Programmable synaptic devices that can achieve timing-dependent weight updates are key components to implementing energy-efficient spiking neural networks (SNNs). Electrochemical ionic synapses (EIS) enable the programming of weight updates with very low energy consumption and low variability. Here, the strongly nonlinear kinetics of EIS, arising from nonlinear dynamics of ions and charge transfer reactions in solids, are leveraged to implement various forms of spike-timing-dependent plasticity (STDP). In particular, protons are used as the working ion. Different forms of the STDP function are deterministically predicted and emulated by a linear superposition of appropriately designed pre- and post-synaptic neuron signals. Heterogeneous STDP is also demonstrated within the array to capture different learning rules in the same system. STDP timescales are controllable, ranging from milliseconds to nanoseconds. The STDP resulting from EIS has lower variability than other hardware STDP implementations, due to the deterministic and uniform insertion of charge in the tunable channel material. The results indicate that the ion and charge transfer dynamics in EIS can enable bio-plausible synapses for SNN hardware with high energy efficiency, reliability, and throughput.