The model family analyzed in this work stems from the classical Hodgkin-Huxley model (HHM). for a single-compartment (space-clamp) and continuous variation of the voltage-gated sodium channels (Nav) half-activation-voltage parameter ΔV1/2, which controls the window of sodium-influx currents. Unlike the baseline HHM, its parametric extension exhibits a richer multitude of dynamic regimes, such as multiple fixed points (FP’s), bi- and multi-stability (coexistence of FP’s and/or periodic orbits). Such diversity correlates with a number of functional properties of excitable neural tissue, such as the capacity or not to evoke an action potential (AP) from the resting state, by applying a minimal absolute rheobase current amplitude. The utility of the HHM rooted in the giant squid for the descriptions of the mammalian nervous system is of topical interest. We conclude that the model’s fundamental principles are still valid (up to using appropriate parameter values) for warmer-blooded species, without a pressing need for a substantial revision of the mathematical formulation. We demonstrate clearly that the continuous variation of the ΔV1/2 parameter comes close to being equivalent with recent HHM ‘optimizations’. The neural dynamics phenomena described here are nontrivial. The model family analyzed in this work contains the classical HHM as a special case. The validity and applicability of the HHM to mammalian neurons can be achieved by picking the appropriate ΔV1/2 parameter in a significantly broad range of values. For such large variations, in contrast to the classical HHM, the h and n gates’ dynamics may be uncoupled - i.e. the n gates may no longer be considered in mere linear correspondence to the h gates. ΔV1/2 variation leads to a multitude of dynamic regimes—e.g. models with either 1 fixed point (FP) or with 3 FP’s. These may also coexist with stable and/or unstable periodic orbits. Hence, depending on the initial conditions, the system may behave as either purely excitable or as an oscillator. ΔV1/2 variation leads to significant changes in the metabolic efficiency of an action potential (AP). Lower ΔV1/2 values yield a larger range of AP response frequencies, and hence provide for more flexible neural coding. Such lower values also contribute to faster AP conduction velocities along neural fibers of otherwise comparable-diameter. The 3 FP case brings about an absolute rheobase current. In comparison in the classical HHM the rheobase current is only relative - i.e. excitability is lost after a finite amount of elapsed stimulation time. Lower ΔV1/2 values translate in lower threshold currents from the resting state.
Citation: Krouchev NI, Rattay F, Sawan M, Vinet A (2015) From Squid to Mammals with the HH Model through the Nav Channels’ Half-Activation-Voltage Parameter. PLoS ONE 10(12): e0143570. https://doi.org/10.1371/journal.pone.0143570
Editor: Vladimir E. Bondarenko, Georgia State University, UNITED STATES
Received: December 2, 2014; Accepted: November 8, 2015; Published: December 2, 2015
Copyright: © 2015 Krouchev 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 relevant data are within the paper and its Supporting Information files.
Funding: The authors have no support or funding to report.
Competing interests: The authors have declared that no competing interests exist.
The Hodgkin-Huxley (HH) model recently turned 60 [1, 2]. It has been the only widely used prototype for microscopic single-compartment models . Other popular models (e.g. the leaky integrate’n fire) do not contend for either detailed and type-specific descriptions of single neurons, or even less of their parts.
More recently, related to the detailed microscopic reconstructions, multi-compartmental neuronal models are built of electrically coupled HH-type compartments. The model has eloquently proven its quantitative capacity to convey an explanatory power to relatively simple neuron models, which account for more and more experimental findings . Yet it has also been criticized, re-parameterized and subjected to multiple attempts to find it incomplete or inefficient [4, 5]. Importantly, the HH model (Table 1 introduces all the commonly used abbreviations) simplicity was blamed, while critics sometimes omitted essential properties themselves . The model’s very applicability to mammalian neurons—with respect to metabolic requirements and flexibility of encoding, has been closely reexamined [2, 7, 8].
An unifying parametric framework is proposed in this work to systematically examine the impact of Nav subtypes’ distributions on neuronal dynamics, and thence on excitability and refractoriness.
We systematically explore the influence of parameter variation on the fundamental biophysical properties of voltage-gated sodium (Nav) channels, within an HH-type modeling framework. The Nav channels produce large and fast membrane currents essential in the generation and propagation of action potentials in excitable tissues. They contain a transmembrane alpha subunit which forms the channel pore. The latter subunit’s genetic expression proved sufficient for the expression of whole functional channels of a given subtype . Hence Na+ channel nomenclature follows closely that of their alpha subunit (in this work we use the Nav1.X notation). With the help of gene engineering and selective expression of specific Nav1.X channels, significant experimental evidence has been accumulated on their (in)activation and localization properties [10–12].
Relatively little is known about the ways in which the different Na+ channel subtypes are distributed and expressed toward functional axons in either a developmental or mature stage [13, 14]. An exceptional wealth of evidence comes from epilepsy research [15–20], where Na+ channel mutations have been associated with either gain-of-function or loss-of-function effects —i.e. increased or decreased neuronal excitability in either excitatory or inhibitory populations (e.g. GABA interneurons or Purkinje cells). The Nav1.1 subtype, which is hypothesized to undergo such mutations, is also involved in an important developmental aspect. Namely, it gradually replaces the low threshold Nav1.3 subtype—which is only expressed during early development or excitable-tissue injury . Interestingly, both the Nav1.1 and Nav1.3 subtypes are encoded on chromosome 2q24. It may also be tempting to speculate that the easily excitable Nav1.3 subtype is desirable during large neural network connectivity formation, but would lead to dynamic stability issues in an adult highly active and interconnected brain.
It is known that action-potentials (AP) are primarily initiated close to the axon hillock . The cerebral cortex is densely packed with an estimated 50,000 neurons/mm3 and at least 100 times as many neural processes . A recent study  used HH models of two cortex-specific Nav1.2 and Nav1.6 subtypes. Nav1.2 sustained AP propagation, while Nav1.6 activation at lower membrane voltage (V) values contributed to AP initiation. The same Nav channel subtypes were then used in a multi-compartmental HH model of a neuron with a single dendrite, soma and axon morphology along a straight axis , which robustly interpreted and predicted the clinical effects of intra-cortical micro-stimulation (ICMS), following up on previous work on the subject . It was found that AP initiation in the axon initial segment (AIS) was more likely and required lower stimulation current. Moreover, this was attributed to the higher density of the Nav1.6 subtype.
Benefitting from a detailed exploration of the literature on experimentally observed Nav channel properties in the central nervous system (CNS), we address the parameter V1/2 controlling the membrane voltage V at which the Na+ conductance attains its half-maximal value. This parameter has a direct impact on a number of fundamental properties. Some of these are straightforward to demonstrate from first biophysical principles. Others required subsequent bifurcation and phase-plane analysis, which are only enabled by models with just a few parameters. Nonetheless, this reductionist approach provides useful generalizations about key aspects under investigation—such as the HH model limits of metabolic efficiency or encoding. We do not claim that the model presented here is a definitive one, or that the different Nav subtypes are only due to simple shifts in V1/2. Nature has access to many complex molecular structures and there may be restrictions we are unaware of. Hence, overly simplistic models may fail to explain some observations. The proper model type depends on its goals and phenomenological scale . Nonetheless, simple models can generate useful predictions and are easy to incorporate as building blocks in novel more elaborate paradigms. Thus, we use our parametric framework—within a physiological V1/2 range—to address metabolic energy savings and other desirable properties such as fast AP transmission or large frequency-encoding capacity. We provide a comprehensive excitability-dynamics description of the Nav subtypes similar to the ones expressed in the mammalian brain (Nav1.X with X = 1, 2, or 6). Such analysis over a range of continuous V1/2 variations seems in order, especially given that the V1/2 values reported by different authors vary significantly [2, 10–12, 15, 17–21, 27–29]. assuming that one (and only) given Nav1.X subtype is expressed in the host cell, on which whole-cell voltage clamping techniques are applied.
