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2016, 13(3): 537-550. doi: 10.3934/mbe.2016006

## Fluctuation scaling in neural spike trains

 1 Department of Statistical Modeling, The Institute of Statistical Mathematics, 10-3 Midoricho, Tachikawa, Tokyo, Japan 2 Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan

Received  March 2015 Revised  November 2015 Published  January 2016

Fluctuation scaling has been observed universally in a wide variety of phenomena. In time series that describe sequences of events, fluctuation scaling is expressed as power function relationships between the mean and variance of either inter-event intervals or counting statistics, depending on measurement variables. In this article, fluctuation scaling has been formulated for a series of events in which scaling laws in the inter-event intervals and counting statistics were related. We have considered the first-passage time of an Ornstein-Uhlenbeck process and used a conductance-based neuron model with excitatory and inhibitory synaptic inputs to demonstrate the emergence of fluctuation scaling with various exponents, depending on the input regimes and the ratio between excitation and inhibition. Furthermore, we have discussed the possible implication of these results in the context of neural coding.
Citation: Shinsuke Koyama, Ryota Kobayashi. Fluctuation scaling in neural spike trains. Mathematical Biosciences & Engineering, 2016, 13 (3) : 537-550. doi: 10.3934/mbe.2016006
##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Dover, (1965). [2] R. M. Anderson and R. M. May, Epidemiological parameters of HIV transmission,, Nature, 333 (1988), 514. [3] B. B. Averbeck, Poisson or not Poisson: Differences in spike train statistics between parietal cortical areas,, Neuron, 62 (2009), 310. doi: 10.1016/j.neuron.2009.04.021. [4] P. Bremaud, Point Processes and Queues,, Springer, (1981). [5] A. N. Burkitt, Balanced neurons: Analysis of leaky integrate-and-fire neurons with reversal potentials,, Biol. Cybern., 85 (2001), 247. doi: 10.1007/s004220100262. [6] A. N. Burkitt, A review of the integrate-and-fire neuron model, I: Homogeneous synaptic input,, Biol. Cybern., 95 (2006), 1. doi: 10.1007/s00422-006-0068-6. [7] M. M. Churchland, et al., Stimulus onset quenches neural variability: A widespread cortical phenomenon,, Nat. Neurosci., 13 (2010), 369. [8] A. K. Churchland, et al., Variance as a signature of neural computations during decision making,, Neuron, 69 (2011), 818. [9] M. M. Churchland and L. F. Abbott, Two layers of neural variability,, Nat. Neurosci., 15 (2012), 1472. [10] D. R. Cox, Renewal Theory,, Chapman and Hall, (1962). [11] D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events,, Chapman and Hall, (1966). [12] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,, Springer Series in Statistics, (1988). [13] M. A. de Menezes and A. L. Barabasi, Fluctuations in network dynamics,, Phys. Rev. Lett., 92 (2004). [14] A. Destexhe, Z. Mainen and T. J. Sejnowski, Kinetic models of synaptic transmission,, in Methods in Neuronal Modeling (eds. C. Koch and I. Segev), (1998), 1. [15] S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model,, Phys. Rev. E., 71 (2005). doi: 10.1103/PhysRevE.71.011907. [16] Z. Eisler, I. Bartos and J. Kertesz, Fluctuation scaling in complex systems: Taylor's law and beyond,, Adv. Phys., 57 (2008), 89. doi: 10.1080/00018730801893043. [17] A. Fronczak