Sensitivity of a model oscillator to shifts in the voltage-dependence of one or more ionic currents

Abstract

The voltage dependences of ion channels are subject to change as a function of activity and neuromodulatory influences. We study the effects of alterations of the activation and inactivation curves in each of seven voltage-dependence currents in a bursting computational model, in control conditions and in simulated high potassium conditions. We find that modifications of the voltage-dependence of one current can result in changes of the contribution of many or all of the currents in the model neuron to its behavior. We construct two-dimensional maps of the regions of bursting, tonic spiking, or silence as a function of the voltage-dependence of participating currents, in both control and simulated high potassium. Consistent with experimental findings, the position of bursting regimes under the two conditions shows that most bursting neurons lose their bursting behavior in high potassium. The maps show that, in most but not all, models, bursting can be recovered in high potassium conditions by shifts in the voltage-dependence of one or more of the neuron’s currents. Surprisingly, in many models, shifts in the activation curves of several different currents can recover bursting behavior. Moreover, multiple degenerate bursting models with different maximal conductances, can recover function in response to changes in the same ionic current. In some models, coordinated shifts in the voltage-dependence of two or more currents are necessary for recovery of bursting activity. These studies demonstrate that modulation of voltage-dependence can both alter a neuron’s activity, and can contribute to resilience subsequent to perturbation.

NEW & NOTEWORTHY

We explored the effects of modifying the activation and inactivation curves of currents in a bursting model neuron, in control conditions and in response to depolarizing perturbations. Modifications of the voltage-dependence of one current can result in changes of the contribution of many or all of the currents in the model neuron to its behavior, and can contribute to recover from perturbations.

Graphical Abstract

I. Introduction

Most excitable cells, such as neurons and muscles, display a large complement of voltage and time-dependent ion channels. It is not uncommon for neurons to express multiple classes of $K^{+}$ and $C{a}^{2+}$ channels that have characteristic channel properties (1; 2), which have been functionally characterized using either single-channel recordings or a variety of voltage clamp protocols. Together these methods have described the effects of voltage and time on the opening and closing of individual classes of channels (3; 4; 5; 6). Since the early days of membrane biophysics, one of the goals has been to provide a mechanistic understanding of how multiple sets of ion channels interact to produce a particular activity profile, either in neurons, cardiac or other tissues (3; 4; 5). To that end, quantitative biophysical data collected on specific membrane currents have been used to construct detailed conductance-based models with which to provide mechanistic understandings of how the dynamics of excitable membranes arise from the ensemble of channels in their membranes (3; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17).

It is now appreciated that multiple, degenerate solutions, or combinations of parameters, exist that can produce similar activity profiles in either single neurons or networks (18; 19). These degenerate solutions have been largely studied in terms of combinations of the maximal conductances of the contributing channels. Nonetheless, many neuromodulatory factors or activity that act through second messenger pathways directly alter the voltage and time-dependence of individual conductances (1; 20; 21; 22; 23; 24).

Despite the considerable body of literature that demonstrates that both the number and properties of channels influence neuronal and network excitability, there is a paucity of computational studies that map the dependence on neuronal firing properties on the regulation of the voltage dependence of conductances. Consequently, here we look at the sensitivity of a bursting model as a function of modifications in channel voltage-dependence.

II. Results

We exploit a semi-realistic conductance-based model of a bursting neuron to examine the robustness of the bursting activity of the neuron in response to perturbation by simulating the application of high extracellular potassium. This model is a modification of models used in previous studies (19; 25). The model is a single compartment with seven voltage- and time-dependent conductances and is described fully in Methods. We characterized the voltage waveforms for all figures in this work by measuring features such as burst frequency and duty cycle; these data are detailed in the tables provided in the supplemental materials (https://doi.org/10.5281/zenodo.20347336).

A. Changes in half-activation voltage of one current changes the contributions of all ionic currents.

The voltage dynamics of neurons is generated by the opening and closing of many of ion channels. The ranges of voltages at which these channels activate and inactivate, as well as the timescales of these processes —that is, the channels’ kinetics —together with the abundance of channels of each type (maximal conductances) determine the neurons’ dynamics. In most cases, the activation and inactivation processes of ion channels can be modeled by sigmoidal functions whose inflection point is located at the half-activation voltage $V_{\frac{1}{2}}$ of the process (Fig. 1A) (26). It is not surprising that altering the half-activation voltages $(V_{\frac{1}{2}})$ can alter neuronal activity.

FIG. 1. A shift in the half-activation voltage of a potassium current triggers changes in the contributions of all currents.

The figure shows the membrane potential $V\left(t\right)$ and the currentscapes of a model neuron as the half-activation voltage $V_{\frac{1}{2}}$ of the $K_{Ca}$ current is shifted, both in control conditions and in response to high $K^{+}$ condition. (A) Voltage-dependence of the activation process of the KCa current. The activation curve is modeled using sigmoid functions whose inflection point is located at $V_{\frac{1}{2}}$. Changing the value of $V_{\frac{1}{2}}$ shifts the activation curves (red, blue). (B) left A model neuron bursting periodically. The membrane potential is shown on top and the currentscapes the percent contribution of each current over time are shown below. A(center) When $V_{\frac{1}{2}}$ is shifted by $\mathrm{\Delta }V=5mV$ the model neuron remains bursting with a different burst duration and different frequency. The currents change noticeably, with a large increase of the contribution of the $K_{Ca}$ (orange). A(right) Further shifting in $V_{\frac{1}{2}}$ does not modify the membrane potential, but changes the contributions of all currents. C(left) When $E_K$ is changed to $-56mV$ from it’s control value ($-80mV$), the model neuron spikes tonically. C (center) The model neuron with shifted $V_{\frac{1}{2}}$ is able to remain bursting in the high $K^{+}$ condition. C (right) Further shifts of $V_{\frac{1}{2}}$ also allow the model neuron to remain bursting in the perturbed condition, but the waveform and currents are different than in the previous case.