Very recent experimental and computational work on the molecular and functional effects of brain trauma points in quite a similar direction. Namely that Nav channels undergo irreversible hyperpolarizing V1/2 shifts [30, 31]. This may for example lead to ectopic excitability and propagation in damaged axons .
The presentation is laid out as follows:
The next section (Methods) presents the model details and key assumptions.
In a structured form the Results section defines a fundamental ensemble of related key biophysical properties of the parameterized single-compartment HH model, such as metabolic efficiency and excitability. The theoretical analysis presents the bifurcation structure as a function of the ΔV1/2 parameter, and as a function of the constant (and hence known as bias) current Ibias, applied intra-cellularly, which is another parameter receiving special attention in the relevant literature
The Discussion provides a succinct summary of the most important findings, as well as a generalization of some results to multi-compartmental models.
To facilitate reading and provide for a wider audience, the most technical sections on automaticity and on the codimension-2 bifurcation structure in the ΔV1/2 × Ibias parameter plane are presented separately as Supplementary Results.
In this work, we explore the nonlinear dynamics for the following 4D system of ordinary differential equations (ODE): (1) where Is stands for the stimulation current injected into the modeled compartment and is the total ionic current. The channel-gate state variables of the HH model m, h and n are described by: Matlab was used to numerically solve the model Eq (1) and AUTO  —for fixed point (FP) or periodic orbit (PO) continuation and bifurcation analysis.
We use the single-compartment Hodgkin-Huxley model as described in . The same model is also used in [24, 25] as a building block. There we noticed an interesting pattern in the description of the Nav1.2 and Nav1.6 channel subtypes. For both the m and h gating variables, all the related V1/2 parameters differed by exactly 13mV.
From whole-cell expression of the Nav1.6 subtype, V1/2 = −29.2±1.8 mV and V1/2 = −29.4±1.6 mV (as reported by  and  respectively). For the Nav1.2 subtype, V1/2 = −18.32 mV . We also noticed that the cited modeling and experimentally observed data preserved a quite similar relative difference in V1/2—between Nav1.6 and Nav1.2. Furthermore,  demonstrated the remarkable effects of the pentapeptide QYNAD. At 200 μM concentration, the Nav1.2 subtype V1/2 was increased to −9.15 mV—an almost 9 mV shift toward decreased neuronal excitability. Since QYNAD (Gln-Tyr-Asn-Ala-Asp) is endogenously present in human cerebrospinal fluid (CSF) , such chemical interactions may also occur naturally and thence have significant effects on neural dynamics.
The Nav1.6 and Nav1.2 channel types can be seen as two instances of a generalized Na+ ionic current model, in which the V1/2 parameter of both the m and h gate variables is controlled by the parameter ΔV1/2 with values of 0 mV and 13 mV, respectively (Fig 1 and Tables 2, 3 and 4; see also the gate-state dynamics maths in Box 1.
A: Asymptotic state y∞(V). B: Time constants τy(V) The subscripts 1 and 2 of the m and h variables refer respectively to the Nav1.2 and Nav1.6 ion channels. Panels C and D illustrate the ionic currents interplay in the model as a function of membrane voltage V, computed for different example values of ΔV1/2. The color traces use a color-code compatible to panels A, B for the two key Nav channel currents; the latter plots are inverted to facilitate the interpretation of equilibria. C: Iion,∞[ΔV1/2](V), computed for the asymptotic state y∞(V) of all gate variables The thin black trace shows the total re-polarization current IK + ILeak Fixed points (FP’s) locations are illustrated through the interplay of Iion,∞(V) components D: Iion,0[ΔV1/2](V)—computed for m = m∞[ΔV1/2](V) similarly to Iion,∞(V), but for the reduced (1D) model at the resting state of the slower h and n gates, i.e. h = hrest = h∞[ΔV1/2](Vrest) and n = nrest = n∞(Vrest), where Vrest = the resting potential. The circle markers in both Panels C and D indicate the resting-state FP Vrest. The diamond markers indicate the second AP-threshold FP VTHR, which for Iion,∞(V) (panel C) exists only for the M3-class models (see text and Fig 4). The gray trace is for ΔV1/2 = -8.7 mV →LP2 (see Fig 4).
Box 1: Gate-state dynamics mathematical details
*1: Temperature dependence assumed linear with slope , where T0 = 23°C.
*2: Opening/closing rate of gates y in the K+ or channels have Boltzmann-like templates (⋅y subscripts dropped to simplify notation): where w = Vm − V1/2, and the rates change in function of w: When both the opening and closing rates share the same V1/2 parameter (e.g. the m and n gates, Table 2), the corresponding V-dependent asymptotic gate state and time constants are: α′(w), β′(w), and y∞(w) are all sigmoidal functions of w. Regardless of a, b, k values, rate-change dependence on w is most important at w = 0, where α′(w) = a/2 and β′(w) = −b/2. Compare to α′(w) ≈ β′(w) ≈ 0 for w ≫ 0.
is reciprocally dependent on k, and its maximum is obtained at w* = −kln(a/b) (hence with a = b, w* = 0). Hence dy∞(w*)/dw is determined by the V1/2 parameter, and y∞(w*) = 1/2—regardless of the actual a, b, k values. Also for Vm ≈ V1/2, τ(V) = τ(w) attains its maximum (τ(w*) ≈ τ(0) = 1/k(a + b)), since: E.g. for a = b = k = 1, the numerator of τ′(w) reduces to u2+2uw − 1 (u = e−w), which vanishes when w = 0.
A Mixture model.
provides a comparative perspective for the effects of parameter variation, and specifically concerning the effects of ΔV1/2 to those of channel density by types and numbers (i.e. gNa variation). In this model extension, a fraction P ≤ 1 of all the Na+ ion channels are of the Nav1.6 type (i.e. ΔV1/2 = 0 mV), while the remaining Na+ channels—i.e. a fraction 1 − P, are of a second different type of channel with a chosen ΔV1/2 ≠ 0. Thus: (2) Such mixture models may provide predictions of the electrophysiological properties for neural processes with a particular experimentally observed distribution of voltage-gated sodium channel subtypes (e.g. as in [10, 12, 14, 27, 35]).
From a compartment to a cell.
The single-node model can be generalized to a multiple-compartment model of an entire axon. In models of higher dimensionality, the spatial distribution (relative proportions) of ionic current types will correspond to better or worse conditions for AP initiation and propagation.
To estimate the effect of ΔV1/2 on AP propagation velocity (through the Nav currents), homogeneous cables of 100 identical compartments—each of length 25 μm (yielding a total cable length of 2.5 mm) were simulated (see also  for a detailed description of this type of cable models). The cable was subject to the no-flux boundary condition on one end, and the single compartment on the opposite end was stimulated (TSTIM = 100 μs or 1 ms). For each value of the ΔV1/2 parameter, the resting threshold was identified. An AP was reliably evoked and propagated to the very end of the cable. A propagating AP was signalled by small deviations of propagation velocity (range < 30% of mean value) in the section occupying the middle 50% of the total cable length.
To avoid overestimating the required stimulation current thresholds, the total duration of simulation was chosen as TSTIM + 1.5 ms. However, this also provided for a little extra variability in the identified ITHR, due to phenomena resulting from the cable’s finite length and no-flux boundary conditions. Namely, lower (near-threshold) stimulation current latently reaches and depolarizes compartments down the cable. As a result, when the AP is finally evoked it propagates faster than in a significantly “supra-threshold” case. Hence, to avoid excess variability in the estimated propagation velocity, a safety factor was applied: .
Our results demonstrate that the continuous variation of the ΔV1/2 parameter comes close to being equivalent with recent HHM ‘optimizations’. The dynamics of the parametric family exhibit a rich multitude of properties, which we have grouped in 4 main categories, namely: excitability, efficiency, frequency-encoding range and conduction velocity. The former three are discussed in a single-compartment (space-clamp) context. Conduction velocity and whole-neuron excitability are presented within a multi-compartment model extension.
