Observers for Phenomenological Models of Epileptic Seizures.

Progress towards effective treatment of epileptic seizures has seen much improvement in the past decade. In particular, the emergence of phenomenological models of epileptic seizures specifically designed to capture the electrical seizure dynamics in the Epileptor model is inspiring new approaches to predicting and controlling seizures. These new models present in various forms and contain important but unmeasurable variables that control the occurrence of seizures. These models have been used mostly as nodes in large networks to study the complex brain behaviour of seizures. In order to use this model for the purposes of seizure forecasting or to control seizures through deep brain stimulation, the states of the model will need to be estimated. Although devices such as EEG electrodes can be related to some of the states of the model, most remain unmeasured and would require an observer (as defined in control theory) for their estimation. Additionally, we would like to consider the case for large nodes of systems where the number of electrodes is far smaller than the number of nodes being estimated. In this paper, we provide methods towards obtaining the full states of these phenomenological models using nonlinear observers. In particular, we explore the effectiveness of the Extended Kalman Filter for small networks of nodes of a smoothed sixth order Epileptor model. We show that observer design is possible for this family of systems and identify the difficulties in doing so.Clinical relevance-The methods presented here can be applied with an individual epileptic patient's EEG to reveal previously hidden biomarkers of epilepsy for seizure forecasting.

Determination of the electrical impedance of neural tissue from its microscopic cellular constituents.

The electrical properties of neural tissue are important in a range of different applications in biomedical engineering and basic science. These properties are characterized by the electrical admittivity of the tissue, which is the inverse of the specific tissue impedance. OBJECTIVE: Here we derived analytical expressions for the admittivity of various models of neural tissue from the underlying electrical and morphological properties of the constituent cells. APPROACH: Three models are considered: parallel bundles of fibers, fibers contained in stacked laminae and fibers crossing each other randomly in all three-dimensional directions. MAIN RESULTS: An important and novel aspect that emerges from considering the underlying cellular composition of the tissue is that the resulting admittivity has both spatial and temporal frequency dependence, a property not shared with conventional conductivity-based descriptions. The frequency dependence of the admittivity results in non-trivial spatiotemporal filtering of electrical signals in the tissue models. These effects are illustrated by considering the example of pulsatile stimulation with a point source electrode. It is shown how changing temporal parameters of a current pulse, such as pulse duration, alters the spatial profile of the extracellular potential. In a second example, it is shown how the degree of electrical anisotropy can change as a function of the distance from the electrode, despite the underlying structurally homogeneity of the tissue. These effects are discussed in terms of different current pathways through the intra- and extra-cellular spaces, and how these relate to near- and far-field limits for the admittivity (which reduce to descriptions in terms of a simple conductivity). SIGNIFICANCE: The results highlight the complexity of the electrical properties of neural tissue and provide mathematical methods to model this complexity.

Slow-Fast Duffing Neural Mass Model.

Epileptic seizures may be initiated by random neuronal fluctuations and/or by pathological slow regulatory dynamics of ion currents. This paper presents extensions to the Jansen and Rit neural mass model (JRNMM) to replicate paroxysmal transitions in intracranial electroencephalogram (iEEG) recordings. First, the Duffing NMM (DNMM) is introduced to emulate stochastic generators of seizures. The DNMM is constructed by applying perturbations to linear models of synaptic transmission in each neural population of the JRNMM. Then, the slow-fast DNMM is introduced by considering slow dynamics (relative to membrane potential and firing rate) of some internal parameters of the DNMM to replicate pathological evolution of ion currents. Through simulation, it is illustrated that the slow-fast DNMM exhibits transitions to and from seizures with etiologies that are linked either to random input fluctuations or pathological evolution of slow states. Estimation and optimization of a log likelihood function (LLF) using a continuous-discrete unscented Kalman filter (CD-UKF) and a genetic algorithm (GA) are performed to capture dynamics of iEEG data with paroxysmal transitions.

Identification of A Neural Mass Model of Burst Suppression.

