Activity-Dependent Regulation Mechanism (τhalf = 6 s and τg = 600 s) was used to assemble a burster that meets the Ca2+ targets from ‘random’ starting parameters (i.e. SP).

Both this set of starting parameters and its resultant model are called SP1. A – Row 1: blow-ups of specific points a, b, and c in the voltage trace in Row 2. Row 2: The neuron’s electrical activity as its channel density and (in)activations curves were being adjusted. α is a measure of how close the model is to satisfying Ca2+ targets (see Methods). It took values between 0 and 1, with 0 meaning the targets were satisfied. Row 3: Plots the adjustments made to the half-activations as the model attempted to satisfy the calcium targets. They are color-coded to match the intrinsic currents shown in the left plot of Row 5. Row 4: Same as Row 3, except for (in)activation curves. Note that not all currents in this model have inactivation curves. Row 5: Same as Row 3, except for maximal conductances (i.e. channel density). Row 6: The final levels of maximal conductances are shown in the left plot. The final half-(in)activations are plotted on the right. The final half-activation of an intrinsic current, , was the sum of the initially posited level of the that current’s half-activation and the shifts the model made to the activation curve, (shown in Row 3). The final half-inactivation of an intrinsic current, , was the sum of the initially posited level of the that current’s half-inactivation and the shifts the model made to the inactivation curve, (shown in Row 4). [The process is illustrated in Figure 1 (supplement 1) for a different set of starting parameters (i.e. initial half-(in)activation shifts and maximal conductance levels).] B – This displays the neuron’s electrical activity if the model was restricted to varying only half-(in)activations (τhalf = 6 s), with maximal conductances fixed (τg = ∞ s). The same starting parameters as in Panel A were used for this simulation.

This model attempts to satisfy the same Ca2+ targets as in Figure 1A, but was started from a different set of ‘random’ initial parameters.

The rows correspond to those in Figure 1A. Note that the maximal conductances of the two potassium currents, IKd and IA, differ from SP1.

All 20 SP’s were analyzed including those illustrated in Figure 1.

A) The top left panel displays the periods for all SP’s. Burst Duration, Interburst Interval, Spike Hight, Maximum Hyperpolarization, and Slow Wave Amplitude measurements for these SP’s are provided in subsequent plots. B) B1 shows maximal conductances, and B2 displays half-(in)activation voltages for each of the 20 SP’s. An asterisk on the right indicates the initially specified level of half-(in)activation from which random deviations were made.

The speed at which the model changed half-(in)activations in response to deviations from Ca2+ target impacted the type of bursters the model assembled.

In all simulation for this figure, τg = 600. A) These plots extend the measurements from Figure 2 to the same 20 SP’s, now with half-(in)activations regulated at a different speed. In blue, τhalf = 6 s; recapitulating results in Figure 2. In green and orange, the SP’s were reassessed with slower half-(in)activation adjustments, τhalf = 600 s, and τhalf = ∞ s (i.e. halted), respectively. Brackets indicate significant differences in medians (p < .05). These pairwise differences were assessed using a Kruskal-Wallis test (to assess whether any differences in medians existed between all groups) and post-hoc Dunn tests with Bonferroni corrections to assess which pairs of groups differed significantly in their medians. B) The model was used to stabilize SP1 to the same Ca2+ target but using different timescales of half-(in)activation alterations. τhalf = 6 s is shown on the left and τhalf = ∞ s (representing the slowest possible response) on the right. Rows 1-4 below each voltage trace show: the total outward current, percentage contribution of each outward intrinsic current to the total outward current, percentage contribution of each inward intrinsic current to the total inward current, and the total inward current. A dashed vertical line across all rows marks the point of maximum hyperpolarization in each activity pattern to guide focused comparison. C) When the model stabilized around a burster, the speed at which the model changed half- (in)activations impacted maximal conductance and half-activation location of the calcium-activated potassium current. Shown here in blue, green, and orange are τhalf = 6 s, τhalf = 600 s, and τhalf = ∞ s (i.e. halted), respectively. SP1 – SP20 are in the colored circles. Trends are illustrated with a black line, with large circles marking median positions. The mauve dot highlights the levels to which maximal conductance and half-(in)activation of SP1 evolved. The asterisks mark the original position of the KCa half-activation . A Kruskal-Wallis test followed by post-hoc Dunn tests (with Bonferroni corrections) were conducted to assess significant (p < .05) pairwise differences in medians (brackets).

