2.10 Action Potential 2.10.1 The physical process 2.10.3 Thermodynamics of Action Potential

2.10.2 The electrical model

In the electrical view, the process can be modeled as follows: a sudden voltage gradient (see Figure 2.19 middle inset) appears on the membrane (as a discrete capacitor), and a current flows out (see Figure 2.19 bottom inset) through the serially connected ion channel array (as a discrete resistor). The neuron forms a simple serially (not parallelly, as assumed by Hodgkin and Huxley [9] and mistakenly claimed by neurophysiology) connected $R C$ oscillator. For the physical and mathematical description, see section 4 in [11]; for its algorithmic specifics [23], for further details [140]. Notice that the difference between the rush-in and the resultant gradients emphasizes the role of the finite resources: without the AIS, no turn-back of the resultant gradient would occur.

Figure 2.19: The physical processes describing the membrane’s operation. The rush-in

Na^{+}

ions instantly increase the membrane’s charge, and the membrane’s capacity discharges (producing an exponentially decaying voltage derivative). The ions are created at different positions on the membrane, so they take different times to reach the AIS, where the current produces a peak in the voltage derivative. The resulting voltage derivative, the sum of the two derivatives (the AIS current is outward), drives the oscillator. Its integration produces the membrane potential. When the membrane potential crosses the threshold, it switches the synaptic currents on or off.

The fundamental difference between the parallel and serial $R C$ circuits is that the parallel circuit cannot produce output voltage with opposite sign, while the serial one changes the sign of its output for rising and falling edges of the input voltage. As discussed, the capacitive current, which by definition changes its direction and so generates an opposite voltage on the resistor represented by the AIS, perfectly describes the so-called ”hyperpolarization”. It is not the effect of a $K^{+}$ current: the resting ion channels do not have sufficient conductance. Not knowing about the AIS leads to assuming that the resting ion channels work also as transient channels (in other words, assuming a parallel $R C$ circuit instead of the serial one, furthermore, that the output current flows through the membrane instead of the axon). Some $K^{+}$ current certainly exists; for the magnitude of currents during AP, see [34].

The shape of the output waveform depends on the ratio of the pulse width to the $R C$ time constant. When $R C$ is much larger than the pulse width, the output waveform resembles the input signal, even with a square wave input. (In the case of the neuronal oscillator, the shape of the front side of the spike is similar to the one derived for the rush-in current, while the back side is very prolonged.)

The sudden rush-in of $Na^{+}$ ions has (at least) a double effect. One effect is that they press the surface of the membrane that elastically increases its diameter [135]. Due to their mutual repulsion, the ions experience a strong electrical force toward the membrane. It represents an impulse $J=F\,\Delta t$ (N s), that enables estimating the time and energy of the change the rush-in causes and explains that the energy needed to generate an AP is suddenly produced electrically, and stored temporarily as elastic energy (the neuron produces the required energy in the background, when ATP produces ions by hydrolysis under the effect of the electrical field near to membrane); see Appendix 2.5.7. The case is practically identical to the deformation of an elastic plate induced by the hydrostatic pressure of a water column. At the beginning, the aluminum plate is suddenly subjected to hydrostatic pressure, and it reaches equilibrium after initial oscillations. The diagram line of the process is shown in the figure [141], and the method of simulation is discussed in [142]. Compare the diagram line to the gradient in Fig. 2.19, middle inset.