We study a one-dimensional system of Brownian particles that interact among themselves through local velocity-lignment forces that are discrete in time and do not affect the particle speed. The local velocity-alignment acts periodically over the system with time period t_a and probability P= t_a/t_0, with t_0 being a control parameter. Depending on the relation between t_a and t_0, the velocity-alignment can affect all of the particles synchronously in time or just a few asynchronously. In any case, as the velocity-alignment events become more frequent, the system is driven from stationary states close to thermal equilibrium, to far-from-equilibrium ones where the system exhibits spontaneous symmetry breaking and self-organization. Therefore, our results show that self-propulsion is not a necessary ingredient to produce the flocking transition even in one-dimensional systems. Above the critical point, for the synchronous case, the velocity distribution of the particles exhibits periodic oscillations in time with period t_a, and their density distribution shows a breathing-like behavior. The effects of these phenomena produce a regular spiking activity in the order parameter as the system approaches to the thermodynamic limit.