Focal seizures have been recognized for more than 3000 years, with descriptions dating back to ancient Mesopotamia (Worthington, 2005). Although focal seizures can present with a plethora of behavioral manifestations that vary according to the affected cortical regions, there are several consistent clinical and large-scale EEG features (Kotagal et al., 2008): propagation from a focal onset location to large brain regions, widespread neuronal synchronization, a transition from tonic to clonic activity, and a slowing pace of neuronal discharging prior to simultaneous seizure termination.

Our recent investigations, utilizing microelectrode array recordings in humans, identified neuronal underpinnings of these common seizure features (Schevon et al., 2012; Smith et al., 2016). Based on these findings and results from animal model studies (Trevelyan et al., 2006; Trevelyan et al., 2007a; Trevelyan et al., 2007b; Wenzel et al., 2017; Wenzel et al., 2019), we proposed a dual spatial structure for focal seizures consisting of a core region of seizing brain bounded by an ictal wavefront surrounded by a passively reactive penumbra. What delineates the boundary of the ictal core is the ictal wavefront, a narrow band of intense, desynchronized multiunit (tonic) firing that marks the transition to seizure at a given brain location. Typically, this tonic firing structure is not detectable in clinical EEG or band-limited local field potentials (LFPs). Evidence from human (Schevon et al., 2012) and animal studies (Trevelyan et al., 2006; Trevelyan et al., 2007b) suggests that collapse of inhibition is the key element causing ictal wavefront propagation, which leads to progressive seizure territory expansion. The slow pace of ictal wavefront propagation (<1 mm/sec) corresponds to the slow evolution of the electrographic seizure and clinical semiology, for example the classic Jacksonian march (York and Steinberg, 2011). As it advances, the ictal wavefront generates fast-moving ictal discharges (Trevelyan, 2009; Smith et al., 2016), directed inward towards the initiation point, at speeds two orders of magnitude higher than the wavefront propagation (Smith et al., 2016; Liou et al., 2017). These fast-moving ictal discharges are the basis of the well-recognized high amplitude field potential deflections that are the hallmark of electrographic seizures. Thus, there are two types of moving waves that characterize seizures: fast, inward-moving ictal discharges and the slow outward-moving wavefront of ictal recruitment.

To date, despite extensive prior work in computational seizure modeling (Soltesz and Staley, 2011), no theoretical study has shown how this dynamic topological structure can arise from basic biophysical mechanisms. Phenomenological models, such as Epileptor (Jirsa et al., 2014; Proix et al., 2018), have successfully utilized large-scale functional EEG features to reproduce the large-area, apparently synchronized EEG activity that characterizes clinical seizure recordings, although identifying smaller scale neurophysiological processes corresponding to each abstract variable in these models can be challenging. Biophysical models have been used to explore the role of specific neurophysiological processes responsible for seizure transitions, such as intracellular and extracellular potassium dynamics (Cressman et al., 2009; Ullah et al., 2009), transmembrane chloride gradients (Buchin et al., 2016), and calcium-activated processes (Yang et al., 2005). However, the evolving, wide-area dynamics that characterize the life cycle of a seizure remain difficult to explain by any single biophysical mechanism. Moreover, given that seizures can simultaneously activate a multitude of pathophysiological processes across extensive interconnected brain regions, biophysical models may be limited to a fragmented or overly specific account of seizure dynamics.

Here, we describe a biophysically-constrained cortical network model designed to link the key pathological cellular mechanisms that underpin seizures to their large-scale spatial structure. Inspired by the spirit of phenomenological models, we adopt an approach with minimal assumptions, aiming to show that complex spatiotemporal dynamics can arise from simple, generalizable, and experimentally validated biophysical principles. Our modeling philosophy is to eschew model features that are inessential or that do not contribute directly or dominantly to the phenomena being considered. This allows us to identify basic mechanisms. In addition, while we, of course, build specific mechanisms into our models, our aim is to highlight the underlying biophysical properties that lead to pathology, so that the lessons learned are more general than the specific models. Using a targeted parameter search, we demonstrate that maintaining the normal transmembrane chloride gradient is critical for inhibition robustness, which is necessary for restricting seizure propagation. Our theory provides a theoretical framework explaining the key clinical features widely observed in focal seizure patients (Kotagal et al., 2008; Ebersole et al., 2014; Extercatte et al., 2015). We also test several predictions arising from the model using our existing dataset of microelectrode recordings of spontaneous human seizures.

