Forbidden ordinal patterns of periictal intracranial EEG indicate deterministic dynamics in human epileptic seizures

Ordinal methods for a characterization of evolving functional brain networks

Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a resulting loss of information, this approach captures meaningful information about the temporal structure of the underlying system dynamics as well as about properties of interactions between coupled systems. This-together with its conceptual simplicity and robustness against measurement noise-makes ordinal time series analysis well suited to improve characterization of the still poorly understood spatiotemporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni- and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments, which would be necessary to advance characterization of evolving functional brain networks.

Assessing serial dependence in ordinal patterns processes using chi-squared tests with application to EEG data analysis

We extend Elsinger's work on chi-squared tests for independence using ordinal patterns and investigate the general class of m-dependent ordinal patterns processes, to which belong ordinal patterns processes derived from random walk, white noise, and moving average processes. We describe chi-squared asymptotically distributed statistics for such processes that take into account necessary constraints on ordinal patterns probabilities and propose a test for m-dependence, with which we are able to quantify the range of serial dependence in a process. We apply the test to epilepsy electroencephalography time series data and observe shorter m-dependence associated with seizures, suggesting that the range of serial dependence decreases during those events.

Fluctuations in EEG band power at subject-specific timescales over minutes to days explain changes in seizure evolutions

Epilepsy is recognised as a dynamic disease, where both seizure susceptibility and seizure characteristics themselves change over time. Specifically, we recently quantified the variable electrographic spatio-temporal seizure evolutions that exist within individual patients. This variability appears to follow subject-specific circadian, or longer, timescale modulations. It is therefore important to know whether continuously recorded interictaliEEG features can capture signatures of these modulations over different timescales. In this study, we analyse continuous intracranial electroencephalographic (iEEG) recordings from video-telemetry units and find fluctuations in iEEG band power over timescales ranging from minutes up to 12 days. As expected and in agreement with previous studies, we find that all subjects show a circadian fluctuation in their iEEG band power. We additionally detect other fluctuations of similar magnitude on subject-specific timescales. Importantly, we find that a combination of these fluctuations on different timescales can explain changes in seizure evolutions in most subjects above chance level. These results suggest that subject-specific fluctuations in iEEG band power over timescales of minutes to days may serve as markers of seizure modulating processes. We hope that future study can link these detected fluctuations to their biological driver(s). There is a critical need to better understand seizure modulating processes, as this will enable the development of novel treatment strategies that could minimise the seizure spread, duration or severity and therefore the clinical impact of seizures.

An Ensemble of Hyperdimensional Classifiers: Hardware-Friendly Short-Latency Seizure Detection With Automatic iEEG Electrode Selection

We propose a new algorithm for detecting epileptic seizures. Our algorithm first extracts three features, namely mean amplitude, line length, and local binary patterns that are fed to an ensemble of classifiers using hyperdimensional (HD) computing. These features are embedded into prototype vectors representing ictal (during seizures) and interictal (between seizures) brain states are constructed. These vectors can be computed at different spatial scales ranging from a single electrode up to many electrodes. This flexibility allows our algorithm to identify the electrodes that discriminate best between ictal and interictal brain states. We assess our algorithm on the SWEC-ETHZ iEEG dataset that includes 99 short-time iEEG seizures recorded with 36 to 100 electrodes from 16 drug-resistant epilepsy patients. Using k-fold cross-validation and all electrodes, our algorithm surpasses state-of-the-art algorithms yielding significantly shorter latency (8.81 s vs. 11.57 s) in seizure onset detection, and higher specificity (97.31% vs. 94.84%) and accuracy (96.85% vs. 95.42%). We can further reduce the latency of our algorithm to 3.74 s by allowing a slightly higher percentage of false alarms (2% specificity loss). Using only the top 10% of the electrodes ranked by our algorithm, we still maintain superior latency, sensitivity, and specificity compared to the other algorithms with all the electrodes. We finally demonstrate the suitability of our algorithm to deployment on low-cost embedded hardware platforms, thanks to its robustness to noise/artifacts affecting the signal, its low computational complexity, and the small memory-footprint on a RISC-V microcontroller.

