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

. The human reaction/response time can be deﬁned as the time elapsed from the onset of stimulus presentation until a response occurs in many sensory and cognitive processes. A reaction time model based on Pi é ron ’ s law is investigated. The model shows a noise-induced transition in the moments of reaction time distributions due to the presence of strong additive noise. The model also demonstrates that reaction times do not follow ﬂuctuation scaling between the mean and the variance but follow a generalized version between the skewness and the kurtosis. The results indicate that noise-induced transitions in the moments govern ﬂuctuations in sensory – motor transformations and open an insight into the macroscopic eects of noise in human perception and action. The conditions that lead to extreme reaction times are discussed based on the transfer of information in neurons.

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.
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 t 0 precedes the asymptotic term or plateau t RT 0 , and both precede the mean reaction time (RT), μ, (0 < t 0 < t RT 0 < μ). The plateau t RT 0 is the irreducible part of Piéron's law and represents a repulsion barrier from the origin located at the encoding time, t 0 , (t RT 0 = t 0 exp(2 ln 2ΔH) > t 0 , ∀ ΔH > 0). At supra-threshold conditions, the mean RT μ in Piéron's law always drifts to the plateau (∀ I > I 0 ⇒ μ → t RT 0 ), and thus, t RT 0 represents a bona fide additive noise term [2][3][4].
In page 8, the multiplicative, D a , and additive diffusion coefficient, D b , should be written as follows: (1) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.  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  10 ) of the RT variance σ 2 as a function of μ for achromatic stimuli. Solid circles and squares indicate those RT stimulus configurations that correspond to a ratio ρ < 1.2 (weak additive noise), and ρ > 3.3 (strong additive noise), respectively. The red solid line corresponds to a linear regression analysis, log 10 (σ 2 ) = log 10 (η) + λ log 10 (μ), being η a coefficient, and λ the corresponding slope. (b) Linear plot of the kurtosis γ 2 as a function of the skewness γ 1 for the same RT data. Blue, and red solid lines indicate the best fit to the symmetric power function model with offset to those RTs in the strong (ρ > 3. 3), and weak additive noise (ρ < 1.2), respectively, being α the scaling exponent in equation (2). In both panels, numbers in parentheses indicate (± standard error). strength is written as: 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 ∼ I 0 ⇒ ρ → 0; being stronger at marked supra-threshold conditions, ∀ I I 0 ⇒ 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.
ε and ν being the amplitude and the scaling exponent respectively [27][28][29][30]. The sample RT standard deviation σ or the variance σ 2 are correlated with the mean RT μ across stimulus conditions and they suer deviations from fluctuation scaling by showing a bi-phasic relation. In general, there is an abrupt transition or knee and σ 2 saturates and reaches an asymptotic value for very large μ values [2,[12][13][14], [31][32][33][34][35]. Figure 1(a) shows an example of manual RTs in the ( ) σ µ , 2 -plane for visual stimuli selected across the cardinal directions of the human colour space. The human colour space is an abstract three-dimensional representation of the colourimetric properties of visual stimuli. In colour coding, current models assert that L (long-), M (middle-) and S (short-) wavelength-sensitive cone photoreceptor signals are re-organized into three post-receptoral mechanisms or orthogonal cone axes: a luminance axis (L+M); and two chromatic-opponent cone axes at a constant luminance level or isoluminance: a redgreen (L-M) axis and a blue-yellow axis [S-(L+M)]. Achromatic stimuli (i.e. black and white) stimulate the luminance system whereas red-green and blue-yellow stimuli at isoluminance stimulate the red-green and blue-yellow vision systems respectively [36], [37][38][39][40]. In figure 1(a), pattern formation in RTs shows a similar trend for both achromatic and isoluminant signals. The mean RT μ is often larger for isoluminance signals and the knee is observed for each visual signal separately [9,36].
