Emergent properties in a V1-inspired network of Hodgkin–Huxley neurons

Mohamed Maama¹ ; Benjamin Ambrosio^2, 3 ; M.A. Aziz-Alaoui² ; Stanislav M. Mintchev⁴

¹ KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia
² Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France
³ The Hudson School of Mathematics, 244 Fifth Avenue, Suite Q224, New York, NY 10001, USA
⁴ Department of Mathematics, The Cooper Union, New York, NY 10003, USA

Mathematical modelling of natural phenomena, Tome 19 (2024), article no. 3

Résumé

This article is devoted to the theoretical and numerical analysis of a network of excitatory and inhibitory neurons of Hodgkin–Huxley (HH) type, for which the topology is inspired by that of a single local layer of visual cortex V1. Our model relies on recent work in this area and thus combines a stochastic drive – which may be interpreted as an ambient drive for each neuron – with recurrent inputs resulting from the network activity. After a review of the dynamics of a single HH equation for both the deterministic and the stochastically driven case, we proceed to an analysis of the network. This analysis reveals emergent properties of the system such as partial synchronization and synchronization (defined here as a state of the network for which all the neurons spike within a short interval of time), correlation between excitatory and inhibitory conductances, and oscillations in the Gamma-band frequency. The collective behavior enumerated herein is observed when the input-amplitude parameter SEE measuring excitatory-to-excitatory coupling (recurrent excitation) increases to within a certain range. Of note, our work indicates a distinct mechanism for obtaining the emergent properties, some of which have been classically observed. As a consequence our article contributes to the understanding of how assemblies of inhibitory and excitatory cells interact together to produce rhythms in the network. It also aims to bring problems from neuroscience to the realm of mathematics, where they can be analyzed rigorously.

Détail

DOI : 10.1051/mmnp/2024001

Bibliographie

[1] B. Ambrosio, M.A. Aziz-Alaoui Synchronization and control of coupled reaction-diffusion systems of the FitzHugh–Nagumo type Comput. Math. Appl. 2012 934 943

[2] B. Ambrosio, M.A. Aziz-Alaoui Synchronization and Control of a network of coupled reaction-diffusion systems of generalized FitzHugh–Nagumo type ESAIM Proc. 2013 15 24

[3] B. Ambrosio, M.A. Aziz-Alaoui Basin of attraction of solutions with pattern formation in slow–fast reaction–diffusion systems Acta Biotheoretica 2013 311 325

[4] B. Ambrosio, M.A. Aziz-Alaoui, A. Balti Propagation of bursting oscillations in coupled non-homogeneous Hodgkin–Huxley reaction-diffusion systems Differ. Equ. Dyn. Syst. 2013 15 24

[5] B. Ambrosio, M.A. Aziz-Alaoui, V.L.E. Phan Large time behaviour and synchronization and control of a network of coupled reaction-diffusion systems of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type IMA J. Appl. Math. 2019 416 443

[6] B. Ambrosio, J.-P. Francoise Propagation of bursting oscillations Philos. Trans. Roy. Soc. A: Math. Phys. Eng. Sci. 2009 4863 4875

[7] B. Ambrosio and L-S. Young, The use of reduced models to generate irregular, broad-band signals that resemble brain rhythms Front. Computat. Neurosci. 16 (2022).

[8] M.A. Aziz-Alaoui, Synchronization of chaos, in Encyclopedia of Mathematical Physics. (2006) 213–226.

[9] B.V. Atallah, M. Scanziani Instantaneous modulation of gamma oscillation frequency by balancing excitation with inhibition Neuron 2009 566 577

[10] A. Balti, V. Lanza, M.A. Aziz-Alaoui A multi-base harmonic balance method applied to Hodgkin–Huxley model Math. Biosci. Eng. 2018 807 825

[11] A.-L. Barabasi, Network Science. Cambridge University Press (2016).

[12] A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks. Cambridge University Press (2008).

[13] V.N. Belykh, I.V. Belykh, M. Hasler, K.V. Nevidin Cluster synchronization in three-dimensional lattices of diffusively coupled oscillators Int. J. Bifurc. Chaos 2003 755 779

[14] V.N. Belykh, I.V. Belykh, M. Hasler, K.V. Nevidin Background gamma rhythmicity and attention in cortical local circuits: a computational study PNAS 2005 7002 7007

[15] C. Borgers, An Introduction to Modeling Neuronal Dynamics. Springer (2017).

[16] N. Brunel Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons J. Computat. Neurosci. 2000 183 208

[17] L. Chariker, L.-S. Young Emergent spike patterns in neuronal populations J. Computat. Neurosci. 2014 203 220

[18] L. Chariker, R. Shapley, L.-S. Young Orientation selectivity from very sparse LGN inputs in a comprehensive model of macaque V1 cortex J. Neurosci. 2018 12368 12384

[19] L. Chariker, R. Shapley, L.-S. Young Rhythm and synchrony in a cortical network model J. Neurosci. 2018 8621 8634

[20] M. Chavez, M. Valencia, V. Latora, J. Martinerie Complex networks: new trends for the analysis of brain connectivity Int. J. Bifurc. Chaos 2010 1 10

[21] J. Cronin, Mathematical Aspects of Hodgkin–Huxley Neural Theory. Cambridge University Press (1987).

