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NEUROSCIENCE |
1 Department of Experimental Ophthalmology, University of Tübingen Eye Hospital, Röntgenweg 11, D-72076 Tübingen, Germany
2 Department of Ophthalmology, University of Erlangen-Nuremberg, Schwabachanlage 6, 91054 Erlangen, Germany
3 Departamento de Fisiologia, Universidade Federal do Pará, 66075-900 Belém, Pará, Brazil
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Abstract |
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(Received 30 January 2007;
accepted after revision 2 April 2007;
first published online 5 April 2007)
Corresponding author J. Kremers: Dept of Ophthalmology, University of Erlangen-Nuremberg, Schwabachanlage 6, 91054 Erlangen, Germany. Email: jan.kremers{at}augen.imed.uni-erlangen.de
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Introduction |
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We showed that cells in the LGN of marmosets displayed response characteristics, caused by receptive field (RF) centre–surround interactions, that could be correlated with the psychophysical data (Kremers et al. 2004). The cells' response amplitudes were large when the centre and surround stimuli modulated in counter-phase, and small when the two modulated in-phase. It was found that the physiological and psychophysical data resembled each other even in detail. For instance, a minimal response and a minimally perceived flicker strength were both reached when the surround stimulus was phase advanced relatively to the centre stimulus. Furthermore, temporal frequency and stimulus contrast had similar effects on the two sets of data. The influence of contrast on the physiological data could be attributed to a non-linear interaction between the RF centre and surround. Based upon the qualitative similarities between physiological and psychophysical data, we proposed that the physiological basis for the described psychophysical centre–surround interactions resides at a subcortical level.
However, there were quantitative differences between the physiological and psychophysical data, which can possibly be attributed to the fact that the stimulus information is not encoded by one neuron, but by an array of LGN neurons, the RFs of which are covered by the stimulus. In the physiological experiments, we used stimuli that spatially matched the RF of the LGN cells. It is impossible to match the stimulus to the RF of every cell in the array. In addition, for linking physiological and psychophysical data, it is important to take into account that the visual cortex decodes the spike pattern that it receives from the LGN, to construct a visual percept.
Here, we present the results of measurements in which we introduce displacements between the stimulus and the cells' RF. On the basis of these data, the output of an array of responding LGN cells, being the input to the visual cortex, is described. Using simple and testable assumptions about cortical processing of this input, we were able to quantitatively link the physiological and the psychophysical data. A schematic overview of the model can be found in Fig. 10 which can be used as a reference at each step of the data analysis.
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Parts of the results have been presented in abstract form (Kremers & Kozyrev, 2003; Kozyrev & Kremers, 2004).
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Methods |
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All animal experiments were carried out in Tübingen where they were approved by the local legal authorities on animal research, and conducted in accordance with the ethical guidelines and with the principles regarding the care and use of animals adopted by the Society for Neuroscience.
The animals (marmosets, Callithrix jacchus) were initially sedated by an intramuscular injection of 15–30 mg kg–1 ketamine hydrochloride (Ketanest, Parke-Davies) and 1 mgkg–1 diazepam (Valium, MM Roche). Main anaesthesia was achieved by a continuous intravenous application of 4–8 µg kg–1 h–1 sufentanil (Sufenta, Janssen) with an initial dose of 5 µg kg–1. Sufentanil is an analgesic compound that is narcotic at the relatively high doses used during the experiment. The animals were respired throughout the experiment with a mixture of 70% N2O and 30% O2 or carbogen, thereby supplementing anaesthesia. To prevent eye movements, 5 mg kg–1 h–1 gallamine triethiodide (Flaxedil, Sigma-Aldrich) was administered intravenously. A warming blanket controlled by a rectal probe was used to maintain the rectal temperature at 37.2°C. Depth of anaesthesia was monitored by continuous recording of ECG and EEG. The criteria for an adequate depth of the anaesthesia were a stable ECG in combination with EEG signals in which large-amplitude oscillations and quieter periods appeared. To be able to react quickly in sudden cases of too light anaesthesia (an unstable heart rate, especially when intervening and an EEG in which high-amplitude oscillations were absent) a vaporizer with enflurane (Ethrane, Abbott) was installed in the respiration circuit. However, intervention with enflurane was not necessary in any of the experiments.
