Echo chambers and opinion polarization have been recently quantified in several sociopolitical contexts, across different social media, raising concerns for the potential impact on the spread of misinformation and the openness of debates. Despite increasing efforts, the dynamics leading to the emergence of these phenomena remain unclear. Here, we propose a model that introduces the phenomenon of radicalization, as a reinforcing mechanism driving the evolution to extreme opinions from moderate initial conditions. Empirically inspired by the dynamics of social interaction, we consider agents characterized by heterogeneous activities and homophily. We analytically characterize the transition between a global consensus and emerging radicalization dynamics in the population, as a function of social influence and the controversialness of the topic discussed. We contrast the model's behavior against empirical data of polarized debates on Twitter, qualitatively reproducing the observed relation between users' engagement and opinions, as well as opinion segregation based on the interaction network. Our findings shed light on the dynamics that may lie at the core of the emergence of echo chambers and polarization in social media.

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models.

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A standardized catheter intervention approach using fixed low-dose USAT for the treatment of intermediate- and high-risk PE was associated with rapid improvement in haemodynamic parameters and low rates of bleeding complications and mortality.