Theory of action potentials | Romain Brette, Exams of Physiology

Download Theory of action potentials | Romain Brette and more Exams Physiology in PDF only on Docsity! Theory of action potentials Romain Brette July 19, 2016 ii Chapter 3 Action potential of an isopotential membrane 3.1 Experimental preparations 3.1.1 The space-clamped squid giant axon The first quantitative model of action potentials was a model of the action potential of the space-clamped squid giant axon, conceived and experimentally tested by Hodgkin and Huxley (Hodgkin and Huxley, 1952a). By space-clamped, we mean that the intracellular potential of the axon is maintained spatially uniform over its length. The Hodgkin-Huxley model was the culmination of a series of 5 papers (120 pages) by Hodgkin, Huxley and Katz, published in the Journal of Physiology in 1952 (Hodgkin et al., 1952; Hodgkin and Huxley, 1952c,b,d,a). As already discussed in chapter ??, the squid giant axon had been introduced in the 1930s as a model of choice for electrophysiology because of its very large diameter (up to 1 mm). It is a rather peculiar axon as it is one of the few exceptions to the neuron doctrine, i.e., it is not a neurite of a cell, but it results from the fusion of hundreds of cells, a syncytium (Young, 1936) (see Fig. ??). In addition, the Hodgkin-Huxley model, which we will present in axon current voltage Figure 3.1: Electrophysiological apparatus used by Hodgkin and Huxley to measure current- voltage relationships in the squid giant axon (Hodgkin et al., 1952). The current wire is exposed over 15 mm. 1 2 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE current voltage Figure 3.2: Experimental configuration for electrophysiology in Paramecium. detail in section 3.4, was established from the space-clamped axon, an experimental preparation where the axon’s intracellular medium is made isopotential by inserting a metal wire inside it. Figure 3.1 is a simplified representation of the experimental apparatus: two wires are inserted into the axon, one in which current is passed and another one used to measure intracellular potential. Reference electrodes are placed outside the axon. The current wire is exposed over 15 mm, and this makes the intracellular medium of the axon isopotential. This configuration was critical to measure current-voltage relationships of the membrane: without the space clamp, currents would be measured coming from different portions of the membrane where the membrane potential is uncontrolled. What textbooks usually refer to as the Hodgkin-Huxley model is thus not exactly a neuron model (the squid giant axon does not belong to a neuron anyway), but a model of the space-clamped squid giant axon. Nonetheless, Hodgkin and Huxley also successfully extended their model into a model of the propagating AP of the squid axon (chapter ??), and the biophysical basis of APs has been shown to be very similar in neurons and all other excitable cells. 3.1.2 Paramecium In this chapter, we will focus mainly on the squid giant axon. But in parallel, we will also present the action potential of Paramecium for two reasons. First, Paramecium is an isopotential cell in its natural state. Second, the biophysical basis of its action potential is also based on the opening of ionic channels, but the depolarizing phase of the AP is due to the entry of Ca2+ ions rather than Na+ ions in the squid giant axon. This will illustrate the diversity of action potentials. As mentioned in chapter ??, calcium APs are also seen in a variety of excitable cells, such as developing neurons, invertebrate muscles and sperm. Calcium channels with similar properties as those of Paramecium are also found in vertebrate neurons. In the 1960s and 70s, there was a community of electrophysiologists interested in Paramecium for several reasons. One is that there is a direct relation between electrical activity and behavior: the action potential triggers a change in swimming direction (see section ??). Another one is that electrophysiological recordings are relatively easy to perform because the cell is large: around 200 µm long (remember that the cell body of a neuron is around 30 µm large). Two glass 3.1. EXPERIMENTAL PREPARATIONS 3 d0 Θ A 0 x x0 R e C e ideal real compensated R e B Figure 3.3: Pipette resistance. A, Truncated cone representing the pipette tip. B, Voltage response of a neuron to a current pulse, as measured at the amplifier end (simulated traces), with intracellular potential shown in dashed. Top: the electrode is seen as an ideal resistance; middle: the electrode also has a capacitance; bottom: electrode voltage corrected by the amplifier (bridge balance or series compensation). microelectrodes, with very sharp tips that can pierce the membrane, are inserted into the cell (Fig. 3.2). One is used for passing current, the other for recording voltage (the third electrode outside the cell is the reference). By inserting a second voltage electrode on the other side of the cell, it can be checked that the membrane potential is the same at different locations inside the cell1 (Eckert and Naitoh, 1970). However, Paramecium is not a spatially homogeneous cell since ionic channels responsible for APs in Paramecium are located in the cilia (Ogura and Takahashi, 1976). It might be that cilia are not isopotential. 3.1.3 Electrophysiological techniques We now present a brief overview of the techniques used to measure electrophysiological properties of membranes. More detail can be found in (Molleman, 2002) for the practical aspects, and in (Brette and Destexhe, 2012) for the technical and modeling aspects. 6 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE cell-attached whole-cell outside-out inside-out Figure 3.4: Patch-clamp technique with different configurations. under the pipette. By applying a strong suction, the membrane is ruptured and the pipette is then in contact with the intracellular medium. This is the whole-cell configuration. It is used in the same way as sharp microelectrodes, with the additional advantage that pipette resistance is much lower and there is no leak due to the microeletrode piercing the membrane. A disadvantage, however, is that the electrolyte in the pipette diffuses into the cell and eventually replaces it (since it has a much larger volume). Therefore, the pipette solution has to be carefully prepared and in any case intracellular pathways are highly disrupted. A variation of the technique has been introduced to deal with this issue, the perforated patch, where the pipette contains antibiotics and is brought in cell-attached mode. The antibiotics progressively forms pores into the membrane, giving electrical access (with higher resistance) without making the membrane permeable to large molecules. The initial motivation of patch-clamp was in fact not to record the intracellular potential of a cell but rather to record transmembrane currents through ionic channels. This can be done in two ways. The first way is to bring the pipette in cell-attached configuration then pull it, bringing with it a small patch of membrane. This is the inside-out configuration. The second way is to bring the pipette in whole-cell configuration then pull it, bringing with it a small disrupted patch of membrane that reseals, with the outside