https://orcid.org/0009-0008-2452-6150
Figures
Abstract
Projection neurons in the dorsal horn relay nociceptive input to supraspinal centers. During central sensitization, a subset of them switches from tonic firing to plateau potentials with sustained afterdischarges, a change that requires intrinsic bistability between a resting and a spiking state. Voltage-gated L-type calcium (CaL) channels can produce bistability, but reach physiological resting states only when paired with voltage-gated potassium channels, most of which simultaneously shrink the bistability window. How robust, physiological bistability arises has therefore remained unclear. Using a minimal conductance-based model, we show that inward rectifier potassium (Kir) channels enlarge the bistability window when combined with CaL channels, while M-type potassium (KM) channels slightly reduce it. Within the parameter region where bistability is both robust and physiological, both channel types can sustain bistability, but the CaL + Kir combination produces a substantially larger window and is more robust to noise and intrinsic variability. This window-enlarging effect traces to a shape feature of the outward Kir steady-state current: like the CaL current, it has a region of negative differential conductance around the spike threshold, a feature absent from KM and from most other voltage-gated potassium currents. Bifurcation analysis further shows that the two pairs support qualitatively distinct excitability: plateau-generating bistability for CaL + Kir and resonator-like dynamics for CaL + KM. These conclusions hold in a two-compartment model of deep projection neurons with realistic ion channel complements, and identify the CaL + Kir pair as a candidate intrinsic mechanism for central sensitization.
Author summary
Many neurons can hold two stable states (resting or firing) at the same input current, depending on their recent history. This bistability underlies a form of short-term memory at the single-cell level and appears in contexts as varied as sleep rhythms, motor control, working memory, and pain signal amplification during central sensitization. How bistability arises in a physiologically plausible way has remained partly unresolved. L-type calcium (CaL) channels generate bistability, but at abnormally low resting states; additional channels must redress these potentials, yet most potassium channels that do so also shrink the range of currents over which bistability exists. Using a conductance-based model, we show that inward rectifier potassium (Kir) channels are more favourable than M-type potassium (KM) channels in this regard: paired with CaL channels, they produce a bistability that is both physiological and stable against noise and cell-to-cell variability over a substantially wider range of parameters than KM channels can. The difference traces to a shape feature of the Kir steady-state current, shared with CaL but absent from most potassium currents. Because Kir channels are regulated by many neuromodulatory pathways in the dorsal horn, the CaL + Kir pair is a candidate mechanism for pain amplification during central sensitization and, more broadly, for single-cell short-term memory.
Citation: De Worm A, Drion G, Sacré P (2026) Inward rectifier potassium channels interact with calcium channels to promote robust and physiological bistability. PLoS Comput Biol 22(7): e1014549. https://doi.org/10.1371/journal.pcbi.1014549
Editor: Hermann Cuntz, Ernst-Strungmann-Institut, GERMANY
Received: May 23, 2025; Accepted: July 8, 2026; Published: July 28, 2026
Copyright: © 2026 De Worm et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data and code used for running experiments and plotting are available at https://github.com/anadew2/cal-kir-bistability/tree/main.
Funding: This work was supported by the Belgian Government through the Federal Public Service Policy and Support (NEMODEI2 to A.D.W., G.D., and P.S.). Computational resources have been provided (to A.D.W.) by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Nociceptive pain protects the body by signaling tissue-damaging stimuli [1]. Under strong or repeated stimulation, sensitization alters the nociceptive system and amplifies pain signaling [2]. In the dorsal horn, projection neurons relay sensory input to supraspinal centers, and sensitization renders a subset of them hyperexcitable [3–5]. Deep projection neurons, in particular, can switch from tonic firing to plateau potentials—a sustained depolarized state interposed between the spiking and resting voltage ranges—with sustained afterdischarges when neuromodulation changes [6]. These afterdischarges substantially increase the signal sent upstream and underlie hyperalgesia [4,7,8]. Because they sustain spiking beyond the end of a stimulus instead of reverting to rest, sustained afterdischarges reveal an intrinsic bistability between resting and spiking states [4].
Neuronal bistability has been observed in many populations [9–14], yet how it emerges in single neurons in a form that is both robust and physiological remains unclear. We call bistability physiological when its resting states lie above the potassium reversal potential, as observed in deep projection neurons [6,15,16]; we call it robust when the bistability window, defined as the range of currents from I1 (onset of spiking from rest) to I2 (cessation of resting from spiking), is wide enough that both states are reliably reached. Computational studies show that voltage-gated calcium channels create bistability, but at resting states that become physiological only when these channels are paired with voltage-gated potassium channels [14,17,18]. Most such potassium channels, however, also shrink the bistability window [13,19,20], producing a trade-off between window size and the plausibility of resting states. Inward rectifier potassium (Kir) channels, a distinct subclass expressed in superficial and deep dorsal horn projection neurons [6,21–24], can themselves induce bistability [25–28]. Whether and how Kir channels interact with calcium channels to shape bistability, and through what mechanism, remain open questions.
Here, we use a minimal conductance-based model to show that Kir channels, when coupled with L-type calcium (CaL) channels, enlarge the bistability window, whereas M-type potassium (KM) channels reduce it. We reproduce this effect in a published two-compartment model of deep projection neurons that already includes CaL and Kir channels alongside other currents [29]. The two pairs produce bistabilities that differ in robustness to noise and intrinsic variability; we trace these differences to the contrasting shapes of the Kir and KM steady-state currents, and relate each pair to a distinct excitability switch.
Results
We begin by asking how each channel pair shapes the bistability window in a minimal model, then validate the effect in a two-compartment model. We then quantify the robustness of each bistability to noise and intrinsic variability, tracing the differences to the steady-state shapes of the three currents. Finally, we link each pair to a distinct excitability switch, which unifies the mechanistic and functional observations.
CaL channels drive bistability; KM channels slightly narrow the window while Kir channels enlarge it
To examine how calcium and potassium channels jointly shape bistability, we built a minimal conductance-based model combining a fast sodium current , a delayed-rectifier potassium current
, a leak current
, and L-type calcium (CaL) channels, and compared the effect of adding either M-type potassium (KM) or inward rectifier potassium (Kir) channels.
CaL channels alone created a bistability window, but at resting states that fell below the potassium reversal potential. Without CaL channels the minimal model did not exhibit bistability (see S1 Appendix). Adding CaL channels produced bistability between a resting and a spiking state, revealed by applying ascending and descending current steps that reached the same final current (Fig 1A-1B). Indeed, for the two middle currents, the final state depended on the initial behavior (Fig 1A-1B, red tones), while the smallest and largest currents yielded resting or spiking regardless of initial conditions (purple and yellow). The frequency-current (fI) curves obtained from initially resting and initially spiking neurons (see Methods) showed a bistability window of 2.1 µA/cm2 delimited by I1 and I2 (Fig 1C, top). Above I2, only spiking was observed (Fig 1, yellow marker). From I1 to I2, both resting and spiking could be observed depending on the neuron initial conditions (Fig 1, markers in red tones). Only resting was observed for currents below I1 (Fig 1, purple marker). Throughout, we use to denote the steady-state membrane potential associated with the resting equilibrium at a given applied current I; this is distinct from the conventional resting potential, which refers to the membrane potential in the absence of stimulation. Within this window,
fell well below the potassium reversal potential (Fig 1C, bottom), so CaL channels must be combined with other currents to produce physiologically plausible bistability.
