2.1.1 Fractional diameter change of arterioles and venules in awake mice.

The model was trained using experimental data extracted from the study of Drew et al. [46], consisting of arteriolar (Fig 3A–3C, red symbols) and venular (Fig 3D–3F, blue symbols) diameter changes induced by a sensory whisker stimulation for three different durations: 20 ms (Fig 3A and 3D), 10 s (Fig 3B and 3E) and 30 s (Fig 3C and 3F). The stimulation consisted of air puffs (8 Hz, 20 ms duration), and as in Barrett et al. [34], we opted to implement this paradigm as one constant stimulation block (i.e. a square wave, combining the period between each air puff, 105 ms, and the air puff itself, 20 ms, with 8 puffs a second equalling 1000ms) to reduce the complexity of the stimulation input to the model. The resulting stimulation has a constant amplitude with the following durations: 125 ms, 10 s, and 30 s. In the model, we calculate the diameter change as the square root of the volume change for each respective compartment. The model parameters were fitted to all experimental time series simultaneously (Fig 3A–3F, colored symbols), achieving a quantitative acceptable agreement to experimental data (Fig 3A–3F, solid, colored lines), evaluated using a χ²-test ( = 292.59, cut-off: χ² (288 data points) = 328.58 for α = 0.05). The degrees of freedom for the confidence interval were df_H1 = 37, which is equal to the number of estimated parameters. The reader is referred to S1 Appendix for the posterior probability profiles (S1 Fig) and the parameter boundaries applied during the estimation (S1 Table). The estimated model uncertainty is depicted in the form of shaded red and blue areas in Fig 3.

The simulations follow the qualitative behavior of the experimental data for each graph except for Fig 3E, which is the venular response to a 10 s stimulation, where the model simulations show a slower rise and decline than the experimental data (Fig 3E, compare blue line with blue symbols). A more accurate fit can be obtained by allowing the viscoelasticity and stiffness coefficients of the capillary and venous compartments to change between the two short (125 ms & 10 s) and the long (30 s) stimulation (S2 Fig, = 199.69 with the parameter profiles depicted in S3 Fig). This improved agreement indicates that a non-linear vascular model is needed to fully capture the dynamics seen in this data set.

2.1.2 The observed dynamic behavior can be explained by two distinct dilation processes.

The arteriolar responses transition from a one-peak (Fig 3A) to a two-peak response (Fig 3C), as the stimulation length increases. In other words, for the short stimulation, the response rises to a peak after 2–3 seconds, followed by a decline to baseline. For the two longer (10 s and 30 s) stimulations, a quick initial dilation is followed by a small cessation to a plateau value around 10 s (Fig 3B and 3C). At 10 s, the stimulation ends in one case, with a minor dilation before returning to baseline diameter (Fig 3B). However, for the 30 s stimulation, a new dilation phase occurs, consistently dilating until the end of the stimulation (Fig 3C, t = 10–30 s). Finally, after stimulus cessation for the longest stimulation, but not for the shorter stimulations, a post-peak undershoot in the arteriolar diameter is observed before returning to baseline (Fig 3C, t ~ 50 s).

These complex dynamic behaviors between different stimulation lengths are captured by the model (Fig 3A–3C). It is therefore relevant to examine by which mechanisms the model produces these behaviors. Model simulations are shown for all three stimulation lengths: 0.125 s (Fig 4A & 4E), 10 s (Fig 4B & 4F), and 30 s (Fig 4C & 4G). Furthermore, Fig 4A–4C correspond to the rapid neuronal responses (Fig 4A–4C), and Fig 4E–4G correspond to the impact of the vasoactive substances: NO (nitric oxide), PGE₂ (prostaglandin E2), NPY (neuropeptide Y). In addition, the simulations are shown for all three neuronal types and corresponding vasoactive substances: pyramidal neurons and PGE₂ (orange), NO interneuron and NO (light green), and NPY interneurons and NPY (green-gray). Finally, the resulting behavior (Fig 4E–4G, black dashed line) is produced by the summation of the three impacts, where NPY is vasoconstrictive (negative), and PGE₂ and NO are vasodilative (positive). To assist navigation between these different simulations, Fig 4D gives a simplified view of the model and the mechanisms which underlie the different dilation behaviors observed in Fig 3A–3C.