now facing the extracellular medium. This is the outside-out configuration. With the patch clamp technique, it is possible to record currents flowing through a single channel (see section ??). In any case, patch-clamp, as all other techniques, is not ideal since the intracellular medium is not preserved (except for perforated patch, which has higher access resistance). It is also not universal, and in particular it cannot be used in an intact Paramecium, because of the cilia and inner membranes. Current clamp and voltage clamp The configuration when we record the voltage response of the membrane to a current, as shown for example on Figure 3.3, is called current clamp. Current clamp is not appropriate to measure the current-voltage relationships of different ionic channels. The first reason is that the mem- brane has a capacitance that produces a current C.dV/dt (C is membrane capacitance) when 3.2. PASSIVE PROPERTIES OF THE ISOPOTENTIAL MEMBRANE 7 the membrane potential varies, and therefore the current flowing through the membrane does not equal the current passing through the electrode. To suppress the capacitive current, the membrane potential must be constant. The second reason is that to measure voltage-dependent changes of permeability, one wants to precisely control voltage. For these two reasons, in the 1940s Marmont and Cole designed the voltage clamp technique (Marmont, 1949; Cole, 1949), where the current necessary to maintain the membrane potential at a given value is measured. The basic principle is a feedback system: current is injected when the potential deviates from the target value. In its simplest form: I = g(Vc − Vm), where Vc is the command potential (target), Vm is the measured membrane potential and g is a large feedback gain. If the gain is large enough, then Vm ≈ Vc and I is recorded. When the membrane potential is clamped, the electrode current exactly matches the membrane current flowing in the opposite direction. Modern amplifiers use additional techniques borrowed from control theory (Astrm and Murray, 2008). As for current clamp, electrode resistance poses an issue for single-electrode voltage clamp. The feedback system ensures that the amplifier end of the electrode is clamped at the command potential Vc. However, if a current I passes through the electrode to maintain that potential, then the actual membrane potential is not Vc but Vc+ReI, where Re is the electrode resistance (more precisely, access resistance as it typically increases when accessing the intracellular medium). Compensation methods are used to deal with this issue, but they do not entirely solve the problem. We can see that the issue is particularly important when measuring large currents, as for example the Na+ currents responsible for AP initiation. For this reason, in the two experimental preparations that we are going to study in this chapter, two electrodes were used for voltage clamp: one to pass current and another one to measure voltage (Figs 3.1 and 3.2). There are a number of variations around current clamp and voltage clamp. One is dynamic clamp, where the injected current is a function of the measured potential, used in general to mimick ionic currents (Bal and Destexhe, 2009). Another one is action potential clamp, where the voltage waveform of an action potential is recorded in current clamp and then used as time-varying command potential in voltage clamp (Carter and Bean, 2009). Optical techniques are also increasingly used to measure the membrane potential and to inject current (Emiliani et al., 2015). However, at the time of writing, these have not reached the level of precision of standard electrophysiological techniques, although they have other advantages, in particular the ability to simultaneous measure membrane potential in the entire cell. Membrane potential can be measured with transmembrane voltage sensors coupled to a fluorescent molecule, which are either injected or genetically expressed. The intensity of the fluorescent signal correlates with membrane potential. The main issues are calibration (mapping optical intensity to membrane potential), noise (high in small structures), response speed and toxicity. Current can be injected by optogenetic techniques, in which a light-sensitive channel is expressed in the membrane. Again calibration and response speed are important issues, because transduced current is not a linear function of light intensity. This chapter will be based mainly on results obtained with standard electrophysiological techniques. 3.2 Passive properties of the isopotential membrane 3.2.1 Resting potential In chapter ??, we have seen that membrane polarization is due to ionic concentration gradients across the membrane combined with membrane permeability to specific ions. In the squid giant axon, Hodgkin and Katz showed that the effect of concentration changes on the resting potential were well predicted by GHK theory (??), assuming the ratios of membrane permeability to K+, 8 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE Intracellular Extracellular ES concentration (mM) concentration (mM) (mV) Squid axon K+ 410 10 -93 Na+ 49 460 56 Cl− 40 540 -65 Paramecium K+ 1 40 -92 Ca2+ 10−4 1 230 Table 3.1: Concentrations of the main ions of the cytosol and extracellular medium in the squid giant axon (Hodgkin, 1951), and estimates in Paramecium in a solution with 1 mM KCl and 1 mM CaCl2 (Machemer, 1998), and corresponding Nernst potentials ES at T = 20◦C (see section ??). Cl− and Na− are around 1:0.45:0.04 (Hodgkin and Katz, 1949). The basis of the resting potential in the squid axon (and other preparations, e.g. the frog muscle) was also confirmed by radioactive tracers, used to measure ionic fluxes at rest (Hodgkin, 1951). The resting potential of the squid giant axon depends on temperature, and is around −60 mV. In Paramecium, the resting potential is of order −30 mV and it also varies with extracellular ionic concentrations, in particular K+ and Ca2+. Table 3.1 gives estimated orders of magnitude for intracellular concentrations of K+ and Ca2+ in Paramecium, in a solution with 1 mM KCl and 1 mM CaCl2 (Machemer, 1998). There is normally very little Ca2+ inside the cell, perhaps of order 10−7 M, as in other cells (including neurons). That concentration rises in the cilia during the action potential. When it exceeds 10−6 M, the cilia beat in the reverse direction (Naitoh and Kaneko, 1972). However, the basis of the resting potential appears more complex in Paramecium than in the squid axon. It varies in complex ways with the concentration of various extracellular cations. Not only does the resting potential vary with extracellular concentrations, but current-voltage relationships appear to shift. This is thought to be due to competitive binding of cations (in- cluding Ca2+) to anionic sites on the outer membrane and the development of a surface potential (Eckert and Brehm, 1979). The phenomenon is sketched on Figure 3.5. Paramecium lives in freshwater, where ionic concentrations are much lower than in the sea or in the extracellular space around neurons. The lipids forming the membrane carry negative charges, which produces an electrostatic potential away from the membrane (Fig. 3.5A). Consequently, the membrane potential is smaller than the potential difference measured away from the membrane. When ex- tracellular ionic concentration is high, cations (for example Ca2+) bind to the anionic sites of the membrane, which act as a “screen” for the surface charges (Fig. 3.5B). The membrane potential is then larger. Because the manipulation of extracellular Ca2+ concentration (and a number of other cations) shifts the current-voltage relationships, it is thought that the membrane is actu- ally mostly permeable to K+ at rest and the effect of extracellular Ca2+ is mostly on surface potential (i.e., a global shift of the membrane potential). Clearly GHK theory is not sufficient to model current-voltage relationships in this case, but in the following, we will assume that the composition of the extracellular medium is fixed, so we will not deal with these complexities. 