A, Schematic diagram of the LSRC procedure (see Materials and Methods for details). C–E, By calculating a cross-correlation between the spike train (D) and spectra of Gaussian-windowed stimuli (C), we obtain a two-dimensional frequency tuning for the given subfield (E). F, By changing iteratively the center position of the Gaussian window, we can obtain a spatial matrix of the two-dimensional frequency tunings, corresponding to the matrix of localized areas of analysis shown in B. The strongest local spectral tuning map is indicated by an asterisk and is shown enlarged on the right. In each of these spectral maps, facilitatory and suppressive responses are shown by red and blue, respectively, according to the scale bar (suppression is essentially absent for this neuron). These conventions are used for subsequent figures. By nature of the Fourier transform, the tunings are symmetric about the origin. 2D, Two-dimensional; freq., frequency; deg, degree.
Results are shown of simulations conducted for four types of model neurons. The leftmost drawings depict structures of the models, and the right drawings show the simulated results from the LSRC and the standard reverse correlation analyses. In linear receptive field (RF) maps shown in the rightmost column, ON and OFF subregions are indicated by green and red, respectively, according to the scale bar at the bottom. The linear receptive fields in these simulations and experiments are obtained from the same data set as that used for LSRC. The vertical dashed lines in C indicate the border between the two complex cell components comprising the final output. D, a and b represent facilitatory and suppressive inputs, respectively. deg, Degree.
Benefits of the LSRC analysis in studying a higher-order neuron with high threshold are illustrated by simulation. A, Schematic diagram for a model neuron. The model is a spatially inhomogeneous neuron with two complex-type subunits as in Figure 2C, but the subunits have a relatively high firing threshold that prevent the cell from firing unless strong excitatory input is given. The simulations of the LSRC analysis were conducted for two different conditions: B, Full-field noise array that stimulates both subunits simultaneously. C, Half-field noise array in which the right half of the stimuli was masked, providing stimulation of only one of the two subunits. The neuron does not respond unless the entire receptive field is stimulated.
Local spectral selectivities are depicted for a simple (A–C) and a complex (D–F) cell. A, A spatial matrix of local spectral selectivity maps for a simple cell in area 17. Each individual plot shows a tuning property in a two-dimensional spatial frequency domain for the corresponding spatial subfield. The spectral selectivity maps are arranged to reflect the spatial positions of the corresponding subfields. B, A detailed profile of the most responsive subfield indicated by an asterisk in A. C, A linear receptive field profile calculated from the same data as for A. D–F, Data for a complex cell in area 17 in the same format as that for A–C. deg, Degree; freq., frequency; RF, receptive field.
Local spectral tuning data are shown of a cell that exhibits substantial spatial inhomogeneity of orientation and spatial frequency tuning within its receptive field. A, Local spectral selectivities for a complex cell in area 17. B, C, Detailed tuning properties in A, as indicated by numbers (1 and 2). D, A linear receptive field (RF) profile of the cell. E, A spatial arrangement of orientation tunings obtained from A, in which the orientation of the bars indicates the preferred orientation of the corresponding subfields. Only the data for subfields that contain significant signals are shown (p < 0.01; Bonferroni corrected). This cell was nondirection selective, based on a test with conventional drifting grating stimuli. deg, Degree; freq., frequency.
A population summary is shown of the spatial inhomogeneity of tuning parameters across the receptive fields. The horizontal axis indicates the maximum difference of the preferred orientations among multiple local spectral tuning maps. The vertical axis indicates the same for spatial frequency. Open and filled symbols indicate data for cells in area 17 (A17) and area 18 (A18), respectively. The arrow indicates the cell shown in Figure 5. For this figure, 178 neurons with bandpass spatial frequency tuning profiles and with more than two significant frequency domain maps are included for computing differences of tuning parameters. Two ellipses represent 95 and 99% confidence limits of variations from simulations as a control, assuming homogeneous properties (see Results for details). deg, Degree.
Spatial phase sensitivities are calculated for a simple cell (A–C) and a complex cell (D–F). A, Fourier coefficients for the maximally effective spatial frequency component in stimuli that led to spikes are shown (red dots) for the optimal correlation delay of 45 ms. The centroid of red dots is offset from the origin indicating selectivity for a given phase. For estimating the distribution for the noise stimuli themselves, Fourier coefficients for the same frequency component are also shown for all frames of the subfield of noise sequence (gray dots). Only the coefficients for one frequency component for the maximally responsive subfield are used. B, Spatial receptive field map obtained by a standard reverse correlation procedure. C, Spatial structure of the phase dependency. The optimal spatial phase, PSI (see Results), and signal amplitude of the LSRC analysis are represented as hue, saturation, and brightness, respectively. D–F, The same as A–C, but for a complex cell. The two cells are the same as those shown in Figure 4. For C and F, the maximum values for the Z-scores were 34.1 and 17.5, respectively. coef., Coefficient; imag, imaginary; deg, degree; RF, receptive field; max, maximum.
A relationship is shown between the PSI and the conventional modulation ratio (F1/F0). The histograms at the top and the right show distributions of these indices separately. There is a significant correlation between the PSI and F1/F0 ratio (p < 0.01; test for the Spearman’s correlation coefficient).
Data from cells with suppressive responses are shown for two complex cells in area 18. A, A result of the LSRC analysis. The reddish regions (Z-score >0) indicate facilitatory components, whereas the bluish regions (Z-score <0) show suppressive components. B, Magnified tuning for a local spectral map, indicated by an asterisk in A. C, Linear receptive field profile. D, An orientation tuning curve obtained by a conventional drifting grating test. The dashed horizontal line indicates the spontaneous firing rate. E–G, Data from another neuron. E, Amplitude and phase map for facilitatory responses in the same format as that of Figure 7, C and F. The facilitation was evaluated at the spatial frequency and orientation of 0.33 cycles/degree and 18°. F, Suppression map is shown using the Z-score only. The suppression was evaluated at the spatial frequency and orientation of 0.23 cycles/degree and 117°. G, Linear receptive field for the second neuron. deg, Degree; freq., frequency; RF, receptive field.
Two-dimensional spatial frequency tunings are shown for four additional cells that exhibit significant suppressions in the LSRC analysis. A, A two-dimensional spatial frequency tuning for the subfield with the strongest suppression for a complex cell in area 17. B, A similar tuning map for a simple cell in area 17. C, D, Similar plots for two complex cells in area 17. OR, Orientation; SF, spatial frequency; deg, degree; cpd, cycles per degree.
Simulation parameters for estimating inhomogeneities of methodological origin
Carrier orientation (°)
Envelope orientation (°)
Carrier phase (°)
Spatial frequency (cycles/stimulus area)
Envelope aspect ratio
Ranges of parameters are shown for simulations for determining confidence intervals of interreceptive field variations (see Fig. 6), that might be induced because of the LSRC procedure. See Results for details. SF, Spatial frequency.