Effects of physiological parameter evolution on the dynamics of tonic-clonic seizures

F. Deeba; P. Sanz-Leon; P. A. Robinson

doi:10.1371/journal.pone.0230510

Abstract

The temporal and spectral characteristics of tonic-clonic seizures are investigated using a neural field model of the corticothalamic system in the presence of a temporally varying connection strength between the cerebral cortex and thalamus. Increasing connection strength drives the system into ∼ 10 Hz seizure oscillations once a threshold is passed and a subcritical Hopf bifurcation occurs. In this study, the spectral and temporal characteristics of tonic-clonic seizures are explored as functions of the relevant properties of physiological connection strengths, such as maximum strength, time above threshold, and the ramp rate at which the strength increases or decreases. Analysis shows that the seizure onset time decreases with the maximum connection strength and time above threshold, but increases with the ramp rate. Seizure duration and offset time increase with maximum connection strength, time above threshold, and rate of change. Spectral analysis reveals that the power of nonlinear harmonics and the duration of the oscillations increase as the maximum connection strength and the time above threshold increase. A secondary limit cycle at ∼ 18 Hz, termed a saddle-cycle, is also seen during seizure onset and becomes more prominent and robust with increasing ramp rate. If the time above the threshold is too small, the system does not reach the 10 Hz limit cycle, and only exhibits 18 Hz saddle-cycle oscillations. It is also seen that the time to reach the saturated large amplitude limit-cycle seizure oscillation from both the instability threshold and from the end of the saddle-cycle oscillations is inversely proportional to the square root of the ramp rate.

Citation: Deeba F, Sanz-Leon P, Robinson PA (2020) Effects of physiological parameter evolution on the dynamics of tonic-clonic seizures. PLoS ONE 15(4): e0230510. https://doi.org/10.1371/journal.pone.0230510

Editor: Daniele Marinazzo, Ghent University, BELGIUM

Received: March 2, 2020; Accepted: March 3, 2020; Published: April 2, 2020

Copyright: © 2020 Deeba et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The NFTsim software and the sample configuration file required to produce seizure are publicly available: https://doi.org/10.1371/journal.pcbi.1006387 https://github.com/BrainDynamicsUSYD/nftsim.

Funding: This work was supported by the Australian Research Council Center of Excellence for Integrative Brain Function (ARC Center of Excellence Grant CE140100007) and by an Australian Research Council Laureate Fellowship (Grant FL140100025).

Competing interests: The authors have declared that no competing interests exist

Introduction

Tonic-clonic seizures, formerly known as grand mal seizures, are the most frequently encountered generalized seizures [1]. These seizures have a tonic phase, which is characterized by an initial increase in tone of certain muscles, followed by a clonic phase, which involves bilateral symmetric jerking of the extremities [2]. Tonic-clonic seizures have markedly different pre- and post-ictal electroencephalograms (EEG) and typically last 1 to 3 minutes. Primary generalized seizures, which is one of the most commonly seen seizures, begin simultaneously across the whole cortex [1].

A number of authors have investigated the mechanisms of seizures using the neural network and neural field approaches [3–14]. Many authors have proposed that transitions from healthy state to the seizure state occur via bifurcations upon changing physiological parameters [3–9, 12, 13]. For example, depending on the instability region, increasing excitatory connection strengths between cortex and thalamus drives the system into ∼ 10 Hz and ∼ 3 Hz seizure oscillations via a subcritical and supercritical Hopf bifurcation, respectively, once a critical value (i.e., a threshold) is passed [3–9, 12, 13]. Results from in vivo studies have also provided evidence that changes in corticothalamic and other connection strengths can induce seizures [12, 15–20], which possibly occur due to changes in GABA_B mediated mechanisms underlying the reduction of the threshold for Ca²⁺ spikes [1, 2], due to the effects of drugs, imbalance in osmotic pressure [20], or excess or deficiency of neurotransmitters or neuromodulators [1, 2, 21]. Although many studies have been done to analyze the transition mechanisms into seizure [3–7, 9, 12, 20, 22–28]. However, the detailed dynamics of generalized tonic-clonic seizure, including its dependence to the changing profile of the corticothalamic connection strength have never been studied in detail; a proper understanding of such features might help in developing seizure prediction and control strategies. Surprisingly, the dependence of the spectral characteristics like the frequencies of the oscillations on the parameters of the changing connection strength have also not been studied, despite their potential to yield precursor signals of seizure onset, for example.

In this study, we apply a widely used neural field model of the corticothalamic system to study the dynamics of tonic-clonic seizures [3–5, 7, 8, 24, 29, 30]. Neural field theory (NFT) is a continuum approach that predicts the average dynamics of large numbers of neurons [31, 32]. The specific model used here [33–36] has reproduced and unified many observed features of brain activity based on the physiology, including evoked response potentials [37], activity spectra [38], arousal state dynamics, age-related changes in the physiology of the brain [39], and many other phenomena [3–5, 7, 8, 24, 29, 30, 40–42]. The above NFT model has also been used in seizure studies [3–5, 7], where it has successfully unified features of tonic-clonic and absence seizures [3–5, 7], and explain the dependence of the dynamics and interictal oscillations during absence seizures on the parameters of the changing connection strength between the cortex and the thalamus [43, 44]. Previous studies have shown that a gradual increase of the connection strength between the cortex and thalamus near the alpha instability boundary shown in [8] in this model can initiate nonlinear dynamics whose characteristics closely resemble those of tonic-clonic seizures as a result of a subcritical Hopf bifurcation that destabilizes the ∼ 10 Hz alpha resonance [3, 4, 24, 41]. Changes in other connection strengths also introduce similar dynamics because of the universality properties of the Hopf bifurcation [12].

The general property and bifurcation mechanism of the resultant tonic-clonic seizure has been studied in detail in [3]. However, the impact of underlying parameter changes of the corticothalamic connectivity strength on tonic-clonic seizure onset, dynamics, and termination have not been studied in detail. In particular, an extensive study like [43] on the dependence of the onset and termination of tonic-clonic seizure on the temporal form of the connection strength is necessary to understand the variability in seizure events, such as difference in the onset time and duration among different subjects, and to help lay the foundations for tonic-clonic seizure control strategies. These analyses are also necessary to explain the changes in harmonic structure seen in previous studies [45–47] during seizure, which might be helpful as inputs to seizure prediction strategies. In short, the aims are to understand the effects of physiological parameters on the temporal and spectral characteristics of seizure dynamics, including saddle-cycle oscillations [24]. We note that there are also other plausible causes or routes to tonic-clonic seizures (e.g., including effects of the cerebellum, basal ganglia, and hormones) [25, 27, 28, 48–53], but we retain our existing model to maintain compatibility with the experimental results against which it has been verified in [3].

The outline of this paper is as follows: In the Results, we explore the general characteristics of seizure as well as the dependence of seizure dynamics on the temporal variation of connection strength. In the Discussion, we provide a summary and discuss possible applications of our outcomes and finally, in the Methods section, we present the corticothalamic neural field model along with the temporal variation function and the numerical methods.

Results

In this section we investigate the dynamical characteristics of model tonic-clonic seizures as well as the effects of the temporal variation of the corticothalamic connection strength, ν_se on the dynamics. For the investigation of general characteristics, we keep a constant maximum connection strength ν_max, characteristic duration t₂ − t₁, and characteristic rise time Δ, and all other parameters listed in Table 1.

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Table 1. Nominal parameters of the neural field model from [3].

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To investigate the effect of the variation of ν_se on seizure dynamics we vary ν_max, Δ, and t₂ − t₁ individually by keeping all other parameters constant. Fig 1(a) shows the variation of ν_se with time for the parameters values specified in Table 1.

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Fig 1. Corticothalamic dynamics for temporally varying ν_se, with Δ = 20 s and rest of the parameters shown in Table 1.

(a) Temporal profile of ν_se varying from ν₀ to ν_max and back. Three different regions are identified as: I = pre-ictal state, II = ictal state, and III = post-ictal state. (b) Cortical excitatory field ϕ_e vs. t, showing a 10 Hz spike-wave oscillation. Individual oscillations can not be distinguished on this scale. (c) Zoom of ϕ_e at seizure onset. (d) Zoom of ϕ_e at seizure offset. (e) Power spectrum of ϕ_e in Region II. An arbitrary dB scaling is used because clinical EEG recordings involve additional attenuation by structures between the cortex and the electrode, which we do not model here.

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General characteristics of tonic-clonic seizures

As in [43], three main regions are distinguished according to the dynamics of the cortical activity ϕ_e (cortical excitatory field) as illustrated in Fig 1(b): Region I from 0–50 s is the pre-ictal state when ν_se is too small to initiate seizure-like oscillations; Region II from 125–175 s is the ictal state when ν_se is around its maximum value, ν_max, and the system oscillates with maximum amplitude; and Region III from 250–330 s is the post-ictal state, where ν_se returns to its baseline value, and oscillations start decreasing in amplitude until they completely cease.

Fig 1(c) and 1(d) show the zoomed seizure onset and offset, respectively, which are the transitions from Region I to II, and from Region II to III, respectively.

The normalized power spectrum in Region II is shown in Fig 1(e). Fig 1(e) shows a dominant resonance at ∼ 10 Hz with multiple harmonics in Region II, where power decreases gradually with frequency.