Together with the mean μ and the variance σ 2 , the skewness γ 1 and the kurtosis γ 2 provide a quantitative description of heavy-tailed non-Gaussian RT distributions [31,36]. RT pdfs are also correlated in the ( ) γ γ , 1 2 -plane across stimulus conditions and follow a U-shaped pattern [36], in the same way as many variables in climate, plasma physics, finance, etc [30,[43][44][45][46]. Figure 1(b) shows the same experimental RT data in the ( ) γ γ , 1 2 -plane. There is not an abrupt transition or knee and the U-shaped pattern is similar for achromatic and isoluminance stimuli. Pattern formation in the ( ) γ γ , 1 2 -plane imposes important constraints for modeling RT pdfs [36]. The U-shaped pattern is quite sensitive to the existence of errors or extreme RT values from the tails of pdfs, specifically, false alarms and misses [36]. False alarms are anticipatory responses that produce very short RTs and left-skewed RT distributions (γ < 0 1 ). Misses are rare RTs at the right tail of pdfs and produce very large kurtosis values (γ 50 2 ) [36]. A generalized version of fluctuation scaling has been proposed in the ( ) γ γ , 1 2 -plane by using a symmetric power function model with oset [36,47]: β, α, S R and K R being the amplitude, the scaling exponent, the center and the oset respectively. The parameters (S R , K R ) indicate a lower bound that aects the overall location of data points. If γ 1 S R and γ 2 K R , the symmetric power function model follows, γ 2 ≃ βγ α 1 . In colour vision the scaling exponent is located between the asymptotic limit / α = 4 3 for very large γ 1 and γ 2 values [46,47] and the quadratic function α = 2 for moderate values [30,43,45], and is higher for achromatic (α ≈ 1.8) than isoluminant stimuli (α ≈ 1.7) [36].
In previous works, we have investigated the functional role of fluctuation scaling in human colour vision and visual masking at threshold. A random multiplicative model with weak additive noise explains pattern formation in the ( ) σ µ , 1 2 -plane as well, with the additive noise term proportional to the contrast of mask stimuli [30]. We have also investigated fluctuation scaling at supra-threshold conditions by using RTs as exemplified in figure 1(b). Although pattern formation was similar in the ( ) γ γ , 1 2 -plane at threshold and supra-threshold conditions [30,36], it is not clear whether fluctuation scaling shares common generative mechanisms in both regimes. Deviations from fluctuation scaling in RTs as shown in figure 1(a) and reported elsewhere [2,14,31,32,34] suggest that fluctuation scaling could be the result of dierent processes in visual-motor communications at supra-threshold conditions. To elucidate these aspects, we will use an RT model based on Piéron's law. Piéron's law describes the hyperbolic decay of the mean RT μ as a function of stimulus strength I [1,2,17,18] , t RT 0 being the asymptotic term or plateau reached at very high I values, and γ and p being coecients, the latter controlling the hyperbolic decay [2,17,18,31,48,49]. Piéron's law is valid in each sensory modality, in both simple and choice RTs and in certain animal models [2,16,18], [49][50][51][52][53][54][55]. In comparison with other approaches to RTs [2,10], the RT model based on Piéron's law uses elements from statistical physics and information theory to define an universal ecient encoder in sensory communications [13,48,49,51,52]. In previous works, the RT model has provided a distinct mechanism that describes the bi-phasic relation type (i), i.e. the shape of RT pdfs as the result of a transition between log-normal and power-law pdfs [14]. We have also discussed the possible functional implications of sustained/transient mechanisms and power functions in RTs [4,9,16,52,56]. In this paper, we will investigate the RT model from a dierent perspective by examining the functional role of additive noise. We will provide a unifying description of bi-phasic relations type (ii) and type (iii), i.e. the bi-phasic behavior of the RT moments and departures from the power function of fluctuation scaling, equation (1), and the origin of equation (2) respectively. In both panels, each point corresponds to dierent stimulus configurations. The total number of experimental RTs was well over 126 000 and they were grouped in 1509 stimulus conditions [36].

