Geometric Understanding of Local Fluctuation Distribution of Conduction Time in Lined-Up Cardiomyocyte Network in Agarose-Microfabrication Multi-Electrode Measurement Assay

We examined characteristics of the propagation of conduction in width-controlled cardiomyocyte cell networks for understanding the contribution of the geometrical arrangement of cardiomyocytes for their local fluctuation distribution. We tracked a series of extracellular field potentials of linearly lined-up human embryonic stem (ES) cell-derived cardiomyocytes and mouse primary cardiomyocytes with 100 kHz sampling intervals of multi-electrodes signal acquisitions and an agarose microfabrication technology to localize the cardiomyocyte geometries in the lined-up cell networks with 100–300 μm wide agarose microstructures. Conduction time between two neighbor microelectrodes (300 μm) showed Gaussian distribution. However, the distributions maintained their form regardless of its propagation distances up to 1.5 mm, meaning propagation diffusion did not occur. In contrast, when Quinidine was applied, the propagation time distributions were increased as the faster firing regulation simulation predicted. The results indicate the “faster firing regulation” is not sufficient to explain the conservation of the propagation time distribution in cardiomyocyte networks but should be expanded with a kind of community effect of cell networks, such as the lower fluctuation regulation.


Introduction
A biological cell network is composed of a group of chemically connected or functionally associated cells. In neuronal networks, a single neuron is linked to other neurons directly with elongated neurites like axons and dendrites. Hence, the conduction time was only dominated by the fast signal conduction of electrochemical impulse on the cell membrane of those neurites with more than a hundred m/s when myelinated, by crossing a synapse, where the impulse is converted from electrical to chemical and finally back to electrical [1][2][3].
In contrast to neuronal networks, propagation of firing in cardiomyocytes is thought to be a homogeneous and synchronized behavior to function the heart as blood pumping. However, when this synchronization is broken, heart function was damaged, such as ventricular arrhythmia.
Coordinated synchronous behavior of electrical conduction among cells was explained as the faster firing regulation of heart beating [4] or is called "overdrive suppression" [5]. This conduction regulation mechanism can also suppress the spontaneous beating of cells, such as Purkinje fibers, to follow the contraction impulse from the upstream sinoatrial node (SA node) with its faster beating intervals.
Massive mathematical models have also been proposed to investigate the mechanism of their beatings. One approach is an elaborated mathematical model composed of a large number of equations, each of which is reflecting the complex electrophysiological processes causing cardiomyocyte beating [6]. Another approach is a simple mathematical model using just a few ordinary equations, which are representing the key phenomenon of the membrane currents and action potentials [7,8], just the same as the famous Hodgkin-Huxley model, the FitzHugh-Nagumo model, and the Van der Pol model.
For investigating the statistical behavior of the beating and synchronization period of cardiomyocytes, we should remark the essence of beating intervals and their synchronization period from the idea that the cardiomyocytes as oscillators. From this viewpoint, starting from the phase model is one of the suitable ways for the construction of the beating interval model [9][10][11]. However, to capture the characteristic features of cardiomyocyte beating, we have to consider and incorporate the conventional stochastic phase models with three essential ideas: irreversible at firing, a refractory period after firing, and induced pulsation associated with the firing of surrounding cells. A part of those ideas was considered in the well-known integrate-and-fire model as a spiking neuron model [12][13][14].
Recently, as a practical application, fluctuation measurement of cell-to-cell conduction has been extensively examined as quasi-in vivo cardiotoxicity measurement assay and found that such conduction measurement can give us a more precise prediction of cardiotoxicity than the conventional in vitro screening assays, like human Ether-a-go-go Related Gene (hERG). assay [15][16][17][18]. Although those cell-to-cell conduction prediction measurements rely on the change of those fluctuations, the propagation manner of electrical conduction from perspectives of traveling distance dependency of fluctuation has not been examined well.
In this study, we examined the traveling distance dependence of conduction time fluctuation. The results showed the conduction time distribution was maintained constant regardless of the propagation distance, which was against the conventional conduction rule "overdrive suppression" only. And this conservation of conduction time distribution was disappeared when Quinidine was applied into the cardiomyocyte networks. The knowledge of length dependence of fluctuation can give us not only about the minimum requirement of cell-to-cell conduction length for reliable cardiotoxicity analysis but also about an insight into the origin of abnormality in conductance in cellular networks.

