Simulation of conducted responses in microvascular networks: role of gap junction current rectification

Abstract

Objective:

Local control of blood flow depends on signaling to arterioles via upstream conducted responses. Here, the objective is to examine how electrical properties of gap junctions between endothelial cells (EC) affect the spread of conducted responses in microvascular networks of the brain cortex, using a theoretical model based on EC electrophysiology.

Methods:

Modeled EC currents are an inward-rectifying potassium current, a non-voltage dependent potassium current, a leak current, and a gap junction current between adjacent ECs. Effects of varying gap junction conductance are considered, including asymmetric conductance, with higher conductance for forward currents (positive currents from upstream to downstream, based on blood flow direction). The response is initiated by a local increase in extracellular potassium concentration. The model is applied to a 45-segment synthetic network and a 4881-segment network from mouse brain cortex.

Results:

The conducted response propagates preferentially to upstream arterioles when the conductance for forward currents is at least 20 times that for backward currents. The response depends strongly on the site of stimulation. With symmetric gap junction conductance, the network acts as a syncytium and the conducted response is dissipated.

Conclusions:

Upstream propagation of conducted responses may depend on asymmetric conductance of EC gap junctions.

INTRODUCTION

The brain depends on continuous blood supply to receive needed nutrients, especially oxygen (Harris et al., 2012; Mink et al., 1981). Increases in local blood flow occur in response to increases in neuronal activity in a process called neurovascular coupling (NVC) (Chaigneau et al., 2003; Fox & Raichle, 1986; Freygang & Sokoloff, 1959; LeDoux et al., 1983; Mosso, 1880; Roy & Sherrington, 1890). While the mechanisms of NVC are debated (Hosford & Gourine, 2019; Iadecola, 2017; Kaplan et al., 2020; Kenny et al., 2018; Leithner & Royl, 2013; Mintun et al., 2001; Petzold & Murthy, 2011), it is known that endothelial cells (ECs) of stimulated capillaries send a hyperpolarizing signal to upstream arterioles, called a conducted response (Collins et al., 1998). The hyperpolarization spreads to smooth muscle cells, causing them to relax and increase blood vessel diameter, with an increase in downstream blood flow.

Cerebral ECs are known to hyperpolarize in response to increases in extracellular potassium $({\mathrm{K}}^{+})$ concentration (Longden et al., 2017). The sensitivity is due to the presence of inward-rectifying ${\mathrm{K}}^{+}({\mathrm{K}}_{\text{IR}})$ channels whose conductance increases in response to rises in extracellular $[{\mathrm{K}}^{+}]$ (Longden et al., 2017; Longden & Nelson, 2015; Moshkforoush et al., 2020). The increased conductance causes a ${\mathrm{K}}^{+}$ current out of the EC, hyperpolarizing the membrane. The hyperpolarization then spreads along vessel walls via gap junction channels connecting ECs (de Wit & Griffith, 2010; Socha et al., 2012).

Crucially, the conducted response has been shown in skeletal muscle to travel from a stimulated capillary preferentially to upstream arterioles, with increases in blood flow occurring at the stimulated capillaries and not in parallel flow pathways (Berg et al., 1997; Twynstra et al., 2012). This phenomenon has not been demonstrated experimentally in the brain. However, without a mechanism to direct the conducted response to upstream arterioles in the brain, the conducted response signal could enter parallel flow pathways and cause diversion of blood flow to inactive areas. Such an uncontrolled spread could also cause the signal to be dissipated and to fail to reach upstream arterioles (Figure 1). How the conducted response could travel preferentially in the upstream direction is not currently known: electrical signals spread equally along all available paths if there is no difference in electrical resistance (Pries et al., 2010). Here, we propose asymmetric gap junction conductance as a possible mechanism for the directionality of the conducted response. According to this hypothesis, gap junction conductance is higher in the forward direction (positive current from upstream to downstream, relative to blood flow, causing hyperpolarization upstream of stimulus) than in the backward direction (positive current from downstream to upstream, which would cause hyperpolarization downstream of stimulus).

