## Abstract

Little is known about how individual cells sense the macroscopic geometry of their tissue environment. Here we explore whether long-range electrical signaling can convey information on tissue geometry to influence electrical dynamics of individual cells. First, we studied an engineered electrically excitable cell line where all voltage-gated channels were well characterized. Cells grown in patterned islands of different shapes showed remarkably diverse firing patterns under otherwise identical conditions, including regular spiking, period-doubling alternans, and arrhythmic firing. A Hodgkin-Huxley numerical model quantitatively reproduced these effects, showing how the macroscopic geometry affected the single-cell electrophysiology via the influence of gap junction-mediated electrical coupling. Qualitatively similar geometry dependent dynamics were experimentally observed in human induced pluripotent stem cell (iPSC)-derived cardiomyocytes. The cardiac results urge caution in translating observations of arrhythmia *in vitro* to predictions *in vivo* where the tissue geometry is very different. We present simulation results and scaling arguments which explore how to extrapolate electrophysiological measurements between tissues with different geometries and different gap junction couplings.

## Introduction

Cells in multicellular organisms sense their location within tissues via diffusible molecules, contact interactions, and mechanical signals. Gap junction-mediated electrical signals can also, in principle, provide long-range positional cues ^{1}, though mechanistic details have been difficult to determine due to the simultaneous presence of, and interactions between, all of the above signaling modalities in physiological tissue. Furthermore, until recently, technical limitations prevented tissue-scale mapping of membrane voltage: point-wise measurements with patch pipettes were slow and laborious, and voltage-sensitive dyes lacked sensitivity and suffered from phototoxicity.

The electrophysiological properties of many isolated cells have been probed in great detail via patch clamp electrophysiology.^{2} In tissues, cells form electrical connections with their neighbors via gap junction channels. One can then ask whether this coupling is a minor perturbation on the individual cells, or whether it fundamentally changes the dynamics. In condensed matter physics, the properties of a bulk solid can differ dramatically from those of its constituent atoms. Similarly, the emergent electrical properties of bulk tissue might differ dramatically from those of individual cells.

One aspect of positional sensing is the detection of boundaries. It has been well established that boundaries can influence paracrine signaling pathways^{3}, but it remains an open question to what extent tissue geometry and topology influence electrical signaling. In the heart, structural defects can act as nuclei for arrhythmias ^{4}, but interpreting these effects is difficult due to the multiple interacting factors that govern dynamics, including electrical coupling between myocytes ^{5-7}, mechano-electrical feedbacks ^{8, 9}, and differences in cell-autonomous properties of the individual myocytes between regions of the heart ^{10-12}. Subtle shifts in any of these parameters can cause discontinuous changes in dynamics, e.g. from a stable beat to a possibly fatal arrhythmia.

It is important to understand how long-range electrical signaling can convey positional cues generally, and how these signals govern cardiac stability specifically. Early models of cardiac dynamics established a stability criterion based solely on the cell-autonomous relation of spike width to beat rate.^{13} Recent theoretical work showed that conduction could dramatically alter the stability conditions.^{14, 15} However, the wide diversity of cardiac models, combined with uncertainty in model parameters, presents a challenge for comparison to experiments.^{15, 16} Only a few experiments have explicitly probed the roles of intercellular coupling in cardiac dynamics.^{17, 18} In complex cells such as cardiomyocytes, one typically cannot vary one parameter without affecting many others. For instance, growing hiPSC-CM on different size islands affects their patterns of gene expression ^{19}, which in turn can affect electrophysiology.

Uncertainties regarding the role of geometry in cardiac stability have an important practical implication: it has been widely claimed that if human induced pluripotent stem cell (iPSC)-derived cardiomyocytes (hiPSC-CM) can be made to show mature patterns of ion channel expression,^{20-23} then *in vitro* cultures will be a useful substrate for studying arrhythmias.^{24-27} However, this claim might need to be reconsidered if one finds that there are fundamental geometry-driven differences in stability between cultured cells and intact tissue, even when all voltage-dependent conductances are identical.

