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). This set of starting parameters is called SP1. A – Row 1: blow-ups of specific points a, b, and c in the voltage trace in Row 2. Row 2: Plots the neuron’s electrical activity as its channel density and (in)activations curves were being adjusted. Alpha was 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 were 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 had 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) 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. C – This also 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). However, different starting parameters were used. The initial half-(in)activation values were identical to those used in Panels A and B. But, the initial maximal conductances were set to a specific configuration: the values obtained at the end of the simulation in Panel A (Row 6, left table).

This model attempts to satisfy the same Ca2+ targets as in Figure 1, Panel A, but was started from a different set of ‘random’ initial parameters. The rows correspond to those in Figure 1, Panel A. 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. Five SP’s (SP1 – SP5) were selected based on their periods being within 2% of each other, with an average period of 416.3 ms. This group was expanded to include an additional 15 model instances (SP6 - SP20) whose periods are within 20% of the mean. A) The top left panel displays the periods for all SP’s, with black dots indicating the center of the circles for SP1 – SP5 within the larger group. Burst Duration, Interburst Interval, Spike Hight, Maximum Hyperpolarization, and Slow Wave Amplitude measurements for these SP’s are provided in subsequent plots. The relative standard deviation of SP1-SP5 vs SP1-SP20 for Period, Burst Duration, Interburst Interval, Spike Hight, Maximum Hyperpolarization, and Slow Wave Amplitude were, respectively: 1.4% vs 6.8%, 10.7% vs 17.9%, 2.8% to 8.1%, 9.9% to 12.0%, 2.3% vs 4.2%, and 25.9% to 28.5%. B) The left panel shows maximal conductances, and the right panel displays half-(in)activation voltages for each of the 20 SP’s. Black dots indicate the center of the circles for SP1 – SP5 within the larger group. An asterisk on the right indicates the initially specified level of half-(in)activation. The relative standard deviation of SP1-SP5 vs SP1-SP20 for the maximal conductances of INa, ICaT, ICaS, IH, IKd, IKCa, and IA were, respectively: 11.4% vs 129.2%, 17.6% vs 35.9%, 15.6% vs 17.6%, 23.2% vs 30.2%, 32.8% vs 96.9%, 10.6% vs 40.4%, and 10.0% vs 51.6%. The relative standard deviation of SP1-SP5 vs SP1-SP20 for the half-activations of INa, ICaT, ICaS, IH, IKd, IKCa, and IA were, respectively: 2.5% vs 10.0%, 2.8% vs 4.1%, 2.3% vs 3.3%, 0.5% vs 1.4%, 10.6% vs 48.9%, 1.8% vs 4.6%, and 1.8% vs 4.8%. The relative standard deviation of the Group of 5 vs the Group of 20 for the half-inactivations of INa, ICaT, ICaS, and IA were, respectively: 0.8% vs 3.3%, 2.1% vs 3.1%, 1% vs 1.5%, and 0.6% vs 1.7%.

The model did not need to alter half-(in)activations to assemble a burster. In each row, the electrical activity of SP1 is shown as the model attempted to meet the Ca2+ targets. In each row, the speed at which the model changed (in)activation curves in response to deviations from Ca2+ target, was altered. From top row to bottom the speeds are: τhalf = 6 s, τhalf = 600 s, and τhalf = ∞ s (i.e. half-(in)activations do not change). The speed at which the model changed channel density is fixed in all rows: τg = 600 s. Superimposed on these traces is the α (alpha) 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 (in)activation curves 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 (in)activation curves 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 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 (in)activation curves 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. Black dots mark the center of the colored circles for SP1 – SP5. Trends are illustrated with a black line, with large circles marking median positions. The mauve dot highlights the level of maximal conductance and half-(in)activation SP1 evolved to. 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 pairwise differences in medians (p < .05).

The model simulated the homeostatic recovery of neuronal activity. A burster (SP1) was perturbed by increasing the extracellular potassium concentration to 2.5 times its baseline level (shown in olive green) for 120 minutes. The model adjusted the neuron’s maximal conductance and half-(in)activations to satisfy the Ca2+ targets (τhalf = 6, τg = 600). The blue arrow indicates the position of inset 3 on the time evolution of the maximal conductances and half-(in)activations. By this point, the model had assembled a burster that satisfied the Ca2+ targets during the extracellular potassium perturbation. The black arrow marks the end of the perturbation.

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 speed of ion channel (in)activation curve adjustments changes in response to deviations from Ca2+ targets, 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 5 Bursters by selecting a putative representative of the entire population and then identifying bursters with similar activity patterns. A - The Group of 5 bursters was created by starting with 111 bursters and narrowing this group to a “selection group” (see Methods). Panel A1 shows the period distribution of this “selection group,” while panel A2 illustrates how closely the periods 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, A2). B - To assemble the Group of 5, we identified neighboring models with waveforms similar to SP1. Depolarizing excursions (spikes) were detected using MATLAB’s findpeaks function with a prominence threshold of 2 (red arrows). Panel B1 shows the neighboring models included in the Group of 5, while B2 shows those excluded. In B2, models a–c were excluded due to small bumps at the end of the slow wave, which SP1 does not have. Model d was excluded for having a different number of spikes than SP1. C – 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

M̅X is a sigmoid function with centering parameter Z:

HX Parameters and Time Constants

H̅X is a sigmoid function with centering parameter Z:

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