**Abstract**:

**Abstract**
Attention to a spatial location or feature in a visual scene can modulate the responses of cortical neurons and affect perceptual biases in illusions. We add attention to a cortical model of spatial context based on a well-founded account of natural scene statistics. The cortical model amounts to a generalized form of divisive normalization, in which the surround is in the normalization pool of the center target only if they are considered statistically dependent. Here we propose that attention influences this computation by accentuating the neural unit activations at the attended location, and that the amount of attentional influence of the surround on the center thus depends on whether center and surround are deemed in the same normalization pool. The resulting form of model extends a recent divisive normalization model of attention (Reynolds & Heeger, 2009). We simulate cortical surround orientation experiments with attention and show that the flexible model is suitable for capturing additional data and makes nontrivial testable predictions.

*not*normalize the response of the center unit. Here our goal is to address the influence of attention in this class of model.

*v*) provides the contextual coordination between filter activations. We denote the filter activations corresponding to center and surround locations,

*x*and

_{c}*x*. Without loss of generality, we discuss the model with regard to two such filters; this generalizes to a set of filters in center and surround locations. The dependency between the two filters arises via a multiplication of the common mixer with two independent Gaussians (

_{s}*x*=

_{c}*vg*;

_{c}*x*=

_{s}*vg*).

_{s}*g*) in light of the statistical dependencies. This is motivated by a number of factors. First, we focus here on experiments that report a fairly local property (i.e., the response of a neuron in a center location or perceived orientation in a center location). In the model, the Gaussian component

_{c}*g*is the local variable, whereas the mixer variable is a more global property across space linking the receptive fields. Also, estimation of the Gaussian amounts to the inverse of the multiplication, i.e., a form of divisive normalization (Schwartz et al., 2009) that is prominent in mechanistic, descriptive, and functional cortical models, and this estimation is also tied to reducing this form of multiplicative statistical dependency.

_{c}*x*is the center unit activation and

_{c}*x*the surround unit (for readability, we assume just one filter in the center and one in the surround, although this can be extended to more filters);

_{s}*m*is the gain signal that includes both center and surround and acts divisively (thus related to divisive normalization);

_{cs}*σ*is a small additive constant as typically assumed in divisive normalization modeling (Heeger, 1992; Schwartz & Simoncelli, 2001; Schwartz et al., 2009; here we fix it at 0.1 for all simulations); and

*ξ*

_{1}indicates inclusion of both center and surround in the normalization pool. The Appendix includes the full form of Equation 1, which contains a function

*f*(

*m*,

_{cs}*n*) to make the equation into a proper probability distribution. The function

*f*(

*m*,

_{cs}*n*) depends on both

*m*and on the number of center and surround filters

_{cs}*n*. We have omitted the function

*f*(

*m*,

_{cs}*n*) in the main text for simplicity and to exemplify that this formulation is similar to the canonical divisive normalization equation (e.g., Heeger, 1992; see this point also in Schwartz et al., 2009; Wainwright & Simoncelli, 2000). However, note that in all simulations and plots in the paper, we compute the full form of the equation according to the Appendix. Also, the exact mixer prior changes the equation details but not the qualitative nature of the divisive normalization and the simulations (see Appendix).

*X*= (

*X*,

_{c}*X*).

_{s}*E*(

*g*|

_{c}*x*,

_{c}*ξ*

_{2}), where

*ξ*

_{2}indicates inclusion of only the center and not the surround normalization pool. Here the divisive gain signal

*m*includes only the center filter responses.

_{c}*p*[

*ξ*

_{1}],

*p*[

*ξ*

_{2}] = 1 −

*p*[

*ξ*

_{1}]). The posteriors for the surround normalizing the center or not given filter activations to an input stimulus are (

*p*[

*ξ*

_{1}|

*x*,

_{c}*x*],

_{s}*p*[

*ξ*

_{2}|

*x*,

_{c}*x*]). In this paper, we treat the priors as a free parameter and calculate the posterior assignments given an input stimulus similar to Coen-Cagli et al. (2012) using Bayes (see Appendix). Figure 2 shows how the posterior co-assignments vary as a function of orientation difference and contrast for different priors, intuitively related to the suggestion above that more homogenous center and surround stimuli are more likely to be in the same normalization pool. When the prior probability of co-assignment is set to 1, then the posterior probabilities become (

_{s}*p*[

*ξ*

_{1}|

*x*,

_{c}*x*] = 1,

_{s}*p*[

*ξ*

_{2}|

*x*,

_{c}*x*] = 0), which is essentially the canonical divisive normalization model in which the surround always normalizes the center for any input stimuli (Figure 1b).

_{s}*i*corresponding either to center or surround location,

*ϕ*to the stimulus orientation, and

*ϕ*to the preferred orientation of the neural unit.

_{i}*E*( $gci$ |

*x*,

_{c}*x*), with 360 preferred orientations. The estimated center angle given by: where

_{s}*ϕ*is the preferred angle of unit

*i*,

*u*(

*ϕ*) is a two dimensional unit vector pointing in the direction of

*ϕ*, and the doubling takes account of the orientation circularity.

