Importance of considering microscopic structures in modeling brain stimulation

Dear Editor,

We are very grateful to Drs. Wang and Aberra for valuable comments conveyed in their letter [1] on our paper “Enabling electric field model of microscopically realistic brain” [2]. Wang and Aberra neatly summarize the convention across nearly all brain stimulation models using a “two-step approach” which 1) solves the macroscopic E-field distribution neglecting microscopic cells and then 2) applying this E-field distribution individually to 1-D neurons. In contrast, our paper provides the first computationally tractable methodology to calculate the E-field distribution in a realistic volume. We report a ~10% change in thresholds compared to the two-step approach. However, the implication of this computational advancement is far from minor or trivial.

1. The two-step approach [3] has been the convention since 1976 [4] when calculations were analytical. Notwithstanding the development of numerical methods and increased sophistication in macroscopic E-field modeling and neuronal/network simulation, the two-step approach remains dogma as the alternative was computationally intractable. Our contribution should be understood as allowing, for the first time, and in the face of hundreds of studies relying on the two-step methodology, an approach to test its reliability.

   It is also important to recognize how far the two-step process is from the known brain microstructure. The majority of brain volume is intercellular compartments, insulated by cell membranes, leaving only a fraction of space (~15-30% by volume) for extracellular current flow. Moreover, this extracellular space is exceptionally tortuous – it is therefore not conceivable that current flows “unimpeded” across the brain and neurons.

2. The two-step approach is therefore dogma not because it is exhaustively tested or informed by brain structure, but because no alternative was available. There is a valid argument on how far the two-step approach has been validated outside of isolated cell models [5], where there is no impact of neighboring cells. To the extent that the two-step approach does fit experimental data, it can be asked how much of this reflects parameter fitting. It has been proposed that even an approximation considering only the macroscopic electric field [6],[7] could be as reliable as the most complex two-step models. In any case, we proposed the two-step process is not a “gold-standard” but a convention.

3. Regardless of the assumptions underlying the two-step process, we recognize that the utility of any model advancement is weighed against any corresponding added complexity and justified by producing distinct predictions (outcomes) [8]. In this sense, we think focusing on the single ~10% in the very first paper on the subject [2] is incomplete for two reasons.

   First, the limits must be considered in an application while taking into account cellular anatomy. Our initial results are from the very small IARPA Phase I L2L3 mouse brain dataset (250 × 140 × 90 μm) with ~60% of axonal terminations at the boundary. As a bridge to the next step, let us consider a single, entirely random cortical cross-section from the much larger IARPA Phase III dataset [9], shown in Fig. 1 with <1% of axonal terminations at the boundary. There, a neuron from the corresponding cortical column is also depicted. We can identify at least three ‘mesoscale’ sources of inhomogeneity, on the order of 100 μm, that might lead to changes in activation thresholds even for spatially constant brain stimulation fields:

   • Differences in cellular density and, consequently, differences in extracellular space (ECS) volume and conductivity across cortical layers on average. These variations may alter electric fields along distal axons traveling from superficial to deep or vice versa. The average ECS per layer reflects an averaging over the true microstructure of the layer.

   • Large, random clusters of somas located both across and along cortical layers. These clusters may alter electric fields along axons traveling in their proximity.

   • Larger blood microcapillaries (bottom left corner of Fig. 1) and astrocytes (red clouds) potentially located anywhere in the cortex. These structures may also alter the electric fields along axons traveling in their proximity.

   While the first source could be homogenized by assigning different conductivities to each layer, the remaining two cannot. We propose that statistical modeling, incorporating representative microscopic/mesoscale field perturbations [10] could quantify their influence on neuronal activating thresholds in a probabilistic sense.

   Second, “small” differences in modeling results between realistic models and the two-step approach may be significant for outcomes. Most brain stimulation approaches are dose titrated so that outcomes can be assumed to follow the most responsive cells. For this reason, much research on the mechanisms of brain stimulation has focused on “which neuronal elements are activated” [24]. A change of a few percent in activation threshold may be significant both for on-target and off-target effects. Even if there is no change in threshold, a change in which neurons are activated may be critically important to outcomes.

   A counter argument to the above points is that surely other modeling parameters and assumptions (used in both the realistic and two-step methods) will influence outcomes. Small changes in macroscopic tissue resistivity or cell biophysics can drastically impact outcomes. To this we would respond, how can one understand the impact of these parameters using a two-step process, if the two-step process is not itself verified in an application specific manner?

4. Wang and Aberra focus on explaining the importance of ensuring meaningful comparisons between the realistic and two-step approaches such as a correction factor to be applied. We applied one such technique in [2], Wang and Aberra proposed another elegant approach [1], and we might suggest a third simple method where the net conductivity of a realistic model is adjusted (simply by considering the volume fraction of the extracellular space) to match bulk resistivity on average. The accurate methods used for correction factors must be carefully considered, with the perspective that the two-step models abstract current flow patterns to unrealistic uniformity, with realistic models eventually representing the actual brain structure and microscopic current flow.

Fig. 1.

Cortical cross-section of an IARPA Phase III sample [9] at z = 813 μm. White spots represent the somas of individual neurons; black indicates the membranes of the neuropil and other cells; red marks the vascular network, including capillaries and astrocytes. Approximate layer boundaries (derived from 3D boundary meshes) are shown in orange. An inhibitory neuron from the corresponding cortical column is depicted in blue, with axonal terminations marked by magenta circles; one of its distal axons extends from deep to superficial layers (from L5 to L2/L3), lying precisely in this cross-sectional plane. Three red arrows indicate the computed locations of initial action potential generation for different directions of an applied brain stimulation field when using Markram’s ion channel model. Note that other channel models may produce different activation sites.

With kind regards,                       The Authors