Alain Destexhe

To evaluate the long-distance magnetic field generated by neurons, we propose a reformulation of the brain’s magneto-quasistatic approximation based on the Jefimenko’s time-dependent generalization of the Biot–Savart law. This differs from the traditional approach relying on Maxwell’s equations and not on their general solution. Instead of a typical length of the medium in the conventional approach, we use the signal traveling distance |r−r′¯|, from the farthest source point r′¯ to the field point r, as the proper length to define the quasistatic dynamics. We consider relatively low frequencies below a typical value fmax=100Hz. The quasistatic approximation is justified since (2πfmax)2|r−r′¯|2ϵμ0≈4×10−7≪1. We take |r−r′¯|≈0.1m, with the gray matter permittivity, previously underestimated, to be ϵ≈107ϵ0. A formalism for the long-distance magnetic field generated by neuronal populations is then developed. Each population is described as a region of small dimensions compared to the average distance from the field point. We split the impressed current density into synaptic and action potential contributions and study their magnetic field. Assuming a small contribution of the impressed currents at the region boundary surface, we obtain two equivalent expressions of the synaptic current dipole moment in terms of the scalar potential and/or its spatial derivatives. Using Maxwell–Wagner’s time, the synaptic current dipole moment of a region is also shown to be related to the electric dipole moment of each part with uniform conductivity and permittivity. Finally, the long-distance magnetic field of action potential currents is expressed in terms of the magnetic dipole moment for these currents.