Exploring Membrane Potential & Excitability: Hodgkin-Huxley Model & Variations, Lecture notes of Neuroscience

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Theory of action potentials

Romain Brette

March 14, 2017

ii

iv CONTENTS

Chapter 4

Excitability of an

isopotential membrane

In this chapter, we address theoretical questions such as: under what conditions is a membrane excitable? what determines the threshold for spike initiation? We focus on the excitability of an isopotential membrane, such as the membrane of the space-clamped squid giant axon. The situation is different in neurons since action potentials are initiated in a small axonal region next to the cell body (the axonal initial segment), as we will see in chapter ??, but the same concepts will be used and some results also apply.

4.1 Elements of dynamical systems

4.1.1 What is a dynamical system?

The relevant mathematical framework to understand excitation is dynamical systems theory. A dynamical system is defined by a space of possible states s, and a rule that describes how the state of the system evolves in time. We call this evolution a trajectory s(t). Importantly, at any given moment t 0 , the future trajectory s(t), t ≥ t 0 , is entirely determined by the present state s(t 0 )^1.

(^1) In a random dynamical system, the future trajectory is not fully determined by the present state; however, we consider that all the information we have about future trajectories is contained in the knowledge of the present state s(t 0 ).

Figure 4.1: Example of a dynamical system.

4.1. ELEMENTS OF DYNAMICAL SYSTEMS 3

dV dt

= f (V )

where f (V ) is the membrane current Im divided by the membrane capacitance C. In general, f is actually also a function of other state variables, for example the gating variables m, n and h in the Hodgkin-Huxley model. Thus, it does not define a one-dimensional dynamical system. We obtain a one-dimensional system in the simple case of a passive membrane, that is, where permeabilities are fixed. For example, with a linear leak current (as in the Hodgkin-Huxley formalism): dV dt

gL C

(EL − V )

where C is membrane capacitance and gL is leak conductance. Here, dV /dt is a function of V only: if we know V (0), then we can deduce the full trajectory V (t) at any future time. One simple consequence is that all trajectories are monotonous, either increasing or decreasing^6 (or constant). This is a general property of all one-dimensional dynamical systems^7. In particular, action po- tentials cannot be modeled by a one-dimensional dynamical system, because an action potential is not monotonous: V increases then decreases. Thus we need at least two state variables to make an action potential model^8.

Spike initiation can be modeled as a one-dimensional dynamical sys- tem

Although a full action potential cannot be modeled by a one-dimensional sys- tem, the initiation of action potentials can, with a few approximations^9. First, we neglect voltage-gated K+ channels. That is, we assume that they do not open on the time scale of interest (the initial rising phase of the action poten- tial). Empirically, this indeed appears to be approximately the case for the squid axon (figure 4.2) and cortical pyramidal cells (figure 4.3). In this chapter, we will simply assume that they are closed. However, to obtain a one-dimensional description, it is sufficient to assume that K+^ permeability is fixed (but not nec- essarily zero). Theoretically, there is a functional reason why the K+^ current should start only near the peak of the action potential: this situation is energet- ically favorable since the K+^ current opposes the Na+^ current (with a positive K+^ current, more Na+^ flux would be needed for the same net current). We will examine the energetic question in more detail in chapter ??; let us simply mention for now that there can be a significant overlap between the Na+^ and K+^ currents, but mostly in the falling phase of the action potential (see Figs 4. and 4.3, and Carter and Bean (2009)). This first approximation implies that we can model spike initiation with a simplified membrane equation:

dV dt

C

IL + IN a

(^6) Indeed, if a trajectory were increasing then decreasing, then there would be a value of V at which we have both f (V ) > 0 and f (V ) < 0. (^7) defined by differential equations. (^8) Two classical two-dimensional models of action potentials are the Fitzhugh-Nagumo model (Nagumo et al., 1962) and the Morris-Lecar model (Morris and Lecar, 1981). (^9) These approximations are not reasonable for all excitable cells, as we will see when we discuss excitability types in section ??.

