October  2021, 14(5): 819-846. doi: 10.3934/krm.2021025

Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

1. 

Laboratory Jacques-Louis Lions (LJLL), UMR 7238 CNRS, Sorbonne Université, 75205 Paris Cedex 06, France

2. 

Laboratory of Computational and Quantitative Biology (LCQB), UMR 7238 CNRS, Sorbonne Université, 75205 Paris Cedex 06, France

* Corresponding author: Pierre Roux

Received  March 2020 Revised  June 2021 Published  October 2021 Early access  August 2021

Fund Project: Delphine Salort was supported by the grant ANR ChaMaNe, ANR-19-CE40-0024

The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.

Citation: Pierre Roux, Delphine Salort. Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence. Kinetic & Related Models, 2021, 14 (5) : 819-846. doi: 10.3934/krm.2021025
References:
[1]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neural activity, Journal of Neurophysiology, 94 (2005), 3637-3642.   Google Scholar

[2]

N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks, J. Comp. Neurosci., 8 (2000), 183-208.   Google Scholar

[3]

N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with long firing rates, Neural Computation, 11 (1999), 1621-1671.   Google Scholar

[4]

M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, J. Math. Neurosci., 1 (2011), Art. 7, 33 pp. doi: 10.1186/2190-8567-1-7.  Google Scholar

[5]

M. J. CáceresJ. A. Carrillo and L. Tao, A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics, J. Comp. Phys., 230 (2011), 1084-1099.  doi: 10.1016/j.jcp.2010.10.027.  Google Scholar

[6]

M. J. Cáceres and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity, J. Theoret. Biol., 350 (2014), 81-89.  doi: 10.1016/j.jtbi.2014.02.005.  Google Scholar

[7]

M. J. Cáceres and A. Ramos-Lora, An understanding of the physical solutions and the blow-up phenomenon for nonlinear noisy leaky integrate and fire neuronal models, arXiv preprint, arXiv: 2011.05860. Google Scholar

[8]

M. J. CáceresP. RouxD. Salort and R. Schneider, Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay, Comm. Partial Differential Equations, 44 (2019), 1358-1386.  doi: 10.1080/03605302.2019.1639732.  Google Scholar

[9]

M. J. Cáceres and R. Schneider, Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods, ESAIM Math. Model. Numer. Anal., 52 (2018), 1733-1761.  doi: 10.1051/m2an/2018014.  Google Scholar

[10]

J. A. CarrilloM. D. M. GonzálezM. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience, Comm. Partial Differential Equations, 38 (2013), 385-409.  doi: 10.1080/03605302.2012.747536.  Google Scholar

[11]

J. A. CarrilloB. PerthameD. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate & fire model in computational neuroscience, Nonlinearity, 25 (2015), 3365-3388.  doi: 10.1088/0951-7715/28/9/3365.  Google Scholar

[12]

J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Process. Appl., 127 (2017), 3870-3912.  doi: 10.1016/j.spa.2017.02.012.  Google Scholar

[13]

J. ChevallierM. J. CáceresM. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Math. Models Methods Appl. Sci., 25 (2015), 2669-2719.  doi: 10.1142/S021820251550058X.  Google Scholar

[14]

F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, First hitting times for general non-homogeneous 1d diffusion processes: Density estimates in small time, Unpublished Notes. Google Scholar

[15]

F. DelarueJ. InglisS. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean–Vlasov type, Ann. Appl. Probab., 25 (2015), 2096-2133.  doi: 10.1214/14-AAP1044.  Google Scholar

[16]

F. DelarueJ. InglisS. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks, Stochastic Process. Appl., 125 (2015), 2451-2492.  doi: 10.1016/j.spa.2015.01.007.  Google Scholar

[17]

F. Delarue, S. Nadtochiy and M. Shkolnikov, Global solutions to the supercooled Stefan problem with blow-ups: Regularity and uniqueness, arXiv preprint, arXiv: 1902.05174. Google Scholar

