Biological Cybernetics

, Volume 73, Issue 1, pp 69–81 | Cite as

Detecting higher-order interactions among the spiking events in a group of neurons

  • L. Martignon
  • H. Von Hassein
  • S. Grün
  • A. Aertsen
  • G. Palm
Original Papers


We propose a formal framework for the description of interactions among groups of neurons. This framework is not restricted to the common case of pair interactions, but also incorporates higher-order interactions, which cannot be reduced to lower-order ones. We derive quantitative measures to detect the presence of such interactions in experimental data, by statistical analysis of the frequency distribution of higher-order correlations in multiple neuron spike train data. Our first step is to represent a frequency distribution as a Markov field on the minimal graph it induces. We then show the invariance of this graph with regard to changes of state. Clearly, only linear Markov fields can be adequately represented by graphs. Higher-order interdependencies, which are reflected by the energy expansion of the distribution, require more complex graphical schemes, like constellations or assembly diagrams, which we introduce and discuss. The coefficients of the energy expansion not only point to the interactions among neurons but are also a measure of their strength. We investigate the statistical meaning of detected interactions in an information theoretic sense and propose minimum relative entropy approximations as null hypotheses for significance tests. We demonstrate the various steps of our method in the situation of an empirical frequency distribution on six neurons, extracted from data on simultaneous multineuron recordings from the frontal cortex of a behaving monkey and close with a brief outlook on future work.


