NeuroBox: Computational Mathematics in Multiscale Neuroscience

Abstract

The brain is a complex organ operating on multiple scales. From molecular events that inform electrical and biochemical cellular responses, the brain interconnects processes all the way up to the massive network size of billions of brain cells. This strongly coupled, nonlinear, system has been subject to research that has turned increasingly multidisciplinary. The seminal work of Hodgkin and Huxley in the 1950s made use of experimental data to derive a coherent physical model of electrical signaling in neurons, which can be solved using mathematical and computational methods, thus bringing together neuroscience, physics, mathematics, and computer science. Over the last decades numerous projects have been dedicated to modeling and simulation of specific parts of molecular dynamics, neuronal signaling, and neural network behavior. Simulators have been developed around a specific objective and scale, in order to cope with the underlying computational complexity. Often times a dimension reduction approach allows larger scale simulations, this however has the inherent drawback of losing insight into structure-function interplay at the cellular level. This paper gives an overview of the project NeuroBox that has the objective of integrating multiple brain scales and associated physical models into one unified framework. NeuroBox hosts geometry and anatomical reconstruction methods, such that detailed three-dimensional domains can be integrated into numerical simulations of models based on partial differential equations. The project further focusses on deriving numerical methods for handling complex computational domains, and to couple multiple spatial dimensions. The latter allows the user to specify in which parts of the biological problem high-dimensional representations are necessary and where low-dimensional approximations are acceptable. NeuroBox offers workflow user interfaces that are automatically generated with VRL-Studio and can be controlled by non-experts. The project further uses uG4 as the numerical backend, and therefore accesses highly advanced discretization methods as well as hierarchical and scalable numerical solvers for very large neurobiological problems.

1. Introduction

Computational Mathematics is playing an ever increasing role in neuroscientific research. The biology of the brain is an extremely complex, multiscalar system of coupled electrical and biochemical signals. Experimental methods have evolved over time, and continue to evolve, so that distinct parts of this vast signaling network can be recorded and visualized. While experimentally recorded data, ranging from molecular events, to cellular and large network dynamics, has been very useful in unraveling processes in the brain, understanding the intricate interplay between a large set of parameters and across multiple scales requires a multidisciplinary research effort involving, e.g., mathematics, physics and computer science. Dating back to the ground breaking work by Hodgkin and Huxley [22, 23, 20, 19, 21], modeling and simulation as a method for studying the brain has evolved alongside experimental tools. Over the last decades, computational tools, such as neuron simulators, have been developed [18, 7, 14]. Given the available morphological data and the computational resources, neurons are often represented as a node (in a network) or by one-dimensional graphs. Stripping morphological detail has the great advantage of reducing the computational problem to a set of ordinary differential equations (ODEs), which can be solved efficiently with ODE-solvers and linear algebra packages.

In the past years, however, the influence of cellular morphology on electrical and biochemical signals has been identified as an important part of cellular and network dynamics involved in learning and plasticity of the brain [32, 41, 17, 40]. Incorporating detailed three-dimensional morphology in modeling and simulation thus shifts from solving a system of ODEs to solving a system of, typically coupled, nonlinear, partial differential equations (PDEs). Developing efficient computational mathematics in multiscale neuroscience becomes a more challenging task, involving the development of adequate multi-physics models, the reconstruction of accurate three-dimensional morphologies, stable domain and PDE discretization methods, as well as efficient numerical solvers.

This paper intends to motivate a multiscale, hybrid-dimensional approach from zero or one-dimensional models to three-dimensional models with detailed morphology representations. The paper further discusses methods for reconstructing computational domains, hybrid-dimensional discretization, and optimized hierarchical numerical solvers. The necessary ingredients for such an approach can be compiled into three main components: 1. the development and implementation of neuroscience-specific models, 2. a hybrid-dimensional framework, in which geometries are represented and coupled at varying space-dimensions, as well as 3. a numerics platform, which harbors the necessary tools to solve the resulting equation systems efficiently. The development of these components, the integration into the workflow-control framework VRL-Studio [24], and the multiphysics platform uG4 [42] define the project NeuroBox [9], which will be discussed in this paper.

The current state of research is introduced in Sec. 2 together with key biological principles, after which solving biological problems at multiple scales with NeuroBox (Sec. 3) will be discussed. Simulations of electrical and biochemical signals in brain cells and neural networks are shown in Sec. 4. In summary, this paper intends to motivate the use of advanced computational mathematics in order to incorporate detailed biological knowledge and data into a unified framework for studying the structure-function interplay in neurons and networks, and to highlight and solve some of the mathematical difficulties that come with this endeavor.

2. Background

This section compiles background information about electrical and biochemical signals in neurons, the associated models and a brief overview of popular methods for solving the underlying model equations. It will then highlight some of the current short-comings, intended to motivate the development of a hybrid-dimensional framework.

