# What Is Required for Neuronal Calcium Waves? A Numerical Parameter Study

**Part of the following topical collections:**

## Abstract

Neuronal calcium signals propagating by simple diffusion and reaction with mobile and stationary buffers are limited to cellular microdomains. The distance intracellular calcium signals can travel may be significantly increased by means of calcium-induced calcium release from internal calcium stores, notably the endoplasmic reticulum. The organelle, which can be thought of as a cell-within-a-cell, is able to sequester large amounts of cytosolic calcium ions via SERCA pumps and selectively release them into the cytosol through ryanodine receptor channels leading to the formation of calcium waves. In this study, we set out to investigate the basic properties of such dendritic calcium waves and how they depend on the three parameters dendrite radius, ER radius and ryanodine receptor density in the endoplasmic membrane. We demonstrate that there are stable and abortive regimes for calcium waves, depending on the above morphological and physiological parameters. In stable regimes, calcium waves can travel across long dendritic distances, similar to electrical action potentials. We further observe that abortive regimes exist, which could be relevant for spike-timing dependent plasticity, as travel distances and wave velocities vary with changing intracellular architecture. For some of these regimes, analytic functions could be derived that fit the simulation data. In parameter spaces, that are non-trivially influenced by the three-dimensional calcium concentration profile, we were not able to derive such a functional description, demonstrating the mathematical requirement to model and simulate biochemical signaling in three-dimensional space.

## Keywords

Calcium waves Endoplasmic reticulum Ryanodine receptors 3D modeling Structure-function interplay Numerical simulation## Abbreviations

- CalB
calbindin-D

_{28k}- CICR
calcium-induced calcium release

- ER
endoplasmic reticulum

- IP
_{3} inositol 1,4,5-trisphosphate

- IP
_{3}R IP

_{3}receptor- LIMEX
linearly-implicit extrapolation

- NCX
Na

^{+}/Ca^{2+}exchanger- PMCA
plasma membrane Ca

^{2+}-ATPase- RyR
ryanodine receptor

- SERCA
sarco-/endoplasmic reticulum Ca

^{2+}-ATPase

## 1 Introduction

Intracellular calcium signals define a transition point between electrical signals and biochemical responses in neurons. While basal calcium concentrations in the cytosol are very low, neurons can modulate local cytosolic calcium concentrations to induce microdomain calcium signals, which can integrate to produce longer ranging signal propagation towards the soma. There they reach the nucleus and trigger gene transcription responses relevant for learning [1, 2, 3, 4] and neuroprotection [5, 6, 7, 8]. Cellular calcium signals are shaped by calcium transport mechanisms embedded in the plasma membrane, across which calcium can be bi-directionally exchanged between cytosol and the extracellular space (see [9] for an overview). In addition, intracellular organelles like mitochondria and the endoplasmic reticulum (ER) function as large calcium stores [10, 11, 12, 13, 14, 15, 16]. Organelle membranes are equipped with calcium exchange mechanisms that allow transport of calcium from the cytosol into the organelles or vice versa. Some of these mechanisms—notably the ryanodine receptor channel (RyR) in the ER membrane, which is able to release large amounts of calcium from the ER into the cytosol—have a positive feedback property: Their opening is facilitated by the presence of cytosolic calcium and will trigger the release of even more calcium through surrounding channels. This calcium-induced calcium release mechanism (CICR) overcomes the limited reach of purely diffusive calcium signals in a buffered regime. CICR in neurons has been implicated in various neurodegenerative diseases [17, 18, 19, 20, 21, 22, 23] and has been studied with respect to CICR wave-like properties [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. The importance of CICR wave-like dynamics therefore merits thorough investigation. One major question is how the cellular and intracellular architecture can shape and regulate calcium signals that are able to propagate over long distances, for instance in the context of synapse-to-nucleus communication. It was shown in [26] that variations in the activation of Ca^{2+} store release through inositol 1,4,5-trisphosphate (IP_{3}) receptors were able to significantly change the calcium wave patterns. Additionally, the distances between distinct IP_{3} receptor clusters and the pump strength were shown to control wave stability and instability [37, 38]. Sequestration properties through SERCA pumps [36] and mitochondrially controlled CICR [27] were further implicated in CICR. Related work has shown that calcium signals can modulate the shape of the cell nucleus [39, 40]. Thus, there seem to exist mechanisms in which calcium signals shape the geometry of organelles and the intracellular architecture shapes calcium signals.