Spike Train Relative Entropy Graphical Scheme Entropy Approximation Minimal Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abeles M (1991) Corticonis. Cambridge University Press, Cambridge, UKGoogle Scholar
  2. Abeles M, Bergman H, Margalit, E, Vaadia E (1993a) Spatiotemporal firing patterns in the frontal cortex of behaving monkeys. Neurophysiol 70:1629–1643PubMedGoogle Scholar
  3. Abeles M, Prut Y, Bergman H, Vaadia E, Aertsen A (1993b) Integration, synchronicity and periodicity. In: Aertsen A (eds) Brain theory: spatio-temporal aspects of brain function. Elsevier, Amsterdam, pp 149–181Google Scholar
  4. Abeles M, Gerstein GL (1988) Detecting spatiotemporal firing patterns among simultaneously recorded single neurons. J Neurophysiol 60:909–924PubMedGoogle Scholar
  5. Aertsen A, Gerstein GL (1991) Dynamic aspects of neuronal cooperativity:fast stimulus-locked modulations of ‘effective connectivity’. In: Krüger J (ed) Neuronal cooperativity. Springer, Berlin Heidelberg New York, pp 52–67Google Scholar
  6. Aertsen A, Bonhoeffer T, Krüger J (1987) Coherent activity in neuronal populations:analysis and interpretation. In: Caianiello ER (eds) Physics of cognitive processes. World Scientific Publishing, Singapore, pp 1–34Google Scholar
  7. Aertsen A, Gerstein GL, Habib MK, Palm G (1989) Dynamics of neuronal firing correlation: modulation of ‘effective connectivity’. J Neurophysiol 61:900–917PubMedGoogle Scholar
  8. Amari S (1982) Differential geometry of curved exponential families — curvatures and information loss. Ann Stat 10:357–385Google Scholar
  9. Amari S (1985) Differential-geometrical methods in statistics. Springer Lecture Notes in Statistics, Vol 28. Springer, Berlin Heidelberg New YorkGoogle Scholar
  10. Amari S (1991) Dualistic geometry of the manifold of higher-order neurons. Neural Networks 4:443–451CrossRefGoogle Scholar
  11. Amari S (1994) Information geometry of the EM and em algorithms for neural networks. Tech report, Department of Mathematical Engineering and Information Physics, Faculty of Engineering, University of TokyoGoogle Scholar
  12. Amari S, Kurata K, Nagaoka H (1992) Information geometry of Boltzmann machines. IEEE Trans Neural Networks 3:260–271CrossRefGoogle Scholar
  13. Amari S. SunHan T (1989) Statistical inference under multiterminal rate restrictions: A differential geometric approach. IEEE Trans Inf Theory 35:217–227CrossRefGoogle Scholar
  14. Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J Stat Soc B 34:75–83Google Scholar
  15. Bishop Y, Fienberg S, Holland P (1989) Discrete multivariate analysis, 10th edn. MIT Press, Cambridge, MassGoogle Scholar
  16. Caianiello E (1975) Synthesis of boolean nets and time behaviour of a general mathematical neuron. Biol Cybern 18:111PubMedGoogle Scholar
  17. Caianiello E (1986) Neuronic equations revisited and completely solved. In: Palm G, Aertsen A (eds) Brain theory, Springer, Berlin Heidelberg New YorkGoogle Scholar
  18. Cover TM, Thomas JA (1991) Elements of information theory, Wiley, New YorkGoogle Scholar
  19. Csiszár I (1975) I-divergence geometry of probability distributions and minimization problems. Ann Probab 3:146–158Google Scholar
  20. Deming WE, Stephan FF (1940) On a least squares adjustment of a sampled frequency table when the expected marginals totals are known. Ann Math Stat 11:427–444Google Scholar
  21. Gerstein G, Aertsen A (1985) Representation of cooperative firing activity among simultaneously recorded neurons. J Ncurophysiol 54:1513–1527Google Scholar
  22. Gerstein GL, Bedenbaugh P, Aertsen A (1989) Neuronal assemblies. IEEE Trans Biomed Eng 36:4–14Google Scholar
  23. Gokhale DV, Kullback S (1978) The information in contingency tables. Dekker, New YorkGoogle Scholar
  24. Griffeath D (1976) Introdction to random fields. Appendix in Knapp A, Kemeny J, Snell J (eds) Denumerablc Markov chains. Springer, Berlin Heidelberg New YorkGoogle Scholar
  25. Grimmett GR (1973) A theorem about random fields. Bull Lond Math Soc 5:81–84Google Scholar
  26. Grün S, Aertsen A, Abeles M, Gerstein G, Palm G (1994) Behaviorrelated neuron group activity in the cortex. Proc 17th Ann Meeting of the European Neurosci Association. Oxford University Press In:ENA, OxfordGoogle Scholar
  27. Grün S, Aertsen A, Abeles M, Gerstein G, Palm G (1994). On the significance of coincident firing in neuron group activity. In: Elsner N, Breer H (eds) Sensory transduction. Stuttgart, Thieme. p558Google Scholar
  28. Hammersley JM, Clifford P (1968) Markov fields on finite graphs and lattices. University of California, BerkeleyGoogle Scholar
  29. Hebb D (1949) The organization of behavior, a neurophysiological theory. Wiley, New YorkGoogle Scholar
  30. Hinton GE, Sejnowski TJ (1986) Learning and relearning in Boltzmann machines. (Parallel distributed processing. Vol 1) MIT Press, Cambridge, Mass pp 282–317Google Scholar
  31. Ireland CT, Kullback SS (1968) Contingency tables with given marginals. Biometrika 55:179–188PubMedGoogle Scholar
  32. Ku HH, Kullback S (1968) Interaction in multidimensional contingency tables: An information theoretic approach. J Res NBS Math 72B:159–199Google Scholar
  33. Ku HH, Kullback S (1969) Approximating discrete probability distributions. IEEE Trans Inf Theory IT-15:444–447CrossRefGoogle Scholar
  34. Kullback S (1968) Information theory and statistics. Dover, New YorkGoogle Scholar
  35. Martignon L, Laskey BK (1995) Statistical inference methods for classifying higher order neural correlations. In: Hermann H (eds) Proceedings of the International Workshop on Supercomputers and the Brain. World Scientific, Singapore (in press)Google Scholar
  36. Martignon L, Hasseln H von, Palm G (1993) Modelling stochastic networks: from data to the connectivity structure. (Informatik aktuell, Subrcihe Künstliche Intelligenz) Gesamtdarstellung des Workshops auf der KI-Jahrestagung, Berlin 1993. Springer, Berlin Heidelberg New York pp 50–58Google Scholar
  37. Martignon L, Hasseln H von, Grün S, Palm G (1994) Modelling the nteraction in a set of neurons implicit in their frequency distribution: a possible approach to neural assemblies. In: Taddei C et al. (eds) Collected lectures of the seminar on biocybernetics. 1st di Cibernetica. Naples. Rosenberg-Sellier, TorinoGoogle Scholar
  38. Miller JW, Goodman RM (1993) Probability estimation from a database. In: Cowan JD, Hanson SJ, Giles CL (eds) Advances in Neural Information Processing Systems, Vol 5. Morgan Kaufman, San Matco. 531–538Google Scholar
  39. Palm G (1981) Evidence, information and surprise. Biol Cybern 42:57–68CrossRefPubMedGoogle Scholar
  40. Palm G, Aertsen A, Gerstein G (1988) On the significance of correlations among neuronal spike trains. Biol Cybern 59: 1–11CrossRefPubMedGoogle Scholar
  41. Pinkas G (1991) Energy minimization and the satisfiability of propositional logic. In: Sejnowsky T, Touretzky D, Ellman H, Hinton G, (eds) Proc of the 1990 Connectionist Models Summer School. Morgan Kaufman, San Mateo, pp 23–31Google Scholar
  42. Vaadia E, Aertsen A (1992) Coding and computing in the cortex: single neuron activity and cooperative phenomena. In: Aertsen A, Braitenberg V (eds) Information processing in the cortex. Springer, Berlin Heidelberg New YorkGoogle Scholar
  43. Vaadia E, Bergman H, Abeles M (1989) Neuronal activities related to higher brain functions theoretical and experimental implications. IEEE Trans Biomed Eng BME 36. 25–35CrossRefGoogle Scholar
  44. Vaadia E, Ahissar E, Bergman H, Lavner Y (1991) Correlated activity of neurons: a neural code for higher brain functions? In: Krüger J (eds) Neuronal cooperatively. Springer. Berlin Heidelberg New York, pp 249–279Google Scholar
  45. Vaadia E, Haalman I, Abeles M, Bergman H, Prut Y, Slovin H, Aertsen A (1995) Dynamics of neuronal interactions in the monkey cortex to behavioral events. Nature 373:515–518CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • L. Martignon
    • 1
  • H. Von Hassein
    • 1
  • S. Grün
    • 2
    • 3
  • A. Aertsen
    • 2
  • G. Palm
    • 1
  1. 1.Department of Neural Information ProcessingUniversity of UlmUlmGermany
  2. 2.Center for Research of Higher Brain Functions, Department of NeurobiologyWeizmann Institute of ScienceRehovotIsrael
  3. 3.Institut für Neuroinformatik, Ruhr UniversitätBochumGermany

Personalised recommendations