Journal of Computational Neuroscience

, Volume 39, Issue 1, pp 29–51 | Cite as

Statistical structure of neural spiking under non-Poissonian or other non-white stimulation

  • Tilo Schwalger
  • Felix Droste
  • Benjamin Lindner


Nerve cells in the brain generate sequences of action potentials with a complex statistics. Theoretical attempts to understand this statistics were largely limited to the case of a temporally uncorrelated input (Poissonian shot noise) from the neurons in the surrounding network. However, the stimulation from thousands of other neurons has various sorts of temporal structure. Firstly, input spike trains are temporally correlated because their firing rates can carry complex signals and because of cell-intrinsic properties like neural refractoriness, bursting, or adaptation. Secondly, at the connections between neurons, the synapses, usage-dependent changes in the synaptic weight (short-term plasticity) further shape the correlation structure of the effective input to the cell. From the theoretical side, it is poorly understood how these correlated stimuli, so-called colored noise, affect the spike train statistics. In particular, no standard method exists to solve the associated first-passage-time problem for the interspike-interval statistics with an arbitrarily colored noise. Assuming that input fluctuations are weaker than the mean neuronal drive, we derive simple formulas for the essential interspike-interval statistics for a canonical model of a tonically firing neuron subjected to arbitrarily correlated input from the network. We verify our theory by numerical simulations for three paradigmatic situations that lead to input correlations: (i) rate-coded naturalistic stimuli in presynaptic spike trains; (ii) presynaptic refractoriness or bursting; (iii) synaptic short-term plasticity. In all cases, we find severe effects on interval statistics. Our results provide a framework for the interpretation of firing statistics measured in vivo in the brain.


Interspike-interval statistics Stochastic integrate-and-fire neuron Non-renewal process Temporal correlations Spontaneous activity 



Research was supported by the European Research Council (Grant Agreement no. 268689, MultiRules), the BMBF (FKZ: 01GQ1001A), and the research training group GRK1589/1.

Conflict of interests

The authors declare that they have no conflict of interest.