The associated calcium model consists of a diffusion-reaction partial differential equation for the propagation of calcium in the cytosol and the reaction of free calcium with the buffer calbindin. Endoplasmic calcium is modeled by a simple diffusion equation. Membrane exchange mechanisms are included via flux boundary conditions at the plasma membrane and the ER membrane: Ca^{2+}-ATPase pumps, Na^{+}/Ca^{2+} exchangers, as well as a leakage term on the plasma membrane; sarco-/endoplasmic reticulum Ca^{2+}-ATPase pumps, ryanodine receptor channels and a leakage flux on the ER membrane.

Our study focused on: (1) how calcium dynamics depend on the parameters dendrite radius, ER radius, and ryanodine receptor density, (2) whether thresholds can be found for stable calcium waves and (3) on gathering information about wave velocities. Simulation results show how cellular architecture, even in this simplified scenario, can determine the shape and fate of calcium waves and their stability. These data were further used to derive empirical laws that explain the behavior of the observed calcium dynamics in distinct geometric regimes. As a side note and in agreement with [26], we also show that the higher spatial dimension (as compared to classical 1D cable models) is indeed required to adequately represent even the most basic properties of calcium wave propagation.

## 2 Model and Methods

### 2.1 Model Domain

We conducted all simulations on a perfectly cylindrical model dendrite with a fixed length of 50 μm and variable radius, containing a cylindrically shaped ER of variable radius positioned exactly at the center of the dendrite; see Fig. 1. We used the rotational symmetry to reduce the problem to two dimensions. Calcium signals were induced by a calcium influx density through the left boundary of the dendrite with an initial strength of \(2.5\times 10^{-18} \ \mathrm{mol}\ \upmu \mathrm{m}^{-2}\ \mathrm{s}^{-1}\) that linearly decreased to zero within 1 ms.

### 2.2 Model Equations

Spatio-temporal calcium dynamics in the intracellular space are modeled by a system of diffusion-reaction equations described in the following. Transmembrane currents through channels and pumps are incorporated into these equations as flux boundary conditions (cf. Sect. 2.3).

_{28k}(CalB) as a calcium buffer. In the equations, we will represent cytosolic and endoplasmic calcium concentrations by the symbols \(c_{c}\), \(c_{e}\), and the (unbound) buffer concentration by

*b*, respectively. Mobility in the cytosol/ER is described by the diffusion equation

^{2+}compound is expressed by the difference of the total concentration of CalB present in the cytosol (\(b^{ \mathrm{tot}}\)) and free CalB, the former of which is assumed to be constant in space and time (this amounts to the assumption that free and calcium-binding CalB have the same diffusive properties). All diffusion and reaction parameters are listed in Table 1.