Embryonic Mouse Primary Cardiomyocytes
Embryonic mouse primary cardiomyocytes (primary) were isolated and purified from 14-day-old Crl:CD1(ICR) mouse embryos using a modified version of a method described in our previous reports [19,20]. All animal protocols and experiments were approved by the animal examination committee of the Institutional of Waseda University (Approval Number: 2019-A072). In brief, the embryos were rapidly removed from pregnant mice (Tokyo Laboratory Animals Science Co., Ltd., Tokyo, Japan) anesthetized with isoflurane (<2%, Wako Pure Chemical Industries, Osaka, Japan), which was volatilized by a vaporizer (Natsume Seisakusho Co., Ltd., Tokyo, Japan). The hearts of the embryos were obtained by trimming their embryos with tweezers and scissors and washed with phosphate-buffered saline (PBS, Takara Bio Inc., Shiga, Japan) containing 0.9 mM CaCl 2 and 0.5 mM MgCl 2 to induce heart contraction and remove corpuscles cells. The hearts were then transferred to PBS without CaCl 2 , and MgCl 2 and the ventricles were separated from the atria, minced into 1 mm 3 pieces with scissors. After that, they were incubated at 37 • C for 30 min in PBS containing 0.2% collagenase (Wako Pure Chemical Industries, Osaka, Japan) to digest the ventricular tissue. After this digestion step was repeated twice, the cell suspension was transferred to primary cultivation buffer (Dulbecco's modified Eagle's medium (DMEM, Invitrogen, Carlsbad, CA, USA) supplemented with 10% heat-inactivated fetal bovine serum (FBS, Invitrogen, Carlsbad, CA, USA), 1% penicillin-streptomycin (Invitrogen, Carlsbad, CA, USA) at 37 • C. In the subsequent experiments, the above primary cultivation medium was used to handle embryonic mouse primary cardiomyocytes. The cells were filtered through a 40 µm-nylon mesh cell strainer (BD Bioscience, Franklin Lakes, NJ, USA) to remove debris that was not able to be digested and then centrifuged at 200 g for 5 min at room temperature. After the precipitation of the cells was resuspended gently in the primary cultivation buffer, the cells were cultivated.

Agarose Microfabrication
The agarose microstructures on the multi-electrode array chip (MEA chip, MED-P530A, AlphaMED scientific Co.Ltd., Osaka, Japan) were prepared as follows. First, the MEA chip was hydrophilized with a plasma ion source device (PIB-20, VACUUM DEVICE, Ibaraki, Japan). Then, the MEA chip was coated with collagen (Cellmatrix Type I-C, KP-4100, Nitta Gelatin Inc., Osaka, Japan) diluted ten times with 1 mM HCL. After drying for 30 min, the MEA chip was covered with a 0.25% agarose (Agarose Low melting point analytical grade, v2111, Promega, Madison, WI, USA) with the spin coater (1H-D7, MIKASA, Tokyo, Japan). The covered thin agarose layer was melted by the spot heating of 1480 nm focused infrared laser (RLM-1-1480, IPG Laser GmBH, Oxford, MA, USA) to form the line patterns with desired width (100, 200, and 300 µm) were fabricated.