J. Stat. Mech. (2016) 093502
Although noise is often considered a nuisance in neural systems, noise-induced phenomena can enhance the dynamics and stability properties in many bi-stable and threshold-based systems. Some examples are stochastic and coherence resonance, noiseinduced synchronization, spatial patterns, etc [57][58][59][60]. Previous works have investigated the functional role of signal-dependent noise in sensory-motor communications [61][62][63] and fluctuation scaling in neurons [62,[64][65][66][67]. Here we report a dierent noise-induced process in mental chronometry. We demonstrate that bi-phasic relations type (ii) and type (iii) are mainly governed by a unique noise-induced transition in the moments of RT pdfs due to strong additive noise. The RT model also derives fluctuation scaling, equations (1) and (2), and explains pattern formation in the ( ) σ µ , 2 -plane and in the ( ) γ γ , 1 2 -plane in figures 1(a) and (b) respectively. We demonstrate that this noiseinduced transition in the RT moments represents a non-trivial eect in sensory-motor communications and opens an insight into how noise-induced phenomena aect stochastic latency mechanisms in the brain at a macroscopic scale [30,36,59,68].
The paper is organized as follows: in section 2, we describe the experimental methods and procedure used for RTs. We focus on RTs in human colour vision by using an extensive RT database that covers a wide range of stimulus conditions from dierent subjects [9,14,36,69], [70]. In section 3 we derive the RT model based on Piéron's law. We discuss some of its basic properties by using a multiplicative growth process with additive noise and showing the existence of an internal threshold mechanism in RTs. Fluctuation scaling in equations (1) and (2) are derived from the RT model. We also present experimental results of a noise-induced transition in the mean RT μ. The RT model of Piéron's law generalizes the bi-phasic relationship in the mean μ type (ii) to higher-order moments and explains the bi-phasic relation type (iii). The implications in the temporal dynamics of parallel visual pathways and the origin of RT false alarms and misses are discussed in section 4. Conclusions are summarized in section 5.

Experimental methods
We have re-analyzed an RT database in colour vision containing more than 126 000 RTs from dierent subjects. The experimental methods and procedure are standard in visual psychophysics and colourimetry and have been reported elsewhere [9,14], [36,[69][70][71]. Briefly, colour coordinates of visual stimuli were selected along the luminance direction and along the red-green and blue-yellow isoluminant directions of the human colour space. Stimuli were generated by using a colour calibrated display connected to a microcomputer equipped with a graphics card. At isoluminance, heterochromatic flicker photometry was used to match the luminance of red-green and blue-yellow stimuli to a reference adapting stimulus. RTs for chromatic variations at isoluminance were performed by using the hue-substitution method. The hue-substitution method avoids any luminance transient changes and RTs were measured for pure hue signals. All stimuli were uniform circular patches and were presented on a dark background. They were centered at the fovea in both monocular and binocular vision by using natural and artificial pupils. Subjects were seated in front of the colour display in a dark room. A chin rest was used for head stabilization. All subjects had normal J. Stat. Mech. (2016) 093502 colour vision according with standard clinical tests and were experienced in RT tasks. For each subject, we collected a number of sessions across dierent days, months and years until reaching a distribution of no less than 70 RTs for each stimulus condition. For each session, subjects were allowed to adapt to darkness and to adapt to a reference stimulus. Dierent adapting stimuli were used at isoluminance. Subjects did not know which stimulus was the next in the sequence and their task consisted only of responding as soon as possible to an intensity change. RTs were measured for manual responses and were taken independently for each cardinal direction of the colour space. The computer clock was programmed to measure RTs with 1 ms accuracy.

Piéron's law as a multiplicative process.