Effects of ΔV1/2 variation on neuronal excitability
Can the parameterized HH models’ exhibit wide variation in their excitability?
It is well known that the HHM’s excitability to brief stimulation pulses is fully dependent on the fast activation of the m gates versus a slower inactivation of the h and activation of the n gates (Fig 1A and 1B). This can be demonstrated by a simplified model in which m is assumed to reach instantaneously m∞(V) while h and n remain at their resting values hrest and nrest. Upon very brief Is duration this is a fair approximation of the system’s behavior (see Ch.3 in ). Stimulating from rest [Vrest, hrest, nrest], the system dynamics is reduced to: (3) For all ΔV1/2, the above approximate ionic current—denoted Iion,0(V)—a function of just V, has 3 zero-crossings: Vrest, Vthr (see Fig 1D) and Vup (not shown). The LOCAL minimum −Irh is reached for a V value located between Vrest and Vthr. While Vrest and Vup remain virtually unchanged for different values of ΔV1/2, Vthr moves toward Vrest and Irh diminishes as ΔV1/2 is decreased. In this 1D approximation, one can see that the membrane will depolarize further toward Vup if at the end of the stimulation V > Vthr. The latter is possible only if Is > Irh. Hence lower ΔV1/2 increases the excitability by reducing both Vthr − Vrest and Irh.
Strength-duration (SD) curves for the ΔV1/2-family, computed through simulations of the full HH dynamics, and for the whole range of ΔV1/2 ∈ [−8, 40] mV, are shown in Panel A of Fig 2. For the short (e.g. TSTIM = 100 μs) the increase of ITHR with ΔV1/2 is close-to-linear, while for the long (e.g. TSTIM = 2000 μs) the increase is steeper than linear.
A Strength/duration (SD) curves; B IS2,THR(t − tup) TS1 = TS2 = 100 μs. ∀ΔV1/2 the test APS1 is obtained by 1.5× the resting ITHR (see Panel A).
Toward approximate analytic SD curves, the model is simplified by assuming that all gate variables follow their asymptotic values, i.e.: (4) This model is valid for low-amplitude stimulation of very long duration. Iion,∞(V) may have one or three zero crossings (Fig 1C). For the one-zero-crossing case, the very existence of a threshold is a transient phenomenon, i.e. a finite Vthr no longer exists, when the h and n gates respectively close and open. With three zero crossings −Iion,∞(V) also has a local minimum—a situation qualitatively similar to the Iion,0(V) case above. Hence, just as with the simplified model of Eq (3), further depolarization of V toward Vup,∞ may be expected when V is beyond Vthr,∞ at the end of the stimulation. In this case Irh is precisely the value of the rheobase current.
Through Iion,∞(V) of Eq (4), a lower bound of the threshold current—the absolute rheobase—can be estimated.
Hence, SD[Vshift] may be estimated for relatively short durations (TSTIM → 0) or long durations (TSTIM → ∞). In the latter case, this is only possible for models like Nav1.6 (ΔV1/2 = 0), but not for models like Nav1.2. This is one of the key properties pointed at in this work.
To evoke an AP, according to Eq (1) the stimulation current Is needs to overcome the dominating opposing ionic currents Iion. In search for patterns of lowest amplitude or highest energy efficiency, Is may just slightly outweigh Iion and still successfully trigger AP’s . However, this is highly dependent also on stimulation duration and NaV class.
Importantly, for the NaV1.6 subtype, but not for the NaV1.2 subtype—as demonstrated through the simplified model Eq (4)—a threshold value of the membrane voltage exists for any stimulation duration. Hence an AP can be evoked by applying almost as little current as the absolute rheobase current of a sufficient duration (Fig 2A).
Refractoriness (absolute or relative) is the other face of excitability. In a standard procedure to assess it, one applies a second stimulus S2 at different times along a test action potential APS1, induced by an initial supra-threshold stimulus S1. The minimal current Itrh,S2, needed for another active response, is then found for each S1 − S2 coupling time, or else the system is diagnosed as absolutely refractory—if current amplitude lies beyond some acceptable limit. Itrh,S2 is a function of t = tS2 − tS1 coupling interval (assume tS1 = 0 for simplicity), as well as of the S2 duration Ts,S2. Whether it also depends on the parameters of the S1 stimulus is to be investigated. Alternatively, refractoriness can be studied by a one-dimensional approximation similar to Eq (3). (5) in which h0 = h(V0) and n0 = n(V0) are the values of the gate variables at each point V0 along the APS1.
As for stimulations from the resting state, a necessary (but not sufficient) condition for S2 to produce an AP is that Eq (5) has 3 FP’s. When Vthr does not exist, the system is absolutely refractory. Otherwise, a minimum stimulus current is needed.
We consider the case where APS1 is evoked by a stimulus applied from the resting state. Both Vthr(t)—calculated from Eq (5), and Ithr,S2(t) depend on the APS1, which in turn varies with Ts,S1 (and the corresponding Is,S1). However, both functions become invariant once the time axis is aligned to tup—the time of maximum dV/dt derivative during the upstroke. Different Is,S1 and Ts,S1 may modify the latency of APS1, but almost none of its time course beyond the upstroke. This invariance is maintained as long as Ts,S1 does not become too long. Conversely, when Ts,S1 → ∞, Is,S1 becomes what is known as bias current. It may induce automatic firing, which in turn precludes the use of the S1/S2 protocol (see also the Supplementary Results subsection on automaticity).
The invariance of the APS1 time course means that a single Ithr,S2(t − tup) curve (excitability is a function on the recovery time t − tup) can be used to capture the system behavior for each Ts,S2 and ΔV1/2. Rabinovitch et al.  referred to this invariant trajectory as hidden structure (HS) and proposed a method that extends to excitable systems the concept of the phase transition curve (PTC), widely used to analyze the forcing of non-linear oscillators [39, 40].
Panel B of Fig 2 shows the variation of Ithr,S2 as a function of t − tup and ΔV1/2 for TSTIM = 0.1 ms. Decreasing ΔV1/2 slightly shortens the absolute refractory period. For any given t − tup, it also reduces the current needed to get a response. For any t − tup, there is a quasi-linear increase of the Ithr,S2 as a function of ΔV1/2. For t − tup > ≈ 2.5 ms, where excitability regained in all model NaV subtype cases, a higher ΔV1/2 may mean up to a two-fold increase of the threshold current.
The metabolic efficiency of the HHM[ΔV1/2] family
Can the parameterized HH models exhibit higher metabolic efficiency?
The Na+ ions that enter the cell during an action potential (AP) will have to be returned back to the extra-cellular space. This is done by the Na+/K+-ATPase enzyme pump, while simultaneously bringing the repolarizing-phase K+ ions back to maintain the resting potential. It binds 3 intracellular Na+ and 2 extracellular K+ ions. One ATP molecule is hydrolyzed, leading to phosphorylation and a conformational change in the pump, which exposes the respective ions to alternate sides of the cell’s membrane. Thus the Na+/K+ pump can be responsible for up to 70% of the neurons’ energy expenditure.
Hence, the metabolic cost for firing a single AP is proportional to the Na+ ionic charge that enters the cell during each AP, and also depends on the overlap of opposing Na+/K+ ionic currents .
