Farey Trees Explain Sequential Effects in Choice Response Time

Abstract

The latencies of successive two-alternative, forced-choice response times display intricately patterned sequential effects, or dependencies. They vary as a function of particular trial-histories, and in terms of the order and identity of previously presented stimuli and registered responses. This article tests a novel hypothesis that sequential effects are governed by dynamic principles, such as those entailed by a discrete sine-circle map adaptation of the Haken Kelso Bunz (HKB) bimanual coordination model. The model explained the sequential effects expressed in two classic sequential dependency data sets. It explained the rise of a repetition advantage, the acceleration of repeated affirmative responses, in tasks with faster paces. Likewise, the model successfully predicted an alternation advantage, the acceleration of interleaved affirmative and negative responses, when a task's pace slows and becomes more variable. Detailed analyses of five studies established oscillatory influences on sequential effects in the context of balanced and biased trial presentation rates, variable pacing, progressive and differential cognitive loads, and dyadic performance. Overall, the empirical patterns revealed lawful oscillatory constraints governing sequential effects in the time-course and accuracy of performance across a broad continuum of recognition and decision activities.

Keywords: bimanual coordination; choice response time modeling; cognitive dynamics; nonlinear dynamics; oscillatory entrainment; sequential effects.

Figure 1

An illustration of the Farey tree branching relationships capturing all 31 permutations of R-leading trial-histories. The branching relationships also depict the mode-locking attractors of generic pairs of coupled oscillators with distinct individual frequencies. The labels terminating each branch indicate the mode-locking ratio in the form of a mediant fraction, and as a specific trial-history sequence. Five levels of trial-history sequences are displayed. The first trial of the trial-histories is always R, and progressively more recent trials are listed in succession. Thus, the first trial in the sequence is the oldest trial, and the trial terminating a sequence is the most recent trial. Notably, the Rs and Ls in popular accounts of the Farey tree are typically reversed, relative to these, indicating paths based on branching to the left or right on the diagram itself. All the mathematics are preserved with the exchange of the letters used here to code sequences of R and L finger movements.

Figure 2

An illustration of the relationship between the Farey tree ratios and the Devil’s staircase. The X-axis of plot (B), labeled Ω, depicts the biasing or driving inputs to the bimanual model. For example, consider the upward pointing arrow on the X-axis indicating 8/13 (0.618), the golden ratio (Φ), as a driving frequency. The left-pointing arrow on the Y-axis then indicates Φ as the likely Farey ratio ($p/q$) at which two suitably coupled oscillators, such as those entailed by the bimanual model, will display mode-locking. The corresponding mediant ratios are indicated on the left, in plot (A). All 31 trial-history permutations are formatted sideways as a Farey tree, and approximate their respective locations on plot (B)’s Y-axis. The solid white line bisecting plot (B) is the Devil’s staircase. Each plateau in the staircase indicates the width of a corresponding Arnold tongue, an attractor basin in which a range of biasing frequencies will likely yield a single mode-locking ratio. The various plateau widths result from the fact that ratios with smaller denominators are more stable than ratios with larger denominators. Notably, the staircase was generated with A = 0.85, and K = 0.95, and driven from Ω = 0 to Ω = 1 in increments of 0.001. For simplicity, the noted A and K parameters were fixed for this and all subsequent plots and statistical contrasts in this article. The mean response times for Remington’s (1969) data set, taken from a table of condition means appearing on Luce (1986, p. 288), were normalized to the unit interval, and plotted against the staircase. Each diamond marker indicates a specific trial-history in terms of the color of its corresponding Farey tree branch on plot (B). The correlation between the normalized response times and their predicted Farey ratios, encoded by the Devil’s staircase is r(29) = 0.89, p < 0.0001. Notable exceptions are the ratios on the golden ratio path. They are related to antiphase bimanual coordination and discussed in The Transition From Repetition to Alternation Dominance section.

