This letter develops a novel fixed-time stable neurodynamic flow (FTSNF) implemented in a dynamical system for solving the nonconvex, nonsmooth model $L1-β2$, $β∈[0,1]$ to recover a sparse signal. FTSNF is composed of many neuron-like elements running in parallel. It is very efficient and has provable fixed-time convergence. First, a closed-form solution of the proximal operator to model $L1-β2$, $β∈[0,1]$ is presented based on the classic soft thresholding of the $L1$-norm. Next, the proposed FTSNF is proven to have a fixed-time convergence property without additional assumptions on the convexity and strong monotonicity of the objective functions. In addition, we show that FTSNF can be transformed into other proximal neurodynamic flows that have exponential and finite-time convergence properties. The simulation results of sparse signal recovery verify the effectiveness and superiority of the proposed FTSNF.