Endothelial-denuded segmented model of mouse penetrating arterioles.

The diameter of a vessel segment is the result of the balance between two opposing processes: passive distension and active constriction. The magnitude of passive distension depends on the vessel’s distensibility and is modulated by circumferential wall stress (WS) [49]. The magnitude of active constriction in each vessel segment depends on the density of surrounding mural cells and is regulated by specific biomechanical forces exerted by the blood on the vessel wall. The exact biomechanical forces and mechanisms through which vascular cells (mural cells and endothelial cells) sense these forces and subsequently modulate their ionic channels are not fully understood, but are thought to mitigate the destructive forces of increased WS and wall shear stress (WSS).

In our earlier study, within the hypothetical autoregulation range for the designed cerebrovascular model, where the artery boundary node pressure (ABNP) varies between 40 and 130 mmHg while the vein boundary node pressure (VBNP) is held constant at 10 mmHg (Fig 1), the constriction force of mural cells is linearly potentiated by the vessel WS [43,50]. According to Laplace’s law, this stress is directly proportional to the intravascular pressure (IP) and the internal diameter (D) of the vessel segment, and inversely proportional to the segment’s wall thickness (), expressed as . We did not account for the cytoskeletal remodeling, which likely occurs in pathological conditions like hypertension [49], and assumed that wall thickness remains constant under varying ABNP values. We quantified WS in all PA segments at ABNP = 130 mmHg (WS_max) and used these values to model a linear relationship between SMC ion channel modulation and WS within the autoregulation range.

Myogenic tone (MT) development is primarily governed by SMC depolarization, activation of voltage-operated calcium channels (VOCCs), and an increase in intracellular calcium concentration [49]. Cell depolarization occurs either through the potentiation of depolarizing currents or the reduction of hyperpolarizing currents. Multiple ion channels have been implicated in the depolarization of SMCs (for review, see references [49] and [23]). Informed by the literature, our proposed SMC model includes WS-activated ion channels, such as TRPC6 [51], TRPM4 [52], and Ca²⁺-activated Cl⁻ channels (TMEM16A) [53,54], which play key roles in potentiating depolarizing currents in response to increase in WS, and WS-deactivated ion channels, including Kir channels [55,56], which inhibit hyperpolarizing currents as WS increases.

Fig 2 presents a schematic of the proposed aSMC-aEC model. Briefly, we assumed a linear relationship between the concentration of certain mechanotransduction signaling molecules (inositol trisphosphate (IP3) and diacylglycerol (DAG) [51]), and the exerted WS that potentiates depolarizing channels. This assumption is supported by several studies showing that increasing IP stimulates the phospholipase C (PLC) activity, which subsequently increases the concentration of second messengers (IP3 and DAG) in SMCs [57,58], with subsequent upregulation of ion channels (TRPC6, TRPM4, and TMEM16A) to potentiate depolarizing currents. Fig 3A depicts the relationship between the concentration of these second messengers and WS in our model, and for Kir channels, we simply assumed that their open probability (Kir_OP) is directly downregulated by WS.

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Fig 2. A schematic of the proposed SMC-EC model.

Created in BioRender. For a detailed description we refer readers to the Methods section.The figure illustrates the channels and pumps regulating intracellular ion concentration and membrane potential in aSMCs, with color coding: potassium channels (blue), calcium channels and pumps (green), chloride channels (purple), and sodium channels (orange). The model includes two microdomains where calcium concentration can exceed the cytoplasm calcium concentration in aSMCs. The first is the mechanosensory microdomain, containing transient receptor potential (TRP) channels (e.g., TRPC6, TRPM4) and calcium-activated chloride channels (TMEM16A), which produce depolarizing currents in response to WS. In this microdomain, SMC mechanosensors are assumed to linearly convert WS into downstream signaling molecules, such as IP3 and DAG, modulating depolarizing currents. The second microdomain is the Glu-BK microdomain, where NMDA receptors, activated by neuron-released glutamate, trigger calcium efflux that activates nearby BK channels. The open probability of aSMC Kir channels is reduced by increasing WS, while larger WSS raises IP3 concentration within aECs, increasing both aEC Kir channel open probability and eNOS activity. The steady-state open probability of aSMC BK channels is increased by eNOS-dependent increases in NO/cGMP. In aSMC constriction mechanics, the rapid modulation of MLCK and MLCP activity is primarily driven by changes in cytoplasm SMC Ca²⁺ concentration and nNOS-dependent increases in NO/cGMP during increased neural activity, respectively.

https://doi.org/10.1371/journal.pcbi.1013113.g002

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Fig 3. Endothelial-denuded segmented model of mouse PAs: (A) Quasi-linear sigmoidal function model the relationship between WS and mechanotransduction signaling molecules (IP3, DAG) and the WS-Kir_OP signaling pathway.

(B) Pressure myography maneuver conducted on an endothelium-denuded vessel segment to calculate MT. (C) MT calculated for this segment, based on the curves plotted in panel (d-top), using . (D) The aSMC I-V curve under various IP values from simulated voltage-clamp analyses. Vertical lines depict the RMP where membrane current is near zero. The reciprocal of the slope of the tangent lines at the intersections of the vertical lines and curves indicates RMR. (E) Calculated RMR and RMP at different IP values. (F) Pressure myography test simulated under the assumption of 50%-inhibition of the main ion channels involved in MT regulation. (G) Simulated vasodynamics of the segmented model in response to a sudden increase in IP from 30 to 80 mmHg, analyzed under various values of the mechanotransduction time constant ().

https://doi.org/10.1371/journal.pcbi.1013113.g003

We evaluated the performance of our proposed endothelial-denuded segmented PA model through simulated pressure myography and patch-clamp tests before integrating it into the cerebrovascular model. Physiologically, external mechanical factors such as extravascular pressure, bath osmolarity [55], and the structural constraints of surrounding tissue impose additional stress on SMCs that scales with intravascular pressure [59]. In our in-silico framework, we omitted these factors to avoid introducing additional model complexity. To facilitate comparison of model outputs with ex-vivo data, we introduced an adjusted intravascular pressure (IP_ex-vivo), defined as a constant 25 mmHg offset of the in-silico IP, to approximate the net mechanical influence of these factors on the force sensed by SMCs.

Initially, we simulated a pressure myography test to evaluate MT development in our model through stepwise increases in IP within a PA segment under calcium-free (passive) and nominal external calcium concentration (active) conditions (Fig 3B). Model parameters for passive distension were set so that the maximum passive diameter at high IP values reached about 130% of the maximum active diameter of the vessel segment [46]. For the active response, as expected, pressure elevation prompted membrane potential depolarization, leading to increased intracellular calcium concentration and subsequent vessel constriction (Fig 3B).

Under low IP conditions (0–25 mmHg), corresponding to unpressurized vessels ex-vivo, WS-dependent depolarizing channels are less active, and a high open probability of Kir channels maintains the membrane potential near the equilibrium potential of potassium. Under these conditions, intracellular [Ca²⁺] is low, aSMCs are less capable of constricting the vessel, and the vessel passively distends until depolarizing channels become more active with increased WS, while simultaneously the open probability of Kir channels decreases. This leads to cell depolarization and a subsequent increase in [Ca²⁺] to a level that induces active constriction. The maximum active diameter of this vessel segment is achieved within the IP range of 30–35 mmHg. This IP range is in agreement with our cerebrovascular model design, where a PA segment at the depth of 250 m reaches its maximum dilation at the IP value between 30–35 mmHg when ABNP is set to 40 mmHg, which is the lower limit of the autoregulation range. Consistent with the in-silico and in-vivo analysis of autoregulation [43,60], vessels must initially constrict steeply after reaching their maximum dilation to be at the optimal point of autoregulation, and as pressure increases, even a small constriction generates significant resistance to blood flow, optimizing the IP distribution in the network at the upper limit of the autoregulation range (Fig 3D). The simulated values of membrane potential and [Ca²⁺] at various IP_ex-vivo levels (Fig 3B) are aligned with ex-vivo measurements for PAs [61–63], and the calculated MT of the modeled vessel segment (Fig 3C) is consistent with the experimental data for PAs in both the maximum tone and profile [46,64], indicating that the active constriction and passive distension models are effectively simulating the myogenic response of a PA vessel segment.