4 CHAPTER 4. EXCITABILITY OF AN ISOPOTENTIAL MEMBRANE

gNa gK

Figure 4.2: Overlap of Na+^ and K+^ conductances in squid axon (adapted from Hodgkin and Huxley (1952b)). The axon is clamped at the voltages indicated on the left, relative to the resting potential (with the convention V = −Vm).

where IL and IN a are leak and Na+^ currents, respectively. The second approximation consists in assuming that the Na+^ current is an instantaneous function of V , IN a(V ). We then obtain a one-dimensional dynamical system:

dV dt

= f (V ) ≡

C

IL(V ) + IN a(V )

There are different ways to justify this approximation. One is phenomenolog- ical: we simply look for the best fitting one-dimensional model. For example, in figure 4.4, a cortical neuron is stimulated with a white noise current I(t), strong enough to make the neuron spike. We then look for f (V ) such that

dV dt

≈ f (V ) +

I(t) C

in a least-square sense^10 (Badel et al., 2008; Harrison et al., 2015). We can see on the figure that f (V ) is approximately linear below −45 mV, which presumably corresponds to the leak current IL(V ) (divided by C), and above −40 mV there is a strong membrane current that presumably corresponds to the Na+ current IN a(V ). As we will see in more detail in section 4.3.3, this current is approximately exponential near spike initiation (see inset in figure 4.4). Another way to justify the one-dimensional approximation is to consider the time scales of the different processes. If we assume that 1) inactivation develops only well after the initiation of the action potential (similarly to the K+ current), and 2) the time constant of Na+^ channel activation is short, compared to the time scale of spike initiation, then we can consider that IN a is effectively an instantaneous function of V. In the Hodgkin-Huxley model, this amounts to replacing the gating variable m by its steady-state value m∞(V ), and the inactivation variable h by its initial value h 0. Figure 4.5A shows that, at 23 ◦C,

(^10) That is, f (V ) is the average of dV /dt − I(t)/C, for all the instants when the membrane potential is V.

6 CHAPTER 4. EXCITABILITY OF AN ISOPOTENTIAL MEMBRANE

A B

Time constant (ms)

Figure 4.5: Time constant of activation (A) and inactivation (B) in prefrontal cortical neurons (adapted from Baranauskas and Martina (2006)).

Figure 4.6: A one-dimensional model of an excitable membrane. A, The model is defined by dV /dt = f (V ). Stable equilibria are marked with filled disks, the unstable equilibrium with an open disk. B, Simulated current-clamp experi- ment, where the membrane potential is moved instantaneously to 4 different values.

the maximum activation time constant is about 0.3 ms in prefrontal cortical neurons (Baranauskas and Martina, 2006). At physiological temperature, one would expect this maximum value to be smaller than 0.1 ms. Inactivation is roughly two orders of magnitude slower (Fig. 4.5B). In the spike initiation region (around −50 mV), inactivation is still quite slow compared to the action potential rise (5-10 ms at 23 ◦C) and therefore we do not expect it to be a major determinant of the initial Na+^ current.

We will look at specific one-dimensional models in section 4.3, but for the moment we can consider for example that C.f (V ) corresponds to the “early current” shown in Fig. ??B and ??B, or to the average current shown in Fig. 4.4.

4.1. ELEMENTS OF DYNAMICAL SYSTEMS 7

4.1.3 Equilibria

Let us consider a one-dimensional dynamical system defined by dV /dt = f (V ), which could be the membrane equation of an excitable cell, with f (V ) = I/C. Figure 4.6A shows an example, which is similar to the early current shown in Fig. ??B for the squid axon and in Fig. ??B for Paramecium, but modified to be more readable. The graph of f can be used to predict the evolution of V (t): if dV /dt = f (V ) > 0, then V (t) will increase; if f (V ) < 0, then V (t) will decrease. We will call this graph the excitability curve. This is illustrated in the simulated current-clamp experiment shown in Fig. 4.6B, where we instanta- neously depolarize or hyperpolarize the membrane at different times. Physically, it corresponds to delivering electrical shocks at different times.

An equilibrium or fixed point is a value of the state variable V such that f (V ) = 0, so that V (t) remains constant. There are three equilibria in the model of Fig. 4.6. The lowest one, V −, corresponds to the resting potential. If the system is moved in the neighborhood of V −, as shown on Fig. 4.6B, then V (t) converges back to V −. This is called a stable equilibrium. In this model, V +^ is also a stable equilibrium, and corresponds to the peak of the action potential. On the contrary, V ∗^ is an unstable equilibrium: when the system is perturbed around V ∗, V (t) moves away from V ∗, towards one of the two stable equilibria V −^ and V +. More precisely, an equilibrium is called unstable when it is not stable, i.e., there is at least one direction of perturbation where the system moves away from the equilibrium.