Estimating topological entropy using ordinal partition networks

We propose a computationally simple and efficient network-based method for approximating topological entropy of low-dimensional chaotic systems. This approach relies on the notion of an ordinal partition. The proposed methodology is compared to the three existing techniques based on counting ordinal patterns-all of which derive from collecting statistics about the symbolic itinerary-namely (i) the gradient of the logarithm of the number of observed patterns as a function of the pattern length, (ii) direct application of the definition of topological permutation entropy, and (iii) the outgrowth ratio of patterns of increasing length. In contrast to these alternatives, our method involves the construction of a sequence of complex networks that constitute stochastic approximations of the underlying dynamics on an increasingly finer partition. An ordinal partition network can be computed using any scalar observable generated by multidimensional ergodic systems, provided the measurement function comprises a monotonic transformation if nonlinear. Numerical experiments on an ensemble of systems demonstrate that the logarithm of the spectral radius of the connectivity matrix produces significantly more accurate approximations than existing alternatives-despite practical constraints dictating the selection of low finite values for the pattern length.

Seizure Onset Zone Lateralization Using a Non-linear Analysis of Micro vs. Macro Electroencephalographic Recordings During Seizure-Free Stages of the Sleep-Wake Cycle From Epilepsy Patients

The application of non-linear signal analysis techniques to biomedical data is key to improve our knowledge about complex physiological and pathological processes. In particular, the use of non-linear techniques to study electroencephalographic (EEG) recordings can provide an advanced characterization of brain dynamics. In epilepsy these dynamics are altered at different spatial scales of neuronal organization. We therefore apply non-linear signal analysis to EEG recordings from epilepsy patients derived with intracranial hybrid electrodes, which are composed of classical macro contacts and micro wires. Thereby, these electrodes record EEG at two different spatial scales. Our aim is to test the degree to which the analysis of the EEG recorded at these different scales allows us to characterize the neuronal dynamics affected by epilepsy. For this purpose, we retrospectively analyzed long-term recordings performed during five nights in three patients during which no seizures took place. As a benchmark we used the accuracy with which this analysis allows determining the hemisphere that contains the seizure onset zone, which is the brain area where clinical seizures originate. We applied the surrogate-corrected non-linear predictability score (ψ), a non-linear signal analysis technique which was shown previously to be useful for the lateralization of the seizure onset zone from classical intracranial EEG macro contact recordings. Higher values of ψ were found predominantly for signals recorded from the hemisphere containing the seizure onset zone as compared to signals recorded from the opposite hemisphere. These differences were found not only for the EEG signals recorded with macro contacts, but also for those recorded with micro wires. In conclusion, the information obtained from the analysis of classical macro EEG contacts can be complemented by the one of micro wire EEG recordings. This combined approach may therefore help to further improve the degree to which quantitative EEG analysis can contribute to the diagnostics in epilepsy patients.

Quantitative EEG biomarkers for epilepsy and their relation to chemical biomarkers

The electroencephalogram (EEG) is the most important method to diagnose epilepsy. In clinical settings, it is evaluated by experts who identify patterns visually. Quantitative EEG is the application of digital signal processing to clinical recordings in order to automatize diagnostic procedures, and to make patterns visible that are hidden to the human eye. The EEG is related to chemical biomarkers, as electrical activity is based on chemical signals. The most well-known chemical biomarkers are blood laboratory tests to identify seizures after they have happened. However, research on chemical biomarkers is much less extensive than research on quantitative EEG, and combined studies are rarely published, but highly warranted. Quantitative EEG is as old as the EEG itself, but still, the methods are not yet standard in clinical practice. The most evident application is an automation of manual work, but also a quantitative description and localization of interictal epileptiform events as well as seizures can reveal important hints for diagnosis and contribute to presurgical evaluation. In addition, the assessment of network characteristics and entropy measures were found to reveal important insights into epileptic brain activity. Application scenarios of quantitative EEG in epilepsy include seizure prediction, pharmaco-EEG, treatment monitoring, evaluation of cognition, and neurofeedback. The main challenges to quantitative EEG are poor reliability and poor generalizability of measures, as well as the need for individualization of procedures. A main hindrance for quantitative EEG to enter clinical routine is also that training is not yet part of standard curricula for clinical neurophysiologists.

Seizure pathways change on circadian and slower timescales in individual patients with focal epilepsy

Personalized medicine requires that treatments adapt to not only the patient but also changing factors within each individual. Although epilepsy is a dynamic disorder characterized by pathological fluctuations in brain state, surprisingly little is known about whether and how seizures vary in the same patient. We quantitatively compared within-patient seizure network evolutions using intracranial electroencephalographic (iEEG) recordings of over 500 seizures from 31 patients with focal epilepsy (mean 16.5 seizures per patient). In all patients, we found variability in seizure paths through the space of possible network dynamics. Seizures with similar pathways tended to occur closer together in time, and a simple model suggested that seizure pathways change on circadian and/or slower timescales in the majority of patients. These temporal relationships occurred independent of whether the patient underwent antiepileptic medication reduction. Our results suggest that various modulatory processes, operating at different timescales, shape within-patient seizure evolutions, leading to variable seizure pathways that may require tailored treatment approaches.

Markov modeling via ordinal partitions: An alternative paradigm for network-based time-series analysis

Mapping time series to complex networks to analyze observables has recently become popular, both at the theoretical and the practitioner's level. The intent is to use network metrics to characterize the dynamics of the underlying system. Applications cover a wide range of problems, from geoscientific measurements to biomedical data and financial time series. It has been observed that different dynamics can produce networks with distinct topological characteristics under a variety of time-series-to-network transforms that have been proposed in the literature. The direct connection, however, remains unclear. Here, we investigate a network transform based on computing statistics of ordinal permutations in short subsequences of the time series, the so-called ordinal partition network. We propose a Markovian framework that allows the interpretation of the network using ergodic-theoretic ideas and demonstrate, via numerical experiments on an ensemble of time series, that this viewpoint renders this technique especially well-suited to nonlinear chaotic signals. The aim is to test the mapping's faithfulness as a representation of the dynamics and the extent to which it retains information from the input data. First, we show that generating networks by counting patterns of increasing length is essentially a mechanism for approximating the analog of the Perron-Frobenius operator in a topologically equivalent higher-dimensional space to the original state space. Then, we illustrate a connection between the connectivity patterns of the networks generated by this mapping and indicators of dynamics such as the hierarchy of unstable periodic orbits embedded within a chaotic attractor. The input is a scalar observable and any projection of a multidimensional flow suffices for reconstruction of the essential dynamics. Additionally, we create a detailed guide for parameter tuning. We argue that there is no optimal value of the pattern length m, rather it admits a scaling region akin to traditional embedding practice. In contrast, the embedding lag and overlap between successive patterns can be chosen exactly in an optimal way. Our analysis illustrates the potential of this transform as a complementary toolkit to traditional time-series methods.

Unraveling the decay of the number of unobserved ordinal patterns in noisy chaotic dynamics

In this paper, we introduce a model to describe the decay of the number of unobserved ordinal patterns as a function of the time series length in noisy chaotic dynamics. More precisely, we show that a stretched exponential model fits the decay of the number of unobserved ordinal patterns for both discrete and continuous chaotic systems contaminated with observational noise, independently of the noise level and the sampling time. Numerical simulations, obtained from the logistic map and the x coordinate of the Lorenz system, both operating in a totally chaotic dynamics were used as test beds. In addition, we contrast our results with those obtained from pure stochastic dynamics. The fitting parameters, namely, the stretching exponent and the characteristic decay rate, are used to distinguish whether the dynamical nature of the data sequence is stochastic or chaotic. Finally, the analysis of experimental records associated with the hyperchaotic pulsations of an optoelectronic oscillator allows us to illustrate the applicability of the proposed approach in a practical context.

