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**Generation of periodic and chaotic bursting in an excitable cell model.**
*(English)*
Zbl 0805.92005

Summary: There are interesting oscillatory phenomena associated with excitable cells that require theoretical insight. Some of these phenomena are: the threshold low amplitude oscillations before bursting in neuronal cells, the damped burst observed in muscle cells, the period-adding bifurcations without chaos in pancreatic \(\beta\)-cells, chaotic bursting and beating in neurons, and inverse periodic-doubling bifurcation in heart cells.

The three variables model formulated by T. R. Chay [Physica D 16, 233-242 (1985; Zbl 0582.92007)] provides a mathematical description of how excitable cells generate bursting action potentials. This model contains a slow dynamic variable which forms a basis for the underlying wave, a fast dynamic variable which causes spiking, and the membrane potential which is a dependent variable.

We use a Chay model to explain these oscillatory phenomena. The Poincaré return map approach is used to construct bifurcation diagrams with the ‘slow’ conductance (i.e., \(g_{K,C}\)) as the bifurcation parameter. These diagrams show that the system makes a transition from repetitive spiking to chaotic bursting as the parameter \(g_{K,c}\) is varied. Depending on the time kinetic constant of the fast variable \((\lambda_ n)\), however, the transition between burstings via period- adding bifurcation can occur even without chaos. Damped bursting is present in the Chay model over a certain range of \(g_{K,C}\) and \(\lambda_ n\). In addition, a threshold sinusoidal oscillation was observed at certain values of \(g_{K,C}\) before triggering action potentials. Probably this explains why the neuronal cells exhibit low- amplitude oscillations before bursting.

The three variables model formulated by T. R. Chay [Physica D 16, 233-242 (1985; Zbl 0582.92007)] provides a mathematical description of how excitable cells generate bursting action potentials. This model contains a slow dynamic variable which forms a basis for the underlying wave, a fast dynamic variable which causes spiking, and the membrane potential which is a dependent variable.

We use a Chay model to explain these oscillatory phenomena. The Poincaré return map approach is used to construct bifurcation diagrams with the ‘slow’ conductance (i.e., \(g_{K,C}\)) as the bifurcation parameter. These diagrams show that the system makes a transition from repetitive spiking to chaotic bursting as the parameter \(g_{K,c}\) is varied. Depending on the time kinetic constant of the fast variable \((\lambda_ n)\), however, the transition between burstings via period- adding bifurcation can occur even without chaos. Damped bursting is present in the Chay model over a certain range of \(g_{K,C}\) and \(\lambda_ n\). In addition, a threshold sinusoidal oscillation was observed at certain values of \(g_{K,C}\) before triggering action potentials. Probably this explains why the neuronal cells exhibit low- amplitude oscillations before bursting.

### Keywords:

periodic bursting; excitable cells; bursting; membrane potential; Poincaré return map approach; bifurcation diagrams; chaotic bursting; action potentials; neuronal cells; low-amplitude oscillations### Citations:

Zbl 0582.92007
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\textit{Y.-S. Fan} and \textit{T. R. Chay}, Biol. Cybern. 71, No. 5, 417--431 (1994; Zbl 0805.92005)

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### References:

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