Vulnerability to paroxysmal oscillations in delayed neural networks: A basis for nocturnal frontal lobe epilepsy?

Abstract

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ∼τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t → 2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.

Do the observed behaviors of complex dynamical systems correspond to the asymptotic behaviors of proposed mathematical models (e.g., fixed-points, limit cycles, chaotic attractors) or to their transient behaviors?^{1, 2, 3} This question is particularly relevant to dynamical systems, such as the nervous system, which contain time delays and which are continuously subjected to the effects of noisy perturbations. Modeling studies demonstrate that a variety of transient behaviors can arise from the interplay between noise and delay including transient stabilization of unstable fixed-points,⁴ noise-induced switching between attractors,^{5, 6} and delay-induced transient oscillations (DITOs).^{7, 8, 9, 10, 11, 12} In particular the duration of DITOs can be so long that these oscillations cannot be easily distinguished from limit cycle oscillations.^{7, 8} Here we ask whether there are situations in which models of neural behaviors are particularly prone to the development of transient behaviors that differ significantly from their asymptotic behaviors. To address this question we study a well known model for neural decision making, namely, two mutually inhibited neurons with delayed feedback.¹² As key parameters are changed this model exhibits a single fixed-point attractor, then a bistable regime with two coexistent fixed-point attractors, and finally a regime with a new fixed-point attractor. DITOs occur only in the bistable regime. This observation may explain why seizures in the epileptic syndrome, nocturnal frontal lobe epilepsy (NFLE), occur during the transition between sleep stages.¹³ During sleep a bistable state can arise transiently as one fixed-point attractor is replaced by another, for example, when a sleeping subject passes from Stage I to Stage II sleep. Our observations anticipate an increased risk for a seizure during the time that the bistable state exists since the dynamical system is naturally brought close to the barrier, or separatrix, that separates the two attractors. In other words this scenario represents a time of vulnerability for a dynamical system to generate novel paroxysmal oscillatory behaviors.

INTRODUCTION

A curious feature of many medical emergencies is that their risk of occurrence exhibits a certain predictability. Well studied examples include the circadian periodicity in the timing of onset of seizures,¹⁴ stroke,¹⁵ lower GI bleeding,¹⁶ and several types of cardiac events, including paroxysmal atrial fibrillation¹⁷ and sudden cardiac death.¹⁸ Timing of onset can also be associated with other physiological rhythms such as the sleep–wake cycle. An exceptional case that motivates this study is NFLE, a familial form of epilepsy associated with mutations in the α-subunit of central nicotinic acetyl-choline receptors.¹⁹ In NFLE seizure onset occurs only during sleep and, in particular, primarily during the transition between Stage I and Stage II sleep.²⁰ The observation that the timing of medical emergencies is not random provides strong reasons to believe that it might be possible to predict their occurrence and hence enable timely preventive strategies.

Dynamical diseases are characterized by qualitative changes in their dynamics as critical parameters are changed. Such changes can arise in a healthy physiological control system when the values of critical control parameters have been changed to values associated with unhealthy dynamics.^{21, 22} It is important to note that in this picture of a dynamical disease the abnormal dynamics are those of an attractor for the control mechanism, most often a limit cycle-attractor.²³ In mathematical models these attractors correspond to the asymptotic stationary states present as t → ∞. However, it is unlikely that real biological dynamics systems ever achieve such stationary states. The concept of a dynamical disease can be extended to disorders characterized by paroxysmal events.^{24, 25, 26, 27} The prototypical example is epilepsy, a disorder characterized by recurring seizures. A number of observations associate bistability with seizure occurrence:^{27, 28} single direct current pulses terminate seizures in rats,³¹ brief electrical stimuli terminate after-discharges in humans,^{32, 33} and sensory stimuli block abort seizures in humans.²⁸ Here we associate an increased risk for seizure onset with bistable states that arise naturally in association with physiological changes in parameters as one attractor is replaced by another (for examples, see Refs. ^{29, 30}). By analogy with NFLE, the changes in state could represent changes in sleep stage.¹³ However, the relevant parameter changes could also occur in relation to hormonal cycles, such as those associated with the menstrual cycle,³⁴ or even to changes related to life’s stresses.³⁵ In all cases the changes in the parameters that produce the bistable state occur on a slower time scale than the time scale of the paroxysmal event. Our point is that in time-delayed dynamical systems, such as those associated with physiological and neural control, the bistable state represents an instance where the dynamics are particularly vulnerable to the generation of relatively long lived transient dynamical states. In particular we suggest that it is not the switching between the attractors that is associated with the medical emergency, but rather the novel transient behaviors that arise on the barrier (“separatrix”) that separates the two attractors.

