A: the center–surround receptive field (RF) organization may constitute a detector for contrast boundaries or contrast envelopes. Solid and broken ellipses indicate the classical receptive field (CRF) and the surround, respectively. A bar inside the solid circle indicates the preferred orientation of the CRF. Overlapping horizontal gratings show the optimal grating component for the neuron. B: the elongated CRF and surround are more suitable to detect orientation of contrast boundaries. C: a partial set of contrast-modulated gratings for probing the center–surround structure used in this study. Not all stimuli used in the experiments are shown. For a given orientation of the contrast envelope, its spatial frequency varied from 0 to near the carrier (fine black and white luminance variations) spatial frequencies.
Main steps for fitting the 2-dimensional (2D) difference-of-Gaussian (DOG) model to the obtained data. See text for the details. A and B: an example of amplitude and phase transfer functions for one selected envelope orientation. C: the amplitude and phase data for each spatial frequency can be represented as points in a polar plane, where they (amplitude and phase) are indicated by radial length and angular orientation of the points, respectively. Since x and y coordinates of data points in this plane can be regarded as real (Re) and imaginary (Im) components of complex numbers, x-axis and y-axis are labeled as Re and Im, respectively. D: amplitude and phase data for A and B are represented in the polar plane. E: a one-dimensional (1D) spatial profile obtained by applying the 1D inverse Fourier transform to data in A and B. F: the spatial profile in E (black curve) was fitted by a 1D DOG function (gray dashed curve). G: predictions of a 2D DOG model (gray curves) are fitted to data shown in D. Note that the 2D fitting was actually conducted for the full data set simultaneously (as shown in Fig. 3), although only data for one envelope orientation are shown here for clarity.
A typical example of response measurement and reconstruction of the center and surround structures. Data for one neuron (complex cell) are shown. A: initial size-tuning measurement using conventional luminance gratings. This neuron showed a strong surround suppression. Error bars show the SE. deg, degrees. Dotted line: spontaneous firing level. B: orientation-tuning measurement for the contrast envelope. Stimuli (top) and responses (bottom) are shown. Orientation of the contrast envelope was varied in 30° steps over 180°. The carriers were always fixed at the optimal parameters for the CRF (90° and 0.22 cpd). The size of the grating patch was 15°, so that it covered both the CRF and the surround. The contrast envelope was drifted. Response amplitude modulated at this drift temporal frequency (F1 components) is plotted. Responses to the 2 opposite directions were averaged. C and D: response measurements to the full set of the contrast-modulated gratings (Fig. 1C). Amplitude and phase of the F1 components of peristimulus time histograms (PSTHs) are plotted as a function of the envelope spatial frequencies in C and D, respectively. The signs of the spatial frequency indicate 2 opposite directions of the envelope drift. Color depicts different envelope orientations: red corresponds to the peak envelope orientation (90° for this cell); green, blue, and black indicate progressively increasing orientations (in 45° step). cpd, cycles per degree; rad: radian. E: a subset of PSTHs for contrast-modulated gratings of vertical contrast envelopes. The PSTHs are ordered for increasing envelope spatial frequencies (0–0.25 cpd) from bottom to top. F1 components of these PSTHs correspond to solid red points in C and D. F: amplitude and phase data shown in C and D are represented by the length and angle of a position vector indicated by each dot (see Fig. 2C). Open and filled symbols indicate, respectively, positive and negative spatial frequencies of the stimulus, as they are in C and D. Solid and broken curves indicate response prediction of the DOG model with its spatial structure shown in G. Broken and solid curves, respectively, correspond to positive and negative spatial frequencies and are expected to fit the open and filled symbols, as they really are. G: the DOG model as a reconstructed center–surround structure. Red and blue regions indicate the excitatory and suppressive regions, respectively. The x-axis of the central panel corresponds to the absolute horizontal axis in the visual field. The scale bar represents the amplitude of a filter. Green lines in the margins show 1D structures for 2 selected axes (x- and y-axes for this neuron) obtained by the 1D inverse Fourier transform applied to the data in C and D. Overlapping broken lines are structures obtained by collapsing the 2D structures along the x- and y-axes. Note that the CRF and surround regions are both elongated along the vertical axis. This center–surround organization is suitable for detecting vertical high-order borders. (cell number: 05-4A_ep015_10)
A–D: center–surround structures are depicted for 4 neurons. Figure convention is the same as that of Fig. 3G, except for color-coding of the maps. Here, bright and dark colors indicate the CRF and surround regions, respectively; x- and y-axes indicate the absolute horizontal and vertical axes, respectively. Preferred (carrier) orientations of the CRF, as indicated by black lines drawn within the CRF regions for A–D, were 90, 13, 30, and 17°, respectively (0 is horizontal). 1D inverse-Fourier reconstructions for 2 axes are shown around the 2D maps by gray curves. The profiles are rotated to indicate the axes along which these profiles were obtained. These axes roughly correspond to the optimal envelope orientations (90, 30, 110, and 30° for A–D) and their perpendicular ones. Note that the positive (CRF) and negative (surround) gray scales are generally different to clarify the weak surrounds, as indicated to the right of scale bars. R2 values of data fitting for reconstruction in A–D are 0.88, 0.86, 0.76, and 0.72, respectively. Note that the center and surround regions of the neurons in A and B are elongated in a parallel manner. Such structures are suitable for encoding orientation of high-order borders. Cell types in A to D, respectively, are simple, simple, complex, and simple.
