Materials and methods

Evidence

Noisy evidence was sampled from the internal representation of the numerical difference. The internal representation was assumed Gaussian, following standard psychophysics of number [27, 28], (1) s(i) is the sample i, taken every R ms (a free parameter). The symbol ∼ stands for “distributed as”. n_L and n_R are the presented numbers on each side. Samples formed the evidence stream s = (s₁, s₂, s₃,…, s_i,…, s_n), with n denoting the total number of samples at a given movement time t. The values for R were bounded to be between 1-100 ms to reflect realistic values of spiking rates of number processing neurons [29]. A negative sample from Eq (1) is evidence favoring the wrong number.

The value of ω determines evidence noise (Eq (1)). Following number cognition literature it will be referred to as the Weber fraction and it is computed by fitting average error rate across participants to, (2) erfc is the complementary error function [30]. A Weber fraction was computed for each participant. The obtained mean Weber fraction from participants’ choices was equal to 0.17 (s.d. = 0.04; mean R² = 0.96, s.d. = 0.02), in line with previous number comparison experiments [31]. We also computed a Weber fraction using all the data and the obtained value was also around 0.17. In the model we used the latter.

The numerical stimulus was on-screen for a brief time which means that evidence samples were produced from memory (see sampling from memory in [32]). We introduced a memory decay to the evidence stream s in order to model a upweighting of initial information in the statistical inference. We used an exponential moving average (EMA) [33] (3)

We initialize EMA(0) = s(1). D is a free parameter between 0 and 1 that determines the strength of the memory decay. For example, if D = 1 the evidence stream s is unmodified (no memory of previous samples). Alternatively, a D close to 0 makes the series gravitate around the initial sample, akin to a system with primacy effects in memory [34] (for visualizations of how the memory decay parameter works see S1 Fig). Other way of interpreting D is as a generalization of different accumulation schemes. A D = 1 is a memory-free stochastic accumulator in which only the noise of the sampling determines choice [35]. A D = 0 is a non-stochastic linear ballistic accumulator [10] in that sampling noise is completely reduced by the initial sample. Values in between can be regarded as continuous variations of these possibilities.

Controller and statistical inference

The model used an optimal feedback control, a framework that has been successful at explaining a wide array of motor behavior [36]. In general, this framework proposes that motor commands try to minimize a cost function along the movement, while concurrently taking into account the present state of the system and task demands. This flexibility was essential for our purposes of generating movements that were affected by continuous number processing.

Specifically, we tackle the kinematic problem with a criterion of jerk (third derivative of position) minimization [37] and an optimal feedback controller derived by [18] (refer to those works for further details on the analytical derivations). At each time step t, changes to the state q of the motor effector were determined by, (4)

The controller used the current position x, velocity , acceleration , the remaining time δ = t_f − t, and the end target x_f, to update the effector state. Here t refers to current movement time and t_f to total movement time. Movement time t_f was randomly sampled at the beginning of each simulated trial from the data of the appropriate ratio. Even though fixed movement times are not ideal, spontaneously generating them is a difficult problem and many influential motor models use a similar strategy of deriving them from data [18, 23, 37, 38]. For the interested reader, derivations of optimal movement time for the jerk model can be found in [39]. The kinematic state q was initialized at 0.

Recent work suggests that motor plans are modulated by certainty [40], an assumption that we incorporate and extend. The model first determined the probability of which side had the largest number of dots with the cumulative probability, (5) (6) with μ = n_L − n_R i.e. the numerical difference. If the left side has the larger number, Eq (5) represents the probability of being correct.

The probabilities p_L and p_R evolved according to the sampled numbers. More concretely, we assumed that subjects were trying to position the effector with the best possible estimate of the numerical difference μ = n_L − n_R, given all the samples e in EMA taken up to time t. The numerical difference μ was inferred with a uniform prior p(μ), and normal likelihood p(e|μ, W, n₁, n₂,); with W as a free parameter representing the likelihood noise. The resulting posterior p(μ|W, n₁, n₂, e) from a uniform prior and normal likelihood is itself normally distributed (due to space considerations we direct the reader interested in this derivation to [41]), (7) where, is the mean of the EMA samples. The free parameter W determines the overall confidence in the obtained mean numerical difference. It should not be confused with ω above, which determines the noise of the internal numerical samples: ω characterizes sampling in perception, while W characterizes confidence in action. Conceptually, Eq 7 tells us that the Bayesian estimate of μ is centered at the mean of the samples e and the noise of that estimate (W) is reduced as more samples (n) arrive. In more direct terms, a Bayesian agent should be more confident of their numerical perception as they get more evidence.

