Multiplexing is an important strategy in multichannel acquisition systems. The per-channel antialiasing filters needed in the traditional multiplexing architecture limit its scalability for applications requiring high channel density, high channel count, and low noise. A particularly challenging example is multielectrode arrays for recording from neural systems. We show that conventional approaches must tradeoff recording density and noise performance, at a scale far from the ideal goal of one-to-one mapping between neurons and sensors. We present a multiplexing architecture without per-channel antialiasing filters. The sparsely sampled data are recovered through a compressed sensing strategy, involving statistical reconstruction and removal of the undersampled thermal noise. In doing so, we replace large analog components with digital signal processing blocks, which are much more amenable to scaled CMOS implementation. The resulting statistically reconstructed multiplexing architecture recovers input signals at significantly improved signal-to-noise ratios when compared to conventional multiplexing with antialiasing filters at the same per-channel area. We implement the new architecture in a 65 536-channel neural recording system and show that it is able to recover signals with performance comparable to conventional high-performance, single-channel systems, despite a more than four-orders-of-magnitude increase in channel density.