Unique spatiotemporal synchronization solutions of heterogeneous coupled oscillators

Spontaneous, stochastic, local Ca2+ oscillations (LCOs) that are heterogeneous in phase, frequency, and amplitude incessantly occur throughout the neuronal-like cytoarchitecture of the heart’s pacemaker, the sinoatrial node (SAN). How information encoded within ensembles of these heterogeneous LCOs is linked to the incessant emergence of the formation of rhythmic SAN impulses, however, is an unsolved problem.

Spatiotemporal Synchronization Dynamics

We applied a wide range of analytical techniques adapted from dynamical systems and neuroscience to map fine details of spatiotemporal distributions of local Ca2+ dynamics in time-series of high-speed acquisition panoramic images of mouse SANs ex vivo. Phase analysis delineated a network of functional pacemaker cell clusters, in which Ca2+ dynamic amplitudes, kinetics, and phases were similar, but still differed from each other, and differed from those of other clusters. Local Ca2+ dynamics within the network of clusters formed rhythmic global Ca2+ impulses, with a mean rate identical to that recorded by the reference sharp electrode in the right atria.

Initial conditions of each impulse initiation differed from each other, due mostly to the stochastic nature of Ca2+ dynamics within a small cluster located near the crista terminalis, having the highest intrinsic power, earliest rotor-like energy transfer, most frequent point-to-point instability, earliest acrophase, and greatest impulse-to-impulse variability within the network of functional clusters. A spontaneous, slowly increasing ensemble Ca2+ signal within this cluster forms a foot of each CaT. Following this foot, the amplitude of the global CaT increased rapidly, due to rapid increases in the kinetics and amplitudes of Ca2+ dynamics in some adjacent clusters.

Topological Structures

In other clusters near to the inter-atrial septum, a delayed, but rapid increase in the Ca2+ dynamic occurred when the relaxation phase of the CaT had already begun. Orbit plots and Lyapunov exponent analyses detected both deterministic and stochastic features within the SAN Ca2+ dynamics.

Correlation analyses detected cross-talk among the Ca2+ dynamics within different clusters that varied, not only throughout an impulse, but also from one impulse to the next, forming unique dynamic small-world networks, assessed as ratios of the highest correlation coefficients and shortest path lengths, ultimately creating a unique spatiotemporal Ca2+ dynamic synchronization solution for each CaT.

Heterogeneous Interactions

Thus, the emergence of each CaT from individual LCOs during each cycle is unique underlies cycle length variability in SAN, recapitulating the emergence of spontaneous CaTs in isolated, single, SAN pacemaker cells. Incessant, stochastic, oscillatory calcium (Ca2+) signals underlie important biologic processes throughout nature, and it has been argued that oscillatory frequency-dependent amplitude control provides a precise, noise-resistant control strategy that allows functional Ca2+ biologic signals that are of graded amplitude to be stable.

Nonlinear Dynamics

Phase analysis delineated a network of functional pacemaker cell clusters, in which Ca2+ dynamic amplitudes, kinetics, and phases were similar, but still differed from each other, and differed from those of other clusters. Local Ca2+ dynamics within the network of clusters formed rhythmic global Ca2+ impulses, with a mean rate identical to that recorded by the reference sharp electrode in the right atrium.

Symmetry Breaking

A spontaneous, slowly increasing ensemble Ca2+ signal within this cluster forms a foot of each CaT. Following this foot, the amplitude of the global CaT increased rapidly, due to rapid increases in the kinetics and amplitudes of Ca2+ dynamics in some adjacent clusters. In other clusters near to the inter-atrial septum, a delayed, but rapid increase in the Ca2+ dynamic occurred when the relaxation phase of the CaT had already begun.

Parameter Variations

Orbit plots and Lyapunov exponent analyses detected both deterministic and stochastic features within the SAN Ca2+ dynamics. Correlation analyses detected cross-talk among the Ca2+ dynamics within different clusters that varied, not only throughout an impulse, but also from one impulse to the next, forming unique dynamic small-world networks, assessed as ratios of the highest correlation coefficients and shortest path lengths, ultimately creating a unique spatiotemporal Ca2+ dynamic synchronization solution for each CaT.

Diffusive Coupling

Thus, the emergence of each CaT from individual LCOs during each cycle is unique underlies cycle length variability in SAN, recapitulating the emergence of spontaneous CaTs in isolated, single, SAN pacemaker cells. Incessant, stochastic, oscillatory calcium (Ca2+) signals underlie important biologic processes throughout nature, and it has been argued that oscillatory frequency-dependent amplitude control provides a precise, noise-resistant control strategy that allows functional Ca2+ biologic signals that are of graded amplitude to be stable.

Phase-Locked States

Phase analysis delineated a network of functional pacemaker cell clusters, in which Ca2+ dynamic amplitudes, kinetics, and phases were similar, but still differed from each other, and differed from those of other clusters. Local Ca2+ dynamics within the network of clusters formed rhythmic global Ca2+ impulses, with a mean rate identical to that recorded by the reference sharp electrode in the right atrium.