Self-Consistent Correlations of Randomly Coupled Rotators in the Asynchronous State

We study a network of unidirectionally coupled rotators with independent identically distributed (i.i.d.) frequencies and i.i.d. coupling coefficients. Similar to biological networks, this system can attain an asynchronous state with pronounced temporal autocorrelations of the rotators. We derive differential equations for the self-consistent autocorrelation function that can be solved analytically in limit cases. For more involved scenarios, its numerical solution is confirmed by simulations of networks with Gaussian or sparsely distributed coupling coefficients. The theory is finally generalized for pulse-coupled units and tested on a standard model of computational neuroscience, a recurrent network of sparsely coupled exponential integrate-and-fire neurons.

Figure 1

System with distributed natural frequencies. Noise spectrum (a) and spectra of three arbitrarily picked oscillators (b) with natural frequencies as indicated; network simulations (squares) compared to theory (lines) obtained from numerically solving Eq. (5) and Fourier transformation of Eqs. (6) and (7), respectively. Parameters: Gaussian coupling coefficients, $\overline{K}=0$, $K=0.5$, ${\omega }_0=1$, ${\sigma }_{\omega }=0.5$, $A_2=1/2$, $A_3=1/2i$, $A_{\ell }=0$ otherwise. In simulations, we used a time step of $\mathrm{\Delta }t=0.1\mathrm{a}.\mathrm{u}.$ in a time window of $T=26T_0$, discarded a transient of $T_0=2500\mathrm{a}.\mathrm{u}.$ and used the remaining 25 pieces of length $T_0$ for averaging.

Figure 2

System with a single natural frequency. Spectra of noise (a) and single-rotator (b), averaged over all units (b) for two system sizes as indicated. Network simulations (squares) compared to theory (lines) [expanded view in (c)]. Deviation $\mathrm{\Delta }$ between theory and simulations for the (network-wide averaged) single-rotator spectrum for different connectivities (d). Parameters as in Fig. 1 except ${\sigma }_{\omega }=0$.

Figure 3

Analytically tractable cases. Power spectra for weak (a) and strong (b) coupling strength; theory [lines, Eqs. (12) and (8), respectively] compared to network simulations (symbols). Quality factor (c), correlation time (d) of rotators and network noise intensity (e); theory [lines, Eqs. (9)–(11) and (13)] and simulations (symbols). Parameters: See Fig. 2 except $A_1=A_{-1}^{*}=1/2i$, $A_{\ell }=0$, $T_0=5000\mathrm{a}.\mathrm{u}.$.

Figure 4

Application to neural network model. Noise spectrum (a) and spectrum of the single neuron spike train, averaged over all units (b). Network simulations of exponential integrate-and-fire (EIF) neurons with Erdős-Rényi topology obeying Dale’s law (circles), perfect integrate-and-fire (PIF) neurons with fully connected Gaussian topology (squares) compared to the theory (lines). Parameters: $N_E=10000$, $N_I=2500$, $ϵ=0.1$, $J_E=0.05\mathrm{mV}$, $g=4.5$, $R{I}_{\mathrm{ext}}=30\mathrm{mV}$, ${\sigma }_{\mathrm{ext}}=0.1R{I}_{\mathrm{ext}}$, $V_{\mathrm{th}}=20\mathrm{mV}$, $V_{\mathrm{leak}}=0\mathrm{mV}$, $V_{\text{reset}}=0\mathrm{mV}$, ${\mathrm{\Delta }}_T=2\mathrm{mV}$, ${\tau }_m=20\mathrm{ms}$, $t_{\mathrm{ref}}=2\mathrm{ms}$, ${\tau }_D=1.5\mathrm{ms}$, $\lambda =0.08$, $dt=0.1\mathrm{ms}$, $T_0=1\mathrm{s}$, and $T=11T_0$.