensitive assay for nonlinearity is the mean membrane potential response to a sinusoidal grating stimulus. The mean luminance of the stimulus integrated over the receptive field does not change with time, and the mean luminance integrated over time is the same at every point in the receptive field. Furthermore, these means are identical to the luminance of the screen in the absence of a grating. Therefore, for a cell that is summing the inputs from different parts of its receptive field linearly, the mean membrane potential should be unaffected by the grating stimulus.
This was not the case for the simple cells in our sample. As is apparent in the membrane potential traces of Figures 1A and 2, the depolarizing events in the responses are larger than the hyperpolarizing events, so that the mean membrane potential is elevated by stimulation. We have observed this elevation previously for stimuli at the preferred orientation (Carandini and Ferster, 1997
In addition to being tuned for stimulus orientation, the mean membrane potential was in many cells higher than at rest for all stimulus orientations. This effect can be observed in Figure 9, which shows the fitted tuning curves of the mean and modulation components in the membrane potential for all our cells. An elevation of the mean potential for all orientations was observed in 14 of 21 simple cells (Fig. 9A). It was also observed in six of seven complex cells (Fig. 9C). A few cells, however, displayed the opposite behavior. These cells (four simple and one complex) exhibited a clear reduction in membrane potential at orientations that are close to orthogonal to the preferred orientation. Finally, in all cells the modulation in membrane potential was close to zero for stimuli orthogonal to the preferred orientation.
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Figure 9. Summary of orientation tuning of the membrane potential. Curves are the fits of the descriptive tuning function (Eq. 1), aligned so that the preferred orientation and direction for the modulated component would be at 0°. A, B, Mean and modulation of membrane potential in 21 simple cells. C, D, Mean and modulation of membrane potential in 7 complex cells.
Having observed a substantial nonlinear component in the responses of simple cells, we can now ask what impact this nonlinearity has on response tuning. To address this issue, we can use the response mean as a measure of the nonlinear component of the response and the response modulation as a measure of the linear component of the response.
In the domain of stimulus orientation, the nonlinear component of the responses amplifies the tuning of the linear component, leaving the preferred orientation and tuning width unchanged. Indeed, we have observed that the mean and the modulation in membrane potential are similarly tuned. This similarity allowed us to obtain high-quality fits of the descriptive function while imposing that the mean and modulation have the same preferred orientation and tuning width. Because the sum of two equally broad Gaussians is a scaled version of the original Gaussians, the sum of the linear and nonlinear components has the same tuning as the linear component alone. In the absence of a model, however, it is not clear how the nonlinear components would behave in determining the tuning to visual stimuli other than drifting gratings. Indeed it has been proposed that they can contribute substantial sharpening in response to flashed bars (Volgushev et al., 1996
In the domain of stimulus direction, the effects of response nonlinearity are less easy to establish. It is nonetheless possible to consider (and rule out) two extreme case scenarios. In a first scenario, direction selectivity would be entirely the result of a nonlinear mechanism. This scenario was ruled out by Jagadeesh et al. (1993
In principle, then, direction selectivity may be enhanced by a nonlinear mechanism. On the other hand, the nonlinear component was often much smaller than the modulated component of the responses. Given this disparity in size, to what extent do the nonlinear components affect the direction selectivity of the cells?
Our results indicate that the nonlinear component of the responses, being much smaller than the modulation component, does not play an important role in the establishment of direction selectivity in simple cells. This result is illustrated in Figure 10, where the direction index for the modulated response alone is compared with the direction index for the sum of the modulated and mean responses for 21 simple cells. The direction indices obtained from the two measures are similar, indicating that the contribution of the nonlinear components to direction selectivity is generally minor.
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Figure 10. Impact of nonlinearity on direction selectivity of the membrane potential responses of 21 simple cells. The direction index obtained from the sum of the mean and modulated components of the responses (ordinate) is plotted against the direction index obtained from the modulated components of the responses (abscissa).