[18]

G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol., 75 (2013), 629-648.  doi: 10.1007/s11538-013-9823-8.  Google Scholar

[19]

R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[20]

B. Hambly and S. Ledger, A stochastic Mckean–Vlasov equation for absorbing diffusions on the half-line, The Annals of Applied Probability, 27 (2017), 2698-2752.  doi: 10.1214/16-AAP1256.  Google Scholar

[21]

B. HamblyS. Ledger and A. Søjmark, A McKean–Vlasov equation with positive feedback and blow-ups, Ann. Appl. Probab., 29 (2019), 2338-2373.  doi: 10.1214/18-AAP1455.  Google Scholar

[22]

J. Hu, J. -G. Liu, Y. Xie and Z. Zhou, A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration, J. Comput. Phys., 433 (2021), Paper No. 110195, 23 pp. arXiv: 1911.07619. doi: 10.1016/j. jcp. 2021.110195.  Google Scholar

[23]

S. Ledger and A. Søjmark, Uniqueness for contagious McKean–Vlasov systems in the weak feedback regime, Bull. Lond. Math. Soc., 52 (2020), 448-463.  doi: 10.1112/blms.12337.  Google Scholar

[24]

J. -G. Liu, Z. Wang, Y. Zhang and Z. Zhou, Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective, arXiv preprint, arXiv: 2005.08285. Google Scholar

[25]

S. MischlerC. Quininao and J. Touboul, On a kinetic Fitzhugh–Nagumo model of neuronal network, Comm. Math. Phys., 342 (2016), 1001-1042.  doi: 10.1007/s00220-015-2556-9.  Google Scholar

[26]

S. Mischler and Q. Weng, Relaxation in time elapsed neuron network models in the weak connectivity regime, Acta Appl. Math., 157 (2018), 45-74.  doi: 10.1007/s10440-018-0163-4.  Google Scholar

[27]

S. Nadtochiy and M. Shkolnikov, Particle systems with singular interaction through hitting times: application in systemic risk modeling, Ann. Appl. Probab., 29 (2019), 89-129.  doi: 10.1214/18-AAP1403.  Google Scholar

[28]

S. Nadtochiy and M. Shkolnikov, Mean field systems on networks, with singular interaction through hitting times, Ann. Probab., 48 (2020), 1520-1556.  doi: 10.1214/19-AOP1403.  Google Scholar

[29]

K. A. NewhallG. KovačičP. R. KramerD. ZhouA. V. Rangan and D. Cai, Dynamics of current-based, Poisson driven, Integrate-and-Fire neuronal networks, Commun. Math. Sci., 8 (2010), 541-600.  doi: 10.4310/CMS.2010.v8.n2.a12.  Google Scholar

[30]

K. PakdamanB. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75.  doi: 10.1088/0951-7715/23/1/003.  Google Scholar

[31]

K. PakdamanB. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM J. Appl. Math., 73 (2013), 1260-1279.  doi: 10.1137/110847962.  Google Scholar

[32]

B. Perthame and D. Salort, On a voltage-conductance kinetic system for integrate & fire neural networks, Kinet. Relat. Models, 6 (2013), 841-864.  doi: 10.3934/krm.2013.6.841.  Google Scholar

[33]

A. V. Rangan, G. Kovačič and D. Cai, Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train, Phys. Rev. E, 77 (2008), 041915, 13 pp. doi: 10.1103/PhysRevE. 77.041915.  Google Scholar

[34]

A. Renart, N. Brunel and X. -J. Wang, Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks, in Computational Neuroscience: A Comprehensive Approach (ed. J. Feng), Chapman & Hall/CRC Mathematical Biology and Medicine Series, 2004. Google Scholar

[35]

D. Sharma and P. Singh, Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods, Internat. J. Modern Phys. C, 31 (2020), 2050041, 25 pp. doi: 10.1142/S0129183120500412.  Google Scholar

