Patterns of membrane potential are thought to play a critical role in many biological patterning processes1,2,3,4,5,6,7,8,9. Less is known about the fate of such patterns when tissues with different resting potential are in contact. In particular, can the interface between two non-excitable tissues be excitable? When cells are gap junction-coupled across the interface, the voltage along a line transverse to the interface must interpolate between steady state voltages of the two halves, including possibly going through unstable or metastable regimes. This continuity requirement on the membrane voltage opens the possibility of robust interfacial action potentials (APs) that would not be supported by the bulk media on either side of the interface.

Robust excitations localized at interfaces are a burgeoning area of basic research and technology10. For example, interfacial excitations have been studied in the context of topological insulators spanning electronic11, photonic12, mechanical13,14,15, hydrodynamic16 and biochemical reaction–diffusion17,18,19 systems. While these systems can often be treated within the topological band theory of linear waves, electrophysiological excitability inherently relies on the topological properties of the underlying non-linear dynamical system.

Historically, spatial structures in bioelectrical signalling have been difficult to investigate experimentally because patch-clamp measurements probed the voltage at only a single point in space. Recent advances in voltage imaging20,21 opened the door to studying the spatial structures of bioelectrical excitations. By combining patterned optogenetic stimulation and high-speed voltage imaging, one can probe the excitability of a complex tissue as a function of space and time. This approach has been used to identify bioelectrical domain walls in engineered tissues22 and also to map bioelectrical signals throughout embryonic development1,20. Engineered and patterned cells have been a powerful tool for dissecting intercellular molecular signalling cascades23,24, but despite some theoretical work25,26, this approach has not previously been applied experimentally to bioelectrical signalling in heterogeneous tissues.

In this Article, we map the excitability at interfaces between non-excitable engineered tissues and observe interface-localized (‘topological’) APs. Detailed numerical simulations and a simple analytical model capture the key features of these excitations. Our simulations predict that topological APs are exceptionally robust to variations in ion channel levels compared to ordinary APs. These findings suggest a mechanism of bioelectrical signalling that may arise in native tissues and that might be used in engineering synthetic bioelectrical circuits with applications in sensing, tissue engineering and unconventional computation. A theoretical analysis of interfacial APs revealing how their properties stem from topological features of the underlying non-linear dynamics (beyond topological band theory) will be presented separately (C.S. et al., manuscript in preparation).


Human embryonic kidney (HEK293) cells become electrically excitable when genetically engineered to express an inward-rectifier potassium channel (for example Kir2.1 or Kir2.3) and a voltage-gated sodium channel (for example NaV1.3 (ref. 27), NaV1.5 (ref. 28) or NaV1.7 (ref.29)). When grown in a confluent monolayer, endogenous gap junctions introduce nearest-neighbour electrical coupling, which can then support propagating electrical waves. If the cells further express a blue-light-activated cation channel such as CheRiff (ref. 30) and either express a far-red voltage-indicator protein21 or are labelled with a red-shifted voltage-sensitive dye31, then one can optically induce APs and simultaneously map their propagation through the engineered tissue22,28,32.

First, we explored whether the sodium and potassium channels needed to be in the same cells. We made separate pools of HEK cells that expressed CheRiff and either NaV1.5 alone or Kir2.1 alone (Fig. 1a). We mixed the cell populations in a 1:1 ratio and mapped the voltage in confluent monolayer cultures using the far-red voltage-sensitive dye BeRST1 (Methods)31. Localized optogenetic stimulation evoked APs that propagated radially outwards (Fig. 1b and Supplementary Video 1). These APs were quantified by the fractional change in fluorescence, ΔF/F, of the BeRST1 indicator. In cultures expressing either NaV1.5 or Kir2.1 alone, optical activation of CheRiff did not lead to APs or wave propagation (Extended Data Fig. 1). These results show that a mixture of non-excitable cells can become excitable by sharing currents through gap junctions.

