Neuronal calcium wave propagation varies with changes in endoplasmic reticulum parameters: a computer model

Abstract

Calcium (Ca²⁺) waves provide a complement to neuronal electrical signaling, forming a key part of a neuron’s second messenger system. We developed a reaction-diffusion model of an apical dendrite with diffusible inositol triphosphate (IP₃), diffusible Ca²⁺, IP₃ receptors (IP₃Rs), endoplasmic reticulum (ER) Ca²⁺ leak, and ER pump (SERCA) on ER. Ca²⁺ is released from ER stores via IP₃Rs upon binding of IP₃ and Ca²⁺. This results in Ca²⁺-induced-Ca²⁺-release (CICR) and increases Ca²⁺ spread. At least two modes of Ca²⁺ wave spread have been suggested: a continuous mode based on presumed relative homogeneity of ER within the cell; and a pseudo-saltatory model where Ca²⁺ regeneration occurs at discrete points with diffusion between them. We compared the effects of three patterns of hypothesized IP₃R distribution: 1. continuous homogeneous ER, 2. hotspots with increased IP₃R density (IP₃R hotspots), 3. areas of increased ER density (ER stacks). All three modes produced Ca²⁺ waves with velocities similar to those measured in vitro (~50–90µm /sec). Continuous ER showed high sensitivity to IP₃R density increases, with time to onset reduced and speed increased. Increases in SERCA density resulted in opposite effects. The measures were sensitive to changes in density and spacing of IP₃R hotspots and stacks. Increasing the apparent diffusion coefficient of Ca²⁺ substantially increased wave speed. An extended electrochemical model, including voltage gated calcium channels and AMPA synapses, demonstrated that membrane priming via AMPA stimulation enhances subsequent Ca²⁺ wave amplitude and duration. Our modeling suggests that pharmacological targeting of IP₃Rs and SERCA could allow modulation of Ca²⁺ wave propagation in diseases where Ca²⁺ dysregulation has been implicated.

1 Introduction

Calcium (Ca²⁺) is an important second messenger signal in many cell types, with diverse roles, from fertilization (Busa and Nuccitelli, 1985; Kretsinger, 1980) to regulating gene expression (West et al., 2001). Ca²⁺ is involved in triggering destructive processes including apoptosis (Orrenius et al., 2003) and ischemia (Lipton, 1999). Cells, including neurons, therefore regulate cytosolic Ca²⁺ concentration via buffers (Stern, 1992) and sequestration into mitochondria (Gunter et al., 2004) or endoplasmic reticulum (ER) (Berridge, 1998). In neurons, sequestration is modulated by neuronal activity (Pozzo-Miller et al., 1997), and elevated Ca²⁺ opens certain ion channels. There is therefore a bidirectional interaction between chemical signaling and electrophysiology (Blackwell, 2013; De Schutter and Smolen, 1998; De Schutter, 2008).

Ca²⁺ is heavily buffered but travels long distances (>100 µm) to reach its targets. This poses a temporal problem if relying purely on diffusion. Ca²⁺ waves increase the rapidity of Ca²⁺ spread via Ca²⁺-induced-Ca²⁺-release (CICR). CICR occurs in neurons and requires stores of Ca²⁺ held in the ER. ER is distributed throughout the cytosol in a connected way, through the dendrites and dendritic spines (Harris, 1994; Spacek and Harris, 1997). The ER SERCA pump pushes Ca²⁺ from the cytosol into the ER. When triggered by IP₃ and Ca²⁺, the ER IP₃ receptors (IP₃Rs) open and release some of the ER’s Ca²⁺ into the cytosol. Regions of elevated Ca²⁺ then spread throughout portions of the dendritic tree (Ross et al., 2005).

Understanding Ca²⁺ wave modulation is difficult since cytosolic Ca²⁺ changes over time and passes between different intracellular compartments and extracellular space (Fall et al., 2004; Wagner et al., 2004). Adding to this complexity is the nonuniform distribution of IP₃Rs, with clusters forming where local variations in IP₃R or ER is heightened (Fitzpatrick et al., 2009).

At least two modes of Ca²⁺ wave spread have been identified: a continuous model that depends on continuous underlying substrate of regenerative potential, and a pseudosaltatory model where Ca²⁺ regeneration occurs at discrete points with diffusion between them. (We call it pseudo-saltatory here to distinguish from classical saltatory conduction in myelinated fibers, involving capacitative effects in addition to electrodiffusion.) It has been hypothesized that these two modes produce downstream functional differences in dendrites, where many mechanisms are responsive to the level of Ca²⁺, for example I_h (Winograd et al., 2008; Neymotin et al., 2013, 2014) and synaptic plasticity (Kotaleski and Blackwell, 2010).

To investigate Ca²⁺ waves in a spatiotemporal context relevant for neurons, we developed a model of Ca²⁺ waves which includes cytosol and ER. Baseline cytosolic Ca²⁺ and IP₃ concentration are set to low values. The model includes ER SERCA pumps (pump Ca²⁺ from cytosol into ER), leak channels (Ca²⁺ leaks out of ER), and IP₃Rs. Our model generates Ca²⁺ waves with realistic physiological properties. We use our model to investigate how IP₃R density and clustering, alterations in SERCA density, alterations in ER stacking, and Ca²⁺ diffusibility alter waves. An additional complexity arises when we consider coupling to plasma membrane calcium channels which will contribute additional calcium flux triggered by rapidly-spreading regenerative voltage changes on the membrane. We have therefore also assessed our results in an extended electrochemical model, which included a variety of ion channels (leak, voltage-dependent calcium channels: VGCC, potassium, sodium) with synaptic activation. We used this model to demonstrate how priming due to AMPA-mediated membrane depolarization would enhance subsequent Ca²⁺ wave amplitude and duration.

2 Materials and methods

All simulations were run in the NEURON (version 7.3) simulation environment (Carnevale and Hines, 2006). NEURON has traditionally supported electrical modeling but has recently been extended to support reaction-diffusion (R×D) modeling as well (McDougal et al., 2013a,b). The full 1D R×D calcium wave model of the neuronal dendrite (depicted in Fig. 1) and the data analysis source-code is available on ModelDB (Peterson et al., 1996; Hines et al., 2004) (http://senselab.med.yale.edu/modeldb).

Figure 1.

Schematic of the dendrite model (1000 µm length; 1 µm diameter) showing the extracellular space, cytosol and endoplasmic reticulum (ER). Thick black lines represent the plasma and ER membranes. The extracellular space provides a source and sink for multiple molecules. Ca²⁺, Na⁺, K⁺, and IP₃ are depicted as black circles. These molecules enter and exit the dendrite and its sub-compartments via the channels/receptors (gray pores) in directions indicated by arrows. SERCA pump moves cytosolic Ca²⁺ into the ER. Lightning symbols represent stimulus locations (AMPA-mediated depolarization; mGluR-mediated IP₃ augmentation). Dotted lines between molecules and receptors indicate the receptor is modulated by the molecule (e.g., Ca²⁺ activates K_Ca and IP₃R).

Our Ca²⁺ dynamics are derived from Wagner et al. (2004), a spatial variant of Li and Rinzel (1994). Parameters are as in Table 1. We modeled a one-dimensional R×D system of intracellular neuronal Ca²⁺ waves in an unbranched apical dendrite of a hippocampal pyramidal neuron (length of 1000 µm and diameter of 1 µm). Within the dendrite, we modeled cytosolic and endoplasmic reticulum (ER) compartments by using a fractional volume for each: suppose that for a given cell volume, f_ER denotes the fraction occupied by the ER (0.17), and f_cyt denotes the fraction occupied by the cytosol (0.83). Necessarily f_cyt + f_ER ≤ 1. The inequality is strict if other structures are present, such as mitochondria.

Table 1.

Baseline parameters for the Ca²⁺ wave model.

fcyt = 0.83

fER = 0.17

p̄IP3R = 120400.0 molecules/mM/ms

kIP3 = 0.00013 mM

kact = 0.0004 mM

kinh = 0.0019 mM

p̄leakER = 18.06 molecules/mM/ms

p̄serca = 1.9565 molecules/ms

kserca = 0.0001 mM

τIP3R = 400 ms

$d_{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}=0.08\mu {\mathrm{m}}^2/\mathrm{m}\mathrm{s}$

$d_{\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}}=0.08\mu {\mathrm{m}}^2/\mathrm{m}\mathrm{s}$

dIP3 = 1.415 μm2/ms

ER-based calcium dynamics were previously modeled (De Young and Keizer, 1992). This previous work showed by a time-scale analysis that the qualitative dynamics of IP₃ receptors (IP₃Rs) could be represented by only considering the slow Ca²⁺ inactivation binding site state (Li and Rinzel, 1994). This work formed the basis of much subsequent work in intracellular Ca²⁺ dynamics (Fall and Rinzel, 2006; Fall et al., 2004; Hartsfield, 2005; Peercy, 2008; Wagner et al., 2004). The model presented here is a variant that neglects dynamic IP₃ production (Wagner et al., 2004).

The ER Ca²⁺ model involves IP₃Rs and SERCA pumps. As with the plasma membrane model, we combine the net effects of all other channels on the ER into a leak channel. We denote by J_IP3R, J_SERCA, and J_leakER the mass flux per unit volume due to the IP₃R, SERCA pump, and leak channels, respectively. Dividing the mass flux by the volume fraction gives the change in concentration.

State Variables

Cytosolic Ca²⁺ concentration, ER Ca²⁺ concentration, IP₃ concentration, and IP₃R gating are denoted by $\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+},\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}$, IP₃, and h_IP3R, respectively.

$$\frac{\partial \mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}{\partial t}=d_{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}·\mathrm{\Delta }\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}+\frac{J_{\mathrm{I}\mathrm{P}3\mathrm{R}}-J_{\mathrm{S}\mathrm{E}\mathrm{R}\mathrm{C}\mathrm{A}}+J_{\text{leak}\mathrm{E}\mathrm{R}}}{f_{\text{cyt}}}+c_{\text{ionic}}$$ (1)

$$\frac{\partial \mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}}{\partial t}=d_{\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}}·\mathrm{\Delta }\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}-\frac{J_{\mathrm{I}\mathrm{P}3\mathrm{R}}-J_{\mathrm{S}\mathrm{E}\mathrm{R}\mathrm{C}\mathrm{A}}+J_{\text{leak}\mathrm{E}\mathrm{R}}}{f_{ER}}$$ (2)

$$\frac{\partial \mathrm{I}{\mathrm{P}}_3}{\partial t}=d_{\mathrm{I}{\mathrm{P}}_3}·\mathrm{\Delta }\mathrm{I}{\mathrm{P}}_3$$ (3)

$$\frac{\partial h_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}}{\partial t}=\frac{h_{\infty \mathrm{I}{\mathrm{P}}_3\mathrm{R}}-h_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}}{{\tau }_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}}$$ (4)

where c_ionic denotes the net flux into the cytosol due to ion channels on the cell membrane and depends on both space and time. The first terms of the right hand sides is the contribution from diffusion. IP₃ diffusion within the cytosol is passive. Equation (1) allows for the coupling between Ca²⁺ that enters the cytosol via the ion channels on the cell membrane and the intracellular Ca²⁺ which is shuttled into ER via SERCA pumps.

Fluxes

Flux from the IP₃R, SERCA pump, and leak channels are denoted by J_IP3R, J_SERCA, and J_leakER, respectively.

$$J_{\mathrm{I}\mathrm{P}3\mathrm{R}}={\overline{p}}_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}·m_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}^3·n_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}^3·h_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}^3·(\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}-\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+})/\mathrm{\Xi }$$ (5)

$$J_{\mathrm{S}\mathrm{E}\mathrm{R}\mathrm{C}\mathrm{A}}=-\frac{{\overline{p}}_{serca}·\mathrm{C}{{\mathrm{a}}_{\text{cyt}}^{2+}}^2}{(k_{serca}^2+\mathrm{C}{{\mathrm{a}}_{\text{cyt}}^{2+}}^2)·\mathrm{\Xi }}$$ (6)

$$J_{\text{leak}\mathrm{E}\mathrm{R}}={\overline{p}}_{\mathit{\text{leak}}ER}·(\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}-\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+})/\mathrm{\Xi }$$ (7)

Here, Ξ = N_A/10¹⁸ ≈ 602214.129, and is the number of molecules in a cubic micron at a concentration of 1 mM, where N_A is the Avogadro constant. The SERCA pump is a pump rather than a channel and so is modeled with Hill-type dynamics. The form of the fluxes J_IP3R and J_leakER parallels the forms for ion channels in the Hodgkin-Huxley equations. Leak is ungated whereas J_IP3R is gated: m_IP3R and n_IP3R are fast gating variables depending on IP₃ and Ca²⁺, and h_IP3R is the slow Ca²⁺ inactivation gating variable.

$$m_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}=\frac{\mathrm{I}{\mathrm{P}}_3}{\mathrm{I}{\mathrm{P}}_3+k_{\mathrm{I}{\mathrm{P}}_3}}$$ (8)

$$n_{\mathrm{I}{\mathrm{P}}_3\mathrm{R}}=\frac{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}+k_{\mathit{\text{act}}}}$$ (9)

$$h_{\infty \mathrm{I}{\mathrm{P}}_3\mathrm{R}}=\frac{k_{inh}}{k_{inh}+\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}$$ (10)

Initial values of $\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}$, C_AVG, IP₃, and h_IP3R were set to 0.0001 mM, 0.0017 mM, 0.1 mM, and 0.8, respectively. $\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}$ was then adjusted based on the cytosolic and ER fractional volumes according to: $\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}=\frac{C_{\text{AVG}}-\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}·f_{\mathit{\text{cyt}}}}{f_{ER}}$

This model supports both bistable and excitable waves. Both types of waves have been observed in the Xenopus oocyte (Fontanilla and Nuccitelli, 1998; Lechleiter et al., 1991), although not clearly defined in neurons.

Electrical dynamics

Electrical dynamics were utilized for the subset of simulations described in the final figure. Electrical dynamics in the dendrite followed the standard parallel conductance model. Ion channels were based on a prior model of a hippocampal pyramidal cell (Safiulina et al., 2010) (http://senselab.med.yale.edu/modeldb/ShowModel.asp?model=126814) and included L-, N-, and T-type Ca²⁺ channels, which allowed extracellular calcium to enter the dendrite at different voltage levels (McCormick and Huguenard, 1992; Kay and Wong, 1987). The equations for the ion channels follow, leak, Na⁺ and K⁺ currents were represented by the conductance approximation: I_ion = g_ion · (V − E_ion) (g conductance; E reversal potential) using E_Na = 50 mV; E_k = −77 mV; E_leak= −64 mV; (g_leak = 39.4 · 10⁻⁶ S/cm²), while the Goldman-Hodgkin-Katz (GHK) flux equation was used for Ca²⁺ currents: I_Ca = p_Ca · GHK_Ca (p permeability). Channel dynamics were corrected for temperature by a Q₁₀ using the factor of $qt=Q_{10}^{(T-25)/10)}$ with T=37°Celsius and 25° was taken to be the temperature at which the experiment was done. Conductances and activation curves were not corrected for temperature (Iftinca et al., 2006). Voltage sensitive channels largely followed variants on the Hodgkin-Huxley formalism, whereby $\frac{dx}{dt}=\frac{x_{\infty }-x}{{\tau }_x}$ using steady-state value: $x_{\infty }=\frac{{\alpha }_{\infty }}{{\alpha }_{\infty }+{\beta }_{\infty }}$, the τ_x forms either

$${\tau }_{\mathrm{E}\mathrm{Q}\mathrm{N}1}:{\tau }_x=\frac{1}{qt·({\alpha }_{\tau }+{\beta }_{\tau })}\mathbf{\text{ or }}{\tau }_{\mathrm{E}\mathrm{Q}\mathrm{N}2}:{\tau }_x=\frac{{\beta }_{\tau }}{qt·a_0·(1+{\alpha }_{\tau })}$$

were where x is m or n for an activation variable and h for an inactivation variable. F is Faraday’s constant; R is the gas constant. Variations on this scheme are noted below.

• L-type Ca²⁺ channel: $p_L={\overline{p}}_L·m_L^2·h_L$ with p̄_L = 10⁻⁶cm/s; h_L (inactivation) was Ca²⁺ dependent: $h_L(\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+})=\frac{k_i}{k_i+\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}$ with k_i = 0.001 mM; $m_L:{\alpha }_{\infty }=\frac{15.69·(-V+81.5)}{\text{exp}((-V+81.5)/10.0)-1.0}$; β_∞ = 0.29 · exp(−V/10.86); τ_EQN2 with α_τ = exp(0.0378 · 2 · (V − 4)), β_τ = exp(0.0378 · 2 · 0.1 · (V − 4)); a₀=0.1; Q₁₀=5.

• T-type Ca²⁺ channel: $p_T={\overline{p}}_T·m_T^2·h_T$ with p̄_T = 10⁻⁶cm/s; $m_T:{\alpha }_{\infty }=\frac{0.2·((-V+19.26)}{\text{exp}(((-V+19.26)/10.0)-1.0)}$; β_∞ = 0.009 · exp(−V/22.03); τ_EQN2 with α_τ = exp(0.0378 · 2 · (V − (−28))); β_τ = exp(0.0378 · 2 · 0.1 · (V − (−28))); h_T: α_∞ = 10⁻⁶ · exp(−V/16.26); ${\beta }_{\infty }=\frac{1}{\text{exp}((-V+29.79)/10)+1}$; τ_EQN2 with α_τ = exp(0.0378 · 3.5 · (V − (−75))); β_τ = exp(0.0378 · 3.5 · 0.6 · (V − (−75))); a₀=0.04; Q₁₀=5.

• N-type Ca²⁺ channel: $p_N={\overline{p}}_N·m_N^2·h_N·h{2}_N$ with p̄_N = 10⁻⁶cm/s; m_N used α_∞ = 0.1967 · (−V +19.88)/(exp((−V +19.88)/10.0)−1.0); β_∞ = 0.046 · exp(−V/20.73); τ_EQN2 with α_τ = exp(0.0378 · 2 · (V − (−14))); β_τ = exp(0.0378 · 2 · 0.1 · (V − (−14))); h_N following: α_∞ = 1.6 · 10⁻⁴ · exp(−V/48.4); β_∞ = 1/(exp((−V +39.0)/10)+1); constant τ_h = 80; with $h{2}_N=\{0.001\}/\{0.001+\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}\}$; a₀=0.03, Q₁₀=5.

The model also contained: a transient sodium channel I_Na and a delayed rectifier channel I_K−DR to allow for action potential generation; a calcium-dependent potassium channel which hyperpolarized the cell after calcium influx; and an A-type potassium channel for rapid inactivation. Equations for these channels follow.

• Na channel: $g_{Na}={ḡ}_{Na}·m_{Na}^3·h_{Na}·s_{Na}$ with ḡ_Na = 0.11 S/cm²; m_Na using τ_EQN1 with: α = 0.4 · (V −(−30+6))/(1−exp(−(V −(−30+6)/7.2))); β = 0.124 · (−V − (30 − 6))/(1 − exp(−(V − (30 − 6)))/7.2) h_Na with special form $h_{\infty }=\frac{1}{1+\text{exp}((V-(-50)-6)/4)}$ and using τ_EQN1 with α_τ = 0.03 · (V −(−45+6))/(1−exp(−(V −(−45+6))/1.5)) and β_τ = 0.01·(V −(45−6))/(1− exp(−(V − (45 − 6))/1.5)); Q₁₀=2. s_Na: s_∞(V) = 1; τ_EQN2 with α_τ = exp(10⁻³ · 12 · (V + 54) · 9.648 · 10⁴/(8.315 · (273.16 + T))) and β_τ = exp(10⁻³ · 12 · 0.2 · (V + 54) · 9.648 · 10⁴/(8.315 · (273.16 + T))).

• Delayed rectifier K channel: g_K−DR = ḡ_K−DR · n_K−DR where ḡ_K−DR = 0.01 S/cm²; n_K−DR following an atypical steady-state: $n_{\infty }=\frac{1}{1+\alpha }$ with α = exp(10⁻³ · −3 · (V − 19) · 9.648·10⁴/(8.315 · (273.16+T))) using the same α in τ_EQN1 with β = exp(10⁻³ · −3 · 0.7 · (V − 19) · 9.648 · 10⁴/(8.315 · (273.16 + T))); a₀=0.02; Q₁₀=1.

• BK-type Ca²⁺dependent potassium channel (K_Ca): g_KCa = ḡ_KCa · o_KCa with ḡ_KCa = 0.009 S/cm²; o_KCa: $\alpha =\{0.28·\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}\}/\{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}+k1·\text{exp}(-2·0.84·F·V/R/(273.15+T)\}$; $\beta =0.48/\{1+\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}/k2·\text{exp}(-2·F·V/R/(273.15+T))\}$ with k1 = 0.48·10⁻³; k2 = 0.13·10⁻⁶; using τ_EQN1.

• A-type potassium channel, K_A: g_A = ḡ_A · n_A · l_A where ḡ_A = 0.02 S/cm²; n_A with τ_EQN2 and an atypical steady-state value: $n_{\infty }=\frac{1}{1+\alpha }$ with α = exp(10⁻³·ζ (V) · (V − 17) · 9.648 · 10⁴/(8.315 · (273.16 + T))); β_n(V) = exp(10⁻³ · ζ(V) · 0.55 · (V − 17) · 9.648 · 10⁴/(8.315 · (273.16+T))); ζ(V) = −1.5+(−1/(1+exp((V +34)/5))); a₀=0.05; Q₁₀=5. l_A: with steady-state: $l_{\infty }=\frac{1}{1+\alpha }$; α = exp(10⁻³ · 3 · (V + 50) · 9.648 · 10⁴/(8.315 · (273.16 + T))); time-constant: τ = 0.26 · (V + 44) · qt; Q₁₀=1.

Synapses

The AMPA receptor was modeled with a double-exponential mechanism: rise 0.05 ms, decay 5.3 ms, E_AMPA 0 mV. A single AMPA synapse was placed at 500 µm – midway on the dendrite; synaptic weight = 0.5 µS. The metabotropic synapse was not modeled in detail but was given as a local 12.5× increase in IP₃ (1.25 µM) in a 4 micron segment at the same location.

Simulation variations

In one set of simulations, we changed the density of IP₃R and SERCA, by scaling p̄_IP3R and p̄_serca, respectively. IP₃R hotspots were modeled by scaling p̄_IP3R at discrete locations, while keeping IP₃R density in between the hotspots at 0.8× baseline IP₃R density. ER stacks were simulated by scaling p̄_IP3R, p̄_serca, and p̄_leakER simultaneously, while keeping ER density in between the stacks at 0.8× baseline density. Changes in Ca²⁺ buffering were simulated by scaling the Ca²⁺ diffusion coefficients, $d_{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}$ and $d_{\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}}$, in equal amounts. We also modulated d_IP3 in a set of simulations. The range of parameter values examined were constrained to generate Ca²⁺ waves with wave features (onset, speed, duration, amplitude) that were around ranges for experimentally-observed limits in neurons (Nakamura et al., 1999; Ross et al., 2005; Fitzpatrick et al., 2009).

Electrochemical simulation variations

One set of simulations (Fig. 11) was run with the electrical dynamics (see Electrical dynamics above) to determine how AMPA-ergic stimulation’s alteration of membrane potential prior to an IP₃ stimulus impacts the subsequent Ca²⁺ waves. Simulations were run for 12 s. Background IP₃ concentration in these simulations was set between 0–0.01 mM to prevent spontaneous oscillations. AMPA activations (150 spikes with interspike interval of 25 ms) finished 250 ms prior to IP₃ stimulus (amplitude: 2.5 mM).

Figure 11.

Electrical stimulation with increased number of AMPA activations enhances Ca²⁺ waves induced by IP₃ (2.5 mM at 7 s). (a) Control simulation: Ca²⁺ wave with no AMPA inputs prior to the IP₃ stimulus. (b) Ca²⁺ wave with train of 150 AMPA inputs (onset: 3 s; interspike interval: 25 ms) prior to the IP₃ stimulus. (c) Comparison of voltage (top), ER Ca²⁺ (middle), and cytosolic Ca²⁺ (bottom) in control (black) and simulation with 150 AMPA inputs (red).

Data analysis

Ca²⁺ and IP₃ concentrations were recorded in a 3D array using temporal and spatial resolution of 5 ms and 1 µm. Cytosolic Ca²⁺ wave features were extracted by thresholding the array using an amplitude threshold of 2× baseline cytosolic Ca²⁺ concentration (2.5× for the electrical simulations). Wave onset was defined as the delay after IP₃ stimulus until the Ca²⁺ passed threshold. Amplitude was maximum Ca²⁺ concentration. Speed was calculated using onset time at stimulation location and final onset time at location of wave termination (note that use of final onset time leads to artifactual lowering of wave speed in pathological condition of multiple Ca²⁺ waves). Duration was calculated as median at each position from wave onset to offset (going below threshold).

3 Results

This study involved over 4800 15–30 second simulations, testing how variations in levels of IP₃R density, SERCA density, IP₃ hotspots, and ER stacks, altered wave initiation, amplitude, speed, and duration. An additional set of 32 twelve-second simulations were run, testing how the number of AMPA inputs prior to an IP₃ stimulus impacted the calcium waves. Simulations were run using the NEURON simulator on Linux on a 2.27 GHz quad-core Intel XEON CPU. Thirty seconds of simulation ran in 1.5–2 minutes, depending on simulation type.

We simulated the arrival of IP₃ from a metabotropic receptor activation cascade as an instantaneous augmentation of IP₃ concentration to 1.25 µM IP₃ (12.5× background) in a 4 micron segment in mid-dendrite (Fig. 2). IP₃ then spread gradually along the dendrite, providing a permissive effect for Ca²⁺ activation of the IP₃ receptors (IP₃Rs). Activation of IP₃Rs permitted release of Ca²⁺ from the endoplasmic reticulum (ER) stores into the cytosol (Fig. 2a). The elevation in cytosolic Ca²⁺ was mirrored by a depression in ER Ca²⁺ as the wave passed (Fig. 2b). The cytosolic Ca²⁺ wave started once adequate Ca²⁺ had built up to spillover and activate neighboring IP₃Rs and initiate sustained positive feedback of Ca²⁺ induced-Ca²⁺-release (CICR). The wave then spread bidirectionally, as Ca²⁺ and IP₃ simultaneously diffused laterally. Ca²⁺ waves were not able to reverse direction and propagate back towards their source due to an IP₃R refractory period provided by inactivation gating. Parameters including d_IP3 (1.415 µm²/ms), $d_{\mathrm{C}{\mathrm{a}}_{\text{cyt}}^{2+}}(0.08\mu {\mathrm{m}}^2/\mathrm{m}\mathrm{s}),d_{\mathrm{C}{\mathrm{a}}_{\mathrm{E}\mathrm{R}}^{2+}}(0.08\mu {\mathrm{m}}^2/\mathrm{m}\mathrm{s})$, p̄_IP3R (120400.0 molecules/mM/ms), p̄_serca (1.9565 molecules/ms), and p̄_leakER (18.06 molecules/mM/ms) were adjusted from modeling (Wagner et al., 2004) and experiments (Allbritton et al., 1992), to produce a wave speed of 77 µm/s, which is comparable to neuronal Ca²⁺ wave speeds measured experimentally (68±22 µm/s in Nakamura et al. (1999)). These parameters provided our baseline simulation for further parameter explorations. Median Ca²⁺ elevation lasted ~ 1 s and peaked at 1.6 µM, both also comparable with experimental results (Ross et al., 2005; Fitzpatrick et al., 2009).

Figure 2.

Ca²⁺ wave propagation with baseline parameters. Elevated IP₃ stimulus placed at mid-dendrite (500 µm on y-axis) after 2 s past start of simulation. (a) cytosolic [Ca²⁺] shows a wave of increased concentration (b) ER [Ca²⁺] shows a mirror image wave of decreased concentration as Ca²⁺ is released to cytosol.

3.1 IP₃R density modulates excitability

IP₃R was necessary for wave generation (Fig. 3). Ca²⁺ waves did not begin below IP₃R density of 92.2% of baseline, when the first wave was obtained (Fig. 3a and left red dots on c,d,e,f). This wave had initiation delays ~100 ms longer compared to the baseline wave (Fig. 2a and middle red dot on Fig. 3c), with only a slightly lowered amplitude, lower speed, and lower duration. The major effect, time to onset, was strongly dependent on IP₃R due to the need to gradually source a sufficient amount of Ca²⁺ to produce enough lateral spillover to initiate a positive feedback cycle, as the lower IP₃R density produced a lower total Ca²⁺ flux from ER to cytosol. ER Ca²⁺ must also be sourced at a rate that exceeds the reuptake governed by SERCA pumping. With increased density, time to onset decreased rapidly to a low of 25 ms.

Figure 3.

Ca²⁺ wave propagation is sensitive to the density of IP₃R. From left to right, the three red dots in c–f correspond to features of the following Ca²⁺ waves: (a) low IP₃R density (0.9× baseline); baseline IP₃R density (Fig. 2); (b) high IP₃R density (1.8× baseline). (c) time to wave onset (0 indicates no wave), (d) wave propagation speed, (e) duration (2× Ca²⁺ elevation; median duration across dendritic locations) (f) peak amplitude. (c–f each show 91 different equally spaced parameter values) (Colorscale in (a) and (b) same as in Fig. 2.)

Speed increased proportionately with increased IP₃R (Fig. 3d). At the lowest value, wave speed was 72 µm/s. Speed depended on both the rate of ER sourcing at each point of wave regeneration, and the rate of diffusion required to reach the next set of IP₃Rs laterally. Proportionate increase in speed with density was due to the reduced requirement for additional Ca²⁺ for positive feedback, since less Ca²⁺ is needed to trigger the release of Ca²⁺. At each location, IP₃R must source adequate Ca²⁺ not only for positive feedback and spillover, but also to exceed the local sink back into ER provided by the SERCA pump. Ca²⁺ wave duration and amplitude both had a small positive association with increases in IP₃R (Fig. 3e,f), due to the higher density of IP₃Rs liberating more Ca²⁺ for the wave, which then remained elevated longer.

3.2 SERCA opposes IP₃R effects

The SERCA pump returns Ca²⁺ back into the ER, opposing the release from IP₃R conductance. Therefore, the parameter dependence of measures was generally opposite to that seen with IP₃R density (Fig. 4). The least excitable wave was produced at the highest SERCA density tested (Fig. 4b). The most excitable wave shown here, at 0.66× baseline SERCA (Fig. 4a), was caused by diminished Ca²⁺ reuptake after release. Further reduction in SERCA resulted first in spontaneous Ca²⁺ waves and then in simultaneous release from all ER sites at once due to regenerative responses starting from baseline levels (not shown).

Figure 4.

Ca²⁺ wave propagation is sensitive to the density of SERCA. Tested by varying SERCA density relative to baseline (n = 45). (a) hyper-excitable Ca²⁺ wave produced with 0.66× baseline SERCA, (b) diminished Ca²⁺ wave produced with 1.03× baseline SERCA, (c) time to Ca²⁺ wave onset (0 indicates no wave at the SERCA level), (d) speed, (e) duration (median duration of Ca²⁺ elevation across dendritic locations), (f) peak amplitude. Left (right) red dots in c–f corresponds to activity shown in (a) ((b)). (Colorscale in (a) and (b) same as in Fig. 2.)

Patterns of wave measures were similar to, but reversed from, those seen with IP₃R increase. Onset to the first wave varied from 30 to 330 ms. Shorter onset times were produced by faster Ca²⁺ accumulation with less reuptake. Similarly, the most excitable wave had a faster propagation speed (84 µm/s) and longer duration due to longer-lasting Ca²⁺ supporting faster spread. Amplitude showed a slight inverse relationship with increasing SERCA density, due to the higher SERCA reuptake of Ca²⁺ into ER stores diminishing Ca²⁺ availability for the wave (Fig. 4f).

At the maximal SERCA consistent with wave propagation, 1.07× baseline, onset to wave initiation was increased dramatically (330 ms), wave speed decreased to ~75 µm/s, and duration decreased to 0.82 s (Fig. 4b). Ca²⁺ waves were no longer able to initiate at higher levels, as Ca²⁺ was sucked back into the ER before a Ca²⁺ wave could ignite.

3.3 IP₃R / SERCA balance regulates wave induction

The balance between IP₃R and SERCA density determined whether Ca²⁺ waves could be elicited via IP₃ stimulation (Fig. 5). Excessive Ca²⁺ or insufficient IP₃R would not allow a wave (regions 1,4). Maximum Ca²⁺ level in region 1 was low compared to regions 2 and 3, since ER Ca²⁺ had already partially leaked out of the ER prior to the IP₃ stimulus, and then diffused within the dendrite, preventing the large, localized rise in cytosolic Ca²⁺ which formed a Ca²⁺ wave.

Figure 5.

Ca²⁺ wave propagation is sensitive to density of IP₃Rand SERCA. 2838 simulations varying IP₃R (86 levels) and SERCA density (33 levels) relative to baseline shows different dynamics: region 1: sustained Ca²⁺ elevation precluded IP₃-evoked wave; region 2: multiple Ca²⁺ waves; region 3: single Ca²⁺ wave; region 4: insufficient IP₃R to source Ca²⁺ wave. (a) time to Ca²⁺ wave onset where present, (b) speed where wave present, (c) duration where wave present (median duration of Ca²⁺ elevation across dendritic locations), (d) maximum calcium level.

Moving from region 1 towards region 2, IP₃R density decreased and SERCA density increased, both of which reduced cytosolic Ca²⁺, which was no longer sustained above threshold throughout the simulation. This allowed the IP₃ stimulus to elicit distinct waves. However, the high levels of IP₃R produced multiple waves without return to a subthreshold state. These repetitive waves caused an artifactual lowering of measured wave speed (see Methods for details of calculation). Ca²⁺ amplitude was higher than in region 1 due to repeated higher elevations of cytosolic Ca²⁺ compared to balanced fluxes in region 1.

In region 3, there was a match across IP₃R and SERCA density, allowing for a single wave to be elicited from the IP₃ stimulus. Within this region, for any given SERCA density, an increase in IP₃ density increased Ca²⁺ flux from ER to cytosol, reducing time to onset, and increasing speed, duration, and amplitude. In region 4, SERCA dominated the dynamics, shuttling Ca²⁺ back into the ER too quickly to allow IP₃ stimulation to initiate a Ca²⁺ wave.

3.4 IP₃ diffusibility regulates wave onset and speed

Altering the IP₃ diffusion constant (d_IP3) had a pronounced effect on Ca²⁺ wave initiation (Fig. 6). With low d_IP3 (0.1415 µm²/ms), the IP₃ stimulus remained localized in the central stimulus region (~500 µm), producing a high prolonged local elevation with more rapid activation of local IP₃Rs and shorter wave onset time (40 ms). Ca²⁺ near the stimulus then remained elevated longer than at other dendritic locations. Since the IP₃ stimulus spread to neighboring dendritic locations more slowly, Ca²⁺ wave speed was slightly reduced (73.5 µm/s).

Figure 6.

Ca²⁺ wave propagation is sensitive to IP₃ diffusion coefficient (d_IP3). From left to right, red dots in c–f correspond to features of Ca²⁺ waves with (a) low (0.1415 µm²/ms); baseline (1.415 µm²/ms; Fig. 2); (b) high (1.981 µm²/ms) d_IP3. (c) time to wave onset, (d) wave propagation speed, (e) duration (2× Ca²⁺ elevation; median duration across dendritic locations) (f) peak amplitude. (Colorscale in (a) and (b) same as in Fig. 2.)

High d_IP3 (1.981 µm²/ms) significantly delayed the onset of the Ca²⁺ wave (230 ms; Fig. 6b), due to IP₃ diffusing quickly away, producing less local IP₃R stimulation and increasing the time required to trigger the wave. However, once the wave was initiated, it spread slightly faster (77.6 µm/s), the pre-arriving IP₃ giving a slight boost at subsequent locations along the dendrite. Amplitude had a minimal inverse relationship with increasing d_IP3 (Fig. 6f) because local IP₃ spread out more quickly, reducing local IP₃R activation. At d_IP3 values larger than 1.981 µm²/ms, IP₃ was so diffuse, that it could not elicit the Ca²⁺ wave.

Overall, increasing d_IP3 caused delayed onet (Fig. 6c), a minor increase in speed (Fig. 6d), and no appreciable change in duration or amplitude (Fig. 6ef).

3.5 Pseudo-saltatory waves via IP₃R hotspots

The prior simulations were performed using uniform density of ER mechanisms at all locations. However, there is evidence of inhomogeneities, for example at dendritic branch-points, where elevations in local IP₃R density (IP₃R hotspots) might contribute to assisted propagation of Ca²⁺ waves at these sites of potential failure (Fitzpatrick et al., 2009). In neurons, hotspots have average center-to-center spacing of approximately 20 µm (edge-to-edge spacing of 10 µm) but show considerable variability in the spacing (Fitzpatrick et al., 2009). In the following simulations, we varied hotspot IP₃R density while keeping the inter-hotspot IP₃R density at 80% of baseline.

Both IP₃R hotspot strength and spacing altered Ca²⁺ wave speeds (Fig. 7). We started with a 20% reduction in density between hotspots (Fig. 7a), with the hotspots having 93% of the IP₃R density used in Fig. 2. This alteration reduced wave speed from 77 to 68 µm/s. However, the propagation pattern differed qualitatively, with its saltatory nature readily seen as spots of high Ca²⁺ concentration which occur at the locations of IP₃R hotspots. Local velocity parallels amplitude in showing an increase at the hotspots, where activation produces a local highly varying gradient (peaked high second spatial derivative) leading to a higher immediate diffusion speed. Slower wave progression occurs between hotspots where lower IP₃R concentration only slightly boosts the wave progression from diffusion. Increased IP₃R density at hotspots increased wave speed further to 90 µm/s (2.0× baseline; Fig. 7b). Further augmentation provided an approximately linear increase in speed due to faster and larger release of Ca²⁺ from the ER stores at the hotspots (Fig. 7c). Below ~93% of hotspot IP₃R density, the waves did not initiate, since there was insufficient Ca²⁺ release to trigger waves (0 speeds at left of Fig. 7c). Wave propagation also did not occur in the absence of IP₃R between hotspots, with the threshold for Ca²⁺ wave initiation having inter-hotspot and hotspot IP₃R densities of ~0.66 and ~0.93× baseline, respectively.

Figure 7.

Comparison of waves (and wave speeds) generated from varying IP₃R hotspot density (a–c) and separation (d–f). (a) 20 µm spacing; 0.93× density. (b) 20 µm spacing; 2× density. (c) wave speed as a function of IP₃R hotspot density, (d) 15 µm spacing; 1.87× density. (e) 100 µm spacing; 1.87× density. (f) wave speed as a function of IP₃R hotspot spacing. (Colorscale in a,b,d,e same as in Fig. 2.) Red points in (c) ((f)) are from waves in (a),(b) ((d),(e)).

Hotspot spacing also modulated Ca²⁺ wave speeds, with larger spacing producing slowing as Ca²⁺ and IP₃ had to travel further via mildly-boosted diffusion between hotspots before being fully reboosted. At 1.87× baseline IP₃R density with a spacing of 15 µm, the wave had a speed of 100 µm/s (Fig. 7d). Increasing center-to-center spacing between hotspots to 100 µm resulted in reduction of wave propagation speed to 66 µm/s. At the densities shown, hotspot spacing had a larger impact on wave speed (~66–100 µm/s) than IP₃R density (~68–90 µm/s).

Intracellular Ca²⁺ concentration is heavily regulated via buffering mechanisms, in part presumed to provide careful regulation of Ca²⁺-triggered signaling cascades. Ca²⁺ buffering also modulates Ca²⁺ diffusion efficacy and apparent Ca²⁺ diffusion coefficient (D_Ca(App)). Because we were not modeling buffering directly in these simulations, we altered D_Ca(App) instead. Changing D_Ca(App) dramatically altered propagation speed over a wide range (Fig. 8). With diminished D_Ca(App), wave speed was substantially reduced to 25 µm/s (Fig. 8a). The converse of this was that Ca²⁺ was relatively immobile, with duration at one location slightly increased (Fig. 8e). This heightened local Ca²⁺ elevation also allowed for a shorter onset to wave initiation (20 ms; Fig. 8c). Increasing D_Ca(App) above baseline level had opposite effects: wave speed was augmented (288 µm/s) but time to onset was delayed (40 ms), both due to faster spread of Ca²⁺. We also note that while onset varied nearly linearly, speed showed much greater alteration.

Figure 8.

Comparison of waves generated from varying D_Ca(App) in the presence of IP₃R hotspots. Ca²⁺ waves generated when using a Ca²⁺ diffusion coefficient of (a) 0.008 µm²/ms and (b) 0.8 µm²/ms. (c) time to Ca²⁺ wave onset (0 indicates no wave at the Ca²⁺ diffusion level), (d) speed, (e) duration (median duration of Ca²⁺ elevation across dendritic locations), (f) peak amplitude. Left (right) red dots in c–f corresponds to activity shown in (a) ((b)). (Colorscale in (a) and (b) same as in Fig. 2.)

3.6 Pseudo-saltatory waves via ER stacks

There are at least two ways that hotspots could occur in dendrites: Type 1. increased density of IP₃R at particular locations on homogeneous distribution of ER; Type 2. increased “density” (accumulation) of ER at particular locations. The former case was explored in the prior section. The latter case has been identified as locations of ER lamellar specialization that are sometimes described as ER stacks. We next explored such stacks as an alternative type of hotspot, noting that this increase in local ER provides increased density of SERCA and leak as well as IP₃R at the hotspot locations. In these simulations, the ER density between stacks was at 0.8× baseline, consistent with continuous ER throughout the dendrite (Martone et al., 1993; Terasaki et al., 1994).

Increasing ER stack density from ~0.8× (Fig. 9a) to ~2× baseline (Fig. 9b), increased wave speed from 68 µm/s to 86 µm/s (Fig. 9c). This is due to more release of Ca²⁺ from the leak and IP₃R channels, which was opposed by heightened SERCA activity. The duration of Ca²⁺ amplitude elevation was reduced at the ER stacks. In addition, as the ER stack density increased, the onset to wave initiation shortened (220 to 30 ms), due to heightened leak and IP₃R extrusion of Ca²⁺ into cytosol. However, Ca²⁺ elevation duration decreased as the ER stack density increased (965 to 795 ms), due to heightened SERCA pumping.

Figure 9.

Comparison of Ca²⁺ waves generated from varying ER stack properties. Simulations were run by varying density and spacing of ER stacks. ER density in between stacks was set at 0.8× baseline. (a) Ca²⁺ wave from 0.8× ER stack density with 20 µm spacing, (b) Ca²⁺ wave from 2.0× ER stack density with 20 µm spacing, (c) Ca²⁺ wave speed as a function of ER stack density (fixed 20 µm spacing), (d) Ca²⁺ wave from 15 µm ER stack spacing with ~1.87× density, (e) Ca²⁺ wave from 100 µm ER stack spacing with ~1.87× density, and (f) wave speed as a function of ER stack spacing. Note the bumps in Ca²⁺ release around ER stacks. (Colorscale in a,b,d,e same as in Fig. 2.) Red points in (c) ((f)) are from waves in (a),(b) ((d),(e)).

Increasing the center-to-center spacing of ER stacks, while maintaining a fixed ER density of 1.86× baseline (Fig. 9d–f), tended to decrease wave speed (93 to 71 µm/s). This was due to lower overall availability of ER for releasing Ca²⁺ and spreading the wave. Interestingly, with heightened center-to-center spacing (15 and 100 µm) of ER stacks, the duration of heightened Ca²⁺ elevation was increased (755 to 960 ms), due to lower overall presence of SERCA pumps.

Waves could not initiate below a minimum of 0.8× ER stack density due to insufficient Ca²⁺ sourcing. Effects of density and spacing of stacks were generally similar to those seen with IP₃R hotspots. Onset time depended primarily on ER stack density. Variation of D_Ca(App) with ER stacks (Fig. 10), produced results very similar to those seen with Type 1 hotspots (Fig. 8), except that the effects on time to onset were more pronounced. With ER stacks, the onsets were significantly larger (35–70 ms) compared to those for IP₃R hotspots (20–40 ms). These heightened onset times with ER stacks were due to the higher SERCA activity, which reduces cytosolic Ca²⁺ availability.

Figure 10.

Comparison of waves generated from varying D_Ca(App) in the presence of ER stacks. (a) Local Ca²⁺ event generated with Ca²⁺ diffusion coefficient of 0.0 µm²/ms. (b) Ca²⁺ waves generated when using a Ca²⁺ diffusion coefficient of 0.8 µm²/ms. (c) time to Ca²⁺ wave onset, (d) speed, (e) duration (median duration of Ca²⁺ elevation across dendritic locations), (f) peak amplitude. Left (right) red dots in c–f corresponds to activity shown in (a) ((b)). (Colorscale in (a) and (b) same as in Fig. 2.)

3.7 Electrical priming enhances Ca²⁺ waves

Adding electrical dynamics (voltage-gated calcium channels – VGCCs, Na⁺ and K⁺ channels, AMPA synapses) to the dendrite allowed for a more complex set of interactions (Fig. 11). VGCCs admitted extracellular calcium into the cytosol, which was then taken up into the ER via SERCA pumps. Compared to the prior simulations, this increased ER steady-state calcium concentration (0.03 mM), allowing for larger contributions to Ca²⁺ waves. Cytosolic calcium equilibrated to 65 nM, due to a balance between SERCA uptake and VGCC calcium influx. In these simulations, the background IP₃ level was set to 0 mM to prevent spontaneous oscillations due to ongoing activation of IP₃R. Calcium waves were more localized due to this absence of background IP₃.

Activation by a 2.5 mM IP₃ stimulus at 7 s combined with the higher background cytosolic calcium levels to trigger IP₃Rs to release calcium from the ER. ER calcium efflux then triggered a localized calcium wave (Fig. 11a middle,bottom), which spread approximately 150 µm, consistent with experimental data (Fitzpatrick et al., 2009; Hong and Ross, 2007; Ross et al., 2005). The spreading calcium wave produced electrical effects, by contributing to the opening of calcium-dependent potassium channels, which hyperpolarized the membrane to −77 mV. Once the calcium wave had passed, the voltage recovered to baseline.

Prior AMPAergic stimulation (150 pulses; 25 ms interspike interval; 3000–6750 ms) produced spikes and further supplemented the ER calcium stores via a sequence of electrochemical interactions (Fig. 11b,c top): Ca²⁺ entry through VGCCs augmented cytosolic Ca²⁺ (early rise of 65–120 nM at center), augmenting ER Ca²⁺ via uptake (Fig. 11b center). This gradually increased Ca²⁺ throughout the dendrite (0.03–0.045 mM at center), priming the ER. Subsequent IP₃ activation was then able to liberate more ER calcium, inducing a higher-amplitude (7.5 µM) calcium wave (Fig. 11b,c bottom). The larger amount of calcium liberated allowed the wave to spread 50 µm further, and produced a longer duration (2 s), compared to the wave produced without synaptic priming. The heightened availability of Ca²⁺ also enhanced the electrical effects, producing greater voltage suppression due to activation of the hyperpolarizing Ca²⁺-dependent potassium channels.

Increasing the number of AMPA stimuli between 0–150 (interspike interval of 25 ms; AMPA activation at 3 s; IP₃ activation at 7 s) produced nearly linear increases in the IP₃-induced calcium wave amplitude (5.5–7.5µM) and duration (1.6–2.0 s) (not shown). Wave speeds tended to decrease slightly (57–63 µm/s). There was no significant impact of AMPA stimulation on the wave onset, which was relatively fast for all simulations, at 5 ms past the IP₃ stimulation. Using increased background IP₃ (0.01 mM) and increasing the number of AMPA stimuli similarly produced higher amplitude, faster Ca²⁺ waves with IP₃ stimulation (not shown).

4 Discussion

We have developed a reaction-diffusion (R×D) model of neuronal Ca²⁺ waves with multiple compartments (ER and cytosol), multiple diffusing species (Ca²⁺ and IP₃), and multiple ER membrane mechanisms (IP₃ receptors, SERCA pumps, and ER leak). IP₃Rs opened in response to a cytosolic Ca²⁺ and IP₃ stimulus, admitting more Ca²⁺ into the cytosol. Ca²⁺ and IP₃ then diffused and bound to neighboring IP₃Rs, triggering further release of Ca²⁺ and initiating the process of Ca²⁺-induced-Ca²⁺-release (CICR). The Ca²⁺ waves had properties matching those observed in vitro (Ross et al., 2005), including wave speed (~50–90 µm/s), amplitude (2 µM), and duration (~1 s; Fig. 2). We used our model to compare wave propagation in 3 scenarios: continuous IP₃R distribution, IP₃R hotspots, and ER stacks.

Because the cytosolic Ca²⁺ required for CICR depended on a supply from ER stores via the action of IP₃Rs, the Ca²⁺ waves demonstrated primary sensitivity to IP₃R density, such that wave initiation required a minimal level of IP₃R. IP₃R density correlated negatively with onset, and positively with amplitude, speed, and duration of Ca²⁺ waves (Fig. 3). SERCA pumps were responsible for setting the pace of reuptake of cytosolic Ca²⁺ into the ER and had a large impact on the properties of Ca²⁺ waves (Fig. 4). Low SERCA triggered the generation of hyper-excitable and spontaneous waves. High SERCA diminished wave spread and speed, while increasing onset to wave initiation.

We compared two hypothesized mechanisms for saltatory waves: 1. heightened density of IP₃R (IP₃R hotspots), and 2. heightened density of ER (ER stacks), where ER stacks heighten local leak and SERCA, as well as IP₃R. IP₃R hotspots were more effective in setting the pace of wave propagation and onset, whereas ER stack augmentation of Ca²⁺ waves was less pronounced due to the opposing effects of SERCA (leak and IP₃R increased excitability, while SERCA decreased excitability). We hypothesize that IP₃R hotspots will predominate since they are the more effective mechanism for boosting Ca²⁺ waves.

Intracellular Ca²⁺ concentration is heavily buffered. Although our model did not contain an explicit representation of buffering mechanisms, we used our model to test the effects of alterations in buffering capacity by modulating the apparent Ca²⁺ diffusion constant (D_Ca(App)). We found that D_Ca(App) had potent effects on Ca²⁺ wave speed and propagation efficacy. Effects were similar for both IP₃R hotspots (Fig. 8) and ER stacks (Fig. 10). Interestingly, at high Ca²⁺ diffusibility, there were competing effects on excitability: higher speed but longer duration to wave initiation (Fig. 8c & d and Fig. 10c & d).

Finally, we extended our chemical signaling model with electrical components, including a set of calcium and voltage-dependent ion channels and synapses. We used this extended model to demonstrate that electrical activation via AMPA synapses and voltage-gated calcium channels primes ER calcium stores, and contributes to enhanced Ca²⁺ waves on subsequent IP₃ stimulation (Fig. 11).

4.1 Predictions

Our model allows us to make the following experimentally-testable predictions.

1. Lowering apparent IP₃R densities will lead to reduced spread of Ca²⁺ waves. This is testable by using IP₃R antagonists (Taylor and Tovey, 2010).

2. Reductions in SERCA with antagonist (e.g., thapsigargin) will lead to hyperexcitability with multiple or spontaneous Ca²⁺ waves.

3. Dendritic branch-point hotspots will be IP₃R hotspots rather than ER stacks, allowing optimal boosting at locations where the wave might otherwise fail. Testable by using immunohistochemistry to look for IP₃Rs and correlating with electron microscopy evaluation of stacking/ER concentration.

4. Increased buffering with added diffusible buffers provided via whole cell patch will reduce Ca²⁺ wave velocity. This is testable by modulating buffering properties and effectiveness via supply ofMg²⁺⁺ (for parvalbumin modulation), BAPTA (for calbindin modulation), or EGTA (ethylene glycol tetraacetic acid) (Storm, 1987). Additionally, the use of dye indicators, used to follow wave progression, will alter velocity and other wave properties.

4.2 Roles of ER and of Ca²⁺ waves

Since neuronal dendritic trees are very large (hundreds of microns), diffusion alone would make it impossible for dendritic Ca²⁺ to reach presumed targets in the soma and other dendritic locations for use in modulation of physiological processes including: synaptic plasticity, transcription regulation, and membrane current regulation (e.g., I_h). Our modeling emphasizes that Ca²⁺ waves can have large variability in speed and distance of propagation, which will have major effects on how Ca²⁺ is distributed and which targets are hit at what concentrations. However, we note that an entirely different role for ER has been proposed by Shemer et al. (2008). Instead of a role for ER in enhancing Ca²⁺ waves, they suggested that the ER forms a “cable-within-a-cable” which is electrically active and would provide more rapid distal to proximal communication via electrical signaling comparable to that of the plasma membrane.

Pathologically, Ca²⁺ dysregulation may occur via multiple pathways, including heightened IP₃R density, lowered SERCA density, or altered buffering. Dysregulation of Ca²⁺ homeostasis has been implicated in Alzheimer’s disease (Lytton et al., 2014; Rowan and Neymotin, 2013; Rowan et al., 2014) and in ischemia, where Ca²⁺ signaling is an important element in the triggering of apoptosis (Green and LaFerla, 2008; LaFerla, 2002; Stutzmann, 2005; Thibault et al., 1998; Zündorf and Reiser, 2011; Taxin et al., 2014).