Spatiotemporal coding of inputs for a system of globally coupled phase oscillators

We investigate the spatiotemporal coding of low amplitude inputs to a simple system of globally coupled phase oscillators with coupling function $g\left(\varphi \right)=-\sin (\varphi +\alpha )+r\sin (2\varphi +\beta )$ that has robust heteroclinic cycles (slow switching between cluster states). The inputs correspond to detuning of the oscillators. It was recently noted that globally coupled phase oscillators can encode their frequencies in the form of spatiotemporal codes of a sequence of cluster states [P. Ashwin, G. Orosz, J. Wordsworth, and S. Townley, SIAM J. Appl. Dyn. Syst. 6, 728 (2007)]. Concentrating on the case of $N=5$ oscillators we show in detail how the spatiotemporal coding can be used to resolve all of the information that relates the individual inputs to each other, providing that a long enough time series is considered. We investigate robustness to the addition of noise and find a remarkable stability, especially of the temporal coding, to the addition of noise even for noise of a comparable magnitude to the inputs.

Figure 1

A system of $N=5$ phase oscillators whose natural frequency is ${\omega }_i$ and phase is ${\theta }_i$. The oscillators are coupled with mean-field (global) coupling where each oscillator receives one-dimensional input in the form of detuning its natural frequency. Each oscillator is coupled equally to all other oscillators by the coupling function $g\left(x\right)=-\sin (x+\alpha )+r\sin (2x+\beta )$.

Figure 2

(Color online) A graphical representation of the robust heteroclinic network generated by the system depicted in Fig. 1 where each node $s_i$ $(i=1,\dots ,30)$ represents a saddle point in phase space. Each directed edge leaving state $s_i$ represents an individual branch of the unstable manifold where differences in the frequencies will drive the system along a particular edge from a given node.

Figure 3

(Color online) A spatiotemporal code generated by the trajectory where ${\omega }_{n+1}={\omega }_n$ for $n=1,\dots ,5$ and $\eta ={10}^{-7}$. Panel (a) represents the trajectory being converted into a spatio-temporal code. Panel (b) shows the state sequence $C_n$ over a set of epochs $n$, where $C_n$ is a reference to state code. Panel (c) shows the residence time sequence $T_n$ representing the time spent in a given epoch.

Figure 4

The spatiotemporal code $(C_n,T_n)$, as a function of epoch number $n$, from a numerical simulation with uniform detuning (7) for $\delta ={10}^{-7}$ and $\eta =0$ after an initial transient period has been allowed to decay. The data shows one of the two period $6\text{cycles}$ shown in Table . Notice that while the system passes through six different states, there are only four possible values for the residence times.

Figure 5

A plot showing residence time $T$ against ${\delta }_u$. Slope fits to $y=A+B\ln \left(x\right)$, where $A=-32.461$ and $B=-6.581\ln \left(x\right)$.

Figure 6

For uniform detuning, ${\omega }_1<{\omega }_2<{\omega }_3<{\omega }_4<{\omega }_5$, the dynamics will drive the system into the sequence of states in Table . This means that only the comparisons shown by lines in graph (a) will be made between oscillator frequencies. Note that at each epoch there is a comparison between one of the two slow oscillators (black) with one of the three fast oscillators (gray) in the slow-fast partition {{1,2},{3,4,5}}. Graph (b) shows a minimum spanning tree for (a) which has four edges, meaning we only need to examine four transitions to determine the ordering of the ${\omega }_i$.

Figure 7

The spatiotemporal code generated by a simulation with $\eta =0$ and a nonuniform detuning $({\omega }_1,\dots ,{\omega }_5)=1.0+(1,10,3,15,6)\sigma$, where $\sigma ={10}^{-7}$ so ${\omega }_1<{\omega }_3<{\omega }_5<{\omega }_2<{\omega }_4$. Note that the resulting sequence is period 6, but the residence times are different than those in Fig. 4.

Figure 8

A spatiotemporal code generated by a numerical simulation over 800 epochs as in Fig. 3 where there is noise present; $\eta ∕\delta =5.0$. Note there is an apparently random switching between the two possible cycles of length six, plus some occasional longer deviations. This is the activity we see in Fig. 9c.

Figure 9

(Color online) $P_{i,j}$ for long simulations with uniform detuning $\delta ={10}^{-7}$ and varying noise $\eta$. The horizontal axis represents $i$, and the vertical axis represents $j$. The dark squares indicate a transition probability close to 1 while the white squares indicate a transition probability close to 0. As before, an initial transient is disregarded. Panel (a) $(\eta ∕\delta =1.0)$ corresponds to the case where noise apparently has no effect, as seen in Fig. 4. Panel (b) $(\eta ∕\delta =2.0)$ shows the effect of low noise that is sufficient to occasionally nudge the system between the two stable period $6\text{cycles}$ present for this detuning. Panel (c) $(\eta ∕\delta =5.0)$ corresponds to the coding from Fig. 8. Panel (d) $(\eta ∕\delta =100.0)$ corresponds to the case where detuning is comparatively weak and all edges are traversed.

Figure 10

The entropy $\xi$ in the spatial coding of the system as a function of $\eta ∕\delta$ from numerical simulations. At $\eta ∕\delta <0.5$, $\xi \approx 0$—there is little uncertainty in the path that the system will follow. The entropy $\xi$ increases quickly between $0.5<\eta ∕\delta <10.0$, where it is close to the theoretical entropy of an unbiased random walk on Fig. 2, namely, $\xi =\ln \left(2\right)$.

Figure 11

(Color online) Each data set shows the cumulative frequency $f\left(T\right)$ of $n$ such that $C_n=18$ and $T_n<T$ from the simulations depicted in Fig. 10, more precisely $f\left(T\right)\mathcal{N}\{n:C_n=18\text{and}{T}_n⩽T\}∕\mathcal{N}\{n:C_n=18\}$. As $\eta ∕\delta$ increases the variance in the sequence of residence times $T_n$ increases and the mean epoch length decreases.