Bioelectric memory: modeling resting potential bistability in amphibian embryos and mammalian cells
Bioelectric gradients among all cells, not just within excitable nerve and muscle, play instructive roles in developmental and regenerative pattern formation. Plasma membrane resting potential gradients regulate cell behaviors by regulating downstream transcriptional and epigenetic events. Unlike neurons, which fire rapidly and typically return to the same polarized state, developmental bioelectric signaling involves many cell types stably maintaining various levels of resting potential during morphogenetic events. It is important to begin to quantitatively model the stability of bioelectric states in cells, to understand computation and pattern maintenance during regeneration and remodeling.
To facilitate the analysis of endogenous bioelectric signaling and the exploitation of voltage-based cellular controls in synthetic bioengineering applications, we sought to understand the conditions under which somatic cells can stably maintain distinct resting potential values (a type of state memory). Using the Channelpedia ion channel database, we generated an array of amphibian oocyte and mammalian membrane models for voltage evolution. These models were analyzed and searched, by simulation, for a simple dynamical property, multistability, which forms a type of voltage memory.
We find that typical mammalian models and amphibian oocyte models exhibit bistability when expressing different ion channel subsets, with either persistent sodium or inward-rectifying potassium, respectively, playing a facilitative role in bistable memory formation. We illustrate this difference using fast sodium channel dynamics for which a comprehensive theory exists, where the same model exhibits bistability under mammalian conditions but not amphibian conditions. In amphibians, potassium channels from the Kv1.x and Kv2.x families tend to disrupt this bistable memory formation. We also identify some common principles under which physiological memory emerges, which suggest specific strategies for implementing memories in bioengineering contexts.
Our results reveal conditions under which cells can stably maintain one of several resting voltage potential values. These models suggest testable predictions for experiments in developmental bioelectricity, and illustrate how cells can be used as versatile physiological memory elements in synthetic biology, and unconventional computation contexts.
The online version of this article (doi:10.1186/s12976-015-0019-9) contains supplementary material, which is available to authorized users.
It is well appreciated that the nervous system implements memory and information processing via electrical communication among its cells. However, bioelectric signaling is not restricted to excitable cells [1, 2]. It has long been known that all kinds of cells both generate and are sensitive to ion currents and electric fields [3–7], using some of the same ion channels and electrical synapses exploited by the CNS, but functioning on a much slower timescale. Recent data have shown that cellular resting potentials control cell behaviors such as proliferation, differentiation, and migration [8–13]. Moreover, spatio-temporal gradients of resting potential (Vmem) are instructive, endogenous regulators of pattern formation in vivo, involved in oogenesis [14, 15], craniofacial patterning [16], left-right asymmetry [17–19], brain development [20], control of innervation [21], eye formation [22, 23], carcinogenesis/metastasis [24–26], regenerative polarity [27], and size control [28]. Manipulation of stable bioelectric states has enabled control of stem cell function [29–31], induction of large-scale regenerative repair [32, 33], and organ-level reprogramming in vivo [23].
Bioelectric gradients control morphogenetic events, regulating cell behavior via changes in downstream gene expression and chromatin state [34, 35]. This occurs via several known transducer mechanisms that convert changes in resting potential to second messenger and ultimately transcriptional responses. The voltage gradients themselves are regulated by two upstream pathways [36]. One is the variable expression of ion channels within cells. However, there is another way for gradients to be established, which does not require pre-existing transcriptional drivers.
Significant spatio-temporal changes in cell voltage distributions can occur without changes in ion channel protein or mRNA levels. This is because ion channels are gated post-translationally: existing channels can open or close due to various physiological signals, even when the transcriptome and proteome had not changed. While this is an unexpected situation in a developmental or cell biology context, it is commonplace in neuroscience, since neural networks conduct spiking dynamics purely based on the physics of ion channel activity. Action potentials do not require regulation of channel expression and networks can conduct complex electrical behavior purely at a physiological level invisible to analysis of protein or mRNA levels. Much as occurs in the brain, non-neural cells can regulate their voltage potentials by post-translational gating of channels. The gating is driven by a range of physiological events, of which perhaps the most fascinating is cell membrane potential itself. Because channels are both gated by, and determine, resting potential, this situation opens the possibility of complex regulatory feedback loops with non-obvious behavior.