2.1. Electrical signals in neurons

A cell is defined by its ability to separate the intracellular from the extracellular space by a double bi-lipid plasma membrane, which prevents the direct exchange of ions and molecules. The accumulation of ions on both sides of the membrane can be compared to a parallel plate capacitor that harbors a certain number of charges on both plates. Thus, there exists a chemical and electrical gradient across the plasma membrane. The electrical gradient V_m is termed the membrane potential. V_m is non-zero at a cell’s resting state in the range of −65mV, and is a consequence of membrane-embedded ion pumps. These pumps are fueled by adenosine triphosphate (ATP) and exchange ions inversely to the chemical and electrical gradients. With respect to V_m the most prominent pumps are sodium-potassium exchangers [20], which play a crucial role in eliciting action potentials, electrical waveforms that travel along the cell membrane over vast distances to communicate biological information within and across neurons. This process was investigated in the ground-breaking work of Hodgkin and Huxley [22, 23, 20, 19, 21], and culminated in a mathematical model called the Hodgkin-Huxley equations. In this model the membrane is interpreted as an ideal capacitor, which yields

$$C_m\frac{\partial V_m}{\partial t}=I_a+I_m,$$ (1)

where V_m is the membrane potential, C_m the membrane capacitance, and I_a and I_m the axial and inward transmembrane electric currents, respectively. In a low-dimensional approximation of a neuron’s geometry, i.e. a neuron represented by a collection of nodes and edges, the local shape of the plasma membrane around a single node can be approximated by a perfect cylinder. This yields a specific capacitance $c_m=\frac{1}{2\pi rl}{C}_m$, where r is the cylinder radius and l the length of the cylinder, so that 2πrl amounts to the surface of the cylinder. Similarly one can define the specific transmembrane current i_m by i_m = 1/(2πrl)I_m, which depends on the ion exchange mechanisms integrated into the cell’s membrane. This is where the Hodgkin-Huxley equations come into play. They describe i_m in the following way:

$$i_m=g_{\mathit{Na}}{m}^{3}h(V_m-V_{\mathit{Na}})+g_K{n}^4(V_m-V_K)+g_L(V_m-V_L),$$ (2)

where g_X are specific maximal membrane conductances for ion species X and V_X are ion-specific reversal potentials defined by the Nernst equation [33, 13]. In the Hodgkin-Huxley equations sodium (Na⁺) and potassium (K⁺) channels are explicitly modeled, while all other effects are compiled into the general leakage term g_L(V_m – V_L). Quantities m, n, and h are channel gating parameters defined by time-dependent ordinary differential equations of the type

$$\frac{\partial z}{\partial t}=\frac{z_{\infty }-z}{{\tau }_z},$$ (3)

where z denotes m, n, and h, respectively and the quantities z_∞ and τ_z are computed by

$$z_{\infty }=\frac{{\alpha }_z}{{\alpha }_z+{\beta }_z}\text{and}{\tau }_z=\frac{1}{{\alpha }_z+{\beta }_z}$$ (4)

with experimentally measured rates α_z and β_z [22]. While the above equations describe the membrane dynamics in a single node an additional axial, purely ohmic, current I_a can be integrated to account for the flow of charged ions in axial direction of the cylinder:

$$I_a=\frac{V_m(x_2)-V_m(x_1)}{r_c∕\pi \cdot {\int }_{x_1}^{x_2}a{(x)}^{-2}\mathit{dx}}=\frac{V_m(x_2)-V_m(x_1)}{r_c∕2\pi \cdot (a_1^{-2}+a_2^{-2})\Vert x_2-x_1\Vert }.$$ (5)

Here, r_c is the specific resistance of the intracellular space (the cytosol), and x is the axial space variable. In summary, equations (1)-(4) define the Hodgkin-Huxley equations for a point-neuron (no spatial extension), and together with equation (5) they yield a one-dimensional model for membrane potential dynamics in neurons.

2.2. Biochemical signals in neurons

The primary purpose of electrical signals is to communicate activity within single neurons and across cell networks. Cells contain ion channels, embedded in the plasma membrane, that exchange ions between the extra- and intra-cellular space. This often occurs through membrane potential-regulated gating mechanisms. These voltage-dependent gating mechanisms couple the electrical and biochemical dynamics in neurons and can be expressed as nonlinear flux boundary conditions for the associated PDEs. The same principle holds for organelle membranes in the intracellular space, but often these nonlinear flux conditions depend more on local concentrations of specific ions and molecules and less on the membrane potential. Thus, flux equations can be imposed on cellular and organelle membranes:

$$j_m=\sum _i{j}_i,$$ (6)

where j_m is the total membrane flux across a given membrane, defined by individual flux densities j_i from ion gating mechanisms i. Many flux densities can be described by the general form

$$j_i={\rho }_i{p}_i^o{I}_i$$ (7)

where ρ_i defines the membrane channel density, $p_i^o$ the probability that a single channel is in the open state and I_i the ion current through a single and open channel. Channel densities and single channel ion currents need to be quantified experimentally and open-state probabilities are derived from the assumption that channels can take on a finite number of states with, potentially concentration-dependent, transition rates. This leads to a set of ordinary differential equations (ODEs) that can be solved for the open state O. An example of a three-state model is:

(8)

with a closed state C, an inactivation state I, and open state O, as well as transition rates r₁,…, r₆. This leads to the system

$$\begin{gathered} \frac{\mathit{dO}}{\mathit{dt}}=r_{1}C-(r_2+r_3)\cdot O+r_{4}I \\ =r_1(1-O-I)-(r_2+r_3)O+r_{4}I, \end{gathered}$$ (9)

$$\frac{\mathit{dI}}{\mathit{dt}}=r_6(1-O-I)-(r_4+r_5)I+r_{3}O.$$ (10)

The complete set of flux-boundary conditions j_m, describing the nonlinear dynamics of ion channel mechanisms, ultimately drives the intracellular ion movement in cells. The diffusion equation

$$\frac{\partial c}{\partial t}=\nabla \cdot (D\nabla c)\text{on}\Omega$$ (11)

forms the basis for most cellular ion processes (presumably because of its energy efficiency). Here, c defines an ion concentration, D defines a diffusion tensor, and Ω the intracellular space. In addition, ions are bound by buffers, molecules that form complexes with free ions. These buffering dynamics can be described by a reaction

$$c+b\underset{{\kappa }_b^{-}}{\overset{{\kappa }_b^{+}}{\rightleftharpoons }}[\mathit{cb}],$$ (12)

such that equation (11) can be extended by a reaction term:

$$\frac{\partial c}{\partial t}=D_c\Delta c+({\kappa }_b^{-}(b_{\infty }-b)-{\kappa }_b^{+}bc),$$ (13)

$$\frac{\partial b}{\partial t}=D_b\Delta b+({\kappa }_b^{-}(b_{\infty }-b)-{\kappa }_b^{+}bc).$$ (14)

D_c and D_b denote the diffusion tensors for ion species c and buffer b respectively, and b_∞ denotes the total buffer concentration present in the system. Eq. (14) can be used for mobile buffers in the intracellular space. Aside from mobile buffers, cells use intracellular organelles to sequester ions from the intracellular space into internal stores. In the case of calcium, the endoplasmic reticulum and mitochondria are large calcium stores that sequester and release calcium, through membrane-embedded and concentration-dependent channels (see eq. (7)). These exchange processes allow cells to generate ion waves that can propagate over long distances, without requiring active transport in the intracellular space.

Note, that electrical and biochemical models are coupled at membrane interfaces by voltage-dependent ion channels. Most current simulation tools treat electrical models with a dimension-reduction approach. Biochemical dynamics, however, need to be resolved three-dimensionally, since they rely heavily on the intracellular architecture, like organelle structure and positioning. The unification of models at multiple dimensional scales is a topic addressed by NeuroBox and will be discussed in Sec. 3.

2.3. Neuron simulators

Over the past decades various simulators have been developed. Very roughly they can be sorted into general purpose simulators – which allow broader variability with respect to the models and computational morphology, potentially at the cost of computational complexity and compute times – and specialized simulators dedicated to a specific model and cell/network structure in order to optimize towards very large systems (e.g. large neural networks), typically at the cost of variability.

2.3.1. General purpose simulators

Likely the most prominent general purpose simulators are NEURON [18] and Genesis [7]. The key idea of these projects is to provide a specialized framework for simulating electrical dynamics in neurons and at the same time to offer flexibility in the design and incorporation of the biophysical models, e.g. cell membrane channel dynamics, into numerical simulations. The premise of NEURON and Genesis is to assume only axial currents along highly anisotropic structures (dendrites and axons can be interpreted as thin cables), leading to the equation

$$C_m\frac{\partial V_m}{\partial t}=\frac{a}{2r_L}\frac{{\partial }^2{V}_m}{\partial x^2}+I_m,$$ (15)

for constant dendrite radius a and axial resistance r_L, which can be derived from eq. (1). The axial current is the one-dimensional term $\frac{{\partial }^2}{d{x}^2}{V}_m$. Assuming that the cable diameter is piecewise constant in the vicinity of each computational node leads to a local cylindrical approximation. Finite Difference discretization of the second order term in an unbranched cable leads to

$$C_m\frac{\partial V_m^i}{\partial t}=\frac{a}{2r_L}\frac{V_m^{i-1}-2V_m^i+V_m^{i+1}}{h^2}+I_m$$ (16)

for each node i. Each discretized node harbors ODEs for any membrane exchange mechanisms involved (I_m) and are coupled via the axial current terms. From a numerical perspective this leads to a system of coupled ODEs. In [18] these systems can be solved using explicit or implicit Euler or a Crank-Nicholson method. Giving top level control of the biophysics through a .hoc-interpreter and a modeling language for channel dynamics (NMODL) allows users to set up simulations without in-depth programming or mathematical knowledge, making these simulators attractive tools for interdisciplinary computational neuroscience.

2.3.2. Specialized simulators

In a push towards addressing the daunting network sizes in the brain (> 10¹²), specialized simulators have been developed with very large networks in mind. One example is the NEST simulator [14], with which electrical signal propagation across network sizes in the range of 10⁷ have been simulated [25]. To reach macroscopic network sizes, single neurons are reduced to dimension-less quantities in space (point neuron model). This leads to integrate and fire models, where weighted synaptic input is integrated to determine the binary cellular response (0: no action potential, 1: action potential):

$$C_m\frac{\partial V_m}{\partial t}=-g{V}_m+I_r+I_c,$$ (17)

where g is a conductance and I_r and I_c are resistive and capacitive currents, respectively. Thus, specific channel gating processes (e.g. sodium and potassium gating) are neglected and replaced by a threshold check to determine the binary cell behavior mentioned above. This reduces the overhead of solving local systems of ODEs.

2.4. Summary

A multitude of neuron simulators that cover the range from more detailed cell models at the single cell or small network level to simple cell models at the large network level have been developed. All approaches rely on a dimension reduction approach, i.e. one-dimensional (compartment models) or zero-dimensional (point neuron models) in space. What is largely missing in the cascade of simulators is a computational framework that includes detailed three-dimensional domain representation in PDE-based models of neurons. Single cell PDE-based approaches, such as Virtual Cell [28], are developed towards handling three-dimensional cell morphologies, yet are somewhat restrictive with biophysical models, and furthermore are not mathematically designed for scalability towards high-performance computing.

3. NeuroBox: Solving biological problems at multiple scales

The tools described in Sec. 2 address important issues for computation-based neuroscience and are optimized towards computability, i.e. dimension and model reduction to increase network size or decrease network size to increase dimensionality and biophysical detail. NeuroBox is designed to integrate these concepts across multiple scales and to solve the resulting PDE-based models efficiently for high-performance computing. The objectives of the NeuroBox project are aligned with the following premises:

1. Detailed morphologies: Integration of morphology reconstruction techniques to define precise computational domains for simulations.

2. Cross-scalar coupling: Unify models at various dimensions and biological detail in a single computational framework.

3. Scalability: Scalable numerical methods for high-performance computing.

4. Usability: Integration into user interface controllable computational workflows.

3.1. Computational Domains

Studying the influence of morphology on biological function at the level of electrical and biochemical signals requires simulations to be run on an accurate domain representation. A natural step is to reconstruct the cellular features from available microscopy data or other anatomical data. In a preliminary step it might however to be advantageous to have a parametrized computational domain to perform systematic studies of the influence of geometric parameters on the underlying signal. This section will focus on the two options

1. parametric geometry design, and

2. geometry reconstruction methods,

that can be used in numerical simulations of biological processes.

3.1.1. Parametric design

Parametric design of computational domains gives the user the ability to systematically generate large numbers of computational domains by varying geometry input parameters across defined intervals. In previous studies tools, such as a bouton generator (see Sec. 4.2) and a synaptic spine generator (Fig. 1), have been developed to study the influence of morphology on biochemical signals. Given the case of synaptic spines, a subset of a neuron that forms chemical contacts with other neurons, geometric features such as the spine neck length and width, the spine head size, as well as the geometric features of the intracellular organelles can be represented by a parametric design, see Fig. 1. Having specified the relevant parameters, geometric information is used to create a computational surface and volume grid, using the grid library of uG4 [42]. Surfaces are represented by triangular grids while volumes can be represented as tetrahedral meshes (using TetGen [38]), or in specific cases as hexahedra, octahedra or even mixed element meshes. Unstructured grid-based geometry representation allows the user to treat very general topologies by fairly simple, piecewise linear, approximations that can be used in grid-based numerical methods.

Fig. 1.

A: Interface of the spine generator integrated in NeuroBox and developed in VRL-Studio [24] and using the grid library of uG4 [42]. It allows the generation of a multitude of computational domains to study structure-function interplay. B and C: Cross-section illustrations of automatically generated computational grids ready for numerical simulations. Surfaces are triangulated and volume tetrahedral grids are generated with TetGen [38].

3.1.2. Microscopy reconstruction

Following a systematic study based on parametric geometry design, that ideally has revealed the link between structure and function, the next logical step is to research whether such structures can be confirmed in experimental data. Microscopy imaging data typically yields stacks of two-dimensional images. Images coming from, e.g. confocal and two-photon microscopy, or 3D tomography are represented, in the simplest case, by a grayscale pixel set. Extracting the surface information from such data sets is often a challenging task. Filtering and segmentation methods have been developed for many applications, with which raw image data is preprocessed for morphology extraction. The Neuron Reconstruction Algorithm (NeuRA) [10, 34] was developed with the purpose of identifying neuronal and organelle structures in confocal microscopy data. Worthy of highlighting here is the issue of geometry conservation during preprocessing. NeuRA employs an inertia-based anisotropic diffusion method to reduce noise and enhance structure identification through directed diffusion on grayscale image data. Assuming that an image stack is made up of a set of voxel values u(x), diffusion on this data can then be represented by

$$\begin{gathered} \frac{\partial u(\mathbf{x},t)}{\partial t}=\nabla (D(u)\nabla u(\mathbf{x},t)\text{on}\Omega \\ u(\mathbf{x},0)=u_0(\mathbf{x})\text{on}\stackrel{‒}{\Omega } \\ (D(u)\nabla u(\mathbf{x},t))\cdot \mathbf{n}=0\text{on}\Gamma \end{gathered}$$ (18)

Introducing time variable t allows for the evolution of grayscale data over time, while the initial condition u₀(x) denotes the original image. The choice of the diffusion tensor D(u) is the important ingredient in this process. The boundary condition assumes no flux across the image borders in normal direction n. In [10, 34] D(u) is defined by the eigenvectors of the moments of inertia of local subsets of the image. Since D(u) is positive definite main axis transformation yields

$$D(u)=S^T\left(\begin{array}{ccc}\hfill {\lambda }_1\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\lambda }_2\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill {\lambda }_3\hfill \end{array}\right)S.$$ (19)

Here, S is the eigenvector matrix of the inertial tensor, and λ₁, λ₂, λ₃ are the corresponding eigenvalues, arranged in decreasing order. Thus, the primary orientation of the identified mass (described by the gray values of the voxels) is in the direction of the first eigenvector belonging to λ₁. D(u) changes from subset to subset and diffusion can be directed in the direction of the structure by resetting the corresponding λ equal to 1, and the ones that are perpendicular to the main structure can be chosen near-zero. Such an image processing approach, together with segmentation algorithms, ultimately allows feature extraction with a marching cubes algorithm [29] (see Fig. 2A).

Fig. 2.

A: Reconstruction of a nucleus from a mouse pyramidal neuron in the CA1 hippocampus using NeuRA [10, 34]. B: Anatomical reconstruction of a neuron [4] (ID: 6B-E17.CNG) in a graph-representation, by recording points, diameters, and connectivity. C: 3D reconstruction of the same cell using AnaMorph [31].

