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Versatile and energy-efficient neural signal processors are in high demand in brain-machine interfaces and closed-loop neuromodulation applications. In this paper, we propose an energy-efficient processor for neural signal analyses. The proposed processor utilizes three key techniques to efficiently improve versatility and energy efficiency. 1) Hybrid neural network design: The processor supports artificial neural network (ANN)- and spiking neural network (SNN)-based neuromorphic processing where ANN is used to support the processing of ExG signals and SNN is used for handling neural spike signals. 2) Event-driven processing: The processor can perform always-on binary neural network (BNN)-based event detection with low-energy consumption, and it only switches to the high-accuracy convolutional neural network (CNN)-based recognition mode when events are detected. 3) Reconfigurable architecture: By exploiting the computational similarity of different neural networks, the processor supports critical BNN, CNN, and SNN operations with the same processing elements, achieving significant area reduction and energy efficiency improvement over those of a naive implementation. It achieves 90.05% accuracy and 4.38 uJ/class in a center-out reaching task with an SNN and 99.4% sensitivity, 98.6% specificity, and 1.93 uJ/class in an EEG-based seizure prediction task with dual neural network-based event-driven processing. Moreover, it achieves a classification accuracy of 99.92%, 99.38%, and 86.39% and energy consumption of 1.73, 0.99, and 1.31 uJ/class for EEG-based epileptic seizure detection, ECG-based arrhythmia detection, and EMG-based gesture recognition, respectively.