Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications

  • Laura SacerdoteEmail author
  • Maria Teresa Giraudo
Part of the Lecture Notes in Mathematics book series (LNM, volume 2058)


Mathematical models are an important tool for neuroscientists. During the last 30 years many papers have appeared on single neuron description and specifically on stochastic Integrate and Fire models. Analytical results have been proved and numerical and simulation methods have been developed for their study. Recent reviews collect the main features of these models but do not focus on the methodologies employed to obtain them. The aim of this paper is to fill this gap by upgrading old reviews. The idea is to collect the existing methods and the available analytical results for the most common one dimensional stochastic Integrate and Fire models to make them available for studies on networks. An effort to unify the mathematical notation is also made. The review is divided in two parts:
  1. 1.

    Derivation of the models with the list of the available closed forms expressions for their characterization.

  2. 2.

    Presentation of the existing mathematical and statistical methods for the study of these models.



Spike Train Wiener Process First Passage Time Jump Diffusion Model Jump Diffusion Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abrahams, J.: A survey of recent progress on level-crossing problems for random processes. In: Blake, I.F., Poor, H.V. (eds.) Communications and Networks. A Survey of Recent Advances, pp. 6–25. Springer, New York (1986)Google Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)Google Scholar
  3. 3.
    Albano, G., Giorno, V., Nobile, A.G., Ricciardi L.M.: A Wiener-type neuronal model in the presence of exponential refractoriness. BioSystems 88, 202–215 (2007)Google Scholar
  4. 4.
    Alili, L., Patie, P., Pedersen, J.L.: Representation of the first hitting time density of an Ornstein–Uhlenbeck process. Stoch. Model 21, 967–980 (2005)Google Scholar
  5. 5.
    Alili, L., Patie, P.: Boundary crossing identities for diffusions having the time-inversion property. J. Theor. Probab. 23, 65–84 (2010)Google Scholar
  6. 6.
    Anderssen, R.S., DeHoog, F.R., Weiss, R.: On the numerical solution of Brownian motion processes. J. Appl. Probab. 10, 409–418 (1973)Google Scholar
  7. 7.
    Baldi, P., Caramellino, L.: Asymptotics of hitting probabilities for general one-dimensional diffusions. Ann. Appl. Probab. 12, 1071–1095 (2002)Google Scholar
  8. 8.
    Bibbona, E., Ditlevsen, S.: Estimation in discretely observed Markov processes killed at a threshold, Early View Scandinavian Journal of Statistics online 28 August 2012 DOI 10.1111/j.1467-9469.2012.00810.xGoogle Scholar
  9. 9.
    Bibbona, E., Lánský, P., Sacerdote, L., Sirovich, R.: Errors in estimation of input signal for integrate and fire neuronal models. Phys. Rev. E 78, Art. No. 011918 (2008)Google Scholar
  10. 10.
    Bibbona, E., Lánský, P., Sirovich, R.: Estimating input parameters from intracellular recordings in the Feller neuronal model. Phys. Rev. E 81, Art. No. 031916 (2010)Google Scholar
  11. 11.
    Bibby, B., Sørensen, M.: On estimation for discretely observed diffusions: A review. Theor. Stoch. Proc. 2, 49–56 (1996)Google Scholar
  12. 12.
    Brette, R., Gerstner, W.: Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophysiol. 94, 3637–3642 (2005)Google Scholar
  13. 13.
    Buonocore, A., Nobile, A.G., Ricciardi, L.M.: A new integral equation for the evaluation of the first-passage-time probability densities. Adv. Appl. Probab. 19, 784–800 (1987)Google Scholar
  14. 14.
    Buonocore, A., Giorno, V., Nobile, A.G., Ricciardi, L.M.: A neuronal modeling paradigm in the presence of refractoriness. BioSystems 67, 35–43 (2002)Google Scholar
  15. 15.
    Buonocore, A., Caputo, L., Pirozzi, E., Ricciardi, L.M.: On a stochastic leaky integrate-and-fire neuronalmodel. Neural Comput. 22, 2558–2585 (2010)Google Scholar
  16. 16.
    Burkitt, A.N.: A review of the integrate and fire neuron model: I. Homogeneous synaptic input. Biol. Cybern. 95, 1–19 (2006)Google Scholar
  17. 17.
