Bi-stability between stationary behaviour and collective oscillations.

(A) Experimental timeline for a monolayer of optogenetically modified NRVMs under constant illumination of its center: 1) observation of a collective stationary state (STA); 2) periodic wave train from the pacing electrode; 3) outcome observation of collective behaviour, either stationary (STA) or oscillatory (OSC). (B) Light intensity influences collective behaviour of excitable systems, transitioning between a stationary state (STA) at low illumination intensities and an oscillatory state (OSC) at high illumination intensities. Bi-stability occurs at intermediate light intensities, where transitions between states are dependent on periodic wave train properties. TR. OSC, transient oscillations.

Frequency dependency of induced collective pacemaker activity.

In vitro monolayers of optogenetically modified NRVMs show that the transition from the stationary state (STA) towards the oscillatory state (OSC) is dependent on the frequency of excitatory waves passing through an illuminated area under a constant number of 4 pulses. (A) Four pulses at high frequency (200 ms) are not enough to induce the transition. (B) Four pulses at medium frequency (600 ms) are sufficient to induce the transition. (C) Four pulses at low frequency (1200 ms) fail once again in inducing the transition.

Accumulation of pulses to induce collective pacemaker activity.

In vitro monolayers of optogenetically modified NRVMs show that the transition from the stationary state (STA) towards the oscillatory state (OSC) is dependent on the number of excitatory waves passing through an illuminated area at a constant electrical pacing frequency of 600 ms. (A) Two pulses are not enough to induce the transition. (B) Four pulses are sufficient to induce the transition.

In silico demonstration of frequency dependency and accumulation of pulses to induce collective pacemaker activity similar to that observed in vitro.

(A) Monolayers of optogenetically modified NRVMs show that the transition from the stationary state (STA) towards the oscillatory state (OSC) is dependent on the frequency of excitatory waves passing through an illuminated area under a constant number of 4 pulses. From left to right: no induction at stimulation periods of 150 and 1000 ms, induction at a stimulation period of 500 ms. (B) Monolayers of optogenetically modified NRVMs show that the transition from the stationary state towards the oscillatory state is dependent on the number of excitatory waves passing through an illuminated area under a constant frequency (600-ms period). From left to right: no induction at 2 pulses, induction at 4 pulses. (C) Three-dimensional parameter scan of all variables (light intensity, pacing period, and number of pulses) showing how dependency on period and on number of pulses relate to each other. Representative slices for fixed light intensities are displayed at the left. Vertical green lines show the natural pacemaker frequency the monolayer settles to after initiation of pacemaker activity. Letters in the middle panel correspond to traces in A and B. (D) Three-dimensional parameter diagram in which the size of the illuminated area was varied instead of the light intensity. Representative slices for fixed area edge lengths are displayed at the left. Vertical green lines show the natural pacemaker frequency the monolayer settles to after initiation of pacemaker activity.

Demonstration of frequency dependency to terminate collective pacemaker activity.

(A) Initiation and termination of collective pacemaker activity in vitro (left panel, 3 pulses of 500-ms period for initiation, 8 pulses of 200-ms period for termination) and in silico (right panel, 4 pulses of 500-ms period for initiation, 7 pulses of 180-ms period for termination). (B) Rightmost slice from Figure 4C showing in silico experiments at a fixed light intensity (1.72 mW/mm2) and size of the illuminated area (67 pixels edge length) with indicated termination (red border) and initiation (magenta border) period ranges. Vertical green line shows the natural pacemaker frequency the monolayer settles to after initiation of pacemaker activity.

Wave train-induced pacemaker activity is a multicellular collective phenomenon.

(A) Illuminated area (under constant light intensity) influences collective behaviour of excitable systems, transitioning between a stationary state for small illuminated areas and an oscillatory state for large illuminated areas. (B) In the oscillatory regime (large illuminated areas), illuminated cells in the center (purple traces) are depolarised (both in silico and in vitro), while oscillatory behaviour still takes place in the bulk of the tissue (orange traces). This discards the simplest model of bi-stable limit cycles at the single cell level.