Figure 1. Schematic diagram illustrating propagation of conducted responses in a network.

The conducted response is initiated by an increase in ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ (purple circle) and travels upstream (arrows). Crossed-out arrows indicate pathways where propagation would reduce the strength or specificity of the upstream conducted response.

To test this hypothesis, a model for the distribution of EC currents is developed and applied to networks of microvessels, to characterize the spread of the conducted response. The model includes EC ${\mathrm{K}}_{\text{IR}}$ channels, whose conductance increases with rises in extracellular $[{\mathrm{K}}^{+}]({[{\mathrm{K}}^{+}]}_{\mathrm{E}})$, resulting in hyperpolarization of the EC membrane. Active neurons release ${\mathrm{K}}^{+}$ into the extracellular space, where it can diffuse rapidly from neurons to capillaries (Djurich & Secomb, 2024). Local neuronal activity is simulated in the model by introducing elevated ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ levels in specific tissue regions, and the resulting spread of EC membrane hyperpolarization in the networks is predicted. The effects on the distribution of EC membrane potentials of varying forward and backward gap junction conductances are examined, to determine conditions under which hyperpolarization is preferentially directed to upstream arterioles.

METHODS AND MATERIALS

A simplified model is used to represent EC electrophysiology. Currents in and out of ECs consist of transmembrane currents (between ECs and the extracellular space) and gap junction currents (between adjacent ECs) (Figure 2). Three transmembrane currents are included in the model: an inward-rectifying ${\mathrm{K}}^{+}$ current $I_{KIR}$ whose conductance depends on membrane potential (Figure 3A), a non-voltage dependent ${\mathrm{K}}^{+}$ current $I_{KNV}$, and a leak current $I_L$. The current $I_{KNV}$ includes effects of changes in ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ on the ${\mathrm{K}}^{+}$ reversal potential and hence on non-voltage-gated potassium currents. The current $I_L$ represents the net effect of currents in other channels. This model is chosen to represent the most relevant aspects of EC physiology, while neglecting possible effects of other voltage-gated or calcium dependent channels (Yang et al., 2015; Delmoe & Secomb, 2023). Asymmetry in gap junction conductance is represented by assuming that the conductance is larger in the forward direction (causing upstream hyperpolarization) than in the backward direction (causing downstream hyperpolarization). The resulting dependence of the gap junction current $I_{GJ}$ on the potential difference between ECs is shown in Figure 3B. Electrical communication with cerebral vascular smooth muscle cells is neglected in the model. Such currents could be represented approximately by including them in the leak current $I_L$. This would not substantially affect the conclusions of the analysis.

Figure 2. Model currents in cerebral capillary ECs.

$I_L$: leak current; $I_{KNV}$: non-voltage dependent ${\mathrm{K}}^{+}$ current; $I_{KIR}$: inward-rectifying ${\mathrm{K}}^{+}$ current; $I_{GJ}$: gap junction current.

Figure 3. Assumed dependence of ${\mathit{I}}_{\mathit{KIR}}$ and ${\mathit{I}}_{\mathit{GJ}}$ on EC membrane potentials.

A. ${\mathrm{K}}_{\text{IR}}$ channel current $I_{KIR}$ per endothelial cell with ${\overline{G}}_{KIR}=0.18{\text{nS}∕\text{mM}}^{0.5}$ and ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}=3\text{mM}$. B. Gap junction current $I_{GJ}$ between two adjacent ECs. Potential difference is difference in membrane potential between ECs.

The network of microvessels is represented as a set of segments and nodes. Each vessel segment connecting two nodes is treated as a discrete element. Its electrical state is characterized by the EC membrane potentials at each node, from which the corresponding transmembrane and gap junction currents are calculated.

The transmembrane currents of a segment depend on the ${\mathrm{K}}_{\text{IR}}$ conductance $g_{KIR}$, the non-voltage-dependent ${\mathrm{K}}^{+}$ conductance $g_{KNV}$, and the leak conductance $g_L$. The $g_{KIR}$ conductance depends on the membrane potential $V$, and is described by

$$g_{KIR}={\overline{\mathrm{G}}}_{KIR}\sqrt{{[K^{+}]}_E}\frac{1}{1+e^{\frac{V-V_{KIR}}{k_{KIR}}}}$$ (1)

where ${\overline{G}}_{KIR}$ is a constant, ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ is the extracellular ${\mathrm{K}}^{+}$ concentration, $V_{KIR}=E_K+25\text{mV}$ is the membrane potential at half-maximal activation, $k_{KIR}$ defines the steepness of the sigmoidal curve describing the relationship between $g_{KIR}$ and $V$, and $E_K$ is the Nernst potential for ${\mathrm{K}}^{+}$, here estimated as −61.5 $\log ({[{\mathrm{K}}^{+}]}_{\mathrm{I}}∕{[{\mathrm{K}}^{+}]}_{\mathrm{E}})\text{mV}$ (Moshkforoush et al., 2020).

The total transmembrane conductance of a segment is proportional to both the number of ECs in parallel along the vessel length and the number around the vessel circumference. A transmembrane conductance factor for segment $j$ is defined as

$$n_{EC,j}^S=\frac{D}{D_{ref}}\frac{L}{L_{EC}}$$ (2)

where $D$ and $L$ are the inner diameter and length of the segment, $D_{ref}$ is a reference diameter, and $L_{EC}$ is the length of an individual EC. Superscripts $S$ and $N$ refer to segments and nodes, respectively. A conductance factor for node $i$ is defined as

$$n_{EC,i}^N=\frac{1}{2}\sum n_{EC,j}^S$$ (3)

where the sum is over the segments $j$ that are connected to node $i$, and the factor of one half denotes that half of a segment’s transmembrane conductance is attributed to the upstream node and half to the downstream node. Total conductances associated with node $i$ are defined as

$$g_{KIR,i}^N=n_{EC,i}^N{\overline{\mathrm{G}}}_{KIR}\sqrt{{[K^{+}]}_E}\frac{1}{1+e^{\frac{V_i-V_{KIR,i}}{k_{KIR}}}}$$ (4)

$$g_{KNV,i}^N=n_{EC,i}^N{g}_{KNV}$$ (5)

$$g_{L,i}^N=n_{EC,i}^N{g}_L$$ (6)

The effective gap junction conductance of a segment is proportional to the number around the vessel circumference, but inversely proportional to the number of ECs in series along the vessel length. A gap junction conductance factor for segment $j$ is defined as

$$n_{GJ,j}^S=\frac{D}{D_{ref}}\frac{L_{EC}}{L}$$ (7)

To allow for asymmetric conductance properties in gap junctions, the gap junction conductance of segment $j$ is defined as

$$g_{GJ,j}^S=\{\begin{array}{cc}{n}_{GJ,j}^S{g}_{GJb}\hfill & \text{if}{V}_{up,j}^N<V_{down,j}^N\hfill \\ n_{GJ,j}^S{g}_{GJf}\hfill & \text{if}{V}_{up,j}^N>V_{down,j}^N\hfill \end{array}\phantom{\}}$$ (8)

where $V_{up,j}^N$ and $V_{down,j}^N$ are the voltages of the nodes at the upstream and downstream ends of the segment $j$, and $g_{GJb}$ and $g_{GJf}$ are the conductances in the backward and forward directions as already defined.

The currents associated with node $i$ are

$$I_{KIR,i}=g_{KIR,i}^N(V_i-E_{K,i})$$ (9)

$$I_{KNV,i}=g_{KNV,i}^N(V_i-E_{K,i})$$ (10)

$$I_{L,i}=g_{L,i}^N(V_i-E_L)$$ (11)

$$I_{GJ,i}=\sum g_{GJ,j}^S(V_i-V_j^{\prime })$$ (12)

where outward currents are positive, $V_i$ is the membrane potential at node $i$ and $E_{K,i}$ is the Nernst potential for potassium at node $i$. The sum in equation (12) is over the segments $j$ that are connected to node $i$ and $V_j^{\prime }$ is the potential at the opposite end of segment $j$ from node $i$. The leak Nernst potential $E_L$ is assumed to be 0 mV (Delmoe & Secomb, 2023; Yang et al., 2015).