A: From an initial resting state, ascending current steps progressively depolarize the membrane (purple and reds). The largest step triggers a switch to spiking (yellow). B: From an initial spiking state, descending current steps progressively decrease the firing frequency (yellow and reds). The largest step triggers a switch to resting (purple). C: The frequency-current (fI) curves obtained from initially resting and initially spiking neurons (top, dashed and solid lines, respectively) identify the bistability window: the current range from I1 to I2 in which both states coexist (red tones, matching A and B). The resting equilibria, noted , depolarize progressively with increasing current (bottom). Markers in A, B, and C identify the final state observed for each of the final applied currents of −2.5, −1.5, −0.5, and 0.5 µA/cm2 (from bottom to top).
Adding an M-type potassium maximal conductance redressed the resting equilibria but slightly narrowed the bistability window. With a baseline current below the new I1, a current pulse elicited a finite afterdischarge: the neuron transitioned to spiking during the pulse and eventually returned to resting after the pulse ended (Fig 2A). The termination of spiking is attributable to the progressive decrease in the CaL current caused by the slow cumulative inactivation of CaL channels during spiking (Fig 2A, bottom). The corresponding fI curves showed a narrower window than without KM channels (Fig 2B, purple, vs. Fig 1C), while
within the window approached physiological values (Fig 2C). The stimulation protocols in Fig 2A and Fig 1A–1B are equivalent approaches that can demonstrate bistability. In the pulse protocol, the pulse amplitude must exceed the bistability window to trigger a transition from resting to spiking, while the baseline current must lie within it to sustain the new state. In the ascending/descending step protocol, the pre-step current must fall above I2 or below I1 to establish the initial state, while the final current must lie within it to maintain that state.
A: After adding an M-type potassium conductance , a current pulse from a baseline below I1 elicits a finite afterdischarge: the neuron spikes during the pulse and returns to rest afterwards (top). The
and
traces show that
declines during spiking, eventually terminating it (bottom). B: The fI curves show that the bistability window created by CaL channels (Fig 1C, top) slightly narrows after adding KM channels. C: The resting equilibria within the window, originally below
(Fig 1C, bottom), are redressed by KM channels. D: After replacing KM with an inward rectifier potassium conductance
, a current pulse from a baseline below the new I1 again elicits a finite afterdischarge. E: The bistability window is preserved after adding Kir channels. F: Kir channels also redress the resting equilibria within the window.
Replacing KM with an inward rectifier potassium (Kir) conductance preserved the correction of resting equilibria while enlarging the bistability window. A current pulse of the same amplitude and a baseline below I1 again produced a finite afterdischarge, with the neuron returning to a resting equilibrium of
(Fig 2D). As with KM channels, spiking termination is attributable to the slow cumulative inactivation of CaL channels (Fig 2D, bottom). The superimposed fI curves showed a larger bistability window than with CaL alone, shifted toward higher currents (Fig 2E). Kir channels also brought
within the window to physiological values (Fig 2F).
To see how these effects evolve across the full conductance space, we swept jointly with either
or
. A systematic sweep over
and
confirmed the observations above. We computed the bistability window size
, its lower limit I1, and the resting equilibrium at that limit
across the two-dimensional parameter space (Fig 3A–3C, respectively). KM channels slightly narrowed the window at high
: with
, raising
to 0.3 mS/cm2 reduced
from
to 2.3 µA/cm2 (Fig 3A). For lower
, KM channels slightly enlarged
, but only in parameter regions where bistability was neither robust (
µA/cm2) nor physiological (
). KM channels raised both I1 (Fig 3B) and
steeply, with
reaching values above
(Fig 3C, blue area); this depolarization already hints at an excitability change driven by KM recruitment, which we examine later.
A–C: Heatmaps of the bistability window size (A), its lower limit I1 (B), and the corresponding resting equilibrium
(C), as a function of
and
. D–F: Same quantities as a function of
and
. Each heatmap contains
data points (
and
in steps of 0.006 mS/cm2;
in steps of
). White lines are contour lines. The red solid line marks
and the red dashed line marks
µA/cm2. The two red lines together delimit the parameter region (top right) in which bistability is simultaneously robust (
µA/cm2) and physiological (
).
In contrast to this small narrowing effect, a sweep over and
showed that Kir channels enlarge the bistability window while anchoring the resting equilibria. For any
, raising
increased both
and I1, and held
close to
(Fig 3D–3F, respectively). These effects of CaL, KM, and Kir channels on window size were independent of the specific CaL current model used (see S2 Appendix). The contrasting behaviors of the CaL + KM and CaL + Kir pairs suggest that each relies on a distinct mechanism and may drive a distinct excitability switch, which we return to later.
A natural question is whether these effects are still observed when KM and Kir are expressed simultaneously. When KM and Kir channels were combined with CaL channels together, each retained its distinct effect. Starting from a baseline bistability window of µA/cm2 (with
), we examined how co-varying
and
(each from 0 to 0.3 mS/cm2) affected
(Fig 4A). Consistent with Fig 3A, increasing
reduced
. For
mS/cm2, increasing
monotonically increased
, while for higher
values, a small reduction in
(up to 0.11 µA/cm2) was observed as
increased. This local reduction likely reflects a transient phenomenon originating from higher-order interactions between the two conductances, as a similar drop (at most 0.05 µA/cm2 was observed for
between 0.15 and 0.21 mS/cm2 at low
, followed by a continuous increase in
as
increased. Thus, while the effect of
on
may become non-monotonic when combined with high
, it generally acts to increase
.
rose more steeply with
than with
(Fig 4B), confirming that Kir channels maintain resting equilibria in a lower voltage range than KM channels, even when combined. This is further illustrated in Fig 4C: at high
and low
,
within the bistability window ranged from −82.6 to
(pink), whereas at low
and high
, it shifted to −64.6 to
with a smaller
(purple). Increasing
in the latter condition slightly lowered
to −67 to
, despite a small reduction in
of 0.08 µA/cm2 (yellow). A similar analysis at half the baseline
(see S3 Appendix) showed that while Kir and KM channels promote distinct ranges of resting equilibria, KM channels can slightly increase
when
is too small to be considered robust (
µA/cm2; largest increase: 0.14 µA/cm2 at
). Altogether, the effects observed in isolation persisted when both channels were combined: Kir channels promote robust bistability by enlarging
, whereas KM channels generally slightly reduce it.
A: Heatmap of the change in bistability window size, , as a function of
and
, with
fixed.
µA/cm2 denotes the window size at
. B: Heatmap of
over the same parameter grid. C: fI curves (top) and corresponding resting equilibria (bottom) for the three parameter sets marked in A and B. Each heatmap contains
data points. White lines are contour lines. The red solid line in A and B marks
.
The effects of CaL and Kir channels on bistability remain in the presence of additional conductances
To test whether these effects generalize beyond the minimal model, we repeated the analysis on a published two-compartment model of deep projection neurons that already includes CaL channels in the dendrite and Kir channels in the soma, alongside calcium-dependent potassium (KCa) channels in both compartments, calcium-dependent nonspecific cation (CAN) channels in the dendrite, and a faster L-type calcium current () in the soma [29]. We varied only
and
and kept the other conductances at their published values.
