**In humans and in foveated animals visual acuity is highly concentrated at the center of gaze, so that choosing where to look next is an important example of online, rapid decision-making. Computational neuroscientists have developed biologically-inspired models of visual attention, termed saliency maps, which successfully predict where people fixate on average. Using point process theory for spatial statistics, we show that scanpaths contain, however, important statistical structure, such as spatial clustering on top of distributions of gaze positions. Here, we develop a dynamical model of saccadic selection that accurately predicts the distribution of gaze positions as well as spatial clustering along individual scanpaths. Our model relies on activation dynamics via spatially-limited (foveated) access to saliency information, and, second, a leaky memory process controlling the re-inspection of target regions. This theoretical framework models a form of context-dependent decision-making, linking neural dynamics of attention to behavioral gaze data.**

*locations*for a given input image with reasonable accuracy (Itti, Koch, & Niebur, 1998; Kienzle, Franz, Schölkopf, & Wichmann, 2009; Torralba, Oliva, Castelhano, & Henderson, 2006; Tsotsos et al., 1995). The models compute so-called

*saliency maps*, highlighting those parts of an input image that stand out relative to the surrounding areas (Itti & Koch, 2001). However, the human visual system is foveated, i.e., it is only able to acquire high-resolution information from a very limited region surrounding the current gaze position (the fovea). Outside the foveal region, visual acuity falls off rapidly, while the effects of visual crowding increase, so that visual processing in the periphery has very limited resolution (Jones & Higgins, 1947; Levi, 2008; Rosenholtz, Huang, & Ehinger, 2012).

*after*exploring the entire image with its fovea—and the dynamic principles of saccadic selection underlying the generation of scanpaths by human observers. Moreover, part of the mismatch between computer-generated saliency maps and actual gaze patterns might be explained by properties of the visuomotor system (Findlay & Walker, 1999). Recently, a number of publications addressed specific aspects of this problem, e.g., different roles for short and long saccades (Tatler, Baddeley, & Vincent, 2006) or return saccades (Ludwig, Farrell, Ellis, & Gilchrist, 2009; Wilming, Harst, Schmidt, & König, 2013). Moreover, behavioral biases might produce an important contribution to eye-movement statistics (Tatler & Vincent, 2009). What is currently missing is an integrative computational model that addresses the key aspects of visuomotor control in a coherent theoretical framework. We set out to develop one possible integrative model.

*N*= {

*x*

_{1},

*x*

_{2},

*x*

_{3},…} (also called a point pattern). The 2D density (or intensity)

*λ*of the spatial point process is given as the expectation or mean value of the number of points in an observation window

*B*, i.e.,

*λ*=

*E*(

*n*(

*B*)), where

*n*(.) is a counting measure. A process is statistically homogeneous (or stationary), if

*N*and the translated set

*N*= {

_{x}*x*

_{1}+

*x*,

*x*

_{2}+

*x*,

*x*

_{3}+

*x*,…} have the same distribution for all

*x*. For a stationary spatial point process, the intensity

*λ*is constant over space. For a nonstationary process, the intensity is a function of location,

*λ*=

*λ*(

*x*). For the computation of densities from experimental data, we used kernel-density estimates with bandwidth parameters chosen according to Scott's rule (Baddeley & Turner, 2005; Scott, 1992). To compute deviations between 2D densities

*P*and

_{kl}*Q*at grid position (

_{kl}*k*,

*l*), we used a symmetric version of the Kullback-Leibler divergence derived from information gain (Beck & Schlögl, 1993), i.e.,

*ρ*(

*x*

_{1},

*x*

_{2}), which gives the probability

*ρ*(

*x*

_{1},

*x*

_{2}) d

*x*

_{1}d

*x*

_{2}of observing points in each of two disks

*b*

_{1}and

*b*

_{2}with linear dimensions d

*x*

_{1}and d

*x*

_{2}, respectively. Point patterns can be characterized by the pair density, which is typically a function of the pair distance, i.e.,