Model parameters and initial values

| ||

\(c_{c}\) | 50 nM | (chosen) |

\(c_{e}\) | 250 μM | (chosen) |

\(c_{o}\) | 1 mM | (chosen) |

\(b^{\mathrm{tot}}\) | 40 μM | [47] |

| ||

\(D_{c}\) | 220 \(\upmu \mathrm{m}^{2}\ \mathrm{s}^{-1}\) | [48] |

\(D_{b}\) | 20 \(\upmu \mathrm{m}^{2}\ \mathrm{s}^{-1}\) | [49] |

\(\kappa _{b}^{-}\) | 19 \(\mathrm{s}^{-1}\) | [47] |

\(\kappa _{b}^{+}\) | 27 \(\upmu \mathrm{M}^{-1}\ \mathrm{s}^{-1}\) | [47] |

| ||

\(k_{a}^{-}\) | 28.8 \(\mathrm{s}^{-1}\) | [50] |

\(k_{a}^{+}\) | 1500 \(\upmu \mathrm{M}^{-4}\ \mathrm{s}^{-1}\) | [50] |

\(k_{b}^{-}\) | 385.9 \(\mathrm{s}^{-1}\) | [50] |

\(k_{b}^{+}\) | 1500 \(\upmu \mathrm{M}^{-3}\ \mathrm{s}^{-1}\) | [50] |

\(k_{c}^{-}\) | 0.1 \(\mathrm{s}^{-1}\) | [50] |

\(k_{c}^{+}\) | 1.75 \(\mathrm{s}^{-1}\) | [50] |

\(I_{R}^{\mathrm{ref}}\) | \(3.5 \times 10^{-18}\ \mathrm{mol}\ \mathrm{s}^{-1}\) | [51] (approx.) |

| ||

\(I_{S}\) | \(6.5 \times 10^{-21}\ \mathrm{mol}\ \upmu \mathrm{M}\ \mathrm{s}^{-1}\) | [52], (adapt.) |

\(K_{S}\) | 180 nM | [24] |

| ||

\(I_{P}\) | \(1.7\times 10^{-23}\ \mathrm{mol}\ \mathrm{s}^{-1}\) | [53] |

\(K_{P}\) | 60 nM | [54] |

\(\rho _{P}\) | 500 \(\upmu \mathrm{m}^{-2}\) | (estim.) |

| ||

\(I_{N}\) | \(2.5 \times 10^{-21}\ \mathrm{mol}\ \mathrm{s}^{-1}\) | [53], (adapt.) |

\(K_{N}\) | 1.8 μM | [53] |

\(\rho _{N}\) | 15 \(\upmu \mathrm{m}^{-2}\) | (estim.) |

| ||

\(v_{l,e}\) | 38 nm \(\mathrm{s}^{-1}\) | (calc.) |

\(v_{l,p}\) | 4.5 nm \(\mathrm{s}^{-1}\) | (calc.) |

### 2.3 Membrane Transport Mechanisms

^{2+}-ATPase pumps (SERCA) for re-uptake as well as a leakage flux on the ER membrane. Moreover, we defined fluxes across the plasma membrane by addition of Ca

^{2+}-ATPase pumps (PMCA), Na

^{+}/Ca

^{2+}exchangers (NCX) and a plasma membrane leakage term. This amounts to the ER and plasma membrane flux density equations

_{3}receptor (IP

_{3}R) channels are another source for intracellular calcium. They are known to require calcium and, more importantly, IP

_{3}in the cytosol to become permeable to luminal calcium. IP

_{3}is produced at the plasma membrane by receptor activation of phospholipase C. Compared to RyR calcium efflux, IP

_{3}R calcium dynamics take place on a slower time scale (governed by the diffusion of IP

_{3}). Preliminary simulations for the present study have shown that IP

_{3}Rs may contribute to the initiation of a traveling wave, when small calcium influxes are not enough to trigger a wave through RyRs alone. The wave dynamics are, however, dominated by RyR. We therefore decided to exclude IP

_{3}R-mediated calcium dynamics.

To incorporate existing single-channel or single-pump models in the partial differential equations formulation, we assume channels and pumps to be continuously distributed along the membranes, using their membrane densities to calculate the flux densities \(j_{\mathrm{ERM}}\) and \(j_{\mathrm{PM}}\).

#### 2.3.1 RyR Channels

^{2+}current. We describe the single channel ionic current by

#### 2.3.2 SERCA Pumps

^{2+}-ATPase pumps is described by a model from [24], which was adapted for the three-dimensional case, and gives rise to the Ca

^{2+}flux density

^{2+}current not only on the cytosolic concentration, but also on the endoplasmic saturation. The density \(\rho_{\mathrm{S}}\) of SERCA pumps in the ER membrane was adapted to the RyR channel density in each simulation to ensure a zero net flux through the membrane in equilibrium conditions.

An interesting question with regard to the modeling of trans-membrane currents is their equilibration at stationary, i.e., resting concentrations. The usual approach is to calibrate a leakage current such that it exactly counters the combined net current of all other involved mechanisms at equilibrium. At the ER membrane, this is only possible as long as the resulting direction of the leakage current points outward. With varying RyR densities, however, this is not necessarily the case (as leakage and RyR current have the same direction). Since the SERCA current is the only inward current through the ER membrane that we consider in our model, we chose to calibrate their density to ensure a zero net flux at equilibrium. This results in variable SERCA densities (larger when the RyR density is larger), which appears plausible from an energy consumption point of view: An increase in leakage currents to counter a reduction in RyR density would only lead to an increase in “futile cycling” through leakage and SERCA pumps. Another approach to avoid unneccessary consumption of energy is to use a bi-directional SERCA model (cf. [55], also for a more detailed view on “futile cycling”). Our simulations showed that the exact equilibration of the membrane currents has only a minuscule impact on the results presented here—the threshold values presented in the Results section only change by about 1% with equilibration by leakage flux and constant SERCA density (data not shown). The reason is that the propagation of the wave front takes place on a much faster time scale than the re-uptake of calcium through SERCA pumps. The same applies to the extrusion by PMCA and NCX pumps, which are therefore modeled in a basic way.

#### 2.3.3 NCX and PMCA Pumps

^{+}/Ca

^{2+}exchanger current, our model assumes a constant Na

^{+}concentration at the plasma membrane. The current densities for both plasma membrane transport mechanisms are expressed as first- and second-order Hill equations, respectively (cf. [53]):

#### 2.3.4 Leakage

^{2+}concentration, which is assumed to be constant throughout all simulations. Values for all model parameters are gathered in Table 1.

### 2.4 Numerical Methods

For numerical simulations, the three equations are discretized in space using a finite volumes method. This allows for a natural integration of current densities across ER and plasma membranes into the reaction-diffusion process.

The system of ordinary differential equations (in time) arising from this procedure is nonlinear (due to the nonlinear reaction term and, more importantly, the highly nonlinear transport terms across the membranes). As these equations, notably the RyR channel dynamics, require a precise calibration of the time step size used for numerical solution, we employed a linearly-implicit extrapolation (LIMEX) scheme to solve the nonlinear system, since LIMEX offers automated error-estimate-based control of integration order and step size [56, 57].

For the results we present here, the linear problems emerging in the LIMEX method were solved using a Bi-CGSTAB [58] linear solver preconditioned by a geometric multigrid method using Gauss–Seidel smoothing and SuperLU [59] as base solver. Computations were facilitated by a domain decomposition parallelization approach and carried out using the UG 4 framework [60] on the JURECA computer system at the Jülich Supercomputing Centre [61].

### 2.5 Implementation

All model components were implemented in a NeuroBox project. NeuroBox [62] is a neuroscientific toolbox that combines 1D, 2D and 3D modeling and simulation of electrical and biochemical signaling in a visual workflow environment. Visual workflows are created with VRL-Studio [63] and the general-purpose numerical framework UG 4 [60] is used to solve the set of coupled nonlinear partial differential equations.

## 3 Results

### 3.1 Range of Calcium Waves

In a first series of simulations, we examined how calcium dynamics depend on the ER radius and the density of RyR channels in the ER membrane. To that end, we created five model dendrite domains with a fixed dendrite radius of 0.2 μm each and an ER radius of 40 nm, 50 nm, 60 nm, 70 nm, and 80 nm, respectively. We then simulated calcium dynamics with RyR densities ranging from 0.5 \(\upmu \mathrm{m} ^{-2}\) to 4.0 \(\upmu \mathrm{m}^{-2}\) on all five domains and measured how far the initial calcium signal was able to travel through the dendrite.