What is the condition for an equilibrium to be stable? This is clear from the excitability curve (graph of f ) shown in Fig. 4.6A. We have f (V ) = dV /dt > 0 just below V −^ and f (V ) = dV /dt < 0 just above V −, so in both cases V (t) evolves towards V −. This situation is obtained when f ′(V ) < 011. Conversely, an equilibrium is unstable when f ′(V ) > 0. In the model shown in Fig. 4.6, the unstable equilibrium V ∗^ splits the state space^12 (all possible values for V ) into the two basins of attraction of the stable equilibria V −^ and V +. The basin of attraction of a stable equilibrium is the set of initial conditions V (0) such that trajectories V (t) converge to that equilibrium. Thus, an unstable equilib- rium defines the notion of a threshold : below V ∗, trajectories converge towards the resting potential V −; above V ∗, trajectories converge towards the action potential peak V +. This definition corresponds to a current-clamp experiment in which an electrical shock is instantaneously applied to the membrane, and then the membrane potential evolves with no stimulus. We will look at other possible definitions of threshold in section 4.2.

(^11) Proof: a Taylor expansion in the neighborhood of V − (^) gives f (V ) ≈ f (V −) + f ′(V −)(V − V −) = f ′(V −)(V − V −). Thus if f ′(V −) < 0 then dV /dt has the sign of V −^ − V. This is a necessary but not sufficient condition, because of the special case f ′(V −) = 0, for example with f (V ) = −V 3. Using a Taylor expansion, one can see that more generally, a stable equilibrium is when there is an odd integer p such that f (k)^ = 0 for all k < p and f (p)^ < 0. However, we will not encounter this situation in this chapter. (^12) In one-dimensional systems, there is always an unstable equilibrium between two stable equilibria, as is graphically intuitive from Fig. 4.6A. Proof: if V −^ is a stable equilibrium, then f (V ) < 0 just above V −; if V +^ > V −^ is a stable equilibrium, then f (V ) > 0 just below V +. Therefore there is a point V ∗^ in between such that f (V ∗) = 0. If we pick the lowest such point, then we must have f (V ) < 0 just below V ∗, therefore V ∗^ cannot be stable.

4.2. THE THRESHOLD 9

occurs, called the bifurcation point^13. The beauty of bifurcation theory is it applies to the change of any parameter. For example, imagine we slowly in- crease the density of Na+^ channels, starting from an inexcitable cell. At the beginning there is a single stable equilibrium, but then at some point a new stable equilibrium appears, together with an unstable equilibrium. This point where the cell becomes excitable is a bifurcation point (again a saddle-node bifurcation). If we keep on increasing channel density, at some point the rest- ing potential will disappear and there will be a single stable equilibrium: the system goes through a second saddle-node bifurcation. Bifurcation is a very general and useful concept: every time we are considering a situation where a graded change in a parameter results in a discrete change in behavior, we are looking at a bifurcation. Bifurcation theory gives us simple mathematical tools to calculate when these changes occur. In the following, we will apply it to stimulus strength, to Na+^ conductance and to properties of the Na+^ channels.

4.2 The threshold

4.2.1 Different ways to excite a membrane

In the previous section, we have seen two ways to excite a membrane. The first way was to deliver an instantaneous electrical shock of charge Q, as in Fig. ??. The effect is to instantaneously shift the potential by Q/C, as shown on Fig. 4.6B^14. If the membrane is initially at rest (V = V −), then this stimulus brings the potential to V −^ + Q/C. Then V (t) will evolve towards one of the two stable equilibria, depending on whether V −^ + Q/C > V ∗. Thus we can define a threshold for voltage, which is the unstable equilibrium V ∗, and a threshold for stimulus strength, in this case a charge threshold, equal to:

Q∗^ = C(V ∗^ − V −)