The model of the brain is the same, but the key corticothalamic parameters place it in the regime where a 10 Hz subcritical Hopf bifurcation occurs, rather than a 3 Hz supercritical one. The new features include the existence of the saddle cycle, the different bifurcation types, the different frequencies, and other features explored and discussed later in the paper.

Dynamics of seizure onset.

Fig 1(b) shows that in Region I, the system remains in the steady state because ν_se is below the bifurcation threshold. A small increase in ϕ_e due to the increase of ν_se is also seen in this region. At t = t_θ, which is the time at which ν_se crosses the linear instability threshold, the fixed point loses its stability, and ∼ 18 Hz oscillations appear. The first few oscillations are too small to be distinguished on this scale, but their envelope increases exponentially until t = t_sc, when the trajectory spirals further outwards to a large amplitude 10 Hz limit cycle, as seen in Fig 1(c); these 18 Hz oscillations are termed saddle-cycle oscillations because they are due to a transient saddle cycle located between the stable steady state and the stable large amplitude limit cycle attractor. The envelope of the 10 Hz oscillations continues to increase from t = t_sc until t = t_lc, when the system reaches the large amplitude limit cycle. At t ≈ t_lc, the amplitude of the oscillations overshoots because ν_se is still rapidly increasing. Then, the amplitude of the oscillations increases gradually until ν_se = ν_max in Region II, then decreases. Fig 1(c) shows a clearer view of saddle-cycle oscillations, and times t_sc and t_lc; where we define t_lc to be the point of inflection. We note these results differ strongly from the ones for absence seizures in Ref [43], where a ∼ 3 Hz spike wave morphology was seen via a supercritical Hopf bifurcation with no saddle cycles.

Dynamics of seizure offset.

In Fig 1(d), we see that the amplitude of the oscillations decreases gradually from its peak during the ramp down of ν_se. More specifically, at t = t_lc2, when ν_se crosses the offset bifurcation threshold ν_lc2 = 0.98 mV s [3], the large limit cycle loses stability and the oscillation amplitude decreases steeply to approach the stable steady state in Region III.

Differences between onset and offset dynamics.

Comparing Fig 1(c) with Fig 1(d), we see that ν_θ > ν_lc2, as expected for transitions due to a subcritical Hopf bifurcation. This is further seen in Fig 2, where we see that the system bifurcates from the fixed point at ν_se = ν_θ and reaches the saturated large amplitude attractor at ν_se = ν_lc. As ν_se decreases, the large amplitude attractor becomes unstable at ν_se = ν_lc2 and the system returns toward the fixed point.

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Fig 2. Hysteresis between seizure onset and offset.

(a) ν_se vs. ϕ_e. Black color shows the variation of ϕ_e during ramp up, i.e. during onset, and gray color shows the variation of ϕ_e during ramp down, i.e. during offset. (b) A schematic diagram of the hysteresis. Solid lines show stable states and dashed lines show unstable ones.

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Analytical prediction of onset and offset transition times.

Paralleling the analytic prediction of the characteristic time required to develop absence seizures [43], we next predict characteristic tonic-clonic onset and offset times.

For ν(t) ≈ ν_θ, the oscillation amplitude A obeys (1) where C is a constant, and ν(t) is the instantaneous value of ν_se. Because ν_se only varies with time t, we can make the approximation ν(t) − ν_θ ≈ c(t − t_θ) near the threshold, when the oscillation starts at A_θ. This yields (2) with ; then A = A_lc at t = t_lc (3) (4) where k = [(2/C)ln(A_lc/A_θ)]^1/2. Similar analysis predicts that the transition time t_lc − t_sc from the saddle-cycle attractor to the larger limit cycle also follows this scaling.

The decrease of oscillation amplitude during the ramp down period can be approximated as (5) (6) where C′ and C″ are constants, and t_lc2 is the offset bifurcation threshold as mentioned in previous sections. This yields (7) which indicates a superexponential decrease during seizure offset.

Dynamics during ictal state plateau.

Fig 3 shows the phase space trajectory of ϕ_e for the default parameters in Table 1, except Δ = 2 s, which we use to see the saddle-cycle more clearly. Fig 3(a) shows the trajectory of ϕ_e on the ϕ_e—dϕ_e/dt plane. In the left edge of the figure, we see the evolving fixed point, which first appears as straight line and then moves towards the right with increasing ν_se. Once the system crosses the linear instability threshold, the fixed point becomes unstable and the trajectory spirals out to a large amplitude limit cycle attractor via the unstable saddle-cycle. The amplitude of the large attractor increases gradually until ν_se = ν_max, then decreases until ν_lc2, where it becomes unstable and the system spirals back to the stable fixed point; no saddle-cycle is seen during the inward spiral. Three segments of the trajectory are shown in Fig 3(b)–3(d), to clarify these dynamics. Fig 3(b) shows ϕ_e spiraling outward from the steady state to the saddle-cycle with amplitude ≈ 30 s⁻¹. Fig 3(c) shows the outward spiral from the transient saddle cycle to the limit cycle attractor with amplitude ≈ 90 s⁻¹. Fig 3(d) shows the inward spiral during ramp down of ν_se.

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Fig 3. Phase space trajectory of ϕ_e for Δ = 2 s, and rest of the default parameters as in Table 1.

(a) Trajectory from from t = 5 s to t = 295 s. Initial small straight line labeled with FP corresponds to the evolving fixed point; small dark gray segment labeled with SC corresponds to the saddle-cycle attractor; black segment labeled with LC corresponds to the large amplitude limit cycle attractor. The fixed point and center of the clockwise limit cycle trajectory move from left to right during ramp up and right to left during ramp down. (b) Trajectory from t = 104 s to t = 107 s. (c) Trajectory from t = 114.5 s to t = 150 s. (d) Trajectory from t = 200 s to t = 295 s.

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Fig 4 shows the dynamic spectrum of ϕ_e from Fig 1(b). A sudden appearance of 10 Hz oscillation with multiple harmonics at t = t_θ is seen. These harmonics resemble with the harmonics seen in [3], both experimentally and theoretically. The power of the harmonics decreases with harmonic number and their duration decreases slightly. We find a frequency broadening during the seizure onset at ∼ 113.5 s, due to the rapid change of the amplitude of the oscillations. Frequency broadening of the first few harmonics during seizure offset is also seen, and there is a slight frequency drop.

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Fig 4. Dynamic spectrum for ν_max = 1.2 mV s with the parameters in Table 1.

A Hanning window of 600 data points, an overlap of 200 points, and sampling frequency of 200 Hz was used. The color bar shows the dB scale.

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Dynamics of corticothalamic seizure propagation.

Fig 5(a) and 5(b) show the time series of the fields ϕ_r during onset and offset, respectively. Similarly, Fig 5(c) and 5(d) show the time series of the fields ϕ_s during onset and offset.

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Fig 5.

Time series of fields during seizure onset and offset: (a) ϕ_r at seizure onset. (b) ϕ_r at seizure offset. (c) ϕ_s at seizure onset. (d) ϕ_s at seizure offset.

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From these plots we observe that (i) during onset ϕ_r reaches much higher amplitudes than ϕ_e; and, (ii) the ratio between the amplitude of the small oscillations that develop after crossing the bifurcation and the amplitude of the saturated limit cycle is smaller for ϕ_e than it is for ϕ_r and ϕ_s.

In order to study the interplay among ϕ_e, ϕ_r, and ϕ_s in more detail, we plot their limit cycle phase space trajectories and time series at ν_se ≈ ν_max in Fig 6. Fig 6(a) and 6(b) show the time series and phase space trajectory of ϕ_e, respectively. Fig 6(c) and 6(d) show the time series and phase space trajectory of ϕ_r, respectively. A t₀/2 time shift between the peaks of ϕ_e and ϕ_r is seen due to the propagation delay between these populations. We also see a wide minimum between two successive peaks of ϕ_r. The phase space in Fig 6(d) shows similar trajectory to Fig 6(d), but with greater amplitude. Fig 6(e) and 6(f) show the time series and phase space of ϕ_s, respectively, and they show an equal amplitude but wider peak than Fig 6(c) and 6(d). Fig 6 shows that all three fields exhibit slightly different trajectories, with the higher amplitudes of ϕ_r and ϕ_s near the maximum firing rate.

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Fig 6. Mid-seizure limit cycle dynamics of ϕ_e, ϕ_s, and ϕ_r from t = 149.7 s to t = 150 s with other parameters as in Table 1.

(a) Time series of ϕ_e at ν_se ≈ ν_max. (b) Phase space trajectory of ϕ_e. (c) ϕ_r at ν_se ≈ ν_max. (d) Trajectory of ϕ_r. (e) ϕ_s at ν_se ≈ ν_max. (f) Trajectory of ϕ_s. P and R are successive minimums and Q is the intermediate maximum.

