The field of bio-inspired computing has established a new Frontier for conceptualizing information processing, aggregating knowledge from disciplines as different as neuroscience, physics, computer science and dynamical systems theory. The study of the animal brain has shown that no single neuron or neural circuit motif is responsible for intelligence or other higher-order capabilities. Instead, complex functions are created through a broad variety of circuits, each exhibiting an equally varied repertoire of emergent dynamics. How collective dynamics may contribute to computations still is not fully understood to date, even on the most elementary level. Here we provide a concise introduction to bio-inspired computing via nonlinear dynamical systems. We first provide a coarse overview of how the study of biological systems has catalyzed the development of artificial systems in several broad directions. Second, we discuss how understanding the collective dynamics of spiking neural circuits and model classes thereof, may contribute to and inspire new forms of 'bio-inspired' computational paradigms. Finally, as a specific set of examples, we analyze in more detail bio-inspired approaches to computing discrete decisions based on multi-dimensional analogue input signals, via k-winners-take-all functions. This article may thus serve as a brief introduction to the qualitative variety and richness of dynamical bio-inspired computing models, starting broadly and focusing on a general example of computation from current research.We believe that understanding basic aspects of the variety of bio-inspired approaches to computation on the coarse level of first principles (instead of details about specific simulation models) and how they relate to each other, may provide an important step toward catalyzing novel approaches to autonomous and computing machines in general.