This is the minimum charge necessary to trigger an action potential. Another way to excite a membrane is to deliver a steady current Ie and increase it slowly, as in section 4.1.4. Again we can define a threshold for stimulus strength, this time a current threshold I∗ e , which can be calculated using the bifurcation equations (see section 4.1.4). This current threshold is called rheobase. We can also define a voltage threshold, as the membrane potential at the bifurcation point. This is the maximum membrane potential that can be reached without eliciting a spike. While this definition also applies to the voltage threshold we have defined for instantaneous shocks, the value of the voltage threshold is different in the two cases. For instantaneous shocks, the voltage threshold is a solution to f (V ) = 0; for steady currents, the voltage threshold is a solution to f ′(V ) = 0. These two equations have different solutions. This can be seen in Fig. 4.7A. A voltage threshold for instantaneous shocks can be defined for a low value of Ie, corresponding to the first curve on the left. As we have seen, this threshold is the unstable equilibrium, the second point

(^13) Note that there are actually two such solutions. (^14) using the physical relation Q = CV that defines capacitance. Alternatively, one can define the injected current as Ie(t) = Qδ(t), where δ(t) is the Dirac function such that δ(t) = 0 when t 6 = 0 and ∫ δ = 1. One can check that indeed ∫ Ie = Q. Then integrating the membrane equation over a small time gives ∆V = Q/C.

10 CHAPTER 4. EXCITABILITY OF AN ISOPOTENTIAL MEMBRANE

A B

Figure 4.8: Spike onset (adapted from Platkiewicz and Brette (2010)). A, Sim- ulated trace of a Hodgkin-Huxley-type model with noisy injected current (rep- resenting synaptic inputs), with spike onset measured by the first derivative method. B, Representation of the trace in (A) in phase space, showing dV /dt vs. V. The first derivative method consists in measuring the membrane po- tential V when the derivative crosses a predefined value (dashed line) shortly before an action potential. The trace is superimposed on the excitability curve dV /dt = (f (V ) + I 0 )/C, which defines the dynamics of the model. I 0 is the mean input current, so that trajectories in phase space fluctuate around this excitability curve.

where the excitability curve crosses the line dV /dt = 0. On the other hand, the voltage threshold for a slowly increasing current Ie corresponds to the point where the excitability curve is tangent to the line dV /dt = 0 (second curve). It appears that this voltage threshold is lower. In other words, in a one-dimensional model of excitable membrane, the voltage threshold for electrical shocks is higher than the voltage threshold for steady currents^15. In the following, we will refer to the voltage threshold for instantaneous shocks as the threshold for fast inputs, the voltage threshold for steady currents as the threshold for slow inputs. Thus, in general the voltage threshold is a concept that depends on the type of stimulation; there is no voltage threshold independent of stimulation (Koch et al., 1995). We will see however that this can be different when we take into account the geometry of the spike initiation system, in particular the fact that spikes are initiated in a small axonal region next to the cell body (chapter ??).

4.2.2 Spike onset

In the experimental literature, the terms spike threshold are often used to refer to a measurement of the voltage at the onset of action potentials. Figure 4.8A shows the simulated membrane potential of model of the Hodgkin-Huxley type (same type of equations, but different models for the ionic currents), where a noisy current representing synaptic inputs is injected. The spike onset is measured using the first derivative method, which consists in measuring the membrane potential V when its derivative dV /dt crosses an empirical criterion (Azouz and Gray, 1999; Kole and Stuart, 2008) (Fig. 4.8B). Other methods have been used to measure spike onset (Sekerli et al., 2004). The second and

(^15) We will see in chapter ?? that this can be different when Na+ (^) channel inactivation is taken into account.

12 CHAPTER 4. EXCITABILITY OF AN ISOPOTENTIAL MEMBRANE

− 10 − 5 0 5 10 dV/dt (V/s)

− 90

− 50

0

50

V (mV)

A

0 100 200 300 Time (ms)

B

Figure 4.10: The sharp model of excitability. A, Excitability curve of the sharp model, with V 1 / 2 = −40 mV. B, Simulated membrane potential trace of an integrate-and-fire model, responding to a step current.