Our hypothesis draws attention to the physiological importance of noisy, bistable dynamical systems;³⁶ the subject of this dedicated issue in memory of the pioneering work of Frank Moss. In Sec. 2 we review the properties of DITOs with particular emphasis to their spontaneous formation in the presence of noisy perturbations. In Sec. 3 we demonstrate that the statistical properties of DITOs in a noisy, time-delayed dynamical system can be studied using a Markov chain model approximation. The formation of long-lived (>100τ) is a low probability event, whose probability of occurrence depends on the time delay. The possibility that medical emergencies may to some extent be predictable has important implications for morbidity and mortality rates.

BACKGROUND

The properties of DITOs have been studied extensively.^{7, 8, 10, 11, 12} Here we illustrate the phenomenon by considerations of a model for two mutually interacting neurons with time-delayed inhibitory feedback (Fig. 1). Models of this form arise in the context of decision making.¹² Specifically the model has the form

$${\alpha }_1\stackrel{·}{x}(t)=-x-F_2(y(t-{\tau }_2))+I_1,{\alpha }_2\stackrel{·}{y}(t)=-y-F_1(x(t-{\tau }_1))+I_2,$$ (1)

where x, y are the neural firing rates, α₁, α₂ are the neural time constants, I₁, I₂ represent constant external inputs, τ₁, τ₂ are the conduction delays between the two neurons, and

$$F_j(u)=\frac{c_j{u}^{n_j}}{{\theta }_j^{n_j}+u^{n_j}}$$

are sigmoidal functions that represent the inhibitory influences. Here we take τ = τ₁ = τ₂. In order to solve 1 it is necessary to define the initial functions Φ(x, y) ≡ (x(s), y(s)), $s\in [t_0-\tau ,t_0]$.

Figure 1.

Schematic representation of the two–neuron network of mutually inhibited neurons whose dynamics are described by Eq. 1.

In the absence of noise and when I₁ and I₂ are positive constants, the fixed points (x*, y*) satisfy

$$x^{*}=I_1-F_2(y^{*}),$$ (2)

$$y^{*}=I_2-F_1(x^{*}).$$ (3)

Provided that I₁ − F₂(I₂) > 0 and I₂ − F₂(I₁) > 0 there will be one, two, or three fixed-points in the first quadrant (Fig. 2). In this study we chose the parameters so that there will be three fixed-points: two of these are stable and the third is unstable. We refer to this parameter range as the bistable regime. The global dynamics in the bistable regime are well understood: depending on the choice of initial condition, trajectories are attracted to one of the co-existent fixed-points.^{9, 12} The unstable fixed-point is associated with a limit cycle; however, this limit cycle is unstable and is not likely to be observed when τ = 0.⁹ When τ > 0 the DITOs represent a transient stabilization of this unstable limit cycle. The exact details of this transient stabilizing mechanism have only partially been worked out.^{4, 8}

Figure 2.

Determination of fixed-points of Eq. 1 for different values of the parameters θ₁ and θ₂. The solid line corresponds to Eq. 2 and the dashed line to Eq. 3. Parameters: c₁ = 0.4, c₂ = 0.6, I₁ = 0.5, I₂ = 0.4, and n₁ = n₂ = 2.

