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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:
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M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Dover, (1965). Google Scholar

[2]

R. M. Anderson and R. M. May, Epidemiological parameters of HIV transmission,, Nature, 333 (1988), 514. Google Scholar

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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. Google Scholar

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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. Google Scholar

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M. M. Churchland, et al., Stimulus onset quenches neural variability: A widespread cortical phenomenon,, Nat. Neurosci., 13 (2010), 369. Google Scholar

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A. K. Churchland, et al., Variance as a signature of neural computations during decision making,, Neuron, 69 (2011), 818. Google Scholar

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M. M. Churchland and L. F. Abbott, Two layers of neural variability,, Nat. Neurosci., 15 (2012), 1472. Google Scholar

[10]

D. R. Cox, Renewal Theory,, Chapman and Hall, (1962). Google Scholar

[11]

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events,, Chapman and Hall, (1966). Google Scholar

[12]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,, Springer Series in Statistics, (1988). Google Scholar

[13]

M. A. de Menezes and A. L. Barabasi, Fluctuations in network dynamics,, Phys. Rev. Lett., 92 (2004). Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

D. H. Johnson, Point process models of single-neuron discharges,, J. Comput. Neurosci., 3 (1996), 275. doi: 10.1007/BF00161089. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

W. S. Kendal, A scale invariant clustering of genes on human chromosome 7,, BMC Evol. Biol., 4 (2004). Google Scholar

[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. Google Scholar

[24]

A. Lerchner, et al., Response variability in balanced cortical networks,, Neural Comput., 18 (2006), 634. doi: 10.1162/neco.2006.18.3.634. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

A. J. F. Siegert, On the first passage time probability problem,, Phys. Rev., 81 (1951), 617. doi: 10.1103/PhysRev.81.617. Google Scholar

[33]

D. L. Snyder, Random Point Processes,, John Wiley & Sons, (1975). Google Scholar

[34]

L. R. Taylor, Aggregation, variance and the mean,, Nature, 189 (1961), 732. doi: 10.1038/189732a0. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

H. C. Tuckwell, Introduction to Theoretical Neurobiology,, Cambridge University Press, (1988). Google Scholar

[38]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry,, 2nd edition, (1992). Google Scholar

[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. Google Scholar

[40]

F. Y. M. Wan and H. C. Tuckwell, Neuronal firing and input variability,, J. Theoret. Neurobiol., 1 (1982), 197. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Dover, (1965). Google Scholar

[2]

R. M. Anderson and R. M. May, Epidemiological parameters of HIV transmission,, Nature, 333 (1988), 514. Google Scholar

[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. Google Scholar

[4]

P. Bremaud, Point Processes and Queues,, Springer, (1981). Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

M. M. Churchland, et al., Stimulus onset quenches neural variability: A widespread cortical phenomenon,, Nat. Neurosci., 13 (2010), 369. Google Scholar

[8]

A. K. Churchland, et al., Variance as a signature of neural computations during decision making,, Neuron, 69 (2011), 818. Google Scholar

[9]

M. M. Churchland and L. F. Abbott, Two layers of neural variability,, Nat. Neurosci., 15 (2012), 1472. Google Scholar

[10]

D. R. Cox, Renewal Theory,, Chapman and Hall, (1962). Google Scholar

[11]

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events,, Chapman and Hall, (1966). Google Scholar

[12]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,, Springer Series in Statistics, (1988). Google Scholar

[13]

M. A. de Menezes and A. L. Barabasi, Fluctuations in network dynamics,, Phys. Rev. Lett., 92 (2004). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

D. H. Johnson, Point process models of single-neuron discharges,, J. Comput. Neurosci., 3 (1996), 275. doi: 10.1007/BF00161089. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

W. S. Kendal, A scale invariant clustering of genes on human chromosome 7,, BMC Evol. Biol., 4 (2004). Google Scholar

[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. Google Scholar

[24]

A. Lerchner, et al., Response variability in balanced cortical networks,, Neural Comput., 18 (2006), 634. doi: 10.1162/neco.2006.18.3.634. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

A. J. F. Siegert, On the first passage time probability problem,, Phys. Rev., 81 (1951), 617. doi: 10.1103/PhysRev.81.617. Google Scholar

[33]

D. L. Snyder, Random Point Processes,, John Wiley & Sons, (1975). Google Scholar

[34]

L. R. Taylor, Aggregation, variance and the mean,, Nature, 189 (1961), 732. doi: 10.1038/189732a0. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

H. C. Tuckwell, Introduction to Theoretical Neurobiology,, Cambridge University Press, (1988). Google Scholar

[38]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry,, 2nd edition, (1992). Google Scholar

[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. Google Scholar

[40]

F. Y. M. Wan and H. C. Tuckwell, Neuronal firing and input variability,, J. Theoret. Neurobiol., 1 (1982), 197. Google Scholar

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