Understanding such bioelectric dynamics would facilitate the construction of comprehensive models of developmental patterning [37–39], the improvement of bioelectrical interventions for regenerative medicine applications [40–42], and the design of artificial constructs for synthetic bioengineering applications [36, 43, 44]. Despite a wealth of information on the electrical properties of neurons, developmental bioelectricity is poorly understood at a quantitative level.
A key property of developmental bioelectricity is the ability of key cells to maintain specific levels of resting potential over time, changing them in response to physiological or genetic signals that trigger new phases of patterning [42, 45, 46]. Unlike most neurons, somatic cells can occupy many different stable levels of Vmem [47]; what features enable a given cell to maintain a specific Vmem range over time (stability) and to switch to a different discrete voltage level when suitably perturbed (lability)? Being able to write and re-write voltage states into cells is a key component of memory elements (e.g., flip-flops) in modern information-processing circuits, and underlies a basic mechanism of bioelectric signaling that could be widely exploited by evolution. Thus, a quantitative analysis of this kind of multi-stability is an important first step toward rational design of circuits with desired bioelectric behavior, and a key component to the formulation of quantitative, predictive models of morphogenetic events that include both physiological and transcriptional dynamics. To this end, here we present analyses designed to understand the voltage memory properties exhibited by two common model systems (mammalian cells and frog oocytes).
Membrane potentials evolve in time due to currents that flow across the membrane through populations of ion channels, and these channel populations themselves activate or inactivate at rates that often depend on the membrane potential. Conductance models are systems of ordinary differential equations that allow one to simulate and interpret these interdependent processes. The elementary theory is due essentially to Hodgkin and Huxley [48], and recapitulated in some detail in Additional file 1: Section 1. The standard form for this system is in terms of current:1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-C\frac{dV}{dt}={I}_{ext}+{\displaystyle \sum_i{g}_i\cdot \left(V-{E}_i\right)}$$\end{document}−CdVdt=Iext+∑igi⋅V−Ei
where i indexes the ion channels, C is the membrane capacitance, V the membrane potential, Iext a source of external current (e.g. a voltage clamp), and E the reversal potential of an ion channel. Each channel conductance gi takes the form2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=\overline{g}{m}^a{h}^b$$\end{document}g=g¯mahb
where a and b are natural numbers and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{g}$$\end{document}g¯ is the maximal conductance for that channel. The functions m(V, t) and h(V, t) are themselves governed by the differential equations:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau}_m\frac{dm}{dt}={m}_{\infty }-m$$\end{document}τmdmdt=m∞−m4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau}_h\frac{dh}{dt}={h}_{\infty }-h$$\end{document}τhdhdt=h∞−h
and the functions x∞(V) and τx(V) are experimentally determined (typically by voltage clamping; [49]). The steady-state activity x∞ is typically fit to a sigmoid function ranging the interval [0,1], and the relaxation time τx is usually approximately Gaussian (e.g. [50]). Together, these two functions describe the voltage-dependence of channel kinetics.
A fixed point in voltage is a point at which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dV}{dt}=0$$\end{document}dVdt=0, and such a point is called asymptotically stable (or an attractor) if, given a nearby initial voltage and time derivative thereof, the system always evolves toward that voltage value. When assessing stability by simulation, we will adopt a looser convention, saying a conductance model is bistable, or has two memories, if for all chosen initial voltages in the physiological range, the dynamics evolve toward one of two final voltages. Mono- and multistability are defined similarly.
Some channel models are tractable to analytic methods that can guarantee multistability. To wit, the membrane timescale is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau}_{membrane}=\frac{{\overline{g}}_{leak}}{C}$$\end{document}τmembrane=g¯leakC, corresponding to the relaxation time of a simple RC circuit with the leak channel as resistor and membrane as capacitor; so-called fast channel variables with τx < < τmembrane are amenable to a reduction of variables x(V, t) ≈ x∞(V) [50]. When all channel variables are fast, the system can be expressed one-dimensionally as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dV}{dt}=f(V)$$\end{document}dVdt=fV, and the dynamics are qualitatively captured by the phase portrait method illustrated in Fig. 2. One may verify by visual inspection that such systems have memories: those roots of f(V) where the slope \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d}{dV}f(V)$$\end{document}ddVfV is negative.
