Model

Since the early seventies, and following the pioneer works of W. Freeman [17] and F. Lopez da Silva [18], NMMs have been extensively used to study LFPs not only in the context of physiological activity (such as brain rhythms [19], and/or visually evoked potentials [20]), but also pathophysiological activity (epilepsy [21–24], Alzheimer’s disease [25], Parkinson’s disease [26]). NMMs are average descriptions of the temporal dynamics of neuronal assemblies, and therefore model parameters are themselves lumped. To some extent, these models are complementary of models of microcircuits in which neurons are explicitly represented. The important point there is that NMMs are neuro-inspired, since they implement interactions between different subpopulations of neurons (glutamatergic–excitatory- neurons and GABAergic–inhibitory- interneurons of various types) and model parameters are physiologically grounded [27,28]. Typically, the time constants governing average post-synaptic potentials (PSPs) match the rising and decaying times of corresponding experimentally recorded PSPs [27]. Regarding average firing rates, sigmoid functions account for threshold and saturation effects that are major features of neuronal physiology [28] and account for non-linearities in neuronal dynamics. Importantly, NMMs have been used to simulate LFPs which closely resemble those experimentally recorded, and led to predictions which could be experimentally verified [29].

We consider the NMM presented in [22] which includes three interacting neuronal subpopulations: pyramidal neurons (divided into two subpopulations) and inhibitory interneurons (SOM+ and PV+, also called “dendrite-projecting slow” and “soma-projecting fast” interneurons, respectively) (see S1 Fig for the block diagram of the model). The average PSP of each subpopulation is determined by two functions: 1) a ‘pulse-to-wave’ function, S(v) = 5/(1+exp(0.56(6−v))), transforming the incoming PSPs into a firing rate; and 2) the input firing rate is converted into the mean PSP of the corresponding subpopulation by a linear transformation, that is h(t) = Wt/τ_w exp(−t/τ_w), where W represents the average synaptic gain and τ_w is the average synaptic time constant, which is related to rise and decay times of PSPs and lumps together the passive membrane time constant and all signal propagation delays. The system reads: (1)

Variables y_i stand for the PSPs generated at the level of pyramidal cells (y₀), excitatory inputs on pyramidal cells (y₁) (or in other words, the pyramidal subpopulation denoted by y₀ excites the pyramidal subpopulation y₁), SOM+ interneurons (y₂), PV+ interneurons (y₃), and inhibitory inputs on PV+ interneurons (y₄). Parameters A, B, G are the synaptic gains, τ_a, τ_b, τ_g are the synaptic time constants, connectivity constants C_is represent the average number of synaptic contacts, and p(t) is the external (noisy) cortical input p(t) = p+ξ, where p is the mean of the external input, and ξ is a random variable following a normal distribution with zero mean and standard deviation σ). Table 1 presents the parameter values used in this manuscript unless otherwise stated. The main difference between the parameter sets of [22] and Table 1 is the connectivity strengths of the circuit involving PV+ interneurons. Note that τ_a is 3.3 times and τ_b is 16.6 times greater than τ_g. These differences would introduce multiple time-scale dynamics in the system. Below, we recall primaries of slow-fast analysis before expressing (1) in slow-fast formulation.

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Table 1. Parameter values during simulated background activity.

https://doi.org/10.1371/journal.pcbi.1008430.t001

Primaries of slow-fast analysis

Complex oscillations, e.g. spiking, bursting and mixed-mode oscillations, are widely present in electrophysiological signals, and are hallmarked by interacting components running in different timescales (for a recent review see [30]). Mathematical models describing these dynamics involve inevitably variables with different timescales. For example, the interaction of two variables running at two different time-scales (1 fast and 1 slow variables) is sufficient for generating spiking, whereas bursting emerges only in higher dimensional systems with at least two different time-scales (2 fast and 1 slow variables/subsystems). Geometric methods in the singular limits (denoted as slow-fast or multiple timescale analysis) simplify the investigation of the model dynamics by breaking into several reduced systems. For instance, bursting patterns can be classified in a singular limit where the slowest variables are treated as parameters [31,32].

A slow-fast system in the general slow form reads, with fast variables x and slow variables z of arbitrary dimensions, time scale parameter 0<ϵ≪1, and dot represents derivation with respect to time t. The dynamics of a slow-fast system can be divided into fast and slow epochs. Each of these epochs can be investigated with the slow-fast analysis in a hybrid manner and then can be concatenated, so that one can understand the underlying structure giving sharp transitions (excitable responses to external inputs) and complex oscillatory patterns (spiking, bursting and subthreshold oscillations) [33].

An important geometrical object for both the slow and the fast dynamics is the critical manifold , defined as the nullcline of the fast variable , eventually obtained by setting ε = 0. For the differential-algebraic system defined for ϵ = 0, the so-called reduced system (slow subsystem) approximates the slow dynamics of the original system. The critical manifold both defines the phase space of the reduced system and equilibrium points of the layer problem expressed in the fast time-scale, that is where (′) denotes derivative with respect to τ = t/ϵ. The stability of the layer problem determines the characteristics of the critical manifold. The critical manifold is normally hyperbolic along the set for which det(f_x(x,z,0))≠0, which can be attracting, repelling or saddle type. The Fenichel theory [15] guarantees that these normally hyperbolic points of the critical manifold perturb smoothly in ϵ and give slow manifolds () of the original system for small enough ϵ>0. If is folded, attracting and repelling branches of C⁰ meet along the fold set , where normal hyperbolicity is lost. Extension of the classical Fenichel theory to non-hyperbolic sets provides a tool to investigate the slow dynamics near , and one can expect canard solutions in the neighborhood of such sets [34].