[36]

D. Sharma, P. Singh, R. P. Agarwal and M. E. Koksal, Numerical approximation for nonlinear noisy leaky integrate-and-fire neuronal model, Mathematics, 7 (2019), 363. doi: 10.3390/math7040363.  Google Scholar

show all references

References:
[1]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neural activity, Journal of Neurophysiology, 94 (2005), 3637-3642.   Google Scholar

[2]

N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks, J. Comp. Neurosci., 8 (2000), 183-208.   Google Scholar

[3]

N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with long firing rates, Neural Computation, 11 (1999), 1621-1671.   Google Scholar

[4]

M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, J. Math. Neurosci., 1 (2011), Art. 7, 33 pp. doi: 10.1186/2190-8567-1-7.  Google Scholar

[5]

M. J. CáceresJ. A. Carrillo and L. Tao, A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics, J. Comp. Phys., 230 (2011), 1084-1099.  doi: 10.1016/j.jcp.2010.10.027.  Google Scholar

[6]

M. J. Cáceres and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity, J. Theoret. Biol., 350 (2014), 81-89.  doi: 10.1016/j.jtbi.2014.02.005.  Google Scholar

[7]

M. J. Cáceres and A. Ramos-Lora, An understanding of the physical solutions and the blow-up phenomenon for nonlinear noisy leaky integrate and fire neuronal models, arXiv preprint, arXiv: 2011.05860. Google Scholar

[8]

M. J. CáceresP. RouxD. Salort and R. Schneider, Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay, Comm. Partial Differential Equations, 44 (2019), 1358-1386.  doi: 10.1080/03605302.2019.1639732.  Google Scholar

[9]

M. J. Cáceres and R. Schneider, Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods, ESAIM Math. Model. Numer. Anal., 52 (2018), 1733-1761.  doi: 10.1051/m2an/2018014.  Google Scholar

[10]

J. A. CarrilloM. D. M. GonzálezM. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience, Comm. Partial Differential Equations, 38 (2013), 385-409.  doi: 10.1080/03605302.2012.747536.  Google Scholar

[11]

J. A. CarrilloB. PerthameD. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate & fire model in computational neuroscience, Nonlinearity, 25 (2015), 3365-3388.  doi: 10.1088/0951-7715/28/9/3365.  Google Scholar

[12]

J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Process. Appl., 127 (2017), 3870-3912.  doi: 10.1016/j.spa.2017.02.012.  Google Scholar

[13]

J. ChevallierM. J. CáceresM. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Math. Models Methods Appl. Sci., 25 (2015), 2669-2719.  doi: 10.1142/S021820251550058X.  Google Scholar

[14]

F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, First hitting times for general non-homogeneous 1d diffusion processes: Density estimates in small time, Unpublished Notes. Google Scholar

[15]

F. DelarueJ. InglisS. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean–Vlasov type, Ann. Appl. Probab., 25 (2015), 2096-2133.  doi: 10.1214/14-AAP1044.  Google Scholar

[16]

F. DelarueJ. InglisS. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks, Stochastic Process. Appl., 125 (2015), 2451-2492.  doi: 10.1016/j.spa.2015.01.007.  Google Scholar

[17]

F. Delarue, S. Nadtochiy and M. Shkolnikov, Global solutions to the supercooled Stefan problem with blow-ups: Regularity and uniqueness, arXiv preprint, arXiv: 1902.05174. Google Scholar

[18]

G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol., 75 (2013), 629-648.  doi: 10.1007/s11538-013-9823-8.  Google Scholar

[19]

R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[20]

B. Hambly and S. Ledger, A stochastic Mckean–Vlasov equation for absorbing diffusions on the half-line, The Annals of Applied Probability, 27 (2017), 2698-2752.  doi: 10.1214/16-AAP1256.  Google Scholar

[21]