Figure 1 demonstrates how altering the half-activation voltage $(V_{\frac{1}{2}})$ of a single current substantially changes the relative contributions of many membrane currents to the dynamics of the neuron. The activation curve of $KCa$ changes as $V_{\frac{1}{2}}$ is shifted (Fig.1A, red and blue curves). The activity with “control” values of $V_{\frac{1}{2}}$ is shown in Fig.2B (left). The currents driving the activity are illustrated in the currentscapes below (19). The currentscapes show the percent contribution of each current (in colors) to the activity over time. The model neuron produces periodic bursts of action potentials with a period $\approx 1sec$. When the half-activation voltage $V_{\frac{1}{2}}$ of $K_{Ca}$ is shifted by $\mathrm{\Delta }{V}_{\frac{1}{2}}=-5mv$ (Fig 2B, middle) the model neuron remains bursting with approximately the same period, but the burst duration is visibly shorter. Further shifts of $V_{\frac{1}{2}}(-10mv)$ do not change the waveform of the membrane potential, but affect the contributions of all currents to the activity (Fig. 1B right). For example, the contribution of the $A$ current (purple) is smaller, while the contribution of the $K_{Ca}$ current increases. There are also changes in the inward currents: the contribution of the Leak current (pink) increases noticeably, while the contribution of the slow calcium current $C_S$ decreases. The contribution of the $H$ current (brown), which was not visible in “control”, and barely visible when $\mathrm{\Delta }V=-5mV$, becomes noticeable. Thus, although the maximal conductances remain fixed, a change in $V_{\frac{1}{2}}$ of one current affects all currents. Interestingly, it can be the case that changing $V_{\frac{1}{2}}$ will not change the waveform of the membrane potential, but will change the relative contributions of each individual current to the voltage dynamics.

FIG. 2. Model neuron activities change as ion channel half-activation voltages vary over a range, in control and in high $K^{+}$.

The behavior of the model neuron changes as the half-activation potentials of a pair of processes $\left(KC{a}_m,K{d}_m\right)$ are varied over a range. A Membrane potential of model neuron in different conditions. (i) Periodic bursting with control half-activation voltages (black dot in panels B and C) and control saline condition ($E_K=-80mV$). (ii) Periodic bursting with “shifted” half-activation voltages (red dot in panels B and C) and control saline condition $\left(E_K=-80mV\right)$. (iii) Tonic spiking with control half-activation voltages in the high $K^{+}$ saline condition $\left(E_K=-56mV\right)$. (iv) Periodic bursting with “shifted” half-activation voltages in the control saline condition. B The horizontal and vertical axis span a range of half-activation voltages for $KC{a}_m$ and $K{d}_m$. The model neuron’s activity type is indicated in color. The black and red dots correspond to the traces in panel A. The cut panels show the activity maps for ranges closer the transition between tonic spiking and bursting. D Activation curves corresponding to control (black) and shifted (red) values of $V_{\frac{1}{2}}$.

Because the currents that drive the activity in Fig. 1B in the three cases are different, it is expected that the excitability properties, and resilience to perturbations, of each model neuron will be different. To reveal this, we subjected the model neurons to a depolarizing perturbation. We exploited changes in the equilibrium potential of potassium currents $(E_K)$ as a way to model the application of high extracellular potassium. Specifically, we changed the equilibrium potential of potassium from its “control” value of $E_K=-80mV$ to the “high K” condition where $E_K=-56mV$. The control model (Fig. 1B) —without changes in any half-activation potential —does not remain bursting and transitions to a tonic spiking regime shown in Fig. 1C (left). When the half-activation of $K_{Ca}$ is shifted by $V_{\frac{1}{2}}=-5mV$ the model neuron remains bursting upon the application of the perturbation (Fig. 1C middle). The waveform changes noticeably with a smaller bursting period and a longer burst duration. The currents also show noticeable changes: the contribution of the $A$ current (purple) becomes negligible, and the contribution of the $K_{Ca}$ current (orange) becomes larger. The contribution of the Leak (pink) current becomes smaller and the contribution of $C_S$ (green) increases noticeably. Finally, Fig. 1C (right) shows the response of the burster in Fig. 1C when subjected to the same perturbation. Despite the waveform being similar to that in Fig. 1B, the response upon perturbation is different.

Together, the examples in Figure 1 illustrate several interesting properties. When the half-activation of one current is modified, the dynamics of all currents are changed in a complicated non-linear fashion. Changes in the half-activations may or may not result in changes of the waveform. Changing the half-activation of a current can enhance the resilience of the model neuron to specific perturbations. Finally, as shown by multiple computational studies, similar waveforms can be generated by different currents’ contributions, and different currents’ contributions result in different excitability properties that are revealed by perturbations.

