Low-dimensional, morphologically accurate models of subthreshold membrane potential

  • Anthony R. Kellems
  • Derrick Roos
  • Nan Xiao
  • Steven J. Cox


The accurate simulation of a neuron’s ability to integrate distributed synaptic input typically requires the simultaneous solution of tens of thousands of ordinary differential equations. For, in order to understand how a cell distinguishes between input patterns we apparently need a model that is biophysically accurate down to the space scale of a single spine, i.e., 1 μm. We argue here that one can retain this highly detailed input structure while dramatically reducing the overall system dimension if one is content to accurately reproduce the associated membrane potential at a small number of places, e.g., at the site of action potential initiation, under subthreshold stimulation. The latter hypothesis permits us to approximate the active cell model with an associated quasi-active model, which in turn we reduce by both time-domain (Balanced Truncation) and frequency-domain (\({\cal H}_2\) approximation of the transfer function) methods. We apply and contrast these methods on a suite of typical cells, achieving up to four orders of magnitude in dimension reduction and an associated speed-up in the simulation of dendritic democratization and resonance. We also append a threshold mechanism and indicate that this reduction has the potential to deliver an accurate quasi-integrate and fire model.


Balanced truncation Krylov methods Quasi-active Dendritic democratization Dendritic resonance Integrate-and-fire Compartmental models 



The work in this paper is supported through the Sheafor/Lindsay Fund via the ERIT program at Rice’s Computer and Information Technology Institute CITI, NSF grant DMS-0240058, and NIBIB Grant No. 1T32EB006350-01A1


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Anthony R. Kellems
    • 1
  • Derrick Roos
    • 1
  • Nan Xiao
    • 1
  • Steven J. Cox
    • 1
  1. 1.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

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