We first examined the response of four illustrative pairs to pulses of current of either 50 or
, each with a baseline current producing a resting equilibrium of
. At low values of both conductances (
and
mS/cm2), no bistability was observed (Fig 5A). Raising
alone to 0.185 mS/cm2 (Fig 5B),
alone to
(Fig 5C), or both together (Fig 5D) all produced bistability. The fI curves resulting from the first and the second pairs of conductances (used in Fig 5A and 5B, respectively), each with
, did not display a significant bistability window, as only the second pair of conductance exhibited a small
of
(Fig 5E, top). Increasing
hyperpolarized
from −47 to
and increased I1 (Fig 5E, bottom). With
raised to
,
increased to
and
for
and 0.185 mS/cm2, respectively, while increasing
again hyperpolarized
from −48 to
and increased I1 (Fig 5F). The difference in pulse amplitude across Fig 5A–5D (50 vs
) reflects the fact that, at high
, a current of only
was sufficient to cross I1 from a resting equilibrium of
.
A: Response of the model with low and low
mS/cm2, initially at
, to a
current pulse. B: Same as A with
raised to 0.185 mS/cm2 and a
pulse. C: Same as A with
raised to
. D: Same as C with
also raised to 0.185 mS/cm2 and a
pulse. E: fI curves (top) and corresponding resting equilibria (bottom) for the parameter sets used in A and C, isolating the effect of raising
. F: Same as E for the parameter sets used in B and D. G–I: Heatmaps of the bistability window size
(G), its lower limit I1 (H), and the corresponding resting equilibrium
(I), as a function of
and
. Each heatmap contains
data points. White lines are contour lines. The red dashed line marks
, the robustness threshold estimated from published experimental data of afterdischarges. All parameter combinations shown in G–I satisfy
, so robust and physiological bistability coincides with the region above the red dashed line.
A systematic sweep over the two conductances reproduced the minimal model pattern (Fig 5G): grew with
, and, for
between
and
, also with
. The increase in
with
appeared to saturate for
mS/cm2. Doubling
from
to
increased
by only
at
and
at
mS/cm2. Above this value,
became more sensitive to
, reaching an increase of
at
mS/cm2. At lower
(between
and
),
still increased overall with
, but exhibited a transient drop of up to
relative to its value at
before rising again.
I1 and followed the same patterns as in the minimal model (Fig 5H–5I). Raising
lowered I1 and hyperpolarized
, while raising
raised I1. The effect of
on
depended on whether bistability was already established. With a well-developed bistability window (
between
and
), increasing
depolarized
by up to
. Where bistability was weak or absent (
,
), it hyperpolarized
by up to
. At intermediate
values, the effect transitioned from hyperpolarization to depolarization as bistability emerged.
These results confirm that the minimal model findings generalize to a more realistic setting: CaL channels create bistability, Kir channels enlarge the window, and both effects persist in the presence of KCa, CAN, and currents. Accounting for the soma surface area (
), a
window in the two-compartment model corresponds to 2 µA/cm2 in the minimal model. Although KM channels were not originally included in the two-compartment model, replacing Kir channels with KM channels in a full parameter sweep reduced
, consistent with the minimal model results, and in some cases eliminated bistability altogether (see S4 Appendix). Having established that the effects generalize, we next ask whether the two pairs also differ in their robustness to noise and intrinsic variability.
Bistability is more robust to noisy inputs and intrinsic variability when CaL and Kir channels are combined
We next asked whether the bistabilities produced by CaL + KM and CaL + Kir pairs differ in robustness to noisy inputs and to intrinsic variability. To compare the two pairs on equal footing, we selected a CaL + KM set and a CaL + Kir set with matching bistability window sizes; because KM channels reduce while Kir channels enlarge it, this required a higher
for the CaL + KM pair, while
and
were set to the same value.
With CaL + KM channels, the resting state was lost under small perturbations while spiking was maintained up to the largest noise levels tested. Applying the central current of the bistability window with superimposed white noise of increasing standard deviation , the initially resting neuron transitioned to spiking within the first few tens of seconds (Fig 6A), whereas the initially spiking neuron kept spiking (Fig 6B). Across 500 trials at each fixed
, the proportion of resting-to-spiking transitions grew rapidly with
, while spiking-to-resting transitions never occurred in our range (Fig 6C).
A: Membrane potential of the CaL + KM model in response to the central current of the bistability window, with superimposed white noise of increasing standard deviation , starting from the resting state. B: Same as A, starting from the spiking state. C: Proportion of 500 trials that transitioned away from the initial state (resting→spiking or spiking→resting) as a function of
held constant throughout each trial, for the CaL + KM model. D–F: Same as A–C for the CaL + Kir model. The two models were matched for bistability window size:
(A–C) and
(D–F) were set to the same value, while
was higher in the CaL + KM model to compensate for the narrowing effect of KM channels.
With CaL + Kir channels, both states could be lost under sufficient noise, and the two states were of comparable robustness. The initially resting neuron transitioned to spiking at moderate (Fig 6D), and the initially spiking neuron transitioned to resting at slightly higher
, with each resting episode lasting a few hundred milliseconds before spiking resumed (Fig 6E). Spiking was more robust than resting in
-bistability, tolerating higher noise levels even when kept constant over time (Fig 6F), in contrast to
-bistability where resting was highly fragile while spiking remained undisrupted (Fig 6C).
This asymmetry in robustness raises the question of whether both states are equally reachable, particularly in -bistability where resting may be almost unreachable. To address this, we simulated the neuron response to a constant current within the bistability window (i.e., between I1 and I2) in noise-free conditions, for both
and
-bistability, starting from initial conditions
and recorded the state to which the model converged (Fig 7). With CaL + Kir channels (Fig 7A), resting was more reachable than with CaL + KM channels (Fig 7B), the latter showing lower reachability of resting throughout the bistability window, while spiking showed the opposite trend. These results were consistent with those in Fig 6, obtained at the center of each bistability window. Near I2, the contrast was particularly striking: in -bistability, only initial conditions close to
converged to resting, making this state extremely difficult to reach and maintain. These results also showed that state reachability varied within each bistability window, suggesting that noise robustness should similarly depend on the applied current. Consistently, repeating the experiment of Fig 6 at I = I1 instead of
showed that resting became more robust than spiking in both bistability types, while resting in
-bistability remained more robust than in
-bistability (see S5 Appendix).
A: Final state reached by the CaL + Kir model as a function of the applied current (within ) and the initial membrane potential
. Blue indicates convergence to resting, orange to spiking. The dark blue line traces
within the bistability window. B: Same as A for the CaL + KM model.
We next investigated how both bistability types responded to intrinsic variability, estimating how evolved across heterogeneous neurons. The membrane capacitance (C), maximal sodium conductance (
), and leak conductance (
) were each sampled uniformly within
,
, or
of their nominal values. The same conductance values were used for both bistability types (
mS/cm2,
), yielding different
. The resulting relative changes in
were computed and represented as a function of C for each variability level (Fig 8A). Changes in C had opposite effects on the two bistability types, but with smaller magnitude for
-bistability. At every level of intrinsic variability tested, relative changes in
-bistability were approximately 1.5 times larger than in
-bistability, which remained within
(Fig 8B).
A: Relative change in bistability window size as a function of the sampled membrane capacitance C, for the CaL + Kir (pink) and CaL + KM (purple) models. Shades of each color correspond to variability levels of
,
, and
applied simultaneously to C, the sodium maximal conductance, and the leak conductance. B: Boxplots of the relative changes in
at each variability level, for the two models.
Ion channels increasing bistability have a steady-state current exhibiting a region of negative differential conductance
The opposing effects of KM and Kir channels on must arise from the shapes of their steady-state currents, since both are voltage-gated potassium channels operating on a similar timescale. We therefore compared the steady-state currents of CaL, KM, and Kir channels, focusing on their differential conductance around the spike threshold (
).
