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Predicting neural responses to intra- and extra-cranial electric brain stimulation by means of the reciprocity theorem

Fig 2

Applying the reciprocity theorem to subthreshold ES of a compartmental model of a cortical neuron.

A: Stimulating a canonical cortical layer 5 pyramidal cell (PC) model [63] with a nearby current (orange dot, 50 μm outside the soma), while recording the somatic membrane potential (green dot). B: responding to ES of 10 Hz and 1 μA (black line). The reciprocal situation, i.e., injecting the same current into the soma while measuring changes in at the orange dot, leads to indistinguishable results (blue dashed line). Note that the reciprocity-based solution predicts the membrane potential response, that is, the deviation from the baseline, but not the resting membrane potential itself. C: Same as in B, but for a 1000 Hz, 10 μA current. D: Same as in B, but for a 10 Hz, 10 μA current. This combination of amplitude and stimulation frequency evokes action potentials, and is therefore outside the linear regime. E: The relative error as a function of at the stimulation frequency, for three different models: the layer 5 pyramidal cell from panel A, a mouse cortical layer 5 GABAergic PV interneuron (IN), and a straight, unbranched myelinated axon with periodic nodes of Ranvier (see Methods). We injected different currents (0.1, 0.5, 1, 5, 10, 50, 100 μA) at different frequencies (1, 10, 100, 1000 Hz) and distances (25, 50, 100 μm) from the target compartment (soma for the PC and IN model; closest axon terminal for the axon model). The instances of panels B, C, and D, are marked, respectively, by a large blue square, diamond, and triangle. The relative error is defined as SD( − )/SD(). F: For each cell model 100 simulations were executed with small increments in the ES amplitude, where the range of amplitudes was chosen so that all models spiked for the highest amplitude. The highest subthreshold amplitude response in the target compartment was identified, and the spike threshold was defined as 0.1 mV above this value. The relative error of applying the RT-based approach is plotted versus the distance of the resulting membrane potential response from spiking threshold (only including subthreshold simulations). The ES was 50 μm outside the target compartment and had a frequency of 100 Hz.

doi: https://doi.org/10.1371/journal.pcbi.1013782.g002