cluding a contrast gain control model as part of a large-scale simulation of the .... surround inhibition spatial frequency inhibition over a b1 octave range and as-.
INCLUDING CONTRAST GAIN CONTROL IN A PARALLEL SIMULATION OF THE VISUAL CORTEX Simon A. J. Winder School of Mathematical Sciences University of Bath, Claverton Down, Bath, BA2 7AY, UK.
Abstract
Neurons in the striate cortex have a compressive contrast response but possess orientation and spatial frequency tuning functions which do not saturate at high contrasts. This suggests that contrast compression is achieved by means of gain control rather than by output saturation. I present the results of including a contrast gain control model as part of a large-scale simulation of the vision system including multiple orientation and spatial frequency channels. In particular, nonlinear interactions within stimulus dimensions are found to occur which are similar to those observed by De Valois and Tootell (1983) among others. I suggest that some of the reported interaction e ects are not isolated events but result from an overall gain control strategy.
Introduction
This paper describes one stage in a larger attempt to simulate neural pathways mediating primate colour and form vision. The present computer model runs on a data-parallel machine and employs image processing techniques to simulate the behaviour of some 40 million neurons in the retina, LGN and V1. The success of such a simulation depends on an accurate model of observed responses at each stage. Here I describe results of simulating contrast gain control interactions in V1. Neurons in the striate cortex are often found to have a saturating response to contrast that is approximately logarithmic (Ma ei and Fiorentini, 1973). Since many receive input from parvocellular geniculate neurons whose responses do not saturate in such a manner, this suggests that a simple linear lter is not
sucient to model this stage. Such a lter can account for the spatial frequency and orientation selectivity possessed by simple cells, but this cannot be followed by a saturating nonlinearity without causing spatial frequency and orientation tuning functions to vary in shape with contrast. Real simple and complex cells do not show this variation (Skottun et al., 1987). This leads to the conclusion that, in order to behave in the way they do, real neurons must be inuenced by their neighbours along all selectivity dimensions so that the linearity of local response relations is preserved but contrast range is compressed (Heeger, 1992). Wilson and Humanski (1993) have developed a theoretical model of this process operating in the orientation domain: Neurons apply an oriented linear lter to input from the geniculate nucleus and the output of each lter passes through a compressive Naka-Rushton function. Responses pooled over local orientations are fed back via interneurons to apply divisive pre-synaptic inhibition. This divisive feedback has three principle e ects: It keeps the response con ned to the linear region of the neuron's own transfer function it produces a saturating response to contrast because division reduces the cell's gain as contrast increases it narrows the orientation tuning of the cell. Gain changes in the cat's cortex were rst investigated by Ohzawa et al. (1985). Here, I adopt a gain control model in the spirit of Wilson and Humanski. I extend their model to include interactions among both orientation and spatial frequency channels as well as including neurons selective for di erent feature types (edge/bar phase).
Methods
The current simulation precedes cortical processing by three stages: a TV model, a nonlinear cone model and a stage which models spatial opponency. The TV model simulates the act of viewing an image displayed on a monitor from a xed distance taking into account ambient illumination|converting pixel values into luminance information. The second stage derives cone excitations and includes the e ects of light adaptation. A third stage simulates the centre-surround organisation of retinal and parvocellular geniculate receptive
elds. The initial cortical ltering stage uses a modied Gabor model (with zero DC response) to introduce orientation and spatial frequency selectivity. The lter bandwidth b was made dependent on centre frequency fc using the relation b = 2 ; log10(fc ) in accordance with observations (Foster et al., 1985). Odd and even phases are modelled, and the lter outputs are split and recti ed to yield four channels responding to edges or bars with positive or negative contrast. Each of these \feature" channel responses, w(x y), is subject to divisive gain control and then passed through a Naka-Rushton function. After rearrangement
Simple Cell Model
LGN Input
Linear Filter
Simple Cell Output
Rectify
Spatial Frequency
Space
Orientation
Phase
Cross-Dimensional Response Pooling
Figure 1: Contrast gain control is assumed to work by feedback pooling of responses from a local neural ensemble.
this gives:
n R(x y) = w(x y)n w+((x y+) kc(x y))n : (1) Here, sets the maximum gain and the exponent n is assigned a value of 2.5 in line with observations (Wilson and Humanski, 1993). For a range of w(x y) below the half-saturation value, a fairly linear contrast-response relation is obtained. Above this value, the function is compressive. The e ect of c(x y) is to vary the e ective half-saturation value of the Naka-Rushton function and hence the slope of the linear region. In the cortex, c(x y) is a local contrast estimate and for any one neuron it is obtained by summing responses from many nearby neurons via horizontalinterconnections. In the current simulation, I assume that these connections combine responses from (1) di erent neurons having receptive elds that are nearby, but not superimposed, (2) neurons with the whole range of orientation preferences, (3) neurons selective for edge and bar features with both contrast signs and (4) neurons with spatial frequency selectivities centred around the channel in question (Figure 1). Seven spatial frequency channels were simulated with centre frequencies ranging from 0.7cpd to 15cpd. For each of these, there were eight orientation channels, each made up of four \feature" channels. Sub-sampling was employed wherever possible to maintain storage economy and increase processing speed. Interpolation was used to match response-map resolutions where necessary. Various types of linear and nonlinear Gaussian-weighted summation were tested for the purposes of pooling across stimulus dimensions. The results in this paper were obtained using a sub-additive form of summation in the spatial domain. Linear summation was found to result in very strong contrast-dependent
surround-inhibition. Large cortical contrast gain changes resulting from extending a grating stimulus into the receptive eld surround have been reported not to occur (Ohzawa et al., 1985). Note that contrast gain control, which is a fairly slow modulatory process, should be distinguished from end- and sidestopping (DeAngelis et al., 1994) which are more likely to be related to form selectivity and result from fast speci c inhibitory connections.
