To understand network computation, we usually focus on the connectivity among neurons. In this talk, however, I demonstrate how the biophysics of action-potential generation can have a decisive impact on network behaviour. Our recent theoretical work shows that, among regularly firing neurons, the less attended homoclinic type (characterized by a spike onset via a saddle homoclinic orbit bifurcation) particularly stands out: First, spikes of this type foster specific network states - synchronisation in inhibitory and splayed-out/frustrated states in excitatory networks. Second, homoclinic spikes can be easily induced by changes in a variety of physiological parameters (like temperature, extracellular potassium, or dendritic morphology). As a consequence, such parameter changes can mediate switches in network states, exclusively based on a modification of cell-intrinsic voltage dynamics. I will discuss functional consequences of homoclinic spikes for the design of efficient pattern-generating motor circuits in Drosophila as well as for mammalian pathologies like febrile seizures. Our work predicts an interesting role for homoclinic action potentials as an integral part of brain dynamics in both health and disease.