Slow-fast formulation of the model

One can notice that the variable y₄ in (1) is equivalent to y₂, thus the dimension of (1) can be reduced by multiplying the PSP variables with C_is before the ‘pulse-to-wave’ conversion. Further, by applying the variable conversion, system (1) can be written as: (2)

Intuitively, system (1), hence system (2), are multiple-time-scale systems which can result in complex epileptogenic patterns for appropriate choices of parameters. Thus, understanding the multiple-time-scale structure of (2) is indispensable for designing brain stimulation protocols aiming at aborting the aforementioned oscillatory patterns. In order to proceed a slow-fast analysis of (2), where the time-scaling in not explicit, and use the standard methods of GSPT, we first normalize time t with respect to τ_g, as , then define two parameters, δ = τ_g/τ_a and ε = τ_a/τ_b. We further assume that ε and δ are independent of the synaptic time constants (τ_a,τ_b,τ_g), similar to the approach followed in [35], that is to say variations in (ε,δ) for theoretical analysis do not require varying (τ_a,τ_b,τ_g). At the end system (2) is expressed in an explicit slow-fast formulation: (3)

System (3) is a three-time-scale system for small enough values of (δ,ε) [27–31], with (v₃, y₈) being fast variables, (v₀, y₅, v₁, y₆) slow variables, and (v₂, y₇) superslow variables. System (3) is written using the (fast) time , and called the fast system. As can be noticed, rescaling of (1) identifies the small parameters such that GSPT can be applied and expresses the importance of timescales. We follow [36,37] to analyze the three time scaled slow-fast structure of (3). Defining gives the slow system: (4) where F_is are as defined for (3), with i representing the system variables’ indices on the left-hand side. S2A Fig presents the bifurcation diagram of the (v₃, y₈, v₀, y₅, v₁, v₆)-subsystem of (4) for ε = 0, where v₂ acts as a parameter, and a periodic bursting orbit of (3) for ε = 0.01. We see that the orbit agrees with the bifurcation diagram when ε is decreased. Details of the bursting behavior are explained in Sec. Bursting analysis.

Defining gives the superslow system, (5)

Systems (3), (4) and (5) are equivalent if ε≠0 and δ≠0, but they give nonequivalent dynamics in the singular limits ε→0 and/or δ→0. The limit δ→0 in the fast system (3) eliminates the slow and superslow dynamics and yields the fast layer problem, (6) which describes the dynamics of the fast variables (v₃, y₈) for fixed values of (v₀, v₂), for instance. The critical manifold is defined by the four-dimensional set of equilibria of the fast layer problem (6), which reads, and S⁰ is eventually in the (y₈ = 0)-space. The stability of S⁰ is determined by deriving the Jacobian of S⁰ with respect to the fast variables, that is,

Since , and the eigenvalues are λ_1,2 = −1, the S⁰ is normally hyperbolic and stable. Hence, S⁰ is perturbed to local invariant slow manifolds for sufficiently small δ>0.

Another singular limit is obtained by letting δ→0 in the slow system (4) gives the algebraic-differential slow reduced problem, (7) which describes the dynamics on S⁰. System (7) is a two-time-scale problem for ε sufficiently small and it gives the slow layer problem in the ε→0 limit, (8) where v₂ appears as a parameter. A periodic orbit of (7) for ε = 0.01 and the bifurcation diagram of (8) as a function of v₂ is projected on the (v₀, v₂)-plane in S2B Fig.

In the slow layer problem (8), the slow variables (v₀, y₅, v₁, y₆) evolve along fibers defined by where are constant and restricted to S⁰. The equilibria of (8) defines the superslow manifold L⁰ which is a subset of S⁰. The superslow manifold L⁰ is reduced to where v₃ = G S(C₅τ_av₀−C₆τ_bv₂) is on S⁰. The curve L⁰ perturbs to locally slow invariant manifolds for ε>0 along the hyperbolic branches of L⁰, while the dynamics of near the non-hyperbolic fold points should be investigated using GSPT. Finally, the superslow reduced problem obtained by setting ε→0 in (5) reads (9)

This algebraic-differential system determines the superslow dynamics restricted to L⁰, and eventually to S⁰.

Strongly perturbed system

Electrical (through direct stimulation) and optical (through optogenetics, using light pulses in genetically modified animals) perturbations can alter action potential firing through modification of the mean membrane potential of the considered neural subpopulation. In the following, we use the word “perturbation” to refer to a strong suprathreshold input. We assumed an additive model for the stimulation effect onto the mean membrane PSP [38]. Thus, the external input I_ext(t) is included in the ‘pulse-to-wave’ functions of the NMM in (3), and the system reads: (10) where k_i with i = {P,SOM,PV} is the coupling coefficient between the stimulation and the considered subpopulation, governing the impact of I_ext(t) on the subpopulation. Same generalization holds for the slow and superslow reduced problems given by (7) and (9), respectively.

