Applicability of cable theory to vascular conducted responses

The conducted vascular response as a mediator of hypercapnic cerebrovascular reactivity: A modelling study

It is well established that the cerebral blood flow (CBF) shows exquisite sensitivity to changes in the arterial blood partial pressure of CO2 ( [Formula: see text] ), which is reflected by an index termed cerebrovascular reactivity. In response to elevations in [Formula: see text] (hypercapnia), the vessels of the cerebral microvasculature dilate, thereby decreasing the vascular resistance and increasing CBF. Due to the challenges of access, scale and complexity encountered when studying the microvasculature, however, the mechanisms behind cerebrovascular reactivity are not fully understood. Experiments have previously established that the cholinergic release of the Acetylcholine (ACh) neurotransmitter in the cortex is a prerequisite for the hypercapnic response. It is also known that ACh functions as an endothelial-dependent agonist, in which the local administration of ACh elicits local hyperpolarization in the vascular wall; this hyperpolarization signal is then propagated upstream the vascular network through the endothelial layer and is coupled to a vasodilatory response in the vascular smooth muscle (VSM) layer in what is known as the conducted vascular response (CVR). Finally, experimental data indicate that the hypercapnic response is more strongly correlated with the CO2 levels in the tissue than in the arterioles. Accordingly, we hypothesize that the CVR, evoked by increases in local tissue CO2 levels and a subsequent local release of ACh, is responsible for the CBF increase observed in response to elevations in [Formula: see text] . By constructing physiologically grounded dynamic models of CBF and control in the cerebral vasculature, ones that integrate the available knowledge and experimental data, we build a new model of the series of signalling events and pathways underpinning the hypercapnic response, and use the model to provide compelling evidence that corroborates the aforementioned hypothesis. If the CVR indeed acts as a mediator of the hypercapnic response, the proposed mechanism would provide an important addition to our understanding of the repertoire of metabolic feedback mechanisms possessed by the brain and would motivate further in-vivo investigation. We also model the interaction of the hypercapnic response with dynamic cerebral autoregulation (dCA), the collection of mechanisms that the brain possesses to maintain near constant CBF despite perturbations in pressure, and show how the dCA mechanisms, which otherwise tend to be overlooked when analysing experimental results of cerebrovascular reactivity, could play a significant role in shaping the CBF response to elevations in [Formula: see text] . Such in-silico models can be used in tandem with in-vivo experiments to expand our understanding of cerebrovascular diseases, which continue to be among the leading causes of morbidity and mortality in humans.

Rude mechanicals in brain haemodynamics: non-neural actors that influence blood flow

Fluctuations in blood oxygenation and flow are widely used to infer brain activity during resting-state functional magnetic resonance imaging (fMRI). However, there are strong systemic and vascular contributions to resting-state signals that are unrelated to ongoing neural activity. Importantly, these non-neural contributions to haemodynamic signals (or 'rude mechanicals') can be as large as or larger than the neurally evoked components. Here, we review the two broad classes of drivers of these signals. One is systemic and is tied to fluctuations in external drivers such as heart rate and breathing, and the robust autoregulatory mechanisms that try to maintain a constant milieu in the brain. The other class comprises local, active fluctuations that appear to be intrinsic to vascular tissue and are likely similar to active local fluctuations seen in vasculature all over the body. In this review, we describe these non-neural fluctuations and some of the tools developed to correct for them when interpreting fMRI recordings. However, we also emphasize the links between these vascular fluctuations and brain physiology and point to ways in which fMRI measurements can be used to exploit such links to gain valuable information about neurovascular health and about internal brain states. This article is part of the theme issue 'Key relationships between non-invasive functional neuroimaging and the underlying neuronal activity'.