*ρ*(

*x*

_{1},

*x*

_{2}) =

*ρ*(

*r*) with

*r*= ||

*x*

_{1}–

*x*

_{2}||, for two arbitrary realizations

*x*

_{1}and

*x*

_{2}. Using a kernel-based method, a estimator for the pair density can be written as where

*k*(.) is an appropriate kernel and

*x*

_{1}–

*x*

_{2}|| (Baddeley & Turner, 2005). For numerical computations we used the Epanechnikov kernel (Illian et al., 2008), i.e.,

*g*(

*r*) is a normalization of the pair density with respect to first-order intensity

*λ̂*, so that the estimator for the pair correlation is given by

*ĝ*(

*r*) =

*ρ*(

*r*) /

*λ̂*

^{2}. The interpretation of the pair correlation function for a given point pattern is straightforward. For a random pattern without clustering, the pair correlation function is

*ĝ*(

*r*) ≈ 1 across the full range of distances

*r*. If

*ĝ*(

*r*) > 1, then pairs of fixations are more abundant than on average at distance

*r*. If

*ĝ*(

*r*)<1, then pairs of fixations are less abundant than on average at a distance

*r*. Thus, the pair correlation function

*ĝ*(

*r*) measures how selection of a particular point location (i.e., fixation position) is influenced by other fixations at distance

*r*.

*ĝ*(

_{inhom}*r*) involves two steps: First, we estimated the overall intensity

*λ̂*(

*x*) for all fixation positions obtained for a given scene. In this procedure we borrow strength from the full set of observations to obtain reliable estimates of the inhomogeneity. Second, we computed the pair correlation function from a single trial with respect to the inhomogeneous density of the full data set.

*ĝ*(

*r*), the scalar quantity (Illian et al., 2008) denoted as PCF deviation in the following, serves as a useful test statistic that quantifies the deviations from randomness for a given point pattern with inhomogeneous density

*λ̂*(

*x*). The integral in Equation (5) was evaluated numerically for pair distances

*r*between 0.1° and 5° (image set 1 and 2) and between 0.1° and 3° (Le Meur, Le Callet, Barba, & Thoreau [2006] data; see below).

*h*

_{density}for the kernel density estimation was computed according to Scott's rule (Scott, 1992; range from 1.8° to 2.2° for

*h*over all images from image set 1). The obtained 2D density

*λ̂*(

*x*,

*y*) is inhomogeneous because of the dependence on position (

*x*,

*y*). A representative sample trajectory from a single trial is given in Figure 1d, where the second and last fixation of the scanpath is highlighted by white color and by their serial numbers. The first fixation was omitted, since all trials started at a random position within an image determined by our experimental procedure (see Methods).

*g*(

*r*) gives a quantitative summary of interactions in fixation patterns by measuring how distance patterns between fixations differ from what we would expect from independently distributed data (see Appendix). A value of

*g*(

*r*) above 1 for a particular distance

*r*indicates clustering, meaning that there are more pairs of points separated by a distance

*r*than we would expect if fixation locations were statistically independent.

*λ̂*(

*x*) used for the inhomogeneous pair correlation function, Equation (4), an optimal bandwidth parameter

*h*is needed to avoid two possible artifacts: First, if

*h*is very small, then spatial correlations might be underestimated due to overfitting of the inhomogeneity of the density. Second, if

*h*is too large, then spatial correlations might be overestimated, since first-order inhomogeneity is not adequately removed from the second-order spatial statistics. We solved this problem by computing the PCF deviation Δ

*for the inhomogeneous point process for varying values of the bandwidth*

_{g}*h*(Figure 2). Since the inhomogeneous point process generates uncorrelated fixations, i.e.,

*g*

_{theo}(

*r*) = 1, the optimal bandwidth for the dataset corresponds to a minimum of the PCF deviation Δ

*(quantifying the deviation from the ideal value*

_{g}*g*(

*r*) = 1). For image set 1, the optimal value was estimated as

*ĥ*

_{1}= 4.0° (Figure 2a).

