Widespread neuronal chaos induced by slow oscillating currents

Abstract

This paper investigates the origin and onset of chaos in a mathematical model of an individual neuron, arising from the intricate interaction between 3D fast and 2D slow dynamics governing its intrinsic currents. Central to the chaotic dynamics are multiple homoclinic connections and bifurcations of saddle equilibria and periodic orbits. This neural model reveals a rich array of codimension-2 bifurcations, including Shilnikov-Hopf, Belyakov, Bautin, and Bogdanov-Takens points, which play a pivotal role in organizing the complex bifurcation structure of the parameter space. We explore various routes to chaos occurring at the intersections of quiescent, tonic-spiking, and bursting activity regimes within this space, and provide a thorough bifurcation analysis. Despite a high dimensionality of the model, its fast-slow dynamics allow a reduction to a one-dimensional return map, accurately capturing and explaining the complex dynamics of the neural model. Our approach integrates parameter continuation analysis, newly developed symbolic techniques, and Lyapunov exponents, collectively unveiling the intricate dynamical and bifurcation structures present in the system.

Figures (21)

• Figure 1: (A) Close-up of the widespread chaotic region (red) in a bi-parametric sweep of $\rm \Delta [Ca]$ and $\rm \Delta V_x$. The colormap illustrates the Lyapunov spectrum, with each RGB component representing a different Lyapunov exponent. The red channel corresponds to positive largest Lyapunov exponents (LLE). Green and blue both correspond to negative second Lyapunov exponents, but at different magnitudes - green indicates values closer to zero, and blue represents more negative values. Chaotic regions appear red, while non-chaotic regions are blue. Representative voltage traces are overlaid on the parameter space: tonic spiking (left), a hyperpolarized quiescent state (bottom), and regular bursting (top-right). The chaotic region is centered around the bifurcation curve $\rm homSF$ (yellow line) corresponding to a Shilnikov saddle-focus. This curve extends from the codimension-2 Shilnikov-Hopf (ShH) bifurcation point located on the subcritical Andronov-Hopf ($\rm AH_{sub}$) bifurcation line (green dashed line). The $\rm homPO_t$ curve (orange line), which separates the chaotic region from the quiescent region below, corresponds to a non-transverse homoclinic to a saddle periodic orbit (PO). In the region marked by the orange dot, situated between the $\rm AH_{\rm sub}$ curve and the $\rm homPO_t$ curve, the model exhibits bistability between chaotic and hyperpolarized quiescent attractors (see Fig. \ref{['fig15']}). Thin white lines within the chaotic region indicate saddle-node bifurcations of periodic orbits, adjacent to stability windows. (B) Chaotic voltage trace corresponding to the parameter values at the red diamond in panel A, showing the irregular bursting dynamics observed within the chaotic region.
• Figure 2: 3D $\left ( V,\, [{\rm Ca}],\, x \right )$-phase space projection of the two critical manifolds that globally determine the slow-fast dynamics in the phase space of the SiN model at $\Delta \rm [Ca]=-4.6$ and $\rm \Delta V_x = -2$. A 2D spiking manifold $\rm M_{PO}$ (yellow) is foliated by fast, round periodic orbits of the fast subsystem and features a characteristic fold. A 2D slow-motion (grey) manifold $\rm M_Q$, also referred here to as the "dune," consists of equilibrium states in the fast subsystem. Three orbits are shown here: chaotic bursting (in blue) switching back and forth between $\rm M_{PO}$ and $\rm M_Q$, a green orbit converging to a stable equilibrium state, and a red saddle periodic orbit whose stable manifold locally separates two coexisting attractors, see Figs. \ref{['fig4']}A and B below.
• Figure 3: Qualitative stability diagrams depicting the positions of characteristic exponents in the complex plane for the two equilibrium states of the SiN model. (A) A saddle-saddle bifurcation of an equilibrium state with characteristic exponents $\lambda_{1,2,3} < \lambda_4 = 0 < \lambda_5$ gives rise to the onset of two saddles: an upper saddle of the topological (4,1)-type (B) and a lower saddle of the (3,2)-type (C) on the 2D slow-motion manifold (the dune) in the 5D phase space of the SiN model. (D) After its positive eigenvalues merge and form a complex conjugate pair next to the imaginary axis in the right open half-plane, the lower saddle transforms into the Shilnikov saddle-focus. (E-F) An Andronov-Hopf bifurcation (sub or super-critical) makes the saddle-focus a stable focus after its complex-conjugate eigenvalues cross over the over imaginary axis to the left open half-plane.
• Figure 4: A sketch of the unfolding of the Shilnikov-Hopf (the ShH point in Fig. \ref{['fig1']}) or Belyakov-I cod-2 bifurcation in a 3D phase space, where a saddle periodic orbit (a red ring labelled by $\rm PO$) with transversely intersecting 2D stable (light-red) and unstable (grey) manifolds, $W^s_{\rm PO}$ and $W^u_{\rm PO}$, collapses into the Shilnikov saddle-focus through a sub-critical AH-bifurcation. The manifolds do not cross initially in panel A, then form a tangency along a homoclinic orbit $\Gamma$ in (B). In (D) the manifolds cross transversally along two homoclinic obits $\Gamma_1$ and $\Gamma_2$ allowing an access to the stable equilibrium state (green dot) within the region bounded by $W_{\rm PO}^s$ (a light-red cylinder). (D) 3D phase projection depicting a Shilnikov saddle-focus with a (red) homoclinic orbit spiraling away and returning to it on the dune $\rm M_Q$ (bent grey surface) from below; this homoclinic bifurcation occurs along the corresponding curve $\rm homSF$ in the parameter plane (see Fig. \ref{['fig1']} above) of the SiN model, at $\rm (\Delta [Ca], \Delta V_x)$-parameter value $(-35.98, -1.1)$.
• Figure 5: Bi-parametric, $\left(\Delta [{\rm Ca}],\,\rm \Delta V_x \right)$-sweeps of the SiN model revealing its dynamics with the aid of three different approaches. Overlaid bifurcation curves are computed using the MATCONT parameter continuation package. (A) Lyapunov exponent sweep, with intensified chaos indicated with the yellow and red colors. (B) Colormap corresponds to the inter-spike-interval (ISI) variance for periodic dynamics (green), and to Lempel-Ziv complexity for chaotic dynamics (red). (C) The symbolic sweep detects homoclinic bifurcations of the upper saddle, with inset panels illustrating the homoclinic U-shaped curves. Numbers $1, 2, \dots$ in (B)-(C) indicate number of spikes per burst in those regions.