Center–surround organizations generally showed elongated structures. A: a distribution of concentricness, which is defined as an angle subtended by the surround (defined as regions with suppression stronger than 10% of maximum suppression) at the peak location of the CRF (θ in the inset). If this value is 360°, the cell has a concentric surround. The majority take values <180°. The median value is 158° (n = 35). B: distribution of elongation index for the CRF. This index is defined as the ratio b/a in the inset, where “a” represents the width of the CRFs along the axis connecting the peaks of the CRF and surround and “b” represents the length of the CRFs along the axis perpendicular to the width axis. Note that the CRFs of the majority of cells are elongated along the length axis (mean = 1.5). Filled bars indicate the index is significantly >1. C: distribution of elongation index for the surround regions (n = 32). The surrounds of the majority of cells are also elongated along the length axis (mean = 1.9). D: relationship between the CRF and surround elongation (n = 32). They are highly correlated, indicating the CRF and surround are elongated to a similar degree (r = 0.67, P < 0.01). E and F: distribution of the absolute difference between the preferred CRF orientations (from conventional orientation-tuning measurements) and the orientation of the elongation axis of the CRF regions (E) and that of the surround regions (F) (measured from the center–surround reconstructed structures). Note the wide spread of these distributions, suggesting that the center–surround elongation axes are not related with the preferred CRF orientations. G and H: relationship between the suppression index and CRF elongation (G) and surround elongation (H). In both cases, there are significant positive correlations.
Neurons that showed highly elongated center–surround structures. Figure conventions are the same as those in Fig. 3. Neurons in A and B had the highest and the second-highest elongation indices of the data samples. Both neurons were simple cells.
A–C: reconstructed center–surround structures can predict neurons' orientation-tuning curves measured in independent orientation-tuning measurements (e.g., Fig. 3B). Predictions are based on the Fourier transform of the reconstructed structures. A: orientation-tuning index. Of 35 reconstructed structures, we analyzed 31 neurons that produced sufficient responses (>4 spikes/s) in independent orientation-tuning measurements. Note that prediction from the reconstructed structures matches actual data, including those for neurons in Fig. 6 (indicated by arrows) (r = 0.87, P < 0.01). B: peak envelope orientation (r = 0.79, P < 0.01, n = 31). C: tuning width at half-height. Only neurons for which minimum response is <50% of the maximum response for both predicted and actual tuning curves were selected for analysis (n = 10, r = 0.71, P < 0.05). D–F: relationship between the elongation of center–surround structures and orientation-tuning properties. D: the tuning width (at half-height) predicted from the 2D structure is plotted against the elongation index (n = 16 for which the predicted orientation-tuning curve showed ≥50% response modulation). Solid and open symbols indicate the CRF and surround elongations, respectively. Note that the more elongated structures generally have narrower orientation tunings. E and F: the orientation-tuning index of each neuron was plotted against the CRF elongation index (n = 31, E) and the surround elongation index (n = 31, F). The elongation of the structures (both CRF and surround) is positively correlated with the orientation-tuning index.
Analysis of envelope-orientation tunings, for a larger population of neurons (n = 99) than that in Fig. 6. Orientation-tuning curves for the contrast envelope obtained in the initial independent measurement (e.g., Fig. 3B) were analyzed. A: a distribution of orientation-tuning index. Filled portions of bars show neurons with tuning curves that were judged to be statistically significant by ANOVA (P < 0.05; n = 55). B: a distribution of peak envelope orientations for 48 neurons with an orientation-tuning index >0.3. Note that these neurons, as a population, represent various orientations for the contrast envelope. C: a distribution of absolute differences between the optimal envelope orientation and preferred orientation of the CRF (for the carrier) (n = 48), indicating that surround is located at various axial positions with respect to the CRF.
