Correlations in ion channel expression emerge from homeostatic tuning rules

Abstract

Experimental observations reveal that the expression levels of different ion channels vary across neurons of a defined type, even when these neurons exhibit stereotyped electrical properties. However, there are robust correlations between different ion channel expression levels, although the mechanisms that determine these correlations are unknown. Using generic model neurons, we show that correlated conductance expression can emerge from simple homeostatic control mechanisms that couple expression rates of individual conductances to cellular readouts of activity. The correlations depend on the relative rates of expression of different conductances. Thus, variability is consistent with homeostatic regulation and the structure of this variability reveals quantitative relations between regulation dynamics of different conductances. Furthermore, we show that homeostatic regulation is remarkably insensitive to the details that couple the regulation of a given conductance to overall neuronal activity because of degeneracy in the function of multiple conductances and can be robust to "antihomeostatic" regulation of a subset of conductances expressed in a cell.

Keywords: computational models; control theory; neuronal excitability; robustness.

Fig. 1.

A toy model of activity-dependent conductance regulation. (A) Schematic of a neuron with regulated inward (g_in) and outward (g_out) conductances. Inward conductances promote Ca²⁺ influx through voltage-gated calcium channels (red) by depolarizing the membrane potential, whereas outward conductances inhibit Ca²⁺ influx. In turn, Ca²⁺ influx up-regulates the outward current and down-regulates the inward current possibly via modulation of transcription rates or ion channel trafficking dynamics (gray circle). (B) A simplified model neuron with three Ohmic conductances g₁, g₂, and g₃, each with a different reversal potential (−90, −30, and +50 mV, respectively). Calcium dynamics are first order with exponential steady-state dependence on membrane potential, (Methods) and each conductance is regulated with a specific regulation time constant, , according to the difference between [Ca²⁺] and a target value, . (C) Behavior of three versions of the model with different sets of regulation rates. The traces show the evolution of the three conductances and internal [Ca²⁺] in 30 simulations of each version of the model. Blue traces, original rates (); green traces, scaled rates (); red traces, g₂ rate flipped . (D) Steady-state conductance distributions. 3D plots showing all three conductances for 300 runs using each of the three sets of regulation rates (orange points, random initial values; blue, original rates; green, scaled rates; red, rate flipped). Each 3D plot is a different view of the same data and the colored curves are the sample trajectories plotted in C. The large plot to the right shows the calculated solution space of conductance values that give target [Ca²⁺] (pink plane). The arrows represent the surface normal of the solution plane (pink), the velocity vector for the trajectory of the mean model trace with the original rate set (light blue), and the vector obtained by projecting this velocity vector onto the solution plane (dark blue). (E) Scatterplot matrices showing pairwise scatterplots (off-diagonals) between the three maximal conductances in each version of the model. Histograms (diagonals) show the distribution of each maximal conductance by itself. Black lines in each scatterplot are the correlations predicted by resolving the model trajectory onto the solution set (Methods) as illustrated in the right plot of D. The schematic to the right of the plots show how the axes in the plots are organized.

Fig. 2.

Behavior of the simple model is recapitulated in a model with active conductances. Behavior of a spiking model neuron with three regulated voltage-dependent conductances controlled by the simple regulatory rule in Fig. 1B. Three versions of the model are shown with the regulation rates at their original values (, blue plots; Methods), scaled (; green), and the regulation rate for g_H flipped (; red). (A) Example traces of [Ca²⁺] in each of the three versions of the model (traces truncated at 1,500 ms). (B) Example membrane potential traces at different time points (1, 2, and 3) for each version of the model. (C) Example traces showing the evolution of the three regulated conductances in 30 simulations of each version of the model. (D) (Upper) 3D scatterplot showing the steady-state conductance distributions for 300 model simulations with each set of rates. The two plots show two rotated views of the same data. (Lower) Correlation plots of steady-state conductances for each version of the model.

Fig. 3.

Structure of the steady-state conductance distribution in a complex homeostatic model neuron. Behavior of a complex bursting model neuron with seven regulated voltage-dependent conductances and a regulation rule that uses three [Ca²⁺] sensors. (A) Evolution of the maximal conductances over time for a single regulated neuron. Example voltage traces at three time points along the evolution trajectory are shown at the right, showing that the model converges to the target bursting behavior (horizontal line = 0 mV). (B) Pairwise scattergrams (off-diagonals) and histograms (diagonals) of the final values for the seven regulated maximal conductances after 1 h of simulated time. Each scattergram is a 2D histogram with color representing count density (red, high; yellow, intermediate; green, low; blue, zero). The conductance ranges plotted are (in μS) 0 through 146 (g_Na), 2.9 (g_CaT), 5.4 (g_CaS), 134 (g_KA), 134 (g_KCa), 69 (g_Kd), and 0.8 (g_H). (C) Pairwise scattergrams and histograms for the same seven conductances as in B, showing randomly sampled solution space of models that satisfy target sensor values within 10%. Ranges for each conductance axis are the same as B. (D) Thinned (5,000 points) pairwise scatterplot between g_Kd and g_Na in converged homeostatic models (regulated) and for random sampling (random). These scatterplots correspond to the subplots outlined with pink boxes in B and C, respectively. Example voltage traces of labeled models are plotted to the right for several different points in the solution space (horizontal line = 0 mV).

Fig. 4.

Antihomeostatic regulation can coexist with homeostatic regulation in a complex model. (A) Regulation coefficients in the original Liu et al. model (Left) and in an alternate version in which regulation coefficients for g_Ka are reversed (Right). Each coefficient determines whether a conductance is up- or down-regulated when sensors are above or below targets (+1, up-regulate if below target/down-regulate if above target; −1, down-regulate if below target/up-regulate if above target.) (B) Pairwise scattergrams and histograms of the final values for the seven regulated maximal conductances after 1 h of simulated time using the original parameters of the Liu et al. model (Left) and the alternate model (Right). Each scattergram is a 2D histogram with color representing count (red, high; yellow, intermediate; green, low; blue, zero). In both sets of scattergrams, the ranges plotted are 0 through 440 (g_Na), 8.6 (g_CaT), 16.2 (g_CaS), 402 (g_KA), 402 (g_KCa), 207 (g_Kd), and 2.3 μS (g_H). (C) Detail of the pairwise relationships between g_Kd/g_Na and g_H/g_CaT (pink boxes in B) shown as thinned scatterplots (5,000 points) for the original model (Upper) and the alternate model (Lower). Example voltage traces of labeled points in solution space are shown to the right (horizontal line = 0 mV).