Multiplication rather than addition of neural signals is believed to underpin a variety of sensory processes, yet the evidence for multiplication is rare. Here we provide psychophysical evidence for neural multiplication in human visual processing of shape. We show that the curvature of a contour is likely detected by a mechanism that multiplies rather than adds the signals from afferent sub-units that detect parts of the curve. Using a novel perceptual after-effect, in which the perceived shape of a sinusoidal-shaped contour is altered following adaptation to a contour of slightly different sinusoidal shape, a pronounced ‘dip’ in the size of the after-effect is found when the adapting contour is broken into segments of a particular length and spacing. Simulations reveal that the presence and shape of the dip is only expected if the afferent sub-units to curvature detectors are multiplied. The after-effect itself is then best explained in terms of the population response of a range of such curvature detectors tuned to different curvatures.

*curvature*. Curvature can be defined at each point along a contour as the rate of change of the slope of the tangent to the contour with respect to the distance along the contour. However in the sinusoidal-shaped contours used here, as with the perturbed-circle radial-frequency patterns commonly used elsewhere (Anderson, Habak, Wilkinson, & Wilson, 2007; Wilkinson, Wilson, & Habak, 1998), curvature, as so-defined, is not a constant along any portion of the curve (which is also true if curvature is defined at each point by 1/radius), unlike for curves that are co-circular. For this reason we define a ‘curve’ as that portion of a contour in which curvature (as defined above) is non-zero everywhere, not necessarily constant, but of constant sign. A curve defined in this way can be characterized by its ‘sag’ and ‘cord’ (Gheorghiu & Kingdom, 2007a, 2008), which leads to a definition of curvature as (proportional to) the product of sag and cord. Both the psychophysics and neurophysiological literature is replete with references to curves, or to the receptive fields of curvature-sensitive neurons, that are not co-circular (Pasupathy & Connor, 1999, 2001, 2002), so our somewhat colloquial definition of curvature is appropriate to existing notions of what are curves.

*detection and discrimination*do not necessarily provide an account of how curvature is

*represented*in the brain. Curvature detection (the task of discriminating a curve from a straight line) and discrimination (the task of discriminating two curves) is probably accomplished with a minimum of neural machinery. For example, single unit recordings in area 17 of the cat visual cortex have shown that end-stopped cells can discriminate between different curves (Dobbins, Zucker, & Cynader, 1987, 1989; Versavel, Orban, & Lagae, 1990). However because end-stopped neurons are univariant with respect to both short straight lines and long curved ones, it is unlikely that they are used to represent curvature. Indeed Zetzsche and Barth (1990) have shown that the Dobbins et al. end-stopped model, which involves the nonlinear combination of pairs of V1 simple-cell-like linear filters with different receptive-field sizes, is unable to fully distinguish between a straight and a curved contour, even if modified by common nonlinearities such as rectification, clipping, and thresholding; although the nonlinearities change the form of the response to a straight line, they never cause it to disappear (see Figures 1a and 1b in Zetzsche & Barth, 1990). Another suggested neural mechanism for curvature discrimination is the comparison of responses from pairs of orientation-selective V1 simple cells positioned at different points along the curve (Anzai, Peng, & Van Essen, 2007; Hedgé & Van Essen, 2000; Kramer & Fahle, 1996; Tyler, 1973; Wilson, 1985; Wilson & Richards, 1989). Again however, this may be insufficient to represent curvature.

*representation*of curvature likely involves more elaborate neural machinery than that needed for detection and discrimination and is probably mediated by neurons in higher visual areas (Connor, Brincat, & Pasupathy, 2007; Gallant, Braun, & Van Essen, 1993; Gallant, Connor, Rakshit, Lewis, & Van Essen, 1996; Pasupathy & Connor, 1999, 2001, 2002) that receive inputs from arrays of V1 simple cells whose receptive fields are arranged in a curvilinear fashion (Gheorghiu & Kingdom, 2007a, 2007b, 2008; Poirier & Wilson, 2006).

^{2}.

*d*is the distance from the midpoint of the contour's luminance profile along a line perpendicular to the tangent,

*L*

_{mean}is mean luminance of 42 cd/m

^{2},

*C*is contrast, and

*σ*is the space constant.

*C*was set to 0.5 and

*σ*to 0.044 deg for all experiments. The ± sign determined the polarity of the contour. Our contours were designed to have a constant cross-sectional width, and the method we used to achieve this is described elsewhere (Gheorghiu & Kingdom, 2006).

*shape phase*of the sine-wave-shaped contour was

*fixed*so that the contour always passed through the curvature detector's receptive field. However, the phase of segmentation was randomized on each iteration. By randomizing segmentation phase the probability that any point along the path of the curvature detector's receptive field was stimulated would be exactly 0.5 per iteration, for all segment lengths. Note that in Figure 2f, which shows an example response to the longest segment length condition, the particular response shown would only occur on a small proportion of iterations. In this segment length condition segment-phase randomization would result in just as many iterations producing no response whatsoever. Example responses from a continuous sequence of iterations (i.e., segmentation phases) can be seen in the movies shown in 1 or at http://www.mvr.mcgill.ca/Fred/research.htm#contourShapePerception.