*g*(

_{inhom}*r*), in which first-order inhomogeneity is removed from the second-order spatial correlations (see Methods). As a result, we obtained pair correlations from individual trials (Figure 1g, gray lines). Deviations from

*g*(

_{inhom}*r*) ≈ 1 indicate spatial clustering at a specific distance

*r*. The mean pair correlation function

*ḡ*(

_{inhom}*r*) provides evidence for clustering at small spatial scales with

*r*< 4° (Figure 1g, red line). Such a scale is greater than the foveal zone (

*r*< 2°) and might provide an estimate of the size of the

*effective*perceptual window in free scene viewing. This result is compatible with earlier findings that the zone of active selection of saccade targets extends beyond the fovea into the parafovea up to eccentricities of 4° (Reinagel & Zador, 1999).

*λ*(

*x*) and for a homogeneous point process with constant intensity

*λ*

_{0}. For the inhomogeneous point process, we sampled from the estimated intensity

*λ̂*(

*x*) (Figure 1b), whereas a constant intensity

*λ̂*

_{0}, obtained from spatial averaging, was used for the homogeneous point process (Figure 1c). Both surrogate datasets are important for checking the reliability of the computation of the pair correlation function for the original data (Figure 1g). First, the inhomogeneous point process gives a flat mean correlation function with

*g*(

*r*) ≈ 1 (Figure 1h), which demonstrates the absence of clustering (except for the divergence at very small scales as an effect of numerical computation issues). Thus, the spatial correlations in the experimental data are not a simple consequence of spatial inhomogeneity. Second, the result for the homogeneous point process (Figure 1h) is the same as for the inhomogeneous point process, which indicates that the correction for inhomogeneity needed for computations in Figure 1g does not produce unwanted artefacts due to possible overfitting of the spatial inhomogeneities. We conclude that our experimental data give a clear indication for spatial clustering at length-scales smaller than 4° of visual angle. Additionally, we checked the hypothesis that this effect of spatial clustering might be due to saccadic undershoot and subsequent short correction saccades by excluding all fixations with durations shorter than 200 ms. A related analysis of the PCF indicates no qualitative differences from the original data (Figure 1h).

*χ*

^{2}(2) = 105.4,

*p*< 0.01. Contrasts revealed that (1) spatially inhomogeneous data-sets were different from the homogeneous data,

*b*= 0.319,

*t*(28) = 22.85,

*p*< 0.01, and that (2) experimental data were significantly different from inhomogeneous surrogate data,

*b*= 0.062,

*t*(28) = 2.56,

*p*= 0.016.

*g*(

*r*) ≈ 1 were computed to obtain a PCF measure indicating the amount of spatial correlation averaged over distances (see Methods). Results indicate that the surrogate data produce—as designed—uncorrelated gaze positions (low PCF deviation), while the experimental data by human observers exhibit spatially correlated gaze positions (Figure 1i). Model type had a significant effect on PCF,

*χ*

^{2}(2) = 98.4,

*p*< 0.01.

*f*(

*x*,

*y*;

*t*) is keeping track of the sequence of fixations by inhibitory tagging (Itti & Koch, 2001). Second, an attention map

*a*(

*x*,

*y*;

*t*) that is driven by early visual processing controls the distribution of attention. Physiologically, the assumption of the dynamical maps is supported by the presence of an allocentric motor map of visual space in the primate entorhinal cortex (Killian, Jutras, & Buffalo, 2012). Moreover, this map is spatially discrete (Stensola et al., 2012) and serves as a biological motivation for the fixation and attention maps in our model.

*L*×

*L*. Lattice points (

*i*,

*j*) have equidistant spatial positions (

*x*,

_{i}*y*) for

_{j}*i*,

*j*= 1, …,

*L*, where

*x*=

_{i}*x*

_{0}+

*i*Δ

*x*and

*y*=

_{j}*y*

_{0}+

*j*Δ

*y*. As a consequence, attention and fixation maps are implemented in spatially discrete forms, {

*a*(

_{ij}*t*)} and {

*f*(

_{ij}*t*)}, respectively. For the numerical simulations, time was discretized in steps of Δ

*t*= 10 ms with

*t*=

*k ×*Δ

*t*and

*k*= 0, 1, 2, …,

*T*.

