references) using comprehensive electronic search strategies combining terms "epilepsy", "seizures", "prediction", "forecasting", "cycles", "patterns", "circadian", and "multidien", with no language restrictions. Identified studies used different outcome measures, but most involved analyses of electroencephalography (EEG) to predict seizures minutes in advance, with variable success. A single prospective trial of an implanted device for chronic EEG demonstrated abovechance accuracy of warnings for imminent seizures in 9 out of 15 enrolled subjects. Two studies in independent cohorts of subjects chronically implanted with intracranial electrodes showed that rates of interictal epileptiform activity oscillate in circadian and multiday (multidien) cycles that help determine seizure likelihood. Circadian cycles and seizure diaries were used in three studies to forecast seizures over short horizons, but we found no results on forecasting seizures several days in advance. Added value of this study:In a large cohort of people with drug-resistant focal epilepsy who had chronic EEG recorded by an approved clinical device, we demonstrate that circadian and multidien cycles can be leveraged to forecast seizures up to three days in advance in some subjects and 24 hours in advance in the majority of subjects. These results highlight the feasibility of seizure forecasting over horizons longer than previously possible. Implications of all the available evidence: Seizures are not entirely random events. Using cyclical patterns of brain activity to forecast seizures hours to days in advance may enable novel seizure warning systems and prevention strategies. Convergence of findings from multiple independent datasets suggests the generalizability of this approach in people with epilepsy, though this will require direct testing in prospective clinical trials.

IMPORTANCEFocal epilepsy is characterized by the cyclical recurrence of seizures, but, to our knowledge, the prevalence and patterns of seizure cycles are unknown.OBJECTIVE To establish the prevalence, strength, and temporal patterns of seizure cycles over timescales of hours to years.

The cyclical organization of seizures in epilepsy has been described since antiquity. However, historical explanations for seizure cycles—based on celestial, hormonal, and environmental factors—have only recently become testable with the advent of chronic electroencephalography (cEEG) and modern statistical techniques. Here, factors purported over millennia to influence seizure timing are viewed through a contemporary lens. We discuss the emerging concept that seizures are organized over multiple timescales, each involving differential influences of external and endogenous rhythm generators. Leveraging large cEEG datasets and circular statistics appropriate for cyclical phenomena, we present new evidence for circadian (day‐night), multidien (multi‐day), and circannual (about‐yearly) variation in seizure activity. Modulation of seizure timing by multiscale temporal variables has implications for diagnosis and therapy in clinical epilepsy. Uncovering the mechanistic basis for seizure cycles, particularly the factors that govern multidien periodicity, will be a major focus of future work.

Paroxysms are sudden, unpredictable, short-lived events that abound in physiological processes and pathological disorders, from cellular functions (e.g., hormone secretion and neuronal firing) to life-threatening attacks (e.g., cardiac arrhythmia, epileptic seizures, and diabetic ketoacidosis). With the increasing use of personal chronic monitoring (e.g., electrocardiography, electroencephalography, and glucose monitors), the discovery of cycles in health and disease, and the emerging possibility of forecasting paroxysms, the need for suitable methods to evaluate synchrony—or phase-clustering—between events and related underlying physiological fluctuations is pressing. Here, based on examples in epilepsy, where seizures occur preferentially in certain brain states, we characterize different methods that evaluate synchrony in a controlled timeseries simulation framework. First, we compare two methods for extracting the phase of event occurrence and deriving the phase-locking value, a measure of synchrony: (M1) fitting cycles of fixed period-length vs (M2) deriving continuous cycles from a biomarker. In our simulations, M2 provides stronger evidence for cycles. Second, by systematically testing the sensitivity of both methods to non-stationarity in the underlying cycle, we show that M2 is more robust. Third, we characterize errors in circular statistics applied to timeseries with different degrees of temporal clustering and tested with different strategies: Rayleigh test, Poisson simulations, and surrogate timeseries. Using epilepsy data from 21 human subjects, we show the superiority of testing against surrogate time-series to minimize false positives and false negatives, especially when used in combination with M1. In conclusion, we show that only time frequency analysis of continuous recordings of a related bio-marker reveals the full extent of cyclical behavior in events. Identifying and forecasting cycles in biomedical timeseries will benefit from recordings using emerging wearable and implantable devices, so long as conclusions are based on conservative statistical testing.

For persons with epilepsy, much suffering stems from the apparent unpredictability of seizures. Historically, efforts to predict seizures have sought to detect changes in brain activity in the seconds to minutes preceding seizures (pre-ictal period), a timeframe that limits preventative interventions. Recently, converging evidence from studies using chronic intracranial electroencephalography revealed that brain activity in epilepsy has a robust cyclical structure over hours (circadian) and days (multidien). These cycles organize pro-ictal states, hours- to days-long periods of heightened seizure risk, raising the possibility of forecasting seizures over horizons longer than the pre-ictal period. Here, using cEEG from 18 subjects, we developed point-process generalized linear models incorporating cyclical variables at multiple time-scales to show that seizure risk can be forecasted accurately over days in most subjects. Personalized risk-stratification days in advance of seizures is unprecedented and may enable novel preventative strategies.