Such AP properties as the INa/IK currents interplay and metabolic efficiency as a function of ΔV1/2 are illustrated in Fig 3. ∀ΔV1/2 the AP is obtained via 1.5× the resting ITHR for duration TSTIM = 100 μs. As observed, larger ΔV1/2 values translate into lower excitability: higher threshold potential VTHR and threshold stimulation currents ITHR. The very large latencies for ΔV1/2 = 0 and -5 mV are due to very low threshold values ITHR, which remain low even after a 50% “safety-factor” increase. The thick color traces in Panel A of the figure show the −INa current profiles, aligned to the AP upstroke (time zero). The Na+ current has a local peak just before the AP upstroke. there is a complete overlap of the Na+ and K+ currents (shown by color and dashed-black traces respectively in Fig 3B). The close INa/IK interplay here gives an electrophysiological and physical perspective onto the nonlinear dynamics concept of the “slow manifold” [37, 42, 43].
A Membrane voltage during the AP; B INa / IK interplay; C Overall Metabolic efficiency Thick color: −INa, aligned to the AP upstroke (time zero); Dashed black: the respective IK QNa—charge transferred into the modeled compartment during a single AP. AP shape and QNa nearly invariant with ISTIM ≫ ITHR.
However, smaller ΔV1/2 values reduce both the Na+ and K+ currents through the AP’s re-polarization stage. First, lower ΔV1/2 implies lower VTHR. Second, the AP upstroke amplitude Vup is invariant to ΔV1/2 and hence h∞(Vup) is significantly more closed for smaller ΔV1/2 values (see Fig 1A).
The resting FP Vrest remains almost identical in the Nav1.2 and Nav1.6 models (-77.03 vs -77.01 mV), but nevertheless corresponds to quite different values of h∞(Vrest) (0.96 vs 0.76). Hence, the Nav1.6 h gates will close more during the AP upstroke, and will recover later and to a lesser value during the post-upstroke re-polarization. Furthermore, the Nav1.6 model has two additional depolarized FP (diamond and square in Fig 1C—denoted Vthr,∞ and Vup,∞).
Compare the equilibria of the gate state h on Fig 4 for the typical models of Nav1.6 (ΔV1/2 = 0), and Nav1.2 (ΔV1/2 = 13). The latter figure contains the big picture of our bifurcation analysis (presented next in more detail), namely that the ΔV1/2 family contains as a superset a wide variety of models with different properties, in which models behaving exactly like the classical HHM archetype are encountered over only a sub-interval of the ΔV1/2 parameter full range of variation.
These bifurcation diagrams illustrate the FP’s and PO’s of the V and h model states as a function of ΔV1/2. Thin black line: stable FP’s; thick cyan: the unstable-FP branches (with one or two (real or complex) positive-real eigenvalues. LP1 and LP2: lower and upper limits of the middle branch. HB: Hopf bifurcation. See the text for more details. Thin and thick dashed lines: minimum and maximum model-state value for the stable and unstable cycles respectively. Unstable cycles are born from FP’s at a subcritical HB and metamorphose to stable at a cycle saddle-node (fold).
Efficiency is estimated by computing the overall Na+ ions’ charge QNa (in micro-Coulomb’s per cm2), which is transferred during a single AP ([41, 44], TSTIM = 100 μs). The integration is carried over the whole time span (30 ms) of the simulation (from AP triggering, incl. latency to the return to rest). The shape of the AP and hence QNa are mostly invariant as long as ISTIM ≫ ITHR. A lower ΔV1/2 clearly associates with a lower estimate of metabolic cost (see Fig 3, Panel C).
Finally, is this effect trivial? Can it be easily deduced by looking at the HHM Eq (1) alone? Here we argue that this is not the case:
Without a significant effect on AP amplitude or duration (Fig 3A and 3B), the overall shape of the AP changes just slightly and a significant gain in metabolic energy expenditure is achieved by an earlier closing of the voltage-gated Na+ channels. Since VTHR is lower in this case, this earlier Na+ inactivation is matched to a significantly lesser activation of the voltage-gated K+ channels.
Bifurcation structure as a function of the ΔV1/2 parameter
The resting potential (Vrest) of the system is a fixed point (FP) of the system set by: (6) It is located at the intersection of the monotonically increasing outward currents IK,∞(V) + Ileak(V) with the bell-shaped ΔV1/2-dependent inward current INa,∞ (Fig 1C). The effect of ΔV1/2 variation is a linear shift of the window-current INa,∞[ΔV1/2](V) toward lower or higher V values. This may create additional FP’s. The number, location and stability of the FP’s is bound to play a significant role in the dynamic properties of the system upon stimulation and re-polarization.
The existence of the two unstable fixed points also changes the organization of the trajectories in the phase plane. As mentioned in the presentation of the bifurcation diagram below, Vthr,∞ has a single unstable direction. Perturbations along this direction (either positive or negative) produce two heteroclinic trajectories ending at Vrest, corresponding to a passive and an active depolarization (Fig 5, the red and green trajectories). The latter is the limiting case of an action potential that would be obtained by a stimulus slightly beyond the rheobase current. Another set of trajectories connect the most depolarized fixed point (unstable focus) Vup,∞ to the stable resting state. All these invariant trajectories and their associated stable manifold do not exist for models like that of the Nav1.2 subtype. They add constraints onto the system dynamics since they cannot be crossed by any other trajectory. As we shall see below, the two additional FP’s also affect the nature of the automatic regimes upon sustained stimulation.
Two heteroclinic trajectories, starting at VTHR (red diamond) and ending at Vrest (green circle): correspond to passive—green and active depolarization—red. Trajectories from VUP (blue square)—thin dotted blue connect into Vr. All trajectories avoid and “slide” on the slow manifold (SM) (wireframe mesh), constructed by solving Iion(V, m∞(V), h, n) = 0 ∀(n, h). Its ‘cubic-like’ shape defines the excitation threshold.
Fig 4A shows the bifurcation structure diagram (BD) of the of the 4D space-clamp model as a function of the ΔV1/2 parameter. In addition to the resting point (the lower branch on the BD, see the “stable FP” arrow), a second depolarized FP (corresponding to Vthr in panel A) appears through a saddle node bifurcation at the limit point LP1: ΔV1/2 = 2.94 mV. From LP2: ΔV1/2 = -9.5 mV, up to LP1, the S-shaped BD FP curve has three coexisting branches. Beyond LP2, a single depolarized FP remains. FP stability is also accounted for in Fig 4A. The middle branch (LP1 to LP2) is half-stable (see Table 5 and ) due to a single positive real eigenvalue. The lower FP branch is stable with four real negative eigenvalues, except for a tiny interval close to LP2 (there is a second subcritical Hopf bifurcation HB2 located 0.005 mV beyond LP2, not shown). The uppermost branch is unstable from LP1 down to the Hopf bifurcation point (HB: ΔV1/2 = -11.05 mV), where it recovers stability. From LP1 to HB the top branch has either two positive real (ΔV1/2 ∈ [1.7 mV, LP1]), or two positive-real complex eigenvalues (ΔV1/2 ∈ [HB, 1.7 mV]).
The HB is subcritical (see Table 5) and produces a branch of unstable PO’s connecting through a saddle-node of cycles (SNC) bifurcation to a high-amplitude stable PO at ΔV1/2 = -35.5 mV. This stable PO branch disappears slightly beyond LP2 (around ΔV1/2 = -9.16 mV) through a second SNC bifurcation (see Table 5, as explained in the Supplementary Results—subsection on automaticity). In this work, the analysis is mostly restricted to ΔV1/2> LP2. There are two distinct model sub-classes by the parameter value range:
- M1. ΔV1/2 > LP1. The corresponding dynamic system has a single stable resting (hyperpolarized) FP, e.g. the Nav1.2 channel subtype.
- M3. ΔV1/2 ∈ [LP2, LP1] yields 1 stable resting FP, and two unstable FP’s, e.g. the Nav1.6 channel subtype. As can be inferred by Fig 1C, increase or decrease of the nominal conductivity gNa offers an alternative way to create or suppress the folding and hence the appearance of the middle and topmost FP branches. Likewise, a variation of the relative proportion of different channel subtypes in a mixture model is equivalent to nominal conductivity variations. Hence, an estimate of the gNa value yielding a transition from one to 3 FP’s is useful to understand the dynamics of the mixed model. From Fig 1C, and with ΔV1/2 = 0 mV, gNa has to be decreased by a third to suppress the BD from folding, but with ΔV1/2 = 13 mV, it has to be increased fourfold to create the fold.