Next, we examined the resting membrane potential (RMP) and resting membrane resistance (RMR) which are critical electrophysiological parameters of mural cells. We simulated several voltage clamp tests under various IP levels and we measured the steady-state membrane current while changing the SMC membrane potential from -100–40 mV (Fig 3D). The generated I–V curves at IP = 20 and 30 mmHg are in good agreement with electrophysiological profiles of freshly isolated SMCs from mouse brain arterioles [55,56,65]. At these low IP values, the high open probability of Kir channels and reduced dominance of WS-depolarizing channels result in an I–V curve for the SMC model that shows the typical characteristics of Kir channels [66,67] including the negative slope and inward rectification. In isolated SMCs, reducing extracellular osmolarity and the consequent cell swelling suppresses Kir channel activity [55], an effect comparable to the shift observed in our simulations when IP increased from 20 to 30 mmHg under 4 mM [K⁺]_ex. As IP increases across the physiological range, the I–V curve of the cell has a shallow slope (reflecting low ion permeability) and maintains a high RMR, which is a characteristic of excitable cells. RMR reaches its maximum value at the lower limit of the physiological range, and as IP increases, RMR decreases and RMP rises (Fig 3D and 3E), indicating that basal IP can modulate both RMP and RMR in SMCs, thereby affecting the sensitivity of the SMC membrane potential to external stimuli such as changes in hemodynamics or membrane potential of aECs.

In vascular SMCs, specific potassium channels operate under physiological conditions, mediating potassium efflux to counteract WS-dependent depolarizing currents, providing negative feedback for MT regulation [67], and maintaining a large RMR in aSMCs to enable rapid responsiveness to external stimuli. Arteriolar SMCs predominantly express four types of potassium channels [67]: ATP-sensitive (K_ATP), large conductance Ca²⁺-activated (BK), inward rectifier (Kir), and voltage-gated (KV). It was suggested that K_ATP channels play a minor role in cortical arterioles [25]. Furthermore, pressure-induced depolarization in arteriole SMCs triggers an amplification of Ca²⁺ sparks due to the facilitated Ca²⁺-induced Ca²⁺ release (CICR) process in ryanodine receptors (RyRs), driven by increased Ca²⁺ efflux via adjacent T-type VOCC [68], forming a negative feedback loop with coupled BK channels that mitigates myogenic vasoconstriction by promoting SMC hyperpolarization. However, unlike cortical surface arteries, BK channel-mediated K⁺ efflux does not inhibit MT significantly in PAs, and under physiological conditions, only Kir and KV channels are highly activated in aSMCs[67]. Consequently, in our proposed SMC model, TMEM16A and TRPM4 channels are key depolarizing channels, while Kir, KV, and BK channels serve as key hyperpolarizing channels, with a less contribution from the latter.

To evaluate the contribution of these channels to SMC constriction, we simulated a pressure myography test under the assumption of 50% channel inhibition (Fig 3F). Kir channel inhibition significantly disrupted MT regulation within the physiological IP range. These simulation results align well with in vitro findings, where vessels exposed to a Kir channel blocker under physiological IP were significantly constricted [55], suggesting that SMC Kir channels play a crucial role in cerebral autoregulation. KV and BK channel inhibition caused MT dysregulation at mid to high IP values due to their voltage-dependent activation mechanism, with BK channels playing a much lesser role in the negative feedback regulation of MT in endothelium-denuded PAs. Inhibition of both key depolarizing channels, TRPM4 and TMEM16A, led to reduced vessel constriction.

Although the steady-state SMC mechanotransduction signaling pathways (such as DAG, IP3, and Kir_OP) are regulated by WS, the cerebral vasculature operates in a dynamic environment where WS continuously fluctuates across vessel segments due to events like FH or changes in mean arteriolar pressure. Because mechanotransduction in SMCs does not occur instantaneously, there is a time delay in the adjustment of WS-dependent ion channel activity in response to rapid changes in WS. We modeled this delay using an exponential function with a mechanotransduction time constant (). This time constant plays a crucial role in shaping the temporal dynamics of the myogenic response, influencing the kinetics of FH and vasomotion, as explored in the following sections.

Various studies affirm that Neuropeptide Y (NPY) can enhance the vasoconstrictor actions of other molecules [69–72]. This facilitating effect is attributed to the synergistic interactions between NPY receptors (NPYRs) and other G-protein-coupled receptors (GPCRs) involved in vasoconstriction [73]. This synergy could potentiate transduction pathways following the mechanoactivation of Gq-coupled receptors in SMCs and lower the . We estimate that under physiological conditions, the value of could range between 1–20 seconds and will provide evidence to support this. However, in-vitro conditions may result in much larger values and a myogenic response that is several orders of magnitude slower than under physiological conditions [46,49,64,74]. Facilitation of the myogenic response by NPY is particularly relevant in cerebral arterioles, as NPY receptors are highly expressed in aSMCs, and there are direct NPY interneuron–aSMC junctions [75]. Fig 3G shows vessel diameter changes in response to an abrupt increase in IP from 30 to 80 mmHg in our model under three different , where the vessel initially passively distends, and the time to reach steady state can vary significantly depending on the value of .

Fast-acting downregulation of SMC constriction in endothelial-denuded vessel segment.

To simulate arteriolar vasodynamics, we need to identify and model the fast-acting downregulators of SMC constriction that induce rapid vasodilation during periods of increased neural activity. The regulation of SMC constriction involves two competing processes: myosin light chain (MLC) phosphorylation by myosin light-chain kinase (MLCK), counteracted by dephosphorylation through myosin light-chain phosphatase (MLCP). A decrease in intracellular calcium concentration inhibits calcium-calmodulin-dependent MLCK activation, thereby promoting muscle relaxation [76]. Additionally, MLCP is activated by protein kinase G (PKG) and MLCK is inhibited by protein kinase A (PKA), with PKG and PKA themselves being activated by secondary messengers cyclic guanosine monophosphate (cGMP) and cyclic adenosine monophosphate (cAMP), respectively. These secondary messengers are primarily produced by NO-dependent sGC [77] and hormone/neurotransmitter-stimulated adenylyl cyclase (AC) [78,79], respectively.

SMC intracellular calcium concentration is predominantly regulated by VOCCs, and reducing the SMC membrane potential or directly inhibiting VOCCs are the primary mechanisms of lowering SMC [Ca²⁺] and MT inhibition. Based on the reported in-vivo SMC [Ca²⁺] [80], we ruled out the possibility that fast VOCC inhibition via NGVC affects SMC constriction state during FH. We only focused on modeling the mediators that can rapidly modulate SMC membrane potential. Modulation of the SMC membrane potential is a multi-faceted process which is affected by hemodynamic changes, synaptic-like transmission between neural axons and SMCs [75], myoendothelial gap junctions [10], NO pathways [81], acidosis [25], and direct extracellular interactions between ion channels, such as astrocyte endfeet BK channels and SMC Kir channels [82]. We ruled out the possibility of the brain tissue acidification during FH, but other pathways were modeled in this study.

Kir channels are key candidates for modulating SMC membrane potential via NGVC, as they play a crucial role in SMC membrane potential autoregulation (Fig 3F).The vasodilatory response to the potassium released from astrocyte endfeet BK channels is mediated by SMC Kir channels [82–84]. Elevation of extracellular potassium increases Kir channel conductance which hyperpolarizes the cell. Fig 4A shows the vasodilatory response of the PA segmented model to the elevation of extracellular potassium ([K⁺]_ex) at discrete levels for three IP and two SMC values. At IP = 50 mmHg, extracellular potassium elevation induces the largest vasodilation compared to higher IPs. In this physiological IP range, the cell’s RMR is large (Fig 3D and 3E), and slight variations in Kir channel conductance can change the SMC membrane potential significantly. At higher IP values, RMR and Kir channel open probability decrease, making Kir channels less capable of inducing hyperpolarization in the SMC.

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Fig 4. Analysis of vasodynamics in an endothelium-denuded model under the influence of vasodilatory mediators.

