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2014, 11(1): 105-123. doi: 10.3934/mbe.2014.11.105

Fano factor estimation

Kamil Rajdl 1, and  Petr Lansky 2,

1. 

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlarska 2a, 611 37 Brno, Czech Republic
2. 

Institute of Physiology, Academy of Sciences of the Czech Republic, Videnska 1083, 142 20 Prague

Received  December 2012 Revised  December 2012 Published  September 2013

Fano factor is one of the most widely used measures of variability of spike trains. Its standard estimator is the ratio of sample variance to sample mean of spike counts observed in a time window and the quality of the estimator strongly depends on the length of the window. We investigate this dependence under the assumption that the spike train behaves as an equilibrium renewal process. It is shown what characteristics of the spike train have large effect on the estimator bias. Namely, the effect of refractory period is analytically evaluated. Next, we create an approximate asymptotic formula for the mean square error of the estimator, which can also be used to find minimum of the error in estimation from single spike trains. The accuracy of the Fano factor estimator is compared with the accuracy of the estimator based on the squared coefficient of variation. All the results are illustrated for spike trains with gamma and inverse Gaussian probability distributions of interspike intervals. Finally, we discuss possibilities of how to select a suitable observation window for the Fano factor estimation.
Keywords: renewal process, Fano factor, refractory period., mean square error, estimation.
Mathematics Subject Classification: Primary: 60K05, 62-07; Secondary: 62P1.
Citation: Kamil Rajdl, Petr Lansky. Fano factor estimation. Mathematical Biosciences & Engineering, 2014, 11 (1) : 105-123. doi: 10.3934/mbe.2014.11.105
References:
[1]

O. Avila-Akerberg and M. J. Chacron, Nonrenewal spike train statistics: Causes and functional consequences on neural coding, Exp. Brain Res., 210 (2011), 353-371. Google Scholar

[2]

M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli, J. Neurosci., 21 (2001), 5328-5343. Google Scholar

[3]

M. H. Chang, K. M. Armstrong and T. Moore, Dissociation of response variability from firing rate effects in frontal eye field neurons during visual stimulation, working memory, and attention, J. Neurosci. Methods, 32 (2012), 2204-2216. doi: 10.1523/JNEUROSCI.2967-11.2012.  Google Scholar

[4]

M. M. Churchland, B. M. Yu, J. P. Cunningham, L. P. Sugrue, M. R. Cohen, G. S. Corrado, W. T. Newsome, A. M. Clark, P. Hosseini, B. B. Scott, D. C. Bradley, M. A. Smith, A. Kohn, J. A. Movshon, K. M. Armstrong, T. Moore, S. W. Chang, L. H. Snyder, S. G. Lisberger, N. J. Priebe, I. M. Finn, D. Ferster, S. I. Ryu, G. Santhanam, M. Sahani and K. V. Shenoy, Stimulus onset quenches neural variability: A widespread cortical phenomenon, Nat. Neurosci., 13 (2010), 369-378. doi: 10.1038/nn.2501.  Google Scholar

[5]

D. R. Cox, "Renewal Theory," Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1962.  Google Scholar

[6]

O. Darbin, J. Soares and T. Wichmann, Nonlinear analysis of discharge patterns in monkey basal ganglia, Brain Res., 1118 (2006), 84-93. doi: 10.1016/j.brainres.2006.08.027.  Google Scholar

[7]

M. Deger, M. Helias, C. Boucsein and S. Rotter, Statistical properties of superimposed stationary spike trains, J. Comput. Neurosci., 32 (2012), 443-463. doi: 10.1007/s10827-011-0362-8.  Google Scholar

[8]

S. Ditlevsen and P. Lansky, Firing variability is higher than deduced from the empirical coefficient of variation, Neural. Comput., 23 (2011), 1944-1966. doi: 10.1162/NECO_a_00157.  Google Scholar

[9]

U. T. Eden and M. A. Kramer, Drawing inferences from fano factor calculations, J. Neurosci. Methods, 190 (2010), 149-152. doi: 10.1016/j.jneumeth.2010.04.012.  Google Scholar

[10]

F. Farkhooi, M. F. Strube-Bloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modeling, and single neuron variability, Phys. Rev. E, 79 (2009), 021905. doi: 10.1103/PhysRevE.79.021905.  Google Scholar

[11]

W. Gerstner and W. M. Kistler, "Spiking Neuron Models. Single Neurons, Populations, Plasticity," Cambridge University Press, Cambridge, 2002.  Google Scholar

[12]

C. Hussar and T. Pasternak, Trial-to-trial variability of the prefrontal neurons reveals the nature of their engagement in a motion discrimination task, P. Natl. Acad. Sci. USA, 107 (2010), 21842-21847. doi: 10.1073/pnas.1009956107.  Google Scholar

