The chapter explores alpha unpredictable and Poisson stable motions within two distinct models of inertial neural networks with discontinuities. Firstly, the investigation focuses on a specific neural network architecture, where the impulse structure replicates that of rates. This choice mirrors real-world system behavior, where voltage typically exhibits smooth continuity but occasionally undergoes sudden changes due to factors like switches, sudden loads, or faults. Representation of these abrupt voltage transitions as discontinuous derivatives provides a more accurate depiction of real-world scenarios. Another model of inertial neural networks is with alpha unpredictable inputs, and a piecewise constant argument. This model enables the exploration of chaotic signal distributions within neural networks. The existence and exponential stability of unique alpha unpredictable and Poisson stable oscillations are demonstrated for both models. The method of included intervals is extended, to analyze the Poisson stability of motions with discontinuities. In the model featuring a piecewise constant argument, solutions appear as continuous functions with discontinuous derivatives. The theoretical findings are bolstered with numerical examples, illustrating the practical feasibility and applicability of the proposed models. They provide concrete demonstrations of alpha unpredictable and Poisson stable behaviors within inertial neural networks, robustness and utility of the theoretical framework.