An intracellular calcium frequency code model extended to the Riemann zeta function

Abstract

We have used the Nernst chemical potential treatment to couple the time domains of sodium and calcium ion channel opening and closing rates to the spatial domain of the diffusing waves of the travelling calcium ions inside single cells. The model is plausibly evolvable with respect to the origins of the molecular components and the scaling of the system from simple cells to neurons. The mixed chemical potentials are calculated by summing the concentrations or particle numbers of the two constituent ions which are pure numbers and thus dimensionless. Chemical potentials are true thermodynamic free Gibbs/Fermi energies and the forces acting on chemical flows are calculated from the natural logarithms of the particle numbers or their concentrations. The mixed chemical potential is converted to the time domain of an action potential by assuming that the injection of calcium ions accelerates depolarization in direct proportion to the amplitude of the total charge contribution of the calcium pulse. We assert that the natural logarithm of the real component of the imaginary term of any Riemann zeta zero corresponds to an instantaneous calcium potential. In principle, in a physiologically plausible fashion, the first few thousand Riemann zeta-zeros can be encoded on this chemical scale manifested as regulated step-changes in the amplitudes of naturally occurring calcium current transients. We show that pairs of Zn channels can form Dirac fences which encode the logarithmic spacings and summed amplitudes of any pair of Riemann zeros. Remarkably the beat frequencies of the pairings of the early frequency terms overlap the naturally occurring frequency modes in vertebrate brains. The equation for the time domain in the biological model has a similar form to the Riemann zeta function on the half-plane and mimics analytical continuation on the complex plane.