A biologically constrained, mathematical model of cortical wave propagation preceding seizure termination

Dynamic evolution of the anterior cingulate-insula network during seizures

OBJECTIVES

In physiological situations, the anterior cingulate cortex (ACC) and anterior insular cortex (AIC) are prone to coactivation. The functional connectivity and interaction between ACC and AIC in the context of epilepsy remain unclear. This study aimed to investigate the dynamic coupling between these two brain regions during seizures.

METHODS

Patients who underwent stereoelectroencephalography (SEEG) recording were included in this study. The SEEG data were visually inspected and quantitatively analyzed. The narrowband oscillations and aperiodic components at seizure onset were parameterized. The frequency-specific non-linear correlation analysis was applied to the functional connectivity. The excitation/inhibition ratio (E:I ratio) reflected by the aperiodic slope was performed to evaluate the excitability.

RESULTS

Twenty patients were included in the study, with 10 diagnosed with anterior cingulate epilepsy and 10 with anterior insular epilepsy. In both types of epilepsy, the correlation coefficient (h2 ) between the ACC and AIC at seizure onset exhibited a significantly higher value than that during interictal and preictal periods (p < 0.05). The direction index (D) showed a significant increase at seizure onset, serving as an indicator for the direction of information flow between these two brain regions with up to 90% accuracy. The E:I ratio increased significantly at seizure onset, with the seizure-onset zone (SOZ) demonstrating a more pronounced increase compared to non-SOZ (p < 0.05). For seizures originating from AIC, the E:I ratio was significantly higher in the AIC than in the ACC (p = 0.0364).

CONCLUSIONS

In the context of epilepsy, the ACC and AIC are dynamically coupled during seizures. The functional connectivity and excitability exhibit a significant increase at seizure onset. By analyzing connectivity and excitability, the SOZ in ACC and AIC can be identified. The direction index (D) serves as an indicator for the direction of information flow from SOZ to non-SOZ. Notably, the excitability of SOZ changes more significantly than that of non-SOZ.

The inhibitory control of traveling waves in cortical networks

Propagating waves of activity can be evoked and can occur spontaneously in vivo and in vitro in cerebral cortex. These waves are thought to be instrumental in the propagation of information across cortical regions and as a means to modulate the sensitivity of neurons to subsequent stimuli. In normal tissue, the waves are sparse and tightly controlled by inhibition and other negative feedback processes. However, alterations of this balance between excitation and inhibition can lead to pathological behavior such as seizure-type dynamics (with low inhibition) or failure to propagate (with high inhibition). We develop a spiking one-dimensional network of neurons to explore the reliability and control of evoked waves and compare this to a cortical slice preparation where the excitability can be pharmacologically manipulated. We show that the waves enhance sensitivity of the cortical network to stimuli in specific spatial and temporal ways. To gain further insight into the mechanisms of propagation and transitions to pathological behavior, we derive a mean-field model for the synaptic activity. We analyze the mean-field model and a piece-wise constant approximation of it and study the stability of the propagating waves as spatial and temporal properties of the inhibition are altered. We show that that the transition to seizure-like activity is gradual but that the loss of propagation is abrupt and can occur via either the loss of existence of the wave or through a loss of stability leading to complex patterns of propagation.

Alteration of Excitation/Inhibition Imbalance in the Hippocampus and Amygdala of Drug-Resistant Epilepsy Patients Treated with Acute Vagus Nerve Stimulation

An imbalance between excitation (E) and inhibition (I) in the brain has been identified as a key pathophysiology of epilepsy over the years. The hippocampus and amygdala in the limbic system play a crucial role in the initiation and conduction of epileptic seizures and are often referred to as the transfer station and amplifier of seizure activities. Existing animal and imaging studies reveal that the hippocampus and amygdala, which are significant parts of the vagal afferent network, can be modulated in order to generate an antiepileptic effect. Using stereo-electroencephalography (SEEG) data, we examined the E/I imbalance in the hippocampus and amygdala of ten drug-resistant epilepsy children treated with acute vagus nerve stimulation (VNS) by estimating the 1/f power slope of hippocampal and amygdala signals in the range of 1-80 Hz. While the change in the 1/f power slope from VNS-BASE varied between different stimulation amplitudes and brain regions, it was more prominent in the hippocampal region. In the hippocampal region, we found a flatter 1/f power slope during VNS-ON in patients with good responsiveness to VNS under the optimal stimulation amplitude, indicating that the E/I imbalance in the region was improved. There was no obvious change in 1/f power slope for VNS poor responders. For VNS non-responders, the 1/f power slope slightly increased when the stimulation was applied. Overall, this study implies that the regulation of E/I imbalance in the epileptic brain, especially in the hippocampal region, may be an acute intracranial effect of VNS.