Results
Figure 2 shows the results of simulating simple cell behaviour using a straightforward linear model (Figure 2B) and with gain control (Figure 2C{D). Responses shown are for simple cells selective for light bars and it is important to note that they have been superimposed across the dimension of orientation for illustration purposes only. Figure 2B shows a large dynamic range of response and there are secondary responses to contours due to the narrow bandwidth of the lter. Figure 2C shows the e ect of gain control operating over space, feature phase and orientation. Many of the contours now elicit similar levels of activity, but the integrity of local response relations is maintained. The simulation results of Figure 2C were generated without including interaction in the frequency domain. This has the e ect of widening the e ective channel bandwidth at high contrasts. The result is an increased sensitivity to the blurred background. When spatial frequency interactions are also included, Figure 2D is obtained. Some of the responses have now been suppressed as the spatial frequency tuning functions of all the channels are narrowed to their correct width. In particular, the responses to the blurred background have been removed. In general, Figure 2D resembles Figure 2B except that the contrast dynamic range has been reduced. Further simulation has shown that spatial contrast surround-inhibition occurs for certain stimulus con gurations. This arises from the limited size of the spatial summation region: A high contrast boundary that excites some neurons strongly will drive down smaller responses produced by nearby cells, but not those further away. With sub-additive summation in the feedback path, this e ect is only noticeable in the presence of a signi cant centre-surround contrast di erential. Simulations were also used to investigate the e ect of gain control operating between spatial frequency channels. Results show that responses to noise are much reduced while responses to features isolated in space or spatial frequency remain strong when compared to straightforward linear ltering. Other nonlinear e ects also arise from frequency-domain surround-inhibition: A neuron can be both excited and inhibited by the same spatial frequency. For example, a neuron A selective for frequency f may be to a lesser extent excited by a frequency of 2f since this is within its excitatory bandwidth. If A is stimulated with f then, the e ect of 2f presented simultaneously will be inhibitory,
(A)
(B)
(C)
(D)
Figure 2: Results of simulating cortical contrast gain control mechanisms: A.
Original bean ower image. B. Simulated responses from simple cells sensitive to light bars using a linear lter model. Note that responses from eight orientations have been superimposed in this gure. C. Results following a gain control stage without spatial frequency interaction. D. As for C, but including spatial frequency interactions over a 1 octave range.
particularly if f has a lower contrast. This is because 2f stimulates other suitably selective neurons that contribute to the inhibitory inuence, reducing A's gain to f more than this frequency would excite via the conventional linear
lter selectivity. These e ects are in line with reports by De Valois and Tootell (1983).
Summary and Conclusions
The primary role for cortical contrast gain control appears to be the maintenance of high contrast sensitivity within a limited cell dynamic range without sacri cing local linearity. Successive gain control stages are likely to be part of
the way that the vision system rst measures contrast, and then progressively removes its e ect. From the literature, it is dicult to separate surround-inhibition produced by speci c inhibitory mechanisms from that produced by gain control. However, some spatial surround-inhibition is evident from this simulation work and has the form observed by De Valois et al. (1985). Simulations predict that such inhibition should be spatial frequency speci c but more broadly tuned than the excitatory response since it has been gathered from a number of frequency channels. Experimental results published by Foster et al. (1985) lend support to these predictions. Gain control is found to reduce noise responses and enhance frequency domain response peaks by spatial frequency surround inhibition. This is a phaseindependent e ect and is evident in both simple and complex cell responses. The assumption of a contrast gain control mechanism tends to tie together a number of otherwise isolated interactions: inter-orientation inhibition spatial surround inhibition spatial frequency inhibition over a 1 octave range and associated phase-independent nonlinear e ects. This simulation has demonstrated how these phenomena may arise from a single processing stage of spatial vision.
References
De Valois, K. K., and Tootell, R. B. H. Spatial-frequency-specic inhibition in cat striate cortex cells. Journal of Physiology, 336:359{376, 1983. Ma ei, L., and Fiorentini, A. The visual cortex as a spatial frequency analyser. Vision Research, 13:1255{1267, 1973. Skottun, B. C., Bradley, A., Sclar, G., Ohzawa, I., and Freeman, R. D. The e ects of contrast on visual orientation and spatial frequency discrimination: A comparison of single cells and behaviour. Journal of Neurophysiology, 57:773{786, 1987. Heeger, D. J. Normalisation of cell responses in cat striate cortex. Visual Neuroscience, 9:181{197, 1992. Wilson, H. R. and Humanski, R. Spatial frequency adaptation and contrast gain control. Vision Research, 33:1133{1149, 1993. Ohzawa, I., Sclar, G., and Freeman, R. D. Contrast gain control in the cat's visual system. Journal of Neurophysiology, 54:651{667, 1985. Foster, K. H., Gaska, J. P., Nagler, M., and Pollen, D. A. Spatial and temporal frequency selectivity of neurones in visual cortical areas V1 and V2 of the macaque monkey. Journal of Physiology, 365:331{363, 1985. DeAngelis, G. C., Freeman, R. D., and Ohzawa, I. Length and width tuning of neurons in the cat's primary visual cortex. Journal of Neurophysiology, 71:347{374, 1994. De Valois, R. L., Thorell, L. G., and Albrecht, D. G. Periodicity of striate-cortex-cell receptive elds. Journal of the Optical Society of America A, 2:1115{1123, 1985.
