Abstract
This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of sparse interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear mean-field process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baladron J, Fasoli D, Faugeras O, Touboul J (2012) Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons. J Math Neurosci 2(1):10
Benachour S, Roynette B, Talay D, Vallois P (1998) Nonlinear self-stabilizing processes—I existence, invariant probability, propagation of chaos. Stoch Process Appl 75(2):173–201. https://doi.org/10.1016/S0304-4149(98)00018-0
Beyeler M, Rounds EL, Carlson KD, Dutt N, Krichmar JL (2019) Neural correlates of sparse coding and dimensionality reduction. PLoS Comput Biol 15(6):1–33. https://doi.org/10.1371/journal.pcbi.1006908
Bolley F, Guillin A, Villani C (2007) Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab Theory Relat Fields 137(3):541–593
Burkitt AN (2006) A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol Cybern 95(1):1–19
Burkitt AN (2006) A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties. Biol Cybern 95(2):97–112
Cáceres MJ, Carrillo JA, Perthame B (2011) Analysis of nonlinear noisy integrate and fire neuron models: blow-up and steady states. J Math Neurosci 1(1):7
Carrillo J, Gvalani R, Pavliotis G, Schlichting A (2020) Long-time behaviour and phase transitions for the McKean–Vlasov equation on the torus. Arch Ration Mech Anal 235(1):635–690
Chichilnisky E (2001) A simple white noise analysis of neuronal light responses. Netw Comput Neural Syst 12(2):199–213
Cortez R, Fontbona J (2016) Quantitative propagation of chaos for generalized KAC particle systems. Ann Appl Probab 26(2):892–916
Davis MHA (1984) Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J R Stat Soc B 46(3):353–388
Delarue F, Inglis J, Rubenthaler S, Tanré E (2015) Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann Appl Probab 25(4):2096–2133
Fournier N, Guillin A (2015) On the rate of convergence in Wasserstein distance of the empirical measure. Probab Theory Relat Fields 162(3–4):707–738
Galloway EM, Woo NH, Lu B (2008) Persistent neural activity in the prefrontal cortex: a mechanism by which BDNF regulates working memory? Prog Brain Res 169:251–266
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500
Kac M (1956) Foundations of kinetic theory. In: Proceedings of the third Berkeley symposium on mathematical statistics and probability, 1954–1955. University of California Press, Berkeley and Los Angeles, vol III, pp 171–197
Lacker D, Ramanan K, Wu R (2022) Local weak convergence for sparse networks of interacting processes. Ann Appl Probab https://doi.org/10.48550/ARXIV.1904.02585
Lapicque L (1907) Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une polarization. Journal de Physiologie et de Pathologie Generalej 9:620–635
Major G, Tank D (2004) Persistent neural activity: prevalence and mechanisms. Curr Opin Neurobiol 14(6):675–684. https://doi.org/10.1016/j.conb.2004.10.017
Malrieu F (2001) Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoch Process Appl 95(1):109–132. https://doi.org/10.1016/S0304-4149(01)00095-3
Mischler S, Mouhot C (2013) KAC’s program in kinetic theory. Invent Math 193(1):1–147. https://doi.org/10.1007/s00222-012-0422-3
Oliveira RI, Reis GH, Stolerman LM (2020) Interacting diffusions on sparse graphs: hydrodynamics from local weak limits. Electron J Probab 25:1–35. https://doi.org/10.1214/20-EJP505
Pillow JW, Paninski L, Uzzell VJ, Simoncelli EP, Chichilnisky E (2005) Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model. J Neurosci 25(47):11003–11013
Pillow JW, Shlens J, Paninski L, Sher A, Litke AM, Chichilnisky E, Simoncelli EP (2008) Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature 454(7207):995–999
Robert P, Touboul J (2016) On the dynamics of random neuronal networks. J Stat Phys 165(3):545–584
Rolls ET, Deco G (2010) The noisy brain: stochastic dynamics as a principle of brain function. Oxford University Press, Oxford
Sznitman A-S (1991) Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Mathematics. Springer, Berlin. vol 1464, pp 165–251. https://doi.org/10.1007/BFb0085169
Touboul J (2012) Mean-field equations for stochastic firing-rate neural fields with delays: derivation and noise-induced transitions. Physica D 241(15):1223–1244
Touboul J (2014) Spatially extended networks with singular multi-scale connectivity patterns. J Stat Phys 156(3):546–573
Touboul J (2014) Propagation of chaos in neural fields. Ann Appl Probab 24(3):1298–1328
Tugaut J (2014) Phase transitions of McKean–Vlasov processes in double-wells landscape. Stochastics 86(2):257–284. https://doi.org/10.1080/17442508.2013.775287
Zylberberg J, Strowbridge BW (2017) Mechanisms of persistent activity in cortical circuits: possible neural substrates for working memory. Ann Rev Neurosci 40(1):603–627. https://doi.org/10.1146/annurev-neuro-070815-014006
Acknowledgements
The third author thanks Martin Carbo Tano for enlightening discussions about the model. We also thank an anonymous reviewer for insightful remarks and comments that have helped us improve the presentation.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The second author was supported by Iniciación Fondecyt Grant 11181082 and by Programa Iniciativa Científica Milenio through Nucleus Millenium Stochastic Models of Complex and Disordered Systems.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Altamirano, M., Cortez, R., Jonckheere, M. et al. Persistence in a large network of sparsely interacting neurons. J. Math. Biol. 86, 16 (2023). https://doi.org/10.1007/s00285-022-01844-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00285-022-01844-x
Keywords
- Biological neural network
- Mean-field limit
- Interacting particle system
- Phase transition
- Propagation of chaos
- Nonlinear Markov process