A random multiplicative model of Piéron’s law and choice reaction times

https://doi.org/10.1016/j.physa.2020.125500Get rights and content

Highlights

  • Differences between simple and two-choice reaction times are due to weak additive noise.

  • A noise-induced transition explains both Hick’s and anti-Hick’s effects.

  • The reciprocal of Piéron’s law leads to Naka–Rushton type equations.

  • The principle of minimum transfer of information leads to shorter reaction times.

Abstract

We present a random multiplicative model with additive noise of human reaction/response times based on the power-law function, Piéron’s law. We study the role of weak additive noise in two different scenarios: in the first case, the multiplicative model describes the differences between simple, and two-choice reaction times in Piéron’s law. In the second case, we investigate how choice reaction times depend on the transfer of information in neurons. A transition is found at 0.5 bits due to weak additive noise. Reaction times follow an U-shaped function that lead to both anti-Hick’s and Hick’s effects. We discuss the implications of random multiplicative processes, and minimum transfer of information in decision making, and neural control.

Introduction

Advances in modern mental chronometry has attracted attention into the potential benefits of fluctuation phenomena, and delayed decision making in the mammalian brain [1], [2], [3]. A reference paradigm concerns the human reaction/response time (RT), and is usually defined as the time elapses from stimulus onset until a response is made (e.g., vocal, manual, saccadic, etc.) [1], [2], [4], [5], [6]. RTs are intrinsically stochastic, and involve fast decision making in a wide variety of everyday tasks (e.g., driving safety, sports and board games, etc.). In a simple RT paradigm, only one stimulus, and one response are possible whereas choice RTs involve selection from among alternatives by using multiple stimuli and responses [1], [5], [6]. At least two psychophysical laws have been investigated extensively in RTs: Piéron’s law, and Hick’s law. Piéron’s law describes the hyperbolic decay of the mean RT, μ, as a function of the stimulus strength, S, (e.g., luminance, loudness intensity, odorant concentration, etc.) by a power-law function [1], [7], [8], [9]: μ=μ0+αSp,being μ0, and α, the asymptotic limit or plateau reached at high S values, and a normalizing factor, respectively. The scaling exponent p is a non-integer value that indicates the steepness of the hyperbolic decay [1], [9]. Piéron’s law is a classical example of psychophysical scaling in perception and action, and holds true in all sensory modalities [1], [7], [8], [9], [10], [11], [12], in both simple and choice RTs [13], [14], and in certain animal models [15], [16], [17].Deviations from Piéron’s law can be found at very high S values and the mean RT could increase from the plateau by following an open U-shaped function [18], [19], [20].

In choice RT tasks, Hick’s law describes the increase of μ with the logarithmic of the number of stimulus–response alternatives, or equivalently, the number of bits involved on a task [1], [21], [22]: μ=ϕ+γH,being ϕ, and γ the intercept, and the slope, respectively, and H is a logarithmic function which denotes an information-theoretic metric measured in bits (e.g., Shannon information) [21], [22].

Hick’s law is considered a benchmark effect in human–computer interactions, and has been verified over a large class of experiments [21], [22], including in certain animal models [17], [22], [23]. The slope of Hick’s law depends on the stimulus–response configuration as well as on individual practice [22]. Departures from Hick’s law have been reported for more than 10 alternatives or 3 bits of information, and the mean RT increases exponentially [17], [22], [24]. Further, a reverse or anti-Hick’s effect have found in saccadic eye movements, and RTs decrease as the number of alternatives increases [22], [25], [26], [27], [28]

A long standing issue concerns the unification of both psychophysical laws, and their functional relevance in decision making as well as in sensory-motor control. A few sequential-sampling models have derived the functional form of both Piéron’s law and Hick’s law simultaneously [17], [29]. Caballero et al. have introduced a Bayesian approach with multiple channels for sequential sampling and decision making [29]. A different approach was presented by Reina et al. by using a nest-site selection model. The model was applied to a colony of honey bees for collective decision making [17]. In that stochastic accumulator models the RT is divided in at least two different stages: a decision, and a non-decision time [1], [3], [6], [9]. The decision stage plays a central role, and is modeled by accumulating noisy evidence or response preparation from stimulus until a threshold criterion or boundary is reached [3], [22]. The non-decision stage is considered a residual fixed additive time offset. The plateau μ0 in Piéron’s law is often identified with that non-decision time, and has been treated as a free parameter or simplified to zero [17], [29], [30], [31].