Quantifying interdependence using the missing joint ordinal patterns

In this paper, we develop the concept of forbidden/missing ordinal patterns into the forbidden/missing joint ordinal patterns and propose the ratio of the number of missing joint ordinal patterns (RMJPs) as a sign of interdependence. RMJP in a surrogate analysis can be used to differentiate the forbidden joint ordinal patterns from the missing joint ordinal patterns due to small sample effects. We first apply RMJP to the simulated time series: a two-component autoregressive fractionally integrated moving average process, the Hénon map, and the Rössler system using active control and discuss the effect of the length of the time series, embedding dimension, and noise contamination. RMJP has been proven to be capable of measuring the interdependence in the numerical simulation. Then, RMJP is further used on the electroencephalogram (EEG) time series for empirical analysis to explore the interdependence of brain waves. With results by RMJP obtained from a widely used open dataset of the sleep EEG time series from healthy subjects, we find that RMJP can be used to quantify the brain wave interdependence under different sleep/wake stages, reveal the overall sleep architecture, and indicate a higher level of interdependence as sleep gets deeper. The findings are consistent with existing knowledge in sleep medicine. The proposed RMJP method has shown its validity and applicability and may assist automatic sleep quantification or bring insight into the understanding of the brain activity during sleep. Furthermore, RMJP can be used on sleep EEG under various pathological conditions and in large-scale sleep studies, helping to investigate the mechanisms of the sleep process and neuron synchronization.

Is human atrial fibrillation stochastic or deterministic?-Insights from missing ordinal patterns and causal entropy-complexity plane analysis

The mechanism of atrial fibrillation (AF) maintenance in humans is yet to be determined. It remains controversial whether cardiac fibrillatory dynamics are the result of a deterministic or a stochastic process. Traditional methods to differentiate deterministic from stochastic processes have several limitations and are not reliably applied to short and noisy data obtained during clinical studies. The appearance of missing ordinal patterns (MOPs) using the Bandt-Pompe (BP) symbolization is indicative of deterministic dynamics and is robust to brief time series and experimental noise. Our aim was to evaluate whether human AF dynamics is the result of a stochastic or a deterministic process. We used 38 intracardiac atrial electrograms during AF from the coronary sinus of 10 patients undergoing catheter ablation of AF. We extracted the intervals between consecutive atrial depolarizations (AA interval) and converted the AA interval time series to their BP symbolic representation (embedding dimension 5, time delay 1). We generated 40 iterative amplitude-adjusted, Fourier-transform (IAAFT) surrogate data for each of the AA time series. IAAFT surrogates have the same frequency spectrum, autocorrelation, and probability distribution with the original time series. Using the BP symbolization, we compared the number of MOPs and the rate of MOP decay in the first 1000 timepoints of the original time series with that of the surrogate data. We calculated permutation entropy and permutation statistical complexity and represented each time series on the causal entropy-complexity plane. We demonstrated that (a) the number of MOPs in human AF is significantly higher compared to the surrogate data (2.7 ± 1.18 vs. 0.39 ± 0.28, p < 0.001); (b) the median rate of MOP decay in human AF was significantly lower compared with the surrogate data (6.58 × 10^-3 vs. 7.79 × 10^-3, p < 0.001); and (c) 81.6% of the individual recordings had a rate of decay lower than the 95% confidence intervals of their corresponding surrogates. On the causal entropy-complexity plane, human AF lay on the deterministic part of the plane that was located above the trajectory of fractional Brownian motion with different Hurst exponents on the plane. This analysis demonstrates that human AF dynamics does not arise from a rescaled linear stochastic process or a fractional noise, but either a deterministic or a nonlinear stochastic process. Our results justify the development and application of mathematical analysis and modeling tools to enable predictive control of human AF.