3.1.3. Anatomical reconstruction

Anatomical reconstruction can be seen as a hybrid approach between parametric design (Sec. 3.1.1) and microscopy reconstruction (Sec. 3.1.2). The idea is to construct detailed representations of neurons or organelles from low-dimensional or incomplete microscopy and anatomical data by enriching these data with parametric design schemes. One example of this concept is the tool NeuGen 2.0 [10, 34] that uses statistical anatomical morphology data of different cell types to synthetically generate cell morphologies and generate detailed networks by connecting these cells synaptically. Typically, cell morphology is recorded by anatomists using semi-automatic tools (for an overview see e.g. [16]) to mark points on the cellular morphology and record the local diameter of the cellular compartment. This yields a collection of coordinate and diameter information, as well as the connectivity graph of the cell. Identifying parameters from these reconstructions, e.g. total dendritic lengths, branching patterns, and total surface and volume of cells, can be measured. In addition to a cellular anatomical fingerprint macroscopic imaging can be used to estimate the intercellular, synaptic connectivity. In [10, 34] this approach was taken for the cortical column and CA1 hippocampus. Synthetically grown networks can be incorporated into network simulations of the type studied in [9]. Towards three-dimensional cell representations, this low-dimensional anatomical cell morphology data can be used as input data for the tool AnaMorph [31]. One of the objectives of AnaMorph is to automatically generate surface triangulations of cell membranes, based on point-diameter information from low-dimensional reconstructions. One large resource of such data is the cell morphology database NeuroMorpho.Org [4], which currently hosts more than 60,000 anatomical neuron reconstructions, another being NeuGen, which generates synthetic neuron morphologies [10, 34]. Such anatomical data can be represented by AnaMorph as a directed graph $\mathcal{T}=(V,E)$, termed morphology tree. Here, V denotes the set of vertices including their cellular specification and diameter information, and E contains the topology of the graph. Given this data, AnaMorph generates a triangular surface mesh following the steps:

1. preconditioning of the morphology tree $\mathcal{T}$,

2. generation of a geometric model,

3. consistency analysis,

4. mesh generation.

For a detailed analysis of the algorithmic procedure, see [31]. The generated surface meshes are guaranteed to be orientable, topological 2-manifolds, and geometrically free of self-intersections. Aspect ratio control in a post-processing step allows the generation of meshes optimized towards numerical simulations. An example of a reconstructed neuron is shown in Fig. 2B, C.

Thus, AnaMorph generates a three-dimensional equivalent of a cell originally represented as a graph. With a 1D and 3D representation of the same cell, hybrid-dimensional simulations can be performed by coupling models at various dimensions, see Sec. 3.2.

3.2. Coupling hybrid-dimensional models and domains

Simulations of biological processes are governed by the detail of the underlying biological models and the resolution of the computational domain. While zero-dimensional domains in space (point-models) imply a systems biology model described by sets of ordinary differential equations, higher-dimensional domain representations in space yield systems of (nonlinear, coupled) partial differential equations. The governing thought behind choosing an appropriate model is to find the simplest model that accurately captures the vital components involved in the process. Even then the resulting model and computational domain can be complex enough to surpass computational feasibility. To give an example, a three-dimensional version of the electrical model defined by eqs. (1)-(5) can be derived from Maxwell’s equations [46, 47], and extends the dimension-reduced model to capture a time-variable intra- and extra-cellular potential, that drives the membrane potential V_m. Such an approach allows the investigation of cellular interaction across the extracellular space, which is not possible with the classical 1D-Hodgkin-Huxley model. The computational problem is, however, significantly larger and sets a limit on the number of cells that can be considered for such a high-dimensional study.

In order to efficiently distribute computational investment to those areas where it is required by the research objective, hybrid modeling and computing can be considered. Electrical and biochemical models can be formulated for one, two, and three dimensions, e.g. the 1D-cable equation for electrical modeling, surface diffusion of proteins on a membrane defined by a two-dimensional manifold, or 3D intracellular dynamics, respectively. While each problem can be formulated for a defined space dimension, biology often provides a highly coupled system, in which many processes interact. For example, changes in the membrane potential influence calcium dynamics and vice versa. While a low-dimensional model for the electrical model might suffice, this is not necessarily true for a coupled biochemical model. This gives rise to hybrid-dimensional methods that allow coupling of models and computational domains formulated in different dimensions.

3.2.1. Coupling 1D electrical models with 3D biochemical models

As an example for hybrid-dimensional modeling and simulation one can consider a 1D model given by eqs. (1)-(5) and a 3D biochemical model defined by eqs. (11)-(14). Since these models are coupled via the membrane current I_m, they need to communicate data in each computational time step. However, the cellular membrane at which the quantity I_m is computed is represented differently in each model. For eqs. (1)-(5), the membrane is not resolved as a two-dimensional manifold and all membrane processes are condensed in the vertices of the graph-representation of the computational domain. For eqs. (11)-(14) the cellular membrane is fully resolved and I_m is defined by eqs. (6)-(7) on the boundary vertices.

In [15] a coupling method that maps information from a one-dimensional to a two-dimensional manifold of the membrane (see Fig. 2B, C) is described. The key idea is to implement a mapping algorithm that associates a set of vertices on the 2D-manifold with a 1D-vertex in the low-dimensional domain. Thus, information can be mapped bi-directionally between the two models.

Based on a kd-tree method the algorithm described in [15] accomplishes this by coupling the NEURON simulator [18] with the multi-physics platform uG4 [42]. In the meantime this hybrid-dimensional method has been fully integrated in NeuroBox. This unified approach enables ideal parallelization and data communication, compared to the initial step of coupling different simulators.

