Spontaneous generation of persistent activity in diffusively coupled cellular assemblies

The spontaneous generation of electrical activity underpins a number of essential physiological processes, and is observed even in tissues where specialized pacemaker cells have not been identified. The emergence of periodic oscillations in diffusively coupled assemblies of excitable and electrically passive cells (which are individually incapable of sustaining autonomous activity) has been suggested as a possible mechanism underlying such phenomena. In this paper we investigate the dynamics of such assemblies in more detail by considering simple motifs of coupled electrically active and passive cells. The resulting behavior encompasses a wide range of dynamical phenomena, including chaos. However, embedding such assemblies in a lattice yields spatiotemporal patterns that either correspond to a quiescent state or to partial or globally synchronized oscillations. The resulting reduction in dynamical complexity suggests an emergent simplicity in the collective dynamics of such large, spatially extended systems. Furthermore, we show that such patterns can be reproduced by a reduced model comprising only excitatory and oscillatory elements. Our results suggest a generalization of the mechanism by which periodic activity can emerge in a heterogeneous system comprising nonoscillatory elements by coupling them diffusively, provided their steady states in isolation are sufficiently dissimilar.

Figure 1

Dependence of the dynamics of a single uncoupled FHN oscillator on the parameter $b$. (a) Real part of the eigenvalues (${\lambda }_R$), obtained through a linear stability analysis, for a range of values of $b$. We find that ${\lambda }_R$ crosses zero (horizontal broken line) at $b=b_{c1}(\simeq 0.136)$ and $b=b_{c2}(\simeq 0.376)$. (b) The system exhibits a low steady state (low SS) for $b<b_{c1}$ and a high steady state (high SS) for $b>b_{c2}$, as can be seen from the fact that the extrema of $V_e$ (red and blue markers) coincide over these values of $b$. (c) In the range $b_{c1}<b<b_{c2}$, we observe oscillatory behavior, with the frequency $\nu$ exhibiting a maximum value at the center of this range. For reference, we display vertical dashed lines at $b=b_{c1}$ and $b=b_{c2}$ in each of the panels.

Figure 2

Decision tree for identifying the dynamical state of a system of coupled excitable cells, each of which are connected to a variable number of passive cells (EP model). The order parameters used for determining the nature of the spatiotemporal pattern observed include the dispersions calculated across space [${\sigma }_i^2\left(\right)$], or across time [${\sigma }_t^2\left(\right)$], for the state variables $Y_i$ ($i=1,...,N$) or the oscillation frequencies $\nu$. Averages calculated over space and over time are denoted by ${\langle \rangle }_i$ and ${\langle \rangle }_t$ respectively, while the deviation of the lowest value of the state variable for any node from its maximum calculated across space is denoted by $Y_i^{maxmin}={max}_i\left[{min}_t\left(Y_i\right)\right]-{min}_t\left(Y_i\right)$ [21]. The threshold values used to distinguish between the different states, viz., oscillation death (OD), inhomogeneous steady state (ISS), chimera (Ch), cluster synchronization (CS), exact synchronization (ES), homogeneous out-of-phase synchronization (HOS), inhomogeneous in-phase synchronization (IIS), and inhomogeneous out-of-phase synchronization (IOS), are: ${\delta }_1={10}^{-3},{\delta }_2={10}^{-5},{\delta }_3={10}^{-3},{\delta }_4={10}^{-7},{\delta }_5={10}^{-5},{\delta }_6={10}^{-7},$ and ${\delta }_7={10}^{-4}$.

Figure 3

Emergent dynamics obtained with different motifs of diffusively coupled excitable and passive cells. (a)–(d) Collective dynamical patterns observed over a range of values of coupling strengths $D$ and $C_r$, obtained using different motifs (shown as insets in the corresponding panels, where the larger and smaller circles represent excitable and passive cells, respectively). The regimes are classified according to the dominant attractor obtained for the given parameter set [see Fig. 2]. (e) Qualitative nature of the patterns in (a)–(d), in terms of the phase plane trajectories and time series of the excitable cells [21]. The black dots represent the instantaneous position of the two cells on the corresponding limit cycle. (f)–(i) Variation of the frequency $\nu$ of the excitable cells on the left (green) and right (maroon) in each of the four motifs in (a)–(d) for $C_r=0.25$, over the range of $D$ indicated by horizontal cyan bars in (a)–(d). As $D$ increases, the frequencies of the two cells merge, and for sufficiently strong coupling the system can either stop oscillating (f), or display a frequency that is between (g), greater than (h) or equal to (i) the maximum of the intrinsic frequencies.

