Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics

Yi Sun, Douglas Zhou, Aaditya V. Rangan, David Cai

Research output: Contribution to journalArticlepeer-review


We present a numerical analysis of the dynamics of all-to-all coupled Hodgkin-Huxley (HH) neuronal networks with Poisson spike inputs. It is important to point out that, since the dynamical vector of the system contains discontinuous variables, we propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network. Three typical dynamical regimes - asynchronous, chaotic and synchronous, are found as the synaptic coupling strength increases from weak to strong. We use the pseudo-Lyapunov exponent and the power spectrum analysis of voltage traces to characterize the types of the network behavior. In the nonchaotic (asynchronous or synchronous) dynamical regimes, i.e., the weak or strong coupling limits, the pseudo-Lyapunov exponent is negative and there is a good numerical convergence of the solution in the trajectory-wise sense by using our numerical methods. Consequently, in these regimes the evolution of neuronal networks is reliable. For the chaotic dynamical regime with an intermediate strong coupling, the pseudo-Lyapunov exponent is positive, and there is no numerical convergence of the solution and only statistical quantifications of the numerical results are reliable. Finally, we present numerical evidence that the value of pseudo-Lyapunov exponent coincides with that of the standard Lyapunov exponent for systems we have been able to examine.

Original languageEnglish (US)
Pages (from-to)247-266
Number of pages20
JournalJournal of Computational Neuroscience
Issue number2
StatePublished - Apr 2010


  • Chaos
  • Hodgkin-Huxley neuron
  • Lyapunov exponents
  • Neuronal network
  • Numerical analysis

ASJC Scopus subject areas

  • Sensory Systems
  • Cognitive Neuroscience
  • Cellular and Molecular Neuroscience


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