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Activity estimation via distributed measurements in an orientation sensitive neural fields model of the visual cortex

Adel Malik Annabi, Dario Prandi, Jean-Baptiste Pomet, Ludovic Sacchelli

TL;DR

The study emphasizes the intrinsic link between the model’s nonlinear nature and its observability and develops a hybrid high-gain observer that achieves, under specific excitation conditions, practical convergence while maintaining asymptotic convergence in cases of biological relevance.

Abstract

This paper investigates the online estimation of neural activity within the primary visual cortex (V1) in the framework of observability theory. We focus on a low-dimensional neural fields modeling hypercolumnar activity to describe activity in V1. We utilize the average cortical activity over V1 as measurement. Our contributions include detailing the model's observability singularities and developing a hybrid high-gain observer that achieves, under specific excitation conditions, practical convergence while maintaining asymptotic convergence in cases of biological relevance. The study emphasizes the intrinsic link between the model's non-linear nature and its observability. We also present numerical experiments highlighting the different properties of the observer.

Activity estimation via distributed measurements in an orientation sensitive neural fields model of the visual cortex

TL;DR

The study emphasizes the intrinsic link between the model’s nonlinear nature and its observability and develops a hybrid high-gain observer that achieves, under specific excitation conditions, practical convergence while maintaining asymptotic convergence in cases of biological relevance.

Abstract

This paper investigates the online estimation of neural activity within the primary visual cortex (V1) in the framework of observability theory. We focus on a low-dimensional neural fields modeling hypercolumnar activity to describe activity in V1. We utilize the average cortical activity over V1 as measurement. Our contributions include detailing the model's observability singularities and developing a hybrid high-gain observer that achieves, under specific excitation conditions, practical convergence while maintaining asymptotic convergence in cases of biological relevance. The study emphasizes the intrinsic link between the model's non-linear nature and its observability. We also present numerical experiments highlighting the different properties of the observer.
Paper Structure (16 sections, 11 theorems, 96 equations, 2 figures)

This paper contains 16 sections, 11 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Modelisation
  3. General model on V1
  4. Finite-dimensional reduction in V1
  5. Main model of study
  6. Notation.
  7. Statement of the results
  8. Observability
  9. Observer design
  10. Technical preliminaries
  11. The observability mapping
  12. Observability mapping extension and differential observability
  13. The pseudo-inverse
  14. Observer convergence
  15. Numerical Simulations
  16. ...and 1 more sections

Key Result

Proposition 2.4

Assume $I\in C^0([0,+\infty))$. The dynamics EqP admit a unique global solution $v$ defined on $\mathbb{R}$ for every initial condition $v(0)\in \mathbb{R}^3$. Moreover, if $\sup_t ||I|| < \infty$, then there exist $R^* > 0$, such that for any $R \geq R^*$, $B_{\mathbb{R}^3}(0,R)$ is an invariant at

Figures (2)

  • Figure 1: Visual cortical mapping of orientation preference and selectivity in V1 region: (a) illustrates a comprehensive view of orientation-selective neuronal distribution through optical imaging, and (b) presents a detailed analysis of orientation preference and selectivity, with emphasis on the normalized selectivity index and key angular orientations.
  • Figure 2: Left-hand side: components of the simulated trajectories of System \ref{['EqP']} (plain lines) and Observer \ref{['Observer 1']} (dashed lines). Right-hand side: plot of the estimation log-error. This particular solution satisfies point \ref{['it:main2']} of the theorem. The switching time occur at $t_1\simeq 1.2$ and $t_2\simeq 1.8$. Peaking phenomena occur at the start of the simulation and at time $t_2$, when the high-gain observer restarts; they are better seen on the log-error plot. Since the error peaking at $t_2$ is proportional to the error at $t_1$, it is much smaller in size. At the beginning of the trajectory, $\hat{v}_1,\hat{v}_2$ are set to 0 due to the cut-off \ref{['E:sat']}, until the observer $\hat{z}$ of the embedded system has sufficiently converged. At time $t_1$, the observer is turned off and the trajectories of $v$ and $\hat{v}$ evolve according to the neural fields dynamics, resulting in a drift of the error over $[t_1,t_2]$.

Theorems & Definitions (31)

  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 21 more