• Previous Article
    Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data
  • DCDS-B Home
  • This Issue
  • Next Article
    Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition
September  2019, 24(9): 5183-5201. doi: 10.3934/dcdsb.2019056

Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size

1. 

Centro de Investigación y Modelamiento de Fenómenos Aleatorios, Valparaíso, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile

2. 

Université Côte d'Azur, Inria, 2004, route des Lucioles, BP93, 06902 Sophia-Antipolis Cedex, France

* Corresponding author

Received  April 2017 Revised  August 2018 Published  April 2019

We study the synchronization of fully-connected and totally excitatory integrate and fire neural networks in presence of Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.

Citation: Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5183-5201. doi: 10.3934/dcdsb.2019056
References:
[1]

P. Bressloff and S. Coombes, Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays, Phys. D, 130 (1999), 232-254. doi: 10.1016/S0167-2789(99)00013-5. Google Scholar

[2]

P. C. Bressloff and S. Coombes, Dynamics of strongly coupled spiking neurons, Neural Comput., 12 (2000), 91-129. doi: 10.1162/089976600300015907. Google Scholar

[3]

N. Brunel and D. Hansel, How Noise Affects the Synchronization Properties of Recurrent Networks of Inhibitory Neurons, Neural Comput., 18 (2006), 1066-1110. doi: 10.1162/neco.2006.18.5.1066. Google Scholar

[4]

E. Catsigeras and P. Guiraud, Integrate and fire neural networks, piecewise contractive maps and limit cycles, J. Math. Biol., 67 (2013), 609-655. doi: 10.1007/s00285-012-0560-7. Google Scholar

[5]

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

[6]

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

[7]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, volume 38 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. doi: 10.1007/978-3-642-03311-7. Google Scholar

[8]

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

[9] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge university press, 2002. doi: 10.1017/CBO9780511815706.
[10]

F. HammondH. Bergman and P. Brown, Pathological synchronization in Parkinson's disease: Networks, models and treatments, Trends Neurosci, 30 (2007), 357-364. doi: 10.1016/j.tins.2007.05.004. Google Scholar

[11]

F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, volume 126, Springer, 1997. doi: 10.1007/978-1-4612-1828-9. Google Scholar

[12]

E. M. Izhikevich, Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models, IEEE Transactions on Neural Networks, 10 (1999), 1171-1266. doi: 10.1142/S0218127400000840. Google Scholar

[13]

E. M. Izhikevich, Weakly pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory, IEEE Transactions on Neural Networks, 10 (1999), 508-526. Google Scholar

[14]

R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662. doi: 10.1137/0150098. Google Scholar

[15]

K. A. Newhall, G. Kovačič, P. R. Kramer and D. Cai, Cascade-induced synchrony in stochastically driven neuronal networks, Phys. Rev. E, 82 (2010), 041903, 17pp. doi: 10.1103/PhysRevE.82.041903. Google Scholar

[16]

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

[17]

L. Sacerdote and M. T. Giraudo, Stochastic integrate and fire models: A review on mathematical methods and their applications, In Stochastic Biomathematical Models, volume 2058 of Lecture Notes in Math., pages 99-148. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32157-3_5. Google Scholar

[18]

L. Sacerdote and C. Zucca, Joint distribution of first exit times of a two dimensional Wiener process with jumps with application to a pair of coupled neurons, Math. Biosci., 245 (2013), 61-69. doi: 10.1016/j.mbs.2013.06.005. Google Scholar

[19]

J. Touboul and O. Faugeras, A markovian event-based framework for stochastic spiking neural networks, J. Comput. Neurosci., 31 (2011), 485-507. doi: 10.1007/s10827-011-0327-y. Google Scholar

[20]

T. S. TurovaW. Mommaerts and and E. C. van der Meulen, Synchronization of firing times in a stochastic neural network model with excitatory connections, Stochastic Process. Appl., 50 (1994), 173-186. doi: 10.1016/0304-4149(94)90155-4. Google Scholar

[21]

C. Van Vreeswijk, Partial synchronization in populations of pulse-coupled oscillators, Phys. Rev. E, 54 (1996), 5522. doi: 10.1103/PhysRevE.54.5522. Google Scholar