3.2.2 Membrane capacitance and resistance The bilipid membrane is a thin insulator between two conducting media (the electrolytes). Elec- trically, it behaves as a capacitor, that is, the membrane can store electrical charge by having positively charged particles (in this case, ions) on one side and an equal amount of negatively charged particles one the other side. Charge is proportional to the potential difference between 3.2. PASSIVE PROPERTIES OF THE ISOPOTENTIAL MEMBRANE 11 A B IIc - - - - - - - +++++++ V0 tτ Vm Figure 3.7: Capacitive current. A, The capacitive current Ic = CdVm/dt represents the redistri- bution of charge around the membrane induced by the transmembrane current I. B, Exponential decay of membrane potential after an initial electrical shock, with time constant τ . conductance are sometimes used, as g corresponds to the slope of the current-voltage relationship at V0. The corresponding resistance R = 1/g is often simply called the membrane resistance. This linear current is sometimes called leak current. Note however that in the Hodgkin-Huxley model (section 3.4), the current named “leak current” actually corresponds to the unspecific current that remains when K+ and Na+ have been discounted. With this linearization, we obtain the following linear differential equation: C. dVm dt = g(V0 − Vm) This type of equation where the capacitive current is matched to the transmembrane currents is called the membrane equation. It is more conveniently expressed in the following form: τ dVm dt = V0 − Vm where τ = RC is called the membrane time constant. This equation predicts that Vm converges to V0 (set dVm/dt = 0), and the convergence is exponential: Vm(t) = V0 + (Vm(0)− V0) exp(−t/τ) The membrane time constant τ can be understood in the following way: if a tangent to the curve is drawn from the initial value after the shock, then the tangent intersects the axis Vm = V0 at time τ (Fig. 3.7B). Figure 3.8 shows the response of Paramecium to a hyperpolarizing current step (from Machemer and Ogura (1979)). The linearized membrane equation then reads: C dVm dt = g(V0 − Vm) + I where I is the injected current. This equation can be rewritten as above in a more convenient way: τ dVm dt = V0 − Vm +RI The membrane potential should then converge to V∞ = V0 + RI (set dVm/dt = 0). This convergence is exponential with time constant τ : 12 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE A B Figure 3.8: Passive response of Paramecium (Machemer and Ogura, 1979). A, Voltage response to a hyperpolarizing current step. B, Normalized relative response in log scale in 4 conditions: non-deciliated (A, B), deciliated (C) and reciliated (D). Vm(t) = V∞ + (Vm(0)− V∞) exp(−t/τ) If we choose the equilibrium value V∞ as the reference potential, that is, if we write ∆V (t) = Vm(t)− V∞ (relative hyperpolarization), then we have: ∆V (t) = ∆V (0) exp(−t/τ) This is perhaps the simplest way to express the solution of a first order linear differential equation: the response relative to the equilibrium value decays exponentially. Figure 3.8B shows ∆V/∆Vmax in log scale. According to the formula above, we should see a straight line (oblique dashed line). At time t = τ , we must have ∆V (t)/∆V (0) = 1/e. Therefore, we obtain an estimate of τ from the intersection of the experimental curve with the horizontal line ∆V (t)/∆V (0) = 1/e (dashed). For the first experimental curve (A), we find τ ≈ 35 ms. The membrane time constant is roughly the same when cilia are removed with ethanol (curve (C)). Since the equilibrium value of Vm is determined by the membrane resistance, the response to a current step can be used to estimate both R and τ , or equivalently R and C. For Paramecium, these estimates are C ≈ 700 pF and R ≈ 65 MΩ. When cilia are removed, membrane capacitance is approximately halved, which suggests that the membrane area of cilia is approximately half the total membrane area. 3.3. ACTIVE PROPERTIES 13 40 mV -70 mV 0 mV Na+ permeable K+ permeable Na+ K+ Figure 3.9: Ionic movements and membrane permeability changes underlying the action potential of the squid axon (adapted from (Hodgkin and Huxley, 1939)). 3.3 Active properties 3.3.1 General view When the membrane potential is depolarized above a certain threshold, an action potential is triggered (Fig. 3.7B). The electrophysiological studies of the 1940s and 1950s converged to the following view of the squid axon’s action potential (Figure 3.9). In addition to the passive properties of the membrane responsible for the resting potential, membrane permeability to Na+ and K+ changes with membrane potential. In modern terms, there are two types of voltage- dependent ionic channels, specific respectively to Na+ and K+. Those channels are closed at rest, but more and more of them open as membrane potential is increased, which increases membrane permeability for specific ions. For example, when an excitable cell is depolarized by an an electrode current, the Na+ channels first start to open. Because there is more Na+ outside than inside, Na+ ions enter the cell, which depolarizes the cell even more. Permeability to Na+ increases, and therefore even more Na+ enters: there is a positive feedback, which causes the explosive nature of the AP — we say that the phenomenon is regenerative. The membrane potential then approaches the reversal potential of Na+. Then the Na+ channels inactivate, that is, Na+ permeability decreases, also through a voltage-dependent process, but slower. Finally K+ channels open through a slower voltage-dependent process, letting K+ flow outside the cell, which hyperpolarizes (or “repolarizes”) the membrane, towards the reversal potential of K+. As the membrane potential goes back to its resting value, permeability to the various ions also returns to its resting state. In Paramecium, the mechanism is similar but varies in detail (Fig. 3.10). The permeability of ciliary membrane to Ca2+ increases with membrane potential (and ciliary membrane only (Ogura and Takahashi, 1976)). Because there is much more Ca2+ outside than inside the cell, Ca2+ flow in when the membrane is depolarized. As for the squid axon, this sets up a positive feedback loop, which produces the explosive, regenerative nature of the action potential, but with a different ionic basis. Calcium APs are seen in other excitable cells, for example the barnacle muscle (Hagiwara et al., 1964), of which there is a classical abstract two-variable mathematical model, 