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Much attention has been dedicated to clinical research of focal epilepsy, but the ability to derive a successful seizure control strategy based on unique dynamical features of the electroencephalogram is an unsolved problem. In this work, we introduce a basic model of spontaneous seizure dynamics and construct from it to a network model of focal-onset seizure dynamics. The full model is composed of coupled oscillators with scale-free network connectivity and a common slow variable. We find that global parameter changes and variation of the connectivity can drive the model from a quiescent state to recurrent seizures, and, eventually, to a permanent-seizure-state. Based on network synchronization features we design a stimulation scheme for the control of the fraction of nodes with strongest phase locking is proposed. Simulations lead to the identification of optimal stimuli for a given type of dynamics. Our results contribute to the development of a rational strategy for the non-surgical treatment of drug-resistant epilepsy.

Objective. Direct electrical stimulation of the brain through intracranial electrodes is currently used to probe the epileptic brain as part of pre-surgical evaluation, and it is also being considered for therapeutic treatments through neuromodulation. In order to effectively modulate neural activity, a given neuromodulation design must elicit similar responses throughout the course of treatment. However, it is unknown whether intracranial electrical stimulation responses are consistent across sessions. The objective of this study was to investigate the within-subject, cross-session consistency of the electrophysiological effect of electrical stimulation delivered through intracranial electroencephalography (iEEG). Approach. We analysed data from 79 epilepsy patients implanted with iEEG who underwent brain stimulation as part of a memory experiment. We quantified the effect of stimulation in terms of band power modulation and compared this effect from session to session. As a reference, we made the same measurements during baseline periods. Main results. In most sessions, the effect of stimulation on band power could not be distinguished from baseline fluctuations of band power. Stimulation effect was consistent in a third of the session pairs, while the rest had a consistency measure not exceeding the baseline standards. Cross-session consistency was highly correlated with the degree of band power increase, and it also tended to be higher when the baseline conditions were more similar between sessions. Significance. These findings can inform our practices for designing neuromodulation with greater efficacy when using direct electrical brain stimulation as a therapeutic treatment.

In presurgical monitoring, focal seizure onset is visually assessed from intracranial electroencephalogram (EEG), typically based on the selection of channels that show the strongest changes in amplitude and frequency. As epileptic seizure dynamics is increasingly considered to reflect changes in potentially distributed neural networks, it becomes important to also assess the interrelationships between channels. We propose a workflow to quantitatively extract the nodes and edges contributing to the seizure onset using an across-seizure scoring. We propose a quantification of the consistency of EEG channel contributions to seizure onset within a patient. The workflow is exemplified using recordings from patients with different degrees of seizure-onset consistency.

Given the complex temporal evolution of epileptic seizures, understanding their dynamic nature might be beneficial for clinical diagnosis and treatment. Yet, the mechanisms behind, for instance, the onset of seizures are still unknown. According to an existing classification, two basic types of dynamic onset patterns plus a number of more complex onset waveforms can be distinguished. Here, we introduce a basic three-variable model with two time scales to study potential mechanisms of spontaneous seizure onset. We expand the model to demonstrate how coupling of oscillators leads to more complex seizure onset waveforms. Finally, we test the response to pulse perturbation as a potential biomarker of interictal changes.

Epilepsy is defined as the brain’s susceptibility to recurrent, hypersynchronous discharges that disrupt normal neuronal function. Over the last decades, progress has been made in using dynamical systems theory and computational analyses to characterise the nature of seizure-like activity. Using simplified models of population dynamics, macroscale features of epileptic seizures can be described as expressions of model interactions. There is a trade-off between complexity of these models and their explanatory power: Models that represent biophysical components of the brain often contain many degrees of freedom and nonlinearities, which can make them challenging to interpret and often means that different model parameterisations can produce similar results. Simple models, on the other hand, do not usually have a direct correlate in brain anatomy or physiology, but rather capture more abstract quantities in the brain. However, the effects of individual parameters are easier to interpret. Here we suggest a design principle to generate the complex rhythmic evolution of tonic-clonic epileptic seizures in a neural population approach. Starting from a simple neuronal oscillator with a single nonlinearity , we show in a step-by-step analysis how complex neuronal dynamics derived from patient observations can be reconstructed.

