# American Institute of Mathematical Sciences

December  2013, 6(4): 841-864. doi: 10.3934/krm.2013.6.841

## On a voltage-conductance kinetic system for integrate & fire neural networks

 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris 2 Institut Jacques Monod UMR 7592 Univ Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France

Received  August 2013 Revised  September 2013 Published  November 2013

The voltage-conductance kinetic equation for integrate and fire neurons has been used in neurosciences since a decade and describes the probability density of neurons in a network. It is used when slow conductance receptors are activated and noticeable applications to the visual cortex have been worked-out. In the simplest case, the derivation also uses the assumption of fully excitatory and moderately all-to-all coupled networks; this is the situation we consider here.
We study properties of solutions of the kinetic equation for steady states and time evolution and we prove several global a priori bounds both on the probability density and the firing rate of the network. The main difficulties are related to the degeneracy of the diffusion resulting from noise and to the quadratic aspect of the nonlinearity.
This result constitutes a paradox; the solutions of the kinetic model, of partially hyperbolic nature, are globally bounded but it has been proved that the fully parabolic integrate and fire equation (some kind of diffusion limit of the former) blows-up in finite time.
Citation: Benoît Perthame, Delphine Salort. On a voltage-conductance kinetic system for integrate & fire neural networks. Kinetic and Related Models, 2013, 6 (4) : 841-864. doi: 10.3934/krm.2013.6.841
##### References:
 [1] D. Arsenio and L. Saint-Raymond, Compactness in kinetic transport equations and hypoellipticity, Journal of Functional Analysis, 261 (2011), 3044-3098. doi: 10.1016/j.jfa.2011.07.020. [2] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [3] 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, Journal of Mathematical Neurosciences, 2 (2012), 10-50. doi: 10.1186/2190-8567-2-10. [4] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976. x+207 pp. [5] F. Bouchut, Hypoelliptic regularity in kinetic equations, Journal de Mathématiques Pures et Appliquées, 8 (2002), 1135-1159. doi: 10.1016/S0021-7824(02)01264-3. [6] F. Bouchut, Non Linear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well Balanced Schemes for Sources, Frontiers in Mathematics. Birkhaüser-Verlag, Basel, 2004. doi: 10.1007/b93802. [7] 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. [8] N. Brunel and N. Fourcaud, Dynamics of the firing probability of noisy integrate-and-fire neurons, Neural Computation, 14 (2002), 2057-2110. [9] N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Computation, 11 (1999), 1621-1671. [10] M. J. Caceres, J. A. Carrillo and B. Perthame, Analysis of Nonlinear Noisy Integrate&Fire Neuron Models: Blow-up and steady states, The Journal of Mathematical Neuroscience, 1 (2011), 33pp. doi: 10.1186/2190-8567-1-7. [11] M. J. Caceres, and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity, Submitted. [12] M. J. Cáceres, J. A. Carrillo and L. Tao, A numerical solver for a nonlinear Fokker-Planck equation representation of network dynamics, Journal of Computational Physics, 230 (2011), 1084-1099. doi: 10.1016/j.jcp.2010.10.027. [13] D. Cai, L. Tao, M. Shelley and D. W. McLaughlin, An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex, PNAS, 101 (2004), 7757-7762. doi: 10.1073/pnas.0401906101. [14] V. Calvez, R. J. Hawkins, N. Meunier and R. Voituriez, Analysis of a nonlocal model for spontaneous cell polarization, SIAM. Journal on Applied Mathematics, 72 (2012), 594-622. doi: 10.1137/11083486X. [15] A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003. [16] A. Coulon, G. Beslon and H. Soula, Enhanced stimulus encoding capabilities with spectral selectivity in inhibitory circuits by STDP, Neural Computation, 23 (2011), 882-908. doi: 10.1162/NECO_a_00100. [17] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean-Vlasov type, 2012. arXiv 1211.0299. [18] G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: Well-posedness, Journal of Mathematical Biology, 67 (2013), 453-481. doi: 10.1007/s00285-012-0554-5. [19] R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. [20] T. Lepoutre, N. Meunier and N. Muller, Cell Polarisation Model: The 1D Case, Journal de Mathématiques Pures et Appliquées, (2013). arXiv:1301.3684. doi: 10.1016/j.matpur.2013.05.006. [21] C. Ly and D. Tranchina, Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling, Neural Computation, 19 (2007), 2032-2092. doi: 10.1162/neco.2007.19.8.2032. [22] K. Pakdaman, M. Thieullen and G. Wainrib, Fluid limit theorems for stochastic hybrid systems and applications to neuron models, Advances in Applied Probability, 42 (2010), 761-794. doi: 10.1239/aap/1282924062. [23] B. Perthame, Transport Equations in Biology, Series 'Frontiers in Mathematics', Birkhauser, 2007. [24] A. V. Rangan, D. Cai and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, Journal of Computational Physics, 221 (2007), 781-798. doi: 10.1016/j.jcp.2006.06.036. [25] 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, Physical Review, 77 (2008), 041915, 13pp. doi: 10.1103/PhysRevE.77.041915. [26] H. Triebel, Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010. 285 pp. [27] C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5. [28] G. Wainrib, M. Thieullen and K. Pakdaman, Reduction of stochastic conductance-based neuron models with time-scales separation, Journal of Computational Neurosciences, 32 (2011), 327-346. doi: 10.1007/s10827-011-0355-7.

