Comptes Rendus
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.

Published online:
DOI: 10.5802/crmath.190
Classification: 62H99, 62M09, 62P10

Patricia Reynaud-Bouret 1; Alexandre Muzy 2; Ingrid Bethus 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Patricia Reynaud-Bouret; Alexandre Muzy; Ingrid Bethus. Towards a mathematical definition of functional connectivity. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 481-492. doi : 10.5802/crmath.190.

[1] Emmanuel Bacry; Jean-François Muzy Second order statistics characterization of Hawkes processes and non-parametric estimation, IEEE Trans. Inf. Theory, Volume 62 (2016) no. 4, pp. 2184-2202 | DOI | MR | Zbl

[2] T. V. Bliss; T. Lømo Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit following stimulation of the perforant path, J. Physiol., Volume 232 (1973), pp. 331-356 | DOI

[3] György Buzsáki Large-scale recording of neuronal ensembles, Nature Neurosci., Volume 7 (2004) no. 5, pp. 446-451 | DOI | Zbl

[4] Natalia Caporale; Yang Dan Spike Timing–Dependent Plasticity: A Hebbian Learning Rule, Annu. Rev. Neurosci., Volume 31 (2008), pp. 25-46 | DOI

[5] Julien Chevallier Mean-field limit of generalized Hawkes processes, Stochastic Processes Appl., Volume 127 (2017) no. 12, pp. 3870-3912 | DOI | MR | Zbl

[6] Rodrigo Cofré; Bruno Cessac Exact computation of the maximum-entropy potential of spiking neural-network models, Phys. Rev. E, Volume 89 (2014) no. 5, pp. 1262-1292 | Zbl

[7] Jozsef Csicsvari; Darrell A. Henze; Brian Jamieson; Kenneth D. Harris; Anton Sirota; Péter Barthó; Kensall D. Wise; György Buzsáki Massively parallel recording of unit and local field potentials with silicon-based electrodes, J. Neurophys., Volume 90 (2003), pp. 1314-1323 | DOI

[8] Vanessa Didelez Graphical models for marked point processes based on local independence, J. R. Stat. Soc., Ser. B, Stat. Methodol., Volume 70 (2008) no. 1, pp. 245-264 | DOI | MR | Zbl

[9] Aline Duarte; Antonio Galves; Eva Löcherbach; Guilherme Ost Estimating the interaction graph of stochastic neural dynamics, Bernoulli, Volume 25 (2019) no. 1, pp. 771-792 | MR | Zbl

[10] S. B. Eickhoff; V. I. Müller Functional connectivity, Brain Mapping, an encyclopedic reference, Volume 2, Academic Press Inc., 2015, pp. 187-201

[11] G. L. Gerstein; P. Bedenbaugh; M. H. Aertsen Neuronal Assemblies, IEEE Trans. Biomed. Eng., Volume 36 (1989), pp. 4-14 | DOI

[12] Sonja Grün; Markus Diesmann; A. Aertsen Unitary events in multiple single-neuron spiking activity: I. detection and significance, Neural Comput., Volume 14 (2002) no. 5, pp. 43-80 | DOI | Zbl

[13] Sonja Grün; Markus Diesmann; Franck Grammont; Alexa Riehle; A. Aertsen Detecting unitary events without discretization in time, J. Neurosci. Methods, Volume 94 (1999)

[14] Niels Richard Hansen; Patricia Reynaud-Bouret; Vincent Rivoirard Lasso and probabilistic inequalities for multivariate point processes, Bernoulli, Volume 21 (2015) no. 1, pp. 83-143 | MR | Zbl

[15] Kenneth D. Harris; Darrell A. Henze; Jozsef Csicsvari; Hajime Hirase; György Buzsáki Accuracy of Tetrode Spike Separation as Determined by Simultaneous Intracellular and Extracellular Measurements, J. Neurophys., Volume 84 (2000) no. 1, pp. 402-413

[16] Donald Olding Hebb The organization of behavior: a neuropsychological theory, John Wiley & Sons; Chapman & Hall, 1949

[17] Darrell A. Henze; Zsolt Borhegyi; Jozsef Csicsvari; Akira Mamiya; Kenneth D. Harris; György Buzsáki Intracellular features predicted by extracellular recordings in the hippocampus in vivo, J. Neurophys., Volume 84 (2000), pp. 390-400 | DOI

[18] M. S. Jog; C. I. Connolly; Y. Kubota; D. R. Iyengar; L. Garrido; R. Harlan; A. M. Graybiel Tetrode technology: advances in implantable hardware, neuroimag-ing, and data analysis techniques, J. Neurosci. Methods, Volume 117 (2002), pp. 141-152 | DOI

[19] James J. Jun; Nicholas A. Steinmetz; Joshua H. Siegle et al. Fully integrated silicon probes for high-density recording of neural activity, Nature, Volume 551 (2017), pp. 232-236

[20] Robert E. Kass; Ryan C. Kelly; Wei-Liem Loh Assessment of synchrony in multiple neural spike trains using loglinear point process models, Ann. Appl. Stat., Volume 5 (2011), pp. 1262-1292 | MR | Zbl

[21] Emmanuel Kowalski An introduction to expander graphs, Contributions in Mathematical and Computational Sciences, Société Mathématique de France, 2019 | MR | Zbl

[22] Régis C. Lambert; Christine Tuleau-Malot; Thomas Bessaih; Vincent Rivoirard; Yann Bouret; Nathalie Leresche; Patricia Reynaud-Bouret Reconstructing the functional connectivity of multiple spike trains using Hawkes models, J. Neurosci. Methods, Volume 297 (2018), pp. 9-21 | DOI

[23] T. Onaga; S. Shinomoto Emergence of event cascades in inhomogeneous networks, Nature Sci. Reports, Volume 6 (2016)

[24] Guilherme Ost; Patricia Reynaud-Bouret Sparse space-time models: concentration inequalities and Lasso, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 4, pp. 2377-2405 | MR | Zbl

[25] V. Pernice; B. Staude; S. Cardanobile; S. Rotter How Structure Determines Correlations in Neuronal Networks, PLoS Comput. Biol., Volume 7 (2011) | DOI | MR

[26] Jonathan Pillow; Jonathan Shlens; Liam Paninski; Alexander Sher; Alan Litke; E. Chichilnisky; Eero Simoncelli Spatio-temporal correlations and visual signalling in a complete neuronal population, Nature, Volume 454 (2008), pp. 995-999 | DOI

[27] Michael Quiquempoix Rôle de la connectivité intracorticale dans le traitement des informations sensorielles, Ph. D. Thesis, Université Paris 6 (2017) (sous la direction de R. Lambert)

[28] Alexa Riehle; Franck Grammont; Markus Diesmann; Sonja Grün Dynamical changes and temporal precision of synchronized spiking activity in monkey motor cortex during movement preparation, J. Physiol., Volume 94 (2000) no. 5-6, pp. 569-582

[29] W. Schultz; A. Dickinson Neuronal coding of prediction errors, Annu. Rev. Neurosci., Volume 23 (2000), pp. 473-500 | DOI

[30] Gordon M Shepherd The synaptic organization of the brain, Oxford University Press, 2004

[31] W. Singer Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol., Volume 55 (1993), pp. 349-374 | DOI

[32] W. Singer; A. K. Engel; A. K. Kreiter; M. H. Munk; S. Neuenschwander; P. R. Roelfsema Neuronal assemblies: necessity, signature and detectability, Trends Cogn. Sci., Volume 1 (1997), pp. 252-261 | DOI

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