B. HamblyS. Ledger and A. Søjmark, A McKean–Vlasov equation with positive feedback and blow-ups, Ann. Appl. Probab., 29 (2019), 2338-2373.  doi: 10.1214/18-AAP1455.  Google Scholar

[22]

J. Hu, J. -G. Liu, Y. Xie and Z. Zhou, A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration, J. Comput. Phys., 433 (2021), Paper No. 110195, 23 pp. arXiv: 1911.07619. doi: 10.1016/j. jcp. 2021.110195.  Google Scholar

[23]

S. Ledger and A. Søjmark, Uniqueness for contagious McKean–Vlasov systems in the weak feedback regime, Bull. Lond. Math. Soc., 52 (2020), 448-463.  doi: 10.1112/blms.12337.  Google Scholar

[24]

J. -G. Liu, Z. Wang, Y. Zhang and Z. Zhou, Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective, arXiv preprint, arXiv: 2005.08285. Google Scholar

[25]

S. MischlerC. Quininao and J. Touboul, On a kinetic Fitzhugh–Nagumo model of neuronal network, Comm. Math. Phys., 342 (2016), 1001-1042.  doi: 10.1007/s00220-015-2556-9.  Google Scholar

[26]

S. Mischler and Q. Weng, Relaxation in time elapsed neuron network models in the weak connectivity regime, Acta Appl. Math., 157 (2018), 45-74.  doi: 10.1007/s10440-018-0163-4.  Google Scholar

[27]

S. Nadtochiy and M. Shkolnikov, Particle systems with singular interaction through hitting times: application in systemic risk modeling, Ann. Appl. Probab., 29 (2019), 89-129.  doi: 10.1214/18-AAP1403.  Google Scholar

[28]

S. Nadtochiy and M. Shkolnikov, Mean field systems on networks, with singular interaction through hitting times, Ann. Probab., 48 (2020), 1520-1556.  doi: 10.1214/19-AOP1403.  Google Scholar

[29]

K. A. NewhallG. KovačičP. R. KramerD. ZhouA. V. Rangan and D. Cai, Dynamics of current-based, Poisson driven, Integrate-and-Fire neuronal networks, Commun. Math. Sci., 8 (2010), 541-600.  doi: 10.4310/CMS.2010.v8.n2.a12.  Google Scholar

[30]

K. PakdamanB. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75.  doi: 10.1088/0951-7715/23/1/003.  Google Scholar

[31]

K. PakdamanB. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM J. Appl. Math., 73 (2013), 1260-1279.  doi: 10.1137/110847962.  Google Scholar

[32]

B. Perthame and D. Salort, On a voltage-conductance kinetic system for integrate & fire neural networks, Kinet. Relat. Models, 6 (2013), 841-864.  doi: 10.3934/krm.2013.6.841.  Google Scholar

[33]

A. V. Rangan, G. Kovačič and D. Cai, Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train, Phys. Rev. E, 77 (2008), 041915, 13 pp. doi: 10.1103/PhysRevE. 77.041915.  Google Scholar

[34]

A. Renart, N. Brunel and X. -J. Wang, Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks, in Computational Neuroscience: A Comprehensive Approach (ed. J. Feng), Chapman & Hall/CRC Mathematical Biology and Medicine Series, 2004. Google Scholar

[35]

D. Sharma and P. Singh, Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods, Internat. J. Modern Phys. C, 31 (2020), 2050041, 25 pp. doi: 10.1142/S0129183120500412.  Google Scholar

[36]

D. Sharma, P. Singh, R. P. Agarwal and M. E. Koksal, Numerical approximation for nonlinear noisy leaky integrate-and-fire neuronal model, Mathematics, 7 (2019), 363. doi: 10.3390/math7040363.  Google Scholar

Figure 1.  Comparison between the lower bound (10) in [4] and the lower bound in Theorem 3.1 for non-existence of stationary states, for different values of a.The new bound is always better when $ V_F-V_R $ is large enough.We set $ V_R =-1 $ for convenience but it does not impact the results
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