Physics-based approaches to human RTs assert the existence of a cascade of random variables that control the time course of RTs in a certain chronological sequence or causal order [2,3,16,49,52,53]. This formulation of RTs plays a central role to define not only an order but also a direction in the time axis. The time direction of RTs makes a parallelism with the definition of an irreversible process and can be characterized by an information entropy function as we will show below [48,51,72]. For a generic RT task, we define the growth of RT in the time axis at discrete steps to indicate the existence of dierent processes. The RT at the step n + 1, x n+1 , depends on the previous step n, x n , by means of a random multiplicative process with additive noise following a discrete-time Langevin equation [16,52,58,73]: (3) a n and b n being the multiplicative and additive noise terms respectively. Equation (3) implies that x n occurs before x n+1 and 0 ⩽ < + x x n n 1 . When a n > 1 and 0 <a n < 1, the magnitude of the RT is amplified or reduced respectively. The solutions of equation (3) are restricted within a lower and an upper bound. In the former, the additive noise term (b n > 0) prevents x n+1 dropping to zero when a n goes to zero and reaches a minimum RT value. In the latter, non-linear terms ( ) O x n 2 , etc keep the solution bounded up to a maximum RT value. Random multiplicative processes as in equation (3) are one of the simplest mechanisms that leads to power functions in such dierent fields as econophysics, noisy on-o intermittency in complex systems, etc [30,58,[74][75][76][77][78][79]. Following the Langeving approach proposed by Nakao [79], it is established that , the brackets ⟨ ⟩ being the time average over many trials or repetitions of the same RT experiment. The factors D a and D b are the diusion coecients and indicate the average strength of interactions of the multiplicative and additive noise terms respectively. Applying the time average to equation (3) and considering only the first two terms on the right-hand side and ⟨ ⟩ ⟨ ⟩ ≡ a x a x n n n 0 , x 0 being a reference value related with an ecient encoder, the mean RT μ is the result of the interplay between the additive and multiplicative noise: It is assumed that both D a and D b are not fixed but elastic and that both are statedependent and stimulus-driven. Equations (3) and (4) are an alternative way to rewrite Piéron's law, µ γ = , by using a certain chronological sequence that cannot be violated [52,53]. In equation (4), fluctuations in D a and D b across stimulus conditions are mapped into t RT 0 , γ and p.

A derivation of Piéron's law.
To elucidate the internal structure of D a and D b as a function of t RT 0 , γ and p, we derive Piéron's law from first principles by using an informational theory proposed by Norwich [16,48,49,51,72]. In this framework, information processing in neurons is not instantaneous and always takes time. The RT can be defined as the time from stimulus presentation needed to gather ∆H bits of information [48,51]: The information entropy H is a Bolztmann-type entropy function that evolves continuously in time and provides a measure of the internal uncertainty state in sensory systems, t 0 and t being the encoding time and the time to react respectively. The encoding time t 0 in ∆H is an important variable and indicates the existence of a maximum entropy classifier or ecient encoder before reaction [49,80]. The gain of information ∆H represents an irreversible process after ecient encoding and is linked with the formation of an internal variable threshold in the sensory system [16,48,49,[51][52][53]. In equation (5), a temporal sequence of events is implicit in the time axis and < < t t 0 0 . A plausible model for the information entropy function H can be written as follows [16,48,49,51,72]: φ being a parameter. The information entropy as defined in equation (6) is a useful approach to examining the drop of internal uncertainty from a maximum entropy value or potential to receive information. Its origin is related to how sensory neurons transmit the information received by the sensory receptors about an external input signal of intensity I. The information entropy in equation (6) provides the basis to explain many empirical laws in human sensation and perception. Its mathematical derivation has been reported elsewhere [48,51,72]. Introducing equation (6) into equation (5): Solving equation (7) for the time t [48,51]: The asymptotic term or plateau in equation (8) The maximum value of t occurs for a just-threshold signal ( = I I 0 ), I 0 being an internal threshold or reference value that depends on the stimulus configuration and ∆H [48,49,51]: Substituting the term ( ) − − ∆H 1 exp 2 ln 2 from equation (10) into equation (8): Equations (8) and (11) show that RTs decay as a hyperbolic function of the stimulus strength I between an upper (t RT MAX ) and a lower bound (t RT 0 ). If the dierence between t RT MAX and t RT 0 is large enough ( t t RT RT 0 M AX ), Piéron's law exists in RTs. In many practical situations, it is assumed that t RT MAX is too large ( → ∞ t RT MAX ), i.e. lack of response at threshold. This issue will be discussed further later. Therefore, a generalized version of Piéron's law can be written as follows [16,48,49,51]: In equation (12), an RT response occurs only if ( > I I 0 ). The classical form of Piéron's law [17] is obtained by taking the first two terms of the geometric series expansion in equation (12) (4), D x a 0 , is related to the asymptotic term t RT 0 of Piéron's law [74,77]: The diusion coecient of additive noise D b in equation (4) also depends on the plateau t RT 0 but is mainly modulated by the reciprocal of the stimulus strength I. Taking the remaining terms of the geometric series expansion in equation

The eects of additive noise
To investigate power-law behavior of moments of RT pdfs, the Langevin approach made by Nakao compares the strength of the additive noise with respect to the multiplicative noise term under external driving [79]. The ratio ρ of the additive to the multiplicative noise strength can be defined as follows [79]: In general, ρ decreases as I increases and the scaling exponent p of Piéron's law controls the decay. The higher the value of p, the faster the decay ρ. The additive noise strength becomes small ( ρ < < 0 1) at marked suprathreshold conditions ( I I 0 ). A balance is obtained when ρ = 1. However, strong additive noise eects (ρ > 1) are persistent in the critical region near the threshold ( ≅ I I 0 ). In this regime, the lower the value of the scaling exponent ( < < p 0 1), the higher the additive noise strength ( figure 2(b)).
