The effect of spatially inhomogeneous extracellular electric fields on neurons

Abstract

The cooperative action of neurons and glia generates electrical fields, but their effect on individual neurons via ephaptic interactions is mostly unknown. Here, we analyze the impact of spatially inhomogeneous electric fields on the membrane potential, the induced membrane field, and the induced current source density of one-dimensional cables as well as morphologically realistic neurons and discuss how the features of the extracellular field affect these quantities. We show through simulations that endogenous fields, associated with hippocampal theta and sharp waves, can greatly affect spike timing. These findings imply that local electric fields, generated by the cooperative action of brain cells, can influence the timing of neural activity.

Figure 1.

Circuit representation of the passive membrane at location x along the cable (Rall, 1962). We here analyze the effect of spatiotemporal variations in the extracellular voltage, V_e, on the membrane voltage, v_m, in terms of the normalized membrane potential, V_m (v_m normalized by the amplitude of v_e).

Figure 2.

The effect of stationary and spatially varying extracellular voltage, V_e, along a passive cable on the normalized membrane potential, V_m, the membrane field, E_m, and the membrane current source density, CSD_m. Distance along the cable is normalized to unity. In the top row (A1, B1, C1), the V_e oscillation with spatial frequency Ω is shown for ΩL/2π = 0.1 (A1), 0.5 (B1), and 3 (C1), whereas the colored lines indicate ϕ_s = 0° (blue), 90° (green), and 180° (red). In the second row (A2, B2, C2), the induced membrane potential (Eq. 5) is plotted for these three cases. Only spatially inhomogeneous fields with a characteristic field length comparable with or smaller than the cable length result in strong V_m deviations. In the third (A3, B3, C3) and the fourth (A4, B4, C4) rows, the induced E_m (Eq. 6) and CSD_m (Eq. 7) are shown. The gray area designates the range for all spatial phases, whereas the dashed lines in the two bottom rows indicate the range of dimensionless extracellular field E_e and current source density CSD_e, respectively. For ϕ_s∈[0°, 360°), the induced |E_m|/Ω- and |CSD_m|/Ω²-range is unity. As observed, the CSD_m is analogous to the V_e profile (compare the individual V_e and CSD_m profiles in the top and bottom rows) for large Ω. Note that in this, and in all remaining figures, no direct current injection or synaptic input is modeled. All changes in V_m are attributable to the spatially varying extracellular field.

Figure 3.

A parent cable bifurcates into two daughter cables (all branches are in the same plane) in the presence of a spatially inhomogeneous extracellular voltage. The external potential V_e that varies over a half-cycle is indicated in color. The cable impedances are matched by applying the 3/2 rule (Rall, 1962). The parent cable is aligned to the external field axis, whereas the angle between the daughter cables and the field axis are as follows: θ_d,1 = 0° and θ_d,2 = −45° (A), θ_d,1 = 45° and θ_d,2 = −45° (B), and θ_d,1 = 90° and θ_d,2 = −45° (C). Below each case, the resulting V_m (second row), E_m (third row), and CSD_m (fourth row) are shown for ϕ_s = 0° (blue), 90° (green), and 180° (red). Note that for X > 0.5L, the dashed colored lines illustrate the trajectory of v_m along the dashed daughter branch (see first row of the figure) for the three ϕ_s, whereas the solid colored lines show the trajectory of v_m along the solid daughter branch. The daughter branches are of equal normalized length as the parent branch, L_parent = L_daughter = L/2. The range of V_m, E_m, and CSD_m along all branches is indicated by the gray areas. Note the increasing attenuation in the induced E_m- and CSD_m-range as the daughter cable becomes perpendicular to the external field axis.

Figure 4.

Morphology of a reconstructed rat CA1 pyramidal neuron. A, The designated sections are a basal dendrite (green) (i), the soma and the proximal apical dendrite connected to it (blue) (ii), a medial apical dendrite (cyan) (iii), and a distal apical dendrite (magenta) (iv). B, The orientation cos(θ_k)cos(σ_k) with respect to the somatodendritic axis (Fig. 4A, y-axis) is shown along each section (i–iv) as a function of the normalized arc length s/l: for the basal section (green), the soma and the proximal apical section (blue; from s/l = 0 to 0.25 soma and then proximal apical dendrite), the medial apical section (cyan), and the distal apical section (magenta). As observed, the basal dendrite is almost perpendicular to the somatodendritic axis, whereas the other three sections are relatively well aligned. The length l of each section is provided in Table 1.

