Enabling electric field model of microscopically realistic brain

Abstract

Background: Modeling brain stimulation at the microscopic scale may reveal new paradigms for various stimulation modalities.

Objective: We present the largest map to date of extracellular electric field distributions within a layer L2/L3 mouse primary visual cortex brain sample. This was enabled by the automated analysis of serial section electron microscopy images with improved handling of image defects, covering a volume of 250 × 140 × 90 μm³.

Methods: The map was obtained by applying a uniform brain stimulation electric field at three different polarizations and accurately computing microscopic field perturbations using the boundary element fast multipole method. We used the map to identify the effect of microscopic field perturbations on the activation thresholds of individual neurons. Previous relevant studies modeled a macroscopically homogeneous cortical volume.

Result: Our result shows that the microscopic field perturbations - an 'electric field spatial noise' with a mean value of zero - only modestly influence the macroscopically predicted stimulation field strengths necessary for neuronal activation. The thresholds do not change by more than 10 % on average.

Conclusion: Under the stated limitations and assumptions of our method, this result essentially justifies the conventional theory of "invisible" neurons embedded in a macroscopic brain model for transcranial magnetic and transcranial electrical stimulation. However, our result is solely sample-specific and is only relevant to this relatively small sample with 396 neurons. It largely neglects the effect of the microcapillary network. Furthermore, we only considered the uniform impressed field and a single-pulse stimulation time course.

Keywords: Biophysical modeling; Boundary element fast multipole method (BEM-FMM); Brain modeling at the microscopic scale; Brain stimulation; Electric field spatial noise; Multiscale brain modeling; Significance statement.

Fig. 1.

Structure of L2L3 sample under study. a) Snapshot of relative position of a neuron (soma) vs. a nearby blood capillary (white area) within the sample [7], [5]. b) Snapshot of different neuronal processes labeled with different colors at a higher resolution. c) Segmented neuronal bodies of the entire sample. d) Centerlines/skeletons of individual neurons and segmented blood vessels in red (with an astrocyte at top right). The blood network of this sample was not the major target and is severely underdeveloped. e,f) Excitatory (pyramidal) and inhibitory (basket) neurons within the sample volume to scale. The axes differ from the mouse-centric coordinate system [5].

Fig. 2.

a) Final high-quality membrane surface triangular mesh for one neuron (#238) which is located approximately in the middle of the sample. The membrane mesh has 2 million facets in total and a 100 nm resolution. b,c) Healed centerlines of a neuron. d) Most common intersections observed – dendrite to dendrite and axon to dendrite intersections. Two different colors correspond to the two different neurons. e) Less common intersections – dendrite to soma and axon to axon, respectively. f,g) Mesh intersection resolution via removal of inner triangles. h) Result of this operation for an axon-to-dendrite intersection. Removed triangular facets are marked dark green.

Fig. 3.

a) Nested iterative solution method. At step 1, every neuron is considered independent. Membrane charge deposition in response to impressed or primary field ${\mathit{E}}^i$ (${\mathit{E}}_e^i$ in the extracellular space) is found via an iterative BEM-FMM solution. At step 2, the primary field ${\mathit{E}}_e^i$ is modified by the field of charges deposited on all other neurons. Membrane charge deposition for every neuron in response to this modified field is found next. The process further repeats itself until a convergence criterion is met. b) Convergence curves for the relative 2-norm surface charge density error. c) Convergence curves for the relative 2-norm error for the collinear electric field projected onto centerlines of neuronal processes.

Fig. 4.

a,b) Induced surface charge density for neuron #321. a) Induced charge density when only neuron #321 is present (iteration 1 or self-interaction). b) Induced charge density when all neurons and microcapillaries of the sample are present (iteration 6). c,d) Induced surface charge density for 200 nearest neighbors of neuron #321. c) Complete network of 200 nearest neighbors. d) Zoomed in area (x33) of the network in c). We were unable to simultaneously plot electromagnetic data for more than 200 neurons due to the limitations of the graphical software used.