Clinical data

Clinical data used for the purpose of this study consisted in SEEG signals collected in a patient with drug-resistant focal epilepsy that required invasive EEG exploration. Recordings were performed using intracranial multichannel electrodes (DIXI Medical, 5–18 contacts; length, 2 mm, diameter, 0.8 mm; 1.5 mm apart), 10 of which are used for investigations. The patient-specific position of each electrode results from a clinical decision process which is based on the many assumptions made from non-invasive data (video EEG recording, semiology, neuro-imaging data, clinical examination). Electrodes were implanted according to the stereotactic method of Talairach [39]. SEEG signals were recorded on a Deltamed system on a maximum number of channels equal to 128, and were sampled at 256 Hz and recorded to hard disk (16 bits/sample) using no digital filter. The only filter present in the acquisition procedure was a hardware analog high-pass filter (cut-off frequency of 0.16 Hz) used to remove very slow variations that sometimes contaminate the baseline. In the patient for which data is displayed in the remainder of the manuscript, a surgical operation was performed 6 month after pre-surgical exploration (cortectomy of the frontal dorsolateral region). The chosen monopolar signal was recorded on one electrode located in the depth of a sulcus located in the dorsolateral frontal cortex. It was chosen because the recorded structure was highly epileptogenic considering the exhibited epileptiform activity (during interictal periods and at the onset of seizures). Histological data revealed the presence of a focal cortical dysplasia (Taylor). After surgery, the patient was seizure free (Engel IA). As a reminder, SEEG is always carried out as part of normal clinical care of patients who give informed consent in the usual way. Patients were informed that their data may be used for research purposes.

Computational methods

The bifurcation analysis was in done with AUTO-07p [40]. Model equations were implemented in XPPaut [41]. Stochastic differential equations were iterated using Euler-Maruyama method with a step size dt = 10⁻⁴ second. All simulation files are available from the GitHub database (https://github.com/elifkoksal/NMM_BurstingDynamics).

Pre-ictal spiking during interictal to ictal transition

In partial (i.e. focal) epilepsies, the onset of seizures is characterized by the appearance of a rapid discharge, typically in the gamma frequency band ([25, 120] Hz) [27]. This fast onset activity has long been recognized as a hallmark of the epileptogenic zone, and a number of methods have been proposed to make use of this pattern to identify the epileptogenic zone [42,43]. Interestingly, fast onset activity can be preceded by a specific electrophysiological pattern consisting of sustained large amplitude bursts with superimposed faster spikes, which can be observed in various etiologies [44]. Two examples of this pre-ictal pattern, as recorded in two different patients during pre-surgical investigation with depth electrodes, are shown in Fig 1. As depicted, this dynamical regime starts with sporadic bursts, which become periodic to change into a sustained discharge of pre-ictal bursts. In the first recording in Fig 1A, the number of spikes of the bursts gradually decreases during the pre-ictal burst phase, which continues approximately for 14 seconds. In the second example in Fig 1D, pre-ictal bursting continues for approximately 35 seconds. In this example, the amplitude of pre-ictal bursts increases. The pre-ictal burst phases are followed by the fast activity that actually marks the onset of the seizure. Patterns of SEEG signals recorded during interictal–ictal transitions for different seizures of a given patient are typical and reproducible [45,46]. As for the patients whose pre-ictal activity are exemplified in Fig 1, such a transition is persistent with quantitative variations in 1) duration of the pre-ictal bursting period, 2) amplitude of discharges, and 3) number of fast spikes in the bursts.

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Fig 1. SEEG signals recorded in two patients with epilepsy during the interictal to ictal transition and simulated signals.

(a) Epileptic seizure recorded in a patient showing the typical pre-ictal spiking pattern with three phases: sporadic spikes, pre-ictal bursts, and fast onset. (b) Zoom into each phase of the actual SEEG signal. The pre-ictal burst type-1 is followed by the pre-ictal burst type-2. (c) Simulated signals corresponding to each phase. (d) Epileptic seizure recorded in a different patient showing the typical pre-ictal spiking pattern with three phases. The region of pre-ictal bursting is zoomed in panel (e).

https://doi.org/10.1371/journal.pcbi.1008430.g001

System (1) represents a physiologically relevant system that includes neuronal subpopulations of excitatory glutamatergic pyramidal neurons, inhibitory SOM+ and PV+ GABAergic interneurons, and makes average PSPs at the level of each subpopulation accessible. This NMM has been extensively explored to establish relationships between model parameters and electrophysiological patterns observed in SEEG recordings [22,47]. For instance, increasing the ratio of the synaptic gain of the excitatory pyramidal cell population and inhibitory SOM+ interneuron population introduces a region in the parameter space where the system can undergo different stages of epileptogenic activity as a function of the synaptic gain of inhibitory SOM+ interneuron population, parameter B, and the synaptic gain of inhibitory PV+ interneuron population, parameter G.

The thorough exploration of system (1) led to the identification of three key parameters: B,G, and the strength of the excitatory synaptic coefficient on the PV+ interneuron subpopulation C₅. Indeed, the tuning of these three parameters enables replicating of the different pre-ictal stages shown in Fig 1A. These results are illustrated by the bifurcation diagram in Fig 2A for the parameter set given in Table 1. As depicted, the decrease of parameter B yields a transition from background activity to fast onset activity, though pre-ictal spiking. Fig 2B shows where these activity regions are localized in the (B,G)-parameter space.

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Fig 2. Bifurcation diagrams of the system (1) showing the different pre-ictal stages numbered 1–3 in Fig 1.

(a) Amplitude of the PSP of the pyramidal cell subpopulation is plotted as a function of the synaptic gain of the SOM+ interneuron subpopulation B. The other system parameters are given in Table 1 with G = 35. Bold and dashed lines correspond to stable and unstable solutions, respectively. Region 1 (blue) corresponds to sporadic bursts, region 2 (orange) to sustained bursts, and region 3 (purple) to low voltage fast onset activity. The system yields large amplitude ≈30 Hz oscillations in the unnumbered green shaded region. The unnumbered yellow shaded area corresponds to high amplitude stable equilibrium points, and white corresponds to physiological background activity. The arrow shows the route from background to low voltage fast onset activity in the parameter space. In order to preserve the readability of the diagram, the bifurcations along the branches of periodic solutions are not shown. (b) Co-dimension 2 diagram of the Hopf (H) and limit point (LP) bifurcations marked on panel (a) in the parameter space of B and G (synaptic gain of the PV+ interneuron subpopulation). The LP₁ and LP₂ points merge on a cusp (CP) bifurcation, and the H₁ and H₂ merge on a zero-Hopf (ZH) bifurcation.

https://doi.org/10.1371/journal.pcbi.1008430.g002

Let us walk through the bifurcation diagram in Fig 2A, starting the equilibrium point at B = 0, and increase B. The system first undergoes a supercritical Hopf bifurcation (H₁) at B≈1.51, giving stable limit cycles at ≈30 Hz (gamma activity). The amplitude of these solutions increases with B and they become unstable via a saddle node bifurcation of periodic orbits at B≈6.56. The branch of periodic orbits connects to a Hopf bifurcation (H₂) of subcritical type at B≈4.56, while the range of oscillations extends until B≈6.56. The branch of equilibrium points undergoes two limit point (LP) bifurcations (LP₁ for B≈7.43 and LP₂ for B≈4.55) between H₁ and H₂. Then, we identify a third Hopf bifurcation (H₃) of supercritical type for B≈7.74, where a branch of stable oscillations around ≈6 Hz appears. As B increases, this branch connects to stable bursting orbits by passing through several limit folds around B≈8.86. At B≈10 the system reaches to the maximum number of spikes per burst orbit (11 spikes for this parameter set). Increasing B decreases the number of spikes via the horizontal up-and-downs in y₀ between B∈(9.5,22.5). The bursts terminate at B≈22.5. The branch holding the unstable equilibrium points forms a Z-shaped curve with two folds at B≈21.3 (LP₃) and B≈35.6 (LP₄), with unstable focus on the upper branch, saddles in the middle and stable nodes on the lower branch after a subcritical Hopf bifurcation (H₄) which gives unstable limit cycles making a heteroclinic connection with the middle branch. For B>35.6, the system only has stable equilibrium points as solutions in the unnumbered white area, which corresponds to physiological background activity.

Continuation of the LP and Hopf bifurcations marked on Fig 2A in the (B,G)-space is shown in Fig 2B. It can be seen that the locations of LP₃, LP₄, H₃, H₄ points do not depend on G, whereas the locations of LP₁, LP₂, H₁ and H_2, which are related to the oscillations at ≈30 Hz, do. The fast onset region does not exist for small values of B if G<5, for which the system only has equilibrium points for B<B_H3. As G increases, the system undergoes Hopf bifurcations (H₁ and H₂) that introduce gamma-band fast oscillations (for G>5). The amplitude of these fast oscillations are smaller closer to H_1, and this difference is more visible in simulated LFPs. Under stochastic inputs, the system can yield fast oscillations for B∈(0, B_H1), and a mixture of fast and slow oscillations of ≈6 Hz for B∈(B_H2, B_H3). Furthermore, G controls the amplitude of the spikes of bursting solutions, which increases with G.

Assume that system (1) is initially in the background activity mode, which corresponds to the white region in Fig 2A for B>35.6, where unique stable equilibrium points on the bifurcation curve is observed. In the blue region between the two folds LP₃ and LP₄, the bifurcation curve takes a Z-form with stable nodes on the lower branch, unstable nodes in the middle and saddle-nodes on the upper branch. For B values in this blue region, system (1) under a stochastic input p(t) undergoes sporadic bursts, with an increasing probability as B approaches to the left fold. As B decreases, the system enters into the bursting region (orange region). Note that further increasing B would increase the number of spikes. However, in the recordings we see that the number of spikes decreases in the course of the pre-ictal bursting regime as the system approached the low voltage fast onset (LVFO), transition from type-1 to type-2 bursting. This change is very subtle to be reproduced in the model because, as detailed in the Sec. Burst analysis, the number of spikes depends on the presynaptic potential on PV+ interneurons: the lower it is, the more spikes within the burst are obtained. Thus, the number of spikes increases when the inhibitory input decreases, or when the excitation onto PV+ interneurons increases. At this stage, transition from the type-1 bursting to type-2 bursting is obtained by keeping B constant, but decreasing the excitatory post-synaptic potential (EPSP) on PV+ by decreasing progressively C₅ to 300 to reduce the number of spikes; and increase G to increase spikes amplitude. Under these variations, the bifurcation diagram in Fig 2A remains qualitatively the same, the most important quantitative change being the location of the Hopf bifurcation point H₁ related to the LVFO. As shown in Fig 2B, increasing G moves the H₁ towards right along the B-axis, and initiates the LVFO for higher values of B≈B_H1. Under stochastic input, one can observe LVFO for B≤B_H1, as well (purple regions in Fig 2). The large amplitude fast oscillations that are obtained for higher B values in (B_H1, B_H2) can also be observed in SEEG recordings during the transition to epileptic seizures.

Bursting analysis

We investigate the bursting dynamics of system (1) using system (3), which is a kind of nondimensionalized version of (1) but expressed in an explicit slow-fast formulation. Fig 3A shows a bursting solution of (3) in the (v₀, v₂, v₃)-space, the critical manifold S⁰ and the superslow manifold L⁰ (see Sec. Slow-fast formulation of the model for definition). The critical manifold S⁰ is normally hyperbolic (not folded) and stable, and stretches between almost horizontal surfaces (lower and upper) with an almost vertical plane. The superslow manifold L⁰ has branches both on the lower horizontal surface and vertical surface of S⁰. While the part of L⁰ on the vertical surface of S⁰ is stable, the part on the lower horizontal surface of S⁰ is divided into stable and unstable sections at two fold points LP₁ and LP₂. The curve L⁰ is stable along the branch that is almost parallel to the v₂-axis, unstable along the branch between LP₁ and LP₂, and then becomes stable again. The stable and unstable branches of L⁰ are normally hyperbolic, whereas the fold points LP₁ and LP₂ are not.

