Parallel Minimum p -Norm Solution of the Neuromagnetic Inverse Problem for Realistic Signals Using Exact Hessian-Vector Products

In the neuromagnetic inverse problem, one is interested in determining the current density inside the human brain from measurements of the magnetic field recorded outside the head. From a numerical point of view, the solution of this inverse problem is challenging not only in terms of nonuniqueness and time complexity but also with respect to numerical stability. An efficient and robust computational technique is presented that finds the minimum $p$-norm solution of the neuromagnetic inverse problem. The approach is based on carefully combining a subspace trust-region algorithm for the solution of an unconstrained nonlinear optimization problem, automatic differentiation for the truncation-error free evaluation of first- and second-order derivatives, and shared-memory parallelization using the OpenMP programming paradigm. Using actual measurements obtained from a head phantom model as well as realistic data sets of middle-latency auditory evoked field data, it is demonstrated that a valid reconstruction of neuromagnetic activity is achieved for values of $p$ less than 2.