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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, Inc., New York, 1966.  Google Scholar

[2]

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

[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), 25-37. doi: 10.1016/S0165-0270(00)00344-7.  Google Scholar

[4]

M. Berman, Inhomogeneous and modulated gamma processes, Biometrika, 68 (1981), 143-152. doi: 10.1093/biomet/68.1.143.  Google Scholar

[5]

J. M. Bernardo, Reference posterior distributions for Bayesian inference. With discussion, J. Roy. Stat. Soc. B, 41 (1979), 113-147.  Google Scholar

[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), 409-413. doi: 10.1007/s10051-001-8691-4.  Google Scholar

[7]

G. S. Bhumbra, A. N. Inyushkin and R. E. J. Dyball, Assessment of spike activity in the supraoptic nucleus, J. Neuroendocrinol., 16 (2004), 390-397. doi: 10.1111/j.0953-8194.2004.01166.x.  Google Scholar

[8]

L. Bonnasse-Gahot and J.-P. Nadal, Perception of categories: From coding efficiency to reaction times, Brain Res., 1434 (2012), 47-61. doi: 10.1016/j.brainres.2011.08.014.  Google Scholar

[9]

A. Borst and F. E. Theunissen, Information theory and neural coding, Nature Neurosci., 2 (1999), 947-958. Google Scholar

[10]

N. Brunel and J.-P. Nadal, Mutual information, Fisher information, and population coding, Neural Computation, 10 (1998), 1731-1757. doi: 10.1162/089976698300017115.  Google Scholar

[11]

R. S. Chhikara and J. L. Folks, "The Inverse Gaussian Distribution: Theory, Methodology, and Applications," Marcel Dekker, New York, 1989. Google Scholar

[12]

M. Cohen, The fisher information and convexity, IEEE Transactions on Information Theory, 14 (1968), 591-592. doi: 10.1109/TIT.1968.1054175.  Google Scholar

[13]

D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events," Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1966.  Google Scholar

[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), 1235-1246. doi: 10.1016/j.neunet.2009.02.004.  Google Scholar

[15]

J. P. Cunningham, B. M. Yu, K. V. Shenoy and M. Sahani, Inferring neural firing rates from spike trains using Gaussian processes, in "Neural Information Processing Systems" (eds. J. C. Platt, D. Koller, Y. Singer and S. Roweis), Vol. 20, (2008), 329-336. Google Scholar

[16]

D. J. Daley and D. Vere-Jones, "An Introduction to the Theory of Point Processes. Vol. I. Elementary Theory and Methods," Second edition, Probability and its Applications (New York), Springer-Verlag, New York, 2003.  Google Scholar

[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), 97-114. doi: 10.1002/neu.20177.  Google Scholar

[18]

R. G. Gallager, "Information Theory and Reliable Communication," John Wiley & Sons, Inc., New York, 1968. Google Scholar

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[20]

I. J. Good, "Probability and the Weighing of Evidence," Charles Griffin & Co., Ltd., London; Hafner Publishing Co., New York, N. Y., 1950.  Google Scholar

[21]

I. J. Good and R. A. Gaskins, Nonparametric roughness penalties for probability densities, Biometrika, 58 (1971), 255-277.  Google Scholar

[22]

P. E. Greenwood and P. Lánský, Optimal signal estimation in neuronal models, Neural Comput., 17 (2005), 2240-2257. doi: 10.1162/0899766054615653.  Google Scholar

[23]

P. E. Greenwood and P. Lánský, Optimum signal in a simple neuronal model with signal-dependent noise, Biol. Cybern., 92 (2005), 199-205. doi: 10.1007/s00422-005-0545-3.  Google Scholar

[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), 4773-4776. doi: 10.1103/PhysRevLett.84.4773.  Google Scholar

[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), 123-135. doi: 10.1016/j.brainres.2011.09.035.  Google Scholar

[26]

P. J. Huber, "Robust Statistics," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[27]

S. Ikeda and J. H. Manton, Capacity of a single spiking neuron channel, Neural Comput., 21 (2009), 1714-1748. doi: 10.1162/neco.2009.05-08-792.  Google Scholar

[28]

S. Iyengar and Q. Liao, Modeling neural activity using the generalized inverse gaussian distribution, Biological Cybernetics, 77 (1997), 289-295. doi: 10.1007/s004220050390.  Google Scholar

[29]

B. Jørgensen, "Statistical Properties of the Generalized Inverse Gaussian Distribution," Lecture Notes in Statistics, 9, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[30]