- (i)
A minimal RyR density is required in all five domains to elicit a calcium wave. The sharp gradient in the curves depicted in Fig. 3 show that once a critical RyR density is surpassed, calcium waves become stable. Otherwise the initial calcium influx causes only a very localized transient that does not propagate along the dendrite. This result is comparable to minimal distance criteria for the IP

_{3}receptor clusters studied in [37]. - (ii)
Dendrites with smaller ER need a higher RyR density to elicit a wave, indicating that the ability to trigger a calcium wave scales with the rate of calcium release from the ER. This seems to agree with [37], where wave stability is controlled by IP

_{3}R pump strength and, as mentioned above, by IP_{3}R cluster distances. The critical threshold for wave stability decreases with increasing ER radius. Since dendrite diameters typically decrease the further they are away from the soma, ER diameters are forced to decrease as well. This would lead to a higher probability for calcium waves to become abortive. Figure 3 demonstrates how such dendritic tapering can be compensated for by an increase in RyR density. - (iii)
Conversely, as calcium release from the ER increases both with ER radius and RyR density, there is also a threshold ER radius for any sufficiently large RyR density above which calcium waves are stable. Thus, larger ER compartments in dendrites will enable neurons to more readily induce long-distance calcium signals.

- (iv)
There are regimes in which abortive calcium waves are initiated (closely below threshold density). Such threshold dependency has previously been introduced in [26]. Our results confirm the existence of such thresholded regimes and are later quantified by deriving empirical laws. While the general behavior will be either a micro-domain calcium event or a calcium wave, there is a critical intermediate region in which abortive waves with variable travel distances can be elicited.

### 3.2 Threshold for Stable Calcium Waves

- (1)
Do such thresholds also exist for larger dendrites?

- (2)
If so, what is the relationship between dendrite radius and ER radius (and RyR density) at the threshold?

- (i)The traces in Fig. 4A (threshold ER radius plotted against RyR density) appear to reach a lower limit for large RyR densities, meaning the ER needs to have a minimal size to produce stable calcium waves. This makes sense as it has to contain a minimal amount of calcium that can actually be released during a wave event. This result is in agreement with [36], where an increase in SERCA density led to an increase in luminal calcium. This was demonstrated to have a positive feedback on the wave dynamics.
- (ii)The traces in Fig. 4B (threshold ER radius plotted against dendrite radius) reach an upper limit for large dendrite radii. In other words, there is a limit to the thickness of the ER required to elicit stable calcium waves and the ER does not need to be bigger than that in, e.g., very thick proximal dendrites. Figure 5 depicts the limit ER threshold values as a function of RyR density. The data is fitted by a model function of the formwith \(a = 0.377\ \upmu \mathrm{m}\), \(b = 0.0115 \ \upmu \mathrm{m}\) and \(c = 0.637\ \upmu \mathrm{m}^{-2}\).$$\begin{aligned} r_{\mathrm{lim}}(\rho ) = \frac{ac+b\rho }{\rho - c} \end{aligned}$$
- (iii)
The trace corresponding to the lowest tested RyR density (0.5 \(\upmu \mathrm{m}^{-2}\)) in Fig. 4B does

*not*reach such an upper limit (the trace corresponding to a RyR density of 1.0 \(\upmu \mathrm{m}^{-2}\) does reach a limit, but outside the depicted range). One can conclude that a minimal RyR density is required for stable calcium waves (see also [26]) if the ER is restricted to specific dendritic volumes. Further investigation of this observation showed that this is due to calcium buffering in the cytosol: If the RyR-supported calcium current through the membrane is too small, too much of the released calcium is buffered and there is not enough left to trigger further release through RyR channels. This results is in agreement with [37], where similar conclusions were made for IP_{3}R-induced waves. Reducing the buffer concentration by a factor of ten allowed for a limit ER threshold radius for all tested RyR densities (cf. Fig. 5).

