Understanding the coordinated activity underlying brain computations requires large-scale, simultaneous recordings from distributed neuronal structures at a cellular-level resolution. One major hurdle to design high-bandwidth, high-precision, large-scale neural interfaces lies in the formidable data streams that are generated by the recorder chip and need to be online transferred to a remote computer. The data rates can require hundreds to thousands of I/O pads on the recorder chip and power consumption on the order of Watts for data streaming alone. We developed a deep learning-based compression model to reduce the data rate of multichannel action potentials. The proposed model is built upon a deep compressive autoencoder (CAE) with discrete latent embeddings. The encoder is equipped with residual transformations to extract representative features from spikes, which are mapped into the latent embedding space and updated via vector quantization (VQ). The decoder network reconstructs spike waveforms from the quantized latent embeddings. Experimental results show that the proposed model consistently outperforms conventional methods by achieving much higher compression ratios (20-500x) and better or comparable reconstruction accuracies. Testing results also indicate that CAE is robust against a diverse range of imperfections, such as waveform variation and spike misalignment, and has minor influence on spike sorting accuracy. Furthermore, we have estimated the hardware cost and real-time performance of CAE and shown that it could support thousands of recording channels simultaneously without excessive power/heat dissipation. The proposed model can reduce the required data transmission bandwidth in large-scale recording experiments and maintain good signal qualities. The code of this work has been made available at https://github.com/tong-wu-umn/spike-compression-autoencoder