Electrical stimulation of large neural networks is a key tool for understanding neural population dynamics and ultimately for developing clinical treatments. However, computational models of cortical networks of spiking neurons are inherently hard to simulate and analyze. We present an efficient mean-field model of excitatory and inhibitory adaptive exponential integrate-andfire (AdEx) neurons of which we analyse the dynamical properties and validate the results using detailed network simulations. Bifurcation diagrams reflecting the network's state reveal asynchronous up and down-states, bistable regimes, and oscillatory regions corresponding to fast excitation-inhibition and slow excitation-adaptation feedback loops. In this dynamical landscape, electrical stimuli can cause state transitions, such as turning on and off oscillations. Oscillatory input can frequency-entrain and phase-lock endogenous oscillations. The effects of time-varying external stimulation are well-predicted by the mean-field model, further underpinning the utility of low-dimensional neural mass models. ResultsThe cortical mass model Here we study a cortical mass model which consists of two populations of excitatory adaptive (E) and inhibitory (I) exponential integrate-and-fire (AdEx) neurons (Fig. 1). Both populations are delay-coupled and the excitatory population has a somatic adaptation feedback mechanism. The low-dimensional mean-field model (Fig. 1 a) is derived from the large network of spiking AdEx neurons (Fig. 1 b).For the construction of the mean-field model, a set of conditions need to be fulfilled: We assume the number of neurons to be very large, all neurons within a population to have equal properties, and the connectivity between neurons to be sparse and random. Additional assumptions about the mathematical nature and a detailed derivation of the mean-field model is presented in the Methods. Bifurcation diagrams: a map of the dynamical landscapeThe E-I motif shown in Fig. 1 can occupy various network states, depending on the baseline inputs to both populations. By gradually changing the inputs, we map out the state space of the system, depicted in the bifurcation diagrams in Fig. 2. Small changes of the parameters of a nonlinear system can cause sudden and dramatic changes of its overall behavior, called bifurcations. Bifurcations separate the state space into distinct regions of network states between which the system can transition from one to the other. In our case, the dynamical state of the E-I system depends on external inputs to both subpopulations, which are directly affected by external electrical stimulation and other driving sources, e.g. inputs from other neural populations such as other brain regions.Comparing the bifurcation diagrams of the mean-field model (Figs. 2 a, c) to the ground truth spiking AdEx network (Figs. 2 b, d) demonstrates the similarity between both dynamical landscapes. Transitions between states take place at comparable baseline input values and in a well-preserved order. 2/25 11/25

During slow-wave sleep, the brain is in a self-organized regime in which slow oscillations (SOs) between up- and down-states travel across the cortex. While an isolated piece of cortex can produce SOs, the brain-wide propagation of these oscillations are thought to be mediated by the long-range axonal connections. We address the mechanism of how SOs emerge and recruit large parts of the brain using a whole-brain model constructed from empirical connectivity data in which SOs are induced independently in each brain area by a local adaptation mechanism. Using an evolutionary optimization approach, good fits to human resting-state fMRI data and sleep EEG data are found at values of the adaptation strength close to a bifurcation where the model produces a balance between local and global SOs with realistic spatiotemporal statistics. Local oscillations are more frequent, last shorter, and have a lower amplitude. Global oscillations spread as waves of silence across the undirected brain graph, traveling from anterior to posterior regions. These traveling waves are caused by heterogeneities in the brain network in which the connection strengths between brain areas determine which areas transition to a down-state first, and thus initiate traveling waves across the cortex. Our results demonstrate the utility of whole-brain models for explaining the origin of large-scale cortical oscillations and how they are shaped by the connectome.

Background: The detection of coronavirus disease 2019 (COVID-19) cases remains a huge challenge. As of April 22, 2020, the COVID-19 pandemic continues to take its toll, with >2.6 million confirmed infections and >183,000 deaths. Dire projections are surfacing almost every day, and policymakers worldwide are using projections for critical decisions. Given this background, we modeled unobserved infections to examine the extent to which we might be grossly underestimating COVID-19 infections in North America. Methods: We developed a machine-learning model to uncover hidden patterns based on reported cases and to predict potential infections. First, our model relied on dimensionality reduction to identify parameters that were key to uncovering hidden patterns. Next, our predictive analysis used an unbiased hierarchical Bayesian estimator approach to infer past infections from current fatalities. Results: Our analysis indicates that, when we assumed a 13-day lag time from infection to death, the United States, as of April 22, 2020, likely had at least 1.3 million undetected infections. With a longer lag time—for example, 23 days—there could have been at least 1.7 million undetected infections. Given these assumptions, the number of undetected infections in Canada could have ranged from 60,000 to 80,000. Duarte’s elegant unbiased estimator approach suggested that, as of April 22, 2020, the United States had up to >1.6 million undetected infections and Canada had at least 60,000 to 86,000 undetected infections. However, the Johns Hopkins University Center for Systems Science and Engineering data feed on April 22, 2020, reported only 840,476 and 41,650 confirmed cases for the United States and Canada, respectively. Conclusions: We have identified 2 key findings: (1) as of April 22, 2020, the United States may have had 1.5 to 2.029 times the number of reported infections and Canada may have had 1.44 to 2.06 times the number of reported infections and (2) even if we assume that the fatality and growth rates in the unobservable population (undetected infections) are similar to those in the observable population (confirmed infections), the number of undetected infections may be within ranges similar to those described above. In summary, 2 different approaches indicated similar ranges of undetected infections in North America. Level of Evidence: Prognostic Level V . See Instructions for Authors for a complete description of levels of evidence.