Cell Culture
After the pre-cultivation of primary and hES on collagen-coated dishes for cell purification and concentration control, the cardiomyocyte networks were formed in the agarose microstructures on the MEA chips. For the pre-cultivation, 35-mm tissue culture dishes were filled with 200 µL of collagen diluted ten times with 1 mM HCL. After drying the collagen solution, the dishes were washed with the cultivation buffer. Then, 2 mL of 1.0 × 10 5 cells/mL of cardiomyocyte suspensions were placed on the prepared collagen-coated 35-mm dishes for three days at 37 • C in 5% CO 2 . For cell collection, 1 mL of 0.25% trypsin-ethylenediaminetetraacetic acid (EDTA). were added to the dishes and incubated for 5 min at 37 • C in 5% CO 2 . Then, the solutions in the dishes were collected and centrifuged at 200 g for 5 min. After the aspiration of the supernatant, 3 mL of cultivation buffers were added as the stop solutions. After cell counting, the cells were centrifuged at 200 g for 5 min again, and the cultivation buffers were added to 3 × 10 6 cells/mL. Ten microliters of the cell suspensions were placed to each microstructure on the MEA chips as the droplets, and the MEA chips were incubated at 37 • C for 3 h. Then, 1 mL of the cultivation buffers were added mildly and exchanged after incubating for one day. The cells were cultivated for 5-7 days, while changing the medium once every two days.

Measurement System
Extracellular potentials were measured using a self-made 64-channel MEA system as described before [17,18]. The MEA system was set at a sampling rate of 100 kHz with a low path filter of 2 kHz and a high path filter of 10 Hz, and signals were amplified by 5000 using an analog amplifier. The MEA chips with cardiomyocyte networks were set in the chip holder of the field potential (FP) measurement device on the stage of the inversed optical microscopy (IX-71 with an x10 phase-contrast objective lens, UPLFLN10X2PH, OLYMPUS, Tokyo, Japan) and incubated at 37 • C in 5% CO 2 .

Data Analysis
The conduction time (CT) is defined as the lower peak interval of the extracellular filed potential spikes measured at each electrode. A histogram of CT between each electrode in the cell network was made from 5-min measurement, and a Gaussian fitting curve was made from the obtained mean time and standard deviation of time. The short term variability (STV) of CT is the mean distance of neighboring CTs calculated as: where CT n represents the CT of n-th beating.

Drug Administration
Dimethyl Sulfoxide (DMSO, Wako Pure Chemical Industries, Osaka, Japan) and Quinidine (Q3625-5G, Sigma-Aldrich Co. LLC., Tokyo, Japan) were purchased for drug measurements. Quinidine was dissolved in DMSO at a thousand-fold concentration to prepare stock solutions. The final concentration of Qinindine was the top critical point of its effective therapeutic concentration (C M ) in consideration of the solubility. The drug administration procedure was modified from previous experiments [15,17]. Briefly, the MEA chips of lined-up cells were selected by their beating frequency (0.6-1.1 Hz) and the waveforms of field potential (FP) recordings. The chip was placed in the holder of the on-chip MEA system and equilibrated for 5 min, and then the control FP waveforms were recorded for 10 min. Subsequently, the drug was applied to the medium at 0.1% (v/v) . dilution in serially increasing additions, and the FP waveforms were recorded for 10 min at each concentration. Finally, the drug-containing medium was replaced with the fresh medium after washing three times. The last 50 beats extracted from 10 min recorded FP waveform data was used for beat rate and CT (the time difference of the first peaks between separately placed electrodes in a lined-up cell-network).

Cell Staining
After measurement of extracellular potential of cardiomyocytes (CMs) , nuclei of CMs were stained with 4',6-Diamidino-2-phenylindole dihydrochloride (DAPI). First, the culture medium was removed from the MEA chip. After washing the MEA chip two times with Phosphate Buffered Saline (PBS), samples were incubated with 5 µL DAPI for 60 min at room temperature. These samples were examined using fluorescence microscopy (IX71, Olympus) with a charge-coupled device (CCD) camera (ORCA-ER, C4742-80-12AG, Hamamatsu Photonics, Shizuoka, Japan), and images were obtained using AQUACOSMOS software (Hamamatsu Photonics, Shizuoka, Japan).

Statistical Analysis
All values are presented as mean±standard deviation (S.D.) (unless stated otherwise). All the interbeat intervals (IBIs) of cells were evaluated using the F test when comparing multiple groups. p < 0.05 was considered as statistically significant. F test was done by R (Ver. 4.0.3., R Core Team, R Foundation for Statistical Computing, Vienna, Austria).