Fig. 5.

Effect of membrane charge deposition on the collinear electric field projected onto the centerlines of neuronal processes for neuron #321. The sagittal plane view is shown corresponding to Fig. 4a,b. a) Electric field projection in homogeneous space, when the neuronal arbor is not present and a uniform impressed electric field of 100 V/m is applied along the $y$ axis (from dorsal to ventral). b) Averaging method over a cross-section. c) Electric field projection when all neurons are present, and the primary field is distorted by the field of all induced charges.

Fig. 6.

a) Equivalent straight fiber density distribution of the sample converted to the form of a rectangular cuboid. The fiber spacing illustrates weak sample anisotropy. b) Artificial computational equivalent of the specimen from a) constructed using ~500 finite cylinders in the form of a 3D lattice. The cylinders are spaced equally but have proportionally different radii to account for the moderate sample anisotropy from a). The cylinders are embedded into an enclosing (yellow) box (250×140×90 μm) whose conductivity contrast (interior vs. exterior conductivity) is varied. The volumetric fill factor of the cylindrical lattice within the box is 0.3, which coincides with the value for the original sample. Cylinder walls (membranes) and their tips, which are protruding the box, are assumed to be non-conducting. c-h) Magnitude (in V/m) and direction of the total extracellular electric field within and around the artificial sample for three different polarizations ($x$, $y$, $z$) of the impressed field ${\mathit{E}}^i$ with the value of 100 V/m. c,e,g) – Field distortion around the heterogeneous artificial sample when its extracellular conductivity is that of the surrounding homogeneous medium. d,f,h) – Field distortion around the heterogeneous artificial sample when its extracellular conductivity is increased with an attempt to achieve a minimum surrounding distortion.

Fig. 7.

a) Equivalent straight fiber density distribution of the sample characterizing its anisotropy. b) Activating threshold reduction factor for the first 90 neurons with the longest axonal arbors (sorted) for three polarizations of the primary electric field: medial-lateral (red), dorsal-ventral (green), and anterior-posterior (blue). c) Absolute activating thresholds for the same 90 neurons in homogeneous space. d) Absolute activating thresholds for the same 90 neurons within the sample. The units in c,d) are given in terms of the base field strength of 100 V/m; one unit corresponds to 100 V/m. These results are given for the case of the unmyelinated axonal arbor and a negative rectangular activation monophasic pulse with the duration of 0.1 ms. Results for all other cases are given in Supplement C.

Fig. 8.

a) Activating threshold reduction factors as functions of rectangular negative pulse duration. The inset shows the corresponding strength-duration curves for neuron #348 within the brain sample. Inset units in are in terms of the initial field strength of 100 V/m; one unit corresponds to 100 V/m. b,c,d) Activating threshold reduction factors as functions of the number of neighboring cells for three representative excitatory neurons with different lengths of the axonal arbor (for the rectangular negative pulse) without the sample polarization correction. All three neurons belong to the set of 90 neurons with the longest axonal arbor. The x-axis is the corresponding volumetric fraction of neuronal mass computed separately for each neuron when sequentially adding its neighboring cells. Red lines show the expected fraction of 70% [50]; dashed lines show expected extrapolation results. Vertical black lines on the right and the corresponding larger circles are the results with the sample polarization correction for volumetric concentration of 0.5.

Fig. 9.

Concept of the electric field spatial noise. a) Straight HH axon from Ref. [43] and the electric field of cathodic stimulation. b) The same case as in a) but with artificial field perturbers – infinite cylinders. c,d) Collinear electric field and extracellular potential with (blue) and without (red) field perturbations. In both cases, the electrode current is 0.94 mA. e-n) Resulting evolution of transmembrane potential and transmembrane ionic current density for both cases at different times. Red curves are those without field perturbations; blue curves correspond to the perturbed extracellular field.