Fig. 1: Topological APs arise at interfaces of non-excitable tissues.
figure 1

a,b, A mixture of non-excitable cells can be excitable. a, Left: sample preparation. HEK cells were engineered to express either Kir2.1 (an inward-rectifier K+ channel) or NaV1.5 (a voltage-gated Na+ channel). All cells also expressed a light-gated ion channel, CheRiff. The two populations were mixed in a 1:1 ratio, plated and grown to form a gap junction-coupled monolayer. Right: distribution of NaV1.5-expressing cells (tagged with a GFP marker) and Kir2.1-expressing cells (tagged with a tdTomato marker). Expression is mutually exclusive. b, Montage of frames from voltage-imaging video (Supplementary Video 1). The monolayer was stimulated by a short pulse of blue light (vertical bar at t = 0). Changes in voltage were then monitored via the fluorescence of the far-red voltage-sensitive dye BeRST1 (baseline signal in grey, ΔF/F in colour). Optogenetic stimulation evoked outward propagating APs. ce, Topological AP at interfaces. c, Sample preparation. HEK cells expressing Kir2.1 or NaV1.5 were plated on opposite sides of a thin plastic divider. After removal of the divider, the cells migrated to fill the gap and formed a gap junction-coupled interface. d, Interface between populations of NaV1.5-expressing cells (tagged with GFP) and Kir2.1-expressing cells (tagged with tdTomato). e, Montage of frames from a voltage-imaging video (Supplementary Video 2). Optogenetic stimulation spanning the interface at t = 0 evoked a topological AP that propagated outwards solely along the interface (baseline signal in grey, ΔF/F in colour). Scale bars, 0.1 mm (a) and 1 mm (b, d and e). See also Extended Data Figs. 1 and 2.

Next, we used a thin plastic divider to physically separate the NaV1.5- and the Kir2.1-expressing cells on opposite halves of the culture dish (Fig. 1c and Methods). Fluorescent tags marked the two populations (Fig. 1d). After the cells had settled, we gently removed the divider and let the cells migrate to fill the gap (Methods). This approach led to a clean interface with the two cell populations on opposite sides.

Optogenetic stimulation of either population alone did not evoke APs. Remarkably, optogenetic stimulation with a stripe that spanned the interface induced electrical excitations that propagated along the interfacial line without entering the bulk on either side (Fig. 1e and Supplementary Video 2). The spikes had a width of 0.76 ± 0.15 mm transverse to the interface, a length of 2.21 ± 0.62 mm along the interface and propagated at 12.9 ± 3.5 mm s−1 (mean ± s.d., n = 5 samples, Fig. 2a). Thus, topological APs can arise between non-excitable tissues. For control experiments, we made interfaces in which one cell population expressed either NaV1.5 or Kir2.1, and the adjoining population expressed neither channel. These interfaces were not excitable, confirming the necessity of both ion channels on opposite sides of the interface (Extended Data Fig. 2a,b). Furthermore, in dishes where the NaV1.5- and Kir2.1-expressing cells had not yet fully closed the gap, topological APs propagated only in regions where the two cell types had come into contact (Extended Data Fig. 2c).

Fig. 2: Quantification and numerical simulations of topological APs.
figure 2

a, Left: spatial profile of a topological AP. Cyan and magenta lines indicate regions of the sample expressing NaV1.5 and Kir2.1, respectively. Scale bar, 1 mm. Right: kymograph constructed by averaging transverse to the interface, showing topological AP propagation. b, Numerical simulations of an interface between a Kir2.1-expressing cell population (gK = 0.04 nS pF−1, EK = −107 mV) and a NaV1.5-expressing cell population (gNa = 3.9 nS pF−1, ENa = 75 mV). Left: image of a topological AP propagating along a tissue interface (Supplementary Video 3). Scale bar, 1 mm. Right: kymograph showing topological AP propagation. The colour scale corresponds to deviation from local resting potential. c, Calculated phase diagram of excitability of Kir2.1–NaV1.5 interface. The parameter space is divided into non-excitable, excitable, and spontaneously active phases, indicated by representative activity patterns (blue lines). The purple region indicates the excitable phase of a homogenous system, where each cell expresses both NaV1.5 and Kir2.1 channels. The interfacial system has a spontaneously active phase (blue region) that is absent in the homogenous system. The orange star corresponds to the parameters used for the simulations in b. d, Steady-state resting voltage and sodium channel availability along a line transverse to the interface. The voltage varies continuously between a polarized state on the Kir2.1 side and a depolarized state on the NaV1.5 side. The available reserve of NaV1.5 conductance (gNa × (1 – m3) h) has a peak near the interface. The variables m and h are the sodium channel activation and inactivation gates, respectively, defined in Supplementary Model 1. The band of high NaV1.5 reserve supports AP propagation. Scale bar, 0.2 mm. e, Numerical simulation of a simplified FHN-like model of topological APs (See Supplementary Model 2 for details.). Left: image of a topological AP propagating along a tissue interface. Right: kymograph showing topological AP propagation. Scale bars omitted because space and time units are arbitrary. Voltage scale, ΔV, is in arbitrary units (a.u.). See also Extended Data Figs. 35.