B. Minimal changes in channels’ kinetics allow recovery in high K

The experiments in (27) and (28) showed that periodically bursting pacemaker neurons transition into tonic spiking upon application of high $K^{+}$ saline. Over the course of minutes, the neurons adapt to the perturbed saline and recover their bursting activity in the perturbed environment. When high $K^{+}$ is washed out, neurons hyperpolarized but their activity remained in a periodic bursting pattern, without displaying any kind of transient activity. Motivated by these experiments, we investigated the possibility that neurons change the kinetics of their ion-channels to adapt to the perturbed high $K^{+}$ environment and recover their periodic bursting activity.

Figure 2A shows the activities of a model neuron under different conditions. Fig. 2A(i) shows a periodic bursting pattern of activity in control saline and control half-activation voltages. Fig. 2A(ii) shows the activity of the same model neuron, but with a “shifted” half-activation voltage $V_{\frac{1}{2}}$ in the $KCa$ current activation. Fig. 2A(iii) shows the activity of the same model neuron with control half-activation voltages in high $K^{+}$ saline. This shows that the control bursting pattern in Fig. 2A (i) turns into tonic spiking upon application of high $K^{+}$ saline. Fig. 2A(iv) shows the activity of the same neuron in high $K^{+}$ saline, but with shifted half activation voltages. Together, the simulations in Fig. 2 show that the model neuron transitions from bursting to tonic spiking if the concentration of extracellular potassium is increased and that it is possible to recover the bursting pattern of activity if the half-activation voltage $V_{\frac{1}{2}}$ of the $KCa$ current activation is changed (Fig 2B). Note that the shifted values of $V_{\frac{1}{2}}$ allow the model neuron to also burst periodically in the control saline (Fig 2A). Thus if the perturbation is removed, a model neuron with shifted $V_{\frac{1}{2}}$ would retain its bursting pattern without any transient behavior as quiescence or tonic spiking. Figure 2C shows that the type of activity changes as the half-activation voltages $V_{\frac{1}{2}}$ of two processes ($KC{a}_m$ and $K{d}_m$) are varied over a range, for both the control saline condition (2B) and the high $K^{+}$ condition (2C). The black dots indicate the control values of the half-activation voltages, and the red dots indicate the values of the shifted half-activation voltages.

Note that in control saline both the black and red dots are in the bursting region (blue) while in the high $K^{+}$ condition the control half-activation voltage (black dot) is in the tonic spiking region (orange). This means that if the model neuron is exposed to a high $K^{+}$ its activity will shift from periodic bursting to tonic spiking. Therefore, a neuron that could actively sense altered activity and produce appropriate changes in channel kinetics could recover its activity. In this case, shifting the $V_{\frac{1}{2}}$ of the $K_{Ca}$ activation process would suffice to recover a bursting pattern of activity in high $K^{+}$ saline. Figure 2D shows the activation curve of the $K_{Ca}$ activation process for both the control half-activation voltage (in black) and the shifted half-activation voltage (in orange) that would allow the model neuron to recover a bursting pattern of activity in the high $K^{+}$ condition. In general, Figure 2 illustrates one process that can be modified to recover bursting, but there are many other possible mechanisms to achieve the same result. Also, as the maps in Fig. 2B and 2C show, the extent of the shifts in $V_{\frac{1}{2}}$ can vary within a range and still recover bursting activity.

C. Model neuron activities as channels’ kinetics change.

To investigate how the activities of the model neurons change as the kinetics of the channels are modified, we performed a systematic exploration of the model’s activities as the $V_{\frac{1}{2}}$ of the different processes change over a range. We took all pairs of processes, computed the model’s solutions, and classified them as quiescent, tonic spiking, or bursting. There are multiple different modifications of channel kinetics that permit recovery of bursting activity in the high $K^{+}$ condition.

Figure 3 shows how the activity of the model neuron changes as several pairs of half-activation voltages are varied over a range. We performed this computation for both the control saline condition (left sub-panels) and the high $K^{+}$ condition (right sub-panels). In all panels, the black dot in the center indicates the control half-activation voltages. We chose a model neuron that displayed periodic bursting in control, and that would display tonic spiking upon application of high $K^{+}$ saline. Therefore, in control saline, the black dot is always in the bursting region (blue), and in high $K^{+}$ saline the black dot is in the tonic spiking region (orange).

FIG. 3. Activity type as half-activation voltages are varied over a range in control and high $K^{+}$.

Different values of ion channel half-activation voltages result in different types of activities (quiescent, gray; tonic firing, orange; bursting, blue). The panels show activity as different pairs of half-activation voltages vary over a range, both in the control saline $\left(E_K=-80mV\right)$ and high $K^{+}$ saline $\left(E_K=-56mV\right)$ conditions. The black dot indicates the reference or control half-activation voltages. In all panels, the model neurons’ activity corresponds to periodic bursting in “control” and tonic spiking in high $K^{+}$.

The experiments in (27) showed that biological neurons regained function during the application of high $K^{+}$. Upon wash, the waveform of the membrane potential changed, but bursting activity was not lost: the neurons did not become quiescent. One way for the model neuron to reproduce the qualitative behavior observed in experiments, is to shift one or more half-activation potentials to a region that is in the intersection of the blue regions, in control and in high $K^{+}$. Based on this, we can classify the pairs of processes according to how many half-activation potentials must be modified to recover bursting in high $K^{+}$ and to ensure that the neuron remains bursting when washing out the high $K^{+}$. Panels A, B, C, in Figure 3, show pairs of $V_{\frac{1}{2}}$ that can be modified to adapt and recover by shifting only one $V_{\frac{1}{2}}$. Panels D, E, and F, correspond to pairs of processes that must be modified together to achieve adaptation and recovery. Panels G and H are examples of pairs of processes in which shifting one half-activation potential results in adaptation and recovery, but adaptation/recovery is not possible when shifting the other half-potential.