With these interpretations in place, we can understand how the model produces the complex dynamic behaviors observed in data. It is easiest to follow the chain of events by starting at the last step before the observed behavior. The response to a short stimulation is produced by the fast release of NO, which produces the initial peak in the arteriolar response (Fig 4F–4G, light green line) and which thereafter simply declines. Note that the vascular effect is the relative difference of released NO compared to the initial condition and a negative value therefore should be interpreted as lower levels compared to the basal state and not as negative (see Eq 8). The other two signaling pathways are slower and therefore negligible. Also, for the longer stimulations, the NO behavior is important. For these stimulations, the NO response culminates quickly, whereafter the impact of NO falls which causes a decrease after the first peak. However, for longer stimulations, this decrease is replaced by a new increase, which is caused by the rise of the slower PGE₂ signaling arm from the pyramidal neuron (Fig 4F–4G, orange line). In other words, the PGE₂ arm is responsible for the second dilation phase observed for the 30 s stimulation (Fig 4G, orange line). Finally, the NPY signaling arm is only important to explain the post-stimulus undershoot observed in the 30 s stimulus paradigm as it is so slow that it requires such long stimulation to become larger than the two other arms (Fig 4G, the green-gray line is above the light green and orange for t ~ 50 s). In summary, the initial peak is explained by the fast NO arm, the second peak is explained by the slower PGE₂ arm, and the post-peak undershoot is explained by the even slower NPY arm.

These different dynamics in the three arms occur at the downstream secretion level but must be initiated by corresponding changes at the rapid neuronal level (Fig 4A–4C). For the shortest stimulation (Fig 4A), the NO interneurons are the fastest, and pyramidal neurons and NPY interneurons are equally fast. For the two longer stimulations (Fig 4B and 4C), the electrophysiology reaches a steady-state within 1 second, whereafter no changes occur until the stimulation ends. Since the secretion of NPY is much slower than PGE₂, and because both these dynamics occur over many seconds, the differences in dynamics at the downstream secretion level are almost exclusively explained by the difference in intracellular signaling. The only difference concerns the NO dynamics, which has a peak and decline at the secretion level caused by a corresponding rise and fall at the electrophysiology level.

2.1.3 The model can still describe and correctly predict arteriolar responses to optogenetic and sensory stimulations.

In our previous computational work (see [42]), we developed a model capable of estimating and predicting the experimental data found in Uhlirova et al. [47]. To verify that this extended model still maintains a good agreement and predictive power with that experimental data, we repeated the model training and model predictions in [42], see S4 Fig. The Uhlirova study reports arteriolar diameter changes in mice evoked by different stimuli. These different stimuli include for instance optogenetic and sensory stimulation, where optogenetics can stimulate either pyramidal neurons or inhibitory interneurons separately. They do this cell-specific stimulation with and without pharmacological perturbations, using the NPY receptor inhibitor BIBP, and the glutamatergic signaling inhibitors CNQX and AP5. These pharmacological perturbations cut off the crosstalk between NPY and the vasculature and the crosstalk between pyramidal neurons and interneurons, respectively. Furthermore, they also study the animals during awake and anesthesia conditions. In [42], our model can explain all these data with the same parameters, and can also predict independent data not used for training. Our updated model has the same capability (see S1 Appendix section 1.3), and these results e.g., support the notion that NPY acts vasoconstrictive, and causes the post-peak undershoot.

2.1.4 Hemoglobin and BOLD measures in awake mice.

Next, we analyzed experimental data from the study of Desjardins et al. [48]. In brief, the study reports hemodynamic measures of blood oxygenation (oxygenated, deoxygenated, and total hemoglobin) from three different stimulations: 1) optogenetic stimulation of inhibitory interneurons; 2) optogenetic stimulation of pyramidal neurons, and 3) sensory whisker stimulation using air puffs. Each of these three paradigms had a short (optogenetic: 100 ms light pulse; sensory: 2 s of air puffs at 3–5 Hz) and long (optogenetic: 20 s of 100 ms light pulses at 1 Hz; sensory: 20 s of air puffs at 3–5 Hz) duration. They also reported BOLD data for the long optogenetic stimulation of pyramidal neurons. We implemented these stimulus paradigms in the model and assumed a 100 ms air puff delivered at 3 Hz for the sensory stimulus. To capture the variability in frequency, which gives rise to an observable difference in stimulation strength between the two sensory experiments (Fig 5C and 5F), we allowed the input parameters (k_u,i) to change between the short and long duration of the sensory stimulation but could keep the same values for both optogenetic paradigms (see Discussion 3.3). Furthermore, in the model, we assumed a baseline blood concentration of 100 μM hemoglobin, in line with previous models [35,37,50].

With these data and model settings, the model was trained to experimental data of hemoglobin dynamics for all three stimulation paradigms simultaneously (Fig 5A–5F, dark-red/light-blue/green symbols), and the BOLD data was left out to be used for model validation (Fig 5J, magenta symbols). The model achieved a quantitative acceptable agreement to experimental data (Fig 5A–5F, red/blue/green lines) after parameter estimation ( = 323.36, cut-off: χ² (354 data points) = 397.87. The degrees of freedom for the confidence interval was df_H4 = 43, which is equal to the number of estimated parameters (Fig 5A–5F, shaded areas). The reader is referred to S1 Appendix for the posterior probability profiles (S5 Fig) and the parameter boundaries applied during the estimation (S3 Table)

To validate the model and avoid overfitting of the parameters, we used the model to generate predictions of a BOLD response for a 20 s optogenetic stimulation of pyramidal neurons. These predictions are depicted as the magenta-shaded area in Fig 5J. The corresponding experimental data (Fig 5J, magenta symbols) lie within the model prediction bounds. The parameter that best describes training data also passes a χ² test for the BOLD validation data ( = 29.23, cut-off: χ² (23 data points) = 35.17).

