Fig 1.
Laplacian associative memory (LAM) model.
Left: associative memory network model with auto-association and hetero-association. The parameter α indicates the auto-associative strength. Right: Equivalent biological neural network model which contains local (assembly-specific) and global (non-specific) inhibition. The parameter α indicates the ratio between local and global inhibition.
Fig 2.
LAM generates multi-scale representations for community structures.
(A) Graph used by Schapiro et al. (2013) [6]. (B) Pattern overlaps of example attractors. (C) Correlation matrices between activity patterns in the attractor states reached from different trigger stimuli (trigger nodes). (D) Maximum pattern overlaps obtained by various values of α. (E) Numbers of active patterns obtained by various values of α. In D and E, we averaged values from all attractors reached from different trigger stimuli. (F) Karate club network[34]. (G-J) Results for Karate-club network. (K) A compartmentalized room structure[5,9,10,36] (four-room graph). (L-O) Results for a four-room graph.
Fig 3.
The relationship between Graph Laplacian eigenvectors and LAM.
(A) Fiedler vector (GL eigenvector with the second smallest eigenvalue) for the graph in Schapiro et al. (2013). (B) Fiedler vector for karate-club network. (C) The comparison of pattern overlaps in LAM (α = −0.9) and Fiedler vector for the four-room graph. (D) A schematic diagram showing that pattern overlaps in LAM (α = −0.5) is mostly explained by the combination of multiple GL eigenvectors with small eigenvalues. (E-G) The explained variance ratio in linear regressions of pattern overlaps by various numbers of GL eigenvectors. The color indicates the value of α. In each condition, we plotted the average value of the explained variance ratio of attractors reached from all trigger stimuli. (E) Results from the graph by Schapiro et al. (2013). (F) Results for the karate-club network. (G) Results for the four-room graph.
Fig 4.
Representations of LAM for a random graph with a hierarchical community structure.
(A) A structure of an example random graph (400 nodes) and three hierarchical levels of communities. (B) Pattern overlaps of attractors in LAM. (C) Pattern correlation matrices between attractors in LAM. (D) The explained variance ratio in linear regressions of pattern overlaps by various numbers of GL eigenvectors. The color indicates the value of α. (E) Average pattern correlations between attractor patterns of node pairs in each hierarchy. A level-h pair is in the same level-h community and not in the same level-(h+1) community if h<H.
Fig 5.
(A) Conversion of images into a weighted graph. We regarded each pixel as a node and determined link weights by spatial proximity and similarity of RGB values. (B,E) Original hi-resolution images used for the segmentation task. We used down-sampled images for the construction of graphs. (C,F) Pattern overlaps obtained after the simulation of LAM with different values of α. (D,G) Representative GL eigenvectors corresponding to segments obtained by LAM.
Fig 6.
Subgoal finding by novelty detection with representations in LAM.
(A) Subgoal finding using low-dimensional (1, 2, or 3) representations constructed from graph Laplacian eigenvectors. The color indicates the novelty index for each node. (B) Subgoal finding using representations in LAM obtained by different values of α.
Fig 7.
Parameter-dependent sequential activities in asymmetric LAM.
(A) A ring structure of hetero-associative links for the simulation of asymmetric LAM. Hetero-associative weights were biased towards one direction. (B) The time course of α (top) and pattern overlaps (bottom) in the simulation of asymmetric LAM. Negative pattern overlaps were truncated to zero. (C) Peak-aligned mean pattern overlap distributions in different ranges of α. (D) Mean speed of the peak shift of the pattern overlap distribution in different ranges of α.
Fig 8.
Chunked sequential activities in asymmetric LAM.
(A) Schematics for the simulation setting. (B) Patten overlaps of simulated neural activities in the uniform model. The red dotted line indicates the actual location of the virtual animal. (C-E) Schematics of the hetero-associative link structure. A uniform ring (C), a ring chunked with local bottlenecks (D), and a ring chunked with local over-representation (E). The details are shown in S8 Fig. (F-H) Mean pattern overlaps at each actual location of a virtual animal in the simulation of a uniform model (F), a bottleneck model (G), and an over-representation model (h). We truncated negative pattern overlaps to zero in these figures. (I-K) Correlations between mean pattern overlaps at different locations in the simulation of a uniform model (I), a bottleneck model (J), and an over-representation model (K). The white dotted lines in G, H, J, and K indicate chunk borders (bottlenecks or over-representations). (L) Experimental data showing segmentation of decoded spatial representations in theta sequences. The labels indicate landmarks on the track. SOM: start of maze, T1 and T2: turns, TC: top corner, F1 and F2: feeders, BC: bottom corner. The image in L was reproduced with permission from ref. [32], Springer Nature.