^{1,a)}, Christoph Kirst

^{1,2}and Marc Timme

^{1,3}

### Abstract

Symmetric networks of coupled dynamical units exhibit invariant subspaces with two or more units synchronized. In time-continuously coupled systems, these invariant sets constitute barriers for the dynamics. For networks of units with local dynamics defined on the real line, this implies that the units’ ordering is preserved and that their winding number is identical. Here, we show that in permutation-symmetric networks with pulse-coupling, the order is often no longer preserved. We analytically study a class of pulse-coupled oscillators (characterizing for instance the dynamics of spiking neural networks) and derive quantitative conditions for the breakdown of order preservation. We find that in general pulse-coupling yields additional dimensions to the state space such that units may change their order by avoiding the invariant sets. We identify a system of two symmetrically pulse-coupled identical oscillators where, contrary to intuition, the oscillators’ average frequencies and thus their winding numbers are different.

Symmetries are an important feature of network dynamical systems, often constraining their dynamics. One such restriction is, e.g., the forced order conservation of oscillators in fully symmetric systems of time-continuously coupled oscillators.

^{1,2}Here, we reveal that symmetric networks of dynamical units coupled via the exchange of pulses, such as networks of spiking neurons, circumvent these restrictions, thus increasing the complexity of the dynamical phenomena emerging in such systems. We show that pulse-coupled oscillators may overtake each other, thus breaking the order conservation in contrast to similar time-continuously coupled systems. We explain the mechanisms behind this overtaking phenomenon and discuss its consequences. Intriguingly, we find that identical and symmetrically pulse-coupled oscillators may exhibit n : m locking (n ≠ m), which is impossible for similar systems that are time-continuously coupled. Our results highlight that the nature of discrete-time pulse-coupling plays a distinct role in network dynamical systems, in particular, for synchronization and phase-locking phenomena.

We thank R.-M. Memmesheimer and T. Kottos for valuable comments. M.T. acknowledges financial support by the German Ministry for Education and Science via the Bernstein Center for Computational Neuroscience (BCCN) Göttingen under Grant No. 01GQ1005B and by the Max Planck Society.

I. IMPACT OF SYMMETRY ON DYNAMICS

II. SYMMETRIES, SYNCHRONY, INVARIANCE, AND ORDER PRESERVATION

III. BREAKDOWN OF ORDER PRESERVATION IN PERMUTATION-SYMMETRIC PULSE-COUPLED SYSTEMS

IV. ANALYTICAL PREDICTION FOR THE BREAKDOWN OF ORDER PRESERVATION

A. Model and numerical simulations

B. Analysis

V. DIFFERENT AVERAGE FREQUENCIES IN PERMUTATION-SYMMETRIC SYSTEMS WITH PULSE-COUPLING

VI. CONCLUSIONS

### Key Topics

- Oscillators
- 79.0
- Coupled oscillators
- 10.0
- Manifolds
- 5.0
- Subspaces
- 5.0
- Eigenvalues
- 3.0

## Figures

(Color online) Real-valued units in a permutation-symmetric smooth network dynamical system [Eq. (5)] cannot pass each other because they cannot cross the invariant manifolds (dashed) which are fixed points of the permutation symmetries [Eq. (3)]. A projection of two trajectories (black, solid) onto the - plane is sketched, one on the invariant set and one that stays on one side of the invariant manifold.

(Color online) Real-valued units in a permutation-symmetric smooth network dynamical system [Eq. (5)] cannot pass each other because they cannot cross the invariant manifolds (dashed) which are fixed points of the permutation symmetries [Eq. (3)]. A projection of two trajectories (black, solid) onto the - plane is sketched, one on the invariant set and one that stays on one side of the invariant manifold.

(Color) Breakdown of order preservation in permutation symmetric pulse-coupled networks: (a) dynamics of pulse-coupled units [Eq. (8)] with for and otherwise, , , , coupling strength , and initial condition near the synchronous state to which the dynamics converges. In the process of synchronization, the units pass each other; (b) magnification.

(Color) Breakdown of order preservation in permutation symmetric pulse-coupled networks: (a) dynamics of pulse-coupled units [Eq. (8)] with for and otherwise, , , , coupling strength , and initial condition near the synchronous state to which the dynamics converges. In the process of synchronization, the units pass each other; (b) magnification.

(Color online) Model. Time evolution [Eq. (8)] of the potential and the corresponding phase of one oscillator [Eqs. (10) and (11)]: (a) the time evolution of the oscillator’s potential, which evolves freely for one period and then receives an inhibitory pulse of strength at time causing a potential jump from to ; (b) the time evolution of the corresponding phase that jumps according to the transfer function (12) from to .