To explore the role of geometry under controlled conditions, we engineered a synthetic excitable tissue where all elements were well understood. This synthetic approach has the further merit of being amenable to rigorous quantitative modeling. We previously introduced Optopatch Spiking Human Embryonic Kidney (OS-HEK) cells^{28} as an engineered excitable cell type with an all-optical electrophysiological interface. The cells expressed just two voltage-dependent channels, the voltage-gated cardiac sodium channel, Na_{V} 1.5, and the inward rectifier potassium channel, K_{ir}2.1. Expression of a channelrhodopsin permitted optogenetic stimulation, and expression of a far-red voltage-sensitive protein (QuasAr2^{29}) or dye (BeRST1^{30}) reported electrical dynamics. When optogenetically stimulated, these cells produced single action potentials. When grown into a confluent syncytium, endogenous gap junctions coupled cells to their neighbors, supporting bulk propagation of electrical waves.

While the OS-HEK cells demonstrated many interesting attributes of excitable tissues (including wave conduction, curvature-dependent wavefront velocity, and re-entrant spiral waves^{28}), they did not show complex dynamical bifurcations (‘arrhythmias’) at fast pacing rates. Irregular and chaotic dynamics have not previously been observed in a synthetic bioelectrical system, suggesting that a necessary ingredient was missing.

Here we describe isradipine-OS-HEK (iOS-HEK) cells, a synthetic bioelectric system which shows dynamical transitions between regimes of regular pacing, complex but repeating patterns, and pseudo-chaotic (non-repeating) dynamics as the drive frequency is changed. We explore in detail how these stability regimes are influenced by the macroscopic tissue geometry. Remarkably, we found that the transitions to complex and irregular patterns depended sensitively on the culture geometry. At a single pacing frequency, we simultaneously observed regular rhythms, alternating patterns, chaos, or depolarization block in islands that were identical in all respects except for their geometry. A biophysically detailed Hodgkin Huxley-style model captured these geometric effects. The iOS-HEK cells further showed second-degree conduction block in regions of high wavefront curvature, demonstrating sensitivity to two-dimensional geometric features. Finally, we show that similar geometry-dependent transitions occur in cultured human iPSC-derived cardiomyocytes.

Together our findings show that macroscopic tissue geometry is a fundamental determinant of bioelectrical dynamics, and not just a perturbation on the cell-autonomous behavior. We discuss which parameters are sensitive or insensitive to tissue geometry, and propose scaling relations that can be used to extrapolate across tissue geometries and intercellular coupling strengths.

## Results

### iOS-HEK cells show alternans and arrhythmias

Our previous studies of OS-HEK cells did not show frequency-dependent stability transitions. We hypothesized that this was due to the absence of any slowly recovering conductances that could provide memory effects between excitations. Upon repolarization after a spike, OS-HEK cell ion channels recovered fully within 4 ms, so each beat was independent of its predecessors. The sodium channel blocker isradipine shows state-dependent block of Na_{V}1.7, with a recovery time of 200 ms at –100 mV.^{31} We reasoned that in the presence of isradipine, the Na_{V}1.5 channels in OS-HEK cells would show a similar slow recovery after a beat, introducing the possibility of complex temporal dynamics.

In a confluent monolayer of OS-HEK cells, we observed regular spiking when the cells were optogenetically paced at 4 Hz. Addition of isradipine (10 μM) converted the spiking to an alternating rhythm between large and small spikes (i.e., ‘alternans’, Fig. 1b), consistent with a previous report.^{31} We refer to the OS-HEK cells with 10 μM isradipine as iOS-HEK cells.

Using microcontact printing,^{32} we patterned cell-adhesive fibronectin features onto cytophobic polyacrylamide surfaces. Features comprised square islands of linear size 100 μm, 200 μm and 500 μm, as well as serpentine tracks of width 100, 200, 500 μm and edge length of 1 mm and 5 mm. Overall track lengths were as long as 7 cm. Cell growth followed the printed patterns (Fig. 1c). We used a digital micromirror device (DMD) to target blue light stimulation to specific regions of the sample. In a change from previous studies on OS-HEK cells, we used the far red dye BeRST1 ^{30} to report membrane voltage (Fig. 1a). This dye had superior brightness to the protein-based reporter, QuasAr2, and showed otherwise similar response. We recorded the voltage dynamics using a custom ultrawide-field ‘Firefly’ microscope.^{33}

Localized stimulation of the serpentine tracks induced propagating electrical waves (**Supplementary Movie 1**). Optogenetically induced waves propagated with a typical conduction velocity of 3.3 cm/s, had an AP width at 50% repolarization of 110 ms, and thus had a depolarized action potential length, *λ*, of *λ* = 3.6 mm. The fluorescence in the paced region of the track showed complex beat rate-dependent dynamics. At 2 Hz pacing, the cells spiked regularly and in phase with the drive (Fig. 1d, left). At 8.5 Hz pacing, the cells showed a complex and irregular fluorescence pattern, indicative of arrhythmia (Fig. 1d, right).