16 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE 3.11B). Let us calculate the impedance. We simply replace I(t) and V (t) by the complex expres- sions above and after simplifications we obtain: 2 = (−2iπfC − g)Z(f) Thus if we measure the impedance Z(f), then we can deduce the membrane capacitance and conductance (specifically, g = Re(−2/Z(f))). This method can be applied if the non-stationary case of an action potential. If at a given instant t the instantaneous current-voltage relationship of the membrane is linear (as in the Hodgkin-Huxley model, see section 3.4), then we can simply replace g by g(t) and V0 by V0(t). We then apply the measurement on a short time window where g(t) and V0(t) do not change much and use a high frequency current I(t). We then obtain g(t), the total membrane conductance at that particular moment — we are assuming that the current is small enough so that it does not perturb the cell. Figure ??A (right) shows the time course of the membrane conductance (fast oscillating trace) superimposed on the action potential of the squid axon: the membrane conductance increases dramatically during the action potential. This is in line with Bernstein’s theory, according to which membrane permeability increases during the action potential. Nonlinear current-voltage relationships As we have mentioned several times, the current-voltage relationship of the membrane is often not linear. Nevertheless, the measurement is still meaningful in this case. Let us consider that the membrane current is f(Vm, t), a function of both membrane potential and time. Then the equality of currents entering and leaving the cell means: I(t) = −C d(Vi − V1) dt − f(Vi − V1, t) = C d(Vi − V2) dt + f(Vi − V2, t) If the current I is small, then so is V = V2−V1. Therefore, adding the two expressions gives: 2I(t) ≈ −C dV dt − ∂f ∂V (Vi − V1, t)V The quantity ∂f/∂V is called the slope conductance, it is the local membrane conductance at particular the membrane potential and time. This is the quantity measured by the transverse impedance measurement technique. It quantifies the effect of a small perturbation of membrane potential on membrane current. In the case of the GHK current model, for example, it is proportional to membrane permeability, assuming ionic concentrations are constant. 3.3.3 Ionic basis of the action potential The measurements of Cole and Curtis showed that membrane conductance increases during the action potential, in agreement with Bernstein’s theory. Shortly after these studies, in 1939, Hodgkin and Huxley made the first intracellular recording of the squid axon action potential, which clearly demonstrated that the membrane potential transiently becomes positive (Hodgkin and Huxley, 1939) (Fig. ??C). This observation contradicts Bernstein’s hypothesis that the action potential is due to a non-specific increase in membrane permeability, because this would result in an action potential peaking near 0 mV. A different theory emerged: the rising phase of the AP is due to an increase in Na+ permeability, while the falling phase is due to an increase in K+ permeability. The basic observations that supported this theory, reviewed in (Hodgkin, 1951), are as follows: 3.3. ACTIVE PROPERTIES 17 • Ionic fluxes for specific species can be measured with radioactive tracers. These measure- ments revealed an influx of Na+ and outflux of K+ during electrical activity. Today similar measurements can be done using with fluorescent probes4, for example for Na+, which show increases in intracellular Na+ concentration associated with APs (Kole et al., 2008; Fleidervish et al., 2010; Baranauskas et al., 2013). A quantitative argument can be made: the amount of Na+ entering the cell during an AP should be at least as large as the amount necessary to charge the membrane capacitance towards the peak of the AP5. This amount of charge can be calculated as explained in section ??. Measurements of Na+ flux with radioactive tracers showed indeed that the influx was 2-3 times larger than the minimum value. Measurements of K+ flux showed that the outflux of K+ was similar to the influx of Na+, as expected from electroneutrality. • Manipulation of Na+ concentration: when sodium is removed from the extracellular medium, action potentials disappear. This observation was in fact made very early by Overton (1902) in frog muscle, and then repeated in other preparations including the squid axon. In addition, when extracellular Na+ concentration is decreased, the peak value of the ac- tion potential decreases (Fig. 3.12A). At high concentration, the variation agrees with the Nernst potential of Na+ (solid line). • The duration of the action potential is very short, on the order of 1 ms (see Fig. ??C). If the falling phase were due to only to the membrane permeability returning to its resting state, then we would expect the membrane potential to decay more or less exponentially with the same time constant as in response to small shocks near rest (Fig. 3.7). The fact that the membrane potential returns to rest very quickly indicates that K+ permeability is much higher during the falling phase than at rest. • After the falling phase, the membrane potential goes beyond the resting potential, towards the Nernst potential of K+ (Fig. ??C). Again this indicates strong membrane permeability to K+. Similar experimental observations have been done in vertebrate neurons (see e.g. Fig. 3.12B). In Paramecium, the membrane potential also changes sign at the peak of the action potential (Fig. 3.10), and the peak value varies with extracellular Ca2+ concentration, while extracellular K+ concentration has small impact (Fig. 3.12C,D). On Fig. 3.10, it can also be noted that the falling phase is faster than expected from passive properties (Fig. 3.8), which indicates that K+ permeability increases after the action potential peak. In Paramecium, action potentials are graded: their amplitude depends on the stimulating current. Graded APs are converted to all-or- none APs when calcium is partly replaced by barium (Ba2+) in the extracellular solution (Naitoh and Eckert, 1968), or when EGTA is injected intracellularly. Calcium channels are permeable to barium, which competes with Ca2+, while EGTA binds with calcium and thus lowers intracellular concentration of Ca2+ (Brehm et al., 1978). These and other experimental observations have led to the following conclusions: Ca2+ concentration increases in the cilia during the rising phase of the AP, and this increase inactivates Ca2+ channels (Eckert and Chad, 1984) and activates K+ channels; there are also K+ channels that are activated by depolarization as in the squid axon. Thus the mechanism of repolarization is quite different in the squid axon and in Paramecium, since in Paramecium it is mostly intracellular Ca2+ and not membrane potential that controls it. 4The general principle is that the fluorescent properties of the probe depends on whether it is bound with the target intracellular ion (ion specificity depends on the probe), so that the fluorescent signal can be related to the ion concentration. 5It could be larger since Na+ influx could occur concurrently with K+ outflux. 