To better understand our results, we simulated a Hodgkin-Huxley (HH)-like conductance-based model of our experiment using established parameters for Kir2.1 and NaV1.5 (refs. 22,33) (Supplementary Model 1). This model reproduced the basic phenomena, showing interface-localized AP initiation and propagation (Fig. 2b and Supplementary Video 3). We then simulated the effect of changes in the expression level of the gap junctions, NaV1.5, or Kir2.1. Modulating the gap junction conductance, Ggj, changed the width, length and conduction speed of the topological AP, with all three parameters scaling as (Ggj)1/2, as one would expect from dimensional analysis (Extended Data Fig. 3). Topological APs arose over at least a 60-fold variation in gNa and at least a 1,000-fold variation in gK, persisting to the highest levels of gK and gNa simulated (Fig. 2c).

We then performed analogous simulations where the NaV1.5 and Kir2.1 were homogeneously distributed through all cells. For cells co-expressing both ion channels, excitability occurred over a much narrower range of expression levels (~30-fold in NaV1.5 but only ~3-fold in Kir2.1; Fig. 2c). This observation highlights an important difference between an interface and a homogeneous mixture of the two types of cell. In the homogeneous case, excitability required a delicate balance of depolarizing (NaV, leak, channelrhodopsin) and polarizing (Kir) currents to drive all the phases of the AP (refs. 34,35). In the interfacial structure, the gap junction coupling created an effective weighted average between the NaV and Kir currents, whose weights depended on the distance from the interface (Fig. 2d). Thus, spatial separation of the NaV and the Kir components, paradoxically, led to a system where the excitability was far more robust than in the system where these two components were mixed.

To facilitate an understanding of the basic requirements for topological APs, we developed a simplified model inspired by the FitzHugh–Nagumo (FHN) model (Supplementary Model 2 and Supplementary Code)36,37. We broke the characteristic N-shaped non-linearity of the FHN model into two components: a non-linear component approximating the shape of a NaV current–voltage (IV) curve and a linear component mimicking the restoring nature of the Kir current. Each component was active in only one half-space. Neither component alone supported AP generation because each had a single zero-crossing and hence a single stable resting potential. However, at the interface, the I–V curve had contributions from both the NaV-like and the Kir-like components and had three zero crossings and hence could support non-linear excitations (Extended Data Fig. 4). The simplified FHN-like model reproduced our observed propagating topological APs (Fig. 2e). The model supports all three excitability states (quiescence, stimulus-triggered APs and spontaneous oscillations) shown by the HH-like model (compare Extended Data Fig. 5 to Fig. 2c). These observations suggest that tissue interfaces might support a rich repertoire of novel bioelectrical phenomena.

We then explored experimentally the interaction of interface topology and sample topology. Using a plastic drinking straw as a separator, we plated NaV1.5-expressing cells in a disc and Kir2.1-expressing cells in the surrounding region (Fig. 3 and Methods). We then removed the straw and let the cells form a circular interface. Optical stimulation in a line across the interface evoked topological APs that propagated bidirectionally around the interface, annihilating each other when they collided. To evoke a directional wave, we applied a tonic stimulation with a line to create a thin zone of NaV channel inactivation. We then turned off the tonic stimulation and stimulated one side of this zone briefly. This stimulus protocol created an AP propagating in only one direction. The inactivated NaV channels recovered by the time the AP had completed one trip around the disc. Thereafter the AP propagated around the interface, circulating stably for many minutes (Extended Data Fig. 6 and Supplementary Video 4). These results demonstrate that topological APs can sense the topology of tissue boundaries and can store information via their handedness of circulation whose sign is set by initial conditions.

