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Reproducibility via neural fields of visual illusions induced by localized stimuli

Cyprien Tamekue, Dario Prandi, Yacine Chitour

Abstract

This paper focuses on the modeling of experiments conducted by Billock and Tsou [V. A. Billock and B. H. Tsou, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 8490--8495] using an Amari-type neural field that models the average membrane potential of neuronal activity in the primary visual cortex (V1). The study specifically focuses on a regular funnel pattern localized in the fovea or the peripheral visual field. It aims to comprehend and model the visual phenomena induced by this pattern, emphasizing their nonlinear nature. The research involves designing sensory inputs that mimic the visual stimuli from Billock and Tsou's experiments. The cortical outputs induced by these sensory inputs are then theoretically and numerically studied to assess their ability to model the experimentally observed visual effects at the V1 level. A crucial aspect of this study is the exploration of the effects induced by the nonlinear nature of neural responses. By highlighting the significance of excitatory and inhibitory neurons in the emergence of these visual phenomena, the research suggests that an interplay of both types of neuronal activities plays a crucial role in visual processes, challenging the assumption that the latter is primarily driven by excitatory activities alone.

Reproducibility via neural fields of visual illusions induced by localized stimuli

Abstract

This paper focuses on the modeling of experiments conducted by Billock and Tsou [V. A. Billock and B. H. Tsou, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 8490--8495] using an Amari-type neural field that models the average membrane potential of neuronal activity in the primary visual cortex (V1). The study specifically focuses on a regular funnel pattern localized in the fovea or the peripheral visual field. It aims to comprehend and model the visual phenomena induced by this pattern, emphasizing their nonlinear nature. The research involves designing sensory inputs that mimic the visual stimuli from Billock and Tsou's experiments. The cortical outputs induced by these sensory inputs are then theoretically and numerically studied to assess their ability to model the experimentally observed visual effects at the V1 level. A crucial aspect of this study is the exploration of the effects induced by the nonlinear nature of neural responses. By highlighting the significance of excitatory and inhibitory neurons in the emergence of these visual phenomena, the research suggests that an interplay of both types of neuronal activities plays a crucial role in visual processes, challenging the assumption that the latter is primarily driven by excitatory activities alone.
Paper Structure (21 sections, 17 theorems, 102 equations, 13 figures, 1 table)

This paper contains 21 sections, 17 theorems, 102 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Billock and Tsou's psychophysical experiments
  3. Neural field model, strategy of study, and presentation of our results
  4. General notations
  5. Assumption on parameters and mathematical modeling of visual stimuli
  6. Assumption on parameters in the Amari-type equation
  7. Coupling kernel:
  8. Response function:
  9. The intra-neural connectivity parameter $\mu>0$:
  10. Mathematical modeling of visual stimuli
  11. Preliminary results on the Amari-type equation
  12. On Billock and Tsou's experiments modelling
  13. Unreproducibility of Billock and Tsou experiments: linear response function
  14. Unreproducibility of Billock and Tsou's experiments with certain nonlinear response functions
  15. Numerical analysis and experiments
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $I\in L^\infty({\mathbb R}^2)$. For any initial datum $a_0\in L^\infty{\mathbb R}^2)$, there exists a unique $a\in C([0,\infty);L^\infty({\mathbb R}^d))$, solution to eq::NF-intro. If $\mu<\mu_0$, there exists a unique stationary state $a_I\in L^\infty({\mathbb R}^2)$ to eq::NF-intro. Moreover,

Figures (13)

  • Figure 1: Visual illustration of the retino-cortical map, redrawn from billock2007. The top-left corresponds to the funnel pattern in the retina, and on the top-right, the corresponding pattern of horizontal stripes is in V1. While the bottom-left corresponds to the tunnel pattern in the retina, and on the bottom-right, the corresponding pattern of vertical stripes is in V1. In particular, these images are regular in shape and symmetrical with respect to a specific subgroup of the plane's motion group bressloff2001.
  • Figure 2: Billock and Tsou's experiments: the presentation of a funnel pattern stimulus in the centre (image on the top-left) induces an illusory perception of tunnel pattern in surround (image on the top-right) after a flickering of the empty region (the white region surrounding the stimulus pattern on the top-left). We have a reverse effect on the bottom. Adapted from billock2007.
  • Figure 3: On the left, nonlinear response functions $f_{m,\alpha}(s) = \max(-m,\min(1,\alpha s))$ for different values of $m$ and $\alpha$. On the right a $1$D DoG kernel $\omega$.
  • Figure 4: Funnel pattern in the centre of the visual field.
  • Figure 5: Horizontal stripes in the left area of V1.
  • ...and 8 more figures

Theorems & Definitions (40)

  • Definition 1
  • Remark 2.1
  • Remark 2.2
  • Definition 2: Stationary state
  • Theorem 3.1: tamekue2024mathematical
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • ...and 30 more