[22]

C. Van VreeswijkL. Abbott and G. B. Ermentrout, When inhibition not excitation synchronizes neural firing, J. Comput. Neurosci., 1 (1994), 313-321. doi: 10.1007/BF00961879. Google Scholar

show all references

References:
[1]

P. Bressloff and S. Coombes, Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays, Phys. D, 130 (1999), 232-254. doi: 10.1016/S0167-2789(99)00013-5. Google Scholar

[2]

P. C. Bressloff and S. Coombes, Dynamics of strongly coupled spiking neurons, Neural Comput., 12 (2000), 91-129. doi: 10.1162/089976600300015907. Google Scholar

[3]

N. Brunel and D. Hansel, How Noise Affects the Synchronization Properties of Recurrent Networks of Inhibitory Neurons, Neural Comput., 18 (2006), 1066-1110. doi: 10.1162/neco.2006.18.5.1066. Google Scholar

[4]

E. Catsigeras and P. Guiraud, Integrate and fire neural networks, piecewise contractive maps and limit cycles, J. Math. Biol., 67 (2013), 609-655. doi: 10.1007/s00285-012-0560-7. Google Scholar

[5]

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

[6]

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

[7]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, volume 38 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. doi: 10.1007/978-3-642-03311-7. Google Scholar

[8]

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

[9] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge university press, 2002. doi: 10.1017/CBO9780511815706.
[10]

F. HammondH. Bergman and P. Brown, Pathological synchronization in Parkinson's disease: Networks, models and treatments, Trends Neurosci, 30 (2007), 357-364. doi: 10.1016/j.tins.2007.05.004. Google Scholar

[11]

F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, volume 126, Springer, 1997. doi: 10.1007/978-1-4612-1828-9. Google Scholar

[12]

E. M. Izhikevich, Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models, IEEE Transactions on Neural Networks, 10 (1999), 1171-1266. doi: 10.1142/S0218127400000840. Google Scholar

[13]

E. M. Izhikevich, Weakly pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory, IEEE Transactions on Neural Networks, 10 (1999), 508-526. Google Scholar

[14]

R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662. doi: 10.1137/0150098. Google Scholar

[15]

K. A. Newhall, G. Kovačič, P. R. Kramer and D. Cai, Cascade-induced synchrony in stochastically driven neuronal networks, Phys. Rev. E, 82 (2010), 041903, 17pp. doi: 10.1103/PhysRevE.82.041903. Google Scholar

[16]

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

[17]

L. Sacerdote and M. T. Giraudo, Stochastic integrate and fire models: A review on mathematical methods and their applications, In Stochastic Biomathematical Models, volume 2058 of Lecture Notes in Math., pages 99-148. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32157-3_5. Google Scholar

[18]

L. Sacerdote and C. Zucca, Joint distribution of first exit times of a two dimensional Wiener process with jumps with application to a pair of coupled neurons, Math. Biosci., 245 (2013), 61-69. doi: 10.1016/j.mbs.2013.06.005. Google Scholar

[19]

J. Touboul and O. Faugeras, A markovian event-based framework for stochastic spiking neural networks, J. Comput. Neurosci., 31 (2011), 485-507. doi: 10.1007/s10827-011-0327-y. Google Scholar

[20]

T. S. TurovaW. Mommaerts and and E. C. van der Meulen, Synchronization of firing times in a stochastic neural network model with excitatory connections, Stochastic Process. Appl., 50 (1994), 173-186. doi: 10.1016/0304-4149(94)90155-4. Google Scholar

[21]

C. Van Vreeswijk, Partial synchronization in populations of pulse-coupled oscillators, Phys. Rev. E, 54 (1996), 5522. doi: 10.1103/PhysRevE.54.5522. Google Scholar

[22]

C. Van VreeswijkL. Abbott and G. B. Ermentrout, When inhibition not excitation synchronizes neural firing, J. Comput. Neurosci., 1 (1994), 313-321. doi: 10.1007/BF00961879. Google Scholar

