Slow brain rhythms, observed during slow-wave sleep or seizures, often coincide with oscillations in extracellular potassium levels. These rhythms have a lower frequency compared to a typical neuron’s tonic firing. Mechanisms organizing these rhythms are believed to rely on synaptic interactions within the network or intrinsic cellular voltage dynamics, likely a combination of both. This study focuses on single neurons that can produce slow rhythms, which in turn can organize slow rhythms in neural networks. Common theories attribute intrinsic slow bursting of neurons to ion channels with slow kinetics or slow-wave bursting relying on multiple slow variables. Here, we propose an alternative mechanism that can manifest in all neuron models with class I excitability and which requires only one slow variable: extracellular potassium concentration. Our mechanism is based on the interplay of fast-spiking voltage dynamics with the slow dynamics of the extracellular potassium concentration, mediated by the activity of the $Na^+/K^+$-ATPase. The slow rhythmic activity manifests as square-wave bursting, a hysteresis loop organized around a bistable region emerging from a saddle-node loop bifurcation, characteristic of class I excitable neurons. Through slow-fast analysis of our system, we show that the hysteresis loop formation requires a shear in the bifurcation diagram induced by the electrogenic pump. Importantly, neglecting the pump's electrogenicity precludes the formation of such rhythmic activity. Additionally, we conduct a comprehensive bifurcation analysis of the complete system. We identify that the transition between tonic spiking and bursting in the complete system occurs through a cascade of period doubling, leading to a chaotic behaviour. We demonstrate that, depending on $Na^+/K^+$-ATPase density, the simple concentration-dependent system exhibits four different regimes: rest, tonic spiking, bursting, and depolarisation block, which have been associated with a range of healthy and/or pathological states. Our study clarifies the sodium-potassium pump’s generic contribution in setting diverse neuronal dynamics in otherwise very simple neuron models. Due to its generic nature in class I neurons, the described mechanism is likely to occur in the brain, potentially contributing to pathologies, including epileptiform activity.