We consider an unstructured neuron network model composed of excitatory and inhibitory neurons. The synaptic connections are supplied with spike timing-dependent plasticity (STDP). We take the STDP model implemented using a memristor. In normal conditions, the network forms so-called bursting discharges typical of unstructured living networks in dissociated neuronal cultures. Incorporating a biologically inspired model, we demonstrate how memristive plasticity emulates spike timing-dependent plasticity, which is crucial for regulating synchronous brain activity. We have found that, when the memristor-based STDP for inhibitory connections is activated, the bursting dynamics are suppressed and the network turns to a random spiking mode. The dependence of bursting properties on the degree of the memristor-based STDP plasticity is analyzed. These findings hold implications for advancing invasive neurointerfaces and for the identification and management of epileptiform activity.