    Burkitt, A.N.: A review of the integrate and fire neuron model: II. Inhomogeneous synaptic input and network properties. Biol. Cybern. 95, 97–112 (2006)Google Scholar
  18. 18.
    Capocelli, R.M., Ricciardi, L.M.: Diffusion approximation and first passage time problem for a model neuron. Kybernetik 8(6), 214–223 (1971)Google Scholar
  19. 19.
    Capocelli, R.M., Ricciardi, L.M.: On the transformation of diffusion processes into the Feller process. Math. Biosci. 29, 219–234 (1976)Google Scholar
  20. 20.
    Cerbone, G., Ricciardi, L.M., Sacerdote, L.: Mean variance and skewness of first passage time for the Ornstein–Uhlenbeck process. Cybern. Syst. 12, 395–429 (1981)Google Scholar
  21. 21.
    Chacron, M.J., Longtin, A., St-Hilaire, M., Maler, L.: Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors. Phys. Rev. Lett. 85, 1576–1579 (2000)Google Scholar
  22. 22.
    Chacron, M.J., Pakdaman, K., Longtin, A.: Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue. Neural Comput. 15(2), 253–278 (2003)Google Scholar
  23. 23.
    Chacron, M.J., Lindner, B., Longtin, A.: Threshold fatigue and information transmission. J. Comput. Neurosci. 23, 301–311 (2007)Google Scholar
  24. 24.
    Clopath, C., Jolivet, R., Rauch, A., Luscher, H.R., Gerstner, W.: Predicting neuronal activity with simple models of the threshold type: adaptive exponential integrate-and-fire model with two compartments. Neurocomputing 70, 1168–1673 (2007)Google Scholar
  25. 25.
    Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. Chapman and Hall, London (1977)Google Scholar
  26. 26.
    Daniels, H.E.: The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Probab. 6, 399–408 (1969)Google Scholar
  27. 27.
    Daniels, H.E.: Sequential tests constructed from images. Ann. Stat. 10, 394–400 (1982)Google Scholar
  28. 28.
    Di Crescenzo, A., Ricciardi, L.M.: On a discrimination problem for a class of stochastic processes with ordered first-passage-times. Appl. Stoch. Model Bus. Ind. 17, 205–219 (2001)Google Scholar
  29. 29.
    Di Crescenzo, A., Di Nardo, E., Ricciardi, L.M.: On certain bounds for first-crossing-time probabilities of a jump-diffusion process. Sci. Math. Jpn. 64(2), 449–460 (2006)Google Scholar
  30. 30.
    Ditlevsen, S., Ditlevsen, O.: Parameter estimation from observations of first-passage times of the Ornstein–Uhlenbeck process and the Feller process. Probabilist. Eng. Mech. 23, 170–179 (2008)Google Scholar
  31. 31.
    Ditlevsen, S., Lánský, P.: Estimation of the input parameters in the Ornstein–Uhlenbeck neuronal model. Phys. Rev. E 71, Art. No. 011907 (2005)Google Scholar
  32. 32.
    Ditlevsen, S., Lánský, P.: Estimation of the input parameters in the Feller neuronal model. Phys. Rev. E 73, Art. No. 061910 (2006)Google Scholar
  33. 33.
    Ditlevsen, S., Lánský, P.: Parameters of stochastic diffusion processes estimated from observations of first hitting-times: application to the leaky integrate-and-fire neuronal model. Phys. Rev. E 76, Art. No. 041906 (2007)Google Scholar
  34. 34.
    Ditlevsen, S., Lánský, P.: Comparison of statistical methods for estimation of the input parameters in the Ornstein–Uhlenbeck neuronal model from first-passage times data. In: Ricciardi, L.M., Buonocore, A., Pirozzi, E. (eds.) American Institute of Physics Proceedings Series, CP1028, Collective Dynamics: Topics on Competition and Cooperation in the Biosciences (2008)Google Scholar
  35. 35.
    Durbin, J.: Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov Smirnov test. J. Appl. Probab. 8, 431–453 (1971)Google Scholar
  36. 36.
    Durbin, J.: The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Probab. 22(1), 99–122 (1985)Google Scholar
  37. 37.
    Durbin, J., Williams, D.: The first-passage density of the Brownian Motion process to a curved boundary. J. Appl. Probab. 29(2), 291–304 (1992)Google Scholar
  38. 38.