When the eects of the additive noise strength are weak ( → ρ 0), Nakao [79] has demonstrated that the moments of the pdf in the Langevin model follow a power law as a function of ρ. The first moment is the mean μ and the second-order moment centered around the mean is the variance σ 2 [79]:

(20b)
A similar treatment can be performed between the higher-order moments. Let τ 3 and τ 4 be the third-and fourth-order moments of the RT pdf, centered around the mean, respectively. Under the assumption of weak additive noise ( → ρ 0) [79]: G 4 , G 5 , G 6 and G 7 being coecients and J 3 and J 4 the corresponding exponents in the same way as in equations (17) and (18). Substituting ρ from equation (21) into equation (22): We assume that sensory systems are symmetric with respect to the osets G 4 , and G 6 in equations (21) and (22) [81]. The skewness γ 1 and the kurtosis γ 2 are the standardized third-and fourth-order moments respectively [82]: τ γ σ = . Therefore, the symmetric power function model in the (γ γ , 1 2 )-plane becomes:

Noise-induced transition
We investigate power-law behavior of equations (17), (18), (21) and (22) by using experimental RT data. The ratio ρ in equation (16) can be evaluated by providing an estimate of I 0 and p. In colour vision and contrast coding, the stimulus strength I is the contrast of stimuli. We have simplified the situation and have focused on RTs for achromatic visual signals. The RTs for red-green and blue-yellow signals at isoluminance follow in a similar way. For achromatic stimuli I can be defined as the standard Michelson contrast: N max and N min being the maximum and minimum luminance of stimuli with respect to the adapting reference stimulus [9,14,36]. Other contrast metrics [71,83] produce similar results. The mean RT μ as a function of I was fitted to the generalized version of Piéron's law in equation (12) and the parameters I 0 and p were estimated. A weighted non-linear least-squares procedure was performed by minimizing the χ 2 statistics [84]. Weights were selected as the reciprocal of the RT standard deviation σ. Figure 3 exemplifies in a double logarithmic plot the mean μ, the variance σ 2 , the absolute value of the third-order moment τ | | 3 and the fourth-order moment τ 4 of RTs as a function of the ratio ρ for achromatic signals. Although there are broadening eects [29], figure 3(a) clearly shows the existence of a noise-induced transition in the mean RT μ. This noise-induced transition represents the bi-phasic relationship type (ii) [20][21][22][23]26], and is generalized to higher-order RT moments in figures 3(b)-(d). At supra-threshold conditions ( I I 0 ) the strength of additive noise is weak or transient ( → ρ 0) (see figure 2(b)). In figure 3, RT moments slightly increase as a function of ρ but the slopes are nearly flat and close to zero. A reliable estimation of the scaling exponents in equations (17), (18), (21) and (22) was not possible due to broadening [29]. However, at near-threshold conditions the additive noise is strong and produces a sustained behavior (ρ 0.8) (see figure 2(b)). In this regime, RT moments in figure 3 increase as the ratio ρ increases and the slopes are higher than unity.

