Fig. 3: Single neural basis of compositionality.

a Illustration of how neural responses in two-dimensional rank subspaces are projected onto each neuron’s single unit vector, thereby deriving a spatial tuning curve. The standard deviation of each rank’s tuning curve denotes the signal strength (A_i), while the preferred location (φ_i) is represented by the phase of the tuning curve. b Illustration of disjoint (top) and overlapping (bottom) neurons. Disjoint neurons aligns primarily with only one single rank subspace, whereas overlapping neurons aligns with both rank subspaces. c The distribution of neuron-to-subspace strength (NSS) indices measures the relationship between each pair of subspaces. Top: in the disjoint neuron dominant scheme, NSS distribution tends to be bimodal. Bottom: in the overlapping neuron dominant scheme, NSS distribution tends to be normal. d Illustration of shared tuning and shifted tuning neurons. φ_diff, differences of preferred locations between different subspaces. e The distribution of φ_diff measures the tuning changes between each pair of subspaces. Top: the shifted neuron dominant scheme with a uniform distribution. Bottom: the shared tuning dominant scheme with a right skew distribution. f NSS and φ_diff distributions pooled across all rank pairs. g Left: percentage of disjoint neurons, error bar represent STD of 100-time samplings (two-sided 1-Proportion z test, n = 4068/ 1596/ 1327/ 832 neurons, z = 10.81/ 5.95/ 9.80/ −0.83, p = 0/ 2.56e−09/ 0/ 0.41). Right: CDFs of φ_diff (K-S test for non-uniformity[dash line], n = 1689/ 679/ 485/ 428, D = 0.08/ 0.08/ 0.09/ 0.11, p = 1.60e−10/ 2.10e−04/ 2.28e−4/ 8.63e−6). h NSS and φ_diff distributions pooled across all length pairs, same format as (f). i Left: percentage of overlapping neurons, error bar represent STD of 100-time samplings (two-sided 1-Proportion z test, n = 4976/1751/1586/1109 neurons, z = 12.50/8.96/3.87/1.65, p = 0/0/1.10e−04/0.10). Right: CDFs of φ_diff (K-S test for non-uniformity[dash line], n = 2929/ 1063/ 870/ 582 neurons, D = 0.41/ 0.37/ 0.39/ 0.30, p = 0/ 1.24e−132/ 9.13e−117/ 2.93e-46). j Summary of the simulation describing how NSS indices and φ_diff affect the relationship between subspaces. See also supplementary Fig. 7. k To maintain generalization between subspaces, an overlapping distribution of NSS indices and a predominant presence of shared tuning neurons are necessary. l Disjoint NSS indices distribution or uniform distributed φ_diff is sufficient to reduce interferences between subspaces.