Multi-stability in neuronal models

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Author: Dr. Gennady S. Cymbalyuk, Dept Physics and Astronomy, Georgia State University, Georgia
Author: Dr. Andrey Shilnikov, Dept Mathematics, GSU, Atlanta, GA

In preparation

1 Types of multistability
2 Bursting and Quiescence
3 Tonic spiking and Quiescence
4 Tonic spiking and Tonic Spiking
5 Tonic spiking and Bursting
6 Canonical leech heart interneuron model
7 Reduced leech heart interneuron model
8 Polyrhythmicity

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Types of multistability

Figure 1: Bifurcation diagram of the reduced model of a leech heart interneuron showing three zones of bistability: tonic spiking and bursting, spiking and hyperpolarized quiescence, depolarized quiescence and bursting

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Bursting and Quiescence

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Tonic spiking and Quiescence

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Tonic spiking and Tonic Spiking

Figure 2: Averaging for finding periodic spikers.

Figure 3: Coexistent tonic spiking attractors in the reduced model of a leech heart interneuron.

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Tonic spiking and Bursting

Figure 4: Tonic spiking and bursting in the phase space of the reduced leech interneuron model.

Figure 5: Coexistent tonic spiking attractors in the reduced model of a leech heart interneuron.

Figure 6: Coexistent tonic spiking attractors in the reduced model of a leech heart interneuron.

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Canonical leech heart interneuron model

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Reduced leech heart interneuron model

Reduced oscillatory heart interneuron model [A. Shilnikov and Cymbalyuk, 2005]:

(1)

$\mathrm{\dot V} = \mathrm{-2\,[30\, m^2_{K2} (V+0.07)+8\,(V+0.046)}+ \mathrm{200\, f^3_{\infty}(-150,\,0.0305\,,V) h_{Na}\,(V-0.045)}+0.0060]$ , $\mathrm{\dot h_{Na}} = \mathrm{[f_{\infty}(500,\,0.0325,\,V)-h_{Na}]/0.0406}$ ,

$\mathrm{\dot m_{K2}} =\mathrm{[f_{\infty}(-83,V_{\frac{1}{2}}+V_{K2}^{shift},V)-m_{K2}]/0.9}$ ,

where $\mathrm{V}$ is the membrane potential, $\mathrm{h}_{\rm Na}$ is inactivation of the fast sodium current, and $\mathrm{m}_{\rm K2}$ is activation of persistent potassium one; a Boltzmann function $\mathrm{f_{\infty}(a,b,V)=1/(1+e^{a(b+V)})}$ describes kinetics of (in)activation of the currents. The bifurcation parameter $\mathrm{V^{shift}_{K2}}$ is a deviation from the canonical value $\mathrm{V_{\frac{1}{2}}}=0.018$ V corresponding to $f_{\infty}=1/2$ , i.e. to the semi-activated potassium channel.