Conceptualizing conduction as a pliant electrical response: impact of gap junctions and ion channels

Vasomotor responses conduct among resistance arteries to coordinate blood flow delivery pursuant to energetic demand. Conduction is set by the electrical and mechanical properties of vascular cells, the former tied to how gap junctions and ion channels distribute and dissipate charge, respectively. These membrane proteins are subject to modulation; thus, conduction could be viewed as "pliant" to the current regulatory state. This study used in silico approaches to conceptualize electrical pliancy and to illustrate how gap junctional and ion channel properties distinctly impact conduction along a single skeletal muscle artery or a branching cerebrovascular network. Initial simulations revealed how vascular cells encoded with electrotonic properties best reproduced spreading behavior; the endothelium's importance as a charge source and a longitudinal conduit was readily observed. Alterations in gap junctional conductance produced unique electrical fingerprints: 1) decreased endothelial coupling impaired longitudinal but enhanced radial spread, and 2) reduced myoendothelial coupling limited radial but enhanced longitudinal spread. Subsequent simulations illustrated how tuning ion channel activity, e.g., inward rectifying- and voltage-gated K⁺ channels, modified charge dissipation, resting membrane potential, and the spread of the electrical phenomenon. Restricting ion channel tuning to a network subregion then revealed how electrical spread could be locally shaped in accordance with the aggregate changes in membrane resistance. In summary, our analysis frames and reimagines electrical conduction as a pliable process, with subtle regulatory changes to membrane proteins shaping network spread and tissue perfusion.NEW & NOTEWORTHY Conducted vasomotor responses depend on initiation and spread of electrical phenomena along arterial walls and their translation into contractile responses. Using computational approaches, we show how subtle but widespread regulation of gap junctions and ion channels can modulate the range and amplitude of electrical spread. Ion channels are regulated by endocrine and mechanical signals and may differ regionally in networks. Subregional electrical changes are not spatially confined but may affect electrical conduction in neighboring regions.

nNOS-expressing interneurons control basal and behaviorally evoked arterial dilation in somatosensory cortex of mice

Cortical neural activity is coupled to local arterial diameter and blood flow. However, which neurons control the dynamics of cerebral arteries is not well understood. We dissected the cellular mechanisms controlling the basal diameter and evoked dilation in cortical arteries in awake, head-fixed mice. Locomotion drove robust arterial dilation, increases in gamma band power in the local field potential (LFP), and increases calcium signals in pyramidal and neuronal nitric oxide synthase (nNOS)-expressing neurons. Chemogenetic or pharmocological modulation of overall neural activity up or down caused corresponding increases or decreases in basal arterial diameter. Modulation of pyramidal neuron activity alone had little effect on basal or evoked arterial dilation, despite pronounced changes in the LFP. Modulation of the activity of nNOS-expressing neurons drove changes in the basal and evoked arterial diameter without corresponding changes in population neural activity.

Conceptualizing Conduction as a Pliant Vasomotor response: Impact of Ca2+ fluxes and Ca2+ Sensitization

Coordinating blood flow to active tissue requires vasomotor responses to conduct among resistance arteries. Vasomotor spread is governed by the electrical and mechanical properties of vessels; the latter being linked to the sigmoid relations between membrane potential (VM), [Ca2+], and smooth muscle contractility. Proteins guiding electrical-to-tone translation are subject to regulation; thus, vasomotor conduction could be viewed as "pliant" to the current regulatory state. Using simple in silico approaches, we explored vasomotor pliancy and how the regulation of contractility impacts conduction along a skeletal muscle artery and a branching cerebrovascular network. Initial simulations revealed how limited electromechanical linearity affects the translation of electrical spread into arterial tone. Subtle changes to the VM-[Ca2+] or [Ca2+]-diameter relationship, akin to regulatory alterations in Ca2+ influx and Ca2+ sensitivity, modified the distance and amplitude of the conducted vasomotor response. Simultaneous changes to both relationships, consistent with agonist stimulation, augmented conduction although the effect varied with stimulus strength and polarity (depolarization vs hyperpolarization). Final simulations using our cerebrovascular network revealed how localized changes to the VM-[Ca2+] or [Ca2+]-diameter relationships could regionally shape conduction without interfering with the electrical spread. We conclude that regulatory changes to key effector proteins (e.g. L-type Ca2+ channels, myosin light chain phosphatase), integral to voltage translation, not only impact conducted vasomotor tone but likely blood flow delivery to active tissues.