The total outward current at each node must be zero. From equations (9-12), this gives:

$$(g_{KIR,i}^N+g_{KNV,i}^N)(V_i-E_{K,i})+g_{L,i}^N{V}_i+\sum g_{GJ,j}^S(V_i-V_j^{\prime })=0$$ (13)

where the sum is as in equation (12). When applied at a boundary node, this equation represents a zero-current boundary condition. This may exaggerate hyperpolarization of segments near the boundary, because currents from outside the domain are neglected.

Because $I_{KIR}$ and $I_{GJ}$ are nonlinear functions of membrane potential, an iterative process is used to determine membrane potentials. Initially, all nodes are assumed to have a membrane potential of −30 mV and the membrane conductances are calculated. These new conductances are used to recalculate the membrane potentials. This process is repeated until the potentials converge within a tolerance of 10⁻³ mV to values from the previous iteration.

The model is applied to two microvascular networks. The first is a synthetic network consisting of 45 segments and 29 nodes, chosen to provide a simple illustration of model behavior. The second is a realistic network reconstructed from 2-photon microscopy imaging of a region in the mouse cerebral cortex (Celaya-Alcala et al., 2021; Gagnon et al., 2015). The tissue volume is 610 × 610 × 660 μm and includes 4881 segments and 4104 nodes.

In the 45-segment network, ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ at three nodes is set to ${[{\mathrm{K}}^{+}]}_{\mathrm{E}\max }=9\text{mM}$, while all other nodes are set at the baseline level, ${[{\mathrm{K}}^{+}]}_{\mathrm{E}0}=3\text{mM}$. In the reconstructed network, a spherically symmetric distribution of ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ is imposed, described by

$${[K^{+}]}_E={[K^{+}]}_{E0}+({[K^{+}]}_{Emax}-{[K^{+}]}_{E0})e^{-d^2∕r_0^2}$$ (14)

where $d$ is the distance of the node from the center of the sphere and $r_0$ is the diameter at which $[{\mathrm{K}}^{+}]$ falls to $e^{-1}$ of its original value. This distribution is chosen to represent the gradient created by diffusion of ${\mathrm{K}}^{+}$ from a region in which multiple neurons are firing and releasing ${\mathrm{K}}^{+}$ (Chen & Nicholson, 2000; Lyall et al., 2021; Ohki et al., 2006).

Calculations were performed using a C++ program in Visual Studio 2022 (Microsoft, Redmond, WA). Parameter values are shown in Table 1. A wide range of gap junction conductances has been reported (Moshkforoush et al., 2020; Sancho et al., 2017). The assumed reference values are within observed ranges.

Table 1.