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Close examination of Fig 6 reveals the signal flow through the populations. A peak of ϕ_e reaches ϕ_r and ϕ_s simultaneously t₀/2 later. The peak of ϕ_e coincides approximately with the bottom of the trough of ϕ_r, and a positive excitation with the maximum firing rate appears, which suppress ϕ_s. This suppression then reduce the excitation of ϕ_e a time t₀/2 later and causes an exponential decay. A negative perturbation to ϕ_e results, which then propagates to the thalamus again and reduces the excitation of ϕ_r after a further time t₀/2, which allows a positive excitation of ϕ_s almost immediately. This positive excitation then flows to ϕ_e and initializes the next cycle of the loop. Unlike the absence seizure case [43], the loop provides direct positive feedback in a single pass, whereas the feedback is negative in the absence case and two passes through the loop are required to yield overall positive feedback, thereby reducing the frequency of the instability [8].

At the cellular level, the imbalance between inhibitory and excitatory conductances induced by blocking synaptic and voltage-gated inhibitory conductances, or by activating synaptic and voltage-gated excitatory conductances, incorporates the positive feedback, which leads to seizures [21, 54]. Seizures are suppressed by the opposite manipulations: increasing inhibition or decreasing excitation [21, 54].

Impact of temporal variation of ν_se on seizure dynamics

In this section, we investigate the effects of the temporal variation of ν_se on the model seizure dynamics by varying the maximum connection strength ν_max, duration t₂ − t₁, and rise time Δ, holding all other parameters at the values in Table 1.

We first analyze the impact of the variation of ν_se on the overall dynamics of ϕ_e, as shown in Fig 7. For ν_max = 1 mV s in Fig 7(a), ϕ_e increases with ν_se as shown in Fig 16, then returns smoothly to the initial steady state value as ν_se returns to ν₀. Fig 7(b) and 7(c) show that increasing ν_max, yields periodic oscillations of increasing magnitude as corticothalamic feedback strengthens; oscillations also start earlier and are damped away later because the system crosses onset threshold earlier and offset threshold later for higher ν_max. However, the system does not return to its initial steady state for ν_max > 1.542 mV s; instead it moves to the high firing steady state of Fig 16.

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Fig 7. Time series for different temporal profiles of ν_se, with other parameters as in Table 1.

(a) ϕ_e vs. t for ν_max = 1 mV s. Individual oscillations cannot be distinguished. (b) ν_max = 1.05 mV s. (c) ν_max = 1.25 mV s. (d) Δ = 2 s. (e) Δ = 20 s. (f) Δ = 60 s. (g) t₂ − t₁ = 20 s. (h) t₂ − t₁ = 40 s. (i) t₂ − t₁ = 60 s.

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Fig 7(d)—7(f) show the effects of varying ramp width Δ from 2 s to 60 s. Fig 7(d) shows that for the step-like variation of ν_se for Δ = 2 s, the oscillations rapidly reach maximum amplitude after the transition to the large amplitude attractor and also decrease sharply from their maximum to the initial steady state once the system crosses the threshold during ramp down. Fig 7(e) and 7(f) show that the slower ramp for larger Δ implies that the amplitude of the oscillations during seizure onset and offset decreases more gradually.

Fig 7(g)–7(i) show the effects of variation of the characteristic time t₂ − t₁ from 20 s to 100 s. As expected, the duration of seizure oscillations increases with t₂ − t₁.

Seizure onset time.

Fig 8 quantifies the effects of ν_max and Δ on seizure onset. We do not revisit the variation with t₂ − t₁ because its effects were already discussed in the previous subsection.

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Fig 8. Effects of temporal variation of ν_se on seizure onset with parameters as in Table 1.

(a) t_θ vs. ν_max. (b) t_θ vs. Δ.

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Fig 8(a) shows that t_θ decreases with increasing ν_max, because the system reaches ν_θ earlier for a higher ν_max. Fig 8(b) shows the variation of t_θ with Δ. For Δ < 10 s, t_θ increases slightly with Δ, because due to the high rate of change, ν_se rapidly approaches its maximum, crossing all the bifurcation values. At longer Δ ≥ 10 s, the temporal profile of ν_se becomes smooth and flat topped like Fig 1(a) and ν_se gradually ramps up to the bifurcation point, so the system crosses the threshold later for a larger Δ, resulting in a decrease in t_θ.

Dynamic spectrum.

In this section we discuss the effects of changing the temporal profile of ν_se on the power spectrum of ϕ_e and use its evolution to further clarify the occurrence of transient saddle cycles.

Fig 9(a) shows the dynamic spectrum for ν_max = 1.05 mV s. During the seizure, we observe a peak at approximately ∼ 10 Hz with several harmonics. We also find lower frequency drop and broadening during seizure onset and offset as in Fig 4. Fig 9(b) shows that for ν_max = 1.15 mV s, harmonics have greater duration and power than Fig 9(a). The frequency broadening is a manifestation of the uncertainty principle, which means, mathematically the frequency content of a rapidly changing nonsinusoidal signal will broaden in order to be able to localize the signal in time. During the change, the system simply does not have a precisely defined frequency, whether or not a Fourier transform is actually applied to resulting data. Fig 9(c) shows that for ν_max = 1.55 mV s, there is no oscillation after t = 143.52 s, because the system moves into the high firing steady state after this time. A detailed investigation shows that the power of the peaks increases significantly with ν_max and t₂ − t₁, but decreases slightly with Δ, especially at higher order harmonics. A small peak around 205 s shows that the system returns to the initial steady state via small oscillation after it crosses the offset bifurcation.

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Fig 9. Dynamic spectrum vs. ν_max for the parameters in Table 1.

The power density of the harmonics is calculated using a Hanning window of 600 data points, an overlap of 200 points, and sampling frequency of 200 Hz, the color bar at top shows the dB scale. (a) Dynamic spectrum for ν_max = 1.05 mV s. (b) ν_max = 1.15 mV s. (c) ν_max = 1.55 mV s.

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Characteristic transition times.

In this section we test the analytic prediction made in earlier sections. Fig 10(a) shows t_lc − t_θ vs. (dν_se/dt)^−1/2. A least-squares fit to these data yields (8) with a = (0.042 ± 0.004) V^1/2 s and b = (0.9 ± 1.4) s, which is consistent with Eq (4). Fig 10(b) shows (dν_se/dt)^−1/2 vs. t_lc − t_sc. A least-squares fit yields (9) with a′ = (0.003 ± 0.001) V^1/2 s and b′ = (0.0 ± 0.2) s, which has the same scaling as Eq (4). The fitting also shows that, despite the different bifurcation mechanisms, both onset transitions follow the same scalings as the onset transition of absence seizures [43].

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Fig 10. Dependence of seizure transition times on (dν_se/dt)^−1/2 with the default parameters as in Table 1 and Δ ranges from 2 s to 50 s.

(a) t_lc − t_θ vs. (dν_se/dt)^−1/2; (b) t_lc − t_sc vs. (dν_se/dt)^−1/2, and (c) ln(A/A_lc2) vs. (t − t_lc2)² for Δ = 10 s and time ranges from 190 s to 250 s. Error bar represent uncertainties of the least-squares fits. Points with no error bars are not considered for the least-squares fit.

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Fig 10(c) shows ln(A/A_lc2) vs. (t − t_lc2)² for Δ = 10 s, which follows Eq (7) until the amplitudes of the oscillations start to decrease super-exponentially towards the steady state. A least-squares fit to the linear decrease yields (10) with a″ = (0.0116 ± 0.0002) s⁻² and b″ = (0.018 ± 0.004). The figure shows that the decrease of the envelope follow the linear fit for a relatively short time, after which the decrease becomes steeper. By using Eqs (2) and (3), it can be also shown that decrease within the linear region also follows the same scaling as Eq (4).

Saddle cycle.

Previously, we mentioned the presence of a small amplitude ∼18 Hz transient saddle cycle. The system orbits there for few seconds, then spirals out towards the large amplitude limit cycle attractor. However, this saddle-cycle is not observed in all cases, for example, a colose zoom near the onset of all subfigures of Fig 7 will show that the small amplitude saddle-cycle oscillations like Fig 1(c) are only prominent in Fig 7(c) and 7(d). Here, we explore the dependence of the transient saddle-cycle oscillations on ν_max and Δ.

Fig 11 shows the variation of saddle-cycle oscillations with respect to ν_max, with other parameters as in Table 1. Fig 11(a) shows the phase space trajectory for ν_max = 1.15 mV s. No saddle-cycle attractor is seen in this figure. Fig 11(b) shows the trajectory for ν_max = 1.25 mV s. A small saddle-cycle attractor is seen between the fixed point and the large amplitude attractor. Fig 11(c) and 11(d) show the trajectories for ν_max = 1.35 mV s and 1.45 mV s, respectively. The transient saddle cycle increases in size with ν_max. A similar investigation shows that similar phenomena occur when Δ is varied, with the transient saddle cycle being most prominent for small Δ, completely disappearing for Δ ≳ 20 s.

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Fig 11. Effects of variation of ν_max on saddle-cycle with rest of the parameters as in Table 1.

(a) Phase space trajectory for ν_max = 1.15 mV s. (b) Trajectory for ν_max = 1.25 mV s. (c) Trajectory for ν_max = 1.35 mV s. (d) Trajectory for ν_max = 1.45 mV s.