Although the stability of the fixed-points is independent of τ,¹² the nature of the basin of attraction associated with the stable fixed-points depends on τ.^{9, 12} Figure 3 shows the dynamics of Eq. 1 for a constant choice of initial function, i.e., Φ(x, y) ≡ (x(s) = X₀, y(s) = Y₀), $s\in [t_0-\tau ,t_0]$. For small delays the solution is attracted to the fixed-point labeled B; however, for larger delays it is attracted to the fixed-point labeled A. When Φ(x, y) is chosen sufficiently close to fixed-point B, solutions are attracted to B even at the higher delay (data not shown).

Figure 3.

Dynamics of Eq. 1 for a single choice of the initial function Φ(x, y) as a function of τ. The left-hand column shows the dynamics of x as a function of time and the right-hand column gives the corresponding phase plane representation, i.e., x versus y. The fixed-points A and B (see Fig. 2) are indicated by •. In all cases Φ(x, y) = (0.4, 0.24). Parameters: c₁ = 0.4, c₂ = 0.6, I₁ = 0.5, I₂ = 0.4, n₁ = n₂ = 2, θ₁ = θ₂ = 0.2.

There are two types of oscillatory transients that occur in Eq. 1. Since these behaviors are not observed when τ = 0 we refer to them as DITO. The first type, referred to as DITO-I, the transient does not cause a switch between attractors. Thus DITO-I resembles the oscillatory transient behavior associated with a stable spiral point. The second type, referred to as DITO-II, is associated with a switch between attractors and is the subject of this study. We refer to the interval between successive maxima of the DITO as the period. Measured in this way the period for the DITO-I and DITO-II oscillations are both ∼2τ. The duration of DITO-II is longer than that of DITO-I presumably because DITO-II arises closer to the separatrix, and hence Φ(x, y) can span the separatrix.

Bistability is not sufficient by itself to ensure that a DITO occurs. There needs to be an additional mechanism which brings trajectories sufficiently close to the unstable fixed-point. Here we consider the possibility that this mechanism is additive noise, i.e., Eq. 1 becomes

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{x}(t)=-x-S_2(y(t-\tau ))+I_1+{\sigma }^2\xi (t),\stackrel{·}{y}(t)=-y-S_1(x(t-\tau ))+I_2+{\sigma }^2\xi (t),}\end{array}$$ (4)

where ξ(t) is uniformly distributed “white noise” with variance σ², and we have set α₁ = α₂ = 1. In writing Eq. 4 we have assumed that both neurons reach a common time varying component and chosen I₁, I₂ to ensure that the dynamics are tuned to the bistable regime.

Figure 4a shows a single realization of a noise-induced switch between fixed-point attractors using Eq. 4. The transition between attractors is associated with an oscillatory transient. The power in the frequency expected for a DITO-II, namely (2τ)⁻¹, increases during the transition. Figure 5a shows the relationship between the power in the DITO frequency range and the variance, or intensity, of the additive noise input: the power in the DITO frequency range (DF power) is maximized when σ² ∼ 0.1. This phenomenon has been referred to as stochastic resonance in the absence of a periodic input.^{10, 11} Figure 5b shows the effect of adding a periodic input of the form K sin ωt, where K is a constant, in addition to the noisy inputs to each of the neurons in Eq. 4. When τ = 0 Eq. 4 exhibits the characteristics of stochastic resonance. In particular the power in the forcing frequency ω goes through a maximum as a function of σ². Since Eq. 1 does not possess a double well potential (see Appendix) this is an example of stochastic resonance in the setting of an asymmetric bistable system.^{37, 40} When τ > 0 no resonance is observed provided that ω ≠ (2τ)⁻¹. We consider only noisy inputs in the following discussion.

Figure 4.

