The goal of neuromorphic computing is to recreate the computational power and efficiency of the human brain with circuitry. The ability of the brain to solve complex real time tasks, while consuming 20 W of power on average, is made possible through its connection density, adaptability, and parallel processing. Recreating these features using traditional electronics circuit elements is incredibly difficult, and therefore, soft-matter memristors made of biomolecules similar to those found in biological synapses and capable of emulating various synaptic features can be used as neuromorphic hardware. In this work, we introduce and experimentally demonstrate an electronic neuron circuit capable of interacting with ionic, soft-matter memristors. These memristors are proven to exhibit short-term plasticity, especially paired-pulse facilitation and depression found in presynaptic terminals - features that are not found in state-of-the-art solid-state memristors. We make use of these features for applications in online learning by developing a synapse-neuron circuit which implements spike-rate-dependent plasticity (SRDP) as a learning function.