×

zbMATH — the first resource for mathematics

An integrate-and-fire model to generate spike trains with long-range dependence. (English) Zbl 1402.92110
Summary: 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.

MSC:
92C20 Neural biology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abry, P., Gonçalvès, P., Flandrin, P. (1995). Wavelets, spectrum analysis and 1/f processes, (pp. 15-29). New York: Springer. · Zbl 0828.62083
[2] 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).
[3] Bair, W; Koch, C; Newsome, W; Britten, K, Power spectrum analysis of bursting cells in area mt in the behaving monkey, Journal of Neuroscience, 14, 2870-2892, (1994)
[4] Beran, J., Feng, Y., Ghosh, S., Kulik, R. (2013). Long-memory processes. Heidelberg: Springer. Probabilistic properties and statistical methods. · Zbl 1282.62187
[5] Bhattacharya, RN; Gupta, VK; Waymire, E, The Hurst effect under trends, Journal of Applied Probability, 20, 649-662, (1983) · Zbl 0526.60027
[6] Bhattacharya, J; Edwards, J; Mamelak, A; Schuman, E, Long-range temporal correlations in the spontaneous spiking of neurons in the hippocampal-amygdala complex of humans, Neuroscience, 131, 547-555, (2005)
[7] Brunel, N; Sergi, S, Firing frequency of leaky intergrate-and-fire neurons with synaptic current dynamics, Journal of Theoretical Biology, 195, 87-95, (1998)
[8] Cardinali, A; Nason, GP, Costationarity of locally stationary time series, Journal of Time Series Econometrics, 2, article 1, (2010) · Zbl 1266.91064
[9] Cardinali, A., & Nason, G.P. (2018). Practical powerful wavelet packet tests for second-order stationarity. Applied and Computational Harmonic Analysis44(3), 558-583. · Zbl 1387.62099
[10] Carmona, P; Coutin, L; Montseny, G, Approximation of some Gaussian processes, Statistical Inference for Stochastic Processes, 3, 161-171, (2000) · Zbl 0986.60036
[11] Chacron, MJ; Pakdaman, K; Longtin, A, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue, Neural Computation, 15, 253-278, (2003) · Zbl 1020.92007
[12] Churilla, AM; Gottschalke, WA; Liebovitch, LS; Selector, LY; Todorov, AT; Yeandle, S, Membrane potential fluctuations of human t-lymphocytes have fractal characteristics of fractional Brownian motion, Annals of Biomedical Engineering, 24, 99-108, (1995)
[13] Coeurjolly, J-F, Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study, Journal of Statistical Software, 5, 1-53, (2000)
[14] Oliveira, RC; Barbosa, C; Consoni, L; Rodrigues, A; Varanda, W; Nogueira, R, Long-term correlation in single calcium-activated potassium channel kinetics, Physica A: Statistical Mechanics and its Applications, 364, 13-22, (2006)
[15] Decreusefond, L; Nualart, D, Hitting times for Gaussian processes, The Annals of Probability, 36, 319-330, (2008) · Zbl 1135.60019
[16] Delorme, M; Wiese, KJ, Maximum of a fractional Brownian motion: analytic results from perturbation theory, Physical Review Letters, 115, 210601, 5, (2015)
[17] Destexhe, A; Rudolph, M; Paré, D, The high-conductance state of neocortical neurons in vivo, Nature Reviews Neuroscience, 4, 739-751, (2003)
[18] Doukhan, P. (1994). Mixing, properties and examples, volume 85 of lecture notes in statistics. New York: Springer.
[19] Drew, PJ; Abbott, LF, Models and properties of power-law adaptation in neural systems, Journal of Neurophysiology, 96, 826-833, (2006)
[20] Fairhall, AL; Lewen, GD; Bialek, W; Ruyter van Steveninck, RR, Efficiency and ambiguity in an adaptive neural code, Nature, 412, 787-792, (2001)
[21] Gerstein, GL; Mandelbrot, B, Random walk models for the spike activity of a single neuron, Biophysical Journal, 4, 41-68, (1964)
[22] Hammond, A; Sheffield, S, Power law Pólya’s urn and fractional Brownian motion, Probability Theory and Related Fields, 157, 691-719, (2013) · Zbl 1311.60044
[23] Hodgkin, AL; Huxley, AF, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117, 500-544, (1952)
[24] Jackson, BS, Including long-range dependence in integrate-and-fire models of the high interspike-interval variability of cortical neurons, Neural Computation, 16, 2125-2195, (2004) · Zbl 1061.92021
[25] Kwiatkowski, D; Phillips, PC; Schmidt, P; Shin, Y, Testing the null hypothesis of stationarity against the alternative of a unit root, Journal of Econometrics, 54, 159-178, (1992) · Zbl 0871.62100
[26] Lewis, CD; Gebber, GL; Larsen, PD; Barman, SM, Long-term correlations in the spike trains of medullary sympathetic neurons, Journal of Neurophysiology, 85, 1614-1622, (2001)
[27] Lindner, B, Interspike interval statistics of neurons driven by colored noise, Physical Review E, 69, 022901, (2004)
[28] Lowen, SB; Cash, SS; Poo, M-m; Teich, MC, Quantal neurotransmitter secretion rate exhibits fractal behavior, Journal of Neuroscience, 17, 5666-5677, (1997)
[29] Lowen, SB; Ozaki, T; Kaplan, E; Saleh, BE; Teich, MC, Fractal features of dark, maintained, and driven neural discharges in the cat visual system, Methods, 24, 377-394, (2001)
