A design principle of spindle oscillations in mammalian sleep

Abstract

Neural oscillations are mainly regulated by molecular mechanisms and network connectivity of neurons. Large-scale simulations of neuronal networks have driven the population-level understanding of neural oscillations. However, cell-intrinsic mechanisms, especially a design principle, of neural oscillations remain largely elusive. Herein, we developed a minimal, Hodgkin-Huxley-type model of groups of neurons to investigate molecular mechanisms underlying spindle oscillation, which is synchronized oscillatory activity predominantly observed during mammalian sleep. We discovered that slowly inactivating potassium channels played an essential role in characterizing the firing pattern. The detailed analysis of the minimal model revealed that leak sodium and potassium channels, which controlled passive properties of the fast variable (i.e., membrane potential), competitively regulated the base value and time constant of the slow variable (i.e., cytosolic calcium concentration). Consequently, we propose a theoretical design principle of spindle oscillations that may explain intracellular mechanisms behind the flexible control over oscillation density and calcium setpoint.

Keywords: Complex system biology; Computer modeling; Neuroscience.

Graphical abstract

Figure 1

The configuration of the averaged-neuron (AN) model and the reduced-AN (RAN) model for the sleep spindle (SS) firing pattern (A) Schematic illustration of the AN model. It is composed of extrinsic components in dendrites and intrinsic components in soma. (B) Membrane potential in the slow-wave sleep (SWS) (top) and SS (bottom) firing patterns. While the intermittent firing pattern is shared between the two firing patterns, the active up and silent down states are not observed in the SS firing pattern. (C) Detailed membrane potential (top) and [Ca²⁺] (bottom) dynamics in the SS firing pattern. The waxing and waning subthreshold oscillations and afterhyperpolarizations (AHPs) are unique to the SS firing pattern. (D) SD of the standardized V (left) and the average of the standardized [Ca²⁺] (right) of the 1,112 parameter sets collected by random parameter search for the AN model. (E) Distinct distributions of several parameters between the SS and the SWS parameter sets. (F) Factor loadings of each component for PC 1 in the AN model. (G) Distributions of the parameters whose factor loadings are highlighted in (F). The components colored yellow or magenta are implied to be characterizing the SS firing pattern. (H) Proportions of parameter sets that bifurcate to other firing patterns after KO of each component. The components colored magenta are implied to be indispensable for generating the SS firing pattern. (I) Table shows channels, receptors, and exchangers/pumps included in each model. (J) Hit rates for the SS firing pattern in each model. Details are listed in Table S6. (K) Schematic illustration of the RAN model

Figure 2

Small changes in $I_{\text{KCa}}$ mediates the transition between phases in the SS firing pattern (A) Membrane potential proportions of each channel's inward and outward current of the representative parameter set in the RAN model over time. (B) Proportion of current through leak K⁺ channel ($I_{\text{KL}}$) (left), slowly inactivating K⁺ channel ($I_{\text{KS}}$) (middle), and calcium-dependent K⁺ channel ($I_{\text{KCa}}$) (right) of 1,166 parameter sets obtained by the random parameter searches in the RAN model (top) and their averages with standard deviations (bottom). (C) Distributions of the proportion of outward current in bursting (left) and silent (right) phases of the SS and SWS firing patterns in the RAN and SAN models, respectively. (D) A trajectory over time (red line) and nullclines of V and $m_{\text{KS}}$ of the representative parameter set of the SS firing pattern in the phase space. The coiled part of the trajectory represents bursting phases, while the linear part represents silent phases. (E) Phase planes and stream plot of the representative parameter set on the V-$m_{\text{KS}}$ plane when [Ca²⁺] is low (67 μM) and high (90 μM). Nullclines determine the streams in the phase plane and therefore the trajectory over time. Each concentration corresponds to the transition from a silent to a bursting phase and a bursting phase to a silent phase, respectively. (F) Fixed points (that is, intersections of V and $m_{\text{KS}}$ nullclines) on the V-[Ca²⁺] plane and the trajectory of the SS firing pattern (left), and a bifurcation diagram representing the linear stability of the fixed points with the arrows indicating the movement of the trajectory (right). The transition from a silent to a bursting phase occurs via the subcritical Andronov-Hopf bifurcation. On the other hand, the transition from a bursting phase to a silent phase occurs theoretically via the fold (saddle-node) limit cycle bifurcation

Figure 3

Balance of background inward and outward currents controls oscillation density and calcium setpoint of the SS firing pattern (A) Relationships between channel conductance (leak K⁺ channel (left) and leak Na⁺ channel (right)) and oscillation density of the SS firing pattern (top). The corresponding representative membrane potential and [Ca²⁺] at each $g_{\text{KL}}$ are shown in the bottom panels. Parameter sets are the representatives chosen from major subgroups of parameter sets shown in (B). (B) The “decreasing” and “increasing” parameter sets are determined by changes in oscillation density when each parameter is modified 2.5$\text{\%}$. Parameter sets that do not change their oscillation density more than 2.5$\text{\%}$ are not shown in this figure. (C) Proportion of the decreasing and the increasing parameter sets among 1,166 parameter sets obtained by the random parameter search. (D) Changes in oscillation frequencies when two sets of channel conductance are modulate at the same time. $g_{\text{KL}}$ and $g_{\text{NaL}}$ modulate oscillation density competitively (left), while $g_{\text{KL}}$ and $g_{\text{KS}}$, and $g_{\text{NaL}}$ and $g_{\text{NaP}}$ modulated cooperatively (middle and right, respectively). (E and F) Relative changes in current through each channel in each phase in response to the shifts in $g_{\text{KL}}$ (E) and $g_{\text{NaL}}$ (F) values in each representative parameter set of the major subgroup. Changes in $I_{\text{KCa}}$ are larger than another current in both cases. (G and H) Trajectories and bifurcation diagrams in response to the parameter shifts of representative parameter sets in major subgroups of $g_{\text{KL}}$ (G) and $g_{\text{NaL}}$ (H), corresponding to (A). In major subgroups, when outward current increases (G, left) or inward current decreases (H, right), the bifurcation structures change, followed by decreases in calcium setpoint and oscillation density

Figure 4

The balance between inward and outward currents through leak channels controls oscillation density of the SS firing pattern based on the AN model (A) Proportion of $I_{\text{KL}}$ (left), $I_{\text{KS}}$ (middle), and $I_{\text{KCa}}$ (right) of 1,112 parameter sets obtained by the random parameter searches in the AN model (top) and their averages with standard deviations (bottom). (B) Distribution of proportions of outward current in bursting (top) and silent (bottom) phases of the SS and SWS firing pattern in the AN model. (C) Relations between channel conductance (leak K⁺ channel (left) and leak Na⁺ channel (right)) and oscillation density of the SS firing pattern based on the AN model (top). The corresponding representative membrane potential and [Ca²⁺] at each $g_{\text{KL}}$ are shown in the bottom panels. Parameter sets are the representatives chosen from subgroups of parameter sets shown in (D). (D) The “decreasing” and “increasing” parameter sets are determined by changes in oscillation density when each parameter is modified 2.5$\text{\%}$. (E) Proportion of the decreasing and the increasing parameter sets among 1,166 parameter sets obtained by the random parameter search. (F) Changes in oscillation density when two sets of channel conductance are modified at the same time. (G) Schematic illustration of the mechanism in which the balance between inward and outward currents through leak channels control oscillation density of the SS firing pattern