D. Switch in channel kinetics can recover activity during depolarizing perturbation.

The experiments in (27) and (28) showed that neurons change their excitability properties over the course of tens of minutes, and recover bursting activity in high $K^{+}$ saline. Can model neurons achieve this by modifying channels $V_{\frac{1}{2}}$ either individually or in pairs?

The present study illustrates that changes in channel kinetics can provide a path to recover bursting activity during the depolarizing perturbation. The maps in Figure 3 show that in most cases, modest shifts in $V_{\frac{1}{2}}$ could result in recovery of bursting function in high $K^{+}$. The notable exception is in the example shown in Fig. 3F, where there is no path back to bursting in high $K^{+}$, for that pair of processes. Once these maps are constructed, it is straightforward to see in each case what changes in $V_{\frac{1}{2}}$ should produce recovery of bursting.

Figure 3 shows that changing the kinetics of the processes would make it possible to recover bursting activity upon a depolarizing perturbation. To achieve this in the model neuron, we devised a simple activity-dependent switch that allows the model neuron to detect changes in its activity. The mechanism monitors the membrane potential over time by computing a quantity that approximates the running average of the membrane potential $\alpha (t)\approx <V(t)>$, and uses this information to shift the $V_{\frac{1}{2}}$ of a given process. The equations and details of the activity-dependent switch are in the Methods section.

This mechanism is illustrated in Figure 4. The simulation was designed to mimic the behavior of the bursting pacemaker neurons in the experiments in (27) and (28). Initially, the model neuron is bursting periodically, (Fig. 4A(i). When the model neuron is exposed to higher concentrations of extracellular potassium, the reversal potential of potassium currents changes from its control value $E_K^C=-80mV$ to $E_K^H=-56mV$. The initial response of the model neuron is to transition to a tonic spiking regime, shown in 4A(ii). Because the membrane potential becomes on average more depolarized, the recovery mechanism begins driving changes in the channels’ kinetics. In this simulation recovery occurs by shifting the half-activation potential $V_{1/2}$ of the $KCa$ activation process (Fig. 2). Changing the kinetics of $KC{a}_h$ allows the model neuron to recover bursting activity while in high $K^{+}$ (Fig. 4A(iii)). We then shifted $E_K$ back to its control value $E_K^C=-80mV$. The initial response of the model neuron upon wash (early wash) is to remain in a periodic bursting regime (Fig. 4A(iv)).

FIG. 4. Activity-dependent changes in channel kinetics recover bursting activity during depolarizing perturbation.

The model neuron fails to produce its control bursting pattern of activity upon exposure to high $K^{+}$, and recovers its bursting activity over the course of minutes while in the perturbed environment. A Membrane potential at different times. (i) Control bursting activity. (ii) Initial response to high $K^{+}$: the activity changes from bursting to tonic spiking. (iii) Bursting pattern in high $K^{+}$. (iv) Bursting pattern at “early wash”. B (top row) Membrane potential over time. (center row) $\alpha$ variable over time. $\alpha (t)$ approximates a running average of $V(t)$. (bottom row) $V_{\frac{1}{2}}$ of the $KCa$ activation process over time(in black) and $V_{\frac{1}{2}}^{\mathrm{\infty }}$ in orange. C Currentscapes for the patterns of activity in A (i-iv).

The complete simulation is shown in Figure 4B. The top panel shows the membrane potential of the model neuron over time. The green shading indicates when the model neuron was in {high $K^{+}$ saline condition. Notice that when the model neuron is exposed to {high $K^{+}$ the membrane potential becomes, on average, more depolarized. The center panel shows the behavior of $\alpha$, the variable that monitors the membrane potential. The value of $\alpha$ does not change in control and begins to change as soon as the model neuron becomes more depolarized. Once the model neuron achieves a bursting pattern of activity, $\alpha$ stays flat until “wash” ($E_K$ is switched back to its control value $E_K=-80mV$), when it begins to decrease back to another steady state. The bottom row in Fig. 4B shows the evolution of the half-activation potential of $KCa$ over time (in black), and the fixed point of eqn. 5 (in orange). Initially $V_{1/2}$ coincides with the fixed point of eqn. 5. When $\alpha$ changes the fixed point of $V_{1/2}$ changes (6), and $V_{1/2}$ evolves towards the new fixed point on a timescale of ${\tau }_{V_{1/2}}$.

The currents that underlie the activities of the model neuron are illustrated in Fig. 4C using currentscapes (19). The currentscapes show the percent contribution of each conductance to the activity over time. During spikes, the dominant currents are the sodium $I_{Na}$ and potassium $I_{Kd}$ currents as expected. When the model neuron is hyperpolarized (not spiking) in control (Fig. 4C(i)), the dominant outward currents are $I_{KCa}$ (orange) and $I_A$ (purple), while the dominant inward currents are $I_{\mathit{leak}}$ and $I_{\mathit{Cas}}$. During the tonic spiking regime (Fig. 4C(ii)) the contribution of $I_{\mathit{CaT}}$ increases and the contribution of $I_A$ decreases. The bursting pattern during the high $K^{+}$ perturbation (Fig. 4C(iii)) is driven by a different combination of ionic currents. Notice that the contribution of $I_A$ is almost negligible while $I_{KCa}$ becomes the main contributor to the outward currents. The inward currents also changed: the leak current contributes less and $I_{CaS}$ contributes more. The currents driving the bursting pattern at the initial wash (Fig. 4C(ii)) are similar to those in control.

