Some theoretical results on neural spike train probability models. (English) Zbl 1129.62101

Summary: This article contains two main theoretical results on neural spike train models, using the counting or point processes on the real line as a model for the spike train. The first part of this article considers template matching of multiple spike trains. \(P\)-values for the occurrences of a given template or pattern in a set of spike trains are computed using a general scoring system. By identifying the pattern with an experimental stimulus, multiple spike trains can be deciphered to provide useful information.
The second part of the article assumes that the counting process has a conditional intensity function that is a product of a free firing rate function \(s\), which depends only on the stimulus, and a recovery function \(r\), which depends only on the time since the last spike. If \(s\) and \(r\) belong to a \(q\)-smooth class of functions, it is proved that sieve maximum likelihood estimators for \(s\) and \(r\) achieve the optimal convergence rate (except for a logarithmic factor) under \(L_{1}\) loss.


62P10 Applications of statistics to biology and medical sciences; meta analysis
60G35 Signal detection and filtering (aspects of stochastic processes)
92C20 Neural biology
62M99 Inference from stochastic processes
Full Text: DOI arXiv Euclid


[1] Berry, M. J. and Meister, M. (1998). Refractoriness and neural precision. J. Neurosci. 18 2200-2211.
[2] Birgé, L. (1983). Approximation dans les espaces métriques et théorie de l’estimation. Z. Wahrsch. Verw. Gebiete 65 181-237. · Zbl 0506.62026
[3] Brillinger, D. R. (1992). Nerve cell spike train data analysis: A progression of technique. J. Amer. Statist. Assoc. 87 260-271.
[4] Brown, E. N., Kass, R. E. and Mitra, P. P. (2004). Multiple neural spike train data analysis: State-of-the-art and future challenges. Nature Neurosci. 7 456-461.
[5] Chan, H. P. (2003). Upper bounds and importance sampling of \(p\)-values for DNA and protein sequence alignments. Bernoulli 9 183-199. · Zbl 1015.62111
[6] Chan, H. P. and Loh, W.-L. (2006). Some theoretical results on neural spike train probability models. Available at www.arxiv.org/abs/math.st/0703829. · Zbl 1129.62101
[7] Chi, Z. (2005). Large deviations for template matching between point processes. Ann. Appl. Probab. 15 153-174. · Zbl 1068.60035
[8] Chi, Z., Rauske, P. L. and Margoliash, D. (2003). Pattern filtering for detection of neural activity, with examples from HVc activity during sleep in zebra finches. Neural Comput. 15 2307-2337. · Zbl 1085.68618
[9] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes 1 . Elementary Theory and Methods , 2nd ed. Springer, New York. · Zbl 1026.60061
[10] Dave, A. S. and Margoliash, D. (2000). Song replay during sleep and computational rules for sensorimotor vocal learning. Science 290 812-816. · Zbl 1203.37138
[11] Dayan, P. and Abbott, L. F. (2001). Theoretical Neuroscience : Computational and Mathematical Modeling of Neural Systems . MIT Press, Cambridge, MA. · Zbl 1051.92010
[12] Dayhoff, J. E. and Gerstein, G. L. (1983). Favored patterns in spike trains. I. Detection. J. Neurophysiol. 49 1334-1348.
[13] Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press. · Zbl 0951.60033
[14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003
[15] Grün, S., Diesmann, M. and Aertsen, A. (2002). Unitary events in multiple single-neuron spiking activity. I. Detection and significance. Neural Comput. 14 43-80. · Zbl 0985.62093
[16] Ibragimov, I. A. and Has’minskii, R. Z. (1981). Statistical Estimation : Asymptotic Theory . Springer, New York.
[17] Johnson, D. H. and Swami, A. (1983). The transmission of signals by auditory-nerve fiber discharge patterns. J. Acoust. Soc. Am. 74 493-501.
[18] Kass, R. E. and Ventura, V. (2001). A spike-train probability model. Neural Comput. 13 1713-1720. · Zbl 0985.92017
[19] Kolmogorov, A. N. and Tihomirov, V. M. (1959). \(\varepsilon\)-entropy and \(\varepsilon\)-capacity of sets in function spaces. Uspekhi Mat. Nauk 14 3-86 [in Russian; English transl. Amer. Math. Soc. Transl. (2) 17 277-364 (1961)]., Mathematical Reviews (MathSciNet):
[20] Lai, T. L. and Shan, J. Z. (1999). Efficient recursive algorithms for detection of abrupt changes in signals and control systems. IEEE Trans. Automat. Control 44 952-966. · Zbl 0956.93060
[21] Miller, M. I. (1985). Algorithms for removing recovery-related distortion from auditory-nerve discharge patterns. J. Acoust. Soc. Am. 77 1452-1464. · Zbl 0579.92005
[22] Mooney, R. (2000). Different subthreshold mechanisms underlie song selectivity in identified HVc neurons of the zebra finch. J. Neurosci. 20 5420-5436.
[23] Nádasdy, Z., Hirase, H., Czurkó, A., Csicsvari, J. and Buzsáki, G. (1999). Replay and time compression of recurring spike sequences in the hippocampus. J. Neurosci. 19 9497-9507.
[24] Ossiander, M. (1987). A central limit theorem under metric entropy with \(L_2\) bracketing. Ann. Probab. 15 897-919. · Zbl 0665.60036
[25] Shen, X. and Wong, W. H. (1994). Convergence rate of sieve estimates. Ann. Statist. 22 580-615. · Zbl 0805.62008
[26] Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673-684. · Zbl 0353.62044
[27] Siegmund, D. (1985). Sequential Analysis . Tests and Confidence Intervals . Springer, New York. · Zbl 0573.62071
[28] Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Statist. 36 546-551. · Zbl 0135.19204
[29] Ventura, V., Carta, R., Kass, R. E., Gettner, S. N. and Olson, C. R. (2002). Statistical analysis of temporal evolution in single-neuron firing rates. Biostatistics 3 1-20. · Zbl 1133.62372
[30] Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339-362. · Zbl 0829.62002
[31] Yatracos, Y. (1988). A lower bound on the error in nonparametric regression type problems. Ann. Statist. 16 1180-1187. · Zbl 0651.62028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.