Burst suppression includes alternating patterns of silent and fast spike activities in neuronal activities observable (in micro or macro scale) electro-physiological recordings. Biological models of burst suppression are given as dynamical systems with slow and fast states. The aim of this paper is to give a method to identify parameters of a mesoscopic model of burst suppression that can provide insights into study underlying generators of intracranial electroencephalogram (iEEG) data. An optimisation technique based upon a genetic algorithm (GA) is employed to find feasible model parameters to replicate burst patterns in the iEEG data with paroxysmal transitions. Then, a continuous-discrete unscented Kalman filter (CD-UKF) is used to infer hidden states of the model and to enhance the identification results from the GA. The results show promise in finding the model parameters of a partially observed mesoscopic model of burst suppression.

First report of aphaenopsoid trechines (Coleoptera: Carabidae: Trechini) from Serbia, with descriptions of new taxa.

The trechine ground beetle taxa, Velesaphaenops gen. n., Velesaphaenops tarensis sp. n., and Acheroniotes lethensis sp. n., are described and diagnosed. The new taxa differ clearly from their closest relatives in a number of external characters and the shape of the genitalia. They probably belong to phyletic lineages of Pliocene age (the age of the palaeokarst of Kamena Gora and Mt. Tara). The new taxa are endemic to western and southwestern Serbia. Keys to the aphaenopsoid trechine genera in Serbia and to species of the genus Acheroniotes Lohaj Lakota, 2010 are appended.

Bifurcation analysis of two coupled Jansen-Rit neural mass models.

We investigate how changes in network structure can lead to pathological oscillations similar to those observed in epileptic brain. Specifically, we conduct a bifurcation analysis of a network of two Jansen-Rit neural mass models, representing two cortical regions, to investigate different aspects of its behavior with respect to changes in the input and interconnection gains. The bifurcation diagrams, along with simulated EEG time series, exhibit diverse behaviors when varying the input, coupling strength, and network structure. We show that this simple network of neural mass models can generate various oscillatory activities, including delta wave activity, which has not been previously reported through analysis of a single Jansen-Rit neural mass model. Our analysis shows that spike-wave discharges can occur in a cortical region as a result of input changes in the other region, which may have important implications for epilepsy treatment. The bifurcation analysis is related to clinical data in two case studies.

Spike history neural response model.

There is a potential for improved efficacy of neural stimulation if stimulation levels can be modified dynamically based on the responses of neural tissue in real time. A neural model is developed that describes the response of neurons to electrical stimulation and that is suitable for feedback control neuroprosthetic stimulation. Experimental data from NZ white rabbit retinae is used with a data-driven technique to model neural dynamics. The linear-nonlinear approach is adapted to incorporate spike history and to predict the neural response of ganglion cells to electrical stimulation. To validate the fitness of the model, the penalty term is calculated based on the time difference between each simulated spike and the closest spike in time in the experimentally recorded train. The proposed model is able to robustly predict experimentally observed spike trains.

Estimation of effective connectivity via data-driven neural modeling.

This research introduces a new method for functional brain imaging via a process of model inversion. By estimating parameters of a computational model, we are able to track effective connectivity and mean membrane potential dynamics that cannot be directly measured using electrophysiological measurements alone. The ability to track the hidden aspects of neurophysiology will have a profound impact on the way we understand and treat epilepsy. For example, under the assumption the model captures the key features of the cortical circuits of interest, the framework will provide insights into seizure initiation and termination on a patient-specific basis. It will enable investigation into the effect a particular drug has on specific neural populations and connectivity structures using minimally invasive measurements. The method is based on approximating brain networks using an interconnected neural population model. The neural population model is based on a neural mass model that describes the functional activity of the brain, capturing the mesoscopic biophysics and anatomical structure. The model is made subject-specific by estimating the strength of intra-cortical connections within a region and inter-cortical connections between regions using a novel Kalman filtering method. We demonstrate through simulation how the framework can be used to track the mechanisms involved in seizure initiation and termination.

Electrical stimulation of neural tissue modeled as a cellular composite: point source electrode in an isotropic tissue.