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To understand the relation between the saddle-cycle oscillation and rate of change of ν_se more clearly, we calculate the power spectrum for different ν_max and Δ. Fig 12(a) shows the variation of the power spectrum with ν_max. For a small ν_max, there is no peak around 18 Hz, but a peak at approximately 18 Hz appears when ν_max ≥ 1.2 mV s and becomes more prominent and strong with increasing ν_max. Fig 12(b) shows that the power of the peak around 18 Hz decreases with Δ and disappears for Δ ≳ 20 s.

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Fig 12. (Color online) Variation in the power of the saddle-cycle oscillations with rest of the parameters in Table 1.

(a) Power spectrum vs. ν_max. (b) Power spectrum vs. Δ. Legends show the corresponding values of ν_max and Δ.

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These results imply that the presence of saddle-cycle oscillations depends on the rate of change of of ν_se. Fig 13 illustrates the presence or absence of saddle-cycle oscillations for 236 different combinations of ν_se and Δ as a function of the value of dν_se/dt. When dν_se/dt < 7 × 10⁻³ mV, there are no saddle-cycle oscillations; for dν_se/dt > 9 × 10⁻³ mV, the system always exhibits saddle-cycle oscillations; while for 7 × 10⁻³ ≲ dν_se/dt ≲ 9 × 10⁻³ mV, there is a narrow mixed region where the presence of transient saddle cycle cannot be predicted solely from the rate of change of ν_se.

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Fig 13. Dependence of saddle-cycle oscillations on dν_se/dt.

Gray crosses show the presence of a saddle-cycle and black crosses show its absence.

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In order to see why transient saddle cycle is only seen for high dν_se/dt, we show the time evolution of 10 Hz and 18 Hz frequency peaks for Δ = 2 s and Δ = 50 s in Fig 14 during seizure onset with other parameters as in Table 1. In Fig 14(a), for Δ = 50 s and dν_se/dt = 0.003 mV, the 10 Hz peak always rise faster than the 18 Hz peak, and hence, always has more power and dominates the spectrum; no saddle cycle is seen in the trajectory. On the other hand, in Fig 14(b), for Δ = 2 s and dν_se/dt = 0.03 mV, the 18 Hz peak rises faster than the 10 Hz peak during onset so there is a ∼ 2 s window in which the 18 Hz peak dominates and hence, the system is seen to exhibit saddle-cycle oscillations during onset in Fig 1, after which the 10 Hz peak dominates. Now, since, ν_θ is a the bifurcation threshold and does not depend on the temporal profile, but ν_lc depends on the temporal profile and the time to reach the 10 Hz limit cycle (i.e., t_lc − t_θ), we conclude that ν_lc is the parameter that defines the existence of the transient saddle cycle. The system will exhibit transient saddle cycle oscillation only if ν_sc > ν_lc at t_sc.

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Fig 14. Temporal variation of frequency peaks during seizure onset; black solid line shows the ∼ 18 Hz peak; gray dashed line shows the ∼ 10 Hz peak with parameters from Table 1.

(a) Δ = 50 s; (b) Δ = 2 s.

https://doi.org/10.1371/journal.pone.0230510.g014

Discussion

We have used an established neural field model of the corticothalamic system [3] to study the dependence of tonic-clonic seizures on the temporal profile of a corticothalamic connection strength ν_se that induces seizures. The effects of varying other connection strengths can also be qualitatively predicted using these outcomes because they will exhibit similar dynamics due to the universality properties of the Hopf bifurcation. Also, our temporal variation of connection strength is an approximation to what seems to occur in living systems, but is an improvement over previous piecewise linear functions with discontinuous derivatives [3]. The parameters and the shape of Eq (20) could be customized in the future using experimental data. The key outcomes are:

The system exhibits ∼ 10 Hz limit cycle oscillations once the connection strength crosses the bifurcation threshold of ν_θ = 1.025 mV s, which is the characteristic frequency of tonic-clonic seizure via a subcritical Hopf bifurcation. The system returns to the resting equilibrium when the connection strength decreases below the offset threshold, ν_lc2 = 0.98 mV s. The difference in onset and offset bifurcation values causes hysteresis; consistent with previously published results that used piecewise linear variation of ν_se, rather than the present more realistic continuous gradual variation.
For V_max ≳ 1.542 mV, the system moves to another steady state near maximum firing rate and only returns to the initial steady state once ν_se returns below an offset threshold.
The amplitude of ϕ_e increases with the maximum connection strength, ν_max, because an increase of the connectivity strength increases the strength of the positive feedback loop between the cortex and the thalamus.
Because increasing the maximum connection strength ν_max increases the amplitudes of the oscillations, it increases the power and the characteristic number of harmonics. The power of the harmonics also increases with the seizure duration t₂ − t₁, but decreases slightly with the ramp duration Δ.
The characteristic transition times required to reach the saturated limit cycle oscillation from the seizure threshold or the end of the saddle-cycle oscillations to the steady state are predicted and verified numerically to be inversely proportional to the square root of the rate of change of the connection strength.
The system can also show transient ∼ 18 Hz saddle-cycle oscillation at the beginning of the seizure for high dν_se/dt before moving to the 10 Hz attractor. These saddle-cycles become more prominent as dν_se/dt increases; a system with dν_se/dt < 7 × 10⁻³ mV never exhibits saddle-cycles, whereas one with dν_se/dt > 9 × 10⁻³ mV always does.

Overall, the present study enables the varying spectral and temporal characteristics of seizures to be related to underlying physiological changes of the brain, such as changes in the connection strength between the cortex and the thalamus. The outcomes can potentially be used to help explain the variability of seizure onset properties and seizure frequency across subjects by examining the temporal and spectral characteristics of seizure [55, 56]. It may thus be possible to constrain the physiological properties of the corticothalamic connection strength dynamics of a subject by comparing the wave properties of seizure oscillations, such as amplitude, and frequency, with theory. A better understanding of the physiological properties of corticothalamic connection strength might also constrain changes in levels of neurotransmitters or neuromodulators. Real-time fitting of the theoretical dynamics to observed waveforms may also be feasible, leading to the possibility of implementing feedback control systems based on the dynamics. Connection strengths can be manipulated experimentally, with varying degrees of specificity, via agonists and antagonists of various neuromodulators, for example, which directly affect synaptic communication. A well known example is the kindling of some types seizures via administration of penicillin. Conversely, antiepileptic medications likely tend to normalize synaptic strengths and more detailed model explorations could help to better target such interventions. Outcomes related to the seizure onsets and saddle-cycle oscillation might also contribute to improved seizure prediction algorithms. Finally, using this model, it is also possible to predict the impact of varying other connection strengths than the corticothalamic one, both via the universality properties of the Hopf bifurcation [3] and through direct simulations.

Methods

In this section, we present a brief description of the corticothalamic neural field model used, along with the form of temporal variation of corticothalamic coupling strength [3, 4, 8].

Corticothalamic field model

To investigate the dynamics of tonic-clonic seizure, we use the neural field model of the corticothalamic system seen in Fig 15. In this study we use the same analytical model of [43], but in different parametric regime suitable to study the tonic-clonic seizure. The neural populations are denoted as: e = excitatory cortical; i = inhibitory cortical; s = thalamic relay neurons; r = thalamic reticular nucleus; and n = external inputs. The dynamical variables within each neural population a are the local mean cell-body potential V_a, the mean rate of firing at the cell-body Q_a, and the propagating axonal fields ϕ_a. The firing rates Q_a are related to the potentials V_a by the response function (11) where S is a smooth sigmoidal function that increases from 0 to Q_max as V_a increases from −∞ to ∞, with (12) where θ is the mean neural firing threshold, σ is the standard deviation of this threshold, and Q_max is the maximum firing rate [3, 8].

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Fig 15. Schematic diagram of the corticothalamic model system.

The neural populations shown are cortical excitatory (e), inhibitory (i), thalamic reticular (r), thalamic relay (s), and n = external inputs. The parameter ν_ab quantifies the connection to population a from population b. Inhibitory connections are shown with dashed lines.

https://doi.org/10.1371/journal.pone.0230510.g015

In each neural population, firing rates Q_a generate propagating axonal fields ϕ_a that approximately obey the damped wave equation [3, 8] (13) where the spatiotemporal differential operator D_a is (14) where γ_a = v_a/r_a is the damping rate, r_a and v_a are the characteristic range and conduction velocity of axons of type a, and ∇² is the Laplacian operator. The smallness of r_i, r_s, and r_r enables us to set γ_a ≃ ∞ except for a = e. The cell-body potential V_a results after postsynaptic potentials have propagated through the dendritic tree and then been summed as their resulting currents charge the soma. For excitatory and inhibitory neurons within the cortex, this is approximated via the second-order delay-differential equation [8] (15) where a = e, i and the temporal differential operator is given by (16) The quantities α and β in Eq (16) are the inverse decay and rise times, respectively, of the cell-body potential produced by an impulse at a dendritic synapse. Note that input from the thalamus to the cortex is delayed in Eq (15) by a propagation time t₀/2. For neurons within the specific and reticular nuclei of the thalamus, it is the input from the cortex that is time delayed, so (17) where a = s, r. The connection strengths are given by ν_ab = N_ab s_ab, where N_ab is the mean number of synapses to neurons of type a from type b and s_ab is the strength of the response in neurons a to a unit signal from neurons of type b. The final term on the right-hand side of Eq (17) describes inputs from outside the corticothalamic system. In order to simplify the model we only include the connections shown in Fig 15, so only 10 of the possible 16 connections between the four neural populations are nonzero [8]. We also assume the random intracortical connectivity and the number of connections between populations is proportional to the number of synapses [57, 58]. This random connectivity assumption provides N_ib = N_eb for all b, so ν_ee = ν_ie, ν_ei = ν_ii and ν_es = ν_is[40].

