• connectivity matrix
• equation
• mode
• modes
• neural field

Neural field equations, studied since the 1950s, describe the activity of a neural network in a continuous spatio-temporal domain. Among the first works that utilized this approach, the paper of Beurle deserves to be mentioned. During the last decades, a lot of effort has been directed to obtain numerical solutions of these equations, and this has allowed to test some of the theoretical assumptions underlying the neural field equations, by comparing the values of these solutions with human brain EEG and MEG recordings.
Neural field equations describe average neural network activity and one very important feature of a neural network is its connectivity, that is how and where neuronal units are connected to each other. It must be noted that each neuronal unit represents a population of homogeneous neurons, according to an approach called neural mass model. Whence it is theoretically easy enough to introduce this quantity, from the experimental side, the procedure for obtaining the brain connectivity is still an active research area of the neuroscience field.
The work of this thesis starts by the delayed integro-differential equation; this equation assumes that the connectivity can be decomposed in the sum of two terms: the first one, called the homogeneous connectivity, contains the spatially invariant connections that surround any neural unit, while the second one, called the heterogeneous connectivity, describes the long-distance connections. Both terms contain temporal delay due to a finite velocity of propagation of the neural signal.
In this thesis, wehave extended the spatial domain to a sphere since it is possible to map the cortical surface using already-implemented algorithms that inflates it to a sphere. Thus, any result obtained in the thesis can be compared, in the future, to experimentally collected data (prevalently, EEG or MEG signals).
In order to optimize the simulation of the neural field equation on the spherical domain, we introduced a truncated mode decomposition of the solution, using spherical harmonics as a complete set of functions.This has proven to be very efficient: actually, as long as this procedure does not affect too much the dynamic of the solution, the simulation is performed with an arbitrary precision in the spatial domain (that is totally described by the spherical harmonics), so that the gain in computational effort is remarkable. We have shown that the first twenty spherical harmonics are sufficient to reproduce the dynamic of the neural field predicted with the complete equation shown above, assuming that a valid spatial resolution of the cortical surface is of the order of one centimeter, that is near to the experimental spatial resolution.
The neural field approach reminds the fluid dynamic division in infinite very small volumes of the system: the methods used in order to analyze and simulate the delayed PDE are very similar to the ones used in fluid dynamic problems.
An extensive use of Matlab as Python has been necessary, and the computational part of the thesis has been developed mostly in Marseille (FR), where the theoretical neuroscience group of Viktor Jirsa works.