The acquisition of novel motor skills requires new patterns of neural activity and changes in synaptic weights, yet the precise rules governing synaptic plasticity in the motor cortex and how they give rise to the observed changes in activity are still unknown. Brain computer interfaces (BCIs) allow for task improvement to be directly linked to changes in neural activity. Recent experiments that pair BCIs with all-optical circuit mapping allow for the simultaneous tracking of in vivo changes in connectivity while task-relevant activity learning occurs. In silico models of learning allow for the mapping of candidate learning rules to the experimentally-observed changes in connectivity and activity. Here, we introduce an RNN model that uses a novel, biologically-realistic rule for synaptic adjustment that explains many aspects of our recent paired BCI-photostimulation experiments. We truncate the exact gradient ascent of a cost function which encourages synaptic changes that lead to improved performance, producing our 3-factor learning rule that only depends upon activity spatially local to a synapse and a reward signal. By comparing to changes in both BCI-relevant neuronal activity and connectivity across all imaged neurons, we show our gradient-approximating learning rule is the most consistent with our data among many other 3-factor alternatives. Furthermore, we show only models with local plasticity within M1 reproduce the observed connectivity changes. This suggests that the remarkable flexibility of mammalian motor systems may, at least partially, be a result of synapses between M1 neurons individually approximating gradient ascent.