[13]

W. S. Jewell, The properties of recurrent-event processes, Oper. Res., 8 (1960), 446-472. doi: 10.1287/opre.8.4.446.  Google Scholar

[14]

L. Kostal, P. Lansky and O. Pokora, Variability measures of positive random variables, PLOS ONE, 6 (2011), e21998. doi: 10.1371/journal.pone.0021998.  Google Scholar

[15]

S. Koyama and S. Shinomoto, Inference of intrinsic spiking irregularity based on the Kullback-Leibler information, BioSystems, 89 (2007), 69-73. doi: 10.1016/j.biosystems.2006.05.012.  Google Scholar

[16]

A. M. Mood, F. A. Graybill and D. C. Boes, "Introduction to the Theory of Statistics," McGraw-Hill, Inc., 1974. Google Scholar

[17]

M. P. Nawrot, Analysis and interpretation of interval and count variability in neural spike trains, in "Analysis of Parallel Spike Trains," Springer, Inc., New York, (2010), 37-58. Google Scholar

[18]

M. P. Nawrot, C. Boucsein, V. R. Molina, A. Riehle, A. Aertsen and S. Rotter, Measurement of variability dynamics in cortical spike trains, J. Neurosci. Methods, 169 (2008), 374-390. doi: 10.1016/j.jneumeth.2007.10.013.  Google Scholar

[19]

T. Omi and S. Shinomoto, Optimizing time histograms for non-Poissonian spike trains, Neural. Comput., 23 (2011), 3125-3144. doi: 10.1162/NECO_a_00213.  Google Scholar

[20]

Z. Pawlas, L. B. Klebanov, M. Prokop and P. Lansky, Parameters of spike trains observed in a short time window, Neural. Comput., 20 (2008), 1325-1343. doi: 10.1162/neco.2007.01-07-442.  Google Scholar

[21]

B. V. D. Pol and H. Bremmer, "Operational Calculus Based on the Two-Sided Laplace Integral," Cambridge University Press, Cambridge, 1950.  Google Scholar

[22]

A. Ponce-Alvarez, B. E. Kilavik and A. Riehle, Comparison of local measures of spike time irregularity and relating variability to firing rate in motor cortical neurons, J. Comput. Neurosci., 29 (2010), 351-365. doi: 10.1007/s10827-009-0158-2.  Google Scholar

[23]

T. Shimokawa and S. Shinomoto, Estimating instantaneous irregularity of neuronal firing, Neural. Comput., 21 (2009), 1931-1951. doi: 10.1162/neco.2009.08-08-841.  Google Scholar

[24]

S. Shinomoto, K. Miura and S. Koyama, A measure of local variation of inter-spike intervals, BioSystems, 79 (2005), 67-72. doi: 10.1016/j.biosystems.2004.09.023.  Google Scholar

[25]

M. C. Teich, D. H. Johnson, A. R. Kumar and R. G. Turcott, Rate fluctuations and fractional power-law noise recorded from cells in the lower auditory pathway of the cat, Hearing Res., 46 (1990), 41-52. doi: 10.1016/0378-5955(90)90138-F.  Google Scholar

show all references

References:
[1]

O. Avila-Akerberg and M. J. Chacron, Nonrenewal spike train statistics: Causes and functional consequences on neural coding, Exp. Brain Res., 210 (2011), 353-371. Google Scholar

[2]

M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli, J. Neurosci., 21 (2001), 5328-5343. Google Scholar

[3]

M. H. Chang, K. M. Armstrong and T. Moore, Dissociation of response variability from firing rate effects in frontal eye field neurons during visual stimulation, working memory, and attention, J. Neurosci. Methods, 32 (2012), 2204-2216. doi: 10.1523/JNEUROSCI.2967-11.2012.  Google Scholar

[4]

M. M. Churchland, B. M. Yu, J. P. Cunningham, L. P. Sugrue, M. R. Cohen, G. S. Corrado, W. T. Newsome, A. M. Clark, P. Hosseini, B. B. Scott, D. C. Bradley, M. A. Smith, A. Kohn, J. A. Movshon, K. M. Armstrong, T. Moore, S. W. Chang, L. H. Snyder, S. G. Lisberger, N. J. Priebe, I. M. Finn, D. Ferster, S. I. Ryu, G. Santhanam, M. Sahani and K. V. Shenoy, Stimulus onset quenches neural variability: A widespread cortical phenomenon, Nat. Neurosci., 13 (2010), 369-378. doi: 10.1038/nn.2501.  Google Scholar

[5]

D. R. Cox, "Renewal Theory," Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1962.  Google Scholar

[6]

