Analysis of Spatial Responsivity Measures in Cat Cortex
Brian C. Madden and Michael Mancini
ARVO 1980
Abstract: The line weighting function (LWF) is a popular measure of the spatial response profile of visual neurons. This stems from the usefulness of the LWF in predicting a (spatially linear) cell’s response. This utility of the LWF is dependent on temporal linearity as well. However, significant temporal nonlinearities are present in cortical responses. In a temporally nonlinear cell the LWF becomes dependent on the temporal nature of the stimulus and varies with the criteria used to summarize the temporal response. If the LWF is formed by summing the responses to a temporal square wave for a neuron that adapts, then the linear portion of the on and off responses will cancel each other. Some nonlinear responses will not cancel and will form a major portion of the response profile. The discrepancies among response profiles are borne out by data obtained with isolated presentation of short, narrow bars at 25 contiguous receptive field locations. LWFs were calculated using a variety of temporal criteria. These results were compared with equivalent Wiener kernel predictions. The predictions displayed adaptive shifts which resulted in cancellation of the linear temporal response to on and off portions of the stimulus.
Simple cell LWFs varied gradually across the receptive field. In contrast, complex cells exhibited sharp discontinuities in responsivity between adjacent positions. Local regions of heightened activity possessed increased temporal nonlinearities, consistent with a subunit organization in complex cell responses. The evidence indicates that both types of cortical neurons are sufficiently nonlinear to preclude the use of LWFs as a basis for superposition.
Separation of Temporal and Spatial Linearity in Cat Visual Cortex
Brian C. Madden and Michael Mancini
Center for Visual Science, University of Rochester, Rochester, NY
Recent Advances in Vision, Optical Society of America 1980
Abstract: Linear systems analysis frequently has been used to characterize the manner in which different parts of visual receptive fields interact to form the response properties of a cell. The utility of these models is derived from their ability to predict complex responses through the use of superposition. For an achromatic, monocular stimulus the flux distribution on the retina may be described completely by the intensity in two dimensions of space and one of time. The linearity of any one dimension is independent of the existence of linearity in any other. Presentation of the exhaustive set of spatial and temporal patterns necessary to test the linearity of these dimensions is almost always prohibitive. Assumptions of structure are frequently made in order to use more efficient tests of linearity. However, with such tests it is possible to confound violations of spatial and temporal linearity should the assumptions turn out not to be true. For example, one assumption in an often used test of linearity, the null test, presumes that a common axis of symmetry exists for each of the summation regions. A linear odd-symmetric cell will fail the null test. Similarly, cells which contain subunits (complex cells, Y-cells) cannot be expected to pass the null test. The varying spatial phase of temporally nonlinear subunits might produce responses which mistakenly appear to be in violation of spatial summation. The use of spatially extensive stimuli (drifting or counterphase sinusoidal gratings) precludes the separation of temporal and spatial response characteristics.
We measured the temporal linearity of local regions of cat cortical receptive fields by using an approximation to binary white noise to control the presentation of a short, narrow bar. Twenty-five contiguous positions were tested. Wiener kernel predictions were obtained for each bar stimulus and also for 25 similarly positioned edges. The zeroth, first and second order temporal Wiener kernels revealed the linear and nonlinear portions of the temporal response. The nonlinear temporal responses included adaptation, response compression and thresholding. The difference of Wiener kernel predictions for adjacent edges was compared to the corresponding bar prediction. In this way spatial linearity could be determined for each millisecond temporal interval of the response at each of the 25 locations across the receptive field. We have obtained results which suggest that some cortical units exhibiting temporal nonlinearities are spatially linear. This cell type would be falsely classified as spatially nonlinear using drifting gratings.
In conclusion, the paradigm presented here, which uses a series of spatially related temporal Wiener kernels, allows the linear and nonlinear spatial and temporal characteristics of a neuron to be identified separately. This is not true for methods which use spatially extensive stimuli (i. e., drifting or counterphase gratings or most conditioning stimuli). Additionally, the method of edge difference (spatial differentiation) is preferable to those techniques using gratings in that it allows localization of the site of spatial nonlinearities.
Linear and Nonlinear Characteristics of Temporal Responses in Cat Visual Cortex
Michael Mancini, Brian C. Madden and Robert C. Emerson
ARVO 1980
Abstract: Several methods of analysis were used to identify temporal properties of simple and complex cells. Averaged responses to flashed presentations of short, narrow bar stimuli ranging in duration from 32 to 128 msec were obtained for each of 25 contiguous locations across the receptive field. Isolated impulse (8 msec) responses were similarly obtained. Convolution of the impulse response to predict the flash data grossly overestimated the actual response due to rectification of the impulse response. Response threshold effects and invariant initial transient responses also contributed to this poor fit. The extent of these temporal nonlinearities is thought to reflect the transition of the cell from quiescence to activity rather than the essential temporal nature of an active neuron.