#### 3.2.1 Empirical Threshold Laws

To complete the investigation of our second question, we tried to describe the relationship between threshold ER radius and dendrite radius/RyR density by deriving appropriate empirical laws for the traces in Fig. 4. The fact that we found a threshold ER radius for wave elicitation in all combinations of dendrite radius and RyR density suggests that there exists a function \(r(R,\rho )\) (*r* for ER radius, *R* for dendrite radius, *ρ* for RyR density) describing a threshold ER radius manifold in the three-dimensional parameter space spanned by *r*, *R* and *ρ*.

*x*, where a cytosolic threshold calcium concentration \(c^{\star }\) is attained near the ER membrane, needs to shift by Δ

*x*to the right within a certain time Δ

*t*(governed by the wave velocity). The amount of calcium necessary to increase the original concentration at \(x + \Delta x\) to \(c^{\star }\) within this time span needs to be released from the ER. It is proportional to the number of releasing RyR channels (see also [26]), which, in turn, is proportional to: (i) the RyR density

*ρ*and (ii) the radius

*r*of the ER membrane, both assumed constant in time. The single-channel current, however, is time-dependent in that it depends on locally available endoplasmic calcium (see Eq. (8)), which decays rapidly during RyR release. While the amount of released calcium is proportional to

*r*, the endoplasmic volume is proportional to \(r^{2}\). Assuming rapid radial equilibration within the ER, the decay of a single-channel current is proportional to \(r^{-1}\). Integrating the decreasing current from

*t*to \(t+\Delta t\) shows that the amount of released calcium is also proportional to (iii) the factor \((1-b/r)\) with some constant

*b*, in a first-order approximation. Putting (i) to (iii) together, an amount proportional to

### Small Distance Between Plasma and ER Membrane

*x*and \(x+\Delta x\) is fix, a constant \(\frac{1}{2c}\) can be introduced to arrive at the relation

*b*and

*c*, usable in a least squares fitting, which very closely fits the 0.1 μm dendrite radius trace in Fig. 4A. The one for the 0.2 μm dendrite radius trace is also acceptable, but using

*R*as an additional parameter in the fitting expression produces substantially better fitting results for the 0.3 μm and 0.4 μm dendrite radius traces, indicating that the model assumption of radially constant calcium concentrations is already largely violated in theses cases.

### Large Distance Between Plasma and ER Membrane

*a*from the ER membrane until the threshold concentration \(c^{\star }\) is reached at axial position \(x+\Delta x\). It is reasonable to assume that the calcium concentration near the ER membrane after release through RyR channels is inversely proportional to the volume it diffuses into, i.e., \((r+a)^{2} - r^{2}\). Like above, with a fixed percentage of the released calcium being bound by the buffer, we can introduce a constant \(\frac{c}{4a}\) (which depends on the buffer concentration) to arrive at the equation

*a*,

*b*, and

*c*can be determined to fit the limit threshold ER radii attained for large dendrite radii (Fig. 5, blue trace), also for buffer concentrations reduced by a factor of ten, in which case the parameter

*c*is significantly smaller (Fig. 5, orange trace). Note that the buffer concentration puts a constraint on the RyR density: If the density falls below a threshold (the constant

*c*), then there can be no calcium wave, regardless of how big the ER is.

### Intermediate Distances

We were not able to find descriptive model functions to fit the threshold values for intermediate distances. In such regimes, the significant radial concentration gradient at the wave front, which varies strongly with the distance between the two membranes, makes it difficult to find expressions similar to the two presented ones.

### 3.3 Wave Velocity

## 4 Discussion

Our computational study finds that calcium waves can be triggered in an idealized morphological setting. If certain geometric and physiological conditions are met, wave propagation is a stable process, similar to the all-or-nothing properties of electrical action potentials. A minimal ryanodine receptor density is required to guarantee sufficient calcium release from the ER and this density scales inversely with ER surface. We also observed abortive regimes in small parameter regions below thresholds for ER size and ryanodine receptor densities. When considering realistic morphologies, spatial variations in dendritic and ER morphology as well as changes in RyR density are likely to occur, thus, abortive regimes may be important when it comes to synaptic cross-communication with respect to calcium signals.