The pupils were dilated with atropine sulphate (1%, Ursapharm) and neosynephrine (5%, Ursapharm). The eyes were refracted and focused with contact lenses on the stimulus monitor at 114 cm distance. The contact lenses also protected the eyes against desiccation. Artificial pupils of 2 mm diameter were placed in front of the eyes.
A craniotomy was performed and tungsten-in-glass electrodes were lowered into the LGN. The layers from which we recorded were identified by the sequence of ocular input of the cells and from small lesions, produced by the injection of small electrical currents through the electrodes, made at the end of the electrode track. After the experiments, the animals were killed by an overdose of sodium pentobarbital (Nembutal). After perfusion, the brains were removed and prepared histologically to visualize the lesions.
The recordings were performed on 12 M-, 9 P-, and 5 K-neurons in the LGNs of three marmosets (two males and one dichromatic female). The population of cells partially overlapped the one used in a previous study (Kremers et al. 2004). The identification of the cell type was mainly based upon a histological reconstruction, and the depth of the microelectrode during the recording relative to its position (using microdrive readings) considering tissue shrinkage during the histological processing. The sequence of ocular input provided additional information.
Visual stimuli
The stimuli were presented on a BARCO monitor (CCID 7751 MKII; 100 Hz refresh rate) controlled by a VSG 2/2 graphic card (Cambridge Research Systems Ltd). The stimuli were very similar to those previously described (Kremers et al. 2004). Briefly, they consisted of a central circular patch and a surrounding annulus. In both centre and surround stimuli a 5 Hz luminance modulation was presented (mean luminance: 66 cd m–2; 50% contrast; mean chromaticity: (0.33,0.32) in commission Internationale de l'E clairage (CIE), 1964, large-field coordinates resulting from a 20, 40 and 6 cd m–2 mean luminance of the red, green and blue phosphors, respectively). Identical modulation was applied in the centre and surround stimuli but the relative phase between them was varied.
The location of the receptive field centre was determined by the position of the common edge between two identical but counter-phase 4 Hz modulating hemi-fields of a bipartite field stimulus that resulted in a minimal response (Kremers & Weiss, 1997; Lee et al. 1998). Bipartite stimuli with horizontal and vertical edges were used. The size of the centre stimulus (ro) was matched to the RF centre by maximizing the cell's response to a 4 Hz luminance modulated circular stimulus inside a counter-phase luminance-modulated annulus (for a detailed description of the procedure see Kremers et al. (2004)). The centre stimulus size for an optimal response generally is substantially larger than the RF centre size when expressed in 
of the centre Gaussian responsivity profile (Kilavik et al. 2003). The radii of the centre stimulus typically varied between 0.2° and 0.8°. The outer radius of the annulus stimulus was 5.1° (we use the notation ‘deg’ for phases and ‘°’ for sizes and positions given in arc-degrees).
The responses of each cell were measured to a series of combined stimuli that matched spatial location (0° displacement) and centre size of the RF. The measurements were performed at 12 different relative phases of the surround. The stimuli were nearly identical to those described before (Kremers et al. 2004). The measurements were repeated at either 8 or 12 different horizontal displacements (including the original zero displacement) between stimulus and RF (see Fig. 1A and B for a sketch of the stimulus configuration with relative sizes of RF and stimuli based upon the measurement with a K-off cell; m34u40). The magnitudes of the displacement were proportional to the stimulus centre size (ro) and varied between 0.35 ro and 2.45 ro (see Fig. 1B). The combined stimulus covered the complete RF at all displacements. The different displacements were presented in a quasi-random order.