Parameters in conducted response model

Variable	Parameter	Value	Units	Source
${\mathit{D}}_{\mathit{ref}}$	Reference diameter of a blood vessel	6	μm	(Arciero et al., 2008)
${\mathit{E}}_{\mathit{K}}$	${\mathrm{K}}^{+}$ Nernst potential	−104 - −74	mV	1.
${\mathit{E}}_{\mathit{L}}$	Leak Nernst potential	0	mV	(Yang et al., 2015)
${\mathit{g}}_{\mathit{L}}$	Leak conductance	0.0355	nS	(Delmoe & Secomb, 2023)2
${\mathit{g}}_{\mathit{GJb}}$	Backward gap junction conductance	1 1-333.333	nS	(Moshkforoush et al., 2020; Sancho et al., 2017)
${\mathit{g}}_{\mathit{GJf}}$	Forward gap junction conductance	20 1-333.333	nS	(Moshkforoush et al., 2020; Sancho et al., 2017)
${\overline{\mathit{G}}}_{\mathit{KIR}}$	${\mathrm{K}}_{\text{IR}}$ channel conductance factor	0.18	nS/mM1/2	(Moshkforoush et al., 2020)
${\mathit{g}}_{\mathit{KNV}}$	Non-voltage dependent ${\mathrm{K}}^{+}$ conductance	0.0145	nS	(Delmoe & Secomb, 2023)2
${[{\mathbf{K}}^{+}]}_{E0}$	Baseline extracellular ${\mathrm{K}}^{+}$ concentration	3	mM	(Dietzel et al., 1982; Walz & Hertz, 1983)
${[{\mathbf{K}}^{+}]}_{\mathbf{Emax}}$	Maximum extracellular ${\mathrm{K}}^{+}$ concentration	9 8-123	mM	(Walz & Hertz, 1983)
${[{\mathbf{K}}^{+}]}_{\mathbf{I}}$	Intracellular ${\mathrm{K}}^{+}$ concentration	145 140-1503	mM	(Adams & Hill, 2004)
${\mathit{k}}_{\mathit{KIR}}$	${\mathrm{K}}_{\text{IR}}$ slope factor	7	mV	(Silva et al., 2007)
${\mathit{L}}_{\mathit{EC}}$	Length of endothelial cell	20 20-333	μm	(Moshkforoush et al., 2020; Sancho et al., 2017)
${\mathit{r}}_0$	Radius at which $[{\mathrm{K}}^{+}]$ falls to $e^{-1}$	100	μm	(Chen & Nicholson, 2000)
${\mathit{v}}_{\mathit{KIR}}$	Voltage at half-maximal activation of ${\mathrm{K}}_{\text{IR}}$	${\mathit{E}}_{\mathit{K}}+25E_K-E_K+{40}^3$ 3	mV	(Silva et al., 2007)

^1.Calculated using the Nernst equation for values of ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}=3$ to 9 mM.

^2.The background conductance value per EC, $G_{bg}=0.05\text{nS}$ (Moshkforoush et al., 2020; Lee, 2022), was partitioned into $g_L$ and $g_{KNV}$ to yield an appropriate EC membrane resting potential (Delmoe & Secomb, 2023).

^3.Reference value is shown in bold with the range of published or considered values.

RESULTS

45-segment network.

Figures 4 and 5 show results for the 45-segment synthetic network, when stimulated with 9 mM extracellular ${\mathrm{K}}^{+}$ at three nodes. If both forward and backward conductance are set to 20 nS (Figure 4A), the network acts approximately as a syncytium with no upstream propagation of the conducted response. If the backward conductance $g_{GJb}$ is reduced to 1 nS (Figure 4B), the network shows a conducted response that propagates preferentially to the upstream arteriole. Downstream segments remain close to a membrane potential of −30 mV. Changes in the membrane potential at the feeding and draining vessels with varying backward gap junction conductance are shown in Figure 5. As $g_{GJb}$ is reduced, the arteriolar ECs become increasingly hyperpolarized, while the venular ECs show small changes in membrane potential.

Figure 4. Effects of backward gap junction conductance on EC membrane potentials in a synthetic network.

Forward conductance $g_{GJf}=20\text{nS}$ in both. Purple circle: node stimulated with 9 mM ${\mathrm{K}}^{+}$. Colored bar: EC membrane potentials. A. Backward conductance $g_{GJb}=20\text{nS}$. B. Backward conductance $g_{GJb}=1\text{nS}$.

Figure 5. Effects of backward gap junction conductance on membrane potentials in synthetic network.

Forward conductance $g_{GJf}=20\text{nS}$ in all cases.

Reconstructed brain network.

The behavior of the 4881-segment network reconstructed from mouse cerebral cortex was examined for several values of forward and backward gap junction conductances, and the effects of varying the stimulus location were also tested. Figure 6 shows results for one stimulus location, for a fixed forward conductance $g_{GJf}=20\text{nS}$, and for three values of $g_{GJb}$. When $g_{GJb}=20\text{nS}$, the effects of a local hyperpolarizing current are dissipated in the surrounding network, and EC membrane potentials are almost unchanged. When $g_{GJb}=1\text{nS}$, the upstream feeding arteriole shows significant hyperpolarization, as do some other nearby vessels outside the region of stimulation. For a much reduced backward conductance, $g_{GJb}=0.01\text{nS}$, the arteriole shows stronger hyperpolarization, with hyperpolarization confined to the simulation region and the upstream flow pathway.