In an alternate approach, an approximation to binary white noise was used to control the presentation of a bar at each spatial position and the zeroth, first and second temporal Wiener kernels were calculated. These kernels were used to create a prediction of the flash response data. Our results indicate that, in an active state, striate neurons exhibit much less temporal nonlinearity, keeping a balance between rectification and compression distortions. The Wiener predictions displayed transient responses which vary with duration, and increments and decrements of response about a raised mean level of activity. In addition, predictions for long duration step stimuli produced an increasingly negative second kernel, indicating neural adaptation. This demonstrated relationship between activity and temporal linearity has important consequences for the predictive validity of data obtained in the traditional manner (isolated presentation).
Temporal Nonlinearities in Simple Cell Responses
Michael Mancini, Brian C. Madden and Robert C. Emerson
ARVO 1982
Abstract: Recordings were made from simple cells in cat cortex to assess temporal linearity. A pseudo-random binary sequence was used to control the contrast of an optimally oriented bar stimulus at each of up to 25 continuous positions across a receptive field. By cross-correlating a cell’s response with the input it was possible to obtain the zeroth, first and second order Wiener kernels at each RF location. The impulse characterizing a linear system corresponds to the sum of the zeroth and first order Wiener kernels. We found that simple cells exhibit significant nonlinear temporal effects as revealed by the presence of prominent second order kernels.
As a further demonstration of this nonlinearity we calculated a more conventional type of histogram from responses to the same stimulus sequence that was used to calculate the Wiener kernels. We search for the stimulus pattern of interest within the random stimulus array and calculated an averaged response histogram to this pattern. A comparison of the response profile of this histogram with the Wiener kernel model again revealed significant second order effects. The linear Wiener model accounted for less of the response variance of the histogram (67%-85%). The presence of significant temporal nonlinearities suggests caution in the use of a linear model in characterizing simple cell responses.
Temporal Nonlinearities in Simple Cell Responses
Michael Mancini
Ph. D. Thesis, University of Rochester, 1983
Abstract: Recordings were made from simple and complex calls in cat visual cortex to assess temporal linearity. A pseudo-random binary sequence was used to control the contrast of an optimally oriented bar stimulus at each of up to 25 adjacent positions across a receptive field. By cross-correlating a cell’s response with the input it was possible to obtain the zeroth, first and second-order Wiener kernels at each RF location. The impulse response characterizing a linear system corresponds to the sum of the zeroth and first-order Wiener kernels. Both simple and complex cells showed pronounced nonlinear temporal effects as revealed by the presence of prominent second-order kernels.
A more convenient type of histogram was calculated from responses to the same stimuli that were used to calculate the Wiener kernels. A search for the stimulus pattern of interest within the random stimulus array was conducted and an averaged response histogram to this pattern was calculated. A comparison of the response profile of this histogram, the time-locked response, with the Wiener kernel model also revealed prominent nonlinear temporal effects for both simple and complex cells.
The second-order temporal properties of simple cells were well represented by a linear/nonlinear cascade model. The nonlinear mechanism could be represented either by a second-order polynomial or a half-wave rectifier. The modeling results suggested that for simple cells, the nonlinearity occurs late and probably is a threshold associated with the spike generating mechanism.
Complex cells showed greater temporal nonlinearities than simple cells as demonstrated by the greater amplitude of their second-order response estimates relative to their first-order response estimates. Complex cells showed full-wave rectification which suggested that fourth-order (and possibly even higher even-order terms) were necessary to account for their response to bright and dark stimuli.
Complex cells were subdivided into B- and C-cells (Henry, 1977) on the basis of their response to statically flashed bars and moving edges. The B- and C-cells had few properties in common. The first-order response estimates of C-cells were the same sign across the RF. The first- and second-order response estimates of C-cells were temporally congruent and the linear/nonlinear cascade model accounted for their second-order responses. B-cells had a first-order response that varied in strength and sign across the RF. The first- and second-order responses of B-cells were dissimilar in shape and the linear/nonlinear cascade model did not represent the second-order responses of B-cells well.
White Noise Analysis of Temporal Properties in Simple Receptive Fields of Cat Cortex
M. Mancini, B. C. Madden and R. C. Emerson
Center for Visual Science, University of Rochester, 14627 Rochester, NY, USA
Biological Cybernetics, 63:209-219
Received: 16 August 1989 Accepted: 28 February 1990
Abstract: We studied the linear and nonlinear temporal response properties of simple cells in cat visual cortex by presenting at single positions in the receptive field an optimally oriented bar stimulus whose luminance was modulated in a random, binary fashion. By crosscorrelating a cell's response with the input it was possible to obtain the zeroth-, first-, and second-order Wiener kernels at each RF location. Simple cells showed pronounced nonlinear temporal properties as revealed by the presence of prominent second-order kernels. A more conventional type of response histogram was also calculated by time-locking a histogram on the occurrence of the desired stimulus in the random sequence. A comparison of the time course of this time-locked response with that of the kernel prediction indicated that nonlinear temporal effects of order higher than two are unimportant. The temporal properties of simple cells were well represented by a cascade model composed of a linear filter followed by a static nonlinearity. These modeling results suggested that for simple cells, the nonlinearity occurs late and probably is a soft threshold associated with the spike generating mechanism of the cortical cell itself. This result is surprising in view of the known threshold nonlinearities in preceding lateral geniculate and retinal neurons. It suggests that geniculocortical connectivity cancels the earlier nonlinearities to create a highly linear representation inside cortical simple cells.