CaL channels carry an inward current (Fig 9A, top) with a region of negative differential conductance near the spike threshold (Fig 9A, shaded area). Previous work has linked this feature to the onset of bistability [13,30]: a small depolarization within this range increases the inward calcium current, which depolarizes the membrane further and triggers spiking at a lower applied current, thereby reducing I1. Because CaL channels barely affect the values of resting equilibria (see Fig 1C and S1 Appendix prior to the addition of CaL channels), resting at a smaller I1 must be balanced by a stronger leak current, that dominates the sum of ionic currents below (the reversal potential of the leak current). This accounts for the hyperpolarization of
observed in Figs 1C and 3C.
A: Steady-state L-type calcium current (top) and its differential conductance (bottom), at an intracellular calcium concentration of 2 mM. The region of negative differential conductance around the spike threshold is shaded. B: Same as A for the KM current
, which has positive differential conductance throughout. C: Same as A for the Kir current
, which has a region of negative differential conductance above
(shaded). The y-scale of the steady-state currents is in µA/cm2 (top) and that of the differential conductance in µA/mV cm2. A,B,C used respectively
,
mS/cm2, and
mS/cm2.
Unlike CaL channels, KM channels carry a small inward and a strong outward current (Fig 9B, top), both with strictly positive differential conductance (Fig 9B, bottom). Empirically, increasing shifts both I1 and I2 upward, but raises I1 more than I2, thereby reducing
(Figs 1C and 2B–2C). This asymmetric shift follows from I1 and I2 being governed by different dynamical states: spiking at I1 and resting at I2. When initially spiking at I1, the membrane potential is overall higher than the resting equilibrium at I2. Since KM channels have positive differential conductance, they produce a stronger outward current in the spiking scenario at I1 than in the resting scenario at I2, requiring a larger compensating current increase at I1. Additionally, increasing
elevates
(Fig 2C), as KM channels provide a non-negligible current at these potentials and, together with the leak current, dominate the sum of ionic currents at rest, causing
and
to rise with I1 and I2 (Fig 3C). For a fixed applied current (e.g.,
µA/cm2), KM channels hyperpolarize the resting equilibrium toward
(from
in Fig 1C to below
in Fig 2C).
In contrast to KM channels, Kir channels carry a strong inward current below and a weaker outward current (Fig 9C, top), and share with CaL channels a region of negative differential conductance around the spike threshold (Fig 9C, bottom). By analogy with the CaL current, this negative-slope region promotes the resting-to-spiking transition: a small depolarization decreases the outward Kir current, which depolarizes the membrane further. Like KM channels, increasing
shifts both I1 and I2 upward, but the shift is larger for I2 than for I1, in contrast to KM channels, thereby increasing
(Figs 1C and 2E–2F). By the same reasoning as for KM channels, Kir channels produce a lower outward current in the spiking scenario at I1 than in the resting scenario at I2, requiring a larger compensating current increase at I2 because of the negative differential conductance of the outward Kir current. Increasing
generally elevates
(Fig 2F), as Kir channels dominate the sum of ionic currents at rest, though their effect is non-linear. Below
, Kir channels display positive differential conductance, which steepens the sum of ionic currents at rest. In this region, the rise in I1 induced by increasing
therefore depolarizes
(Figs 2F and 3F, upper half). Above
, the Kir current increasingly dominates the sum of ionic currents as the potential approaches
, since the leak current diminishes near its reversal potential. In this region, however, the Kir current displays negative differential conductance, creating a negative-slope region in the sum of ionic currents near
. To understand the contribution of Kir channels to
, consider a neuron with the same conductances as in Fig 2F, initially resting at I1, with the applied current progressively increased. As the membrane potential rises, the positive differential conductance of the Kir current initially allows the sum of ionic currents to compensate for the depolarization. This compensation reaches its limit at
, just before the negative-slope region of the sum of ionic currents, determining
. As
increases, the negative-slope region grows, shifting this limit to more hyperpolarized potentials and thus progressively hyperpolarizing
, as observed when
for
and
in Fig 3F.
Both KM and Kir channels produce an asymmetric shift in I1 and I2, which respectively narrows or enlarges . This asymmetry arises because I1 and I2 are set by the appearance and disappearance of two different behaviors, governed by distinct dynamical mechanisms that rely on different types of bifurcations (examined later, Fig 11). These mechanisms are potentially shaped by the sign of the differential conductance around the spike threshold, suggesting that its sign predicts whether a potassium channel enlarges or narrows
. We test this hypothesis in the next subsection by selectively blocking the inward and outward components of each current.
The inward currents redress resting equilibria and maintain bistability, but the outward potassium currents either increase or decrease bistability
If the sign of the differential conductance around the spike threshold dictates the effect on , then only the outward component of each potassium current, which carries that differential conductance, should reshape the bistability window, while the inward component, being zero above
, should not interfere with spiking. We tested this by blocking either the inward or the outward component of each current in turn, measuring the resulting
and the resting equilibria
within the bistability window, the latter being determined by applying hyperpolarizing current steps between I1 and I2.
To isolate the inward KM current, the outward component was blocked above (Fig 10A, left). With only its inward component active, the KM current did not alter
(Fig 10A, center). Hyperpolarizing steps within the bistability window showed that it only slightly raised the resting equilibria that lay below
(Fig 10A, right). Conversely, isolating the outward KM current by blocking the inward component (Fig 10B, left) showed that increasing
narrowed
while efficiently adjusting resting equilibria (Fig 10B, center–right).
A: Steady-state KM current with the outward component blocked (left), bistability window size as a function of the inward conductance
(center), and hyperpolarizing steps of current across the window at a fixed
(right). B: Same as A with the inward component of the KM current blocked instead. C: Same as A for the Kir current, with the outward component blocked. D: Same as A for the Kir current, with the inward component blocked. In the right-hand panels, the maximum and minimum applied currents equal I1 and I2 for each blocked condition, at a conductance of 0.15 mS/cm2.
Blocking the outward Kir current isolated the inward component (zero above , Fig 10C, left), which left
unchanged as
increased (Fig 10C, center), but efficiently hyperpolarized resting equilibria up to
(Fig 10C, right). This correction was more efficient than for the inward KM current, due to the larger amplitude of the inward Kir current. Conversely, blocking the inward Kir current isolated the outward component (non-zero above
, Fig 10D, left), which enlarged the bistability window as
increased, while only adjusting resting equilibria above
(Fig 10D, center–right).
These four experiments together show a clear partition of roles. The inward components of both KM and Kir currents act only on resting equilibria below and leave
unchanged, since the spike threshold lies above
and the inward currents vanish there. The outward components redress the remaining resting equilibria and, in addition, act on
: the outward KM current narrows it while the outward Kir current enlarges it. This difference is consistent with the sign of their differential conductance around the spike threshold, which is positive for KM and negative for Kir, confirming the hypothesis of the previous subsection.
CaL and Kir channels facilitate the generation of plateau potentials, whereas KM channels induce a distinct excitability switch
The qualitative changes in and
observed as
,
, or
increases (Fig 3) suggest that each channel type triggers a distinct excitability switch, that is, a qualitative change in how the neuron approaches or leaves the spiking regime. To identify these switches, we used the minimal conductance-based model containing only
,
, and
as a reference, and compared it to three variants in which CaL, Kir, or KM channels were added individually. For each condition we recorded the response to a hyperpolarizing current step crossing I1 (Fig 11 A–11D) and computed the bifurcation diagram of the membrane potential as a function of the applied current (Fig 11 E–11H).
