Integrability of a Globally Coupled Complex Riccati Array: Quadratic Integrate-and-Fire Neurons, Phase Oscillators, and All in Between

We present an exact dimensionality reduction for dynamics of an arbitrary array of globally coupled complex-valued Riccati equations. It generalizes the Watanabe-Strogatz theory [Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70, 2391 (1993).] for sinusoidally coupled phase oscillators and seamlessly includes quadratic integrate-and-fire neurons as the real-valued special case. This simple formulation reshapes our understanding of a broad class of coupled systems—including a particular class of phase-amplitude oscillators—which newly fall under the category of integrable systems. Precise and rigorous analysis of complex Riccati arrays is now within reach, paving a way to a deeper understanding of emergent behavior of collective dynamics in coupled systems.

Figure 1

$N=8$ quadratic integrate-and-fire neurons with global coupling interacting via Gaussian pulses. The dynamics is exactly described by the low dimensional Eqs. (9). Left panel: time series of the input current $I(t)$, alongside the individual neurons’ firing events (top). Right panel: trajectory of the macroscopic variable $Q(t)$ determining voltages via (10).

Figure 2

Complex generalization of coupled quadratic integrate-and-fire neurons (13). Unlike the real QIF model, the voltage resetting in this complex generalization is redundant since the dynamics everywhere (except the special case of real line with real coupling) loops back and stays finite; see also example (I) in Fig. 1 where voltages diverge, but the resetting naturally results from the transformation (2). $N=8$ units settle into periodic motion with a nontrivial limit cycle. Trajectories of oscillators on the limit cycle in the complex plane are depicted with colored lines. Initial conditions are marked with (color-coded) points. The fixed point $x_0^{+}$ is emphasized with a black dot.

Figure 3

Array of $N=8$ Josephson junctions (16) with complex initial conditions inside the unit disk (colored points) generalize pure phase oscillators bound on the unit circle, see Fig. 4(c) in [2] and Fig. 2 in [21]. Trajectory of one oscillator in the complex plane is shown in orange. Small blue scatter points depict the value of $Z_1$ on the Poincaré section where the oscillator on the unit circle passes phase $\pi /2$.