Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

An integrate-and-fire model to generate spike trains with long-range dependence

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. A limitation of the point process approach is that it is far from the biological reality.

References

  • Abry, P., Gonçalvès, P., Flandrin, P. (1995). Wavelets, spectrum analysis and 1/f processes, (pp. 15–29). New York: Springer.

    Google Scholar 

  • Baddeley, R., Abbott, L.F., Booth, M.C., Sengpiel, F., Freeman, T., Wakeman, E.A., Rolls, E.T. (1997). Responses of neurons in primary and inferior temporal visual cortices to natural scenes, (Vol. 264 pp. 1775–1783).

  • Bair, W., Koch, C., Newsome, W., Britten, K. (1994). Power spectrum analysis of bursting cells in area mt in the behaving monkey. Journal of Neuroscience, 14(5), 2870–2892.

    Article  PubMed  CAS  Google Scholar 

  • Beran, J., Feng, Y., Ghosh, S., Kulik, R. (2013). Long-memory processes. Heidelberg: Springer. Probabilistic properties and statistical methods.

    Book  Google Scholar 

  • Bhattacharya, R.N., Gupta, V.K., Waymire, E. (1983). The Hurst effect under trends. Journal of Applied Probability, 20(3), 649–662.

    Article  Google Scholar 

  • Bhattacharya, J., Edwards, J., Mamelak, A., Schuman, E. (2005). Long-range temporal correlations in the spontaneous spiking of neurons in the hippocampal-amygdala complex of humans. Neuroscience, 131(2), 547–555.

    Article  PubMed  CAS  Google Scholar 

  • Brunel, N., & Sergi, S. (1998). Firing frequency of leaky intergrate-and-fire neurons with synaptic current dynamics. Journal of Theoretical Biology, 195(1), 87–95.

    Article  PubMed  CAS  Google Scholar 

  • Cardinali, A., & Nason, G.P. (2010). Costationarity of locally stationary time series. Journal of Time Series Econometrics, 2(2), Article 1.

    Google Scholar 

  • Cardinali, A., & Nason, G.P. (2018). Practical powerful wavelet packet tests for second-order stationarity. Applied and Computational Harmonic Analysis 44(3), 558–583.

  • Carmona, P., Coutin, L., Montseny, G. (2000). Approximation of some Gaussian processes. Statistical Inference for Stochastic Processes, 3(1-2), 161–171. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998).

    Article  Google Scholar 

  • Chacron, M.J., Pakdaman, K., Longtin, A. (2003). Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue. Neural Computation, 15(2), 253–278.

    Article  PubMed  Google Scholar 

  • Churilla, A.M., Gottschalke, W.A., Liebovitch, L.S., Selector, L.Y., Todorov, A.T., Yeandle, S. (1995). Membrane potential fluctuations of human t-lymphocytes have fractal characteristics of fractional brownian motion. Annals of Biomedical Engineering, 24(1), 99–108.

    Article  Google Scholar 

  • Coeurjolly, J.-F. (2000). Simulation and identification of the fractional brownian motion: a bibliographical and comparative study. Journal of Statistical Software, 5(1), 1–53.

    Google Scholar 

  • de Oliveira, R.C., Barbosa, C., Consoni, L., Rodrigues, A., Varanda, W., Nogueira, R. (2006). Long-term correlation in single calcium-activated potassium channel kinetics. Physica A: Statistical Mechanics and its Applications, 364, 13–22.

    Article  CAS  Google Scholar 

  • Decreusefond, L., & Nualart, D. (2008). Hitting times for Gaussian processes. The Annals of Probability, 36 (1), 319–330.

    Article  Google Scholar 

  • Delorme, M., & Wiese, K.J. (2015). Maximum of a fractional Brownian motion: analytic results from perturbation theory. Physical Review Letters, 115(21), 210601, 5.

    Article  CAS  Google Scholar 

  • Destexhe, A., Rudolph, M., Paré, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews Neuroscience, 4(9), 739–751.

    Article  PubMed  CAS  Google Scholar 

  • Doukhan, P. (1994). Mixing, properties and examples, volume 85 of lecture notes in statistics. New York: Springer.

    Google Scholar 

  • Drew, P.J., & Abbott, L.F. (2006). Models and properties of power-law adaptation in neural systems. Journal of Neurophysiology, 96(2), 826–833.

    Article  PubMed  Google Scholar 

  • Fairhall, A.L., Lewen, G.D., Bialek, W., de Ruyter van Steveninck, R.R. (2001). Efficiency and ambiguity in an adaptive neural code. Nature, 412, 787–792.

    Article  PubMed  CAS  Google Scholar 

  • Gerstein, G.L., & Mandelbrot, B. (1964). Random walk models for the spike activity of a single neuron. Biophysical Journal, 4(1), 41–68.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Hammond, A., & Sheffield, S. (2013). Power law Pólya’s urn and fractional Brownian motion. Probability Theory and Related Fields, 157(3–4), 691–719.

