Abstract
An improved understanding of the mechanisms underlying neuromodulatory approaches to mitigate seizure onset is needed to identify clinical targets for the treatment of epilepsy. Using a Wilson–Cowanmotivated network of inhibitory and excitatory populations, we examined the role played by intrinsic and extrinsic stimuli on the network’s predisposition to sudden transitions into oscillatory dynamics, similar to the transition to the seizure state. Our joint computational and mathematical analyses revealed that such stimuli, be they noisy or periodic in nature, exert a stabilizing influence on network responses, disrupting the development of such oscillations. Based on a combination of numerical simulations and meanfield analyses, our results suggest that high variance and/or high frequency stimulation waveforms can prevent multistability, a mathematical harbinger of sudden changes in network dynamics. By tuning the neurons’ responses to input, stimuli stabilize network dynamics away from these transitions. Furthermore, our research shows that such stabilization of neural activity occurs through a selective recruitment of inhibitory cells, providing a theoretical undergird for the known key role these cells play in both the healthy and diseased brain. Taken together, these findings provide new vistas on neuromodulatory approaches to stabilize neural microcircuit activity.
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Introduction
Epilepsy is a condition in which the brain paroxysmally and repeatedly transitions into a seizure, a short lived state characterized by excessive neural activity, varying degrees of exaggerated synchronization and desynchronization, and oscillatory dynamics^{1,2}. Medication is the first clinical option for patients with epilepsy, although when not effective such individuals are deemed to be medically refractory. Although half of medically refractory patients are not candidates for resective surgery, they may benefit from the use of neuromodulatory devices^{3,4,5,6,7}. Neuromodulation in epilepsy is used to prevent the transition to seizure, or extinguish pathological oscillatory activity associated with seizures^{8}. Current approaches seek to reduce seizures using either chronic open loop stimulation to alter circuit excitability^{9} (i.e. vagus nerve stimulation^{10}), or closedloop stimulation where seizures are detected in real time and electrical stimulation delivered in an attempt to abort an imminent seizure^{11,12,13,14}. To improve the efficacy of neuromodulation for epilepsy, it is important to understand how such electrical stimulation impacts recurrent neural circuits^{15,16,17,18}, particularly how neuromodulation may make these circuits more resilient to the dynamics characterizing seizures.
Experiments using optogenetic tools to selectively activate and inhibit specific celltypes are starting to reveal celltype specific roles in seizure initiation and propagation. In contradistinction to these celltype specific experiments that can be performed in vitro and in vivo in transgenic animals, the clinical application of neuromodulation is currently limited to the nonspecific delivery of electrical current to the brain. While the realworld clinical benefits of neuromodulation have been demonstrated both for epilepsy^{3,4,5,6,7} and for movement disorders using deep brain stimulation (DBS)^{16,19,20}, our understanding of how these effects are realized in neuronal circuits remains speculative at best^{17,18}. In this context it has been opined that “neurostimulation for epilepsy often has moved retrograde, from pilot clinical studies back to the laboratory for validation and modification of stimulation methods”^{21}.
In this vein we sought to understand the effects of neuromodulation on seizurelike transitions from a computational and mathematical perspective, allowing a principled understanding of dynamical interactions between modeled neurons and stimulus parameters. There are numerous existing computational models of ictogenesis^{9,22,23,24,25} and seizure propagation^{26,27,28}. Irrespective of the different neurophysiological mechanisms involved in ictogenesis^{29}, these in silico models generally represent seizure onset via spontaneous transitions between irregular spiking dynamics and hyperactive oscillatory states. Mathematically speaking, analogous transitions typically occur via passage through a cascade of dynamical instabilities called bifurcations^{30}, mediated by an ad hoc control parameter (often via a parameter broadly encapsulating “neuronal excitability”). Such systems tend to be nonlinear, meaning that the relationship between the inputs to the system and the outputted dynamics is not always proportional. These nonlinear features are generic and essential ingredients of bifurcations and play a fundamental role in seizurelike state transitions in these models^{24,31}; in fact, without nonlinearities, many existing models lose the ability to exhibit these state transitions. Such nonlinearities interact with noise to give rise to a plethora of noiseinduced effects^{32,33,34,35}.
Given the very different roles that celltypes have in inctogenesis^{27,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52}, we chose a simplified yet fully mathematically tractable model built of both excitatory and inhibitory cells with nonlinear recurrent connections. Such twopopulation models have a long history of applications in neuroscience and brain modeling^{53,54} and generically display stereotyped dynamics such as synchronous oscillatory dynamics and multistability, which are core features observed in many seizure models^{55}. The relative simplicity and analytical tractability of such models makes them an ideal candidate to study how neuromodulatory approaches engage recurrent cortical microcircuits in a theoretical setting, and similarly how stimulation interferes with the machinery involved in ictogenesis.
Using this framework, we explored how seizurelike sudden transitions into oscillatory activity are influenced by both intrinsic and extrinsic inputs, where these extrinsic inputs can be viewed as a model of a neuromodulatory intervention. Building on previous theoretical results^{33,55,56,57,58}, our analysis demonstrates that stimulation, be it noisy or periodic in time, suppresses nonlinear state transitions by preferentially recruiting the activity of inhibitory neurons without otherwise significantly altering the system’s dynamics. We provide mathematical evidence for the viability of this mechanism and show that extrinsic stimulation can be tuned to optimize this stabilizing effect. We note that a similar dynamic, a bistability in a purely inhibitory network, provides a mechanism underlying the increased interneuronal activity observed immediately prior to seizure onset in recent work^{45}. Viewed together with this previous work, our results support a general hypothesis that stabilization of neural network dynamics away from sudden state transitions reduce their susceptibility to dynamics associated with the transition into seizure.
