. 2021 Oct 6;17(10):e1009386. doi: 10.1371/journal.pcbi.1009386

Table 1. Summary of studies detailing mathematical models of the neurovascular unit (NVU) components and their input (I) and output (O).

S. No.	Study	NVU system	Description
1	Riera et al., 2005 [80]	Synaptic space, neuron-astrocyte interface, capillary, post capillary; I: stimulus (afferent pathways); O: vessel volume	The study used a bottom-up physiological model for fusing EEG and fMRI data depicting the electro-vascular coupling. Synaptic activity and hemodynamics were governed by mesoscopic dynamic equations system.
2	Riera et al., 2006 [81]	Canonical neural mass component, electrovascular coupling component, vascular state dynamics component; I: excitatory synaptic inputs; O: Cerebral blood volume and cerebral blood flow	The study presented the cerebral architecture, electro-vascular coupling and energy considerations with respect to EEG and fMRI data fusion. The study extended balloon approach for modeling the vascular changes.
3	Bennett et al., 2008 [87]	Synaptic space, astrocyte, SMC; I: Glutamate released from glutamatergic synapse; O: Cerebral blood volume and cerebral blood flow	The study showed that the coupling between glutamatergic synapses to arteriolar SMC is mediated by astrocytic epoxyeicosatrienoic acids (EETs). Results depicted a linear rise in blood flow with an increase in numbers of activated astrocytes; however, the response was non-linear with respect to the release of glutamate.
4	Banaji et al., 2008 [88]	Cerebral circulation component, mitochondrial metabolism component; I: Blood pressure changes, changes in arterial oxygen and carbon-dioxide levels, functional activation; O: cerebral metabolic rate of oxygen (CMRO2), changes in oxidation level	The study modeled the brain circulation and metabolism to analyze the experimental fNIRS data in response to various stimuli.
5	Farr and David, 2011 [83]	Synaptic space, astrocyte, perivascular space, SMC, and endothelial cells; I: Glutamate and K⁺ in synaptic space; O: arteriolar diameter	The study showed that the coupling between neuronal activation (due to K⁺ and glutamate) to arteriolar dilation was mediated by astrocytic K⁺, EET, and Ca²⁺.
6	Witthoft and Karniadakis, 2012 [45]	Synaptic space, astrocyte, perivascular space, SMC; I: Glutamate and K⁺ in synaptic space; O: arteriolar radius	The study showed the bidirectional communication between cerebral astrocytes and the microvessels. Major signaling pathways considered were: neural synaptic K⁺ and glutamate to astrocytes, K⁺ signaling between astrocytes and microvasculature, and microvasculature to astrocytes via astrocyte perivascular endfoot.
7	Chander and Chakravarthy, 2012 [89]	Neuron, astrocyte, vessel, and interstitium; I: Synaptic current and Adenosine Triphosphate (ATP); O: vessel radius via EET signaling	The study presented a model for the neuron-astrocyte-vessel loop based on neuronal and metabolic activity.
8	Witthoft et al., 2013 [90]	Synaptic space, astrocyte, perivascular space, SMC; I: Glutamate and K⁺ in synaptic space; O: arteriole radius	Extended version of the model from Witthoft and Karniadakis, 2012⁴²with K⁺ buffering across all components of NVU.
9	Chang et al., 2013 [91]	Soma, dendrite, extracellular space, vascular tree compartment, glial compartment; I: extracellular K⁺; O: vessel radius	The study demonstrated the coupling between the vascular diameters and neuronal activity mediated by K⁺ concentrations in extracellular space in the vicinity to dendritic processes that were assumed to be buffered through astrocytes.
10	Dormanns et al., 2015 [92]	Neuron, synaptic cleft, astrocyte, perivascular space, SMC, endothelial cell, and arteriolar lumen; I: synaptic K⁺; O: arteriole radius	The study used lumped model of the NVU for depicting the connection between a neuron and the perfusing arteriole through the astrocytic perivascular K⁺ signaling and the SMC’s Ca²⁺ dynamics. The study showed the significance of luminal agonists in flowing blood influencing the endothelial and SMC dynamics.
11	Dormannset al., 2016 [84]	Neuron, synaptic cleft, astrocyte, perivascular space, SMC, endothelial cell, and arteriolar lumen; I: synaptic K⁺; O: arteriole radius	An extended version of Dormannset al., 2015 model⁶⁷with NO signaling pathway. The model considered the production of NO in the neuron and the endothelial cell compartments and its diffusion in the other compartments.
12	Blanchard et al., 2016 [93]	Pyramidal cells, interneurons, extracellular space, astrocytes, vessels; I: Excitatory postsynaptic potential (EPSP) and Firing rates (FR) from pyramidal cells and interneurons; O: local field potential and regional cerebral blood flow	The study demonstrated the connection between the neuronal activity and regional CBF via neuro-glio-vascular link at the population scale (voxel). The model evaluated the role of astrocytes in glutamate and GABA recycling, which then influences adjoining vessels
13	Mathias et al., 2017 [94]	Soma, dendrite, extracellular space, synaptic space, astrocyte, perivascular space, SMC, endothelial cell and lumen; I:neural activation through ion channels; O: fMRI BOLD signal	The study demonstrated the signaling method of neurovascular coupling through a model of pyramidal neurons and its analogous fMRI BOLD response. The study extended the NVU to include a complex neuron system with Na/K ATPase pump mechanism, which provides CBF and CMRO2.
14	Kenny et al., 2018 [85]	Neuron, synaptic cleft, astrocyte, perivascular space, endothelial cell, SMC, and lumen; I: Glutamate and K⁺ in synaptic space; O: arteriolar radius	The model used lumped parameter systems to depict the connection between a neuron and perfusing arteriole through the astrocytic perivascular K⁺ and the SMC’s Ca²⁺ dynamics mediated by astrocytic EETs and TRPV4. Results indicated that K⁺ mediated pathway drives the quick start of vaso-dilation compared to the NO-mediated pathway.
15	Mathias et al., 2018 [86]	Neuron (soma, dendrite), extracellular space, synaptic space, astrocyte, perivascular space, SMC, endothelial cell and lumen; I:neuronal current; O: fMRI BOLD signal	The model simulated NVU mechanisms and BOLD signal by extending the previous models by Mathias et al., 2017⁶⁹ and Kenny et al., 2018⁵⁹. The study included a transient sodium ion channel in the neuron compartment.
16	Sten et al., 2020 [95]	Pyramidal neuron, GABAergic interneuron, astrocyte, SMC, arteriole; I: neuronal pulse mediating vaso-agents; O: arteriolar diameter	The study modeled the interplay between pyramidal neurons and GABAergic interneurons in the NVU. The study evaluated the role of cell-specific contributions in NVU due to the effect of an anesthetic agent.