December  2019, 12(6): 1329-1358. doi: 10.3934/krm.2019052

Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network

Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, UPS IMT, F-31062 Toulouse Cedex 9 France

* Corresponding author: joachim.crevat@math.univ-toulouse.fr

Received  March 2019 Revised  June 2019 Published  September 2019

We consider a spatially-extended model for a network of interacting FitzHugh-Nagumo neurons without noise, and rigorously establish its mean-field limit towards a nonlocal kinetic equation as the number of neurons goes to infinity. Our approach is based on deterministic methods, and namely on the stability of the solutions of the kinetic equation with respect to their initial data. The main difficulty lies in the adaptation in a deterministic framework of arguments previously introduced for the mean-field limit of stochastic systems of interacting particles with a certain class of locally Lipschitz continuous interaction kernels. This result establishes a rigorous link between the microscopic and mesoscopic scales of observation of the network, which can be further used as an intermediary step to derive macroscopic models. We also propose a numerical scheme for the discretization of the solutions of the kinetic model, based on a particle method, in order to study the dynamics of its solutions, and to compare it with the microscopic model.

Citation: Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
References:
[1]

J. BaladronD. Fasoli and O. Faugeras, Three applications of GPU computing in neuroscience, Computing in Science and Engineering, 14 (2012), 40-47. 

[2]

J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, The Journal of Mathematical Neuroscience, 2 (2012), Art. 10, 50 pp. doi: 10.1186/2190-8567-2-10.

[3]

P. W. BatesP. C. FifeX. F. Ren and X. F. Wang, Traveling Waves in a Convolution model for phase transitions, Archive for rational Mechanics and Analysis, 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[4]

F. BolleyJ. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Mathematical Models and Methods in Applied Sciences, 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.

[5]

M. Bossy, O. Faugeras and D. Talay, Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons", The Journal of Mathematical Neuroscience, 5 (2015), Art. 19, 23 pp. doi: 10.1186/s13408-015-0031-8.

[6]

M. BossyJ. Fontbona and H. Olivero, Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels, Journal of Mathematical Biology, 78 (2019), 1771-1820.  doi: 10.1007/s00285-019-01326-7.

[7]

P. C. Bressloff, Spatially periodic modulation of cortical patterns by long-range horizontal connections, Physica D: Nonlinear Phenomena, 185 (2003), 131-157.  doi: 10.1016/S0167-2789(03)00238-0.

[8]

T. Cabana and J. D. Touboul, Large deviations for randomly connected neural networks: I. Spatially extended systems, Advances in Applied Probability, 50 (2018), 944-982.  doi: 10.1017/apr.2018.42.

[9]

M. Campos PintoE. SonnendrückerA. FriedmanD. P. Grote and S. M. Lund, Noiseless Vlasov-Poisson simulations with linearly transformed particles, Journal of Computational Physics, 275 (2014), 236-256.  doi: 10.1016/j.jcp.2014.06.032.

[10]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[11]

P. Carter and A. Scheel, Wave train selection by invasion fronts in the FitzHugh-Nagumo equation, Nonlinearity, 31 (2018), 5536-5572.  doi: 10.1088/1361-6544/aae1db.

[12]

J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Processes and Their Applications, 127 (2017), 3870-3912.  doi: 10.1016/j.spa.2017.02.012.

[13]

J. ChevallierM. J. CaceresM. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2669-2719.  doi: 10.1142/S021820251550058X.

[14]

J. ChevallierA. DuarteE. Löcherbach and G. Ost, Mean-field limits for nonlinear spatially extended hawkes processes with exponential memory kernels, Stochastic Processes and Their Applications, 129 (2019), 1-27.  doi: 10.1016/j.spa.2018.02.007.

[15]

H. Chiba and G. S. Medvedev, The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas, Discrete & Continuous Dynamical Systems-A, 39 (2019), 131-155.  doi: 10.3934/dcds.2019006.

[16]

J. CrevatG. Faye and F. Filbet, Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system, SIAM J. Math. Anal., 51 (2019), 346-373.  doi: 10.1137/18M1178839.

