In this work, we present a scheme for implementing learning on a digital non-volatile memory (NVM) based hardware accelerator for Spiking Neural Networks (SNNs). Our design estimates across three prominent non-volatile memories - Phase Change Memory (PCM), Resistive RAM (RRAM), and Spin Transfer Torque RAM (STT-RAM) show that the STT-RAM arrays enable at least 2× higher throughput compared to the other two memory technologies. We discuss the design and the signal communication framework through the STT-RAM crossbar array for training and inference in SNNs. Each STT-RAM cell in the array stores a single bit value. Our neurosynaptic computational core consists of the memory crossbar array and its read/write peripheral circuitry and the digital logic for the spiking neurons, weight update computations, spike router, and decoder for incoming spike packets. Our STT-RAM based design shows ~20× higher performance per unit Watt per unit area compared to conventional SRAM based design, making it a promising learning platform for realizing systems with significant area and power limitations.