E. Multiple recovery strategies.

The recovery strategy described in Figure 4 is not the only way to recover bursting activity. As shown in Figure 3 there are multiple different modifications of the channels’ kinetics that allow a similar feat.

In Figure 5 we show alternative modifications to the channels’ kinetics of the same model as in Figures 2 and 4. For example, another way to recover activity is to shift the half-activation voltage of $Kd$. Figure 5A shows the sequence of states the model neuron goes through as it is subject to the same perturbation protocol as in Figure 4, but now the recovery mechanism is changing the kinetics of the $Kd$ current. The upper panels show the membrane potential of the model neuron over time, with the same time stamps as in Fig. 4. The bottom panels show the corresponding currentscapes for the patterns of activity in the top panels. By construction, the “control” and “initial response” traces and currentscapes are identical to those in Fig. 4 since we use the same model. The adapted bursting pattern of activity in “late ‘high $K^{+}$’” is similar to the one in Fig. 4A (iii). The “early wash” pattern of activity (instantaneous response to removing high $K^{+}$) is different to that in Fig. 4 (iv) and the currents are also different, with a larger contribution of $I_A$.

FIG. 5. Multiple recovery strategies for the same model neuron.

The model neuron can recover bursting activity during the perturbation by modifying the channel kinetics of the $Kd$ current A and the $Na$ current A. The sub panels show the waveforms and the currentscapes of the model neuron in control (i), when it is initially exposed to the high $K^{+}$ perturbation (early high $K^{+}$), when it recovers bursting activity (iii) (late high $K^{+}$), and when the perturbation is removed (iv) (early wash).

Figure 5B shows the same model subject to the same protocol, but now the half-activation of the ${Na}^{+}$ is allowed to change in an activity-dependent manner to recover bursting during high $K^{+}$. The control activity and initial response are reproduced again for completeness in panels (i) and (ii). The model neuron eventually recovers bursting activity as before, but the currents that drive the activity are different from the two previous cases (Fig. 4 and Fig.5A): there is a small contribution of $Kd$ (yellow) during the hyper polarized phase of the bursting pattern and there is a visible increment in the contribution of the transient calcium current $I_{CaT}$ (blue). As before, the model neuron remains bursting when the perturbation is removed (iv).

F. Different model neurons, same recovery strategy.

Neurons and neuronal circuits display similar activities even when driven by different combinations of ionic currents (18; 19; 29). One consequence of this is that model neurons with similar bursting patterns of activity may respond differentially to perturbations, and may also require different recovery strategies to adapt to a perturbed environment. This raises the question whether any given recovery strategy will work in different models. To investigate this, we produced three bursting model neurons with different values of maximal conductances, and implemented the recovery mechanism described in Fig. 4.

Figure 6 shows how three different models can recover using the same mechanism. The control activity is shown on the left panels. Because the maximal conductances are different, the contributions of each ionic current to the activity are different. The currents in model (A) show a large contribution of $CaS$ (green), while in model (B) $CaT$ (blue) contributes more. The model in (C) utilizes the $H$ current (in brown), not present in the previous two models. When the model neurons are exposed to high $K^{+}$ they all transition into tonic spiking regimes with slightly different firing rates. When the model neurons depolarize the activity-dependent switch shifts the half-activation voltage $V_{\frac{1}{2}}$ of the $K_{Ca}$ current allowing the recovery of bursting activity in the perturbed environment. The currents driving the bursting regime are different across models. As before, all models remain bursting upon removal of the perturbation.

FIG. 6. Different model neurons recover activity using the same mechanism.

Model neurons (A-C) displaying similar bursting patterns of activity, but using different ionic currents, are subjected to the high $K^{+}$ perturbation and recover in the perturbed environment, by modifying the kinetics of the same ion channel. The figure shows the traces and currentscapes of the model neurons at their control (i), early high $K^{+}$ (ii) responses, late high $K^{+}$ (iii), and early wash (iv).

G. The general case

Most models of neuronal homeostasis assume that biological neurons can modify their properties in an activity-dependent manner to stabilize a prescribed pattern of activity (30; 31; 32; 33; 34; 35). We showed that the channel kinetics can also be modified to adapt to a perturbed environment by shifting the half-activation voltages of one process at a time. A more general hypothesis is that neurons can actively modify several properties to adapt to perturbations in their environments. The mechanism explored in this work can be generalized to two or more processes.

Figure 7 shows a model neuron recovering during a depolarizing perturbation by modifying two half-activation voltages. As before, we monitor the membrane potential using a state variable $\alpha$ that approximates the running average of $V(t)$ over time: $\alpha (t)\approx \le V(t)\ge$. As the model neuron depolarizes due to changes in $E_K,\alpha$ changes its value, and we use that to change the fixed points of the dynamic equations for $V_{\frac{1}{2}}$. See Methods for additional details.

FIG. 7. Activity-dependent changes in the kinetics of two ionic channels allow recovery during depolarizing perturbation.

