Title: How neurons add spikes: From simple rhythms to complex bursting patterns

Abstract: Many neurons and other excitable cells display bursting activity—periods of rapid firing separated by quiet phases. Understanding how these rhythmic patterns arise and change helps us connect cellular mechanisms to networks of neurons and the complex behaviors they generate. In this talk, I will explore how neurons transition from generating single spikes to producing bursts with multiple spikes. Using a simplified mathematical model, I will show how slow rhythmic inputs can shape neuronal activity, leading to gradual “spike-adding” transitions as the input frequency and amplitude change. I will describe how geometric ideas from dynamical systems help explain where new spikes come from and why they disappear. To connect theory with biology, I will also show that similar spike-adding patterns appear in more detailed neuronal models. Overall, the talk will illustrate how mathematics can reveal the hidden structures that organize complex neuronal rhythms.