Bioelectricity: action potential

Principle

As a type of a propagating disturbance of the nerve (or muscle) membrane potential the electrotonic propagation has been described in Bioelectricity: electrotonic propagation. It happens in dendrites, axons and muscle fibers. These structures can be some mm long, but axons and muscle fibers are mostly some cm and axons may reach many meters (as in whales). However, mostly an electrotonic potential is strongly reduced along distances of centimeters, is propagating slow and, due to temporal filtering, the peak time at the nerve terminal will not be very well defined. Therefore it is not very appropriate to transmit information over long distances in living organisms. There exist a better way of propagating nerve information. This is via action potentials (spikes), an active way of propagation. With spikes information transport is faster, with higher temporal resolution and more information can be transmitted. Spikes arise from the somatic potential, the sum of the dendritic potentials, at the axon hillock and then propagate along axons (sometimes certain types of dendrites), muscle fibers and also heart muscle fibers. Fig. 1 visualizes this propagation.

Fig. 1 Principle of propagation.

Fig. 2 depicts the 3 main types. Spies are found in vertebrates and invertebrates, but also some plants (relying on K⁺ and Ca⁺⁺, with the phloem as channels).

Fig. 2 From left to right action potential of axon, muscle fiber and heart muscle cell.

Depolarization and repolarization

Below the axonal spike, the ‘common’ one is described, first for an unmyelinated axon. Fig. 3 gives the various phases which can be distinguished during its time coarse. Often one dendritic potential can give rise to a couple of spikes, depending on its amplitude. A propagating spike generally maintains its waveform and amplitude. This is caused by the fact that the membrane conductance g_m (= 1/r_m) of the axon is not constant. An excellent way to investigate the changes of the conductances is the voltage clamp technique (holding the membrane potential at a constant value whatever injected current is needed, see Electrophysiology: clamping techniques). Essential for this technique is that there flows no axial current through the axon. In a thick axon (squid), this is achieved by inserting a fine silver wire longitudinally in an axon. By an intracellular electrode and an extracellular electrode current is injected into the axon to compensate for changes in current through the membrane. The current needed in the clamp technique compensates the ionic currents (and the initial capacitive current) and is measured as a function of time. According to Ohm's law the total ionic current at any time is proportional with the total membrane conductance since the membrane potential is kept constant. The Na⁺ and K⁺ conductances can be measured separately by applying certain drugs which make either the Na⁺ or the K⁺ conductance zero. The Na⁺ and K⁺ have different types of pores, i.e. selective channels. The permeability for Na⁺ and K⁺ appears to be a function of the membrane potential. In rest, most Na⁺ channels are closed, but the K⁺ channels open, causing a constant leaking out of K⁺. This is way the rest potential is mainly determined by K⁺ with its 75 times higher conductivity. The outflow of K⁺ is constantly compensated by Na⁺ inflow.

If now the membrane is stimulated and either depolarized or hyperpolarized the membrane potential will change with time, resulting in time-and-voltage dependent ionic resistances. Now r_m is composed of a resistance r_Na formed by the Na⁺ channels, a resistance r_K formed by the K⁺ channels and r_L formed by other passive ionic channels, mainly for Cl^─, each of them connected with the Nernst equilibrium potential (see Bioelectricity). The channel mechanisms involved in the generation of a spike are the rapid increase in the permeability for Na⁺ and the delayed and slower increase in the permeability for K⁺ if the cell is depolarized (so-called cathodal stimulation). Na⁺ moves in under the influence of the driving force, the difference between the membrane potential and the equilibrium potential of Na⁺. In the squid axon this occurs in a few ms. As soon as the Na⁺ permeability (or conductance) increases, more Na⁺ streams into the axon or soma, diminishing the membrane potential still further. This causes a further increase of the Na⁺ permeability and at a certain critical value, the firing threshold (about 15 mV more positive than E_rest) this becomes so strong a positive feedback that the cell even reverses its potential to positive values in the direction of the Na⁺ equilibrium potential. Actually, the threshold is reached when the inward Na⁺ current exceeds the outward K⁺ current. Very shortly afterwards the Na⁺ permeability returns to its original value and the K⁺ permeability increases temporarily. As with the Na⁺ influx, it is not the movement of K⁺ that changes E. It is the value for g_K rising above that for g_Na, dragging E back towards the equilibrium constant for K⁺. Since the voltage-gated K⁺ channels have a delayed response, such that K⁺ continues to flow out of the cell even after the membrane has fully repolarized. This causes the undershoot (short hyperpolarization).