[22] P. Dayan and L.F. Abott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. The MIT Press (2001).

[23] M. Chavez, M. Valencia, V. Latora, J. Martinerie Canards, clusters and synchronization in a weakly coupled interneuron model SIADS 2019 253 278

[24] G.B. Ermentrout, N. Kopell Fine structure of neural spiking and synchronization in the presence of conduction delays PNAS 1998 1259 1264

[25] G.B. Ermentrout and D.H. Terman, Mathematical Foundations of Neuroscience. Springer, New York (2010).

[26] E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press (2006).

[27] R. Fitzhugh Impulses and physiological states in theoretical models of nerve membrane Biophys. J. 1961 445 466

[28] W. Gerstner, W.M. Kistler, R. Naud, M. Richard and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press (2014).

[29] J.L. Hindmarsh, R.M. Rose A model of neuronal bursting using three coupled first order differential equations Proc. Roy. Soc. Lond. Ser. B 1984 87 102

[30] A.L. Hodgkin, A.F. Huxley A quantitative description of membrane current and its application to conduction and excitation in nerve J. Physiol. 1952 500 544

[31] E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience). The MIT Press (2006).

[32] E.R. Kandel, J.H. Schwartz, T.M. Jessell, S.A. Siegelbaum, A.J. Hudspeth and S. Mack, Principles of Neural Science. McGraw-Hill (2013).

[33] I.C. Lin, D. Xing, R. Shapley Integrate-and-fire vs Poisson models of LGN input to V1 cortex: noisier inputs reduce orientation selectivity J. Comput. Neurosci. 2012 559 572

[34] B. Hassard Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon J. Theor. Biol. 1978 401 420

[35] J. Guckenheimer, J.S. Labouriau Bifurcation of the Hodgkin and Huxley equations: a new twist Bull. Math. Biol. 1993 937 952

[36] J. Guckenheimer, R.A. Oliva Chaos in the Hodgkin–Huxley Model SIAM J. Appl. Dyn. Syst. 2002 105 114

[37] C. Morris, H. Lecar Voltage oscillations in the barnacle giant muscle fiber Biophys. J. 1981 193 213

[38] J. Nagumo, S. Arimoto, S. Yoshizawa An active pulse transmission line simulating nerve axon Biophys. J. 1962 2061 2070

[39] M.E.J. Newman, Networks: An Introduction, Oxford University Press (2010).

[40] K.P. O’Keeffe, H. Hong and S.H. Strogatz, Oscillators that sync and swarm., Nat. Commun. 8 (2017).

[41] M. Okun, I. Lampl Instantaneous correlation of excitation and inhibition during ongoing and sensory-evoked activities Nat. Neurosci. 2008 535 537

[42] J. Pantaleone Oscillators that sync and swarm Am. J. Phys. 2002 992 1000

[43] L.M. Pecora, F. Sorrentino, A.M. Hagerstrom, T.E. Murphy, R. Roy Cluster synchronization and isolated desynchronization in complex networks with symmetries Nat. Commun. 2014 4079

[44] A.V. Rangan, L.-S. Young Emergent dynamics in a model of visual cortex J. Computat. Neurosci. 2013 155 167

[45] J. Rinzel, R. N. Miller Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations Math. Biosci. 1980 27 59

[46] Y. Shu, A. Hasenstaub, D. Mccormick Turning on and off recurrent balanced cortical activity Nature 2003 288 293

[47] S. Stogatz and I. Stewart, Coupled Oscillators and biological synchronization. Am. Sci. (1993) 68–75.

[48] Y. Sun, D. Zhou, A.V. Rangan, D. Cai Library-based numerical reduction of the Hodgkin–Huxley neuron for network simulation J. Computat. Neurosci. 2009 369 390

[49] A.Y.Y. Tan, L.I. Zhang, M.M. Merzenich, C.E. Schreiner Evoked excitatory and inhibitory synaptic conductances of primary auditory cortex neurons J. Neurophysiol. 2004 630 643

[50] R.D. Traub, J.G. Jefferys, R. Miles, M.A. Whittington, K. Tóth A branching dendritic model of a rodent CA3 pyramidal neuron J. Physiol. 1994 79 95

[51] R.D. Traub, M.A. Whittington, E.H. Buhl, J.G.R. Jefferys, H.J. Faulkner On the mechanism of the γ → β frequency shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation J. Neurosci. 1999 1088 1105

[52] M.A Whittington, R.D. Traub, N. Kopell, B. Ermentrout, E.H. Buhl Inhibition-based rhythms: experimental and mathematical observations on network dynamics Int. J. Psychophysiol. 2000 315 336

[53] H.R. Wilson, J.D. Cowan Excitatory and inhibitory interactions in localized populations of model neurons Biophys. J. 1972 1 24

[54] D. Zhou, A.V. Rangan, D.W. Mclaughlin, D. Cai Spatiotemporal dynamics of neuronal population response in the primary visual cortex PNAS 2013 9517 9522