*x*,

_{g}*y*) at time

_{g}*t*, then a position-dependent activation change

*F*(

_{ij}*x*,

_{g}*y*) and a global decay proportional to the current activation –

_{g}*ωf*(

_{ij}*t*) are added to all lattice positions to update the activation map at time

*t*+ 1, i.e., where the activation change

*F*(

_{ij}*x*,

_{g}*y*) ≡

_{g}*F*(

_{ij}*t*) is implicitly time-dependent because of the time-dependence of gaze positions,

*x*(

_{g}*t*),

*y*(

_{g}*t*). The constant

*ω <*1 determines the strength of the decay of activation. For the spatial distribution of the activation change

*F*(

_{ij}*t*) we assume a Gaussian profile, i.e., with the free parameters

*σ*

_{0}and

*R*

_{0}controlling the spatial extent of the activation change and the strength of the activation change, respectively. In our model, the build-up of activation in the fixation map is a mechanism of inhibitory tagging (Itti & Koch, 2001) to reduce the amount of refixations on recently visited image patches.

*a*(

_{ij}*t*) we assume similar dynamics, however, the width of Gaussian activation change

*A*(

_{ij}*t*) is assumed to be proportional to the static saliency map

*ϕ*. The updating rule for the attention map is given by with decay constant

_{ij}*ϕ*is accessed locally through a Gaussian aperture with size

_{ij}*σ*

_{1}and scale parameter

*R*

_{1}, similar to Equation (7). Using the local read-out mechanism, information is provided for the attention map to identify regions of interest for eye guidance.

*i*,

*j*), then a position-dependent activation change

*F*in the form of a Gaussian profile is added locally in each time step, while a global decay proportional to the current activation is applied to all lattice positions. The width of the Gaussian activation

_{ij}*σ*

_{0}and the decay

*ω*are the two free parameters controlling activation in the fixation map. For the attention map

*a*(

_{ij}*t*) we assume similar dynamics, including local increase of activation with size

*σ*

_{1}and global decay

*ρ*. However, the amount of activation change

*A*is assumed to be proportional to the time-independent saliency map

_{ij}*ϕ*, so that the local increase of activation is +

_{ij}*ϕ*/ ∑

_{ij}A_{ij}*.*

_{kl}A_{kl}*σ*

_{1}>

*σ*

_{0}, since attention is the process driving eye movements into new regions of visual space, while the inhibitory tagging process should be more localized. A similar expectation can be formulated on the decay constants. Since inhibitory tagging is needed on a longer time scale as a foraging facilitator, we expect a slower decay in the fixation map compared to the attention map, i.e.,

*ω*<

*ρ*.

*t*, both maps are evaluated to select the next saccade target. First, we apply a normalization of both attention and fixation maps as a general neural principle to obtain relative activations (Carandini & Heeger, 2011). Second, we introduce a potential function as the difference of the normalized maps, where the exponents

*λ*and

*γ*are free parameters. However, a value of

*λ*= 1 is a necessary boundary condition to obtain a model that accurately reproduces the densities of gaze positions. In a qualitative analysis of the model (see Appendix), pilot simulations showed that

*γ*is an important control parameter determining spatial correlations, where

*γ ≈*0.3 was used to reproduce spatial correlations observed in our experimental data.

*u*(

_{ij}*t*), Equation (9), can be positive or negative at position (

*i*,

*j*). Lattice positions with a positive potential,

*u*> 0, are excluded from saccadic selection, since corresponding regions were visited recently with high probability. Among the lattice positions with negative activations, we implemented stochastic selection of saccade targets proportional to relative activations, also known as Luce's choice rule (Luce, 1959). We implemented this form of stochastic selection from the set

_{ij}*π*(

_{ij}*t*) to select lattice position (

*i*,

*j*) at time

*t*as the next saccade target is given by

*η*is an additional parameter controlling the amount of noise in target selection.

