(A) Left: distribution of caches across different distances from the previous site. Right: distribution of site checks across different distances from the previous site. Black trace: mean probability across birds. Gray: 95% confidence interval calculated by bootstrapping birds. In both tasks, probability decreases with distance, but is also low at small distances. In the main text, the previous site scaling factor is modeled using a Gaussian with a notch as 0 (Equation 5), which captures the basic features of these traces. (B) Schematic of two alternate functions that were tested instead of a notched Gaussian. Left: Notched exponential function, parametrized by decay . Right: Mexican hat function parameterized by two widths for the positive and negative Gaussian components ( and , respectively) and the amplitude of the negative Gaussian (). The amplitude of the exponential on the left and the amplitude of the positive Gaussian on the right are both 1; however, the entire function is scaled by in the models (see below). (C) Performance of Models #14 and # 15 applied to the Caching task, plotted as in Figure 4A. These models are identical to Model #1, but instead of the notched Gaussian function use a notched exponential scaled by or the Mexican hat function scaled by , respectively. Model #14 did not improve the fit (p = 0.99), whereas Model #15 did (p < 0.001). Therefore, a Mexican hat function better describes the proximity effect than a notched Gaussian. (D) Best-fit values of parameters of , and applied to individual birds in the Caching task, plotted as in Figure 4B. Dashed line on the left: distance between cache sites. (E–F) Same as (C–D), but applied to the Retrieval task. p = 1, p < 0.005 respectively. (G) Performance of Models #16 and # 17 in the Caching task, plotted as in Figure 4A. These models used a Mexican hat function to model the proximity effect and introduced the mnemonic parameters and to quantify the spatial extent of the effect of occupied and checked-empty sites on behavior, respectively. In other words, these models were like Models #2 and #3 (see main text), but used a Mexican hat function instead of a Gaussian for the previous site scaling factor. Both models improved the fit (p < 0.005). (H) Values of the mnemonic parameters and across birds, compared between Model #2 and Model #16, and Model #3 and Model #17 respectively, applied to the Caching task. In both cases, the value of the parameters is unchanged between models. Therefore, using a simpler Gaussian scaling factor for the proximity effect (as in Models #2 and #3) did not distort the values of the mnemonic parameters compared to those produced by the more complex Mexican hat scaling factor (Models #16 and #17). (I–J) Same as (G–H), but applied to the Retrieval task.