A central challenge in today's care of epilepsy patients is that the disease dynamics are severely under-sampled in the currently typical setting with appointment-based clinical and electroencephalographic examinations. Implantable devices to monitor electrical brain signals and to detect epileptic seizures may significantly improve this situation and may inform personalized treatment on an unprecedented scale. These implantable devices should be optimized for energy efficiency and compact design. Energy efficiency will ease their maintenance by reducing the time of recharging, or by increasing the lifetime of their batteries. Biological nervous systems use an extremely small amount of energy for information processing. In recent years, a number of methods, often collectively referred to as brain-inspired computing, have also been developed to improve computation in non-biological hardware. Here, we give an overview of one of these methods, which has in particular been inspired by the very size of brains' circuits and termed hyperdimensional computing. Using a tutorial style, we set out to explain the key concepts of hyperdimensional computing including very high-dimensional binary vectors, the operations used to combine and manipulate these vectors, and the crucial characteristics of the mathematical space they inhabit. We then demonstrate step-by-step how hyperdimensional computing can be used to detect epileptic seizures from intracranial electroencephalogram (EEG) recordings with high energy efficiency, high specificity, and high sensitivity. We conclude by describing potential future clinical applications of hyperdimensional computing for the analysis of EEG and non-EEG digital biomarkers.