Early Warning Signal with Power Spectrum in Neuronal Firing

Brain diseases often coincide with critical transitions in neural system and abnormal neuronal firing. Studying early warning signals (EWS) of critical transitions can offer a promising avenue for predicting neuronal firing behaviors, which can potentially aid in the early diagnosis and prevention of brain diseases. Conventional EWS, such as autocorrelation and variance, have been widely used to detect the critical transitions in various dynamical systems. However, these methods are limited in distinguishing different types of bifurcations. In contrast, EWS with power spectrum have shown a significant advantage in not only predicting bifurcation points but also distinguishing the types of bifurcations involved. Previous studies have demonstrated its predictive power in climate and ecological models. Based on this, this study applies the EWS with power spectrum to neuronal systems in order to predict the neuronal firing behaviors and distinguish different classes of neuronal excitability. Specifically, we compute the EWS before the occurrence of saddle-node bifurcation on the invariant circle and subcritical Hopf bifurcation in the Morris-Lecar neuron model. Additionally, we extend the analysis to the Hindmarsh-Rose model, calculating EWS before both saddle-node bifurcation and supercritical Hopf bifurcation. The study contains the four types of codimension-1 bifurcations corresponding to the neuronal firing. For comparison, we also calculate two types of conventional EWS: lag-1 autocorrelation and variance. In numerical simulations, the stochastic differential equations are simulated by the Euler-Maruyama method. Then, the simulated responses are detrended by the Lowess filter. Finally, the EWS are calculated using the rolling window method to ensure the detection of EWS before bifurcation points. Our results show that the EWS with power spectrum can effectively predict the bifurcation points, which mean that it can predict neuronal firing activities. Comparing with the lag-1 autocorrelation and the variance, the EWS with power spectrum not only accurately predict the neuronal firing, but also distinguish the classes of excitability in neurons. That is, according to the different characteristics of the power spectrum frequencies, the EWS with power spectrum can effectively distinguish saddle-node bifurcations and Hopf bifurcations during neuronal firing. This work presents a novel approach for predicting the critical transitions in neural system, with potential applications in diagnosing and treating brain diseases.