### Geometry dependent instabilities in iOS-HEK cells

We characterized in detail the dependence of the electrical dynamics on local geometry. Cells were grown either in small square ‘islands’ or on adjacent linear ‘tracks’, and paced simultaneously in both geometries at frequencies between 2 and 11 Hz. The islands were paced with spatially homogeneous illumination, so cells across each island spiked synchronously and gap junction-mediated conduction did not contribute to the dynamics. We refer to these as zero-dimensional (0D) dynamics.

Tracks were stimulated in small regions (200 μm wide, 100 μm long) to induce 1D propagating waves. We observed the response in both the directly stimulated region (near field) and in the conductively stimulated region (far field; Fig. 2a). Electrical waveforms stabilized to their far-field dynamics within a distance d ≈ 0.05 *λ* from the stimulus (d = 180 μm in our experiments), so we characterized the far-field dynamics at a distance 750 μm from the stimulus. Island and track features were intermixed within each dish, were seeded with the same stock of OS-HEK cells, and were measured simultaneously. Thus any differences in dynamics between regions could be ascribed to the island geometry.

At low pace frequencies (≤ 3 Hz), all three regions spiked with a stable rhythm in synchrony with the pacing (Fig. 2b). To our surprise, at higher pace frequencies we observed dramatically different dynamics in the three regions (**Supplementary Movie 2**). The 0D islands always produced regular electrical oscillations at the pace frequency. As the pace frequency increased, the amplitude of these oscillations diminished, up to 8 Hz, beyond which the responses were undetectable (Fig. 2).

The 1D near-field response developed a 1:1 alternans pattern at frequencies > 3 Hz. Additional bifurcations arose at 6 Hz and 8 Hz. At 10 Hz the dynamics no longer showed any repeating pattern, suggesting a transition to chaos (Fig. 2).

Remarkably, the 1D far-field showed no transitions to alternans or chaos (Fig. 2b). All spikes in the far-field had nearly equal amplitude and waveform. At pace frequencies between 3 and 6 Hz, only every other spike propagated to the far field, i.e. the local beat frequency was half the pace frequency. At higher pace frequencies, a smaller portion of spikes reached the far field. These spikes had irregular timing in the near-field but conducted at velocities which gradually evened out timing variations, such that the spiking appeared regular in the far-field, always at a frequency < 3 Hz. The 1D track acted as a filter which converted high-frequency arrhythmic spiking in the near-field into lower frequency rhythmic spiking in the far field. Together these experiments gave the unanticipated result that under regular pacing, iOS-HEK cells showed irregular dynamics only in the 1D near-field, not in the 0D or 1D far-field regions.

### A Hodgkin-Huxley model captures iOS-HEK dynamics

We developed a conductance-based Hodgkin Huxley-type model to simulate the dynamics of iOS-HEK cells. The properties of Na_{V}1.5, K_{ir}2.1, and channelrhodopsin CheRiff are all well known, so we were able to constrain the model with a small number of free parameters (Fig. 3a). The sodium channel model comprised Hodgkin-Huxley activation and inactivation gates *m* and *h* with dynamics taken from the literature.^{34, 35} To capture the use-dependent block by isradipine, we introduced an additional slowly activating and slowly recovering gate, *j* (Methods). The K_{ir}2.1 conductance was modeled as an instantaneous function of voltage, inferred from the shape of the action potential repolarization (Methods). The channelrhodopsin was modeled as a linear conductance with a 0 mV reversal potential, modulated in space and time by the blue light illumination. A diffusive term captured the nearest-neighbor gap junction coupling. The governing equation is:
where *C _{m}* is the membrane capacitance of a cell, and

*G*=

_{Cxn}*g*×

_{Cxn}*l*, where

^{2}*g*is the gap junction conductance between cells, and

_{Cxn}*l*is the linear dimension of a cell. The electrical diffusion coefficient is given by

*G*. To simulate 0D dynamics,

_{Cxn}/ C_{m}*g*was set to zero. Eq. 1 represents a continuum model, which treats the domain as homogeneous tissue. In simulations, the cell discreteness is recovered by setting the spatial discretization equal to cell length.