Predicting Surgery Targets in Temporal Lobe Epilepsy through Structural Connectome Based Simulations

Temporal lobe epilepsy (TLE) is a prevalent neurological disorder resulting in disruptive seizures. In the case of drug resistant epilepsy resective surgery is often considered. This is a procedure hampered by unpredictable success rates, with many patients continuing to have seizures even after surgery. In this study we apply a computational model of epilepsy to patient specific structural connectivity derived from diffusion tensor imaging (DTI) of 22 individuals with left TLE and 39 healthy controls. We validate the model by examining patient-control differences in simulated seizure onset time and network location. We then investigate the potential of the model for surgery prediction by performing in silico surgical resections, removing nodes from patient networks and comparing seizure likelihood post-surgery to pre-surgery simulations. We find that, first, patients tend to transit from non-epileptic to epileptic states more often than controls in the model. Second, regions in the left hemisphere (particularly within temporal and subcortical regions) that are known to be involved in TLE are the most frequent starting points for seizures in patients in the model. In addition, our analysis also implicates regions in the contralateral and frontal locations which may play a role in seizure spreading or surgery resistance. Finally, the model predicts that patient-specific surgery (resection areas chosen on an individual, model-prompted, basis and not following a predefined procedure) may lead to better outcomes than the currently used routine clinical procedure. Taken together this work provides a first step towards patient specific computational modelling of epilepsy surgery in order to inform treatment strategies in individuals.

Temporal lobe epilepsy (TLE) is a disorder characterised by unpredictable seizures, where surgical removal of brain tissue is often the final treatment option. In roughly 30% of cases surgery procedures are unsuccessful at preventing future seizures. This paper shows the application of a computational model which uses patient derived brain connectivity to predict the success rates of surgery in people with TLE. We consider the brains of 22 patients as networks, with brain regions as nodes and the white matter connections between them as edges. The brain network is unique to each subject and produced from brain imaging scans of 22 patients and 39 controls. Seizures are simulated before and after surgery, where surgery in the model is the removal of nodes from the network. The model successfully identifies regions known to be involved in TLE, and its predicted success rates for surgery are close to the results found in reality. The model additionally provides patient specific recommendations for surgical procedures, which in simulations show improved results compared to standard surgery in every case. This is a first step towards designing personalised surgery procedures in order to improve surgery success rates.

Resected Brain Tissue, Seizure Onset Zone and Quantitative EEG Measures: Towards Prediction of Post-Surgical Seizure Control

BACKGROUND

Epilepsy surgery is a potentially curative treatment option for pharmacoresistent patients. If non-invasive methods alone do not allow to delineate the epileptogenic brain areas the surgical candidates undergo long-term monitoring with intracranial EEG. Visual EEG analysis is then used to identify the seizure onset zone for targeted resection as a standard procedure.

METHODS

Despite of its great potential to assess the epileptogenicty of brain tissue, quantitative EEG analysis has not yet found its way into routine clinical practice. To demonstrate that quantitative EEG may yield clinically highly relevant information we retrospectively investigated how post-operative seizure control is associated with four selected EEG measures evaluated in the resected brain tissue and the seizure onset zone. Importantly, the exact spatial location of the intracranial electrodes was determined by coregistration of pre-operative MRI and post-implantation CT and coregistration with post-resection MRI was used to delineate the extent of tissue resection. Using data-driven thresholding, quantitative EEG results were separated into normally contributing and salient channels.

RESULTS

In patients with favorable post-surgical seizure control a significantly larger fraction of salient channels in three of the four quantitative EEG measures was resected than in patients with unfavorable outcome in terms of seizure control (median over the whole peri-ictal recordings). The same statistics revealed no association with post-operative seizure control when EEG channels contributing to the seizure onset zone were studied.

CONCLUSIONS

We conclude that quantitative EEG measures provide clinically relevant and objective markers of target tissue, which may be used to optimize epilepsy surgery. The finding that differentiation between favorable and unfavorable outcome was better for the fraction of salient values in the resected brain tissue than in the seizure onset zone is consistent with growing evidence that spatially extended networks might be more relevant for seizure generation, evolution and termination than a single highly localized brain region (i.e. a "focus") where seizures start.