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Data acquisition
Times of spike occurrences were recorded with 0.25 ms accuracy and stored on a CED 1401 data acquisition system (Cambridge Electronic Design Ltd). Synchronization between the stimulus presentation and data acquisition was provided by TTL-pulses from the VSG-card, which triggered the CED 1401. To avoid stimulus-onset artifacts, the responses to the first period of the sinusoidal stimulus presentation were disregarded. Spike occurrences were recorded during 6 s of stimulus presentation. During the measurements, the recording program constructed peristimulus time histograms (PSTHs) of the cells' responses.
Psychophysics
The results of the psychophysical measurements have been presented before (Kremers et al. 2004), where a full description of the stimuli and procedures can be found. Briefly, in these experiments combined centre and surround stimuli modulating at different relative phases were presented, and the perceived flicker strength in the centre stimulus was measured by comparing it with the perceived flicker strength in a test stimulus of the same size, mean luminance and mean chromaticity as the centre of the combined stimulus. The following procedure was employed: after a presentation of the combined stimulus, the observer could, by pressing a button, exchange it for the test stimulus. The observer was then required to communicate, by pressing buttons on a computer keyboard, whether the perceived flicker in the test stimulus was stronger or weaker than the perceived flicker in the centre of the combined stimulus. Then the combined stimulus was displayed again. The contrast in the next presentation of the test stimulus was decreased when the perceived flicker in the test stimulus was stronger than in the centre of the combined stimulus and vice versa. Initially the contrasts were changed in steps of 60% (from 100% to 40% or from 0% to 60%). At every reversal of direction of contrast change, the step size was halved (from 60% to 30% to 15%, etc). The perceived contrasts in the centre of the combined stimulus and in the test stimulus were considered to be equal when the step size in contrast change was less than 0.14x the actual contrast in the test stimulus. Using a PEST (Parameter estimation by sequential testing) procedure with two randomly interleaved staircases, the contrast in the test stimulus was changed until the perceived flicker strengths in the test stimulus and the centre of the combined stimulus were equal. The measurements were repeated for different temporal frequencies (4, 8 and 20 Hz), different sizes of the centre stimulus (1° and 0.4° diameter) and all combinations of 50% and 25% contrast in the two stimuli.
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Results |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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An example of original PSTHs of a typical response of an on-centre M-cell to counter-phase modulated centre and surround stimuli (5 Hz; 50% contrast) at 12 different displacements relative to its RF (specified above each PSTH) are displayed in Fig. 2. The response strongly depends upon the displacement between the RF and the stimulus: the response amplitude decreases with increasing displacement, is minimal at a displacement of about 0.36° and increases for larger displacements with a 180 deg shift in the response phase.
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Simulation of neuronal responses
The response pattern shown in Fig. 3 can be explained on the basis of a linear model that assumes rotationally symmetric Gaussian sensitivity profiles of the cell's RF centre and surround, and a phase delay between the RF centre and surround responses (Rodieck & Stone, 1965; Enroth-Cugell & Robson, 1966; Kremers et al. 2001; White et al. 2001; Kilavik et al. 2003). The total response of the cell,
, is assumed to be a vector addition of the RF centre (
) and surround (
) responses and, on the other hand, of the responses to the centre (
) and surround (
) stimuli:
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| (1) |
The components of the RF response to the combined stimulus were treated as complex numbers; then the complex total response R represents a sum of the respective complex response components:
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| (2) |
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| (3) |
is the phase difference between the RF centre and surround responses. By definition, is positive if the RF surround response is delayed relative to the centre response.