[30] Lundstrom, BN; Higgs, MH; Spain, WJ; Fairhall, AL, Fractional differentiation by neocortical pyramidal neurons, Nature Neuroscience, 11, 1335-1342, (2008)
[31] Mandelbrot, BB, 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, (1965) · Zbl 0127.09501
[32] Mandelbrot, BB, Limit theorems on the self-normalized range for weakly and strongly dependent processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31, 271-285, (1975) · Zbl 0288.60033
[33] Mandelbrot, BB; Wallis, JR, Noah, Joseph, and operational hydrology, Water Resources Research, 4, 909-918, (1968)
[34] Mandelbrot, BB; Ness, JW, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10, 422-437, (1968) · Zbl 0179.47801
[35] Metzler, R; Klafter, J, 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, r161-r208, (2004) · Zbl 1075.82018
[36] Middleton, JW; Chacron, MJ; Lindner, B; Longtin, A, Firing statistics of a neuron model driven by long-range correlated noise, Physical Review E, 68, 021920, (2003)
[37] Nason, G, 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, 879-904, (2013) · Zbl 1411.62259
[38] Peng, CK; Buldyrev, SV; Goldberger, AL; Havlin, S; Sciortino, F; Simon, M; Stanley, HE, Long-range correlations in nucleotide sequences, Nature, 356, 168-170, (1992)
[39] Peng, C-K; Mietus, J; Hausdorff, JM; Havlin, S; Stanley, HE; Goldberger, AL, Long-range anticorrelations and non-Gaussian behavior of the heartbeat, Physical Review Letters, 70, 1343-1346, (1993)
[40] Peng, C-K; Buldyrev, SV; Havlin, S; Simons, M; Stanley, HE; Goldberger, AL, Mosaic organization of DNA nucleotides, Physical Review E, 49, 1685-1689, (1994)
[41] Pozzorini, C; Naud, R; Mensi, S; Gerstner, W, Temporal whitening by power-law adaptation in neocortical neurons, Nature neuroscience, 16, 942-948, (2013)
[42] Priestley, MB; Subba Rao, T, A test for non-stationarity of time-series, Journal of the Royal Statistical Society. Series B. Methodological, 31, 140-149, (1969) · Zbl 0182.51403
[43] Rangarajan, G., & Ding, M. (Eds.). (2003). Processes with long-range correlations: theory and applications, volume 621 of Lecture Notes in Physics. Berlin: Springer.
[44] 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.
[45] 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. · Zbl 1390.92032
[46] Samorodnitsky, G. (2016). Stochastic processes and long range dependence. Springer Series in Operations Research and Financial Engineering. Cham: Springer. · Zbl 1376.60007
[47] Schwalger, T; Schimansky-Geier, L, Interspike interval statistics of a leaky integrate-and-fire neuron driven by Gaussian noise with large correlation times, Physical Review E, 77 , 031914, (2008)
[48] Schwalger, T; Fisch, K; Benda, J; Lindner, B, How noisy adaptation of neurons shapes interspike interval histograms and correlations, PLoS Computational Biology, 6, e1001026, 25, (2010)
[49] Schwalger, T; Droste, F; Lindner, B, Statistical structure of neural spiking under non-Poissonian or other non-white stimulation, Journal of Computational Neuroscience, 39, 29-51, (2015) · Zbl 1382.92085
[50] Segev, R; Benveniste, M; Hulata, E; Cohen, N; Palevski, A; Kapon, E; Shapira, Y; Ben-Jacob, E, Long term behavior of lithographically prepared in vitro neuronal networks, Physical Review Letters, 88, 118102, (2002)
[51] Sobie, C; Babul, A; Sousa, R, Neuron dynamics in the presence of 1/f noise, Physical Review E, 83, 051912, (2011)
[52] Sottinen, T, Fractional Brownian motion, random walks and binary market models., Finance and Stochastics, 5, 343-355, (2001) · Zbl 0978.91037
[53] Taqqu, MS, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31, 287-302, (1975) · Zbl 0303.60033
[54] Taqqu, MS; Teverovsky, V; Willinger, W, Estimators for long-range dependence: an empirical study, Fractals, 03, 785-798, (1995) · Zbl 0864.62061
[55] 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.
[56] Teich, MC; Turcott, RG; Siegel, RM, Temporal correlation in cat striate-cortex neural spike trains, IEEE Engineering in Medicine and Biology Magazine, 15, 79-87, (1996)
[57] Teich, MC; Heneghan, C; Lowen, SB; Ozaki, T; Kaplan, E, Fractal character of the neural spike train in the visual system of the cat, Journal of the Optical Society of America A, 14, 529-546, (1997)
[58] Teka, W; Marinov, TM; Santamaria, F, Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model, PLoS Computational Biology, 10, e1003526, (2014)
[59] Weron, R, Estimating long-range dependence: finite sample properties and confidence intervals, Physica A. Statistical Mechanics and its Applications, 312, 285-299, (2002) · Zbl 0997.91030
[60] Willinger, W; Taqqu, MS; Sherman, R; Wilson, DV, Self-similarity through high-variability: statistical analysis of Ethernet lan traffic at the source level, IEEE/ACM Transactions on Networking, 5, 71-86, (1997)
[61] Zilany, MS; Bruce, IC; Nelson, PC; Carney, LH, 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, 390-2412, (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.