A. M. Kagan, I. V. Linnik and C. R. Rao, "Characterization Problems in Mathematical Statistics," John Wiley & Sons, New York, 1973. Google Scholar

[31]

R. E. Kass and V. Ventura, A spike-train probability model, Neural Computation, 13 (2001), 1713-1720. doi: 10.1162/08997660152469314.  Google Scholar

[32]

S. M. Kay, "Fundamentals of Statistical Signal Processing: Estimation Theory," Prentice Hall, New Jersey, 1993. Google Scholar

[33]

L. Kostal, Information capacity in the weak-signal approximation, Phys. Rev. E, 82 (2010), 026115. doi: 10.1103/PhysRevE.82.026115.  Google Scholar

[34]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code, Brain Res., 1434 (2012), 136-141. doi: 10.1016/j.brainres.2011.07.007.  Google Scholar

[35]

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

[36]

L. Kostal and O. Pokora, Nonparametric estimation of information-based measures of statistical dispersion, Entropy, 14 (2012), 1221-1233. doi: 10.3390/e14071221.  Google Scholar

[37]

S. Koyama, Coding efficiency and detectability of rate fluctuations with non-Poisson neuronal firing, in "Neural Information Processing Systems," Vol. 25, The Institute of Statistical Mathematics, 2013. Google Scholar

[38]

S. Koyama and R. E. Kass, Spike train probability models for stimulus-driven leaky integrate-and-fire neurons, Neural Computation, 20 (2008), 1776-1795. doi: 10.1162/neco.2008.06-07-540.  Google Scholar

[39]

S. Kullback, "Information Theory and Statistics," Dover Publications, Inc., Mineola, New York, 1968.  Google Scholar

[40]

E. L. Lehmann and G. Casella, "Theory of Point Estimation," Second edition, Springer Texts in Statistics, Springer-Verlag, New York, 1998.  Google Scholar

[41]

M. W. Levine, The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells, Biol. Cybern., 65 (1991), 459-467. doi: 10.1007/BF00204659.  Google Scholar

[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), 1325-1343. doi: 10.1162/neco.2007.01-07-442.  Google Scholar

[43]

D. H. Perkel and T. H. Bullock, Neural coding, Neurosci. Res. Prog. Sum., 3 (1968), 405-527. Google Scholar

[44]

J. W. Pillow, Time-rescaling methods for the estimation and assessment of non-Poisson neural encoding models, in "Neural Information Processing Systems" (eds. Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams and A. Culotta), Vol. 22, (2008), 1473-1481. Google Scholar

[45]

E. J. G. Pitman, "Some Basic Theory for Statistical Inference," Monographs on Applied Probability and Statistics, Chapman and Hall, London; A Halsted Press Book, John Wiley & Sons, New York, 1979.  Google Scholar

[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), 119-144. doi: 10.1016/j.jneumeth.2009.01.037.  Google Scholar

[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), 10090-10104. Google Scholar

[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), 147-161. Google Scholar

[49]

J. J. Rissanen, Fisher information and stochastic complexity, IEEE Trans. Inf. Theory, 42 (1996), 40-47. doi: 10.1109/18.481776.  Google Scholar

[50]

L. J. Savage, "The Foundations of Statistics," John Wiley & Sons, Inc., New York; Chapman & Hill, Ltd., London, 1954.  Google Scholar

[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), 10749-10753. doi: 10.1073/pnas.90.22.10749.  Google Scholar

[52]

C. E. Shannon and W. Weaver, "The Mathematical Theory of Communication," University of Illinois Press, Urbana, Illinois, 1949.  Google Scholar

[53]

R. B. Stein, The information capacity of nerve cells using a frequency code, Biophys. J., 7 (1967), 797-826. doi: 10.1016/S0006-3495(67)86623-2.  Google Scholar

[54]

F. Theunissen and J. P. Miller, Temporal encoding in nervous systems: A rigorous definition, J. Comput. Neurosci., 2 (1995), 149-162. doi: 10.1007/BF00961885.  Google Scholar

[55]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology, Vol. 2. Nonlinear and Stochastic Theories," Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988.  Google Scholar

[56]

A. W. van der Vaart, "Asymptotic Statistics," Cambridge Series in Statistical and Probabilistic Mathematics, 3, Cambridge University Press, Cambridge, 1998.  Google Scholar

[57]

K. Zhang and T. Sejnowski, Neural tuning: To sharpen or broaden?, Neural Computation, 11 (1999), 75-84. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables," Dover Publications, Inc., New York, 1966.  Google Scholar