2.1.5 Summary of all mice data.

In summary, all results up until now taken together show that our model structure provides an acceptable explanation for a wide variety of NVC aspects observable in mice. Figs 3 and 4 unravel the role of three regulatory arms: NO interneurons are responsible for the rapid rise, PGE₂ from pyramidal neurons are responsible for the second peak that occurs for long stimulations, and NPY interneurons for the post-peak undershoot. In these data, vessel diameter is used as a proxy for the NVC control. In Fig 5, we confirm that our model can explain the NVC control expressed in hemoglobin measures: total hemoglobin serves as a proxy for blood volume, oxygenated- and deoxygenated hemoglobin examines the balance between metabolism and blood flow, and the BOLD signal provides a link to the most common non-invasive measure in primates and humans. For a comparison of the posterior probability profiles for Figs 3 and 5 –see S1 Appendix section 1.7.1. Finally, just like our previous model [42], our new model can explain the highly informative optogenetic-data from Uhlirova et al. (2016) [47].

2.2.1 The model agrees with BOLD and LFP data.

We analyzed experimental data from the study of Shmuel et al. [10], which presents unique data for higher primates i.e., macaques. We extracted electrophysiological and BOLD responses induced by a visual task in anesthetized macaques. The task was presented in two ways, inducing both positive and negative responses in the same area (see section 4.4.3 for details). The electrophysiological response was related to the local field potential (LFP), which is to a large extent influenced by pyramidal neuron firing (see review in [51]). The model was trained to experimental data for both positive (Fig 6A and 6B, BOLD, magenta and LFP, pink symbols) and negative (Fig 6A and 6B, BOLD, purple and LFP, dark-purple symbols) responses simultaneously. The different experimental stimulus was applied as a square pulse to the model (see Eq 2). For the negative experimental setup, sign parameters were added to interpret if the stimulus should act positively or negatively upon the neurons (see Eq 3), resulting in a negative square pulse. The model achieved a quantitative acceptable agreement to experimental data (Fig 6A and 6B, red and blue solid lines) after parameter estimation ( = 176.13, cut-off: χ² (160 data points) = 190.52). The degrees of freedom for the confidence interval were df_H4 = 52, which is equal to the number of estimated parameters (Fig 6A and 6B, shaded areas). The reader is referred to S1 Appendix for the posterior probability profiles (S6 Fig) and the parameter boundaries applied during the estimation (S4 Table).

2.2.2 The model translates the mechanistic insights and qualitative behaviors obtained from mice data.

Finally, we translated the mechanistic insights obtained from studies in mice (Figs 3–5) to this analysis on primates (Fig 6C–6F). More specifically, for the sensory stimulation (Fig 6C), we know aspects of the order and relative function of the three signaling arms (Fig 4): NO (light-green) displays only a fast initial response followed by a rapid decline; PGE₂ (orange) displays a slower and more prolonged response covering the main part of the NVC control; NPY (green-gray) has the slowest response, and is only prominent in the creation of the post-peak undershoot. Here, there is no visible peak on NO, and the response almost directly goes to the rapid decline. This behavior is needed for the model to be able to generate the rapid response in LFP, which is predominantly caused by the pyramidal cells, which are inhibited by the NO-interneurons. In other words, if the NO-interneurons would have had a substantial rapid increase, the LFP simulation would not be fast or high enough. Note that there is no known information about the relationship between the three arms for the negative BOLD signal (Fig 6D). Apart from these mechanisms, we also translated mechanistic knowledge from Fig 5, concerning the relative dynamics of total hemoglobin (HbT), oxygenated hemoglobin (HbO), and deoxygenated hemoglobin (HbR) (Fig 6E and 6F). Just as in the mice data, for the positive BOLD response, the model is showing an increase in HbT (green), an even greater increase in HbO (dark red), and a decrease in HbR (light blue), all with the same dynamics as in the BOLD response. For a further comparison of the posterior probability profiles for Figs 3–7 –see S1 Appendix section 1.7.2.

2.3.1 The model agrees with both estimation and validation data for CBV, CBF, and BOLD data, from both arterioles and venules.