The recovery mechanisms we found can be generalized to the case of modifying two or more channels to recover. In this case, when the model neuron loses its target bursting pattern of activity, the half-activation voltages $V_{\frac{1}{2}}$ of the $X$ and $Y$ channels are shifted to recover bursting activity in the perturbed environment. A Membrane potential at different times. (i) Control bursting activity. (ii) Initial response to high $K^{+}$: the activity changes from bursting to tonic spiking. (iii) Bursting pattern in high $K^{+}$. (iv) Bursting pattern at “early wash”. B (top row) Membrane potential over time. (middle row) $\alpha$ variable over time. $\alpha (t)$ approximates a running average of $V(t)$: i.e. $\alpha (t)\approx <V(t)>$. (bottom row) $V_{\frac{1}{2}}$ of the $X$ activation process over time (in black) and $V_{\mathit{frac}12}^{\mathrm{\infty }}$ in orange, $V_{\frac{1}{2}}$ of the $Y$ activation process over time (in gray) and $V_{\frac{1}{2}}^{\mathrm{\infty }}$ in yellow C Currentscapes for the patterns of activity in A (i-iv).

The upper panels of Figure 7A show snippets of the activity at different time stamps of interest. The control activity for this model is shown in (A i). The initial response to high $K^{+}$ is shown in snippet (A ii), the adapted membrane potential is shown in (A iii) and the initial response to wash or removal of the perturbation is shown in (a iv). As in previous examples, the neuron is bursting normally in control. The activity is disrupted initially by the depolarizing perturbation and the model neuron becomes quiescent. Shortly after that the neuron goes into a single spike bursting mode. Over time, due to changes in the half-activation voltages $V_{\frac{1}{2}}$ of channels $KCa$ and $Kd$, the model neuron regains the ability to burst regularly. When the perturbation is removed, the model neuron stays bursting periodically with a slightly different waveform. Over time, the values of the half-activation voltages return to values close to those in control.

The complete course of the simulation is shown in Figure 7B. The top panel shows the membrane potential of the model neuron. The green shading indicates the time interval over which the depolarizing perturbation is applied. The time series in the center panel corresponds to the average membrane potential ($\alpha$), and the bottom panels show both the fixed points $\left(V_{1/2}^{X\mathrm{\infty }}\right)$ of both processes under modification, and the time evolution of the corresponding state variables $V_{1/2}(t)$.

III. Discussion

Neuronal excitability depends on the interaction of numerous classes of ion channels that contribute to neuronal dynamics as a function of their density and their voltage dependence (36; 1). Starting in the era of Hodgkin and Huxley (3), voltage clamp methods and single channel recordings have been used to characterize the voltage and time-dependence of channel opening and closing. Using these methods, it was demonstrated that channel function, including shifts in voltage-dependence can be modified by intracellular signal transduction pathways activated by GPCR receptors. (37; 38; 39; 40; 20; 41; 42; 43; 44; 2).

Although the dynamics of channel function depend on the properties of all membrane currents at any moment in time (19), experimentally it is almost impossible to measure the biophysical properties (either maximal conductances or voltage-dependences) of all of the conductances that contribute to dynamics at the same time. Moreover, because even neurons of the same cell type showing quite similar behavior can be degenerate and display substantially different values of their maximal conductances (45; 46; 47; 48; 49; 50; 51; 19; 52), it is difficult and/or impossible to measure subsets of parameters in different neurons and then assume those values are accurate measures in all neurons (46; 19). Consequently, many of the insights that we produce in these computational studies are exceptionally informative, as at present it is not possible to do the equivalent measurements experimentally.

Part of the motivation for this study comes from a series of experimental studies in which high potassium solutions were applied to the pyloric rhythm of the crab stomatogastric ganglion. In response to the high potassium, the neurons depolarized, lost their bursting activity, and then slowly over minutes, recovered activity and bursting (27; 28; 53). Interestingly, repeated applications of the high potassium solution showed a persistent acceleration in the recovery process, although the control activity appeared unchanged (27; 28; 53). We were therefore curious to ask whether the recovery in high potassium could result from changes in the voltage-dependence of one or more currents that were triggered by the perturbation and resulted from an activity-dependent process that finds degenerate solutions characterized by shifts in membrane current voltage-dependence. The models in this paper constitute an existence proof of the plausibility of that mechanism, and also provide new insights into degenerate neuronal models that come not from different combinations of membrane conductances (54; 18; 19), but from voltage-dependence.

In this study we generated maps of bursting activity as a function of the voltage dependences of their activation and inactivation curves. In these maps we show the parameters that show bursting and tonic firing activity under control conditions and in response to perturbations by high potassium (28; 53). Because we could manipulate the properties of the same model neuron in different ways as well as to compare the same manipulation across different models with very similar activity, but different underlying parameter values, we were able to make a number of intuitively surprising conclusions that would not have been possible to discern in experimental work.

The first surprising finding was that even shifts in the voltage-dependence of a single current could result in alterations of many membrane currents (Fig. 1). Figures 4 and 5 together show that individually moving the $V_{\frac{1}{2}}$ of three different currents, KCa, Kd, and Na, can result in recovery of bursting activity late in the high $K^{+}$ perturbation, and the similarity in the activity patterns carries over into the early wash. Given the different roles and timing of these three currents in the burst, it is surprising that shifts in any of these three currents can achieve approximately the same result. When the same changes in $V_{\frac{1}{2}}$ to the KCa in three different bursting models (Fig. 6) are applied to the same channel type in different models, all three models essentially recovered their bursting activity, demonstrating that in some cases, sterotyped changes can result despite the degeneracy in the underlying conductances of the initial bursting neurons. It is easy to imagine that in some neurons, more than one $V_{\frac{1}{2}}$ is modified at the same time. An example of this is shown in Figure 7, although we did not exhaustively explore simultaneous alterations of all possible pairs, triples, etc of conductances.

