Equilibrium States and Their Stability in the Head-Direction Ring Network - Wang, Caixia; Zhang, Kechen (Frontiers in Computational Neuroscience, 2020)
Head-direction neural network is made of head-direction cells which are found in the limbic system in the brains of many mammals and they increase their firing rates above baseline levels only when the animal's head points in a specific direction. So head-direction neurons are mostly orientation specific and location invariant. By combining analytical techniques and numerical simulations, we have analyzed common properties and the equilibrium states of the head-direction neural network, a ring attractor network, which is the leading model of the head-direction system. By using mathematical analysis we found that under specific conditions all solutions of head-direction neural network are bounded. By Fourier analysis we get the relationship of Fourier series coefficient in head-direction neural network, and then we solve the form of equilibrium solutions. Meanwhile we proved the stability of head direction neural network strictly through building Lyapunov Function. This result shows that the average output of head direction neural network will finally converge to one of equilibrium states as t tends infinity whether which kind of self-motion information for inertially based updating or what kind of visual land marks for calibration.To further explore these equilibrium states and their stability we choose representative synaptic weight and gain function. Thus we have ever obtained all equilibrium solutions of this neural network, including both the single-peaked and double-peaked activity patterns, together with a perturbation analysis of their stability. To our surprise, we also discovered an asymmetric equilibrium activity pattern even when the network connectivity pattern is strictly symmetric. At last we show the parameter space of head-direction system.