Erratum: Noise-induced transition in human reaction times (2016 J. Stat. Mech.: Theory Exp. 9 093502)

Due to an error, the two limiting regimes of the theory were interchanged. We regret to inform the readers that equation (16) in [1], that describes the ratio of the additive to the multiplicative noise in human reaction times, is flawed. Here, we report the correct expression, and, accordingly the modified figures 2(b), 3, and 4.

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In section 3.1.2, the random multiplicative model of Piéron's law implies a chronological order that must be preserved. That is, the encoding time t₀ precedes the asymptotic term or plateau ${t}_{{\text{RT}}_{0}}$, and both precede the mean reaction time (RT), μ, $\left(0{< }{t}_{\text{0}}{< }{t}_{{\text{RT}}_{0}}{< }\mu \right)$. The plateau ${t}_{{\text{RT}}_{0}}$ is the irreducible part of Piéron's law and represents a repulsion barrier from the origin located at the encoding time, t₀, $\left({t}_{{\text{RT}}_{0}}={t}_{\text{0}}\enspace \mathrm{exp}\left(2\enspace \mathrm{ln}\enspace 2{\Delta}H\right){ >}{t}_{\text{0}},\enspace \forall \enspace {\Delta}H{ >}0\right)$. At supra-threshold conditions, the mean RT μ in Piéron's law always drifts to the plateau $\left(\forall \enspace I{ >}{I}_{\text{0}}{\Rightarrow}\mu \to {t}_{{\text{RT}}_{0}}\right)$, and thus, ${t}_{{\text{RT}}_{0}}$ represents a bona fide additive noise term [2–4].

In page 8, the multiplicative, D_a , and additive diffusion coefficient, D_b ', should be written as follows:

$$\begin{equation}{D}_{a}\cong \mathrm{exp}\left(2\enspace \mathrm{ln}\enspace 2{\Delta}H\right)\left[{\left(\frac{{I}_{0}}{I}\right)}^{p}+{\left(\frac{{I}_{0}}{I}\right)}^{2p}+\cdots \enspace \right]{\geqslant}0.\end{equation} \tag{ 1 }$$

$$\begin{equation}{D}_{b}^{\prime }=\mathrm{exp}\left(2\enspace \mathrm{ln}\enspace 2{\Delta}H\right){\geqslant}1,\end{equation} \tag{ 2 }$$

Equations (1) and (2) replace equations (14b) and (15) in [1], respectively. Then, it follows in page 9, section 3.2, that the ratio ρ of the additive to the multiplicative noise strength is written as:

$$\begin{equation}\rho =\sqrt{\frac{{D}_{b}^{\prime }}{{D}_{a}}}={\left[\sqrt{{\left(\frac{{I}_{0}}{I}\right)}^{p}+{\left(\frac{{I}_{0}}{I}\right)}^{2p}+\cdots }\right]}^{-1}=\sqrt{{\left(\frac{I}{{I}_{0}}\right)}^{p}-1}.\end{equation} \tag{ 3 }$$

Therefore, equation (3) is the reciprocal of equation (16) in [1], and replaces it.

Here, the additive noise becomes small at near-threshold conditions, ∀ I ∼ I₀ ⇒ ρ → 0; being stronger at marked supra-threshold conditions, ∀ I ≫ I₀ ⇒ ρ ≫ 1. Therefore, when we said 'strong additive noise', it should be said, 'weak additive noise' and vice versa across the entire text in [1]. There is a noised-induced transition and the transition zone is now found at ρ ≈ 2. Accordingly, the modified figures 2(b), 3 and 4 are provided below. The authors want to point out that these corrections do not affect the rest of analyses and discussion except the above cited changes. We apologize to the editor, and to the readers for any inconvenience caused.