Receptive field classification from modulation indices
A large body of spike response data from both cats and monkeys indicates that simple and complex cells correspond to two clear modes in the distribution of an index of linearity (Skottun et al., 1991
Both simple and complex cells, however, show some degree of nonlinearity, and complex cells show some degree of linearity, so the question naturally arises of whether these types of cells constitute two separate classes. This separation into classes has been questioned altogether (Dean and Tolhurst, 1983
These issues can be fruitfully investigated intracellularly, by studying the subthreshold membrane potential responses. To this effect, we have considered a potential modulation index, defined as above but with the mean and modulation of the membrane potential responses substituted for those of the firing rate responses. A perfectly linear simple cell would have a potential modulation index equal to infinity (because the mean would be zero), and a perfectly nonlinear complex cell would have a potential modulation index of zero (because the modulation would be zero).
The complex cell in Figures 1B, 3, and 5 and the simple cell in Figures 1A, 2, and 4 differed widely in both their spike modulation index and their potential modulation index. The complex cell had a spike modulation index of 0.44 (one-half of 7.9 spikes/sec peak-to-peak modulation divided by a mean component of 9.0 spikes/sec), which placed it solidly in the complex group. The simple cell had a spike modulation index much higher than one, 1.73, which placed it solidly in the simple group. The potential modulation indices were 0.14 for the complex cell and 1.92 for the simple cell. These values are rather extreme: many other cells, such as the complex cell in Figure 6E-H and the simple cell in Figure 6I-L had intermediate potential modulation indices (in the range of ~0.5).
The potential modulation index is plotted against the spike modulation index for all our cells in Figure 11. The two are clearly correlated. For potential modulation indices of ~1, the spike modulation index is approximately half the potential index. For potential modulation indices >1, the spike modulation index saturates at ~2. The vertical line indicates a spike modulation index of 1: cells to the left are defined as complex, and cells to the right are defined as simple. This criterion level for the spike modulation index corresponds roughly to a level of 0.5 (horizontal line) for the potential modulation index. At the simple-complex criterion level, thus, the threshold appears to enhance the modulation index and may therefore enhance the difference between simple and complex cells.
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Figure 11. Distribution of the modulation indices for the membrane potential and for the firing rate. The vertical line indicates a standard criterion for classifying simple and complex cells based on the spike responses (Skottun et al., 1991
At the value of 0.5 for the potential modulation index, the peak-to-peak membrane potential modulation equals the mean potential increase. So for most simple cells, the membrane potential modulation was larger than the mean potential increase, whereas for complex cells, the modulation was smaller than the mean increase.
Plotted as they are in Figure 11, simple and complex cells appear to lie in a continuum of response linearity. The results of Skottun et al. (1991)
Rectification model of the firing rate
Having investigated the tuning of the membrane potential responses and its relationship to the tuning of the spike responses, we turn to a more basic question, namely, the nature of the relation between membrane potential and firing rate. Our main motivation is to test whether the observed differences in tuning width between the spike responses and the membrane potential can be ascribed entirely to the spike threshold.
One of the simplest models for the firing rate is the rectification model, which was perhaps first proposed quantitatively by Granit et al. (1963)
The rectification model can be perhaps better formulated as receiving a coarse (slow-varying, spike-free) membrane potential as input (Carandini et al., 1996
To test the rectification model, we obtained coarse membrane potential traces by removing the spikes from the membrane potential responses and low-pass filtering the resulting traces (see Materials and Methods). We then low-pass filtered the corresponding spike trains, obtaining firing rate traces that did not contain information about the precise timing of spikes. The effects of these manipulations on the membrane potential traces and on the spike trains are illustrated in Figure 12 for the responses of the simple and complex cells that we showed in Figure 1. The coarse membrane potentials, V(t), are shown in Figure 12, C and F. Above those panels, in Figure 12, B and E, are the corresponding firing rates, R(t).
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Figure 12. Coarse potentials and firing rates and fits of the rectification model. A-C, Results for the simple cell in Figures 1A, 2, and 4 (cell 61). D-F, Results for the complex cell in Figures 1B, 3, and 5 (cell 24). The results are plotted for the responses to stimuli having the preferred orientation and drifting in the preferred direction (left panels) or the opposite direction (right panels). The coarse potentials are plotted in C and F. The firing rates are plotted in B and E, and their estimation from the coarse potential, using the rectification model, is shown in A and D. The lines over the coarse potential traces indicate the estimated thresholds.
