Abstract
We are interested in understanding the functioning of the fly’s nervous system, more specifically the central complex of Drosophila, in relation to its orientation behavior. To model this dynamics, we study a neural network described by a system of ordinary differential equations (ODEs). We analytically analyze explicit solutions in different regions of the system, studying their stability, and complete this analysis with numerical simulations. We introduce a stochastic term in order to model the random changes of direction observed in the fly. The model then becomes a system of stochastic differential equations (SDEs), where the input vector varies randomly over time and Brownian motion modulates fluctuations in neural activity. This approach makes it better to capture the spontaneous dynamics of the neural system under uncertainty. Finally, we derive the associated Fokker–Planck equation, a one-dimensional advection–diffusion type partial differential equation, which governs the temporal evolution of the probability density of neural activity. This transformation enables a more detailed study, both theoretical and numerical, of the bump dynamics of activity, used to model the fly’s perceived direction.