A complex-cell receptive-field model

The time course of the response of a single cortical neuron to counterphase-grating stimulation may vary as a function of stimulation parameters, as shown in the preceding paper. The poststimulus-time histograms of the response amplitudes against time are single or double peaked, and where double peaked, the two peaks are of equal or unequal amplitudes. Furthermore, the spatial-phase dependence of cortical complex-cell responses may be a function of spatial frequency, so that the receptive field appears to have linear spatial summation at some spatial frequencies and nonlinear spatial summation at others. In the first part of this paper, we analyze a model receptive field that displays this behavior, and in the second part experimental data are presented and analyzed with regard to the model. The model cortical receptive field in its simplest form contains (two rows) of geniculate X-cell-like, DOG (difference-of-Gaussians)-shaped, center-surround antagonistic, circular-input subunits. We propose nonlinear summation between these two subunits, by introducing a half-wave rectification stage before pooling. The model is tested for the responses it predicts for the application of counterphase-grating stimulation. This simple model predicts the appearance of three response forms as a function of counterphase-stimulation parameters. At periodic spatial frequencies the expected-response histogram has a single peak, whose amplitude has a sinusoidal dependence on spatial phase. At spatial frequencies halfway between these, the expected-response histogram has two equal peaks whose amplitudes have a full-wave rectified sinusoidal dependence on spatial phase. At all intermediate spatial frequencies the expected-response histogram has a 'mixed' form; the histogram appears sometimes with one peak, sometimes with two equal peaks, and generally with two peaks of unequal amplitude, as a function of spatial phase. Null responses are expected to appear at specific spatial phases only for the periodic spatial frequencies that give 'pure' response time courses as in paragraph 5 above, and not in the more common mixed response case of paragraph 6. The analysis procedure described in the preceding paper is used, separating the odd and even Fourier components of the response histograms reflecting the receptive-field intrasubunit linear summation and intersubunit nonlinear summation, respectively. We propose that this model may be used as a working hypothesis for the analysis of these aspects of the various cortical receptive-field types. Experimental data are described and discussed in terms of the model. Simple cells have a single receptive-field subunit, i.e., they have linear spatial summation of the excitatory and inhibitory influences in different parts of their receptive fields. 'Mixed' cells have two spatially disparate subunits. Their responses show all the phenomena predicted by the model including the spatial-phase and spatial-frequency dependences of the response time course and the variation with spatial frequency of the spatial-phase dependence of the amplitudes of the odd and even Fourier portions of the response. 'B' cells have two entirely overlapping subunits, one on-center and one off-center. The summation between these subunits is nonlinear (pooling follows rectification) leading to the on-off nature of the responses, but the spatial coincidence of the subunits makes the spatial-phase dependence appear similar to that of cells with linear spatial summation. 'Intermediate' cells have two nonlinearly summing spatially disparate subunits each of which is similar to a B-cell receptive field. Thus the response of these cells is always on-off, but the variation with stimulation parameters is complex. As a final test of the model, the dependence on spatial frequency of the ratio of the amplitude of the even Fourier portion to that of the odd Fourier portion is measured for the different cortical cell types. As predicted by the model, this ratio varies with spatial frequency in mixed cells, and from the periodicity of the variation the apparent distance between the subunits may be inferred.