In the reference condition (), reducing the applied current below I1 causes the neuron to settle directly to a resting equilibrium within the range of potentials spanned by spiking (Fig 11 A). The bifurcation diagram confirmed this, showing that resting equilibria near I1 exceeded the minimum of the spiking limit cycle (Fig 11 E). The overlap between I1 and I2 reflects the absence of bistability, as spiking appeared precisely when resting disappeared. As the applied current increased, the resting state disappeared at I2 through a saddle-node (SN) bifurcation, where the stable equilibrium collided with a saddle point. Since the stable equilibrium at I2 lay within the spiking voltage range and spiking frequency near I1 was of the order of a few hertz (Fig 11 A), the SN bifurcation occurred on an invariant circle, giving rise to a saddle-node on invariant circle (SNIC) bifurcation. By nature, the SNIC bifurcation is associated with low-frequency spiking at onset [31].
A–D: Response of the model to a hyperpolarizing current step crossing I1, for four conditions. A: reference model (,
,
). B: reference model + CaL channels. C: reference model + Kir channels. D: reference model + KM channels. In each panel the lower and upper current values are
for the corresponding conductances. E–H: Bifurcation diagrams for the four conditions. Bifurcation types are labelled directly on each panel: SNIC (saddle-node on invariant circle), SN (saddle-node), SH (saddle-homoclinic), and Hopf.
Adding CaL channels alone () rendered the neuron bistable, and produced a plateau upon hyperpolarization below I1 (Fig 11 B). This plateau represented a clear separation between the range of potentials encompassed by spiking and the resting equilibria below I1, that are typically below the range of potentials covered by spiking. Bistability arose because the addition of CaL transformed the SNIC bifurcation into a separate SN bifurcation at I2 and a saddle-homoclinic (SH) bifurcation at I1 (Fig 11 F). The saddle-homoclinic bifurcation is defined by the collision between a saddle point and a homoclinic loop, which is defined by the high-dimensional model trajectory during spiking, at I1, where spiking appears (for more details see [31]). However, the bifurcation diagram also revealed that the separation between resting equilibria and the spiking voltage range exists only near I1, so the plateau observed upon hyperpolarization may not persist for currents closer to I2 (Fig 11 F).
Adding Kir channels alone ( mS/cm2) produced a smaller plateau (Fig 11 C) but generated the same qualitative bifurcation structure as CaL channels, transforming the SNIC bifurcation into a separate SN bifurcation and a SH bifurcation (Fig 11 G). This shared bifurcation structure likely underlies the enlarging effect of Kir channels on
. Notably, the voltage separation between resting equilibria and the spiking voltage range persisted throughout the entire bistability window, up to
(Fig 11 G).
Adding KM channels alone ( mS/cm2) produces a qualitatively different switch. Instead of a plateau, the hyperpolarizing step elicits damped subthreshold oscillations as the neuron approaches a higher resting equilibrium (Fig 11 D). The bifurcation diagram revealed that the resting equilibria largely overlapped the spiking voltage range (Fig 11 H). The resting state disappeared through a Hopf bifurcation rather than a SN bifurcation, as the stable equilibrium lost stability before colliding with the saddle point. The Hopf bifurcation confers resonator behavior, whereby the neuron preferentially responds to inputs at the resonant frequency of its subthreshold oscillations, in contrast to integrators associated with SN or SNIC bifurcations, which perform temporal integration of input pulses [32]. Intuitively, this lack of voltage separation means that a neuron transitioning from spiking to rest remains within the voltage range visited during spiking, making the resting state far easier to escape than in the SN + SH case, where the transition requires crossing a voltage region not visited during spiking.
When CaL and Kir channels are combined, the hyperpolarizing step produces a pronounced plateau (Fig 12A), and the bifurcation diagram preserves the SN + SH structure observed for each channel alone (Fig 12C). In contrast, combining CaL with KM channels suppresses the plateau (Fig 12B), as resting equilibria are pulled into the spiking voltage range and disappear at I2 through a Hopf bifurcation (Fig 12D). The resonant behavior is less visible in Fig 12B than in Fig 11 D because the hyperpolarizing step is applied at I1, which lies further from the Hopf bifurcation than the current used in Fig 11 D. The two channel combinations thus differ not only in bistability window size and robustness, but also in the type of excitability they support: plateau-generating in CaL + Kir and tonic firing with resonant behavior in CaL + KM. In CaL + Kir, the plateau directly reflects resting state robustness: the larger the voltage separation between resting and spiking, the harder it is for noise to drive the neuron out of rest and back into spiking. This separation is largest near I1 and shrinks toward I2, consistent with the current-dependent reachability and robustness observed in Fig 7A and Fig 6C vs Fig A in S5 Appendix. In CaL + KM, the Hopf bifurcation eliminates this voltage separation and confers resonator behavior, both of which contribute to the lower robustness and reachability of the resting state compared to CaL + Kir, especially near the Hopf bifurcation (Figs 6, 7, and Fig A in S5 Appendix). The shared SH bifurcation at spiking onset in both cases also provides further insight into the asymmetric shift of I1 and I2 induced by Kir and KM channels. The spiking scenario discussed earlier depends on the membrane potential at which the saddle point lies, which is always higher than the resting equilibrium at I2, supporting the conclusion that the outward Kir (resp. KM) current contribution at I1 is lower (resp. higher) than at I2, thereby increasing (resp. decreasing) .
A: Response of the CaL + Kir model to a hyperpolarizing current step crossing I1. B: Same as A for the CaL + KM model. In A and B the lower and upper current values are for the corresponding conductances. The brief spiking following the step in B is an afterdischarge, not a plateau potential. C: Bifurcation diagram of the CaL + Kir model as a function of the applied current. D: Same as C for the CaL + KM model. Bifurcation types are annotated directly on the panel.
Discussion
Cooperation between L-type calcium (CaL) channels and inward rectifier potassium (Kir) channels produces a bistability between resting and spiking states that is physiological, robust to noise and intrinsic variability, and capable of supporting plateau potentials and sustained afterdischarges. This combination provides a candidate intrinsic pathway by which deep dorsal horn projection neurons amplify nociceptive input during central sensitization. Our analysis also shows why this pair is special: Kir channels are unusual among voltage-gated potassium channels in that their outward steady-state current, like the CaL inward current, has a region of negative differential conductance around the spike threshold, the same shape feature that enlarges the bistability window.
Most voltage-gated potassium channels reduce the bistability window size in exchange for redressing the resting equilibria within the window, producing the trade-off that has made physiological bistability via CaL channels alone hard to explain [13,19,20]. Kir channels avoid this trade-off through the negative slope of their outward current, which asymmetrically shifts the boundary behaviors of the bistability window to expand it, while anchoring the resting equilibria below the local negative-slope region that Kir channels introduce in the sum of ionic currents. The result is that the region of parameter space where bistability is both robust and physiological is larger for CaL + Kir than for CaL + KM, even though physiological bistability is achievable with either channel combination given sufficient CaL conductance. Selectively blocking either the inward or the outward Kir current reinforces this picture: the inward component exclusively redresses subthreshold resting equilibria, while the outward component alone is responsible for enlarging the bistability window. This clarifies a proposal by Amarillo and colleagues [27] that Kir channels support intrinsic bistability, and generalizes it beyond thalamocortical neurons. The bifurcation analysis tells the same story. Both CaL and Kir channels produce a saddle-node + saddle-homoclinic structure, with Kir channels shifting the saddle point outside the spiking limit cycle range and thereby creating a plateau that separates resting from spiking. KM channels, by contrast, drive the resting equilibrium through a Hopf bifurcation, preventing such a separation.
