Abstract
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.
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Acknowledgements
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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Appendix
Appendix
The following tables contain all the information pertaining to the ion channels and gating variable kinetics used in this paper. Table 2 is for the uniform channel model, which uses the Hodgkin–Huxley squid giant axon parameters. Table 3 is for the non-uniform channel model, whose non-uniformity comes from an A-type K + current following Connor-Stevens kinetics and consistent with the graded channel distribution of (Hoffman et al. 1997).
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Kellems, A.R., Roos, D., Xiao, N. et al. Low-dimensional, morphologically accurate models of subthreshold membrane potential. J Comput Neurosci 27, 161–176 (2009). https://doi.org/10.1007/s10827-008-0134-2
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DOI: https://doi.org/10.1007/s10827-008-0134-2