Encyclopedia of Computational Neuroscience

2015 Edition
| Editors: Dieter Jaeger, Ranu Jung

Dynamical Systems: Overview

Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-6675-8_768

Definition

Many models of computational neuroscience are formulated in terms of nonlinear dynamical system (sometimes the dynamical system with noise) and include a large number of parameters. Therefore, it is difficult to analyze dynamics and find correspondence between parameter values and dynamical mode. Mathematical theory of dynamical systems and bifurcations provide a valuable tool for a qualitative study and finding regions in parameter space corresponding to different dynamical modes (e.g., oscillations or bistability). Thus, the dynamical systems (or nonlinear dynamics) approach to analysis of neural systems has played a central role for computational neuroscience for many years (summarized, e.g., in recent textbooks, Izhikevich (2007) and Ermentrout and Terman (2010)).

Detailed Description

Theory of dynamical systems and bifurcations provides a list of universal scenarios of dynamics changes under parameter variation. For example, one scenario explaining the onset of...

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References

  1. Ermentrout GB, Terman D (2010) Mathematical foundations of neuroscience. Springer, New YorkGoogle Scholar
  2. Fitzhugh R (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys 17(4):257–278Google Scholar
  3. Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-medullated axon. J Physiol Lond 107:165–181PubMedCentralPubMedGoogle Scholar
  4. Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge, MAGoogle Scholar
  5. Rinzel J, Ermentrout B (1998) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neuronal modeling: from ions to networks, 2nd edn. MIT Press, Cambridge, MA, pp 251–291Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA