Differential polarization of cortical pyramidal neuron dendrites through weak extracellular fields

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Abstract

The rise of transcranial current stimulation (tCS) techniques have sparked an increasing interest in the effects of weak extracellular electric fields on neural activity. These fields modulate ongoing neural activity through polarization of the neuronal membrane. While the somatic polarization has been investigated experimentally, the frequency-dependent polarization of the dendritic trees in the presence of alternating (AC) fields has received little attention yet. Using a biophysically detailed model with experimentally constrained active conductances, we analyze the subthreshold response of cortical pyramidal cells to weak AC fields, as induced during tCS. We observe a strong frequency resonance around 10-20 Hz in the apical dendrites sensitivity to polarize in response to electric fields but not in the basal dendrites nor the soma. To disentangle the relative roles of the cell morphology and active and passive membrane properties in this resonance, we perform a thorough analysis using simplified models, e.g. a passive pyramidal neuron model, simple passive cables and reconstructed cell model with simplified ion channels. We attribute the origin of the resonance in the apical dendrites to (i) a locally increased sensitivity due to the morphology and to (ii) the high density of h-type channels. Our systematic study provides an improved understanding of the subthreshold response of cortical cells to weak electric fields and, importantly, allows for an improved design of tCS stimuli.

Figures

  • Fig 1. In a passive pyramidal cell model subject to an electric field, soma and basal dendrites get oppositely polarized than apical dendrites. This later are the most responsive to low frequency stimulation. (A) Considered neuron morphology with color coding the distance of each segment to the soma. (B) (Bottom) Membrane polarization of the passive cell due to positive and negative steps of DC electric field (orange, top). (C) Polarization due to a positive 1 V/m field plotted as the function of the distance from the soma. For clarity basal dendrites are plotted with negative distance. (D) Example polarization at the apical dendrites (blue star, bottom) and soma (green circle, middle) due to an an oscillating field of diverse frequencies (orange, top). (E) Frequency-dependent sensitivity to AC fields of different cell segments, namely basal, proximal dendrites and soma (left panel) and distal apical dendrites (right). Colors of the polarization (B) and field sensitivity (E) correspond to the distance from the soma as depicted in A. The red dashed lines correspond to the soma.
  • Fig 2. The polarization of a passive straight cable due to an extracellular field is anti-symmetric and decreases with the field frequency. (A) Schematic representation of a straight cable subject to a parallel electric field. The variables are detailed in Methods. (B) (top) Field sensitivity, i.e. ratio between membrane polarization and field amplitude, along the cable for different field frequencies (0.5, 50, 200 and 1000Hz, color coded) for τ = 40ms and L = 1λ. (bottom) Phase shift between the field and the membrane polarization oscillations. (C) Field sensitivity at the cable ends as a function of frequency (x axes), for various cable’s electrotonic length L (top: 0.5λ, bottom: 3λ) and time constant τ (color coded). (D) Field sensitivity at the cable ends as a function of electrotonic length L, for various frequencies (color coded) and membrane time constants (solid lines: 5ms, dashed lines: 40ms).
  • Fig 3. Passive cable with an acute bending angle can display a resonance in their sensitivity to spatially uniform fields. (A) Schematic representation of a bent cable. The unbent branch of the cable, of length H, is parallel to the field axis. The bent branch, of length D, have an angleΘ with the field. L is the total cable length. (B,C) Sensitivity (in V/(V/λ)) at both cable ends, i.e. at the main (B) and bent (C) branches ends, as function of the field frequency. The sensitivities are displayed for various main (H, columns) and bent (D color coded) branches lengths. (D) Distribution of the sensitivity (top) and phase (bottom) along the bent cable for different field frequencies (0.5, 50, 200 and
  • Fig 4. The presence of several basal dendrites increases the field sensitivity at the apical dendrites while decreasing the field sensitivity at the soma and basal dendrites. (A) Schematic representation of the simplified neuron model. The model consists of a main branch of length H, parallel to the extracellular field. Several branches of length D are attached to one end of the main cable. At least one of the attached branch is parallel to the main cable axis, the others form an angleΘ with that axis. (B,C) Sensitivity (in V/(V/λ)) at the end of the main cable (red star), at the branching point (green square) and at the end of the parallel branching cable (purple circle), as function of the field frequency. The field sensitivities are displayed for various number N (color coded) of branches with an angle ofΘ and various main cable lengths (H, columns) and branch lengths (D, columns). In all the plots the bending angle isΘ = π/2 (rad) and the membrane time constant τ = 40 ms. H and D are electrotonic lengths.
  • Fig 5. Unlike the soma and basal dendrites, the apical dendrites of an active cell have an increased subthreshold response to AC field of 10-20Hz. We consider a pyramidal cell model of Hay et al. [16] with all active channels. (A) Membrane polarization around the resting state due to a positive step current electric field (orange). (B) Frequency-dependent sensitivity of the cell to AC fields measured at different location on the whole cell. (C) Sensitivity to sinusoidal fields of frequency 0.5Hz (left) and 10.3Hz (right) as a function of distance to the soma. For clarity basal dendrites are plotted with negative distance. Colors of the polarization (A) and field sensitivity (B) correspond to the distance from the soma as depicted in A. The red dashed lines correspond to the soma.
  • Fig 6. The hyperpolarization-activated inward current, Ih, is responsible for the frequency resonance in the sensitivity of apical dendrites to AC fields. The subplots display the field sensitivity (in mV/(V/m)) of the cell at given locations depending on the channels included in the model: without any active channels (passive) or with all active channels present in the model of Hay et al. [16] active (Active). We also consider the model with frozen channels, i.e. their gating variable is fixed to their resting value. We freeze either all the channels (Fully frozen) or all except the Ih which is fully active (Frozen + Ih active) or linearized following the quasi-active approximation (Frozen + Ih QA).
  • Fig 7. The field sensitivity of a pyramidal cell decreases in a high-conductance state. The subplots display the field sensitivity (in mV/(V/m)) of the pyramidal cell model in different conductance states. The low-conductance state corresponds to the model with the same leak conductance as the original Hay et al. [16] model. In the high- (blue lines) and higher-conductance (orange lines) states, we uniformly increased the passive conductance of the original model by adding respectively 350μS/cm2 and 900μS/cm2 uniformly throughout the model. To remove the effects of changes in resting membrane potential, we uniformly set the resting membrane potential of all 3 models to −65 mV by adjusting the leak reversal potential (see Methods). The somatic input resistances of the low-, high- and higher-conductance state model are respectively: 53 MO, 15.62 MO, and 9.05 MO.
  • Fig 8. The quasi-active channel conductance distribution affects the cell field sensitivity depending of the local conductance at the considered location. We consider a neuron model which includes solely a leak conductance and a single quasi-active channel (QA). The QA channel have no active dynamics (μ = 0) and acts as an additional leak current. We consider 3 different QA conductance distributions: uniform and linearly increasing/decreasing with distances from the soma. For each distribution, the sum over the whole cell of the QA conductances at rest is equal to the sum of the leak conductances. The subplots display the cell field sensitivity (in mV/(V/m)) for the different conductance distributions.

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CITATION STYLE

APA

Aspart, F., Remme, M. W. H., & Obermayer, K. (2018). Differential polarization of cortical pyramidal neuron dendrites through weak extracellular fields. PLoS Computational Biology, 14(5). https://doi.org/10.1371/journal.pcbi.1006124

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