Vasodilatory responses were measured for all mediators under two and three IP values. (A) Changes in vasodynamics, SMC membrane potential, and [Ca²⁺] when increasing the SMC extracellular potassium concentration from a baseline of 4 mM up to 60 mM in discrete steps. The transient hyperpolarization overshoot observed after washout of extracellular K⁺ (60 to 4 mM) is primarily Kir channel-dependent. The abrupt negative shift in the K⁺ reversal potential (E_K) transiently relieves rectification block, enhancing Kir channel conductance and producing a brief surge of outward current and hyperpolarization. In our simulations, elimination of Kir channel conductance abolished this overshoot, confirming its Kir channel origin. The ensuing dilation increases wall stress, after which WS-activated depolarizing currents and suppression of Kir channel activity restore V_m toward baseline. (B) Changes in vasodynamics, SMC membrane potential, and [Ca²⁺] when increasing the glutamate concentration from a baseline of 0 to 1.2 AU (arbitrary unit) in four discrete steps. (C) Changes in vasodynamics and [Ca²⁺] when increasing SMC NO/cGMP concentrations from a baseline of 0 to 10 AU in four discrete steps. Numbers considered for the concentration of glutamate and NO were hypothetical since the exact physiological values are unknown. Model parameters were adjusted accordingly to produce the maximum dilatory response at the highest hypothetical concentrations.

https://doi.org/10.1371/journal.pcbi.1013113.g004

The SMC mechanotransduction time constant () can also affect the magnitude of vasodilation. When is small, SMCs respond rapidly to WS changes. The initial vasodilation increases WS according to Laplace’s law, leading to cell depolarization that counteracts Kir channel-induced hyperpolarization. This myogenic response is more pronounced at lower IP values because, under these conditions, the cell’s RMR is large, and slight changes in WS-dependent currents have a large impact on the SMC membrane potential.

At IP = 50 mmHg, the largest vasodilation was observed at [K⁺]_ex = 8 mM, but the peak shifts to higher potassium concentrations at IP = 60 mmHg because altering extracellular potassium not only impacts Kir channel conductance but also changes the potassium Nernst potential. Increasing IP also raises intracellular potassium concentration, further affecting the potassium Nernst potential [85]. Elevating extracellular potassium up to 16 mM induces vasodilation at IP = 50 and 60 mmHg, but it results in slight constriction at IP = 75 mmHg. Therefore, the direction and extent of diameter change depend on both extracellular potassium concentration and the value of IP [82]. At a high [K⁺]_ex, the potassium Nernst potential is much larger than its physiological level, leading to a significant SMC depolarization (∼-20 mV at [K⁺]_ex 60 mM) and causing maximum constriction in the vessel, which typically does not occur at any IP value under physiological extracellular potassium concentration.

BK channels are modulated by the membrane potential and Ca²⁺ sparks near the cell membrane, which makes them key candidates for fast hyperpolarization and relaxation of aSMCs [86]. Studies have shown that BK channel activation can induce pronounced vasodilation in PAs when exogenous activators are present. For example, brain tissue acidification dilated PAs rapidly due to a dramatic increase in Ca²⁺ spark activity and BK channel function, driven by the enhanced RyR open probability [25]. BK channels in PA SMCs are also colocalized with GluN1, a subunit of NMDA receptors, and glutamatergic neurons have been shown to dilate arterioles through synaptic-like transmission at neural–aSMC junctions (NsMJs). Activation of NMDA receptors in Glu-NsMJs leads to a Ca²⁺ ion influx, which binds to and activates BK channels, resulting in increased potassium ion efflux, membrane hyperpolarization, and subsequent relaxation of aSMCs. In-vivo disruption of Glu-NsMJs inhibits NGVC substantially [75]. To incorporate this NGVC mechanism, we included a Glu-BK microdomain in our model, where activation of glutamate receptors locally increases [Ca²⁺] in the microdomain and activates BK channels (Fig 2).

Fig 4B shows the vasodilatory response of the PA segmented model to elevated glutamate at four discrete hypothetical concentrations, across three IP and two SMC values. The highest hypothetical glutamate concentration induces a 3.5% steady-state dilation at IP = 50 mmHg (after balancing neurogenic and myogenic responses when = 10 s), while this level is about 5% at IP = 60 mmHg. This occurs because BK channels are mediated by both Ca²⁺ and membrane potential. At IP = 60 mmHg, the higher SMC membrane potential enhances the vasodilatory effect of glutamate. The myogenic response following initial vasodilation is less pronounced at IP = 75 mmHg due to small RMR, where changes in WS-dependent currents are less capable of altering the membrane potential and the [Ca²⁺]. The post-stimulus undershoot in the diameter change (observed in the blue curves of Fig 4A and 4B) has the same origin as the initial overshoot and is associated with the potentiation/inhibition of myogenic response following vasodilation/constriction. Once glutamate or [K⁺]_ex return to the baseline, the membrane potential is expected to return to its baseline. However, since MT is larger than its basal level in the dilation phase, following the vasoconstriction due to the removal of the stimulus, the delay in the adjustment of MT causes the vessel to further constrict, which produces a post-stimulus undershoot.

Nitric oxide (NO) is a potent vasodilator produced rapidly by both endothelial (eNOS) and neural nitric oxide synthase (nNOS) and plays a crucial role in NGVC [87]. In-vivo evidence shows that neuronally produced NO acts directly on arterioles [9,88,89]. Blockade of neural NO synthase has the most significant inhibitory effect on the feedforward mechanisms [87], which can be compensated by feedback mechanisms [90]. NO relaxes vascular SMCs and dilates blood vessels by increasing the intracellular cGMP concentration, which stimulates cGMP-dependent PKG activity. However, the rapid vasodilatory signalling pathway mediated by PKG is not fully understood. This complexity arises from the diverse ways that PKG can influence the SMC constriction state. PKG modulates various SMC ion channels, including the activation of BK channels [91], inhibition of TRPM4 channels [81], and inhibition of VOCCs [92], collectively leading to decreased SMC [Ca²⁺]. PKG also directly activates MLCP [93,94]; however, it has remained unclear whether PKG-dependent MLCP activation or PKG-SMC ion channel modulation plays a more significant role during the rapid dynamics of FH. Here, as one possible mechanistic scenario, we assume that the fast-acting effect of NO on aSMCs is primarily governed by the increased MLCP activity rather than by SMC ion channel modulation. This assumption is supported by ex-vivo studies showing that the early phase of NO-induced vasodilation involves the suppression of CPI-17 phosphorylation which is an MLCP inhibitory protein [95–97], and will be further evaluated against in-vivo experimental evidence in the following sections. Therefore, in our vessel wall mechanical model adopted from [98–100], variations in NO/cGMP/PKG activity directly affect the kinetics of actin-myosin interactions, ultimately leading to SMC relaxation.

Fig 4C shows the vasodilatory response of the PA segmented model to elevated NO/cGMP at four discrete hypothetical levels, across three IP and two SMC values. Despite the immediate rise in NO/cGMP concentration, we assumed a degradation rate for these molecules that delays their decline under physiological conditions. This degradation was modeled by a decaying exponential function with a 5-sec time constant, regardless of any dose dependency that might exist in reality. The relative diameter change of NO-induced dilation is larger at a higher basal IP. This occurs because the increase in MLCP activity correlates with a decrease in MLC phosphorylation sensitivity to [Ca²⁺] [95], resulting in [Ca²⁺] having a less pronounced effect on SMC contraction during periods of elevated NO/cGMP. Therefore, during NO/cGMP dose-dependent relaxation of SMCs, vessels with higher basal IP experience larger passive distention, leading to a more pronounced diameter change.

If the myogenic response is defined as the intrinsic ability of SMCs to modulate their membrane potential and [Ca²⁺] by WS, then increased MLCP activity due to the NO/cGMP/PKG pathway can be seen as a mechanism of dampening this response by reducing the sensitivity of MLC phosphorylation to MLCK activity. On the other hand, hyperpolarizing SMCs with mediators can be viewed as the inhibition of myogenic response. During NO-induced dilation, the increase in diameter and WS leads to cell depolarization, causing a rise in [Ca²⁺] and potentiation of the myogenic response (Fig 4C). However, since the NO/cGMP/PKG pathway dampens the myogenic response, the effect of the potentiated myogenic response is small and negligibly counteracts the vasodilatory effect of NO. Comparing the NO vasodilatory response under two analyzed mechanotransduction time constants at IP = 50 mmHg shows that a smaller leads to a sooner and more pronounced increase in [Ca²⁺], but this potentiation of the myogenic response during the dilatory phase only negligibly reduces the peak amplitude of NO-induced dilation compared to the case of a larger .