Figure 5.

The effect of a spatially harmonic one-dimensional electric field (along the somatodendritic axis) on the V_m of the reconstructed CA1 neuron (top row; white lines indicate the same four sections as in Fig. 4A) for three spatial frequencies: λ_s = 6.25 mm (f_s = 0.16 mm⁻¹; A), 1 mm (1 mm⁻¹; B), and 0.2 mm (5 mm⁻¹; C). The color map illustrates the (spatially) one-dimensional extracellular V_e oscillation along the somatodendritic axis for a phase ϕ_y. The distance along each section is shown as dimensionless arc length s/l. i is the basal; ii, the soma and the proximal apical; iii, the medial apical; and iv, the distal apical section (Fig. 4A, Table 1). The range of V_m is indicated in gray, whereas the individual traces are ϕ_y = 0° (blue), 45° (red), and 90° (green). Condition (II) is only satisfied for λ_s = 0.2 mm (5 mm⁻¹; C) along the proximal (Cii), medial (Ciii), and distal apical (Civ) sections (Table 1) as observed from the induced V_m range.

Figure 6.

A, The maximum and minimum of the normalized membrane potential V_m in the presence of a spatially inhomogeneous and oscillating extracellular field V_e. The colored areas indicate the V_m amplitude along the cable for dc (gray area), and f_t = 100 (green area) and 200 Hz (red area). As expected for a low-pass membrane, for increasing values of the temporal frequency f_t (arrows), V_m follows less and less the V_e oscillation. B, The effect of the time-dependent harmonic excitation on V_m as described through the normalized deviation β(V_m). As long as β(V_m) ≈ 0, the membrane response remains quasistationary. Once β deviates from zero, low-pass filtering affects the overall process. The results are shown for the basal dendrite (crosses), the soma (circles), the proximal apical dendrite (squares), the medial apical dendrite (diamonds), and the distal apical dendrite (x) in black color. The excitation was spatially centered at the middle of each section. For a typical value of τ_m = 20 ms, (2π f_t τ_m)^1/2 = 1 and 5 are equivalent to f_t ≈ 8 and 200 Hz, respectively. β(V_m) is also shown for the unbranched cable case (red line) for comparison.

Figure 7.

The effect of theta (left column) and SPW (right column) extracellular field activity on the membrane potential v_m of a CA1 pyramidal neuron. A1, B1, The individual extracellular recordings (black traces) from equally spaced recording sites (8 of the 16 electrodes are shown) (supplemental Figs. S6, S7, available at www.jneurosci.org as supplemental material) are shown during two 1 s epochs, starting from the stratum lacunosum moleculare (y = 0 μm) toward the stratum oriens (y = 700 μm). The color map shows the current source density csd_e (units, millivolts per square millimeter) (Mitzdorf, 1985). Based on experimental evidence suggesting that the endogenous field is strongest along the somatodendritic axis of CA1 neurons, we applied the v_e recordings (A1, B1) along the y-axis of the realistic neuron (Fig. 4A) and calculated the resulting spatiotemporal evolution of v_m (supplemental section 7, available at www.jneurosci.org as supplemental material). Ai–Aiv, LFP-induced deviations of v_m along each section (i–iv) (Table 1) during theta. The gray areas indicate the range, whereas the three individual traces show v_m − v_rest along each section for t = 0.13 s (black), 0.22 s (blue), and 0.30 (cyan) and are indicated by the arrows in the top row. The periodic fluctuations of the extracellular potential induces a location-specific fluctuation of the membrane potential that increases in amplitude toward stratum lacunosum moleculare. The antiphase relationship between the somatic and apical dendritic v_e fluctuations results in an antiphase v_m fluctuation; that is, compare v_m at the theta peak (black arrow and lines) and at the trough of theta (blue arrow and lines) at the soma (Aii) and the distal apical section (Aiv), respectively. Bi–Biv, LFP-induced changes in v_m during the SPW. The three individual traces show v_m − v_rest along each section for t = 0.30 s (black), 0.38 s (blue), and 0.48 (cyan). Note the pronounced antiphase relationship in v_m at the SPW negativity (blue arrow and lines) between the soma (Bii) and the medial apical section (Biii). Unlike theta, the somatic membrane potential is significantly, but transiently, entrained during the SPW: compare v_m deviations immediately before (black arrow and lines) and after (cyan arrow and lines) the SPW (supplemental movies, available at www.jneurosci.org as supplemental material).