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2014, 11(1): 63-80. doi: 10.3934/mbe.2014.11.63

The effect of interspike interval statistics on the information gain under the rate coding hypothesis

1. 

The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan

2. 

Institute of Physiology, Academy of Sciences of the Czech Republic, Videnska 1083, 14220 Prague, Czech Republic

Received  December 2012 Revised  April 2013 Published  September 2013

The question, how much information can be theoretically gained from variable neuronal firing rate with respect to constant average firing rate is investigated. We employ the statistical concept of information based on the Kullback-Leibler divergence, and assume rate-modulated renewal processes as a model of spike trains. We show that if the firing rate variation is sufficiently small and slow (with respect to the mean interspike interval), the information gain can be expressed by the Fisher information. Furthermore, under certain assumptions, the smallest possible information gain is provided by gamma-distributed interspike intervals. The methodology is illustrated and discussed on several different statistical models of neuronal activity.
Citation: Shinsuke Koyama, Lubomir Kostal. The effect of interspike interval statistics on the information gain under the rate coding hypothesis. Mathematical Biosciences & Engineering, 2014, 11 (1) : 63-80. doi: 10.3934/mbe.2014.11.63
References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables,", Dover Publications, (1966).

[2]

E. D. Adrian, The basis of sensation,, Br. Med. J., 1 (1954). doi: 10.1136/bmj.1.4857.287.

[3]

R. Barbieri, M. C. Quirk, L. M. Frank, M. A. Wilson and E. N. Brown, Construction and analysis of non-Poisson stimulus-response models of neural spiking activity,, Journal of Neuroscience Methods, 105 (2001), 25. doi: 10.1016/S0165-0270(00)00344-7.

[4]

M. Berman, Inhomogeneous and modulated gamma processes,, Biometrika, 68 (1981), 143. doi: 10.1093/biomet/68.1.143.

[5]

J. M. Bernardo, Reference posterior distributions for Bayesian inference. With discussion,, J. Roy. Stat. Soc. B, 41 (1979), 113.

[6]

A. Bershadskii, E. Dremencov, D. Fukayama and G. Yadid, Probabilistic properties of neuron spiking time-series obtained in vivo,, Eur. Phys. J. B, 24 (2001), 409. doi: 10.1007/s10051-001-8691-4.

[7]

G. S. Bhumbra, A. N. Inyushkin and R. E. J. Dyball, Assessment of spike activity in the supraoptic nucleus,, J. Neuroendocrinol., 16 (2004), 390. doi: 10.1111/j.0953-8194.2004.01166.x.

[8]

L. Bonnasse-Gahot and J.-P. Nadal, Perception of categories: From coding efficiency to reaction times,, Brain Res., 1434 (2012), 47. doi: 10.1016/j.brainres.2011.08.014.

[9]

A. Borst and F. E. Theunissen, Information theory and neural coding,, Nature Neurosci., 2 (1999), 947.

[10]

N. Brunel and J.-P. Nadal, Mutual information, Fisher information, and population coding,, Neural Computation, 10 (1998), 1731. doi: 10.1162/089976698300017115.

[11]

R. S. Chhikara and J. L. Folks, "The Inverse Gaussian Distribution: Theory, Methodology, and Applications,", Marcel Dekker, (1989).

[12]

M. Cohen, The fisher information and convexity,, IEEE Transactions on Information Theory, 14 (1968), 591. doi: 10.1109/TIT.1968.1054175.

[13]

D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events,", Methuen & Co., (1966).

[14]

J. P. Cunningham, V. Gilja, S. I. Ryu and K. V. Shenoy, Methods for estimating neural firing rates, and their application to brain-machine interfaces,, Neural Networks, 22 (2009), 1235. doi: 10.1016/j.neunet.2009.02.004.

[15]

J. P. Cunningham, B. M. Yu, K. V. Shenoy and M. Sahani, Inferring neural firing rates from spike trains using Gaussian processes,, in, (2008), 329.