Electroencephalography (EEG) allows recording of cortical activity at high temporal resolution. EEG recordings can be summarised along different dimensions using network-level quantitative measures, e.g. channel-to-channel correlation, or band power distributions across channels. These reveal network patterns that unfold over a range of different time scales and can be tracked dynamically.Here we describe the dynamics of network-state transitions in EEG recordings of spontaneous brain activity in normally developing infants and infants with severe early infantile epileptic encephalopathies (n=8, age: 1-8 months). We describe differences in measures of EEG dynamics derived from band power, and correlation-based summaries of network-wide brain activity.We further show that EEGs from different patient groups and controls can be distinguished based on a small set of the novel quantitative measures introduced here, which describe dynamic network state switching. Quantitative measures related to the smoothness of switching from one correlation pattern to another show the largest differences between groups.These findings reveal that the early epileptic encephalopathies are associated with characteristic dynamic features at the network level. Quantitative network-based analyses like the one presented here may in future inform the clinical use of quantitative EEG for diagnosis.

‘Big Data’ and a surge in quantitative methodologies increasingly urge bioresearchers to train and improve their mathematical and computational skills. In response to this challenge the Biotechnology and Biological Sciences Research Council in the United Kingdom offered funds to develop web-based training in mathematics and computation for life scientists in all fields. The SysMIC consortium won the bid and implemented a new type of online course, which so far has been delivered to more than 1400 scientists, mostly in the UK. Researchers train self-paced or in cohorts in a media rich environment with materials based around intuitive biological examples. We describe the course, its trainee spectrum, and the modes of trainee engagement. We propose that tailored interactive online training is a scalable model for continuing professional development in the interdisciplinary life sciences in the 21st century.

Brain-derived neurotrophic factor (BDNF) has recently been implicated in the modulation of receptor activation leading to dynamic state transitions in temporal lobe epilepsy (TLE). In addition, the crucial role of neuronal noise in these transitions has been studied in electrophysiological experiments. However, the precise role of these factors during seizure generation in TLE is not known. Building on a previously proposed model of an epileptogenic hippocampal network, we included the actions of BDNF-regulated receptors and intrinsic noise. We found that the effects of both BDNF and noise can increase the activation of N-methyl-D-aspartate receptors leading to excessive C a 2 + flux, which induces abnormal fast spiking and bursting. Our results indicate that the combined effects have a strong influence on the seizure-generating network, resulting in higher firing frequency and amplitude. As correlations between firing increase, the synchronization of the entire network increases, a marker of the ictogenic transitions from normal to seizures-like dynamics. Our work on the effects of BDNF dynamics in a noisy environment might lead to an improved model-based understanding of the pathological mechanisms in TLE.

In the postgenomic era, systems biology has rapidly emerged as an exciting field predicted to enhance the molecular understanding of complex biological systems by the use of quantitative experimental and mathematical approaches. Systems biology studies how the components of a biological system (e.g. genes, transcripts, proteins, metabolites) interact to bring about defined biological function or dysfunction. Living systems may be divided into five dimensions of complexity: (i) molecular; (ii) structural; (iii) temporal; (iv) abstraction and emergence; and (v) algorithmic. Understanding the details of these dimensions in living systems is the challenge that systems biology aims to address. Here, we argue that the hair follicle (HF), one of the signature features of mammals, is a perfect and clinically relevant model for systems biology research. The HF represents a stem cell-rich, essentially autonomous mini-organ, whose cyclic transformations follow a hypothetical intrafollicular "hair cycle clock" (HCC). This prototypic neuroectodermal-mesodermal interaction system, at the cross-roads of systems and chronobiology, encompasses various levels of complexity as it is subject to both intrafollicular and extrafollicular inputs (e.g. intracutaneous timing mechanisms with neural and systemic stimuli). Exploring how the cycling HF addresses the five dimensions of living systems, we argue that a systems biology approach to the study of hair growth and cycling, in man and mice, has great translational medicine potential. Namely, the easily accessible human HF invites preclinical and clinical testing of novel hypotheses generated with this approach.

Neuroblastoma (NB) is a common pediatric malignancy characterized by clinical and biological heterogeneity. A host of prognostic markers are available, contributing to accurate risk stratification and appropriate treatment allocation. Unfortunately, outcome is still poor for many patients, indicating the need for a new approach with enhanced utilization of the available biological data. Systems biology is a holistic approach in which all components of a biological system carry equal importance. Systems biology uses mathematical modeling and simulation to investigate dynamic interactions between system components, as a means of explaining overall system behavior. Systems biology can benefit the biomedical sciences by providing a more complete understanding of human disease, enhancing the development of targeted therapeutics. Systems biology is largely contiguous with current approaches in NB, which already employ an integrative and pseudo-holistic approach to disease management. Systems modeling of NB offers an optimal method for continuing progression in this field, and conferring additional benefit to current risk stratification and management. Likewise, NB provides an opportunity for systems biology to prove its utility in the context of human disease, since the biology of NB is comprehensively characterized and, therefore, suited to modeling. The purpose of this review is to outline the benefits, challenges and fundamental workings of systems modeling in human disease, using a specific example of bottom-up modeling in NB. The intention is to demonstrate practical requirements to begin bridging the gap between biological research and applied mathematical approaches for the mutual gain of both fields, and with additional benefits for clinical management. © 2010 Expert Reviews Ltd.

In multivariate time series analysis, the equal-time cross-correlation is a classic and computationally efficient measure for quantifying linear interrelations between data channels. When the cross-correlation coefficient is estimated using a finite amount of data points, its non-random part may be strongly contaminated by a sizable random contribution, such that no reliable conclusion can be drawn about genuine mutual interdependencies. The random correlations are determined by the signals' frequency content and the amount of data points used. Here, we introduce adjusted correlation matrices that can be employed to disentangle random from non-random contributions to each matrix element independently of the signal frequencies. Extending our previous work these matrices allow analyzing spatial patterns of genuine cross-correlation in multivariate data regardless of confounding influences. The performance is illustrated by example of model systems with known interdependence patterns. Finally, we apply the methods to electroencephalographic (EEG) data with epileptic seizure activity.

We study diffusively coupled excitable systems where local perturbations induces propagation waves only within finite frequency bands. Resonance-associated complex wave trains are found for parameters fixed near the subcritical Hopf bifurcation, and crisis-associated regular spiking near a supercritical Hopf bifurcation. Whereas in the subcritical case the qualitative excitation pattern of the coupled system is similar to that of an individual element, in the supercritical case the coupling works as a narrow band-pass filter for the external perturbation.

We study the spatio-temporal dynamics of a qualitative model of the glycolytic reaction catalyzed by phosphofructokinase. Adding the nonlinear regulation by fructose-2,6-bisphosphate, the model is found to exhibit complex oscillations and two types of chaos. In particular, a region of excitable chaos is found in parameter space. In a spatially extended system, this excitable chaos gives rise to spontaneous irregular spiking as well as self-organized formation of spots of high ATP concentration with “center-surround” characteristics. The deterministic spiking has noise-like temporal properties but shows a pronounced “memory”, which we attribute to the transient aparition of a Turing-pattern that exists nearby in parameter space. The model thus provides an example of an excitable medium that prohibits (rather than permits) the propagation of concentration waves. This could be exploited by the living cell to self-organize pronounced local amplification of ATP.

Active brain stimulation to abate epileptic seizures has shown mixed success. In spike-wave (SW) seizures, where the seizure and background state were proposed to coexist, single-pulse stimulations have been suggested to be able to terminate the seizure prematurely. However, several factors can impact success in such a bistable setting. The factors contributing to this have not been fully investigated on a theoretical and mechanistic basis. Our aim is to elucidate mechanisms that influence the success of single-pulse stimulation in noise-induced SW seizures. In this work, we study a neural population model of SW seizures that allows the reconstruction of the basin of attraction of the background activity as a four dimensional geometric object. For the deterministic (noise-free) case, we show how the success of response to stimuli depends on the amplitude and phase of the SW cycle, in addition to the direction of the stimulus in state space. In the case of spontaneous noise-induced seizures, the basin becomes probabilistic introducing some degree of uncertainty to the stimulation outcome while maintaining qualitative features of the noise-free case. Additionally, due to the different time scales involved in SW generation, there is substantial variation between SW cycles, implying that there may not be a fixed set of optimal stimulation parameters for SW seizures. In contrast, the model suggests an adaptive approach to find optimal stimulation parameters patient-specifically, based on real-time estimation of the position in state space. We discuss how the modelling work can be exploited to rationally design a successful stimulation protocol for the abatement of SW seizures using real-time SW detection. Published version

A time series from a human electroencephalogram (EEG) is used as a local perturbation to a reaction-diffusion model with spatiotemporal chaos. For certain finite ranges of amplitude and frequency it is observed that the strongly irregular perturbations can induce transient coherence in the chaotic system. This could be interpreted as "on-line" detection of an inherently correlated pattern embedded in the EEG.