Setting all spatial and temporal derivatives in Eqs (12)–(17) to zero determines spatially uniform corticothalamic steady states. The steady state firing rate, of ϕ_e is then given by [29] (18) The properties of steady states in the corticothalamic model have been studied extensively in [8, 29], and we use the outcomes to identify the stable and unstable regions of the steady state. Fig 16 shows the steady state dependence of on ν_se with other parameters as in Table 1. It is seen that there are two stable steady state solutions: one corresponds to low mean firing rate and another to very high mean firing rate [29]. The low firing steady state was identified with normal states of brain activity in previous studies [8, 36]. The low firing-rate fixed point loses its stability at ν_se = ν_θ. A steep increase in is seen near ν_i because the increasing ν_se push the sigmoid from its minimum by increasing the in Eq (18), which results in an increase of the gain between the thalamus and the cortex. With further increase of ν_se, the system eventually moves to a steady state with near-maximum firing rate. This high firing steady state is beyond the scope of our model because it will lead to effects such as hypoxia, which are not included here.

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Fig 16. (Color online) Steady states solution of the corticothalamic system for the variation of ν_se for tonic-clonic seizure.

Black lines and the letter ‘S’ represent the stable steady state, and red lines and the letter ‘U’ represent the unstable steady states. Here ν_θ is the threshold value when the stable steady state becomes unstable. The inset shows zoomed view of the area around ν_θ.

https://doi.org/10.1371/journal.pone.0230510.g016

Temporal ramping

Brain activity propagates via the coupling of the various neuronal populations. Previous studies have shown that a gradual ramp-up of the coupling strength between the neuronal populations can lead from a stable steady state to periodic seizure oscillations [3, 43]. It is also seen that the dynamical and spectral characteristics of the resultant seizure-like oscillations depend on the physiological properties of the ramp of the coupling strength, such as, the maximum amplitude of the ramp, ramp rate, and characteristic duration [43].

In this paper, we ramp the coupling strength ν_se from an initial value ν₀ to a maximum value ν_max and back to see the impact of the ramp characteristics on tonic-clonic seizures, with [43] (19) (20) where t is the time. The ramp rise is centered on t₁, and the ramp fall is centered on t₂, and Δ is the characteristic rise time. Now, 0 ≤ f(t) ≤ π, so we normalize by dividing by f_max − f_min as seen in Eq (19), where f_max and f_min are the maximum and minimum values of f(t) actually encountered in a given instance.

Numerical methods

We use NFTsim [59], which is a publicly available neural field software, to solve Eqs (11)–(17) numerically for the spatially uniform case in which the ∇² term in Eq (14) is zero. To vary ν_se temporally, we use Eqs (19) and (20). This involves solving ordinary delay differential equations, because there is a propagation time delay t₀/2 between the different neural populations present in Eqs (15) and (17). Hence, a fourth-order Runge-Kutta integration is employed to solve these equations, with an integration time step of 10⁻⁴ s and store time histories of the delay terms t₀/2 into the past.

Because extensive comparisons with experiment have demonstrated that the normal brain operates close to stable fixed points [3, 8, 29, 40, 42], we start our simulations from a corticothalamic steady state with low firing rate. However, because of the delay time t₀/2, we must specify these initial steady-state conditions to apply for times −t₀/2 < t ≤ 0.

We use the parameters in Table 1 as the initial parameters, which are taken from [3] with ν₀ = 0.8 mV s in all cases. A constant input ν_sn ϕ_n = 2 mV is used and no external noise is applied in the simulations as the seizure onset occurs spontaneously. Simulations are 300 s long, and we record the output time series every 5 ms. For all simulations, we use the default parameters shown in Table 1 unless otherwise specified. The default parameters we used are the corresponding parameter set of [3] for tonic-clonic seizure which push the system into the vicinity of alpha instability. For the dynamic spectrum and power spectrum analysis, we employ the FFT (fast Fourier transform) algorithm with a Hanning window of 600 data points with an overlap of 200 points and sampling frequency of 200 Hz.