Heading direction was then determined with the largest probability e.g. if p_L ≥ p_R, move to the left. The certainty was defined to be a rescaling of the entropy over p_L and p_R that gives a measure between 0 and 1: (8) where H(p) = −(p log p + (1 − p)log(1 − p)). Thus, Eq (8) effectively measures how much information there is favoring the current heading direction.

The certainty is then used to modulate the motor update, (9)

The incorporation of certainty is equivalent to delaying the updates of the effector without altering the nature of the optimal control (minimizing jerk). The end result is that velocity and the other kinematic changes are slowed down according to the available level of information.

Traditional threshold model

We also implemented a more traditional threshold model [5] with five parameters: threshold (thr), drift of samples (dft), variance of samples (sig2), non-decision time (ndt), collapse rate of threshold (k). We actually tested two versions: a) one drift that functionally changed with difficulty level and b) five drifts. Results were practically similar with both versions and we decided to present the results of the simpler first version. The functional form that we used was based on a traditional formula in number cognition literature that reflects how numerical ratio affects difficulty [30], (10)

The threshold collapsed every ms as follows, (11)

Fitting

The model has three free parameters: sampling rate (R), confidence/acuity (W), and memory decay (D). Fitting them allows us to test the influence of these cognitive variables. Parameter values were determined with a global search of the parameter space with the DIRECT-L algorithm [42] as implemented in nloptR [43]. The algorithm minimized the sum of two mean-squared errors (MSE): 1) MSE of mean horizontal positions along the trajectory (Fig 1C), and 2) MSE of accuracy (Fig 1C). We simulated 480 trajectories per numerical ratio and computed an average simulated trajectory for the MSE. As with data, we normalized time to 101 points and use positions up to 99% of the end target to account for the gap that was not recorded (see Task and apparatus).

We did not model a button and the forces required to release it. We used the median amount of movement before participants released the button. The median value was 0.28 mm across participants. We used that reference to define RT. RT was the amount of time required to move 0.28 mm plus non-decision time. Non-decision time refers to a constant time, mainly visual and motor response delays [5]. This was set manually, after the fitting procedure, with values found in the threshold literature: 200-500 ms [7, 15].

Changes of mind were defined as correct or incorrect trajectories that penetrated 1 cm of the opposite side of the last recorded position of the index finger.

Accuracy and response times for the traditional threshold model were fitted using BADS algorithm [44] (again minimizing a function of MSE). After obtaining the best parameters we then produced positions as explained along the main text.

Interim summary

The number comparison task (Fig 1) was modeled by combining key ideas from three influential approaches in the sensorimotor domain (Fig 2).

1. Accumulation of evidence [4]: it was assumed that subjects sampled evidence about the numerosity from each display according to Weber’s law [27, 28, 30, 45, 46]. This evidence was sampled at a rate (R) and it was affected by a memory decay (D).
2. Bayesian inference [47]: the most likely location of the correct option was inferred using the accumulated evidence and the precision of the inference was determined by an acuity parameter (W).
3. Optimal feedback control [18]: a minimizing jerk movement was weighted by certainty [40] computed from the probabilities of the inference.

Changes of mind

Changes of mind are deviations of the motor effector toward a target different from the initial selection. They are of interest because they confirm the presence of online processing during reach. Even though the model and the static nature of the stimulus were not designed to probe changes of mind, the model was able to produce them (Fig 7D).

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Fig 7. Changes of mind without post-lift off thresholds.

(A) Changes of mind in human participants. Thick color lines are averages colored by numerical ratio. Arrows on the x-axis are a mean proxy of where the change of mind first appeared (the x coordinates of the minimum points of the thick lines). (B) Changes of mind produced by a parametrization of the model. (C) They increased with cognitive difficulty, both in the model and human data. The minority of changes of mind were from correct to incorrect (gold). (D) Examples of double changes of mind in the model and human data.

https://doi.org/10.1371/journal.pone.0195188.g007

Also, we observed more than one change of mind in some trials (Fig 7D). In the model, changes of mind are the result of continuous statistical fluctuations. This means that multiple changes of mind are possible without additional assumptions (Fig 7D; they are highly scarce in human data and the model (<0.3%).