The critical manifold S⁰ and superslow manifold L⁰ perturb for small enough values of time-scale parameters, hence the three-time-scale dynamics of (3) approximate to S⁰ and L⁰. During the superslow time-scale under (9), the bursting orbit follows the stable branch of L⁰ almost parallel to the v₂-axis. Near the fold point LP₁, the trajectory bends in the v₃-direction along the vertical plane of S⁰ and enters into the spiking regime, which runs in fast time-scale. The spiking terminates close to LP₂ and the trajectory jumps back to the stable branch of L⁰ almost parallel to the v₂-axis in slow time-scale under (8). Fig 3B shows the time series in of the orbit in Fig 3A.

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Fig 3. Bursting orbit of system (3).

(a) Solution of (3) (blue orbit) and L⁰ (red curve) on the critical surface S⁰(green surface) projected on the (v₀, v₂, v₃)-space. Single-headed, double-headed and triple-headed arrows indicate the flow direction during superslow, slow and fast time-scales, respectively. LP denotes limit point bifurcation. The L⁰ curve changes stability at two limit points, LP₁ and LP₂ (red dots). The middle branch of the L⁰ curve between these limit points is unstable (dashed). (b) Time course of the variables (v₃, v₀, v₂) of the orbit plotted in panel (a). (c) Solution of (3) projected on the bifurcation diagram (black curve) of (4) for ε = 0 where v₂ is threaded as a parameter. Arrows show the direction of the flow with respective time-scales. Bold and dashed lines correspond to stable and unstable solutions, respectively. H donates a Hopf bifurcation, LP a limit point bifurcation. The equilibrium points along the black Z-shaped curve are unstable on the middle branch of the curve, between LP₁ at and LP₂ at (black dots), and on the upper branch between H₁ at and H₂ at (green dots). The amplitude of the stable limit cycles is bounded by the green continuous curves connecting the H₁ and H₂ in the ε = 0 limit.

https://doi.org/10.1371/journal.pcbi.1008430.g003

For a better understanding of the bursting dynamics, we consider system (4) at ε = 0 for which the variables of the slowest subsystem (v₂, y₇) act as parameters of the (v₃, y₈, v₀, y₅, v₁, y₆)-subsystem. Since only v₂ appears in the (v₃, y₈, v₀, y₅, v₁, y₆)-subsystem, its dynamics depend on v₂. In Fig 3C, the bursting orbit in Fig 3A is superimposed on the bifurcation diagram of the (v₃, y₈, v₀, y₅, v₁, y₆)-subsystem in (4) at ε = 0 as a function of v₂. Although the fastest variables of (4) are (v₃, y₈), we chose v₀ vs v₂ for a clearer visualization (the same trajectory and the bifurcation diagram are given on the (v₃, v₂)-plane Fig 4). We see that the corresponding system poses a Z-shaped bifurcation diagram as a function of v₂ with two folds, and . The equilibrium points are stable on the lower branch of the Z-shaped curve for unstable along the middle branch between and . The upper branch has two supercritical Hopf bifurcations, at and , with stable limit cycles in between. Along the upper branch, equilibrium points are stable for and . The bursting behavior resulting from this bifurcation structure in the (v₃, y₈, v₀, y₅, v₁, y₆)-subsystem is classified as ‘fold/Hopf bursting’ by Izhikevich [32] due to the presence of a ‘fold/Hopf’ hysteresis in the bifurcation diagram.

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Fig 4. Spike number as a function of C₅.

(a) Solution of (4) with 3 spikes for C₅ = 500 projected on the bifurcation diagram of the fast system (5) as a function of v₂. Arrows indicate the direction of the flow. (b) Solution of (4) with 1 spike for C₅ = 300 projected on the bifurcation diagram of the fast system (5) as a function of v₂. Arrows indicate the direction of the flow. (c) Co-dimension 2 diagrams of the Hopf (H) points (green) and the limit points (LP, black) in the (v₂, C₅) parameter space marked on the left and middle panels. As C₅ decreases, H₂ moves leftwards and eventually the spike number decreases. For C₅ = 139, H₂ and LP₁ are aligned at v₂ = 4.778. A further decrease in C₅ places H₂ on the left of LP₁ and leaves no chances for a bursting solution.

https://doi.org/10.1371/journal.pcbi.1008430.g004

System (4) may undergo through these bifurcations in a repetitive manner for ε≠0, which results eventually in the bursting solutions for small enough values of ε. As the arrows on Fig 3C and the traces on Fig 3B demonstrate, the trajectory follows the lower stable branch during the quiescence phase of the bursting, which terminates near . Then, it jumps to the region of the stable limit cycles on the upper branch, which initiates the active phase of the bursting. The spiking frequency during the active phase is faster at the beginning than the end due to the fact that the Hopf bifurcation at gives limit cycles with ≈30 Hz frequency whereas the Hopf bifurcation at gives limit cycles with ≈10 Hz. The spiking terminates at , but the active phase continues until the trajectory jumps back to the stable lower branch at . We underline that as ε→0, the bursting orbit attaches more and more the bifurcation diagram obtained for ε = 0 (see S2 Fig for an example).