    Favella, L., Reineri, M.T., Ricciardi, L.M., Sacerdote, L.: First-passage-time problems and some related computational methods. Cybern. Syst. 13, 95–128 (1982)Google Scholar
  39. 39.
    Fortet, R.: Les fonctions aléatoires du type Markoff associées à certaines èquations linéaires au dérivées partielles du type parabolique. J. Math. Pure Appl. 22(9), 177–243 (1943)Google Scholar
  40. 40.
    Gerstein, G.L., Mandelbrot, B.: Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41–68 (1964)Google Scholar
  41. 41.
    Gerstner, W., Kistler, W.M.: Spiking Neuron Models Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002)Google Scholar
  42. 42.
    Giorno, V., Nobile, A.G., Ricciardi, L.M., Sacerdote, L.: Some remarks on the Raleigh process. J. Appl. Probab. 23, 398–408 (1986)Google Scholar
  43. 43.
    Giorno, V., Lánský, P., Nobile, A.G., Ricciardi, L.M.: Diffusion approximation and first-passage-time problem for a model neuron: III. A birth-and-death process approach. Biol. Cybern. 58, 387–404 (1988)Google Scholar
  44. 44.
    Giorno, V., Nobile, A.G., Ricciardi, L.M.: A symmetry based constructive approach to probability densities for one dimensional diffusion processes. J. Appl. Probab. 27, 707–721 (1989)Google Scholar
  45. 45.
    Giorno, V., Nobile, A.G., Ricciardi, L.M.: On the asymptotic behavior of first-passage-time densities for one dimensional diffusion processes and varying boundary. Adv. Appl. Probab. 22, 883–914 (1990)Google Scholar
  46. 46.
    Giorno, V., Nobile, A.G., Ricciardi, L.M.: Instantaneous return process and neuronal firings. In: Trappl, R. (ed.) Cybernetics and Systems Research 1992, pp. 829–836. World Scientific, New York (1992)Google Scholar
  47. 47.
    Giorno, V., Nobile, A.G., Ricciardi, L.M.C.: On the moments of firing numbers in diffusion neuronal models with refractoriness. In: Mira, J., Alvarez, J.R. (eds.) IWINAC 2005. Lecture Notes in Computer Sciences 3561, pp. 186–194. Springer, New York (2005)Google Scholar
  48. 48.
    Giraudo, M.T.: A similarity solution for the Ornstein–Uhlenbeck diffusion process constrained by a reflecting and an absorbing boundary. Ricerche Matemat. 49(1), 47–63 (2000)Google Scholar
  49. 49.
    Giraudo, M.T., Mininni, R., Sacerdote, L.: On the asymptotic behavior of the parameter estimators for some diffusion processes: application to neuronal models. Ricerche Matemat. 58(1), 103–127 (2009)Google Scholar
  50. 50.
    Giraudo, M.T., Sacerdote, L.: Some remarks on first-passage-time for jump-diffusion processes. In: Trappl, R. (ed.) Cybernetics and Systems ’96, pp. 518–523. University of Wien Press, Wien (1996)Google Scholar
  51. 51.
    Giraudo, M.T., Sacerdote, L.: Jump-diffusion processes as models for neuronal activity. BioSystems 40, 75–82 (1997)Google Scholar
  52. 52.
    Giraudo, M.T., Sacerdote, L.: Simulation methods in neuronal modeling. BioSystems 48, 77–83 (1998)Google Scholar
  53. 53.
    Giraudo, M.T., Sacerdote, L.: An improved technique for the simulation of first passage times for diffusion processes. Comm. Stat. Simulat. Comput. 28(4), 1135–1163 (1999)Google Scholar
  54. 54.
    Giraudo, M.T., Sacerdote, L., Zucca, C.: Evaluation of first passage times of diffusion processes through boundaries by means of a totally simulative algorithm. Meth. Comp. Appl. Probab. 3, 215–231 (2001)Google Scholar
  55. 55.
    Giraudo, M.T., Sacerdote, L., Sirovich, R.: Effects of random jumps on a very simple neuronal diffusion model. BioSystems 67, 75–83 (2002)Google Scholar
  56. 56.
    Giraudo, M.T.: An approximate formula for the first-crossing-time density of a Wiener process perturbed by random jumps. Stat. Probab. Lett. 79, 1559–1567 (2009)Google Scholar
  57. 57.