Figure 4

Spontaneous emergence of oscillatory activity upon coupling a pair of quiescent units. (a) Variation of $\nu$ with $D$ for $C_r=0.6$, obtained using the motif shown as an inset, i.e., a pair of coupled excitable cells that are attached to zero and two passive cells, respectively. Although both cells are quiescent for low $D$, they exhibit oscillations for sufficiently large $D$. (b) Dependence of ${\lambda }_R$, corresponding to the real parts of the five eigenvalues of the system, on $D$. We note that two pairs of eigenvalues (EV) are complex conjugates and hence their real parts overlap, as indicated above the corresponding branches. The inset displays a zoomed-in segment of the panel, indicating the value of $D$ at which two of the eigenvalues changes sign. (c) Bifurcation diagram of the extrema values of $V_e$ for this system, showing the transition from a stable steady state to oscillations, where the maximum and minimum values for each of the two excitable cells are indicated by red and blue markers, respectively.

Figure 5

Coexistence of chaotic and nonchaotic dynamical activity in a system comprising three diffusively coupled excitable cells, each connected to a different number ($n_p$) of passive cells. (a) Collective dynamical patterns observed over a range of values of diffusive coupling strengths $D$ between excitable cells and the coupling $C_r$ between excitable and passive cells for the system shown as an inset. The dynamical regimes are classified according to the dominant attractor obtained for the given parameter set, and are the same as those detailed in Figs. 3–3. (b) Time series of membrane potential $V_e$ of the excitable cells coupled to (top) $n_p=1$, (middle) $n_p=2$ and (bottom) $n_p=3$ passive cells, for $D=0.02$ and $C_r=0.19$. (c)–(d) Phase plane trajectories and power spectral densities of the three excitable cells, colored as in the corresponding panels of (b).

Figure 6

Collective dynamics of a lattice of diffusively coupled elements that can be either excitable or oscillatory. (a) Schematic diagram of uterine tissue, modeled as a two-dimensional lattice where each site comprises an excitable cell coupled to $n_p$ passive cells (top, the EP model), where the value of $n_p$ at each site is drawn from a Poisson distribution. The dynamics at each site can equivalently be described through cells that are either oscillatory or excitable (bottom, the OE model). The latter cell type could be in one of two possible steady states (SS), characterized by low and high values of the state variable $V$. Note that the state of a cell in the bottom panel (high SS, low SS, or oscillatory) is chosen such that the uncoupled dynamics at that site is qualitatively the same as that of the corresponding site in the top panel. (b) Snapshots of the activity $V$ in a planar simulation domain comprising an equal mixture of excitable and oscillatory elements ($f=0.5$) diffusively coupled with strength $D$ to nearest neighbors, showing (top row, $D=0.1$) cluster synchronization (CS) and (bottom row, $D=0.3$) global synchronization (GS). (c) Varying the diffusion coefficient $D$ and the fraction of oscillatory cells $f$ in the lattice for the case of the OE model, several distinct dynamical regimes are observed [see Fig. 7 for details]. (d)–(e) Variation of the mean oscillatory frequency $\langle \nu \rangle$ with $D$. Each curve is obtained by starting from different random initial states at low $D$ and then gradually increasing $D$ over time. In (d), the cell at each site can be either oscillatory (with probability $f=0.7$) or excitable (with probability $1-f$). In (e), we associate each lattice site with either an excitable or an oscillatory cell, following the procedure outlined in the main text. All simulations are performed on square lattices comprising $64\times 64$ cells, with periodic boundary conditions.

Figure 7

Decision tree for identifying the dynamical state of a heterogeneous system of coupled cells, each of which can either be in the oscillatory or the excitable regime (OE model). The order parameters used for determining the nature of the spatiotemporal pattern observed include (i) $f_{\mathrm{osc}}$: the number of oscillating nodes in the lattice, (ii) ${max}_t\left[f_{\mathrm{act}}\left(t\right)\right]$: the maximum number of nodes that were active (i.e., above the excitation threshold $\alpha$) together over the period of observation, and (iii) ${\sigma }_i^2\left(\nu \right)$ is the dispersion of frequencies of all oscillating nodes calculated across space [21]. The threshold values used to distinguish between the states, viz., no oscillations (NO), cluster synchronization (CS), local synchronization (LS), global synchronization (GS), and coherence (COH), are ${\epsilon }_1={10}^{-3},{\epsilon }_2=0.999,{\epsilon }_3=0.995$, and ${\epsilon }_4={10}^{-10}$.

Figure 8

Comparison of the spatiotemporal activity in two-dimensional lattices of cells described by the EP and OE models. We find that similar behavior can be seen for systems characterized by a large number of coupled units arranged in two-dimensional lattices, where each cell is described by the EP model (top row) or the OE model (bottom row). The two models can yield (a), (b) cluster synchronization, as evident from the spatial distribution of frequencies $\nu$. In addition, both systems exhibit (c), (d) traveling wavefronts, and (e), (f) spiral waves, as can be seen from the instantaneous spatial distribution of $V_e$ (top row) and $V$ (bottom row), respectively.