Rapid myogenic response potentiation ( =10 s) can help the vessel return to its basal level sooner after the stimulus; however, the slow degradation rate of NO/cGMP and the dampened myogenic response in the presence of these molecules does not lead to a pronounced post-stimulus undershoot. As NO/cGMP degrades and returns to its basal state, the myogenic response may cause a post-stimulus undershoot if [Ca²⁺] is above pre-stimulus levels. In the =100 s case, where the myogenic response occurs more slowly, once NO/cGMP returns to its basal state and Ca²⁺ sensitivity of MLC phosphorylation increases, the post-stimulus undershoot period is longer. However, due to a smaller increase in myogenic response-induced [Ca²⁺], the degree of post-stimulus undershoot is only marginally larger than in the fast myogenic response scenario, where a significant rise in [Ca²⁺] occurs alongside the reduced Ca²⁺ sensitivity of MLC phosphorylation. Similar to the vasodilatory response of other mediators, like [K⁺]_ex and glutamate, the degree of post-stimulus undershoot decreases with increasing IP. This occurs because, under higher IP, cell RMR is small, and WS-dependent depolarizing currents have a reduced impact on the membrane potential, resulting in a smaller increase in [Ca²⁺] (Fig 4C, bottom panel). Consequently, smaller post-stimulus undershoots are observed at IP = 75 mmHg and IP = 60 mmHg compared to IP = 50 mmHg. In the following sections, as we analyze the neurogenic impulse response of the CBF regulatory system, we will present strong experimental evidence to support our analysis of the dynamics of NO-induced vasodilation.

Integrating segmented model of mouse penetrating arterioles into cerebral vasculature model.

Thus far, we have developed a segmented model of an endothelium-denuded vessel and simulated the primary vasodilatory signaling pathways. Endothelial cells are also well recognized for their crucial role in FH dynamics. In this section, we propose a simplified model of aECs to be added to the PA segment model, which we then integrate into the PA segments of the cerebral vasculature to capture their physiological contribution to FH dynamics. ECs are known to serve as sensors of neural activity within the CBF regulatory system, relaying this information retrogradely up the vascular tree via electrical coupling [10]. Active neurons discharge potassium ions, which enhance the conductance of Kir channels in ECs, leading to fast hyperpolarization of arteriolar ECs in both proximal and distal regions of the activated area, which is transmitted to SMCs through MGJs and results in their dilation [101,102].

ECs sense changes in potassium concentration and translate that to changes in their membrane potential. A large RMR, in the range of several Gigaohms [103,104], makes ECs sensitive to such changes, allowing them to respond quickly and precisely. [66] proposed an abstract mathematical model for ECs, which includes a model for the Kir current (I_Kir) while combining all other transmembrane currents into one nonspecific linear background current, I_bg(v) = G_bg(V_EC − E_bg), where E_bg = −30 mV was assumed to be constant. In our proposed model, we replaced I_bg with . Here, R_MGJ represents the resistance of gap junctions between an EC and SMC which was assumed to be 1 G [66,105]. This replaced depolarizing current opposes I_Kir-induced hyperpolarization to maintain a large RMR in ECs, and couple the EC membrane potential with the membrane potential of mural cells for autoregulation.

Fig 5A shows that, unlike the endothelium-denuded model, increasing the value of IP does not fully depolarize the SMC due to large Kir currents in coupled aEC. Consequently, SMC [Ca²⁺] does not exceed 150 nM, allowing passive distention to counteract the low contractile force of SMCs, which results in dilation at high IP values. In this pressure myography test, IP was increased with no blood flow in the vessel, resulting in zero WSS. The Kir current in aECs is also dependent on WSS, with WSS activating this channel [55]. At low mean arteriolar pressure, where blood flow is minimum and vessel diameter is maximum, WSS is minimum, nullifying the hyperpolarizing effect of EC Kir channels. As mean arteriolar pressure increases, WS-dependent currents in SMCs can constrict the vessel. As SMCs depolarize, the EC membrane potential also depolarizes due to gap junction coupling and small EC Kir channel activity under small WSS. With increasing SMC constriction force and vessel contraction, WSS rises which activates EC Kir channels; however, the EC is now less polarized due to coupled depolarized SMC. Therefore, Kir channel conductance is reduced at physiological extracellular potassium concentrations. Thus, EC Kir channel activity has a negligible effect on SMC membrane potential, allowing SMCs to autoregulate while maintaining EC Kir channels functional [55]. Increasing extracellular potassium of ECs by neuronally discharged potassium during FH causes a rightward and upward shift of the I_Kir curve [66], inducing EC hyperpolarization, which in turn hyperpolarizes SMCs.

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Fig 5. Integration of Intact segmented model of mouse PAs into the cerebral vasculature model: (A) pressure myography test performed on an intact vessel segment with no blood flow and exerted WSS.

Note: these simulation results are not physiologically valid, as no WSS dependency for EC-Kir channel activation was considered in the model. In the absence of flow, EC-Kir channels are deactivated (panels B, C), and MT is likely correctly developed in vessels without flow [106]. (B) A quasi-linear sigmoidal function models the relationship between WSS and Kir_EC open probability. (C) Analysis of the autoregulatory performance of the intact PA model. Purple curves are associated with a segment of the final PA model, where aEC respond to increased WSS caused by the rising ABNP. This response results in an increase in the open probability of EC Kir channels and the potentiation of eNOS-based signaling. NO produced by eNOS diffuses into the SMC, potentiating BK channels to enhance MT negative feedback and mitigate the elevated WSS. (D) Analysis of dysregulation with a 50% inhibition of primary ion channels in the PA model. (E) Vasodynamic analysis of the PA model under two different SMC and three ABNP values, in response to an increase in EC extracellular potassium concentration from a baseline of 3 mM up to 7 mM in four discrete steps.

https://doi.org/10.1371/journal.pcbi.1013113.g005

To run simulations with this model, blood needs to flow into the vessel segment to exert WSS. This requires a circulatory network where blood flows through the network based on the IP gradient between its entry and outlet nodes. Taking a stepwise approach, we first integrated the endothelium-denuded segmented model of PA into the designed cerebral vascular model. Our earlier study demonstrated that, morphological differences exist between PA segments [43]. Superficial segments generally have larger diameters to accommodate higher blood flow and have higher expression of contractile elements to counteract passive distension for cerebral autoregulation. The relative expression of contractile elements in each segment was modeled by introducing a relative contractility index. Model parameters associated with the baseline diameter, passive distension, and expression of contractile elements (relative contractility) were tuned in the proposed endothelium-denuded segmented model, extending the segmented model to all 28 segments of each PA in the designed cerebral vasculature.

Once the endothelium-denuded PA model established, we set the VBNP to 10 mmHg and increased the ABNP from 40 to 130 mmHg in 10 mmHg increments, where PA segment diameters were automatically adjusted by the integrated endothelium-denuded model, while diameters of other segments, including surface arteries, sphincters, TZ, and capillaries, were manually optimized based on the myogenic response for each ABNP [43]. At ABNP = 130 mmHg, PA segments experience maximum WSS, as the blood flow peaks and vessel diameters are minimum. We quantified WSS values in all PA segments at ABNP = 130 mmHg and saved them as the maximum WSS (WSS_max) of each segment. These WSS_max values were then used to model the modulation of EC Kir channel open probability in response to an imposed WSS in intact PA model, using a quasi-linear sigmoidal function(Fig 5B). In this intact PA model, the homocellular gap junctions between adjacent aSMCs and aECs have a resistance of 100 mega [103,107,108].

To evaluate the developed intact PA model, we analyzed its performance in autoregulation. In our previous work, we showed that with optimal MT potentiation in all segments of our cerebral vasculature model, when ABNP increases within the autoregulation range, blood flow in non-bifurcating capillaries (Cap1 in Fig 1) remains around 0.5 nl/min. We also demonstrated that MT in TZ vessels and sphincters primarily autoregulates superficial blood flow, while MT in PAs plays a more pronounced role in deeper layers [43]. Building on this, we calculated the steady-state values of deep capillary (L4_Cap1) blood flow and other model variables for a PA segment at a depth of 250 to assess the intact PA model’s autoregulation performance.

