Learning is a fundamental component of creating intelligent machines. Biological intelligence orchestrates synaptic and neuronal learning at multiple time scales to self-organize populations of neurons for solving complex tasks. Inspired by this, we design and experimentally demonstrate an adaptive hardware architecture Memristive Self-organizing Spiking Recurrent Neural Network (MEMSORN) . MEMSORN incorporates resistive memory (RRAM) in its synapses and neurons which configure their state based on Hebbian and Homeostatic plasticity respectively. For the first time, we derive these plasticity rules directly from the statistical measurements of our fabricated RRAM-based neurons and synapses. These "technologically plausible” learning rules exploit the intrinsic variability of the devices and improve the accuracy of the network on a sequence learning task by 30%. Finally, we compare the performance of MEMSORN to a fully-randomly-set-up spiking recurrent network on the same task, showing that self-organization improves the accuracy by more than 15%. This work demonstrates the importance of the device-circuit-algorithm co-design approach for implementing brain-inspired computing hardware.