In the last step, we move to humans and looked at the study conducted by Huber et al. [49]. We extracted the experimental measurements of CBV, CBF, and BOLD responses in humans, evoked by multiple visual stimuli. The study employed a set of three different flickering checkerboard patterns as the visual stimuli (see section 4.4.5. for more details). The model was trained on experimental data consisting of both positive and negative CBV (Fig 7A) and BOLD responses (Fig 7D), as well as experimental data for the total CBV changes for the excitatory (Fig 7B, black) and inhibitory (Fig 7C, black) tasks. Parameter estimation was employed to achieve an acceptable fit to the data ( = 54.64, cut-off: χ² (122 data points) = 148.78). The reader is referred to S1 Appendix for the posterior probability profiles (S7 Fig) and the parameter boundaries applied during the estimation (S5 Table). The simulation uncertainty can be seen as the colored shaded areas in Fig 7A–7E. As seen, the model simulations have a good agreement with the experimental data (Fig 7A and 7D, shaded areas, and Fig 7B and 7C, black shaded areas), as the model uncertainty area overlaps well with the experimental data points. To test the model’s predictive qualities, we used the model to generate predictions of CBF for both the positive and negative responses (Fig 7E), as well as predictions of arterial CBV and venous CBV for both the excitatory (Fig 7B, red and blue) and inhibitory flickering checkerboard tasks (Fig 7C, red and blue). As can be seen, there is a good agreement between the model predictions (Fig 7B, 7C and 7E, areas without a line) and the experimental data (error bars).

2.3.2 The model has translated all insights obtained from both rodent and primate data.

We translated all mechanistic insights obtained so far i.e., from both the mice (Figs 3–5) and the primate (Fig 6) data. For the positive BOLD response (Fig 7D), we know and preserve the order of the three arms: the NO-response (light green) is fastest and followed by a rapid decline, the PGE₂ response (orange) is slower and covers the main response, and the NPY-response (grey-green) is slowest and is only dominant during the post-peak undershoot (Fig 7F). For the hemoglobin dynamics (Fig 7H) in the positive BOLD response, we know that the total hemoglobin (HbT, green) increases due to an increased blood flow, which increases oxygenated hemoglobin (HbO, dark red) and decreases deoxygenated hemoglobin (HbR, light-blue) (Fig 5F). Finally, from the primate data (Fig 6), we know that the LFP response (Fig 7J) is faster than the BOLD response, has an overshoot that goes down to an intermediate plateau throughout the stimulation, and has a corresponding undershoot below baseline post-stimulation. We show corresponding predictions for these three properties (the three arms of vascular control, Fig 7G; hemoglobin dynamics, Fig 7I; and LFP, Fig 7K), but it is only for LFP that a known behavior is translated. For a further comparison of the posterior probability profiles for Figs 3–7 –see S1 Appendix section 1.7.2.

4.2.1. Presynaptic activity and calcium influx.

Glutamate and GABA are released from different types of neurons upon membrane depolarization and bind to specific ion channel-coupled receptors located in the neuronal plasma membrane. Glutamate, an excitatory neurotransmitter, binds to α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-D-aspartate (NMDA) receptors. Activation of these ion channel-coupled receptors triggers ion-conducting pores to open, triggering an influx of Na⁺ and Ca²⁺ ions which cause a depolarization of the neuron. Depolarization opens voltage-gated calcium channels, allowing a further influx of Ca²⁺ ions. In contrast, γ-aminobutyric acid (GABA) acts to prevent depolarization of neurons as it binds to the ion channel-coupled GABA_A or GABA_B receptor, which opens Cl^- ion-conducting pores. Pyramidal neurons are glutamatergic, meaning that glutamate is released upon depolarization, and acts on e.g., astrocytes and interneurons. GABAergic interneurons release GABA upon depolarization and target pyramidal neurons or other interneurons. This forms a simple relationship, where interneurons regulate the activity of pyramidal neurons. In the model, we represent these interplays between pyramidal and inhibitory interneuron activity and their combined effect on respective neuronal Ca²⁺ levels as

Where N_NO, N_NPY, and N_Pyr describe the neuronal activity of the GABAergic nitric oxide (NO) and neuropeptide Y (NPY) interneurons, and the pyramidal neurons, respectively; where represents intracellular calcium in each respective neuron; k_u,i is the input strength of the stimulation to the respective neuron; where k_PF1 and k_PF2 are the kinetic rate parameters governing pyramidal to GABAergic interneuron signaling; where k_INF1 and k_INF2 are the kinetic rate parameters describing the negative feedback from GABAergic interneurons to pyramidal neurons; where k_IN1 and k_IN2 are the kinetic rate parameters describing the negative feedback between GABAergic interneurons; where sink_N,i are kinetic rate parameters governing the degradation of the activity of respective neurons; where k_Ca is the basal inflow rate of Ca²⁺ (k_Ca = 10 for all three neurons), and finally, where sink_Ca,i is the elimination rate of intracellular Ca²⁺, and i = (Pyr, NO, NPY) denote the three different neuronal types.

