Population coding and self-organized ring attractors in recurrent neural networks for continuous variable integration

Roman Kononov^1,2Vasily Tiselko^1,3,4Oleg Maslennikov^1,2*Vladimir Nekorkin^1,2

• ¹Institute of Applied Physics (RAS), Nizhny Novgorod, Russia
• ²Nacional'nyj issledovatel'skij Nizegorodskij gosudarstvennyj universitet imeni N I Lobacevskogo, Nizhny Novgorod, Russia
• ³Laboratory of Complex Networks, Center for Neurophysics and Neuromorphic Technologies, Moscow, Russia
• ⁴Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Dolgoprudny, Russia

Representing and integrating continuous variables, like spatial orientation, is a fundamental capability of the brain. This process often relies on ring attractors—specialized neural circuits that maintain a persistent “bump” of activity to encode a circular variable. Here, we investigate how such structures can self-organize in a recurrent neural network (RNN) trained to perform path integration on a ring. We show that by providing the network with velocity inputs encoded by a population of neurons, it autonomously develops a modular architecture. One subpopulation learns to form a stable ring attractor that accurately tracks and maintains the integrated position. A second, distinct subpopulation organizes into a dissipative structure that acts as a dynamic control unit, translating velocity inputs into directional signals for the ring attractor. Through systematic perturbations, we demonstrate that the precise topological alignment between these two modules is essential for reliable integration. Our findings illustrate how functional specialization and biologically plausible representations can emerge from a general learning objective, offering insights into the principles of self-organization in neural circuits and providing a framework for designing more interpretable and robust neuromorphic systems for navigation and control.