Figure 6. Effects of backward gap junction conductance on EC membrane potentials in reconstructed network.

Coordinate axes are shown meeting at the origin. The length of each axis represents the corresponding dimension of the tissue region, which is 610 × 610 × 660 μm. The network is stimulated by a localized increase of ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$ centered at (150, 305, 330) μm. Forward conductance $g_{GJf}=20\text{nS}$ in all cases. Purple sphere: region of increased ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$. Color bar: EC membrane potential. A. Backward conductance $g_{GJb}=20\text{nS}$. B. Backward conductance $g_{GJb}=1\text{nS}$, C. Backward conductance $g_{GJb}=0.01\text{nS}$.

To test the effect on the network of overall gap junction conductance levels, the forward and backward gap junction conductances were varied across a wide range of values while maintaining a 20:1 forward to backward ratio (Figure 7). The changes in membrane potential were predicted at two consecutive segments of the feeding arteriole (Figure 7A). The downstream segment shows stronger polarization than the upstream segment, illustrating the decay of the signal with distance from the stimulus. The results show hyperpolarization of the feeding arteriole for a wide range of gap junction conductances. However, if both conductances are high enough, then the network acts as a syncytium and the upstream signal dissipates. Conversely, if both conductances are very low, then electrical signals are not propagated to the arteriole.

Figure 7. Effects of varying forward and backward gap junction conductances, with a fixed 20:1 ratio.

The network is stimulated by a localized increase of $[{\mathrm{K}}^{+}]$ in the ECS centered at (150, 305, 330) μm. A. EC membrane potentials of two consecutive arteriole segments. Segment 1 is upstream and segment 2 is downstream. B. Average EC membrane potentials of segments whose midpoints lie within the indicated ranges of distance $d$ from the center of the stimulus.

The spread of the conducted response was further analyzed by computing the average EC membrane potentials of segments with midpoints lying in specified ranges of distance from the center of stimulation, as shown in Figure 7B. A 20:1 conductance ratio was again assumed. If $g_{GJf}>50\text{nS}(g_{GJb}>2.5\text{nS})$, the entire network acts as a syncytium with no hyperpolarization. For lower conductance values, vessels lying within the characteristic radius of the stimulus $(d<100\mu \mathrm{m})$ consistently show hyperpolarization. The upstream conduction of hyperpolarization is reflected in the partial hyperpolarization of vessels in the range $100<d<150\mu \mathrm{m}$, except when conductances are very high or very low.

Due to the heterogeneous structure of the reconstructed brain network, the evoked conducted response depends on the location of the stimulation. For stimuli centered at 48 points placed in a regular array within the region, average EC membrane potentials were computed for all segments with midpoints lying in specified ranges of distance from the stimulus. The spread of the conducted response shows strong dependence on stimulus location, as indicated by the scatter of the results (Figure 8). When averaged over the 48 stimulus locations, the hyperpolarization decreases with increasing distance from the stimulus (Figure 8A). The simulations were repeated for $g_{GJb}=5\text{nS}$, i.e. a 4:1 ratio of forward to backward conductance. In this case, the degree of hyperpolarization decreased on average, but remained highly variable with stimulus location, with some stimuli producing strong hyperpolarization. The degree of asymmetry necessary to produce a conducted response was also affected by the location of the stimulus (see Supplementary Figure S1). To test the possibility that the observed heterogeneity of response was sensitively dependent on the density of ${\mathrm{K}}_{\text{IR}}$ channels, further simulations were performed with decreased and increased values of the ${\mathrm{K}}_{\text{IR}}$ conductance factor ${\overline{G}}_{KIR}$. The results obtained were qualitatively similar (see Supplementary Figure S2).

Figure 8. Effects of stimulus location and distance from stimulus center on EC membrane potentials.