4.2.2. Pyramidal neuron signaling.

The rise in intracellular Ca²⁺ levels in pyramidal neurons activates phospholipases which metabolize, through intermediary enzymatic steps, membrane phospholipids into intracellular arachidonic acid (AA) [12,69,91]. This is described by: where k_PL and k_COX are kinetic rate parameters. In pyramidal neurons, AA is metabolized into prostaglandin E₂ (PGE₂) through a cyclooxygenase-2 (COX-2) and PGE synthase rate-limiting reaction [69,70]. PGE₂ evokes vasodilation through the activation of EP2 and EP4 receptors expressed on the surface of vascular smooth muscle cells (VSM) cells [69,92]. In the model, this mechanism described as

Where PGE_2,vsm represents PGE₂ acting on the VSM cells, and k_PGE2 and sink_PGE2 are kinetic rate parameters.

4.2.3. GABAergic interneuron signaling.

The rise in intracellular Ca²⁺ levels in GABAergic interneurons evokes the release of different vasoactive messengers and substances. For instance, NO has previously been shown to be a potent vasodilator at the level of arteries and arterioles in both in vitro and in vivo studies [87,92–94]. NO is released by specific NO-interneurons through a Ca²⁺ dependent nitric oxide synthase (NOS) rate-limiting reaction. Here, we represent this process as where NO_vsm represents NO acting on the VSM cells, and k_NOS, k_NO and sink_NO are kinetic rate parameters. Here, sink_NO>1 s⁻¹ following experimental work by [95].

Another potent vasoactive messenger is NPY. NPY is released by specific NPY producing subtypes of GABAergic interneurons and has previously been shown to induce vasoconstriction of vessels in vitro [92,96] and in vivo [47]. NPY binds to the NPY receptor Y1, a G_αi-protein coupled receptor expressed on the surface of VSM cells enwrapping arteries and arterioles [97–99]. Activation of this receptor inhibits the synthesis of adenosine monophosphate (cAMP) and increases the intracellular Ca²⁺ [100], leading to VSM contraction and vasoconstriction. In the model, these mechanisms are represented as: where NPY_vsm represents NPY acting on the VSM cells, and k_NPY and sink_NPY are kinetic rate parameters. The conversion between intracellular NPY and NPY in the VSM, NPY_vsm, is governed by Michaelis-Menten kinetics [101,102], where K_M is the Michaelis constant (the concentration of the substrate at which half max reaction rate is achieved) and V_max is the maximal reaction rate.

The expression of the different vasoactive substances is scaled with a parameter, ky_i, i = {NO, PGE2, NPY}, and summarized to a total vascular influence G:

4.2.4. Cerebrovascular dynamics described electrical circuit analog model by Barrett et al. (2012).

We describe the dynamics of three cerebrovascular compartments, corresponding to arteries/arterioles (a), capillaries (c), and veins/venules (v), respectively. The compartments are represented by an electrical circuit analogy, as originally presented in the work by Barrett et al. [34,43,44]. We use the derivative work of [103], and only present the fully derived equations to improve the clarity for the reader. We refer the reader to the original articles for an in-depth description of the equations [34,43,44,103]. The following equations describe the volume change for each respective cerebrovascular compartment, V_i, with i = {a, c, v}.

Here, K are stiffness coefficients; k_vis,i are viscoelastic parameters; C_i represents the compliance of the vessel; R represents the vessel resistance; f represents the flow of blood between the cerebrovascular compartments; the baseline value is indicated by the subscript 0, and finally, G is the vasoactive function translating the actions of the vasoactive arms (Eq. 8) into hemodynamic changes.

In short, the arterial compartment is actively regulated by the vasoactive arms (Eq. 8 & 9a), and the evoked hemodynamic changes in the arterial vessels are propagated through capillary and venous compartments.

Using conservation of mass, the rate at which the blood volume change in a compartment is given by the difference between the in- and outflow of blood:

Restructuring the equation gives the following relationship:

Next, the pressure drop over a compartment is given by:

Which is subject to the pressure boundary conditions:

Using these conditions, we can solve for the inflow of blood into the first compartment f₀:

The equations above form a DAE system, comprised of three volumes and four flows. To further simplify, the scales of the variables are normalized to:

Using this, the rest of the physiological terms can be expressed as: where P is the pressure at the entry point to each compartment, C is the compliance of the vascular vessel, L is the length of the vascular segment, and R is the vessel resistance to flow.

Given Eqs 9–16, the initial conditions for the cerebrovascular volumes V_i and the vascular resistance R_i must be specified. We set these initial conditions in accordance with the work of Barrett et al. [34], which based these values on the available literature, resulting in V_i,0 = [0.29, 0.44, 0.27] and R_i,0 = [0.74, 0.08, 0.18].