Fig. 3: Persistently propagating topological APs at a circular interface.
figure 3

a, Sample preparation. HEK cells expressing NaV1.5 and CheRiff were seeded into the circular region defined by a thin plastic cylinder. HEK cells expressing Kir2.1 and CheRiff were then added and settled in the unoccupied periphery. Image at the right shows NaV1.5-expressing cells (cyan) and Kir2.1-expressing cells (magenta). b, Stimulation protocol for evoking a unidirectional topological AP wave. A long (450 ms) illumination pulse was delivered to a portion of the interface, to drive local NaV1.5 inactivation. A short (30 ms) pulse was then delivered to a nearby interface region, evoking a topological AP that propagated unidirectionally. By the time the AP had completed one cycle, the inactivated NaV channels had recovered, permitting cyclical propagation. c, Montage showing application of the inactivation stimulus, the trigger stimulus and persistent AP propagation along the interface. d, Time-dependent fluorescence in the region indicated by the green polygon, showing a persistently propagating topological AP. Scale bar, 1 mm. Panels c and d show same field of view as a. See also Extended Data Fig. 6 and Supplementary Video 4.

In the presence of voltage-gated Ca2+ channels, topological APs might lead to locally elevated Ca2+, which could then turn on downstream patterns of gene expression. First, we tested whether co-expression of a T-type calcium channel, CaV3.2, and Kir2.3 in HEK cells were sufficient, on their own, to support AP propagation. We used HEK cells with constitutive expression of Kir2.3, doxycycline-inducible expression of CaV3.2 (ref. 38) and lentiviral expression of CheRiff for optogenetic stimulation. We induced CaV3.2 expression with doxycycline and then mapped the voltage in confluent monolayer cultures of these cells (Extended Data Fig. 7a). Local optogenetic stimulation induced APs that propagated radially outwards and then broke up into self-sustaining spiral waves (Supplementary Video 5). Nifedipine is primarily known as an L-type calcium channel blocker, but at high concentrations (>22 μM) nifedipine blocks T-type channels such as CaV3.2 as well39. Incubation with nifedipine (50 μM) abolished the excitability (Extended Data Fig. 7b). Samples where CaV3.2 expression was not induced were also not excitable (Extended Data Fig. 7c). Together, these results establish that CaV3.2 and Kir2.3 are sufficient to sustain propagating APs in confluent HEK cell monolayers.

We then asked whether topological APs could arise at the interface between cells expressing CaV3.2 and cells expressing Kir2.1. As with the NaV channels, optogenetic stimulation with a stripe that spanned the interface induced excitations in membrane voltage that propagated along the interface (Fig. 4a and Supplementary Video 6). In separate samples, we used a far-red calcium-sensitive dye (Methods) and observed propagating Ca2+ signals along the interface too (Fig. 4b and Supplementary Video 7). These results show that topological APs can be driven by multiple types of depolarizing channel (NaV, CaV) and can drive increases in intracellular Ca2+ in cells within an electrical coupling length of a tissue interface.

Fig. 4: Topological APs driven by Ca2+ currents.
figure 4

a, Montage of frames from a voltage-imaging video (Supplementary Video 6). Optogenetic stimulation spanning the interface at t = 0 evoked a topological AP that propagated outwards solely along the interface. b, Montage of frames from a calcium-imaging video (from a different sample, Supplementary Video 7). Optogenetic stimulation at the interface evoked a calcium signal that propagated along the interface. Scale bar, 1 mm. Field of view in b is the same size as in a. Baseline signal in grey, ΔF/F in colour. See also Extended Data Fig. 7.


The non-linear behaviour of ion channels, together with gap junctional sharing of currents, gives tissue interfaces unique excitability properties that are different from those of the bulk. In this Article, we have shown that: (1) mixtures of non-excitable cells can be excitable, (2) an interface between non-excitable tissues can be excitable, (3) topological APs at tissue interfaces are far more robust to variations in ion channel expression than are APs in homogeneous tissues, (4) a circular tissue interface can support stably circulating topological APs and (5) topological APs can drive local elevations in Ca2+ concentration.

It is useful to compare the observed excitable interfaces to the phenomenology of edge modes in topological insulators10. The two effects are analogous in the following sense: topological insulators host a conducting boundary at the interface between two distinct bulk insulators. Likewise, the topological APs are excitations at the interface between two tissues that alone are not excitable. However, the theory of edge modes can be often formulated within topological band theory: the amplitude of the relevant waves is proportional to the amplitude of the drive and the waves obey the superposition principle. For the excitable interfaces studied in this Article, the modes are either self-exciting or require a finite amplitude perturbation and they do not obey a superposition principle because of a finite refractory period. Non-linear topological excitations have been studied in a mechanical context40, but not previously, to our knowledge, in the context of bioelectricity.