The stimulus center was placed at 48 locations within the region. For each location, an open circle indicates the average of EC membrane potentials for segments in the indicated range of distances $d$ from the stimulus center. Black bars indicate the average over all stimulus locations. Forward conductance $g_{GJf}=20\text{nS}$. A. Backward conductance $g_{GJb}=1\text{nS}$. B. Backward conductance $g_{GJb}=5\text{nS}$.

DISCUSSION

Increase of local blood flow and oxygen level in response to increased local neuronal activity is the physiological basis of blood-oxygen-level dependent functional magnetic resonance imaging (BOLD-fMRI) in the brain. However, the degree to which the fMRI signal reflects underlying neuronal activity is not well understood. According to the concept of the neurovascular unit (Iadecola, 2017), a strict overlap between increased neuronal activity and increased blood oxygenation would be expected. In reality, the structure, cellular composition and activity of the brain vasculature and parenchyma are heterogeneous, and the relationship between neuronal activity and blood flow is more complex than implied by the concept of the neurovascular unit (Alkayed & Cipolla, 2023; Schaeffer & Iadecola, 2021). Improved understanding of the mechanisms of neurovascular coupling, and particularly the spatial and temporal relationship between brain activity and local blood flow regulation, could be helpful in better understanding the results of fMRI imaging.

Among the several mechanisms that have been identified as contributing to regulation of blood flow, the role of upstream conducted responses in coordinating blood flow with metabolic needs is well recognized (Chen et al., 2014; Mironova et al., 2024; Segal & Duling, 1986). A number of theoretical models have been developed describing conducted responses in terms of electrical currents in vessel walls (Crane et al., 2001; Hald et al., 2012; Kapela et al., 2018) and analyzing the role of conducted responses in local flow regulation (Arciero et al., 2008; Kapela et al., 2010; Lee, 2022; Moshkforoush et al., 2020). Preferential propagation of conducted responses in the upstream direction relative to blood flow has been observed experimentally in skeletal muscle (Berg et al., 1997; Twynstra et al., 2012) and its theoretical necessity throughout the microcirculation has been pointed out (Pries et al., 2010). However, the mechanistic basis for such behavior has received little attention in the published literature.

Here, a mechanism is proposed for preferential propagation of conducted responses to upstream arterioles, based on asymmetric conductance properties of gap junctions connecting ECs in vessel walls. A theoretical model is developed for the spread of EC currents in networks with arbitrary topology. The model takes into account transmembrane currents including ${\mathrm{K}}_{\text{IR}}$ channel currents and asymmetric gap junction conductivity between ECs.

The behavior of the model system is explored by considering effects of varying gap junction conductances, while holding the transmembrane conductances constant. However, the distribution of electrical potentials in the network depends only on the ratios of the conductances, and not on their absolute values. Therefore, the effects of variations in gap junction conductances as shown in Figure 7 can equally be interpreted as the effects of the inverse variations applied to the transmembrane conductances.

According to the results of this model, preferential upstream propagation of conducted responses can result from asymmetrical gap junction conductance, where hyperpolarizing currents in ECs are conducted more readily in the downstream direction than in the upstream direction, based on the direction of blood flow. If, by contrast, gap junction conductance is assumed to be equal in both directions, the simulated network acts like a syncytium in response to a local EC membrane hyperpolarization. The bidirectional spread of current into nearby connected vessels results in dissipation of the signal, without significant propagation either to upstream arterioles or downstream venules (Figures 4 and 6).

The strength of the conducted response depends on the location of the stimulus that evokes the response. Stimuli at some points produced a strong conducted response while stimuli at other points caused little or no response (Figure 8). The variability in response is a consequence of the heterogeneity of the network structure, and the relatively small size of the assumed stimulus region, with a characteristic radius of 100 μm. This finding implies that neuronal activity that is restricted to small regions in the brain would result in highly variable fMRI signals, and would often be undetectable. If the activity is spread over larger regions, so that a large number of microvessels are exposed to elevated extracellular potassium levels, then a more consistent conducted response would be expected, with a higher probability of a detectable fMRI signal.