Adding the WSS-dependency of Kir channel activity to the endothelium-intact PA model (yellow curves) provides a similar autoregulatory performance to the endothelium-denuded model (blue curves) and also corrects the dysregulation observed in the absence of WSS-dependency in the EC Kir current (orange curves)(Fig 5C). Within the physiological ABNP range of 40–80 mmHg (corresponding to 30–65 mmHg IP within PA), in scenarios where MT is correctly adjusted in vessels (blue and yellow curves), blood flow in L4 non-bifurcating capillaries remains relatively constant. When ABNP increases above this range, even though the myogenic induced constriction in PA segments could not sustain plateaued autoregulation, L4_Cap1 blood flow moderately increases, which is in agreement with the computational and experimental observations [43,60]. SMC membrane potential and [Ca²⁺] are also in the range reported in in-vitro studies [61]. The maximum MT level at ABNP = 130 mmHg is about 42% () for the analyzed PA segment, which matches the reported maximum MT in PAs[46].

WSS-dependent activation of eNOS is also known to inhibit MT in vessels [109]. In mouse PAs, this NO-induced dilation has been linked to increased SMC BK channel activity [46]. We adjusted the model parameters associated with the WSS/NO/BK pathway to reduce MT_max by approximately 14% compared to the case without the WSS-eNOS pathway (decreasing from 42% to 36%). As expected, when the WSS-NO pathway is included (purple curves) and the vessel is less constricted at the upper end of the autoregulation range, L4_Cap1 blood flow increases at a higher rate. Additionally, while WS increases slightly compared to the case of no WSS-NO-BK pathway ( changes from 1.75 to 1.9), WSS increases much less in the presence of the WSS-NO pathway ( changes from 3.5 to 2.65). Within the autoregulation range, as mean arteriolar pressure increases, MT potentiation mitigates the WS increase associated with the rising ABNP, while MT inhibition via WSS mitigates the increased WSS. This regulation implies that vessels potentially adapt through both WS-dependent mitigation of WS via SMCs, and WSS-dependent mitigation of WSS via ECs, preventing excessive mechanical stress that could compromise vascular integrity and optimal function.

Next, we aimed to qualitatively demonstrate how disruption of different ion channels in our PA model can lead to cerebral dysregulation (Fig 5D). The simulation tested a 50% inhibition of various ion channels in the PA model. Focusing on the PA, and optimally adjusting other segments’ MT for autoregulation, the most dysregulated scenario occurs when Kir channels are disrupted in vascular cells, as SMC Kir current inhibition leads to more constricted vessels within the physiological pressure range (Fig 3F). Oxidative stress and inflammation in pathological conditions such as Alzheimer’s disease (AD) can impair Kir channel function [110–112], which might explain the hypoperfusion previously reported in AD mouse models [113,114]. As expected, the inhibition of TMEM16A channels and KV channels results in the hyperperfusion and hypoperfusion, pointing to their crucial roles in MT adjustment at different WS levels in aSMCs. Inhibition of BK channels also led to less WSS-dependent inhibition of MT in the upper range, and consequently, a smaller increase in blood flow.

After evaluating the intact PA model, we aimed to demonstrate that its EC Kir channels are functional in converting neural activity into vessel dilation. We addressed the question whether active neurons discharge potassium around ECs in specific vascular zones to hyperpolarize the endothelial network through gap junctions, or if all parenchymal ECs experience increased extracellular potassium during neural activity. In both scenarios, it seems logical that the EC basal membrane potential is mediated by MT in mural cells to maintain the autoregulation process. An increase in aEC [K⁺]_ex, leads to endothelial network hyperpolarization in proportion to the amount of potassium released, which is subsequently transferred to SMCs, inducing vasodilation (Fig 5E). This proportionality is less evident in the case of ABNP = 50 mmHg due to the instability of both SMC and EC membrane potentials during the test, which continuously decline and impair the correlation between [K⁺]_ex and vasodilation. Under very low mean arterial brain pressure (ABNP = 50 mmHg), aSMCs operate at low IP (30–40 mmHg depending on cortical depth), small WS-depolarizing currents, and large RMR (Fig 3D and 3E), which makes them sensitive to minor changes in the dominant SMC Kir currents. The negative slope region of the Kir channel I-V curve explains this continuous downward trend (Fig 3D - green curve). A slight initial hyperpolarization can trigger a positive feedback loop, increasing Kir channel conductance and driving the membrane potential further toward negative values until the Kir current saturates at the peak of the negative slope region. This persistent hyperpolarization and vasodilation might serve as a protective mechanism under low mean arterial pressure, maximizes vessel dilation, and increases arterial pressure to counteract the low perfusion.

Modeling arteriolar vasodynamics in response to neuronal impulse activity.

To refine the proposed computational model to better reflect physiology, we incorporated in-vivo experimental data.The essential experimental data for our analysis in this section were sourced from in-vivo vasodynamic recordings derived from 0.5 sec optogenetic (OG) stimulation, captured using two-photon microscopy from the somatosensory cortex of mice [124]. Previous studies have shown that OG-evoked vasodilations predominantly initiate below cortical layer IV [125–127]. Thus, analyzing the upward propagation of OG-evoked vasodynamics from deeper to more superficial cortical layers provides a valuable lens to examine the impact of neuronal impulse input on vasodynamics across different cortical depths. This approach allows us to identify depth-dependent deterministic features in neurogenic-myogenic interactions, which can be validated against the predictions of our proposed model to evaluate its accuracy.

Brief OG stimulation triggers delayed and attenuated arteriolar dilations in superficial cortical layers compared to deeper ones, with a more pronounced post-stimulus undershoot in the deeper layers ([125,128]) (Fig 9A and 9B). We quantified the propagation speed of arteriolar dilation by measuring the onset and peak times of dilation at different cortical depths of each PA (as detailed in Methods), and identified vasodilation propagation speeds of 300–400 m/sec (Fig 9C), significantly slower than the retrograde electrical propagation speed of 2 mm/sec reported by [10]. nNOS-expressing interneurons evoke arterial dilation in the somatosensory cortex of mice [129], and the delayed vasodilation in superficial arterioles might result from slower neurovascular communication kinetics in superficial layers rather than from the delayed propagation of neural activity from the deeper layers to the superficial area [125,128]. Taking these observations together, we assumed, as a hypothesis, that instantaneous neuronal activity leads to the release of equal amounts of neuronally produced NO across all cortical depths, with delayed modulation of the myogenic response and vasodilation in superficial layers due to slower neurovascular communication kinetics. We then evaluated whether this scenario could recapitulate the depth-dependent deterministic features observed in PA vasodynamics, including the attenuated vasodilation during upward propagation and the more pronounced post-stimulus undershoot in deeper layers.

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Fig 9. Neurogenic impulse response of the CBF regulatory system: (A-C) OG-evoked vascular reactivity in PAs of awake mouse brain across different cortical depths, sourced from [124,125].

(A-left) Example of a two-photon image of the mouse somatosensory cortex vasculature. (A-right) Depth-resolved illustration of the ROI shown on the left. (B) Averaged OG-evoked arteriolar vasodynamics measured in three different cortical depth regions (colored cross-sections in (A-right)), highlighting three depth-dependent features of the in-vivo OG-evoked vasodynamics: 1. Delayed arteriolar vasodilation in superficial layers, 2. Attenuated arteriolar dilation in superficial layers, and 3. More pronounced post-stimulus constrictions in deeper layers. (C) Violin plot of the calculated speed of vasodilation propagation along each PA in the dataset, determined using onset-to-onset and peak-to-peak times from concurrently recorded vasodynamics across different depths, assuming PAs penetrate perpendicularly into the tissue. (D) Left: Modeled PA consisting of 28 coupled segments, each including EC, SMC, and passive distension models. Right: Spatiotemporal changes in model parameters at three cross-sections of the PA, located at depths of 405, 225, and 45 m, in response to instantaneous neuronal activity. The dots in the zoomed-in area represent timestamps used for calculating Ca²⁺ wave propagation and electrical signaling. (E) Spatiotemporal changes in model parameters in response to instantaneous neuronal activity, assuming there exists no electrical coupling between PA segments (removal of gap junctions), resulting in inaccurate predictions of vasodynamics compared to in-vivo recordings.

https://doi.org/10.1371/journal.pcbi.1013113.g009

We initially set the ABNP of the cerebrovascular model to 60 mmHg, and we allowed the oscillations to fully damp before applying neurogenic input to the cellular-level PA, allowing us to analyze the average neurogenic impulse response of the PA, without interference from oscillations. We then assumed that instantaneous neuronal activity induced a sudden rise in NO concentration in PA segments located below 600 m depth, whereas in PA segments above 600 m, this rise propagated at a speed of 400 m/sec. This delayed adjustment was designed to capture the depth-dependent onset of vasodilation observed in Fig 9B.