Thirdly, a manually-tuned parametrization of the model replicates known characteristics of changes of mind reported in previous literature [15]: 1) highly scarce and more prevalent in hard trials (Fig 7C); 2) they tend to fix an error i.e. they go from incorrect to correct (Fig 7C); and 3) most of them appear relatively far from the target depending in trial difficulty (see arrows in Fig 7A and 7B). Therefore, the model in Fig 7 represents a plausible average cognitive state present when changes of mind occurred. This was obtained with a manually tuned model because fitting the model with the global method by adding changes of mind (see Methods) produced a mildly informative compromise between positions, accuracy, and changes of mind. This may only testify about the difficulty to fit 3 dependent variables at once. More relevant is that the manually-tuned parameter values were insightful about changes of mind as continuous in nature. Note, however, that the parameter values obtained by fitting positions and accuracy (Fig 4) produced too few changes of mind (< 1%). Subjects’ cognitive parameters may have fluctuated to produce the rate of observed changes of mind (∼3%). The details of such fluctuation are beyond this work but could be caused by drops in attention or fatigue, leading to a higher Weber fraction or an altered likelihood noise.

Importantly, changes of mind can occur in the proposed continuous regime and comply with features found in previous behavioral work [15]. For example, in the model the reason that changes of mind tend to be prevalent in hard trials and correct an error is that an initial stream of incorrect evidence is more likely to occur when numbers are close (Eq 1) but as more samples arrive, and because there is a correct answer, the initial stream of incorrect evidence is overrun and the inference changes. Similarly, the point where a change of mind occurs along the reaching trajectory depends on numerical distance (see colored triangles in Fig 7A and 7B) because difficult comparisons generate low quality information taking longer to overrun an initial stream of incorrect evidence.

Weber fractions

So far we used the Weber fraction ω estimated from accuracy, following standard practices in number cognition literature (Eq 2). With the model it is possible to estimate it using subjects’ positions and accuracy (see Methods). That is, we add ω as a free parameter in the fitting procedure. We found parameter values that reproduced sensitivity to numerical ratio in reach, accuracy, and response time (Fig 8). The obtained Weber fraction ω was higher than the one obtained from accuracy alone (0.17 vs 0.39). A perceptual system that is noisier, as revealed by a higher Weber fraction, reproduces accuracy. In other words, even though it is possible to estimate a low Weber fraction from accuracy the underlying uncertainty, as revealed by reaching, is not captured by just using percentage of correct answers. However, it is worth to highlight that the larger Weber fraction may also be a reflection of a strong memory effect in sampling (D = 0.06). Future experiments could try to separate memory effects from purely perceptual noise.

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Fig 8. Weber fraction estimation from dynamic (reach) and discrete (accuracy) dependent variables.

In all the previous figures the Weber fraction was a fixed parameter obtained from subjects’ accuracy, as traditionally done in most psychophysics papers of numerosity. In this figure, we fitted a model with the Weber fraction as a free parameter. This means that the parameter is obtained both from dynamic reach and discrete accuracy. By including reach, we obtained a higher Weber (0.17 vs 0.39) i.e. a noisier perceptual system that reproduces the same accuracy and reach. Error bars are s.e.m.

https://doi.org/10.1371/journal.pone.0195188.g008

Number cognition

There is a considerable amount of work exploring number representations and their effects on movement plans [17, 54–56]. The novelty of our work is a formalization of how psychophysical representations of number may be projected to reach. Two relevant theoretical insights are worth noting. First, the model never directly projected the details of the numerical representation. Previous work has proposed otherwise and suggests that the motor system forces logarithmic scales or numerical formats in how the motor effector, i.e. index finger, is moved in physical space [25, 54, 57]. In our model the displacement of the index finger is not forced to follow precise aspects of the number representation; it is the state of the inference and confidence that is being reflected along the reaching trajectory (see a similar discussion in [56]). In a broader sense, movement may only be a window to a domain-general process, such as confidence, rather than domain-specific aspects of cognition (log. scaling of mental space).