18 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE A B C D Figure 3.12: Effect of ionic concentrations on action potentials. A, Resting potential (1, top) and peak of action potential (2, bottom) as a function of extracellular Na+ concentration in the squid axon (Hodgkin, 1951). The difference between extracellular and intracellular potential is plotted (i.e., −Vm). B, Same as A in frog myelinated fiber. C, Same as A in Paramecium, except extracellular Ca2+ concentration is varied (Naitoh et al., 1972). The convention for potential is reversed (Vm is plotted). D, Same as C except extracellular K+ concentration is varied. 3.3. ACTIVE PROPERTIES 21 voltage is instantaneously switched from resting potential to a target value. In the hyperpolarized range, the current switches to a different value and stays constant. When the target voltage is 20 mV above resting potential or more, the current increases then decreases and changes sign: first it is inward, consistent with positive ions entering the cell (possibly Na+), then it is outward, consistent with positive ions leaving the cell (possibly K+). At very depolarized voltages, above ENa, the current is outward. On Figure 3.13B, we observe that the early current (here at 0.63 ms) is non-monotonous and reverses around 100 mV. This suggests that the early current is due to Na+ permeability increasing with membrane potential (note that −Vm is on the horizontal axis). The late current (steady-state value) reverses at the resting potential and the voltage- dependence is much steeper at depolarized voltages. This suggests that the late current is due to K+ permeability increasing with membrane potential, but more slowly than Na+ permeability. Similar observations can be made on Paramecium (Fig. 3.14; note the opposite conventions for current and voltage). Separation of ionic currents in the squid axon To understand these currents in terms of changes of Na+ and K+ permeabilities (or Ca2+ and K+ for Paramecium), one needs to isolate the currents carried by each of these ion species. The method used by Hodgkin and Huxley was to measure the membrane current in voltage-clamp with different extracellular Na+ concentrations. To that effect, sodium was partly or entirely replaced by choline in the extracellular medium, to which the membrane is not permeable (Hodgkin and Huxley, 1952c). Entirely replacing sodium by choline makes the axon completely inexcitable with almost no change in the resting potential. Let us consider two extracellular solutions with different Na+ concentration. Assuming the K+ current is not affected Na+ concentration (which is approximately but not exactly true), the currents recorded in voltage-clamp are I = INa + IK and I ′ = I ′Na + IK , that is, only the Na+ current should differ between the two traces. Therefore the difference I − I ′ = INa− I ′Na is only due to the Na+ current. The Na+ current is the time-dependent permeability of the membrane times a factor that depends on membrane potential and ionic concentrations (e.g. in the GHK model, section ??). Since the membrane potential is fixed, it follows that the two currents INa and I ′Na are proportional: INa = kI ′Na. Therefore, I − I ′ = (1 − k)INa. The proportionality constant k can be deduced by assuming that initially I ≈ INa and I ′ ≈ I ′Na. We can then deduce both INa and IK . An example is shown on Figure 3.15. With this separation technique, we observe that when the axon is depolarized (here by 56 mV above rest), the Na current first increases (inward), reaches a peak, then decreases and returns near its initial value (Fig. 3.15B). As pointed out above, this means that the membrane first becomes permeable to Na+, then impermeable again. The first process is called activation and the second inactivation. The K+ current does not show inactivation (Fig. 3.15c): an outward current develops and converges to a steady-state. We also note that the activation of the K+ current is slower than that of the Na+ current. The inactivation of the Na+ current is voltage-dependent. To demonstrate it, Hodgkin and Huxley used more complex voltage-clamp protocols, where the membrane potential is first clamped to a small depolarized value V1 for a variable duration, then clamped to a larger depolarized value V2 (Fig. 3.16A). With V1 = 8 mV and V2 = 44 mV above resting potential, the inward current is only seen in response to the second voltage step, but its amplitude depends on the duration of the first step — the longer the first step, the smaller the inward current in response to the second step. Since there is no measurable current in response to the first step, the inactivation is unlikely due to ionic movements but rather voltage-dependent. The voltage-dependence can be measured by varying the value V1: the relative amplitude of the inward current decreases as V1 22 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE 1 ms Figure 3.15: Separation of membrane currents in the squid axon (Hodgkin and Huxley, 1952c). The membrane potential is stepped from resting potential to 56 mV above it, in two experimental configurations: in sea water (Ii, INa, IK), and in sea water with lowered sodium concentration (10%). (a) Total membrane current (I > 0 means inward current). (b) Sodium current. (c) Potassium current. is increased (Fig. 3.16B; remember potential is given with the inverse convention extracellular minus intracellular potential = −Vm). To interpret these currents in terms of permeability changes, Hodgkin and Huxley define the chord conductance as the current divided by the driving force. For example, for Na+, the chord conductance is INa/(Vm−ENa). This is a definition that can always be made, independently of whether the instantaneous relation between membrane current and driving force is linear — but of course, it is more meaningful in that case, which turns out to be approximately the case in the squid axon in normal physiological conditions. Figure 3.17 shows the dynamics of Na+ and K+ conductances when the axon is depolarized to various membrane potential values. The general time course of the conductances is of course similar as that of the ionic currents, but additionally we note that the maximum conductance increases monotonously with membrane potential, for both Na+ and K+. These experimental observations form the basis of the Hodgkin-Huxley model of the action potential of the squid axon (section 3.4). Separation of ionic currents is now typically done in a different way, using pharmacological blockers. These are molecules that bind selectively to certain ionic channels and block them. For example, tetrodoxin (TTX) is a toxin produced by some fish, which blocks a large class of Na channels when applied extracellularly (Narahashi et al., 1964). There are many other blockers, for different types of ionic channels, which bind either on the extracellular or intracellular side of the membrane. The effect of blocking Na channels is similar to removing Na+ is the extracellular medium, but not exactly equivalent, because the ionic selectivity of channels is not perfect (for example, some calcium can pass through Na+ channels (Baker et al., 1971)). This topic will be discussed further when we introduce ionic channels in chapter ??. Separation of ionic currents in Paramecium In Paramecium, the separation technique used by Hodgkin and Huxley is unfortunately not effective. In addition to the dependence of surface charges on extracellular Ca2+ concentration that we have previously discussed (section 3.1), part of the K+ current is activated not by voltage but by intracellular Ca2+. Besides, the Ca2+ current is inactivated not by voltage but also by intracellular Ca2+. Therefore, when extracellular Ca2+ concentration is manipulated, the Ca2+ current changes in a nonlinear way (because inactivation is modified) and the K+ current is also affected. Therefore, alternative techniques have been used in Paramecium. First, cesium (Cs) and tetraethylammonium (TEA) block a large class of K+ channels (mostly when intracellular in- 3.3. ACTIVE PROPERTIES 23 A B Figure 3.16: Inactivation of the Na+ current in squid axon (Hodgkin and Huxley, 1952d). A, Two-step voltage-clamp protocol: membrane potential is first depolarized by V1 = 8 mV for variable duration, then by 44 mV above resting potential. Note that potentials are given with the convention V = Ve−Vi = −Vm. B, Relative amplitude of the inward current for initial steps of long duration, as a function of V1. 