Interictal discharges in the human brain are travelling waves arising from an epileptogenic source

While seizure activity may be electrographically widespread, increasing evidence has suggested that ictal discharges may in fact represent travelling waves propagated from a focal seizure source. Interictal epileptiform discharges (IEDs) are an electrographic manifestation of excessive hypersynchronization of cortical activity that occur between seizures and are considered a marker of potentially epileptogenic tissue. The precise relationship between brain regions demonstrating IEDs and those involved in seizure onset, however, remains poorly understood. Here, we hypothesize that IEDs likewise reflect the receipt of travelling waves propagated from the same regions which give rise to seizures. Forty patients from our institution who underwent invasive monitoring for epilepsy, proceeded to surgery and had at least one year of follow-up were included in our study. Interictal epileptiform discharges were detected using custom software, validated by a clinical epileptologist. We show that IEDs reach electrodes in sequences with a consistent temporal ordering, and this ordering matches the timing of receipt of ictal discharges, suggesting that both types of discharges spread as travelling waves. We use a novel approach for localization of ictal discharges, in which time differences of discharge receipt at nearby electrodes are used to compute source location; similar algorithms have been used in acoustics and geophysics. We find that interictal discharges co-localize with ictal discharges. Moreover, interictal discharges tend to localize to the resection territory in patients with good surgical outcome and outside of the resection territory in patients with poor outcome. The seizure source may originate at, and also travel to, spatially distinct IED foci. Our data provide evidence that interictal discharges may represent travelling waves of pathological activity that are similar to their ictal counterparts, and that both ictal and interictal discharges emerge from common epileptogenic brain regions. Our findings have important clinical implications, as they suggest that seizure source localizations may be derived from interictal discharges, which are much more frequent than seizures.

Multiple Sources of Fast Traveling Waves during Human Seizures: Resolving a Controversy

During human seizures, organized waves of voltage activity rapidly sweep across the cortex. Two contradictory theories describe the source of these fast traveling waves: either a slowly advancing narrow region of multiunit activity (an ictal wavefront) or a fixed cortical location. Limited observations and different analyses prevent resolution of these incompatible theories. Here we address this disagreement by combining the methods and microelectrode array recordings (N = 11 patients, 2 females, N = 31 seizures) from previous human studies to analyze the traveling wave source. We find, inconsistent with both existing theories, a transient relationship between the ictal wavefront and traveling waves, and multiple stable directions of traveling waves in many seizures. Using a computational model that combines elements of both existing theories, we show that interactions between an ictal wavefront and fixed source reproduce the traveling wave dynamics observed in vivo We conclude that combining both existing theories can generate the diversity of ictal traveling waves.SIGNIFICANCE STATEMENT The source of voltage discharges that propagate across cortex during human seizures remains unknown. Two candidate theories exist, each proposing a different discharge source. Support for each theory consists of observations from a small number of human subject recordings, analyzed with separately developed methods. How the different, limited data and different analysis methods impact the evidence for each theory is unclear. To resolve these differences, we combine the unique, human microelectrode array recordings collected separately for each theory and analyze these combined data with a unified approach. We show that neither existing theory adequately describes the data. We then propose a new theory that unifies existing proposals and successfully reproduces the voltage discharge dynamics observed in vivo.

Interictal SEEG Resting-State Connectivity Localizes the Seizure Onset Zone and Predicts Seizure Outcome

Localization of epileptogenic zone currently requires prolonged intracranial recordings to capture seizure, which may take days to weeks. The authors developed a novel method to identify the seizure onset zone (SOZ) and predict seizure outcome using short-time resting-state stereotacticelectroencephalography (SEEG) data. In a cohort of 27 drug-resistant epilepsy patients, the authors estimated the information flow via directional connectivity and inferred the excitation-inhibition ratio from the 1/f power slope. They hypothesized that the antagonism of information flow at multiple frequencies between SOZ and non-SOZ underlying the relatively stable epilepsy resting state could be related to the disrupted excitation-inhibition balance. They found flatter 1/f power slope in non-SOZ regions compared to the SOZ, with dominant information flow from non-SOZ to SOZ regions. Greater differences in resting-state information flow between SOZ and non-SOZ regions are associated with favorable seizure outcome. By integrating a balanced random forest model with resting-state connectivity, their method localized the SOZ with an accuracy of 88% and predicted the seizure outcome with an accuracy of 92% using clinically determined SOZ. Overall, this study suggests that brief resting-state SEEG data can significantly facilitate the identification of SOZ and may eventually predict seizure outcomes without requiring long-term ictal recordings.

Fractional-Order Traveling Wave Approximations for a Fractional-Order Neural Field Model

In this work, we establish a fractional-order neural field mathematical model with Caputo's fractional derivative temporal order α considering 0 < α < 2, to analyze the effect of fractional-order on cortical wave features observed preceding seizure termination. The importance of this incorporation relies on the theoretical framework established by fractional-order derivatives in which memory and hereditary properties of a system are considered. Employing Mittag-Leffler functions, we first obtain approximate fractional-order solutions that provide information about the initial wave dynamics in a fractional-order frame. We then consider the Adomian decomposition method to approximate pulse solutions in a wider range of orders and longer times. The former approach establishes a direct way to investigate the initial relationships between fractional-order and wave features, such as wave speed and wave width. In contrast, the latter approach displays wave propagation dynamics in different fractional orders for longer times. Using the previous two approaches, we establish approximate wave solutions with characteristics consistent with in vivo cortical waves preceding seizure termination. In our analysis, we find consistent differences in the initial effect of the fractional-order on the features of wave speed and wave width, depending on whether α <1 or α>1. Both cases can model the shape of cortical wave propagation for different fractional-orders at the cost of modifying the wave speed. Our results also show that the effect of fractional-order on wave width depends on the synaptic threshold and the synaptic connectivity extent. Fractional-order derivatives have been interpreted as the memory trace of the system. This property and the results of our analysis suggest that fractional-order derivatives and neuronal collective memory modify cortical wave features.