  1. Alijani, A., & Richardson, M. (2011). Rate response of neurons subject to fast or frozen noise: from stochastic and homogeneous to deterministic and heterogeneous populations. Physical Review E, 84, 011,919–1.CrossRefGoogle Scholar
  2. Baddeley, R., Abbott, L.F., Booth, M.C.A., Sengpiel, F., Freeman, T., Wakeman, E.A., & Rolls, E.T. (1997). Responses of neurons in primary and inferior temporal visual cortices to natural scenes. Proceedings of the Royal Society of London B, 264, 1775.CrossRefGoogle Scholar
  3. Badel, L., Lefort, S., Brette, R., Petersen, C.C., Gerstner, W., & Richardson, M.J. (2008). Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. Journal of Neurophysiology, 99(2), 656.PubMedCrossRefGoogle Scholar
  4. Bair, W., Koch, C., Newsome, W., & Britten, K. (1994). Power spectrum analysis of bursting cells in area MT in the behaving monkey. Journal of Neurophysiology, 14, 2870.Google Scholar
  5. Bauermeister, C., Schwalger, T., Russell, D., Neiman, A., & Lindner, B. (2013). Characteristic effects of stochastic oscillatory forcing on neural firing statistics: theory and application to paddlefish electroreceptor afferents. PLoS Computational Biology, 9(8), e1003,170.CrossRefGoogle Scholar
  6. Brenner, N., Agam, O., Bialek, W., & de Ruyter van Steveninck R. (2002). Statistical properties of spike trains: universal and stimulus-dependent aspects. Physical Review E, 66, 031,907.CrossRefGoogle Scholar
  7. Brunel, N. (2000). Sparsely connected networks of spiking neurons. Journal of Computational Neuroscience, 8, 183.PubMedCrossRefGoogle Scholar
  8. Brunel, N., & Sergi, S. (1998). Firing frequency of leaky integrate-and-fire neurons with synaptic currents dynamics. Journal of Theoretical Biology, 195, 87.PubMedCrossRefGoogle Scholar
  9. 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.PubMedCrossRefGoogle Scholar
  10. Bulsara, A., Lowen, S.B., & Rees, C.D. (1994). Cooperative behavior in the periodically modulated Wiener process: noise-induced complexity in a model neuron. Physical Review E, 49, 4989.CrossRefGoogle Scholar
  11. Burkitt, A.N. (2006). A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biological Cybernetics, 95, 1–19.PubMedCrossRefGoogle Scholar
  12. Buzsáki, G., & Draguhn, A. (2004). Neural oscillations in cortical networks. Science, 304, 1926.PubMedCrossRefGoogle Scholar
  13. Câteau, H., & Reyes, A.D. (2006). Relation between single neuron and population spiking statistics and effects on network activity. Physical Review Letters, 96, 058,101.CrossRefGoogle Scholar
  14. Chacron, M.J., Longtin, A., St-Hilaire, M., & Maler, L. (2000). Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors. Physical Review Letters, 85, 1576.PubMedCrossRefGoogle Scholar
  15. Chacron, M.J., Longtin, A., & Maler, L. (2005). Delayed excitatory and inhibitory feedback shape neural information transmission. Physical Review E, 72, 051,917.CrossRefGoogle Scholar
  16. Compte, A., Constantinidis, C., Tegner, J., Raghavachari, S., Chafee, M.V., Goldman-Rakic, P.S., & Wang, X.J. (2003). Temporally irregular mnemonic persistent activity in prefrontal neurons of monkeys during a delayed response task. Journal of Neurophysiology, 90(5), 3441.PubMedCrossRefGoogle Scholar
  17. Cox, D.R., & Lewis, P.A.W. (1966a). The statistical analysis of series of events. London: Chapman and Hall.CrossRefGoogle Scholar
  18. Cox, D.R., & Lewis, P.A.W. (1966b). The statistical analysis of series of events. London: Chapman and Hall. chap 4.6.CrossRefGoogle Scholar
  19. Cox, D.R., & Miller, H.D. (1965). The theory of stochastic processes: Chapman and Hall.Google Scholar
  20. Destexhe, A., Rudolph, M., & Paré, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews Neuroscience, 4(9), 739–751.PubMedCrossRefGoogle Scholar
  21. Dittman, J.S., Kreitzer, A.C., & Regehr, W.G. (2000). Interplay between facilitation, depression, and residual calcium at three presynaptic terminals. Journal of Neuroscience, 20, 1374.PubMedGoogle Scholar
  22. Droste, F., & Lindner, B. (2014). Integrate-and-fire neurons driven by asymmetric dichotomous noise. Biological Cybernetics, 108(6), 825–843.PubMedCrossRefGoogle Scholar