Symmetry of the center–surround structures. A: a distribution of spatial phases of Gabor functions fitted to 1D sections of the center–surround structures that pass the peak and trough locations. Zero degrees indicates an even-symmetric structure, whereas 90° indicates an odd-symmetric structure. The Gabor function always provided a good fit (R2 was >0.95 for 33 neurons). B: relationship between the symmetry and the concentricness index.
Spatial-frequency tunings of the center–surround structure. Among the 8 or 4 amplitude tuning curves for different envelope orientations obtained in the main run (Fig. 3C), we picked one tuning curve in which maximum responses were obtained. A: a distribution of the index for the degree of band-pass tuning, defined as (peak response − response at 0 cpd)/(peak response + response at 0 cpd) (mean = 0.44, SD = 0.23, n = 98). B: a distribution of the peak spatial frequency (mean = 0.22, SD = 0.15, n = 71). C: ratio of carrier to envelope spatial frequency (mean = 2.1, SD = 0.9, n = 71). D: a distribution of the direction tuning index, which is defined as (Rp − Rnp)/(Rp + Rnp), where Rp and Rnp are peak amplitudes for the preferred and nonpreferred (null) directions, respectively. The mean value of the index for the population was 0.13 (SD = 0.11, n = 98). These data show that neurons with surround suppression can encode a sufficient range of spatial frequencies of high-order borders, but are insensitive to their direction of movement.
Comparison of the reconstructed structures with response profiles to small patches in the surround. Results of 2 neurons are shown. Each row shows data for one neuron. The neuron on the top row is the one shown in Fig. 6B. A and C: stimulus configuration is overlaid over the reconstructed structures. The surround patch was presented at one of 8 locations indicated by the broken lines, whereas another optimal stimulus patch was always presented to the CRFs. Solid circles indicate the center-patch location. Diameters of the center and surround patches for A and C are 4 and 6.5°, respectively. B and D: response profiles to the surround patches are shown in polar planes where the angular orientation of each data point (squares) corresponds to the location of the surround patches and its radial length indicates the response magnitude. The firing rate indicated by the outer circle is shown at the bottom right. The inner circle represents the firing rate for the CRF stimulus presented alone. Therefore the points inside the inner circles indicate suppression from the surround patches. Predictions for the response profiles based on the reconstructed structures are also indicated by the broken curves. They were calculated by summing volumes of the reconstructed structures within the center patch regions and surround patch regions (60 evenly separated locations with equal eccentricity, including 8 real locations). The overall size of the prediction profiles is adjusted so that prediction for the center patch alone matches the real response. Note that the reconstructed structures shown in A and C are highly consistent with the response profiles (solid squares) shown in B and D, respectively, which is indicated by similarity between the responses profiles and their predictions (broken curves).
Comparison of the size-tuning curves actually measured and predicted from the reconstructed structures. A–C: results for 3 neurons. Actual data and predictions are shown by thin black and thick gray lines, respectively. A: a neuron with a small CRF (1.2°). B and C: neurons with medium to large CRFs. Generally, the 2 curves are similar with peak (white and black squares) and asymptotic diameters (circles) that are often matched, respectively. Error bars show the SE. D: comparison of the peak diameters of the predicted and measured size-tuning curves for the population. These 2 values are mostly similar. E: comparison of the diameters at the asymptotic response between the predicted and actual size-tuning curves.
Cover: Electrophysiological and morphological measurements were obtained simultaneously from a single corticospinal neuron. These data served as constraints on evolutionary optimization, generating a family of corticospinal models. A three-dimensional reconstruction serves as the backbone for a pseudo-color visualization of synaptic efficiency as a function of dendritic location, simulated in a single biophysical model selected from the family of optimal individuals. Excitatory synapses at yellow dendritic locations resulted in the largest depolarizations at the soma, while the same synaptic activation at purple locations generated only weak somatic depolarizations. This visualization is surrounded by scatter plots representing the evolutionary optimization: biophysical models optimized across different fitness functions demonstrate tradeoffs between full high-dimensional error (y-axis) and individual error scores (individual x-axes; clockwise order from top
left: subthreshold error, instantaneous firing rate error, spike-shape error, average firing rate error). Color based on 5 error percentiles in increasing instantaneous firing-rate error (purple, red, dark orange, light orange, yellow). From Neymotin SA, Suter BA, Dura-Bernal S, Shepherd GMG, Migliore M, Lytton WW. Optimizing computer models of corticospinal neurons to replicate in vitro dynamics. J Neurophysiol; doi:10.1152/jn.00570.2016.