zbMATH — the first resource for mathematics

Maximum likelihood identification of neural point process systems. (English) Zbl 0658.92007
The theory of stochastic point processes is used to describe and to detect functional relationships between two neurons. The representation of the point processes involved by the stochastic intensity is used. For the form of the stochastic intensity assumed the maximum likelihood estimates are derived and their asymptotic properties are presented in the form of two theorems. The model is a version of the additive risk model presented by O. Aalen, Ann. Stat. 6, 701-726 (1978; Zbl 0389.62025), in the context of survival analysis.
A computational method is also devised to make the maximum likelihood method applicable. An algorithm, derived using the procedure proposed by A. P. Dempster, N. M. Laird and D. B. Rubin, J. R. Stat. Soc., Ser. B 39, 1-38 (1977; Zbl 0364.62022), is proposed for the iterative solution of the likelihood equations. Convergence results for the algorithm as well as a sequence of upper bounds on the log-likelihood values of iterates are shown. The results of the paper are illustrated on simulated experiments.
Reviewer: P.Lánský

92Cxx Physiological, cellular and medical topics
62M99 Inference from stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text: DOI
[1] Aalen O (1978) Nonparametric inference for a family of counting processes. Ann Stat 6:701–726 · Zbl 0389.62025 · doi:10.1214/aos/1176344247
[2] Aertsen Ad MHJ, Gerstein GL (1985) Evaluation of neuronal connectivity: sensitivity of cross-correlation. Brain Res 340:341–354 · doi:10.1016/0006-8993(85)90931-X
[3] Billingsley P (1961) Statistical inference for Markov processes. The University of Chicago Press, Chicago · Zbl 0106.34201
[4] Boogaard H van den (1986) Maximum likelihood estimations in a nonlinear self-exciting point process model. Biol Cybern 55:219–225 · Zbl 0623.62080 · doi:10.1007/BF00355597
[5] Borisyuk GN, Borisyuk RM, Kirillov AB, Kovalenko EI, Kryukov VI (1985) A new statistical method for identifying interconnections between neuronal network elements. Biol Cybern 52:301–306 · Zbl 0576.92016 · doi:10.1007/BF00355752
[6] Breŕnaud P (1981) Point processes and queues: martingale dynamics. Springer, Berlin Heidelberg New York
[7] Brillinger DR (1975) The identification of point process systems. Ann Probab 3:909–929 · Zbl 0348.60076 · doi:10.1214/aop/1176996218
[8] Chornoboy ES (1986) Maximum likelihood techniques for the identification of neural point processes, Ph.D. Thesis. The Johns Hopkins School of Medicine, Baltimore, Md
[9] Cover TM (1984) An algorithm for maximizing expected log investment return. IEEE Trans IT-30:369–373 · Zbl 0541.90007 · doi:10.1109/TIT.1984.1056869
[10] Cox DR (1972) The statistical analysis of dependencies in point processes. In: Lewis PA (ed) Stochastic point processes. Wiley, New York, pp 55–66
[11] Csiszár I, Tusnády G (1984) Information geometry and alternating minimization procedures. Statistics and decisions. [Suppl.] 1:205–237 · Zbl 0547.60004
[12] Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc, Ser B 39:1–38 · Zbl 0364.62022
[13] Habib MH, Sen PK (1985) Non-stationary stochastic point-process models in neurophysiology with applications to learning, In: Sen PK (ed) Biostatistics: statistics in biomedical. Public health and environmental sciences. Elsevier, North-Holland
[14] Hawkes AG (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58:83–90 · Zbl 0219.60029 · doi:10.1093/biomet/58.1.83
[15] Karr AF (1986) Point processes and their statistical inference. Dekker, New York · Zbl 0601.62120
[16] Karr AF (1987) Maximum likelihood estimation in the multiplicative intensity model. Ann Stat 15:473–490 · Zbl 0628.62086 · doi:10.1214/aos/1176350356
[17] Kirkwood PA (1979) On the use and interpretation of cross-correlation measurements in the mammalian central nervous system. J Neurosci Methods 1:107–132 · doi:10.1016/0165-0270(79)90009-8
[18] Luenberger DG (1984) Linear and nonlinear programming. Addison-Wesley, Reading, Mass
[19] McKeague IW (1986) Nonparametric inference in additive risk models for counting processes, Florida State University Technical Report No. M741
[20] Marmarelis PZ, Marmarelis VZ (1978) Analysis of physiological systems: the white-noise approach. Plenum Press, New York
[21] Ogata Y (1978) The asymptotic behavior of maximum likelihood estimators for stationary point processes, Ann Inst Stat Math Part A 30:243–261 · Zbl 0451.62067 · doi:10.1007/BF02480216
[22] Peters C, Coberly WA (1976) The numerical evaluation of the maximum likelihood estimate of mixture proportions, Commun Statist Theor Methods A 5:1127–1135 · Zbl 0364.62023 · doi:10.1080/03610927608827429
[23] Pyatigorskii B Ya, Kastyukov AI, Chinarov VA, Cherkasskii VL (1984) Deterministic and stochastic identification of neurophysiological systems. Neirofiziologiya 16:419–434
[24] Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans MI-1:113–121 · doi:10.1109/TMI.1982.4307558
[25] Vardi Y, Shepp LA, Kaufman L (1983) A statistical model for positron emission tomography. J Am Stat Assoc 80:8–20 · Zbl 0561.62094 · doi:10.2307/2288030
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.