Optimal control based seizure abatement using patient derived connectivity

Epilepsy is a neurological disorder in which patients have recurrent seizures. Seizures occur in conjunction with abnormal electrical brain activity which can be recorded by the electroencephalogram (EEG). Often, this abnormal brain activity consists of high amplitude regular spike-wave oscillations as opposed to low amplitude irregular oscillations in the non-seizure state. Active brain stimulation has been proposed as a method to terminate seizures prematurely, however, a general and widely-applicable approach to optimal stimulation protocols is still lacking. In this study we use a computational model of epileptic spike-wave dynamics to evaluate the effectiveness of a pseudospectral method to simulated seizure abatement. We incorporate brain connectivity derived from magnetic resonance imaging of a subject with idiopathic generalized epilepsy. We find that the pseudospectral method can successfully generate time-varying stimuli that abate simulated seizures, even when including heterogeneous patient specific brain connectivity. The strength of the stimulus required varies in different brain areas. Our results suggest that seizure abatement, modeled as an optimal control problem and solved with the pseudospectral method, offers an attractive approach to treatment for in vivo stimulation techniques. Further, if optimal brain stimulation protocols are to be experimentally successful, then the heterogeneity of cortical connectivity should be accounted for in the development of those protocols and thus more spatially localized solutions may be preferable.

Ordinal symbolic analysis and its application to biomedical recordings

Ordinal symbolic analysis opens an interesting and powerful perspective on time-series analysis. Here, we review this relatively new approach and highlight its relation to symbolic dynamics and representations. Our exposition reaches from the general ideas up to recent developments, with special emphasis on its applications to biomedical recordings. The latter will be illustrated with epilepsy data.

Discriminating chaotic and stochastic dynamics through the permutation spectrum test

In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism.

Ngram-derived pattern recognition for the detection and prediction of epileptic seizures

This work presents a new method that combines symbol dynamics methodologies with an Ngram algorithm for the detection and prediction of epileptic seizures. The presented approach specifically applies Ngram-based pattern recognition, after data pre-processing, with similarity metrics, including the Hamming distance and Needlman-Wunsch algorithm, for identifying unique patterns within epochs of time. Pattern counts within each epoch are used as measures to determine seizure detection and prediction markers. Using 623 hours of intracranial electrocorticogram recordings from 21 patients containing a total of 87 seizures, the sensitivity and false prediction/detection rates of this method are quantified. Results are quantified using individual seizures within each case for training of thresholds and prediction time windows. The statistical significance of the predictive power is further investigated. We show that the method presented herein, has significant predictive power in up to 100% of temporal lobe cases, with sensitivities of up to 70-100% and low false predictions (dependant on training procedure). The cases of highest false predictions are found in the frontal origin with 0.31-0.61 false predictions per hour and with significance in 18 out of 21 cases. On average, a prediction sensitivity of 93.81% and false prediction rate of approximately 0.06 false predictions per hour are achieved in the best case scenario. This compares to previous work utilising the same data set that has shown sensitivities of up to 40-50% for a false prediction rate of less than 0.15/hour.

Detecting functional hubs of ictogenic networks

Quantitative EEG (qEEG) has modified our understanding of epileptic seizures, shifting our view from the traditionally accepted hyper-synchrony paradigm toward more complex models based on re-organization of functional networks. However, qEEG measurements are so far rarely considered during the clinical decision-making process. To better understand the dynamics of intracranial EEG signals, we examine a functional network derived from the quantification of information flow between intracranial EEG signals. Using transfer entropy, we analyzed 198 seizures from 27 patients undergoing pre-surgical evaluation for pharmaco-resistant epilepsy. During each seizure we considered for each network the in-, out- and total "hubs", defined respectively as the time and the EEG channels with the maximal incoming, outgoing or total (bidirectional) information flow. In the majority of cases we found that the hubs occur around the middle of seizures, and interestingly not at the beginning or end, where the most dramatic EEG signal changes are found by visual inspection. For the patients who then underwent surgery, good postoperative clinical outcome was on average associated with a higher percentage of out- or total-hubs located in the resected area (for out-hubs p = 0.01, for total-hubs p = 0.04). The location of in-hubs showed no clear predictive value. We conclude that the study of functional networks based on qEEG measurements may help to identify brain areas that are critical for seizure generation and are thus potential targets for focused therapeutic interventions.