ObjectiveEpilepsy is characterized by spontaneous seizures that recur at unexpected times. Nonetheless, using years‐long electroencephalographic (EEG) recordings, we previously found that patient‐reported seizures consistently occur when interictal epileptiform activity (IEA) cyclically builds up over days. This multidien (multiday) interictal–ictal relationship, which is shared across patients, may bear phasic information for forecasting seizures, even if individual patterns of seizure timing are unknown. To test this rigorously in a large retrospective dataset, we pretrained algorithms on data recorded from a group of patients, and forecasted seizures in other, previously unseen patients.MethodsWe used retrospective long‐term data from participants (N = 159) in the RNS System clinical trials, including intracranial EEG recordings (icEEG), and from two participants in the UNEEG Medical clinical trial of a subscalp EEG system (sqEEG). Based on IEA detections, we extracted instantaneous multidien phases and trained generalized linear models (GLMs) and recurrent neural networks (RNNs) to forecast the probability of seizure occurrence at a 24‐h horizon.ResultsWith GLMs and RNNs, seizures could be forecasted above chance in 79% and 81% of previously unseen subjects with a median discrimination of area under the curve (AUC) = .70 and .69 and median Brier skill score (BSS) = .07 and .08. In direct comparison, individualized models had similar median performance (AUC = .67, BSS = .08), but for fewer subjects (60%). Moreover, calibration of pretrained models could be maintained to accommodate different seizure rates across subjects.SignificanceOur findings suggest that seizure forecasting based on multidien cycles of IEA can generalize across patients, and may drastically reduce the amount of data needed to issue forecasts for individuals who recently started collecting chronic EEG data. In addition, we show that this generalization is independent of the method used to record seizures (patient‐reported vs. electrographic) or IEA (icEEG vs. sqEEG).

Inferring the topology of a network using the knowledge of the signals of each of the interacting units is key to understanding real-world systems. One way to address this problem is using datadriven methods like cross-correlation or mutual information. However, these measures lack the ability to distinguish the direction of coupling. Here, we use a rank-based nonlinear interdependence measure originally developed for pairs of signals. This measure not only allows one to measure the strength but also the direction of the coupling. Our results for a system of coupled Lorenz dynamics show that we are able to consistently infer the underlying network for a subrange of the coupling strength and link density. Furthermore, we report that the addition of dynamical noise can benefit the reconstruction. Finally, we show an application to multichannel electroencephalographic recordings from an epilepsy patient.

Empirical data on real complex systems are becoming increasingly available. Parallel to this is the need for new methods of reconstructing (inferring) the structure of networks from time-resolved observations of their node-dynamics. The methods based on physical insights often rely on strong assumptions about the properties and dynamics of the scrutinized network. Here, we use the insights from machine learning to design a new method of network reconstruction that essentially makes no such assumptions. Specifically, we interpret the available trajectories (data) as features, and use two independent feature ranking approaches -Random Forest and RReliefF -to rank the importance of each node for predicting the value of each other node, which yields the reconstructed adjacency matrix. We show that our method is fairly robust to coupling strength, system size, trajectory length and noise. We also find that the reconstruction quality strongly depends on the dynamical regime. arXiv:1902.03896v2 [math.DS] 26 Aug 2019This problem is in literature formulated in several ways. Typically, one considers the nodes to be individual dynamical systems, with their local dynamics governed by some difference or differential equation [56]. The interaction among these individual systems (nodes) is then articulated via a mathematical function that captures the nature of interactions between the pairs of connected nodes (either by directed or non-directed links). In this setting, the problem of network reconstruction reduces to estimating the presence/absence of links between the pairs of nodes from time-resolved measurements of their dynamics (time series), which are assumed available. It is within this formulation that we approach the topic in this paper, i.e., we consider the structure of the studied network to be hidden in a "black box", and seek to reconstruct it from time series of node dynamics (i.e., discrete trajectories).Within the realm of physics literature, many methods have been proposed relying on above formulation of the problem, and are usually anchored in empirical physical insights about network collective behavior [46,52,74]. This primarily includes synchronization [3], both theoretically [2,58] and experimentally [6,36], and in the presence of noise [72]. Other methods use techniques such as compressive sensing [80,82] or elaborate statistics of derivative-variable correlations [39,43]. Some methods are designed for specific domain problems, such as networks of neurons [55,59] or even social networks [76]. There are also approaches specifically intended for high-dimensional dynamical system, mostly realized as phase space reconstruction methods [33,41,45,53]. While many methods in general refer to non-directed networks, some aim specifically at discerning the direction of interactions (infer the 'causality network'). One such method is termed Partial Mutual Information from Mixed Embedding (PMIME [35]) and will be of use later in this work.However, a severe drawback of the existing physical reconstruction paradigms i...