| (A) Microscopic view of cardiac muscle cells (longitudinal section) stained with hematoxylin (nuclei) and eosin (cytoplasm). The myocardial cells can be recognized by their elongated aspect (50-100 µm long, with sections ∼ 10µm), a longitudinal striated organization, multiple branchings and connections at their extremities via intercalated disks (IDs). (B) Schematic GJCs at the onset of depolarization. Depolarizing (resp. polarized) cells are stained in red (resp. blue). Surface charges are depicted with ± symbols, on the inner and outer side of the depolarizing (gray) and polarized (black) membranes and on the dysfunctional GJCs (red) with capacitive charge loading Q g such that Q g = ρ g dx. Voltage-gated channel ionic flows are marked with green vertical arrows. Normal GJCs: red resistor symbol, dysfunctional GJCs: gray capacitor symbol. Total currents I (blue horizontal arrow) circulate in opposite directions inside and outside the cell, splitting into the normally flowing ohmic current I (red), and residual closed GJCs current I g (dashed red) building up charge. Typical membrane and intercalated disk dimensions are l m ≈ l g ∼ 5 nm.

Contexts in source publication

Context 1

... us consider an idealized elongated fiber of excitable cells (Figure 1), along which a traveling depolarization front (AP upstroke) is classically modeled by a 1D cable equation, assuming Kirchhoff 's law of conservation of currents ( Plonsey and Barr, 2007;Niebur, 2008;Macfarlane et al., 2011): ...

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... takes a time τ d ∼ 1 to 10 ms to depolarize one cell membrane. Actually, τ d acts as a cut-off time scale for continuous models as Equation (2) Figure 1B), which acts as a spatial cut-off scale in such continuous models. Note that for very slow conduction velocity and rapid upstroke, the upstroke length scale can decrease down to the longitudinal size of one cell. ...

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... cardiac cell-cell contacts, where electrical signal conduction occurs, are found at intercalated discs located mostly at the narrow end of elongated cardiomyocytes. On their lateral side, the cardiomyocytes are ensheathed by cell-matrix contacts (Figure 1) with weaker electrical coupling. This organization favors a synchronized unidirectional propagation of electrical signals through serial strands of cardiomyocytes. ...

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... the local ohmic current flowing through open GJCs experiences losses due to closed GJCs. Thus, adding the GJC current as I g = c −1 g gρ g , to the unpertubed conduction current I = − κ ∂ ∂x U m , gives a total longitudinal current I = I + I g ( Figure 1B). From charge conservation hypothesis, taking the divergence of I yields Equation (6a). ...

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... 2 shows typical pseudo bipolar potential time series numerically simulated with our 1D system of PDEs (Equation 6) with, as boundary condition at x = 0 (Equation 7), an automatically beating source of frequency αγ ∼ 5 Hz so as to match the cardiac pulse trains observed experimentally. Tuning the newly introduced parameters ω 2 and ν 1 in Equation (6), we have found quite easily paths leading from a phase of coherent propagation of AP pulses to a phase of quite incoherent and intermittent electrical activity (Figure 2) that strongly reminds the very irregular behavior of electric potential time series recorded during AF (see for comparison Figure 1 in our companion paper I Attuel et al., 2017). Besides the obvious interest of analyzing the succession of bifurcations and transition events encountered along these paths in parameter space, we will focus in this paper on a comparative study of the complex and highly intermittent modulation of cardiac pulse trains simulated numerically with our model of cardiac AP conduction and GJC dynamics and the one observed experimentally in the coronary sinus during episodes of AF ( Attuel et al., 2017). ...

Context 6

... n ψ > h(t 0 ), where h(t 0 ) is the point-wise Hölder exponent that characterizes the maximum regularity of the signal E at point t 0 . As experienced in the companion paper I (Attuel et al., 2017) for experimental local impulse energy time-series, we will use in this work the third derivative of a Gaussian function as analyzing wavelet with n ψ = 3 (Muzy et al., 1994; Arneodo et al., 1995b) ( Figure S1 in the companion paper I): ...

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... the proportionality coefficient c 2 is the intermittency coefficient defined in Equations (21) and (22) [Note that C(a, t = 0) ≡ C 2 (a) ∼ −c 2 ln a]. Thus, by computing C(a, t) from Equation (25) and plotting it as a function of ln t, inferences can be made about long-range dependence and consistency with a multiplicative cascading process (Arneodo et al., 1998a,b). Applications of the two-point magnitude correlation method have already provided insight into a wide variety of problems, e.g., the validation of the log-normal cascade phenomenology of fully developed turbulence ( Arneodo et al., 1998aArneodo et al., ,c, 1999) and of high resolution temporal rainfall ( Venugopal et al., 2006;Roux et al., 2009), and the demonstration of the existence of a causal cascade of information from large to small scales in financial time series (Arneodo et al., 1998d;Muzy et al., 2001). ...

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... results of the two-point magnitude correlation analysis of the local impulse energy time series numerically generated with our 1D PDE system (Equations 6-8) with the set of parameter values defined in Simul #2 are shown in Figure 10D), for time-lag t a, C(a, )/C(a, 0) drops to zero as a clear indication that the magnitudes are uncorrelated. As a reference, we put in each panel in Figure 10, a dashed straight line of slope −c 2 as predicted by Equation (26) for multifractal signals exhibiting a cascading multiplicative structure along a time-scale tree ( Arneodo et al., 1998a,b). ...

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... results of the two-point magnitude correlation analysis of the local impulse energy time series numerically generated with our 1D PDE system (Equations 6-8) with the set of parameter values defined in Simul #2 are shown in Figure 10D), for time-lag t a, C(a, )/C(a, 0) drops to zero as a clear indication that the magnitudes are uncorrelated. As a reference, we put in each panel in Figure 10, a dashed straight line of slope −c 2 as predicted by Equation (26) for multifractal signals exhibiting a cascading multiplicative structure along a time-scale tree ( Arneodo et al., 1998a,b). The slow decay predicted by the "multiplicative" lognormal model with intermittency coefficient c 2 is definitely (6)- (8) with the parameters defined in Simul #6 (Table 1) not observed. ...