_{Cxn}The unknown parameters (*C _{m}, g_{Cxn}, g_{cKir}, g_{ChR}*) were determined by fitting to observed fluorescence waveforms, conduction velocities, and patch clamp measurements (Methods). The free parameters in the model enter in the dynamics of the isradipine variable,

*j,*which was assumed to bind sodium channels in their open state with rate

*α,*and unbind with rate

*μ*: where

*m*

_{∞}is the steady state value of the

*m*gate. The kinetic parameters in Eq. 2 were chosen to fit the experimentally observed dynamical transitions.

Action potential waveforms were simulated in a 0D geometry and in a linear 1D track comprising 2000 cells, with pacing delivered to 40 cells on one end. The near-field response was monitored in the paced zone, and the far-field response was monitored at a distance of 3 mm from the stimulus.

Simulated electrical waveforms (Fig. 3b) captured the main geometry and frequency-dependent features of the data. Specifically, the simulations in 0D showed a smooth and monotonic decrease in spike amplitude with increasing pace frequency. Simulations in the 1D near-field showed a series of frequency-dependent bifurcations that became increasingly irregular at high pace frequency. In the 1D far-field, these bifurcations were suppressed: spikes that propagated to the far-field had full amplitude, regular spacing, and were within the narrow band of frequencies that supported far-field propagation. These simulations confirm that cells with identical conductances, paced at the same frequency, can show widely divergent behavior depending upon the geometry of the surrounding excitable tissue.

### Mapping the transition between the near field and far field response

We next investigated the transition from near-field to far-field behavior. How does alternans, or even chaos, in the near-field lead to regular spiking in the far-field? We mapped the fluorescence dynamics of 1D tracks as a function of distance from the stimulated zone, across a range of stimulus frequencies (Fig. 4a,b). There was a clear bifurcation in the ability of spikes to propagate into the far-field. Near-field spikes with amplitude lower than a critical threshold decayed as a function of distance, while spikes above this threshold grew to become propagating far-field spikes. Spatially resolved simulations yielded similar results (Fig. 4d). The simulations showed that the conduction velocity of each far-field spike decreased as it approached the preceding spike such that irregularities spike spacing gradually evened out.

The propagation of spikes in the far-field is governed by the nonlinear dispersion relation, which defines the relationship between wavenumber and frequency (Supplement and Supplementary Fig. 3). The numerically simulated dispersion relation indicated that the far-field dynamics only supported frequencies up to f_{max} = 3.85 Hz. Waves of greater frequency could not propagate into the far field.

At pace frequencies just above the transition to near-field alternans (experimentally observed between 3 and 4 Hz), one might expect period-doubling deviations from a regular spike train to arise slowly. We defined an alternans decay length, d, as the distance over which a near-field alternans beat decayed to 50% of its initial height (Fig. 4e). Simulations near the alternans transition indeed revealed a divergence in *d* near the critical frequency. At pace frequencies far above the critical frequency, the alternans decay length became a small fraction of the far-field action potential length, d ≈ 0.04 *λ* (Fig. 4f), consistent with experimental results (Fig. 4c).

### Second-degree conduction block in iOS-HEK cells

Conduction block and consequent arrhythmias can arise when a region of the heart acts as a partial barrier to conduction.^{36} This effect can arise from spatial variations in either ion channel expression ^{37}, or gap junctional coupling.^{38, 39} We thus explored the effect of local defects on spike propagation in iOS-HEK cell tracks. In a serpentine track with sharp turns, we observed that stably propagating far-field waves sometimes failed at the turns (Fig. 5a, **Supplementary Movie 3**). Furthermore, failures occurred in a regular temporal sequence, e.g. Fig. 5a shows a pattern that after the first few beats stabilized into a 3:2 block (3 upstream spikes triggered 2 downstream spikes). The waves developed a curved wavefront and slowed repolarization at the turns, a purely geometrical consequence of the increased electrotonic loading associated with bending a wavefront around a corner. Thus geometrical effects alone are sufficient to cause conduction block, even in a background of homogeneous ion channel levels and gap junction strengths. A related effect has been reported in cultured cardiomyocytes, where a junction of a thin strand of cells to a large island showed unidirectional conduction block due to the inability of the thin strand to drive the large island.^{40}

