Routes into bursting

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Author: Dr. Andrey Shilnikov, Dept Mathematics, GSU, Atlanta, GA
Author: Dr. 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.

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Onset of elliptic bursting

We begin consideration with the FitzHugh-Rinzel model

(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.


Figure 1: Elliptic bursting orbit switches between the tonic spiking and quiescent manifold in the FitzHugh-Rinzel model.
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Onset of square-wave bursting in the Hindmarsh-Rose model

Figure 2: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
Figure 3: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
Figure 4: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
Figure 5: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.


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Onset of square-wave bursting

Figure 6: Bursting manifold in the reduced leech heart interneuron model.
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Transitions between tonic spiking and bursting

Figure 7: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
Figure 8: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
Figure 9: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
Figure 10: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.


Figure 11: Tonic spiking orbit on the tonic spiking manifold in the 3D Hindmarsh-Rose model.
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