Our in silico results suggest that the activity of inhibitory interneurons, in particular their response to both intrinsic and extrinsic noise, acts to stabilize neural activity away from a seizurelike state in vivo. We additionally extend these results to other periodic stimulation waveforms to show that highfrequency stimuli of different types can also be used to generate the same effect. Taken together, this research provides mathematical support for further investigation into an expanded variety of neuromodulatory paradigms with the potential to increase a neural network’s resilience to seizure.
Methods
Oscillatory network of spiking excitatory and inhibitory cells
Since the focus of this work is primarily the interplay between excitatory and inhibitory populations, rather than the dynamics of individual cells, we use Wilson–Cowan models as opposed to other phenomenological or cellularbased models. Consider the following Wilson–Cowanlike network of excitatory and inhibitory populations (\(u^e\) and \(u^i\) respectively) subjected to recurrent connections and intrinsic noise^{33,53,57}
with the linear operator \(L[u]=u\) and the membrane rate constants \(a_e\) and \(a_i\). The spike trains of excitatory and inhibitory cells \(X_k^{e,i}(t)\) are governed by
where \(t_i\) are the firing times of the cells whose firing follows a Poisson process with rate \(f[u_k^{e,i}]\). The firing rate response function f is a sharp sigmoid of the form
The constants \(I^e\) and \(I^i\) correspond to baseline currents to excitatory and inhibitory cells, respectively, and \(I^o\) is an additional external excitatory input. The parameters \(w_{jk}^{nm}\), \(n,m=e,i\), are the connectivity weights between cells j and k of the populations n and m. Each cell is subjected to zero mean intrinsic Gaussian white noise \(\xi _j^{e,i}(t)\), which is further uncorrelated in time and across cells, i.e. \(<\xi _j^{e,i}(t)\xi _k^{e,i}(s)>=\delta (ts)\delta (kj)\), and D is the noise variance. The external stimulation S(t) is global, i.e. identical at each cell, and can take various forms (to be defined in the following). Parameters chosen here are listed in Table 1: these follow previous work of the authors in cortical networks made of excitatory and inhibitory populations^{33,57} and were chosen based on previous cortical modeling work in the literature^{59,60,61}. An investigation regarding the robustness of our main findings to varying E/I balances can be found in the Supplemental Information.
For the set of parameters chosen (see Table 1), the network generically displays inputdriven shifts between asynchronous dynamics and oscillatory activity also consisting of synchronous neuron firing. Representative network activity is plotted in Fig. 1A, both before (\(I^o=0\)) and after (\(I^o=0.1\)) tonic excitation of the excitatory cells. There are parallels between these transitions and seizure onset, considering seizure onset is similarly sudden and seizure dynamics are typically oscillatory in nature (more specifically, seizures can be viewed as a type of oscillatory activity, although not all oscillations are seizurelike). Considering the similarities between the resulting dynamics and some features of seizurelike activity observed experimentally^{1,48,49,50}, we study these transitions as theoretical approximations of seizure onset. A bifurcation analysis of this system is straightforward, and highlights not only the influence of network parameters on resulting dynamics, but also how different forms of stimulation might prevent sudden state transitions in the network.
We note two potential biophysical interpretations of the \(I^o\) term initiating the seizurelike behavior of our model. Indeed, while there is consensus that epilepsy is a dynamical disease in which a neural network evolves as an “ictal attractor”^{62,63}, there is no consensus as to how the system transitions into this attractor and the corresponding seizure state. One hypothesis is that an underlying neuromodulatory mechanism may cause this transition by inducing increased excitatory drive to the ictogenic zone^{62}. Another is found in reflex seizures, where a sensory input represents the trigger initiating seizure^{64} if the stimulus statistics support synchronous neural activity^{65,66}. In either interpretation, the \(I^o\) term is related to biophysical processes that activate specific neural populations resulting in increased input to the seizure onset zone. This term can also be viewed as analogous to the “slow” or “control” parameters that trigger the transition into seizure in other seizure models (see for example the z slow variable in the Epileptor^{26}).
Stochastic meanfield reduction
We here follow the lines of previous work^{55,56} while extending it to two population spiking networks^{57}. We assume that the firing rate of cells is sufficiently high (i.e. the population firing rate \(N_{e,i}f\) is larger than the membrane rate \(a_{e,i}\)) to make use of the diffusion approximation^{67} appropriate, and gain
to obtain the selfconsistent set of nonlinear stochastic differential equations
We also introduce the mean membrane activities \(\bar{u}^{e,i}\)
with average period T that is much shorter than the time scale of the dynamics of the population mean. We further assume that the system primarily evolves in the meandriven regime and local activity obeys
where deviations \(v_j^{e,i}\) are small. Then the mean activities for excitatory and inhibitory populations obey^{57}
where \(\bar{w}^{nm}=\sum _{k=1}^{N_n}\sum _{j=1}^{N_m}w_{kj}^{nm}/N_nN_m\), \(\mu _s=\frac{1}{t}\int _t^{t+T}S(t')dt'\) and the effective transfer function
over some domain \(\Omega (v)\) of deviations which is determined by the stimulation waveform. Indeed, the deviations from the mean \(v_j^{e,i}\) follow a waveformdependent probability distribution \(\rho ^{e,i}(v)\)^{57,58}. This holds if \(\bar{u}^{e,i}\) evolves on a time scale that is much larger than the average period T. Under this condition, and given an extrinsic stimulus S(t), deviations from the mean obey
Without stimulation, i.e. \(S(t)=0\) (see below for the various cases with stimulation), the network dynamics are fully characterized by its parameters and intrinsic noise statistics. In this case, local deviations from the mean \(v^{e,i}_j\) obey the independent and identically distributed (IID) Ornstein–Uhlenbeck processes
and thus \(\rho ^{e,i}(v)=N(0,a_{e,i}D)\), where N is the normal distribution with zero mean and variance \(a_{e,i}D\).
