Universal differential equations as a unifying modeling language for neuroscience.

The rapid growth of large-scale neuroscience datasets has spurred diverse modeling strategies, ranging from mechanistic models grounded in biophysics, to phenomenological descriptions of neural dynamics, to data-driven deep neural networks (DNNs). Each approach offers distinct strengths as mechanistic models provide interpretability, phenomenological models capture emergent dynamics, and DNNs excel at predictive accuracy but this also comes with limitations when applied in isolation. Universal differential equations (UDEs) offer a unifying modeling framework that integrates these complementary approaches. By treating differential equations as parameterizable, differentiable objects that can be combined with modern deep learning techniques, UDEs enable hybrid models that balance interpretability with predictive power. We provide a systematic overview of the UDE framework, covering its mathematical foundations, training methodologies, and recent innovations. We argue that UDEs fill a critical gap between mechanistic, phenomenological, and data-driven models in neuroscience, with potential to advance applications in neural computation, neural control, neural decoding, and normative modeling in neuroscience.