A dynamical system approach to modeling neural network activity in Drosophila orientation
We introduce and analyze a class of neural network models motivated by the Drosophila central complex nervous system, designed to capture the emergence and dynamics of orientation-selective activity bumps. Starting from a biologically inspired ring-connectivity model, we derive a simplified reduced model of recurrent neural activity that supports stable, localized patterns encoding angular position during the fly's flight orientation. We first study the deterministic dynamics and identify parameter regimes ensuring existence and global stability of bump solutions. We then extend the framework to a stochastic setting, incorporating both additive Brownian noise and a Markovian switching mechanism representing time-varying external cues. The resulting system is a switching diffusion with piecewise linear drift, for which we establish well-posedness, characterize the infinitesimal generator, and prove the existence of an invariant measure. Numerical simulations in low and high dimensions illustrate the robustness of the bump attractor under noise and switching stimuli, as well as the convergence toward the predicted stationary states. These results provide a mathematically tractable framework for understanding how population activity in the insect central complex encodes heading direction in the presence of variability.