[16]

D. J. Daley and D. Vere-Jones, "An Introduction to the Theory of Point Processes. Vol. I. Elementary Theory and Methods,", Second edition, (2003).

[17]

P. Duchamp-Viret, L. Kostal, M. Chaput, P. Lánsky and J.-P. Rospars, Patterns of spontaneous activity in single rat olfactory receptor neurons are different in normally breathing and tracheotomized animals,, J. Neurobiology, 65 (2005), 97. doi: 10.1002/neu.20177.

[18]

R. G. Gallager, "Information Theory and Reliable Communication,", John Wiley & Sons, (1968).

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0.

[20]

I. J. Good, "Probability and the Weighing of Evidence,", Charles Griffin & Co., (1950).

[21]

I. J. Good and R. A. Gaskins, Nonparametric roughness penalties for probability densities,, Biometrika, 58 (1971), 255.

[22]

P. E. Greenwood and P. Lánský, Optimal signal estimation in neuronal models,, Neural Comput., 17 (2005), 2240. doi: 10.1162/0899766054615653.

[23]

P. E. Greenwood and P. Lánský, Optimum signal in a simple neuronal model with signal-dependent noise,, Biol. Cybern., 92 (2005), 199. doi: 10.1007/s00422-005-0545-3.

[24]

P. E. Greenwood, L. M. Ward, D. F. Russell, A. Neiman and F. Moss, Stochastic resonance enhances the electrosensory information available to paddlefish for prey capture,, Phys. Rev. Lett., 84 (2000), 4773. doi: 10.1103/PhysRevLett.84.4773.

[25]

A. Grémiaux, T. Nowotny, D. Martinez, P. Lucas and J.-P. Rospars, Modelling the signal delivered by a population of first-order neurons in a moth olfactory system,, Brain Res., 1434 (2012), 123. doi: 10.1016/j.brainres.2011.09.035.

[26]

P. J. Huber, "Robust Statistics,", Wiley Series in Probability and Mathematical Statistics, (1981).

[27]

S. Ikeda and J. H. Manton, Capacity of a single spiking neuron channel,, Neural Comput., 21 (2009), 1714. doi: 10.1162/neco.2009.05-08-792.

[28]

S. Iyengar and Q. Liao, Modeling neural activity using the generalized inverse gaussian distribution,, Biological Cybernetics, 77 (1997), 289. doi: 10.1007/s004220050390.

[29]

B. Jørgensen, "Statistical Properties of the Generalized Inverse Gaussian Distribution,", Lecture Notes in Statistics, 9 (1982).

[30]

A. M. Kagan, I. V. Linnik and C. R. Rao, "Characterization Problems in Mathematical Statistics,", John Wiley & Sons, (1973).

[31]

R. E. Kass and V. Ventura, A spike-train probability model,, Neural Computation, 13 (2001), 1713. doi: 10.1162/08997660152469314.

[32]

S. M. Kay, "Fundamentals of Statistical Signal Processing: Estimation Theory,", Prentice Hall, (1993).

[33]

L. Kostal, Information capacity in the weak-signal approximation,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.026115.

[34]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code,, Brain Res., 1434 (2012), 136. doi: 10.1016/j.brainres.2011.07.007.

[35]

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

[36]

L. Kostal and O. Pokora, Nonparametric estimation of information-based measures of statistical dispersion,, Entropy, 14 (2012), 1221. doi: 10.3390/e14071221.

[37]

S. Koyama, Coding efficiency and detectability of rate fluctuations with non-Poisson neuronal firing,, in, (2013).

[38]

S. Koyama and R. E. Kass, Spike train probability models for stimulus-driven leaky integrate-and-fire neurons,, Neural Computation, 20 (2008), 1776. doi: 10.1162/neco.2008.06-07-540.

[39]

S. Kullback, "Information Theory and Statistics,", Dover Publications, (1968).

[40]

E. L. Lehmann and G. Casella, "Theory of Point Estimation,", Second edition, (1998).

[41]

M. W. Levine, The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells,, Biol. Cybern., 65 (1991), 459. doi: 10.1007/BF00204659.