Stochastic stability analysis
We here follow the standard stability analysis of the Wilson–Cowan system^{52,53,57}, but adapt it to account for the presence of additive noise. In the absence of stimulation, i.e. \(S(t)=0\), a steady state \((\bar{u}_o^e,\bar{u}_o^i)\) is implicitly determined by
We note that intrinsic noise is still present here implicitly since the steady states of the network and their associated dynamics—are strongly impacted by the amplitude of intrinsic noise in the transfer function.
Linearizing about the steady state \((\bar{u}_o^e,\bar{u}_o^i)\) one obtains the linear system
with \(R^{e,i}=R(\bar{u}_o^e,\bar{u}_o^i)=\int _{\Omega (v)}f'[\bar{u}_o^{e,i}]\rho ^{e,i}(v)dv\). The stability of this system is given by the eigenvalues of matrix \(\mathbf{A}\)
with
The damping rate (a) and eigenfrequency (\(\omega\)) of the system are the real and imaginary parts of the eigenvalues, respectively. The damping rate determines the stability of the steady state \((\bar{u}_o^e,\bar{u}_o^i)\) and allows us to investigate the role of model parameters and stimuli statistics in the population dynamics.
Notably, the influence of various parameters on the network stability can readily be examined. While our work focuses on the effect of stimulation, the analysis above is amenable to the examination of the role played by all parameters in dictating seizurelike behavior.
Extrinsic noise stimulation
The case of extrinsic noise stimulation can be analyzed using a similar approach as intrinsic noise. In contrast to the case of intrinsic noise however, the external stimulus is global and both excitatory and inhibitory populations receive the same input. Hence it must be treated accordingly when deriving meanfield approximations and associated convolutions. Since realistic external stimuli do not have infinite power as white noise, their power spectral density has to decay for large frequencies. Consequently the noise stimulation S(t) is low passfiltered Gaussian noise with \(\mu _s=0\)
and with white noise \(<\eta >=0\), \(<\eta (t)\eta (t')>=\delta (tt')\), time scale \(\tau\) and noise variance \(D_s\). The cutoff frequency of this low passfilter is \(f_c=1/\tau\), i.e. at this frequency the filter power corresponds to half its maximum value. Then the stationary probability density functions of \(v_j^{e,i}\) (see Eq. 13) are \(\rho ^{e,i}(v)=N(0, \bar{D}^{e,i})\) with effective noise variances
We see that the white noiselimit \(f_c \rightarrow \infty\) yields the maximum effective noise strength \(\bar{D}^{e,i} \rightarrow a_{e,i}(D+D_s)\) and retaining very low frequencies only reduces the impact of external stimulation since \(\bar{D}^{e,i} \rightarrow a_{e,i}D\) for \(f_c \rightarrow 0\).
Recalling that \(f[u]=(1+\text {exp}(\beta u))^{1}\), if we set \(\beta>>1\), then the firing rate function in Eq. (11) can be written explicitly as
Extrinsic periodic stimulation
Conversely to the extrinsic noise stimulation, periodic stimulation was analyzed numerically only. We refer the reader to our previous work^{58} for a more thorough mathematical analysis of the effects of the waveforms on the network response function.
Positive and negative pulse train stimulation
Pulse train stimuli correspond to successive, periodic and brief (1 ms duration, i.e. a single time step) excitatory and/or inhibitory inputs. The associated waveform is given mathematically by
where \(\delta (0)=1\) and zero otherwise. The pulse train has an angle frequency \(\omega _s\) and an intensity of \(S_0\). For \(S_0<0\) and \(S_0>0\) the stimulation is inhibitory and excitatory, respectively.
Biphasic pulse train stimulation
Biphasic pulse train stimuli combine excitatory and inhibitory pulse train stimuli, so they have a net zero mean. Brief, 1 ms pulses are alternated between positive and negative amplitudes. The associated waveform is given here by
where \(\delta (0)=1\) and zero otherwise, and \(\Delta t=1\) ms, our normal time step.
Sinusoidal stimulation
Extrinsic sinusoidal stimulation is a zero mean input whose waveform is given by
with angle frequency \(\omega _s\) and amplitude \(S_0\).
Measure of neuromodulatory effectiveness
To quantify the impact of waveform parameters on state transitions into oscillatory activity, we simulated networks with stimuli of various amplitudes and frequencies (visualized using a heatmap) and calculated the spike coherence^{68}. The larger the spike coherence in the population, the more synchronously the cells fire. A thorough investigation into the dynamics of this system revealed a clear correspondence between synchronous spiking and oscillatory network dynamics, and as such this spike coherence measure is used to quantify the network’s tendency to oscillate.