Impact of gyral geometry on cortical responses to surface electrical stimulation: insights from experimental and modeling studies

Objective.Invasive simultaneous stimulation and recording from intracranial electrodes and microwire arrays were used to investigate direct cortical responses to single pulses of electrical stimulation in humans.Approach.Microwire contacts measured surface potentials in cortical microdomains at a distance of 2-6 mm from the intracranial electrode. Direct cortical responses to stimulation (<20 ms) consisted of a larger surface negative potentials.Main results. The latencies of these responses were directly or inversely correlated with distances between the intracranial electrode and microwire contacts. We hypothesize that surface negative potentials reflected local synchronous depolarization of apical dendrites of pyramidal neurons in cortical microdomains in the superficial cortical layer and resulted from the activation of gray matter axons that delivered excitatory inputs to apical dendrites after cortical stimulation. We further hypothesized that the positive or inverse distance-latency correlations of the recorded negative responses were measured depending on whether activation of neurons originated at one (crown) or multiple (crown, lip, bank) sites throughout the gyrus simultaneously. The inverse distance-latency correlations then reflected the spatiotemporal superposition of different nearby sources of neuronal recruitment in the gyrus. To prove this hypothesis, we built an anatomically informed and biophysically realistic cortical network model and simulated early responses of cortical neurons to electrical stimulation in this cortical network model. The model simulations yielded negative potentials in simulated microdomains in the cortical model consistent with those recorded from humans. The model predicted sensitivity of cortical responses to the alignment of the stimulating electrode and microwire array with respect to the cortical gyrus and confirmed that gyral geometry has a major impact on direct neuronal recruitment, the timing, and the time course of neuronal activation in cortical microdomains.Significance.In this work, we demonstrated how the high-resolution forward network models can be used for better understanding and detailed prediction of cortical stimulation effects. Accurate predictive modeling tools are needed for the progress of brain stimulation therapies.

The impact of a closed-loop thalamocortical model on the spatiotemporal dynamics of cortical and thalamic traveling waves

Propagation of activity in spatially structured neuronal networks has been observed in awake, anesthetized, and sleeping brains. How these wave patterns emerge and organize across brain structures, and how network connectivity affects spatiotemporal neural activity remains unclear. Here, we develop a computational model of a two-dimensional thalamocortical network, which gives rise to emergent traveling waves similar to those observed experimentally. We illustrate how spontaneous and evoked oscillatory activity in space and time emerge using a closed-loop thalamocortical architecture, sustaining smooth waves in the cortex and staggered waves in the thalamus. We further show that intracortical and thalamocortical network connectivity, cortical excitation/inhibition balance, and thalamocortical or corticothalamic delay can independently or jointly change the spatiotemporal patterns (radial, planar and rotating waves) and characteristics (speed, direction, and frequency) of cortical and thalamic traveling waves. Computer simulations predict that increased thalamic inhibition induces slower cortical frequencies and that enhanced cortical excitation increases traveling wave speed and frequency. Overall, our results provide insight into the genesis and sustainability of thalamocortical spatiotemporal patterns, showing how simple synaptic alterations cause varied spontaneous and evoked wave patterns. Our model and simulations highlight the need for spatially spread neural recordings to uncover critical circuit mechanisms for brain functions.

Travelling waves reveal a dynamic seizure source in human focal epilepsy

Treatment of patients with drug-resistant focal epilepsy relies upon accurate seizure localization. Ictal activity captured by intracranial EEG has traditionally been interpreted to suggest that the underlying cortex is actively involved in seizures. Here, we hypothesize that such activity instead reflects propagated activity from a relatively focal seizure source, even during later time points when ictal activity is more widespread. We used the time differences observed between ictal discharges in adjacent electrodes to estimate the location of the hypothesized focal source and demonstrated that the seizure source, localized in this manner, closely matches the clinically and neurophysiologically determined brain region giving rise to seizures. Moreover, we determined this focal source to be a dynamic entity that moves and evolves over the time course of a seizure. Our results offer an interpretation of ictal activity observed by intracranial EEG that challenges the traditional conceptualization of the seizure source.

Investigating the role of gap junctions in seizure wave propagation

The effect of gap junctions as well as the biological mechanisms behind seizure wave propagation is not completely understood. In this work, we use a simple neural field model to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination. We consider a voltage-based neural field model consisting of an excitatory and an inhibitory population as well as both chemical and gap junction-like synapses. We are able to approximate important properties of cortical wave propagation previously observed in vivo before seizure termination. This model adds support to existing evidence from models and clinical data suggesting a key role of gap junctions in seizure wave propagation. In particular, we found that in this model gap junction-like connectivity determines the propagation of one-bump or two-bump traveling wave solutions with features consistent with the clinical data. For sufficiently increased gap junction connectivity, wave solutions cease to exist. Moreover, gap junction connectivity needs to be sufficiently low or moderate to permit the existence of linearly stable solutions of interest.

