Encyclopedia of Computational Neuroscience

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Delay-Induced Transient Oscillation (DITO) and Metastable Behavior

  • Clodoaldo Grotta-Ragazzo
  • Coraci P. MaltaEmail author
  • Khashayar Pakdaman
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_100678-1


Delay-induced transient oscillations (DITOs) are oscillations that occur in bi- or multi-stable system of delayed differential equations as it converges to one of its stable equilibria, oscillations that do not exist when the delay vanishes.

Metastable oscillations are transient oscillatory patterns (or states) that attract neighboring initial conditions and last for a “long time” almost without change. These oscillatory patterns seem to be stable: to detect their instability, it is necessary to observe the system over a seemingly endless time window. Such long-lasting transients would be undistinguishable from (nearly) periodic solutions. A mathematical definition of metastability needs a quantification of “long time”. The most common approach is to consider a one-parameter family of systems and to compare the transient time duration for different values of the parameter. For instance, if ε > 0 is a real parameter, then an oscillatory pattern is said to be metastable if the time duration of the transient oscillations grows exponentially as ε → 0.

Long-lasting DITOs could be very important in understanding the dynamics of the system.

Detailed Description

Delays are ubiquitous and have many different sources. In neurons the delay corresponds to the time it takes for a signal to be transmitted (see entry “Time-Delayed Neural Networks: Stability and Oscillations”). Experimental studies of the behavior of self-connected single neurons have shown that the discharge pattern of biological neurons (Vibert et al. 1979; Diez-Martínez and Segundo 1983) can be influenced by the delay. Different ranges of delay correspond to different patterns of neural activities (an der Heiden 1981; Plant 1981; Chapeau-Blondeau and Chauvet 1992; Lourenço and Babloyantz 1994; Vibert et al. 1994; Menon 1995; Hangartner and Cull 1995; Pakdaman et al. 1996).

In some artificial neural network (ANN) applications, such as content addressable memories, information is stored as stable equilibrium points of the system. Retrieval occurs when the system is initialized within the basin of attraction of one of the equilibria and the network is allowed to stabilize in its steady state (Hopfield 1984; Hirsch 1989; Foss et al. 1996). Delayed interunit transmissions may render some equilibria unstable, thus deteriorating information retrieval. Many studies have derived conditions that ensure that the asymptotic behavior of networks with and without delay is similar. (i) The system with delay and the system without delay should have exactly the same stable equilibria (the delay does not alter local stability). (ii) The system without delay and the system with delay are both almost convergent, so that the delay does not induce spurious stable undamped oscillations. Constraints (i) and (ii) avoid delay-induced changes in the asymptotic behavior of the system. Yet, even under such constraints, the delay may still influence important features in the system’s dynamics. For instance, the delay can alter the boundary of the basins of attraction of the stable equilibrium points of the network as shown in Pakdaman et al. (1998a). Moreover, the delay can also produce transient oscillations that end up vanishing with time. In Pakdaman et al. (1997a, 1998a), it was studied the influence of the delay on the behavior of a network composed of graded-response neurons (GRNs) forming a ring where each unit is connected unidirectionally to the next one. The GRN has constant decay rate, and its output is a sigmoidal function (for instance, tanh function) that depends on its activation. The connection is said to be excitatory, when the sigmoidal function is increasing, and it is said to be inhibitory when it is decreasing. The GRN ring network satisfies the constraints (i) and (ii), so the system will converge to one of its equilibrium points if the interunit connections are instantaneous or delayed. However, the transient regime of the system with instantaneous interunit transmission greatly differs from that of the system with delayed interunit transmission. In the case of a pair of interconnected GRNs sketched in Fig. 1, when the connections are instantaneous, the system evolves rapidly (without oscillations) to the equilibrium which basin of attraction contains the initial condition. The smaller the decay rate (charge-discharge time) of the neurons, the faster it will reach the equilibrium: the time evolution of the neuron activation x, y, represented by the solid line in Fig. 1 (graph on the right), corresponds to a pair of neurons with decay rate smaller than the decay rate of the pair of neurons corresponding to the dashed lines in Fig. 1 (thick dashed line represents activation x; thin dashed line represents activation y).
Fig. 1