Spike coherence was measured after an increase in the tonic drive to the excitatory population \(I^o\) that triggers ocillatory dynamics in the absence of neuromodulatory stimulus. The stimulation paradigm of interest was applied continuously both before and after this change in input. To do so, we randomly selected two excitatory cells i, j and binned their respective responses over a given time window \(\Delta t\)=20 ms. For cell j, \(X_j^{\text {bin}}(k\Delta t)=1\) if a spike occurred during the interval \([k\Delta t, (k+1)\Delta t]\), and zero otherwise, where \(k=0,\ldots ,T/\Delta t1\) and T is the duration of the spike train. Spike coherence was then computed using
where \(\sum _T\) denotes the sum over time intervals. For \(C(\Delta T)=0\) two spike trains in the excitatory population are independent and no seizure is present, whereas \(C(\Delta T)=1\) reflects maximum coherence.
We note that in the heatmap visualizations in Figs. 5 and 6, the stimulation frequencies on the xaxis are not uniformly spaced and do not exceed 200 Hz. This was necessary since in order to to maintain computational efficiency, our model resolved differences between stimulation paradigms whose periods were greater than 4 Hz and differed by at least 1 Hz. Thus, we simulated stimuli with periods of [1000, 500, 300:25:100, 90:10:50, 45:5:30, 28:2:20, 19:1:5] ms to reasonably sample a range of frequencies between 1 and 200 Hz.
Mean firing rate
The mean firing rate of a given population over a short time period \(\Delta T\) is given by the mean of the spiking activity over the population:
Single cell power spectrum analysis
Excitatory and inhibitory cells have different intrinsic time scales^{61} and hence respond to external stimuli differently. For a decoupled cell, Eqs. (1) and (2) read
The external input \(\eta _i(t)=\xi _i(t)+S(t)\) is a linear superposition of the intrinsic noise \(\xi _i(t)\) at node i and the extrinsic global stimulation S(t). For a time series of duration T, the power spectral density of \(u^{e,i}_i\) is
with discrete angle frequency \(\omega _n=2\pi n/T\) and integer \(n\in \mathbb {N}_0\). Here, we consider the intrinsic white noise process \(\xi _i(t)\) with variance D that is uncorrelated to the extrinsic stimulation input S(t) and \(\tilde{S}_n\) is the discrete Fourier transform of the global stimulus
Equation (29) describes a lowpassfilter with input \(\eta _i\). It has a cutoff angle frequency \(\omega _c=a_{e,i}\) defined by \(S^2_{e,i}(\omega _c)/S^2_{e,i}(0)=1/2\) (see Eq. 30). This means that input with angle frequency \(\omega <\omega _c\) is said to pass the filter, whereas input with angle frequency \(\omega >\omega _c\) is filtered out. This cutoff frequency is larger for inhibitory than for excitatory cells due to \(a_i>a_e\) and hence \(S^2_i(\omega )>S^2_e(\omega )\). Since this time scale separation and hence the different cutoff frequencies reflect physiological properties, inhibitory cells respond stronger to external stimuli than excitatory cells. However, we point out that the neural populations in our network are coupled and highly nonlinear. Thus, while this reasoning does not apply directly to the network but to individual cells, and as such additional couplinginduced effects should be expected in network dynamics, it may still provide insight into a potential mechanistic explanation for this result.
More specifically, the power spectra of

the extrinsic noise stimulation S(t) with \(\langle S(t) \rangle =0,~\langle S(t)S(t\tau )\rangle =D_s\delta (\tau )\), \(\langle \cdot \rangle =\int _{T/2}^{T/2}\cdot dt/T\) with the \(\delta\)distribution \(\delta (\cdot )\) reads
$$\begin{aligned} S^2_{e,i}(\omega _n)= & {} \frac{a_{e,i}^2\left( D+D_s\right) }{a_{e,i}^2+\omega _n^2}~. \end{aligned}$$(32) 
the positive and negative pulse train stimulation, cf. Equation (24), now described by \(S(t)=S_0\sum _{l}\delta (tt_l),~S_0\in \mathbb {R}\), reads
$$\begin{aligned} S^2_{e,i}(\omega _n)= & {} \lim _{T\rightarrow \infty }\frac{a_{e,i}^2}{a_{e,i}^2+\omega _n^2} \left( D+TS_0^2\left( \frac{\omega _s}{4\pi }\sum _{l=N/2}^{N/2}(\delta _{n,lk} +\delta _{n,lk})+\frac{1}{T}\right) ^2\right) \end{aligned}$$(33)with \(k=\omega _sT/2\pi\) and \(t_l=2\pi l/\omega _s\) and N is the number of pulses in time T. Here we have applied the Poisson summation formula.