*a*and

_{c}x_{c}*a*, where

_{s}x_{x}*a*and

_{c}*a*are center and surround attention weights and

_{s}*x*and

_{c}*x*are the filter outputs without attention as before. The output of each spatial location may be multiplied by its own attention weight, but to simplify the notation, we show the equations for a single surround location and attention weight.

_{s}*g*=

_{c}v*a*;

_{c}x_{c}*g*=

_{s}v*a*. The model neuron estimates the Gaussian corresponding to the center unit, which depends on both

_{s}x_{s}*a*and

_{c}x_{c}*a*: Here

_{s}x_{s}*m*is the gain signal with attention that includes both center and surround filter activations. The divisive signal weights more heavily the location that is more strongly attended. In addition, attention to the center location weights both the numerator and the divisive gain signal, while attention to the surround weights only the divisive signal.

_{cs}*P*is the likelihood of the observed filter values under the assumption that center and surround are coordinated (

_{cs}*ξ*

_{1}). Similarly,

*P*is the likelihood under the assumption that center and surround are independent (

_{c}P_{s}*ξ*

_{2}). The equations for

*P*,

_{cs}*P*, and

_{c}*P*are given in the Appendix and depend on the gain signals for center and surround

_{s}*m*, center alone

_{cs}*m*, and the surround alone

_{c}*m*, as well as the number of filters in the center and surround. Note that attention also influences these gains since it acts multiplicatively on the filter outputs, and so can also affect these co-assignment probabilities. Intuitively, when the gains of center and surround are more similar, they are more likely to be co-assigned.

_{s}*w*such that

_{c}w_{s}*m*= $wcxc2ac2+wsxs2as2+\sigma $. Both versions with and without this added (fixed) flexibility yielded similar results. We specifically fit the weighting factors, the additive constant, and the attend yes and no conditions:

_{cs}*w*,

_{c}*w*,

_{s}*σ*,

*a*, and

_{yes}*a*(the fits were 2.0294, 0.5395, 0.0891, 0.3148, and 0.2073). For surround attention, we set

_{no}*a*=

_{s}*a*and

_{yes}*a*=

_{c}*a*; for center attention

_{no}*a*=

_{s}*a*and

_{no}*a*=

_{c}*a*; and for distant attention,

_{yes}*a*=

_{s}*a*and

_{no}*a*=

_{c}*a*. For the flexible normalization model we fit

_{no}*σ*,

*a*,

_{yes}*a*, and

_{no}*ξ*(the fits were 0.01, 0.354, 0.14, and 0.32), where the last term is the prior for assignment.

*repulsive biases*, which are apparent, for instance, in the classical tilt illusion for surround stimuli, the center orientation is perceived as

*tilted away*from the surround more than it actually is. Such repulsive effects occur for small orientation differences between the center and surround stimuli. Such effects without attention can be explained by canonical divisive normalization (Schwartz et al., 2009). Surround stimuli can also induce weaker attractive effects for large orientation differences between the center and surround, an effect that arises in our model without attention from flexible normalization pools (Schwartz et al., 2009).

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*Network: Computation in Neural Systems**v*, we assume a loglogistic prior. The exact form of the Gaussian estimate in the GSM depends on the mixer variable prior (see also Wainwright, Simoncelli, & Willsky, 2001). This does not affect the divisive form and qualitative nature of the simulations reported here, but influences the exponent of the numerator and denominator and therefore the saturation of the contrast response function. Here we use a loglogistic prior, which leads to a saturating curve. In previous modeling we have used a Rayleigh prior (Coen-Cagli et al., 2012; Schwartz et al., 2009), for which the exponent of the denominator is smaller; this can accommodate all the results reported here except for the needed saturation of the contrast response function for obtaining the results of Figure 4 on changes in contrast versus response gain with attention. Different neurons have different saturations, and from our perspective here (as in other divisive normalization formulations), we assume saturation via this choice of mixer prior.

*x*to

_{c}*a*and

_{c}x_{c}*x*to

_{x}*a*in all the equations below.

_{s}x_{s}*m*is the gain signal for center and surround (this is the same signal as in the main text, but we write it out more generally with the covariance matrix Σ

_{cs}*);*

_{cs}*k*are the number of filters in center and surround;

_{cs}*Ei*( ) is an exponential integration function; and Γ( ) is the Gamma function.

*cs*subscripts with

*c*subscripts (since center does not depend on surround and has its own mixer variable).

*P*, replacing the subscripts

_{c}P_{s}*cs*with

*c*and

*s*respectively, and therefore reflecting the scenario in which center and surround are not co-assigned.

*x*to

_{c}*a*and

_{c}x_{c}*x*to

_{s}*a*in all the equations. In this case, the posterior probabilities are defined as

_{s}x_{s}*p*(

*ξ*

_{1}|

*x*,

_{c}*x*,

_{s}*a*,

_{c}*a*) and

_{s}*p*(

*ξ*

_{2}|

*x*,

_{c}*x*,

_{s}*a*,

_{c}*a*).

_{s}