The rectification model attempts to predict the firing rates from the coarse membrane potentials with the assistance of just two free parameters. The first parameter is a threshold, Vthresh, and is expressed in millivolts. The second parameter is a gain, Rgain, and is expressed in spikes per second per millivolt. It expresses the predicted firing rate per each millivolt of potential above the threshold. In mathematical notation, then, the prediction of the rectification model is simply:
where [x]+ = x for x > 0, and is 0 otherwise.
![R(t)=R<SUB><UP>gain</UP></SUB>[V(t)−V<SUB><UP>thresh</UP></SUB>]<SUP>+</SUP>,](/content/vol20/issue1/fulltext/470/img002.gif)
(2) We fitted Equation 2 to the coarse potentials and firing rates, leaving the parameters Rgain and Vthresh free to vary across cells but not across stimulus orientations and contrasts within one cell. Having established the values of these parameters, for each cell we could apply the rectification model to the coarse membrane potential traces and obtain predicted firing rates.
The performance of the model in predicting the firing rate from the coarse membrane potentials can be seen in Figure 12. The firing rates predicted by the model are illustrated in Figure 12, A and D, and are quite similar to the actual firing rates of the cells (Fig. 12B,E). Overall, the model predicted 74% of the variance of the firing rates of the simple cell and 58% of the variance of the firing rates of the complex cell. The median across the cell population of the percentage of the firing rate variance accounted for by the rectification model was 52%.
These values for the percentage of the variance indicate that the fits were acceptable but far from perfect. Indeed the predicted firing rates are in some points incorrect. For example the predicted firing rates in Figure 12D seem rather more tonic than those observed in Figure 12E. These defects, however, are not as impressive if one considers that the model has only two parameters, and the fits were performed on much larger data sets than the traces shown in Figure 12. For each cell, the data sets consisted of 100-200 sec of responses to randomly interleaved gratings of different orientations and blank stimuli. The cells in which the rectification model performed worst were invariably those in which a slow drift in the mean potential was not accompanied by a similar trend in the firing rate. The extracellular potential recorded on exiting these cells indicated that the drift was a result of polarization in the electrode.
The estimated values of the two parameters of the model, threshold and gain, were remarkably constant across the population. The mean value of the threshold across the population was Vthresh =
The thresholds estimated by the rectification model were clearly related to the mean of the spike thresholds measured directly from the intracellular records. At 0.95, the correlation between the two was very high. The thresholds estimated by the rectification model, however, were 6.0 ± 0.4 mV (mean ± SEM; n = 28) lower than the mean threshold measured directly from the individual spikes. This difference was systematic and is easily explained: within a given cell, the threshold of the individual spikes typically varied over a range of ~10 mV. The threshold estimated by the rectification model was always at the low end of this range. A higher threshold estimate would predict zero firing rates when the neuron was actually firing. The model therefore chooses a lower threshold and corrects for possible excessive firing by lowering the gain parameter. The variation in the threshold of individual spikes, in turn, is to be expected from the known properties of the spiking mechanism: the precise potential at which an individual spike is generated depends on factors such as the rate of variation of the membrane potential, the time since the last spike. We did not observe any systematic dependence of this threshold on stimulus condition.
The rectification model can be used to predict the firing rate responses not only as a function of time but also as a function of orientation. Indeed the rectification model does an excellent job at predicting the orientation tuning of the firing rate responses. For example, the predictions of the model for the simple cell in Figure 12 are illustrated in Figure 4. Superimposed on the tuning of the mean (Fig. 4A) and modulation (Fig. 4B) of the firing rate response are thick lines obtained by computing the mean and modulation of the firing rates predicted by the rectification model. The agreement of the fits with the descriptive tuning curve defined in Equation 1 is such that the two can hardly be distinguished.