Overview of mathematical modeling of myocardial blood flow regulation

The oxygen consumption by the heart and its extraction from the coronary arterial blood are the highest among all organs. Any increase in oxygen demand due to a change in heart metabolic activity requires an increase in coronary blood flow. This functional requirement of adjustment of coronary blood flow is mediated by coronary flow regulation to meet the oxygen demand without any discomfort, even under strenuous exercise conditions. The goal of this article is to provide an overview of the theoretical and computational models of coronary flow regulation and to reveal insights into the functioning of a complex physiological system that affects the perfusion requirements of the myocardium. Models for three major control mechanisms of myogenic, flow, and metabolic control are presented. These explain how the flow regulation mechanisms operating over multiple spatial scales from the precapillaries to the large coronary arteries yield the myocardial perfusion characteristics of flow reserve, autoregulation, flow dispersion, and self-similarity. The review not only introduces concepts of coronary blood flow regulation but also presents state-of-the-art advances and their potential to impact the assessment of coronary microvascular dysfunction (CMD), cardiac-coronary coupling in metabolic diseases, and therapies for angina and heart failure. Experimentalists and modelers not trained in these models will have exposure through this review such that the nonintuitive and highly nonlinear behavior of coronary physiology can be understood from a different perspective. This survey highlights knowledge gaps, key challenges, future research directions, and novel paradigms in the modeling of coronary flow regulation.

A Tool to Select FES Parameters for chronic SCI

Functional electrical stimulation has been used in rehabilitation programs for patients with chronic spinal cord injury. When used correctly it is able to improve the well-being of patients. However, when the stimulus is not adequate it can accelerate the process of fatigue, reducing the time available for training the programmed motor activity. To optimize the configuration of the stimulatory parameters, we developed a tool capable of simulating the muscle strength performance in response to different stimulatory profiles. The tool was able to reproduce the behavior of motoneurons in chronic spinal cord injury and to estimate the muscular strength resulting from the application of different stimuli. We consider that this FES Simulator is a promising tool to design and simulate different profiles of electrical stimulation, optimizing the decision process of the stimulation parameters.

An assessment of KIR channel function in human cerebral arteries

In the rodent cerebral circulation, inward rectifying K⁺ (K_IR) channels set resting tone and the distance over which electrical phenomena spread along the arterial wall. The present study sought to translate these observations into human cerebral arteries obtained from resected brain tissue. Computational modeling and a conduction assay first defined the impact of K_IR channels on electrical communication; patch-clamp electrophysiology, quantitative PCR, and immunohistochemistry then characterized K_IR2.x channel expression/activity. In keeping with rodent observations, computer modeling highlighted that K_IR blockade should constrict cerebral arteries and attenuate electrical communication if functionally expressed. Surprisingly, Ba²⁺ (a K_IR channel inhibitor) had no effect on human cerebral arterial tone or intercellular conduction. In alignment with these observations, immunohistochemistry and patch-clamp electrophysiology revealed minimal K_IR channel expression/activity in both smooth muscle and endothelial cells. This absence may be reflective of chronic stress as dysphormic neurons, leukocyte infiltrate, and glial fibrillary acidic protein expression was notable in the epileptic cortex. In closing, K_IR2.x channel expression is limited in human cerebral arteries from patients with epilepsy and thus has little impact on resting tone or the spread of vasomotor responses. NEW & NOTEWORTHY K_IR2.x channels are expressed in rodent cerebral arterial smooth muscle and endothelial cells. As they are critical to setting membrane potential and the distance signals conduct, we sought to translate this work into humans. Surprisingly, K_IR2.x channel activity/expression was limited in human cerebral arteries, a paucity tied to chronic brain stress in the epileptic cortex. Without substantive expression, K_IR2.x channels were unable to govern arterial tone or conduction.

Biophysical properties of microvascular endothelium: Requirements for initiating and conducting electrical signals

OBJECTIVE

Electrical signaling along the endothelium underlies spreading vasodilation and blood flow control. We use mathematical modeling to determine the electrical properties of the endothelium and gain insight into the biophysical determinants of electrical conduction.

METHODS

Electrical conduction data along endothelial tubes (40 μm wide, 2.5 mm long) isolated from mouse skeletal muscle resistance arteries were analyzed using cable equations and a multicellular computational model.

RESULTS

Responses to intracellular current injection attenuate with an axial length constant (λ) of 1.2-1.4 mm. Data were fitted to estimate the axial (r_a ; 10.7 MΩ/mm) and membrane (r_m ; 14.5 MΩ∙mm) resistivities, EC membrane resistance (R_m ; 12 GΩ), and EC-EC coupling resistance (R_gj ; 4.5 MΩ) and predict that stimulation of ≥30 neighboring ECs is required to elicit 1 mV of hyperpolarization at distance = 2.5 mm. Opening Ca²⁺ -activated K⁺ channels (K_Ca ) along the endothelium reduced λ by up to 55%.

CONCLUSIONS

High R_m makes the endothelium sensitive to electrical stimuli and able to conduct these signals effectively. Whereas the activation of a group of ECs is required to initiate physiologically relevant hyperpolarization, this requirement is increased by myoendothelial coupling and K_Ca activation along the endothelium inhibits conduction by dissipating electrical signals.

Integrative model of coronary flow in anatomically based vasculature under myogenic, shear, and metabolic regulation

Coronary blood flow is regulated to match the oxygen demand of myocytes in the heart wall. Flow regulation is essential to meet the wide range of cardiac workload. The blood flows through a complex coronary vasculature of elastic vessels having nonlinear wall properties, under transmural heterogeneous myocardial extravascular loading. To date, there is no fully integrative flow analysis that incorporates global and local passive and flow control determinants. Here, we provide an integrative model of coronary flow regulation that considers the realistic asymmetric morphology of the coronary network, the dynamic myocardial loading on the vessels embedded in it, and the combined effects of local myogenic effect, local shear regulation, and conducted metabolic control driven by venous O₂ saturation level. The model predicts autoregulation (approximately constant flow over a wide range of coronary perfusion pressures), reduced heterogeneity of regulated flow, and presence of flow reserve, in agreement with experimental observations. Furthermore, the model shows that the metabolic and myogenic regulations play a primary role, whereas shear has a secondary one. Regulation was found to have a significant effect on the flow except under extreme (high and low) inlet pressures and metabolic demand. Novel outcomes of the model are that cyclic myocardial loading on coronary vessels enhances the coronary flow reserve except under low inlet perfusion pressure, increases the pressure range of effective autoregulation, and reduces the network flow in the absence of metabolic regulation. Collectively, these findings demonstrate the utility of the present biophysical model, which can be used to unravel the underlying mechanisms of coronary physiopathology.

KIR channels tune electrical communication in cerebral arteries

The conducted vasomotor response reflects electrical communication in the arterial wall and the distance signals spread is regulated by three factors including resident ion channels. This study defined the role of inward-rectifying K⁺ channels (K_IR) in governing electrical communication along hamster cerebral arteries. Focal KCl application induced a vasoconstriction that conducted robustly, indicative of electrical communication among cells. Inhibiting dominant K⁺ conductances had no attenuating effect, the exception being Ba²⁺ blockade of K_IR. Electrophysiology and Q-PCR analysis of smooth muscle cells revealed a Ba²⁺-sensitive K_IR current comprised of K_IR2.1/2.2 subunits. This current was surprisingly small and when incorporated into a model, failed to account for the observed changes in conduction. We theorized a second population of K_IR channels exist and consistent with this idea, a robust Ba²⁺-sensitive K_IR2.1/2.2 current was observed in endothelial cells. When both K_IR currents were incorporated into, and then inhibited in our model, conduction decay was substantive, aligning with experiments. Enhanced decay was ascribed to the rightward shift in membrane potential and the increased feedback arising from voltage-dependent-K⁺ channels. In summary, this study shows that two K_IR populations work collaboratively to govern electrical communication and the spread of vasomotor responses along cerebral arteries.

A generative modeling approach to connectivity-Electrical conduction in vascular networks

The physiology of biological structures is inherently dynamic and emerges from the interaction and assembly of large collections of small entities. The extent of coupled entities complicates modeling and increases computational load. Here, microvascular networks are used to present a novel generative approach to connectivity based on the observation that biological organization is hierarchical and composed of a limited set of building blocks, i.e. a vascular network consists of blood vessels which in turn are composed by one or more cell types. Fast electrical communication is crucial to synchronize vessel tone across the vast distances within a network. We hypothesize that electrical conduction capacity is delimited by the size of vascular structures and connectivity of the network. Generation and simulation of series of dynamical models of electrical spread within vascular networks of different size and composition showed that (1) Conduction is enhanced in models harboring long and thin endothelial cells that couple preferentially along the longitudinal axis. (2) Conduction across a branch point depends on endothelial connectivity between branches. (3) Low connectivity sub-networks are more sensitive to electrical perturbations. In summary, the capacity for electrical signaling in microvascular networks is strongly shaped by the morphology and connectivity of vascular (particularly endothelial) cells. While the presented software can be used by itself or as a starting point for more sophisticated models of vascular dynamics, the generative approach can be applied to other biological systems, e.g. nervous tissue, the lymphatics, or the biliary system.