Neural field models for latent state inference: Application to large-scale neuronal recordings

Large-scale neural recording methods now allow us to observe large populations of identified single neurons simultaneously, opening a window into neural population dynamics in living organisms. However, distilling such large-scale recordings to build theories of emergent collective dynamics remains a fundamental statistical challenge. The neural field models of Wilson, Cowan, and colleagues remain the mainstay of mathematical population modeling owing to their interpretable, mechanistic parameters and amenability to mathematical analysis. Inspired by recent advances in biochemical modeling, we develop a method based on moment closure to interpret neural field models as latent state-space point-process models, making them amenable to statistical inference. With this approach we can infer the intrinsic states of neurons, such as active and refractory, solely from spiking activity in large populations. After validating this approach with synthetic data, we apply it to high-density recordings of spiking activity in the developing mouse retina. This confirms the essential role of a long lasting refractory state in shaping spatiotemporal properties of neonatal retinal waves. This conceptual and methodological advance opens up new theoretical connections between mathematical theory and point-process state-space models in neural data analysis.

Developing statistical tools to connect single-neuron activity to emergent collective dynamics is vital for building interpretable models of neural activity. Neural field models relate single-neuron activity to emergent collective dynamics in neural populations, but integrating them with data remains challenging. Recently, latent state-space models have emerged as a powerful tool for constructing phenomenological models of neural population activity. The advent of high-density multi-electrode array recordings now enables us to examine large-scale collective neural activity. We show that classical neural field approaches can yield latent state-space equations and demonstrate that this enables inference of the intrinsic states of neurons from recorded spike trains in large populations.

Estimating neuronal conductance model parameters using dynamic action potential clamp

BACKGROUND

Parameterization of neuronal membrane conductance models relies on data acquired from current clamp (CC) or voltage clamp (VC) recordings. Although the CC approach provides key information on a neuron's firing properties, it is often difficult to disentangle the influence of multiple conductances that contribute to the excitation properties of a real neuron. Isolation of a single conductance using pharmacological agents or heterologous expression simplifies analysis but requires extensive VC evaluation to explore the complete state behavior of the channel of interest.

NEW METHOD

We present an improved parameterization approach that uses data derived from dynamic action potential clamp (DAPC) recordings to extract conductance equation parameters. We demonstrate the utility of the approach by applying it to the standard Hodgkin-Huxley conductance model although other conductance models could be easily incorporated as well.

RESULTS

Using a fully simulated setup we show that, with as few as five action potentials previously recorded in DAPC mode, sodium conductance equation parameters can be determined with average parameter errors of less than 4% while action potential firing accuracy approaches 100%. In real DAPC experiments, we show that by "training" our model with five or fewer action potentials, subsequent firing lasting for several seconds could be predicted with ˜96% mean firing rate accuracy and 94% temporal overlap accuracy.

COMPARISON WITH EXISTING METHODS

Our DAPC-based approach surpasses the accuracy of VC-based approaches for extracting conductance equation parameters with a significantly reduced temporal overhead.

CONCLUSION

DAPC-based approach will facilitate the rapid and systematic characterization of neuronal channelopathies.

Multiscale recordings reveal the dynamic spatial structure of human seizures

The cellular activity underlying human focal seizures, and its relationship to key signatures in the EEG recordings used for therapeutic purposes, has not been well characterized despite many years of investigation both in laboratory and clinical settings. The increasing use of microelectrodes in epilepsy surgery patients has made it possible to apply principles derived from laboratory research to the problem of mapping the spatiotemporal structure of human focal seizures, and characterizing the corresponding EEG signatures. In this review, we describe results from human microelectrode studies, discuss some data interpretation pitfalls, and explain the current understanding of the key mechanisms of ictogenesis and seizure spread.

A Proposed Mechanism for Spontaneous Transitions between Interictal and Ictal Activity

Epileptic networks are characterized by two outputs: brief interictal spikes and rarer, more prolonged seizures. Although either output state is readily modeled in silico and induced experimentally, the transition mechanisms are unknown, in part because no models exhibit both output states spontaneously. In silico small-world neural networks were built using single-compartment neurons whose physiological parameters were derived from dual whole-cell recordings of pyramidal cells in organotypic hippocampal slice cultures that were generating spontaneous seizure-like activity. In silico, neurons were connected by abundant local synapses and rare long-distance synapses. Activity-dependent synaptic depression and gradual recovery delimited synchronous activity. Full synaptic recovery engendered interictal population spikes that spread via long-distance synapses. When synaptic recovery was incomplete, postsynaptic neurons required coincident activation of multiple presynaptic terminals to reach firing threshold. Only local connections were sufficiently dense to spread activity under these conditions. This coalesced network activity into traveling waves whose velocity varied with synaptic recovery. Seizures were comprised of sustained traveling waves that were similar to those recorded during experimental and human neocortical seizures. Sustained traveling waves occurred only when wave velocity, network dimensions, and the rate of synaptic recovery enabled wave reentry into previously depressed areas at precisely ictogenic levels of synaptic recovery. Wide-field, cellular-resolution GCamP7b calcium imaging demonstrated similar initial patterns of activation in the hippocampus, although the anatomical distribution of traveling waves of synaptic activation was altered by the pattern of synaptic connectivity in the organotypic hippocampal cultures.SIGNIFICANCE STATEMENT When computerized distributed neural network models are required to generate both features of epileptic networks (i.e., spontaneous interictal population spikes and seizures), the network structure is substantially constrained. These constraints provide important new hypotheses regarding the nature of epileptic networks and mechanisms of seizure onset.

