Robustness of a dynamical systems model with a plastic self-organising vector field to noisy input signals

We investigate the robustness with respect to random stimuli of a dynamical system with a plastic self-organising vector field, previously proposed as a conceptual model of a cognitive system and inspired by the self-organised plasticity of the brain. This model of a novel type consists of an ordinary differential equation subjected to the time-dependent “sensory” input, whose time-evolving solution is the vector field of another ordinary differential equation governing the observed behaviour of the system, which in the brain would be neural firings. It is shown that the individual solutions of both these differential equations depend continuously over finite time intervals on the input signals. In addition, under suitable uniformity assumptions, it is shown that the non-autonomous pullback attractor and forward omega limit set of the given two-tier system depend upper semi-continuously on the input signal. The analysis holds for both deterministic and noisy input signals, in the latter case in a pathwise sense.