4.2.5. Oxygen transportation.

To describe the oxygen transportation through the cerebrovascular system, we describe three vascular compartments and a cerebral-tissue compartment. The amount of oxygen in these compartments is given by: where the flow of blood f transports oxygen in and out from each compartment. The permeability in the vessel wall allows for oxygen to diffuse to the tissue compartment, jO2_i, where i = {a, c, v}. An arterio-venous diffusion shunt is described by jO2_s, where oxygen moves from the arteriole to the venous compartment. Oxygen is metabolized in the tissue compartment, CMRO₂. The metabolism of oxygen is given by: where CMRO_2,0 is the basal metabolism of oxygen, which is increased by the activity level of each neuron (see Eq. 3) scaled by the parameter k_met.

The concentration of oxygen, is given by the amount of oxygen, , divided by the volume of each compartment, V_i, except for the oxygen concentration that enters the arteries, , which is assumed to be constant, minus a loss of oxygen to the surrounding tissue .

Where the tissue volume fraction V_t = 34.8 [43] and (calculated using data from [43,104]).

As in the original work, by ignoring the minor fraction of oxygen directly dissolved in the blood plasma, blood oxygen concentration can be related to oxygen partial pressures through the oxygen-hemoglobin saturation curve:

Where p₅₀ = 36 mmHg is the oxygen partial pressure at the halfway saturation point, is the maximum oxygen concentration in blood, and h = 2.6 is the Hill exponent. These values are taken from [105].

The oxygen pressure in the tissue, , is calculated using Henry’s law:

Where σ_Oxygen = 1.46 μM/mmHg is the coefficient for solubility of oxygen in tissue [106].

The average pressure, , in a compartment is obtained by averaging over the input and output pressures:

The diffusion of oxygen between the vessels and the tissue, , as well as between the arteries and the veins, , is driven by the difference in partial oxygen pressure: where g_i, i = {a, c, v, s} are rate constants.

We employed the identical strategy of Barrett and Suresh [43] (see supplementary material in [43]), and used the experimental pO₂ data from Vovenko [104] to calculate g_i, , and CMRO_2,0. This leaves the parameter k_met to be estimated where applicable.

The oxygen saturation of blood in the different vascular compartments, S_iO₂, is approximated by dividing the average oxygen concentration of each compartment with :

Lastly, using the blood oxygen saturation S_iO₂ and vascular volumes (V_i), corresponding changes in oxygenated hemoglobin (HbO), deoxyhemoglobin (HbR), and total hemoglobin (HbT) are given by:

4.2.6. BOLD signal derivation.

Finally, the oxygen saturation of blood in the different vascular compartments, S_iO₂ (Eq. 24) and the vascular volumes, V_i (Eq. 9) are used to calculate the BOLD signal. The following derivation of the BOLD signal equations and parameter values are taken from the work of [45], and we refer the reader to the original article for a detailed view of the equations. The BOLD signal is expressed as a summation of the contributions from each vascular compartment {a, c, v} and an extracellular compartment {e} symbolizing the tissue:

Where S_j is the intrinsic signal from each compartment; is the extravascular volume; is the intravascular volume scaled by the baseline intravascular volume fraction (V_iv,0 = 0.05, [107]); ε_i is the intrinsic signal ratio of blood and λ is the intravascular to extravascular spin density ratio (we assume λ = 1.15); is the transverse signal relaxation rate for the four compartments, and TE is the echo time used during image acquisition in each study.

The transverse signal relaxation rate for the extravascular compartment () was assumed to be 25.1s⁻¹ [108], and was calculated for the intravascular compartments by utilizing the following quadratic expression taken from [109]: where

These equations are governed by S_iO2 (Eq. 24) and the resting hematocrit in blood (Hct; Hct_a,v = 0.44, Hct_c = 0.33 [110,111]).

The respective signal contribution from each compartment is given by:

Where (i.e., the change in MR signal relaxation rate) is given by the following expressions:

Here, Δχ = 2.64×10⁻⁷ is the susceptibility of fully deoxygenated blood [112]; γ = 2.68×10⁸ is the gyromagnetic ratio of protons; B₀ is the magnetic field strength used during image acquisition, and finally, S_offO₂ = 0.95 is the blood saturation which gives no magnetic susceptibility difference between blood and tissue [112].

4.2.7. LFP simulations.

Finally, for the fitting to the LFP data in macaques, we use the following expression: where N_Pyr is the phenomenological activity state of pyramidal neurons, and is an unknown scaling constant.

