Tasks and Behavioural Performance.

A: Participants performed three tasks: two numerical (symbolic and non-symbolic) and one perceptual (detection). B: Example participant’s psychometric function obtained during the calibration procedure. C: The proportion of gratings reported as ‘present’ in the detection task. In all subplots C-F, black dots are individual participants and red dots are the grand mean over all participants. D: Criterion and d’ in the detection task. E: Reaction times for absent and present responses in the detection task. F: Accuracy in the symbolic and non-symbolic numerical tasks.

Distinct Representations of Perceptual and Numerical Absence.

A: Temporal generalisation decoding results. Representations of perceptual (top-left), non-symbolic (centre), and symbolic (bottom-right) absence were all decodable within each task, respectively. Numerical absences generalised across numerical formats (centre-right, bottom-centre) indicative of format-invariant representations of numerical zero. Representations of perceptual absence did not generalise to numerical tasks, except for a small cluster of significant decoding when the perceptual absence decoder was tested on non-symbolic empty sets (top-right). Black outlines represent areas of significant decoding as computed by cluster-based permutation tests. B: Bayes factor analyses on diagonals of temporal generalisation matrices from A. log10(Bayes factors) above 1 indicate strong evidence for above chance decoding and those below −1 indicate strong evidence for chance-level decoding. Black dotted lines indicate this threshold for strong evidence in both directions. There was decisive evidence for decoding of absences within each task (light blue). There was very strong evidence for shared representations of absence between symbolic and non-symbolic formats (pink). There was very strong evidence for distinct representations of perceptual and numerical absence (red). C: Top: training a decoder to classify non-symbolic empty sets from non-symbolic numerosities and testing it on symbolic numbers in a one-vs-all process revealed increasing discriminability as distance from zero increased (left). The same cross-format distance effect is observed when training a classifier on symbolic zero and testing it on non-symbolic numerosities (right). Bottom: there was no evidence for larger numerosities being more discriminable from perceptual absence than smaller numbers in either the non-symbolic (top) or symbolic (bottom) task. Shaded areas represent 95% Cls.

Stimulus Features Drive Generalisation Between Detection and Empty Set Stimuli.

A: A decoder trained to classify Hits vs. Correct Rejections could decode the presence and absence of grating stimuli from around 100ms after stimulus onset. B: Bayes factors exceeding the upper black dashed line (light blue) reflect strong evidence in favour of above-chance decoding. Bayes factors beneath the lower black dashed line (dark blue) reflect strong evidence in favour of chance-level decoding. C: Cross-decoding between grating presence and absence (A) and empty set vs. non-zero dot stimuli. Left: Training on the detection stimuli and testing on non-symbolic stimuli. A cluster of significant generalisation was observed between 100 and 300ms post stimulus onset. Right Training on empty set vs. non-zero stimuli and testing on grating presence vs. absence in the detection task. Again, a cluster of significant cross-decoding was observed between 100 and 300ms following stimulus onset. D: Left: Bayes factors indicated moderate to strong evidence in favour of generalisation when testing on non-symbolic stimuli. Right: Bayes factors indicated strong evidence in favour of successful generalisation (light blue) when testing on perceptual presence vs. absence. Black outlines represent areas of significant decoding as computed by cluster-based permutation tests.

Visual Features of Non-Symbolic Dot Stimuli.

Total dot area was measured as the number of pixels covered by all dots. Density was computed as the negative median Euclidean distance between dots, such that higher distances between dots results in lower density scores. Luminance was computed using Weber contrast. Convex hull area was computed using the convhull and polyarea functions in MATLAB. Values are Pearson correlation (r) coefficients.

Multidimensional Scaling of Neural Numerical Representations.

Multidimensional scaling reveals numerical representations to be ordered along a single axis (Dim. 1) almost immediately after stimulus onset (left column) through to 600ms after stimulus onset (right column). Both symbolic numerals and non-symbolic numerosities are aligned along the same neural dimension (Dim 2, bottom row).

Colour Can Be Decoded Within and Across All Tasks.

Diagonal: Within-Task decoding of blue vs. orange stimuli was successful within all three tasks. Off Diagonal: CrossTask decoding of blue vs. orange stimuli was successful across all pairwise combinations of tasks and train-test direction. Clusters represent areas of significant decoding as computed by cluster-based permutation tests.

Source-level Decoding Weights for Non-Numerical Absence, Empty Set, and Symbolic Zero Decoders.

Source reconstruction of decoding weights reveals the brain regions utilised by different decoders to classify different neural representations of absence. All decoding weights have undergone the appropriate transformation (Haufe et al, 2014), which in the case of a binary LDA is a contrast between the mean activity in each class. Colour bars represent the t statistic of this contrast. Maps are unthresholded for illustrative purposes.

Representation of Absence are Domain-Specific Within the Alpha Band.

Decoding the amplitude of the alpha rhythms identified representations of absence within all three tasks, however none of these representations generalised across tasks. Clusters represent areas of significant decoding as computed by cluster-based permutation tests.

Bayes Factors Show Strong Evidence for Domain-Specific Representations of Absence in the Alpha Band.

Bayes factors computed for the diagonal of the temporal generalisation matrices in Supplemental Figure 3. There is strong evidence for domainspecific (and strong evidence against domain-general) representations of absence in the alpha band.