^{1,a)}, Ivan Osorio

^{2}, Toru Ohira

^{3,b)}and John Milton

^{3,c)}

### 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* → 2*sτ*, 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?

^{1–3}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).

^{7–12}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.

J.M. acknowledges a long friendship with Frank Moss without whose encouragement and support much of the early work of J.M. on delays and noise, especially with Jennifer Foss, would never have been undertaken. We acknowledge useful discussions with S. A. Campbell and Peter Jung and support from the William R. Kenan, Jr. Foundation (JM), the National Sciences Foundation (Grant No. NSF-1028970 (J.M. and T.O.) and UBM-0634592 (A.Q.)) and the National Institutes of Health (Grant No. NIH/NINDS-5R21NS5056022 (A.Q. and I.O.)).

I. INTRODUCTION

II. BACKGROUND

III. RESULTS

IV. DISCUSSION

### Key Topics

- Attractors
- 28.0
- Markov processes
- 14.0
- Epilepsy
- 7.0
- Time series analysis
- 6.0
- Stochastic processes
- 5.0

## Figures

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

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

Determination of fixed-points of Eq. (1) for different values of the parameters *θ* _{1} and *θ* _{2}. The solid line corresponds to Eq. (2) and the dashed line to Eq. (3). Parameters: *c* _{1} = 0.4, *c* _{2} = 0.6, *I* _{1} = 0.5, *I* _{2} = 0.4, and *n* _{1} = *n* _{2} = 2.

Determination of fixed-points of Eq. (1) for different values of the parameters *θ* _{1} and *θ* _{2}. The solid line corresponds to Eq. (2) and the dashed line to Eq. (3). Parameters: *c* _{1} = 0.4, *c* _{2} = 0.6, *I* _{1} = 0.5, *I* _{2} = 0.4, and *n* _{1} = *n* _{2} = 2.

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: *c* _{1} = 0.4, *c* _{2} = 0.6, *I* _{1} = 0.5, *I* _{2} = 0.4, *n* _{1} = *n* _{2} = 2, *θ* _{1} = *θ* _{2} = 0.2.

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: *c* _{1} = 0.4, *c* _{2} = 0.6, *I* _{1} = 0.5, *I* _{2} = 0.4, *n* _{1} = *n* _{2} = 2, *θ* _{1} = *θ* _{2} = 0.2.

(a) A single realization of a switch between one fixed-point attractor (*S* _{0}) and the other (*S* _{2}) with an intervening DITO-II (*S* _{1}) 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 ((2*τ*)^{−1}), DF Power, was determined from Eq. (1). Parameters are the same as in Fig. 3 with *τ* = 6 and *σ* ^{2} = 0.05. (b) A three-state Markov chain model for the transitions, such as shown in (a).

(a) A single realization of a switch between one fixed-point attractor (*S* _{0}) and the other (*S* _{2}) with an intervening DITO-II (*S* _{1}) 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 ((2*τ*)^{−1}), DF Power, was determined from Eq. (1). Parameters are the same as in Fig. 3 with *τ* = 6 and *σ* ^{2} = 0.05. (b) A three-state Markov chain model for the transitions, such as shown in (a).

(a) Power in the DITO frequency range, *DF* = (2*τ*)^{−1}, as a function of the variance, *σ* ^{2}, of the input noise intensity. Parameters are the same as in Fig. 3 with *τ* = 6 and *σ* ^{2} = 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.

(a) Power in the DITO frequency range, *DF* = (2*τ*)^{−1}, as a function of the variance, *σ* ^{2}, of the input noise intensity. Parameters are the same as in Fig. 3 with *τ* = 6 and *σ* ^{2} = 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.

(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 2.5*τ*, where *τ* = 6. The horizontal dashed lines denote the *P* = 0.05 significance level which is equal to , where *N* is the number of data points. See text for discussion.

(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 2.5*τ*, where *τ* = 6. The horizontal dashed lines denote the *P* = 0.05 significance level which is equal to , where *N* is the number of data points. See text for discussion.

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 *S* _{1} correlate highly with oscillations in the output. The parameters are the same as in Fig. 2 with *τ* = 8 and *σ* ^{2} = 0.05.

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 *S* _{1} correlate highly with oscillations in the output. The parameters are the same as in Fig. 2 with *τ* = 8 and *σ* ^{2} = 0.05.

The estimated probability of remaining in the *S* _{1} state, *p* _{1}, as a function of *τ*. The parameters are the same as in Fig. 2 with *σ* ^{2} = 0.05. The solid line represents the mean value obtained from 1000 realizations and the dashed lines are ± the standard error of the mean.

The estimated probability of remaining in the *S* _{1} state, *p* _{1}, as a function of *τ*. The parameters are the same as in Fig. 2 with *σ* ^{2} = 0.05. The solid line represents the mean value obtained from 1000 realizations and the dashed lines are ± the standard error of the mean.

Comparison of the distribution of *S* _{1} 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 *S* _{1} states were identified by taking the discrete Fourier transform of sequential intervals and estimating *S* _{1} duration by isolating the power in the DITO-II frequency of (2τ)^{−1} above a preset threshold chosen to identify an oscillating state.

Comparison of the distribution of *S* _{1} 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 *S* _{1} states were identified by taking the discrete Fourier transform of sequential intervals and estimating *S* _{1} duration by isolating the power in the DITO-II frequency of (2τ)^{−1} above a preset threshold chosen to identify an oscillating state.

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