The main difference between the type-1 and type-2 bursting is the number of spikes during the active phase of bursting. In the model, the variations in the number of spikes can be met by changing the excitation level on the PV+ interneurons: as aforementioned, the number of spikes increases with the amount of excitation received by PV+ interneurons. This can be achieved by either decreasing inhibition or by increasing excitation. For instance, decreasing B in region-2 in Fig 2 increases the number of spikes. In (4) at ε = 0, the excitation on PV+ depends on two synaptic coupling coefficients, C₅ and C₆. The effect of C₆ will be similar to the one of B, since they both scale the PSP of SOM+ interneurons given by the variable v₂ in (4). Below, the role of the excitatory synapses in the (v₃, y₈, v₀, y₅, v₁, y₆)-subsystem by changing C₅ is investigated.

As displayed by Figs 3C, 4A and 4B, the spikes are bounded by LP₁ and H₂ in the bifurcation diagram of the (v₃, y₈, v₀, y₅, v₁, y₆)-subsystem as a function of v₂. The distance between LP₁ and H₂ in v₂ affects the number of spikes; the further they are, the more spikes the burst has. In Fig 4C, LP and Hopf bifurcations are continued in the parameter space of (v₂, C₅). While the LP₁ and LP₂ lie along almost vertical lines, the Hopf bifurcation points form a V-shaped curve along which the left arm locates the H₁ points and the right arm the H₂ points. The distance between H₂ and LP₁ increases with C₅, hence, the spike number. At C₅ = 139, H₂ and LP₁ are aligned. Further decrease in C₅ places H₂ on the left of LP₁ and leaves no chances for a bursting solution. The system yields only relaxation type of oscillations for C₅<139.

Overall, the aforementioned analysis shows that pre-ictal bursting runs in three-time-scales. The system sustains the bursting regime for a certain range of parameter B denoting SOM+ synaptic gain. The complex pre-ictal bursting pattern can be accurately adjusted by tuning parameters G, which controls the PV+ synaptic gain, and the connectivity coefficient C₅, which controls PV+ excitability. In particular, the number of spikes and their amplitude can be adjusted by tuning C₅ and G, respectively.

Strong perturbation analysis

The mean membrane potential of a neuronal subpopulation can be altered by electrical and optical stimulations. Under the assumption of an additive model for the stimulation effect, an external input can be included in the ‘pulse-to-wave’ functions, S(v), of the NMM after being scaled by the subpopulation specific impact coefficients (see Sec. Strongly perturbed system).

A pulse input (biphasic or monophasic) changes the PSP of the perturbed subpopulation by shifting it above its base level S(0). We assume that a neural mass block, given by , receives biphasic stimulation. The PSP of the neural mass block increases during the anodal pulse (positive perturbation or depolarization of the membrane potential), but decreases (discharges) during the cathodic pulse (negative perturbation or hyperpolarization of the membrane potential) and between the inter-pulse intervals of the biphasic input. Depending on the pulse width, pulse amplitude, and mostly on the synaptic time constant of the neural mass block, this shift may be sustained or not. For instance, the discharge takes longer in a neural mass block with slow synaptic kinetics than the one with fast synaptic kinetics. If the pulse frequency is sufficiently high to stimulate the neural mass before it completely resumes to its base level, then the PSP of the neural mass can oscillate above the base level. As visualized in S3 Fig, the same perturbation shifts the PSP of a neural mass with slow synaptic kinetics, while the neural mass with fast synaptic kinetics decays to its base level during the inter-pulse intervals of the stimulation. Increasing the stimulation frequency can keep the PSP of the neural mass with fast kinetics above the base level, and therefore the firing rate and PSP of the fast neural mass increase with the stimulation frequency.

The bursting solution is driven by the slow oscillations in system (3) (see Sec. Bursting Analysis). The slow dynamics of (3) (subsystems representing the pyramidal cell and SOM+ interneuron subpopulations) can be approximated by (7), which preserves the burst-driver slow oscillations behavior for the same parameter values yielding bursting oscillations in (3) (see Sec. Slow-fast formulation and S2 Fig). Thus, it is sufficient to investigate the response of (7) under perturbation to understand the impact of the perturbation on the burst-driver slow dynamics. The most common signal delivered to brain tissue in the field of deep brain stimulation (DBS) is bi-phasic pulses with balanced anodic/cathodic phases of brief durations (approximately 100 μs). Below, the impact of anodic and cathodic constant external inputs is considered without taking into account their duration, to simply understand how they alternate the phase space of system (7). For this purpose, a constant input (I_ext = 1) scaled with the impact coefficients k_P and k_SOM is applied to (7) by following the formulism given by (10) (see Sec. Strongly perturbed system).

In Fig 5, subpopulations representing pyramidal cells and inhibitory SOM+ interneurons are perturbed. The left panels of Fig 5 show the y₅-nullsurface Θ, v₂-nullsurface Σ and the superslow manifold L⁰ projected on the (y₅, v₂, v₀)-space. The solution of (7) is visible on the left panels, and the solution of (3) for the same parameters is given on the right panels of Fig 5. Fig 5A shows the case where only the SOM+ interneuron subpopulation described by the (v₂, y₇)-subsystem in (7) is subject to the constant external input (k_P = 0). In the absence of any perturbation (k_SOM = 0), Θ and Σ intersect for v₂>20. System (7) has a limit cycle which flows on Θ and (3) a burst orbit (black solutions Fig 5A and 5A1, respectively). The quiescence phase of the burst corresponds to the slow passage following L⁰ where v₀≈0, and the active phase correspond to the trajectory on the upper sheet of Θ.

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Fig 5. Geometrical analyses of a constant input.