Except for a small interval of ΔV1/2 ∈ [HB2, -9.16 mV] where a stable FP and a stable PO coexist, the two model sub-classes necessarily return to their resting state after a sub-theshold stimulus. However, the supra-threshold dynamics (leading to an AP) will be dramatically influenced by the existence or not of the 2 additional FP’s (see the Supplementary Results’ subsection on automaticity).
Lower ΔV1/2 brings about a large frequency-encoding range
Can the parameterized classical HH model exhibit the low-frequency automatic firing needed to account for certain regimes observed experimentally in mammalian species?
Before we answer this question, let us introduce another preliminary bifurcation analysis result. It is known , that in the HH model periodic firing may be obtained by injecting a constant Ibias—known as bias current within a certain range (see also Fig A in S1 File). In the literature one can frequently encounter bifurcation analysis of the HHM with respect to Ibias.
Panels A and C in Fig 6 result from exactly such bifurcation analysis for the typical models of Nav1.6 (ΔV1/2 = 0), and Nav1.2 (ΔV1/2 = 13). Here a higher ΔV1/2 translates into incapacity for low-frequency automatic firing and a narrow overall frequency-encoding range (see Panel B in Fig 4), consistently with earlier observations .
Panel B in Fig 6 illustrates the period length of automatic firing as a function of Ibias. In the Nav1.6 case (see also Panel C), stable automatic firing (blue dots in Fig 6B) is lost through a cycle saddle node, which accounts for the finitely increased length of the related cycle period, which is a desirable property of the parametric model family. On the other hand, the unstable cycle (green dots in Fig 6B) disappears into an homoclinic bifurcation, when Ibias tends to the forementioned absolute rheobase value. As is well known, homoclinic bifurcations may be associated with infinite increase of the related period.
Fig 7 illustrates the limits of automatic firing and of periodic stimulation maintaining the 1:1 response pattern (also known as pacing) as a function of ΔV1/2. In Panel A of the figure, the periodically-applied stimulation current ISTIM required for an 1:1 response—at the given minimal pacing period τ11, depends on pacing frequency. Clearly, ISTIM → ITHR (the resting threshold, see also Fig 2), as τ11 → ∞. Notice that, with near-threshold current values, higher ΔV1/2 allow periodic stimulation at higher frequency.
A Stimulation current (also function of frequency); B Min. Ibias (→ Max. period length) for stable cycles ISTIM required for an 1:1 response at pacing period τ11. TSTIM = 100 μs; ISTIM → ITHR as τ11 ≫ 0; Semi-transparent quasi-planar surface = 1.5× the resting ITHR.
However, the situation is reversed for automatic regimes. The period of automatic firing is a monotonously decreasing function of Ibias with a minimum well below 5 ms. This means that some (less excitable) models may not have automatic firing at a frequencies below 200Hz. Importantly, this is a monotonicity property specific to the model at hand, contrasting it to other models (e.g. the Fitzhugh-Nagumo model)—(see Fig 7B—itself, a simplified presentation of Fig D.B in S1 File. The full detail of periodic firing variability and bifurcation structure as a function of ΔV1/2 and Ibias are provided as Supplementary Results).
How about the maximum period that can be achieved?
Applying the same monotonicity property for the period, it would be longest for the lowest Ibias for which automatic firing is still possible. Lower ΔV1/2 provide for such regimes with progressively lower firing frequency (longer inter-AP periods). As seen in Panel B of Fig 7, a very reasonable ΔV1/2 parameter variation translates into a five-fold variation of period lengths. Thus, the model family is capable of handling automatic firing at a frequency less than 20Hz.
Hence lower-ΔV1/2 models bring about a significantly larger frequency-coding range, which in addition requires less metabolic resources.
On Hogkin’s maximum-velocity hypothesis
Can the parameterized HH models conduct AP’s faster?
The effect of ΔV1/2 on AP propagation velocity (through the Nav currents), was studied in a multiple-compartment model as described in the Methods. To reliably evoke propagating AP’s for all values of the ΔV1/2 parameter, the resting stimulation-current thresholds ITHR were identified. Fig 8 illustrates this for two post hoc runs of the central step of the ITHR-search algorithm (ΔV1/2 = 2 mV, please see Methods for a detailed interpretation of the obtained voltage-distribution patterns in either case).
A When ISTIM is lower, it latently reaches and depolarizes compartments down the cable. This is clearly visible from the voltage traces, which are depolarized up to halfway down. As a result, when the AP is finally evoked it propagates faster. B “supra-threshold” case with a safety factor applied .
According to Hogkin’s maximum-velocity hypothesis, the ion channels’ ion-types and density evolved through natural selection to maximize the AP conduction velocity. However, it has been argued that gating current limits the increase of Na+ channel density as a means to gain in conduction velocity. Moreover, the computed optimal density is more than twice higher than in the real squid, and with capacitance reduced by neglecting gating current, optimal density is even higher .
A summary of resting threshold ITHR and mid-axon mean AP conduction velocity as a function of ΔV1/2 is presented in Fig 9. Notice the clear linear trend of increasing ITHR with ΔV1/2 for both TSTIM cases. Despite the large estimation variability (as illustrated and discussed in Fig 8) a significant trend of deccelerating AP propagation with ΔV1/2 can be acknowledged.
Notice the clear linear trend of ITHR increasing with ΔV1/2 for both TSTIM cases. A significant trend of accelerating AP propagation with ΔV1/2 can be acknowledged. To prevent excessive results variability (see Fig 8 and text), conduction velocity was computed for ISTIM = 1.5 × ITHR.
Hence, evolution may play another card to solve many a problem at once—avoid the overhead of having too many channels, while simultaneously gaining in AP transmission speed, and even in efficiency (as just shown above). All of it by a simple shift of the sodium current window along the membrane potential dimension.
From the preceding analysis, the single-channel model has 3 FP’s for ΔV1/2 < LP1 (Fig 4A). This section analyzes mixed models with sodium current. For ease of reference Eq (2) is recalled from the Methods section: (7) The B.D. on Fig 10A shows that, for ΔV1/2 = 13 mV, the 2 upper FP branches appear at saddle-node bifurcation (see Table 5) for P ≈ 0.648, which is close to the minimum relative conductivity (gNa/gK = 0.663) needed for the single-channel model with ΔV1/2 = 0 to enter the M3 class. In the voltage range of peak INa,∞[ΔV1/2 = 0](V), the contribution of INa,∞[ΔV1/2](V) is so low that it does not change much the total INa,∞ profile. Hence, the upper FP’s rely almost exclusively on the of the M3 current contribution. In codimension-2 (see Fig 10C), as ΔV1/2 decreases, the contribution of INa,∞[ΔV1/2](V) becomes larger, thereby reducing the proportion of M3 current needed to preserve the 3 FP’s. In Fig 10C, the required proportion P of Nav1.6 channels (recall the Methods section) decreases abruptly as ΔV1/2 < 5 mV. Clearly, the latter is due to the intuitive fact that such ΔV1/2 values mean that the second channel type is itself tending to the M3 class—a tendency which is completely realized with ΔV1/2 < LP1.