There is a common misconception that the Na⁺/K⁺ pump restores the resting potential during the spike falling phase by actively pumping Na⁺ out and K⁺ into the neuron. This (along with the misconception that sodium 'floods' the cell to cause the spike), is not correct. The Na⁺/K⁺ ATPase (the pump) does ultimately maintain the resting potential by maintaining the concentration gradients for Na⁺ and K⁺, but does so on a much slower time scale.

Fig. 3 Basic time coarse of action potential.

Refractory period

During a short period after the occurrence of a spike the cell cannot be stimulated. This is the refractory (1-5 ms) period consisting of an absolute and relative phase. In the former, the Na⁺ channels cannot be opened by a stimulus irrespective of applied voltage. In the subsequent relative phase, spikes can be initiated, since Na⁺ channels are reactivated (in a stochastic manner) but the threshold is greater. This is caused by the slightly hyperpolarized state due to still higher than resting value for g_K, so more voltage is required to reach threshold, and also the threshold itself is higher than usual because some of the Na⁺ channels will still be inactivated. (Note that Na⁺ channel has at least three states: closed, open and inactivated - closed and not able to open). The refractory period is important because it ensures unidirectional (one way) propagation of the spike.

The basic theory of spike propagation, the Hodgkin-Huxley (HH) theory, is described in More Info.

Application

Spikes, mostly in the form of “spike trains”, are used most extensively by the nervous system for communication between neurons and for transmitting information from neurons to other body tissues such as muscles and glands (neurohypophysis).

Spikes are measured with the recording techniques of electrophysiology and more recently with neurochips containing EOSFETs (electrolyte-oxide-semiconductor field effect transistor). Such chips are applied in retinal and cortical implants to record and stimulate neuronal activity. (A cochlear implant is formally not a neurochip since it is only used for stimulation; it is a neuroprosthesis). An oscilloscope showing the membrane potential recording from a single point on an axon shows each stage of the spike as the wave passes. A speaker is very useful to listen to the elicited spike (trains).

Spikes in general cannot be measured at distance, since, due o its dipole nature, it diminishes with the third power of distance. The electrotonic potential changes caused by synaptic transmission which, if strong enough, give rise to the spike, have a less strong decay with distance. They also last longer. If there is enough geometrical coordination between a group of excited neurons, so-called slow or graded potentials can be recorded for instance on the skull of man. They are always sign of mass action. If they are spontaneous we speak of the EEG, if they are excited by light, sound or peripheral nerve stimulation we speak of visual, auditory or somatosensory evoked potentials (EPs) respectively. Also the electroretinography (ERG) reflects graded potentials and not the spikes of the optic nerve.

Some diseases reduce the speed of spike conductance. The most well-known of these diseases is multiple sclerosis, in which the breakdown of myelin impairs coordinated movement.