[42]

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.

[43]

D. H. Perkel and T. H. Bullock, Neural coding,, Neurosci. Res. Prog. Sum., 3 (1968), 405.

[44]

J. W. Pillow, Time-rescaling methods for the estimation and assessment of non-Poisson neural encoding models,, in, (2008), 1473.

[45]

E. J. G. Pitman, "Some Basic Theory for Statistical Inference,", Monographs on Applied Probability and Statistics, (1979).

[46]

C. Pouzat and A. Chaffiol, Automatic spike train analysis and report generation. An implementation with R, R2HTML and STAR,, J. Neurosci. Methods, 181 (2009), 119. doi: 10.1016/j.jneumeth.2009.01.037.

[47]

D. S. Reich, J. D. Victor and B. W. Knight, The power ratio and the interval map: Spiking models and extracellular recordings,, Journal of Neuroscience, 18 (1998), 10090.

[48]

B. J. Richmond and L. M. Optican, Temporal encoding of two-dimensional patterns by single units in primate inferior temporal cortex. II. Quantification of response waveform,, Journal of Neurophysiology, 57 (1987), 147.

[49]

J. J. Rissanen, Fisher information and stochastic complexity,, IEEE Trans. Inf. Theory, 42 (1996), 40. doi: 10.1109/18.481776.

[50]

L. J. Savage, "The Foundations of Statistics,", John Wiley & Sons, (1954).

[51]

H. S. Seung and H. Sompolinsky, Simple models for reading neuronal population codes,, Proceedings of the National Academy of Sciences of the United States of America, 90 (1993), 10749. doi: 10.1073/pnas.90.22.10749.

[52]

C. E. Shannon and W. Weaver, "The Mathematical Theory of Communication,", University of Illinois Press, (1949).

[53]

R. B. Stein, The information capacity of nerve cells using a frequency code,, Biophys. J., 7 (1967), 797. doi: 10.1016/S0006-3495(67)86623-2.

[54]

F. Theunissen and J. P. Miller, Temporal encoding in nervous systems: A rigorous definition,, J. Comput. Neurosci., 2 (1995), 149. doi: 10.1007/BF00961885.

[55]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology, Vol. 2. Nonlinear and Stochastic Theories,", Cambridge Studies in Mathematical Biology, 8 (1988).

[56]

A. W. van der Vaart, "Asymptotic Statistics,", Cambridge Series in Statistical and Probabilistic Mathematics, 3 (1998).

[57]

K. Zhang and T. Sejnowski, Neural tuning: To sharpen or broaden?,, Neural Computation, 11 (1999), 75.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables,", Dover Publications, (1966).

[2]

E. D. Adrian, The basis of sensation,, Br. Med. J., 1 (1954). doi: 10.1136/bmj.1.4857.287.

[3]

R. Barbieri, M. C. Quirk, L. M. Frank, M. A. Wilson and E. N. Brown, Construction and analysis of non-Poisson stimulus-response models of neural spiking activity,, Journal of Neuroscience Methods, 105 (2001), 25. doi: 10.1016/S0165-0270(00)00344-7.

[4]

M. Berman, Inhomogeneous and modulated gamma processes,, Biometrika, 68 (1981), 143. doi: 10.1093/biomet/68.1.143.

[5]

J. M. Bernardo, Reference posterior distributions for Bayesian inference. With discussion,, J. Roy. Stat. Soc. B, 41 (1979), 113.

[6]

A. Bershadskii, E. Dremencov, D. Fukayama and G. Yadid, Probabilistic properties of neuron spiking time-series obtained in vivo,, Eur. Phys. J. B, 24 (2001), 409. doi: 10.1007/s10051-001-8691-4.

[7]

G. S. Bhumbra, A. N. Inyushkin and R. E. J. Dyball, Assessment of spike activity in the supraoptic nucleus,, J. Neuroendocrinol., 16 (2004), 390. doi: 10.1111/j.0953-8194.2004.01166.x.