Fig 9D shows the changes in various parameters of the analyzed PA at three different depths. As shown in the bottom panel, the NO/cGMP concentrations were modeled to increase with a delay corresponding to cortical depth and to decay exponentially with a 5 sec time constant. The NO/cGMP/PKG pathway primarily induces rapid vasodilation by activating PKG, which in turn activates MLCP and desensitizes MLC to Ca²⁺ which leads to partial dampening of the myogenic response (Section 1). Therefore, when the PA segment located at 405 m (black curves) experiences this NO-induced vasodilation, the damped myogenic response is followed by an increase in vessel segment diameter and elevated IP during the dilation phase, which increases the WS and potentiates the myogenic response. This, in turn, causes a subsequent increase in SMC membrane potential and SMC [Ca²⁺]. Due to the electrical coupling between vascular cells, the increased SMC membrane potential in deeper PA segments, which experience the rise in NO concentration earlier, is transmitted to the upper vascular cells via gap junctions, causing rapid depolarization in the upper layers’ SMCs even before the NO concentration rises there. Based on the timing of the mid and peak [Ca²⁺] increases in analyzed segments, the electrical propagation speed in our model was approximately 1200 m/s. In-vivo findings showed that during instantaneous sensory stimulation, the propagation of vasodilation in superficial vessels lagged behind the rise in SMC [Ca²⁺], indicating that vasodilation occurred at a speed of around 400 m/sec, consistent with the values calculated in Fig 9C, where the SMC Ca²⁺ wave was propagating at approximately 1200 m/sec [130].

During the time interval when SMC [Ca²⁺] is increasing almost simultaneously across all segments but before reaching the peak, the NO/cGMP concentration is higher in the deeper layers due to the earlier rise. As a result, the dilation of deeper PA segments during this phase is more pronounced than in the superficial segments. In the superficial segments, the equal concentration of NO/cGMP is accompanied by a more potentiated myogenic response evoked by earlier dilations in deeper segments, which is quickly transmitted electrically upstream (Fig 9D, top panel). This dynamic explains the attenuation observed in the upward propagation of vasodilation observed experimentally (Fig 9B). Furthermore, the potentiated myogenic response results in rapid vasoconstriction following the initial NO-induced partial dampening of the myogenic response, rather than the vasodynamic response that follows the slow degradation of NO/cGMP. Therefore, during the interval when SMC [Ca²⁺] is near its peak in all segments, the NO/cGMP concentration in deeper cortical layers is lower due to its earlier degradation following the initial rise. At this interval, the increased MLC Ca²⁺ sensitivity in the deeper segments leads to a more pronounced post-stimulus undershoot. This dynamic also explains the more pronounced post-stimulation undershoot observed in the deeper segments experimentally (Fig 9B).

Fig 9E shows the simulation results under similar neurogenic input, but without coupling between adjacent vascular cells in the modeled PA. The basal SMC membrane potential at different depths is not within the same range as it was in the coupled segments scenario depicted in Fig 9D. Additionally, the myogenic response begins to potentiate independently in each PA segment following its delayed NO-induced vasodilation, rather than in a synchronized manner, despite the delay in dilation. In this simulation, the depth-dependent deterministic features observed experimentally and computationally are absent. Therefore, Fig 9 collectively demonstrates that the observed experimental features can be recapitulated by a scenario in which instantaneous neuronal activity triggers NO production to act on SMCs, momentarily dampening the myogenic response and inducing vasodilation. In this scenario, the NO-induced dilation subsequently potentiates the myogenic response, leading to the attenuation of the initial vasodilation and the emergence of a post-stimulus undershoot in PA segments. Depending on the cortical depth of the PA, the degree of attenuation in the initial vasodilation and the magnitude of the post-stimulus undershoot vary.

Next, we analyzed the effects of different values on vasodynamics in response to neuronal impulse input. Increasing —which corresponds to lower activation of SMCs’ NPYRs—resulted in a slower myogenic response, where adjustments allowed a less immediate and pronounced reaction to changes in the diameter and WS (Fig 10A). Consequently, the initial dilation induced by NO-dependent partial dampening of the myogenic response was smallest with the fastest myogenic response and largest with the slowest. However, the magnitude of the post-stimulus undershoot depends not only on the degree of the potentiated myogenic response, which is more pronounced with smaller , but also on the degradation rate of NO/cGMP molecules. The post-stimulus undershoot was most pronounced at values of 25 and 50 secs, compared to both smaller and larger values. The reason is, within this range, and with a 5 sec time constant for the exponential decay of NO/cGMP concentration, the increased Ca²⁺ sensitivity of MLC from 10 to 15 sec after the initial NO/cGMP rise coincides with a delayed myogenic response-induced Ca²⁺ peak. This, in turn, resulted in a more pronounced post-stimulus undershoot compared to faster myogenic response scenarios, where even larger Ca²⁺ peaks occurred within the first 10 sec, when MLC sensitivity to Ca²⁺ was reduced due to higher NO/cGMP concentration. Furthermore, the analyzed dynamics shows that both the duration of the initial vasodilation and the post-stimulus undershoot increase with increasing , indicating a slower myogenic response. These modeled and comparative dynamics were similarly observed in instantaneous sensory-evoked vasodynamics in adult and aged mice [131]. Based on the reported vasodynamics and concurrent SMC Ca²⁺ monitoring in that study, and the vasodynamics we analyzed under various values in Fig 10A, we can conclude that the myogenic response in aged mice is slower than in adults, since the duration of both vasodilation and the myogenic response-induced elevation of SMC [Ca²⁺] were longer in aged mice.

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Fig 10. Analyzing arteriolar vasodynamics in response to neuronal impulse input under various model parameter values.

(A) Vasodynamics and SMC Ca²⁺ dynamics under various SMC mechanotransduction time constant (). Decreasing leads to a more attenuated vasodilation, along with a shorter duration of vasodilation and post-stimulus undershoot. (B) Predicted vasodynamics under various NO/cGMP degradation time constants. Faster degradation results in more pronounced post-stimulus constriction and oscillatory behavior, which is absent in in-vivo signals, suggesting that NO/cGMP degradation occurs slowly under physiological conditions. (C) In-vivo recorded vasodynamics from Uhlirova’s study [125], showing significant changes in vasodynamics upon the application of the Y1 receptor blocker BIBP 3226. Data were digitized from Uhlirova’s study and replotted here, under the terms of the CC BY 4.0 license (https://creativecommons.org/licenses/by/4.0/). (D) Model-predicted vasodynamics assuming that, during instantaneous neuronal activity, neuronally released mediators also hyperpolarize vascular cells, leading to biphasic SMC [Ca²⁺] and larger vasodilation. (E) Model-predicted vasodynamics based on the assumption that instantaneous neuronal activity occurs at four different phases of ongoing vasomotion.

https://doi.org/10.1371/journal.pcbi.1013113.g010

Next, we analyzed vasodynamics in response to neuronal impulse input with a fixed of 5 sec, while changing the time constant of the assumed NO/cGMP concentration exponential decay. As shown in Fig 10B, reducing the time constant, which accelerates the decay of NO/cGMP concentration, affected the duration of vasodilation, the degree of post-stimulus undershoot, and the timing of its occurrence. Comparing these simulation results with the experimental data shown in Fig 9B suggests that a slower decay in NO/cGMP concentration, with sec degradation time constant, replicates the experimentally observed vasodynamics more accurately. Therefore, the 5-sec time constant is a reasonable estimate for the NO/cGMP concentration exponential decay.

Our systems biology approach demonstrated that the potentiated myogenic response is primarily responsible for the biphasic vascular reactivity observed in neuronally-evoked vasodynamics, characterized by an initial vasodilation followed by rapid constriction, often leading to a post-stimulus undershoot. However, the constriction phase is blunted following the administration of the NPY receptor antagonist, BIBP ([125]), suggesting a potential interaction between NPY-positive inhibitory neurons and vascular SMCs. This interaction would facilitate inhibitory response within the CBF system (myogenic response), effectively counterbalancing initial blood flow increases triggered by excitatory responses. By comparing the vasodynamics under various SMC values (Fig 10A) with those observed after BIBP administration (Fig 10C), where neither the attenuation of initial vasodilation nor the post-stimulus undershoot is present, we can infer that the NPY receptor antagonist significantly slows the myogenic response. In our model, to replicate vasodynamics under BIBP application, the SMC would have to increase by a factor of 40–80 or more compared to sec. Therefore, the vasodynamics observed with BIBP likely represent pure NO-induced vasodilation, which follows the decay of NO/cGMP concentration. This approximately 16–20 sec constriction phase suggests that the NO/cGMP degradation time constant is around 4–5 sec.

