On the φ⁴ field theoretical model for the action potential

Abstract

In this short letter we report on the possibility of a field-theoretical model of the action potential in biological membranes. In order to give a qualitative description of the (K ⁺ andNa ⁺ driven) process of propagation of the action potential we introduce two classical scalar fields φ and ψ representingNa ⁺ andK ⁺ ions, respectively. These fields are described by the Lagrangian densities$\mathcal{L}_\phi$ and$\mathcal{L}_\psi$. Moreover, we add the interaction term$\mathcal{L}_{\left( {\phi ,\psi } \right)}$ between the fields. The Lagrangian densities$\mathcal{L}_\phi$ and$\mathcal{L}_\psi$ include a double-well potential that leads to a spontaneous symmetry breaking (SSB) which may produce topologically non-trivial structures (i.e. structures with non-zero topological charge). In order to describe the transversal motion ofK ⁺ andNa ⁺ ions we have to assume non-uniform solutions. Eventually, after deriving the Euler-Lagrange system of equations of motion we perform the Lorentz transformation (boost) on the static solution.