    Giraudo, M.T., Greenwood, P.E., Sacerdote, L.: How sample paths of Leaky Integrate and Fire models are influenced by the presence of a firing threshold. Neural Comput. 23(7), 1743–67 (2011)Google Scholar
  58. 58.
    Grün, S., Rotter, P.: Analysis of Parallel Spike Trains. Springer, New York (2010)Google Scholar
  59. 59.
    Gutierrez, R., Ricciardi, L.M., Román, P., Torres, F.: First-passage-time densities for time-non-homogeneous diffusion processes. J. Appl. Probab. 34, 623–631 (1997)Google Scholar
  60. 60.
    Hampel, D., Lánský, P.: On the estimation of refractory period. J. Neurosci. Meth. 171, 288–295 (2008)Google Scholar
  61. 61.
    Helias, M., Deger, M., Diesmann, M., Rotter, S.: Equilibrium and response properties of the integrate-and-fire neuron in discrete time. Front. Comput. Neurosci. 3, Article 29 (2010)Google Scholar
  62. 62.
    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)Google Scholar
  63. 63.
    Honerkamp, J.: Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis. VCH, New York (1994)Google Scholar
  64. 64.
    Hopfner, R.: On a set of data for the membrane potential in a neuron. Math. Biosci. 207(2), 275–301 (2007)Google Scholar
  65. 65.
    Inoue, J., Sato, S., Ricciardi, L.M.: On the parameter estimation for diffusion models of single neuron’s activities. I. Application to spontaneous activities of mesencephalic reticular formation cells in sleep and waking states. Biol. Cybern. 73(3), 209–221 (1995)Google Scholar
  66. 66.
    Jahn, P., Berg, R.W., Hounsgaard, J., Ditlevsen, S.: Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process. J. Comput. Neurosci. 31, 563–579 (2011)Google Scholar
  67. 67.
    Jolivet, R., Lewis, T.J., Gerstner, W.: Generalized integrate-and-fire models of neuronal activity. Approximate spike trains of a detailed model to a high degree of accuracy. J. Neurophysiol. 92(2), 959–976 (2004)Google Scholar
  68. 68.
    Jolivet, R., Roth, A., Schurmann, F., Gerstner, W., Senn, W.: Special issue on quantitative neuron modeling. Biol. Cybern. 99, 237–239 (2006)Google Scholar
  69. 69.
    Kallianpur, G.: On the diffusion approximation to a discontinuous model for a single neuron. In: Contributions to Statistics, pp. 247–258. North-Holland, Amsterdam (1983)Google Scholar
  70. 70.
    Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic, New York (1981)Google Scholar
  71. 71.
    Kistler, W.M., Gerstner, W., vanHemmen, J.L.: Reduction of the Hodgkin–Huxley equations to a single-variable threshold model. Neural Comput. 9(5), 1015–1045 (1997)Google Scholar
  72. 72.
    Kloeden, P., Platen, P.: The numerical solution of Stochastic differential equations. Springer, New York (1992)Google Scholar
  73. 73.
    Kobayashi, R., Tsubo, Y., Shinomoto, S.: Predicting spike times of any cortical neuron. Frontiers in Systems Neuroscience. Conference Abstract: Computational and systems neuroscience. doi: 10.3389/conf.neuro.06.2009.03.196 (2009)Google Scholar
  74. 74.
    Kobayashi, R., Tsubo, Y., Shinomoto, S.: Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Front. Comput. Neurosci. 3, Article 9 (2009)Google Scholar
  75. 75.
    Lánský, P.: Inference for the diffusion models of neuronal activity. Math. Biosci. 67, 247–260 (1983)Google Scholar
  76. 76.
    Lánský, P., Lánská, V.: Diffusion approximations of the neuronal model with synaptic reversal potentials. Biol. Cybern. 56, 19–26 (1987)Google Scholar
  77. 77.
    Lánský, P., Rodriguez, R.: Coding range of a two-compartmental model of a neuron. Biol. Cybern. 81, 161 (1999)Google Scholar
  78. 78.
    Lánský, P., Sacerdote, L., Tomassetti, F.: On the comparison of Feller and Ornstein–Uhlenbeck models for neural activity. Biol. Cybern. 73, 457–465 (1995/1996)Google Scholar
  79. 79.
