# American Institute of Mathematical Sciences

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

## Fano factor estimation

 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.
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. [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. [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. doi: 10.1523/JNEUROSCI.2967-11.2012. [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. doi: 10.1038/nn.2501. [5] D. R. Cox, "Renewal Theory,", Methuen & Co., (1962). [6] O. Darbin, J. Soares and T. Wichmann, Nonlinear analysis of discharge patterns in monkey basal ganglia,, Brain Res., 1118 (2006), 84. doi: 10.1016/j.brainres.2006.08.027. [7] M. Deger, M. Helias, C. Boucsein and S. Rotter, Statistical properties of superimposed stationary spike trains,, J. Comput. Neurosci., 32 (2012), 443. doi: 10.1007/s10827-011-0362-8. [8] S. Ditlevsen and P. Lansky, Firing variability is higher than deduced from the empirical coefficient of variation,, Neural. Comput., 23 (2011), 1944. doi: 10.1162/NECO_a_00157. [9] U. T. Eden and M. A. Kramer, Drawing inferences from fano factor calculations,, J. Neurosci. Methods, 190 (2010), 149. doi: 10.1016/j.jneumeth.2010.04.012. [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). doi: 10.1103/PhysRevE.79.021905. [11] W. Gerstner and W. M. Kistler, "Spiking Neuron Models. Single Neurons, Populations, Plasticity,", Cambridge University Press, (2002). [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. doi: 10.1073/pnas.1009956107. [13] W. S. Jewell, The properties of recurrent-event processes,, Oper. Res., 8 (1960), 446. doi: 10.1287/opre.8.4.446. [14] L. Kostal, P. Lansky and O. Pokora, Variability measures of positive random variables,, PLOS ONE, 6 (2011). doi: 10.1371/journal.pone.0021998. [15] S. Koyama and S. Shinomoto, Inference of intrinsic spiking irregularity based on the Kullback-Leibler information,, BioSystems, 89 (2007), 69. doi: 10.1016/j.biosystems.2006.05.012. [16] A. M. Mood, F. A. Graybill and D. C. Boes, "Introduction to the Theory of Statistics,", McGraw-Hill, (1974). [17] M. P. Nawrot, Analysis and interpretation of interval and count variability in neural spike trains,, in, (2010), 37. [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. doi: 10.1016/j.jneumeth.2007.10.013. [19] T. Omi and S. Shinomoto, Optimizing time histograms for non-Poissonian spike trains,, Neural. Comput., 23 (2011), 3125. doi: 10.1162/NECO_a_00213. [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. doi: 10.1162/neco.2007.01-07-442. [21] B. V. D. Pol and H. Bremmer, "Operational Calculus Based on the Two-Sided Laplace Integral,", Cambridge University Press, (1950). [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. doi: 10.1007/s10827-009-0158-2. [23] T. Shimokawa and S. Shinomoto, Estimating instantaneous irregularity of neuronal firing,, Neural. Comput., 21 (2009), 1931. doi: 10.1162/neco.2009.08-08-841. [24] S. Shinomoto, K. Miura and S. Koyama, A measure of local variation of inter-spike intervals,, BioSystems, 79 (2005), 67. doi: 10.1016/j.biosystems.2004.09.023. [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. doi: 10.1016/0378-5955(90)90138-F.

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. [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. [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. doi: 10.1523/JNEUROSCI.2967-11.2012. [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. doi: 10.1038/nn.2501. [5] D. R. Cox, "Renewal Theory,", Methuen & Co., (1962). [6] O. Darbin, J. Soares and T. Wichmann, Nonlinear analysis of discharge patterns in monkey basal ganglia,, Brain Res., 1118 (2006), 84. doi: 10.1016/j.brainres.2006.08.027. [7] M. Deger, M. Helias, C. Boucsein and S. Rotter, Statistical properties of superimposed stationary spike trains,, J. Comput. Neurosci., 32 (2012), 443. doi: 10.1007/s10827-011-0362-8. [8] S. Ditlevsen and P. Lansky, Firing variability is higher than deduced from the empirical coefficient of variation,, Neural. Comput., 23 (2011), 1944. doi: 10.1162/NECO_a_00157. [9] U. T. Eden and M. A. Kramer, Drawing inferences from fano factor calculations,, J. Neurosci. Methods, 190 (2010), 149. doi: 10.1016/j.jneumeth.2010.04.012. [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). doi: 10.1103/PhysRevE.79.021905. [11] W. Gerstner and W. M. Kistler, "Spiking Neuron Models. Single Neurons, Populations, Plasticity,", Cambridge University Press, (2002). [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. doi: 10.1073/pnas.1009956107. [13] W. S. Jewell, The properties of recurrent-event processes,, Oper. Res., 8 (1960), 446. doi: 10.1287/opre.8.4.446. [14] L. Kostal, P. Lansky and O. Pokora, Variability measures of positive random variables,, PLOS ONE, 6 (2011). doi: 10.1371/journal.pone.0021998. [15] S. Koyama and S. Shinomoto, Inference of intrinsic spiking irregularity based on the Kullback-Leibler information,, BioSystems, 89 (2007), 69. doi: 10.1016/j.biosystems.2006.05.012. [16] A. M. Mood, F. A. Graybill and D. C. Boes, "Introduction to the Theory of Statistics,", McGraw-Hill, (1974). [17] M. P. Nawrot, Analysis and interpretation of interval and count variability in neural spike trains,, in, (2010), 37. [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. doi: 10.1016/j.jneumeth.2007.10.013. [19] T. Omi and S. Shinomoto, Optimizing time histograms for non-Poissonian spike trains,, Neural. Comput., 23 (2011), 3125. doi: 10.1162/NECO_a_00213. [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. doi: 10.1162/neco.2007.01-07-442. [21] B. V. D. Pol and H. Bremmer, "Operational Calculus Based on the Two-Sided Laplace Integral,", Cambridge University Press, (1950). [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. doi: 10.1007/s10827-009-0158-2. [23] T. Shimokawa and S. Shinomoto, Estimating instantaneous irregularity of neuronal firing,, Neural. Comput., 21 (2009), 1931. doi: 10.1162/neco.2009.08-08-841. [24] S. Shinomoto, K. Miura and S. Koyama, A measure of local variation of inter-spike intervals,, BioSystems, 79 (2005), 67. doi: 10.1016/j.biosystems.2004.09.023. [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. doi: 10.1016/0378-5955(90)90138-F.
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