Linear control systems on a 4D solvable Lie group used to model primary visual cortex $V1$
Adriano Da Silva, Eyüp Kizil, Victor Ayala
TL;DR
This work analyzes linear control systems on the 4D solvable Lie group $G=SE(2)\times S^1$, motivated by a cortical model of V1 where fixed-frequency subspaces yield the state space. By deriving derivations and automorphisms of the Lie algebra and decomposing vector fields into $SE(2)$ and $S^1$ components, the authors connect control properties on $G$ to those on $SE(2)$ via projection and conjugation. The key finding is a dichotomy for control sets: if $\det A=0$, there are infinitely many control sets with empty interiors; if $\det A\neq 0$, there is a unique control set with nonempty interior given by $\mathcal{C}_{SE(2)}\times S^1$, and controllability occurs exactly when the Lie algebra rank condition holds with $\det A\neq 0$ and $\mathrm{tr}(A)=0$. These results provide a rigorous link between geometric models of V1 and dynamical control properties, with implications for understanding how neural activity could propagate across cortical space under structured stimuli.
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 $θ\in S^{1}$, spatial frequency $ω\in \mathbb{R}^{+}$, and phase $φ\in S^{1}$ as intrinsic parameters. For each fixed frequency $ω$, this model defines a Lie group $G(ω) = \mathbb{R}^{2} \times S^{1} \times S^{1}$, 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.