O. Darbin, J. Soares and T. Wichmann, Nonlinear analysis of discharge patterns in monkey basal ganglia, Brain Res., 1118 (2006), 84-93. doi: 10.1016/j.brainres.2006.08.027.  Google Scholar

[7]

M. Deger, M. Helias, C. Boucsein and S. Rotter, Statistical properties of superimposed stationary spike trains, J. Comput. Neurosci., 32 (2012), 443-463. doi: 10.1007/s10827-011-0362-8.  Google Scholar

[8]

S. Ditlevsen and P. Lansky, Firing variability is higher than deduced from the empirical coefficient of variation, Neural. Comput., 23 (2011), 1944-1966. doi: 10.1162/NECO_a_00157.  Google Scholar

[9]

U. T. Eden and M. A. Kramer, Drawing inferences from fano factor calculations, J. Neurosci. Methods, 190 (2010), 149-152. doi: 10.1016/j.jneumeth.2010.04.012.  Google Scholar

[10]

F. Farkhooi, M. F. Strube-Bloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modeling, and single neuron variability, Phys. Rev. E, 79 (2009), 021905. doi: 10.1103/PhysRevE.79.021905.  Google Scholar

[11]

W. Gerstner and W. M. Kistler, "Spiking Neuron Models. Single Neurons, Populations, Plasticity," Cambridge University Press, Cambridge, 2002.  Google Scholar

[12]

C. Hussar and T. Pasternak, Trial-to-trial variability of the prefrontal neurons reveals the nature of their engagement in a motion discrimination task, P. Natl. Acad. Sci. USA, 107 (2010), 21842-21847. doi: 10.1073/pnas.1009956107.  Google Scholar

[13]

W. S. Jewell, The properties of recurrent-event processes, Oper. Res., 8 (1960), 446-472. doi: 10.1287/opre.8.4.446.  Google Scholar

[14]

L. Kostal, P. Lansky and O. Pokora, Variability measures of positive random variables, PLOS ONE, 6 (2011), e21998. doi: 10.1371/journal.pone.0021998.  Google Scholar

[15]

S. Koyama and S. Shinomoto, Inference of intrinsic spiking irregularity based on the Kullback-Leibler information, BioSystems, 89 (2007), 69-73. doi: 10.1016/j.biosystems.2006.05.012.  Google Scholar

[16]

A. M. Mood, F. A. Graybill and D. C. Boes, "Introduction to the Theory of Statistics," McGraw-Hill, Inc., 1974. Google Scholar

[17]

M. P. Nawrot, Analysis and interpretation of interval and count variability in neural spike trains, in "Analysis of Parallel Spike Trains," Springer, Inc., New York, (2010), 37-58. Google Scholar

[18]

M. P. Nawrot, C. Boucsein, V. R. Molina, A. Riehle, A. Aertsen and S. Rotter, Measurement of variability dynamics in cortical spike trains, J. Neurosci. Methods, 169 (2008), 374-390. doi: 10.1016/j.jneumeth.2007.10.013.  Google Scholar

[19]

T. Omi and S. Shinomoto, Optimizing time histograms for non-Poissonian spike trains, Neural. Comput., 23 (2011), 3125-3144. doi: 10.1162/NECO_a_00213.  Google Scholar

[20]

Z. Pawlas, L. B. Klebanov, M. Prokop and P. Lansky, Parameters of spike trains observed in a short time window, Neural. Comput., 20 (2008), 1325-1343. doi: 10.1162/neco.2007.01-07-442.  Google Scholar

[21]

B. V. D. Pol and H. Bremmer, "Operational Calculus Based on the Two-Sided Laplace Integral," Cambridge University Press, Cambridge, 1950.  Google Scholar

[22]

A. Ponce-Alvarez, B. E. Kilavik and A. Riehle, Comparison of local measures of spike time irregularity and relating variability to firing rate in motor cortical neurons, J. Comput. Neurosci., 29 (2010), 351-365. doi: 10.1007/s10827-009-0158-2.  Google Scholar

[23]

T. Shimokawa and S. Shinomoto, Estimating instantaneous irregularity of neuronal firing, Neural. Comput., 21 (2009), 1931-1951. doi: 10.1162/neco.2009.08-08-841.  Google Scholar

[24]

S. Shinomoto, K. Miura and S. Koyama, A measure of local variation of inter-spike intervals, BioSystems, 79 (2005), 67-72. doi: 10.1016/j.biosystems.2004.09.023.  Google Scholar

[25]

M. C. Teich, D. H. Johnson, A. R. Kumar and R. G. Turcott, Rate fluctuations and fractional power-law noise recorded from cells in the lower auditory pathway of the cat, Hearing Res., 46 (1990), 41-52. doi: 10.1016/0378-5955(90)90138-F.  Google Scholar

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