    Article  Google Scholar 

  • Hodgkin, A.L., & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500–544.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Jackson, B.S. (2004). Including long-range dependence in integrate-and-fire models of the high interspike-interval variability of cortical neurons. Neural Computation, 16(10), 2125–2195.

    Article  PubMed  Google Scholar 

  • Kwiatkowski, D., Phillips, P.C., Schmidt, P., Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54(1), 159–178.

    Article  Google Scholar 

  • Lewis, C.D., Gebber, G.L., Larsen, P.D., Barman, S.M. (2001). Long-term correlations in the spike trains of medullary sympathetic neurons. Journal of Neurophysiology, 85(4), 1614–1622.

    Article  PubMed  CAS  Google Scholar 

  • Lindner, B. (2004). Interspike interval statistics of neurons driven by colored noise. Physical Review E, 69, 022901.

    Article  CAS  Google Scholar 

  • Lowen, S.B., Cash, S.S., Poo, M.-m., Teich, M.C. (1997). Quantal neurotransmitter secretion rate exhibits fractal behavior. Journal of Neuroscience, 17(15), 5666–5677.

    Article  PubMed  CAS  Google Scholar 

  • Lowen, S.B., Ozaki, T., Kaplan, E., Saleh, B.E., Teich, M.C. (2001). Fractal features of dark, maintained, and driven neural discharges in the cat visual system. Methods, 24(4), 377–394.

    Article  PubMed  CAS  Google Scholar 

  • Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L. (2008). Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience, 11(11), 1335–1342.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Mandelbrot, B.B. (1965). Une classe de processus stochastiques homothétiques à soi; application à la loi climatologique de H. E. Hurst. C. R. Acad. Sci. Paris, 260, 3274–3277.

    Google Scholar 

  • Mandelbrot, B.B. (1975). Limit theorems on the self-normalized range for weakly and strongly dependent processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31, 271–285.

    Article  Google Scholar 

  • Mandelbrot, B.B., & Wallis, J.R. (1968). Noah, Joseph, and operational hydrology. Water Resources Research, 4(5), 909–918.

    Article  Google Scholar 

  • Mandelbrot, B.B., & Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review, 10(4), 422–437.

    Article  Google Scholar 

  • Metzler, R., & Klafter, J. (2004). The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics. A. Mathematical and General, 37 (31), R161–R208.

    Article  Google Scholar 

  • 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, 021920.

    Article  CAS  Google Scholar 

  • Nason, G. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. Journal of the Royal Statistical Society. Series B. Statistical Methodology, 75(5), 879–904.

    Article  Google Scholar 

  • Peng, C.K., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Sciortino, F., Simon, M., Stanley, H.E. (1992). Long-range correlations in nucleotide sequences. Nature, 356, 168–170.

    Article  PubMed  CAS  Google Scholar 

  • Peng, C.-K., Mietus, J., Hausdorff, J.M., Havlin, S., Stanley, H.E., Goldberger, A.L. (1993). Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Physical Review Letters, 70, 1343–1346.

    Article  Google Scholar 

  • Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L. (1994). Mosaic organization of DNA nucleotides. Physical Review E, 49, 1685–1689.

    Article  CAS  Google Scholar 

  • Pozzorini, C., Naud, R., Mensi, S., Gerstner, W. (2013). Temporal whitening by power-law adaptation in neocortical neurons. Nature neuroscience, 16(7), 942–948.

    Article  PubMed  CAS  Google Scholar 

  • Priestley, M.B., & Subba Rao, T. (1969). A test for non-stationarity of time-series. Journal of the Royal Statistical Society. Series B. Methodological, 31, 140–149.

    Google Scholar 

  • Rangarajan, G., & Ding, M. (Eds.). (2003). Processes with long-range correlations: theory and applications, volume 621 of Lecture Notes in Physics. Berlin: Springer.

  • Richard, A., & Talay, D. (2016). Hölder continuity in the Hurst parameter of functionals of Stochastic Differential Equations driven by fractional Brownian motion. arXiv:1605.03475.

  • Sacerdote, L., & Giraudo, M.T. (2013). Stochastic integrate and fire models: a review on mathematical methods and their applications. In Stochastic biomathematical models, volume 2058 of Lecture Notes in Math. (pp. 99–148). Heidelberg: Springer.

  • Samorodnitsky, G. (2016). Stochastic processes and long range dependence. Springer Series in Operations Research and Financial Engineering. Cham: Springer.

    Book  Google Scholar 

  • 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, 031914.

    Article  CAS  Google Scholar 

  • 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), e1001026, 25.