Figure 1.  Simulation of the model for $ N = 3 $, $\gamma = 100$ $ \text{s}^{-1} $, $ \beta = -52 $ mV, $\theta = -55$ mV, $ V_r = -70 $ mV, $ \varepsilon = 2.25\times10^{-4} $ $ \text{V}^2/\text{s} $ and $ H_{ji} = 1.5 $ mV for all $ i\neq j\in\{1,2,3\} $
Figure 2.  $\mathbb{P}\left(S_{n+1}^\varepsilon\middle|S_{n}^\varepsilon\right)$ versus $\varepsilon$ for $N = 1599$, $\gamma = 100$ $\text{s}^{-1}$, $\theta = -55$ mV, $V_r = -70$ mV, $m = 0.75$ mV. Red plot $\beta = -52$ mV, green plot $\beta = -54.7$ mV and blue plot $\beta = -55.3$ mV. $\varepsilon\in[0, 2.5\times 10^{-3}\text{V}^2/\text{s}]$
Figure 3.  Zoom on Fig.2 for $\varepsilon\in[0,2.25\times 10^{-4}\text{V}^2/\text{s}]$.
Figure 4.  $ \mathbb{P}\left(BS_{n}^\varepsilon\right) $ versus $ n $. The values of the parameters are: $ N = 1599 $, $ \gamma = 100 $ $ \text{s}^{-1} $, $ \theta = -55 $ mV, $ V_r = -70 $ mV, $ \beta = -52 $ mV and $ \varepsilon = 0.225\times 10^{-4}\text{V}^2/\text{s} $. Magenta $ m = 0.0150 $ mV, green $ m = 0.0225 $ mV, blue $ m = 0.0270 $ mV and orange $ m = 0.0300 $ mV. The initial distribution of the membrane potentials are i.i.d. with uniform law on \([-100\text{ mV}, -55\text{ mV}]\)
Figure 5.  $ \mathbb{P}\left(BS_{n}^\varepsilon \setminus BS^\varepsilon_{n-1}\right) $ probability of first synchronization at the $ n $-th firing instant versus $ n $. The values of the parameters are: $ N = 1599 $, $ \gamma = 100 $ $ \text{s}^{-1} $, $ \theta = -55 $ mV, $ V_r = -70 $ mV, $ \beta = -52 $ mV and $ \varepsilon = 0.225\times 10^{-4}\text{V}^2/\text{s} $. Magenta $ m = 0.0150 $ mV, green $ m = 0.0225 $ mV, blue $ m = 0.0270 $ mV and orange $ m = 0.0300 $ mV. The initial distribution of the membrane potentials are i.i.d. with uniform law on \([-100\text{ mV}, -55\text{ mV}]\)
[1]

Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651

[2]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 189-201. doi: 10.3934/mbe.2014.11.189

[3]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013

[4]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451

[5]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43

[6]

Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049

[7]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[8]

Giuseppe Da Prato. Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 637-647. doi: 10.3934/dcdss.2013.6.637

[9]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871

[10]

Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285

[11]

Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525

[12]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

[13]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049

[14]

Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385

[15]

Stefano Fasani, Sergio Rinaldi. Local stabilization and network synchronization: The case of stationary regimes. Mathematical Biosciences & Engineering, 2010, 7 (3) : 623-639. doi: 10.3934/mbe.2010.7.623

[16]

Maria Francesca Carfora, Enrica Pirozzi. Stochastic modeling of the firing activity of coupled neurons periodically driven. Conference Publications, 2015, 2015 (special) : 195-203. doi: 10.3934/proc.2015.0195

[17]

David J. Aldous. A stochastic complex network model. Electronic Research Announcements, 2003, 9: 152-161.

[18]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[19]

Rumi Ghosh, Kristina Lerman. Rethinking centrality: The role of dynamical processes in social network analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1355-1372. doi: 10.3934/dcdsb.2014.19.1355

[20]

Nikolai Dokuchaev. On strong causal binomial approximation for stochastic processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1549-1562. doi: 10.3934/dcdsb.2014.19.1549

2018 Impact Factor: 1.008

Article outline

Figures and Tables

[Back to Top]