[17]

R. L. Dobrushin, Vlasov equations, Funktsional. Anal. i Prilozhen, 13 (1979), 48–58, 96.

[18]

O. Faugeras, J. Touboul and B. Cessac, A constructive mean-field analysis of multi population neural networks with random synaptic weight and stochastic inputs, Frontiers in computational neuroscience, 3 (2009), 1. doi: 10.3389/neuro.10.001.2009.

[19]

G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Advances in Mathematics, 270 (2015), 400-456.  doi: 10.1016/j.aim.2014.11.005.

[20]

F. Filbet and L. M. Rodrigues, Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field, SIAM J. Numer. Analysis, 54 (2016), 1120-1146.  doi: 10.1137/15M104952X.

[21]

R. FitzHugh, Impulses and physiological sates in theoretical models of nerve membrane, Biophysical journal, 1 (1961), 445-466. 

[22]

F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Lecture Notes in Applied Mathematics and Mechanics, Springer, Cham, 3 (2016), 1–144. doi: 10.1007/978-3-319-26883-5_1.

[23]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking Dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[24]

F. H. Harlow, The particle-in-cell computing method for fluid dynamics, Method in Computational Physics, 3 (1964), 319-343. 

[25]

M. Hauray and P.-E. Jabin, N-particles approximation of the vlasov equations with singular potential, Arch. Ration. Mech. Anal., 183 (2007), 489-524.  doi: 10.1007/s00205-006-0021-9.

[26]

D. W. Hewett, Fragmentation, merging, and internal dynamics for PIC simulation with finite size particles, Journal of Computational Physics, 189 (2003), 390-426.  doi: 10.1016/S0021-9991(03)00225-0.

[27]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500-544. 

[28]

E. Luçon and W. Stannat, Mean-field limit for disordered diffusions with singular interactions, The Annals of Applied Probability, 24 (2014), 1946-1993.  doi: 10.1214/13-AAP968.

[29]

S. MischlerC. Quiñinao 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.

[30]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[31]

I. Omelchenko, B. Riemenschneider, P. Hövel, Y. Maistrenko and F. Schöll, Transition from spatial coherence to incoherence in coupled chaotic sytems, Physical Review E, 85 (2012), 026212.

[32]

D. Parker, Variable properties in a single class of excitatory spinal synapse, The Journal of Neuroscience, 23 (2003), 3154-3163.  doi: 10.1523/JNEUROSCI.23-08-03154.2003.

[33]

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.

[34]

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.

[35]

K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, Journal of Mathematical Neuroscience, 4 (2014), Art. 14, 26 pp. doi: 10.1186/2190-8567-4-14.

[36]

C. Quiñinao and J. Touboul, Clamping and synchronization in the strongly coupled FitzHugh-Nagumo model, preprint, arXiv: 1804.06758.

[37]

P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1127 (1985), 243–324. doi: 10.1007/BFb0074532.

[38]

J. TouboulG. Hermann and O. Faugeras, Noise-induced behaviors in neural mean field dynamics, SIAM Journal on Appplied Dynamical Systems, 11 (2012), 49-81.  doi: 10.1137/110832392.

[39]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[40]

G. Wainrib and J. Touboul, Topological and dynamical complexity of random neural networks, Phys. Rev. Lett., 110 (2013), 118101. doi: 10.1103/PhysRevLett.110.118101.

show all references

References:
[1]

J. BaladronD. Fasoli and O. Faugeras, Three applications of GPU computing in neuroscience, Computing in Science and Engineering, 14 (2012), 40-47. 

[2]

J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, The Journal of Mathematical Neuroscience, 2 (2012), Art. 10, 50 pp. doi: 10.1186/2190-8567-2-10.

[3]

P. W. BatesP. C. FifeX. F. Ren and X. F. Wang, Traveling Waves in a Convolution model for phase transitions, Archive for rational Mechanics and Analysis, 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[4]

F. BolleyJ. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Mathematical Models and Methods in Applied Sciences, 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.