*σ*

_{0}= 2.2°, is considerably smaller than the corresponding size of the build-up function for the attention map,

*σ*

_{1}= 4.9°. Our second expectation was related to the decay constants, which turned out to be larger for the attention map,

*ρ*= 0.066, than for the fixation map,

*ω*= 9.3 × 10

^{−5}, so

*ρ*was greater than

*ω*, again as expected. Finally, the noise level in the target map is

*η*= 9.1 × 10

^{−5}.

Parameter | Symbol | Mean | Error | Min | Max | Reference |

Fixation map | ||||||

Activation span [°] | σ_{0} | 2.16 | 0.11 | 0.3 | 10.0 | Eq. (7) |

Decay | log_{10} ω | –4.03 | 0.28 | –5.0 | –1.0 | Eq. (6) |

Attention map | ||||||

Activation span [°] | σ_{1} | 4.88 | 0.25 | 0.3 | 10.0 | Eq. (8) |

Decay | log_{10} ρ | –1.18 | 0.08 | –3.0 | –1.0 | Eq. (8) |

Target selection | ||||||

Additive noise | log_{10} η | –4.04 | 0.07 | –9.0 | –3.0 | Eq. (10) |

*predict*saccade-length distributions. The statistical control model approximated the distribution of saccade lengths

*l*and 2D densities of gaze positions

*x*by sampling from the joint probability distribution

*p*(

*x*,

*l*) under the assumption of statistical independence of saccade lengths and gaze positions, i.e.,

*p*(

*x*,

*l*) =

*p*(

*x*)

*p*(

*l*). This model, by construction, approximates the distribution of saccade lengths and 2D density of gaze positions (Figure 5c). The simulations indicate, however, that even the combination of inhomogeneous density of gaze positions and non-normal distribution of saccade lengths used by the statistical control model cannot explain spatial correlations in the experimental data characterized by the pair correlations function (Figure 5d).

*χ*

^{2}(4) = 109.4,

*p*< 0.01. Post hoc comparisons indicated significant effects between all models (

*p*< 0.01) except for the comparison between experimental data and the dynamical model (

*p*= 0.298) and for the comparison between dynamical model and inhomogeneous point process (

*p*= 0.679). In an analysis of the PCF estimated from the same set of simulated data (Figure 6b), the dynamical model (blue) produced deviations from an uncorrelated point process that are in good agreement with the experimental data (red). Model type had a significant effect on PCF deviation,

*χ*

^{2}(4) = 91.6,

*p*< 0.01. Post hoc comparisons indicated significant effects for all comparisons (

*p*< 0.01) except for the comparison between experimental data and the dynamical model (

*p*= 0.990) and between homogeneous and inhomogeneous point processes (

*p*= 0.996).

*χ*

^{2}(4) = 162.6,

*p*< 0.01. Post hoc comparisons indicated significant effects between all models (

*p*< 0.01) except for the comparison between experimental data and the dynamical model (

*p*= 0.251), for the comparison between dynamical model and inhomogeneous point process (

*p*= 0.784), and for the comparison between inhomogeneous point process and experimental data (

*p*= 0.012). For the pair correlation function (Figure 6d), model type had a significant effect,

*χ*

^{2}(4) = 140.6,

*p*< 0.01, and post hoc comparisons indicated significant effects for all comparisons (

*p*< 0.01) except for the comparison between experiment and dynamical model (

*p*= 0.453) and between homogeneous and inhomogeneous point processes (

*p*= 0.700). Thus the main statistical results obtained from image set 1 were reproduced for images of abstract natural patterns (image set 2). These results lend support to the hypothesis that scene content does not have a strong influence on second-order spatial statistics of gaze patterns.