Constant input is applied to SOM+ interneurons (a), to pyramidal cells (b) and to both SOM+ interneurons and pyramidal cells (c). Left panels show the projection of the nullsurfaces, critical slow manifold and the orbit of the reduced model (7). Right panels show the LFP signal of the full system (3) subject to the constant inputs analyzed on the left. All parameters are as given in Table 1, except B = 15. (a) The y₅-nullsurface Θ (blue surface for k_P = 0), and y₇-nullsurface Σ (red surface for k_SOM = −1, black surface for k_SOM = 0, green surface for k_SOM = 1) are projected on the (v₂, y₅, v₀)-space. The blue curve L⁰ (stable on the bold, unstable on the dashed) is on the intersection between Θ and the {y₅ = 0}-hyperplane. The black and red orbits are the solutions of the system for k_SOM = 0 and k_SOM = −1, respectively. For k_SOM = 1, the solution approaches to the green stable equilibrium point on the intersection between Σ(k_SOM = 1) and L⁰. Panel (a1) shows the time series for k_SOM = {0, 1}, and panel (a2) for k_SOM = −1. (b) The y₅-nullsurface Θ (red surface for k_P = 1, green surface for k_P = −1), and y₇-nullsurface Σ (black surface for k_SOM = 0) are projected on the (v₂, y₅, v₀)-space. The red curve L⁰ (stable on the bold, unstable on the dashed) is on the intersection between Θ(k_P = 1) and the {y₅ = 0}-hyperplane. The green curve L⁰ (stable on the bold, unstable on the dashed) is on the intersection between Θ(k_P = −1) and the y₅ = 0 hyperplane. The green and red orbits are the solutions of the system for k_P = 1 and k_P = −1, respectively. Panel (b1) shows time series for k_P = 1, and panel (b2) for k_P = −1. (c) The y₅-nullsurface Θ (red surface for k_P = 1) and y₇-nullsurface Σ (green surface for k_SOM = 1, blue surface for k_SOM = 2) are projected on the (v₂, y₅, v₀)-space. The red curve L⁰ (stable on the bold, unstable on the dashed) is on the intersection between Θ(k_P = 1) and the y₅ = 0 hyperplane. The green curve L⁰ (stable on the bold, unstable on the dashed) is on the intersection between Θ(k_P = 1) and the {y₅ = 0}-hyperplane. The green orbit is the solution of the system for (k_P, k_SOM) = (1,1). For (k_P, k_SOM) = (1,2) the solution approaches to the cyan stable equilibrium point on the intersection between Σ(k_SOM = 2) and L⁰. Panel (c1) shows time series for (k_P, k_SOM) = (1,1), and panel (c2) for (k_P, k_SOM) = (1,2).

https://doi.org/10.1371/journal.pcbi.1008430.g005

A key point in terms of controlling bursting activity through direct stimulation is that an input leading to a bifurcation from the stable limit cycle to an equilibrium point can prevent the system from bursting by keeping the system in the silent phase. This can be achieved by an input that ensures an intersection between Σ hyperplane and the lower branch of L⁰. Indeed, for k_SOM = 1, (7) possesses a stable equilibrium point near the left fold of L⁰ which traps the trajectory (green dot in Fig 5A). For the same input, the bursting in (3) is aborted (green solution in Fig 5A1). On the other hand, a negative constant input (k_SOM = −1), moves Σ away from the left fold of L⁰. Being Σ closer to the upper branch of L⁰ prolongs the active phase of the burst and increases the number of spikes, as seen in Fig 5A2. These observations indicate that increasing the excitation on SOM+ interneurons can abort bursting.

In Fig 5B, only the subsystem representing the pyramidal cells receives the perturbation (k_SOM = 0). The input on the pyramidal cell subpopulation acts on Θ. While positive constant input (k_P = 1) increases the distance between the lower fold of L⁰ and Σ, negative constant input (k_P = −1) decreases this distance. Both systems (7) and (3) preserve the oscillatory behavior for these values of k_P, yet, the oscillation frequency decreases for k_P = −1 due to the decreased distance between the lower fold of L⁰ and Σ). Thus, hyperpolarization of pyramidal cells by increasing inhibition on them can abort bursting.

As aforementioned above, pulsed stimulation increases the firing rate of a neuronal population. However, a stimulation applied to one specific region might not affect all neural populations in the same manner. This can be due to the relative position of electrodes with respect to neurons, cell specific firing thresholds, or synchronization level within neural subpopulations. However, such features can bring certain advantages in aborting bursting. Fig 5C shows the response of the system when both subpopulations of pyramidal cells and SOM+ interneurons are perturbed, the oscillatory behavior in systems (7) and (3) continues under the same positive constant input (k_P = k_SOM = 1). With such input, the number of spikes during the active phase is decreased (Fig 5C1). If the subpopulation of SOM+ interneurons is perturbed more strongly than the subpopulation of pyramidal cells (k_P = 1, k_SOM = 2), the system can bifurcate to the resting state (Fig 5C2).