In our experiments, the conduction block was attributable to a 2D wavefront curvature effect. To capture this effect in computationally tractable 1D simulations, we simulated linear tracks in which a small region (10 cells) had a reduced gap junctional coupling (*G’ _{Cxn} =G_{cxn}* / 3). The simulated waves showed a 1:1 conduction block (Fig. 5C), qualitatively similar to that observed experimentally. These experiments and simulations show that local perturbations in the electrotonic coupling, are sufficient to lead to second-degree conduction block.

### Geometry dependent instabilities in human iPSC cardiomyocytes

Finally, we explored whether the geometry-dependent effects observed in iOS-HEK cells also occurred in human iPSC-derived cardiomyocytes (hiPSC-CM). * ^{41}* Due to the significant commercial interest in using these cells as an

*in vitro*model for cardiotoxicity testing

^{24, 25, 42-44}, the correspondence (or lack of) between

*in vitro*and

*in vivo*arrhythmias has some practical importance.

We used microcontact printing to define side-by-side patterns of 0D islands and 1D tracks and then plated hiPSC-CM onto these patterns (Fig. 6b). CheRiff was expressed using a lentiviral vector to allow patterned blue light stimulation^{19}, and BeRST1 was used to image changes in membrane potential. We optogenetically paced 0D islands and 1D tracks simultaneously across a range of frequencies. In the 1D tracks, waves propagated with a conduction velocity of 7.2 cm/s and had an AP duration of 360 ms, corresponding to a depolarized action potential length of 2.6 cm.

As with the iOS-HEK cells, we observed that the dynamics depended strongly on the geometry (Fig. 6c, **Supplementary Movie 4**). At high pace frequency (3.3 Hz), the 0D islands showed small regular oscillations, the 1D near-field showed an erratic pattern of large beats with small oscillations superposed, and the 1D far-field only showed the large beats at a sub-harmonic of the pace frequency. As in the iOS-HEK cultures, the alternans beats decayed over a distance much less than the action potential length (Figure 6e; decay length *d* = 535 μm, corresponding to *d* = 0.2 *λ*). Thus the qualitative geometry-dependent behavior of the hiPSC-CM largely mirrored the behavior of the iOS-HEK cells.

The hiPSC-CM cells differed from the iOS-HEK cells in several important regards. First, the hiPSC-CM were spontaneously active, imposing a minimum on the optogenetic pace frequency. Second, the 0D hiPSC-CM islands showed a transition to alternans (Fig. 6c,d) which disappeared at high drive frequencies (3 Hz). Third, the 1D bifurcation to alternans was continuous in the iOS-HEK cells but discontinuous in the hiPSC-CM. In iOS-HEK simulations, these last two differences can be captured simply by tuning the isradipine unbinding rate (Fig. S1b). Simulations of the Noble model also showed clear geometry-dependent differences in the onset of instabilities (Fig. S1c).^{45} Finally, real cardiac tissue can support alternans in far-field traveling waves^{46}, while both the iOS-HEK cells and the hiPSC-CM seemed only to support this phenomenon in the near field. Thus there remain important dynamical features of real cardiac tissue which appear not to be captured by either the iOS-HEk cells or the hiPSC-CM.

## Discussion

Gap junction-mediated currents convey the effects of boundaries to all cells in the tissue. The highly simplified iOS-HEK ‘toy’ model revealed that the qualitative dynamics depend in a sensitive way on the overall geometry. Islands of composed of identical cells and differing only in geometry showed vastly different dynamics under identical pacing frequencies, including regular spiking, complex but repeating multi-spike patterns, and irregular spiking. The iOS-HEK system was simple enough to model with biophysically realistic numerical simulations, which confirmed that these observations could be explained by geometry alone. Moreover, the observation of similar geometry-dependent bifurcations in hiPSC-CM cultures suggests that similar principles apply to physiological tissues.