and P. Fronczak, Origins of Taylor's power law for fluctuation scaling in complex systems,, Phys. Rev. E, 81 (2010). doi: 10.1103/PhysRevE.81.066112. [18] J. Inoue, S. Sato and L. M. Ricciardi, 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 (1995), 209. [19] D. H. Johnson, Point process models of single-neuron discharges,, J. Comput. Neurosci., 3 (1996), 275. doi: 10.1007/BF00161089. [20] J. Keilson and H. F. Ross, Passage time distribution for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes,, in Selected Tables in Mathematical Statistics, (1975), 233. [21] W. S. Kendal and P. Frost, Experimental metastasis: A novel application of the variance-to-mean power function,, J. Natl. Cancer Inst., 79 (1987), 1113. [22] W. S. Kendal, A scale invariant clustering of genes on human chromosome 7,, BMC Evol. Biol., 4 (2004). [23] P. Lansky and V. Lanska, Diffusion approximation of the neuronal model with synaptic reversal potentials,, Biol. Cybern., 56 (1987), 19. doi: 10.1007/BF00333064. [24] A. Lerchner, et al., Response variability in balanced cortical networks,, Neural Comput., 18 (2006), 634. doi: 10.1162/neco.2006.18.3.634. [25] B. Lindner and A. Longtin, Comment on "Characterization of Subthreshold Voltage Fluctuations in Neuronal Membranes," by M. Rudolph and A. Destexhe,, Neural Comput., 18 (2006), 1896. doi: 10.1162/neco.2006.18.8.1896. [26] A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities,, J. Appl. Probab., 22 (1985), 360. doi: 10.2307/3213779. [27] Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes,, J. Amer. Statist. Assoc., 83 (1988), 9. doi: 10.1080/01621459.1988.10478560. [28] L. M. Ricciardi and S. Sato, First-passage-time density and moments of the Ornstein-Uhlenbeck process,, J. Appl. Probab., 25 (1988), 43. doi: 10.2307/3214232. [29] M. J. Richardson and W. Gerstner, Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance,, Neural Comput., 17 (2005), 923. doi: 10.1162/0899766053429444. [30] B. K. Roy and D. R. Smith, Analysis of the exponential decay model of the neuron showing frequency threshold effects,, Bull. Math. Biophys., 31 (1969), 341. doi: 10.1007/BF02477011. [31] N. M. Shadlen and W. T. Newsome, The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding,, J. Neurosci., 18 (1998), 3870. [32] A. J. F. Siegert, On the first passage time probability problem,, Phys. Rev., 81 (1951), 617. doi: 10.1103/PhysRev.81.617. [33] D. L. Snyder, Random Point Processes,, John Wiley & Sons, (1975). [34] L. R. Taylor, Aggregation, variance and the mean,, Nature, 189 (1961), 732. doi: 10.1038/189732a0. [35] D. J. Tolhurst, J. A. Movshon and I. D. Thompson, The dependence of response amplitude and variance of cat visual cortical neurones on stimulus contrast,, Exp. Brain Res., 41 (1981), 414. doi: 10.1007/BF00238900. [36] J. B. Troy and J. G. Robson, Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance,, Vis. Neurosci., 9 (1992), 535. doi: 10.1017/S0952523800001784. [37] H. C. Tuckwell, Introduction to Theoretical Neurobiology,, Cambridge University Press, (1988). [38] N. G. van Kampen, Stochastic Processes in Physics and Chemistry,, 2nd edition, (1992). [39] R. D. Vilela and B. Lindner, Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?,, J. Theor. Biol., 257 (2009), 90. doi: 10.1016/j.jtbi.2008.11.004. [40] F. Y. M. Wan and H. C. Tuckwell, Neuronal firing and input variability,, J. Theoret. Neurobiol., 1 (1982), 197.

