Recurrent neural networks underlie the astounding information processing capabilities of the brain, and play a key role in many state-of-the-art algorithms in deep reinforcement learning. But it has remained an open question how such networks could learn from rewards in a biologically plausible manner, with synaptic plasticity that is both local and online. We describe such an algorithm that approximates actor-critic policy gradient in recurrent neural networks. Building on an approximation of backpropagation through time (BPTT): e-prop, and using the equivalence between forward and backward view in reinforcement learning (RL), we formulate a novel learning rule for RL that is both online and local, called reward-based e-prop. This learning rule uses neuroscience inspired slow processes and top-down signals, while still being rigorously derived as an approximation to actor-critic policy gradient. To empirically evaluate this algorithm, we consider a delayed reaching task, where an arm is controlled using a recurrent network of spiking neurons. In this task, we show that reward-based e-prop performs as well as an agent trained with actor-critic policy gradient with biologically implausible BPTT.