References

1. Engel J, Pedley TA. Epilepsy: A comprehensive textbook. vol. 1. Lippincott-Raven, Philadelphia; 1997.
2. Browne TR, Holmes GL. Handbook of epilepsy. Lippincott Williams & Wilkins, Philadelphia; 2000.
3. Breakspear M, Roberts JA, Terry JR, Rodrigues S, Mahant N, Robinson PA. A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb Cortex. 2006;16(9):1296–1313. pmid:16280462
- View Article
- PubMed/NCBI
- Google Scholar
4. Marten F, Rodrigues S, Benjamin O, Richardson MP, Terry JR. Onset of polyspike complexes in a mean-field model of human electroencephalography and its application to absence epilepsy. Philos Trans R Soc A. 2009;367(1891):1145–1161.
- View Article
- Google Scholar
5. Roberts JA, Robinson PA. Modeling absence seizure dynamics: implications for basic mechanisms and measurement of thalamocortical and corticothalamic latencies. J Theor Biol. 2008;253(1):189–201. pmid:18407293
- View Article
- PubMed/NCBI
- Google Scholar
6. Rodrigues S, Barton D, Szalai R, Benjamin O, Richardson MP, Terry JR. Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model. J Comput Neurosci. 2009;27(3):507–526. pmid:19499316
- View Article
- PubMed/NCBI
- Google Scholar
7. Kim JW, Robinson PA. Compact dynamical model of brain activity. Phys Rev E. 2007;75(3):031907.
- View Article
- Google Scholar
8. Robinson PA, Rennie CJ, Rowe DL. Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Phys Rev E. 2002;65(4):041924.
- View Article
- Google Scholar
9. Chen M, Guo D, Wang T, Jing W, Xia Y, Xu P, et al. Bidirectional control of absence seizures by the basal ganglia: a computational evidence. PLoS Comput Biol. 2014;10(3):e1003495. pmid:24626189
- View Article
- PubMed/NCBI
- Google Scholar
10. Velazquez JLP, Dominguez LG, Gaetz W, Cheyne D, Snead OC III, Wennberg R. Fluctuations in phase synchronization in brain activity: physiological interpretations of phase locking patterns. In: Topical Problems of Nonlinear Wave Physics. International Society for Optics and Photonics; 2006. p. 597510.
11. Lytton WW, Orman R, Stewart M. Computer simulation of epilepsy: implications for seizure spread and behavioral dysfunction. Epilepsy Behav. 2005;7(3):336–344. pmid:16105749
- View Article
- PubMed/NCBI
- Google Scholar
12. Jirsa VK, Stacey WC, Quilichini PP, Ivanov AI, Bernard C. On the nature of seizure dynamics. Brain. 2014;137(8):2210–2230. pmid:24919973
- View Article
- PubMed/NCBI
- Google Scholar
13. Dodrill CB. Correlates of generalized tonic-clonic seizures with intellectual, neuropsychological, emotional, and social function in patients with epilepsy. Epilepsia. 1986;27(4):399–411. pmid:3087744
- View Article
- PubMed/NCBI
- Google Scholar
14. Baier G, Rosch R, Taylor PN, Wang Y. Design Principle for a Population-Based Model of Epileptic Dynamics. In: Complexity and Synergetics. Springer; 2018. p. 333–347.
15. Williams D. A study of thalamic and cortical rhythms in petit mal. Brain. 1953;76(1):50–69. pmid:13041922
- View Article
- PubMed/NCBI
- Google Scholar
16. Steriade M, Amzica F, Neckelmann D, Timofeev I. Spike-wave complexes and fast components of cortically generated seizures. II. Extra-and intracellular patterns. J Neurophysiol. 1998;80(3):1456–1479. pmid:9744952
- View Article
- PubMed/NCBI
- Google Scholar
17. Steriade M, Contreras D. Spike-wave complexes and fast components of cortically generated seizures. I. Role of neocortex and thalamus. J Neurophysiol. 1998;80(3):1439–1455. pmid:9744951
- View Article
- PubMed/NCBI
- Google Scholar
18. Voss LJ, Jacobson G, Sleigh JW, Steyn-Ross A, Steyn-Ross M. Excitatory effects of gap junction blockers on Cereb. Cortex seizure-like activity in rats and mice. Epilepsia. 2009;50(8):1971–1978. pmid:19486358
- View Article
- PubMed/NCBI
- Google Scholar
19. Gloor P, Fariello RG. Generalized epilepsy: some of its cellular mechanisms differ from those of focal epilepsy. Trends Neurosci. 1988;11(2):63–68. pmid:2465601
- View Article
- PubMed/NCBI
- Google Scholar
20. Andrew RD. Seizure and acute osmotic change: clinical and neurophysiological aspects. J Neurol Sci. 1991;101(1):7–18. pmid:2027029
- View Article
- PubMed/NCBI
- Google Scholar
21. Staley K. Molecular mechanisms of epilepsy. Nature Neurosci. 2015;18(3):367. pmid:25710839
- View Article
- PubMed/NCBI
- Google Scholar
22. Hall D, Kuhlmann L. Mechanisms of seizure propagation in 2-dimensional centre-surround recurrent networks. PLoS One. 2013;8(8):e71369. pmid:23967201
- View Article
- PubMed/NCBI
- Google Scholar
23. Kramer MA, Truccolo W, Eden UT, Lepage KQ, Hochberg LR, Eskandar EN, et al. Human seizures self-terminate across spatial scales via a critical transition. Proc Natl Acad Sci USA. 2012;109(51):21116–21121. pmid:23213262
- View Article
- PubMed/NCBI
- Google Scholar
24. Yang DP, Robinson PA. Critical dynamics of Hopf bifurcations in the corticothalamic system: Transitions from normal arousal states to epileptic seizures. Phys Rev E. 2017;95(4):042410. pmid:28505725
- View Article
- PubMed/NCBI
- Google Scholar
25. Suffczynski P, Lopes da Silva FH, Parra J, Velis DN, Bouwman BM, Van Rijn CM, et al. Dynamics of epileptic phenomena determined from statistics of ictal transitions. IEEE Trans Biomed Engg. 2006;53(3):524–532.
- View Article
- Google Scholar
26. Suffczynski P, Kalitzin S, Lopes da Silva FH. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network. Neurosci. 2004;126(2):467–484.
- View Article
- Google Scholar
27. Suffczynski P, Kalitzin S, Pfurtscheller G, Lopes da Silva FH. Computational model of thalamo-cortical networks: dynamical control of alpha rhythms in relation to focal attention. Int J Psychophysiol. 2001;43(1):25–40. pmid:11742683
- View Article
- PubMed/NCBI
- Google Scholar
28. Suffczynski P, Lopes da Silva FH, Parra J, Velis D, Kalitzin S. Epileptic transitions: model predictions and experimental validation. J Clin Neurophysiol. 2005;22(5):288–299. pmid:16357634
- View Article
- PubMed/NCBI
- Google Scholar
29. Robinson PA, Rennie CJ, Rowe DL, O’Connor SC. Estimation of multiscale neurophysiologic parameters by electroencephalographic means. Hum Brain Mapp. 2004;23(1):53–72. pmid:15281141
- View Article
- PubMed/NCBI
- Google Scholar
30. Kim JW, Roberts JA, Robinson PA. Dynamics of epileptic seizures: evolution, spreading, and suppression. J Theor Biol. 2009;257(4):527–532. pmid:19150365
- View Article
- PubMed/NCBI
- Google Scholar
31. Deco G, Jirsa VK, Robinson PA, Breakspear M, Friston KJ. The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Comput Biol. 2008;4(8):e1000092. pmid:18769680
- View Article
- PubMed/NCBI
- Google Scholar
32. Pinotsis DA, Moran RJ, Friston KJ. Dynamic causal modeling with neural fields. NeuroImage. 2012;59(2):1261–1274. pmid:21924363
- View Article
- PubMed/NCBI
- Google Scholar
33. Nunez PL. The brain wave equation: A model for the EEG. Math Biosci. 1974;21(3-4):279–297.
- View Article
- Google Scholar
34. Freeman WJ. Mass action in the nervous system. Academic Press, New York; 1975.
35. Jirsa VK, Haken H. Field theory of electromagnetic brain activity. Phys Rev Lett. 1996;77(5):960–963. pmid:10062950
- View Article
- PubMed/NCBI
- Google Scholar
36. Robinson PA, Rennie CJ, Wright JJ. Propagation and stability of waves of electrical activity in the Cereb. Cortex. Phys Rev E. 1997;56(1):826.
- View Article
- Google Scholar
37. Rennie CJ, Robinson PA, Wright JJ. Unified neurophysical model of EEG spectra and evoked potentials. Biol Cybern. 2002;86(6):457–471. pmid:12111274
- View Article
- PubMed/NCBI
- Google Scholar
38. Robinson PA, Whitehouse RW, Rennie CJ. Nonuniform corticothalamic continuum model of electroencephalographic spectra with application to split-alpha peaks. Phys Rev E. 2003;68(2):021922.
- View Article
- Google Scholar
39. van Albada SJ, Kerr CC, Chiang AKI, Rennie CJ, Robinson PA. Neurophysiological changes with age probed by inverse modeling of EEG spectra. Clin Neurophysiol. 2010;121(1):21–38. pmid:19854102
- View Article
- PubMed/NCBI
- Google Scholar
40. Abeysuriya RG, Rennie CJ, Robinson PA. Physiologically based arousal state estimation and dynamics. J Neurosci Methods. 2015;253:55–69. pmid:26072247
- View Article
- PubMed/NCBI
- Google Scholar
41. Breakspear M, Terry JR, Friston KJ. Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. Network: Comp Neural Sci. 2003;14(4):703–732.
- View Article
- Google Scholar
42. Abeysuriya RG, Rennie CJ, Robinson PA. Prediction and verification of nonlinear sleep spindle harmonic oscillations. J Theor Biol. 2014;344:70–77. pmid:24291492
- View Article
- PubMed/NCBI
- Google Scholar
43. Deeba F, Sanz-Leon P, Robinson PA. Dependence of absence seizure dynamics on physiological parameter evolution. J Theor Biol. 2018;454:11–21. pmid:29807025
- View Article
- PubMed/NCBI
- Google Scholar
44. Deeba F, Sanz-Leon P, Robinson PA. Unified dynamics of interictal events and absence seizures. Phys Rev E. 2019;100(2):022407. pmid:31574631
- View Article
- PubMed/NCBI
- Google Scholar
45. Schiff SJ, Colella D, Jacyna GM, Hughes E, Creekmore JW, Marshall A, et al. Brain chirps: spectrographic signatures of epileptic seizures. Clin Neurophysiol. 2000;111(6):953–958. pmid:10825700
- View Article
- PubMed/NCBI
- Google Scholar
46. Jiruska P, De Curtis M, Jefferys JGR, Schevon CA, Schiff SJ, Schindler K. Synchronization and desynchronization in epilepsy: controversies and hypotheses. J Physiol. 2013;591(4):787–797. pmid:23184516
- View Article
- PubMed/NCBI
- Google Scholar
47. Schiff SJ, Milton JG, Heller J, Weinstein SL. Wavelet transforms and surrogate data for electroencephalographic spike and seizure localization. Optical Engineering. 1994;33(7):2162–2170.
- View Article
- Google Scholar
48. Lopes da Silva FH, Pijn JP, Velis D, Nijssen PCG. Alpha rhythms: noise, dynamics and models. Int J Psychophysiol. 1997;26(1):237–249. pmid:9203006
- View Article
- PubMed/NCBI
- Google Scholar
49. Blumenfeld H, Varghese GI, Purcaro MJ, Motelow JE, Enev M, McNally KA, et al. Cortical and subcortical networks in human secondarily generalized tonic–clonic seizures. Brain. 2009;132(4):999–1012. pmid:19339252
- View Article
- PubMed/NCBI
- Google Scholar
50. Aminoff MJ, Simon RP, Wiedemann E. The hormonal responses to generalized tonic-clonic seizures. Brain. 1984;107(2):569–578. pmid:6144354
- View Article
- PubMed/NCBI
- Google Scholar
51. Dam AM. Epilepsy and neuron loss in the hippocampus. Epilepsia. 1980;21(6):617–629. pmid:6777154
- View Article
- PubMed/NCBI
- Google Scholar
52. Bauer J, Stefan H, Schrell U, Uhlig B, Landgraf S, Neubauer U, et al. Serum prolactin concentrations and epilepsy. Eur Arch Psyc Clin Neurosci. 1992;241(6):365–371.
- View Article
- Google Scholar
53. Koppert M, Kalitzin S, Velis D, Lopes da Silva FH, Viergever MA. Reactive control of epileptiform discharges in realistic computational neuronal models with bistability. Int J Neural Sys. 2013;23(01):1250032.
- View Article
- Google Scholar
54. Soltesz I, Staley K. Computational neuroscience in epilepsy. Academic Press; 2011.
55. Tzallas AT, Tsipouras MG, Fotiadis DI. Epileptic seizure detection in EEGs using time–frequency analysis. IEEE Trans Inf Technol Biomed. 2009;13(5):703–710. pmid:19304486
- View Article
- PubMed/NCBI
- Google Scholar
56. Cohen L. Time-frequency distributions-a review. Proc IEEE. 1989;77(7):941–981.
- View Article
- Google Scholar
57. Liley DTJ, Cadusch PJ, Wright JJ. A continuum theory of electro-cortical activity. Neurocomputing. 1999;26:795–800.
- View Article
- Google Scholar
58. Braitenberg V, Schüz A. Cortex: statistics and geometry of neuronal connectivity. Springer Science & Business Media; 2013.
59. Sanz-Leon P, Robinson PA, Knock SA, Drysdale PM, Abeysuriya RG, Fung PK, et al. NFTsim: theory and simulation of multiscale neural field dynamics. PLoS Comput Biol. 2018;14(8):e1006387. pmid:30133448
- View Article
- PubMed/NCBI
- Google Scholar

[ref1] 1. Engel J, Pedley TA. Epilepsy: A comprehensive textbook. vol. 1. Lippincott-Raven, Philadelphia; 1997.

