Biological Cybernetics

, Volume 81, Issue 5, pp 381–402

Exact digital simulation of time-invariant linear systems with applications to neuronal modeling

  • Stefan Rotter
  • Markus Diesmann

DOI: 10.1007/s004220050570

Cite this article as:
Rotter, S. & Diesmann, M. Biol Cybern (1999) 81: 381. doi:10.1007/s004220050570


An efficient new method for the exact digital simulation of time-invariant linear systems is presented. Such systems are frequently encountered as models for neuronal systems, or as submodules of such systems. The matrix exponential is used to construct a matrix iteration, which propagates the dynamic state of the system step by step on a regular time grid. A large and general class of dynamic inputs to the system, including trains of δ-pulses, can be incorporated into the exact simulation scheme. An extension of the proposed scheme presents an attractive alternative for the approximate simulation of networks of integrate-and-fire neurons with linear sub-threshold integration and non-linear spike generation. The performance of the proposed method is analyzed in comparison with a number of multi-purpose solvers. In simulations of integrate-and-fire neurons, Exact Integration systematically generates the smallest error with respect to both sub-threshold dynamics and spike timing. For the simulation of systems where precise spike timing is important, this results in a practical advantage in particular at moderate integration step sizes.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Stefan Rotter
    • 1
  • Markus Diesmann
    • 1
  1. 1.Neurobiologie und Biophysik, Institut für Biologie III, Universität Freiburg, Freiburg, GermanyDE

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