(a) A single realization of a switch between one fixed-point attractor (S₀) and the other (S₂) with an intervening DITO-II (S₁) generated by Eq. 1. All simulations used PyDelay with an Euler algorithm and time step of 0.01 and were run using Python in combination with the SciPy and NumPy libraries. The power in the DITO frequency range ((2τ)⁻¹), DF Power, was determined from Eq. 1. Parameters are the same as in Fig. 3 with τ = 6 and σ² = 0.05. (b) A three-state Markov chain model for the transitions, such as shown in (a).

Figure 5.

(a) Power in the DITO frequency range, DF = (2τ)⁻¹, as a function of the variance, σ², of the input noise intensity. Parameters are the same as in Fig. 3 with τ = 6 and σ² = 0.05. (b) Power in the forcing frequency, ω, in the output of Eq. 4 as a function of σ for different choices of the time delay: τ = 0 (solid line), τ = 2 (dashed line), and τ = 8 (dotted line). These simulations were performed by adding an input K sin ωt to both neurons in Eq. 4 with K = 0.15, ω = 1, and τ = 6.

RESULTS

We develop a three-state Markov chain approximation to describe the dynamics of Eq. 1 as illustrated in Fig. 4b: two states correspond to the fixed-points and the other to the DITO-II (specifically the DITO whose amplitude crosses the separatrix). Since there is a single discrete delay, the probability that the system is in a given state at time t₁ can be deduced, at least in part, from knowledge of its state at t₁ – τ.^{10, 11} Thus we have a higher order Markov process.³⁸ In the discussion which follows we describe how to convert the higher order Markov process into a Markov chain model, namely, a stochastic process whose development can be treated as a series of transitions between states which have the property that the probabilistic law of the future development of the system, once it is in a given state, depends only on the state and not how it arrived in the state. This model contains three states: two correspond to the stable fixed-point attractors and are denoted as S₀ and S₂. The third state, S₁, corresponds to the DITO-II. The definition of the transition probabilities is also shown. It is necessary to consider a three-state model since DITOs can be formed starting from S₀ or S₁ with differing transition probabilities (p₀ ≠ p₂ and p₁₀ ≠ p₁₂).

The first step in passing from Eq. 1 to the Markov chain model is to subdivide the time series into intervals whose length represents the maximum time that the time delay has an effect. Figure 6a shows the partial autocorrelation function determined for Eq. 1 when τ = 8 and σ² = 0.1. The partial autocorrelation function is the autocorrelation between x(t) and x(t + k) with the linear dependence of x(t + 1) through to x(t + k − 1) removed, where k is the lag. Thus, the advantage of the partial autocorrelation function over the auto-correlation function is that it removes the linear effects of earlier dependencies (smaller lags) from the calculation of correlation of later values,³⁹ thus painting a more accurate picture of dependence on past states. As can be seen in Fig. 6a, the point at which no significant correlation is seen occurs at a lag between 2τ and 2.5τ. It follows that if we rescale time as t → 2.5τ so that one step of the Markov chain model corresponds to an interval of ≥2.5τ, then all dependencies will be contained within these intervals. Figure 6b shows that when the data is down-sampled so that the interval between successive points is 2.5τ that there is no significant correlation. Thus, under these conditions, we expect the Markov chain property to be satisfied.

Figure 6.

(a) The partial autocorrelation function, PACF, for Eq. 1. (b) PACF for the time series after down-sampling so that the interval between successive points was 2.5τ, where τ = 6. The horizontal dashed lines denote the P = 0.05 significance level which is equal to $\pm 1.96/\sqrt{N}$, where N is the number of data points. See text for discussion.

