Memories may be encoded in the brain via assemblies, groups of neurons that coactivate upon memory recall. The concept of Hebbian plasticity suggests that these assemblies are generated through synaptic plasticity, strengthening the recurrent connections within select groups of neurons that receive correlated stimulation. To remain stable in absence of such stimulation, these assemblies need to be self-reinforcing under the plasticity rule. Previous models of such assembly generation and maintenance required mechanisms of heterosynaptic competition or homeostatic plasticity often with biologically implausible timescales. Here we provide a model of neuronal assembly generation and maintenance without homeostatic plasticity. We assume that the neural networks are in a state of asynchronous irregular activity, allowing to model them as networks of linear Poisson neurons. Synaptic strengths change according to spike timing-dependent plasticity (STDP), with a simple pairwise, symmetric STDP function with negative integral and bounded weights. Our mathematical analysis and numerical simulations show that no further plasticity rules are required for stable network activity without pathological assembly evolution. Depending on the choice of parameters, the networks exhibit stationary assemblies, which consist of the same neurons over time, or drifting assemblies, which exchange neurons between each other. Further we show that neurons can be part of multiple assemblies at the same time for appropriate individual background firing rates.