The Channelpedia database [51] contains a large number of models of voltage-gated ion channels derived from measurements in a variety of cell types and animals. Such a model typically consists of (at least) the constants a, b and the functions τx, x∞ as well as the temperature T at which the experiments to determine these were performed. We retrieved 45 of these ion channel models in ChannelML format; 17 of these channels were suitable for our purposes [52–65]; see also Additional file 1: Section 2.3). It was necessary to make several corrections to database temperature values (Additional file 1: Section 2.1). We imported the channel data using the myokit toolbox [66], which also serves as a Python interface to simulations using the CVODE library [67].
The ion channel models were recalibrated to one of two sets of mock experimental conditions, the first of these matching an amphibian oocyte preparation at 23°C and the second matching a standard mammalian preparation at 36°C. Ion concentrations and reversal potentials are given in Tables 1 and 2, respectively. We rescaled τx for each channel to reflect the model temperature using the standard temperature coefficient Q10 = 3 (see Additional file 1: Section 2.2).Table 1Chemical gradients and reversal potentials for amphibian oocyte modelsIonIntracellular (mM)Extracellular (mM)Reversal potential (mV)Na+ 2110−19K+ 900.2−156Cl− 6010.445Ca++ 0.50.2−12Na + K (for HCN channels)−78Table 2Chemical gradients and reversal potentials for mammalian cell modelsIonIntracellular (mM)Extracellular (mM)Reversal potential (mV)Na+ 1514560K+ 1455−89Cl− 10125−67Na+ + K+ (for HCN channels)−15
We let the membrane capacitance C = 1 μF/cm2 and the leak conductance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{g}}_{leak}=200\mu S/c{m}^2$$\end{document}g¯leak=200μS/cm2, corresponding to a linear membrane time constant τmembrane = 5 ms. Timescale ranges were computed for the transition region of all channel variables in the following way: the voltages at which x∞ = 0.05 and x∞ = 0.95 were computed, and τx was then evaluated at these voltages to obtain an effective timescale range for membrane variables. We classified as fast all variables whose effective timescale range fell below the membrane time constant (Table 3) non-leak channels had maximum conductance set to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{g}}_i=2mS/c{m}^2$$\end{document}g¯i=2mS/cm2: ten times the leak conductance.Table 3Simulated ion channelsChannelRemarksReferenceCav2.1Fast, Persistent[57]Cav2.2[55]Cav2.3[57]Cav3.3[63]HCN1Persistent[58]HCN2Persistent[58]HCN3Persistent[58]HCN4Persistent[58]Kir2.1Fast activation[56]Kv1.1[52]Kv1.2[61]Kv1.4[62]Kv1.6[54]Kv2.1[64]Kv2.2[59]Nav1.3Fast activation[53]Nav1.6Fast, Persistent[60]See text for definition of fast variables and persistent channels
We generated models in the form of Equations (1) through (4) for a number of combinations of ion channels. We adopt a summative notation to refer to models and model classes: for instance, one arbitrary channel coexpressed with a leak channel would be X + leak and pairs of arbitrary channels coexpressed with a leak channel and an inward rectifying potassium channel would be X + Y + Kir2.1 + leak. For each channel combination, we first simulated one continuous 30-s period divided into one–second epochs. Each epoch consisted of a 50 ms strongly-clamped (modeled at 1000 mS/cm2) voltage followed by a relaxation period of 950 ms. Voltages were clamped in the interval [−140 mV, 150 mV], descending with epoch, with a 10 mV difference between consecutive epochs. Channel combinations forming finite-state memories were then tabulated based on visual inspection of the simulation results.
We examined two basic model systems: mammalian cells and Xenopus oocytes, which are distinct due to quite different concentrations of ions inside the cells and in their respective culture media. The mammalian case will be most useful for human tissue bioengineering applications, as well as studies of channelopathy-induced birth defects [68–70] and regenerative medicine approaches. The Xenopus model is particularly well-suited for mechanistic investigations into biophysical controls of pattern formation [20, 21, 40, 71, 72], and frog oocytes are also an attractive substrate for unconventional computing or synthetic biology applications.
