RNNs show a novel order-dependent behavior having two distinct versions.

A, Schematic of the end order effect, a response-time (RT) behavioral pattern. The behavioral pattern is observed in RNNs in two qualitatively different versions: 1st-faster vs. 2nd-faster. A schematic showing the trial types used to quantify the effect is in Fig 3A (fourth column). B, Histograms of end-order behavior across RNNs (counts: RNN instances). The behavior was quantified as the difference of RTs divided by their sum (end order index; RTs calculated for trials where end items (A and G) occurred either 1st vs. 2nd). RNN variants (f-RNN vs. r-RNN and higher to lower constraint regimes) follow that diagramed in Fig 1D. These results summarize those in the last column of Fig 3.</p

A simple neural solution to delay TI.

A, Population activity trajectories in an example RNN (high-constraint f-RNN) that performs TI. Top and bottom plots show two different views. Shown are trajectories from all 42 trial types (see Fig 1B). To clarify the operation of the network, three trial times are highlighted: (i) presentation of item 1 (green circles; shade indicating item rank: A (dark green) to G (white)), (ii) the last time point of the delay period (green stars; same color convention), (iii) last time point of the trial (red/blue symbols; red: choice 1 trials, blue: choice 2 trials, light to dark shading indicating symbolic distance (1 to 6); diamonds: training trials, triangles: test trials). Also shown: cross-condition mean (XCM; the average trajectory across all trial types) (yellow line) and fixed points (FPs) (crosses). Two FPs were attractors (black crosses), one FP was a saddle point (black cross with pink line indicating axis of unstable mode), and one FP (orange cross) was located near trajectories during the delay period (‘early-trial’ FP). Note two prominent activity patterns: linearly arranged rank-ordered activity upon presentation of item 1 (green circles) and the oscillatory evolution of trajectories in the delay period (circles to stars). B, Linear dynamics of RNN in panel A. Left, eigenvalue spectra of the RNN. The spectra were calculated in two ways: first, from delay-period neural activity (black points; inferred via least-squares linear fit, R2 = 0.65) and second, from linearization of the network with respect to the early-trial FP (orange circles; FP shown as orange cross in panel A). Right, population activity trajectories of the RNN plotted in the plane of the oscillation inferred from delay period activity (filled arrowheads in spectra plot). C, Linear dynamics of higher-constraint f-RNNs (n = 400 instances; 200 highest and 200 high). Eigenvalue spectra of delay-period neural activity (grey translucent points; inferred via least-squares linear fit, R2 ∼0.6–0.9; see S6A Fig). Note the density of oscillatory modes with frequency ∼0.5 cycles / delay (filled arrowhead). D, Activity trajectories in the oscillatory mode of the linearized RNN. The oscillatory mode is that of the linearization of the early-trial FP (open arrowheads in panel B). Plotted in background are flow field vectors (not to scale; shown to indicate motion direction). To clarify how the activity evolves, trajectories are plotted for two successive trial periods (left and right panels; schematic at bottom of each): early trial (left) and presentation of item 2 (right). Three trial times are highlighted: (i) presentation of item 1 (circles; color indicating item rank: A (dark green) to G (white)), (ii) the last time point of the delay period (stars; same color convention), (iii) presentation of item 2 (red/blue symbols; red: choice 1 trials, blue: choice 2 trials; diamonds: training trials, triangles: test trials). Activity states for (iii) solely reflect the application of (feedforward) item 2 input. Note the separation of choice 1 vs. 2 trials (red vs. blue symbols), indicating that correct responses were evolved in the activity space. E, Diagram of solution expressed in networks: population-level “subtraction”. Plotted are activity trajectories generated by simulating a 2D linear dynamical system defined by an oscillation of frequency ∼0.5 cycles / delay, with initial condition at the origin and input vectors encoding task items (A, B, C, etc.) in ordered collinear arrangement in state space (compare to panels a and d). Trial-based input (item 1—delay—item 2, see S1A Fig) was applied to the system. Plotting conventions are the same as in panel D. For further detail of the solution, see S1 Appendix.</p

Delay TI: Transitive inference (TI) with a requirement for working memory.

A, Trial structure. Each trial consists of three periods: rest, delay, and choice. The duration of the delay was 2τ to 6τ and either of fixed or variable length. Note that subjects respond on the basis of item order: if the correct response in trial type X vs. Y (item 1: X, item 2: Y) is choice 1, then the correct response in trial type Y vs. X (item 1: Y, item 2: X) is choice 2. B, Target values of RNNs output units (zi(t, m), where t is time and m is trial type; see Methods). (TIF)</p

Responses to debriefing question.

Examples of typed responses to a question (“How did you decide which item to choose?”) in a debriefing questionnaire given to subjects after completing all trials in the delay TI task. Example responses are sorted by performance on test trial types averaged over the last three blocks of trials. See S12 Fig for word count summary. Subjects performing at high levels tended to use words indicating understanding of the transitive relationship (e.g. “hierarchy”, “order”, “higher”, “ranked”). Lower-performing subjects appeared to use these words less often, rather using phrases like “random”, “tried to remember”, or referring to other strategies or a lack thereof (e.g. “didn’t have a specific strategy”, “the more attractive image”, or strategies based on a single item, “choose one and if it was right I would keep choosing the same one”). Words denoting a comparative relationship between items were commonly used (e.g. “beat”, “win”, “lost”). A number of subjects at various performance levels mentioned that some items were always correct. (PDF)</p

The “subtractive” solution to delay TI.