Standard volume conductor models of neural electrical stimulation assume that the electrical properties of the tissue are well described by a conductivity that is smooth and homogeneous at a microscopic scale. However, neural tissue is composed of tightly packed cells whose membranes have markedly different electrical properties to either the intra- or extracellular space. Consequently, the electrical properties of tissue are highly heterogeneous at the microscopic scale: a fact not accounted for in standard volume conductor models. Here we apply a recently developed framework for volume conductor models that accounts for the cellular composition of tissue. We consider the case of a point source electrode in tissue comprised of neural fibers crossing each other equally in all directions. We derive the tissue admittivity (that replaces the standard tissue conductivity) from single cell properties, and then calculate the extracellular potential. Our findings indicate that the cellular composition of tissue affects the spatiotemporal profile of the extracellular potential. In particular, the full solution asymptotically approaches a near-field limit close to the electrode and a far-field limit far from the electrode. The near-field and far-field approximations are solutions to standard volume conductor models, but differ from each other by nearly an order or magnitude. Consequently the full solution is expected to provide a more accurate estimate of electrical potentials over the full range of electrode-neurite separations.

A neural mass model of spontaneous burst suppression and epileptic seizures.

The paper presents a neural mass model that is capable of simulating the transition to and from various forms of paroxysmal activity such as burst suppression and epileptic seizure-like waveforms. These events occur without changing parameters in the model. The model is based on existing neural mass models, with the addition of feedback of fast dynamics to create slowly time varying parameters, or slow states. The goal of this research is to establish a link between system properties that modulate neural activity and the fast changing dynamics, such as membrane potentials and firing rates that can be manipulated using electrical stimulation. Establishing this link is likely to be a necessary component of a closed-loop system for feedback control of pathological neural activity.

Estimating the unmeasured membrane potential of neuronal populations from the EEG using a class of deterministic nonlinear filters.

We present a model-based estimation method to reconstruct the unmeasured membrane potential of neuronal populations from a single-channel electroencephalographic (EEG) measurement. We consider a class of neural mass models that share a general structure, specifically the models by Stam et al (1999 Clin. Neurophysiol. 110 1801-13), Jansen and Rit (1995 Biol. Cybern. 73 357-66) and Wendling et al (2005 J. Clin. Neurophysiol. 22 343). Under idealized assumptions, we prove the global exponential convergence of our filter. Then, under more realistic assumptions, we investigate the robustness of our filter against model uncertainties and disturbances. Analytic proofs are provided for all results and our analyses are further illustrated via simulations.

Electrical probing of cortical excitability in patients with epilepsy.

Standard methods for seizure prediction involve passive monitoring of intracranial electroencephalography (iEEG) in order to track the 'state' of the brain. This paper introduces a new method for measuring cortical excitability using an electrical probing stimulus. Electrical probing enables feature extraction in a more robust and controlled manner compared to passively tracking features of iEEG signals. The probing stimuli consist of 100 bi-phasic pulses, delivered every 10 min. Features representing neural excitability are estimated from the iEEG responses to the stimuli. These features include the amplitude of the electrically evoked potential, the mean phase variance (univariate), and the phase-locking value (bivariate). In one patient, it is shown how the features vary over time in relation to the sleep-wake cycle and an epileptic seizure. For a second patient, it is demonstrated how the features vary with the rate of interictal discharges. In addition, the spatial pattern of increases and decreases in phase synchrony is explored when comparing periods of low and high interictal discharge rates, or sleep and awake states. The results demonstrate a proof-of-principle for the method to be applied in a seizure anticipation framework. This article is part of a Supplemental Special Issue entitled The Future of Automated Seizure Detection and Prediction.

On extremum seeking in bioprocesses with multivalued cost functions.

Finding optimal operating modes for bioprocesses has been, for a long time, a relevant issue in bioengineering. The problem is of special interest when it implies the simultaneous optimization of competing objectives. In this paper, we address the problem of finding optimal steady states that achieve the best tradeoff between yield and productivity by using nonmodel-based extremum-seeking control with semiglobal practical stability and convergence properties. A special attention is paid to processes with multiple steady states and multivalued cost functions.