41 55 70 84 99 113 127 Figure 3.17: Ionic conductances of the squid axon (Hodgkin and Huxley, 1952c). Dynamics of Na+ (a) and K+ (b) chord conductances when the membrane potential is displaced from rest to a depolarized value (in mV on the left, relative to resting potential). 26 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE A B V (mV) C Wild Type Pawn Dierence Figure 3.19: Genetic separation of ionic currents in Paramecium. A, Response to current steps in wild-type Paramecium and Pawn mutant (Kung and Eckert, 1972). B, Early current in wild type Paramecium (squares) vs. membrane potential, and late current wild type (triangles) and mutant Paramecium (disks) (Oertel et al., 1977). C, Membrane current measured in voltage- clamp with a 20 mV depolarization in wild type (top) and mutant (middle) Paramecium, and difference between the two current waveforms (bottom) (Oertel et al., 1977). 3.4. THE HODGKIN-HUXLEY MODEL 27 B C A Figure 3.20: Refractoriness of excitable cells. A, Two short current pulses (2 ms) are injected in Paramecium with varying interpulse interval (Naitoh et al., 1972). Membrane potential (bottom) and its derivative (top) are shown. The arrows point a second peak in dVm/dt corresponding to the inward Ca2+ current. B, Current recorded in voltage-clamp for two 44 mV voltage pulses with varying interpulse interval (Hodgkin and Huxley, 1952d); inward current (Na+) is positive (note that there are also capacitive transients). Top trace: response to a single pulse. C, Relative peak current in the second pulse as a function of interpulse interval. and in the squid axon, inactivation actually persists much longer than the action potential. This is shown for the squid axon in Fig. 3.20B, where the current is recorded in voltage-clamp in response to two 44 mV voltage pulses lasting about 2 ms (Hodgkin and Huxley, 1952d). Each voltage pulse triggers an inward current, but the amplitude of the current in response to the second pulse is reduced if the interval between two pulses is shorter than about 20 ms — much longer than the duration of the action potential. Figure 3.20C shows that the inward current is reduced by about 60% after an action potential, and then recovers approximately exponentially with a time constant of about 12 ms. 3.4 The Hodgkin-Huxley model The series of experiments done by Hodgkin and Huxley on the space-clamped squid axon culmi- nated in the design of a quantitative model of the action potential, now called the Hodgkin-Huxley model (Hodgkin and Huxley, 1952a). There is one quantitative model of Paramecium action po- tential that reproduces some of its qualitative features (Hook and Hildebrand, 1979), but it is 28 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE less tightly bound to experimental data. The complete model is presented in section 3.4.4, but as we also want to explain how the model was built, we will introduce its ingredients one by one, together with the corresponding hypotheses and experiments. 3.4.1 The equivalent circuit First, the membrane is considered isopotential, that is, the membrane potential Vm is the same everywhere. This assumption does not hold in any neuron, except possibly in developing neurons that do not have neurites7. As discussed in section 3.1, the assumption holds for the experiments of Hodgkin and Huxley because a metal wire is inserted inside the axon, but this is not the physiological condition (unlike in Paramecium). The Hodgkin-Huxley model is thus a model of the action potential of the space-clamped squid giant axon. Second, the total membrane current is modelled as a sum of a Na+ current INa, a K+ current IK , a non-specific “leak” current IL carried by Cl− and other ions (i.e., what remains when the two other ionic currents have been removed), and a capacitive current C.dVm/dt. At first sight this may seem obvious, but it is a hypothesis that the currents carried by different ionic species are independent. It is a logical assumption to make if the Na+ and K+ currents are thought to be carried by two physically separate types of ionic channels, each being specific for only one ionic species, but of course this was only a hypothesis. The independence of ionic currents was consistent with the experiments where ionic concentrations are manipulated. In an experiment where a current I is passed through an intracellular electrode, conservation of charge then implies: C dVm dt + INa + IK + IL = I This equation is called the membrane equation. Here we have adopted the following convention: for the electrode current, I > 0 means positive current is injected into the cell (inward); for the ionic currents, INa > 0 means current is leaving the cell (outward). This is the modern convention, but Hodgkin and Huxley used the opposite convention. This equation corresponds to an equivalent electrical circuit with different elements in parallel: a capacitor and several sources of current. 3.4.2 The linear model of currents In section 3.3.4, we have seen that Hodgkin and Huxley introduced the conductance gS of the membrane for a given ion species S, defined by the formula IS = gS(Vm − ES) where IS is the current (e.g. Na+ current) and ES is the reversal potential for S. At this point, this is just a definition, called the chord conductance, which can be used whether the current-voltage relationship of the membrane is linear or not. For example, one could define the chord conductance for a membrane with fixed permeability governed by GHK theory, where the current-voltage relation is nonlinear (section ??). In that case, the conductance would be a function of permeability but also of membrane potential, which makes the definition less useful. The basic postulate of the Hodgkin-Huxley model is that chord conductances depend only on permeability and not on instantaneous voltage. As we have seen in the previous section, membrane permeability can change with membrane potential, but not instantaneously. Therefore 7Even in that case, there can be spatial gradients of membrane potential across the cell, i.e., electrical fields within the cell, see for example De Loof (1986) and Levin (2014). 