The second step is to identify the states and estimate the transition probabilities. For each choice of τ we estimated the separatrix numerically for Eq. 1 by changing the initial functions and then determining which attractor the solution converged to in the absence of noisy inputs. Next we determined the states by first dividing up the series into consecutive, non-overlapping intervals of length 2.5τ and then for each interval determined which basin of attraction the time series was confined. If the interval time series was confined to the basin of attraction of one of the stable fixed points, we classified it as belonging to either S₀ or S₂. On the other hand, if the time series contained segments that fell within both basins of attraction we classified as S₁. Figure 7 shows an example of this procedure. The transition probabilities were determined by setting the initial function to either one of the stable fixed points (states S₀ and S₂) and then integrating Eq. 1 for a sufficiently long time (t_final = 10 000 time steps) so that a steady state was achieved. This procedure was repeated 500 times. In each case the states were classified as above for consecutive 2.5τ intervals as a function of time and the transition probabilities were estimated from the pooled data.

Figure 7.

Example of the three-state filtering of a time series generated by Eq. 1 showing the duration of a single DITO. As can be seen intervals classified as S₁ correlate highly with oscillations in the output. The parameters are the same as in Fig. 2 with τ = 8 and σ² = 0.05.

Figure 8 shows the probability of remaining in S₁ in the next time step, p₁, as a function of τ when σ² = 0.05. This probability goes through a maximum when τ ∼ 3 − 4. The expected duration, T_i, of a state S_i with probability p_i of remaining in the state in the next time step is

$$E[T_I]=\sum _{j=1}^{\infty }j{p}_i{(1-p_i)}^{j-1}=\frac{1}{p_i}.$$ (5)

For the states S₀ and S₂ Eq. 5 defines the average resident time in that state, as well as the expected first passage time. With respect to S₁, Eq. 5 roughly defines the expected number of oscillations for a given transient. Remaining in S₁ for one time step in the Markov chain representation is equivalent to one oscillation in Eq. 1. For example, if p₁ = 0.5 then from Eq. 5 the expected number of oscillations is 1/(1 − 0.5) or 2 oscillations. Each time step in the Markov chain model is 2.5τ. Thus when τ = 1 the oscillation lasts 5 time steps and when τ = 10 to 25 time steps. Figure 9 shows that the distribution of the durations of S₁ measured from time series (method given in figure legend) when τ = 6 compares very well to that obtained from simulating the three-state Markov chain using the estimates we obtained for the transition probabilities. The agreement with the distribution of DITO duration times determined from simulation of Eq. 1 supports the validity of our procedure for constructing the Markov chain model.

Figure 8.

The estimated probability of remaining in the S₁ state, p₁, as a function of τ. The parameters are the same as in Fig. 2 with σ² = 0.05. The solid line represents the mean value obtained from 1000 realizations and the dashed lines are ± the standard error of the mean.

Figure 9.

Comparison of the distribution of S₁ durations predicted using the Markov chain approximation developed in the text (lines) versus the distribution estimated using time series generated from Eq. 1 (•). The solid line represents the mean value obtained from 1000 realizations and the dashed lines are ± the standard error of the mean. From (1) the S₁ states were identified by taking the discrete Fourier transform of sequential intervals and estimating S₁ duration by isolating the power in the DITO-II frequency of (2τ)⁻¹ above a preset threshold chosen to identify an oscillating state.

DISCUSSION

Here we have investigated the transient oscillations, namely DITO-IIs, that arise in bistable, time-delayed models of a two-neuron network that is tuned near the separatrix that separates two attractors. Our goal was to demonstrate that DITO-IIs can occur in the presence of random perturbations (“noise”). The surprising result was that it was possible to obtain some insight into the statistical properties of these transients. Whereas the analysis of nonlinear delay differential equations is typically a formidable task, their analysis in the presence of noise appears to be easier in certain contexts. This is because the autocorrelation function, a measure of the effect of the past on the future, decays quite rapidly and becomes negligible for lags ≥2.5τ. This observation makes it possible to use a Markov chain approximation to model the dynamics.

