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

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.