Compared with digital methods, sparse recovery based on spiking neural networks has great advantages like high computational efficiency and low power-consumption. However, current spiking algorithms cannot guarantee more accurate estimates since they are usually designed to solve the classical optimization with convex penalties, especially the $\ell_{1}$-norm. In fact, convex penalties are observed to underestimate the true solution in practice, while non-convex ones can avoid the underestimation. Inspired by this, we propose an adaptive version of spiking sparse recovery algorithm to solve the non-convex regularized optimization, and provide an analysis on its global asymptotic convergence. Through experiments, the accuracy is greatly improved under different adaptive ways.