Dendritic diameters affect the spatial variability of intracellular calcium dynamics in computer models

Abstract

There is growing interest in understanding calcium dynamics in dendrites, both experimentally and computationally. Many processes influence these dynamics, but in dendrites there is a strong contribution of morphology because the peak calcium levels are strongly determined by the surface to volume ratio (SVR) of each branch, which is inversely related to branch diameter. In this study we explore the predicted variance of dendritic calcium concentrations due to local changes in dendrite diameter and how this is affected by the modeling approach used. We investigate this in a model of dendritic calcium spiking in different reconstructions of cerebellar Purkinje cells and in morphological analysis of neocortical and hippocampal pyramidal neurons. We report that many published models neglect diameter-dependent effects on calcium concentration and show how to implement this correctly in the NEURON simulator, both for phenomenological pool based models and for implementations using radial 1D diffusion. More detailed modeling requires simulation of 3D diffusion and we demonstrate that this does not dissipate the local concentration variance due to changes of dendritic diameter. In many cases 1D diffusion of models of calcium buffering give a good approximation provided an increased morphological resolution is implemented.

Keywords: active dendrite; calcium buffering; calcium concentration; compartmentalization; dendritic diameter; diffusion; intracellular calcium; morphology.

Figure 1

Cytosolic compartmentalization for diffusion from membrane toward the center of a cylindrical compartment. Schematic diagram shows two different ways of dividing the compartment volume into concentric shells. The DM mechanism (left) has a fixed number of shells and the depths of all shells vary so that the sum equals the compartment diameters, the outer shell also has a smaller depth than subsequent shells. In the DM_FD mechanism (right) all shells have an identical, fixed depth except for the core shell whose depth is adjusted to get the correct compartment diameter. The number of shells is given by compartment diameter.

Figure 2

Spatial Ca²⁺ gradients strongly depend on type of model implementation. Panels (A–C) show maps of the integrated calcium levels in the dendrite during a spontaneous burst of Ca²⁺ spikes (panel D). The dendritic branches are color coded to show the integrated calcium levels using a 20 ms window around the peak Ca²⁺ concentration of the first dendritic Ca²⁺ spike. The color scales used in these maps are nonlinear (using histogram equalization) to enhance the contrast. (A) Single Ca²⁺ pool model using SP_old mechanism results in homogenous Ca²⁺ levels. (B) Single Ca²⁺ pool model using SP_new mechanism results in variable Ca²⁺ levels. (C) Detailed Ca²⁺ dynamics model with buffering and 1D diffusion results in variable Ca²⁺ levels with larger Ca²⁺ gradients. (D) Voltage traces show the first spike of the Ca²⁺ burst for each model in all dendritic compartments for the 3 different models (see color code in Figure). The inset shows complete traces. (E) The underlying Ca²⁺ and K_Ca currents (recorded from all dendritic compartments) for the Ca²⁺ spike of the three different models (see color code in Figure).

Figure 3

Errors introduced by incorrect submembrane volumes of single pool models. (A) Comparison between cylindrical dendritic compartments with diameters of 1 μm (left) and 0.5 μm (right) with submembrane shells with a depth of 0.1 μm. A correct implementation of the volume of the submembrane shell representing the single Ca²⁺ pool (SP_new mechanism) results in a SVR that depends on the compartment diameter. (B) For the same compartments using the SP_old mechanism results in volumes that are too large and have a constant SVR. The cross-sectional area of each compartment (black disks shown in A) is unfolded and drawn to show that the actual volume of the submembrane shell (SP_new) is smaller than the volume used in the SP_old mechanism. The red triangles represent extra cross-sectional area included in the volume of SP_old. (C) Ca²⁺ transients generated using a “ramp-like” voltage command in single compartments with diameters ranging from 0.2 to 6 μm in steps of 0.1 μm. P-type Ca²⁺ channel with P_max of 5.2 × 10⁻⁵ cm/s was used for Ca²⁺ influx. Inset: comparison of peak amplitudes of Ca²⁺ transients using SP_old and SP_new show that the first mechanism causes exactly the same transient in all compartments, whereas, SP_new causes transients with varying peak Ca²⁺ amplitudes. (D) Error in peak Ca²⁺ levels caused by using the SP_old mechanism [error = (max([Ca²⁺]_{SP_old}) − max([Ca²⁺]_{SP_new})/max([Ca²⁺]_{SP_new}))]. Pool models used β-values of 0.02, 6.86, and 10 ms⁻¹; and depth (d) values of 0.05, 0.1, 0.15, 0.2, and 0.25 μm. The lower edge of shaded areas of each color shows error in peak calcium for β-value of 10 ms⁻¹, whereas, the upper edge of shaded areas of each color show error for β-value of 0.02 ms⁻¹. The colored asterisks show corresponding error for β-value (used to model PC dendrites) of 6.86 ms⁻¹. Inset highlights large errors for branches with small diameters (diam ≤ 1 μm).