3.2.2. Coupling high-dimensional regions of interest with 1D extensions

Dividing a computational model into low- and high-dimensional parts as described above is possible as long as each subset of PDE models is formulated for a specific dimension, e.g. electrical model exclusively in 1D and biochemical model exclusively in 3D. In situations where a full 3D biochemical model is computationally infeasible on a given scale, e.g. single cell or network, coupling high-dimensional regions of interest (at smaller spatial scales) to lower dimensional extensions (at larger spatial scales), can function as a compromise between necessary problem resolution and computational cost. As an example one can consider the Poisson-Nernst-Planck (PNP) equations, that define an ultra-structural model of ion movement through electrically charged spaces:

$$\begin{gathered} 0=-\nabla (-D_i\nabla c_i-D_i\frac{z_{i}F}{\mathit{RT}}{c}_i\nabla \varphi )\text{on}{\Omega }_i \\ -\nabla ({ϵ}_r{ϵ}_0\nabla \varphi )=\sum _i{z}_{i}F{c}_i+\rho f\text{on}{\Omega }_i\cup {\Omega }_f \end{gathered}$$ (20)

This model of electrodiffusion describes the spatio-temporal dynamics of ion concentrations c_i, based on a diffusive flux −D_i∇_ci, with diffusion coefficient D_i, and an electric flux $-D_i\frac{z_{i}F}{RT}{c}_i\nabla \varphi$, with z_i being the ionic charge, and F, R, and T defined by the Faraday constant, gas constant, and system temperature, respectively. The Nernst-Planck equation is coupled to the Poisson equation through the potential ϕ, which itself depends on the distribution of free ions and any charged surfaces, e.g. biological membranes containing a surface charge density ρ_f. Here, ϵ₀ and ϵ_r are the dielectric constant in vacuum, and the dielectric constant of the material, respectively.

The complexity of solving these equations numerically becomes apparent when considering the ultra-structural composition of the intracellular space, filled with a dense network of charged protein filaments [26] and other organelles [36]. At each of these interfaces, separating Ω_i (intracellular space) and Ω_f (intra-organelle and intra-filament space), electrical double layers form at a nanometer range with large ion concentration gradients that need to be resolved numerically. While adaptive grid refinement strategies can reduce the number of degrees of freedom in the discrete problem, solving the PNP equations at an ultrastructural level for a whole cell is not possible with current and direct methods.

In order to resolve specific regions of interest, e.g. local subsets of a cell, it is not sufficient to crop the computational domain to that region and impose boundary conditions, since these will strongly influence the physical behavior of ion movement. The boundary conditions would effectively clamp the stationary solution of ϕ. A low-dimensional extension of the computational domain would prevent the boundary condition problem, but requires a reformulation of the model equations for the low dimensional case. As long as this can be done, transfer operators need to be formulated for the 2D/1D or 3D/1D interface. While transfer operators will be problem specific ([30] shows this principle for the cardio-vascular system), hybrid-dimensional PNP coupling is content of a paper in preparation, and relies on the introduction of constrained degrees of freedom to formulate the transfer (Fig. 3D).

Fig. 3.

A—C: Time series of a calcium simulation on a neuron using adaptive grid refinement. Residuum-based error estimation allows a reduction of degrees of freedom from 1.7 million (global refinement) to 33,000 (adaptive refinement). D: Sketch of a 2D to 1D transfer operation. The interface layer is reflected by an additional layer of constrained nodes, that are not geometrically resolved but allow an averaging step to determine the value at the first low-dimensional node. (Simulations and illustration by M. Breit)

3.3. Numerical backend

NeuroBox is designed for the purpose of defining and solving biological models formulated as ODEs and PDEs. Deriving a discretized problem can be accomplished with the standard procedure of domain discretization and discretization of the differential equations system. The multiphysics platform uG4 [42] offers a flexible numerical backend for NeuroBox, that allows Finite Element and Finite Volume methods and supports a list of standard 1D, 2D, and 3D grid elements for spatial discretization. Fast solvers, such as geometric and algebraic multigrid methods, as well as Krylov subspace methods, can be used to solve very large systems of linear equations on highly parallel computing infrastructures [35]. In the context of solving systems of PDEs on complex domains, one focus of the NeuroBox project is the efficient numerical treatment of such complex-domain problems.

3.3.1. Subdivision geometric multigrid

Standard linear refinement of volume grids, constrained by complex interfaces, can significantly reduce the efficiency of a geometric multigrid (GMG) solver. This has to do with the fact that volume element quality is directly tied to the matrix condition of the linear system [6, 45, 37]. Therefore, a project towards robust numerical treatment of complex-domain PDEs is the development of grid refinement strategies ideal for complex computational domains, that can be embedded into a geometric multigrid (GMG) framework. Subdivision theory [11, 12], originally developed in the area of computer graphics to optimize surface triangulations for visual purposes, has been extended to the three-dimensional case. The general idea of subdivision methods is to reposition vertices in order to minimize local energy (if one thinks of a system of springs that interconnects vertices). This translates into using B-splines to determine optimal vertex-repositioning. In the three-dimensional case this can be generalized to box-spline functions (B-splines multiplied by directional vectors). A subdivision surface or volume is then defined as p^∞, the unique limit of refinements

$$p^{\infty }≔\underset{i\to \infty }{\lim }{p}^{i+1}=S{p}^i$$ (21)

where the subdivision matrix S governs the vertex insertions and repositioning according to above mentioned box-spline interpolation. A thorough description of this subdivision geometric multigrid method is given in [39].