However, that models have ignored the range of empirical values spanned by μ0 in each sensory modality and in different experimental conditions [1], [7], [8], [9], [10], [11], [14], [22], [32], [33]. An implicit assumption is that the non-decision time in μ0 includes the rest of components such as the encoding time, and the motor latency [3], [17], [22], [29], [34], but this is without preserving any chronological order. Further, sequential-sampling models often link neural activity with RTs by associating spike rates with noisy stimulus information during stochastic accumulation [3], [17], [29], [30], [34], [35].

In previous studies, Pins and Bonnet [14] have found that the scaling exponent p remains unchanged, and the plateau μ0 was higher when comparing two-choice versus simple RTs regardless of the complexity of the task [14]. Although sequential-sampling models can mimic the shape of Piéron’s law and Hick’s law [17], [29], [30], it is not clear whether they can describe the effects found by Pins and Bonnet in two-choice RTs [14], and whether they are also able, on one hand, to unify both Hick’s and anti-Hick’s effects and, in the other hand, to predict testable changes in the asymptotic plateau μ0.

In the present study, we address the question whether both Piéron’s power-law function and, Hick’s and anti-Hick’s effects can be derived from a common generative mechanism by treating the RT as a random multiplicative process with additive noise. Multiplicative growth processes are one common way to generate power-law functions with a wide number of applications in physics, biology, finance, etc. [36], [37], [38], [39]. In our approach, we focus on an information-theoretic framework that derives Piéron’s law from an optimal decision process within sensory systems [40]. We pose that the transfer of information in neurons leads to an internal threshold criterion by power-law scaling and modulates a form of signal-dependent neural noise. Contrary to sequential-sampling models [17], [29], [30], [31], we show that the asymptotic term μ0 depends on both sensory and decisional factors. In previous works, we have demonstrated the functional role of random multiplicative processes to derive the shape of RT distributions and Zipf’s law, the effect of Weber’s law in RTs, deviations from Piéron’s law at high S values and deviations from fluctuation scaling in human color vision [41], [42], [43], [44], [45], [46], [47].

Here, we report that differences between simple and two-choice reaction times in Piéron’s law [14], arise from an increase in the transfer of information at the threshold leading to weak additive noise. We found a transition around 0.5 bits [44], and Piéron’s law diverges under the presence of weak additive noise as well. This noise-induced transition leads to both Hick’s and anti-Hick’s effects. In comparison to sequential-sampling models, we also discuss that power-law behavior in Piéron’s law is intimately linked with neural activity by a form of symmetry or scale invariance [39], [48], [49], [50]. We also discuss a plausible basis for optimal decision making based on the minimum transfer of information in sensory systems [44], [50], [51].

Section snippets

A derivation of Piéron’s law

For generic RT task, we define the growth of RT in the time axis as an irreversible process that arises from a cascade of local stages. This implies a hierarchical organization or chronological order in the RT that must be compatible with the principle of causality (i.e., every effect or event has a possible cause) [52]. For instance, a response selection cannot be before in time than the stimulus encoding because the act of encoding contributes to produce that response selection, etc. We will

The asymptotic term of Piéron’s law

Fig. 1(a) represents ΔH as a function of the sensory threshold βS0 (in normalized units) for different values of the scaling exponent p of Piéron’s law. There are two different regimes in Eq. (19). For a fixed threshold value lower than unity, βS0<1, the lower the value of p, 0<p<1, the higher the transfer of information ΔH in neurons (e.g., simulated by p<1, dashed and dash-dotted lines in Fig. 1(a)). However, an inverse tendency is found for threshold values higher than unity βS0>1, and ΔH

Discussion

RTs are a standard tool in human performance such as in skilled chess and soccer players, etc. We have extended previous works and proposed that a signal-dependent multiplicative process with weak additive noise rules Piéron’s power-law function and provides a common basis for both Hick’s and anti-Hick’s effects.

A key point is the time delay produced by the asymptotic term or plateau, μ0, in Piéron’s law. The plateau μ0 is the irreducible RT, and contains not only those time delays associated

Conclusion

Human RTs are a fundamental approach to elucidate the stochastic latency mechanisms in decision making [1], [4], [5], [6]. Here we have shown that a random multiplicative process with weak additive noise provides a unifying description of Piéron’s power-law function in simple and two-choice RTs, and both Hick’s and anti-Hick’s effects.

In our approach, we have used an information entropy or the H-function that describes the internal uncertainty state of neurons as a function of time [40], [53].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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