We modeled the neocortex as a 2-dimensional neuron sheet (Bressloff, 2014). Model neurons are pyramidal cell-like and contain two intrinsic conductances – leak and slow potassium – and two synaptic conductances – excitatory (AMPA) and inhibitory (GABA-A). Neuronal mean firing rates are calculated by passing the difference between membrane potentials and thresholds through a sigmoid function. Model neurons incorporate spike-frequency adaptation mechanisms – neuronal firing increases the spike threshold and activates an additional slow afterhyperpolarization (sAHP) conductance. Model neurons are recurrently connected by direct excitatory projections (Figure 1A). They also inhibit each other indirectly through di-synaptic pathways via interneurons, whose dynamics are simplified in this study (see Materials and methods for more information regarding interneuron simplification). The effects of recurrent projections between model neurons are hypothesized to be distance-dependent, with the range of di-synaptic recurrent inhibition longer than the mono-synaptic excitatory range, thereby making the spatial distribution of the effective synaptic weights from a model neuron follows a ‘Mexican hat’ structure (Prince and Wilder, 1967; Coombes, 2005; Bressloff, 2014). In addition, our model includes a distance-independent recurrent inhibition pathway (Figure 1A, part γ), inspired by recent observation of large-scale inhibitory effects of focal seizure activity (Eissa et al., 2017; Liou et al., 2018). Biophysical parameters are set based on previously reported values from recorded neocortical pyramidal neurons (Table 1). In its original form, the model does not spontaneously seize. Instead model seizures are initiated by transient focal excitatory inputs. This model can be considered a model of a healthy neural circuit, with the degree of resistance to seizure controlled by its parameters.

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Throughout the manuscript, variations of the primary 2D rate model are introduced to provide more in-depth assessment of key features. Simulation results of the primary model in a noise-free environment are first presented and qualitatively compared to patient recordings (Figure 1). Next, the model is simplified to 1-dimensional space to better illustrate physiological mechanisms that account for the slow advancement of the ictal wavefront and the generation of fast inward traveling waves (Figures 2 and 3). A corresponding 1-dimensional spiking model is subsequently shown in Figures 4 and 5 to validate our rate-based approach and explore the effects of spike-timing dependent plasticity on spontaneous seizure occurrence. Finally, the main model is revisited with the addition of background current noise to explain bi-stable seizure evolution endpoints and the origin of complex spatial configurations of focal seizure activities (Figures 6 and 7). While such variations of the primary model are used to more clearly illustrate key dynamical features, the dynamics presented here are common across model variations.

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We have presented a biophysically constrained computational model of seizures that, despite being limited to a reduced set of neural properties, reproduces many key aspects of human focal seizures. We showed that the collapse of inhibition strength ahead of the ictal wavefront, followed by the emergence of hyperpolarizing currents, accounts for the slow expansion of the ictal core, the tonic-to-clonic transition, and the experimentally observed generation of inward traveling waves. These features are closely aligned with properties of human seizures that were previously described using combined clinical EEG and microelectrode recordings (Smith et al., 2016). The interplay of usage-dependent inhibition exhaustion and adaptation, which drives wavefront propagation during early stages of the seizure life cycle, also characterizes seizure-prone versus seizure-resistant tissues. Our study supports the idea that recurrent inhibition that is spatially non-specific terminates seizures, creating the slowing frequency of discharging and increased traveling wave variability during the pre-termination stage. Additionally, the model demonstrates that distinct seizure onset patterns can result from topologically variant recurrent inhibitory projections. The inclusion of STDP in the model results in progressive, localized pathological enhancement of excitatory connectivity, reducing seizure threshold in the model and enabling spontaneous seizure generation. Our results on plasticity emphasize the importance of the discovery and modeling of rapid inwardly directed traveling waves (Smith et al., 2016; Liou et al., 2017), as these may induce plasticity that increases susceptibility to future seizures. Finally, the model provides candidate scenarios for both spontaneous seizure termination and ongoing status epilepticus, based on stochastic events following dissipation of the ictal wavefront.