The total response depends on the spatio-temporal configuration of the stimulus, and the location of the stimulus relative to the RF, as well as the RF's intrinsic properties. Figure 1 displays an example of the configuration of the stimulus and the model description. The relative sizes are realistic and based upon data obtained from a K-off cell. To calculate amplitudes of each response component in eqn (2) at any displacement between the RF and the stimulus, the centre and surround Gaussian sensitivities (GC and GS, respectively) are convolved with the strength of the centre and the surround stimulus:
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| (4) |
) describes a polar coordinate system concentric with the stimulus; r' is the displacement (°) of the RF relative to the midpoint of the stimulus; ro is the radius (°) of the central stimulus; CC and CS are contrasts (%) of modulation in the central and surround stimuli, respectively.
The Gaussian sensitivity functions of the RF centre and surround (GC and GS) can be described in polar coordinates as follows:
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| (5) |
C and S are the RF centre and surround size (°) expressed as the standard deviations of the two Gaussians. The outer radius of the surround stimulus (5.1°) was large enough to cover about 2–5 x
S at the largest stimulus displacement (about 2°) and therefore can be considered to completely cover the rest of the RF.
Thus, our model (eqns (2–4)) provides a linear description of the responses to the combined stimulus with five independent parameters (AC, AS, C, S and ). The model assumes that the response depends linearly on stimulus contrast, thereby ignoring contrast-dependent non-linearities such as response saturation (Lee et al. 1990), contrast gain controls (Shapley & Victor, 1978, 1979), contrast-dependent non-linear interaction between the RF centre and surround (Kremers et al. 2004) and the effect of the inhibitory extraclassical RF (Solomon et al. 2002; Webb et al. 2002). The effects of some of the contrast-dependent non-linearities on the model are considered below.
Solving eqn (3) for the RF centre results in eqn (5):
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| (6) |
was introduced considering the substitution 
. Similarly the responses of the RF surround can be calculated. These equations resemble eqn (5) of Einevoll & Heggelund (2000). However, there are two differences between their model and ours. First, our stimuli are not a simple circular stimulus as was used in their experiments, but a combination of two stimuli that can have different phases relative to each other. Second, Einevoll and Heggelund assume that the RF centre and surround are completely antagonistic. We take into account an additional phase delay between the two.
The RF parameters AC, AS, C, S and are estimated for each individual cell on the basis of response data. There are two possibilities to obtain these estimates. First, the model can be fitted to the measured responses. This method is direct, but requires a sophisticated fitting procedure, with a risk that the fit gets ‘trapped in local minima’. The second possibility is that the parameters are estimated by fitting the model to an additional but simpler set of data and are subsequently imported into eqns (2–5). Although this procedure involves a second set of data, it is much simpler and less prone to errors (see below). We therefore chose the latter procedure and estimated the parameters on the basis of additional measurements using stimuli described by Kilavik et al. (2003) (in which the stimuli are centred on the RF and the size of the centre stimulus is varied; see Methods). In Fig. 4, the response amplitudes and phases of the same cell as in Figs 2 and 3 to these stimuli are shown, together with fits of the DOG model. The curves are fits of the DOG model to the data. The RF parameters were obtained from these fits.
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, which is the case when the first derivative equals zero. Thus, it can be calculated that:
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| (7) |
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Model predictions and response data closely resemble each other at small (0–0.14°) and large (0.36–064°) displacements. For intermediate displacements (close to the radius of the stimulus centre) the descriptions deviate more strongly, because in this range the response amplitudes are very sensitive to small mismatches between stimulus and RF.
To study how close the resulting model description for this cell is to the optimal description, we varied C, S and , and calculated the sum of squared distances in the complex plain (defined by eqn (2)) between model description and actual data. Figure 5 displays the sum of squared distances given as a function of the values of the three parameters. Because the parameters are not completely independent, the real optimal values may deviate slightly from the values at the minima. The arrows display the actual estimates. Clearly, the estimates are close to the minima, suggesting that the described procedure yielded adequate model descriptions of the cell data.