[2]

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

[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), 25-37. doi: 10.1016/S0165-0270(00)00344-7.  Google Scholar

[4]

M. Berman, Inhomogeneous and modulated gamma processes, Biometrika, 68 (1981), 143-152. doi: 10.1093/biomet/68.1.143.  Google Scholar

[5]

J. M. Bernardo, Reference posterior distributions for Bayesian inference. With discussion, J. Roy. Stat. Soc. B, 41 (1979), 113-147.  Google Scholar

[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), 409-413. doi: 10.1007/s10051-001-8691-4.  Google Scholar

[7]

G. S. Bhumbra, A. N. Inyushkin and R. E. J. Dyball, Assessment of spike activity in the supraoptic nucleus, J. Neuroendocrinol., 16 (2004), 390-397. doi: 10.1111/j.0953-8194.2004.01166.x.  Google Scholar

[8]

L. Bonnasse-Gahot and J.-P. Nadal, Perception of categories: From coding efficiency to reaction times, Brain Res., 1434 (2012), 47-61. doi: 10.1016/j.brainres.2011.08.014.  Google Scholar

[9]

A. Borst and F. E. Theunissen, Information theory and neural coding, Nature Neurosci., 2 (1999), 947-958. Google Scholar

[10]

N. Brunel and J.-P. Nadal, Mutual information, Fisher information, and population coding, Neural Computation, 10 (1998), 1731-1757. doi: 10.1162/089976698300017115.  Google Scholar

[11]

R. S. Chhikara and J. L. Folks, "The Inverse Gaussian Distribution: Theory, Methodology, and Applications," Marcel Dekker, New York, 1989. Google Scholar

[12]

M. Cohen, The fisher information and convexity, IEEE Transactions on Information Theory, 14 (1968), 591-592. doi: 10.1109/TIT.1968.1054175.  Google Scholar

[13]

D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events," Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1966.  Google Scholar

[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), 1235-1246. doi: 10.1016/j.neunet.2009.02.004.  Google Scholar

[15]

J. P. Cunningham, B. M. Yu, K. V. Shenoy and M. Sahani, Inferring neural firing rates from spike trains using Gaussian processes, in "Neural Information Processing Systems" (eds. J. C. Platt, D. Koller, Y. Singer and S. Roweis), Vol. 20, (2008), 329-336. Google Scholar

[16]

D. J. Daley and D. Vere-Jones, "An Introduction to the Theory of Point Processes. Vol. I. Elementary Theory and Methods," Second edition, Probability and its Applications (New York), Springer-Verlag, New York, 2003.  Google Scholar

[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), 97-114. doi: 10.1002/neu.20177.  Google Scholar

[18]

R. G. Gallager, "Information Theory and Reliable Communication," John Wiley & Sons, Inc., New York, 1968. Google Scholar

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[20]

I. J. Good, "Probability and the Weighing of Evidence," Charles Griffin & Co., Ltd., London; Hafner Publishing Co., New York, N. Y., 1950.  Google Scholar

[21]

I. J. Good and R. A. Gaskins, Nonparametric roughness penalties for probability densities, Biometrika, 58 (1971), 255-277.  Google Scholar

[22]

P. E. Greenwood and P. Lánský, Optimal signal estimation in neuronal models, Neural Comput., 17 (2005), 2240-2257. doi: 10.1162/0899766054615653.  Google Scholar

[23]

P. E. Greenwood and P. Lánský, Optimum signal in a simple neuronal model with signal-dependent noise, Biol. Cybern., 92 (2005), 199-205. doi: 10.1007/s00422-005-0545-3.  Google Scholar

[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), 4773-4776. doi: 10.1103/PhysRevLett.84.4773.  Google Scholar

[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), 123-135. doi: 10.1016/j.brainres.2011.09.035.  Google Scholar

[26]

P. J. Huber, "Robust Statistics," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[27]

S. Ikeda and J. H. Manton, Capacity of a single spiking neuron channel, Neural Comput., 21 (2009), 1714-1748. doi: 10.1162/neco.2009.05-08-792.  Google Scholar

[28]

S. Iyengar and Q. Liao, Modeling neural activity using the generalized inverse gaussian distribution, Biological Cybernetics, 77 (1997), 289-295. doi: 10.1007/s004220050390.  Google Scholar

[29]

B. Jørgensen, "Statistical Properties of the Generalized Inverse Gaussian Distribution," Lecture Notes in Statistics, 9, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[30]