Given the importance of gap junction strength and sample geometry, it is interesting to ask whether one can scale both parameters to preserve the overall dynamics. Such scaling could be useful, for instance, in modeling *in vivo* cardiac dynamics in a cell culture system. We explored this question by systematically varying the gap junction strength in iOS-HEK simulations of a 1-D track. As anticipated from dimensional analysis, the far-field conduction velocity scaled as (Fig. S2a). The action potential duration (APD) was largely insensitive to *g _{cxn},* so the action potential length

*λ*scaled as (Fig. S2b). Together these results imply that scaling the size of a system by some factor

*k*and the gap junction strength by will preserve the overall dynamics.

The transition frequency for alternans, *f ^{*}*, was largely insensitive to

*g*varying by < 2% over a 100-fold change in

_{cxn},*g*(Fig. S2c), but

_{cxn}*f*depended sensitively on the dynamics of the slow recovery variable (Fig. S3). Thus the maximum stable frequency is a parameter that should be largely independent of gap junction strength, and by extension, tissue geometry—provided that the cells are paced via gap junction-mediated conduction rather than by direct pacing.

^{*}Can one extrapolate from *in vitro* measurements of small samples with relatively weak gap junction coupling to larger tissues with stronger gap junctional coupling? In iOS-HEK simulations, the alternans decay length, *d*, scaled almost linearly with *λ* (Fig. S2d). These simulations suggest that the action potential length, *λ*, sets a natural length scale for a bioelectric tissue. Moreover, using *λ* as a scaling parameter is advantageous since this parameter can be experimentally measured without knowledge of gap junction conductance or other system parameters.

These results have clear implications for ongoing efforts to predict cardiac stability *in vivo* from *in vitro* systems. Intact myocardium differs in many ways from cultured hiPSC-CM. However, even if one could create hiPSC-CM with fully adult single-cell properties, their behavior in a cultured syncytium would differ dramatically from their behavior in the intact heart. Thus one cannot directly infer *in vivo* behavior from *in vitro* measurements.

The conduction velocity in the atria and ventricles of the human heart *in vivo* is approximately 50 cm/s ^{47, 48}, and action potential durations are typically 350 ms, implying an action potential length of *λ* = 18 cm. An adult human heart is approximately *L* = 12 cm long, so L ≈ 0.7 *λ*. Pacemaker-triggered action potentials *in vivo* are thus primarily in the near-field and close far-field regimes. Remarkably, the near-field is the only regime in which we observed arrhythmias in either the iOS-HEK cells or in the hiPSC-CMs, either in experiment or in simulation.

In the cultured hiPSC-CM the action potential length was *λ* = 2.6 cm, so to best match the geometrical regime of the heart, the culture should have size *L* ≈ 0.7 *λ*, or ^{~}1.7 cm. Others have reported conduction velocities *in vitro* ranging from 3.5 to 20 cm/s,^{21, 33, 49} corresponding to *λ*= 1.2 to 7 cm (assuming a 350 ms AP width). While cultures of size *L* ≈ 0.7 *λ* are easily accessible on the smaller end of the *λ* range, 5 cm-wide hiPSC-CM cultures are impractical. This challenge can be addressed either by cell patterning to produce serpentine tracks, or by adding gap junction blockers such as 2-APB to diminish the strength of the gap junction coupling and thereby to slow the conduction velocity.

To best mimic activation via the sinoatrial node, cultures of hiPSC-CM should be stimulated locally to launch propagating waves. Whole-field stimuli, e.g. as delivered by field stimulation electrodes or wide-area optogenetic stimulation, will induce synchronous depolarization of all cells, mimicking the 0D case and missing possibly important gap junction-mediated dynamical instabilities. One should ideally perform spatially-resolved voltage measurements to reveal the distinct dynamics in the near- and far-field regimes. While it is not possible to recapitulate the full spatial structure of connectivity of the adult myocardium *in vitro,* one can use the *in vitro* measurements to benchmark dynamical regimes according to biophysical parameters (e.g. *λ* and *L*) which can be measured *in vivo.* We expect that these considerations will be important for ongoing efforts to develop *in vitro* models of cardiac dynamics.

