Membranotronic devices are artificial neural membranes mimicing the functionality of biological neural networks. These devices rely on the emergence of negative differential resistance (NDR). A minimalistic physical model for membranotronic devices capable of generating NDR is presented. The model features a deformable membrane with holes that facilitate ion currents. The deformation of the membrane, induced by electrostatic pressure from an applied voltage, modulates these currents. The model comprises a well-established mechanical framework for describing deformable membranes with holes, alongside a model for ionic current that considers temperature-dependent ion mobilities. It is demonstrated that the model can faithfully reproduce NDR across a wide and physically realistic range of parameter combinations. Furthermore, the simulations reveal that the temperature of the electrolyte can exceed its boiling point, resulting in bubble formation. To mitigate this issue, materials with high heat transfer coefficients and low conductivity are recommended. In essence, the work bridges the gap between artificial membranotronic devices and biological neural networks by providing a robust physical model capable of emulating NDR, a key feature in the operation of such systems. This advancement in membranotronics holds great promise for the development of bioinspired soft artificial neuromimetic systems that closely mimic their biological counterparts.