A Probabilistic Approach to the Analysis of Propagation Delays in Large Cortical Axonal Trees

  • Israel Nelken


The most important means by which a neuron passes information to its postsynaptic partners is the propagation of action potentials from the soma to the synapses. Since the synapses are located at different distances from the soma, the action potential gets to them after different propagation delays. These propagation delays cause desynchronization in the computations performed by the cortical network. This paper presents an analysis of the propagation delays along a randomly branching axon. The model can describe propagation delays caused by both the randomness of the structure of the axon and by the changing electrical properties along its branches. The statistics of the branching axon are described by a subcritical branching process. It is shown that such a model does describe well some populations of axons. The propagation delays along the branching structure are random functions of the type of the branches. The main result is that the delays along the axon have an exponential or related distribution, which is wider than the Gaussian distribution with the same variance. Using some numerical estimates it follows that the coherence length in the cortex is in the order of magnitude of 1 mm. This result has implications concerning some current theories of brain function.


Branch Point Propagation Delay Forward Problem Terminal Segment Probability Generate Function 
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  1. Abeles M (1982) Local cortical circuits. Springer, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  2. Anderson CH, van Essen DC (1987) Shifter circuits: a computational strategy for dynamic aspects of visual processing. Proc Nat Acad Sci 84:6297–6301PubMedCentralPubMedCrossRefGoogle Scholar
  3. Abeles M (1991) Corticonics: neural networks of the cerebral cortex. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  4. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc B 39:1–38Google Scholar
  5. Feldman ML (1984) Morphology of the Neocortical Pyramidal Neuron. In: Peters A, Jones EG (ed) Cerebral Cortex vol 1. Plenum PressGoogle Scholar
  6. Feller W (1957) An Introduction to Probability Theory and its Applications. John Wiley & Sons, New York LondonGoogle Scholar
  7. Hellwig B, Schüz A, Aertsen AMHJ (1991) In preparationGoogle Scholar
  8. Khodorov BI, Timin YN (1975) Nerve impulse propagation along nonuniform fibers (investigation using mathematical models). Prog Biophys Molec Biol 30:145–184CrossRefGoogle Scholar
  9. Kliemann W (1987) A stochastic dynamical model for the characterizations of the geometrical structure of dendritic processes. Bull Math Bio 49:135–152CrossRefGoogle Scholar
  10. Manor Y (1990) Theoretical analysis of the propagation of action potentials in branching axon by computerized simulation. M.Sc. Thesis (in Hebrew)Google Scholar
  11. Manor Y, Koch C, Segev I (1991) The effect of geometrical irregularities on propagation delay in axonal trees. Biophys J (in press)Google Scholar
  12. Martin KAC (1984) Neuronal Circuits in Cat Striate Cortex. In: Peters A, Jones EG (ed) Cerebral Cortex vol 2. Plenum PressCrossRefGoogle Scholar
  13. Nelken I (1991) In preparationGoogle Scholar
  14. Parnas I (1979) Propagation in nonuniform neuntes: form and function in axons. In: Schmidt FO, Worden FG (ed) The Neurosciences, Fourth Study Section. M.I.T. pressGoogle Scholar
  15. Parnas I, Segev I (1979) A mathematical model for conduction of action potentials along bifurcating axons, J Physiol (Lond.) 295:323–343Google Scholar
  16. Poggio T, Koch C, Torre V (1981) Microelectronics of nerve cells: dendritic morphology and information processing. M.I.T. A.I. memo no. 650, October 1981Google Scholar
  17. Schüz A, Palm G (1989) Density of Neurons and Synapses in the Cerebral Cortex of the Mouse. J Comp Neurol 286:442–455CrossRefGoogle Scholar
  18. Schüz A (this volume)Google Scholar
  19. Triller A, Kom H (1986) Variability of Axonal Aiborizations Hides Simples Rules of Construction: a Topological Study From HRP Intracellular Injections. J Comp Neurol 253:500–513PubMedCrossRefGoogle Scholar
  20. van Pelt J, Verwer RWH, Uylings HBM (1989) Centrifugal-order distributions in binary topological trees. Bull Math Biol 51:511–536CrossRefGoogle Scholar
  21. Verwer RWH, van Pelt J, Noest AJ (1987) Parameter estimation in topological analysis of binary tree structures. Bull Math Biol 49:363–378PubMedCrossRefGoogle Scholar
  22. White EL (1989) Cortical Circuits: Synaptic Organization of the Cerebral Cortex, Structure, Function and Theory. Birkhauser, Boston BaselCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Israel Nelken
    • 1
  1. 1.Department of PhysiologyHebrew University Medical SchoolJerusalemIsrael

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