In this work, the use of a simplified synthetic system was crucial to identifying geometry as a determinant of dynamical stability. By comparing experimental results to a biophysically based numerical model, we demonstrated that changing tissue geometry alone could produce dramatically different responses in systems of otherwise identical excitable cells. We anticipate that this synthetic strategy may be useful in clarifying interactions in other complex excitable tissues.^{50}

## Materials and methods

### iOS-HEK cell line generation, culture, and patterning

Clonal OS-HEK cell lines were generated as described previously^{31} and maintained in DMEM-10 supplemented with antibiotics to maintain transgene expression. For functional imaging experiments, cells were first incubated for 30 minutes in Tyrode’s solution (containing 125 mM NaCl, 2 mM KCl, 2 mM CaCl_{2}, 1 mM MgCl_{2}, 10 mM HEPES, 30 mM glucose; pH 7.3, osmolality 305-310 mOsm) supplemented with 10 μg/mL isradipine and 1 μM BeRST1 voltage-sensitive dye.

Patterned cell growth was achieved using previously described methods^{28, 32}. Cytophilic patterns of fibronectin were deposited on cytophobic polyacrylamide gels using microcontact printing. Following pattern definition, cells were gently seeded at high density (>500k cells/mL) and allowed to adhere and proliferate throughout the pattern over several days before imaging experiments.

### Wide-field all-optical electrophysiology

All-optical electrophysiology of iOS-HEK and hiPSC-CM cells was performed using an adapted ‘Firefly’ ultrawidefield inverted microscope.^{33} Spatially patterned blue excitation for optogenetic stimulation was achieved using digital micromirror device (DMD). Near-infrared voltage sensors were excited using widefield 635 nm illumination (DILAS 8 W diode laser, M1B-638.3-8C-SS4.3-T3) configured in a near-TIRF configuration Custom LabView software allowed for synchronization of time-modulated signals controlling blue and red light excitation, DMD patterns, and camera acquisitions over experimental runs.

All data were processed and analyzed using custom software (Supplementary Methods). To investigate geometry-dependent dynamical regimes, spatial regions of interest (ROIs) were defined and averaged across pixels to give fluorescence time-traces. A single set of ROIs was defined for a given sample dish and was applied uniformly across movies at different drive frequencies to systematically investigate dynamical responses (Figs. 2B and 6C). Using these spatially resolved measurements, we extract information from multiple replicates of 0D, 1D near-field, and 1D far-field responses all in a single dish.

### Numerical modeling of iOS-HEK cells

In the conductance-based model, the voltage dynamics are governed by the equation

Dynamics of individual currents were modeled according to well-established numerical equations with conductance magnitudes fit to experimental data, and an additional slow gating variable was added to capture activity-dependent sodium channel blockade by isradipine (Supplementary Methods). Numerical simulations were run on isolated cells and on 1D tracks under identical model parameters for comparison to experimental results.

## Disclosure

AEC is a co-founder of Q-State Biosciences.

## Contributions

HMM and AEC designed the study. HMM conducted and analyzed experiments. HMM and SD developed and simulated the iOS-HEK numerical models. SD and BS conducted dispersion analysis and Noble model simulations. YLH and EWM provided BeRST1 dye reagent. AEC and HMM wrote the manuscript, with input from SD and BS. AEC and BS oversaw the research.

## Supplementary Movies

**Movie S1. Electrical waves propagating through patterned iOS-HEK cells**. Optical recordings of optogenetically triggered action potentials propagating through confluent monolayers of patterned electrically excitable iOS-HEK cells. See Fig. 1 for details.

**Movie S2. Geometry dependent action potential dynamics in iOS-HEK cells**. The square islands on the right produced spikes in response to each stimulus. The 1D tracks on the left produced an alternans pattern in the near-field and conducting waves only on alternating stimuli.

**Movie S3. Second degree conduction block near sharp turns in an iOS-HEK serpentine track**. The first four action potentials conducted into the far-field, but the fifth failed to conduct at a defect in the middle of the track. Thereafter the waves showed a 3:2 conduction block at the defect in the track.

**Movie S4. Geometry dependent action potential dynamics in hiPSC-CM**. The square islands on the right produced spikes in response to each stimulus. The 1D tracks on the left produced an alternans pattern in the near-field and conducting waves only on alternating stimuli.

## Acknowledgments

We thank Christopher Werley and Miao-Ping Chien for the hiPSC-CM image in Fig. 6a. This work was supported by the Howard Hughes Medical Institute. HMM was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) program. SD was supported by the NSF Graduate Research Fellowship Program under Grant No 1644760. Y-LH and EM were supported by NIH grant R35GM119855.

## Footnotes

↵* cohen{at}chemistry.harvard.edu