A finite volume method for stochastic integrate-and-fire models
Tóm tắt
The stochastic integrate and fire neuron is one of the most commonly used stochastic models in neuroscience. Although some cases are analytically tractable, a full analysis typically calls for numerical simulations. We present a fast and accurate finite volume method to approximate the solution of the associated Fokker-Planck equation. The discretization of the boundary conditions offers a particular challenge, as standard operator splitting approaches cannot be applied without modification. We demonstrate the method using stationary and time dependent inputs, and compare them with Monte Carlo simulations. Such simulations are relatively easy to implement, but can suffer from convergence difficulties and long run times. In comparison, our method offers improved accuracy, and decreases computation times by several orders of magnitude. The method can easily be extended to two and three dimensional Fokker-Planck equations.
Tài liệu tham khảo
Apfaltrer, F., Ly, C., & Tranchina, D. (2006). Population density methods for stochastic neurons with realistic synaptic kinetics: Firing rate dynamics and fast computational methods. Network: Computation in Neural Systems, 17(4), 373–418.
Barth, T., & Ohlberger, M. (2004). Finite volume methods: Foundation and analysis. Encyclopedia of computational mechanics. New York: Wiley.
Bruneau, C. H., Marpeau, F., & Saad, M. (2005). Numerical simulation of the miscible displacement of radionuclides in a heterogeneous porous medium. International Journal for Numerical Methods in Fluids, 49(10), 1053–1085.
Brunel, N., Chance, F. S., Fourcaud, N., & Abbott, L. F. (2001). Effects of synaptic noise and filtering on the frequency response of spiking neurons. Physical Review Letters, 86, 2186–2189.
Brunel, N., & Hakim, V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Computation, 11, 1621–1671.
Brunel, N., & Latham, P. E. (2003). Firing rate of the noisy quadratic integrate-and-fire neuron. Neural Computation, 15(10), 2281–2306.
Burkitt, A. N. (2006). A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biological Cybernetics, 95(1), 1–19.
Courant, R., Friedrichs, K., & Lewy, H. (1928). Uber die partiellen differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32–74.
Doiron, B., Chacron, M. J., Maler, L., Longtin, A., & Bastian, J. (2003). Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature, 421(6922), 539–43.
Ermentrout, G. B., & Kopell, N. (1986). Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM Journal on Applied Mathematics, 46(2), 233–253.
Galán, R. F., Ermentrout, G. B., & Urban, N. N. (2007). Stochastic dynamics of uncoupled neural oscillators: Fokker-Planck studies with the finite element method. Physical Review E, 76(5), 56110.
Gardiner, C. W. (1985). Handbook of stochastic methods. New York: Springer.
Godlewski, E., & Raviart, P. A. (1990). Hyperbolic systems of conservation laws. Mathématiques et applications (Vol. 3/4). Paris: Ellipses.
Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49, 357–392.
Kloeden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. New York: Springer.
Lapicque, L. (1907). Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une polarisation. Journal de Physiologie et de Pathologie Générale (Paris), 9, 622–635.
Lascaux, P., & Théodor, R. (1986–1987). Analyse numérique matricielle appliquée à l’Art de l’Ingénieur (Vol. 1,2). Paris: Masson.
Latham, P. E., Richmond, B. J., Nelson, P. G., & Nirenberg, S. (2000). Intrinsic dynamics in neuronal networks. I. Theory. Journal of Neurophysiology, 83(2), 808–827.
Lindner, B. (2001). Coherence and stochastic resonance in nonlinear dynamical systems. Ph.D. thesis, Humboldt University.
Lindner, B., & Longtin, A. (2006). Comment on “Characterization of subthreshold voltage fluctuations in neuronal membranes” by M. Rudolph and A. Destexhe. Neural Computation, 18(8), 1896.
Lindner, B., & Schimansky-Geier, L. (2001). Transmission of noise coded versus additive signals through a neronal ensemble. Physical Review Letters, 86, 2934–2937.
Mattia, M., & Del Giudice, P. (2002). Population dynamics of interacting spiking neurons. Physical Review E, 66(5), 51917.
Mattia, M., & Del Giudice, P. (2004). Finite-size dynamics of inhibitory and excitatory interacting spiking neurons. Physical Review E, 70(5), 52903.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical recipes: The art of scientific computing (3rd ed.) Cambridge: Cambridge University Press.
Rall, W. (1995). The theoretical foundation of dendritic function: Selected papers of Wilfrid Rall with commentaries. Boston: MIT.
Renart, A., Brunel, N., & Wang, X. J. (2004). Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. In Computational neuroscience: A comprehensive approach (pp. 431–490). Boca Raton: Chapman & Hall.
Risken, H. (1989). The Fokker-Planck equation: Methods of solution and applications. Berlin: Springer.
Roe, P. L. (1984). Generalized formulation of TVD Lax-Wendroff schemes. ICASE Rep. 53–84.
Rudolph, M., & Destexhe, A. (2005). An extended analytic expression for the membrane potential distribution of conductance-based synaptic noise. Neural Computation, 17(11), 2301–2315.
Strang, G. (1968). On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis, 5, 506–517.
Sweby, P. K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 21 995–1011.
Toro, E. F. (2001). Godunov methods: Theory and applications. Dordrecht: Kluwer/Plenum.
Tuckwell, H. C. (1988). Introduction to theoretical neurobiology. Cambridge: Cambridge University Press.
Tuckwell, H. C. (1989). Stochastic processes in the neurosciences. Philadelphia: SIAM.