Numerical models of blood regulation provide

Numerical models of blood regulation provide insight into the interaction of the cellular-level and macro-scale phenomena studied in silico, a term referring to computer simulations of the dynamics of complex biological systems as opposed to in vivo or in vitro experimental studies. These in silico simulations can provide a deeper understanding of experimental results. A complex numerical model of a single NVU has been presented in the work of Mathias et al. (2018) and previously developed by Kenny et al. (2018a), Dormanns, Brown, David, 2016, Dormanns, van Disseldorp, Brown, David, 2015b and Farr and David (2011). This lumped parameter model contains compartments for each cell and distinct extracellular space in the NVU, and is based on ion exchange with experimentally validated ion channel parameters. The model contains what are believed to be the crucial components of NVC but allows for the inclusion of additional pathways. A previous NVU model has been implemented on the macro scale as described by Kenny et al. (2018b) by embedding multiple NVU in a two dimensional (2D) tissue slice. Each NVU was coupled to its nearest four neighbours by extracellular K diffusion and globally coupled via a space filling binary H-tree simulating a perfusing arterial tree (vasculature). This NVU model did not contain the nitric oxide (NO) signalling pathway (Dormanns et al., 2016), astrocytic calcium (Ca) dynamics (Kenny et al., 2018a), or complex neuron/ECS dynamics and the blood-oxygen-level dependent (BOLD) response in Nordihydroguaiaretic acid to the current numerical model of Mathias et al. (2018). In this paper, the complex NVU model of Mathias et al. (2018) has been implemented on the macro scale and extended by the addition of astrocyte to astrocyte communication via K gap junctions. In addition, extracellular electrodiffusion of K and Na has been implemented rather than the simpler approximation of Fickian diffusion. This state of the art model is the first of its kind to be able to simulate spatial phenomena such as CSD and astrocytic spatial buffering in a 2D tissue slice while also containing the complex dynamics of the NVU. Importantly, it contains full neuron/ECS dynamics and the vascular dynamics which in turn control the vessel radius and hence the rate of perfusion. This is in contrast to various models detailing only the neuron/ECS subsystem (Chang et al., 2013), neuron/astrocyte subsystem Conte et al. (2018); Huguet et al. (2016); Øyehaug et al. (2012), or in particular the previous spatial NVC model of Kenny et al. (2018b) which did not include the complex neuron/ECS dynamics necessary for modelling pathologies such as CSD. In addition, the model is embedded in a 2D tissue slice which allows for the simulation of 2D propagating wavefronts; as CSD waves travel across the cortical surface (Leâo, 1944), this is preferable to simpler one dimensional (1D) models simulating a row or array of cells (Chang, Brennan, He, Huang, Miura, Wilson, Wylie, 2013, Conte, Lee, Sarkar, Terman, 2018, Huguet, Joglekar, Messi, Buckalew, Wong, Terman, 2016, O’Connell, Mori, 2016).
Method and model development An overview of the single NVU model is described below in Section 2.1, and a schematic diagram is given in Fig. 1. The implementation of a tissue slice containing multiple NVUs on the macro scale is given below, with an explanation of the two extensions to the model: extracellular electrodiffusion and astrocytic gap junctions. A schematic overview of communication within the tissue slice is given in Fig. 2. For a full list of model equations and parameters, the reader is referred to the supplementary material.
Results While regular NVC leads to vasodilation following neuronal activity, dysfunctional NVC can lead to various pathologies such as CSD where a wave of high extracellular K moves throughout the cortex. The parameters of our model were chosen in order to produce a response typical of CSD as follows: the maximum rate of the neuronal ATPase pump was set at a lower rate of  mA cm (following the work of Kager et al. (2000)), and the number of active synapses per astrocytic process was set as (since communication between the neuron and astrocyte during CSD happens primarily through the ECS rather than the SC). These parameters are used for all simulations in this section.