Phase plane projections of pulse-dependent collective state transitions.

(A) Phase space trajectories of the NRVM computational model show a limit cycle (OSC) that is not lying around a stable fixed point (STA). (B) Parameter space slice showing the relationship between stimulation period and number of pulses for a fixed illumination intensity (1.72 mW ∕mm2) and size of the illuminated area (67 pixels edge length). Letters correspond to the graphs shown in C. (C) Phase space trajectories for different combinations of stimulus train period and number of pulses. A low stimulus frequency (800-ms period) and a low number of pulses (2), as well as a high stimulus frequency (180-ms period) with a high number of pulses (8) do not result in a transition from the resting state to ectopic pacemaker activity, as under these circumstances the system moves towards the stationary stable fixed point from outside and inside the stable limit cycle, respectively. However, at a low stimulation frequency with a high pulse number (4), and at a high stimulation frequency and a low pulse number (3), the stable limit cycle is approached from outside and inside, respectively, and ectopic pacemaker activity is induced.

Single cell “hidden” bi-stability and its relation to high frequency pacing.

(A) Phase space trajectory and corresponding voltage trace at short pacing period (180 ms). (B) Formation of the quasi-steady-state in the center of an irradiated region during a train of 17 and 37 high frequency (180-ms period) pulses. (C) Single cell bi-stability. (1) Single cell IV curve for an for illuminated cell (1.72 mW ∕mm2) subject to a bias current. (2) Membrane voltage traces for stimulation periods of 1000 and 180 ms and stimulation amplitudes of −20 and 50 mV. A transition to the depolarised stable state occurred for short stimulation periods and low stimulation amplitudes. (D) Effect of spatially uniform voltage perturbations of different amplitudes (−58, −20, and 50 mV) and a period of 180 ms in cardiomyocyte monolayers. The top trace and phase space projection show the effect of perturbations when oscillation is ongoing, while the bottom ones show how perturbations affect the tissue is in the stationary resting state. At an amplitude of −20 mV (2) oscillations were terminated when ongoing and could not be initiated when the tissue was in the resting state. In the other two cases (1,3), the opposite happened.

Collective pacemaker activity in a simplified in silico NRVM model.

(A) Phase space trajectories of a point inside the illuminated region for different combinations of stimulus train period and number of pulses. A low stimulus frequency (2250-ms period) and a low number of 2 pulses, as well as a high stimulus frequency (540-ms period) with a high number of 8 pulses both result in no transition from the resting state to ectopic pacemaker activity, moving towards the stationary stable fixed point from outside and inside the stable limit cycle, respectively. However, when applying 3 pulses with a low frequency (2250-ms period) or 2 pulses with a high frequency (540-ms period), ectopic pacemaker activity is induced and the stable limit cycle is approached from outside and inside, respectively. (B) Under illumination, long period stimulation (1000 ms) shows a repolarised state after pacing, while short period stimulation shows a depolarised state (500 ms). (C) 1 Evolution of the voltage variable as a result of high frequency voltage perturbations of high (blue trajectories) and low (green, orange and red) magnitudes. The dashed parts of graphs demonstrate the transient non-periodic parts of the trajectories, the solid lines show the periodic and stationary states. 2 Corresponding trajectories in the (V, xr) phase plane.

Experimental set-up.

(A) Diagrams of the in vitro model of NRVMs expressing a light-gated ion channel (top) and of the depolarising light-gated ion channel CheRiff (bottom) together with a representative example of the cell surface fluorescence produced by the enhanced yellow fluorescent protein tag attached to the C terminus of the light-gated ion channel in a CheRiff-expressing NRVM monolayer. (B) Optical voltage mapping setup including mapping computer, patterned illuminator (Polygon 400), and optical mapping camera.

Optogenetic depolarising tools and ectopy.