A: Red trace: Existence/creation P parameter-range for the 2 additional fixed points (saddle-node in the middle branch, and unstable center/focus in the top branch of the B.D.’s) With ΔV1/2 = 13 mV, P ≥ 0.648, which is very close to the minimum relative Na/K conductivity ratio (0.663) needed for 3 FP’s in the single-Nav-subtype model with ΔV1/2 = 0 mV. In the voltage range of peak M3 current, the contribution from the Nav1.2 channels is low and does not change much the total current profile. Hence, the upper branch of FP’s is due the contribution of the Nav1.6 channels. B: Resting threshold of the mixed models as a function of the P parameter. The figure shows κ(P) = ITHR(P, ΔV1/2)/ITHR(1,ΔV1/2) of the mixed model for TSTIM = 100 μs. Notice that there is an approximately twofold difference between the thresholds of the M3 and M1 single-Nav-subtype models More than 60% of this difference vanishes for P as low as 0.1. The difference becomes minimal for ΔV1/2 < 4 mV (see also Panel C) or P ≥ 0.1. C: As the ΔV1/2 parameter is decreased (from 13 mV down to 0), the M1 contribution becomes larger, thereby reducing the proportion of M3 current needed for 3FP-dynamics. Notice that the proportion of Nav1.6 channels may decrease significantly when ΔV1/2 < 4 mV. Importantly, this is also about where the transition from M1 to M3 occurs (ΔV1/2 = LP1 ≈ -2.94 mV.
The relative resting threshold κ(P) = ITHR(P, ΔV1/2)/ITHR(1,ΔV1/2) of the mixture model is a function of the P parameter (TSTIM = 0.1 ms). There is an approximately twofold difference between the thresholds of the M3 and M1 single-channel models (Fig 10B, see also Fig 2), but more than 70% of this difference vanishes for P as low as 0.2. This could be understood by considering the 1D resting version of the mixed model—see Eq (3) with the m gates of the two channel subtypes at their saturation values, and all other gates at their resting values. As in section on neuronal excitability, both Ithr and Vthr − Vrest provide measure of excitability. Both Ithr (Fig 10B) and Vthr − Vrest (data not shown) have decreased by more that 70% at P = 0.2.
A generalized model
The model family analyzed in this work stems from the classical Hodgkin-Huxley model (HHM). The latter has been criticized and its very applicability to mammalian neurons has been questioned. Various related HHM optimization directions have been suggested.
Here, we argue that:
- The model family analyzed in this work contains the classical HHM as a special case
- That the validity and applicability of the HHM to mammalian neurons can be achieved simply by picking the ΔV1/2 parameter value in an appropriate range.
With respect to the first claim above, from Fig 1 the knowledgeable reader can rapidly see that the quantitative (and hence the qualitative) picture may also be very unlike that of the classical HHM. Namely the interplay between the asymptotic states y∞(V) for the sodium and potassium channels is far more involved: In contrast to the classical HHM, the h and n gates’ dynamics are uncoupled. I.e. the n gates can no longer be considered in mere linear correspondence to the h gates as is often assumed when working with the HHM and its simplifications such as the Fitzhugh-Nagumo model [45–47].
In the latter n = ≈ 1 − h lead to a model of reduced dimensionality. Such linear correspondence would however be inadequate over the whole range of ΔV1/2 parameter variations. For the model considered here, regressing the n(t) variation during an AP (data not shown) by a polynomial in 1 − h requires at least order 2. I.e. there is nonlinear (parabolic) relationship between the n and h dynamic variables during an AP. Moreover, around the resting state, the K+ channels are fully closed (nr est = 0), which is quite unlike the classical HHM (where nr est ≈ 0.2). Hence, the resting FP is fully determined by the leak parameters (conductance and Nernst potential). This explains the invariance of Vr est with ΔV1/2.
A similar perspective stems also from Fig 4, where the classical HHM corresponds to a narrow range of values for the ΔV1/2 parameter out of the significantly broader variation range considered by this work.
As to the second claim, in the Results section we dedicated paper space and work effort to demonstrate that indeed essential model family properties, in function of the given ΔV1/2 parameter value, covary in similar directions as the HHM optimizations published in the literature (see also the Summary and Conclusions below).
An general discussion point is also that many of the observed neural dynamics phenomena are nontrivial—i.e. cannot be explained singlehandedly by the model equations. They are the complex outcome from the interplay of the NaV and KV channels, their properties and the related multitude of key parameter choices.
Parametric-model relevance in the light of previous work
How does our parametric analysis of the HH model compare to previous work? Where does it bring new insight and where is it similar? A body of previous work examined bifurcation structure in the Hodgkin-Huxley model. One arguably popular parameter has been Ibias. Rinzel and Miller examined the Ibias range for which stable and unstable periodic solutions exist and their findings are qualitatively similar to our own results .
The two most relevant other parametric models, which we believe provide parallels and grounds for contrasts are the codimension-2 analysis in the parametric plane Ibias × EK [49, 50] and the two-dimensional INa,p + IK HH-type model  with its either high- or low-threshold IK (i.e. two EK levels). The latter model is equivalent in many respects to the well-known INa,p + IK Morris and Lecar model .
The particular interest of these two alternative parameter models is in their choice of “mirror” parameters: The potassium’s Nernst potential EK is manipulated toward changing the IK sign-switching point. This may have similar effects to ΔV1/2 variation, which affects the position of the sodium current window relative (closer or farther) to the inversion point of the opposing K+ current. However, the two parameters are also qualitatively different as the ΔV1/2 variation affects the onset (and offset) of sodium current through the dynamics of Na+ channel opening and closing gates.
Both [49, 50] and  contain a detailed analysis of the related nonlinear dynamics structure, as well as introduce the “menagerie” of codimension-1 and codimension-2 bifurcation types. This provides an excellent baseline toward interpreting the results of our work. Finally, other previous work (e.g.) examined the variational effects of nominal gK conductance, relative to nominal gNa. As demonstrated in the Results’ section on mixture models, this may also be qualitatively similar to a ΔV1/2 variation.
The key difference brought about by ΔV1/2 variation is that it affects the interplay of sodium and potassium currents not only directly (like compatible EK or gK variation) but also indirectly through the V-dependent temporal dynamics of the HH gates.
Summary and Conclusions
This work is about bridging theory to practice in an attempt to go all the way from the model mathematics (nonlinear dynamics), through simulations and to the experimental practice. Here we summarize the gist of our key findings.
Summary of ΔV1/2-related dynamic properties
In this work we explore the effects of Na+ ion-channel properties on neuronal dynamics. Starting from two well-studied channel types ([24, 25]) we observed that a key difference between the Nav1.2 and Nav1.6 types is their half-activation-voltage parameter V1/2. Hence, we introduced a generalization parameter. This resulted in a parametric family of models exhibiting a continuous variation of sodium current-influx properties.
The introduction of the ΔV1/2 parameter provides for a comprehensive analysis of neuronal excitability, refractoriness and periodic-stimulation dynamics for a space-clamp model. As seen, this analysis can also be extended further to multiple-compartment models of an entire neuron. In models of higher dimensionality, the spatial distribution of ionic current types and their relative proportions will yield specific spatio-temporal patterns, corresponding to better or worse conditions for AP initiation and propagation. The approach used in this paper can be generalized further to produce these profiles of threshold stimulation-field and stimulation-current values, that lead to successful AP initiation and propagation .
The following key effects were determined and explored:
- Continuous ΔV1/2 variation leads to a bifurcation diagram (Fig 4A) containing a multitude of dynamic regimes. One of the most noteworthy differences is that for ΔV1/2 ∈ [LP2,LP1] 3 FP’s coexist. These may also coexist with stable periodic orbits. This means that, depending on the initial conditions, the system may behave as either purely excitable or as an oscillator.
- Absolute vs relative rheobase current: In the range LP2 < ΔV1/2 < LP1 (the M3 class), the simplified model of Eq (5) can be applied, and −Iion,∞(V) has a local minimum −Irh between Vrest and VTHR,∞ (Fig 1C and 1D). Hence an AP can be evoked by applying as little current as the absolute rheobase current Irh for a sufficient amount of time TSTIM (Fig 2).
- In general, lower ΔV1/2 values translate into lower threshold current ITHR[ΔV1/2](TSTIM) from the resting state (Fig 2).