3.4. THE HODGKIN-HUXLEY MODEL 31 BA Figure 3.23: Fitting the K+ conductance of the squid axon (Hodgkin and Huxley, 1952a). A, K+ conductance measured in voltage-clamp for a 25 mV depolarization and back to resting potential. The curves are fits from the Hodgkin-Huxley model. B, Opening (αn) and closing (βn) rates as a function of voltage (convention: V = −Vm). The data are collected for different axons and adjusted for temperature. increase of the conductance, which cannot be obtained with an exponential model. Hodgkin and Huxley then proposed a simple modification, where gK is proportional to a power of a variable obeying a first-order equation, and found out that an exponent of 4 gave a good empirical fit8. The model is then: gK = ḡKn 4 (3.2) τn dn dt = n∞ − n where ḡK is the maximum conductance and n is a variable between 0 and 1, called activation variable; τn is the time constant and n∞ is the steady-state value. The continuous curves in Fig. 3.23A are the fits of this model to the experimental data. This model can be given the following physical interpretation, which will be elaborated in chapter ??. K+ ions can pass through specific ionic channels in the membrane. Each of these channel is made of 4 identical and independent molecules, which can be in two configurations, open and closed, and the channel is open when the 4 molecules are open. A molecule switches stochastically between these two configurations with a transition rate that depends on mem- brane potential, which can be explained by postulating that the molecules are charged. This is summarized by the following kinetic scheme: C αn βn O where C and O represent the closed and open state of a molecule, respectively, αn is the opening rate and βn is the closing rate (in s−1). This notation means that a closed molecule has probability αn.dt of opening in time dt, and an open molecule has probability βn.dt of closing. Then the probability n that the molecule is in the open state follows the following equation: dn dt = αn(1− n)− βnn (3.3) 8Not the best one, however, as a sixth exponent gave a better fit but was considered not worth the extra computing cost, given the machines of the time. 32 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE Indeed, the probability n(t + dt) that the molecule will be in the open state at time t + dt is the probability 1 − n(t) that it was closed at time t times the probability that it opens in time dt (αn.dt), minus the probability n(t) that it was open times the probability that it closes (βn.dt). Opening and closing rates can be given an interpretation in terms of the initial currents measured in the activation and deactivation protocols (Fig. 3.23A). If the channels are initially closed (n = 0) and the membrane potential is clamped to a depolarized value, then initially (small t) n(t) ≈ αnt and therefore gK ≈ ḡK(αnt) 4. The fourth exponent accounts for the “delay” in the current seen in Fig. 3.23A (left), which is scaled by the activation rate αn. If the channels are initially open (n = 1) and the membrane potential is clamped to resting potential, then initially n(t) ≈ 1 − βnt and therefore gK ≈ ḡK(1 − βnt) 4 ≈ ḡK(1 − 4βnt) (Taylor expansion). Thus there is no delay in deactivation, as seen in Fig. 3.23A (right), and the initial rate of decay is proportional to the closing rate βn. This kinetic scheme is equivalent to the first-order linear equation above, where n∞ = αn αn + βn τn = 1 αn + βn The fact that the equations provide a good fit to the experimental data of course does not prove that this physical interpretation is correct. In particular, nothing in the experiments of Hodgkin and Huxley indicated the binary and stochastic nature of channel opening (see chapter ??). In fact, Hodgkin and Huxley did not use term channel as it was not at all obvious that ions passed through channels; another hypothesis was that ions cross the membrane by interacting with carriers, an interpretation also consistent with the model (Armstrong, 2007). The Hodgkin- Huxley model is thus better seen as a phenomenological model of membrane permeability rather than a model of ionic channels. Values of the rates αn and βn can then be obtained by fitting conductance measurements from voltage-clamp experiments as shown in Fig. 3.23A, for different values of the membrane potential Vm. Results over a large potential range are shown on Fig. 3.23B, with data collected on a number of axons. These axons were recorded at different temperatures, and temperature has a strong impact on the currents: essentially, it appears that the effect of temperature change is to compress or expand the currents in the time domain9 (Hodgkin et al., 1952). In the model, this corresponds to scaling the rates αn and βn by a constant factor. Empirically, Hodgkin, Huxley and Katz found that the scaling factor to apply is: Q ∆T 10 10 where ∆T is the temperature change in degrees (Kelvin or Celsius) and Q10 ≈ 3 in this case. With this correction, there is a clear relation between rates and voltage. These relations were fitted by the following formulae: αn(Vm) = 0.01 −Vm + 10 exp((−Vm + 10)/10)− 1 (3.4) βn(Vm) = 0.125 exp(−Vm/80) (3.5) where Vm is in mV and rates are in ms−1. These are empirical formulae, chosen as the simplest mathematical expressions that fit the data; they have no strong biophysical basis. It should 9This is of course a rough approximation. Biophysics suggests that the amplitude of the currents should also depend on temperature (see section ??). 3.4. THE HODGKIN-HUXLEY MODEL 33 BA Figure 3.24: The steady-state K+ conductance of the squid axon. A, Maximum K+ conduc- tance in log scale vs. voltage (V = −Vm) (Hodgkin and Huxley, 1952c). B, Steady-state value n∞(V ), with the continuous curve representing the Hodgkin-Huxley model of the K+ conduc- tance (Hodgkin and Huxley, 1952a). be clear from Figure 3.23B that other choices could have been made, especially for βn. In particular, an alternative approach would be to fit simple mathematical expressions to n∞ and τn, rather than to αn and βn. In electrophysiogical studies, the equilibrium value of the relative conductance (gK/ḡK), which in the Hodgkin-Huxley model corresponds to n4∞(Vm), is typically fitted to a Boltzmann function of Vm: n4∞(Vm) = 1 1 + e(V1/2−Vm)/k where V1/2 is called the half-activation voltage (value at which n∞ = 1/2) and k is called the Boltzmann factor or slope factor. This function has similar shape than the function obtained with equations (3.4) and (3.5) (sigmoidal) but is not identical. In the hyperpolarized range (most channels are closed), this expression predicts: n4∞(Vm) ∝ eVm/k Empirically, Hodgkin and Huxley found indeed an exponential relationship with slope k ≈ 5 mV (Fig. 3.24A). If n∞(Vm) is calculated from equations (3.4) and (3.5), one finds n4∞(Vm) ∝ −V 4 me 4Vm/k ′ with k′ ≈ 9 mV, i.e., k′/4 ≈ 2.25, a different relationship. There is indeed some (small) discrep- ancy between the experimental data for n∞ and the Hodgkin-Huxley model (Fig. 3.23B). The Na+ conductance model The timecourse of the Na+ conductance cannot be explained by a first-order linear differential equation, because it is not monotonous (Fig. 3.17A). Therefore Hodkgin and Huxley proposed to model it with two first-order equations, as follows: 36 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE Constant Value C 1 µF/cm2 ENa 115 mV EK −12 mV EL 10.613 mV gNa 120 mS/cm2 gK 36 mS/cm2 gL 0.3 mS/cm2 Table 3.2: Main constants of the Hodgkin-Huxley model. 