*average*response of the curvature detector estimated over a large number (1000) of iterations (each with random segmentation phase), as a function of segment length. As one can see, the mean response is similar at both short and long segment lengths, but for a narrow range of intermediate segment lengths, the response collapses to zero. Additional simulations reveal that the position and width of the dip is determined by two factors: (a) the number of sub-units, and (b) the length of the sub-unit receptive field. We simulated the effect of these two factors separately. As an illustration, Figure 3a shows the effect of the number of sub-units (

*n*= 2, 3, 4, and 5) for a constant sub-unit length (

*s*= 40). Figure 3b shows the effect of sub-unit length for a constant number of sub-units (

*n*= 4). These examples show that by increasing either the number or the length of the sub-units, the position of the dip shifts toward longer segment lengths.

*n*= 4 sub-units and sub-unit length

*s*= 40. Similar

*SEM*s were obtained for other combinations of number and length of sub-units.

*pronounced near-symmetrical dip*is found at a particular intermediate segment length.

*single*curvature detector whose global receptive-field structure is closely matched to a half-cycle cosine part of the adapting contour. As such, the model is

*not*a model of the SFAE or SAAE. These after-effects presumably result from changes in the gain of a sub-set of curvature-sensitive neurons whose

*population*response contains the code for curvature. Only some of these neurons will have a global receptive field structure matched to the half-cycle cosine parts of the contour. Therefore although all responding curvature detectors will be subject to the effects of segmentation as described in our matched single-detector model, the model can only make

*qualitative,*not quantitative, predictions about the SFAE and SAAE.

*position*of the dip for most subjects is about 0.33 of the length along the path of a single cycle, allows us to constrain our model simulations if we make an important caveat. The model simulations are for a single curvature detector, whereas the after-effects we are dealing with presumably result from the operation of multiple curvature detectors, as explained above. The caveat is that any estimate of the length of the curvature detector, which is a product of the number and length of the sub-units, will likely represent an

*average*value of a range of curvature detectors varying in either or both of length and number of sub-units.

*s*= 25 pixels.

*n*= 4 sub-units of length

*s*= 25 (see Figure 1g). The range of curvature detectors could receive input either from (a) the same number,

*n*= 4 of afferent sub-units but with different sub-unit lengths, or (b) different numbers of sub-units having the same sub-unit length (

*s*= 25). Figure 6 shows both individual responses (dark gray symbols) as well as average responses (light gray symbols) for (a) 3 stimulated curvature detectors whose sub-unit lengths are

*s*= 23, 25, and 27, respectively; (b) 4 stimulated curvature detectors whose sub-unit lengths are

*s*= 20, 25, 30, and 35, respectively. As can be seen the dips in the combined average responses do not reach zero and are broader than the individual responses, in line with the data.

*same*receptive-field length but which receive input from different combinations of number and length of sub-units. Figure 6c shows the average response (light gray symbols) from a range of curvature detectors, whose individual responses (dark gray symbols) are from 100 pixel length curvature detectors with

*n*= 4 sub-units of length

*s*= 25,

*n*= 3 sub-units of length

*s*= 33.3, and

*n*= 5 sub-units of length

*s*= 20. As can be seen the dips in the combined average responses do not reach zero and are broader than the individual responses, again in line with the data.

*not*models of either the SAAE or SFAE. Assuming these after-effects are caused by internal gain changes to a sub-set of curvature detectors whose population response signals curvature, a sub-set that not only differs in receptive-field length but also curvature, the psychophysical data are clearly insufficient to constrain a fully fledged multi-filter model of the after-effects that would inevitably contain many free parameters. Thus our model simulations are ultimately qualitative not quantitative. Nevertheless, we were able to show that two features of the single-filter model simulation that did not accord with the psychophysical data, namely a dip that was both very narrow and reached a minimum of zero, could be dealt with simply by combining responses across a range of curvature receptive-field lengths. This is not to claim that this is the only explanation of these features of the data, merely that in principle it could be.

*thresholding*. Let us assume that the sub-units have a threshold (

*T*) such that below

*T*their response is zero and above

*T*their response is a linear function of stimulation. Simulations of linear summation of sub-unit responses are shown in Figures 8a and 8b.

*T*is expressed as a proportion of the maximum sub-unit response, for

*T*= 0.25 (white),

*T*= 0.35 (light gray), and

*T*= 0.5 (dark gray). Figure 8a shows the results for different numbers of sub-units (

*n*= 2, 3, 4, and 5) with constant sub-unit length (

*s*= 40). Figure 8b shows the results for various sub-unit lengths (

*s*= 40, 60, 80, and 100) with a fixed number of sub-units (

*n*= 4). The simulations reveal that a sharp dip at an intermediate segment length only occurs

*for high thresholds*(

*T*= 0.5). A threshold of 0.5 produces a dip comparable in size to that obtained experimentally for the SAAE (∼0.44) but not for the SFAE. However, as Figures 8a and 8b also show, the modeled shape of the response function for these high thresholds is

*asymmetric,*with the left side of the function prominently reduced, unlike the experimental data. A high threshold, such as

*T*≥ 0.5, is also implausible. It would predict that after-effects would only be obtained at medium and high contrasts. In a previous study we found similar sized SFAEs with adaptor and test contrasts of 0.05, 0.15, and 0.45 (Gheorghiu & Kingdom, 2006). Therefore model simulations of the effects of thresholding, together with considerations of plausibility, argue against high thresholding as the cause of the dip in the after-effects we have observed.

*a*·

*b*, or via another mathematical operation that is equivalent to multiplication). For example, multiplication can be implemented via a Log–Exp transform

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