- The mere FP count corresponds to nontrivial differences in dynamic organization and its complexity. E.g. supernormality of ITHR,S2 as a function of t − tup, esp. for the lowest ΔV1/2 (data not shown).
- A complete picture of refractoriness is given by:
- the application of the S1-S2 simulation protocol (Fig 7, panel A and Fig 2, panel B). Fig 7 presents the two faces of periodic AP firing in the model. Importantly, and as noted in the Results’ subsection, the M1 model sub-class is not only less excitable. Its lowest periodic firing frequency does not get much below 100 Hz. Hence, at stimulation durations of about 10 ms the models of this sub-class will already behave as oscillators. We return to this point in the summary on ‘relative’ refractoriness below.
- the continuation (Fig A in S1 File) of the bifurcation diagram of Fig 4A, with respect to bias current and periodic regimes (see the Supplementary Results section on the codimension-2 bifurcation structure in the ΔV1/2 × Ibias parameter plane). Fig A in S1 File demonstrates transitions between pure excitability and oscillation determined by the Ibias and ΔV1/2 values.
The automatic-firing region of Ibias variation becomes gradually larger as ΔV1/2 is increased from 0 to 13 mV (Fig A in S1 File). However the corresponding inter-spike period shows very little variation (see Panel B in Fig 4).
- Absolute values of ISTIM as a function t − tup and ΔV1/2 (Fig 7, panel A) imply that “1:1” responses of lower-ΔV1/2 models can be maintained for either shorter pacing periods or lower stimulation currents.
However, maintaining ∀ΔV1/2 a similar proportion of ISTIM relative to the resting threshold ITHR,0[ΔV1/2](TSTIM) (see also Fig 2), Fig 7B predicts that the higher-ΔV1/2 models would be the first to behave as oscillators as TSTIM is increased.
- ’Relative’ refractoriness may be the consequence of more complex dynamic organization. Thus low ΔV1/2 means higher excitability and the existence of an absolute rheobase (Figs 1C and 2). However, the higher number of FP’s implies more features and—hence, constraints in phase space (Fig 5).
The the semi-transparent surface in Fig 7A represents the (1D) dependence of resting threshold ISTIM on ΔV1/2 for TSTIM = 100 μs, applying ∀ΔV1/2 a 150% safety factor for robust AP-triggering—i.e. This is intersects into the main 3D surface, which shows the minimum 1:1 pacing ISTIM currents as a function of ΔV1/2 and the pacing interval τ11 (the black-yellow line in Fig 7A).
For ISTIM close to the ITHR value (for a given model), the M1 sub-class is paced at a faster frequency. Such higher ‘relative’ refractoriness is also clearly present in Fig 7B where the longest automatic-firing period is substantially shorter for ΔV1/2 = 13 mV, than it is for ΔV1/2 = 0 mV.
Sometimes this may not be desired (see the Discussion points about a large frequency-encoding range, above).
However, when ISTIM is much above the ITHR value, the trend can be completely reversed, given that the M3 model sub-class is both more excitable and has a very large frequency range (Fig 7B and Fig D.B in S1 File). The higher ‘relative’ refractoriness of the M3 sub-class may be in part due to the h gates’ dynamics, since the position of the resting FP (Vrest) does not vary much with ΔV1/2. Hence, the h gates’ may need a longer-lasting recovery phase of the membrane to regain excitability (Fig 3A and 3B).
- Simplified models such as the ones relying on the concept of “hidden structure” (a generalization of the PTC concept for non-linear oscillators, data not shown) capture the essential parts of the complex dynamics—such as the invariance of the evoked AP and the latency dependence on ISTIM compared to ITHR.
On the need for a methodology to study and predict neuronal dynamics determined by ion-channel distributions
Nonlinear dynamics mechanisms may explain some experimental observations in a predictive way. A methodology to study the effect of ion-channel distributions on neuronal dynamics, excitability and refractoriness holds promise for experimental neuroscience practice (e.g. ).
Hence, such modeling and analysis are worth pursuing. Functional electric stimulation is at the threshold of a new realm of possibilities. The paradigm is rapidly shifting from classical work ([54, 55]), as rapid stimulation (typically through trains of ∼ 300 Hz) to evoke single AP’s in isolated neurons is largely suboptimal.
Recent experimental ICMS work (e.g. ) confirmed the modeling predictions (e.g. [22, 25, 26]) that AP’s are evoked almost exclusively from axonal targets, even if this results antidromic AP propagation. Due to refractoriness, an outcome of ‘ad hoc’ high-frequency stimulation trains may be depression instead of activation. A given electrode(s) geometry and relative position with respect to the targeted excitable tissue will imply a given Nav channel density (as the latter depends on the underlying cell(s) morphology). From our analysis, it may be inferred that particular positions and stimulation protocols may be appropriate for a given desired outcome.
Together with critical aspects such as the long-term compatibility, functional visual prosthetics require effective and reliable (1:1) stimulation of select (precisely targeted) neural sub-populations. Such stimulation not only uses optimally little energy (from an optimal-control point of view) and has minimum undesirable side-effects (e.g. tissue damage or involuntary saccades, ). Selective stimulation with lowest possible currents also means higher possibility for conveying information through visual percepts and specific visual features: e.g. oriented lines conveying shape or colored stimuli versus bright but color-less and feature-less phosphenes (). Finally, a very large impact on the type of possibilities is determined by the size, type and number of the stimulation electrodes that are used, as well as by the precision of their guidance to their neuronal targets. We demonstrated here that with reasonably high ISTIM values the limit periods of 1:1 pacing may be as short as ∼ 5 ms. Hence stimulation train frequency can be as high as 200 Hz. The latter result is consistent with the experimentally observed fact that past such frequencies the subjective visual percept reaches a plateau, beyond which it starts getting weaker. This is suggestive that the neuronal targets of stimulation will respond with activation ratios lower than unity to such ‘overdrive’.
Theory needs to build bridges to practice—from modeling (mathematics, nonlinear dynamics, simulations) to the applications such as functional electric stimulation.
Appendix A: Supplementary Results
The effect of ΔV1/2 on automaticity
Applying the S1/S2 protocol to study refractoriness implicitly assumes that the system does not reach an automatic regime—i.e. produce multiple AP’s during either of the S1 or S2 stimuli. For brief stimulation, the system reverts to the non-stimulated dynamics during re-polarization. However, as TSTIM increases, new attractors associated to long stimulation cannot be ignored—the system may now approach them. Hence, the outcome depends on the duration of the transient trajectories leading to these attractors. The latter are revealed by the bifurcation structure as a function of a constant bias current parameter (Ibias) Such current is known to change the attractors of the system (e.g. , Chapter 4).
Figs B and C in S1 File present examples of the rich bifurcation structure created by a constant stimulation current Ibias for different values of ΔV1/2. As ΔV1/2 decreases, the FP curve changes from a monotonous increasing single-branch (Fig C in S1 File, top panel: ΔV1/2 = 13 mV) to a three-branch organization (e.g ΔV1/2 = 3 and -8 mV). The stability of the hyperpolarized FP is lost through a subcritical Hopf bifurcation (see Table 5). The most depolarized FP also becomes stable by a supercritical Hopf (HB) at a very high value of bias current. High amplitude stable PO’s also exist in an Ibias range specific to each ΔV1/2. These stable cycles are created by a saddle-node of cycles bifurcation (see Table 5), which produces also unstable cycles. The unstable PO’s may connect with the subcritical HB (of the lower branch), as in the case of ΔV1/2 = 13 mV. The interaction of the unstable PO’s with the low-ΔV1/2 (yielding a three-branch FP structure) is interesting. They may end through a homoclinic bifurcation (see Table 5), when hitting the middle FP branch, as in the case of ΔV1/2 = 3 mV. In the latter case, the unstable cycles created at the HB1 also disappear through a similar homoclinic bifurcation, but at a different (higher) bias current. Hence, different situations are possible depending on the values of ΔV1/2 and Ibias: a unique stable FP; bi-stability between a stable FP and a stable PO, which in turn are separated by an unstable cycle and/or two unstable fixed points; a unique stable cycle with one or three unstable fixed points (see the zoomed-inset in Fig C in S1 File, the respective phase-plane trajectories, corresponding to the “menagerie” of possibilities, are shown in Figs H,I and J in S1 File).
