Coupling from a theoretical perspective

Historically, networks of autonomous self-sustained oscillators have been classified into two distinct states: (i) desynchronized or incoherent (oscillators cycle independently of each other, with individual periods and phases) and (ii) synchronized

or coherent (oscillators cycle with locked or similar periods and phases) [208]. With regard to circadian systems, the concept of two oscillator states was expanded by Christoph Schmal et al. (2017). Authors describe the state of network oscillations as function of intercellular coupling strength between individual oscillators. Therefore, classification of network synchrony becomes gradual rather than binary and can be characterized by the distributions of the circadian parameters amplitude, phase, and period [172]. In this model three biologically relevant states (even though mixed states may exist) arise that can serve to explain experimentally observed behavior of circadian oscillators on single cell and population/network level:

(i) Uncoupled (incoherent): virtually no intercellular coupling, oscillators cycle independently with very broad phase/period distributions

(ii) Undercritically coupled: weak intercellular coupling, oscillators show some degree of phase coherence, frequency-locking, and amplitude expansion if a critical coupling threshold is reached

(iii) Overcritically coupled (coherent): strong intercellular coupling, complete synchronization of oscillators leading to network-wide frequency-locking and high amplitude rhythms, phase distributions may be become narrower in response to strong synchronizing signals

Moreover, since intercellular coupling is difficult to quantify in absolute numbers, it may be inferred from behavior of single cell oscillators within a population or from circadian parameters of the ensemble. Changes of parameters, which define circadian oscillations, in dependence of intercellular coupling are described below:

Phase synchronization

In the 1960s, Arthur Winfree introduced a model of self-sustained oscillator populations and their mutual interactions, incorporating a phase dependent “Influence Function”

(X(f)), as well as a periodically varying “Sensitivity Function” (Z(f)) [209]. This model describes how each oscillator in a network exerts a phase dependent influence (X(f)) on all other oscillators, as well as how their resulting response is constrained by their sensitivity to this influence (Z(f)). Consistent with concepts of non-parametric entrainment, X(f) and Z(f) are assumed to determine de- or acceleration of oscillator period in a phase dependent fashion. This idea was picked up by Yoshiki Kuramoto in

the 1970s. He described that self-entrainment of oscillator populations is achieved when all mutual interactions X(f) lead to increased oscillator synchrony. Thus, as a function of interoscillator coupling strength (k) and network period distribution, synchrony (or phase coherence) between oscillators is expected to induce progressive phase transition from incoherent to coherent network states once a critical coupling threshold is reached [210]–[212]. In coherent networks, oscillators with shorter free-running periods are expected to phase-lead, while oscillators with longer free-free-running periods are expected to phase-lack relative to the average phase of the network. This phenomenon is comparable with phase of entrainment (for details see 1.3) [155].

Period- (frequency-)locking

In line with the Kuramoto model, increased coupling strength is expected to result in the convergence of free-running periods of individual oscillators. Again, above a critical coupling threshold, oscillators will become “frequency-locked” to the population mean if their intrinsic periods are similar enough [172]. Additionally, if clusters of frequency-locked oscillators arise, they can exert “period-pulling” effects on the remaining oscillators until the entire network remains locked to the mean period (as for overcritical coupling). Ultimately, progressive frequency-locking will results in narrowing of the period distribution due to decreased period dispersion between oscillators until network-wide frequency-locking is achieved (overcritically coupled networks) [172].

The critical coupling threshold, which has to be reached to initiate frequency-locking of the network, depends on the spread of the free-running period distribution [172].

Whether or not coupling affects the mean period of a coupled network is not clear.

Intuitively, period shortening with increased coupling strength appears plausible since amplitude resonance between coupled oscillators (see below) may promoter high frequency oscillations. However, for decoupled SCN neurons, i.e. upon knock-out of VIP or VPAC2 (see below), means of period distributions did not change compared to controls [53]. Moreover, theoretical models have reported both, no effect of coupling on the mean period, as well as shortening and lengthening of the mean period upon increased intercellular coupling strength [213], [214].

Amplitude expansion and relaxation

By definition resonance is described as “vibration of large amplitude in a mechanical or electrical system, caused by a relatively small periodic stimulus of the same or nearly

the same period as the natural vibration period of the system” [215]. For a network of oscillators, it can be interpreted as amplification of the amplitude of individual oscillators if their intrinsic frequency approaches the frequency of the population mean.

In physics, this relationship is described by the so-called “Lorenztkurve” (or resonance curve), which predicts that resonance is maximized as the frequencies of a forcing and a forced oscillator approach each other. Thus, as a consequence of frequency-locking, coupling affects the amplitude of a network in such a way that increased coupling strength results in amplitude expansion by resonance effects [172]. Additionally, amplitude relaxation rate (λ) of the network, i.e. the return of the amplitude to its initial state following a perturbation, and intercellular coupling are interdependent entities.

One the one hand, coupling renders oscillator networks more robust against deviations from the synchronized state [57], [157], [216], thereby making coupled networks more rigid (faster amplitude relaxation). On the other hand, amplitude relaxation rate of individual oscillators is inversely correlated with amplitude resonance [57], [172]. This means that rigid oscillators display almost no change in amplitude upon coupling, while non-rigid oscillators display relatively strong amplitude expansion. Additionally, amplitude relaxation rate, but not intrinsic amplitudes of individual oscillators affect the critical coupling threshold [57], [172]. This implies that increased amplitudes of single cell oscillators may increase the network amplitude independently of coupling dependent amplitude resonance effects.

Damping rates

Based on the inverse relationship between amplitude resonance and damping rate [217], damping of circadian oscillations (on network level) is a reflection of decreasing synchrony among individual oscillators. This is because a reduction of the network amplitude can be explained by increased dispersion of intrinsic oscillator frequencies or in other words: by decreased frequency-locking. Hence, logically increased coupling strength will result in decreased damping and vice versa.

Entrainment to Zeitgebers

As mentioned above (for details see 1.3), entrainment to a rhythmic Zeitgeber is characterized by the phase of entrainment (y) between intrinsic and extrinsic oscillations. The period mismatch (t - T) between the free-running circadian clock period (t) and the period of the extrinsic Zeitgeber (T) modulates the phase of

entrainment (y) [156]. Moreover, strength of an external Zeitgeber [155], [218], [219]

relative to the amplitude of the endogenous oscillator [59], [220], [221] have been shown to impact the phase of entrainment. Thereby, these parameters govern the range of entrainment for which adaptation to a given Zeitgeber cycle is still possible.

Since circadian parameters of oscillator networks, such as amplitude (based on resonance effects), amplitude relaxation rate, mean phase, and mean period, depend on oscillator synchrony, the concepts of entrainment and coupling are inherently interconnected. This idea was supported by Abraham et al. (2010), who pointed out that intercellular coupling, via changes of amplitude and amplitude relaxation rates, affects the response to entrainment signals, rendering more strongly coupled networks harder to entrain (smaller range of entrainment) and more robust against perturbation by Zeitgeber pulses [57]. Additionally, if frequency-locking results in altered network periods (even though this is still unclear), this may modify entrainment range due to altered relationships (t - T).