Second, the model allowed us to estimate Weber fractions using both accuracy and movement patterns. We found a larger Weber fraction than the one estimated from accuracy alone (0.39 vs 0.17, respectively). This suggests that human numerical sensitivity may be misestimated if confidence and decision dynamics are not taken into account. This has direct implications for math education given the efforts to connect perceptual numerical estimation, number metacognition, and formal math learning [31, 58, 59]. By bringing forth the idea that numerical estimation may be affected by confidence or other decision parameters, such as memory in sampling, it may be possible to explain why some researchers find positive and others negative results regarding the importance of approximate numerical estimation in formal/symbolic mathematics; an argument also extended in [11]. In brief, the observed relation between the approximate number system and mathematics may be mediated by aspects unrelated to numerical Weber ratios (e.g. perhaps the quality of the inference varies between students, as indexed by the parameter W in our model).

Thresholds and action

In most domains decision thresholds are useful for describing a large number of brain and behavioral responses [4, 5]. However, our results show that in some very simple computational systems, threshold-like behavior may arise as essentially an epiphenomenon: behavior may be effectively delayed due through uncertainty over motor plans, not inherent cognitive or neural mechanisms that set bounds for action. This suggests that in some domains thresholds may be replaceable with other rational processes.

Thresholds are a dominant framework in the field. However, there are prior reports of their failure to capture some aspects of behavior. In a recent neurophysiology paper, one of the monkeys failed to show neural signatures of a threshold when selecting one of the options [60]. The monkey’s neurons behaved as if they were not accumulating evidence, even though the accuracy of the monkey in the task was high. In another study testing for sources of noise in the accumulation process, it was found that subjects (rats and humans) were best fit by models with infinite thresholds [35]—bounds were not necessary. In work that used dynamical systems to model behavior it was found that very low thresholds were required to produce changes of mind [61], and that fast response times and high spike rates were not correlated as predicted by a rise-to-threshold theory [62]. Finally, neural responses in canonical accumulation areas can be explained by discrete steps rather than ramping accumulation [63], plausible indicating the production of discrete samples instead of a rise-to-threshold.

Our study complements these findings and strongly suggests that once motor models are combined with statistical inference models, threshold-like behavior may emerge as a consequence of motor planning. This work therefore points towards models which make explicit the interrelation of decision and action, and show how interesting behavior may emerge from the confluence of simple processes in inference, choice, and movement.

An interesting result was that micro movements before button release were modulated by numerical ratio, suggesting information processing early on. There are precedents on this idea. Micro eye motions during fixation have been connected to active information processing that transforms seemingly static visual space to temporal signals interpretable by the retina [64]. Studying early signals becomes important in a continuous decision framework but can also inform on whether choice is discrete or not.

Decision making is highly likely to be a mixture of discrete and continuous mechanisms. The results herein are not directly undermining the whole idea behind thresholds. In fact, it should be possible to come up with a discrete scheme that explains the observed human behavior in our task, perhaps by adding multiple post-lift off thresholds. Rather, we think the main insight is that decisions can be placed in a continuous framework, even for simple static comparison tasks. It would be interesting to extend this to classic dynamic contexts such as the random dot motion task, usually modeled with a discrete framework.

Choice error and movement uncertainty

The task design (stimulus on-screen for only 200 ms) and model results (D = 0.06) suggest that subjects gathered evidence from memory or, equivalently, that they accumulate non-stochastically and linearly [10]. In general, the literature suggests that both of them are plausible [10, 32]. However, it is important to clarify that this implicates that for the model the source of error made by subjects is mostly due to a bias in the mean of the internal memory representation. The initial samples strongly bias subjects and when wrong they produce mistakes. That is, the source of error is driven mostly by a bias in the perceived mean of the internal representation, rather than its variance.

Another intriguing aspect of our results is that it is unclear how to move when there is ambiguous evidence. In our task, ambiguous information appears in trials with equal number in both sides (S6 Fig). A simple average of parallel motor plans to each option will make the movement land in between the options (e.g. [65]). Similarly, weighting by amount of information, as we did here, will fail to fully drive the index finger to the target (S6A Fig). Dealing with uncertainty gridlocks is a difficult challenge for future decision-making models that try to account for movement patterns, accuracy, and response times in 2AFC. Still, we present a possible solution using a satisficing strategy in the supplemental material (S6 Fig).

A signature of uncertainty is the presence of changes of mind [15]. We propose that they result from an early incorrect inference due to initial numerical evidence favoring the non-selected target. Also, because the model produced changes of mind with slightly different parameters than the best fitted ones, they may require fluctuations in cognitive parameters; perhaps brought about by external variables, such as drop in attention or fatigue. More interesting is that changes of mind in number comparisons can be framed as continuous events rather than the arrival to a discrete post lift-off threshold [15].