2.9.4 Speeds in the unified model

A way to estimate the ions’ rush-in speed is that one assumes that the Stokes-Einstein speed describes the ions’ speed (Fig. 3 of [9], quantitatively underpins the hypothesis, see [137]) both in axons and ion channels. If so, the ratio of the potential gradients equals the ratio of the speeds of ions. One can estimate that the $100mV$ AP on the $50\mu m$ AIS generates a $2\ast {10}^4\frac{V}{m}$ electrical gradient and [9] measured $20\frac{m}{s}$ for the speed of the AP. By assuming that the electrical field across the membrane is ${10}^7\frac{V}{m}$, one can estimate the speed at the exit of the ion channels as ${10}^4\frac{m}{s}$.

For thermodynamic distributions, one can interpret the ”temperature of individual particles” with mean kinetic energy $E$ as $T=\frac{2\ast E}{3\ast k_B}$ (where $k_B$ is the Boltzmann constant and $T$ is the thermodynamic temperature of the bulk quantity of the substance); that is, the temperature is directly proportional to the average kinetic energy. The energy arrives at the neuron in the form of ${10}^7$ $N{a}^{+}$ ions, so the average energy a single ion consumes is $2.5\ast {10}^{-14}[J]$. One can assume that most (99%) of that energy is consumed for moving the ion against the Stokes-Einstein force in the viscous medium, so after exiting the membrane, the speed of the ions is $\sqrt{\frac{2\ast 2.5\ast 10-16}{3.82\ast {10}^{-26}}}\approx {10}^5[m/s]$, which is well above the $\approx 500m/s$ average speed. The ions on the two sides of the membrane are in the state with temperature $T$ before and after the AP. When the $N{a}^{+}$ ions rush into the intracellular space, they gain energy through electrostatic acceleration by the membrane’s electrical potential, so the same ions appear on the intracellular side as having more energy, i.e., slightly higher temperature (the temperature is more than two orders of magnitude higher, but the proportion of those ”hot” ions is about six orders of magnitude lower, so one can expect temperature change up to the range of fragments of up to millikelvin, depending on the measuring conditions). In the second phase of the AP, those more energetic ions that provide excess voltage (and so: excess local potential) above the resting potential leave the membrane’s proximity through the AIS, so the temperature decreases in the second phase to its original value; another effect that can contribute to the general heat production and adsorption discussed above. In the original publication on solitons (essentially pressure waves) [70], a wide range of measured and theoretical data is discussed. They are in line with our conclusions, among others, $80\mu K$ measured temperature change. Furthermore, the electrical energy is more than an order of magnitude smaller than the elastic energy (called soliton energy there).