In our analysis of vasodynamics in response to neuronal impulse input, we assumed that the brief duration of neuronal activity was insufficient to induce SMC hyperpolarization, either through neuronally released potassium acting on the endothelial network or by direct neuronal activation of SMC glutamate receptors colocalized with BK channels (Section 1). However, if either pathway is activated, the resultant vasodynamics would not change significantly. Fig 10D shows the vascular dynamics where the SMC membrane potential is immediately hyperpolarized due to neuronally mediated increases in EC [K⁺]_ex and the activation of SMC BK channels for a brief period (1.5 sec). This rapid hyperpolarization of SMCs caused an initial decrease in SMC Ca²⁺ levels, followed by a myogenic response-dependent elevation in SMC Ca²⁺, resulting in a biphasic Ca²⁺ pattern—an initial decline followed by a subsequent increase. Such biphasic Ca²⁺ dynamics have been reported in short-duration (<4–5 s) stimulus-evoked vascular responses [130,131]. In the 1.5 s neuronal activity scenario (Fig 10D), the initial neurogenic network hyperpolarization and associated SMC Ca²⁺ decrease are smaller than the subsequent myogenic post-stimulus depolarization and Ca²⁺ increase (also evident in the Supplementary Material of [130] (Fig 4 and videos)). For slightly longer duration of neuronal activity (3–5 s), the neurogenic hyperpolarization and the subsequent myogenic depolarization become comparable (Fig 3 of [131]). Both our simulations and in-vivo data consistently show post-stimulus uncoupling between Ca²⁺ and vascular tone: significant increases in Ca²⁺ occur after the stimulus, while only marginal or no post-stimulus constriction is observed. This provides validation for our assumption that NO-mediated PKG→MLCP activation reduces Ca²⁺–tone coupling during the phases of FH.

Next, we analyzed vasodynamics in response to neuronal impulse input, assuming that there is ongoing vasomotion in the arteriolar segments. As previously mentioned, the cellular-level PA model oscillates for two to three minutes after the simulation onset, until oscillations are fully damped (Fig 10E, top panel). Here, we assumed that neuronal input was applied to the PA model at four distinct points in time during oscillations including the: oscillation trough, mid-dilation, peak, and mid-constriction. As shown in Fig 10E, the ongoing oscillatory myogenic response influences both the magnitude of the initial dilation and the post-stimulus undershoot of vasodynamics depending on the timing of the neurogenic input to the PA. Furthermore, since NO suppresses the myogenic response for 10–20 sec post-stimulation due to its slow degradation rate, the ongoing vasomotion is damped after the stimulus. This occurs because the vessel is less constricted following the initial stimulus-evoked vasodilation, and the reduced constriction leads to a weaker delayed inhibition of the myogenic response during the dilation phase. As a result, oscillations remain damped and do not recover in this simulation, since we only modeled one PA with no synchronization in the cerebrovascular model. In reality, synchronization in the vascular network may help the oscillation recover post-stimulation, albeit with a new phase compared to the pre-stimulation oscillation, and this new phase is determined by the timing of the stimulus relative to the ongoing oscillation (Fig 10E).

Modeling arteriolar vasodynamics in response to neuronal step activity.

Here, we analyze vasodynamic output in response to a neuronal step input, assuming a relatively stable overall neural activity and minimal neuronal feedback inhibition during sustained stimulation. If the stimulation persists for an extended duration, the neuronally mediated SMC hyperpolarization, in addition to the dampening effect of NO on the myogenic response, remains effective for the entire period of neuronal activity. Fig 11A shows key PA model variables in response to a 10 sec sustained neurogenic input under three different neuronal activity levels with sec. We assumed that the hypothetical values of all neuronally released mediators, including NO, glutamate, and discharged potassium, increase linearly from level 1 to level 3. As shown, the SMC membrane potential initially hyperpolarizes in proportion to the level of neuronally released SMC hyperpolarization mediators. Simultaneously, NO-induced dilation significantly potentiates the myogenic response, which counteracts the neuronally induced SMC hyperpolarization, causing the SMC [Ca²⁺] to revert toward its pre-stimulation level or even higher after several seconds, until the equilibrium between neurogenic hyperpolarization and myogenic depolarization is established.

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Fig 11. (A) Neurogenic step response of the CBF regulatory system.

Arteriolar vasodynamics, vascular cells membrane potential, and SMC [Ca²⁺] are plotted for three hypothetical neuronal activity levels, increasing from level 1 to level 3. The corresponding hypothetical level of neuronal mediators are as follows: EC external potassium concentration changes from 3 mM at the baseline to 4.5, 6, and 7.5 mM; Glutamate concentration changes from 0 at baseline to 0.4, 0.8, and 1.2 AU; NO levels change from 0 at baseline to 2.5, 5, and 7.5 AU. In the right panel, all mediators corresponding to level 3 neuronal activity were applied except NO production. (B–E) Role of astrocytes in modulating arteriolar vasodynamics: (B) In-vivo recorded integrated gamma-band power and concurrent vasodynamics of an awake mouse as reported by Mateo et al. [26]. (C) Magnified view of two range-normalized epochs of in-vivo signals. Single-sided arrows and lowercase letters indicate timestamps for the initiation of astrocytic responses triggered by vessel constriction. Double-sided arrows and capital letters indicate time intervals for delayed potentiated myogenic responses in reaction to astrocytic responses. (D) Predicted vasodynamics by the proposed NVC model. (E) Predicted vasodynamics by the NVC model under both slow and fast degradation rates of NO/cGMP.

https://doi.org/10.1371/journal.pcbi.1013113.g011

In sustained FH in awake mice, such a pronounced SMC [Ca²⁺] undershoot in the first 5 s of stimulation is evident in experimental data [12,132]. Notably, as shown in Fig 1 of [12], during the initial 5 s of a 30 s sensory stimulation, while ∼60% of the initial neurogenic decrease in SMC [Ca²⁺] was reverted by the myogenic response, only ∼20% of the vasodynamic response was reverted. This observation further supports our assumption that NO-mediated PKG→MLCP activation reduces Ca²⁺–tone coupling during the phases of FH. In both [12,132], after myogenic depolarization and constriction within the first 5 s, a secondary phase of SMC [Ca²⁺] reduction is observed during the remaining stimulation interval, which has been attributed to astrocytic responses and is discussed in the next section.

The myogenic response should get more potentiated in neuronal activity level 3 compared to level 2, as the dose-dependent NO-induced vasodilation in level 3 is more pronounced. Consequently, we would expect a larger myogenic response-induced Ca²⁺ elevation in level 3 compared to level 2, although both reach the same [Ca²⁺] and SMC membrane potential by the end of the stimulation. This occurs because the more pronounced potentiated myogenic response in level 3 is counteracted by a stronger neuronally induced SMC hyperpolarization (inhibition of the myogenic response). As a result, the steady-state myogenic tone at the end of stimulation in both levels 2 and 3 are approximately equal. This dynamic is evident when comparing the level of SMC membrane potential depolarization immediately after the stimulus ends. Upon the sudden cessation of neuronal inputs, the SMC depolarization in level 3 is more than level 2, indicating that neuronally mediated inhibition of the myogenic response during the stimulation period was larger in level 3 than level 2.

Next, we analyzed the vasodynamics in neuronal activity level 3, assuming that nNOS is inhibited and NO is not produced during the stimulation period (Fig 11A, right panel). As shown, the neuronally mediated SMC hyperpolarization induced initial dilation which was followed by the myogenic response potentiation and a constriction phase. These simulation results demonstrate that delayed myogenic response potentiation is not solely attributed to NO-induced dilation, and any form of sudden constriction or dilation is followed by the subsequent dilation or constriction, respectively. For example, ATP puffing by micropipette close to vessels in-vivo induced dilation followed by constriction, and this constriction was abolished by P2 receptor antagonists [133], GPCRs that are largely associated with the myogenic tone regulation in the cerebral microvasculature [134].

