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Transcranial magnetic stimulation (TMS) is a popular method for thenoninvasive stimulation of neurons in the brain. It has become a standardinstrument in experimental brain research and is approved for a range ofdiagnostic and therapeutic applications. Various applications have beenestablished or approved for specific coil designs with their correspondingspatial electric field distributions. However, the specific coil implementationmay no longer be appropriate from the perspective of material and manufacturingopportunities or considering the latest understanding of how to achieve inducedelectric fields in the head most efficiently. Furthermore, in some cases, fieldmeasurements of coils with unknown winding or a user-defined field areavailable and require an actual implementation. Similar applications exist formagnetic resonance imaging coils. This work aims at introducing a formalismthat is completely free from heuristics, iterative optimization, and ad-hoc ormanual steps to form practical stimulation coils with a winding consisting ofindividual turns to either equivalently match an existing coil or produce agiven field. The target coil can reside on practically any sufficiently largeor closed surface adjacent to or around the head. The method derives anequivalent field through vector projection. In contrast to other coil design oroptimization approaches recently presented, the procedure is an explicitforward Hilbert-space vector projection or basis change. For demonstration, wemap a commercial figure-of-eight coil as one of the most widely used devicesand a more intricate coil recently approved clinically for addiction treatment(H4) onto a bent surface close to the head for highest efficiency and lowestfield energy. The resulting projections are within 4% of the target field andreduce the necessary pulse energy by more than 40%.

Formalism for Anatomy-Independent Projection and Optimization of Transcranial Magnetic Stimulation Coils