the biphasic pulse train stimulation, cf. Equation (25), now described by \(S(t)=S_0\sum _{l}(\delta (tt_l)\delta (tt_l\tau )),~S_0\in \mathbb {R}\) and \(\tau >0,~\tau \rightarrow 0\), reads
$$\begin{aligned} S^2_{e,i}(\omega _n)= & {} \lim _{T\rightarrow \infty }\frac{a_{e,i}^2}{a_{e,i}^2+\omega _n^2} \left( D+\frac{S_0^2\omega _s^2\tau ^2}{4T}n^2\sum _{l=N/2}^{N/2}\left( \delta _{n,lk} +\delta _{n,lk}\right) \right) ~. \end{aligned}$$(34) 
the sinusoidal stimulus, cf. Equation (26), reads
$$\begin{aligned} S^2_{e,i}(\omega _n)= & {} \lim _{T\rightarrow \infty }\frac{a_{e,i}^2}{a_{e,i}^2+\omega _n^2} \left( D+\frac{TS_0^2}{4}\left( \delta _{n,k}+\delta _{n,k}\right) \right) ~. \end{aligned}$$(35)
Results
Intrinsic noise, endogenous activity and susceptibility to state transitions
Given the chosen parameters (see Table 1; “Methods”), our model generically exhibits sudden shifts between asynchronous dynamics and oscillatory activity, typically including more synchronized neural spiking, mediated by increased excitatory input. Although our chosen network parameters and variables of interest may be “phenomenological” in nature^{69}, the network exhibits the dynamical transition of interest that has important parallels with seizure onset. This assertion relies on the abrupt nature of this transition and the fact that seizure is itself an oscillatory state with specific features. Maintaining these phenomenological elements of the model allows for the necessary mathematical analysis to explore mechanisms of action for neuromodulation on these microcircuits with strong theoretical support.
Our model demonstrates a sharp transition to oscillatory dynamics (and corresponding synchronous neural firing) arising from relatively small increase in drive to the excitatory cells, as shown in Fig. 1. Oscillatory activity is common in networks with interconnected excitatory and inhibitory cells, colloquially termed “EI networks”^{70,71,72,73,74,75,76}, making these sharp transitions from irregular activity into oscillatory dynamics expected for large portions of the parameter space. More importantly in the context of this study, and the mathematical exploration of this phenomenon, is that such behaviour is a signature of multistability: the existence of multiple stable solutions to a dynamical system. Indeed, such multistability represents a key feature of potential seizureinducing mechanisms, often represented by a sudden transition into some form of oscillatory dynamics, both in vivo and in silico^{45,55,62,77,78}. Both excitatory and inhibitory cells fire synchronously following the transition into oscillatory dynamics in our spiking networks (Fig. 1A). Motivated by these observations and building on previous findings on the interaction between noise and oscillatory activity^{33,56,79}, we chose to further investigate the influence of intrinsic noise on the system’s predisposition to this multistability by examining the mathematical features necessary for bifurcations and seizurelike transitions to occur. We note that in Fig. 1A the frequency of the seizurelike oscillations are approximately in the highbeta/lowgamma range, paralleling the frequencies observed clinically in low voltage fast seizure onsets^{80,81,82,83}.
We first numerically examined the influence of intrinsic noise on input driven oscillatory activity. Figure 1B demonstrates that increasing the intrinsic noise level (represented by the variable D) suppresses the transition from asynchronous dynamics into oscillatory activity. When subjected to an increase in excitatory drive, neural activity remained asynchronous and irregular with larger levels of intrinsic noise, with the network responding to the increase in this drive with only a small uptick in excitatory activity. In contrast, clear oscillatory dynamics are observed for the low values of D shown in Fig. 1A, corresponding with more substantial increases in excitatory activity. This suggests that intrinsic fluctuations driven by noise make the network’s response to increased external drive smooth, monotonic, and commensurate to the change in external drive. In Fig. 2A we show that this effect scales, i.e. that increases in the noise variance D gradually decrease the salience of the oscillatory activity (and correspondingly the nonlinearity of the response to an increase in drive) until such oscillations are entirely absent.
This result may appear counterintuitive, considering that the randomness inherent in noisy input may be expected to make a network’s dynamics less, rather than more, predictable. Detailed mathematical analysis is uniquely suited to uncover potential mechanisms explaining such results, especially in complex systems such as neural circuits. In this vein, we performed a stochastic mean field analysis (see “Methods”) on our network in search of a convincing explanation of this behavior. Fig 2B shows the steady state(s) of the system as a function of input drive (see “Methods”). For weak noise (i.e. low values of D), the system exhibits the multistable behaviour observed previously: the populations possess multiple stable states between which the system can transition abruptly due to shifts in the tonic excitatory drive. In the spiking network, these jumps are accompanied by a shift from irregular to oscillatory activity (as seen in Fig. 2A).
However, as illustrated by Fig. 2B, increasing the level of intrinsic noise gradually reduces the width of the multistable region in the parameter space until it vanishes completely. This corresponds with a decrease in oscillatory activity seen following an increase in the tonic excitatory drive in the spiking networks (Fig. 2A). Another important observation is the suppressing effect of noise on the level of activity in the excitatory resting states. Indeed, both within and outside the multistability region, increased intrinsic noise intensity shifts steady states to lower values of the mean potential. This indicates that, in the present model, noise serves to damp down excitatory activity. This is consistent with the noiseinduced reduction in excitatory firing rates seen in Fig. 1.
Intrigued by the stabilizing role of intrinsic noise, we pushed our mean field analysis further and performed a linear stochastic stability analysis (see “Methods”). Figure 3 depicts the system’s steady state stability as a function of the excitatory tonic drive for two different noise variances, detailing how increases in input to the excitatory population impacts state transitions between asynchronous and oscillatory activity. For weak noise (Fig. 3A), increased drive leads to passage through the multistable states, in which the system undergoes a saddle nodehomoclinic bifurcation^{84}: increasing the excitatory drive causes the system to jump from a stable node to a remaining unstable oscillatory steady state and the population dynamics jump abruptly to a highfrequency limit cycle. In contrast, when intrinsic noise is increased in the system (Fig. 3B), the same change in excitatory drive does not cause such a transition: the unique steady state remains stable and at a lower level of the mean potential when compared to the results in Fig. 3A, conforming with the results of Fig. 2B.