Spatial Embedding Imposes Constraints on Neuronal Network Architectures

Recent progress towards understanding circuit function has capitalized on tools from network science to parsimoniously describe the spatiotemporal architecture of neural systems. Such tools often address systems topology divorced from its physical instantiation. Nevertheless, for embedded systems such as the brain, physical laws directly constrain the processes of network growth, development, and function. We review here the rules imposed by the space and volume of the brain on the development of neuronal networks, and show that these rules give rise to a specific set of complex topologies. These rules also affect the repertoire of neural dynamics that can emerge from the system, and thereby inform our understanding of network dysfunction in disease. We close by discussing new tools and models to delineate the effects of spatial embedding.

A neural mass model to predict electrical stimulation evoked responses in human and non-human primate brain

OBJECTIVE

Deep brain stimulation (DBS) is a valuable tool for ameliorating drug resistant pathologies such as movement disorders and epilepsy. DBS is also being considered for complex neuro-psychiatric disorders, which are characterized by high variability in symptoms and slow responses that hinder DBS setting optimization. The objective of this work was to develop an in silico platform to examine the effects of electrical stimulation in regions neighboring a stimulated brain region.

APPROACH

We used the Jansen-Rit neural mass model of single and coupled nodes to simulate the response to a train of electrical current pulses at different frequencies (10-160 Hz) of the local field potential recorded in the amygdala and cortical structures in human subjects and a non-human primate.

RESULTS

We found that using a single node model, the evoked responses could be accurately modeled following a narrow range of stimulation frequencies. Including a second coupled node increased the range of stimulation frequencies whose evoked responses could be efficiently modeled. Furthermore, in a chronic recording from a non-human primate, features of the in vivo evoked response remained consistent for several weeks, suggesting that model re-parameterization for chronic stimulation protocols would be infrequent.

SIGNIFICANCE

Using a model of neural population activity, we reproduced the evoked response to cortical and subcortical stimulation in human and non-human primate. This modeling framework provides an environment to explore, safely and rapidly, a wide range of stimulation settings not possible in human brain stimulation studies. The model can be trained on a limited dataset of stimulation responses to develop an optimal stimulation strategy for an individual patient.

Multivariate regression methods for estimating velocity of ictal discharges from human microelectrode recordings

OBJECTIVE

Epileptiform discharges, an electrophysiological hallmark of seizures, can propagate across cortical tissue in a manner similar to traveling waves. Recent work has focused attention on the origination and propagation patterns of these discharges, yielding important clues to their source location and mechanism of travel. However, systematic studies of methods for measuring propagation are lacking.

APPROACH

We analyzed epileptiform discharges in microelectrode array recordings of human seizures. The array records multiunit activity and local field potentials at 400 micron spatial resolution, from a small cortical site free of obstructions. We evaluated several computationally efficient statistical methods for calculating traveling wave velocity, benchmarking them to analyses of associated neuronal burst firing.

MAIN RESULTS

Over 90% of discharges met statistical criteria for propagation across the sampled cortical territory. Detection rate, direction and speed estimates derived from a multiunit estimator were compared to four field potential-based estimators: negative peak, maximum descent, high gamma power, and cross-correlation. Interestingly, the methods that were computationally simplest and most efficient (negative peak and maximal descent) offer non-inferior results in predicting neuronal traveling wave velocities compared to the other two, more complex methods. Moreover, the negative peak and maximal descent methods proved to be more robust against reduced spatial sampling challenges. Using least absolute deviation in place of least squares error minimized the impact of outliers, and reduced the discrepancies between local field potential-based and multiunit estimators.

SIGNIFICANCE

Our findings suggest that ictal epileptiform discharges typically take the form of exceptionally strong, rapidly traveling waves, with propagation detectable across millimeter distances. The sequential activation of neurons in space can be inferred from clinically-observable EEG data, with a variety of straightforward computation methods available. This opens possibilities for systematic assessments of ictal discharge propagation in clinical and research settings.

The effect of inhibition on the existence of traveling wave solutions for a neural field model of human seizure termination

In this paper we study the influence of inhibition on an activity-based neural field model consisting of an excitatory population with a linear adaptation term that directly regulates the activity of the excitatory population. Such a model has been used to replicate traveling wave data as observed in high density local field potential recordings (González-Ramírez et al. PLoS Computational Biology, 11(2), e1004065, 2015). In this work, we show that by adding an inhibitory population to this model we can still replicate wave properties as observed in human clinical data preceding seizure termination, but the parameter range over which such waves exist becomes more restricted. This restriction depends on the strength of the inhibition and the timescale at which the inhibition acts. In particular, if inhibition acts on a slower timescale relative to excitation then it is possible to still replicate traveling wave patterns as observed in the clinical data even with a relatively strong effect of inhibition. However, if inhibition acts on the same timescale as the excitation, or faster, then traveling wave patterns with the desired characteristics cease to exist when the inhibition becomes sufficiently strong.

