Network constraints on learnability of probabilistic motor sequences

Abstract

Human learners are adept at grasping the complex relationships underlying incoming sequential input¹. In the present work, we formalize complex relationships as graph structures² derived from temporal associations^3,4 in motor sequences. Next, we explore the extent to which learners are sensitive to key variations in the topological properties⁵ inherent to those graph structures. Participants performed a probabilistic motor sequence task in which the order of button presses was determined by the traversal of graphs with modular, lattice-like or random organization. Graph nodes each represented a unique button press, and edges represented a transition between button presses. The results indicate that learning, indexed here by participants' response times, was strongly mediated by the graph's mesoscale organization, with modular graphs being associated with shorter response times than random and lattice graphs. Moreover, variations in a node's number of connections (degree) and a node's role in mediating long-distance communication (betweenness centrality) impacted graph learning, even after accounting for the level of practice on that node. These results demonstrate that the graph architecture underlying temporal sequences of stimuli fundamentally constrains learning, and moreover that tools from network science provide a valuable framework for assessing how learners encode complex, temporally structured information.

Figure 1.. Experimental Setup.

(A) An example of the first few steps of a graph traversal defined by a walk on the graph. Top: Each node is uniquely associated with a key combination, and the sequence of key combinations is determined by a walk on the graph. Bottom: A series of trials are presented to the participant. The red squares indicate which keys to press on that trial. Colored arrows illustrate the edge from the graph at top being traversed. However, the participant only is shown the five squares. (B) The mapping between fingers and keys, and average reaction times for each key press. Top: A schematic of the mapping between visual stimuli (squares) and response effectors (fingers). Bottom: The average reaction time (RT) for each key or pair of keys across all data. The diagonal elements of the matrix represent trials in which a single key was pressed, and the off-diagonal elements of the matrix represent trials in which a pair of keys was pressed. (C) The three graph structures that we examine in this study. From left to right, we show a modular graph, a lattice graph, and a random graph with N = 15 nodes connected by E = 30 edges.

Figure 2.. Modular Graph Learning Effects.

(A) Mean reaction times (RTs) as a function of trial number for stage 1 of Experiment 1, among participants exposed to the modular graph. The red line indicates the mean for cross-cluster trials, and the black line indicates the mean for all other trials, each binned in sets of 30 trials (n=30 subjects). (B) Mean reaction times on correct trials for the modular graph. An increase in reaction time across cluster boundaries can be seen, here visualized by yellower colors in the matrix elements that sit between the larger blocks. (C) Mean reaction times collapsed across the symmetric structure of the modular graph. All three clusters were structurally identical and starting position was randomized between subjects, so we combine reaction times across the three clusters into one ‘canonical’ cluster for visualization purposes only. The mean increase in reaction time between clusters is more apparent, here visualized by yellower colors on the edges that connect the top cluster with the two bottom clusters. (D) Relationship between surprisal effect on stage 1 (random walk) and surprisal effect on stage 2 (Hamiltonian walk) for each subject of Experiment 2. Subjects that displayed a strong surprisal effect in stage 1 likewise do so when the walk structure is changed (n=59).

Figure 3.. Learning Rate and Edge Surprisal.

Impact of new edges on reaction time. (A) Mean RT increases in stage 2 when new edges are added to the graph (trials 1501–2000). Included for comparison are Experiments 2 and 3, where – respectively – only the walk or a subset of edges were changed. In both cases the increase in RT is much smaller. (B) Per-subject learning rate correlated with the novel edge effect, defined as the mean difference in reaction time for a subject learning the second graph when responding to a novel edge versus a familiar edge (see Methods; n=109). Learning rate, the model coefficient for log(trial), was scaled amongst all subjects to the range [0,1]. The blue line is the least squares fit, with the gray envelope indicating the 95% confidence interval. (C) Individual correlations shown for the three types of graphs trained on in the first stage. Subjects exposed to the modular and lattice graphs show a significant relationship (p < 0.01, n = 30 and p < 0.03, n = 43, respectively), while those exposed to the random graph do not (p < 0.2, n = 36). Solid lines represent least squares fits, and gray envelopes represent the respective 95% confidence intervals. (D) Differences in reaction time by graph type, across graphs learned in sequence. Each bar shows the number of milliseconds by which the modeled effect for the top listed graph is faster. The increase in RT from lattice to modular, and from random to modular graphs, are both significant to p = 0.02 and p = 0.001, respectively (See Supplementary Table 5). Error bars indicate standard error as estimated in the mixed effects model. Asterisks indicate significance in the mixed effects model. Group sizes: Lattice-Random: n=70, Modular-Lattice: n=72, Modular-Random: n=71. (E) Examples of the graph types.

Figure 4.. Relation Between Small and Large Scale Graph Statistics and Reaction Time.

(A) Illustration of node betweenness centrality. We show shortest paths from a number of nodes on the far left to a node on the far right, which all pass through the blue node. (B) Relationship between node degree and reaction time (RT), after regressing out the number of visits to a node, with each point representing a separate node, e.g., 15 points per subject. The regression line shows least squares fit, and the gray envelope is the 95% confidence interval. Reported correlation is based on Kendall’s τ (n=177 subjects). (C) Relationship between node betweenness centrality and reaction time using the same approach as used with node degree. (D) Mean reaction time shown as a function of degree, where the mean was z-scored across the 15 nodes for a given subject. Error bars represent bootstrapped 95% confidence intervals. (E) Mean reaction time as a function of node betweenness centrality using the same approach as used with node degree.