A. M. Kagan, I. V. Linnik and C. R. Rao, "Characterization Problems in Mathematical Statistics," John Wiley & Sons, New York, 1973. Google Scholar

[31]

R. E. Kass and V. Ventura, A spike-train probability model, Neural Computation, 13 (2001), 1713-1720. doi: 10.1162/08997660152469314.  Google Scholar

[32]

S. M. Kay, "Fundamentals of Statistical Signal Processing: Estimation Theory," Prentice Hall, New Jersey, 1993. Google Scholar

[33]

L. Kostal, Information capacity in the weak-signal approximation, Phys. Rev. E, 82 (2010), 026115. doi: 10.1103/PhysRevE.82.026115.  Google Scholar

[34]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code, Brain Res., 1434 (2012), 136-141. doi: 10.1016/j.brainres.2011.07.007.  Google Scholar

[35]

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

[36]

L. Kostal and O. Pokora, Nonparametric estimation of information-based measures of statistical dispersion, Entropy, 14 (2012), 1221-1233. doi: 10.3390/e14071221.  Google Scholar

[37]

S. Koyama, Coding efficiency and detectability of rate fluctuations with non-Poisson neuronal firing, in "Neural Information Processing Systems," Vol. 25, The Institute of Statistical Mathematics, 2013. Google Scholar

[38]

S. Koyama and R. E. Kass, Spike train probability models for stimulus-driven leaky integrate-and-fire neurons, Neural Computation, 20 (2008), 1776-1795. doi: 10.1162/neco.2008.06-07-540.  Google Scholar

[39]

S. Kullback, "Information Theory and Statistics," Dover Publications, Inc., Mineola, New York, 1968.  Google Scholar

[40]

E. L. Lehmann and G. Casella, "Theory of Point Estimation," Second edition, Springer Texts in Statistics, Springer-Verlag, New York, 1998.  Google Scholar

[41]

M. W. Levine, The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells, Biol. Cybern., 65 (1991), 459-467. doi: 10.1007/BF00204659.  Google Scholar

[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), 1325-1343. doi: 10.1162/neco.2007.01-07-442.  Google Scholar

[43]

D. H. Perkel and T. H. Bullock, Neural coding, Neurosci. Res. Prog. Sum., 3 (1968), 405-527. Google Scholar

[44]

J. W. Pillow, Time-rescaling methods for the estimation and assessment of non-Poisson neural encoding models, in "Neural Information Processing Systems" (eds. Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams and A. Culotta), Vol. 22, (2008), 1473-1481. Google Scholar

[45]

E. J. G. Pitman, "Some Basic Theory for Statistical Inference," Monographs on Applied Probability and Statistics, Chapman and Hall, London; A Halsted Press Book, John Wiley & Sons, New York, 1979.  Google Scholar

[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), 119-144. doi: 10.1016/j.jneumeth.2009.01.037.  Google Scholar

[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), 10090-10104. Google Scholar

[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), 147-161. Google Scholar

[49]

J. J. Rissanen, Fisher information and stochastic complexity, IEEE Trans. Inf. Theory, 42 (1996), 40-47. doi: 10.1109/18.481776.  Google Scholar

[50]

L. J. Savage, "The Foundations of Statistics," John Wiley & Sons, Inc., New York; Chapman & Hill, Ltd., London, 1954.  Google Scholar

[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), 10749-10753. doi: 10.1073/pnas.90.22.10749.  Google Scholar

[52]

C. E. Shannon and W. Weaver, "The Mathematical Theory of Communication," University of Illinois Press, Urbana, Illinois, 1949.  Google Scholar

[53]

R. B. Stein, The information capacity of nerve cells using a frequency code, Biophys. J., 7 (1967), 797-826. doi: 10.1016/S0006-3495(67)86623-2.  Google Scholar

[54]

F. Theunissen and J. P. Miller, Temporal encoding in nervous systems: A rigorous definition, J. Comput. Neurosci., 2 (1995), 149-162. doi: 10.1007/BF00961885.  Google Scholar

[55]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology, Vol. 2. Nonlinear and Stochastic Theories," Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988.  Google Scholar

[56]

A. W. van der Vaart, "Asymptotic Statistics," Cambridge Series in Statistical and Probabilistic Mathematics, 3, Cambridge University Press, Cambridge, 1998.  Google Scholar

[57]

K. Zhang and T. Sejnowski, Neural tuning: To sharpen or broaden?, Neural Computation, 11 (1999), 75-84. Google Scholar

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