Bifurcation analysis provides theoretical guidelines and a powerful analytical framework to tune the activity of neuronal models. A bifurcation is a qualitative change in the activity of a dynamical system, such as a transition from silence to spiking. Bifurcation analysis identifies general dynamical laws governing transitions between activity regimes in response to changes in controlling parameters. By using these quantitative laws, the temporal characteristics of a neuronal model can be tuned to reproduce the characteristics of experimentally recorded activity by navigating the parameter space in the vicinity of bifurcations. In most of the bifurcations described here, the general shape of the waveform – for example, the shape of action potentials in a time series – is preserved, while a specific temporal characteristic changes.
This work was supported by National Science Foundation grant PHY-0750456.
- Ermentrout G, Terman D (2010) Mathematical foundations of neuroscience, vol 35, Interdisciplinary applied mathematics. Springer, New YorkGoogle Scholar
- Izhikevich E (2007) Dynamical systems in neuroscience. MIT Press, Cambridge, MAGoogle Scholar
- Lukyanov V, Shilnikov L (1978) On some bifurcations of dynamical systems with homoclinic structures. Soviet Math Dokl 19:1314Google Scholar
- Rinzel J (1987) Mathematical topics in population biology, morphogenesis, and neuroscience, vol 71, Lecture notes in biomathematics. Springer, BerlinGoogle Scholar
- Rinzel J, Ermentrout B (1998) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neural modeling. The MIT Press, Cambridge, MA, pp 251–292Google Scholar
- Shilnikov L, Turaev D (2000) A new simple bifurcation of a periodic orbit of blue sky catastrophe type. Am Math Soc Transl II Ser 200:165–188Google Scholar