Multiple timescale dynamics of conductance-based models of brainstem locomotor neurons

Abstract

The pedunculopontine nucleus (PPN) is a heterogeneous brainstem locomotor hub implicated in Parkinson's disease and potentially relevant for its treatment. We propose single-compartment, conductance-based models for three classes of PPN neurons, such that each model reproduces relevant experimentally observed stimulus-dependent responses, including post-inhibitory rebound dynamics, transient low-threshold activity, and gamma band oscillations. To understand the mechanisms underlying these transient responses to current stimulation, we leverage the models' intrinsic multi-timescale structure and apply dynamical system methods designed for multiple timescale systems. By separating fast membrane and channel-gating dynamics from slower gating and calcium processes, we identify specific ionic mechanisms underlying hallmark dynamics across cell types. We also generate new predictions about PPN behavior under a post-inhibitory facilitation protocol.

Figures (12)

• Figure 1: Comparison of low gamma-band oscillations in response to ramping applied current in the NC model and the experiment by Luster et al. luster2016intracellular. A: Model voltage trace (black) with ramping current injection $I_{\mathrm{App}}(t), \; t \in [100,700]$ (blue), for $I_{\mathrm{App}}(t)=0.0085\cdot (t-100) \cdot H(t-100) \cdot H(700-t)$ where $H(t)$ is the Heaviside step function. After a gradual rise in voltage, oscillations emerge when $I_{\mathrm{App}}=0.16$ pA/pF. B: Experimental data reproduced from Fig. 2E (left panel) of Luster et al. luster2016intracellular, licensed under CC BY 4.0. Voltage traces show intrinsic membrane oscillations during 1-second current ramps from in vitro patch-clamp recordings in rat PPN neurons. C: Comparison of interspike-intervals (ISIs) between NC model (green, see panel A) and experiment luster2016intracellular (pink, see panel B). Oscillations in the model and the experiment both increase in frequency over time as the ramp current is increased. ISIs are normalized to the time window during which the voltage exhibits oscillatory activity. The model and the experimental data (dots) are fit to an exponential function (solid line).
• Figure 2: Slow dynamics of $I_{\mathrm{CaT}}$ gating variable and intracellular calcium concentration drive oscillation onset in NC model under depolarizing ramp protocol. A: Voltage trace (black) overlaid with the bifurcation diagram of the full NC model computed with respect to $I_{\mathrm{App}}$. A branch of stable equilibria (red) persists until $I_{\mathrm{App}}=1.41$, where a supercritical AH bifurcation (pink) gives rise to stable limit cycles (green, minima/maxima). The voltage follows the stable equilibrium branch initially, with oscillations emerging before the AH point. After the bifurcation, $v_{\mathrm{NC}}$ remains oscillatory and generally aligns with the limit cycle extrema. $\textbf{B}$: Same trajectory (black), full system stable equilibria (red), and family of stable periodic orbits (POs) (green) as in A, projected onto the $(\mathrm{Ca}$, $h_{\mathrm{CaT}})$-plane. The trajectory initially tracks the stable equilibrium branch before crossing a branch of fast subsystem AH bifurcations (pink) as applied current increases, moving from a region where the fast subsystem converges to stable equilibria (gray) into a region where the fast system follows stable POs (light green). The inset shows a zoomed view of the boxed region in the main diagram, where the full system AH bifurcation point from A occurs. C: Voltage traces (black) are shown for $t\leq 181$ (top), $\leq 186$ (center), and $\leq 210$ (bottom), overlaid with the $\mathcal{M}_1^{181}|_{\hat{y}(181)}$, $\mathcal{M}_1^{186}|_{\hat{y}(186)}$, and $\mathcal{M}_1^{210}|_{\hat{y}(210)}$, respectively, for $\hat{y}(t)=[h_{\mathrm{CaT}}(t)]$. The gray-green curve shows the future evolution of the trajectory. Stable (red) and unstable equilibrium (gray) branches are shown in addition to an AH bifurcation (pink) and stable limit cycle extrema (green, minima/maxima). Initially, the fast voltage dynamics drive the trajectory toward a stable fixed point. As applied current increases over time, the AH bifurcation shifts to higher $\mathrm{Ca}$ values and the trajectory follows stable limit cycles.
• Figure 3: Comparison of voltage responses to a small stimulus application in NC model and experimental recordings kang1990electrophysiological. Left: Model voltage trace (black) with current step to $I_{\mathrm{App}}(t)=185$ pA/pF (blue). The excitatory stimulus depolarizes the cell, inducing two action potentials, presumed to be mediated by the T-type calcium current, before the injected current is removed and the voltage returns to rest. Right: Experimental data adapted from Kang and Kitai kang1990electrophysiological in PPN cells exhibiting the LTS property, reproduced with permission from Elsevier. The membrane potential slowly depolarizes after the onset of the stimulus. After approximately $150$ ms, the cell fires two action potentials, after which the stimulus is removed and voltage returns to rest around $-70$ mV.
• Figure 4: NC model spikes at a fast subsystem saddle-node on an invariant circle (SNIC) bifurcation. A: Solution to full NC system (black) from time 0 until the time indicated in each panel, along with the future trajectory path (gray-green). Plots include invariant sets of the fast subsystem at $t=150,186,$ and $212$ ms: projections of critical manifolds $\mathcal{M}_1^e\vert_{\hat{y}(150)}$, $\mathcal{M}_1^e\vert_{\hat{y}(186)}$ and $\mathcal{M}_1^e\vert_{\hat{y}(212)}$ for $\hat{y}(t)=[m_{\mathrm{CaT}}(t),\mathrm{ca}(t)]$ (stable red branches and unstable gray branches) and max/min voltages along periodic orbits (green). In each panel, the position of the trajectory at the time indicated is marked with a black triangle. This position aligns with a fast subsystem SNIC bifurcation (cyan) near $t=186$ ms (center panel). After this time, the fast subsystem no longer has a stable branch of the critical manifold nearby, and stable POs become the only attractors of the fast variables. Fast variables follow the PO family for two large-amplitude oscillations as slow variable $h_{\mathrm{CaT}}$ continues to slowly decrease. B: Illustration of system solution during excitation (black) and after its removals (dotted, black). Black triangles mark the trajectory positions at the indicated times, matching the left panels. Once the trajectory crosses the curve of SNIC bifurcations of $\mathcal{M}_1^e$ with excitation on (solid, cyan), the fast subsystem exhibits two large-amplitude oscillations. After the stimulus is removed, the trajectory ends up above the SNIC curve for $\mathcal{M}_1^b$ (dotted, cyan) and follows a stable branch of equilibria (not explicitly shown) on $\mathcal{M}_1^b$.
• Figure 5: Comparison of delayed firing after removal of inhibitory input in C model and experimental recordings from rat PPN cells by Takakusaki and Kitai takakusaki1997ionic. Left: Model voltage trace (black) with inhibitory current (blue) during $t \in (100,400)$ and $(660,800)$. The voltage fires rhythmically before the onset of the inhibition at $t=100$ ms. During the inhibition, the voltage remains hyperpolarized near $-80$ mV. Once the inhibition is removed, the voltage experiences a gradual depolarization before returning to regular firing activity. Right: Experimental data from Takakusaki and Kitai takakusaki1997ionic, reproduced with permission from Elsevier. The onset of inhibitory current (blue) around $t=90$ ms causes the voltage (black) to remain hyperpolarized near $-90$ mV. After removal of the inhibition near $t=400$ ms, a delayed return to rhythmic firing suggests the presence of the A-current.