show all references

##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Dover, (1965). [2] R. M. Anderson and R. M. May, Epidemiological parameters of HIV transmission,, Nature, 333 (1988), 514. [3] B. B. Averbeck, Poisson or not Poisson: Differences in spike train statistics between parietal cortical areas,, Neuron, 62 (2009), 310. doi: 10.1016/j.neuron.2009.04.021. [4] P. Bremaud, Point Processes and Queues,, Springer, (1981). [5] A. N. Burkitt, Balanced neurons: Analysis of leaky integrate-and-fire neurons with reversal potentials,, Biol. Cybern., 85 (2001), 247. doi: 10.1007/s004220100262. [6] A. N. Burkitt, A review of the integrate-and-fire neuron model, I: Homogeneous synaptic input,, Biol. Cybern., 95 (2006), 1. doi: 10.1007/s00422-006-0068-6. [7] M. M. Churchland, et al., Stimulus onset quenches neural variability: A widespread cortical phenomenon,, Nat. Neurosci., 13 (2010), 369. [8] A. K. Churchland, et al., Variance as a signature of neural computations during decision making,, Neuron, 69 (2011), 818. [9] M. M. Churchland and L. F. Abbott, Two layers of neural variability,, Nat. Neurosci., 15 (2012), 1472. [10] D. R. Cox, Renewal Theory,, Chapman and Hall, (1962). [11] D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events,, Chapman and Hall, (1966). [12] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,, Springer Series in Statistics, (1988). [13] M. A. de Menezes and A. L. Barabasi, Fluctuations in network dynamics,, Phys. Rev. Lett., 92 (2004). [14] A. Destexhe, Z. Mainen and T. J. Sejnowski, Kinetic models of synaptic transmission,, in Methods in Neuronal Modeling (eds. C. Koch and I. Segev), (1998), 1. [15] S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model,, Phys. Rev. E., 71 (2005). doi: 10.1103/PhysRevE.71.011907. [16] Z. Eisler, I. Bartos and J. Kertesz, Fluctuation scaling in complex systems: Taylor's law and beyond,, Adv. Phys., 57 (2008), 89. doi: 10.1080/00018730801893043. [17] A. Fronczak and P. Fronczak, Origins of Taylor's power law for fluctuation scaling in complex systems,, Phys. Rev. E, 81 (2010). doi: 10.1103/PhysRevE.81.066112. [18] J. Inoue, S. Sato and L. M. Ricciardi, 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 (1995), 209. [19] D. H. Johnson, Point process models of single-neuron discharges,, J. Comput. Neurosci., 3 (1996), 275. doi: 10.1007/BF00161089. [20] J. Keilson and H. F. Ross, Passage time distribution for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes,, in Selected Tables in Mathematical Statistics, (1975), 233. [21] W. S. Kendal and P. Frost, Experimental metastasis: A novel application of the variance-to-mean power function,, J. Natl. Cancer Inst., 79 (1987), 1113. [22] W. S. Kendal, A scale invariant clustering of genes on human chromosome 7,, BMC Evol. Biol., 4 (2004). [23] P. Lansky and V. Lanska, Diffusion approximation of the neuronal model with synaptic reversal potentials,, Biol. Cybern., 56 (1987), 19. doi: 10.1007/BF00333064. [24] A. Lerchner, et al., Response variability in balanced cortical networks,, Neural Comput., 18 (2006), 634. doi: 10.1162/neco.2006.18.3.634. [25] B. Lindner and A. Longtin, Comment on "Characterization of Subthreshold Voltage Fluctuations in Neuronal Membranes," by M. Rudolph and A. Destexhe,, Neural Comput., 18 (2006), 1896. doi: 10.1162/neco.2006.18.8.1896. [26] A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities,, J. Appl. Probab., 22 (1985), 360. doi: 10.2307/3213779. [27] Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes,, J. Amer. Statist. Assoc., 83 (1988), 9. doi: 10.1080/01621459.1988.10478560. [28] L. M. Ricciardi and S. Sato, First-passage-time density and moments of the Ornstein-Uhlenbeck process,, J. Appl. Probab., 25 (1988), 43. doi: 10.2307/3214232. [29] M. J. Richardson and W. Gerstner, Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance,, Neural Comput., 17 (2005), 923. doi: 10.1162/0899766053429444. [30] B. K. Roy and D. R. Smith, Analysis of the exponential decay model of the neuron showing frequency threshold effects,, Bull. Math. Biophys., 31 (1969), 341. doi: 10.1007/BF02477011. [31] N. M. Shadlen and W. T. Newsome, The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding,, J. Neurosci., 18 (1998), 3870. [32] A. J. F. Siegert, On the first passage time probability problem,, Phys. Rev., 81 (1951), 617. doi: 10.1103/PhysRev.81.617. [33] D. L. Snyder, Random Point Processes,, John Wiley & Sons, (1975). [34] L. R. Taylor, Aggregation, variance and the mean,, Nature, 189 (1961), 732. doi: 10.1038/189732a0. [35] D. J. Tolhurst, J. A. Movshon and I. D. Thompson, The dependence of response amplitude and variance of cat visual cortical neurones on stimulus contrast,, Exp. Brain Res., 41 (1981), 414. doi: 10.1007/BF00238900. [36] J. B. Troy and J. G. Robson, Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance,, Vis. Neurosci., 9 (1992), 535. doi: 10.1017/S0952523800001784. [37] H. C. Tuckwell, Introduction to Theoretical Neurobiology,, Cambridge University Press, (1988). [38] N. G. van Kampen, Stochastic Processes in Physics and Chemistry,, 2nd edition, (1992). [39] R. D. Vilela and B. Lindner, Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?,, J. Theor. Biol., 257 (2009), 90. doi: 10.1016/j.jtbi.2008.11.004. [40] F. Y. M. Wan and H. C. Tuckwell, Neuronal firing and input variability,, J. Theoret. Neurobiol., 1 (1982), 197.
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