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The simulation of response amplitudes for different stimuli displacements was performed for 26 neurons. The model can be considered as a non-linear regression of the RF parameters with two independent variables (S and r'). The goodness of data description (simultaneously for the whole set of displacements between the RF and the stimulus) was compared with a simple average using an F test (Lvovsky, 1988):
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| (8) |
The total and residual mean squares are equal to, respectively
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| (9) |
1
=
n
– 1 and 2
=
n
– 3 are total and residual degrees of freedom, respectively. The values of F were calculated for each cell (see Table 1) and then compared to the critical value of F distribution, 
(Zar, 1999). The corresponding significance levels 
(1) are also given in Table 1. For an adequate prediction, the 5% level of significance was chosen.
The model could describe the data adequately for the majority of cells. Inadequate predictions ((1) > 0.05) were found for five cells (four M-cells and one K-cell). Possibly, response non-linearities are present in the responses of these neurons that cannot be captured by the linear model. These five cells were excluded from subsequent analysis.
No significant differences between the parameters C and S of M-, P- and K-cells were found. A strong overlap in these parameters for the different cell classes was found previously for marmoset LGN cells (Kremers & Weiss, 1997; Kremers et al. 1997), and the relatively small number of cells prohibits more definite conclusions on differences between cell types on the basis of the presented data. We neither found significant differences between the mean values of AC and AS of the different cell types. Probably the 50% stimulus contrast used in the present paper is high enough to induce response saturation in M-cells possibly counteracting the larger responsivity. Because of the small differences in response characteristics between the different cell types it seems likely that the responses of all cell types contribute to the perception of similar stimuli used in our psychophysical tasks (Kremers et al. 2004).
The model description not only describes the response of one ideal (noise-free) cell for different locations of the stimulus. It can also be regarded as a simulation of the response output of an array of identical and ideal cells with different RF locations selective to a fixed stimulus. As mentioned before, the main reason for the simulation is to obtain an estimate of the response output of an LGN cell array and to link physiological and psychophysical data.
Because in the psychophysical experiments, the subjects were asked to fixate the central stimulus, a foveal cell array should be considered. The averaged properties of a foveal array were calculated from the cell parameters with retinal eccentricities smaller than 5.5°. Parameters of 7 M-, 3 P- and 4 K-cells that were included in the analysis are highlighted in bold in Table 1. The mean values for the foveal array, given in the bottom row of Table 1, were implemented in our model (eqns (2–5)). The model responses can be considered as the output of the LGN and the input of the visual cortex during the psychophysical experiments. We averaged data of M-, P- and K-cells because the response amplitudes were not significantly different in these cell types.
As the model predicted the cell responses for 5 Hz stimuli (see Methods), we recalculated the surround phase lag for 4, 8, and 20 Hz stimuli, which were used in the psychophysical experiments. Based on the data of Kilavik et al. (2003) and Kremers et al. (2004), a fixed 6 ms time delay of the RF surround relative to the centre was assumed here. The recalculated values of are shown in the upper row of Table 2. The predicted response amplitude of an averaged foveal array to 4 Hz stimuli with 0.5° centre stimulus radius, a stimulus size used during the psychophysical experiments, is displayed in Fig. 6. This stimulus size is much larger than the RF centre size expressed as C. But, as discussed in the Methods, the optimal centre stimulus radius is generally substantially larger than C. With this stimulus, the responses of the cells strongly depend on relative phase at zero displacement. The model predicts that when the centre stimulus is very large, thereby covering a larger part of the RF, the response is scarcely modulated by the relative phase at small displacements (for the same reason it is not modulated at large displacements in Fig. 6). As a result, in the LGN cell array, responses only of the RFs located closely to the border between centre and surround are modulated by the relative stimulus phase.
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In the above-presented cell data, all recordings were performed with equal (50%) contrast in the centre and surround. However, the psychophysical measurements presented by Kremers et al. (2004) included all combinations of 25% and 50% contrasts in the stimulus centres and surrounds.