4.3.1 Optimization of parameters.

Once the model structure has been formulated (Fig 2) and data has been collected (data acquisition described below), the parameters, θ, need to be determined. This is commonly done by evaluating the negative log-likelihood function. Assuming independent, normally distributed additive measurement noise, the negative logarithm of the likelihood of observing data D given θ is: where n_e is the number of experiments e; where is the number of observables o per e; where is the number of samples per o and e; where is the standard deviation of the data point; where is the measured data point; where is the corresponding simulated data point. By maximizing L, the maximum likelihood estimate of the unknown parameters, θ can be obtained. However, it is more common and more numerically efficient to minimize the equivalent negative log-likelihood function. If the measurement noise is known, J(θ) share the same optimal parameters with the least-squares function J_lsq:

In addition to this quantitative comparison with data, qualitative insights are added to the function to be optimized, converting J_lsq(θ) to J_opt(θ) (see below). In practice, the parameters are determined by optimizing J_opt(θ) over θ by using various optimization algorithms (see Section 4.5 below), i.e. they are given by: where are the optimal parameters.

4.3.2 Simulation uncertainty.

The model uncertainty was estimated using a Markov chain Monte Carlo (MCMC)-sampling procedure with 10⁵ samples (see section 4.5 for details), generating posterior distributions of the parameters, and collecting all χ² acceptable parameters encountered. Using the acceptable χ² parameters, a confidence interval: was drawn where the threshold is the α quantile of the statistics [113,114]. We also require that all acceptable parameters should pass a χ²-test.

For a small fraction of the parameters identified using the optimization approach described above, certain types of unrealistic behavior may appear. For instance, in Fig 5, regarding the Desjardins mice data, less than 1% of the parameters display a rapid spike after stimuli has ceased. Such unrealistic spikes are not prohibited by the cost function because the simulation has returned to the next data point. However, we judge such rapid spikes to be unrealistic and thus remove them from the uncertainties.

4.3.3 Preservation of qualitative features between the species and experiments.

The optimization is done using a cost function that minimizes the quantitative agreement between each dataset studied, while also fulfilling the qualitative demands and mechanistic insights obtained from the analysis of other datasets (Section 4.3.1). This is done by the addition of extra terms to the cost function, which are zero if the qualitative demand is fulfilled, and high otherwise. On a top-level description, those qualitative demands are as follows:

• Insights from Figs 3 and 4: The vasoactive substance NO, produced by the NO-interneurons, has initially fast dynamics compared to the PGE₂-release by the pyramidal cells. After this fast initial response, the vasoactive control by the NO-interneurons is quickly turned off again, leaving the pyramidal cells as the primary dilator for prolonged stimulations, sometimes causing a two-peak response. The post-peak undershoot is caused by the slow NPY-release caused by the NPY-interneurons, which is vasoconstricting also after the vasodilating controls have subsided.
• Insight from Fig 5: The oxygenated and deoxygenated hemoglobin responses should have the same timing, and the difference in amplitude between these two responses should not be too big.
• Insight from Fig 6: Following a stimulation, a fast peak in LFP is produced, which thereafter diminishes down to a lower plateau, where LFP remains for the rest of the stimulation. After the stimulation has been turned off, LFP quickly declines to values below baseline, whereafter it recovers back to the baseline again. For a negative BOLD response, this behavior in the LFP is inverted.

Quantitative specifications of these qualitative demands are given in S2 Appendix, Eqs S1–15, and all optimization scripts are provided, together with the analyzed models and data.

4.3.3 Handling differences between negative and positive BOLD response.

For the data with both a positive and a negative BOLD (Figs 6 and 7), we allow the input stimulus parameters (k_u,i) to change. Furthermore, we also allow the parameters governing the inter-neuronal interactions (k_PF1, k_PF2, k_INF1, k_INF2, k_IN1, k_IN2), and the parameters for neuronal recovery (sink_N,NO, sink_N,NPY, sink_N,Pyr) to change between positive and negative responses. These differences are allowed for, and implemented, due to a similar need as was done in our previous model for positive and negative BOLD [39]. The main limitation of this model regarding negative BOLD is that such dynamics also involve neuronal crosstalk and inhibition between different brain areas, which only is described in a highly simplified way in our models.

4.4.1 Fractional diameter change of arterioles and venules in mice.

The work of Drew et al. [46] presents experimental data of fractional diameter change in arterioles and venules upon different lengths of sensory stimulation in mice (n = 21), measured using 2-photon imaging. The sensory stimulation consisted of trains of air puffs (delivered at 8 Hz, 20 ms duration per puff) directed at the vibrissae, with three different durations: 20 ms, 10 s, and 30 s. The mice were awake during the experiment but were briefly anesthetized before the imaging session using isoflurane (2% in air). We have chosen to consider the period after each puff as a part of the stimulation cycle. This makes the shortest stimulation length of one air puff 125 ms long. The vascular responses were extracted from Fig 2C in the original publication [46].