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##### References:
 [1] D. Arsenio and L. Saint-Raymond, Compactness in kinetic transport equations and hypoellipticity, Journal of Functional Analysis, 261 (2011), 3044-3098. doi: 10.1016/j.jfa.2011.07.020. [2] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [3] 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, Journal of Mathematical Neurosciences, 2 (2012), 10-50. doi: 10.1186/2190-8567-2-10. [4] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976. x+207 pp. [5] F. Bouchut, Hypoelliptic regularity in kinetic equations, Journal de Mathématiques Pures et Appliquées, 8 (2002), 1135-1159. doi: 10.1016/S0021-7824(02)01264-3. [6] F. Bouchut, Non Linear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well Balanced Schemes for Sources, Frontiers in Mathematics. Birkhaüser-Verlag, Basel, 2004. doi: 10.1007/b93802. [7] 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. [8] N. Brunel and N. Fourcaud, Dynamics of the firing probability of noisy integrate-and-fire neurons, Neural Computation, 14 (2002), 2057-2110. [9] N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Computation, 11 (1999), 1621-1671. [10] M. J. Caceres, J. A. Carrillo and B. Perthame, Analysis of Nonlinear Noisy Integrate&Fire Neuron Models: Blow-up and steady states, The Journal of Mathematical Neuroscience, 1 (2011), 33pp. doi: 10.1186/2190-8567-1-7. [11] M. J. Caceres, and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity, Submitted. [12] M. J. Cáceres, J. A. Carrillo and L. Tao, A numerical solver for a nonlinear Fokker-Planck equation representation of network dynamics, Journal of Computational Physics, 230 (2011), 1084-1099. doi: 10.1016/j.jcp.2010.10.027. [13] D. Cai, L. Tao, M. Shelley and D. W. McLaughlin, An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex, PNAS, 101 (2004), 7757-7762. doi: 10.1073/pnas.0401906101. [14] V. Calvez, R. J. Hawkins, N. Meunier and R. Voituriez, Analysis of a nonlocal model for spontaneous cell polarization, SIAM. Journal on Applied Mathematics, 72 (2012), 594-622. doi: 10.1137/11083486X. [15] A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003. [16] A. Coulon, G. Beslon and H. Soula, Enhanced stimulus encoding capabilities with spectral selectivity in inhibitory circuits by STDP, Neural Computation, 23 (2011), 882-908. doi: 10.1162/NECO_a_00100. [17] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean-Vlasov type, 2012. arXiv 1211.0299. [18] G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: Well-posedness, Journal of Mathematical Biology, 67 (2013), 453-481. doi: 10.1007/s00285-012-0554-5. [19] R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. [20] T. Lepoutre, N. Meunier and N. Muller, Cell Polarisation Model: The 1D Case, Journal de Mathématiques Pures et Appliquées, (2013). arXiv:1301.3684. doi: 10.1016/j.matpur.2013.05.006. [21] C. Ly and D. Tranchina, Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling, Neural Computation, 19 (2007), 2032-2092. doi: 10.1162/neco.2007.19.8.2032. [22] K. Pakdaman, M. Thieullen and G. Wainrib, Fluid limit theorems for stochastic hybrid systems and applications to neuron models, Advances in Applied Probability, 42 (2010), 761-794. doi: 10.1239/aap/1282924062. [23] B. Perthame, Transport Equations in Biology, Series 'Frontiers in Mathematics', Birkhauser, 2007. [24] A. V. Rangan, D. Cai and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, Journal of Computational Physics, 221 (2007), 781-798. doi: 10.1016/j.jcp.2006.06.036. [25] 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, Physical Review, 77 (2008), 041915, 13pp. doi: 10.1103/PhysRevE.77.041915. [26] H. Triebel, Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010. 285 pp. [27] C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5. [28] G. Wainrib, M. Thieullen and K. Pakdaman, Reduction of stochastic conductance-based neuron models with time-scales separation, Journal of Computational Neurosciences, 32 (2011), 327-346. doi: 10.1007/s10827-011-0355-7.
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