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*μ*= 275 ms. Second, we assumed that the build-up of activation is considerably faster in the attention map than in the fixation map by choosing

*R*

_{0}= 0.01 and

*R*

_{1}= 1, i.e.,

*R*

_{1}/

*R*

_{0}∼ 100.

*p*and

^{e}*p*are the experimental and simulated distributions of pair distances between all data points for a given image and

^{s}*q*and

^{e}*q*are the distributions of saccade lengths for experimental and simulated data, respectively. The minimum of the objective function Λ was determined by a genetic algorithm approach (Mitchell, 1998) within a predefined range (Table 1). Mean values and standard errors of the means were computed from five independent runs of the genetic algorithm.

^{s}*γ*in the fixation map of the potential, Equation (9). We performed numerical simulations with the value of parameter

*γ*fixed at different values between 0 and 1 to investigate the dependence of the spatial correlations on this parameter qualitatively (Figure A1). While

*γ*= 1 produces negatively correlated scanpaths,

*g*(

*r*) < 1, at short pair distances

*r*, it is possible to produce even stronger PCF value than in the experimental data for

*γ*< 0.3. Thus, a single parameter in our model can generate a broad range of second-order statistics.

*point process*is a probability distribution that generates random point patterns: a sample from a point process is a set of observed locations (i.e., fixations, in our case). Therefore, taking two different samples from the same point process will result in two different sets of locations, although the locations may be similar (Figure A2).

*First-order statistics: the intensity function*. The first-order statistics of a point process are given by its intensity function

*λ*(

*x*). The higher the value of

*λ*(

*x*), the more likely we are to find points around location

*x*. Figure A2c shows the theoretical intensity function for the point process generating the points in Figures A2a and A2b.

*c*that counts how many points fall within area

_{A}*A*, for a given realization of the point process. The expectation of

*c*(how many points fall in

_{A}*A*on average) is given by the intensity function, i.e., where the integral is computed over area

*A*. A slightly different viewpoint is given by the

*density function*, which is a normalized version of the intensity function, defined as where the integral in the denominator is over the

*observation window*Ω

*,*which in our case corresponds to the monitor (we cannot observe points outside of the observation window). The density function integrates to 1 over the observation window and represents a probability density: If we now define a random variable

*z*that is equal to one, when a (small) area

_{A}*A*contains one point and 0 otherwise, we obtain where

*x*

_{A}is the center of area

*A*and d

*A*its area. If

*A*is small, then λ̄(

*x*) will be approximately constant over

*A*, and the integral simplifies to λ̄(

*x*) times the volume.

_{A}*Second-order properties*. The first-order properties inform us about how many points can be expected to find in an area, or, in the normalized version, whether we can expect to find a point at all. Second-order properties tell us about

*interaction between areas:*whether for example we are more likely to find a point in area

*A*if there is a point in area

*B*.

*pair density function ρ*(

*x*,

_{A}*x*), which gives the probability of finding points at

_{B}*both*location

*x*and location

_{A}*x*. Let us consider two random variables

_{B}*z*and

_{A}*z*, which are equal to 1, if there are points in their respective areas

_{B}*A*and

*B*, and 0 otherwise (again we assume that the areas are small). The probability that

*z*= 1 and that

_{A}*z*= 1 individually is given by the density function, Equation (A13). The probability that

_{B}*both*are equal to one is given by the pair density function,

*x*and two dimensions for

*x*′) function and in practice it is preferable to use a summary measure, which is the pair correlation function expressing how often pairs of points are found at a distance of

*ε*from each other. The pair correlation function is explained informally in Figure A3.

*x*′ that are on a circle of radius

*r*around

*x*(but still in the observation window Ω).

*ρ*(

*r*) from data, we need an estimate of the intensity function, as it appears as a correction in Equation (A17). In addition, since we have only observed a discrete number of points, the estimated pair density function can only be estimated by smoothing, which is why a kernel function needs to be used. We refer readers to (Illian et al., 2008) for details on pair correlation functions.