The spatial seizure topology inferred from human multiscale recordings (Schevon et al., 2012; Smith et al., 2016) and explicitly reproduced in the computational model described here has not previously been studied in its entirety. The propagation of the ictal wavefront is often invisible to standard EEG, and even to wideband microelectrode recordings (Schevon et al., 2012). In contrast, the fast-moving inward traveling waves manifest as field potential discharges, forming the classic EEG signature of seizures (Smith et al., 2016; Martinet et al., 2017). Due to the speed at which these discharges travel (up to 100 cm/sec) (Liou et al., 2017), they appear to be synchronized across large brain areas, leading to suppositions of large-area network origin (Kramer et al., 2010; Jirsa et al., 2014; Khambhati et al., 2016). Our model demonstrates that these spatiotemporal dynamics can result from well-established neuronal processes occurring at the level of localized cellular interactions. Further, persistent traveling wave direction preferences can result from random fluctuations in background neural activity. A caution applies, however, in interpreting effects that occur near the boundary of the neural sheet, as this area may be artificially hyperexcitable in the model due to reduction in the maximal inhibitory conductances.

Without modification by STDP, the network is reminiscent of a healthy, non-epileptic brain – it maintains a non-trivial baseline firing rate, does not spontaneously seize, but can be provoked into a full-blown seizure. Varying the external stimulus’ strength produces a range of responses from post-stimulation depression, short-run after-discharges, to full-blown seizures. In this sense, perhaps the most straightforward real-world interpretation of the external stimuli in our simulations is the focal electrical zap given during intra-operative cortical mapping. The suprathreshold stimuli are therefore analogous to the electrical shock given during convulsive therapies (Spellman et al., 2009). More generally, the stimuli are qualitatively equivalent to any factor that cause a neighborhood of neurons to depolarize, such as anoxic depolarization caused by local ischemia (Somjen, 2001) or discharges triggered by transcranial magnetic stimulation (Lisanby et al., 2003). Accordingly, responses to the external stimuli, in return, may reveal the network’s intrinsic propensity to seize.

Both modern in vitro brain slice studies (Trevelyan et al., 2007b) and in vivo human recordings (Schevon et al., 2012) support the classical hypothesis of surround inhibition (Prince and Wilder, 1967), which motivated the Mexican-hat pattern of connectivity and the usage-dependent exhaustible inhibition mechanism employed in our study (Eissa et al., 2017; Liou et al., 2018). Theoretically, massive GABAergic activity provoked by ictal events can overwhelm chloride buffering and clearance mechanisms. The collapsed transmembrane chloride gradient compromises the strength of surround inhibition, leading to seizure onset and propagation (Lillis et al., 2012; Alfonsa et al., 2015). Indeed, excitatory effects of GABAergic transmission have been found in brain slices taken from epilepsy patients (Cohen et al., 2002). The paradoxical excitatory effects of GABA in epilepsy patients may be attributed to clearance defects, as tissues from temporal lobe epilepsy (Huberfeld et al., 2007) and tumor-associated epilepsy patients (Pallud et al., 2014) both showed reduced KCC2 for chloride extrusion. Experimentally knocking down KCC2 also leads to epileptiform discharges (Zhu et al., 2005), and an in-vitro fluorescence study showed that chloride accumulation preceded seizure onset (Lillis et al., 2012). In agreement with previous studies, our model suggests that tissues with slow chloride clearance are seizure-prone. Inhibition robustness is critical for preventing, restraining, and shaping focal seizures.

