2.9.3 Mechanical force and pressure wave

The pressure of electrical origin enables one to compare theoretical pressure and force values with the experimental ones. That pressure on a $10\times 10\mu m$ tooltip is converted to a force $2\ast {10}^3\ast 100\ast {10}^{-12}=200pN$ force. The measured force value [116] is about $600pN$, so our estimation is in the correct range. The measured pressure value is [134] $5\frac{dyn}{c{m}^2}=0.5\frac{N}{m^2}$, which is an average value for some period. If we assume a $1Hz$ frequency, and a $1ms$ period for the duration of the AP peak, it means $1\ast {10}^3\left[\frac{N}{m^2}\right]$. So, the measured mechanical change values [135] do not contradict our hypothesis that the increased pressure increases the neuron’s size via its elasticity.

This change is large enough to explain why pressure wave and other mechanical changes [95] also start at the beginning of the AP, and other (such as optical, density) changes are accompanied with it. Nature invests energy also in the conventional way, a term $P\mathrm{\Delta }V$, into the thermodynamics of neural operation; not only in the form of storing energy in the changed electric field. Thermodynamics and electricity must not be separated also from the point of view of the appearance of energy investment.

The consequences of the changes in the electrical charge are large enough to explain why pressure wave and other mechanical changes [95] also start at the beginning of the AP, and other (such as optical, density) changes are accompanied by it. Nature invests energy also in the conventional way, a term $V\mathrm{\Delta }P$, into the thermodynamics of neural operation; not only in the form of storing energy in the changed electrical field. Thermodynamics and electricity, not to mention elasticity, must not be separated when discussing the neuronal energy business.

Given that [134] measured $5\frac{dyn}{c{m}^2}=0.5\frac{N}{m^2}$, we have good reason to presume that this changed pressure increases the neuron’s size and the neuron’s elasticity. The peak of swelling was shown to coincide fairly accurately with the peak of the action potential (i.e. when the concentration reaches its peak value) [136].

Notice an important difference. The acceleration of an ion is unbelievably large. It is sufficiently large to keep the potential at the same value, provided that the ions must follow a small change, such as due to the ”leaking” current through the AIS. Similarly, the ions can ’instantly’ follow quick changes such as a square wave gradient. However, if many similar ions are ahead, their repulsion decreases the acceleration, and the ion travels only at a few $m/s$ speed. The huge forces and accelerations means that a potential change acts immediately. However, the force decays quickly. The ions start to move ’instantly’, but the charge carriers can move only with a limited speed, much below the interaction speed of EM interactions. The effect can propagate only with that lower speed.

See the case of axon: there exist a mechanical constraint that the ions cannot spread through the wall (they must keep the direction), and the ion package propagates as Equs.2.13 and 2.16 describe it.

A more complete discussion of the energatucs of the neuron is given in section

We can also estimate the work done when the rush-in charge extends the neuron, as the $\mathrm{\Delta }P\times \mathrm{\Delta }V$. We consider that $\mathrm{\Delta }P$ remains constant, and we calculate the change of the volume as neuron’s surface ${10}^{-8}{m}^2$ multiplied by the change of the radius $\mathrm{\Delta }r={10}^{-9}[m]$ multiplied by the pressure $\mathrm{\Delta }P$, that gives $E_{mech}=2\ast {10}^{-14}[J]$. If one assumes that the pressure is of entirely mechanical origin [135], one must assume that the pressure is ${10}^6\frac{N}{m^2}$, which is well above the measured value. We can also estimate the electrical energy $E_{electr}=\frac{1}{2}C\ast {(\mathrm{\Delta }V)}^2=1.4\ast {10}^{-12}[J]$, see also [78]. To estimate the time required to change the pressure, one can assume that the collision force of a $N{a}^{+}$ ion provides the pressure (the shock wave), and this can be calculated as $F=m\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$. Given that Eq. (2.78) provides the force, and one must assume that the ion loses its ${10}^{4}to{10}^5\frac{m}{s}$ speed, one concludes that it generates a shock wave in $\mathrm{\Delta }t={10}^{-8}to{10}^{-7}seconds$.

In line with the experimentally measured energy consumption and our calculation in section 2.10.3, the AP cycle consumes energy in the range of ${10}^{-7}[J]$. The five orders of magnitude between the total and the electrical energy seems to underpin that in addition to the ${10}^{-3}$ concentration of ions, only a small fraction of the ions (the ones in the layers near the membrane) in the volume participate in the electric activity of issuing an AP, while the entire volume in the mechanical activity. That is, practically all the energy is invested as elastic energy, but the elasticity modulus of the membrane is exceptionally high. The primary reason for neuronal operation is the appearance of new ions in the vicinity of the membrane, but this creates a secondary, vast mechanical shock wave. By considering the electrical repulsion force between particles in a slow current, which is entirely omitted in the classic theory, we open the way for explaining the observed thermodynamic and mechanical changes. Instead of alternative disciplinary theories, cooperation between the classic disciplines is needed.