We previously found that non-linear interactions between RF centre and surround should be taken into account when the contrast in one of the two subfields is altered (Kremers et al. 2004). We described two types of non-linearities. First, an increase in the contrast of the surround leads to a decrease of the response amplitude of the RF centre (AC). An explanation for this decrease may be that the large surround stimulus activates the inhibitory extra-classical RF (Einevoll & Heggelund, 2000; Solomon et al. 2002). The second non-linearity involves a decrease in the phase lag of the RF surround response when the response amplitude of the RF centre is increased (caused by an increase of contrast in the centre stimulus). In other words, when the response amplitude in the RF centre is increased, the RF surround responds earlier. This non-linearity therefore has an effect on the phase difference between RF centre and surround responses ().
As a result of these non-linearities, the estimated RF parameters of AC and , used in the simulations of the output of the LGN cell array, should be corrected when stimulus contrast in the centre or the surround stimuli is altered (as is the case for the stimuli with unequal contrasts that were used in the psychophysical measurements). On the basis of our previous data (Kremers et al. 2004), we estimated that when the surround stimulus contrast is decreased from 50% to 25% contrast while keeping the centre stimulus at 50%, the RF surround will respond 14.1 deg earlier. Decreasing the centre stimulus contrast from 50% to 25% contrast and keeping the surround stimulus at 50% results in a decrease of the RF centre response by a factor of 1.6. Table 2 gives the estimated RF parameters of the foveal LGN cell array for the different contrast combinations used in the psychophysical experiments. The RF parameters, that were changed on the basis of the mentioned non-linearities, are shown in bold in the second and third rows.
A cortical detector
Next, a cortical decision mechanism has to be introduced so that the array responses can be linked with the psychophysical data. A simple possible detector will be described in this section.
Figure 7 (left plot) shows the output of the proposed LGN array as a function of the displacement between the positions of the cells within the array and the stimulus plotted for four different phase differences between centre and surround stimuli (both containing 50% contrast modulation). Thus, these functions correspond to slices of the 3-D plot shown in Fig. 6. When the centre and surround stimuli are modulated in-phase (0 deg phase difference), the responses are identical at all positions because in that case the combined stimulus is identical to a full field modulation that covers the complete RF for all displacements. The responses depend most strongly on the spatial position when the stimuli are modulated in counter-phase (180 deg phase difference). The two other curves correspond to relative stimulus phases of –60 deg and +60 deg. Please note that they are not identical. This is caused by phase delay of the RF surround response.
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The right plot in Fig. 7 shows the output of the detector as a function of the relative stimulus phase (S). The output is not symmetric around the y axis, and is similar to the psychophysically measured perceived flicker strength at similar stimulus conditions (Fig. 8).
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The results of the psychophysical measurements have been presented before by Kremers et al. (2004). We summarize here the data to compare them with the output of the cortical peak-to-trough detector.
In Fig. 8, the perceived flicker strengths (means ± S.D. of the results of measurements with three different observers) are displayed as a function of relative phase between centre and surround stimuli, plotted separately for two different temporal frequencies (4 Hz and 8 Hz; upper and lower plots, respectively) and for the different contrast combinations in the centre and surround stimuli. Centre size (diameter) was 1°. The data for the 20 Hz stimuli and for the 4 Hz and 8 Hz data with the 0.4° centre stimulus were very similar.
It was assumed that the output of the described detector (expressed in spikes s–1) was followed by a Naka-Rushton-type of saturation and a threshold mechanism, after which the output was proportional to the perceived flicker strength in the central stimulus.
The psychophysical data was fitted using the Solver routine of the Excel 2000 program with the following formula:
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| (10) |
is the threshold. Three free parameters (Qm, b and ) were used to fit predicted values of Q to the psychophysical data. Qm was constrained to values between 50% and 200%, because values outside this range were not realistic. The solid curves in Fig. 8 display the fits. For the conditions in which the contrast in the centre and the surround stimuli were equal, the fits were performed on the basis of the linear model output of the LGN array with parameters given in the upper row of Table 2 (continuous curves). For those conditions in which the two stimuli had unequal contrasts, fits based on both the linear model (continuous curves) and on the model corrected for the non-linear centre–surround interactions (dotted curves) were performed (parameters of the latter are shown in the second and third rows of Table 2). It is obvious that in these conditions the goodness of fit increases when the non-linearities are included.
