Volume 9 Supplement 1

## Seventeenth Annual Computational Neuroscience Meeting: CNS*2008

# A novel method for modelling nonlinear dynamical systems applied to the Hodgkin-Huxley neuron

- Caitriona Boushel
^{1}Email author and - Paul Curran
^{1}

**9(Suppl 1)**:P39

**DOI: **10.1186/1471-2202-9-S1-P39

© Boushel and Curran; licensee BioMed Central Ltd. 2008

**Published: **11 July 2008

*u*and

*μ*as the bifurcation parameters,

with r ≥ 0, *φ* ∈ [0, 2*π*] and *σ*_{0}(*μ*) = -*μ*2, *τ*_{0}(*μ*) = 1-*μ*, *σ*_{1}(*μ*) = -2*μ* and ${\tau}_{1}(\mu )=\{\begin{array}{ll}\frac{2{\mu}_{2}}{{u}_{2}},\hfill & \text{for}\mu \ge {\mu}_{2}\hfill \\ \frac{2\mu +2\sqrt{\left(\mu -({u}_{2}-1)\right)\left(\mu -{\mu}_{2}\right)}}{{u}_{2}+{\mu}_{2}-\mu},\hfill & \text{for}\mu \le {\mu}_{2}\hfill \end{array}$

_{>K}) as the bifurcation parameters (see Fig. 1). It also reproduces much of the input-output behaviour of the HH neuron. When single input pulses are applied to the second order neuron and the HH neuron, both neurons switch from their stable equilibria to stable limit cycles and back again at the same time. When a continuous train of input pulses or a sinusoidal input are applied, both neurons oscillate at the frequency of the input signal once its amplitude is large enough (see Fig. 2). In addition, the oscillations of two second order neurons synchronise when the neurons are coupled, replicating the behaviour of two coupled HH neurons. These results indicate that nonlinear dynamical systems, which have topologically equivalent bifurcation diagrams, will have qualitatively similar input-output behaviour and qualitatively similar network behaviour as well as the same bifurcation behaviour.

## Authors’ Affiliations

## References

- Hodgkin AL, Huxley AF: A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 1952, 117: 500-544.PubMed CentralView ArticlePubMedGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd.