Mathematical Framework to Enable Adaptive Neuron Transitions

This paper introduces a novel mathematical framework for adaptive neural transfer functions that dynamically reconfigure input-output state spaces based on structural gradients. Unlike traditional weight-centric approaches, our Adaptive State-Space Transfer Function (ASSTF) enables neurons to evolve their topology during learning and inference. We formalize continuous gradient-driven dimensionality adaptation, establish theoretical guarantees for state-space continuity and Lyapunov stability, and present a regularized bilevel optimization scheme. This framework provides a mathematical foundation for adaptive neurons that transcend weight-centric learning.