The goodness of the fits was assessed by an F test as described above (eqns (8) and (9). The values of F were calculated for each stimulus condition and given next to each plot in Fig. 8. The critical value 
for the significance level (1) = 0.05 is 2.66 (Zar, 1999). The values of F' quantify the goodness of the fits based on the non-linear model.
In Fig. 9 the values of F and F' are given as a function of temporal frequency for all fits of the detector output to the psychophysical data. Adequate fits were obtained for all conditions where the two stimuli had equal contrast. However, most fits with unequal contrasts in centre and surround did not reach significance level (the drawn horizontal lines in the plots) when the fits were based on the linear model. After correction for the non-linear RF centre and surround interactions, the goodness of most fits improved substantially and was above significance level for nearly all conditions. The goodness of fits generally decreases with increasing temporal frequencies. The physiological data were based upon 5 Hz stimuli. Thus, the use of higher temporal frequencies (8 Hz and 20 Hz) may have introduced additional factors that were not captured by our model.
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Discussion |
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In accordance with our previous proposition, the data suggest that the physiological basis for the psychophysical lateral interactions in the perception of flicker may already be present at the level of the LGN. In agreement with this proposal, we have recently found that the spatial extent of the interactions in flicker perception is similar to that of the response interactions between centres and surrounds of LGN cells (Kremers & Rimmele, 2007). The proposal that the psychophysical data are determined by the interactions between RF centres and surround responses of subcortical neurons implies that the psychophysically measured lateral interactions are not present when the two stimuli are presented dichoptically. The results of preliminary experiments performed in our lab are in agreement with this prediction. This might be another distinction between the studied lateral interactions and the contrast induction effects; the latter can clearly be transferred interocularly (Singer & D'Zmura, 1994).
To be able to link LGN cell data with psychophysical data, a cortical detection mechanism has to be introduced. We propose a cortical spatial peak-to-trough detector. We do not claim that this detector is the only possible cortical mechanism that will be able to link the two data sets. Instead, any mechanism, the output of which is based upon a difference in response amplitude between cells with different positions within the LGN array, may be conceivable. The exact nature and locus of the cortical spatial peak-to-trough detector is not clear. There are different possibilities of spatial interactions playing a role in the responses of cortical cells even in the first area of the visual cortex (V1). The spatial interactions in area V1 can be based upon mechanisms within the classical RF of the neurons or through interactions between the classical RF and a modulatory non-classical RF surround (Sceniak et al. 1999; Angelucci et al. 2002). We have recently provided psychophysical evidence (Kremers & Rimmele, 2007) that the spatial extent of the lateral interactions in the perception of flicker is of the size of the RF surround of LGN cells as well as of the classical RF of V1 cells (Angelucci et al. 2002). Thus, even if the proposed model is only partially valid, it may be a working hypothesis that can be tested by cortical physiology.
The spatial location of the response maximum and minimum in the foveal cell array is about 0.5° (see Fig. 7, left plot). This is within the range of the sizes of the classical RF of V1 neurons at this location and for the stimulus contrasts used in the present paper (Sceniak et al. 1999; Angelucci et al. 2002). It is conceivable that a double opponent process within the classical RF of these cells may represent (a part of) the physiological basis of the spatial peak-to-trough detector. A model similar to that proposed by Heeger et al. (1996) may be an adequate model for the spatial-peak-to-trough detector or differentiator, because it assumes that neural inputs over a local spatial region are linearly processed. Furthermore, the model of Heeger et al. (1996) includes a threshold and normalization process (in which the response of a neuron is divided by the pooled activity of a large number of neurons in the cortical vicinity). As a result of the normalization, the responses of the cortical cells saturate. Thus, the proposed model may be a special case of the more generalized model of Heeger et al. (1996). Figure 10 displays a scheme of the model we employed to link physiology with psychophysics. A qualitative similarity with model of Heeger et al. (1996) can be observed.