Post-stimulus (Fig 11A, right panel), with the cessation of neuronal activity, the SMC depolarizes, and the vessel immediately constricts since there is no NO to desensitize the coupling between changes in Ca²⁺ and the constriction force of SMCs. Subsequently, this immediate constriction is followed by a dilation phase caused by the delayed myogenic response, which is then followed by another constriction phase. Similar dynamics were observed in Fig 1 of [135], where application of a NOS inhibitor produced an abrupt post-stimulus constriction followed by oscillatory responses. However, that study examined short (2 s) FH dynamics, in which neurogenic hyperpolarization can evoke the initial dilation and the myogenic response is not fast enough to counteract it, leading to the conclusion that NO is not responsible for the initial sensory-evoked hyperemia. Our simulations highlight that NO plays a critical role in both sustained (>2 s) and instantaneous (<2 s) FH dynamics. In sustained FH, NO dampens the myogenic response to promote dilation and prevents it from being reversed by subsequent potentiated myogenic activity. In instantaneous FH, where neuronally released hyperpolarizing factors are not sustained long enough to fully hyperpolarize the network and drive a pronounced Ca²⁺ reduction in SMCs, NO again serves as the primary vasodilator.

Astrocytic Regulation of Arteriolar Vasodynamics.

Ultra-slow fluctuations in neuronal signaling, occurring as an envelope over -band activity, entrain vasomotion [26]. Therefore, we aimed to test whether modulating arteriolar myogenic response with the envelope of -band activity would result in model-predicted vasodynamics that align with experimental observations. Fig 11B shows an example of recorded integrated gamma-band power over 0.4 sec intervals (the envelope of -band activity) and concurrent arteriolar vasodynamic measurements are as reported in Mateo et al.’s study. For this test, we used the simplified macro-scale arteriolar segment model, where vessel diameter was determined by delayed myogenic response in coupled segments and passive distension, which resulted in vasomotion in resting state. To include neuronal modulation in this model, we needed to determine the time lag between changes in neuronal activity and the corresponding changes in the myogenic response, whether through inhibition or dampening. Vasodynamics lag behind the envelope of -band activity by approximately 1.9 sec ([26]). Therefore, we can modify Eq. 3 as follows:

(6)

Here, represents a factor of the range-normalized -band signal, where the real-time myogenic response is inhibited by the -band signal from 1.9 sec earlier. We showed that the myogenic response is damped by NO produced during neuronal activity, and that NO/cGMP degradation occurs gradually. Therefore, we initially smoothed the decreasing trend of the range-normalized -band signal and then used this smoothed signal to dampen the myogenic response. Fig 11E (blue dashed line) shows the smoothed range-normalized -band signal, while Fig 11E (red dashed line) depicts the non-smoothed version. Thus, the real-time vessel diameter can be updated as:

(7)

The term models the dampening effect of neuronally-produced NO on the myogenic response by the smoothed and delayed -band signal.

Fig 11C displays a magnified view of two range-normalized epochs of in-vivo signals, and Fig 11D shows the range-normalized predicted vasodynamic by our macro-scale NVC model, plotted alongside the in-vivo -band signal. Comparing the model-predicted vasodynamic with the in-vivo vasodynamic shows that, in some cases, the model output shows good similarity to the in-vivo signal. For instance, in epoch 1, between the 145–170 sec interval, the smoothed decay of NO-induced dampening of the myogenic response prevents the immediate resumption of oscillatory myogenic activity after a sudden decrease in neuronal activity. As shown in the red curves in Fig 11E, if the NO/cGMP concentration closely follow the -band signal without smoothing (red dashed curve), the rapid decline in diameter would trigger a pronounced delayed myogenic response, leading to a subsequent dilation and the onset of oscillations. Therefore, the slow decay of NO-induced dampening of myogenic response prevents the emergence of myogenic oscillations in arteriolar vasodynamics in awake mice and helps vascular dynamics more closely follow neuronal demand, thereby stabilizing FH. Supporting this, injection of an NO inhibitor reduced the correlation between hemodynamic and LFP signals [136] and increased myogenic oscillations (vasomotion) in mice [135].

Even with the above refinements, significant discrepancies exist between the vasodynamics predicted by the NVC model and the in-vivo vasodynamics, which are associated with the astrocytic response in the CBF regulatory system. For example, take the timestamp ‘b’ in epoch 1. At this point, our model predicted that, due to the decline in the -band signal, the vasodynamics would follow a delayed downward trend. However, during the initial moments of this decline, we observed a sudden and unexpected rise. This unexpected deterministic feature is not exclusive to this timestamp; similar events could be seen at all timestamps marked by red single-sided arrows and lowercase letters. For instance, at timestamps ‘a’ and ‘c’ in epoch 1, similar unexpected events occur, and in epoch 2, we observed at least nine similar events within the 70 sec magnified view.

We attributed this deterministic feature of vasodynamics to astrocytes activities based on studies investigating astrocytes contributions to FH. A key study demonstrating that astrocytes possess mechanosensitive TRPV4 channels, which enable them to detect mechanical strain caused by changes in vessel diameter [137]. If, due to ongoing neuronal and synaptic activities, astrocytic endfeet TRPV4 channels are preconditioned with their endogenous ligands, such as epoxyeicosatrienoic acids (EET) [12,138], and if IP3 receptors on the endoplasmic reticulum within astrocytic endfeet are primed for a Ca²⁺ wave [139], paired with astrocytes depolarization caused by potassium removal from brain tissue, then TRPV4-evoked Ca²⁺ signaling in astrocyte endfeet could activate nearby BK channels. This would trigger the release of retained potassium into the surrounding perivascular space (PVS), amplifying the conductance of SMC Kir channels and resulting in SMC hyperpolarization (Fig 4A). Thus, if astrocyte endfeet are primed for significant Ca²⁺ signaling and subsequently detect vessel constriction during FH, they can actively induce arteriolar vasodilation. This mechanism aligns with findings from multiple studies, where Ca²⁺ signaling in astrocyte endfeet was observed post-stimulation in short-duration FH experiments, specifically when PAs began to constrict [140–144].

At timestamp ‘b’ in epoch 1, after astrocyte endfeet detected vessel constriction and released a large amount of potassium near the SMC Kir channels, the vessel dilated. Following this immediate rise in vessel diameter, the delayed myogenic response induced a subsequent constriction phase. Although our model predicted that vasodynamics should remain relatively unchanged during time interval ’B’, the delayed potentiated myogenic response triggered by astrocytic activity at timestamp ‘b’ induced a constriction phase, despite neuronal activity remaining relatively high during the ‘B’ interval. Similarly, during time interval ‘C’ in epoch 1, while our proposed NVC model predicted that vasodynamics would follow the rising trend of the -band signal with a delay, the delayed potentiated myogenic response triggered by astrocytic inhibition of the myogenic response at timestamp ‘c’ had a significant impact on vasodynamics. Once this delayed potentiated myogenic response passed, the increased -band signal caused vessel dilation, but this dilation was quickly terminated due to the subsequent decline in the -band signal. At timestamp ‘f’ in epoch 2, while our NVC model predicted constriction, astrocytic inhibition of the myogenic response caused the vessel to dilate. However, neuronal activity also showed an immediate decline during this time. As a result, during time interval ‘F,’ both the delayed potentiated myogenic response triggered by astrocytic activity at timestamp ‘f’ and the small neuronal inhibition of the myogenic response significantly constricted the vessels. Even the subsequent increase in the -band signal could not easily elevate the vessel diameter during this interval, giving the impression that the -band signal would lag behind the vasodynamics. However, this was not the case, since the delayed potentiated myogenic response triggered by astrocytic activity drove this unexpected vasodynamics.

Other indicated timestamps for the initiation of astrocytic responses and the corresponding time intervals for delayed potentiated myogenic responses in reaction to these astrocytic signals are well-justified with the proposed NGVC model. As demonstrated, astrocytes periodically inhibit the myogenic response, particularly in response to its potentiation, while neurons continuously send signals to modulate the myogenic response in vessels. Study of these interactions provide a clearer understanding of how astrocytes introduce additional low-frequency components to vasodynamics [137]. Vasodynamics generally should have stochastic low-frequency components from the envelope of the -band signal, as the myogenic response is continuously inhibited or dampened by this signal. Additionally, the non-singular frequency of vasomotion, which is more pronounced during periods of low and steady -band signal amplitude, also emerges in vasodynamics. Our findings show that astrocytic activity further modulates arteriolar vasodynamics, thereby adding more low-frequency components to the vasodynamics.