Fig 1.
Schematic representation of the 2-D tissue.
Schematic representation of the 2-D square lattice of uterine myocytes (shown in red), each myocyte coupled to a random number of passive cells (shown in blue). Neighboring myocytes are electrically coupled with strength Gm and the coupling strength between a myocyte and a passive cell is Gp.
Fig 2.
Possible dynamical regimes of a myocyte coupled to a passive cell.
Dynamical behaviour of a myocyte coupled to a passive cell for different values of the gap junction conductance Gp, and with and np = 1. (a) At Gp = 0.174nS, the period is ∼ 1min. (b) At Gp = 0.5nS, the period has reduced to ∼ 12s. (c) At Gp = 1nS, the period reduces further to ∼ 7s. Here we observe that the sudden uncoupling of the myocyte and passive cell, at the time indicated by the vertical broken line, immediately terminates activity. (d) Above a critical value G0 = 0.164nS, we observe a logarithmic divergence in the oscillatory time period: T ∼ log[(Gp − G0)/G0] as indicated by the broken line.
Fig 3.
Time periods T of the oscillation as a function of coupling.
The time period T (in sec) of the membrane potential, for a range of values of and Gp. Regions shown in white correspond to the absence of oscillatory activity. The bifurcation from oscillatory activity to a quiescent dynamical regime occurs at the interface indicated by the continuous curve obtained by fitting numerical data.
Fig 4.
Time periods T as a function of the number of coupled passive cell.
Time periods of oscillation, T (in sec) at two different values of , for a range of values of np and Gp. Regions shown in white correspond to the absence of oscillatory activity. Activity is seen when np ≥ A + B/Gp (indicated by the solid line), a functional form that can be justified from elementary considerations, see text. (a) (A ≈ 0.28, B ≈ 0.27) (b) (A ≈ 0.17, B ≈ 0.15).
Fig 5.
Patterns of electrical activity.
Pattern observed for different coupling strengths Gm between myocytes on a 50 × 50 lattice, where each myocyte interacts on average with f(= 0.2) passive cells. The upper row shows snapshots of the membrane potential while the lower row shows the corresponding effective time periods of oscillatory activity. The horizontal bar in the upper right-most panel indicates the length corresponding to 10 cells, which is of the order of ∼ 1 − 2mm. (a) Cluster Synchronization (CS) is observed at Gm = 0.48nS where cells group into several synchronously oscillating clusters, each characterized by a different frequency, coexist with regions in which the tissue is at rest. (b) At Gm = 1.8nS all cells in the lattice that oscillate do so with a single frequency. However, we also observe a few non-oscillating cells indicating that this corresponds to the LS regime. (c) At Gm = 2.4nS, which lies in the GS regime, every cell in the lattice oscillates with the same frequency.
Fig 6.
Phase space describing the dynamical systems of the model.
Dynamical regimes observed in the 2-D lattice of coupled myocytes and passive cells for a range of coupling strengths. Three distinct synchronization regimes are observed: CS at low Gm, LS at intermediate Gm and GS at high Gm. For every value of Gm, there exists a critical value of Gp below which no oscillations (NO) are observed. The symbols indicate numerically determined points lying on the boundaries between the various dynamical regimes.
Fig 7.
Regular periodic activity in the 2-D lattice of coupled myocytes and passive cells.
The regime presented here corresponds to an inter-myocyte coupling strength, Gm = 20nS. (a) Waves are emitted periodically from a single, dominant region characterized by high passive cell density. The snapshots are separated in time by 75 ms. The horizontal bar in the lowest, right-most panel indicates the length corresponding to 10 cells, which is of the order of ∼ 1 − 2mm. (b) This behaviour causes each cell in the system to exhibit a periodic pattern of activity with a period T ∼ 50 s. The only difference between the recorded time series of any two cells in the system is a temporal shift, dependent on the proximity of the cell to the source.
Fig 8.
Irregular patterns of activity in the 2-D lattice of coupled myocytes and passive cells.
The regime observed here corresponds to an inter-myocyte coupling strength, Gm = 12nS. (a) The membrane potential of a single cell exhibits recurrent patterns of activity, each pattern arising after an interval Ta. (b) Each pattern is characterized by an initial quiescent phase, followed by a series of action potentials with period Tr ∼ 20s and a brief duration of fast oscillations with periods Tf ∼ 1s. The profiles of a representative action potential (c) and fast oscillations (d) are also shown.
Fig 9.
Occurrence of spiral waves of activity in a 50 × 50 lattice of coupled myocytes and passive cells, observed for inter-myocyte coupling strength, Gm = 12nS. (a) Snapshots of membrane potential are shown at intervals of T = 100ms, in sequence from left to right and from top to bottom. (b) Trajectory indicating the motion of the tip of the spiral on the 2-D lattice. The two arrows indicate the locations of the spiral when it emerges and disappears, respectively The thick segment corresponds to the sequence shown in (a). The horizontal bars in the lowest, left-most panel of (a), and in (b) indicate the length corresponding to 10 cells, which is of the order of ∼ 1 − 2mm.
Fig 10.
Characterisation of the irregular regimes.
Dependence on Gm of the quantities characterizing the patterns of activity shown in Fig. 8, namely the mean values of (a) Ta, (b) Tr, and (c), Tf (c); and of the number of (d) action potentials, Nr, and (e) fast oscillations, Nf. The error bars indicate the standard deviation of the fluctuations of each individual quantity. Each quantity is measured over an interval of at least 6000 s. The quantities shown here have been determiend after averaging over 30 independent realizations of the passive cell distribution.