The application of a Markov chain approach to the study of SR in discrete models is often facilitated by using estimates of the transition probabilities obtained by either equating Kramer’s rate with the theoretical switching rate or by choosing probabilities proportional to the height of the potential barrier.^{10, 11, 40} However, Eq. 1 corresponds to a three-state Markov chain model, and it does not possess a potential function (Appendix). Consequently it was necessary to estimate the transition probabilities using numerical simulations. However, all of the required information for the Markov chain approximation can readily be estimated from numerical simulations, and by implication, experimentally. In particular, no knowledge of the strength of the inhibition, F, was required.

The possibility that the dynamics of biological systems functioning in real world environments are dominated by transients has long attracted interest. Recently emphasized examples arise in the context of neural encoding,^{41, 42} epileptic neural populations,^{43, 44, 45, 46} tolerance in immune systems⁴⁷ and transient growth, or reactivity, in ecosystems.^{48, 49} Historically chaotic systems attracted interest because time series were transiently dominated by intervals in which dynamics appeared almost periodic.^{50, 51, 52} An alternate hypothesis was that the transient nature of human physiological dynamics reflected the operation of discontinuous control mechanisms.² Discontinuous feedback appears to be mandatory for control in the setting of noisy perturbations and time delay.^{53, 54} Finally, long duration transients arise as attractors appear and disappear.^{55, 56} DITO-IIs are a particularly important type of transient behavior since they are anticipated to occur whenever one attractor is gradually replaced by another. This observation may, at least in part, explain the observation that “micro-seizures” can be recorded from the cortex of patients with and without a history of epilepsy.⁴⁴ Moreover, it has recently been shown the intermittent dynamics suggestive of DITOs can arise in a spatially extended neural mass model containing both slow and fast time scale inhibition.⁴⁵

Prediction of certain medical emergencies, e.g., epileptic seizures, is as valuable as it is elusive. Typically clues for prediction have been sought by examining the changes in the variables that characterize the medical emergency, e.g., the EKG for cardiac arrhythmias, the EEG for epileptic seizures, and so on. At first sight this appears to be a hopeless task given the large dynamical space that must be searched. Thus an important first strategy is to search for clinical situations which reduce the size of the temporal search space and the complexity of the state space during which it must be performed. NFLE is unique among the various epileptic syndromes given the almost exclusive occurrence of seizures during sleep and in particular during sleep stage transitions (e.g., Stage I to Stage II). Our observations support the relevance of bistability for understanding seizure onset in NFLE. However, unlike previous suggestions^{27, 28, 31, 32, 33} we suggest that seizure occurrence is not associated with a switch between two attractors, i.e., a seizure and a non-seizing state, but rather by the occurrence of transient behaviors that arise in association with the barrier, or separatrix, that separates two or more co-existing attractors. The required bistable states may themselves represent transient phenomena evolving on slower time scales that arise as attractors are gradually replaced by others. The hope is that by studying NFLE it will be possible to extend our knowledge of the dynamics of brain bistability in such a way that it will be possible to not only to devise strategies to benefit its sufferers, but also tools that can be applied to study the predict other types of seizures as well.

APPENDIX: NON–EXISTENCE OF POTENTIAL FUNCTION FOR EQ. 1(1)

The existence of a potential function for Eq. 1 requires that

$$\frac{\partial f_1}{\partial y}=\frac{\partial f_2}{\partial x},$$

where

$$\begin{array}{c}\multicolumn{1}{c}{f_1=-x(t)-F_2(y(t))+I_1(t)f_2=-y(t)-F_1(x(t))+I_2(t).}\end{array}$$

We obtain

$$\frac{\partial f_1}{\partial y}=-\frac{2c_2{\theta }_2^{2}y}{{({\theta }_2^2+y^2)}^2},$$

$$\frac{\partial f_2}{\partial x}=-\frac{2c_1{\theta }_1^{2}x}{{({\theta }_1^2+x^2)}^2}.$$

Since c_i > 0 and θ_i > 0 we see that a potential function does not exist.