Youhua Qian

Continuous representations in brain navigation system are manifested as spatially structured patterns of population activity, such as a single-peaked bump moving along a ring manifold in head-direction system and hexagonal lattice patterns underlying spatial representation in grid-cell systems. These phenomena are commonly modelled within the framework of continuous attractor networks (neural dynamical field), yet the mechanisms by which activation-function nonlinearities interact with connectivity structure to determine pattern selection and dynamics remain incompletely understood. This paper separately analyses the interactions between non-resonant and resonant modes using a multiscale unfolding approach. We show that, when the critical modes satisfy a resonance condition, the quadratic nonlinearity of the activation function induces a three-mode coupling that fundamentally alters the structure of the amplitude equations and becomes the dominant mechanism governing spatial pattern selection. Building on this analysis, we introduce a weak asymmetric component in the connectivity and analytically derive the resulting pattern drift velocity, which is subsequently confirmed by numerical simulations. Finally, we apply these dynamical mechanisms to input-driven scenarios, illustrating that similar dynamical mechanisms can account for activity-bump tracking in head-direction models and lattice translations in grid-cell models. Overall, this work provides an analytically tractable framework for studying pattern dynamics in neural field models relevant to spatial representations, and may inform biomimetic approaches to spatial representation and navigation.

Studying a semi-analytical solution for time-delay coupled nonlinear systems contributes to the advancement of the nonlinear dynamics theory. In this paper, based on the Time-domain Minimum Residual Method (TMRM), we introduced a time-delay term and obtained the semi-analytical solutions for two types of time-delay nonlinear systems. First, the periodic solutions of the time-delay coupled van der Pol–Duffing system were solved in three different cases based on different values of the time delay. The results were compared with numerical solutions and the Multifrequency Homotopy Analysis Method (MFHAM), showing good consistency among the three methods. Second, the quasi-periodic solutions of the time-delay coupled van der Pol–Duffing system were similarly solved in three different cases and compared with the numerical solutions, showing excellent agreement as well. Finally, the slow–fast analysis method was used to analyze the equilibrium points and bifurcation points of the time-delayed nonlinear oscillations of magnetic levitation, and the periodic bursting solutions were solved analytically, which demonstrates strong consistency with the numerical results. This shows that time delay has a significant impact on the behavior and performance of the system. Introducing a time-delay term based on the TMRM is highly valuable for obtaining the analytical solutions of time-delay nonlinear systems.