  23. Droste, F., Schwalger, T., & Lindner, B. (2013). Interplay of two signals in a neuron with heterogeneous synaptic short-term plasticity. Frontiers in Computational Neuroscience, 7, 86.PubMedCentralPubMedCrossRefGoogle Scholar
  24. Dummer, B., Wieland, S., & Lindner, B. (2014). Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity. Front Comp Neurosci, 8, 104.Google Scholar
  25. Ermentrout, G.B., & Terman, D.H. (2010). Mathematical foundations of neuroscience: Springer.Google Scholar
  26. Fisch, K., Schwalger, T., Lindner, B., Herz, A., & Benda, J. (2012). Channel noise from both slow adaptation currents and fast currents is required to explain spike-response variability in a sensory neuron. Journal of Neuroscience, 344(48), 17,332– 17.Google Scholar
  27. Fortune, E.S., & Rose, G.J. (2001). Short-term synaptic plasticity as a temporal filter. Trends in Neurosciences, 24, 381.PubMedCrossRefGoogle Scholar
  28. Fourcaud, N., & Brunel, N. (2002). Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Computation, 14, 2057.PubMedCrossRefGoogle Scholar
  29. Fourcaud-Trocmé, N., Hansel, D., van Vreeswijk, C., & Brunel, N. (2003). How spike generation mechanisms determine the neuronal response to fluctuating inputs. Journal of Neuroscience, 640(37), 11,628–11.Google Scholar
  30. Franklin, J., & Bair, W. (1995). The effect of a refractory period on the power spectrum of neuronal discharge. SIAM Journal on Applied Mathematics, 55, 1074.CrossRefGoogle Scholar
  31. Gerstein, G.L., & Mandelbrot, B. (1964). Random walk models for the spike activity of a single neuron. Biophysical Journal, 4, 41.PubMedCentralPubMedCrossRefGoogle Scholar
  32. Gerstner, W., & Kistler, W.M. (2002). Spiking neuron models: single neurons, populations, plasticity. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  33. Hänggi, P., & Jung, P. (1995). Colored noise in dynamical systems. Advances in Chemical Physics, 89, 239.Google Scholar
  34. Holden, A.V. (1976). Models of the stochastic activity of neurones. Berlin: Springer.CrossRefGoogle Scholar
  35. Kamke, E. (1965). Differentialgleichungen, Lösungsmethoden und Lösungen II: partielle Differentialgleichungen erster Ordnung für eine gesuchte Funktion. Leipzig: Geest & Portig.Google Scholar
  36. van Kampen, N.G. (1992). Stochastic processes in physics and chemistry. North-Holland, Amsterdam.Google Scholar
  37. Koch, C. (1999). Biophysics of computation: information processing in single neurons: Oxford University Press.Google Scholar
  38. Lerchner, A., Ursta, C., Hertz, J., Ahmadi, M., Ruffiot, P., & Enemark, S. (2006). Response variability in balanced cortical networks. Neural Compution, 18(3), 634.Google Scholar
  39. Lindner, B (2004). Interspike interval statistics of neurons driven by colored noise. Physical Review E, 69, 022,901.Google Scholar
  40. Lindner, B. (2006). Superposition of many independent spike trains is generally not a Poisson process. Physical Review E, 73, 022,901.CrossRefGoogle Scholar
  41. Lindner, B., Gangloff, D., Longtin, A., & Lewis, J.E. (2009). Broadband coding with dynamic synapses. Journal of Neuroscience, 29(7), 2076–2088.PubMedCrossRefGoogle Scholar
  42. Liu, Y.H., & Wang, X.J. (2001). Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron. Journal of Computational Neuroscience, 10, 25.PubMedCrossRefGoogle Scholar
  43. London, M., Roth, A., Beeren, L., Häusser, M., & Latham, P.E. (2010). Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature, 466(7302), 123.PubMedCentralPubMedCrossRefGoogle Scholar
  44. Lowen, S.B., & Teich, M.C. (1992). Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales. Journal of the Acoustical Society of America, 92, 803.PubMedCrossRefGoogle Scholar
  45. Merkel, M., & Lindner, B. (2010). Synaptic filtering of rate-coded information. Physical Review E, 921(4 Pt 1), 041,921–041.CrossRefGoogle Scholar
  46. Middleton, J.W., Chacron, M.J., Lindner, B., & Longtin, A. (2003). Firing statistics of a neuron model driven by long-range correlated noise. Physical Review E, 68, 021,920.CrossRefGoogle Scholar
