Microscopic mechanism for self-organized quasiperiodicity in random networks of nonlinear oscillators

Self-organized quasiperiodicity is one of the most puzzling dynamical phases observed in systems of nonlinear coupled oscillators. The single dynamical units are not locked to the periodic mean field they produce, but they still feature a coherent behavior, through an unexplained complex form of correlation. We consider a class of leaky integrate-and-fire oscillators on random sparse and massive networks with dynamical synapses, featuring self-organized quasiperiodicity, and we show how complex collective oscillations arise from constructive interference of microscopic dynamics. In particular, we find a simple quantitative relationship between two relevant microscopic dynamical time scales and the macroscopic time scale of the global signal. We show that the proposed relation is a general property of collective oscillations, common to all the partially synchronous dynamical phases analyzed. We argue that an analogous mechanism could be at the origin of similar network dynamics.

Figure 1

Dynamics of the local and global fields. Black continuous lines are the average fields $F\left(t\right)$ while dashed and dotted lines are the local fields $f_i\left(t\right)$. The top panel refers to the $\alpha$ model with $\alpha =10$, $g=0.4$, and $N={10}^3$. The central panel refers to the TUM model with $g=21$ defined on an inhomogeneous massive network with $N={10}^4$ and $P_N\left(k\right)$ a Gaussian distribution with $\langle k\rangle =0.7N$ and ${\sigma }_k=0.06\langle k\rangle$. For these values, $k_{c1}\simeq 4500$ and $k_{c2}\simeq \langle k\rangle =7000$. The bottom panel describes the TUM model with $g=20$ on a sparse graph with $k=20$ and $N=500$. The same values of the parameters are used also in the following figures.

Figure 2

Microscopic field (continuous blue line) $f_i\left(t\right)$ for a neuron in a globally coupled network for the $\alpha$ model. In the inset we show a zoom with the fast oscillations of the microscopic dynamics. The dashed red curve is the sequence of ${\mathrm{ISI}}_m$ at each $m\mathrm{th}$ spike of the considered neuron which has been suitably rescaled in order to make it visible.

Figure 3

Escape statistics for the TUM model in a sparse network of $N=500$ neurons, $k=20$, and $g=10$ (upper panel) and $g=60$ (lower panel). In the upper inset of each panel we plot ${\mathrm{ISI}}_m$ for a generic neuron. We define the time lapse between two escapes $\tau$ as the positive difference between consecutive peaks of ${\mathrm{ISI}}_m$. The blue histogram is the probability distribution of these events obtained on a sample of ${10}^6$ escapes. In the lower inset of each panel we plot in log-linear scales the same distribution (squares) and an inverse Gaussian with the same average and variance of the data (continuous black line).

Figure 4

Modulus of the Fourier transform of the macroscopic fields $F\left(t\right)$ (black circles) and of the local fields relevant to different neurons $f_i\left(t\right)$ (other symbols). Top, middle, and bottom panels refer to the $\alpha$ model on a fully connected graph, and the TUM model on heterogeneous massive and sparse random networks, respectively.

Figure 5

Phases ${\varphi }_i\left(t\right)$ of the Fourier transform of the local fields. Top, middle, and bottom panels refer to the $\alpha$ model on a fully connected graph, and the TUM models on heterogeneous massive and sparse random networks, respectively. $\Omega$ is the characteristic frequency of the global field.

Figure 6

Phase differences ${\varphi }_i(m_1,m_2)-{\varphi }_j(m_1,m_2)$ for different couples of neurons as a function of ${\omega }_1{m}_1$. Lines connect points related to the same couple. Asterisks and circles represent peaks in the Fourier transform in Eq. (6) having the same value of $m_2$ but different values of $m_1$.

Figure 7

Map $C\left[t\right]$ for the globally coupled $\alpha$ model. Continuous (blue) line is the bisector of the square and dashed (red), dotted (black), and dash-dotted (green) lines are the maps of single neurons for $\alpha =9$, $\alpha =10$, and $\alpha =15$, respectively. The zoom in the upper inset clarifies that the map never intersects the bisector of the square. In the lower inset we plot the map $C^M$, i.e., $t_{n+M}=C^M\left[t_n\right]$ (see the text for the definition of $M$).

Figure 8

Upper panel: Behavior of the characteristic frequencies of the fully connected $\alpha$ model as a function of $\alpha$. In the inset we the plot shape of ${\omega }_2$ as a function of $\alpha$. Lower panel: the same frequencies for the TUM model on a sparse graph. The inset shows the shape of ${\omega }_2$ as a function of $g$.

Figure 9

Kuramoto parameter (see text) as a function of connectivity $k$ ($g=30$) and coupling $g$ ($k=20$) for sparse networks with different sizes $N=500$ and $N=1000$.

Figure 10

Neuron average escape frequency $\langle \nu \rangle \equiv 1/\langle \tau \rangle$ as a function of the noise variance on the received field. Each value has been obtained by averaging over five single-neuron dynamics. The continuous line is the curve $\langle \nu \rangle \sim \exp (-W/{\sigma }^2)$, with $W=0.05$. The in-degree is $k=20$ while the coupling $g$ changes in order to obtain different values of ${\sigma }^2$.