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Towards a mathematical definition of functional connectivity
Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 481-492.

Functional connectivity is a neurobiological notion, informally stating that there would be a strong dependence between neurons and that this dependence might be useful in understanding the way the brain encodes stimuli, programs actions, etc. However, in practice such strong dependencies are often reconstructed via Hawkes processes based on an amazingly small number of neurons, because of the very scarce observation of this very complex and huge network. We derive new simple equations, which explain how the ideal Hawkes reconstruction is linked to the covariance between the observed neurons. These equations help us in particular to understand what the Hawkes reconstruction does in two settings, synchronization and classical point process asymptotics. Moreover they might help us to also understand what is qualitatively happening at the scale of the huge unobserved network, paving the path for a possible mathematical definition of functional connectivity.

La connectivité fonctionnelle est une notion neurobiologique, qui affirme informellement qu’il y aurait une forte dépendance entre neurones et que cette dépendance pourrait être utilisée pour comprendre comment le cerveau encode les stimuli, programme les actions, etc. Cependant, en pratique, ces fortes dépendances sont souvent reconstruites, grâce aux processus de Hawkes, sur un nombre incroyablement faible de neurones, parce que l’observation du réseau sous-jacent est excessivement partielle. Nous prouvons de nouvelles équations qui expliquent comment la reconstruction idéale par processus de Hawkes est liée à la covariance entre neurones observés. Ces équations nous aident à comprendre ce que fait exactement la reconstruction par processus de Hawkes dans deux cadres asymptotiques, la synchronization et le cadre classique des processus ponctuels. De plus, elles pourraient nous permettre de comprendre qualitativement ce qui se passe dans l’immense réseau non observé, ouvrant la voie à une possible définition mathématique de la connectivité fonctionnelle.

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DOI: 10.5802/crmath.190
Classification: 62H99, 62M09, 62P10
Reynaud-Bouret, Patricia 1; Muzy, Alexandre 2; Bethus, Ingrid 3

1 Université Côte d’Azur, CNRS, LJAD, France
2 Université Côte d’Azur, CNRS, I3S, France
3 Université Côte d’Azur, CNRS, IPMC, France
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Reynaud-Bouret, Patricia; Muzy, Alexandre; Bethus, Ingrid. Towards a mathematical definition of functional connectivity. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 481-492. doi : 10.5802/crmath.190. http://www.numdam.org/articles/10.5802/crmath.190/

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