The response of the LGN cell array is defined by the first harmonic component of the firing rate of the cells, because we used a periodically modulating stimulus. We found that on- and off-centre cells respond in a very similar manner when using this definition of response amplitude. But on- and off-centre cells respond in counter-phase. Our analysis demonstrates that a cortical detector that is solely based upon the response amplitudes of the LGN cells may give an appropriate output. Any detector that takes into account the response phases of on- and off-centre cells in a ‘push–pull’ fashion might give an equivalent output.
As is often the case in linking physiological with psychophysical data, our previous proposition (Kremers et al. 2004) is based upon similarities in the data. As is argued by Teller (1984), three aspects make such a linking proposition problematic. These aspects are: range of applicability, the assumption of homogeneity and ‘peripherality’. These three aspects are addressed in the proposed model so that the proposition is much stronger now.
The first aspect is the range of applicability: it is important to know for which stimulus conditions the linking proposition holds. A proposition is more important and, possibly, more fundamental if the range of applicability is extended. Otherwise, the similarity could be based upon mere coincidence or the proposition is relatively ad hoc. In the present paper we have been able to link physiological and psychophysical data for an extended range of conditions. By including contrast-dependent non-linearities, the range of applicability was substantially extended. Therefore our proposition is not ad hoc but can hold for a relatively extended range of stimuli.
The second aspect, homogeneity, addresses the fact that cells are not all the same (Gomes et al. 2005). The effects of noise and of differences between individual cells should be considered. The LGN cells within the array were implicitly considered to be identical and noise free, and the array was completely homogeneous. In reality, there are individual differences between cells (Table 1 gives an impression of this variability) and the responses are not completely noise free. But, the response properties of nearly all LGN cells can be described by a DOG model. As a result, the basic output of an LGN array will be similar to the one described by Fig. 6 despite the variability in RF parameters. It therefore can be expected that the psychophysical tasks that are based upon this output will also have characteristic properties independent of the variability in cell properties. Nevertheless, a more systematic analysis of the expected output of the model, by changing single properties of the cell array, may give a better insight into the applicability of the model. For instance, the effect of a change of RF size may provide a prediction of the influence of eccentricity on the psychophysical data. Moreover, the question of whether the different subcortical pathways (M, P or K) have different function in coding the stimulus has only briefly been addressed. A more extended analysis should also consider the differences between the different pathways. Such an analysis is, however, beyond the scope of the present paper.
The third aspect, which Teller (1984) called ‘peripherality’, addresses the problem that linking propositions are often based upon response similarities, and implicitly assumes that a central mechanism maintains this response pattern. Of course this problem especially concerns linking propositions based upon responses of subcortical cells. We propose a simple cortical process before the ‘bridge locus’ (defined by Teller & Pugh (1983) as a set of neurons whose activities form the immediate substrate of visual perception) that indeed can maintain the response properties found in the LGN. With a simple cortical peak-to-trough detector, together with subsequent threshold and saturation mechanisms, it is possible to describe the psychophysical data that showed a close resemblance with the LGN physiology.
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) and standard deviations (vertical bars). For each subject, the thresholds were measured six times. The continuous curves are the fits of the cortical peak-to-trough detector, including saturation and threshold, to the psychophysical data. The results of F tests are given next to respective graphs. Predictions with F > 2.66 were at the 5% level of significance and qualified as adequate. The dotted curves in the plots for which the centre and surround contrast were not equal (lower four plots) are fits after recalculations with corrected RF parameters; the goodness of these fits is given by the value of F'.