The observations from our spiking network, confirmed by the use of the mathematical tool of stochastic stability analysis, provide a key insight: intrinsic noise stabilizes neural population activity by making responses to increases in external drive more commensurate to the degree of this change, i.e. the dynamics become more linear. In turn, this makes abrupt, major shifts in network dynamics (like those that occur during the sudden transition between asynchronous and oscillatory dynamics) unlikely. This wellknown noisedriven phenomenon^{56,58,79,85,86,87} shapes the stability of this nonlinear system.
With the effects of intrinsic noise on our neural microcircuit thoroughly explored, we asked whether similar stabilizing effects can be achieved by an extrinsic stimulus. Such a stimulus could be provided, in the in vivo setting, via a neuromodulatory device. We extend our mathematical and theoretical understanding from the intrinsic to the extrinsic setting in the following.
Stabilizing network activity via external modulatory stimuli
Our analysis of the effects of intrinsic noise on this model microcircuit, combined with previous work, provides some intuition that extrinsic noise might have a similar stabilizing effect as intrinsic noise in this in silico setting. Indeed, highfrequency exogeneous inputs can be used to modulate endogenous neural oscillatory activity^{58}, although it remains to be seen whether they can also be used to stabilize the system to prevent sudden dynamical transitions. We thus examined the role of various stimulation waveforms on the transition into oscillatory dynamics. Specifically, to test the effect of these waveforms on the multistability observed in our model, we suddenly increased the tonic drive to the excitatory population (which induces a transition between asynchronous and oscillatory dynamics in systems with low intrinsic noise in the absence of external modulation) while continuously (i.e. pre and post increased drive) subjecting the network to stimuli with various waveforms. Fig. 4 presents stimuli (see “Methods”) that exert a waveformdependent effect on the oscillatory properties of our spiking networks. All five stimuli of interest can, for informed choices of the stimulus amplitude and frequency, exert a stabilizing effect on the network and prevent major changes in network dynamics following an increase in excitatory drive. We examine the differences between the effects of these waveforms in the following.
Response to extrinsic noise stimulation
Neuromodulation by noisy stimuli, or extrinsic noise, corresponds to the use of electric fields of irregular amplitudes and fluctuating frequencies to modulate the activity of superficial cortical neurons. Either noninvasive or intracranial, extrinsic noise stimulation is used in a variety of clinical settings to modulate cortical excitability^{88}. It further has numerous and varied clinical applications, including reducing fatigue in multiple sclerosis patients^{89}, improving creativity^{90} and alleviating tinnitus symptoms^{91}. Its effect on ongoing alpha activity has been both computationally and mathematically characterized^{58}.
We emphasize that, as discussed in detail in “Methods”, the primary difference between the intrinsic and extrinsic noise in this research is that the intrinsic noise is independently generated for each cell, while the extrinsic noise is uniform for each modeled cell.
Figure 4D shows an example of how extrinsic noise stimulation prevents a sudden shift into oscillatory dynamics. The effect of extrinsic noise on the tendency for the network to exhibit such dynamics following an increase in excitatory drive is further examined in Fig. 5E. Neuromodulation with the largest noise variances and cutoff frequencies (see “Methods”) yield diminished spike coherence following the seizure stimulus, indicating a tendency for less synchronous network dynamics, where network synchrony is used as a computationally efficient quantification of oscillatory behavior (see “Methods”). This implies that population synchrony can be suppressed, and sudden transitions into oscillatory activity mitigated, for multiple combinations of relatively high amplitudes and cutoff frequencies of this extrinsic noise stimulation.
Response to periodic neurostimulation
As shown in earlier work^{58}, stochastic stability analysis is amenable to highfrequency (nonstochastic) stimuli, whose effect can be examined and investigated using similar meanfield techniques. The influence of stimuli on ongoing neural activity depends on the statistical properties of their waveform, and can alter network dynamics by tuning its stability. These results suggest that highfrequency stimuli such as sinusoidal waves and pulse trains could interact with the neural populations and interfere with transitions into oscillatory dynamics.
Confirming this hypothesis, various nonnoisy waveforms were found to disrupt the onset of oscillatory dynamics. We repeated the above analysis for inputs consisting of different waveforms, computing spike coherence in the network following the seizure stimulus as a function of stimulation amplitude and frequency. The results in Fig. 5A–D reveal a codependent role of amplitude and frequency in preventing oscillatory dynamics. Minimal spike coherence was achieved for high amplitude, high frequency stimuli for most waveforms. Different response patterns are expected given the waveformdependent effect of these stimuli on the system response function^{58}.
Differential excitatory and inhibitory responses to neurostimulation
To better understand the mechanism by which these stimulation paradigms, both noisy and periodic, decrease spike coherence and correspondingly disrupt the transition into oscillatory activity, we simultaneously computed the mean firing rates of excitatory and inhibitory cells during these explorations. The results plotted in Fig. 6 show that, under a wide variety of frequency and amplitude combinations, stimulation enables a differential recruitment of excitatory and inhibitory cells. Specifically, the simulations show that highfrequency, highamplitude stimuli predominantly recruit inhibitory cells by increasing their firing rates, which in turn further suppresses excitatory activity. This occurs in the biphasic and positive pulsatile stimuli (Figs. 5A,C; 6A,C), in the sinusoidal stimuli (Figs. 5B, 6B) and in the noise stimulation (Figs. 5E, 6E). The negative pulsatile stimulus (Figs. 5D, 6D) is the lone exception, in which neuromodulation suppresses firing activity in both excitatory and inhibitory cells. In sum, stimulusinduced actions diminish excitatory neural firing and minimize spike coherence (Figs. 5, 6), showcasing a potential mechanism of action for the antiictogenic effects of neuromodulation through the selective activation of inhibitory cells.
