A specific geometric glitch in the mathematics of nerve cells is the hidden engine behind the rhythmic patterns of the brain.
Neurons often fire in complex, alternating patterns of small and large spikes known as mixed-mode oscillations. These patterns were previously thought to be random or the result of incredibly complex feedback loops. A mathematical feature called a cusped singularity acts as a universal organizer for these rhythms in inhibitory networks. This geometry dictates exactly how and when a neuron switches between different firing modes. Understanding this mathematical core helps explain how the brain coordinates everything from walking to breathing.
Cusped singularities organize mixed-mode oscillations in mutually inhibitory slow-fast systems
arXiv · 2605.03606
Mutual inhibition is a common motif in neural systems. Here, we establish that cusped singularities - folded singularities located at cusp points of critical manifolds - provide a universal organizing mechanism for mixed-mode oscillations (MMOs) in coupled slow-fast systems with mutual inhibition. We show that the geometric setup of these systems generically satisfies the conditions required by established geometric singular perturbation theory and blow-up methods, guaranteeing that such cusped