(A) Single cell computational data for application of an optogenetic depolarising tool under constant light illumination, showing a steady depolarised state and lack of sustained oscillations in transmembrane voltage. (B) Computational and experimental data showing emergence of ectopic activity in a coupled multicellular system. Sustained ectopic activity emanates periodically from the inside (center) of the illuminated area (light blue box) in both the computational and the experimental model. The sampling positions inside and outside of the illuminated area are indicated by purple and orange squares, respectively. The in silico light intensity was the same as in (A).

A range of collective phenomena are observed in a multicellular in silico NRVM model.

Exposure of a squared region in the center of the NRVM monolayer to light of different intensities generates of a photocurrent of different strength by the light-gated depolarising ion channel. Application of a periodic wave train to these NRVM monolayers results in 5 different outcomes, which are each visualised by representative snapshots and a linear time-extract. Going from a high (top) to low (bottom) light intensity these outcomes are: (1) monostable surface oscillations, S.OSC, (2) monostable ectopic oscillations, OSC, (3) bi-stability, (4) transient oscillations, TR.OSC, and (5) stationary state with no activity, STA.(1). The bi-stability region (STA->OSC) and the transient oscillations (STA->TR.OSC) (green box) are described in the main manuscript and supplemental material of this manuscript, respectively.

Frequency dependency of transient pacemaker activity induction.

(A) In vitro monolayers of optogenetically modified NRVMs show that the transition from the stationary state (STA) towards the transient oscillatory state (TR.OSC) is dependent on the frequency of excitatory waves passing through an illuminated area under a constant number of 4 pulses. Four pulses at a high frequency (300-ms period) or low frequency (1200-ms period) do not induce the transition in contrast to 4 pulses with an intermediate period of 600 ms. (B) Qualitative correspondence of this behaviour is shown in a computational NRVM model for low (1000-ms period), intermediate (450-ms period), and high (150-ms period) stimulation frequencies. (C) Slices of the full parameter plot for a constant size of the illuminated central region of the optogenetically modified NRVM monolayer (80 × 80 pixels) with indications of the selected frequency visualisations.

Influence of the size of the illuminated central square on transient pacemaker activity induction.

(A) Slices of the full parameter plot for constant illumination (1.72 mW ∕mm2) with indications of selected frequency visualisations. (B) Transient oscillations can be induced with a small-sized illumination region (64 × 64 pixels) for intermediate frequencies. (C) The oscillatory state can be induced with a large-sized illumination region (67 × 67 pixels) for intermediate frequencies.

Trajectory reconstruction using Takens time delay embedding.

(A) The optical voltage signal shows induced initiation and termination of pacemaker activity. The initial stationary state is shown in grey, the initiating stimulating pulses in blue, periodic oscillations in orange, the terminating high frequency stimulation in purple, and the stationary state after termination of oscillatory pacemaker activity in light green. (B) Trajectories in phase space in (V (t), V (t + τ)) coordinates showing multiple attractors (grey/green [STA]) and orange [OSC]) and the transition between them (blue/purple).

The effect of ionic changes on the termination of pacemaker activity

The mechanism that moves the oscillating illuminated tissue back to the stationary state after high frequency pacing is dependent on the ionic properties of the tissue, i.e. higher repolarisation reserves (20% IKs + 50% Ito) are associated with longer transition times.

Qualitative similarities between the simplified and complex reaction-diffusion model.

(a-c) Five pulses of low and high frequency fail to induce pacemaker activity in the simplified in silico NRVM model, while intermediate frequencies do. (d-f) Under lower illumination intensity, five pulses of low and high frequency fail to induce pacemaker activity, while intermediate frequencies induce transient oscillations. (g) Termination of pacemaker activity by high frequency travelling waves of excitation. (h-j) Failure to terminate pacemaker activity by spatially uniform voltage perturbations of high frequency and low (−30 mV) or high (50 mV) magnitude, while termination is successful at an intermediate magnitude of −20 mV).