Fast Voltage Dynamics of Voltage–Conductance Models for Neural Networks

  • Jeongho Kim
  • Benoît PerthameEmail author
  • Delphine Salort


We present the conductance limit of the voltage–conductance model with random firing voltage when conductance dynamics are slower than the voltage dynamics. The result of the limiting procedure is a transport/Fokker–Planck equation for conductance variable with a non-linear drift which depends on the total firing rate. We analyze the asymptotic behavior of the limit equation under two possible rescalings which relate the voltage scale, the conductance scale and the firing rate. We provide the sufficient framework in which the limiting procedure can be rigorously justified. Moreover, we also suggest a sufficient condition on the parameters and firing distribution in the limiting conductance equation under which we are able to obtain a unique stationary state and its asymptotic stability. Finally, we provide several numerical illustrations supporting the analytic results.


Voltage–conductance model Integrate-and-Fire Asymptotic behavior Neuron assemblies 

Mathematics Subject Classification

35Q92 35Q84 92B20 



BP has received funding from the European Research Council (ERC) under the European Union’s c innovation programme (Grant agreement no. 740623).


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Copyright information

© Sociedade Brasileira de Matemática 2020

Authors and Affiliations

  • Jeongho Kim
    • 1
  • Benoît Perthame
    • 2
    Email author
  • Delphine Salort
    • 3
  1. 1.Department of Mathematical SciencesSeoul National UniversitySeoulKorea
  2. 2.Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis LionsParisFrance
  3. 3.Sorbonne Université, CNRS, Laboratoire de Biologie Computationnelle et Quantitative, UMR 7238ParisFrance

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