We formulate the Fast-Weights Homeostatic Reentry Network (FHRN) as a continuous-time neural-ODE system, revealing its role as a norm-regulated reentrant dynamical process. Starting from the discrete reentry rule $x_t = x_t^{(\mathrm{ex})} + γ\, W_r\, g(\|y_{t-1}\|)\, y_{t-1}$, we derive the coupled system $\dot{y}=-y+f(W_ry;\,x,\,A)+g_{\mathrm{h}}(y)$ showing that the network couples fast associative memory with global radial homeostasis. The dynamics admit bounded attractors governed by an energy functional, yielding a ring-like manifold. A Jacobian spectral analysis identifies a \emph{reflective regime} in which reentry induces stable oscillatory trajectories rather than divergence or collapse. Unlike continuous-time recurrent neural networks or liquid neural networks, FHRN achieves stability through population-level gain modulation rather than fixed recurrence or neuron-local time adaptation. These results establish the reentry network as a distinct class of self-referential neural dynamics supporting recursive yet bounded computation.

翻译：我们将快速权重稳态再入网络（FHRN）表述为一个连续时间神经-常微分方程系统，揭示了其作为范数调节的再入动力学过程的作用。从离散再入规则 $x_t = x_t^{(\mathrm{ex})} + γ\, W_r\, g(\|y_{t-1}\|)\, y_{t-1}$ 出发，我们推导出耦合系统 $\dot{y}=-y+f(W_ry;\,x,\,A)+g_{\mathrm{h}}(y)$，表明该网络将快速联想记忆与全局径向稳态耦合。该动力学允许由能量泛函控制的有界吸引子，产生一个环状流形。雅可比谱分析识别出一种反射机制，其中再入诱导稳定的振荡轨迹而非发散或崩溃。与连续时间递归神经网络或液态神经网络不同，FHRN通过群体水平的增益调制而非固定递归或神经元局部时间适应来实现稳定性。这些结果确立了再入网络作为一类独特的自指神经动力学，支持递归但有界的计算。