Predicting the spatiotemporal diversity of seizure propagation and termination in human focal epilepsy

Recent studies have shown that seizures can spread and terminate across brain areas via a rich diversity of spatiotemporal patterns. In particular, while the location of the seizure onset area is usually invariant across seizures in an individual patient, the source of traveling (2–3 Hz) spike-and-wave discharges during seizures can either move with the slower propagating ictal wavefront or remain stationary at the seizure onset area. Furthermore, although many focal seizures terminate synchronously across brain areas, some evolve into distinct ictal clusters and terminate asynchronously. Here, we introduce a unifying perspective based on a new neural field model of epileptic seizure dynamics. Two main mechanisms, the co-existence of wave propagation in excitable media and coupled-oscillator dynamics, together with the interaction of multiple time scales, account for the reported diversity. We confirm our predictions in seizures and tractography data obtained from patients with pharmacologically resistant epilepsy. Our results contribute toward patient-specific seizure modeling.

A major goal of epilepsy research is understanding the spatiotemporal dynamics of seizure. Here, the authors extend the Epileptor neural mass model into a neural field model, in order to provide a unified and patient-specific model of seizure initiation, propagation, and termination.

Predicting seizure by modeling synaptic plasticity based on EEG signals - a case study of inherited epilepsy

This paper explores the internal dynamical mechanisms of epileptic seizures through quantitative modeling based on full brain electroencephalogram (EEG) signals. Our goal is to provide seizure prediction and facilitate treatment for epileptic patients. Motivated by an earlier mathematical model with incorporated synaptic plasticity, we studied the nonlinear dynamics of inherited seizures through a differential equation model. First, driven by a set of clinical inherited electroencephalogram data recorded from a patient with diagnosed Glucose Transporter Deficiency, we developed a dynamic seizure model on a system of ordinary differential equations. The model was reduced in complexity after considering and removing redundancy of each EEG channel. Then we verified that the proposed model produces qualitatively relevant behavior which matches the basic experimental observations of inherited seizure, including synchronization index and frequency. Meanwhile, the rationality of the connectivity structure hypothesis in the modeling process was verified. Further, through varying the threshold condition and excitation strength of synaptic plasticity, we elucidated the effect of synaptic plasticity to our seizure model. Results suggest that synaptic plasticity has great effect on the duration of seizure activities, which support the plausibility of therapeutic interventions for seizure control.

Spatiotemporal evolution of focal epileptiform activity from surface and laminar field recordings in cat neocortex

New devices that use targeted electrical stimulation to treat refractory localization-related epilepsy have shown great promise, although it is not well known which targets most effectively prevent the initiation and spread of seizures. To better understand how the brain transitions from healthy to seizing on a local scale, we induced focal epileptiform activity in the visual cortex of five anesthetized cats with local application of the GABA_A blocker picrotoxin while simultaneously recording local field potentials on a high-resolution electrocorticography array and laminar depth probes. Epileptiform activity appeared in the form of isolated events, revealing a consistent temporal pattern of ictogenesis across animals with interictal events consistently preceding the appearance of seizures. Based on the number of spikes per event, there was a natural separation between seizures and shorter interictal events. Two distinct spatial regions were seen: an epileptic focus that grew in size as activity progressed, and an inhibitory surround that exhibited a distinct relationship with the focus both on the surface and in the depth of the cortex. Epileptiform activity in the cortical laminae was seen concomitant with activity on the surface. Focus spikes appeared earlier on electrodes deeper in the cortex, suggesting that deep cortical layers may be integral to recruiting healthy tissue into the epileptic network and could be a promising target for interventional devices. Our study may inform more effective therapies to prevent seizure generation and spread in localization-related epilepsies. NEW & NOTEWORTHY We induced local epileptiform activity and recorded continuous, high-resolution local field potentials from the surface and depth of the visual cortex in anesthetized cats. Our results reveal a consistent pattern of ictogenesis, characterize the spatial spread of the epileptic focus and its relationship with the inhibitory surround, and show that focus activity within events appears earliest in deeper cortical layers. These findings have potential implications for the monitoring and treatment of refractory epilepsy.

Cross-scale effects of neural interactions during human neocortical seizure activity

Small-scale neuronal networks may impose widespread effects on large network dynamics. To unravel this relationship, we analyzed eight multiscale recordings of spontaneous seizures from four patients with epilepsy. During seizures, multiunit spike activity organizes into a submillimeter-sized wavefront, and this activity correlates significantly with low-frequency rhythms from electrocorticographic recordings across a 10-cm-sized neocortical network. Notably, this correlation effect is specific to the ictal wavefront and is absent interictally or from action potential activity outside the wavefront territory. To examine the multiscale interactions, we created a model using a multiscale, nonlinear system and found evidence for a dual role for feedforward inhibition in seizures: while inhibition at the wavefront fails, allowing seizure propagation, feedforward inhibition of the surrounding centimeter-scale networks is activated via long-range excitatory connections. Bifurcation analysis revealed that distinct dynamical pathways for seizure termination depend on the surrounding inhibition strength. Using our model, we found that the mesoscopic, local wavefront acts as the forcing term of the ictal process, while the macroscopic, centimeter-sized network modulates the oscillatory seizure activity.