Figure 3

For continuity, plot (A) again depicts the Remington’s (1969) data as color-coded diamond markers, depicting the 31 ratios, just as in Figure 2B, now atop a detailed bifurcation diagram portraying the potential range of the relative phase relations ($\varphi$) that arise in the model’s dynamics. Rather than the whole-orbit counts of $p/q$, relative phase ($\varphi$) relates two oscillators as an angular difference in their respective cycles. Regions in white indicate the absence of a phase attractive region, turquoise shades (lighter gray) indicate relatively weaker phase attractive regions, and pink shades (darker gray) indicate stronger phase attractive regions. The bifurcation diagram illustrates how the attractors encoded by the staircase can be achieved through a variety of relative phase relations. The Devil’s staircase is an asymptotic summary of these dynamics, in integer counts of both oscillator’s respective cycles. Plot (B) depicts (Kirby’s, 1976) delayed ITI version of a two-button task. The raw data was unavailable. It was estimated from measurements of an enlarged photograph of a plot in the original published article and transformed to the millisecond scale, based on the graph’s axes. The correlation between the normalized latencies and their predicted Farey ratios is weaker than Remington’s data, r(29) = 0.52, p < 0.003. While the markers deviate more from the staircase than the Remington data set, the bifurcation diagram indicates the deviations are generally consistent with the model’s phase attractive topology, as indicated in the bifurcation diagram. Both right-hand Y-axes in this and all subsequent response time plots depict the trial-histories’ untransformed latencies in milliseconds (ms).

Figure 4

Plot (A) depicts unit-normalized response times for the (Jones et al. 2013) condition emphasizing the repetition pattern as color-coded markers, indicating the 31 Farey ratios, over the relative-phase bifurcation diagram. The X-axis tracks the input bias (Ω), the left-hand Y-axis indicates the resulting relative phases ($\varphi )$. Again, the range of the untransformed mean response times is on right-hand Y-axis in ms. The trial-histories are generally consistent with the predictions of the bimanual model. Correlations indicate the left-side ratios approximate the staircase and deviations tend to occupy phase attractive regions of the bifurcation diagram. The primary exceptions to the staircase on the plots right side are the golden ratio and the RLLLL patterns. The normalized response times for the alternation condition appear in plot (C). Trial histories on the golden ration path show a clear advantage, and the underrepresented repetition ratio (RRRRR), and its neighbors on the plot’s left are notably slower. Plots (B,D) depict the error rates in terms of a sideways Farey tree; as in Figure 2A, the root ratio of 1/2 is on the left, rather than at the top of the plot. The error rates for each of the 31 trial histories are again plotted as a function of their color and branching relationships. To the extent the color-coded branching tracks the qualitative sideways tree-structure, as depicted in Figure 2A, the error-rates track the relative stability of the Farey ratios. In plot (B), the repetition pattern has the lowest error rates, and is most stable. In plot (D), the golden ration path has a clear stability advantage. Error-rates for ratios deviating from the path are comparable to the remaining ratios. This suggests the learning resulting from the trial-history bias manipulation does not generalize to closely related ratios.

Figure 5

Plot (A) depicts normalized response times for the −200ms response deadline version of the two-choice Hick task. The X-axis is bias (Ω), and the left-hand Y-axis indicates relative phase ($\varphi )$. The pattern is similar to that of the (Jones et al. 2013) repetition condition, in Figure 4A. Ratios <0.5, emphasizing the right hand are fastest, and drawn to staircase-coincident phase attractive regions; trial-history ratios >0.5 are compressed below the diagonal. Plot (B) depicts the corresponding error rates of the trial-history ratios in terms of their color-coded Farey branching relationships. While the −200ms deadline yielded relatively high error rates, the ratios closely tracked the Farey tree’s denominator-stability predictions. Plot (C), displays the normalized response times for the −100ms response deadline version of the task. In this case, they are comparable over the range of trial-history sequences. By contrast, in plot (D), the error rates again tracked the Farey branching and ratio denominators. Excepting the golden ratio path, the normalized response times in plot (E) are relatively similar. The marker dispersion is almost identical to the (Jones et al. 2013) alternation condition in Figure 4C. Apparently, the slower, variable ITI version of the task favored the intrinsic antiphase bimanual coordination attractor. The error rates indicate increasing stability for the golden ratio path (F). The stability advantage for repetition, evident in the previous versions of the task, is maintained, but more repetitions are required to achieve stability.