Unveiling the complex organization of recurrent patterns in spiking dynamical systems

Complex systems displaying recurrent spike patterns are ubiquitous in nature. Understanding the organization of these patterns is a challenging task. Here we study experimentally the spiking output of a semiconductor laser with feedback. By using symbolic analysis we unveil a nontrivial organization of patterns, revealing serial spike correlations. The probabilities of the patterns display a well-defined, hierarchical and clustered structure that can be understood in terms of a delayed model. Most importantly, we identify a minimal model, a modified circle map, which displays the same symbolic organization. The validity of this minimal model is confirmed by analyzing the output of the forced laser. Since the circle map describes many dynamical systems, including neurons and cardiac cells, our results suggest that similar correlations and hierarchies of patterns can be found in other systems. Our findings also pave the way for optical neurons that could provide a controllable set up to mimic neuronal activity.

Towards a large-scale model of patient-specific epileptic spike-wave discharges

Clinical electroencephalographic (EEG) recordings of the transition into generalised epileptic seizures show a sudden onset of spike-wave dynamics from a low-amplitude irregular background. In addition, non-trivial and variable spatio-temporal dynamics are widely reported in combined EEG/fMRI studies on the scale of the whole cortex. It is unknown whether these characteristics can be accounted for in a large-scale mathematical model with fixed heterogeneous long-range connectivities. Here, we develop a modelling framework with which to investigate such EEG features. We show that a neural field model composed of a few coupled compartments can serve as a low-dimensional prototype for the transition between irregular background dynamics and spike-wave activity. This prototype then serves as a node in a large-scale network with long-range connectivities derived from human diffusion-tensor imaging data. We examine multivariate properties in 42 clinical EEG seizure recordings from 10 patients diagnosed with typical absence epilepsy and 50 simulated seizures from the large-scale model using 10 DTI connectivity sets from humans. The model can reproduce the clinical feature of stereotypy where seizures are more similar within a patient than between patients, essentially creating a patient-specific fingerprint. We propose the approach as a feasible technique for the investigation of patient-specific large-scale epileptic features in space and time.

Nonrandomness, nonlinear dependence, and nonstationarity of electroencephalographic recordings from epilepsy patients

To derive tests for randomness, nonlinear-independence, and stationarity, we combine surrogates with a nonlinear prediction error, a nonlinear interdependence measure, and linear variability measures, respectively. We apply these tests to intracranial electroencephalographic recordings (EEG) from patients suffering from pharmacoresistant focal-onset epilepsy. These recordings had been performed prior to and independent from our study as part of the epilepsy diagnostics. The clinical purpose of these recordings was to delineate the brain areas to be surgically removed in each individual patient in order to achieve seizure control. This allowed us to define two distinct sets of signals: One set of signals recorded from brain areas where the first ictal EEG signal changes were detected as judged by expert visual inspection ("focal signals") and one set of signals recorded from brain areas that were not involved at seizure onset ("nonfocal signals"). We find more rejections for both the randomness and the nonlinear-independence test for focal versus nonfocal signals. In contrast more rejections of the stationarity test are found for nonfocal signals. Furthermore, while for nonfocal signals the rejection of the stationarity test increases the rejection probability of the randomness and nonlinear-independence test substantially, we find a much weaker influence for the focal signals. In consequence, the contrast between the focal and nonfocal signals obtained from the randomness and nonlinear-independence test is further enhanced when we exclude signals for which the stationarity test is rejected. To study the dependence between the randomness and nonlinear-independence test we include only focal signals for which the stationarity test is not rejected. We show that the rejection of these two tests correlates across signals. The rejection of either test is, however, neither necessary nor sufficient for the rejection of the other test. Thus, our results suggest that EEG signals from epileptogenic brain areas are less random, more nonlinear-dependent, and more stationary compared to signals recorded from nonepileptogenic brain areas. We provide the data, source code, and detailed results in the public domain.