[ref2] 2. Browne TR, Holmes GL. Handbook of epilepsy. Lippincott Williams & Wilkins, Philadelphia; 2000.

[ref3] 3. Breakspear M, Roberts JA, Terry JR, Rodrigues S, Mahant N, Robinson PA. A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb Cortex. 2006;16(9):1296–1313. pmid:16280462
View Article
PubMed/NCBI
Google Scholar

[4] View Article

[5] PubMed/NCBI

[6] Google Scholar

[ref4] 4. Marten F, Rodrigues S, Benjamin O, Richardson MP, Terry JR. Onset of polyspike complexes in a mean-field model of human electroencephalography and its application to absence epilepsy. Philos Trans R Soc A. 2009;367(1891):1145–1161.
View Article
Google Scholar

[8] View Article

[9] Google Scholar

[ref5] 5. Roberts JA, Robinson PA. Modeling absence seizure dynamics: implications for basic mechanisms and measurement of thalamocortical and corticothalamic latencies. J Theor Biol. 2008;253(1):189–201. pmid:18407293
View Article
PubMed/NCBI
Google Scholar

[11] View Article

[12] PubMed/NCBI

[13] Google Scholar

[ref6] 6. Rodrigues S, Barton D, Szalai R, Benjamin O, Richardson MP, Terry JR. Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model. J Comput Neurosci. 2009;27(3):507–526. pmid:19499316
View Article
PubMed/NCBI
Google Scholar

[15] View Article

[16] PubMed/NCBI

[17] Google Scholar

[ref7] 7. Kim JW, Robinson PA. Compact dynamical model of brain activity. Phys Rev E. 2007;75(3):031907.
View Article
Google Scholar

[19] View Article

[20] Google Scholar

[ref8] 8. Robinson PA, Rennie CJ, Rowe DL. Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Phys Rev E. 2002;65(4):041924.
View Article
Google Scholar

[22] View Article

[23] Google Scholar

[ref9] 9. Chen M, Guo D, Wang T, Jing W, Xia Y, Xu P, et al. Bidirectional control of absence seizures by the basal ganglia: a computational evidence. PLoS Comput Biol. 2014;10(3):e1003495. pmid:24626189
View Article
PubMed/NCBI
Google Scholar

[25] View Article

[26] PubMed/NCBI

[27] Google Scholar

[ref10] 10. Velazquez JLP, Dominguez LG, Gaetz W, Cheyne D, Snead OC III, Wennberg R. Fluctuations in phase synchronization in brain activity: physiological interpretations of phase locking patterns. In: Topical Problems of Nonlinear Wave Physics. International Society for Optics and Photonics; 2006. p. 597510.

[ref11] 11. Lytton WW, Orman R, Stewart M. Computer simulation of epilepsy: implications for seizure spread and behavioral dysfunction. Epilepsy Behav. 2005;7(3):336–344. pmid:16105749
View Article
PubMed/NCBI
Google Scholar

[30] View Article

[31] PubMed/NCBI

[32] Google Scholar

[ref12] 12. Jirsa VK, Stacey WC, Quilichini PP, Ivanov AI, Bernard C. On the nature of seizure dynamics. Brain. 2014;137(8):2210–2230. pmid:24919973
View Article
PubMed/NCBI
Google Scholar

[34] View Article

[35] PubMed/NCBI

[36] Google Scholar

[ref13] 13. Dodrill CB. Correlates of generalized tonic-clonic seizures with intellectual, neuropsychological, emotional, and social function in patients with epilepsy. Epilepsia. 1986;27(4):399–411. pmid:3087744
View Article
PubMed/NCBI
Google Scholar

[38] View Article

[39] PubMed/NCBI

[40] Google Scholar

[ref14] 14. Baier G, Rosch R, Taylor PN, Wang Y. Design Principle for a Population-Based Model of Epileptic Dynamics. In: Complexity and Synergetics. Springer; 2018. p. 333–347.

[ref15] 15. Williams D. A study of thalamic and cortical rhythms in petit mal. Brain. 1953;76(1):50–69. pmid:13041922
View Article
PubMed/NCBI
Google Scholar

[43] View Article

[44] PubMed/NCBI

[45] Google Scholar

[ref16] 16. Steriade M, Amzica F, Neckelmann D, Timofeev I. Spike-wave complexes and fast components of cortically generated seizures. II. Extra-and intracellular patterns. J Neurophysiol. 1998;80(3):1456–1479. pmid:9744952
View Article
PubMed/NCBI
Google Scholar

[47] View Article

[48] PubMed/NCBI

[49] Google Scholar

[ref17] 17. Steriade M, Contreras D. Spike-wave complexes and fast components of cortically generated seizures. I. Role of neocortex and thalamus. J Neurophysiol. 1998;80(3):1439–1455. pmid:9744951
View Article
PubMed/NCBI
Google Scholar

[51] View Article

[52] PubMed/NCBI

[53] Google Scholar

[ref18] 18. Voss LJ, Jacobson G, Sleigh JW, Steyn-Ross A, Steyn-Ross M. Excitatory effects of gap junction blockers on Cereb. Cortex seizure-like activity in rats and mice. Epilepsia. 2009;50(8):1971–1978. pmid:19486358
View Article
PubMed/NCBI
Google Scholar

[55] View Article

[56] PubMed/NCBI

[57] Google Scholar

[ref19] 19. Gloor P, Fariello RG. Generalized epilepsy: some of its cellular mechanisms differ from those of focal epilepsy. Trends Neurosci. 1988;11(2):63–68. pmid:2465601
View Article
PubMed/NCBI
Google Scholar

[59] View Article

[60] PubMed/NCBI

[61] Google Scholar

[ref20] 20. Andrew RD. Seizure and acute osmotic change: clinical and neurophysiological aspects. J Neurol Sci. 1991;101(1):7–18. pmid:2027029
View Article
PubMed/NCBI
Google Scholar

[63] View Article

[64] PubMed/NCBI

[65] Google Scholar

[ref21] 21. Staley K. Molecular mechanisms of epilepsy. Nature Neurosci. 2015;18(3):367. pmid:25710839
View Article
PubMed/NCBI
Google Scholar

[67] View Article

[68] PubMed/NCBI

[69] Google Scholar

[ref22] 22. Hall D, Kuhlmann L. Mechanisms of seizure propagation in 2-dimensional centre-surround recurrent networks. PLoS One. 2013;8(8):e71369. pmid:23967201
View Article
PubMed/NCBI
Google Scholar

[71] View Article

[72] PubMed/NCBI

[73] Google Scholar

[ref23] 23. Kramer MA, Truccolo W, Eden UT, Lepage KQ, Hochberg LR, Eskandar EN, et al. Human seizures self-terminate across spatial scales via a critical transition. Proc Natl Acad Sci USA. 2012;109(51):21116–21121. pmid:23213262
View Article
PubMed/NCBI
Google Scholar

[75] View Article

[76] PubMed/NCBI

[77] Google Scholar

[ref24] 24. Yang DP, Robinson PA. Critical dynamics of Hopf bifurcations in the corticothalamic system: Transitions from normal arousal states to epileptic seizures. Phys Rev E. 2017;95(4):042410. pmid:28505725
View Article
PubMed/NCBI
Google Scholar

[79] View Article

[80] PubMed/NCBI

[81] Google Scholar

[ref25] 25. Suffczynski P, Lopes da Silva FH, Parra J, Velis DN, Bouwman BM, Van Rijn CM, et al. Dynamics of epileptic phenomena determined from statistics of ictal transitions. IEEE Trans Biomed Engg. 2006;53(3):524–532.
View Article
Google Scholar

[83] View Article

[84] Google Scholar

[ref26] 26. Suffczynski P, Kalitzin S, Lopes da Silva FH. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network. Neurosci. 2004;126(2):467–484.
View Article
Google Scholar

[86] View Article

[87] Google Scholar

[ref27] 27. Suffczynski P, Kalitzin S, Pfurtscheller G, Lopes da Silva FH. Computational model of thalamo-cortical networks: dynamical control of alpha rhythms in relation to focal attention. Int J Psychophysiol. 2001;43(1):25–40. pmid:11742683
View Article
PubMed/NCBI
Google Scholar

[89] View Article

[90] PubMed/NCBI

[91] Google Scholar

[ref28] 28. Suffczynski P, Lopes da Silva FH, Parra J, Velis D, Kalitzin S. Epileptic transitions: model predictions and experimental validation. J Clin Neurophysiol. 2005;22(5):288–299. pmid:16357634
View Article
PubMed/NCBI
Google Scholar

[93] View Article

[94] PubMed/NCBI

[95] Google Scholar

[ref29] 29. Robinson PA, Rennie CJ, Rowe DL, O’Connor SC. Estimation of multiscale neurophysiologic parameters by electroencephalographic means. Hum Brain Mapp. 2004;23(1):53–72. pmid:15281141
View Article
PubMed/NCBI
Google Scholar

[97] View Article

[98] PubMed/NCBI

[99] Google Scholar

[ref30] 30. Kim JW, Roberts JA, Robinson PA. Dynamics of epileptic seizures: evolution, spreading, and suppression. J Theor Biol. 2009;257(4):527–532. pmid:19150365
View Article
PubMed/NCBI
Google Scholar

[101] View Article

[102] PubMed/NCBI

[103] Google Scholar

[ref31] 31. Deco G, Jirsa VK, Robinson PA, Breakspear M, Friston KJ. The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Comput Biol. 2008;4(8):e1000092. pmid:18769680
View Article
PubMed/NCBI
Google Scholar

[105] View Article

[106] PubMed/NCBI

[107] Google Scholar

[ref32] 32. Pinotsis DA, Moran RJ, Friston KJ. Dynamic causal modeling with neural fields. NeuroImage. 2012;59(2):1261–1274. pmid:21924363
View Article
PubMed/NCBI
Google Scholar

[109] View Article

[110] PubMed/NCBI

[111] Google Scholar

[ref33] 33. Nunez PL. The brain wave equation: A model for the EEG. Math Biosci. 1974;21(3-4):279–297.
View Article
Google Scholar

[113] View Article

[114] Google Scholar

[ref34] 34. Freeman WJ. Mass action in the nervous system. Academic Press, New York; 1975.

