User:Andrey Shilnikov/Proposed/Routes into bursting

• Andrey Shilnikov, Dept Mathematics, GSU, Atlanta, GA

• Gennady S. Cymbalyuk, Dept Physics and Astronomy, Georgia State University, Georgia

Bifurcation scenarios of onset of bursting in various neuronal models including Hodgkin-Huxley and FitzHugh-Rinzel models of an elliptic burster, the Hindmarsh-Rose model and the reduced leach heart interneuron of some square-wave bursters.

Onset of elliptic bursting

We begin consideration with the FitzHugh-Rinzel model \[\tag{1} \begin{array}{rclcl} \dot v &=& v-v^3/3-w+y+I,\\ \dot w &=& 0.08(0.7+v-0.8w),\\ \dot y &=& \mu(-0.775-v-y), \quad \mu=10^{-4}, \end{array} \]

with a single control parameter - an applied current I; when I = 0.3125 the model exhibits the elliptic bursting shown in Fig. 1.

Onset of square-wave bursting in the Hindmarsh-Rose model

Onset of square-wave bursting

Transitions between tonic spiking and bursting