More Info

The conductances for Na+ and K+ change according to:

g_Na+ = ğ_Na+m³h,

g_K+ = ğ_K+n⁴, (1)

where ğ_Na+ and ğ_K+ are the maximal conductances. The variables n, m, and h have a value between 0 and 1. In the equation for the K⁺ conductance n⁴ denotes the fraction of the K⁺ channels which are open. If all channels are open then n⁴ ≈ n ≈ 1. If all are closed n = 0. Apparently four events of equal nature have to coincide to open the four folded locked K⁺ channel. For the Na⁺ channel two kinds of keys (events) are used. Three identical keys are needed to open the three folded m lock. Another lock (h) is open in rest, but closes when the membrane is depolarized. These fractions m, h and n are voltage and consequently time dependent. They can be found by solving three experimentally found differential equations. For n this equation is:

dn/dt = α_n (1-n) ─ β_nn (2a)

where α_n denotes the open condition and β_n the closed condition. This is visualized in Fig. 4a. After solving, α_n appears to be a positive exponential function of the membrane potential E and β_n is a negative one (Fig. 4b1). Since during excitation E changes with time the two variables also change with time what finally results in the initially progressive increase of n⁴, for a stepwise change of E (Fig. 4c1). Fig. 4c2 gives the final value of n and n⁴ measured during voltage clamp.

For Na⁺ we have to do with a change of m and h. Both can be calculated. The differential equations are:

dm/dt = α_m (1-m) ─ β_mm and (2b)

dh/dt = α_h (1-h) ─ β_hh. (2c)

Fig. 4 The gating model for potassium and sodium. b2) and e2) depict the dependency of n and n⁴, and m,h and hm³, when a voltage step is applied very long (infinite).

The time course of m looks like that of n but is faster. However, h behaves differently. It decreases instead of increases due to the decrease of β_h and increase of α_h when the membrane is depolarized (Fig. 4e2). Therefore, also for long lasting depolarization (voltage clamp) the Na⁺ conductance restores to its originally low rest value within about 5 ms (Fig. 4e2). The opposite behavior of h and m during long lasting depolarization is clearly shown.

The currents which flow through the membrane are composed of the capacitive current i_c and three ionic currents of Na⁺ (through the variable g_Na) of K⁺ (through the variable g_K) and the anion current (mainly Cl^─, through the fixed g_I), together i_i. Fig. 5 gives the time coarse of a spike, together with i_c, i_i, there sum i_m, and also the underlying g_K and g_Na. For the 4 composing currents together the relation is (see equation (2) and (3) of Bioelectricity: electrotonic propagation):

i_m = (1/r_a)∂²E/∂x²= c_m ∂E/∂t + n⁴ğ_K(E─E_K) + m³h ğ_Na(E─E_Na) + ğ_L (E─E_L). (3)

Fig. 5 Time coarse of conductances and currents during an action potential. At time A and B the slopes of the action potential are maximal and i_m zero. At B both reach their extrema.

Suppose that by summation of dendritic and somatic potentials at the axon hillock a spike arises. At that site the local currents become so strong that also the next part of the (axonal) membrane becomes enough depolarized to be excited and a propagated spike without decrement runs along the axon. The refractory period makes that the spike will not reverse and occur only once for a short lasting stimulus. If a nerve is stimulated in the middle of an axon the impulse will propagate to both sides. Longer lasting stimuli may cause trains of spikes. Just like for the propagation of a dendritic potential it can be shown that the conduction velocity Θ is a parameter of the equation:

Θ ^─2·d²E/dt² = c_m dE/dt + n⁴ğ_K(E─E_K) + m³h ğ_Na(E─E_Na) + ğ_L(E─E_L-). (4)

The numerical solution of this non-linear differential equation gives a quite complicated expression of Θ.

Spikes recorded close to the soma (or axon hillock) are biphasic, as the one of Fig. 3, but when recorded in the vicinity of an axon they are triphasic.

One could think that a nerve impulse which reverses the nerve potential would bring about an important depletion of K⁺ which leaves the cell because there is no potential gradient anymore keeping it in the interior, and that the inflow of Na⁺ would cause a permanent disturbance of the membrane potential. However, the amounts of ions displaced are small compared to the actual number present. Even in very small nerves several thousands of spikes can be generated without a significantly increased metabolism to expel Na⁺.

The behavior of the channels has extensively been studied by clamping techniques (see Electrophysiology: clamping techniques), in which the i/E (so conductance) is influenced by administrating all kind of drugs. A discussion of these phenomena is beyond the scope of this chapter.

