Spirals in the stochastic Fitzhugh-Nagumo equation
Channel: Nils Berglund
Date: 2022-02-08

This is a simulation of another favorite equation of mine, the FitzHugh-Nagumo equation with noise. The original version of this equation has only one spatial dimension, corresponding to the distance along the axon of a neuron, or nerve cell. It involves two variables, the electric potential across the axon's membrane, and the "recovery variable" which is related to how permeable the membrane is. The FitzHugh-Nagumo system is a simplification of the more realistic Hodgkin-Huxley system. 
Here we look at a two-dimensional version of the FitzHugh-Nagumo equation, which can be thought of as modeling the behavior of a large number of interconnected neurons. The colors correspond to the membrane potential, with yellow indicating a higher potential that describes neuron that "spike" (emit a signal). The initial state is random, and after a while, the yellow spiking patterns start behaving as spiral waves. Similar spiral patterns can be observed in some chemical reactions, such as the Belousov–Zhabotinsky reaction.
Fun fact: to be mathematically well-posed when space-time white noise is added to the equation as is the case here, the system has to undergo a so-called renormalisation procedure: https://www.researchgate.net/publication/274967288_Regularity_structures_and_renormalisation_of_FitzHugh-Nagumo_SPDEs_in_three_space_dimensions
https://www.researchgate.net/publication/336379291_Corrigendum_to_Regularity_structures_and_renormalisation_of_FitzHugh-Nagumo_SPDEs_in_three_space_dimensions

Render time: 42 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap

Music: "The Gentlemen" by DivKid@DivKid

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on  a few previous simulations of wave equations. 

The simulation solves the Fitzhugh-Nagumo equation by discretization. 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html