[5]

M. Bossy, O. Faugeras and D. Talay, Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons", The Journal of Mathematical Neuroscience, 5 (2015), Art. 19, 23 pp. doi: 10.1186/s13408-015-0031-8.

[6]

M. BossyJ. Fontbona and H. Olivero, Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels, Journal of Mathematical Biology, 78 (2019), 1771-1820.  doi: 10.1007/s00285-019-01326-7.

[7]

P. C. Bressloff, Spatially periodic modulation of cortical patterns by long-range horizontal connections, Physica D: Nonlinear Phenomena, 185 (2003), 131-157.  doi: 10.1016/S0167-2789(03)00238-0.

[8]

T. Cabana and J. D. Touboul, Large deviations for randomly connected neural networks: I. Spatially extended systems, Advances in Applied Probability, 50 (2018), 944-982.  doi: 10.1017/apr.2018.42.

[9]

M. Campos PintoE. SonnendrückerA. FriedmanD. P. Grote and S. M. Lund, Noiseless Vlasov-Poisson simulations with linearly transformed particles, Journal of Computational Physics, 275 (2014), 236-256.  doi: 10.1016/j.jcp.2014.06.032.

[10]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[11]

P. Carter and A. Scheel, Wave train selection by invasion fronts in the FitzHugh-Nagumo equation, Nonlinearity, 31 (2018), 5536-5572.  doi: 10.1088/1361-6544/aae1db.

[12]

J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Processes and Their Applications, 127 (2017), 3870-3912.  doi: 10.1016/j.spa.2017.02.012.

[13]

J. ChevallierM. J. CaceresM. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2669-2719.  doi: 10.1142/S021820251550058X.

[14]

J. ChevallierA. DuarteE. Löcherbach and G. Ost, Mean-field limits for nonlinear spatially extended hawkes processes with exponential memory kernels, Stochastic Processes and Their Applications, 129 (2019), 1-27.  doi: 10.1016/j.spa.2018.02.007.

[15]

H. Chiba and G. S. Medvedev, The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas, Discrete & Continuous Dynamical Systems-A, 39 (2019), 131-155.  doi: 10.3934/dcds.2019006.

[16]

J. CrevatG. Faye and F. Filbet, Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system, SIAM J. Math. Anal., 51 (2019), 346-373.  doi: 10.1137/18M1178839.

[17]

R. L. Dobrushin, Vlasov equations, Funktsional. Anal. i Prilozhen, 13 (1979), 48–58, 96.

[18]

O. Faugeras, J. Touboul and B. Cessac, A constructive mean-field analysis of multi population neural networks with random synaptic weight and stochastic inputs, Frontiers in computational neuroscience, 3 (2009), 1. doi: 10.3389/neuro.10.001.2009.

[19]

G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Advances in Mathematics, 270 (2015), 400-456.  doi: 10.1016/j.aim.2014.11.005.

[20]

F. Filbet and L. M. Rodrigues, Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field, SIAM J. Numer. Analysis, 54 (2016), 1120-1146.  doi: 10.1137/15M104952X.

[21]

R. FitzHugh, Impulses and physiological sates in theoretical models of nerve membrane, Biophysical journal, 1 (1961), 445-466. 

[22]

F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Lecture Notes in Applied Mathematics and Mechanics, Springer, Cham, 3 (2016), 1–144. doi: 10.1007/978-3-319-26883-5_1.

[23]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking Dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[24]

F. H. Harlow, The particle-in-cell computing method for fluid dynamics, Method in Computational Physics, 3 (1964), 319-343. 

[25]

M. Hauray and P.-E. Jabin, N-particles approximation of the vlasov equations with singular potential, Arch. Ration. Mech. Anal., 183 (2007), 489-524.  doi: 10.1007/s00205-006-0021-9.

[26]

D. W. Hewett, Fragmentation, merging, and internal dynamics for PIC simulation with finite size particles, Journal of Computational Physics, 189 (2003), 390-426.  doi: 10.1016/S0021-9991(03)00225-0.

[27]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500-544. 