A spatially extended model of diffusively coupled (bio)chemical oscillators creates spatio-temporal disorder (hyperchaos) characterized by complex and weakly correlated concentration patterns in space and time. Applying different types of local stimulation we show that the system forms stable patterns with well-defined global order for certain windows of parameters. Spatio-temporal hyperchaos thus supports efficient pattern control with minimum effort. A possible relevance of this mechanism for experimentally observed calcium wave patterns is pointed out.

We report current oscillations during the potentiostatic oxidation of iodide in aqueous solution. The oscillations become visible as columns of descending liquid with periodic structure colored by iodine (or iodine−starch complex). They can be attributed to a periodically varying flow of solution in front of the electrode due to a density gradient. The instability is explained using a qualitative model with a threshold function for the onset of the flow.

The kinetics of the peroxidase-catalyzed oxidation of iodide was found to be strongly nonlinear under certain conditions. At slightly acidic pH, the reaction behaves like an enzymatic “iodine clock...

The effect of various phenols and aromatic amines on the dynamics of the peroxidase-catalyzed aerobic oxidation of NADH was investigated. We demonstrate that several aromatic compounds may substitute for 2,4-dichlorophenol in inducing sustained oscillations and complex dynamics in the reaction. These aromatic compounds can also act as substrates in the classical peroxidase reaction. Since some of the aromatic compounds studied are naturally occurring substrates for peroxidases, we conjecture that oscillations and complex dynamics may well occur in the intact plant.

Spike-wave discharges (SWD) are a striking phenomena detectable on the electroencephalogram (EEG) of all patients during absence seizures. There is experimental and clinical evidence to suggest that seizures can be terminated early through the use of short auditory stimulation [1], however, stimulation protocols for seizure abatement are underdeveloped and their varied success is poorly understood. In this work we extend the model of [2] to account for known thalamocortical connectivity which has previously been implicated in SWD [3]. This model is capable of producing transient spike-wave trains upon perturbation, for example, through the inclusion of noise. We show that a single pulse perturbation during a simulated seizure can, if applied with the correct timing and amplitude, successfully terminate the seizure early (Figure. 1a). Furthermore, if the same stimulus is applied incorrectly (e.g. at a different time) the seizure could be prolonged (Figure. 1b). The complex phase and amplitude dependency of successful stimulation can be explained in the model with its nontrivial phase space configuration. The complex and sensitive dependency could account for the variations in success of different clinical and experimental stimulation studies. Our modeling approach makes the prediction that these optimal stimuli can be predicted through the use of a learning algorithm included in a closed-loop stimulation device as suggested by [1]. Successful clinical implementation and application of such a learning algorithm could have dramatic impact on epileptic patients and offer a potential alternative to anti-epileptic drug based therapy. The combination of animal experiments on seizure control of SWD [4,5] and the current model predictions regarding strength and timing could lead to improved translation into the clinical setting. Figure 1 (A) Successful single pulse stimulation applied at 0.15 seconds after seizure onset. Blue dashed line indicates the SWD duration without a stimulus. (B) Unsuccessful single pulsestimulation applied 0.2 seconds after seizure onset.