These functional differences translate into distinct modes of bistability with different implications for neuronal computation. In CaL + Kir bistability, resting and spiking are comparably robust to noise at the center of the bistability window, allowing a projection neuron to reliably hold either state and transition between them under stimulus control. CaL + KM bistability, by contrast, features robust spiking but fragile resting, so once the neuron begins spiking, it tends to remain in that state. This asymmetry is consistent with the Hopf-type loss of stability of the KM resting equilibrium and with our reachability analysis, which showed that the CaL + KM resting state near I2 is accessible only from a narrow band of initial conditions, becoming increasingly fragile as the applied current approaches I2. Near I1, the resting state of CaL + KM becomes more robust than spiking, but both states show comparable robustness, so spiking remains generally dominant throughout the bistability window. For pain processing, where afterdischarges must eventually terminate to allow the system to return to baseline between stimuli, CaL + Kir is therefore the functionally more appropriate mechanism. This conclusion is further supported by our finding that CaL + Kir bistability is more reliable than CaL + KM bistability across a heterogeneous neuronal population exhibiting intrinsic bistability.
The occurrence of long-lasting afterdischarges depends critically on whether the baseline current between pulses falls within the bistability window. Modulatory inputs that shift the window boundaries can therefore gate afterdischarge occurrence: if the window shifts upward (e.g., due to increased ), a fixed baseline current may fall below I1, preventing afterdischarges. Conversely, if the window shifts downward (e.g., due to decreased
), the same baseline current may fall within the window, facilitating them. These predictions are consistent with the experimental findings of [6].
The absolute size of the bistability window matters equally for observing long-lasting afterdischarges, but it has not been directly quantified experimentally. However, studies using rats typically applied current pulses of 20 to [4,6,15,16], suggesting that the bistability window in these preparations falls within this range. Based on the estimated soma diameter of deep dorsal horn projection neurons of 20 µm [6], this corresponds to approximately 1.6 to 11.9 µA/cm2, consistent with the range over which we defined our robust-bistability criterion in the minimal model. The upper bound remains less constrained and would require direct experimental measurements of I1 and I2. The resting states and firing frequencies observed experimentally could provide additional information to estimate the physiological range of bistability window sizes. However, these properties depend strongly on the full set of ionic conductances present in the neuronal membrane, making it difficult to infer physiological window sizes from a conductance-based model alone.
The firing frequencies in the bistability window of the minimal model (100 to ) exceed those reported experimentally for deep projection neurons (5 to
) [3,4,6,15,16]. To assess whether this discrepancy limits the applicability of our results, we repeated the analysis of bistability in a two-compartment model of deep projection neurons incorporating additional ion channels [29], which produced firing frequencies more consistent with the experimentally reported range of
[33]. This supports the generalizability of our findings to more comprehensive models and their robustness to the presence of additional conductances. The other ion channels may further modulate the bistability window. CAN channels have been linked to afterdischarge maintenance [15,29,34,35] and may therefore also promote bistability. KCa channels, which contribute to afterdischarge termination [15] and oppose CaL channel effects on excitability [13,30], may reduce the bistability window and could account for the termination of persistent firing. The faster CaL subtype (
) enlarged the window much less than the slower subtype in the minimal model (S6 Appendix), suggesting that channel timescale matters as much as channel type. Finally, both CaL and CAN channels have been associated with windup [29,34–36], a form of short-term sensitization in projection neurons. Whether windup shares a mechanistic basis with bistability remains an open question.
In the dorsal horn, Kir channels are modulated by several signaling cascades, including GABA binding to receptors, neuropeptide Y (NPY) binding to its Y1 receptor, and gastrin-releasing peptide (GRP) binding to its receptor (GRPR) [37–39]. Bistability may therefore be a more general feature of dorsal horn circuitry, not restricted to projection neurons but potentially extending to interneuron populations that co-express Kir and calcium channels and are targeted by these signaling pathways. In this context, the persistent depolarization and spontaneous activity observed for minutes after synaptic stimulation in GRPR-expressing interneurons [39] may reflect underlying bistability. More broadly, Kir-supported bistability at the single-cell level may be useful wherever a neuron must retain a short-term memory of past input, as noted in thalamocortical and other systems [26–28].
In summary, a minimal conductance-based model shows that CaL and Kir channels cooperate to produce robust and physiological bistability, plateau potentials, and sustained afterdischarges in deep dorsal horn projection neurons. The mechanism relies on a shape feature shared by both steady-state currents, namely a region of negative differential conductance around the spike threshold, which enlarges the bistability window without compromising the physiological plausibility of resting states. Because Kir channels are regulated by multiple neuromodulatory inputs in the dorsal horn, the CaL + Kir pathway is a plausible intrinsic target by which central sensitization switches the functional state of nociception.
Materials and methods
Software
All simulations and experiments were performed in the Julia programming language. The code and packages used in this work are available on GitHub.
Minimal conductance-based model
We used a single-compartment Hodgkin-Huxley model, in which the membrane potential V evolves in response to an applied current I according to:
where C is the membrane capacitance, is the leak current,
are the intrinsic ionic currents with
the set of all ionic channels. In this work, the ionic currents flowing in each type of voltage-dependent ion channels are defined by:
where denotes the ion channels maximal conductance,
and
correspond to the voltage-dependent activation and inactivation gates of the ion channels,
is an integer between 1 and 4,
is an integer between 0 and 1, and
denotes the reversal potential of the considered ionic current.
The L-type calcium channels, embedding a calcium-dependency, were modeled using a Goldman–Hodgkin–Katz formalism [40] such that:
with
and
where F is the Faraday constant, R is the perfect gas constant, T is the temperature, is the intracellular calcium concentration, and
is the extracellular calcium concentration, considered constant.
The evolution of the intracellular calcium concentration was modeled as:
where is the intrinsic ionic current flowing in the L-type calcium channels F is the Faraday constant, d is the depth of the shell beneath the membrane,
is the initial intracellular calcium concentration, and
is the buffering time constant.
The evolutions of the activation and inactivation gates of a given ion channel are defined as:
where and
denote the steady-state values of the activation and inactivation gates at a given membrane potential V, and
and
the time constants of the activation and inactivation gates at a given membrane potential V, respectively.
The minimal model includes a voltage-gated fast sodium current , a slow delayed-rectifier potassium current
, an L-type calcium current
, an M-type potassium current
, an inward rectifier potassium current
, and a leak current
. The models of
and
follow [41]. The model of
corresponds to the slower L-type calcium current of [29], with its steady-state activation slightly modified to follow a Boltzmann equation; the faster L-type calcium current, used only in the Appendix, is described there. The models of
and
are also from [29]. The model of
follows [42], with the half-activation voltage lowered to -76 mV and the slope factor increased to 25.7 mV. These modifications bring the activation range of KM closer to that of Kir, to better contrast their effects. An unmodified version is used in the Appendix and yields the same conclusions (see S7 Appendix).
Throughout, ionic currents are expressed in µA/cm2, maximal conductances in mS/cm2, the L-type calcium permeability in cm s-1, calcium concentrations in mM, the membrane potential in mV, and time in ms. Time constants, steady-state activations, and inactivations for ,
, and
are given in Table 1. The nominal values of the fixed parameters are given in Tables 2 and 3.
