Recent experiment by the group of Tzlil (Nitsan et al., Nat. Phys. 2016) have shown that nearby cardiac cells seeded (∼ 100 micrometers apart) on an elastic gel, synchronize their beating phase and frequency even without direct contact. By introducing an inert probe that induced periodic elastic deformations in the substrate, the experiments showed that one can pace beating cardiac cells that are relatively far from the probe. The time required to pace the cell was on the order of ∼ 15 min, and the cell maintained the new beating frequency for as long as ∼ 1 hr after the probe was removed. These long time scales are in complete contrast to the very short time scales (∼ 1 sec) that characterize relaxation after electrical stimulation is removed. We construct a simple, analytical model based on the works of Julicher and Duke (Duke et al., PNAS, 2000), and treat the deterministic dynamics of beating. The model predicts spontaneous, entrained beating (with the probe frequency) and “bursting” (short periods of entrainment to the probe separated by quiescence) of paced cells, and how these depend on the probe amplitude and frequency, in agreement with experiment (Nitsan et al., Nat. Phys. 2016). We further consider the interesting effects of small noise on the non-linear oscillator model of the beating cell, and show how it affects the coherence of beating. Finally, we predict the dependence of time required for a cell to transition from spontaneous to entrained beating once the probe is applied as well as its dependence on the probe amplitude. We account for the origin of the much longer time scale (minutes) required to entrain spontaneously beating cells by considering biological adaptation (which delays the response of the cell to the external signal).
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