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Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs
June  2019, 39(6): 3037-3067. doi: 10.3934/dcds.2019126

## A mean-field model with discontinuous coefficients for neurons with spatial interaction

 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy 2 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, Torino, Italy 3 Institute of Science and Technology Austria (IST Austria), Am Campus 1, Klosterneuburg, Austria 4 LUISS Università Guido Carli, Viale Romania 32, Roma, Italy

* Corresponding author: Giovanni Zanco

Received  February 2018 Published  February 2019

Fund Project: The second author has been partially supported by INdAM through the GNAMPA Research Project (2017) "Sistemi stocastici singolari: buona posizione e problemi di controllo". The third author was partly funded by the Austrian Science Fund (FWF) project F 65.

Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.

Citation: Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
##### References:
 [1] 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. [2] 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. [3] A. De Masi, A. Galves, E. Löcherbach and E. Presutti, Hydrodynamic limit for interacting neurons, J. Stat. Phys., 158 (2015), 866-902.  doi: 10.1007/s10955-014-1145-1. [4] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks, Stochastic Processes and their Applications, 125 (2015), 2451-2492.  doi: 10.1016/j.spa.2015.01.007. [5] F. Delarue, J. Inglis, S. 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. [6] N. Fourcaud-Trocmé and N. Brunel, Dynamics of the instantaneous firing rate in response to changes in input statistics, J. Comput. Neurosci., 18 (2005), 311-321.  doi: 10.1007/s10827-005-0337-8. [7] A. Friedman, Stochastic Differential Equations and Applications. Vol. 1, Probability and Mathematical Statistics, Vol. 28, Academic Press, New York-London, 1975. [8] W. Gerstner and W. M. Kistler, Spiking Neuron Models, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511815706. [9] I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Prob. Theory Relat. Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981. [11] B. W. Knight, D. Manin and L. Sirovich, Dynamical models of interacting neuron populations in visual cortex, Robot Cybern, 54 (1996), 4-8. [12] T. J. Lewis and J. Rinzel, Dynamics of spiking neurons connected by both inhibitory and electrical coupling, J. Comput. Neurosci., 14) (2003), 283-309.  doi: 10.1023/A:1023265027714. [13] Y. S. Mishura and A-Y. Veretennikov., Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, arXiv: 1603.02212. [14] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probability Theory and Related Fields, 82 (1989), 565-586.  doi: 10.1007/BF00341284. [15] S. Ostojic, N. Brunel and V. Hakim, Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities, J. Comput. Neurosci., 26 (2009), 369-392.  doi: 10.1007/s10827-008-0117-3. [16] T. Schwalger, M. Deger and W. Gerstner, Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size, PLoS Comput. Biol., 13 (2017), e1005507. doi: 10.1371/journal.pcbi.1005507. [17] J. Simon, Compact sets in the space $L(0, T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [18] A.-S. Sznitman, Topics in propagation of chaos, vol. 1464 of Lecture Notes in Mathematics., Springer-Verlag, Berlin, 1991, 165–251. doi: 10.1007/BFb0085169. [19] H. C. Tuckwell, Introduction to Theoretical Neurobiology. Vol. 1, vol. 8 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1988. [20] A. J. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR-Sb. (N.S.), 39 (1981), 387-403.

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##### References:
 [1] 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. [2] 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. [3] A. De Masi, A. Galves, E. Löcherbach and E. Presutti, Hydrodynamic limit for interacting neurons, J. Stat. Phys., 158 (2015), 866-902.  doi: 10.1007/s10955-014-1145-1. [4] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks, Stochastic Processes and their Applications, 125 (2015), 2451-2492.  doi: 10.1016/j.spa.2015.01.007. [5] F. Delarue, J. Inglis, S. 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. [6] N. Fourcaud-Trocmé and N. Brunel, Dynamics of the instantaneous firing rate in response to changes in input statistics, J. Comput. Neurosci., 18 (2005), 311-321.  doi: 10.1007/s10827-005-0337-8. [7] A. Friedman, Stochastic Differential Equations and Applications. Vol. 1, Probability and Mathematical Statistics, Vol. 28, Academic Press, New York-London, 1975. [8] W. Gerstner and W. M. Kistler, Spiking Neuron Models, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511815706. [9] I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Prob. Theory Relat. Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981. [11] B. W. Knight, D. Manin and L. Sirovich, Dynamical models of interacting neuron populations in visual cortex, Robot Cybern, 54 (1996), 4-8. [12] T. J. Lewis and J. Rinzel, Dynamics of spiking neurons connected by both inhibitory and electrical coupling, J. Comput. Neurosci., 14) (2003), 283-309.  doi: 10.1023/A:1023265027714. [13] Y. S. Mishura and A-Y. Veretennikov., Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, arXiv: 1603.02212. [14] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probability Theory and Related Fields, 82 (1989), 565-586.  doi: 10.1007/BF00341284. [15] S. Ostojic, N. Brunel and V. Hakim, Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities, J. Comput. Neurosci., 26 (2009), 369-392.  doi: 10.1007/s10827-008-0117-3. [16] T. Schwalger, M. Deger and W. Gerstner, Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size, PLoS Comput. Biol., 13 (2017), e1005507. doi: 10.1371/journal.pcbi.1005507. [17] J. Simon, Compact sets in the space $L(0, T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [18] A.-S. Sznitman, Topics in propagation of chaos, vol. 1464 of Lecture Notes in Mathematics., Springer-Verlag, Berlin, 1991, 165–251. doi: 10.1007/BFb0085169. [19] H. C. Tuckwell, Introduction to Theoretical Neurobiology. Vol. 1, vol. 8 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1988. [20] A. J. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR-Sb. (N.S.), 39 (1981), 387-403.
Simulation of the system (2) for $3$ neurons with no spatial interaction
Solution of system (2) for 10 neurons with strong interaction uniform in space. A spike from one neuron propagates to all other neurons in the network
A network made of two subnetworks with localized strong interaction. Each blue line corresponds to the presence of interaction; only one neuron of the first subnetwork interacts with a single neuron of the second one. The vertical axis represents time; the network is drawn on the $t = 0$ plane and to each spike of a neuron corresponds a sphere above it, with the same color as the neuron. Near time $t = 1$ many neurons in the first network spike and the signal is propagated to the second network, while at time $t = 2$ the signal does not propagate to the second network.
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