J. Stat. Mech. (2016) 093502
In all RT moments, there is a transition zone around ρ ≈ 0.5 that separates the transient/sustained behavior. It is interesting to note that the transition zone is smoother for the mean RT μ ( figure 3(a)) and becomes sharper for higher-order moments. Transition zones are similar between the variance, σ 2 and the third-τ | | 3 and fourthorder τ 4 moments (figures 3(b)-(d)). These dierences between transition zones explain the existence of a knee in the ( ) σ µ , 2 -plane ( figure 1(a)) and thus, the bi-phasic relation type (iii), whereas a similar knee in the (γ γ , 1 2 )-plane is absent ( figure 1(b)). Figure 4(a) exemplifies in a double logarithmic plot the same RT data for achromatic signals in the (σ µ , 2 )-plane. RTs at dierent stimulus conditions were classified into two groups: those σ 2 and μ values that correspond to weak additive noise (ρ < 0.3) and those that correspond to strong additive noise (ρ > 0.8). The remaining RT conditions around the transition zone (ρ ≈ 0.5) were excluded. In figure 4(a) the two dierent regimes of the bi-phasic relation type (iii) are clearly discerned and avoid the existence of a single power function or the conventional version fluctuation scaling in equation (1) [28][29][30]. The transient or weak additive noise regime (ρ < 0.3) produces the lowest σ 2 and μ values. Figure 4(b) represents the same RT data in the (γ γ , 1 2 )-plane. Those stimulus conditions for weak and strong additive noise lead to two dierent ). The RT model with weak additive noise produces power-law behavior in the moments of pdfs as shown in equations (17), (18), (21) and (22) [79]. Power-law behavior of moments could also be extended to strong additive noise because Piéron's law is valid under both supra-threshold and near-threshold conditions ( figure 2(a)). We investigate this issue in figure 3 in the strong additive noise regime (ρ 0.8). A linear leastsquared regression analysis for each RT moment was performed in a log-log plot. The scaling exponents for μ, σ 2 , τ | | 3 and τ 4 for strong additive noise were J 1 = 1.38, J 2 = 2, J 3 = 1.5 and J 4 = 1.9 respectively. In figure 4(a), a linear least-squared regression analysis was performed between σ 2 and μ in the strong additive noise regime and for the slope λ = 1.5. From equation (20b), the scaling exponent of fluctuation scaling ν leads to, ( / ) ν ≡ = J J 1.44 2 1 , which is a very good approximation to the slope λ. In figure 4(b), a non-linear least-squares procedure to the generalized version of fluctuation scaling, equation (2), was performed by minimizing the χ 2 statistics [84]. The scaling exponent α in the strong additive noise regime was α ≈ 1.73. From equation (25a), the scaling exponent of the symmetric power function model leads to ( / ) α ≡ = J J 1.26 4 3 , which is a reasonable approximation. It is interesting to note that the U-shaped pattern for achromatic signals found using raw RT data was α ≈ 1.8 [36]. This scaling exponent nearly matches the results obtained in the weak additive noise condition in figure 4(b) (α = 1.77) [36]. However, raw RTs for both red-green and blue-yellow isoluminant signals show U-shaped patterns (α ≈ 1.7) that are closer to the strong additive noise condition in figure 4(b) (α ≈ 1.73) [36].

Implications in neurophysiology
It has been argued that the bi-phasic relationship in the mean RT μ type (ii) for achromatic signals is mediated by dierent sub-cortical pathways [23,25,26], whereas the same bi-phasic relation is absent for chromatic signals at isoluminance [23][24][25][26]. We have extended previous works and generalized the bi-phasic relation type (ii) to higherorder RT moments, i.e. σ 2 , τ 3 and τ 4 by using a RT database that contains a huge number of RTs spanning a broad range of stimulus conditions and several subjects [36]. The analyses of RT pdfs [9] and RT moments (figure 1) [36] are not compatible with a bi-phasic relation only for achromatic signals. The RT model based on Piéron's law demonstrates that bi-phasic relations type (ii) and deviations from fluctuation scaling or type (iii) are the result of a generic noise-induced transition due to strong additive noise. In both cases, a transient/sustained dynamics co-exist when using both achromatic and isoluminant stimuli (figures 1, 3 and 4).