    Lánský, P., Sacerdote, L.: The Ornstein–Uhlenbeck neuronal model with signal-dependent noise. Phys. Lett. A 285, 132–140 (2001)Google Scholar
  80. 80.
    Lánský, P., Sato, S.: The stochastic diffusion models of nerve membrane depolarization and interspike interval generation. J. Periph. Nerv. Syst. 4, 27–42 (1999)Google Scholar
  81. 81.
    Lánský, P., Musila, M.: Generalized Stein’s model for anatomically complex neurons. Biosystems 25, 179–191 (1991)Google Scholar
  82. 82.
    Lánská, V., Lánský, P., Smith, C.E.: Synaptic transmission in a diffusion model for neural activity. J. Theor. Biol. 166, 393–406 (1994)Google Scholar
  83. 83.
    Lánský, P., Sanda, P., He, J.F.: The parameters of the stochastic leaky integrate-and-fire neuronal model. J. Comput. Neurosci. 21, 211–223 (2006)Google Scholar
  84. 84.
    Lánský, P., Ditlevsen, S.: A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models. Biol. Cybern. 99, 253–262 (2008)Google Scholar
  85. 85.
    Lapique, L.: Reserches quantitatives sur l’excitation électrique des nerfs traitée comme une polarization. J. Physiol. Pathol. Gen. 9, 620–635 (1907)Google Scholar
  86. 86.
    Lerche, H.R.: Boundary Crossing of Brownian Motion. Lecture Notes in Statistics, vol. 40. Springer, Heidelberg (1986)Google Scholar
  87. 87.
    Lindner, B., Chacron, M.J., Longtin, A.: Integrate and fire neurons with threshold noise: A tractable model of how interspike interval correlations affect neuronal signal transmission. Phys. Rev. E 72, 021911 (2005)Google Scholar
  88. 88.
    Marpeau, F., Barua, A., Josic, K.: A finite volume method for stochastic integrate-and-fire models. J. Comput. Neurosci. 26, 445–457 (2009)Google Scholar
  89. 89.
    Mullowney, P., Iyengar, S.: Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data. J. Comput. Neurosci. 24, 179–194 (2008)Google Scholar
  90. 90.
    Nobile, A.G., Ricciardi, L.M., Sacerdote, L.: Exponential trends of first passage time densities for a class of diffusion processes with steady-state distribution. J. Appl. Probab. 22, 611–618 (1985)Google Scholar
  91. 91.
    Nobile, A.G., Ricciardi, L.M., Sacerdote, L.: Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Probab. 22, 360–369 (1985)Google Scholar
  92. 92.
    Nobile, A.G., Pirozzi, E., Ricciardi, L.M.: On the Estimation of First-Passage Time Densities for a Class of Gauss-Markov Processes EUROCAST 2007. In: Diaz, M. (ed.) LNCS 4739:146–153, Springer, Berlin, 2007Google Scholar
  93. 93.
    Pakdaman, K., Mestivier, D.: External noise synchronizes forced oscillators. Phys. Rev. E 64, 030901 (2001)Google Scholar
  94. 94.
    Paninski, L., Haith, A., Szirtes, G.: Integral equation methods for computing likelihoods and their derivatives in the stochastic integrate and fire model. J. Comput. Neurosci. 24, 69–79 (2008)Google Scholar
  95. 95.
    Pawlas, Z., Klebanov, L.B., Prokop, M., Lánský, P.: Parameters of spike trains observed in a short time window. Neural Comput. 20, 1325–1343 (2008)Google Scholar
  96. 96.
    Peskir, G.: Limit at zero of the Brownian first-passage density. Probab. Theor. Relat. Field 124, 100–111 (2002)Google Scholar
  97. 97.
    Picchini, U., Lánský, P., De Gaetano, A., Ditlevsen, S.: Parameters of the diffusion leaky integrate-and fire neuronal model for a slowly fluctuating signal. Neural Comput. 20, 2696–2714 (2008)Google Scholar
  98. 98.
    Plesser, H.E., Diesmann, M.: Simplicity and efficiency of integrate-and-fire neuron models. Neural Comput. 21, 353–359 (2009)Google Scholar
  99. 99.
