SOLITARY TRANSMISSION OF NEURONAL SIGNALS APPROACHED VIA HE'S SEMI INVERSE METHOD

An investigation of neuronal signal transmission is intended in this paper through the establishment of solitary wave solutions for the improved Heimburg Jackson model governing the propagation of the mechanical wave in biomembranes. The computation of soliton solutions is carried out employing He's semi inverse variational principle. The role of nonlinearity and dispersive effects in the solitonic propation is correlated to the role of compressibility, elasticity, and inertia over the neuronal signal transmission in the unilamellar DPPC vesicles at T = 45o. The study reveals that He's semi inverse method is a direct and effective algebraic method to study the experimental features of the nerve pulse in the biomembranes.

Comment on “On biological signaling” by G. Nimtz and H. Aichmann, Z. Naturforsch. 75a: 507–509, 2020

In 2005, we proposed that the nerve pulse is an electromechanical soliton (T. Heimburg and A. D. Jackson. “On soliton propagation in biomembranes and nerves,” Proc. Natl. Acad. Sci. U.S.A., vol. 102, pp. 9790–9795, 2005). This concept represents a challenge to the well-known electrochemical Hodgkin–Huxley model. The soliton theory was criticized by Nimtz and Aichmann in a recent article in Zeitung für Naturforschung A (G. Nimtz and H. Aichmann. “On biological signaling,” Z. Naturforsch. A, vol. 75, pp. 507–509, 2020). Here, we wish to comment on some statements that we regard as misinterpretations of our views.

On biological signaling

Presently, nerve pulse propagation is understood to take place by electric action pulses. The theoretical description is given by the Hodgkin-Huxley model. Recently, an alternative model was proclaimed, where signaling is carried out by acoustic solitons. The solitons are built by a local phase transition in the lyotropic liquid crystal (LLC) of a biologic membrane. We argue that the crystal structure arranging hydrogen bonds at the membrane surface do not allow such an acoustic soliton model. The bound water is a component of the LLC and the assumed phase transition represents a negative entropy step.

Nerve pulse propagation in biological membranes — Solitons and other invariant solutions

We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry (GCS) method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.

Catastrophe and Hysteresis by the Emerging of Soliton-Like Solutions in a Nerve Model

The nonlinear problem of traveling nerve pulses showing the unexpected process of hysteresis and catastrophe is studied. The analysis was done for the case of one-dimensional nerve pulse propagation. Of particular interest is the distinctive tendency of the pulse nerve model to conserve its behavior in the absence of the stimulus that generated it. The hysteresis and catastrophe appear in certain parametric region determined by the evolution of bubble and pedestal like solitons. By reformulating the governing equations with a standard boundary conditions method, we derive a system of nonlinear algebraic equations for critical points. Our approach provides opportunities to explore the nonlinear features of wave patterns with hysteresis.