Although the reduced system (7) does not include the fast dynamics of the PV+ interneurons, the effect of the perturbation on the subpopulation of PV+ interneurons can be understood geometrically. First, let us notice that increasing the inhibition on SOM+ interneurons encourages spiking (Fig 5A and 5A2), while increasing the excitation on SOM+ aborts bursting (Fig 5A and 5A1). Perturbing (stimulating) the subpopulation of PV+ interneurons increases the PSP from PV+ interneurons to pyramidal cells, reduces the PSP from pyramidal onto SOM+, and in turn favors bursting. Another way to illustrate the impact of perturbing the PV+ on bursting is to examine the diagram in Fig 4. Anodic pulses can shorten the quiescent phase in the ‘fold/Hopf’ hysteresis loop by kicking the trajectory to the region of stable limit cycles between the two Hopf bifurcations H₁ and H₂. Hence, such pulses applied periodically can increase the bursting frequency by shortening the quiescent phase. On the other hand, cathodic pulses can lengthen the quiescent phase by hooking the trajectory near the down state of the hysteresis loop. One can also think of inhibitory effect of PV+ subpopulation on the pyramidal cell population which could abort bursting. Indeed, it would be possible if an input was capable of depolarizing the PV+ subpopulation during the quiescence phase of the bursting where it is inhibited by the SOM+ subpopulation. To overcome the inhibition from the SOM+ subpopulation to the PV+ subpopulation, such an input should be very high in amplitude (order of 20 if a constant input on the PV+ subpopulation is considered), which could induce undesired discharges.

Overall, this geometric perturbation analysis helps to clarify the role of hyperpolarizing and depolarizing inputs on ongoing bursting activity. In particular, depolarization of the subpopulation of SOM+ interneurons or hyperpolarization of the subpopulation of pyramidal cells can abort bursting by keeping the sum of PSPs at low levels. Depolarization of the subpopulation of PV+ interneurons contributes to bursting.

Stimulation applied during the pre-ictal burst regime

The analysis in Sec. Perturbation Analyses has shown that a positive constant input applied to the subpopulation of SOM+ interneurons can bifurcate the limit cycle (oscillating epilepsy-like activity) to an equilibrium point (background activity), while a positive constant input on the subpopulations of pyramidal cells and PV+ interneurons preserve bursting and high frequency oscillations (Fig 5). Hence, an appropriate strategy for pre-ictal bursting abortion consists in the excitation of the SOM+ interneuron subpopulation.

In this section, the results obtained from the mathematical analysis are translated into an in silico set-up mimicking experimental conditions. Typically, charge-balanced bi-phasic pulses (pulse width = 0.5 ms and total duration 1 ms) with an arbitrary unit (arb. unit) amplitude are applied during the pre-ictal bursting/spiking regime in the presence of a stochastic input. In order to test our predictions on the role of different neural populations, only SOM+ interneurons are perturbed in Fig 6A (k_SOM = 1, k_P = 0, k_PV = 0), whereas in Fig 6B all neural subpopulations are perturbed homogenously (coupling coefficients k_SOM = k_P = k_PV = 1).

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Fig 6. System (1) under stimulation.

Biphasic stimulation with a 0.5 ms pulse width (total pulse duration is 1 ms) is applied to the system in type-1 bursting. Panels (a) and (b) show the energy map of the simulated LFP signal that is lower in the blue region than the yellow region (see the color bar on the right). The energy of the LFP signal was computed by using where n stands for the index of and N for the size of the discrete signal x(n). (a) Only the SOM+ interneurons receive the biphasic perturbation. (b) The pyramidal cell, SOM+ interneurons and PV+ interneurons receive the same biphasic perturbation. Panels (c), (d) and (e) show the time course of the marked stimulation on panels (a) and (b). (c) 15 Hz biphasic stimulation with 10 arb. unit amplitude is applied to the SOM+ interneurons (k_SOM = 1, k_P = k_PV = 0). (d) 15 Hz biphasic stimulation with 10 arb. unit amplitude is homogenously applied to all subpopulations (k_SOM = k_P = k_PV = 1). (e) 25 Hz biphasic stimulation with 10 arb. unit amplitude is homogenously applied to all subpopulations (k_SOM = k_P = k_PV = 1).

https://doi.org/10.1371/journal.pcbi.1008430.g006

Results indicate that pre-ictal bursts frequency decreases when the stimulation is switched on at, typically at the instant t = 5s in both cases. The bursting regime can be aborted if the stimulation frequency and amplitude are sufficiently high. The minimum values of the stimulation frequency and amplitude to abort bursting depend on which neuronal subpopulation receives the stimulation. When only the SOM+ interneuron subpopulation is stimulated, the minimum stimulation frequency and amplitude required to abort bursting are lower than the case where all neural subpopulations are stimulated homogenously. As exemplified in Fig 6C, bursting is suppressed at f = 15 Hz for an amplitude of 10 arb. unit when only SOM+ interneurons are stimulated. While the same stimulation can considerably decreases the frequency of bursting events (Fig 6D) when all subpopulations are impacted, the stimulation frequency should be increased to 25 Hz for a complete bursting suppression (Fig 6E). For a sufficiently high stimulation frequency for which bursting is aborted, increasing the stimulation amplitude can depolarize the PV+ subpopulation. However, such an input induces high frequency components to the burstless response observed in the LFP, as opposed to the low amplitude response which is closer to the physiological background activity (data now shown). While the purpose of the stimulation is not only aborting bursting but also keeping the system as close as possible to background activity, low amplitude stimulations result in more beneficial outputs for clinical applications.

The difference between type-1 and type-2 bursting is the number of spikes during the active phase that is related to the excitatory input onto PV+ interneurons. In particular, the EPSP is larger in the former case. Despite this difference, the bursting mechanisms in both types are the same; i.e. slow oscillations in the SOM+ interneurons drive sequentially and periodically the same type of bifurcations in the subsystem of pyramidal cells and PV+ interneurons. Hence, the strategy for aborting bursting relying on aborting oscillations in the SOM+ subsystem does not depend on the bursting type. The estimations on the stimulation parameters (in terms of frequency and amplitude) given in Fig 6C, which are for type-1 bursting, are capable of aborting type-2 bursting and sporadic bursting, as well, because both of the regimes are less excited than type-1 bursting.