    Ricciardi, L.M.: On the transformation of Diffusion Processes into the Wiener process. J. Math. Anal. Appl. 54(1), 185–199 (1976)Google Scholar
  100. 100.
    Ricciardi, L.M.: Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics, vol. 14. Springer, Berlin (1977)Google Scholar
  101. 101.
    Ricciardi, L.M., Sacerdote, L.: The Ornstein–Uhlenbeck process as a model for neuronal activity. Biol. Cybern. 35, 1–9 (1979)Google Scholar
  102. 102.
    Ricciardi, L.M., Sacerdote, L., Sato, S.: Diffusion approximation and first passage time problem for a model neuron II Outline of a computational method. Math. Biosci. 64, 29–44 (1983)Google Scholar
  103. 103.
    Ricciardi, L.M., Sacerdote, L., Sato, S.: On an integral equation for first-passage-time probability densities. J. Appl. Probab. 21(2), 302–314 (1984)Google Scholar
  104. 104.
    Ricciardi, L.M., Sato, S.: Diffusion processes and first-passage-time problems. In: Ricciardi, L.M. (ed.) Lectures Notes in Biomathematics and Informatics. Manchester University Press, Manchester (1989)Google Scholar
  105. 105.
    Ricciardi, L.M., Di Crescenzo, A., Giorno, V., Nobile, A.G.: On the instantaneous return process for neuronal diffusion models. In: Marinaro, M., Scarpetta, G. (eds.) Structure: From Physics to General Systems, pp. 78–94. World Scientific, New York (1992)Google Scholar
  106. 106.
    Ricciardi, L.M., Di Crescenzo, A., Giorno, V., Nobile, A.: An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling. Math. Jpn. 50(2), 247–321 (1999)Google Scholar
  107. 107.
    Ricciardi, L.M., Esposito, G., Giorno, V., Valerio, C.: Modeling neuronal firing in the presence of refractoriness. In: Mira, J., Alvarez, J.R. (eds.) IWANN 2003. Lecture Notes in Computer Sciences 2686, pp. 1–8. Springer, New York (2003)Google Scholar
  108. 108.
    Rodriguez, R., Lánský, P.: Two-compartment stochastic model of a neuron with periodic input. Lecture Notes in Computer Science 1606 “Foundations and Tools for Neural Modeling (IWANN’99)”. Springer, New York (1999)Google Scholar
  109. 109.
    Rodriguez, R., Lánský, P.: A simple stochastic model of spatially complex neurons. Biosystems 58, 49 (2000)Google Scholar
  110. 110.
    Rodriguez, R., Lánský, P.: Effect of spatial extension on noise-enhanced phase locking in a leaky integrate-and-fire model of a neuron. Phys. Rev. E 62, 8427 (2001)Google Scholar
  111. 111.
    Roger, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales. Wiley Series in Probability and Mathematical Statistics, New York (1987)Google Scholar
  112. 112.
    Román, P., Serrano, J.J., Torres, F.: First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal. 52, 4132–4146 (2008)Google Scholar
  113. 113.
    Sacerdote, L.: Asymptotic behavior of Ornstein–Uhlenbeck first-passage-time density through boundaries. Appl. Stoch. Mod. Data Anal. 6, 53–57 (1988)Google Scholar
  114. 114.
    Sacerdote, L.: On the solution of the Fokker–Planck equation for a Feller process. Adv. Appl. Probab. 22(1), 101–110 (1990)Google Scholar
  115. 115.
    Sacerdote, L., Ricciardi, L.M.: On the transformation of diffusion equations and boundaries into the Kolmogorov equation for the Wiener process. Ricerche Matemat. 41(1), 123–135 (1992)Google Scholar
  116. 116.
    Sacerdote, L., Sirovich, R.: Multimodality of the interspike interval distribution in a simple jump-diffusion model. Scientiae Mathematicae Japonicae Online 8, 359–374 (2003)Google Scholar
  117. 117.
    Sacerdote, L., Sirovich, R.: Noise induced phenomena in jump-diffusion models for single neuron spike activity. IJCNN Proc., Budapest (2004)Google Scholar
  118. 118.
    Sacerdote, L., Smith, C.E.: New parameter relationships determined via stochastic ordering for spike activity in a reversal potential model. BioSystems 58, 59–65 (2000)Google Scholar
  119. 119.