3.3.2. Adaptive grid refinement

In addition to a robust refinement strategy, summarized in the previous section, adaptive grid refinement can significantly reduce the degrees of freedom in the linear problem. While residuum-based a posteriori estimation for finite element discretization has been studied extensively, much of the theory does not directly translate to the finite volume case. There has been selective work on finite volume based error estimation for adaptive grid refinement, e.g. [2, 1]. Aimed at biological simulations, a finite volume a posteriori error indicator has been developed and implemented into the framework uG4. Considering a system of diffusion-reaction equations

$$\frac{\partial u_i}{\partial t}-D_i\Delta u_i-\sum _{j,l=1}^N\left(k_{ijl}^{-}{u}_l-k_{ijl}^{+}{u}_i{u}_j\right)=0\text{on}{\Omega }_i$$ (22)

with $D_i\in \mathbb{R}$ and reactions

$$u_i+u_j\underset{{\kappa }_{ijl}^{-}}{\overset{{\kappa }_{ijl}^{+}}{\rightleftharpoons }}{u}_l,$$ (23)

and imposing Lipschitz continuous flux boundary conditions

$$D_i\nabla u_i{\mathbf{n}}_i=j_i(u)\text{on}\partial {\Omega }_i,$$ (24)

residuum-based a posteriori estimation can be employed for the above equations with nonlinear boundary conditions in a finite volume setting to control local grid refinement. Since signals in biological systems often travel in form of waves, such moving fronts can be readily treated with local grid refinement (see Fig. 3A-C for an example) and thus leads to a significant reduction in degrees of freedom and therefore in computation time.

3.4. Workflow control

The diversity of biological problems and continued research in all related areas (see introduction to Sec. 3), as well as the interdisciplinary character of this research field, controlling the modularity and adaptivity of the underlying workflow becomes the centerpiece of an efficient simulation framework. NeuroBox makes use of the Visual Reflection Library (VRL) [24] and the interactive development environment VRL-Studio [24], designed to develop workflow components with automatic user interface (UI) representation. VRL thus significantly reduces the developmental overhead for UI code representation, and allows users to combine individual UI components into a workflow with an integrated data flow. Visual workflows can then be designed in a way that is accessible to an interdisciplinary audience, and adapted for specific projects or research problems (see Fig. 4).

Fig. 4.

NeuroBox is developed using VRL-Studio to provide easy to control and modify workflows that are accessible beyond the core research area. Illustrated is an example workflow of intracellular calcium simulations, using the spine generator, a calcium simulation framework, and uG4 as the numerical backend for solving the resulting system of nonlinear partial differential equations. VTK output can be generated for visualization and analysis purposes.

The neuroscientific section of the simulation framework uG4 can thus be integrated via VRL-Studio into a coherent simulation toolbox (NeuroBox). Typically the workflow consists of the four main parts (a) geometry specification (e.g., parametric design, import of reconstructed geometries), (b) physiology specification (e.g., model equations, ion channels), (c) computational specification (e.g., discretization, solver), and (d) data visualization (e.g. plots, movie rendering).

4. Results

Sec. 2 summarized that biology relies on electrical and biochemical signals at multiple, coupled, scales to communicate and store information. Sec. 3 gave an introduction to the driving ideas behind NeuroBox and how model precision can be achieved, while keeping the problem size computationally feasible. Leaning on the multiscalar architecture of the brain, the following examples are intended to illustrate bottom-up integration, starting with single contact points between two brain cells (synapses), a cluster of contact points (boutons), single cells, and cell networks. Simulations were carried out with NeuroBox and illustrated using Paraview [3, 5].

4.1. Synaptic spine simulation

Synaptic spines define a cell’s region that is triggered electrically upon neurotransmitter release, and diffusion through the synaptic cleft from a connected presynaptic cell. Neurotransmitters binding to postsynaptic receptors initiate ion flux across the postsynaptic membrane that defines the boundary of the cleft, leading to changes in the postsynaptic cell’s membrane potential and subsequently ion entry into the intracellular space. Importantly, synaptic events trigger calcium release into the spine, such that an intracellular calcium wave can be initiated towards the cell body and cell nucleus (synapse-to-nucleus communication). In order for the cell to induce long-range calcium signals (over 100 μm and more), it relies on exchange mechanisms embedded in the cell’s plasma membrane and intracellular organelle membranes, e.g. the endoplasmic reticulum and mitochondria. A detailed model for intracellular calcium signaling can be found in [8]. The example in Fig. 5 illustrates the propagation of boundary-flux driven calcium signals. A synaptic spine, consisting of a spine neck and head, and a part of the cell’s dendrite was created using the NeuroBox spine generator. Embedded in the intracellular space is an endoplasmic reticulum that extends from the dendrite into the spine. This example nicely illustrates the necessity of resolving the three-dimensional architecture of the cell, since the nature of the calcium signal strongly depends on it. In [8], the authors show how reorganization of the intracellular domain leads to optimized coupling between the spine head and the rest of the cell.

Fig. 5.

Simulation of intracellular calcium dynamics. Using the spine generator a piece of dendrite was created with an attached spine with neck and head. Inside the cell, an endoplasmic reticulum (red) was placed inside the dendrite, and extended into the spine neck and head. Triggering calcium entry at the top of the spine head, calcium diffuses, and is buffered in the cytosol (blue). Once it reaches the endoplasmic reticulum calcium exchangers in the plasma membrane are activated, generating a calcium induced calcium release wave that propagates to and along the dendrite. A detailed study, using this method, is described in [8].