How long does the transient take to reach a stable cycle from the resting state? Some clue is provided by the cycle period. As seen in Fig D.B in S1 File, the period is a monotonously decreasing function of Ibias with a minimum well below 5 ms.
The codimension-2 bifurcation structure in the ΔV1/2 × Ibias parameter space
Ibias stands for the magnitude a constant current injected into the modeled compartment and is known as bias current.
Already from the phase portrait of typical sub- and supra-threshold trajectories (Fig 5) one gets a pretty colorful idea about the dynamic-structure richness of the parametric HH-type model. Consistently to [49, 50], in this analysis we found a very rich a codimension-2 bifurcation structure.
- Cyan trace shows the codimension-2 positions of the two Hopf bifurcations HB1 and HB2 as a function of the model parameters. Note that for the low ΔV1/2 there is an only-apparent “collision” of the two (but see also Fig D.A in S1 File). While for very high ΔV1/2 the two HB’s indeed coalesce, which in codimension-2 is known as Generalized Hopf (see Table 5). This case is further illustrated by Fig F in S1 File.
An important feature here is that, slightly before the two HB’s indeed coalesce, their type changes from subcritical to supercritical, which is accompanied by the disappearance of two extra branches of stable/unstable PO’s (see Fig F in S1 File and SNC5 in Fig B.A in S1 File).
- Dashed-black lines These illustrate the parameter-range for the saddle-nodes LP1 and LP2 through which are created the 2 additional FP branches (saddle in the middle branch and unstable center/focus in the top branch of the B.D.’s, see Fig 4A and Fig C in S1 File). Here, there are also 2 related codimension-2 bifurcation phenomena. First, the two LP’s coalesce in what is known as Cusp (see Table 5). Second, for low-enough ΔV1/2 LP1 coalesces with HB1 through a codimension-2 bifurcation known as Takens-Bogdanov (see Table 5).
- Red and blue lines the ΔV1/2 × Ibias range of stable/unstable PO’s determined by the saddle-nodes for cycles SNC1 and SNC2 (the red locus) and SNC3 and SNC4 (the blue locus, see also Fig B in S1 File)
Notice that outside (below and over) the HB area, there is at least bi-stability of a stable fixed point with stable PO (or multi-stability between the stable FP and two stable PO’s, see also Fig B.A in S1 File and the case of ΔV1/2 = 3 mV in Fig C in S1 File). For very large Ibias currents (beyond the blue trace) only the stable depolarized fixed point remains (apart when the island described next exists).
For very large ΔV1/2 the stable/unstable PO’s between SNC1 and SNC2 detach from the FP’s locus to form an island (see Fig G in S1 File).
S1 File. Supplementary Figures.
Fig A The ΔV1/2 × Ibias parameter plane, bistability and automatic regimes Bifurcation structure in the ΔV1/2 × Ibias parameter plane Cyan trace: the Hopf bifurcations (HB) as a function of the Ibias and ΔV1/2 model parameters (see also Fig D.A). Dashed-black lines: Existence/creation parameter-range of the 2 additional fixed points (saddle-node in the middle branch, and unstable center/focus in the top branch of the B.D.’s, see Fig 4A and 4C) Red and blue lines: the ΔV1/2 × Ibias range of stable/unstable PO’s (see also Fig G) Notice that outside (below and over) the HB area, there is bistability with a stable fixed point, while for very large Ibias currents (beyond the blue trace) only the stable depolarized fixed point remains (see also Fig C). For very large ΔV1/2 the stable/unstable PO’s between SNC1 and SNC2 detach from the FP’s locus to form an island (see also Fig F and G)
Fig B The Bifurcation Glossary (Codimension 1 and possibly 2) terms illustrated for two ΔV1/2 cases black lines: stable FP’s. They lose (or recover) stability via Hopf Bifurcations red lines: unstable FP’s. In A, unstable FP’s appear (or disappear) via Saddle node (SN) bifurcations at LP1 and LP2. blue lines: Minimum and Maximum voltage of stable periodic orbits. These can appear (or disappear) through SNC bifurcations or supercritical HB’s (e.g. the low amplitude stable cycles in B). cyan lines: Minimum and Maximum voltage of unstable periodic orbits. These can appear (or disappear) through SNC bifurcations or subcritical HB (e.g. HB2 in A), or homoclinic bifurcation (e.g. the lower open ends in A).
Fig D Combined BD’s and periods of the high-amplitude cycle—in the ΔV1/2 × Ibias parameter space A: Combined BD’s B: max. cycle-periods as a function of ΔV1/2 × Ibias parameters (B) Note that the periods of the low-amplitude cycles (around SNC3 and SNC4) have not been continued.
Fig E BD’s for a set more of ΔV1/2 values (see the panel legends, by SPO color) (see also Fig C)
Fig F Generalized Hopf in codimension-2: Collision of two Hopf Bifurcations (HB) As the two Hopf’s become closer in parameter space, one unstable PO (UPO) is lost. Moreover, the HB type changes from sub-critical to super-critical (see also Fig G)
Fig G The creation of an island in codimension-2 A little before the 2 HB’s collide, the small-amplitude SPO detaches from the UPO. The latter forms an island together with the large-amplitude SPO. (see also Fig F)
Fig H Very complex dynamic organization for ΔV1/2 = 3 mV and Ibias = 9.5 μA/cm2 A: Temporal evolution of membrane voltage V for the SPO (magenta, green trace on Panel C) and the UPO (black, thick cyan trace on Panel C). B: Gate states dynamics for the SPO (green trace on Panel C). Notice the very limited range of the n gate’s variation. C: The unstable FP invariant directions yield transient trajectories which converge mostly to the SPO. This is due to the UPO (thick cyan trace). The transient trajectories starting at the UPO’s single unstable invariant direction converge respectively to the resting FP and to the SPO (thinner dashed cyan traces). See also Fig H.A
Fig I Examples of dynamic objects (phase-space trajectories) corresponding to bifurcations in codimension-2. Ibias variation for ΔV1/2 = 3 mV results in a complex organization (see the B.D. in Fig H—middle row and Inset). The figure presents the rich diversity of dynamic objects (e.g. heteroclinic, homoclinic, stable and unstable periodic orbits) by simulating them for 3 values of the Ibias parameter: 7.44, 7.49 and 9.068 μA/cm2. Heteroclinic (dashed red), homoclinic (blue), stable (green) and unstable (cyan and black, Panel A only) periodic orbits
Fig J An unstable limit cycle as a phase-space separatrix A: Due to the existence of an UPO (cyan), all trajectories (black and red) initiated on the unstable FP’s (red triangle) invariant directions deviate toward the resting FP (blue circle) and not to the SPO (green). Incidently (and like in the B.D.’s on Fig F.C), the UPO’s own invariant direction yields 2 transient trajectories (data not shown) which converge respectively to the resting FP and to the SPO. B: An UPO is absent here unlike Panel A. Hence, the same initial conditions as in Panel A yield transient trajectories half of which (2 out of 4) converge to the SPO.
We thank Paul Werginz and Cornelia Wenger for their helpful and constructive comments, and for providing extra motivation for this work.
Conceived and designed the experiments: NK AV FR. Performed the experiments: NK AV. Analyzed the data: NK AV. Contributed reagents/materials/analysis tools: NK AV. Wrote the paper: NK AV FR MS.
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