Abstract
Using the theory of random point processes, a method is presented whereby functional relationships between neurons can be detected and modeled. The method is based on a point process characterization involving stochastic intensities and an additive rate function model. Estimates are based on the maximum likelihood (ML) principle and asymptotic properties are examined in the absence of a stationarity assumption. An iterative algorithm that computes the ML estimates is presented. It is based on the expectation/maximization (EM) procedure of Dempster et al. (1977) and makes ML identification accessible to models requiring many parameters. Examples illustrating the use of the method are also presented. These examples are derived from simulations of simple neural systems that cannot be identified using correlation techniques. It is shown that the ML method correctly identifies each of these systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aalen O (1978) Nonparametric inference for a family of counting processes. Ann Stat 6:701–726
Aertsen Ad MHJ, Gerstein GL (1985) Evaluation of neuronal connectivity: sensitivity of cross-correlation. Brain Res 340:341–354
Billingsley P (1961) Statistical inference for Markov processes. The University of Chicago Press, Chicago
Boogaard H van den (1986) Maximum likelihood estimations in a nonlinear self-exciting point process model. Biol Cybern 55:219–225
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
Breŕnaud P (1981) Point processes and queues: martingale dynamics. Springer, Berlin Heidelberg New York
Brillinger DR (1975) The identification of point process systems. Ann Probab 3:909–929
Chornoboy ES (1986) Maximum likelihood techniques for the identification of neural point processes, Ph.D. Thesis. The Johns Hopkins School of Medicine, Baltimore, Md
Cover TM (1984) An algorithm for maximizing expected log investment return. IEEE Trans IT-30:369–373
Cox DR (1972) The statistical analysis of dependencies in point processes. In: Lewis PA (ed) Stochastic point processes. Wiley, New York, pp 55–66
Csiszár I, Tusnády G (1984) Information geometry and alternating minimization procedures. Statistics and decisions. [Suppl.] 1:205–237
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
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
Hawkes AG (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58:83–90
Karr AF (1986) Point processes and their statistical inference. Dekker, New York
Karr AF (1987) Maximum likelihood estimation in the multiplicative intensity model. Ann Stat 15:473–490
Kirkwood PA (1979) On the use and interpretation of cross-correlation measurements in the mammalian central nervous system. J Neurosci Methods 1:107–132
Luenberger DG (1984) Linear and nonlinear programming. Addison-Wesley, Reading, Mass
McKeague IW (1986) Nonparametric inference in additive risk models for counting processes, Florida State University Technical Report No. M741
Marmarelis PZ, Marmarelis VZ (1978) Analysis of physiological systems: the white-noise approach. Plenum Press, New York
Ogata Y (1978) The asymptotic behavior of maximum likelihood estimators for stationary point processes, Ann Inst Stat Math Part A 30:243–261
Peters C, Coberly WA (1976) The numerical evaluation of the maximum likelihood estimate of mixture proportions, Commun Statist Theor Methods A 5:1127–1135
Pyatigorskii B Ya, Kastyukov AI, Chinarov VA, Cherkasskii VL (1984) Deterministic and stochastic identification of neurophysiological systems. Neirofiziologiya 16:419–434
Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans MI-1:113–121
Vardi Y, Shepp LA, Kaufman L (1983) A statistical model for positron emission tomography. J Am Stat Assoc 80:8–20
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chornoboy, E.S., Schramm, L.P. & Karr, A.F. Maximum likelihood identification of neural point process systems. Biol. Cybern. 59, 265–275 (1988). https://doi.org/10.1007/BF00332915
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00332915