Left: sketch of the system formed by two interconnected GRN (I1 and I2 are external inputs). Right: temporal evolution of the neuron activation x, y in the case of identical neurons (I1 = I2) with instantaneous connections. The smaller the decay rate (charge-discharge time) of the neurons (solid line), the faster it will reach the equilibrium

When the connections are delayed, the GRN activation will oscillate before converging to the equilibrium point as shown in Fig. 2 for two values of the delay. As the delay increases, the longer is the duration of the oscillations of the activations x and y. Also, as displayed in Fig. 2, the shape of the transient oscillations assumes a square-wave-like shape for large delays, the square-wave-like oscillations outlasting the observation window. Only the a priori knowledge that the point is in the basin of attraction of one of the equilibria allows us to state that these oscillations are transient. In Pakdaman et al. (1998b), the term DITO was coined for this phenomenon both as an acronym for “delay-induced transient oscillation” and as a means to emphasize its repetitive nature.
Fig. 2

Transient oscillations when the interunit connections are delayed. The larger the delay, the longer the duration of the transient. Time is measured in units of delay. Left: thick and thin lines represent the activations x and y, respectively. Right: larger delay, only the activation x is displayed

A ring network of GRN is inhibitory only if it has an odd number of inhibitory connections. Therefore, a pair of two interconnected GRNs is excitatory when both connections are excitatory (like in Pakdaman et al. 1997a, 1998b) or both inhibitory (like in Milton et al. 2010). So the time evolution of the activations x and y, displayed in Figs. 1 and 2, describe the behavior of a pair of reciprocally excitatory neurons or a pair of reciprocally inhibitory neurons.

Experimentalists have long recognized the importance of transients in neural behavior as a means to convey information about environmental as well as internal changes (e.g., Segundo et al. 1994). Long-lasting DITOs could strongly alter the dynamics of the system and, for instance, interfere with information retrieval in neural networks. The information contained in the transient regime is all the more important when the system evolves in rapidly changing environments where the neural networks involved in information processing do not dispose of the time lapse necessary to reach a stationary regime. Overall oscillatory patterns are frequently observed in the activity of nervous systems. Their roles are either clear as in respiration, where they control motor activity, or unexplained as in the electroencephalogram where they are apparently related to the brain information processing in a still obscure way. Oscillatory patterns are observed when units discharge periodically and synchronously. In living nervous systems, delays are ubiquitous, ranging from a few to several hundreds of milliseconds. They are due to action potential propagation along axons, synaptic delay, etc. Delay-induced long-lasting transient oscillations could thus take part in various nervous system operations: epileptic seizure, for instance, could be due to a DITO (see Milton et al. 2017 and entry “Dynamic Diseases of the Brain Dynamics of disease states: Overview”). In ANN applications relying on the convergence of the network to a steady state, control over the transient regime is also an important issue. Large increase in the transient regime duration can seriously deteriorate the network performance by slowing down the system.

Example 1

Consider the following system of a single delayed differential equation
$$ \frac{da}{dt}(t)=-a(t)+\tanh \left[2a\left(t-\tau \right)\right], $$
where the delay τ ≥ 0 is constant. This equation is used to describe a single neuron with delayed self-excitatory positive-feedback connection. This equation has three equilibria: a = a < 0, a = 0, and a = a+ > 0. If τ = 0 then the equation is an ordinary differential equation without delay. In this case, if the initial condition a(0) = a0 is strictly negative (positive), then the solution converges to a (a+). So, all solutions but a = 0 converge to either a or a+, and the system is said to be bistable. The equilibrium point a = 0 is the boundary of the basin of attraction of a and a+. If τ > 0, the initial condition is a continuous function a0: [−τ, 0] R that gives the history of the variable a up to time −τ (beyond −τ past cannot affect future). In this case the equation has a delay, and the space of initial conditions is infinite dimensional. It is possible to show (see Pakdaman et al. 1997b and references therein) that the system with delay is still bistable and most solutions converge to either equilibrium a or equilibrium a+. The difference to the case τ = 0 is that the boundary of the basins of attraction of a and a+ is an infinite dimensional set in the space of initial conditions. Any solution that starts at the basin boundary must oscillate forever either with a vanishing amplitude, as the solution tends to the unstable equilibrium a = 0, or with a finite amplitude, as the solution tends to an unstable periodic orbit which is in the basin boundary (see illustration in the Fig. 3). DITOs appear whenever the initial condition a0 changes sign within the interval [−τ, 0] in a way that a0 is close to but not in the basin boundary. These are the delay-induced transient oscillations (DITOs).
Fig. 3