[28]

E. Luçon and W. Stannat, Mean-field limit for disordered diffusions with singular interactions, The Annals of Applied Probability, 24 (2014), 1946-1993.  doi: 10.1214/13-AAP968.

[29]

S. MischlerC. Quiñinao 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.

[30]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[31]

I. Omelchenko, B. Riemenschneider, P. Hövel, Y. Maistrenko and F. Schöll, Transition from spatial coherence to incoherence in coupled chaotic sytems, Physical Review E, 85 (2012), 026212.

[32]

D. Parker, Variable properties in a single class of excitatory spinal synapse, The Journal of Neuroscience, 23 (2003), 3154-3163.  doi: 10.1523/JNEUROSCI.23-08-03154.2003.

[33]

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.

[34]

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.

[35]

K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, Journal of Mathematical Neuroscience, 4 (2014), Art. 14, 26 pp. doi: 10.1186/2190-8567-4-14.

[36]

C. Quiñinao and J. Touboul, Clamping and synchronization in the strongly coupled FitzHugh-Nagumo model, preprint, arXiv: 1804.06758.

[37]

P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1127 (1985), 243–324. doi: 10.1007/BFb0074532.

[38]

J. TouboulG. Hermann and O. Faugeras, Noise-induced behaviors in neural mean field dynamics, SIAM Journal on Appplied Dynamical Systems, 11 (2012), 49-81.  doi: 10.1137/110832392.

[39]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[40]

G. Wainrib and J. Touboul, Topological and dynamical complexity of random neural networks, Phys. Rev. Lett., 110 (2013), 118101. doi: 10.1103/PhysRevLett.110.118101.

Figure 1.  Bistable regime. (A)-(B) Spatio-temporal evolution of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with $ \tau = 0 $, and different values of the parameter $ \varepsilon $, fixed at (A) $ 10^{-1} $, (B) $ 10^{-3} $. (C) Profile of the macroscopic function $ V_f(t, \cdot) $ at different fixed times, computed with $ \varepsilon = 10^{-3} $ and with $ \tau = 0 $
Figure 2.  Bistable regime. Numerical approximation of the density function $ f $ solution of the kinetic equation (5) at fixed time (A) $ t = 0 $, (B) $ t = 75 $ and (C) $ t = 150 $, computed with the parameters $ \varepsilon = 10^{-3} $ and $ \tau = 0 $
Figure 3.  Oscillatory regime. (A)-(B)-(C) Spatio-temporal evolution of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with three different values of the parameter $ \varepsilon $, fixed at (A) $ \varepsilon = 10^{-1} $, (B) $ \varepsilon = 10^{-3} $ and (C) $ \varepsilon = 10^{-5} $. (D)-(E) Profile of the macroscopic function $ V_f(t, \cdot) $ computed with $ \varepsilon = 10^{-5} $ at time $ t = 60 $ and $ t = 400 $ respectively. (F) Trajectory in the phase space $ (v, w) $ of the couple $ (V_f, W_f) $ at fixed position $ \mathbf{x} = 0.2 $ between times $ 0 $ and $ t = 400 $ computed with $ \varepsilon = 10^{-5} $. The other parameters are fixed at $ a = -0.25 $, $ b = 3 $, and $ \tau = 0.02 $
Figure 4.  Excitable regime. (A) Spatio-temporal evolution of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with $ \varepsilon = 10^{-5} $. (B) Corresponding profile of the macroscopic function $ V_f(t, \cdot) $ computed at different times. (C) Trajectory in the phase space $ (v, w) $ of the couple $ (V_f, W_f) $ at fixed position $ \mathbf{x} = 0.2 $ between times $ 0 $ and $ t = 1000 $ computed with $ \varepsilon = 10^{-5} $. The other parameters are fixed at $ a = 0 $, $ b = 7 $, and $ \tau = 0.002 $
Figure 5.  Profile of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with $ \varepsilon = 10^{-4} $, and with the points $ ( \mathbf{x}_i, v_i)_{1\leq i \leq n} $ from the solution of the FHN system (3), at fixed time $ t = 225 $. The other parameters are the same as in Figure 3
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