Abstract: Visual representations become progressively more abstract along the cortical hierarchy. These abstract representations define notions like objects and shapes, but at the cost of spatial specificity. By contrast, low-level regions represent spatially local but simple input features. How do spatially non-specific representations of abstract concepts in high-level areas flexibly modulate the low-level sensory representations in appropriate ways to guide context-driven and goal-directed behaviors across a range of tasks? We build a biologically motivated and trainable neural network model of dynamics in the visual pathway, incorporating local, lateral, and feedforward synaptic connections, excitatory and inhibitory neurons, and long-range top-down inputs conceptualized as low-rank modulations of the input-driven sensory responses by high-level areas. We study this ${\bf D}$ynamical ${\bf C}$ortical ${\bf net}$work ($DCnet$) in a visual cue-delay-search task and show that the model uses its own cue representations to adaptively modulate its perceptual responses to solve the task, outperforming state-of-the art DNN vision and LLM models. The model's population states over time shed light on the nature of contextual modulatory dynamics, generating predictions for experiments. We fine-tune the same model on classic psychophysics attention tasks, and find that the model closely replicates known reaction time results. This work represents a promising new foundation for understanding and making predictions about perturbations to visual processing in the brain.