Our approach is dierent from the random multiplicative model with weak additive noise used in visual masking at threshold [30]. First, the RT model derives Piéron's law which describes sensory-motor transformations by using a power J. Stat. Mech. (2016) 093502 function at supra-threshold conditions. Second, the ratio ρ in RTs as defined in equation (16), decreases as the stimulus strength I increases ( figure 2(b)). However in visual masking, that ratio increases as the stimulus strength increases [30]. Third, the additive noise term of the RT model is not always weak as in visual masking [30,79] but it becomes stronger at near-threshold conditions ( figure 2(b)). Therefore, we conclude that there are distinct generative mechanisms that exhibit transient/ sustained dynamics and modify fluctuation scaling in RTs at supra-threshold conditions. Dierences between achromatic and isoluminant signals can be investigated by analyzing the scaling exponent α in the (γ γ , 1 2 )-plane [30,36]. Figure 4(b) demonstrates that both transient and sustained mechanisms produce U-shaped patterns with distinct α values. This is a crucial aspect that characterizes RTs for each visual signal. The results indicate that RTs for achromatic signals mainly contribute to the development of weak additive noise or transient dynamics (α ≈ 1.8) [36]. However, isoluminant signals often provoke strong additive noise or sustained dynamics in RTs (α ≈ 1.7) [36].
Piéron's law is invariant under transformations of scale in the time axis [14,49,85]. This property is similar to deflation or block renaming by means of a renormalization group approach in statistical physics. Self-similarity in Piéron's law at dierent time scales leads to an analogy with the reciprocal of the Naka-Rushton equation in neurophysiology [86]. The Naka-Rushton equation is considered a canonical form of gain control in neurons as a function of the stimulus strength [86][87][88]. Let / = R t 1 RT and / = ′ R t 1 RT 0 . Then, the Naka-Rushton equation can be derived from Piéron's law as follows [14,49]: In equation (27), the neural response R (in spikes per second) increases as the stimulus strength I increases and then it saturates at high I values until reaching the asymptotic limit ′ R . The exponent of the Naka-Rushton equation p has the same role as in Piéron's law and controls the raise or the hyperbolic growth in equation (27). In RTs, p is related with microscopic neural interactions and dierent interpretations have been proposed [13,14,48,49,51,89,90]. In our case, gain control mechanisms as modeled by Piéron's law have a fundamental role and modulate the strength of the additive noise term under external stimulus driving (equation (16)). RTs for achromatic signals favour weak additive noise and, thus, their corresponding neural responses in equation (27) saturate sooner ( > p 1 achromatic ) [91,92] (see also figure 2(b)). However, RTs for isoluminant signals promote strong additive noise. In accordance with equation (27), their associate neural responses are more linear and sustained and for the scaling exponent ( < p p isoluminant achromatic ) ( figure 2(b)). These RT dierences between achromatic and isoluminant signals support the notion that noise-induced transitions in RT moments are mediated by the retinocortical magno-, parvo-and konio-cellular parallel pathways [39,40,93]. Magno cells combine L-and M-cone signals (L+M). They exhibit transient activity and respond better to achromatic stimuli. Their responses are faster and they saturate at high stimulus contrasts. However, parvo cells combine L-cones opposed to M-cones (L-M). They are more sustained and respond stronger to red-green stimuli at isoluminance. Their responses are slower and they do not saturate, and increase linearly as the stimulus contrast increases. Konio cells combine S-cones opposed to L-and M-cone signals [S-(L+M)]. They exhibit a sluggish delay in the visual cortex. They respond better to blue-yellow stimuli at isoluminance and their responses are heterogeneous at high stimulus contrasts [39,40], [91][92][93][94][95][96][97][98][99].