(Color online) Model. Time evolution [Eq. (8)] of the potential and the corresponding phase of one oscillator [Eqs. (10) and (11)]: (a) the time evolution of the oscillator’s potential, which evolves freely for one period and then receives an inhibitory pulse of strength at time causing a potential jump from to ; (b) the time evolution of the corresponding phase that jumps according to the transfer function (12) from to .

(Color) Order preservation and its breakdown in the time evolution of a network of pulse-coupled oscillators [Eqs. (10) and (11)] with , [Eq. (21)]. The phases of all oscillators (color-coded) are shown vs. time . Insets show magnifications as indicated: (a) time evolution for and with approach to stable synchronous state where the ordering of the oscillators is conserved; (b) similar dynamics that synchronize more slowly for and ; (c) time evolution for and shows that oscillators pass each other; (d) similar dynamics with slower synchronization for and . Notice that in (d) all oscillators exchange their ordering, while in (c) only the blue oscillator overtakes the red and yellow oscillator.

(Color) Order preservation and its breakdown in the time evolution of a network of pulse-coupled oscillators [Eqs. (10) and (11)] with , [Eq. (21)]. The phases of all oscillators (color-coded) are shown vs. time . Insets show magnifications as indicated: (a) time evolution for and with approach to stable synchronous state where the ordering of the oscillators is conserved; (b) similar dynamics that synchronize more slowly for and ; (c) time evolution for and shows that oscillators pass each other; (d) similar dynamics with slower synchronization for and . Notice that in (d) all oscillators exchange their ordering, while in (c) only the blue oscillator overtakes the red and yellow oscillator.

(Color online) Parameter dependence of the transition between order conservation and its breakdown in - -parameter space of the pulse-coupled network used in Figure 4. For each parameter set ( in steps of , in steps of ), the system was initialized 250 times with uniform distributed phases . Shown is the percentage of these runs where the ordering is not conserved. The solid red line, given by Eq. (22), indicates the theoretically predicted transition between the regimes where the order is conserved or units pass each other. The parameter values used for Figure 4 for the cases of order conservation (•, ▴) and overtaking (+, ▾) are indicated. Because eigenvalues are close to zero near the transition line, the dynamics synchronize faster for ▴ and ▾.

(Color online) Parameter dependence of the transition between order conservation and its breakdown in - -parameter space of the pulse-coupled network used in Figure 4. For each parameter set ( in steps of , in steps of ), the system was initialized 250 times with uniform distributed phases . Shown is the percentage of these runs where the ordering is not conserved. The solid red line, given by Eq. (22), indicates the theoretically predicted transition between the regimes where the order is conserved or units pass each other. The parameter values used for Figure 4 for the cases of order conservation (•, ▴) and overtaking (+, ▾) are indicated. Because eigenvalues are close to zero near the transition line, the dynamics synchronize faster for ▴ and ▾.

(Color online) Dynamics of the ordering of the oscillator phases (, , ). Without loss of generality, we labeled all possible orderings of the oscillators from to . We start the system close to the synchronous state and plot the ordering index at discrete times just after the reception of all pulses of one cycle (return map) for (), (), and (▪).

(Color online) Dynamics of the ordering of the oscillator phases (, , ). Without loss of generality, we labeled all possible orderings of the oscillators from to . We start the system close to the synchronous state and plot the ordering index at discrete times just after the reception of all pulses of one cycle (return map) for (), (), and (▪).

(Color online) (a) Phase response curve [Eq. (25)] and (b) interaction function [Eq. (26)] used in the simulations shown in Figure 8.

(Color online) (a) Phase response curve [Eq. (25)] and (b) interaction function [Eq. (26)] used in the simulations shown in Figure 8.

(Color) Symmetrically pulse-coupled identical oscillators with different average frequencies: (a) dynamics of phase oscillators [Eqs. (24)–(26)] with and numerically integrated using Euler’s method with step size . Dashed lines indicate the beginning of one period of a frequency locked state; (b) pulse generation times of the two oscillators in (a) indicated by vertical bars showing the convergence of the dynamics to the periodic frequency locked state from initial condition , . By permutation symmetry of the network, the exchange of the initial phases leads to frequency locked state.

(Color) Symmetrically pulse-coupled identical oscillators with different average frequencies: (a) dynamics of phase oscillators [Eqs. (24)–(26)] with and numerically integrated using Euler’s method with step size . Dashed lines indicate the beginning of one period of a frequency locked state; (b) pulse generation times of the two oscillators in (a) indicated by vertical bars showing the convergence of the dynamics to the periodic frequency locked state from initial condition , . By permutation symmetry of the network, the exchange of the initial phases leads to frequency locked state.

## Tables

The phases of the *N* units directly after the events of the sequence

The phases of the *N* units directly after the events of the sequence

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