4.4.2 Arteriolar diameter responses for optogenetic and sensory stimuli in mice.

Uhlirova et al. [47] report arteriolar diameter responses to both optogenetic and sensory stimulations measured using in vivo two-photon imaging, during both awake and under anesthesia conditions in mice. Briefly, four different stimulation protocols were used in their study: three conducted under the administration of the anesthetic α-chloralose, and the fourth during the awake condition. Below, summarized descriptions of each stimulation protocol are presented, and we refer the reader to the original publication [47] and our previous modeling paper [42] for a complete description.

4.4.3 Hemoglobin and BOLD responses for optogenetic and sensory stimuli in mice.

Desjardins et al. [48] present experimental data consisting of optical intrinsic imaging (OIS) of blood oxygenation for both wild-type mice (n = 16) and two mice-lines transfected with (ChR2). The mice were kept in an awake state during the data acquisition. In the transfected mice, ChR2 was expressed in pyramidal neurons (Emx1-Cre/Ai32) or all inhibitory interneurons (VGAT-ChR2(H134R)-EYFP). For the wild-type mice, a sensory stimulus consisting of air puffs (3–5 Hz) delivered to a whisker pad (duration 2 or 20 s) was used to induce hemodynamic responses. The optogenetic-transfected mice were exposed to an optogenetic stimulus consisting of a single 100 ms light pulse (473 nm) or a 20 s block of 100 ms light pulses delivered at 1 Hz. The blood oxygenation data were analyzed using the baseline assumption of 60 μM oxygenated hemoglobin and 40 μM deoxygenated hemoglobin. In addition, some Emx1-Cre/Ai32 mice underwent BOLD fMRI imaging for the 20 s optogenetic stimulus using gradient-echo echo-planar-imaging (GE-EPI) readout (TE/TR = 11–20 ms / 1 s).

4.4.4 Electrophysiological and BOLD responses in macaque monkeys.

The study by Shmuel et al. [10] report simultaneously measured electrophysiological responses by invasive recording, and non-invasively measured BOLD responses. These measurements were carried out in the visual cortex of anesthetized macaques at 4.7 T (n = 7). The stimulus paradigm consisted of blocks of 20 s visual stimulus featuring flickering radial checkers rotating at 60°/s. A grey background persisted for 5 s before and 25 s after the stimuli. If the visual stimuli overlapped with the receptive field of V1, it induced positive BOLD responses in close vicinity to the electrode. In contrast, if the visual stimuli were presented outside the receptive field of V1, it induced negative BOLD responses in the same area. The electrophysiological recordings were processed by averaging the fractional change of the power spectrum over the frequency range of 4–3000 Hz, using a temporal resolution of 1 s. BOLD data were acquired using a gradient-echo echo-planar-imaging (GE-EPI) readout (TE = 20 ms, TR = 1 s, in-plane spatial resolution 0.75 x 0.75 x 2 mm³). Positive and negative BOLD responses evoked by the two visual stimuli were sampled and averaged over the same voxels located around the electrode. These processed responses were extracted from Figs 1D (BOLD) and 2A (electrophysiological responses) in the original publication [10], respectively. The SEM in all extracted LFP time series with SEM measurements smaller than the mean SEM was set to the mean SEM, to avoid overfitting to a few extreme data points.

4.4.5 MR-based hemodynamic measurements in humans.

The study by Huber et al. [49] presents in vivo MR-based measurements of CBF, CBV, and BOLD changes in humans (n = 17) evoked using multiple visual stimuli. The study used three different visual stimulation paradigms (alternating 30 s rest vs. 30 s stimulation), consisting of two different full-field flickering checkerboards and one small circle flickering checkerboard paradigm. The full-field flickering checkerboard paradigms were used to measure arterial CBV, venous CBV, and total CBV data for both an excitatory and an inhibitory task. The small circle paradigm was used to evoke strong negative BOLD responses, and to concurrently measure both positive and negative BOLD responses, as well as the associated total CBV changes. These data were acquired using a Slice-Saturation Slab-Inversion Vascular Space Occupancy (SS-SI-VASO) sequence [115] with a multi-gradient echo EPI readout (TE₁/TE₂/TE₃/TI₁/TI₂/TR = 12/32/52/1000/2000/2500/3000 ms, nominal resolution = ~1.3 x ~1.3 x 1.5 mm³). In addition, CBF changes were acquired from ROIs of positive and negative BOLD responses using a pulsed arterial spin label (PASL) sequence in four subjects (TE₁/TE₂/TI₁/TI₂/TR = 8.2/19.4/700/1700/3000 ms, nominal resolution = 3 x 3 x 3 mm³). BOLD and total CBV responses were extracted from Fig 4A and 4B; arterial CBV, venous CBV, and total CBV responses were extracted from Fig 5B and 5C, and finally, CBF responses were extracted from Fig 6C in the original publication.