The tonic-clonic transition has previously been theorized to arise from intrinsic neuronal processes such as sAHP (Beverlin et al., 2012). In our model, we further explore the hyperpolarizing current’s role throughout a seizure’s life cycle, including controlling seizure onset, mediating tonic-clonic transition, and participating in the evolution of pre-termination seizure activities. Clinical studies have confirmed the importance of sAHP in prohibiting epileptic activity. Mutations of KCNQ2 and KCNQ3 channels, which contribute to sAHP currents, are associated with familial neonatal epilepsy (Tzingounis and Nicoll, 2008).

We adopt a simplistic approach to ionic dynamics during seizures. Ion concentrations are held constant except intracellular chloride, which is modelled as the mechanism underpinning usage-dependent exhaustion of inhibition. This model’s approach, however, should not be misinterpreted. Other than chloride, significant ionic shifts, including sodium, potassium, calcium, hydrogen and etc., have also been observed during peri-ictal periods (Raimondo et al., 2015). The tortuosity of extracellular space further complicates the picture, as ions may distribute unevenly due to diffusion limits (Syková and Nicholson, 2008). Dysregulation of neuronal excitability due to extracellular potassium accumulation, for example, has been proposed a candidate mechanism of seizure initiation (González et al., 2019). A potential extension of this model may include multiple ion species and their regional variance.

In our model, spontaneous seizure termination is caused by progressive build-up of inhibition, which is used to model the widespread effect of focal seizures (Burman and Parrish, 2018; Liou et al., 2019). This widespread inhibition has not been extensively described, but nevertheless has been demonstrated in both an acute animal seizure model (Liou et al., 2018) and in a computational model validated with human microelectrode recordings (Eissa et al., 2017). Experimentally, calcium imaging has confirmed seizure-induced cross-areal activation of PV(+) interneurons, for example in visual cortex in response to a focal seizure triggered in somatosensory cortex (Wenzel et al., 2017; Liou et al., 2018). Anatomically, such widespread inhibition may be mediated by long-distance, cross-areal projections that preferentially project to inhibitory interneurons in their target circuits (Zhang et al., 2014; Sun et al., 2019). In addition, focal seizures may depress brain-wide activities via disrupting subcortical structures (Feng et al., 2017) and ascending activation systems (Blumenfeld, 2012). Large-scale non-neural mechanisms that are not specifically modelled in our study may also contribute to seizure termination. For example, global hypoxia secondary to seizure-induced cardiopulmonary compromise may activate ATP-sensitive potassium channels in pyramidal neurons in a spatially non-specific way (Ching et al., 2012), thereby terminating seizures by building resistance ahead of the ictal wavefronts and generating termination dynamics, analogous to the global recurrent inhibition effect modelled in our study. In other words, the global inhibition assumption may be interpreted as a hypothesis that, as a seizure territory enlarges, principle neurons tend to hyperpolarize and are therefore progressively harder to recruit, with the specific neurological mechanisms varying between patients.

In this study, we simplified inhibitory interneurons to reduce model parameter space. This should not be misinterpreted as implying that we mean to diminish the role of interneuron dynamics in seizure evolution. Recent studies have shown that interneurons may participate in seizure initiation (Librizzi et al., 2017), and their depolarization block can be critical for seizure propagation (Eissa et al., 2017). However, complex inhibitory interneuron dynamics, as shown in our study, may not be required for the replication of key features of seizure dynamics.

The biophysical processes proposed here are certainly not the only mechanisms causing seizure evolution. Instead, they should be considered as representative candidates, with the suggestion that other mechanisms should share the features of the ones we have proposed. Processes that produce similar effects on network dynamics and that operate over the same time scales may also contribute to the generation of seizure dynamics, as exemplified by our phenomenological model, in which the general variable z controls local effectiveness of inhibition rather than the transmembrane chloride gradient. For example, in addition to intracellular chloride accumulation, depolarization block of GABAergic neurons may serve a similar role in contributing to the breakdown of inhibition (Meijer et al., 2015). Similarly, aside from hyperpolarizing currents, intense neuronal bursts can inactivate sodium channels, quickly reducing neuronal excitability in a couple of seconds (Fleidervish et al., 1996). Additional depressing mechanisms, including excitatory synaptic depletion (Beverlin et al., 2012) and inhibitory cell recovery (Ziburkus et al., 2006) may collectively contribute to tonic-clonic transitions, slowing, and termination. However, key similarities in seizure dynamics observed across different brain regions and pathological conditions suggest that some mechanisms may be universally present, and these are primary candidates for physiological processes underlying seizure evolution. Developing therapies targeting at these mechanisms could therefore achieve broad spectrum anti-seizure effects.