[ref35] 35. Jirsa VK, Haken H. Field theory of electromagnetic brain activity. Phys Rev Lett. 1996;77(5):960–963. pmid:10062950
View Article
PubMed/NCBI
Google Scholar

[117] View Article

[118] PubMed/NCBI

[119] Google Scholar

[ref36] 36. Robinson PA, Rennie CJ, Wright JJ. Propagation and stability of waves of electrical activity in the Cereb. Cortex. Phys Rev E. 1997;56(1):826.
View Article
Google Scholar

[121] View Article

[122] Google Scholar

[ref37] 37. Rennie CJ, Robinson PA, Wright JJ. Unified neurophysical model of EEG spectra and evoked potentials. Biol Cybern. 2002;86(6):457–471. pmid:12111274
View Article
PubMed/NCBI
Google Scholar

[124] View Article

[125] PubMed/NCBI

[126] Google Scholar

[ref38] 38. Robinson PA, Whitehouse RW, Rennie CJ. Nonuniform corticothalamic continuum model of electroencephalographic spectra with application to split-alpha peaks. Phys Rev E. 2003;68(2):021922.
View Article
Google Scholar

[128] View Article

[129] Google Scholar

[ref39] 39. van Albada SJ, Kerr CC, Chiang AKI, Rennie CJ, Robinson PA. Neurophysiological changes with age probed by inverse modeling of EEG spectra. Clin Neurophysiol. 2010;121(1):21–38. pmid:19854102
View Article
PubMed/NCBI
Google Scholar

[131] View Article

[132] PubMed/NCBI

[133] Google Scholar

[ref40] 40. Abeysuriya RG, Rennie CJ, Robinson PA. Physiologically based arousal state estimation and dynamics. J Neurosci Methods. 2015;253:55–69. pmid:26072247
View Article
PubMed/NCBI
Google Scholar

[135] View Article

[136] PubMed/NCBI

[137] Google Scholar

[ref41] 41. Breakspear M, Terry JR, Friston KJ. Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. Network: Comp Neural Sci. 2003;14(4):703–732.
View Article
Google Scholar

[139] View Article

[140] Google Scholar

[ref42] 42. Abeysuriya RG, Rennie CJ, Robinson PA. Prediction and verification of nonlinear sleep spindle harmonic oscillations. J Theor Biol. 2014;344:70–77. pmid:24291492
View Article
PubMed/NCBI
Google Scholar

[142] View Article

[143] PubMed/NCBI

[144] Google Scholar

[ref43] 43. Deeba F, Sanz-Leon P, Robinson PA. Dependence of absence seizure dynamics on physiological parameter evolution. J Theor Biol. 2018;454:11–21. pmid:29807025
View Article
PubMed/NCBI
Google Scholar

[146] View Article

[147] PubMed/NCBI

[148] Google Scholar

[ref44] 44. Deeba F, Sanz-Leon P, Robinson PA. Unified dynamics of interictal events and absence seizures. Phys Rev E. 2019;100(2):022407. pmid:31574631
View Article
PubMed/NCBI
Google Scholar

[150] View Article

[151] PubMed/NCBI

[152] Google Scholar

[ref45] 45. Schiff SJ, Colella D, Jacyna GM, Hughes E, Creekmore JW, Marshall A, et al. Brain chirps: spectrographic signatures of epileptic seizures. Clin Neurophysiol. 2000;111(6):953–958. pmid:10825700
View Article
PubMed/NCBI
Google Scholar

[154] View Article

[155] PubMed/NCBI

[156] Google Scholar

[ref46] 46. Jiruska P, De Curtis M, Jefferys JGR, Schevon CA, Schiff SJ, Schindler K. Synchronization and desynchronization in epilepsy: controversies and hypotheses. J Physiol. 2013;591(4):787–797. pmid:23184516
View Article
PubMed/NCBI
Google Scholar

[158] View Article

[159] PubMed/NCBI

[160] Google Scholar

[ref47] 47. Schiff SJ, Milton JG, Heller J, Weinstein SL. Wavelet transforms and surrogate data for electroencephalographic spike and seizure localization. Optical Engineering. 1994;33(7):2162–2170.
View Article
Google Scholar

[162] View Article

[163] Google Scholar

[ref48] 48. Lopes da Silva FH, Pijn JP, Velis D, Nijssen PCG. Alpha rhythms: noise, dynamics and models. Int J Psychophysiol. 1997;26(1):237–249. pmid:9203006
View Article
PubMed/NCBI
Google Scholar

[165] View Article

[166] PubMed/NCBI

[167] Google Scholar

[ref49] 49. Blumenfeld H, Varghese GI, Purcaro MJ, Motelow JE, Enev M, McNally KA, et al. Cortical and subcortical networks in human secondarily generalized tonic–clonic seizures. Brain. 2009;132(4):999–1012. pmid:19339252
View Article
PubMed/NCBI
Google Scholar

[169] View Article

[170] PubMed/NCBI

[171] Google Scholar

[ref50] 50. Aminoff MJ, Simon RP, Wiedemann E. The hormonal responses to generalized tonic-clonic seizures. Brain. 1984;107(2):569–578. pmid:6144354
View Article
PubMed/NCBI
Google Scholar

[173] View Article

[174] PubMed/NCBI

[175] Google Scholar

[ref51] 51. Dam AM. Epilepsy and neuron loss in the hippocampus. Epilepsia. 1980;21(6):617–629. pmid:6777154
View Article
PubMed/NCBI
Google Scholar

[177] View Article

[178] PubMed/NCBI

[179] Google Scholar

[ref52] 52. Bauer J, Stefan H, Schrell U, Uhlig B, Landgraf S, Neubauer U, et al. Serum prolactin concentrations and epilepsy. Eur Arch Psyc Clin Neurosci. 1992;241(6):365–371.
View Article
Google Scholar

[181] View Article

[182] Google Scholar

[ref53] 53. Koppert M, Kalitzin S, Velis D, Lopes da Silva FH, Viergever MA. Reactive control of epileptiform discharges in realistic computational neuronal models with bistability. Int J Neural Sys. 2013;23(01):1250032.
View Article
Google Scholar

[184] View Article

[185] Google Scholar

[ref54] 54. Soltesz I, Staley K. Computational neuroscience in epilepsy. Academic Press; 2011.

[ref55] 55. Tzallas AT, Tsipouras MG, Fotiadis DI. Epileptic seizure detection in EEGs using time–frequency analysis. IEEE Trans Inf Technol Biomed. 2009;13(5):703–710. pmid:19304486
View Article
PubMed/NCBI
Google Scholar

[188] View Article

[189] PubMed/NCBI

[190] Google Scholar

[ref56] 56. Cohen L. Time-frequency distributions-a review. Proc IEEE. 1989;77(7):941–981.
View Article
Google Scholar

[192] View Article

[193] Google Scholar

[ref57] 57. Liley DTJ, Cadusch PJ, Wright JJ. A continuum theory of electro-cortical activity. Neurocomputing. 1999;26:795–800.
View Article
Google Scholar

[195] View Article

[196] Google Scholar

[ref58] 58. Braitenberg V, Schüz A. Cortex: statistics and geometry of neuronal connectivity. Springer Science & Business Media; 2013.

[ref59] 59. Sanz-Leon P, Robinson PA, Knock SA, Drysdale PM, Abeysuriya RG, Fung PK, et al. NFTsim: theory and simulation of multiscale neural field dynamics. PLoS Comput Biol. 2018;14(8):e1006387. pmid:30133448
View Article
PubMed/NCBI
Google Scholar

[199] View Article

[200] PubMed/NCBI

[201] Google Scholar

Figures

Abstract

Introduction

Results

General characteristics of tonic-clonic seizures

Dynamics of seizure onset.

Dynamics of seizure offset.

Differences between onset and offset dynamics.

Analytical prediction of onset and offset transition times.

Dynamics during ictal state plateau.

Dynamics of corticothalamic seizure propagation.

Impact of temporal variation of νse on seizure dynamics

Seizure onset time.

Dynamic spectrum.

Characteristic transition times.

Saddle cycle.

Discussion

Methods

Corticothalamic field model

Temporal ramping

Numerical methods

References

Impact of temporal variation of ν_se on seizure dynamics