We used spatially constant RyR densities and membrane radii throughout the study, since we were interested in obtaining functions expressing their relationships with respect to the potential for the elicitation of stable calcium waves. Of course, in real dendrites neither dendrite radius nor RyR density are spatially constant [64], nor can the ER be expected to be a perfectly shaped cylinder that is centrally positioned within the dendrite. Future research, which is outside the scope of this study, could integrate heterogeneous membrane composition and detailed three-dimensional morphologies.

In the context of spike-timing-dependent plasticity, calcium wave velocity may be a controlling parameter. Our study shows that calcium wave velocities vary depending on ER size and RyR density. This merits a closer investigation of the timing of calcium signals in connection to, e.g., back-propagating action potentials.

A more precise understanding of the intracellular architecture may be critical in wave timing and stability. For example, mitochondria are an important sink for excess cytosolic calcium. They can quickly absorb considerable amounts of calcium through mitochondrial calcium uniporter channels [65] and extend through dendrites as filamentous networks [41, 42]. It would therefore be interesting for future work to extend this study to a fully three-dimensional, non-symmetric case, to investigate the interplay between mitochondria and ER in the context of dendritic calcium waves.

In addition to the stability and velocity of dendritic calcium waves, changes in the cellular and intracellular architecture may promote direction selectivity of calcium signals. Since the radius of dendrites is variable, typically large in proximity of the soma and smaller in more distal regions, we wondered whether there could be some kind of direction selectivity in the propagation of calcium waves. Due to the increasing amount of releasable calcium in the growing ER, we suspected that stable waves might be better supported when propagating from thin dendrites to larger ones with larger ER than in the direction of decreasing radii. We tested this hypothesis with three simulations (data not shown), the first on a medium-size dendrite with constant radius, the second on the same dendrite with increasing radius, and the last one on the same dendrite with decreasing radius. In all cases, the ER radius at each location was chosen to be just below the threshold for stable calcium waves at the chosen (constant) RyR density, so that one would expect the waves to terminate at some point. This indeed happened in all cases, however, the distance the waves were able to travel differed: shortest in the shrinking dendrite and longest in the expanding dendrite. The results confirmed our hypothesis. Yet, the effect was relatively small (approx. 11 μm difference in travel distance between small-to-large and large-to-small scenarios) and required very precise calibration of the ER radius (too small and all waves would have terminated even earlier, too large and all waves would have been stable). We therefore believe that this effect is unlikely to be relevant under realistic conditions.

Our study further highlights the question of model dimensionality. While in many cases, in which electrical properties of neurons are studied, one-dimensional multi-compartment models are employed, this may not always be possible for intracellular biochemical modeling. The fact that we could not find a simple descriptive model for regimes with neither very small nor very large distance between ER and plasma membrane reflects the non-trivial dynamics of the calcium signal with respect to space. Even in this relatively simple model setup with rotational symmetry in the geometry, it is not generally possible to reduce dimensionality (by ignoring the radial concentration profile in the cytosol) to a model that is essentially 1D. A fortiori, such a model would not be able to generate the correct stability thresholds and wave velocities in medium-size and large dendrites. For very thin dendrites, however, a simplification of 3D calcium models to 1D is conceivable provided the symmetry prerequisites of this study are met.

## Notes

### Acknowledgements

We wish to thank Arne Nägel for the implementation of the LIMEX scheme in UG 4 and for his advice on how to use it. We gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Jülich Supercomputing Centre (JSC).

### Availability of data and materials

Please contact the authors for data requests.

### Authors’ contributions

MB and GQ formulated the scientific questions and designed the simulations. The simulations were executed and analyzed by MB. The research project was supervised by GQ. MB and GQ wrote the manuscript. All authors read and approved the final manuscript.

### Funding

This research was funded by BMBF (Collaborative Research in Computational Neuroscience 01GQ1410B to GQ).

### Ethics approval and consent to participate

Not applicabale.

### Competing interests

The authors declare that they have no competing interests.

### Consent for publication

Not applicabale.

## Supplementary material

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