Complete conductance-based model
To compute the results shown in Fig 5, we reimplemented a published two-compartment conductance-based model of deep projection neurons (see [29] for more details). The ion currents and the fixed parameters described in the previous section are identical to those used in the two-compartment model. This model includes two types of L-type voltage-gated calcium channels as described in [29]. The CaL current described in the previous section corresponds to the slower type, denoted “CaLs” in this paper. In addition, the two-compartment model includes calcium-dependent potassium (KCa) channels and calcium-dependent nonspecific cation (CAN) channels.
Current step and current pulse simulations
A time-dependent applied current was used to observe the neuron response to a current step and pulse. To create a step of current, the time dependency was introduced such that the applied current I is equal to:
where H(x) is the unit step function such that it equals 1 if and 0 if x < 0. This current step was depolarizing when
and hyperpolarizing otherwise.
To create a pulse of current, the time dependency was introduced such that the applied current I is equal to:
meaning that this current pulse was depolarizing when and hyperpolarizing otherwise.
Both protocols probe bistability. In the step protocol, ascending steps from rest and descending steps from spiking to the same final currents reveal the bistability window by showing which currents lead to different final states depending on initial conditions (Figs 1A–1B). In the pulse protocol, a depolarizing pulse with baseline and amplitude above I2 triggers a transition to spiking that terminates when the pulse ends, tracing the same window boundaries (Figs 2A,2D). We use the step protocol for parameter sweeps and the pulse protocol to illustrate afterdischarges.
In Fig 1A, we applied four hyperpolarizing steps of current at with
-3 µA/cm2 and
taking one of the four values in the ensemble
µA/cm2 from bottom to top. In Fig 1B, we applied four depolarizing steps of current at
with
1 µA/cm2 and
taking one of the four values in the ensemble
µA/cm2 from bottom to top. For Figs 1A and 1B, we used
and
.
In Fig 2A, we applied a depolarizing pulse of current from to
with
1 µA/cm2 and
4 µA/cm2 to the model defined by the set of maximal conductances:
,
mS/cm2, and
mS/cm2. In Fig 2D, the same depolarizing pulse of current was adapted such that
-0.8 µA/cm2 and
2.2 µA/cm2 to the model defined by the set of maximal conductances:
,
mS/cm2, and
mS/cm2.
In Fig 5A-5D, we applied several depolarizing pulses of current from to
. The values of
,
,
and
used are given in the bottom part of each of these figures.
In Fig 11, a hyperpolarizing step was applied at around I1, with
and
for each set of conductances. Panel A used
µA/cm2 and
; panel B used
and
; panel C used {3.6;3.58} and {0.;0.;0.3} ; panel D used {6.6;6.58} and {0.;0.3;0.}. The same step protocol was applied in Fig 12A–12B: panel A used {2.86;2.84} and
; panel B used {6.05;6.03} and
.
Computation of the frequency-current curves
A frequency-current (fI) curve represents the steady-state spiking frequency observed at a constant applied current. In the context of a bistable neuron, an overlap exists between a zero-frequency state (i.e., a resting state) and a non-zero-frequency state (i.e., a spiking state) within a range of current values in which bistability is observed. In this work, we designate this range of current the bistability window, which is defined between the onset of spiking at current I1 and the cessation of resting at I2.
Thus, the initial conditions used when computing the steady-state spiking frequency for currents within the bistability window must allow the neuron to converge to its spiking state. To determine them, we initialized the neuron at resting and applied a depolarizing current step from the ith current (in the range of current tested) to a higher current, out of the bistability window. Then, the final state observed was used as the initial conditions to simulate the ith constant current and determine the corresponding steady-state spiking frequency. This ensures that the ultra-slow inactivation gate of calcium channels is still open, and facilitates the convergence toward spiking, even with low steady-state frequencies. The model is then simulated for windows of 120 s until the spiking frequency reaches steady-state. To compute the line of zero-frequency of the fI curve, we computed the model fixed points and incorporated a zero-frequency point if a stable fixed point was identified. The combination of these 2 computations was used in Fig 1C (top) and Figs 2B and 2E. In Fig 1C (top), we used the maximal conductances: and
mS/cm2, and a range of applied current between −2.65 and 1 µA/cm2, with a total of 129 points. In Fig 2B, the maximal conductance
was increased to 0.15 mS/cm2, with
and
mS/cm2, and a range of applied current between 0.7 and 4 µA/cm2, with a total of 125 points. Similarly, the conductance
was increased to 0.15 mS/cm2 in Fig 2E, with
and
mS/cm2, and a range of applied current between −1.1 and 2.4 µA/cm2, with a total of 127 points.
In Fig 4C, the minimal model was used with and three
pairs (in mS/cm2): {0.02;0.28} (pink, 0.6–5.6 µA/cm2, 133 points), {0.28;0.02} (purple,
, 132 points), and {0.28;0.28} (yellow,
, 120 points).
In Fig 5E,5F (top), the complete model was used with four pairs:
(blue,
, 1030 points),
(pink, 161–250, 893 points),
(orange, −134–149, 1038 points), and
(green, 14–247, 1028 points).
In Fig 1C (bottom), and Figs 2C and 2F, we also represented the resting equilibria observed for the range of current used to display the fI curve. The resting equilibria were computed together with the line of zero-frequency, by computing the membrane potential associated with the stable fixed point identified for an input range of applied current. The maximal conductances used in Figs 1C (bottom), 2C and 2F were identical to the maximal conductances used in Figs 1C (top), 2B and 2E, respectively. Because the computation of the resting equilibria was much faster than the computation of the fI curve, we were able to use an even more refined range of current for these figures. Indeed, we used a range of applied current between −3 and −0.1 µA/cm2 with a total of 8914 points for Fig 1C (bottom), between 0 and 3 µA/cm2 with a total if 2273 points for Fig 2C, and between −1.6 and 1.7 µA/cm2 with a total of 2920 points for Fig 2E. Note that there were no stable fixed points existing above these ranges of current.
In Fig 4C(bottom), the curves defining the resting equilibrium observed for a given current were computed on ranges of current between 0 and 3.6 µA/cm2 with a total of 2207 points for the pink curve, between 0 and 6.2 µA/cm2 with a total of 2278 points for the purple curve, and between 0 and 8.8 µA/cm2 with a total of 2989 points for the yellow curve. The set of conductances used are identical to those used in Fig 4C(top) for each color.
In Fig 5E–5F(bottom), the curves defining the resting equilibrium observed for a given current were computed on ranges of current between −150 and with a total of 492 points for the blue curve, between −150 and
with a total of 284 points for the pink curve, between −150 and
with a total of 283 points for the green curve, and between −150 and
with a total of 480 points for the orange curve. The set of conductances used are identical to those used in Fig 5E–5F(top) for each color.
Computation of the bistability window heatmaps
The computation of the fI curve is associated with a given set of conductances. To capture the evolution of the bistability window size, we varied the pair of conductances, and either
or
, and computed for each pair of conductances the (absolute) size of the bistability window, noted:
The current I2, which marks the end of the bistability window, was determined by identifying the highest current with a stable resting state. The stable resting states were computed using the same methodology to compute the fI curve zero-frequency line.
In contrast to the methodology employed in the computation of the descending fI curve, which determined the steady-state spiking frequency for each current within a specified range, in this instance, only the initial current, I1, situated at the beginning of the bistability window, was subjected to analysis. To compute I1 for a given pair of conductances in a reasonable execution time, this algorithm was modified by employing an ascending range of current and initiated at a current from the resting-only regime. In this manner, the current I1 was still identified as the lowest current at which spiking with a constant frequency can be observed over time intervals of 120 s, and with the same methodology employed to compute the neuron initial conditions (extracted from the end of a pulse simulation). The results of this analysis are shown in Fig 3A. In Fig 3D, we performed the same experiment by varying the maximal conductance instead of
.
