Chaotic Dynamics in Neural Systems

Introduction

Several basic mechanisms of chaotic dynamics in phenomenological and biologically plausible models of individual neurons are discussed. We show that chaos occurs at the transition boundaries between generic activity types in neurons such as tonic spiking, bursting, and quiescence, where the system can also become bi-stable. The bifurcations underlying these transitions give rise to period-doubling cascades, various homoclinic and saddle phenomena, torus breakdown, and chaotic mixed-mode oscillations in such neuronal systems.

Neurons exhibit various activity regimes and state transitions that reflect their intrinsic ionic channel behaviors and modulatory states. The fundamental types of neuronal activity can be broadly defined as quiescence, subthreshold, and tonic spiking oscillations, as well as bursting composed of alternating periods of spiking activity and quiescence. A single neuron can endogenously demonstrate various bursting patterns, varying in response to the...

Appendix

Leech Heart Interneuron Model

The reduced leech heart model is derived using the Hodgkin-Huxley formalism:

$$ {\displaystyle \begin{array}{c}C{V}^{\prime }=-{I}_{\mathrm{Na}}-{I}_{\mathrm{K}2}-{I}_{\mathrm{leak}}+{I}_{\mathrm{app}},\\ {}{\tau}_{\mathrm{Na}}{h}_{\mathrm{Na}}^{\prime }={h}_{\mathrm{Na}}^{\infty }(V)-h,\\ {}{\tau}_{\mathrm{K}2}{m}_{\mathrm{K}2}^{\prime }={m}_{\mathrm{K}2}^{\infty }(V)-{m}_{\mathrm{K}2},\end{array}} $$

(3)

with

$$ {I}_{leak}=8\left(V+0.046\right),\, {I}_{\mathrm{K}2}=30{m}_{\mathrm{k}2}^2\left(V+0.07\right),\, {I}_{\mathrm{Na}}=200{\left[{m}_{\mathrm{Na}}^{\infty }(V)\right]}^3{h}_{\mathrm{Na}}\left(V-0.045\right), $$

and where V is the membrane potential, C = 0.5; h_Na is a fast (τ_Na = 0.0405 sec) activation of I_Na, and m_K2; I_L describes the slow (τ_K2 = 0.25 sec) activation of I_K2, I_app is an applied current. The steady states ${h}_{\mathrm{Na}}^{\infty }(V)$, ${m}_{\mathrm{Na}}^{\infty }(V)$, ${m}_{\mathrm{K}2}^{\infty }(V)$, of the of the gating variables are given by the Boltzmann equations given by

$$ {\displaystyle \begin{array}{c}{h}_{\mathrm{Na}}^{\infty }(V)=\left[1+\exp \left(500(0.0333)+V\right)\right)\Big]{}^{-1},\\ {}{m}_{\mathrm{Na}}^{\infty }(V)=\left[1+\exp \left(-150(0.0305)+V\right)\right)\Big]{}^{-1},\\ {}{m}_{\mathrm{K}2}^{\infty }(V)=\left[1+\exp \left(-83(0.018)+{\mathrm{V}}_{\mathrm{K}2}^{\mathrm{shift}}+V\right)\right)\Big]{}^{-1}.\end{array}} $$

(4)

The bifurcation parameter ${\mathrm{V}}_{\mathrm{K}2}^{\mathrm{shift}}$ of the model is a deviation from the experimentally determined voltage V_1/2 = 0.018 V corresponding to the half-activated potassium channel, i.e., to ${m}_{\mathrm{k}2}^{\infty }(0.018)=1/2$. In its range, ${\mathrm{V}}_{\mathrm{K}2}^{\mathrm{shift}}$ is [−0.025; 0.0018]V the upper boundary corresponds to the hyperpolarized quiescent state of the neuron, whereas the model produces spiking oscillations at the lower end ${\mathrm{V}}_{\mathrm{K}2}^{\mathrm{shift}}$ values and bursts in between.

Chay Model

The 3D Hodgkin-Huxley type Chay model reads as follows:

$$ {\displaystyle \begin{array}{c}{V}^{\prime }=-{g}_I{m}_{\infty}^3{h}_{\infty}\left(V-{V}_I\right)-{g}_{K,V}{n}_{\infty}^4\left(V-{V}_K\right)-{g}_{K,C}\frac{C}{1+C}\left(V-{V}_K\right)-{g}_L\left(V-{V}_L\right),\\ {}{n}^{\prime }=\left({n}_{\infty}\left[V\right]-n\right)/{\tau}_n\left[V\right],\\ {}{C}^{\prime }=\rho \left\{{m}_{\infty}^3{h}_{\infty}\left({V}_C-V\right)-{k}_CC\right\},\end{array}} $$

(5)

where n represents the gating variable of the voltage-sensitive K⁺ channel and C represents the intracellular free calcium concentration. See (Chay 1985) for the detailed description.