    Sacerdote, L., Smith, C.E.: A qualitative comparison of some diffusion models for neural activity via stochastic ordering. Biol. Cybern. 83(6), 543–551 (2000)Google Scholar
  120. 120.
    Sacerdote, L., Smith, C.E.: Almost sure comparisons for first passage times of diffusion processes through boundaries. Meth. Comput. Appl. Probab. 6(3), 323–341 (2004)Google Scholar
  121. 121.
    Sacerdote, L., Tomassetti, F.: On evaluations and asymptotic approximations of first-passage-time probabilities. Adv. Appl. Probab. 28(1), 270–284 (1996)Google Scholar
  122. 122.
    Sacerdote, L., Zucca, C.: Threshold shape corresponding to a Gamma firing distribution in an Ornstein–Uhlenbeck neuronal model. Scientiae Mathematicae Japonicae 58(2), 295–30 (2003)Google Scholar
  123. 123.
    Sacerdote, L., Zucca, C.: On the relationship between interspikes interval distribution and boundary shape in the Ornstein-Uhlenbeck neuronal model. In: Mathematical Modelling and Computing in Biology and Medicine (Capasso, V. ed.): 161–168, the MIRIAM Project Series, Progetto Leonardo, Esculapio Pub. Co., Bologna, 2003Google Scholar
  124. 124.
    Sacerdote, L., Villa, A.E.P., Zucca, C.: On the classification of experimental data modeled via a stochastic leaky integrate and fire model through boundary values. Bull. Math. Biol. 68(6), 1257–1274 (2006)Google Scholar
  125. 125.
    Sato, S.: Evaluation of the First-Passage Time Probability to a Square Root Boundary for the Wiener Process, J. Appl. Probab. 14(4), 850–856 (1977)Google Scholar
  126. 126.
    Sato, S.: Note on the Ornstein–Uhlenbeck process model for stochastic activity of a single neuron. Lect. Note Biomath. 70, 146–156 (1987)Google Scholar
  127. 127.
    Segundo, J., Vibert, J.-F., Pakdaman, K., Stiber, M., Diez, Martinez O.: Noise and the neuroscience: a long history, a recent revival and some theory. In: Pribram, K.H. (eds.) Origins: Brain & Self Organization. Erlbaum, Hillsdale, NJ (1994)Google Scholar
  128. 128.
    Shimokawa, T., Pakdaman, K., Sato, S.: Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise. Phys. Rev. E 59, 3427–3443 (1999)Google Scholar
  129. 129.
    Siegert, A.J.F.: On the first passage time probability problem. Phys. Rev. 81, 617–623 (1951)Google Scholar
  130. 130.
    Sirovich, R.: Mathematical models for the study of synchronization phenomena in neuronal networks, Ph.D. Thesis, University of Torino and Université de Grenoble (2006)Google Scholar
  131. 131.
    Stein, R.B.: A theoretical analysis of neuronal variability, Biophys. J. 5, 385–386 (1965)Google Scholar
  132. 132.
    Taillefumier, T., Magnasco, M.O.: A fast algorithm for the first-passage times of Gauss–Markov processes with Hölder continuous boundaries. J. Stat. Phys. 140, 1130–1156 (2010)Google Scholar
  133. 133.
    Tuckwell, H.C.: Introduction to Theoretical Neurobiology. Linear Cable Theory and Dendritic Structure, vol. 1. Cambridge University Press, Cambridge (1988)Google Scholar
  134. 134.
    Tuckwell, H.C.: Introduction to Theoretical Neurobiology. Nonlinear and Stochastic Theories, vol. 2. Cambridge University Press, Cambridge (1988)Google Scholar
  135. 135.
    Uhlenbeck, G.E., Ornstein, L.S.: On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930)Google Scholar
  136. 136.
    Wang, L., Potzelberger, K.: Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Probab. 34, 54–65 (1997)Google Scholar
  137. 137.
    Zhang, X., You, G., Chen, T., Feng, J.K.: Maximum likelihood decoding of neuronal inputs from an interspike interval distribution. Neural Comput. 19(4), 1319–1346 (2009)Google Scholar
  138. 138.
    Zucca, C., Sacerdote, L.: On the inverse first-passage-time problem for a wiener process. Ann. Appl. Probab. 19(4), 1319–1346 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorinoTorinoItaly

Personalised recommendations