Myelinated axons

Propagation speed Θ can be increased by increasing the axon diameter. Taking for simplicity equation (8) of Bioelectricity: electrotonic propagation (Θ = 2λ/τ = 0.5C_m^─1(d/(R_m·R_i))^0.5 ) this speed is proportional with the square root of diameter (d).

However, for metabolic reasons, the diameter is limited (only the cold blooded squid reaches a value of 1 mm).

Unmyelinated fibers (about 2 µm) are generally found in the autonomic nervous system of vertebrates where speeds of about 1 m/s are sufficient.

In vertebrates, sensory and motor ones are generally myelinated. This is more effective to increase Θ. The effect of myelin can also be evaluated since all above considerations can be applied in principle to the myelinated nerve. Myelin can be considered as the dielectricum between two condenser plates. It decreases membrane capacitance, since myelin has a lower relative dielectric constant ε_m (about 7-18) than interstitial fluid (ε_m close to ε_m of water, being about 80; C_m ~ ε_m/d). Now, C_m is only about 4 nF/cm². and R_m is about 10⁵ Ωcm². Increasing the effective membrane thickness by using myelin (leaving the inner fiber diameter constant) also decreases C_m, so Θ. When myelin thickness and inner diameter increase with the same factor, then Θ increases linear with this factor as follows from (5). This shows the efficiency of diameter increase. Experimentally this has been found indeed for vertebrate peripheral fibers.

However, a myelinated fiber longer than some 100 μm does not work properly. Myelin allows the rapid (essentially instantaneous) conduction of ions, but prevents the regeneration of spikes. Therefore, the cylindrical shape of the myelin sheath is interrupted every 0.01 – 0.1 mm by a node of Ranvier, a naked piece of ca. 0.5 μm of axon. Their C_m is about 4 F/cm² and R_m is only about 15 Ωcm². An abundance of voltage-gated Na⁺ channels on these bare segments (up to 10⁴ more than their density in unmyelinated axons) allows spikes to be efficiently regenerated at the nodes of Ranvier. The excitation jumps from one node to the other, which is a passive, so electrotonic transmission (see Bioelectricity: electrotonic propagation) implying some decrement. Basically, this can go in either directions, but the spike travels unidirectional because the node behind the propagating spike is refractory. This way of propagation of the spike is called saltatory conduction: at the myelinated segments the propagation is very fast (due to the insulation), whereas at the nodes there is a small delay of 0.01 to 0.1 ms. The length of the internodal segments are such that one, or sometimes even two nodes can be passed and that the amplitude is still sufficient to reach the threshold for restoring the amplitude of the spike. Thus, the safety factor of saltatory conduction is high, allowing transmission to bypass nodes in case of injury.

Mammalian myelinated motor neurons can reach 100m/s. Saltatory conduction increases nerve conduction velocity without having to dramatically increase axon diameter. Without saltatory conduction, conduction velocity would need large increases in axon diameter, resulting in organisms with nervous systems too large for their bodies.

Alternative models

A few observations are not easily reconciled with the model. A signal traveling along a neuron is accompanied by a slight local thickening of the membrane and a force acting outwards.

Also, a spike traveling along a neuron results in a slight increase in temperature followed by a decrease in temperature, whereas electrical charges traveling through a resistor always produce heat.

The recent soliton model explains the above observations and possibly all properties of the HH-model. A soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed; solitons are caused by a cancelation of nonlinear and dispersive effects in the medium. This theory attempts to explain signals in neurons as pressure (or sound) solitons traveling along the membrane, accompanied by electrical field changes resulting from Piezoelectricity.

Literature

HH-model:

Noble, Physiol. Review, 46, 1-50, 1966.

Soliton model:

Heimburg T, Jackson AD. On soliton propagation in biomembranes and nerves. PNAS, 102, 12 2005.

Heimburg T, Jackson AD. The thermodynamics of general anesthesia. Biophysical Journal, 9, February 2007.