Gap junctions suppress electrical but not [Ca(2+)] heterogeneity in resistance arteries

Despite stochastic variation in the molecular composition and morphology of individual smooth muscle and endothelial cells, the membrane potential along intact microvessels is remarkably uniform. This is crucial for coordinated vasomotor responses. To investigate how this electrical homogeneity arises, a virtual arteriole was developed that introduces variation in the activities of ion-transport proteins between cells. By varying the level of heterogeneity and subpopulations of gap junctions (GJs), the resulting simulations shows that GJs suppress electrical variation but can only reduce cytosolic [Ca(2+)] variation. The process of electrical smoothing, however, introduces an energetic cost due to permanent currents, one which is proportional to the level of heterogeneity. This cost is particularly large when electrochemically different endothelial-cell and smooth-muscle-cell layers are coupled. Collectively, we show that homocellular GJs in a passively open state are crucial for electrical uniformity within the given cell layer, but homogenization may be limited by biophysical or energetic constraints. Owing to the ubiquitous presence of ion transport-proteins and cell-cell heterogeneity in biological tissues, these findings generalize across most biological fields.

Conducted vasoreactivity: the dynamical point of view

Conducted vasodilation is part of the physiological response to increasing metabolic demand of the tissue. Similar responses can be elicited by focal electrical or chemical stimulation. Some evidence suggests an endothelial pathway for nondecremental transmission of hyperpolarizing pulses. However, the underlying mechanisms are debated. Here, we focus on dynamical aspects of the problem hypothesizing the existence of a bistability-powered mechanism for regenerative pulse transmission along the endothelium. Bistability implies that the cell can have two different stable resting potentials and can switch between those states following an appropriate stimulus. Bistability is possible if the current–voltage curve is N shaped instead of monotonically increasing. Specifically, the presence of an inwardly rectifying potassium current may provide the endothelial cell with such properties. We provide a theoretical analysis as well as numerical simulations of both single- and multiunit bistable systems mimicking endothelial cells to investigate the self-consistence and stability of the proposed mechanism. We find that the individual cell may switch readily between two stable potentials. An array of coupled cells, however, as found in the vascular wall, requires a certain adaptation of the membrane currents after a switch, in order to switch back. Although the formulation is generic, we suggest a combination of specific membrane currents that could underlie the phenomenon.

Origins of variation in conducted vasomotor responses

Regulation of blood flow in the microcirculation depends on synchronized vasomotor responses. The vascular conducted response is a synchronous dilatation or constriction, elicited by a local electrical event that spreads along the vessel wall. Despite the underlying electrical nature, however, the efficacy of conducted responses varies significantly between different initiating stimuli within the same vascular bed as well as between different vascular beds following the same stimulus. The differences have stimulated proposals of different mechanisms to account for the experimentally observed variation. Using a computational approach that allows for introduction of structural and electrophysiological heterogeneity, we systematically tested variations in both arteriolar electrophysiology and modes of stimuli. Within the same vessel, our simulations show that conduction efficacy is influenced by the type of cell being stimulated and, in case of depolarization, by the stimulation strength. Particularly, simultaneous stimulation of both endothelial and vascular smooth muscle cells augments conduction. Between vessels, the specific electrophysiology determines membrane resistance and conduction efficiency-notably depolarization or radial currents reduce electrical spread. Random cell-cell variation, ubiquitous in biological systems, only cause small or no reduction in conduction efficiency. Collectively, our simulations can explain why CVRs from hyperpolarizing stimuli tend to conduct longer than CVRs from depolarizing stimuli and why agonists like acetylcholine induce CVRs that tend to conduct longer than electrical injections. The findings demonstrate that although substantial heterogeneity is observed in conducted responses, it can be largely ascribed to the origin of electrical stimulus combined with the specific electrophysiological properties of the arteriole. We conclude by outlining a set of "principles of electrical conduction" in the microcirculation.