Toward a Mechanistic Understanding of Epileptic Networks

Focal epileptic seizures have long been considered to arise from a small susceptible brain area and spread through uninvolved regions. In the past decade, the idea that focal seizures instead arise from coordinated activity across large-scale epileptic networks has become widely accepted. Understanding the network model's applicability is critical, due to its increasing influence on clinical research and surgical treatment paradigms. In this review, we examine the origins of the concept of epileptic networks as the nidus for recurring seizures. We summarize analytical and methodological elements of epileptic network studies and discuss findings from recent detailed electrophysiological investigations. Our review highlights the strengths and limitations of the epileptic network theory as a metaphor for the complex interactions that occur during seizures. We present lines of investigation that may usefully probe these interactions and thus serve to advance our understanding of the long-range effects of epileptiform activity.

Human seizures couple across spatial scales through travelling wave dynamics

Epilepsy—the propensity toward recurrent, unprovoked seizures—is a devastating disease affecting 65 million people worldwide. Understanding and treating this disease remains a challenge, as seizures manifest through mechanisms and features that span spatial and temporal scales. Here we address this challenge through the analysis and modelling of human brain voltage activity recorded simultaneously across microscopic and macroscopic spatial scales. We show that during seizure large-scale neural populations spanning centimetres of cortex coordinate with small neural groups spanning cortical columns, and provide evidence that rapidly propagating waves of activity underlie this increased inter-scale coupling. We develop a corresponding computational model to propose specific mechanisms—namely, the effects of an increased extracellular potassium concentration diffusing in space—that support the observed spatiotemporal dynamics. Understanding the multi-scale, spatiotemporal dynamics of human seizures—and connecting these dynamics to specific biological mechanisms—promises new insights to treat this devastating disease.

The authors record both local and long-range neural activity during human epileptic seizures to study the underlying multi-scale dynamics. They find that coupling of activity across spatial scales increases during seizures through propagating waves that are fit by a model that combines neural activity and potassium concentration dynamics.

Spatiotemporal canards in neural field equations

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely unexplored. We identify and describe a type of coherent structure in which a spatial pattern displays temporal canard behavior. Using interfacial dynamics and geometric singular perturbation theory, we classify spatiotemporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatiotemporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatiotemporal canards with octahedral symmetry in a neural field model posed on the unit sphere.

Neural Field Models with Threshold Noise

The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate function so prevalent in large scale neuronal modelling. In this paper we explore the effects of such threshold noise without recourse to averaging and show that spatial correlations can have a strong effect on the behaviour of waves and patterns in continuum models. Moreover, for a prescribed spatial covariance function we explore the differences in behaviour that can emerge when the underlying stationary distribution is changed from Gaussian to non-Gaussian. For travelling front solutions, in a system with exponentially decaying spatial interactions, we make use of an interface approach to calculate the instantaneous wave speed analytically as a series expansion in the noise strength. From this we find that, for weak noise, the spatially averaged speed depends only on the choice of covariance function and not on the shape of the stationary distribution. For a system with a Mexican-hat spatial connectivity we further find that noise can induce localised bump solutions, and using an interface stability argument show that there can be multiple stable solution branches.

Interaction between synaptic inhibition and glial-potassium dynamics leads to diverse seizure transition modes in biophysical models of human focal seizures

How focal seizures initiate and evolve in human neocortex remains a fundamental problem in neuroscience. Here, we use biophysical neuronal network models of neocortical patches to study how the interaction between inhibition and extracellular potassium ([K (+)] o ) dynamics may contribute to different types of focal seizures. Three main types of propagated focal seizures observed in recent intracortical microelectrode recordings in humans were modelled: seizures characterized by sustained (∼30-60 Hz) gamma local field potential (LFP) oscillations; seizures where the onset in the propagated site consisted of LFP spikes that later evolved into rhythmic (∼2-3 Hz) spike-wave complexes (SWCs); and seizures where a brief stage of low-amplitude fast-oscillation (∼10-20 Hz) LFPs preceded the SWC activity. Our findings are fourfold: (1) The interaction between elevated [K (+)] o (due to abnormal potassium buffering by glial cells) and the strength of synaptic inhibition plays a predominant role in shaping these three types of seizures. (2) Strengthening of inhibition leads to the onset of sustained narrowband gamma seizures. (3) Transition into SWC seizures is obtained either by the weakening of inhibitory synapses, or by a transient strengthening followed by an inhibitory breakdown (e.g. GABA depletion). This reduction or breakdown of inhibition among fast-spiking (FS) inhibitory interneurons increases their spiking activity and leads them eventually into depolarization block. Ictal spike-wave discharges in the model are then sustained solely by pyramidal neurons. (4) FS cell dynamics are also critical for seizures where the evolution into SWC activity is preceded by low-amplitude fast oscillations. Different levels of elevated [K (+)] o were important for transitions into and maintenance of sustained gamma oscillations and SWC discharges. Overall, our modelling study predicts that the interaction between inhibitory interneurons and [K (+)] o glial buffering under abnormal conditions may explain different types of ictal transitions and dynamics during propagated seizures in human focal epilepsy.

Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools

The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.

Analytically tractable studies of traveling waves of activity in integrate-and-fire neural networks

In contrast to other large-scale network models for propagation of electrical activity in neural tissue that have no analytical solutions for their dynamics, we show that for a specific class of integrate and fire neural networks the acceleration depends quadratically on the instantaneous speed of the activity propagation. We use this property to analytically compute the network spike dynamics and to highlight the emergence of a natural time scale for the evolution of the traveling waves. These results allow us to examine other applications of this model such as the effect that a nonconductive gap of tissue has on further activity propagation. Furthermore we show that activity propagation also depends on local conditions for other more general connectivity functions, by converting the evolution equations for network dynamics into a low-dimensional system of ordinary differential equations. This approach greatly enhances our intuition into the mechanisms of the traveling waves evolution and significantly reduces the simulation time for this class of models.

Disinhibition-Induced Delayed Onset of Epileptic Spike-Wave Discharges in a Five Variable Model of Cortex and Thalamus

Based on a modified neural field network model composed of cortex and thalamus, we here propose a computational framework to investigate the onset control of absence seizure, which is characterized by the spike-wave discharges. Firstly, we briefly demonstrate the existence of various transition types in Taylor's model by increasing the thalamic input. Furthermore, after the disinhibitory function is reasonably introduced into the Taylor's model, we can observe the occurrence of various transition states of firing patterns with different dominant frequencies as the thalamic input is varied under different disinhibitory effects onto the pyramidal neural population. Interestingly, it is found that the onset of spike-wave discharges can be delayed as the disinhibitory input is considered. More importantly, we explore bifurcation mechanism of firing transitions as some key parameters are changed. And also, we observe other dynamical states, such as simple oscillations and saturated discharges with different spatial scales, which are consistent with previous theoretical or experimental findings.

The ictal wavefront is the spatiotemporal source of discharges during spontaneous human seizures

The extensive distribution and simultaneous termination of seizures across cortical areas has led to the hypothesis that seizures are caused by large-scale coordinated networks spanning these areas. This view, however, is difficult to reconcile with most proposed mechanisms of seizure spread and termination, which operate on a cellular scale. We hypothesize that seizures evolve into self-organized structures wherein a small seizing territory projects high-intensity electrical signals over a broad cortical area. Here we investigate human seizures on both small and large electrophysiological scales. We show that the migrating edge of the seizing territory is the source of travelling waves of synaptic activity into adjacent cortical areas. As the seizure progresses, slow dynamics in induced activity from these waves indicate a weakening and eventual failure of their source. These observations support a parsimonious theory for how large-scale evolution and termination of seizures are driven from a small, migrating cortical area.

Epileptic brains display inhibitory restraint as manifested by the spread of synchronized activities being delayed in timing. Here, Elliot Smith and colleagues show fast-moving traveling wave that originates from the edge of ictal wavefront with subsequent depolarization and multiunit firing in the seizing brain regions in epileptic patients.

Microscale spatiotemporal dynamics during neocortical propagation of human focal seizures

Some of the most clinically consequential aspects of focal epilepsy, e.g. loss of consciousness, arise from the generalization or propagation of seizures through local and large-scale neocortical networks. Yet, the dynamics of such neocortical propagation remain poorly understood. Here, we studied the microdynamics of focal seizure propagation in neocortical patches (4×4 mm) recorded via high-density microelectrode arrays (MEAs) implanted in people with pharmacologically resistant epilepsy. Our main findings are threefold: (1) a newly developed stage segmentation method, applied to local field potentials (LFPs) and multiunit activity (MUA), revealed a succession of discrete seizure stages, each lasting several seconds. These different stages showed characteristic evolutions in overall activity and spatial patterns, which were relatively consistent across seizures within each of the 5 patients studied. Interestingly, segmented seizure stages based on LFPs or MUA showed a dissociation of their spatiotemporal dynamics, likely reflecting different contributions of non-local synaptic inputs and local network activity. (2) As previously reported, some of the seizures showed a peak in MUA that happened several seconds after local seizure onset and slowly propagated across the MEA. However, other seizures had a more complex structure characterized by, for example, several MUA peaks, more consistent with the succession of discrete stages than the slow propagation of a simple wavefront of increased MUA. In both cases, nevertheless, seizures characterized by spike-wave discharges (SWDs, ~2-3 Hz) eventually evolved into patterns of phase-locked MUA and LFPs. (3) Individual SWDs or gamma oscillation cycles (25-60 Hz), characteristic of two different types of recorded seizures, tended to propagate with varying degrees of directionality, directions of propagation and speeds, depending on the identified seizure stage. However, no clear relationship was observed between the MUA peak onset time (in seizures where such peak onset occurred) and changes in MUA or LFP propagation patterns. Overall, our findings indicate that the recruitment of neocortical territories into ictal activity undergoes complex spatiotemporal dynamics evolving in slow discrete states, which are consistent across seizures within each patient. Furthermore, ictal states at finer spatiotemporal scales (individual SWDs or gamma oscillations) are organized by slower time scale network dynamics evolving through these discrete stages.