[8]

L. Bonnasse-Gahot and J.-P. Nadal, Perception of categories: From coding efficiency to reaction times,, Brain Res., 1434 (2012), 47. doi: 10.1016/j.brainres.2011.08.014.

[9]

A. Borst and F. E. Theunissen, Information theory and neural coding,, Nature Neurosci., 2 (1999), 947.

[10]

N. Brunel and J.-P. Nadal, Mutual information, Fisher information, and population coding,, Neural Computation, 10 (1998), 1731. doi: 10.1162/089976698300017115.

[11]

R. S. Chhikara and J. L. Folks, "The Inverse Gaussian Distribution: Theory, Methodology, and Applications,", Marcel Dekker, (1989).

[12]

M. Cohen, The fisher information and convexity,, IEEE Transactions on Information Theory, 14 (1968), 591. doi: 10.1109/TIT.1968.1054175.

[13]

D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events,", Methuen & Co., (1966).

[14]

J. P. Cunningham, V. Gilja, S. I. Ryu and K. V. Shenoy, Methods for estimating neural firing rates, and their application to brain-machine interfaces,, Neural Networks, 22 (2009), 1235. doi: 10.1016/j.neunet.2009.02.004.

[15]

J. P. Cunningham, B. M. Yu, K. V. Shenoy and M. Sahani, Inferring neural firing rates from spike trains using Gaussian processes,, in, (2008), 329.

[16]

D. J. Daley and D. Vere-Jones, "An Introduction to the Theory of Point Processes. Vol. I. Elementary Theory and Methods,", Second edition, (2003).

[17]

P. Duchamp-Viret, L. Kostal, M. Chaput, P. Lánsky and J.-P. Rospars, Patterns of spontaneous activity in single rat olfactory receptor neurons are different in normally breathing and tracheotomized animals,, J. Neurobiology, 65 (2005), 97. doi: 10.1002/neu.20177.

[18]

R. G. Gallager, "Information Theory and Reliable Communication,", John Wiley & Sons, (1968).

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0.

[20]

I. J. Good, "Probability and the Weighing of Evidence,", Charles Griffin & Co., (1950).

[21]

I. J. Good and R. A. Gaskins, Nonparametric roughness penalties for probability densities,, Biometrika, 58 (1971), 255.

[22]

P. E. Greenwood and P. Lánský, Optimal signal estimation in neuronal models,, Neural Comput., 17 (2005), 2240. doi: 10.1162/0899766054615653.

[23]

P. E. Greenwood and P. Lánský, Optimum signal in a simple neuronal model with signal-dependent noise,, Biol. Cybern., 92 (2005), 199. doi: 10.1007/s00422-005-0545-3.

[24]

P. E. Greenwood, L. M. Ward, D. F. Russell, A. Neiman and F. Moss, Stochastic resonance enhances the electrosensory information available to paddlefish for prey capture,, Phys. Rev. Lett., 84 (2000), 4773. doi: 10.1103/PhysRevLett.84.4773.

[25]

A. Grémiaux, T. Nowotny, D. Martinez, P. Lucas and J.-P. Rospars, Modelling the signal delivered by a population of first-order neurons in a moth olfactory system,, Brain Res., 1434 (2012), 123. doi: 10.1016/j.brainres.2011.09.035.

[26]

P. J. Huber, "Robust Statistics,", Wiley Series in Probability and Mathematical Statistics, (1981).

[27]

S. Ikeda and J. H. Manton, Capacity of a single spiking neuron channel,, Neural Comput., 21 (2009), 1714. doi: 10.1162/neco.2009.05-08-792.

[28]

S. Iyengar and Q. Liao, Modeling neural activity using the generalized inverse gaussian distribution,, Biological Cybernetics, 77 (1997), 289. doi: 10.1007/s004220050390.

[29]

B. Jørgensen, "Statistical Properties of the Generalized Inverse Gaussian Distribution,", Lecture Notes in Statistics, 9 (1982).

[30]

A. M. Kagan, I. V. Linnik and C. R. Rao, "Characterization Problems in Mathematical Statistics,", John Wiley & Sons, (1973).

