Nonlinear Estimation of Synaptic Conductances via Piecewise Linear Systems

  • Antoni Guillamon
  • Rafel Prohens
  • Antonio E. Teruel
  • Catalina Vich
Conference paper
Part of the Trends in Mathematics book series (TM, volume 8)


We use the piecewise linear McKean model to present a proof-of-concept to address the estimation of synaptic conductances when a neuron is spiking. Using standard techniques of non-smooth dynamical systems, we obtain an approximation of the period in terms of the parameters of the system which allows to estimate the steady synaptic conductance of the spiking neuron. The method gives also fairly good estimations when the synaptic conductances vary slowly in time.


Periodic Orbit Flight Time Unique Fixed Point Interspike Interval Slow Manifold 
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.



This work is partially supported by the Spanish Ministry of Economy and Competitiveness through project MTM2015-71509-C2-2-R (AG), by the MCYT/FEDER grant number MTM2014-54275-P (RP, AT and CV) and by the Government of Catalonia under grant 2014-SGR–504 (AG).


  1. 1.
    L.F. Abbott, A network of oscillators. J. Phys. A: Math General 23(16), 3835 (1990)CrossRefzbMATHGoogle Scholar
  2. 2.
    S. Coombes, Neuronal networks with gap junctions: a study of piecewise linear planar neuron models. SIAM J. Appl. Dyn. Syst. 7(3), 1101–1129 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. Coombes, R. Thul, K.C.A. Wedgwood, Nonsmooth dynamics in spiking neuron models. Phys. D 241(22), 2042–2057 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    G.B. Ermentrout, D.H. Terman, Mathematical Foundations of Neuroscience (Springer, New York, 2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    S. Fernández-García, M. Desroches, M. Krupa, F. Clément, A multiple time scale coupling of piecewise linear oscillators. Application to a neuroendocrine system. SIAM J. Appl. Dyn. Syst. 14, 643–673 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Guillamon, D.W. McLaughlin, J. Rinzel, Estimation of synaptic conductances. J. Physiol.-Paris 100(1–3), 31–42 (2006)CrossRefGoogle Scholar
  7. 7.
    A. Tonnelier, W. Gerstner, Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models. Phys. Rev. E 67, 021908 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    C. Vich, A. Guillamon, Dissecting estimation of conductances in subthreshold regimes. J. Comput. Neurosci. 39(3), 271–287 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Antoni Guillamon
    • 1
  • Rafel Prohens
    • 2
  • Antonio E. Teruel
    • 2
  • Catalina Vich
    • 2
  1. 1.Departament de Matemàtiques-EPSEBUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Departament de Matemàtiques i InformàticaUniversitat de les Illes BalearsPalmaSpain

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