Schematic phase portrait representation of the boundary of the basins of attraction of the delayed self-excited neuron (Eq. 1). The gray surface represents the boundary separating the basins of attraction of the two stable equilibria r1 = a and r3 = a+. The boundary contains the unstable equilibrium r2 = 0, together with an unstable closed orbit. Some solutions are transiently attracted to the oscillatory solutions on the boundary before approaching one of the stable equilibria r1 = a, r3 = a+

Example 2

The space of initial conditions of Eq. (1), a0: [−τ, 0] R, changes with the delay parameter τ. When the goal is to analyze the dependence of the solution on the delay, it is convenient to fix the space of initial conditions to a0: [1, 0] R by rescaling time as t = τs. Eq. (1) becomes
$$ \varepsilon \frac{da}{ds}(s)=-a(s)+\tanh \left[2a\left(s-1\right)\right], $$
where ε = 1/τ > 0. The time duration of the transient oscillations of Eq. (2) increases as ε decreases (τ increases). These oscillations last for times of order exp(c/ε), for some c > 0, as ε → 0 (see Pakdaman et al. 1997b; Grotta-Ragazzo et al. 2010), so they are said to be metastable. In terms of the original time t, the metastable oscillatory pattern persists for a time of the order exp(). When ε is small enough, the metastable oscillatory pattern is close to a square- wave function (see Fig. 2, right panel).

DITO and the Boundary of the Basins of Attraction: Mathematical Tools

DITOs appear in various systems of delayed differential equations: one equation (Pakdaman et al. 1997b; Pakdaman and Malta 1998), two equations (Pakdaman et al. 1998b), and many equations (see Pakdaman et al. 1997a and entry “Time-Delayed Neural Networks: Stability and Oscillations”). In all these systems, almost all initial condition leads to a solution that converges to a stable equilibrium point. DITOs are generated by initial conditions close to the boundary of the basins of attraction, as in the case of Eq. (1). So, the geometric understanding of DITO is related to the geometric understanding of the boundaries of the basins of attraction of the equilibria. The boundary of the basins of attraction of simple systems with a few stable equilibria may be very complicated. (For instance, for almost all initial condition, Newton’s method converges to one of the three roots of the complex polynomial z3 1 = 0. The boundary of the basin of attraction of each attracting fixed point has the “Wada property,” namely, a point is in the boundary of one of the basins of attraction if, and only if, it is simultaneously in the other two basin boundaries.) So it is remarkable that it is possible to geometrically describe the boundary of the basin of attraction of delay differential equations (Pakdaman et al. 1997b, c, 1998a, 1999).

The mathematical property that allows for this description is the monotonicity of the semiflows generated by the equation. This property means that Eq. (2) (see Pakdaman et al. 1997b) is order preserving: if a0: [1, 0] R and \( {\widehat{a}}_0 \): [1, 0] R are two different continuous initial conditions such that \( {\widehat{a}}_0 \) (η) ≥ a0(η) for η ∈ [1, 0], then for s > 2 the solutions \( \widehat{a} \) and a satisfy \( {\widehat{a}}_0 \) (s + η) > a(s + η) for all η ∈ [1, 0]. This is illustrated in Fig. 4. This order-preserving property implies that for a given continuous function a0: [1, 0] R, the initial condition a0 + c, where c is a real constant, generates a solution to Eq. (2) that converges to the equilibrium a+, if c is sufficiently large, or converges to a, if c is sufficiently small. Moreover, for a unique value of c, it converges neither to a nor to a+ but remains oscillating in the boundary of the basin of attraction of a+ and a. So the basin boundary is a subset of the set of initial conditions that is transversal to the one-dimensional subspace of constant functions in [1, 0]. Similar ideas are used to describe the boundary of the basins of attraction for systems with two or more equations.
Fig. 4

Illustration of the order-preserving property

Are DITOs Always Metastable?