[31]

R. E. Kass and V. Ventura, A spike-train probability model,, Neural Computation, 13 (2001), 1713. doi: 10.1162/08997660152469314.

[32]

S. M. Kay, "Fundamentals of Statistical Signal Processing: Estimation Theory,", Prentice Hall, (1993).

[33]

L. Kostal, Information capacity in the weak-signal approximation,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.026115.

[34]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code,, Brain Res., 1434 (2012), 136. doi: 10.1016/j.brainres.2011.07.007.

[35]

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

[36]

L. Kostal and O. Pokora, Nonparametric estimation of information-based measures of statistical dispersion,, Entropy, 14 (2012), 1221. doi: 10.3390/e14071221.

[37]

S. Koyama, Coding efficiency and detectability of rate fluctuations with non-Poisson neuronal firing,, in, (2013).

[38]

S. Koyama and R. E. Kass, Spike train probability models for stimulus-driven leaky integrate-and-fire neurons,, Neural Computation, 20 (2008), 1776. doi: 10.1162/neco.2008.06-07-540.

[39]

S. Kullback, "Information Theory and Statistics,", Dover Publications, (1968).

[40]

E. L. Lehmann and G. Casella, "Theory of Point Estimation,", Second edition, (1998).

[41]

M. W. Levine, The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells,, Biol. Cybern., 65 (1991), 459. doi: 10.1007/BF00204659.

[42]

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.

[43]

D. H. Perkel and T. H. Bullock, Neural coding,, Neurosci. Res. Prog. Sum., 3 (1968), 405.

[44]

J. W. Pillow, Time-rescaling methods for the estimation and assessment of non-Poisson neural encoding models,, in, (2008), 1473.

[45]

E. J. G. Pitman, "Some Basic Theory for Statistical Inference,", Monographs on Applied Probability and Statistics, (1979).

[46]

C. Pouzat and A. Chaffiol, Automatic spike train analysis and report generation. An implementation with R, R2HTML and STAR,, J. Neurosci. Methods, 181 (2009), 119. doi: 10.1016/j.jneumeth.2009.01.037.

[47]

D. S. Reich, J. D. Victor and B. W. Knight, The power ratio and the interval map: Spiking models and extracellular recordings,, Journal of Neuroscience, 18 (1998), 10090.

[48]

B. J. Richmond and L. M. Optican, Temporal encoding of two-dimensional patterns by single units in primate inferior temporal cortex. II. Quantification of response waveform,, Journal of Neurophysiology, 57 (1987), 147.

[49]

J. J. Rissanen, Fisher information and stochastic complexity,, IEEE Trans. Inf. Theory, 42 (1996), 40. doi: 10.1109/18.481776.

[50]

L. J. Savage, "The Foundations of Statistics,", John Wiley & Sons, (1954).

[51]

H. S. Seung and H. Sompolinsky, Simple models for reading neuronal population codes,, Proceedings of the National Academy of Sciences of the United States of America, 90 (1993), 10749. doi: 10.1073/pnas.90.22.10749.

[52]

C. E. Shannon and W. Weaver, "The Mathematical Theory of Communication,", University of Illinois Press, (1949).

[53]

R. B. Stein, The information capacity of nerve cells using a frequency code,, Biophys. J., 7 (1967), 797. doi: 10.1016/S0006-3495(67)86623-2.

[54]

F. Theunissen and J. P. Miller, Temporal encoding in nervous systems: A rigorous definition,, J. Comput. Neurosci., 2 (1995), 149. doi: 10.1007/BF00961885.

[55]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology, Vol. 2. Nonlinear and Stochastic Theories,", Cambridge Studies in Mathematical Biology, 8 (1988).

[56]

A. W. van der Vaart, "Asymptotic Statistics,", Cambridge Series in Statistical and Probabilistic Mathematics, 3 (1998).

[57]

K. Zhang and T. Sejnowski, Neural tuning: To sharpen or broaden?,, Neural Computation, 11 (1999), 75.

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