# Excitability, chaos and mixed-mode oscillations in neuron models

Excitability is a phenomenon that involves a large reaction of a system to a small but sufficiently large stimulus. A typical manifestation is one or several pulse-like responses in the state variables followed by a decay to a resting state. How many pulses are generated (if any) from a single stimulus depends on the initial condition being beyond certain excitability threshold. It has also been found that excitable behaviour is governed by a mathematical mechanism which is typical of homoclinic bifurcations. While excitability was first studied in biological systems, it has also been found in laser systems, neuron dynamics and certain chemical reactions, often in presence of different time scales (i.e., specified by a slow-fast dynamical system).

On the other hand, dynamical systems with different time scales may exhibit mixed oscillations which are solutions that alternate between large and small amplitude oscillations. Mixed mode oscillations are useful to explain the response from a neuron system to an external stimulus, and can be periodic, aperiodic, and even chaotic. Today, advanced computational techniques allow us to find and visualize the way these models behave. Together with analytical tools from bifurcation theory and geometrical singular perturbation theory we can also discover what are the geometrical mechanisms that produce mixed oscillations, excitable dynamics, and how they can become chaotic.

**Researcher AM2V:** Pablo Aguirre.

**Chilean collaborator:** Patricio Orio (Centro Interdisciplinario de Neurociencia de Valparaíso).