4.2. Vesicle dynamics in presynaptic boutons

Moving up towards a scale where multiple synapses are combined into a structure termed synaptic bouton, the example in Fig. 6 shows the schematics of multisynapse bouton formations. These can be found at the Neuromuscular Junction (NMJ) of Drosophila larvae and were studied computationally in [27]. There, the distinct bouton sizes found at the NMJ were explained using a three-dimensional computational model. Using a bouton generator developed for NeuroBox, synaptic boutons with an extracellular domain, synaptic release sites with tabletop-like T-bar structures and an inner vesicle pool can be generated. Upon vesicle/neurotransmitter release at synapses, calcium influx is reflected at the T-bars, see Fig. 6. The vesicle depletion dynamics at these multi-synapse sites has been studied in [27].

Fig. 6.

Geometry construction of synaptic boutons and calcium simulations. A: Illustration of the bouton generator interface, cross-section of the generated volume grid, cross-section of the surface subsets and zoom-in on the T-bar region at a single vesicle release site (from left to right). B: Time series of calcium dynamics around a release site.

4.3. 1D-3D simulation of electrical and calcium signaling in single cells

Synaptic input as seen in Figs. 5 and 6 is integrated at the cellular level. Both electrical and biochemical signals are controlled by the spatio-temporal synaptic activity pattern. To simulate the effects on coupled electrical/biochemical signals, a 1D-3D simulation approach allows to efficiently compute the membrane potential in 1D and coupling to the necessary 3D model of calcium dynamics. Using AnaMorph, 1D cell reconstructions can be used for 3D reconstructions of a cell, such that two computational domains (one for the electrical model the other for the calcium model) can be integrated into a hybrid-dimensional simulation, using an approach comparable to [15], see Sec. 3.2.1. Using a cell reconstruction from the database NeuroMorpho.Org [4], and 3D-reconstructing the 1D geometry with AnaMorph, Fig. 7 illustrates the hybrid-dimensional representation of the neuron. Stimulation of the neuron via multiple synaptic inputs triggers an electrical response, followed by a slower calcium response. The electrical model is based on the Hodgkin-Huxley model described in Sec. 2.1 and the calcium model is based on [8].

Fig. 7.

Time series of electrical and calcium signaling in a single, reconstructed, neuron. This example illustrates the 1D-3D coupling mechanism in which electrical signals are computed on the 1D neuron reconstruction (left panels) and the coupled calcium dynamics are computed on the equivalent 3D geometry (right panels). The 3D equivalent geometry was generated from the original 1D reconstruction using AnaMorph.

4.4. Network simulation with 3D coupling

Single cell activity contributes to a massive neural network. In order to simulate the electrical and biochemical activity in large networks, computational complexity demands a dimension reduced model, i.e. 1D cell representations. In Fig. 8 a network of 1,200 neurons were generated with NeuGen [44] and connected synaptically. The network was stimulated by exciting synapses in one layer of this cortical column. While the electrical signal, that travels through the network, can be modeled relatively well in this dimension-reduced approach, intracellular dynamics can not be resolved. In a hybrid-dimensional approach (see Sec. 3.2) a single cell from the network was isolated and reconstructed three-dimensionally with AnaMorph [31]. Fig. 8 shows the wave of electrical activity passing through the cortical column, while activating intra-cellular calcium dynamics in the 3D-representation of a single neuron (inside the network).

Fig. 8.

Visualized are the electrical responses of a cortical column network (A–C, right), after synaptic stimulation of the network in layer 4 (approx. center of the network), and the calcium response of one cell in the network (A–C, left). The network was generated with NeuGen, and the isolated cell reconstructed with AnaMorph, allowing for hybrid-dimensional simulations on the network and single-cell scale.

5. Discussion

Neuroscience has evolved into a multidisciplinary discipline to better understand the multiscale architecture of the brain. Computational Mathematics plays a central role in such interdisciplinary methods, as mathematical formulation of biological systems enables a link between modeling, numerical mathematics, and computer science to systematically investigate neurobiological phenomena. There are typically two opposing forces when designing a computational approach, the inclusion of a detailed model and the computational complexity to solve the model. This leads to compromises where either biological detail is reduced in order to solve large network problems, or the size of the region of interest (e.g. single cell studies vs. network studies) is reduced in order to include necessary biological detail. In a sense, this is equivalent to adaptive refinement in grid-based numerical methods, where additional degrees of freedom are added to areas where a finer, more detailed resolution is necessary to reduce the error.

This paper introduced the two main components in cellular signaling, electrical and biochemical dynamics, followed by a far from complete overview of neuron simulators that are widely used in the computational neuroscience community. This overview also highlighted the importance of integrating modeling and simulation techniques developed for a specific model and biological scale into a framework that can make use of space-dimension adaptivity and hybrid-dimensional coupling, in order to bridge biological processes at multiple scales. The NeuroBox project, introduced in Sec. 3 has this adaptivity in mind. The ability to reconstruct the three-dimensional details of neurons, either from microscopy data, anatomical fingerprints, or lower dimensional anatomical reconstructions (see Sec. 3.1), and include it in computational studies (see Sec. 4), enables a systematic approach to study structure-function interplay in neurobiological systems, which is significantly limited with classical methods. The development of detailed 3D calcium models [8] that have been integrated in NeuroBox has furthered the understanding in morphology-controlled intracellular signaling and is beginning to shed light on the importance of geometric cell adaptation in long-term information storage [43].

Current and future research will concentrate on continued optimization of mathematical methods to efficiently handle complex computational domains, numerical solvers for biology-inspired problems, large-scale computing, and cross-disciplinary usability. NeuroBox may also have the potential to function as a conceptual model for other interdisciplinary research areas.