Our model demonstrates that the spatial distribution of recurrent inhibition can shape neural network dynamics, explaining the variance of focal seizure onset subtypes. Seizures manifesting with focal low voltage fast activity, a positive predictor of seizure freedom following resection of the focus (Alarcon et al., 1995), occurred in the setting of relatively strong local inhibition in our model. In contrast, relatively weak local inhibition allows seizure-related activity to spread quickly in the form of periodic outward traveling waves moving ahead of the ictal wavefront (Figure 4B). These outward traveling waves result in the EEG appearance of rhythmic discharging at seizure onset. Previous studies have proposed that this ‘hypersynchronous’ seizure onset pattern depends on long-range cortical connections (Perucca et al., 2014; Weiss et al., 2016), or increased surrounding tissue excitability (Wang et al., 2017). Our model instead highlights the role of localized inhibition and demonstrates that both types of electrographic onset signatures can arise from focal sources, depending primarily on how the ictal foci are established.

The model predicts that physiological plasticity mechanisms (STDP in this model) can be hijacked by pathological seizure dynamics (Mehta et al., 1993). The fast inward traveling waves during each seizure create self-enhancing pathological centripetal connectivity, resulting in the emergence of pre-ictal large amplitude LFP discharges that manifest as centripetally propagating waves. Such single or repetitive sharp waves are often seen just prior to LVFA seizure onsets in clinical recordings (Lee et al., 2000) and were also present in one of our human microelectrode recordings (Figure 5D–E). We also reported similar spatial attraction of bursts in a rodent model during interictal periods (Liou et al., 2019). Our model therefore offers a theoretical explanation that synaptic plasticity results in spatial rewiring which increases seizure susceptibility and manifests as interictal discharges or herald spikes that travel towards the site of eventual seizures. This model prediction has potential implications in clinical seizure management. If patterns of interictal discharge traveling waves could be used to reduce uncertainty about the location of subsequent seizures, they could be used as a valuable seizure prediction and diagnostic tool. Furthermore, preventing or reversing traveling wave-induced spatial rewiring may break the pathological, self-enhancing loop, eventually leading to slowdown or reversal of seizure progression.

It has been proposed that seizure evolution has bistable endpoints, and that failure of seizure termination can be a stochastic event (Kramer et al., 2012). In the presence of noise, the cortex may become trapped in the spiral wave scenario leading to an indefinitely persistent seizure, that is status epilepticus. Animal experiments, both in brain slice (Huang et al., 2004) and in vivo (Petsche et al., 1974; Huang et al., 2010) have shown that spiral waves can develop in acute seizure models. Multielectrode array recordings have also revealed spiral epileptiform discharges during feline picrotoxin-induced seizures (Viventi et al., 2011). Such spiral waves have yet to be detected in humans, where they may serve as a marker of risk for status epilepticus.

Our model predicts such spiral waves may be terminated by a brief synchronizing input. In agreement with this prediction, synchronization has been shown to promote seizure termination in status epilepticus patients (Schindler et al., 2007). Analogously, spiral waves also develop during ventricular fibrillation, a fatal form of cardiac arrhythmia. Delivering a brief pulse that widely activates cardiomyocytes at once has been the standard therapy to rescue such patients. This suggests that a stimulation strategy involving wide, synchronous excitatory input delivery may be an effective approach for seizures characterized by spiral waves.

