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 Baspinar et al. (J Math Neurosci 10:11, 2020), 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 R2), with orientation θ∈S1, spatial frequency ω∈R+, and phase ϕ∈S1 as intrinsic parameters. For each fixed frequency ω, this model defines a Lie group G(ω)=R2×S1×S1, 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.