A potential explanation of this phenomenon is found by studying the effects of such stimuli on individual uncoupled cells. In Fig. 7 these effects were quantified via the power spectral density for an individual modeled excitatory and inhibitory cell; the underlying mathematical derivations for these curves can be found in “Methods” section. For all four stimulus paradigms, inhibitory cells respond more strongly than excitatory cells (shown by the larger power spectral densities) for all frequencies. We note that the stimulation paradigms in Fig. 7A,B yield several clear spectral peaks at equidistant frequencies, which are multiples of the pulse repetition frequency and their powerspectral density is proportional to the repetition frequency. The sinusoidal stimulation paradigm presented in Fig. 7C yields a single spectral peak with a low magnitude, since this magnitude is not frequency dependent (see “Methods” section for more details). The noise stimulation paradigm in Fig. 7D does not exhibit any spectral peak since the input is white noise whose frequencies have identical magnitude.
As derived in “Methods” section, these differential responses at the single cell level can be explained by the different time scales of the excitatory and inhibitory cells, which represent experimentally observed differences in the membrane time constants of cortical pyramidal neurons and parvalbumin (PV) positive interneurons^{61}. Taken together, these results indicate that these differential time scales promote an enhanced response of inhibitory cells over excitatory cells to the type of stimuli studied here, in turn potentially explaining the finding in Fig. 6 that, at high enough stimulus amplitudes, inhibitory cells become more active than excitatory cells.
Discussion
In this study, we use a variety of computational techniques, including numerical simulation and meanfield mathematical analysis, to probe the influence of both intrinsic noise and extrinsic stimulation (both noisy and periodic) on the stability of a spiking neural network. Our results show that both intrinsic and extrinsic noisy inputs stabilize our microcircuit, making it less susceptible to the sudden transition into oscillatory dynamics that is often used to model seizure onset. Building on previous results^{57,58}, we have extended these findings to other periodic stimuli and hence gleaned a better understanding of the relationship between sudden transitions into oscillatory activity and neuromodulation using a mathematical and computational lens.
Our analysis shows that noise serves to suppress multistable behavior, leading the system into a regime in which the network’s response to changing input is commensurate to the magnitude of that change. Furthermore, noise minimizes the activity of excitatory cells in these driven states (see Figs. 2, 3) by repositioning the system’s equilibrium point. Simulations in the spiking model show that various combinations of stimulation waveforms, frequencies and amplitudes can stabilize network activity in a similar manner, with the coherence following an increase in external drive (the seizure stimulus) typically minimized for highfrequency and highamplitude stimulation. This regime of minimal coherence represents the stimulation paradigms with antiictogenic effects, and effective paradigms with minimal frequency or amplitude can be determined by analyzing Fig. 5. Detailed analysis of the activity of the excitatory and inhibitory populations further reveal a selective and predominant recruitment of inhibitory cells: more strongly driven by highfrequency inputs, the inhibitory population suppresses excitatory activity and biases the system towards a desynchronized state antithetical to seizure.
The models used here are chosen purposefully to retain both mathematical tractability and the ability to draw parallels between related population and spiking models. The Wilson–Cowan equations^{53} are a frequently used formulation in this endeavor. These choices allow us to simultaneously consider three key points in our in silico investigations: first, the correspondence between mathematical analysis of the population model and the simulated behavior of the related spiking model (we note the correspondence of the predictions of our analyses of these population models with the computational results of the related spiking model); second, the differential role of excitatory and inhibitory cell populations in seizure onset in both settings; and third, the effects of neuromodulatory intervention on both populationlevel and celllevel dynamics related to seizure onset, using both mathematical and computational tools. In this manner, we have provided theoretical and mathematical support for our hypotheses regarding the relationships between intrinsic noise, neuromodulatory paradigms, and seizure onset. Work using more detailed neural models, potentially including the implementation of features such as spike timing dependent plasticity (STDP), is a fertile ground for future research to probe these hypotheses in a more biophysically constrained setting. Additional future work could investigate how in vitro neuromodulatory stimuli can be optimized so as not to induce oscillatory activity, with our analysis here providing initial evidence that stimulation with a sufficiently high frequency may accomplish this through selectively activating inhibitory cells in a nonrhythmic fashion. The use of biohybrid systems^{92,93,94,95} could provide an useful tool in such research.
A paramount use of these mathematical findings is towards the articulation of a theoretical understanding of how neuromodulation acts to mitigate seizure onset, and in turn potential pathways through which these treatments might be improved. While the oscillatory dynamics observed in our simulations are not meant as “model seizures” (indeed, there are crucial features that differentiate seizures from general oscillatory dynamics, such as the “glissandi”^{96}), fully encapsulating all of these features is outside the realm of this study considering the desire to preserve the mathematical tractability of this model. Instead, the conclusions drawn from this work remain focused on seizure onset. This focus is motivated by the similarities between the transitions between asynchronous and oscillatory dynamics in this model and important features of the welldescribed lowvoltage fastonset transition into seizure. These include maintaining “intact inhibition” during this transition^{82,83} and the initial oscillatory frequencies in the beta/gamma range^{80,81,82}. Considering the focus on seizure onset, this model can be viewed as analogous to a mesoscopic population making up a few millimeters of tissue, given experimental evidence that ictal states emerge in brain regions of this scale^{55,97,98,99}.
