**Analyzing sequential eye movement patterns is difficult**

Eye movements are an important data source in cognitive science, providing a rich description of how attentional resources are deployed. However, they are difficult to analyze and the vast majority of eye movement studies ignore sequential information in the data and utilize only first-order statistics. The main reason that sequential information is ignored is that the number of possible eye movement patterns grows exponentially with their length. In Hayes, Petrov & Sederberg (2011) we demonstrate how temporal difference learning can be used to construct a successor representation (SR; Dayan, 1993) that captures the statistical regularities in temporally extended eye movement sequences. The resulting SRs are interpretable and are a uniform size, allowing the SR matrices from different observers and/or trials to be analyzed using standard statistical methods. As a result, SRSA is an excellent tool for studying individual differences in attentional control.

**Illustration of SRSA**

As an example consider a standard matrix reasoning task with a 3 x 3 problem matrix and 8 response alternatives. The relational reasoning problem space was separated into 10 areas of interest (AOIs). The 10 AOIs included the 9 cells of the problem matrix (top row = 1 2 3; middle row = 4 5 6; bottom row = 7 8 9) and the entire response area (R). Sample fixation sequences were generated according to (a) systematic and (b) toggling strategies.

Successor representations (c & d) were computed for each trial scanpath, resulting in one 10 x 10 SR matrix ** M**. Each trial SR matrix is initialized with zeros and then updated for each transition in the scanpath sequence. Consider a transition from state

*i*to state

*j*. The

*i*th column of the matrix- the column corresponding to the “sender” AOI is updated according to:

*ΔM*_{i} = α (I_{j} + γM_{j} – M_{i})

_{i}= α (I

_{j}+ γM

_{j}– M

_{i})

where I is the identity matrix, each subscript picks a column in the matrix, alpha is the learning-rate parameter, and gamma is a temporal discount parameter. The latter term is the key to extending the event horizon to encompass both immediate and long-range state transitions. Note the entries in the SR matrix are not probabilities, they are discounted/expected numbers of visits. When gamma is set to zero the SR is equivalent to a first-order transition matrix and as gamma increases the event horizon is extended farther into the future.

The diagonal box structure in (c) reflects the row-by-row scanning pattern in (a), whereas the bottom-heavy matrix in (d) reflects the toggles to the response area. The matrix (e) is the mean of (c) and (d). The deviations from the mean are shown in (f & g). These extracted regularities can then be used to test cognitive theories (Hayes, Petrov & Sederberg, 2011, 2015 ; Hayes & Henderson 2017).