Mechanistic Mathematical Modeling Tests Hypotheses of the Neurovascular Coupling in fMRI

Functional magnetic resonance imaging (fMRI) measures brain activity by detecting the blood-oxygen-level dependent (BOLD) response to neural activity. The BOLD response depends on the neurovascular coupling, which connects cerebral blood flow, cerebral blood volume, and deoxyhemoglobin level to neuronal activity. The exact mechanisms behind this neurovascular coupling are not yet fully investigated. There are at least three different ways in which these mechanisms are being discussed. Firstly, mathematical models involving the so-called Balloon model describes the relation between oxygen metabolism, cerebral blood volume, and cerebral blood flow. However, the Balloon model does not describe cellular and biochemical mechanisms. Secondly, the metabolic feedback hypothesis, which is based on experimental findings on metabolism associated with brain activation, and thirdly, the neurotransmitter feed-forward hypothesis which describes intracellular pathways leading to vasoactive substance release. Both the metabolic feedback and the neurotransmitter feed-forward hypotheses have been extensively studied, but only experimentally. These two hypotheses have never been implemented as mathematical models. Here we investigate these two hypotheses by mechanistic mathematical modeling using a systems biology approach; these methods have been used in biological research for many years but never been applied to the BOLD response in fMRI. In the current work, model structures describing the metabolic feedback and the neurotransmitter feed-forward hypotheses were applied to measured BOLD responses in the visual cortex of 12 healthy volunteers. Evaluating each hypothesis separately shows that neither hypothesis alone can describe the data in a biologically plausible way. However, by adding metabolism to the neurotransmitter feed-forward model structure, we obtained a new model structure which is able to fit the estimation data and successfully predict new, independent validation data. These results open the door to a new type of fMRI analysis that more accurately reflects the true neuronal activity.


Mm1
The metabolic model Mm1 assumes that the only mechanism that controls the shape of the BOLD response is that the blood vessels increase the blood flow in response to a lack of oxygen during the stimulus.

States and reactions
Stimulus input signal

Change in oxygen level
Delay state

Mm2
The metabolic model Mm2 assumes that the only mechanism that controls the shape of the BOLD response is that the blood vessels increase the blood flow in response to a lack of glucose during the stimulus.
Stimulus input signal

Mm3
The metabolic model Mm3 assumes that the only mechanism that controls the shape of the BOLD response is that the blood vessels increase the blood flow in response to a lack of glucose during the stimulus. The delay states are placed in the glucose feedback, not between stimulus and metabolism. Whole squares = states, dashed squares = variables (summed from states), whole arrows = transformations, dashed arrows = interactions, green area = astrocyte, blue area = neuron, grey area = blood. All states starting with S and a number are delay states. Stimulus is the input to the model. Gbody, O2body, oHbbody, dHbbody, and PL are variables. Stimulus = input signal. oHb and dHb are oxyhemoglobin and deoxyhemoglobin, respectively.

Rate of releasing oxyhemoglobin into oxygen and deoxyhemoglobin
Rate of binding oxygen and deoxyhemoglobin into oxyhemoglobin

Mn1
The neurotransmitter model Mn1 assumes that the mechanism that controls the shape of the BOLD response is the vessel response to signaling substances released by neurons and astrocytes in response to a stimulus.

Interaction graph
Fig. C: Interaction graph of the neurotransmitter feed-forward model. The neurotransmitter feed-forward hypothesis is described in more detail in Attwell 2010. Whole squares = states, dashed squares = variables (summed from states), whole arrows = transformations, dashed arrows = interactions, green area = astrocyte, blue area = neuron. All states starting with S and a number are delay states. Stimulus is the input to the model. Calcium neuron and calcium Astrocyte = calcium ion (Ca 2+ ) level in the cell, NO = nitric oxide, cGMP = cyclic guanosine monophosphate, AA = arachidonic acid, EET = epoxyeicosatrienoic acids, PG = prostaglandins and HETE = hydroxyeicosatetraeonic acid (20-HETE).

States and reactions
State Interpretation Steady State value p6 Change in AA level 768.92 EET effecting the blood vessels

Stimulus input
Glucose breakdown and reuptake Calcium influx in the astrocyte Calcium outflux in the astrocyte Calcium induced AA AA turning into HETE AA turning into PG AA turning into EET Calcium influx in the neuron Calcium outflux in the neuron Calcium induced NO

Mn2
Minimized version of the neurotransmitter model Mn1. The main mechanism is the balance between the vasoconstricting and the vasodilating arm of the model structure.
The states, variables and parameters of this model do not have a biological interpretation.

Fig E:
Fit of the minimal model Mn2 to data. Red dots = data mean and SE, blue line = model simulation.

Mnm1 Model structure
The combined model Mnm1 is a merge between the metabolic model Mm3 and the neurotansmitter model Mn1. It assumes that the vessel response to signaling substances released by neurons and astrocytes in response to a stimulus controls the blood flow. The metabolism controls the balance of dHb and oHb. Therefore, both the metabolism and the intracellular signaling controls the shape of the BOLD response. The metabolism of O2 is the main mechanism behind the intial dip while the blood flow controls the peak and the post-peak undershoot.

Change in deoxyhemoglobin level
14.45 12.79 Change in NO level 127.69 0.09 EET effecting the blood vessels

Rate of releasing oxyhemoglobin into oxygen and deoxyhemoglobin
Rate of binding oxygen and deoxyhemoglobin into oxyhemoglobin

Glucose influx Glucose outflux
Oxygen influx Oxygen outflux Stimulus input to the metabolic module Delay state reaction Stimulus input to the neurotransmitter module Glucose breakdown and reuptake Calcium influx in the astrocyte Calcium outflux in the astrocyte Calcium induced AA AA turning into HETE AA turning into PG AA turning into EET Calcium influx in the neuron Calcium outflux in the neuron Calcium induced NO Signal substance effect on blood flow ŷ unitless ! × Output signal

Mnm2 Model structure
Mnm1 is a minimized version of the model structure Mnm1.

Fig. H:
Interaction graph of the final model structure Mnm2. The model structure has two modules: the neurotransmitter module, which controls the blood flow, and the metabolic module, which controls the oxygen and glucose metabolism. Whole squares = states, dashed squares = variables (dependent on states), whole arrows = transformations, dashed arrows = interactions, green area = astrocyte, blue area = neuron, grey area = blood. All states starting with "delay" and a number (e.g delay2s) are delay states. Stimulus is the input to the model. Stimulus = input signal. oHb and dHb are oxyhemoglobin and deoxyhemoglobin, respectively. Glu = glucose, Calcium neuron and calcium astrocyte = calcium ion (Ca 2+ ) level in the cell, AA = arachidonic acid. All terms starting with k (e.g k1), are parameters and in most cases represent rate constants. PL is a parameter representing phospholipase A2, which is present in abundance. Gbody, O2body, oHbbody and dHbbody, are variables representing the glucose, oxygen and hemoglobin delivered into the area. "Constrict" and "dilate" are states representing the vasoactive substances which control the blood flow.

States and reactions
Change in deoxyhemoglobin level 16.24 16.72 Change in AA level 386.51 124.27

Rate of releasing oxyhemoglobin into oxygen and deoxyhemoglobin
Rate of binding oxygen and deoxyhemoglobin into oxyhemoglobin

Parameters and parameter values
Parameter sets for model Mnm1 used in Fig. 9 and Fig. 10. Cost: p8 = 27.5 and p10 = 18.8