Diagrams presenting the solution in greater detail (compare to Fig 5). Top, diagram of each of the population-level components comprising the solution (top: the specific form of the component; bottom: the network implementation). Bottom, activity trajectories across trial periods (columns; diagram of each period at bottom) and across different trial types (rows; top row: all trials; single trial types in rows below). Trajectories were generated by simulating a 2D linear dynamical system defined by an oscillation of frequency ∼0.5 cycles / delay, with initial condition at the origin and input vectors encoding task items (A, B, C, etc.) in ordered collinear arrangement in state space. Trial-based input (item 1—delay—item 2, see S1A Fig) was applied to the system. (TIF)</p

Main predictions of representative models.

••, a prediction unique to this model. •, two predictions that jointly are unique to this model. “End order” effect: Fig 4. Collinearity: Fig 7A. Mean angle change: Fig 7B. Oscillation in delay period: Fig 5C and S6 and S7A–S7C Figs. Choice axis encoding: Fig 7C. All neural predictions refer to the top PCs of population activity.</p

A neural basis for order-dependent behavior.

A, Neural encodings (axis-projected activity) in four example RNNs (rows). Left column, projection of neural activity along the readout axis (output unit weights). Middle column, item 1 vs. 2 encoding, corresponding to projected neural activity at the end and beginning of the delay, respectively. Right column, sum of item 1 and item 2 encodings. Note that the sum yields large values for trials containing end items, either when the end item is item 1 (rows; examples 2–4) or item 2 (columns; example 1). B, Left, histograms of the encoding index across RNNs. The encoding index was defined as the multiplicative gain in the magnitude of end items’ neural encodings (>0: item 1 encoding larger than item 2, ‘1st-dominant’; 0, respectively). The analogous analyses for the choice axis are presented in S10C Fig.</p

RNNs performing delay TI: Linear dynamics.

RNN activity during the delay period was fit to a linear dynamics model (least-squares). Rows show results for RNN variants differing by learnable connectivity (f-RNN: fully-trainable RNN (all weights trainable), r-RNN: recurrent-trainable RNN (only recurrent weights trainable), ff-RNN: feedforward-trainable RNN (only feedforward weights trainable), r-RNN with trainable output weights). Column 1: R2 values of the fit. Constraint regime variants plotted by color. Columns 2–6: eigenvalue spectra (grey points; calculated for each RNN instance using top 10 PCs; numbers of instances reported at bottom right), with each column corresponding to a different RNN variant (higher to lower constraint regime, indicated by color). A, RNNs trained on basic delay TI. Note that the spectra shown in Fig 5C corresponds to two of the spectra here combined (f-RNN highest (black, column 1) and high (blue, column 2)), and that spectra shown in S8C Fig is the same as that shown in row 3, column 1. B, RNNs trained on extended and variable delay TI. (TIF)</p

Cognitive task and neural approach.

A, Diagram of the relational structure (schema) underlying transitive inference (TI). Subjects learn correct responses to premise pairs (A vs. B, B vs. C, C. vs. D, etc.; training trials), and must infer correct responses in previously unobserved (novel) pairs (C vs. E, etc.; testing trials). For every possible pair, the “higher” item should be chosen. Items (A, B, etc.) can correspond to arbitrary stimuli. B, Trial types and their correct responses in 7-item TI. Each trial type consists of a permutation of two items (item 1 and item 2). C, Item presentation formats: traditional TI vs. delay TI. In traditional TI, items are presented simultaneously and are chosen on the basis of their presented position (e.g. left vs. right). The present study proposes “delay TI”, in which items are presented with an intervening delay and are chosen on the basis of presentation order (1st vs. 2nd). This task format explicitly requires working memory (WM). D, Neural models in the present study. Among neural network (NN) architectures, recurrent neural networks (RNNs) are able to implement WM. Two variants of RNNs were studied: fully-trainable RNNs (f-RNNs), for which both feedforward and recurrent synaptic weights were modifiable in training, and recurrent-trainable RNNs (r-RNNs), for which only recurrent synaptic weights were modifiable in training (feedforward weights randomly generated and fixed; trainable vs. fixed weights diagramed as black vs. pink arrows, respectively). In conjunction, RNNs were trained with different levels of regularization and initial synaptic strengths (‘constraint regime’, indicated with colors shown; parameters in Table 1). All networks had 100 recurrent units, which all had the same non-linear activation function (tanh). See Methods for additional details.</p

Traditional TI in feedforward models.

A, Schematic of feedforward model architecture (see Methods). B, Example LR and MLP model instances that perform traditional TI (i.e. no explicit delay between items, with choice made on basis of position (left vs. right); Fig 1B). C, Schematic of behavior patterns. D, Behavior of feedforward models (n = 100 instances / model). All plots show average performance (proportion correct, averaged across 500 simulations of every trial type). Column 1: Averages across model instances by trial type. Columns 2–4: Averages across trials for each model instance by trial type. Trial types follow that defined for each behavioral pattern in panel C (column 2: symbolic distance; column 3: end item), in addition to distinguishing between choice 1 vs. choice 2 trial types (red vs. blue, respectively; diagramed in panel C, transitivity). E and F, feedforward models express a ‘subtractive’ solution to TI. E, Analysis of an example LR. At left, activation of readout ‘units’ (see Methods) as a function of input position (y-axis) and rank (x-axis). At right, relationship between position of inputs and readout unit activation. Note that activations by item position (left vs. right) were sign-inverted versions of each other. F, Analysis of an example MLP. At left, activation of hidden units as a function of input position (y-axis) and rank (x-axis). At right, relationship between position of inputs and unit activation, plotted for all hidden units (N = 100 tanh units). Note that activations by item position (left vs. right) were approximately sign-inverted versions of each other, akin to the LR model. (TIF)</p