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 and 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. 

[2]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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. 

[2]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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