Figure 6

Plot (A) portrays the trial-histories for easy lexical decision as a function of the input bias (Ω) and their corresponding relative phase ($\varphi$). The association between the normalized response times and the staircase was strong, and several points that over- or undershoot the staircase are consistent with the bifurcation diagram’s phase attractive regions. Plot (B) indicates the error rates closely track the Farey branching. The harder lexical decision trial-histories appear on plot (C). Again, the correlation between the normalized latencies and the staircase was strong. Plot (D) depicts error rates for harder lexical decision as a function of the Farey tree branching. The association between the relative magnitude of the error rates and the Farey tree branching was robust. Plot (E) portrays the response times for the very hard version of the task; they closely tracked the staircase. The error rates in plot (F) illustrate a continued trend in which ratios with larger denominators garner more errors, while the stability of R-leading repetition and alternation is largely preserved.

Figure 7

Plot (A) depicts the letter rotation trial-histories as a function bias (Ω) and relative phase ($\varphi$). The bulk of the trial-history ratios conformed to the staircase prediction. Plot (B) depicts the error rates in terms of the Farey tree branching. The error rates track the shape and details of the Farey tree, but not as well as for lexical decision. Instead, the R-leading repetition and golden ratio trial histories are more stable than the remaining sequences. Plot (C) depicts the normalized response times for the L-leading ratios, L, LL, LR, and so on. The location of three key trial-histories are highlighted on the figure. L-leading repetition LLLLL has among the largest relative phases, and LRRRR the shortest. The L-leading error rates generally conform to the R-leading Farey tree. Plot (D) indicates that L-leading repetition is the most stable ratio, and L-leading alternation is a close second.

Figure 8

Plot (A) depicts the 1-2-4 letter rotation trial histories as a function of bias and relative phase. The repetition and golden ratio trial histories were faster than the remaining sequences. Plot (B) depicts the error Farey tree for the 1-2-4 condition. It retained consistent branching, despite the manipulation’s impact on the relative speed of R and L responses. Plot (C) depicts the R-leading 2-4-8 letter rotation trial histories, in terms of the L-leading ratios and bifurcation diagram. Increasing the duration of “yes” responses slowed the R-leading ratios, and they expressed the relative phases typical of the L-leading ratios. Plot (D) indicates the error branching remained consistent with the stability of the R-leading ratios. Plot (E) depicts the L-leading ratios for the 2-4-8 task. The speeded “no” responses conformed to the staircase, as predicted by their new role as the relatively faster oscillator. In Plot (F) the color-coded Farey branching for the L-leading ratios is inverted in a manner consistent with the L-leading portion of the Stern-Brocot tree. The LRRRR pattern, and the related brown and tan branches are the least error prone. The LLLLL pattern has lower errors, relative to related branches, indicating stability. Likewise, the bulk of the blue branches now fall above the regression line.

Figure 9

Plot (A) depicts the normalized response times for the dyadic easy lexical decision condition. The trial histories express some compression for the sequences on the RLLLL path. Also, the ratios between 0.25 and 0.5 tend to occupy phase attractive regions just above the diagonal. These outcomes closely mirror the solo version of this task depicted in Figure 6A. The dyad’s error rates appear on plot (B) as a function of their Farey tree branching. The repetition pattern associated is with self-resonance and favored relative to the other ratios. Likewise, the alternation pattern associated with the antiphase coordination was favored relative to nearby ratios. The response times for the hard lexical decision appear in Plot (C). The trial histories are again strongly associated with the Devil’s staircase. The corresponding error rates are depicted in plot (D), note the Y-axis expansion, relative to Plot (B). Again, the self-reinforcing repetition and the alternating golden ratio path are favored relative to the bulk of the remaining ratios. Likewise, the same basic outcomes were evident for the very-hard response times and errors appearing in plots (E,F).