Changes to the timescale of half-(in)activation alterations steers the model through different trajectories in intrinsic parameter space. Each row shows the behavior of Starting Parameter 1 (SP1) as the model adjusts toward a configuration that satisfies the calcium activity target, using different timescales for half-(in)activation regulation.

A – τhalf = 6 s, B – τhalf = 600 s, C –τhalf = ∞ s (i.e. half-(in)activations do not change). In all panels, the speed at which the model changed channel density is fixed in all rows: τg = 600 s. Superimposed on these traces is the α parameter, which indicates whether the activity patterns were meeting the Ca2+ targets. Blow-ups of the activity pattern when the calcium sensors were satisfied are displayed to the right.

The speed at which the model changed half-(in)activations in response to deviations from Ca2+ targets impacted the type of bursters the model assembled following perturbation.

A) Voltage traces from SP1 before (blue) and 265 minutes after (red) a simulated increase in extracellular potassium concentration. Each panel shows recovery under a different timescale for half-(in)activation regulation: fast (τhalf = 6 s), intermediate (τhalf = 600 s), and infinitely slow (i.e. off, τhalf = ∞ s). In all condition, alteration of maximal conductances was held fixed (τg = 600 s). B) Summary of activity features for all SP1–SP20 models that recovered to a stable burst pattern. The number of models that successfully recovered varied with the homeostatic mechanism: fast (n = 17), intermediate (n = 18), and no half-(in)activation modulation (n = 20). Gray points indicate pre-perturbation measurements of SP1-20 (from Figure 3A). Significant pairwise differences in median values were assessed using a Kruskal–Wallis test followed by post hoc Dunn tests with Bonferroni correction. Significant comparisons (p < 0.05) are marked with horizontal brackets. C) The speed of half-(in)activation regulation influenced both the maximal conductance and half-activation voltage of KCa. Data are grouped by as they are in B). Statistical comparisons were performed as in B).

The impact of adjusting the timescale of ion channel (in)activation curve alterations can be illustrated conceptually using the space of activity characteristics (right) and underlying intrinsic parameters (left).

The regions in green and blue are the activity patterns that are consistent with the Ca2+ targets in the space of activity characteristics and underlying intrinsic parameters, respectively. As the rate of ion channel half-(in)activation adjustments in response to deviations from Ca2+ targets is changed, the regions targeted by the model also shift (shown in orange). Regardless of the timescale, all targeted regions reside within a larger region encompassing all activity profile measurements and intrinsic parameters consistent with the Ca2+ targets.

We constructed the Group of 20 Bursters by selecting a putative representative of the entire population and then identifying bursters with similar activity patterns.

A) - 111 bursters were narrowed to 80 bursters by excluding certain characteristics (see Methods). These models had a period distribution shown in A. Highlighted in red are the models chosen for further selection. B) This plot illustrates how closely the periods of the models chosen in A align with their two neighbors after they were ordered from smallest to largest. The first model, SP1, was chosen as the neuron with neighbors that also have close periods (red arrow). C) - To assemble the Group of 20, we symmetrically expanded the inter-instance period range around SP1’s period by 20%. SP2-SP5’s wave forms are shown. Depolarizing excursions (spikes) were detected using MATLAB’s findpeaks function with a prominence threshold of 2 (red arrows). D) – Troughs were identified using the findpeaks function on the negative waveform of a cycle period, applying the same prominence threshold. The detected troughs are indicated by red arrows. The example displayed here represents the inverted voltage trace of the burster assembled by the model from SP1. The period was measured as the time difference between successive points of maximum hyperpolarization.

Activation Curve Exponents & Equilibrium Potentials

Normative values of Half-Activation and Half-Inactivation used to generate all starting parameter sets by randomized shifts. Final parameter sets are converted to the normative value minus the shift.

Activation Curves and Associated Time Constants

Inactivation Curves and Associated Time Constants

MX Parameters and Time Constants

M̄X is a sigmoid function with centering parameter Z:

HX Parameters and Time Constants

H̄X is a sigmoid function with centering parameter Z:

Parameters describing Homeostatic Modulation of Intrinsic Current Properties (Li Matrix)