Fig 1.
The reciprocity theorem in the context of electric brain stimulation.
A, B: The traditional usage of the RT is to link extracellular current sources to extracellular potentials (or sometimes current dipoles to electric fields), without explicitly considering neurons and their properties [52–56]. C, D: We here point out that by moving the current source into the cell the RT can be used to directly account for the effect of ES on membrane potentials. Note that the intracellular stimulation current in panel D should be treated as part of the membrane current (see Methods).
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.
Fig 3.
Extracellular potentials surrounding different neural models are informative about the effect of ES.
A: following somatic white-noise current input, where each frequency component has an amplitude of 1 nA (see Methods), to a passive rat cortical layer 5 pyramidal cell (PC) model (passive version of the model shown in Fig 2A).
is calculated on a dense grid around the neuron. Through Fourier analysis, the 1 Hz (top) and 1000 Hz (bottom) components of
are extracted (see Methods). B: Same as in panel A, except for a passive mouse cortical layer 5 interneuron model (IN). C: Same as in panel B, except for a passive myelinated axon model, with the current input at the terminal end. The axon is oriented vertically, parallel to the apical dendrite in panel A. D: Same as in C, except that the current input is in the middle of the axon. E:
as a function of distance from the site of current input, calculated along the dotted lines in A-D (at -30° relative to the horizontal direction). Different colors correspond to the four different scenarios in panels A-D, as well as
from a point source (PS). Full and dashed lines correspond to 1 Hz and 1000 Hz input respectively. F:
as a function of frequency for white noise current input, calculated at two different distances along the dotted lines in A-D. Full and dashed lines correspond to 10 μm and 10,000 μm distance respectively. All amplitudes are normalized to the value at 1 Hz.
Fig 4.
Analytical expressions for gives insight into the parameter-dependence of ES.
A: Illustration of how decays with distance from the soma of a ball and stick neuron (see schematic) receiving a somatic current input with a frequency of 1 Hz and an amplitude of 1 nA. The bottom panel shows results from a numerical simulation of changes in
with increasing distance. Grey dotted lines indicate expected trends in the near (1/r) and far-field (1/r2) regimes. B-G: The amplitude of
as a function of frequency for a white-noise input current where each frequency component has an amplitude of 1 nA, and
is measured either very near (top) or very far away from (bottom) the soma. The three lines correspond to different parameter choices for a single parameter (columns), either default (black), increased by a factor of two (blue), or decreased by a factor of two (light blue). The parameters are membrane resistance
in B, axial resistance
in C, membrane capacitance
in D, length of the stick l in E, dendritic diameter d in F, and somatic diameter
in G. The grey dotted lines show the expected trends in the high-frequency limit in the near- (1/
) and far-field (1/f) [73].
Fig 5.
Intrinsically active cell model.
A: Macaque STN neuron from Miocinovic et al. (2006) [43]. B: The intrinsic spiking activity of the model in the absence of input. C: An electric stimulation (location marked by orange dot in panel A) at 10 Hz with a current amplitude of 100 μA, 500 μm away from the soma of the cell model. D: The somatic membrane potential of the cell under the influence of the ES in panel C, simulated through the traditional approach. E: Through the RT, we can calculate the effect of the ES in isolation, by inserting the current in panel C into the soma of a passive version of the cell model, and calculating at the ES location. The resulting amplitude of
was 2.1 mV. From the same simulation, we can calculate the input impedance of the model, and from this the equivalent somatic current amplitude of 67.6 pA needed to cause this 2.1 mV oscillation. F: The original active cell model is stimulated with an “ES proxy”, that is, by injecting this current into the soma (amplitude 67.6 pA) designed to mimic the
in panel E. This closely approximates the direct simulation (panel D).
Fig 6.
The effect of transcranial electric stimulation (tES) on individual pyramidal cells.
A: A passive pyramidal cell model receiving a somatic current input (orange dot). B: The input current with a frequency of 10 Hz and a 1 nA amplitude. C: The z-component of the resulting current-dipole moment, perpendicular to the cortical surface. D: The amplitude of this z-component as a function of the input frequency for different cell models. E: The current-dipole (panel C) is placed in the middle frontal gyrus of the New York head model. The orange dot marks the location, the black arrow marks the orientation, and the green ellipsoid marks the nearest EEG electrode. F: The induced voltage signal of 38 pV at the nearest EEG electrode (green ellipsoid in panel E), from the single pyramidal cell at the location marked in panel E. G: The EEG signal at the time of the maximum signal (time indicated by gray dashed lines in panels B, C, and F). The location of the dipole is marked by the orange dot (partially covered by an EEG electrode). H: Through the RT we can invert the situation and calculate the somatic for the pyramidal neuron (located at the green dot) responding to tES (located at the orange ellipsoid). I: A 10 Hz, 1 mA current input through an EEG electrode (marked in panel E and H), that is, 106 times larger amplitude than the original intracellular current input. J: A cross section of the head model’s lead field, displaying the amplitude of the electric field along the cortical normal direction (
). For a tES current amplitude of 1 mA through the considered electrode (orange ellipsoid), the maximum value of
is 0.21 mV/mm. This is the unmodified lead field as given by the New York head model [52]. The somatic
response for the cell model in panel A at different locations in the cortex, in response to the tES in panels H and I, can be found by a scaling of the lead field, in this case by a factor of 186 μm. K: The somatic
response to tES for the cell model in panel A at the location marked in panel E is
38 pA = 38 μV. As a control, we also simulate the effect of a uniform electric field of amplitude 0.21 mV/mm along the z-axis of the cell model, and see that the somatic
response is indistinguishable.
Fig 7.
Illustration of the reciprocity theorem.
A: The equivalent circuit diagram for a passive neuron model stimulated by an extracellular current source , while the membrane potential of compartment 1 is measured. B: The equivalent circuit diagram for a passive neuron model stimulated by an intracellular current source
, while the extracellular potential is measured. According to the reciprocity theorem, the measured potential in the two equivalent circuits should be the same.