To see intuitively how an interface enables excitability, first note that for a single cell (or a homogeneous tissue) to be excitable, the ratio of the conductances of the two channels gNa/gK must be neither too small nor too large: the sodium current must overwhelm the potassium current on the upstroke, but the potassium current must overwhelm the sodium current on the downstroke. When the cells are gap junction coupled, the ionic currents introduce an averaging effect since each cell seeks to match the mean voltage of its neighbours. A cell that contains only one type of ion channel is nonetheless influenced by the other type of current, which spreads along the chain of gap junction couplings transverse to the interface. In the presence of an interface between pure NaV and pure Kir conductances, one can always find an averaging window with a ratio of sodium to potassium currents conducive to excitability. However, simply having an interface is not always sufficient (even though the averaging will occur). In multicellular models, the interface plays a crucial role in controlling topological features of the dynamical system, such as the number of fixed points and their Morse indexes that ultimately decide the fate and nature of interfacial excitability. A detailed topological analysis of excitable interfaces is in preparation (C.S. et al., manuscript in preparation).

The existence of topological APs at tissue interfaces raises the possibility of other types of topological effect in electrophysiology. For instance, our numerical simulations predicted a regime in which a NaV1.5–Kir2.1 interface shows pacemaking activity, whereas homogeneous monolayers of cells co-expressing NaV1.5 and Kir2.1 do not show pacemaking activity under any conditions. In the interfacial case, the membrane voltage can transition through a region where there is no stable fixed point for the voltage. By contrast, in the homogeneous case, there is always a global fixed point near either the K+ or Na+ reversal potential. This explains the absence of pacemaking activity. Other combinations of ion channels could lead to additional topological effects. For instance, expression of pacemaker channels on one side of the interface might lead to spontaneous spiking that starts only when two tissues touch, providing a means to signal first contact. Depending on the other ion channels on both sides of the interface, these contact-evoked spikes could either propagate just along the interface, into the bulk on one side or into the bulk on both sides. Topological electrophysiology mechanisms could enable cells to sense the presence of nearby tissue interfaces, to determine the topology of their tissue environment or to communicate over long distances.

The ways in which bioelectrical signals complement chemical signals in embryonic development, morphogenesis and wound healing1,5,8,9,41,42 are still being worked out. It is not yet known whether topological excitations arise in native tissues. Recent work has highlighted the importance of voltage-gated Ca2+ channels in development43,44,45. These effects occur in nominally non-excitable cells. However, our results highlight that excitability need not be an intrinsic feature of a cell. Instead, it can emerge as a collective property of the whole tissue. Interactions between zebrafish skin melanophores and xanthophores also suggest a contact-dependent bioelectrical signalling mechanism46. Our results suggest that analyses of such effects should not just examine the properties of individual cell types, but should also consider tissue interfaces as distinct electrophysiological entities. Finally, we note that due to the parallel structure of the Hodgkin–Huxley and Turing reaction–diffusion equations, analogous interfacial excitations could also occur in chemical reaction–diffusion systems. More broadly, we suggest that interfacial excitations merit study as distinct entities in spatially extended active matter47 and other non-linear dynamical systems, ranging from ecology to epidemiology.


Cell line generation and culture

Cells stably co-expressing NaV1.5 and CheRiff-eGFP are described in ref. 29. For cells expressing Kir2.1 and CheRiff, cells constitutively expressing CheRiff were transiently transfected with pCAG-Kir2.1-T2A-tdTomato (from Massimo Scanziani, Addgene 60598) using CalFectin transfection reagent (SignaGen Laboratories) following the manufacturer’s directions. Cells were seeded in imaging plates 24–48 h after transfection. HEK cells were maintained in Dulbecco’s modified Eagle medium with 10% fetal bovine serum (DMEM-10), penicillin (100 U ml−1) and streptomycin (100 mg ml−1). Cells stably expressing human CaV3.2, along with cells constitutively expressing Kir2.3 and doxycycline-inducible CaV3.2, were a generous gift from T. Snutch and are described in ref. 38. These cells were infected with a lentiviral vector expressing pCMV-CheRiff-CFP (Addgene 136636) to render them light sensitive.