There is an important difference between the implementation of the $V_{\frac{1}{2}}$ modifications in this manuscript, in which the changes in $V_{\frac{1}{2}}$ are implemented by tracking the membrane potential, and the mechanism implemented in a new set of studies in which $V_{\frac{1}{2}}$’s and conductance densities are coordinately regulated homeostatically (34; 35) using intracellular $C{a}^{2+}$ concentrations and a target activity level. The present study uses the activity rule to find a path towards recovery that is restricted to changes in specified voltage-dependences, while a homeostatic rule allows all parameters to vary simultaneously. and therefore provides less information about the structures of the parameter maps that are consistent with recovered function. In other words, the present study shows the relationships in parameter space of the $V_{\frac{1}{2}}$ parameters, while the homeostatic model does not reveal the actual structure of the parameter spaces traveled during the homeostatic recovery process. Thus, this model shows qualitative recovery of function similar to what was seen in the motivating biological experiments, but is specifically not a homeostatic model. Instead, these approaches provide complementary information about the role that modifications of voltage dependence can play in controlling a neuron’s activity profile.

A variety of processes may allow neurons to adapt to perturbations, and for alterations in intrinsic membrane properties that persist over various time courses from minutes to days. It is likely that these different processes can be gated on and off under different physiological conditions, as needed (33).

Quantitative measurements for all simulations are provided in the Supplementary Materials: https://doi.org/10.5281/zenodo.20347336.

IV. Methods

A. Model neuron

The dynamics of the model neurons in this study is given by a Hodgkin-Huxley equation for a single compartment with eight currents: a sodium current, $I_{Na}$; transient and slower calcium currents, $I_{CaT}$ and $I_{CaS}$; a transient potassium current, $I_A$; a calcium-dependent potassium current, $I_{KCa}$; a delayed rectifier potassium current, $I_{Kd}$; a hyperpolarization-activated inward current, $I_H$; and a leak current $I_{\mathit{leak}}$. The membrane potential $V$ of a model cell containing channels $N$ and membrane capacitance $C$ is given by the following:

$$C\frac{dV}{dt}=I_e-\sum _{i=1}^8{I}_i.$$ (1)

Each term in the sum corresponds to an ionic current $I_i=g_i{m}_i^{p_i}{h}_i^{q_i}\left(V-E_i\right)$ and $I_e$ is an externally applied current. The maximal conductance of each channel is given by $g_i,m_i$ and $h_i$ are the activation and inactivation variables, the integers $p_i$ and $q_i$ are the number of gates in each channel, and $E_i$ is the reversal potential of the ion associated with the $i-th$ current. The reversal potential of the Na, K, H and leak currents $E_{Na}=30\mathrm{m}\mathrm{V},E_K=-80\mathrm{m}\mathrm{V},E_H=-20\mathrm{m}\mathrm{V}$ and $E_{\mathit{leak}}=-50\mathrm{m}\mathrm{V}$. The calcium reversal potential $E_{Ca}$ was computed dynamically using the Nernst equation assuming an extracellular calcium concentration of 3 × 10³μM. The kinetic equations describing the seven voltage-gated conductances were modeled as in reference (31),

$$\begin{array}{rr}{\tau }_{m_i}(V)\frac{d{m}_i}{dt}& =m_{{\mathrm{\infty }}_i}(V)-m_i\\ {\tau }_{h_i}(V)\frac{d{h}_i}{dt}& =h_{{\mathrm{\infty }}_i}(V)-h_i.\end{array}$$ (2)

The functions ${\tau }_{m_i}(V)$, $m_{{\mathrm{\infty }}_i}(V)$, ${\tau }_{h_i}(V)$ and $h_{{\mathrm{\infty }}_i}(V)$ are based on the experimental in (55). The activation functions of the $K_{Ca}$ current require a measure of the internal calcium concentration $\left[{Ca}^{+2}\right]$ (31). The dynamics of the intracellular calcium concentration are given by,

$${\tau }_{Ca}\frac{d\left[C{a}^{+2}\right]}{dt}=-C{a}_F\left(I_{CaT}+I_{CaS}\right)-\left[C{a}^{+2}\right]+C{a}_0.$$ (3)

Here $C{a}_F=0.94\frac{\mu M}{nA}$ is a current-to-concentration factor and $C{a}_0=0.05\mu M$. Finally, the total capacitance of the cell is $C=1nF$.

In all figures, the model was integrated using an exponential Euler scheme (56) with step $dt=0.1\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$. The model and the integration routine were implemented in Matlab. Code is available at https://github.com/Liaro0903/HighK-Marder/tree/paper.

Table I summarizes the maximum conductance $(\bar{g})$ values for various ion channels used in different simulation sets (A through E). These parameter sets correspond to the figures referenced for specific modeling scenarios.

TABLE I.