Distribution of Conduction Time in Width Controlled Cardiomyocyte Networks
We examined the characteristics of the propagation of excitation conduction in width-controlled cardiomyocyte networks. We measured the extracellular field potentials in the linearly lined-up mouse primary cardiomyocytes and human embryonic stem (hES) cell-derived cardiomyocytes with 100 kHz sampling intervals multi-electrodes signal acquisitions ( Figure 1A). The width of lined-up cardiomyocyte networks was controlled geometrically by the rectangular shapes of agarose microchambers fabricated with an agarose microfabrication technology ( Figure 1B). In this experiment, we constructed 100, 200, and 300 µm-width mouse primary cardiomyocyte networks (Figure 2Aa-Ca) and hES cell-derived cardiomyocyte networks (Figure 2Da-Fa). Each network was cultivated in the MEA chip for 5-7 days. Then, we obtained the waveforms of extracellular field potentials of cardiomyocytes on microelectrodes and measured the local conduction time in each network. Figure 2Ac-Fc show typical waveforms of each electrode in the control condition. The peak of the waveform was used for the timing of propagation at each electrode. Table 1 Figure 2Ad-Fd show the distance-dependent conduction time distributions between two microelectrodes from 300 µm distance to those multiples. For example, Figure 2Ad shows the propagation from the microelectrode 3 just after the pacemaker area (between microelectrodes 2 and 3) to the microelectrode 4 (blue histogram, 300 µm distance), 3 to 5 (orange histogram, 600 µm distance), and 3 to 6 (grey histogram, 900 µm distance) in 100 µm width primary cardiomyocyte networks. As shown in the graph, the Gaussian distribution was observed even the propagation lengths were increased from 300 µm to 600 µm and 900 µm. However, there was no significant difference in variance of conduction time of 300 µm distance (Figure 2Ad3      Conduction distance dependence of the average conduction time and standard deviation (S.D.) of primary and hES networks in 100, 200, and 300 µm width were also summarized in Figure 3Aa-f. As shown in these graphs, the mean conduction time increased in proportion to the conduction distance (a unit length d represents 300 µm, the distance of neighboring microelectrodes). In contrast, S.D. of conduction time did not diffuse (increase) or diffused slightly, while conduction unit length increased both in primary and hES (Figure 3Aa-f). These results suggest that the fluctuation of conduction time is maintained regardless of conduction unit length, or at least tends to preserve their fluctuation constant regardless of conduction distance. This tendency seems to be enhanced when the cell network width decreased in primary cardiomyocytes and increased in hES cardiomyocytes in our results. These results also indicate that the wider contaminated cell network will reduce their coordinated propagation ability, and the wider homogeneous cell network can enhance their coordinated propagation manner.

Effect of Fast Inward-Sodium-Current Blocking on the Distribution of Conduction Time
Next, we administered Quinidine to hES cardiomyocyte networks according to the time-course of drug administration procedure ( Figure 1C). Quinidine is a class I antiarrhythmic agent (Ia) in patients with short QT syndrome or Brugada syndrome with 'use-dependent block (the block increases at higher heart rates, while the block decreases at lower heart rates)' of voltage-gated sodium channels and is also well known for severe side effects, such as ventricular arrhythmias (the therapeutic concentration range, C M = 3.8 µm -10.2 µm [21][22][23]. Hence, to control (reduce) the cell-to-cell conduction efficiency by blocking the fast inward sodium current, Quinidine was applied into the cardiomyocyte networks.
After setting to the measurement system, cells in the MEA chip were placed in the cultivation medium for 30 min ('wash' in Figure 1C). Then, the blank medium (DMSO diluted into the cultivation medium with 0.1 % (v/v), 'Ct' in Figure 1C) was applied and recorded the waveforms five min after the administration. Then, 3 µm (lowest critical point of C M ) and 9 µm (highest critical point of C M ) Quinidine were applied and recorded the waveforms for five min after each administration ('a1' and 'a2' in Figure 1C, respectively).
As shown in Figures 2Dd-f and 3Ad,g,j, the average conduction time in 100 µm width increased with conduction distance regardless of administration concentration concentration of Quinidine. The same tendency was observed in 200 µm (Figures 2Ed-f and 3Ae,h,k) and 300 µm (Figures 2Fd-f and 3Af,i,l)   To check the internal correlation in those temporal fluctuations, the short-term variability (STV) between the endpoints of the conduction section was also calculated (Figure 3Am). The STVs of three width hES networks in control and 3 µm Quinidine did not show apparent differences. However, in 9 µm Quinidine, the STV increased significantly especially in 100 µm width network (138.2%, 52.2%, and 51.2% compared to control in 100 µm width, 200 µm width, and 300 µm width network, respectively). The results suggest that the ability of cell-to-cell correlation for synchronous beating decreased significantly, especially in 100 µm width in 9 µm Quinidine cell networks.