Pathological micro-domains have been proposed to underpin the generation of local pathological activity, as seen in focal seizures in the epileptic cortex [1-3]. Specifically, so-called micro-seizures have been suggested to be markers for these micro-domains [2,3]. Astonishingly, micro-seizures have also been observed in non-epileptic control patients [3]. This suggests that local activity, such as micro-seizures, only become pathological when in a specific arrangement. We hypothesize that pathological dynamics could be due to an increased density of micro-domains. To test this, we introduce a computational model on the mesoscopic scale of a 5 × 5 mm2 cortical sheet [4]. The units are modelled as excitable minicolumns. This model also incorporates realistic connectivity schemes observed at this spatial scale [5]. The model shows occasional, non-pathological micro-seizure occurrences, as well as recruitment of normal tissue into full-blown seizure activity in the presence of dense clusters of hyperactive micro-seizure domains. A specific prediction of this model is that the transition to full-blown seizures can be prevented by using micro-incisions to separate the clusters of abnormally active micro-domains (Figure. ​(Figure.1)1) [6]. Figure 1 First column indicated the location of the pathological clusters in black and the position of the micro-incision using the labeled line. The second column shows the temporal average activity in 10000 simulation steps. The third column indicates the spatial ...

Recent clinical and experimental evidence suggests that the spike-wave discharges (SWD) of absence seizures result from local activity within a hyperexcitable cortical region with rapid generalization through thalamocortical networks. The cortical focus is said to react more strongly to stimulation than other areas. We seek to develop a model which is in agreement with these recent experimental findings and suggest a possible explanation. In this study we extend an existing neural field model of thalamocortical interaction to account for multiple cortical regions which are connected according connectivity inferred from a clinically diagnosed epileptic patient. We stimulate at different model electrodes and investigate the resulting seizure duration. We observe that stimulation of only a small subset (11%) of model electrodes can lead to the rapid generalisation of SWD via both corticocortical and thalamocortical pathways. We find that the resulting model dynamics (seizure duration) varies significantly dependent upon the nodes stimulated and the amplitude of the stimulus. Our model indicates that heterogeneities in corticocortical connectivity could serve as a possible reason for the cortical focus and provides a platform for in silico hypothesis generation in complement to in vivo hypothesis validation. © 2013 IEEE.

We derive two kinetic models based on commonly occurring, simple enzymatic reactions. The first belongs to the class of activator‐inhibitor models, whereas the second is a Selkov‐type substrate‐depletion model. The bifurcation behavior of both models is studied in a spatially homogeneous environment. We consider one‐dimensional arrays of N oscillatory reaction cells coupled by diffusion. For small N we find two kinds of hyperchaos depending on a bifurcation parameter and the ratio of the diffusion coefficients of activator and inhibitor (Da/Di). For large N and Da/Di≳1, we observe spatiotemporally chaotic states characterized by phase defects. For Da/Di

The nonlinear behavior of the peroxidase-oxidase reaction was studied using structurally different peroxidases. For the first time sustained oscillations with peroxidases other than horseradish peroxidase in a single-enzyme system were observed. All peroxidases that showed significant oxidase activity were able to generate sustained oscillations. When adjusting the overall reaction rate, either of the two modifiers 2,4-dichlorophenol or Methylene blue could be omitted from the reaction. Due to the observation of different enzyme intermediates when using different peroxidases, we conclude that the mechanisms responsible for oscillatory kinetics may vary from one peroxidase to the other.

This application overview summarizes the ideas, techniques, and potential application domains of the sonification of the human electroencephalogram (EEG). We start by explaining why sonification is a particularly promising tool to understand EEG data; then we review early uses of sound for the investigation of the electric brain activity, followed by a design-oriented explanation of different techniques for sonifying EEG data, many of which have been introduced by the authors. We supply sonification examples on an accompanying Web site so that the information content and aesthetics of the sonifications can be directly perceived. Finally, we give an outlook on how EEG sonification can change clinical procedures in the near future, focusing particularly on the potential of interactive sonification.

We investigate the model of an enzymatic reaction with Michaelis-Menten kinetics and bell-shaped pH-dependence of the reaction rate. In the case of proton consumption the reaction can generate oscillations in a homogeneous reactor and turbulent spatio-temporal patterns in a reaction-diffusion environment. I n s t a b il it i e s in a S im p le E n z y m e R e a c tio n C a u s e d b y p H -D e p e n d e n c e *

The dynamics of the lateral leaflet movement of Desmodium motorium is studied. Simple periodic, quasiperiodic and aperiodic time series are observed. The long-scale dynamics may either be uniform or composed of several prototypic oscillations (one of them reminiscent of homoclinic chaos). Diffusively coupled nonlinear oscillators may account for the variety of ultradian rhythms.