Figure 4

Biophysically detailed Ca²⁺ dynamics model causes larger differences in calcium levels in adjacent dendritic branches than single pool models. Histograms of ratios between integrated calcium from adjacent dendritic branches for 12 different PCs using SP_new and DM. To make the differences between cells more visible only the range of ratios 1–3 is shown, for the two cells that have significantly larger ratios the full distribution is shown in the inset. PC1 is shown in Figure 2. Integrated Ca²⁺ was computed for 20 ms around the first peak of Ca²⁺ transients for all PCs.

Figure 5

Large changes in diameters of unbranched dendritic segments exist in Purkinje and Pyramidal neurons. Stacked histograms show the distribution of CV values for the diameters changes over unbranched dendritic segments in Purkinje cells (A, N = 12) and in neocortical and hippocampal pyramidal neurons (B, N = 322). Notice the presence of large variability of diameters (CVs > 0.2 or more) in many neurons and the large neuron to neuron differences which are mostly caused by lab to lab differences in reconstruction quality (see text).

Figure 6

Inaccuracies of different calcium 1D diffusion models result in erroneous calcium levels. (A) Errors introduced by making the number of concentric shells independent of compartment diameter, for 4, 8, or 12 shells respectively. Two mechanisms are implemented: the standard NEURON scheme with variable depths for all shells (circles) and an FD scheme where the submembrane shell has a constant depth d₁ = 0.1 μm and the rest of the shells has variable depth (triangles). The DM_FD mechanism is used as reference. Note that for both mechanisms the errors become large for diameters beyond 2 μm if only four shells are used (as is the case in some NEURON models). (B) Ca²⁺ transients generated using a “ramp-like” voltage command in single compartments (see Figure 3C for details) comparing the responses of the DM and DM_FD models. Both models show very similar behavior with only small numerical differences. (C) Errors due to discretization of radial shells in DM, which may result in variable d₁ resulting in rapid changes of submembrane shell volume for increasing compartment diameter. The broken line with asterisks shows errors related to conversion of Ca²⁺ influx to Ca²⁺ concentration with variable depth d₁ of the submembrane shell (it varies between 0.075 and 0.125 μm due to discretization) as compared to fixed d₁ of 0.1 μm (DM_FD). The solid lines with diamonds shows the actual error in free Ca²⁺ in the submembrane shell for DM models for different sizes of Ca²⁺ influx as indicated. Note that these errors are much smaller than predicted by the Ca²⁺ influx conversion.

Figure 7

Different Ca²⁺ buffering model respond variably to changes in dendrite diameters. Predicted ratio of integrated Ca²⁺ concentration (100 ms window) for different combinations of diameters of pairs of dendritic compartments using (A) SP_new, (B) DM, and (C) DM_FD. The maps are derived from the data shown in Figure 3C (A) and Figure 6B (B,C). The color scales used in these maps are nonlinear (using histogram equalization) to enhance the contrast.

Figure 8

Large differences in calcium levels in adjacent dendritic branches persist in presence of 3D diffusion. (A) STEPS model using 3D buffered diffusion to compute the Ca²⁺ concentration resulting from the burst of Ca²⁺ spikes. Spatial map of integrated calcium (140 ms window) in a piece of carefully reconstructed PC dendritic arbor (part of PC 1). Every colored dot drawn at the center coordinates of each tetrahedron belonging to the mesh in which 3D diffusion was simulated shows the integrated Ca²⁺ in that particular tetrahedron. Only tetrahedrons representing the submembrane region are plotted. The color scales used in these maps are nonlinear (using histogram equalization) to enhance the contrast. (B) Spatial map of dendritic diameters in the dendrite shown in (A,D,F). (C) Normalized histograms compare the ratios of adjacent diameters in the original morphological reconstruction with similar ratios of diameters of adjacent compartments in the NEURON model (1 segment per unbranched section). (D,E) NEURON simulation with many compartments for each unbranched segment, carefully reflecting the variability of dendrite diameter. Data for the STEPS simulation are averaged over all tetrahedrons representing the corresponding NEURON compartment. (F,G) NEURON simulation with a single compartment for each unbranched segment, data for the STEPS simulation averaged for corresponding NEURON compartments. (D,F) Spatial maps of integrated submembrane Ca²⁺ concentration using the detailed calcium dynamics model with 3D diffusion (STEPS) and 1D radial diffusion (NEURON) are shown for the different compartmentalization schemes. (D) and (F) Use same color as in A. (E,G) Normalized histograms show the ratios of integrated Ca²⁺ concentration between every adjacent compartment using simulations with 3D diffusion (STEPS) and 1D diffusion (NEURON) for the results shown in (D,F) respectively.