Beat Frequency Induced Transitions in Synchronization Dynamics

TL;DR

This study investigates how beat-frequency interactions govern intermittent synchronization in a heterogeneous network of Izhikevich neurons with mean-field coupling. By combining isolated-neuron bifurcation analysis with network-level dynamics, the authors show that a new frequency mode emerges near a spike-gain bifurcation, enabling partial synchronization while preserving individual frequency traits. The core finding is that transition times between unsynchronized and partially synchronized states align with beat-period statistics, determined by the frequency gaps $\Delta f$ induced by heterogeneity, rather than by the coupling alone. This beat-frequency framework yields characteristic transition times and offers a quantitative tool to predict state switches in problems where beat frequencies shape dynamics. The results also indicate limitations in larger or irregularly distributed networks, guiding future work on topology and frequency distributions for robust beat-driven control of synchronization.

Abstract

In neurosciences, the brain processes information via the firing patterns of connected neurons operating across a spectrum of frequencies. To better understand the effects of these frequencies in the neuron dynamics, we have simulated a neuronal network of Izhikevich neurons to examine the interaction between frequency allocation and intermittent phase synchronization dynamics. As the synchronized population of neurons passes through a bifurcation, an additional frequency mode emerges, enabling a match in the mean frequency while retaining distinct most probable frequencies among neurons. Subsequently, the network intermittently transits between two patterns, one partially synchronized and the other unsynchronized. Through our analysis, we demonstrate that the frequency changes on the network lead to characteristic transition times between synchronization states. Moreover, these transitions adhere to beat frequency statistics when the neurons' frequencies differ by multiples of a frequency gap. Finally, our results can improve the performance in predicting transitions on problems where the beat frequency strongly influences the dynamics.

Figures (6)

• Figure 1: Burst frequency (inverse of the bursting period) and time evolution of the membrane potential $v(t)$ for different values of the bifurcation control parameter $a$ and synaptic input $I$ of the Izhikevich neuron model ($b=0.20$, $c=-50.0$, $d=2.0$). In panel (a), we show the frequency dependence on $a$ for two relevant values of the synaptic input $I$; in solid black for $I=10.0$ and in dotted gray for $I=8.2$. The values $I=10.0$ and $I=8.2$ represent the constant input for the uncoupled neurons. The bottom panels depict the representative time evolution for the isolated neurons near the bifurcation, with parameters: (b) $a=0.016$ and $I=10.0$; (c) $a=0.018$ and $I=10.0$; (d) $a=0.016$ and $I=8.2$; and (e) $a=0.018$ and $I=8.2$. The bursting dynamics displays one additional spike per burst for $a > a^* = 1.678 \times 10^{-2}$ when $I=10.0$, which is followed by a sudden decrease in frequency marghoti2022intermittent.
• Figure 2: Panel (a) shows the time evolution of the mean-field during intermittent dynamics and the mean-field distribution. Panels (b) and (c) display representative synaptic inputs (as described in Eq. \ref{['eq:I']}), collected at periods indicated in panel (a), representing unsynchronized and partially synchronized states, respectively. Panel (d) shows the FFT of the signals using a sampling rate of 200 $Hz$ and a duration period of 200 $s$. The intermittent section is represented in black, the unsynchronized state in yellow, and the partially synchronized state in magenta.
• Figure 3: Intermittent transition between unsynchronized and partially synchronized states. We integrate $N=60$ coupled neuronal equations with the following parameters: $b=0.20$, $c=-50.0$, $d=2.0$, $I_b =10.0$ and $\gamma = 0.03$. Additionally, the parameter $a$ is uniformly distributed from $a_{min}=0.013$ to $a_{max}=0.024$ for each neuron We show the Kuramoto order parameter time evolution for (a) several transitions in a larger time scale, (b) the amplification of one transition to a partially synchronized state, (c) one transition to an unsynchronized state, in (d) and (e) the respective raster plots of these transitions. The curves in panel (a) correspond to partial and total order parameters, $R_T(t)$ in solid black for the entire network, $R_1(t)$ in dotted blue for the cluster of neurons with $a<a^*$, and $R_2(t)$ in dashed red for the cluster of neurons with $a>a^*$. In panels (d) and (e) the black slashes are the beginning of each burst and the color scale gives the instantaneous frequency of each burst. We notice that the unsynchronized state retains the distinct natural frequencies while the partially synchronized state experiences fluctuations in instantaneous frequencies, eventually leading the network to experience phase synchronization.
• Figure 4: Probability distribution density of residence times for (a-b) partially synchronized and (c-d) unsynchronized states of the network, using random (black) or arranged by discrete gaps (red) of parameter $a$ for $a_{min}<a<a_{max}$. The oscillations in $p(T_r)$ arising from frequency gaps due to beat times are discernible when compared to the random scenario.
• Figure 5: Burst frequency distributions of each neuron in the network. We present the partially phase-synchronized state in magenta and the unsynchronized state in yellow. The network comprises 60 distinct neurons with uniform parameter allocation between $a_{min}=0.013$ and $a_{max}=0.024$, coupling strength $\gamma=0.03$ and current bias $I_b=10.0$. Color shades show the burst frequency distribution, while dots indicate the mean frequency of each neuron. In the unsynchronized state, the mean frequency increases with $a$, while in the partially synchronized state, the synchronized neuron cluster has the same mean but different most probable frequencies due to the bimodal distribution.