Neuromorphic computing — brainlike computing in hardware — typically requires myriad CMOS spiking neurons interconnected by a dense mesh of nanoscale plastic synapses. Memristors are frequently cited as strong synapse candidates due to their statefulness and potential for low-power implementations. To date, plentiful research has focused on the bipolar memristor synapse, which is capable of incremental weight alterations and can provide adaptive self-organisation under a Hebbian learning scheme. In this paper we consider the Unipolar memristor synapse — a device capable of switching between only two states (conductive and resistive) through application of a suitable input voltage — and discuss its suitability for neuromorphic systems. A self-adaptive evolutionary process is used to autonomously find highly fit network configurations. Experimentation on a dynamic-reward scenario shows that unipolar memristor networks evolve task-solving controllers faster than both generic bipolar memristor networks and networks containing nonplastic connections whilst performing comparably.