The critical antiictogenic role of inhibitory interneurons is well established in the epilepsy literature^{96,100}, with many antiseizure interventions aiming to provide additional inhibition via the activation of inhibitory receptors^{101,102} or directly rescuing the activity of inhibitory interneurons^{103,104}. Neuromodulation has been shown to be an effective treatment for many individuals that do not respond to these treatments^{3,4,5,6,7}, with the commonly hypothesized mechanism of action for analogous DBS systems being the creation of a “functional” lesion, potentially via depolarization blockade^{16,19}. Our results present theoretical support for a distinct mechanism of action for the effects of neuromodulation via the upregulation of inhibitory activity. Simultaneously, these findings highlight a very important practical aspect of clinical neuromodulation which is that neuromodulatory devices need not be overly precise (i.e. selectively target only inhibitory cells), given the different responses observed in excitatory and inhibitory cells to global stimulation in our in silico explorations.
However, the seemingly paradoxical role of interneurons in seizure must be taken into account when interpreting these findings. While interneurons serve a clear role in restraining seizure propagation, there is also evidence that interneurons are complicit in seizure initiation, potentially serving a causal role^{38,43,44,105,106,107,108}. One hypothesized pathway through which interneuron hyperactivity may contribute to seizure is through the accumulation of chloride in these circuits which can make GABAergic signalling excitatory instead of inhibitory^{37,109}, a behavior not implemented in our model to preserve mathematical tractability. Nonetheless, when considering neuromodulatory paradigms in the in vivo setting these factors must still be considered.
Thus, it is likely that in vivo neurostimulation paradigms motivated by this work would have to be temporally limited to avoid potentially initiating, rather than suppressing, seizure by overstimulating inhibitory cells. Indeed, while neuromodulation is a relatively recent clinical treatment to prevent seizure, it has long been known to have ictogenic potential^{110}. Our results may correspondingly help design stimulation paradigms that increase interneuronal firing intermittently and transiently in order to avoid hyperactivity in interneurons and preserve their stabilizing effects. In this vein, our previous work explored how a hyperexcitable purely interneuronal network is more likely to serve as a potential “trigger” starting a seizure than a less excitable interneuronal network^{45}. Together, our findings suggest that neuromodulatory strategies will need to be tuned to avoid the proictogenic effects of excessively increasing interneuronal activity (potentially by triggering a nonlinear state transition amongst interneuronal networks), while exploiting the antiictogenic effects we describe here.
We note that our goal here was not to explore our entire model parameter space, but instead provide an example of how a specific type of state transition can be suppressed by neurostimulation and what potential mechanisms underlie this phenomenon. While various bifurcations scenarios can lead to seizurelike dynamics^{26,111}, we argue that the waveforms explored here can stabilize neural activity and thus provide a generic mechanism to prevent multistability and other types of nonlinear properties arising in a variety of neural networks.
Despite the theoretical nature of this work and its corresponding neuroscientific applications, there are a plethora of known similar noiseinduced effects in physical systems that can be seen as support for the feasibility of translating these in silico findings to the physical brain. The generic mechanism by which additive noise stabilizes the dynamics of nonlinear systems is seen in systems with thresholdlike nonlinearities such as digital audio, image processing, optics, communications systems and stock market fluctuations (see^{112} and related literature on dithering). In neuroscience specifically, this occurs in individual neurons^{113}, delayedfeedback systems^{56,79} and plays a role in stochastic^{87} and coherenceresonance^{114}, which are widespread in across physical and neural systems.
Furthermore, there are existing experimental examples within the field of neuroscience where the theoretical mechanisms explored here have been practically implemented in order to suppress pathological dynamics. Sensory stimulation has long been shown to halt or shorten seizurelike events^{31,115}, and it makes intuitive sense that techniques to stop an existing seizure may have parallels to techniques to prevent the onset of seizure altogether. Other types of stimuli, such as vagus nerve stimulation, activate upstream neural populations, which translates as an increased afference of spiking inputs at the epileptic focus^{10}. While our simplistic model is far from capturing the full complexity of the mechanisms involved in the examples above, our analysis provides theoretical support for the hypothesis that noise, either intrinsic or extrinsic, serves to stabilize the dynamics of neural networks in a way may oppose seizure initiation, and articulates pathways through which the effects of these extrinsic inputs can be optimized.
Data availability
The code used in this study will be made available upon request to the corresponding author.
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Acknowledgements
We thank the National Research Council of Canada (NSERC Grants RGPIN201706662 and RGPIN201606182) and the Krembil Foundation (Krembil Seed Grant and Krembil Postdoctoral Fellowship) for support of this research.
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J.L. conceived the study. J.L., S.R. and A.H. conducted the experiments and analyses and analyzed the results. J.L., S.R. and A.H. wrote the initial drafts of the manuscript. All authors reviewed and extensively revised the manuscript.
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Rich, S., Hutt, A., Skinner, F.K. et al. Neurostimulation stabilizes spiking neural networks by disrupting seizurelike oscillatory transitions. Sci Rep 10, 15408 (2020). https://doi.org/10.1038/s41598020723356
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DOI: https://doi.org/10.1038/s41598020723356
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