3.4.4 The full model Putting equations (3.1)–(3.12) together, we obtain a complete model of the space-clamped squid axon, recapitulated below: C dVm dt = ḡNam 3h(ENa − Vm) + ḡKn 4(EK − Vm) + gL(EL − Vm) + I dm dt = αm(1−m)− βmm dh dt = αh(1− h)− βhh dn dt = αn(1− n)− βnn αm = 0.1 −Vm + 25 exp((−Vm + 25)/10)− 1 βm = 4 exp(−Vm/18) αh = 0.07 exp(−Vm/20) βh = 1 exp((−Vm + 30)/10− 1) αn = 0.01 −Vm + 10 exp((−Vm + 10)/10)− 1 βn = 0.125 exp(−Vm/80) where rates are in ms−1 and Vm is in mV in the expression of rates. The remaining constants are listed in table 3.2. Conductance densities where obtained from the voltage-clamp measure- ments discussed previously. Reversal potentials for Na+ and K+ were obtained from previous experiments and are relative to the resting potential (i.e., Vm = 0 mV means the neuron is at rest). The leak potential EL is chosen so that the resting potential of the model is 0 mV. The full model predicts the membrane current measured in voltage-clamp, as well as the membrane potential in current-clamp. A remarkable achievement of this model is that it makes excellent predictions for both sets of experiments with no further optimization. In addition, the model can be readily extended to a model of action potential propagation (by adding the axial current, see chapter ??), and again with no further optimization, it correctly predicts action potential shape, conduction velocity (to about 20%) and total conductance (as shown in Fig. ??A). Figure 3.26 shows action potentials produced by the model, compared with measured ones. Some discrepancies are noticeable, for example in the repolarization phase, but otherwise the match is excellent. The model also accounts for Na+ and K+ fluxes, subthreshold oscillations and other phenomena. 3.4. THE HODGKIN-HUXLEY MODEL 37 MODEL SQUID Figure 3.26: Simulated and measured action potential of the space-clamped squid axon (Hodgkin and Huxley, 1952a). Top: action potential of the Hodgkin-Huxley model for different initial de- polarizations (numbers in mV). Bottom: measured action potential of the space-clamped squid axon for different initial shocks (numbers in nC/cm2, correponds to expected initial depolariza- tion in mV). 38 CHAPTER 3. ACTION POTENTIAL OF AN ISOPOTENTIAL MEMBRANE 3.4.5 The squid axon after Hodgkin and Huxley The Hodgkin-Huxley model was a great achievement, and it turned out that the same formalism could be applied to virtually all excitable cells. The example of Paramecium shows that important modifications can be necessary, for example inactivation of the inward current can depend on the entry of current rather than on voltage. Although the Hodgkin-Huxley model is the historical model of the squid axon action potential, research on this topic continued after 1952. Not surprisingly, a number of modifications have been introduced since then, which are summarized in (Clay, 2005). Here we shall mention a small number of them. The K+ current When discussing their results, Hodgkin and Huxley pointed out that their model of the K+ current did not fully account for the activation delay seen in voltage-clamp experiments. In the model, this delay is tuned by the exponent 4 in the K+ current equation (n4). The reason for choosing that exponent was mainly technical: it was computationally difficult to use larger exponents. Later work showed that an exponent as large as 25 (n25) better accounts for the experimental data (Cole and Moore, 1960). More fundamentally, the K+ current actually follows the GHK theory, rather than a linear model (Clay, 1991). In GHK theory, the current is the product of permeability and of a rec- tifying function of voltage. Using the GHK equation for the K+ model makes the equilibrium permeability n4∞(Vm) a much steeper function of voltage. K+ ions can also accumulate around the axon during the repolarisation phase, which then reduces the K+ current. The Na+ current In the frog node of Ranvier, it was found that the Na+ current follows GHK theory, not a linear model (Frankenhaeuser, 1960). One may wonder why it would be linear in the squid axon. This seems to be a serendipitous consequence of the interaction of external Ca2+ with Na+ channels; in calcium-free medium, the Na+ current of the squid axon does follow the GHK formula (Vandenberg and Bezanilla, 1991a). Considerable research on Na+ channels has taken place since the 1950s, and many new models have been proposed, including for the squid axon. These models generally take the form of kinetic models of channels with a number of states and voltage-dependent transition rates (see chapter ??). They depart from the Hodgkin-Huxley model in several respects, reviewed in (Patlak, 1991) (see Tables 1 and 3 thereof). One is that activation and inactivation are not independent processes (Bezanilla and Armstrong, 1977; Armstrong and Bezanilla, 1977; Vandenberg and Bezanilla, 1991b,a). Rather, several state transitions toward the open state must occur before the channel can be inactivated. Other currents Other channels than Na+ and K+ channels are known to exist in the squid axon. For example, there are calcium channels that let extracellular calcium ions enter the axon when an action potential is produced (Baker et al., 1971). A different type of Na+ channels, with distinct electrophysiological properties, has also been found, although it accounts for a small proportion of the current (Gilly and Armstrong, 1984). 3.6. SUMMARY AND EPISTEMOLOGICAL NOTES 41 3. The membrane current can be separated in specific ionic currents that do not directly interact with each other. It could have been, for example, that the passage of Na+ and K+ ions through the membrane is mediated by a shared mechanism, so that the two fluxes might not be independent. 4. Ionic currents can be expressed as the product of ion specific permeability with driving force (Vm − E). This was shown with deactivation experiments. As we have previously discussed, this is not the generic case, as currents often follow GHK theory. 5. Permeability depends only on membrane potential. This is not the case in Paramecium, for example, where the inactivation of the Ca2+ current is mediated by intracellular Ca2+. It could also be, for example, that mechanical changes in the membrane have an effect on permeability, through conformational changes in ionic channels (Anishkin et al., 2014). 6. Ion concentrations do not vary in either space or time. In principle, the entry of Na+ and K+ ions should change the concentrations of these ions, which in turn would change the amplitude of the ionic currents. This is neglected in the Hodgkin-Huxley model, on the basis that these changes should be very small in the large squid axon (see section ??). From the assumption of uniformity of the squid axon membrane, it is also assumed that ionic concentrations are spatially constant. If this assumption were not valid, then one should consider axial diffusion currents in addition to the electrical term present in the cable equation. This set of non-trivial assumptions forms the basis of the model; more specifically, of the first equation of the model (the membrane equation). 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