Linear control systems on a 4D solvable Lie group used to model primary visual cortex $V1$

TL;DR

This paper models the primary visual cortex V1 using linear control systems on a 4D solvable Lie group, providing new insights into controllability and control sets within this geometric framework.

Contribution

It introduces a novel application of a 4D solvable Lie group to model V1 and characterizes controllability and control sets for the associated control systems.

Findings

Established controllability conditions for the system.

Characterized the control sets in the model.

Linked the geometric framework to visual cortex modeling.

Abstract

In this article, we study linear control systems on a 4-dimensional solvable Lie group. Our motivation stems from the model introduced in \cite{baspinar}, which presents a precise geometric framework in which the primary visual cortex $V1$ is interpreted as a fiber bundle over the retinal plane $M$ (identified with $\mathbb{R}^{2}$), with orientation $\theta \in S^{1}$, spatial frequency $\omega \in \mathbb{R}^{+}$, and phase $\phi \in S^{1}$ as intrinsic parameters. For each fixed frequency $\omega$, this model defines a Lie group $G(\omega) = \mathbb{R}^{2} \times S^{1} \times S^{1}$, which we adopt in this work as the state space group $G$ of our linear control system. We also present new results concerning controllability and characterize the control sets associated with this class of systems.