Fluctuation scaling in the visual cortex at threshold

José M. Medina and José A. Díaz

Phys. Rev. E 93, 052403 – Published 6 May 2016

Abstract

Fluctuation scaling relates trial-to-trial variability to the average response by a power function in many physical processes. Here we address whether fluctuation scaling holds in sensory psychophysics and its functional role in visual processing. We report experimental evidence of fluctuation scaling in human color vision and form perception at threshold. Subjects detected thresholds in a psychophysical masking experiment that is considered a standard reference for studying suppression between neurons in the visual cortex. For all subjects, the analysis of threshold variability that results from the masking task indicates that fluctuation scaling is a global property that modulates detection thresholds with a scaling exponent that departs from 2, $β = 2.48 \pm 0.07$ . We also examine a generalized version of fluctuation scaling between the sample kurtosis $K$ and the sample skewness $S$ of threshold distributions. We find that $K$ and $S$ are related and follow a unique quadratic form $K = (1.19 \pm 0.04) S^{2} + (2.68 \pm 0.06)$ that departs from the expected 4/3 power function regime. A random multiplicative process with weak additive noise is proposed based on a Langevin-type equation. The multiplicative process provides a unifying description of fluctuation scaling and the quadratic $S - K$ relation and is related to on-off intermittency in sensory perception. Our findings provide an insight into how the human visual system interacts with the external environment. The theoretical methods open perspectives for investigating fluctuation scaling and intermittency effects in a wide variety of natural, economic, and cognitive phenomena.

Received 1 November 2015
Revised 7 March 2016

DOI:https://doi.org/10.1103/PhysRevE.93.052403

Physics Subject Headings (PhySH)

Fluctuations & noise Scaling laws of complex systems

Visual system

Physics of Living Systems

Authors & Affiliations

José M. Medina^* and José A. Díaz

Departamento de Óptica, Facultad de Ciencias, Universidad de Granada, Edificio Mecenas, 18071, Granada, Spain

^*jmedinaru@cofis.es

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Vol. 93, Iss. 5 — May 2016

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Images

Figure 1
(a) Example of a horizontally oriented achromatic test Gabor patch, orthogonal mask, and both superimposed as a plaid at high contrasts. In the example the spatial frequency is 0.375 cpd. (b) Red-green stimuli at isoluminance. The luminance level remains constant but producing a chromatic signal. The same spatial frequency follows as in (a).
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Figure 2
Typical examples of a 2AFC staircase procedure at different stimulus configurations. Semilogarithmic plot ( ${log}_{10}$ ) of the stimulus contrast threshold as a function of the trial number. The vertical dashed lines indicate the first reversal point. Arrows indicate subsequent reversal points.
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Figure 3
Example of cross-orientation masking in human vision. Double logarithmic plot ( ${log}_{10}$ ) of the test contrast detection threshold versus mask contrast $C$ for achromatic Gabor gratings at 0.375 cpd, 8 Hz. Solid circles indicate the mean threshold value in binocular vision from one subject (P2). The solid square at the left and the corresponding horizontal dashed line indicate the threshold level that denotes no effect of the mask on the test Gabor grating. Points above and below the dashed line indicate threshold elevation and facilitation, respectively. Error bars are one standard deviation. The solid line indicates the best fit to the equation $y = (6 \times 10^{- 14} / C^{4.9}) + 0.12 C^{0.5}$ (see Sec. 4 D). Vertical arrows show a schematic representation of the vertical mask Gabor at different mask contrasts $C$ .
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Figure 4
(a) Double logarithmic plot ( ${log}_{10}$ ) of the sample variance $σ^{2}$ as a function of the sample mean µ from all subjects together. Solid line indicates a linear fit ${log}_{10} (σ^{2}) = {log}_{10} (α) + β {log}_{10} (μ)$ , where α is a coefficient and β is the slope, $β = 2.48 \pm 0.07$ , where the error is the standard error. (b) Linear plot of the kurtosis $K_{1}$ versus skewness $S_{1}$ from all subjects together. The solid line indicates the best fit to a power function with an exponent of 2, $K_{1} = (1.19 \pm 0.04) S_{1}^{2} + (2.68 \pm 0.06)$ , where errors are standard errors. The dashed line indicates the statistical limit $K = S^{2} + 1$ [52, 53]. Vertical and horizontal dotted lines indicate $S = 0$ and $K = 3$ , respectively. The intersection of dotted lines corresponds to the Gaussian point. In (a) and (b) data points represent a total of 392 and 318 stimulus conditions, respectively, that result from the combination of all spatial and temporal frequencies, mask contrast values, viewing conditions (binocular and monocular), type of stimuli (achromatic and red-green isoluminant), and masking effects (threshold elevation and facilitation). Solid circles and squares represent the values for red-green isoluminant and achromatic stimuli, respectively.
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Figure 5
(a) Semilogarithmic plot ( ${log}_{10}$ ) of the time average of the multiplicative $〈 λ_{i} 〉$ and the additive noise term $〈 η_{i} 〉$ of the Langevin model. The parameters are $x_{0} = 0.1, ω = 1, ɛ = 2, J_{λ} = 0.02, J_{η} = 0.18$ . (b) Double logarithmic plot ( ${log}_{10}$ ) of the TvC functions for achromatic gratings for several values of the temporal and the spatial frequency: $x_{0} = 0.032, ω = 2, ɛ = 1, J_{λ} = 0.05, J_{η} = 0.053$ (squares), $J_{η} = 0.1$ (circles), $J_{η} = 0.55$ (triangles). (c) TvC functions for binocular and monocular vision: $x_{0} = 0.032$ (binocular), $x_{0} = 0.1$ (monocular), $ω = 1, ɛ = 2, J_{λ} = 0.02, J_{η} = 0.19$ . (d) TvC functions for achromatic and red-green isoluminant stimuli: $x_{0} = 0.032, ω = 2, ɛ = 2, J_{λ} = 0.07, J_{η} = 0.07$ (achromatic), $J_{η} = 0.9$ (red-green).
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Figure 6
Double logarithmic plot ( ${log}_{10}$ ) of the first four moments $m_{r}$ as a function of the strength of the additive noise $ρ$ . (a) Mean µ. (b) Variance $σ^{2}$ . (c) Absolute value of the third moment $m_{3}$ . (d) Fourth moment $m_{4}$ . In each panel the solid line corresponds to a linear regression analysis. The corresponding slopes (± standard error) for µ, $σ^{2}, | m_{3} |$ , and $m_{4}$ are $0.35 \pm 0.01, 0.83 \pm 0.05, 1.13 \pm 0.08$ , and $1.7 \pm 0.1$ , respectively.
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