where H is the Heavyside function^16 and V 1 / 2 is called the activation voltage or half-activation voltage (the reason for these terms will be clearer in the next section). Figure 4.9 shows m(V ) vs. V , which we will call the activation curve of the Na+^ channels, for granule cells of the hippocampus (from Schmidt-Hieber and Bischofberger (2010)). To obtain this curve, the peak Na+^ current is mea- sured with an activation protocol in voltage-clamp, and is then divided by the driving force (EN a − V ) to obtain a conductance g(V )^17. The conductance is then divided by the maximum conductance to obtain m(V ) ≡ g(V )/ max g. The measured function m(V ) is clearly not a step function, but if we consider the entire relevant voltage range, from −80 mV to the reversal potential of Na+ (EN a ≈ 50 mV), we may (roughly) approximate m(V ) by a step function with activation voltage Va ≈ −40 mV (for the somatic channels) or −50 mV (for the axonal channels). This is of course a very crude approximation, but it will help us develop some intuition. To be more specific, we will consider a model with linear currents as in the Hodgkin-Huxley model, so that the membrane equation is:

C

dV dt

= gL(EL − V ) + gN aH(V − V 1 / 2 )(EN a − V )

where the first term is the leak current IL and the second term is the Na+ current. The excitability curve is represented on Figure 4.10A. Most of our analysis is directly applicable to other permeability models, in particular GHK theory. In the sharp model, the various definitions of voltage threshold and spike onset match and correspond to the activation voltage V 1 / 2. This corresponds to a simple neuron model known as the integrate-and-fire model. The integrate- and-fire model is a phenomenological neuron model, first introduced by Lapicque

(^16) such that H(x) = 1 if x > 0 and H(x) = 0 otherwise. (^17) Note that it is implicitly assumed that the Na+ (^) current follows a linear current-voltage relation as in the Hodgkin-Huxley model, but this may not be correct.

4.3. THREE SIMPLIFIED MODELS OF EXCITABILITY 13

Figure 4.11: Activation properties of Na+^ channels, showing Boltzmann slope factor k vs. half-activation voltage V 1 / 2 in a number of rat Na+^ channels, collected from various channel subtypes and experimental preparations (adapted from Angelino and Brenner (2007)). Black: neuronal channels; red: muscular channels.

(1907), which is defined by a membrane equation without voltage-dependent ionic channels:

C dV dt

= gL(EL − V ) + Ie

where Ie is an injected current. A spike is triggered when V reaches some threshold value Vt (corresponding to V 1 / 2 in the sharp model), then the mem- brane potential is instantaneously reset to a value Vr and maintained there for a refractory period ∆ (Fig. 4.10B). Since in this chapter we are only interested in spike initiation, we will discuss only the spike initiation component of the integrate-and-fire model, which we call the sharp model of spike initiation.

4.3.2 The Boltzmann model

A more accurate model is the Boltzmann model, where the Na+^ channel acti- vation curve m(V ) is modelled as a Boltzmann function. On Figure 4.9, the curves are fits of Boltzmann functions to patch-clamp measurements of the ac- tivation curve (peak conductance vs. V , relative to maximum conductance). Measurements are indeed generally well fitted by a Boltzmann function, which is defined as:

m(V ) ≡

1 + e(V^1 /^2 −V^ )/k

where V 1 / 2 is the half-activation voltage, such that m(V 1 / 2 ) = 1/2, and k is the Boltzmann slope factor. Figure 4.6A shows the excitability curve of a Boltzmann model (including a leak current). The slope factor k quantifies the voltage range over which the channels open. Specifically, the proportion of open channels goes from about 1/4 at voltage V 1 / 2 − k to about 3/4 at voltage V 1 / 2 + k. The sharp model is obtained is the limit k → 0 mV. Figure 4.11 gives typical values for V 1 / 2 and k. It is important to realize that these channel properties can vary between cells and even within a cell, because there is no such thing as “the Na+^ channel”. First, there are 9 subtypes of voltage-gated Na+^ channels (named Nav1.1 to Nav1.9), each corresponding to one gene coding the main protein, called α- unit. A channel is made of an α-unit and additional proteins called β-units.