Although microelectrode recording plays an important role in validating our model, the key dynamics which are commonly seen behaviorally and observed in macroelectrode recordings, in our opinion, provides an equal, if not more important, support the generality of this model. The question therefore arises why relatively few human microelectrode recordings have demonstrated the existence of ictal wavefronts. Our model, indeed, provides a straightforward explanation – a microelectrode array not only needs to be positioned at a region that is recruited into the seizure territory but also needs to be close enough to the onset spot. Otherwise, the seizure could have evolved into its pre-termination stage, during which the seizure activity is still slowly propagating but the wavefront has been annihilated.

Finally, this model is not built to exactly duplicate the microelectrode recordings. Hyper-tuning the model parameters, in our opinion, is not fruitful as seizure dynamics vary significantly even within the same patient from one episode to another. Instead, we focus on how the cellular neurophysiological principles generate the key spatiotemporal dynamics in a larger scale. Similar bottom-up approaches have also been adopted to study the effects of network connectivity on seizure initiation and subtypes (Wang et al., 2017; Jacob et al., 2019). Alternatively, seizure EEG and ECoG databases have allowed a top-down, machine learning-based approach (Karoly et al., 2018). The availability of macroelectrode data might reduce patient selection biases. However, making mechanistic inference based on LFP-derived signals is challenging because of their limited resolution and sensitivity to geometric configurations (Einevoll et al., 2013). In our perspective, the two mutually complementary approaches are both indispensable in order to fully understand seizure dynamics.

The human seizure data used in this manuscript is currently restricted from open sharing by university policy, but may be shared with qualified investigators under appropriate institutional protections upon request. Source data files with some anonymised data have been provided. The MATLAB code for the computational model is available at https://github.com/jyunyouliou/LAS-Model (copy archived at https://github.com/elifesciences-publications/LAS-Model).

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Author details

1. Jyun-you Liou

   1. Department of Physiology and Cellular Biophysics, Columbia University, New York, United States
   2. Department of Anesthesiology, NewYork-Presbyterian Hospital/Weill Cornell Medicine, New York, United States
   3. Department of Neurology, Columbia University Medical Center, New York, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Investigation, Methodology

   For correspondence

   jyl9010@nyp.org

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4851-3676

2. Elliot H Smith

   Department of Neurological Surgery, Columbia University Medical Center, New York, United States

   Present address

   Department of Neurosurgery, University of Utah, Salt Lake City, United States

   Contribution

   Conceptualization, Resources, Data curation, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4323-4643

3. Lisa M Bateman

   Department of Neurology, Columbia University Medical Center, New York, United States

   Contribution

   Resources, Writing - review and editing

   Competing interests

   No competing interests declared

4. Samuel L Bruce

   Vagelos College of Physicians & Surgeons, Columbia University, New York, United States

   Contribution

   Software, Formal analysis

   Competing interests

   No competing interests declared

5. Guy M McKhann

   Department of Neurological Surgery, Columbia University Medical Center, New York, United States

   Contribution

   Resources, Data curation

   Competing interests

   No competing interests declared

6. Robert R Goodman

   Department of Neurological Surgery, Columbia University Medical Center, New York, United States

   Present address

   Department of Neurosurgery, Donald and Barbara Zucker School of Medicine at Hofstra/Northwell, Hempstead, United States

   Contribution

   Resources, Data curation

   Competing interests

   No competing interests declared

7. Ronald G Emerson

   Department of Neurology, Columbia University Medical Center, New York, United States

   Present address

   Hospital for Special Surgery, Weill Cornell Medical College, New York, United States

   Contribution

   Conceptualization, Resources, Data curation

   Competing interests

   No competing interests declared

8. Catherine A Schevon

   Department of Neurology, Columbia University Medical Center, New York, United States

   Contribution

   Conceptualization, Resources, Data curation, Software, Supervision, Funding acquisition, Validation, Investigation, Project administration, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-4485-7933

9. LF Abbott

   1. Department of Physiology and Cellular Biophysics, Columbia University, New York, United States
   2. Mortimer B. Zuckerman Mind Brain Behavior Institute, Department of Neuroscience, Columbia University, New York, United States

   Contribution

   Conceptualization, Resources, Supervision, Funding acquisition, Investigation, Methodology, Project administration, Writing - review and editing

   Competing interests

   No competing interests declared

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