  47. Moreno-Bote, R., & Parga, N. (2004). Role of synaptic filtering on the firing response of simple model neurons. Physical Review Letters, 92(2), 028102.PubMedCrossRefGoogle Scholar
  48. Moreno-Bote, R., & Parga, N. (2006). Auto- and crosscorrelograms for the spike response of leaky integrate-and-fire neurons with slow synapses. Physical Review Letters, 96(2), 028,101.  10.1103/PhysRevLett.96.028101.CrossRefGoogle Scholar
  49. Moreno-Bote, R., Beck, J., Kanitscheider, I., Pitkow, X., Latham, P., & Pouget, A. (2014). Information-limiting correlations. Nature Neuroscience, 17(10), 1410.PubMedCentralPubMedCrossRefGoogle Scholar
  50. Nawrot, M.P., Boucsein, C., Rodriguez-Molina, V., Aertsen, A., Grun, S., & Rotter, S. (2007). Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro. Neurocomp, 70, 1717.CrossRefGoogle Scholar
  51. Peterson, A.J., Irvine, D.R.F., & Heil, P. (2014). A model of synaptic vesicle-pool depletion and replenishment can account for the interspike interval distributions and nonrenewal properties of spontaneous spike trains of auditory-nerve fibers. Journal of Neuroscience, 34(45), 15,097.CrossRefGoogle Scholar
  52. Pozzorini, C., Naud, R., Mensi, S., & Gerstner, W. (2013). Temporal whitening by power-law adaptation in neocortical neurons. Nature Neuroscience, 16(7), 942–948.PubMedCrossRefGoogle Scholar
  53. Richardson, M.J.E. (2008). Spike-train spectra and network response functions for non-linear integrate-and-fire neurons. Biological Cybernetics, 99(4–5), 381.PubMedCrossRefGoogle Scholar
  54. Risken, H. (1984). The Fokker-Planck equation. Berlin: Springer.CrossRefGoogle Scholar
  55. Rosenbaum, R., Rubin, J., & Doiron, B. (2012). Short term synaptic depression imposes a frequency dependent filter on synaptic information transfer. PLoS Computational Biology, 8(6).Google Scholar
  56. Salinas, E., & Sejnowski, T.J. (2002). Integrate-and-fire neurons driven by correlated stochastic input. Neural Compution, 14, 2111.CrossRefGoogle Scholar
  57. Schwalger, T., & Lindner, B. (2013). Patterns of interval correlations in neural oscillators with adaptation. Frontiers in Computational Neuroscience, 7(164).Google Scholar
  58. Schwalger, T., & Schimansky-Geier, L. (2008). Interspike interval statistics of a leaky integrate-and-fire neuron driven by Gaussian noise with large correlation times. Physical Review E, 77, 031,914–9.CrossRefGoogle Scholar
  59. Schwalger, T., Fisch, K., Benda, J., & Lindner, B. (2010). How noisy adaptation of neurons shapes interspike interval histograms and correlations. PLoS Computational Biology, 6(12), e1001,026. doi: 10.1371/journal.pcbi.1001026.CrossRefGoogle Scholar
  60. Schwalger, T., Miklody, D., & Lindner, B. (2013). When the leak is weak – how the first-passage statistics of a biased random walk can approximate the ISI statistics of an adapting neuron. European Physical Journal Spec Topics, 222(10), 2655.CrossRefGoogle Scholar
  61. Sobie, C., Babul, A., & de Sousa R. (2011). Neuron dynamics in the presence of 1/f noise. Physical Review E, 83(5), 051,912.CrossRefGoogle Scholar
  62. Stratonovich, R.L. (1967). Topics in the theory of random noise, vol 1. New York: Gordon and Breach.Google Scholar
  63. Wang, X.J. (1998). Calcium coding and adaptive temporal computation in cortical pyramidal neurons. Journal of Neurophysiology, 79(3), 1549–1566.PubMedGoogle Scholar
  64. Wang, X.J., Liu, Y., Sanchez-Vives, M.V., & McCormick, D.A. (2003). Adaptation and temporal decorrelation by single neurons in the primary visual cortex. Journal of Neurophysiology, 89(6), 3279–3293.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Tilo Schwalger
    • 1
    • 2
  • Felix Droste
    • 2
    • 3
  • Benjamin Lindner
    • 2
    • 3
  1. 1.Brain Mind InstituteÉcole Polytechnique Féderale de Lausanne (EPFL) Station 15LausanneSwitzerland
  2. 2.Bernstein Center for Computational NeuroscienceBerlinGermany
  3. 3.Department of PhysicsHumboldt Universität zu BerlinBerlinGermany

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