The channel models considered included four major families: HCN channels as well as voltage-gated sodium, potassium, and calcium channels. While the Xenopus embryo has abundant intracellular calcium, the internal calcium concentration in mammalian cells is typically low enough that depletion effects could introduce additional nonlinearities into the system’s behavior [73, 74]. However, calcium is not itself usually a voltage modifier because calcium channel expression is typically quite low compared to other channels. In fact, the classic squid giant axon model of Hodgkin and Huxley ignores calcium entirely and fits the data quite well. Unless calcium channels are overexpressed, changes in calcium concentration act as one of the many responses to Vmem state, alongside transduction mechanisms that include voltage-directed movement of serotonin and butyrate, activity of voltage-sensitive phosphatases, clustering of certain membrane proteins, etc. We therefore did not consider calcium channels in the mammalian models. HCN channels, while permeable to all cations, have conductances that were reported to vary with the type of transmitted ion [75]. Calcium conductance, in particular, was reported by these authors to be significantly lower than sodium or potassium conductance. We assume here that they conduct no calcium at all, but we note that simulations (not shown) suggested that HCN channels are more conducive to bistability with increased calcium conductance, at least in amphibian models.
We first investigated in some detail the known bistability in the mammalian Nav1.6 + leak model (cf. the Na,p + leak model so-called in [50]), both by simulation and through the phase portrait method. After the fast channel reduction, the model is:5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-C\frac{dV}{dt}=\frac{{\overline{g}}_{Na}}{1+{e}^{-0.1565\cdot \left(V+17\right)}}\cdot \left(V-{E}_{Na}\right)+{\overline{g}}_{leak}\cdot \left(V-{E}_{leak}\right)+{I}_{ext}$$\end{document}−CdVdt=g¯Na1+e−0.1565⋅V+17⋅V−ENa+g¯leak⋅V−Eleak+Iext
where ENa = 60 mV and Eleak = − 67 mV. Transmembrane currents near one equilibrium are visualized directly in Fig. 1b, c and voltage evolution near clamp release is shown in Fig. 1d. The former illustrates one part of the simple underlying mechanism: at the high-voltage memory, the Na channels are largely open and overwhelm the leak channel current so that the voltage approaches the sodium reversal potential. At the low-voltage memory, the Na channels are closed, and the voltage approaches an equilibrium near the leak reversal. This system’s dynamics are further elucidated in the phase portrait in Fig. 2a, where the voltage evolves in the direction of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dV}{dt}$$\end{document}dVdt (black arrowheads). Two memory voltages may be ascertained from this diagram, and simulations from 30 clamped initial conditions (Fig. 2b) confirmed that voltages evolve toward one of these two equilibria. Thus, in mammals, if Nav1.6 and leak channels are the ion channels predominantly expressed in a cell, and if the sodium channels are overexpressed relative to the leak channels, one might expect two stable memory states: one near the sodium reversal and one near the leak reversal.Fig. 1Modeling of voltage stability for ion channel combinations. Our approach for retrieving, calibrating, and simulating is diagrammed in (a).
Currents (suppresing a factor of 2) for an arbitrary ion channel are schematized geometrically in (b); inward or outward currents are represented by wedge areas inside or outside the circle, respectively (c) Currents diagrammed using the method illustrated in b at three voltages above (top), below (bottom) and at (middle) a stable equilibrium for the Nav1.6 + leak model system, in mammals, with a 10:1 ratio of maximal conductances between Nav1.6 and leak channels. Arrows indicate the direction of voltage evolution. Note that in these diagrams, V and E have been shifted by +150 mV to be strictly positive. d Simulations of voltage evolution in the mammalian Nav1.6 + leak model after a 50 ms initial voltage-clamping period, with corresponding currents as indicated in c Fig. 2Phase-space diagrams and simulations demonstrate Nav1.6 + leak is bistable for mammalian cells but not for amphibian oocytes. We compared the phase spaces and membrane voltage dynamics for the Nav1.6 + leak model (with 10:1 maximal conductance ratio) under mammalian and amphibian oocyte conditions. a Phase plane for the timescale-reduced Nav1.6 + leak model for mammals and b corresponding simulations of voltage evolution. c Phase plane of the same ion channels expressed in amphibian oocyte models with corresponding simulation in (d). Black arrows in (a) and (c) indicate the direction of voltage evolution. Red curves indicate dV/dt as a function of V for the combined system, while blue (resp. green) indicates dV/dt supposing the Nav1.6 channel (resp. leak channel) were expressed alone. In the simulations, 30 initial voltages were chosen ranging from −140 to 150 mV and clamped for 50 ms before release. The model is bistable in the mammalian case, but monostable in the amphibian case
We then examined the same model using amphibian oocyte reversal potentials ENa = −19 mV and Eleak = 45 mV. Simulations at a number of initial conditions demonstrated the existence of only one stable fixed point, confirmed by the fact that Equation (1) has only one root with these reversal values (Fig. 2c, d). We conclude here that, unlike typical mammalian cells, amphibian oocytes will not exhibit bistability when Nav1.6 channels are overexpressed relative to leak channels, but rather will always evolve toward one fixed potential slightly above the sodium reversal.