Less is more: minimal expression of myoendothelial gap junctions optimizes cell-cell communication in virtual arterioles

Dysfunctional electrical signalling within the arteriolar wall is a major cause of cardiovascular disease. The endothelial cell layer constitutes the primary electrical pathway, co-ordinating contraction of the overlying smooth muscle cell (SMC) layer. As myoendothelial gap junctions (MEGJs) provide direct contact between the cell layers, proper vasomotor responses are thought to depend on a high, uniform MEGJ density. However, MEGJs are observed to be expressed heterogeneously within and among vascular beds. This discrepancy is addressed in the present study. As no direct measures of MEGJ conductance exist, we employed a computational modelling approach to vary the number, conductance and distribution of MEGJs. Our simulations demonstrate that a minimal number of randomly distributed MEGJs augment arteriolar cell-cell communication by increasing conduction efficiency and ensuring appropriate membrane potential responses in SMCs. We show that electrical coupling between SMCs must be tailored to the particular MEGJ distribution. Finally, observation of non-decaying mechanical conduction in arterioles without regeneration has been a long-standing controversy in the microvascular field. As heterogeneous MEGJ distributions provide for different conduction profiles along the cell layers, we demonstrate that a non-decaying conduction profile is possible in the SMC layer of a vessel with passive electrical properties. These intriguing findings redefine the concept of efficient electrical communication in the microcirculation, illustrating how heterogeneous properties, ubiquitous in biological systems, may have a profound impact on system behaviour and how acute local and global flow control is explained from the biophysical foundations.

Programming strategy for efficient modeling of dynamics in a population of heterogeneous cells

MOTIVATION

Heterogeneity is a ubiquitous property of biological systems. Even in a genetically identical population of a single cell type, cell-to-cell differences are observed. Although the functional behavior of a given population is generally robust, the consequences of heterogeneity are fairly unpredictable. In heterogeneous populations, synchronization of events becomes a cardinal problem-particularly for phase coherence in oscillating systems.

RESULTS

The present article presents a novel strategy for construction of large-scale simulation programs of heterogeneous biological entities. The strategy is designed to be tractable, to handle heterogeneity and to handle computational cost issues simultaneously, primarily by writing a generator of the 'model to be simulated'. We apply the strategy to model glycolytic oscillations among thousands of yeast cells coupled through the extracellular medium. The usefulness is illustrated through (i) benchmarking, showing an almost linear relationship between model size and run time, and (ii) analysis of the resulting simulations, showing that contrary to the experimental situation, synchronous oscillations are surprisingly hard to achieve, underpinning the need for tools to study heterogeneity. Thus, we present an efficient strategy to model the biological heterogeneity, neglected by ordinary mean-field models. This tool is well posed to facilitate the elucidation of the physiologically vital problem of synchronization.

AVAILABILITY

The complete python code is available as Supplementary Information.

CONTACT

bjornhald@gmail.com or pgs@kiku.dk

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

The vascular conducted response in cerebral blood flow regulation

Despite recent advances in our understanding of the molecular and cellular mechanisms behind vascular conducted responses (VCRs) in systemic arterioles, we still know very little about their potential physiological and pathophysiological role in brain penetrating arterioles controlling blood flow to the deeper areas of the brain. The scope of the present review is to present an overview of the conceptual, mechanistic, and physiological role of VCRs in resistance vessels, and to discuss in detail the recent advances in our knowledge of VCRs in brain arterioles controlling cerebral blood flow. We provide a schematic view of the ion channels and intercellular communication pathways necessary for conduction of an electrical and mechanical response in the arteriolar wall, and discuss the local signaling mechanisms and cellular pathway involved in the responses to different local stimuli and in different vascular beds. Physiological modulation of VCRs, which is a rather new finding in this field, is discussed in the light of changes in plasma membrane ion channel conductance as a function of health status or disease. Finally, we discuss the possible role of VCRs in cerebrovascular function and disease as well as suggest future directions for studying VCRs in the cerebral circulation.