Correlation of Beating Intervals and Conduction Time
The relationship between interbeat intervals (IBIs) and conduction time (CT) in hES cardiomyocyte networks was also examined ( Figure 3B). There was no apparent correlation between the fluctuation of IBIs and conduction times in all the width of hES cardiomyocyte network without Quinidine (Figure 3Ba-c), and also in all of the width of hES cardiomyocyte network with 3 µm Quinidine (Figure 3Bd-f). Besides, two groups of plots were observed, and there was no strong correlation between IBIs and conduction time in all the width of hES cardiomyocyte networks with 9 µm Quinidine (Figure 3Bg-i). Those results indicate that the conduction times were independent of those IBIs. Hence, the fluctuation of conduction times was not caused by the instability of IBIs. It should be noted that the IBI dependence of QT length change was well known as Bazett and Fridericia corrections [24,25]. However, the conduction time did not show such an apparent relationship with IBIs.
Two groups of IBI-dependent distributions were also observed in 200 µm and 300 µm width samples in 9 µm Quinidine (Figure 3Bh,i). The faster IBI groups' conduction time distribution was wider than the slower groups' distribution. It might reflect the 'use-dependent block' function of Quinidine. Even in the 100 µm-order conduction propagation, the faster beating cardiomyocyte networks caused more blocking of fast inward-sodium ion channels. Hence, the larger fluctuation of conduction time was observed in the faster beating cardiomyocyte networks.

Correlation of Conduction Time and Its Fluctuation between Neighboring Units
The relationship of conduction times among neighboring unit sections in hES cardiomyocyte network was examined ( Figure 3C). In control hES cardiomyocyte network in 200-300 µm width, there was no a significant correlation in CTs between neighboring units (Figure 3Cb-c, R = 0.004 and 0.103, respectively). However, there was an apparent correlation in 100 µm width (Figure 3Ca, R = 0.723). It is consistent with the results shown in Figure 3Ad-f, in which the distribution of their fluctuation of conduction time (S.D.) was not increased even though the conduction lengths were increased. In contrast, when 3 µm Quinidine was applied to hES cardiomyocyte networks, the correlation of conduction times between the adjacent units was appeared or significantly improved (Figure 3Cd-f, R = 0.860, 0.249, and 0.766 for 100 µm, 200 µm, and 300 µm width, respectively). The appearance of those direct correlations in conduction time among unit lengths increases fluctuation of conduction time, which means the conduction distance-dependent increase of S.D. appeared as shown in Figure 3Ag-i. It indicates that the contribution of 3 µm Quinidine was the improvement of cooperatively of conduction between neighboring units and hence the conduction length-depended conduction time fluctuation increase appeared. This result also may explain the reason why Quinidine is the class I antiarrhythmic agent in the therapeutic concentration range. Moreover, when 9 µm Quinidine was introduced to hES cardiomyocyte networks, the correlation of conduction times was dispersedly disappeared (Figure 3Cg-i, R = 0.027, 0.145 and 0.001 in 100 µm, 200 µm, and 300 µm width, respectively)). Hence, the significant increase of S.D. in Figure 3Aj-l should be caused by the disappearance of their correlations and just by the increase of individual fluctuations. The side effect of Quinidine, such as ventricular arrhythmias, also might be explained by this significant increase of S.D. in a higher concentration of Quinidine.