Sample preparation

Dishes (35 mm) with 14 mm glass coverslip bottoms (Cellvis, #D35-14-1.5-N) were coated with polyethylenimine (PEI). Two days prior to imaging, linear interfaces were created by seeding NaV1.5- and Kir2.1-expressing cells separated by a 100 µm-thick plastic divider, 4 × 105 cells of each type. After 2 h, the divider was carefully removed, the plate was gently washed with phosphate-buffered saline (PBS) to remove unattached cells and the plate was incubated in culture medium. For the circular interfaces, the central island was created by seeding 5 × 105 NaV1.5-expressing cells inside a region defined by a circular 5 mm-diameter plastic cylinder (a cut drinking straw). After 2 h, the cylinder was removed, unattached cells were removed by gentle washing with PBS and the plate was incubated overnight with culture medium. Approximately 106 Kir2.1-expressing cells were then seeded onto the dish. The Kir2.1 cells were primarily attached in the annular region not already occupied by NaV1.5-expressing cells.

Imaging and stimulation

For voltage imaging, cells were washed to remove culture medium and then incubated with 1–2 μM BeRST1 dye in PBS for 30 min in a tissue culture incubator. Immediately before imaging, samples were washed twice and immersed in imaging solution containing 125 mM NaCl, 15 mM HEPES, 25 mM glucose, 2.5 mM KCl, 1 mM MgCl2 and 2 mM CaCl2 with the pH adjusted to 7.3 with NaOH. Sample preparation for Ca2+ imaging was similar, except that the cells were washed and incubated with 3 μM BioTracker 609 Red Ca2+ AM Dye (EMD Millipore #SCT021) in PBS for 30 min in a tissue culture incubator, then washed and immersed in imaging solution. Widefield imaging was performed using an Axiovert 200 (Zeiss) inverted microscope equipped with a light source (xenon arc, XBO 75 W) and filter sets for the GFP, tdTomato, BeRST1 and calcium dye channels. Light was collected using a 2×, numerical aperture 0.1 objective lens (ThorLabs #TL2X-SAP) into an electron-multiplying charged-coupled device camera (Andor iXon EM+ 897, 512 × 512 pixels, 35 frames per s). The field of view (FOV) was expanded to 9.9 mm in the sample plane by placing a 0.5× image reducer (View Solutions, #MA513302) in the light collection path. CheRiff was excited by a 470 nm light-emitting diode (LED) (ThorLabs #M470L4-C4) connected to the topside illumination path. The stimulation region was defined by a thin slit inserted in the excitation beam and projected onto the sample plane by the microscope condenser. The camera was controlled using the Andor SOLIS software. The LED illumination was controlled by a National Instruments DAQ card. The camera and LED were synchronized using a custom MATLAB code. For imaging and patterned optogenetic stimulation of CaV3.2–Kir2.3 monolayers and CaV3.2–Kir2.1 interfaces we used an ultrawidefield microscope as described in ref. 48.

Data analysis

All data were processed and analysed using custom software (MATLAB). The baseline fluorescence, F, of the BeRST1 signal was calculated for each pixel from the mean intensity over the time series. The video was then converted into units of ΔF/F and smoothed by spatially filtering with a 12 × 12 pixels kernel.

For the linear interface experiments (Fig. 1), ten consecutive trials were averaged before converting to ΔF/F units. To calculate the spatial shape of the propagating AP (Fig. 2a), images acquired at different times were laterally shifted to cancel the motion of the AP wavefront. Background noise from variation between rows was suppressed by setting pixels below the 20th percentile of their row to the minimum intensity of the frame. Finally, the video was averaged along the time axis with a 12-frames window.

Kymographs were constructed by averaging rectangles of 40 × 10 pixels along the interface.

For the circular interface experiment (Fig. 3) the time series of the ΔF/F video was reconstructed from a detrended vectorized ΔF/F sequence to account for changes in baseline signal due to photobleaching. In addition to spatial filtering, videos were averaged along time with a six-frame sliding window. For the time-trace plot (Fig. 3d) a spatial region of interest was defined and averaged across pixels.

GFP and tdTomato images (Fig. 1a,b) were processed by subtracting a background image (no illumination) and dividing pixel-wise by a reference image (a dish with no cells) to account for spatial variations in illumination intensity. A gamma correction was applied to the tdTomato image to permit visualization of cells with widely varying expression levels.

Numerical modelling

Model equations and simulation methods can be found in Supplementary Models 1 and 2 and Supplementary Code.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.