Formal Neuron Models: Delays Offer a Simplified Dendritic Integration for Free

  • Ophélie GuinaudeauEmail author
  • Gilles Bernot
  • Alexandre Muzy
  • Daniel Gaffé
  • Franck Grammont
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1024)


We firstly define an improved version of the spiking neuron model with dendrites introduced in [8] and we focus here on the fundamental mathematical properties of the framework. Our main result is that, under few simplifications with respect to biology, dendrites can be simply abstracted by delays. Technically, we define a method allowing to reduce neuron shapes and we prove that reduced forms define equivalence classes of dendritic structures sharing the same input/output spiking behaviour. Finally, delays by themselves appear to be a simple and efficient way to perform an abstract dendritic integration into spiking neurons without explicit dendritic trees. This overcomes an explicit morphology representation and allows exploring many equivalent configurations via a single simplified model structure.


Spiking neuron models Dendrites Formal models Behavioural equivalence classes Delays Leaky integrate-and-fire 


  1. 1.
    Brette, R., Gerstner, W.: Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophysiol. 94(5), 3637–3642 (2005)CrossRefGoogle Scholar
  2. 2.
    Buchera, D., Goaillard, J.M.: Beyond faithful conduction: short-term dynamics, neuromodulation, and long-term regulation of spike propagation in the axon. Prog. Neurobiol. 94(4), 307–346 (2011)CrossRefGoogle Scholar
  3. 3.
    Byrne, J.H., Roberts, J.L.: From Molecules to Networks. Academic Press, Cambridge (2004)zbMATHGoogle Scholar
  4. 4.
    Dayan, P., Abbott, L.F.: Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. Massachusetts Institute of Technology Press, Cambridge (2001)zbMATHGoogle Scholar
  5. 5.
    Debanne, D.: Information processing in the axon. Nat. Rev. Neurosci. 5, 304–316 (2004)CrossRefGoogle Scholar
  6. 6.
    Gerstner, W., Naud, R.: How good are neuron models? Science 326(5951), 379–380 (2009)CrossRefGoogle Scholar
  7. 7.
    Gorski, T., Veltz, R., Galtier, M., Fragnaud, H., Telenczuk, B., Destexhe, A.: Inverse correlation processing by neurons with active dendrites. bioRxiv, Forthcoming (2017)Google Scholar
  8. 8.
    Guinaudeau, O., Bernot, G., Muzy, A., Gaffé, D., Grammont, F.: Computer-aided formal proofs about dendritic integration within a neuron. In: BIOINFORMATICS 2018–9th International Conference on Bioinformatics Models, Methods and Algorithms (2018)Google Scholar
  9. 9.
    Guinaudeau, O., Bernot, G., Muzy, A., Grammont, F.: Abstraction of the structure and dynamics of the biological neuron for a formal study of the dendritic integration. In: Advances in Systems and Synthetic Biology (2017)Google Scholar
  10. 10.
    Häusser, M., Mel, B.: Dendrites: bug or feature? Curr. Opin. Neurobiol. 13(3), 372–383 (2003)CrossRefGoogle Scholar
  11. 11.
    Huguenard, J.R.: Reliability of axonal propagation: the spike doesn’t stop here. Proc. Natl. Acad. Sci. 97(17), 9349–9350 (2000)CrossRefGoogle Scholar
  12. 12.
    Izhikevich, E.M.: Polychronization: computation with spikes. Neural Comput. 18(2), 245–282 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lapicque, L.: Recherches quatitatives sur l’excitation electrique des nerfs traitee comme polarisation. J. Physiol. Pathol. Gen. 9, 620–635 (1907)Google Scholar
  14. 14.
    Maass, W., Schnitger, G., Sontag, E.D.: On the computational power of sigmoid versus Boolean threshold circuits. In: Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, pp. 767–776. IEEE (1991)Google Scholar
  15. 15.
    Maass, W.: Networks of spiking neurons: the third generation of neural network models. Neural Netw. 10(9), 1659–1671 (1997)CrossRefGoogle Scholar
  16. 16.
    Maass, W.: On the relevance of time in neural computation and learning. Theoret. Comput. Sci. 261(1), 157–178 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5(4), 115–133 (1943)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mel, B.W.: Information processing in dendritic trees. Neural Comput. 6(6), 1031–1085 (1994)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Paulus, W., Rothwell, J.C.: Membrane resistance and shunting inhibition: where biophysics meets satate dependent human neurophysiology. J. Physiol. 594(10), 2719–2728 (2016)CrossRefGoogle Scholar
  20. 20.
    Rall, W.: Branching dendritic trees and motoneuron membrane resistivity. Exp. Neurol. 1(5), 491–527 (1959)CrossRefGoogle Scholar
  21. 21.
    Rall, W.: Theory of physiological properties of dendrites. Ann. N. Y. Acad. Sci. 96(1), 1071–1092 (1962)Google Scholar
  22. 22.
    Segev, I., London, M.: Untangling dendrites with quantitative models. Science 290(5492), 744–750 (2000)CrossRefGoogle Scholar
  23. 23.
    Stern, E.A., Jaeger, D., Wilson, C.J.: Membrane potential synchrony of simultaneously recorded striatal spiny neurons in vivo. Nature 394(6692), 475–478 (1998)CrossRefGoogle Scholar
  24. 24.
    Stuart, G., Spruston, N., Häusser, M.: Dendrites. Oxford University Press, Oxford (2016)CrossRefGoogle Scholar
  25. 25.
    Thorpe, S., Imbert, M.: Biological constraints on connectionist modelling. In: Connectionism in Perspective, pp. 63–92 (1989)Google Scholar
  26. 26.
    Thorpe, S., Delorme, A., Van Rullen, R.: Spike-based strategies for rapid processing. Neural Netw. 14(6), 715–725 (2001)CrossRefGoogle Scholar
  27. 27.
    Van Rullen, R., Thorpe, S.J.: Rate coding versus temporal order coding: what the retinal ganglion cells tell the visual cortex. Neural Comput. 13(6), 1255–1283 (2001)CrossRefGoogle Scholar
  28. 28.
    Williams, S.R., Stuart, G.J.: Dependence of EPSP efficacy on synapse location in neocortical pyramidal neurons. Science 295(5561), 1907–1910 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ophélie Guinaudeau
    • 1
    Email author
  • Gilles Bernot
    • 1
  • Alexandre Muzy
    • 1
  • Daniel Gaffé
    • 2
  • Franck Grammont
    • 3
  1. 1.Université Côte d’Azur, CNRSSophia-AntipolisFrance
  2. 2.Université Côte d’Azur, CNRSSophia-AntipolisFrance
  3. 3.Université Côte d’Azur, CNRSNiceFrance

Personalised recommendations