Influence of Cell Density for Conduction Velocity in Unit Length
In this experiment, we cultivated cardiomyocyte networks in the rectangle microchambers having different widths (Figure 2Aa-Fa). As described above, we controlled the concentration of cells 1 × 10 3 -3 × 10 3 cells/mm 2 to form the homogeneous monolayered two-dimensional (2D) sheet of cells in the rectangular microchambers. Figure 4 shows the cell concentration dependence of unit length conduction velocity in 100 µm, 200 µm, and 300 µm width primary (A) and hES (B) networks. Both plots showed weak correlations between cell density and conduction velocity in primary and hES cardiomyocytes (correlation coefficient = 0.39 in primary and 0.31 in hES). In both cell types, the fluctuation (S.D.) of conduction velocity was large when conduction velocity was large. In addition, when cell density and conduction velocity was lower, the conduction velocity was stable. However, in primary cardiomyocytes, the fluctuation was large when conduction velocity was small in low cell density.

Influence of Cell Density for Conduction Velocity in Unit Length
In this experiment, we cultivated cardiomyocyte networks in the rectangle microchambers having different widths (Figure 2Aa-Fa). As described above, we controlled the concentration of cells 1 × 10 3 −3 × 10 3 cells/mm 2 to form the homogeneous monolayered two-dimensional (2D) sheet of cells in the rectangular microchambers. Figure 4 shows the cell concentration dependence of unit length conduction velocity in 100 µm, 200 µm, and 300 µm width primary (A) and hES (B) networks. Both plots showed weak correlations between cell density and conduction velocity in primary and hES cardiomyocytes (correlation coefficient = 0.39 in primary and 0.31 in hES). In both cell types, the fluctuation (S.D.) of conduction velocity was large when conduction velocity was large. In addition" when cell density and conduction velocity was lower, the conduction velocity was stable. However, in primary cardiomyocytes, the fluctuation was large when conduction velocity was small in low cell density.

Can the Faster Firing Regulation Explain These Conduction Characteristics?
When we assume the origin of cell-to-cell conduction was caused only by the faster firing regulation as a probabilistic one-way propagation model, traveling distance dependence of conduction time was increased gradually and showed the Gaussian distributions. For example, if the distribution of propagation time in the unit length (d = 300 µm) can be described as a Gaussian distribution (3.3 s as mean propagation time µ 0 , and 0.36 s as S.D. of propagation time σ 0 for unit length), both the mean value and S.D. of propagation time increased as the propagation distance increased. Figure 5 shows the numeric simulation of propagation time distribution for 10 4 samples based on the one-way random walk model. As shown in the graphs, the propagation distance dependence of the mean value µ t (m) and S.D. of propagation time σ t (m) are also described as: where m is the number of unit lengths for propagation, i.e., m × d is the propagation distance.

