Order parameter expansion and finite-size scaling study of coherent dynamics induced by quenched noise in the active rotator model

We use a recently developed order parameter expansion method to study the transition to synchronous firing occurring in a system of coupled active rotators under the exclusive presence of quenched noise. The method predicts correctly the existence of a transition from a rest state to a regime of synchronous firing and another transition out of it as the intensity of the quenched noise increases and leads to analytical expressions for the critical noise intensities in the large coupling regime. It also predicts the order of the transitions for different probability distribution functions of the quenched variables. Using numerical simulations and finite-size scaling theory to estimate the critical exponents of the transitions, we found values which are consistent with those reported in other scalar systems in the exclusive presence of additive static disorder.

Figure 1

Graphical analysis of the solutions of the equation $⟨\omega ⟩=\left(1-{\sigma }^2/2{\left(\text{cos}{\Psi }^{\ast }+C\right)}^2\right)\text{sin}\left({\Psi }^{\ast }\right)\equiv {\Omega }_2^{\ast }\left({\Psi }^{\ast }\right)\text{sin}\left({\Psi }^{\ast }\right)$ for $C=4.0$ and $\sigma =0.5,3.0,7.5$ (dashed, dotted, and dash dotted). The horizontal line marks $⟨\omega ⟩=0.97$. Note that the line corresponding to $\sigma =3.0$ does not cut the horizontal line, thus no stable steady state exists for this value of $\sigma$, whereas two solutions exist for the other values of $\sigma$.

Figure 2

Maximum of the rhs of Eq. (7) for $C=4.0$ as continuous line and $⟨\omega ⟩$ marked as dashed line. When the curve of the maximum lies above the dashed line a fixed point solution exists. Otherwise not.

Figure 3

Phase diagram: fixed points of Eqs. (7, 8, 9) (for $⟨\omega ⟩=0.97$) exist below and above the two continuous lines (gray region). In between no fixed points exist and the global phase rotates, i.e., the individual rotators oscillate in a coherent manner. Approximate values of the critical diversity for $C⪢1$ according to Eqs. (10, 11) are plotted with dashed lines. Symbols show values taken from numerical simulations of the set of Eq. (1) with a Gaussian distribution. The dotted line marks the approximate solution of the critical diversity given in [23].

Figure 4

Phase diagram for the exponential distribution of natural frequencies which satisfies $\sigma =⟨\omega ⟩$. As in Fig. 3, fixed points of Eqs. (7, 8, 9) exist below and above the two continuous lines (gray region). Approximate values of the critical diversity for $C⪢1$ according to Eqs. (12, 13) are plotted with dashed lines. Symbols show numerical simulations with exponential distribution.

Figure 5

Simulation results for Gaussian distributed frequencies: (a) the order parameter $⟨⟨m⟩⟩$ for $C=4.0$ and various values of $⟨\omega ⟩$. The location of the second transition changes little by small variations in $⟨\omega ⟩$ if it is close to one. Simulations were done with $N=102400$. (b) Ensemble fluctuations (performed over 1000 realizations of the quenched noise variables and initial conditions) of the order parameter increase with system size ($⟨\omega ⟩=0.97$ and $C=4.0$).

Figure 6

Finite size analysis for Gaussian distributed frequencies with $⟨\omega ⟩=0.97$ and $C=4$: (a) linear fits of maximal fluctuations yield $\text{ln}\left({\chi }_{max}\right)\sim c\text{ln}\left(N\right)$ with $c=0.65\pm 0.07$ for the first transition (circles) and $c=0.6\pm 0.1$ for the second (squares). (b) Rescaled order parameter $⟨⟨m\left(\sigma ,N\right)⟩⟩N^{b/2}$ collapses as a function of $ϵN^b$ with exponent $b=1/3$, $\left(N=12800,\dots ,204800\right)$.

Figure 7

Simulation results for exponentially distributed ${\omega }_i$s: (a) the order parameter is nonzero for finite disorder. Fluctuations show that the first transition takes place around $\sigma \approx 1$, as Eq. (12) predicts for large $C$. (b) Ensemble fluctuations (for $C=5.0$) diverge at the first transition for increasing $N$.

Figure 8

Finite size analysis for exponentially distributed frequencies $\left(C=5\right)$ (a) linear fit of maximal fluctuations yields $c=0.65\pm 0.02$. (b) Rescaled order parameter with scaling exponent $b=1/3$.

Figure 9

Histogram of 1000 steady states from simulations of Eqs. (1) with exponentially distributed frequencies. At the transition into coherent firing (left column) the values are distributed around one single value, broadening near the critical disorder. The destruction of the ordered state is a first-order transition (right column), the values are distributed narrowly around zero or the nonzero value, $C=5.0$.

Figure 10

Results from the integration of Eqs. (4, 5, 6) in the vicinity of the transition points with $⟨\omega ⟩=0.97$ and $C=4.0$. Both transitions, into the ordered state and into asynchronous firing, panels (a) and (b), respectively, are of second order.

Figure 11

Results from the integration of Eqs. (4, 5, 6) in the vicinity of the transition points with $⟨\omega ⟩=\sigma$. Whereas the transition into the ordered state is of second order [panel (a)] the second transition is discontinuous [panel (b)].