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

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

Benoît Perthame 1, and  Delphine Salort 2,

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.
Keywords: Integrate-and-fire networks, neural networks, smoothness., voltage-conductance Vlasov equation, Fokker-Planck kinetic equation.
Mathematics Subject Classification: 35B65, 35Q84, 62M45, 82C32, 92B2.
Citation: Benoît Perthame, Delphine Salort. On a voltage-conductance kinetic system for integrate & fire neural networks. Kinetic & 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.  doi: 10.1016/j.jfa.2011.07.020.  Google Scholar

[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 (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

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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.  doi: 10.1186/2190-8567-2-10.  Google Scholar

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J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, 223 (1976).   Google Scholar

[5]

F. Bouchut, Hypoelliptic regularity in kinetic equations,, Journal de Mathématiques Pures et Appliquées, 8 (2002), 1135.  doi: 10.1016/S0021-7824(02)01264-3.  Google Scholar

[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, (2004).  doi: 10.1007/b93802.  Google Scholar

[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.   Google Scholar

[8]

N. Brunel and N. Fourcaud, Dynamics of the firing probability of noisy integrate-and-fire neurons,, Neural Computation, 14 (2002), 2057.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1186/2190-8567-1-7.  Google Scholar

[11]

M. J. Caceres, and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity,, Submitted., ().   Google Scholar

[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.  doi: 10.1016/j.jcp.2010.10.027.  Google Scholar

[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.  doi: 10.1073/pnas.0401906101.  Google Scholar

[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.  doi: 10.1137/11083486X.  Google Scholar

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A. Cohen, Numerical Analysis of Wavelet Methods,, Studies in Mathematics and its Applications, (2003).   Google Scholar

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A. Coulon, G. Beslon and H. Soula, Enhanced stimulus encoding capabilities with spectral selectivity in inhibitory circuits by STDP,, Neural Computation, 23 (2011), 882.  doi: 10.1162/NECO_a_00100.  Google Scholar

[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., (1211).   Google Scholar

[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.  doi: 10.1007/s00285-012-0554-5.  Google Scholar

[19]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[20]

T. Lepoutre, N. Meunier and N. Muller, Cell Polarisation Model: The 1D Case,, Journal de Mathématiques Pures et Appliquées, (2013).  doi: 10.1016/j.matpur.2013.05.006.  Google Scholar

[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.  doi: 10.1162/neco.2007.19.8.2032.  Google Scholar

[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.  doi: 10.1239/aap/1282924062.  Google Scholar

[23]

B. Perthame, Transport Equations in Biology,, Series 'Frontiers in Mathematics', (2007).   Google Scholar

[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.  doi: 10.1016/j.jcp.2006.06.036.  Google Scholar

[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).  doi: 10.1103/PhysRevE.77.041915.  Google Scholar

[26]

H. Triebel, Theory of Function Spaces,, Modern Birkhäuser Classics, (2010).   Google Scholar

[27]

C. Villani, Hypocoercivity,, Memoirs of the American Mathematical Society, 202 (2009).  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  doi: 10.1007/s10827-011-0355-7.  Google Scholar

show all references

References:
[1]

D. Arsenio and L. Saint-Raymond, Compactness in kinetic transport equations and hypoellipticity,, Journal of Functional Analysis, 261 (2011), 3044.  doi: 10.1016/j.jfa.2011.07.020.  Google Scholar

[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 (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[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.  doi: 10.1186/2190-8567-2-10.  Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, 223 (1976).   Google Scholar

[5]

F. Bouchut, Hypoelliptic regularity in kinetic equations,, Journal de Mathématiques Pures et Appliquées, 8 (2002), 1135.  doi: 10.1016/S0021-7824(02)01264-3.  Google Scholar

[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, (2004).  doi: 10.1007/b93802.  Google Scholar

[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.   Google Scholar

[8]

N. Brunel and N. Fourcaud, Dynamics of the firing probability of noisy integrate-and-fire neurons,, Neural Computation, 14 (2002), 2057.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1186/2190-8567-1-7.  Google Scholar

[11]

M. J. Caceres, and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity,, Submitted., ().   Google Scholar

[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.  doi: 10.1016/j.jcp.2010.10.027.  Google Scholar

[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.  doi: 10.1073/pnas.0401906101.  Google Scholar

[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.  doi: 10.1137/11083486X.  Google Scholar

[15]

A. Cohen, Numerical Analysis of Wavelet Methods,, Studies in Mathematics and its Applications, (2003).   Google Scholar

[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.  doi: 10.1162/NECO_a_00100.  Google Scholar

[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., (1211).   Google Scholar

[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.  doi: 10.1007/s00285-012-0554-5.  Google Scholar

[19]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[20]

T. Lepoutre, N. Meunier and N. Muller, Cell Polarisation Model: The 1D Case,, Journal de Mathématiques Pures et Appliquées, (2013).  doi: 10.1016/j.matpur.2013.05.006.  Google Scholar

[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.  doi: 10.1162/neco.2007.19.8.2032.  Google Scholar

[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.  doi: 10.1239/aap/1282924062.  Google Scholar

[23]

B. Perthame, Transport Equations in Biology,, Series 'Frontiers in Mathematics', (2007).   Google Scholar

[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.  doi: 10.1016/j.jcp.2006.06.036.  Google Scholar

[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).  doi: 10.1103/PhysRevE.77.041915.  Google Scholar

[26]

H. Triebel, Theory of Function Spaces,, Modern Birkhäuser Classics, (2010).   Google Scholar

[27]

C. Villani, Hypocoercivity,, Memoirs of the American Mathematical Society, 202 (2009).  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  doi: 10.1007/s10827-011-0355-7.  Google Scholar

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