In this talk I will first present a simple design for an implantable neuro-glial-vascular unit optimized to perform some fundamental processes that could facilitate monitoring and supporting the proper functionality of the brain. The name I give to this engineered unit is brain-on-a-chip. Then I will focus on the mathematical modeling of the coupled mechano-electrochemistry of a neuron and its membrane. I will present constrained Lagrangian formulations that link the Hodgkin-Huxley model of the electronic membrane and the mechanics of the inside of the neuron. Multiple time scales will be modeled using fractional order derivatives with variable order. The effects of traumatic injuries on neurons will be investigated through numerical simulations.