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.
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 ((2τ)−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 = (2τ)−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 2.5τ, 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.
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.
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