Can the Faster Firing Regulation Explain These Conduction Characteristics?
When we assume the origin of cell-to-cell conduction was caused only by the faster firing regulation as a probabilistic one-way propagation model, traveling distance dependence of conduction time was increased gradually and showed the Gaussian distributions. For example, if the distribution of propagation time in the unit length (d = 300 µm) can be described as a Gaussian distribution (3.3 s as mean propagation time µ 0 , and 0.36 s as S.D. of propagation time σ 0 for unit length), both the mean value and S.D. of propagation time increased as the propagation distance increased. Figure 5 shows the numeric simulation of propagation time distribution for 10 4 samples based on the one-way random walk model. As shown in the graphs, the propagation distance dependence of the mean value µ t (m) and S.D. of propagation time σ t (m) are also described as: where m is the number of unit lengths for propagation, i.e., m × d is the propagation distance. When we applied 3 µM Quinidine for sodium-ion channel blocking (Figure 3Ag-i), S.D. of conduction time was increased as the propagation distance increase. That might indicate that the cell-to-cell conduction of fast inward-sodium-current was not only caused by the probabilistic faster-firing regulation but also by some more cooperative contribution of cell-to-cell networks to maintain the lower constant S.D. distributions regardless of the propagation distances. In other words, it might indicate that, when the ability of fast inward-sodium-current regulation decreased, the fluctuation of conduction propagation increases.
In our previous studies, we also found that the faster firing regulation was not sufficient for understanding the synchronous behavior of spontaneously beating cardiomyocyte networks and proposed the lower fluctuation regulation rule to explain their beating synchronization tendencies [20]. The fluctuation-dissipation simulation model was proposed to explain the community effect of cardiomyocyte synchronization [26,27]. The results in Figure 3Ad-f showed that the wider the cell network width increases, the lower the fluctuation of conduction time, indicating the lower fluctuation regulation and fluctuation-dissipation model can explain a part of these experimental results. However, just the same as the faster firing model, our lower fluctuation regulation model only considered neighboring cells. It is still difficult to explain why the fluctuation was not increased during conduction.
Recently Bub's group proposed a cellular automaton model to explain the origin of spiral formation in ventricles [28][29][30] based on the Greenberg-Hastings model [31], in which they assumed the firing of individual cells were determined by the summation of all the surrounding cells within the radius r. When the radius r was sufficiently large, that means the correlated total number of cells is sufficient, the distribution did not widen and maintained their width regardless of propagation distance. Experimental results in 3 µM Quinidine might indicate the importance of long-range communication of cells. It can be explained by reducing the radius r caused by blocking of sodium ion channels. If we can consider such long-range communication of surrounding cells for the decision of individual cells, we may understand those dynamics as the community effect of cell networks.

Conclusions
We reported the distance dependence of conduction time and conduction time fluctuation in linearly connected cardiomyocyte networks using agarose microstructures and the MEA device. The conservation of conduction time distribution was observed regardless of the distance in the experiments. In contrast, when Quinidine was applied to the network, the conduction When we applied 3 µm Quinidine for sodium-ion channel blocking (Figure 3Ag-i), S.D. of conduction time was increased as the propagation distance increase. That might indicate that the cell-to-cell conduction of fast inward-sodium-current was not only caused by the probabilistic faster-firing regulation but also by some more cooperative contribution of cell-to-cell networks to maintain the lower constant S.D. distributions regardless of the propagation distances. In other words, it might indicate that, when the ability of fast inward-sodium-current regulation decreased, the fluctuation of conduction propagation increases.
In our previous studies, we also found that the faster firing regulation was not sufficient for understanding the synchronous behavior of spontaneously beating cardiomyocyte networks and proposed the lower fluctuation regulation rule to explain their beating synchronization tendencies [20]. The fluctuation-dissipation simulation model was proposed to explain the community effect of cardiomyocyte synchronization [26,27]. The results in Figure 3Ad-f showed that the wider the cell network width increases, the lower the fluctuation of conduction time, indicating the lower fluctuation regulation and fluctuation-dissipation model can explain a part of these experimental results. However, just the same as the faster firing model, our lower fluctuation regulation model only considered neighboring cells. It is still difficult to explain why the fluctuation was not increased during conduction.
Recently Bub's group proposed a cellular automaton model to explain the origin of spiral formation in ventricles [28][29][30] based on the Greenberg-Hastings model [31], in which they assumed the firing of individual cells were determined by the summation of all the surrounding cells within the radius r. When the radius r was sufficiently large, that means the correlated total number of cells is sufficient, the distribution did not widen and maintained their width regardless of propagation distance. Experimental results in 3 µm Quinidine might indicate the importance of long-range communication of cells. It can be explained by reducing the radius r caused by blocking of sodium ion channels. If we can consider such long-range communication of surrounding cells for the decision of individual cells, we may understand those dynamics as the community effect of cell networks.

Conclusions
We reported the distance dependence of conduction time and conduction time fluctuation in linearly connected cardiomyocyte networks using agarose microstructures and the MEA device. The conservation of conduction time distribution was observed regardless of the distance in the experiments. In contrast, when Quinidine was applied to the network, the conduction distance-dependent conduction time distribution increase was observed. The origin and mechanism of this conservation phenomenon are not exact yet. However, faster firing regulation is not sufficient to explain this phenomenon.