Maximum Conductance $(\bar{g})$ Values for Different Parameter Sets in (nS)

Parameter Set	${\bar{g}}_{\mathrm{A}}$	${\bar{g}}_{\mathrm{C}\mathrm{a}\mathrm{S}}$	${\bar{g}}_{\text{CaT}}$	${\bar{g}}_{\mathrm{H}}$	${\bar{g}}_{\mathrm{K}\mathrm{C}\mathrm{a}}$	${\bar{g}}_{\mathrm{K}\mathrm{d}}$	${\bar{g}}_{\text{Leak}}$	${\bar{g}}_{\mathrm{N}\mathrm{a}}$
Set A (Figs 1–5)	500	60	25	0.1	100	1000	0.1	4000
Set B (Fig 6a)	500	60	25	0.1	50	1000	0.1	1000
Set C (Fig 6b)	262.76	5.03	149.94	0.02	59.91	4448.59	0.08	6631.51
Set D (Fig 6c)	1267.82	210.68	92.00	0.17	355.5	3186.61	0.014	12253.03
Set E (Fig 7)	113.04	6.28	8.164	0.1256	12.56	118.44	0.03	1210.78

The timescale functions are identical to those in (18) with the exception of Set E with timescale functions identical to (31).

B. Activity Maps

The activities of different half-activation shifts were organized into two-dimensional maps. The half-activations were organized into pairs, and with 11 activations from 8 channels, a total of 55 pairs were created based on the combinatorial calculation of choosing 2 from 11. To simulate the half-activation kinetics, the Xolotl (57) code was modified to include the half-activation as a varied parameter. Each half-activation shift was simulated over a range from −30 mV to +30 mV relative to their respective half-activation, with 101 simulations conducted across this range. After integrating the differential equations, the simulated results were classified into silence, tonic spiking, or bursting using a custom classifier. The classifier first determined the inter-spike interval (ISI) by finding the spike times using a spike threshold of 0mV. If there were no spikes detected, the activity was classified as silence. If the standard deviation of the ISI exceeded 10, the simulation was classified as bursting, while the remaining cases were categorized as tonic spiking. Each pair was run in a parallel for-loop to reduce simulation time.

C. Activity-dependent switch in channel kinetics

The strategy behind the activity-dependent switch is to monitor the membrane potential over time by introducing a quantity that approximates the running average of the membrane potential $V(t)$ using a linear differential equation,

$${\tau }_{\alpha }\dot{\alpha }=V(t)-\alpha .$$ (4)

$$\tau \frac{d\mathrm{\Delta }{V}_{1/2}}{dt}=\mathrm{\Delta }{V}_{1/2}^{\mathrm{\infty }}(\alpha )-\mathrm{\Delta }{V}_{1/2}$$ (5)

$$\mathrm{\Delta }{V}_{1/2}^{\mathrm{\infty }}(\alpha )=\frac{a}{1+e^{\frac{-\left(\alpha -b\right)}{c}}}.$$ (6)

The fixed point of eqn. (5) is $\mathrm{\Delta }{V}_{1/2}^{\mathrm{\infty }}(\alpha )$. Let $V_{1/2}^C$ be the value (values) of the half-activation potentials in “control” (i.e. when the model neuron is bursting periodically), and $V_{1/2}^R$ the value (values) of the the half-activation potentials necessary for recovery (see Fig. 1D for example). By choosing the parameters of the sigmoid function $a,b$ and $c$, it is possible to produce the shifts required to recover. Specifically, the value of $a$ determines the total shift in $V_{1/2}$ in milivolts. When the activity changes from periodic bursting to tonic spiking and consequently $\alpha$ changes from control ${\alpha }_C$ to hyperpolarized ${\alpha }_H$, the fixed point (points) of $V_{1/2}(t)$ changes from $V_{1/2}^C$ to $V_{1/2}^C+\mathrm{\Delta }{V}_{1/2}$ and effectively reaches that value on a “recovery timescale” set by the parameter $\tau$. The timescale for recovery in the experiments in (27) and (28), was on the order of minutes, so we set $\tau =100sec$.

This mechanism can be generalized to the case where more than one process is modified (Figure 7). In the case of two processes X and Y we change the half-activations as follows,

$$\tau \frac{d\mathrm{\Delta }{V}_{1/2}^X}{dt}=\mathrm{\Delta }{V}_{1/2}^{X\mathrm{\infty }}(\alpha )-\mathrm{\Delta }{V}_{1/2}^X$$ (7)

$$\tau \frac{d\mathrm{\Delta }{V}_{1/2}^Y}{dt}=\mathrm{\Delta }{V}_{1/2}^{Y\mathrm{\infty }}(\alpha )-\mathrm{\Delta }{V}_{1/2}^Y,$$ (8)

with,

$$\mathrm{\Delta }{V}_{1/2}^{X\mathrm{\infty }}(\alpha )=\frac{a_X}{1+e^{\frac{-\left(\alpha -b_X\right)}{c_X}}}$$ (9)

$$\mathrm{\Delta }{V}_{1/2}^{Y\mathrm{\infty }}(\alpha )=\frac{a_Y}{1+e^{\frac{-\left(\alpha -b_Y\right)}{c_Y}}}.$$ (10)

The values of parameters used in eq. (6) are listed in table II.

TABLE II.

Parameters for activity dependent switch in eq. 6.

Parameter Set	a	b	c
Fig. 4	3	−48.5	0.1
Fig. 5a	−7	−45	0.5
Fig. 5b	−2	−45	0.5
Fig. 6a	3	−43.5	0.5
Fig. 6b	8	−45	0.5
Fig. 6c	3	−50	0.1
Fig. 7 (KCa)	8	−44	0.1
Fig. 7 (Kd)	8	−44	0.1

All units are milivolts (mV). Parameter $a$ corresponds to the total shift in $V_{1/2}$ after a transient period of time $\tau$.