The effects of the degree of gap junction asymmetry on the propagation of the conducted response are examined in Figures 5 and 8. As the ratio of forward to backward conductance is increased, the amount of hyperpolarization and the specificity of the conducted response are enhanced. In the reconstructed network, the average hyperpolarization of nodes in the regions further from the center of the stimulus is lower than that of nodes closer to the center of stimulus. Some hyperpolarization is seen at more distant points, consistent with the spread of the conducted response to upstream arterioles.

If a 20:1 forward to backward ratio of conductances is imposed, then upstream propagation of conducted responses to feeding arterioles is predicted for a wide but finite range of forward conductances (Figure 7A). When total conductance is too high, the network acts like a syncytium, and when total conductance is too low, hyperpolarization cannot spread. The predicted arteriolar hyperpolarization shows an irregular variation with forward conductance, with a maximal response at a forward conductance of about 1 nS. Figures 7 and 8 show that some nodes further from the stimulus are more hyperpolarized than nodes closer to the stimulus. As a result of the nonlinear behavior of ${\mathrm{K}}_{\text{IR}}$ channels, ECs can exhibit bistability of membrane potential for some values of ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$, which can cause non-decaying propagation of conducted responses (see Supplementary Figure S3) (Delmoe & Secomb, 2023; Lee, 2022; Moshkforoush et al., 2020; Postnov et al., 2015). These phenomena may account for the unexpected behaviors seen in the present simulations, for certain combinations of parameter values and levels of ${[{\mathrm{K}}^{+}]}_{\mathrm{E}}$. Further work is needed to clarify the role of non-decaying conducted responses under normal physiological conditions.

The mechanistic basis for the proposed asymmetric gap junction conductance is not known. Asymmetric expression of connexins is a possible mechanism. Differences in voltages between two connected ECs can modulate gap junction conductance, and connexin subtypes have varying sensitivities to voltage differences (Bukauskas & Verselis, 2004; Palacios-Prado & Bukauskas, 2009), with connexins on each side of the gap junction having different sensitivities. Such behavior might result from shear-force-dependent transcription causing differential upstream and downstream expression of connexins in ECs. Connexin expression is observed to vary in the vasculature, with more connexin-43 at areas of disturbed or fluctuating flow (Hautefort et al., 2019). Preferential upstream propagation of conducted responses could result from asymmetric gap junction conductances at bifurcation points. Also, chemical gating of gap junctions (Bukauskas & Verselis, 2004) may contribute to asymmetric conductance.

Experimental evidence for asymmetry of gap junction conductance is inconsistent. Asymmetric gap junction conductance has been observed in amacrine cells in the mammalian retina (Veruki & Hartveit, 2002), suggesting that asymmetry could occur in blood vessels. However, a study in the posterior cerebral artery did not show an asymmetrical electrical response, with equal spread in both directions (Hakim & Behringer, 2023). Asymmetry may occur in capillaries and arterioles but not arteries, or be present in bifurcations and not within vascular segments (Twynstra et al., 2012). Variations in transmembrane ion channel expression in pre- versus post-capillary blood vessels may also affect the spread of the conducted response (Delmoe & Secomb, 2023; Lee, 2022), although this cannot explain how spread of responses into parallel flow pathways is inhibited. The results presented here emphasize the importance of EC gap junctions for local control of blood flow, and suggest that their role may be more complex than has been considered previously.

PERSPECTIVES.

A theoretical model is used to analyze a possible mechanism by which conducted responses to upstream arterioles result in local increases in blood flow in response to neuronal activity. Asymmetric gap junction conductance between endothelial cells may be necessary for conducted responses to propagate preferentially to upstream arterioles.

List of Abbreviations

EC

Endothelial cell

${\mathrm{K}}^{+}$

Potassium ion

${\mathrm{K}}_{\text{IR}}$

Inward-rectifying potassium

KNV

Non-voltage dependent potassium

NVC

Neurovascular coupling

SMC

Smooth muscle cell