Similar observations can be made for the complex cell in Figure 12, whose tuning is illustrated in Figure 5, as well as for the four cells in Figure 6. For each of these cells, the thick curves on the tuning of the firing rate responses were derived from the coarse membrane potential responses using the rectification model. The fits between tuning curves derived from the real and modeled spike rates are excellent. They are often better than the fits by the purely descriptive curve of Equation 1, which is not constrained by the membrane potential responses. That the model fits the spike tuning curves so well suggests that the threshold does not change substantially from one stimulus orientation to another, and that it is indeed threshold that accounts for much of the sharpening of the tuning of spike responses relative to the tuning of the membrane potential responses.
Contrast responses and pattern adaptation
We have seen that the spike threshold is essential in establishing the sharpness of tuning exhibited by cells in the primary visual cortex and that it is largely constant for all stimulus orientations. Less quantitative investigations in the domains of temporal frequency and spatial frequency revealed that the iceberg effect is strong in those domains as well. We now report on the role played by the threshold in the contrast responses of the cells and in their modification by sensory experience through pattern adaptation.
Examples of contrast responses are illustrated in Figure 13 for three simple cells. As usual, for each cell, the four panels report the mean and modulation for firing rate and membrane potential. The only difference with similar previous graphs is that rather than stimulus orientation, the abscissa here represents stimulus contrast. Let us for now concentrate on the filled symbols, which were obtained with a protocol similar to that used in orientation tuning experiments. For each cell, in general, increasing contrast increased both the mean and the modulation of the membrane potential responses. At the highest contrasts, however, mean potential responses often started to decrease again. This effect was mild for the cell in Figure 13A, absent for the cell in Figure 13B, and pronounced for the cell in Figure 13C.
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Figure 13. Contrast responses of three simple cells and effects of pattern adaptation. For each cell, mean (left) and modulation (right) are plotted for the firing rate (top) and membrane potential responses (bottom) as a function of stimulus contrast. Filled symbols indicate responses obtained while adapting to low contrast (1%); open symbols indicate responses obtained while adapting to high contrast (47%). Solid curves are predictions of the rectification model, obtained from the membrane potential responses. Error bars are twice the SE of the measurements. The cell in A is the same as in Figures 1A, 2, and 4. Data in C were published by Carandini and Ferster (1997)
The rectification model accounted for the firing rate dependence on stimulus contrast. This can be observed in Figure 13, top panels. The curves fitted to the data are predictions of the rectification model derived from the coarse potential responses of the cells. The model captures the dependence of the spike train on contrast, both in the mean firing rate and in the firing rate modulation. For example, it correctly predicts that there are different portions of the contrast responses: first one in which the firing rate is zero, then one in which the responses grows smoothly, and finally one in which the responses saturate or even start decreasing (Li and Creutzfeldt, 1984
As a final demonstration of the role of spike threshold in determining the visual properties of the spike responses in cat V1, we consider the effects of pattern adaptation. We have recently reported that in cat V1 cells, the main effect of prolonged visual stimulation with an adequate stimulus is a tonic hyperpolarization (Carandini and Ferster, 1997
The adaptation state of our cells was controlled by preceding the measurement runs with a long (e.g., 20 sec) presentation of the adapting stimulus. The adapting stimulus was also presented for 4 sec before each test stimulus to provide a "top-up" of adaptation (Movshon and Lennie, 1979
To determine whether the rectification model predicts the effects of adaptation on the firing rate responses, we obtained the parameters of the model by fitting only the responses obtained during adaptation to low contrasts. We then asked whether these parameters would also fit the responses obtained during adaptation to high contrasts. As shown in Figure 13 by the curves fitted to the open and closed symbols, the model captured the main aspects of the responses in both adaptation conditions. The effects of adaptation are accounted for by the changes in membrane potential, including the adaptation-induced tonic hyperpolarization. Thus the parameters of the rectification model, the gain and the threshold, are unaffected by pattern adaptation. Because they are also unaffected by stimulus contrast and stimulus orientation, we suggest that they are constant under all conditions.
DISCUSSION