Similarly, the resting equilibrium at the beginning of the bistability window () was determined as the membrane potential of the stable fixed point exhibited at I1, following the variation of either pair of conductances
or
. This methodology was used for Fig 3C and Fig 3F, respectively.
The heatmaps shown in Fig 4A–4B and Fig 5G–5I were computed in the same manner as what is described here above, with the minimal model and the complete model, respectively. The conductances and
were varied in ranges each defined between 0 and 0.3 mS/cm2 including 31 points (Fig 4A–4B). The conductances
and
were varied in ranges defined between 0 and
(with 41 points) and 0 and 0.3 mS/cm2 (with 31 points), respectively (Fig 5G–5I).
Response to noisy input currents
To evaluate the robustness of resting and spiking within the bistability window, we applied a constant current centered on the window of each pair of conductances and added white noise to the voltage equation. For this purpose, we simulated the ODE described in 1 with additive white noise, following a Normal distribution with zero-mean and a standard deviation of . The parameter
was varied between 0 and 9 mV to modify the noise amplitude. The system of equations formed by this stochastic differential equation and the ordinary differential equation associated with the evolution of the ion channels gates and the intracellular calcium concentration was solved with a fixed time step of 0.01 ms.
In Figs 6A, 6B, 6D and 6E, was increased at each time step, for a total simulation time of 5 s. In Figs 6A and 6D, the set of initial conditions of the experiment was the resting state observed in the center of the bistability window of the pair of conductances considered (either
or
, respectively). Reciprocally, in Figs 6B and 6E, the set of initial conditions of the experiment was a point of the spike trajectory obtained after applying the center of the bistability window of the pair of conductances considered (either
or
, respectively), without noise for 80 s, such that spiking has reached steady-state before applying any noise.
In Figs 6C and 6F, we evaluated the global neuron ability to sustain either a resting or a spiking state in the presence of a fixed noise amplitude for each type of bistability, which was produced by coupling either or
, respectively. To that end, the stochastic model was simulated for a period of 50 s with a constant value of
and initialized to either the resting or the spiking state, both computed from a 80 s noise-free simulation. If the neuron behavior, initially at resting (resp. spiking) transitioned to spiking (resp. resting) at the end of the simulation, a value of 1 was stored as a marker of this state transition, and 0 otherwise. This experiment was repeated 500 times for each standard deviation value tested. The ability to sustain the initial behavioral pattern was quantified by the proportion of state transition and calculated as the mean value of the state transition markers recorded for each of the 500 simulations. Both Figs 6C and 6F display the proportion of state transition given the constant value of
when the neuron is initially spiking or resting. In Fig 6C, bistability relies on the pair
while in Fig 6F, bistability relies on the pair
.
The two sets of conductances were matched for window size. Fig 6A–6C used , giving
µA/cm2; Fig 6D–6F used
, giving
µA/cm2.
Assessment of the reachability of the resting and spiking states
To complement our analysis of the robustness of the resting and spiking states within the bistability window for each pair of conductances considered, we assessed the reachability of each state. For that purpose, we simulated the system of ODE described in 1 with a constant value of applied current I sampled from the range , delimiting each bistability window obtained with
or
. For each value of I tested, we used a range of initial membrane potential V0 to define different sets of initial conditions written as:
where each activation and inactivation gate were initialized to the value of their steady-state function in V0 (i.e., and
, respectively). The initial calcium concentration
was set to
mM. For each set of initial conditions and current, we determined the state toward which the neuron converged to, which was either resting or spiking.
In Fig 7A, the following conductances were used: ,
mS/cm2, and
mS/cm2, which created a bistability window of
µA/cm2 with
µA/cm2 and
µA/cm2. In Fig 7B, the following set of conductances was used
,
mS/cm2, and
mS/cm2, which created a bistability window of
µA/cm2 with
µA/cm2 and
µA/cm2. For both Fig 7A and 7B, we used a range of current from I1 to I2 with a step of 0.035 µA/cm2 such that it contained 50 data points, and a range of V0 defined between
and
with a step of
containing 701 data points.
Robustness to different levels of intrinsic variability
Each pair of conductances or
give rise to two types of bistability. We tested their robustness by measuring the relative change in the bistability window size
for each pair considering neurons with heterogeneous intrinsic membrane properties.
To achieve this, three levels of intrinsic variability were introduced, comprising 10, 20, and 30% in three parameters: the membrane capacitance C, the voltage-gated sodium channels maximal conductance , and the leak conductance
. Each parameter was randomly chosen in an interval centered on the corresponding nominal value used in the model. More specifically, in the case of a level of 10% of variability, the membrane capacitance was sampled according to a uniform distribution from
to C (1 + 0.1). The same procedure was repeated to sample
and
. Subsequently, we computed I1 and I2, the limits of the bistability window, using the same algorithm as for the bistability window heatmaps, and computed the relative change in the bistability window size
to its nominal value. This computation was realized for both pairs
and
. The results of the KM model (in purple tones) and the Kir model (in pink tones) in Fig 8 were obtained with the sets of conductances
and
, respectively.
Computation of the steady-state currents and the differential conductances
Fig 9 (top) illustrates the steady-state currents of ,
, and
. At a given membrane potential, these steady-state currents are defined by the current amplitude with the activation and inactivation gates at steady-state, that is:
for voltage-dependent ion channels, and
for the L-type calcium channels.
Based on this definition, the differential conductance shown in Fig 9 (bottom) is the derivative of the steady-state current to the membrane potential () (method inspired from [30]). Each derivative of the steady-state ionic currents were computed with an Euler method, and the intracellular calcium concentration was set to a constant. In Fig 9, we used
.
Analysis of the contribution of inward and outward currents
To investigate the reasons behind the differing impacts of KM and Kir channels on the bistability window size, we modified these channels separately to block their outward (resp. inward) currents. This approach allowed us to analyze the contribution of their inward (resp. outward) currents alone. The resulting inward currents flowing in these modified channels are designated by and
, while the outward currents are denoted
and
. This modification is applied to the steady-state functions of the activation gates of both KM and Kir channels, both defined without an inactivation gate, realized as follows:
Fig 10 (left) represents the steady-state inward and outward KM and Kir currents. To compute them, we replaced the steady-state activation gates previously used by their inward/outward form, defined above, in the definition of the steady-state currents. In Fig 10 (center), we represented the evolution of the bistability window limits (I1 and I2) as the maximal conductance of the current taken into account increased. The same methodology employed for the bistability window heatmaps was used to compute those limits. In Fig 10 (right), we applied hyperpolarizing steps of currents to a neuron with a fixed maximal conductance for the current considered (i.e., the inward KM current in the case of Fig 10A (right) from I2 to I1, with , starting from the resting state in I2, for a total time of 1 s. The permeability
used in Fig 10 was
.
Bifurcation analyses around the excitability switches
In Figs 11E–11H and Figs 12C–12D, we computed bifurcation diagrams with respect to the applied current I. Fixed points were computed with the NLsolve package for each tested I. Limit-cycle extrema during spiking were computed using the same methodology as for the fI curves. Conductance sets used (as ): Fig 11 E, {0.;0.;0.}; Fig 11 F,
; Fig 11 G, {0.;0.;0.3} ; Fig 11 H, {0.;0.3;0.}; Fig 12C,
; Fig 12D,
.
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