    Article  CAS  Google Scholar 

  • Schwalger, T., Droste, F., Lindner, B. (2015). Statistical structure of neural spiking under non-Poissonian or other non-white stimulation. Journal of Computational Neuroscience, 39(1), 29–51.

    Article  PubMed  Google Scholar 

  • Segev, R., Benveniste, M., Hulata, E., Cohen, N., Palevski, A., Kapon, E., Shapira, Y., Ben-Jacob, E. (2002). Long term behavior of lithographically prepared in vitro neuronal networks. Physical Review Letters, 88, 118102.

    Article  PubMed  CAS  Google Scholar 

  • Sobie, C., Babul, A., de Sousa, R. (2011). Neuron dynamics in the presence of 1/f noise. Physical Review E, 83, 051912.

    Article  CAS  Google Scholar 

  • Sottinen, T. (2001). Fractional Brownian motion, random walks and binary market models. Finance and Stochastics, 5(3), 343–355.

    Article  Google Scholar 

  • Taqqu, M.S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31, 287–302.

    Article  Google Scholar 

  • Taqqu, M.S., Teverovsky, V., Willinger, W. (1995). Estimators for long-range dependence: an empirical study. Fractals, 03(04), 785–798.

    Article  Google Scholar 

  • Teich, M.C. (1992). Fractal neuronal firing patterns. In McKenna, T., Davis, J., Zornetzer, S. F. (Eds.), Single Neuron Computation, Neural Networks: Foundations to Applications (pp. 589–625). San Diego: Academic Press.

  • Teich, M.C., Turcott, R.G., Siegel, R.M. (1996). Temporal correlation in cat striate-cortex neural spike trains. IEEE Engineering in Medicine and Biology Magazine, 15(5), 79–87.

    Article  Google Scholar 

  • Teich, M.C., Heneghan, C., Lowen, S.B., Ozaki, T., Kaplan, E. (1997). Fractal character of the neural spike train in the visual system of the cat. Journal of the Optical Society of America A, 14(3), 529–546.

    Article  CAS  Google Scholar 

  • Teka, W., Marinov, T.M., Santamaria, F. (2014). Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Computational Biology, 10(3), e1003526.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Weron, R. (2002). Estimating long-range dependence: finite sample properties and confidence intervals. Physica A. Statistical Mechanics and its Applications, 312(1-2), 285–299.

    Article  Google Scholar 

  • Willinger, W., Taqqu, M.S., Sherman, R., Wilson, D.V. (1997). Self-similarity through high-variability: statistical analysis of ethernet lan traffic at the source level. IEEE/ACM Transactions on Networking, 5(1), 71–86.

    Article  Google Scholar 

  • Zilany, M.S., Bruce, I.C., Nelson, P.C., Carney, L.H. (2009). A phenomenological model of the synapse between the inner hair cell and auditory nerve: long-term adaptation with power-law dynamics. The Journal of the Acoustical Society of America, 126(5), 390–2412.

    Article  Google Scholar 

Download references

Acknowledgements

Part of this work was carried out while A.R. was a postdoc at Inria Sophia-Antipolis and at Ecole Polytechnique (the support from ERC 321111 Rofirm is gratefully acknowledged). A.R. and E.T. acknowledge the support from the ECOS-Sud Program Chili-France C15E05 and from the European Union’s Horizon 2020 Framework Program for Research and Innovation under Grant Agreement No. 720270 (Human Brain Project SGA1). P.O acknowledges the support from the Advanced Center for Electrical and Electronic Engineering (Basal Funding FB0008, Conicyt) and the project P09-022-F from the Millennium Scientific Initiative of the Chilean Ministry of Economy, Development, and Tourism.

We thank the reviewers for their remarks which helped to improve significantly the quality of this paper.

Author information

Authors and Affiliations

  1. CentraleSupélec, Université Paris-Saclay, Laboratoire MICS et Fédération CNRS - FR3487, Gif-sur-Yvette, France

    Alexandre Richard

  2. Instituto de Neurociencia, Facultad de Ciencias, Universidad de Valparaíso, Valparaiso, Chile

    Patricio Orio

  3. Centro Interdisciplinario de Neurociencia de Valparaíso, Universidad de Valparaíso, Valparaiso, Chile

    Patricio Orio

  4. Université Côte d’Azur, Inria, 2004 Route des Lucioles BP 93, 06902, Sophia-Antipolis, France

    Etienne Tanré

Authors
  1. Alexandre Richard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patricio Orio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Etienne Tanré
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre Richard.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Action Editor: A. Compte

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Richard, A., Orio, P. & Tanré, E. An integrate-and-fire model to generate spike trains with long-range dependence. J Comput Neurosci 44, 297–312 (2018). https://doi.org/10.1007/s10827-018-0680-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-018-0680-1

Keywords

Search

Navigation