4.3. THREE SIMPLIFIED MODELS OF EXCITABILITY 15

m(V)

Figure 4.12: Fitting the Na activation curve to a Boltzmann function (adapted from Platkiewicz and Brette (2010)). A, The Na channel activation curve of a Hodgkin-Huxley type model (black line) was fit to a Boltzmann function on the entire voltage range (dashed blue line) and on the spike initiation range only (−60 mV to −40 mV, red line). The green line shows the exponential fit on the spike initiation range. B, In the hyperpolarized region (zoom of the dashed rectangle in A), the global Boltzmann fit (dashed blue line) is not accurate, while the local Boltzmann fit and the local exponential fit better match the original curve.

a Boltzmann function (m^3 ∞(V )). In addition, the activation curve is measured from the peak current in a voltage-clamp experiment, which means that the measurements may be influenced by the development of inactivation (see for example Fig. 10 in Platkiewicz and Brette (2010)). In this chapter, we will use the Boltzmann model with a linear current- voltage relation, that is:

C

dV dt

= gL(EL − V ) + gN am(V )(EN a − V )

where m(V ) is a Boltzmann function. It is of course possible to consider the Boltzmann model of Na+^ channel activation with the GHK model of currents.

4.3.3 The exponential model

The third simplified model we will consider is the exponential model. It was introduced by Fourcaud-Trocme et al. (2003) as an approximation of models of the Hodgkin-Huxley type that allows analytical calculations. It is based on the Boltzmann model. In the hyperpolarized range, that is, when V << V 1 / 2 , the Boltzmann function can be approximated by an exponential function:

m(V ) ≈ e(V^ −V^1 /^2 )/k

as shown on Fig. 4.12 (green curve). However, as noted in section 4.3.2, Boltz- mann functions are fitted to empirical measurements over a large voltage range, not on the hyperpolarized range. Nevertheless, Figure 4.13 shows that in the squid axon, the peak Na+^ conductance is indeed relatively well fitted by an ex- ponential function of voltage in the low voltage range, here between the resting potential and about 20 mV above it (Hodgkin and Huxley, 1952b). In cortical

16 CHAPTER 4. EXCITABILITY OF AN ISOPOTENTIAL MEMBRANE

464 A. L. HODGKIN AND A. F. HUXLEY

104 o¢<t t° $ @ 2 o to + \

C Do

o ~ Axon Temp. (CC.) D

o 0 1 _- o^17 6
u o^20

A21 85

V 21 8 5 0 E (^0) 01_ E

00001 -120-110-100 -90-80 -70-60-50-40-30-20-10 0 Displacement (^) of membrane potential (mV.) Fig. 9. Maximum sodium conductance reached during a voltage (^) clamp. Ordinate: peak conduc- tance relative to value reached with depolarization of (^100) mV., logarithmic scale. Abscissa: displacement of membrane potential from resting value (^) (depolarization negative).

-= Axon Temp. (° C.) } 0 -1 + 15 11 E 0 18 22 r oA 20 6 21 8- EA, 0@ _

~0-

o.oo1 (^) -120, (^) -110-100I^ ,90-80-70-60-50-40 , , , I^ , (^) -30 -20£, (^) -10 10 + Displacement of membrane potential (mV.) Fig. 10. Maximum potassium conductance reached during a voltage clamp. Ordinate: maximum

conductance relative to value reached with depolarization of 100 mV., logarithmic scale.

Abscissa: displacement of membrane potential from resting value (depolarization negative).

TABLE 2. Peak (^) values of (^) sodium and (^) potassium conductance at (^) a depolarization of 100 mV. Same experiments as Figs. 9 and 10. In^ each case, the value given in this table is represented as (^) unity in (^) Fig. 9 or Fig. 10. Peak conductances at -100 mV. Temp. Sodium Potassium Axon no. (O C.) (m.mho/cm.2) (m.mho/cm.2) 15 11 21 17 6 18 20 18 21 28 20 6 22 23

Mean 20 25

Figure 4.13: Na+^ activation curve of the squid axon shown in logarithmic scale, fitted by an exponential function (dashed line) (Hodgkin and Huxley, 1952b), with a slope factor k ≈ 4 mV. The horizontal axis shows V 0 − Vm, where V 0 is the resting potential.

A B

Figure 4.14: Na+^ activation curve of a pyramidal neuron of the prefrontal cortex (A), with the hyperpolarized region shown in logarithmic scale (B) (adapted from Baranauskas and Martina (2006)). Relative permeability is obtained based on the GHK model.