We then searched for other ion channel combinations that would lead to multistable memories under mammalian ionic conditions by simulating all X + leak and X + Y + leak models. In the X + leak scenario, the only channel combination leading to bistability was the Nav1.6 + leak model above. Bistable channel sets in X + Y + leak models are shown in Table 4. Inclusion of Nav1.6 was necessary, but not sufficient, for bistability of all of these models: introducing certain potassium channels (n.b. we considered only those from the Kv1.x and Kv2.x families; see Section 2.3 in the Additional file 1) tended to disrupt bistability. Combinations X + Y + Nav1.6 + leak and X + Y + Z + Nav1.6 + leak did not lead to multistability with more than two memories. Bistability was preserved after introducing more channels in many cases, for instance in Kv1.1 + Nav1.6 + leak (Fig. 3a). Note that the convergence rate is slow in this case owing to the slow dynamics of the Kv1.1 channel. These simulations suggest that bistability may be experimentally realized in mammalian cells simply by increasing the expression of Nav1.6 channels.Table 4Channel sets forming bistable memories in mammalian cell models and corresponding memory lociX + Nav1.6 + leakMemory loci (mV)-−67,48HCN1−60,48HCN2−65,48HCN3−60,48HCN4−60,48Nav1.3−67,48Kv1.1−70,48Kv1.4−71,48Kv2.1−67,48Kv2.2−67,48Kir2.1−83,48Fig. 3Simulations demonstrating bistability in two mammalian models. We searched for voltage bistability in X + Y + leak mammalian models (10:10:1 maximal conductance ratio) by simulation. Two such channel combinations uncovered by this search were the a Kv1.1 + Nav1.6 + leak and b HCN1 + Nav1.6 + leak models. As before, simulations were initialized at 30 voltages ranging from −140 to 150 mV and clamped for 50 ms before release.
In both of these models, voltages initialized in the physiological range converge on one of two locally stable voltages
We repeated the simulations from the previous section under amphibian oocyte conditions (Table 5). Kir2.1 in the amphibian case plays a very similar role to Nav1.6 in the mammalian case. Inward rectifying potassium channels formed memories when combined with a leak channel (Kir2.1 + leak; Fig. 4a). Among the channel pairs (X + Y + leak) we considered, the presence of inward-rectifying potassium channels was necessary, but again insufficient, for bistability. Here, the presence of other potassium channels had a strong tendency to disrupt bistability. Again, attempts to construct higher-order memories using 3-tuples of these channels with Kir2.1 (up to X + Y + Z + Kir2.1 + leak) did not yield any n-state memory with n > 2. Consideration of timescales pointed to a correspondence between fast channels and memory location in the amphibian case. When memory-forming pairs included a fast channel (indicated with an superscript ‘a’ in Table 5), one memory was found near the reversal potential of that channel’s corresponding ion (e.g. the Cav2.1 + Kir2.1 + leak model in Fig. 4b).Table 5Channel sets forming bistable memories in amphibian oocyte models and corresponding memory lociX + Kir + leakMemory Loci (mV)-−117,39Cav2.1a −117,−9Cav2.2−117,39Cav2.3−117,39Cav3.3−117,39HCN1−93,39HCN2−96,39HCN3−94,39HCN4−96,39Kv1.4−117,39Nav1.3−117,39Nav1.6a −117,–16 aIndicates a fast channelFig. 4Simulations demonstrating bistability in three amphibian models. We searched for voltage bistability in X + Y + leak amphibian oocyte models (10:10:1 maximal conductance ratio) by simulation. Simulations were initialized at 30 voltages ranging from −140 to 150 mV and clamped for 50 ms before release. Three such channel combinations were the a Kir2.1 + leak, b Cav2.1 + Kir2.1 + leak, and c Cav2.2 + Kir2.1 + leak models.
Note that Cav2.1 is a fast channel, and that A and C have effectively identical memory loci. In all of these models, voltages initialized in the physiological range converge on one of two locally stable voltages. The basins of attraction for these stable voltages are however, disconnected (in V), as initial hyperpolarization to –140 mV resulted in evolution to the higher-voltage memory