The answer is no. DITOs seem to appear very often in multi-stable delay equations, but the oscillations do not necessarily persist for a time t of the order of exp(β), for some constants c > 0 and β > 0, as τ → ∞. The mathematical question of existence of metastable solutions is delicate as the following examples show. Consider a scalar equation of the form
$$ \varepsilon \frac{da}{ds}(s)=-a(s)+f\left(a\left(s-1\right)\right). $$

Let us assume that

f (a) = l if a < 0 (left value), f (0) = 0, f (a) = r if a > 0, (right value).

If l < 0 and r > 0 (positive feedback), then the equation is bistable with two attracting equilibria a = l and a = r. Since f is a piecewise constant, function an explicit computation (Grotta-Ragazzo et al. 1999) shows that DITOs are metastable if, and only if, −l = r. If l > 0 and r < 0 (negative feedback), then the equation has a single attracting periodic orbit, which attracts most initial conditions, with period p(ε) > 2 and exactly two sign changes within a period (a “slowly oscillating periodic orbit”). In this case, DITOs are transient oscillatory solutions with more than two sign changes in a time interval of length 2. In contrast to the positive-feedback case, in the negative-feedback case, metastable DITOs always exist regardless of the values of l and r. When f is a smooth function, the mathematical analysis of the existence of metastable oscillations for Eq. (3) is difficult. In Grotta-Ragazzo et al. (2010), using “transition layer equations” (Chow and Mallet-Paret 1983), in the positive-feedback case \( \left(\frac{df}{da}>0\right) \), it was possible to show that odd symmetry of the positive-feedback f is a sufficient condition for the existence of metastable oscillations. However, the oddness of positive-feedback f is not a necessary condition for metastability as there are positive-feedback functions f that are not odd but satisfy instead a condition formulated in terms of transition layer solutions, for which metastable oscillations occur. In the negative-feedback case \( \left(\frac{df}{da}<0\right) \), metastable oscillations always exist (see Grotta-Ragazzo et al. 2010 for details), provided a “slowly oscillating periodic orbit” exists.

Metastable oscillations were detected not only in scalar equations but also in systems (Pakdaman et al. 1997a, b, 1998b). In all these works, the existence of metastable oscillations was shown by both numerical investigation and theoretical analysis using symmetries of the equations. For scalar equations or systems, if f is smooth or not, the short-time stability of metastable oscillatory patterns is a consequence of the behavior of the system in the singular limit ε = 0 (the analysis of the singular limit is a classical idea present in several previous results, for instance, Chow and Mallet-Paret 1983; Ikeda and Matsumoto 1987; Sharkovsky et al. 1993).

In physics, metastable behavior is mostly related to phase transitions as some parameter (for instance, temperature) crosses some threshold value (Fusco and Hale 1989). In a bistable system, the phase transition value separates parameter regions where either one of the stable states dominates. In multi-stable complex networks modeled by differential equations with delay, the analysis of transient time duration may be an important and practical tool to detect important global changes in the system due to variation of a parameter that is not the delay. For instance, for the piecewise constant f with fixed l < 0 and varying r > 0, metastable oscillations occur at r = −l, where the system has a reflection symmetry that interchanges the two equilibria. As the analysis in the positive feedback shows (Grotta-Ragazzo et al. 2010), at a phase transition threshold, the system does not have to exhibit an explicit symmetry like the oddness of the function f.


DITOs seem to appear quite often in multi-stable systems of delayed differential equations, for constant delay, and also for state-dependent delay (Pellegrin et al. 2014). The transient oscillations occur for initial conditions close to the boundary of the basins of attraction. The time duration of transient oscillations may be very long and, depending on the system, may increase exponentially with increasing delay. In this case DITOs could be misidentified as stable oscillations. Metastable patterns may appear at a particular value of a parameter of the system, and this can be an indication of a dynamical “phase transition” at that value.



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Copyright information

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Authors and Affiliations

  • Clodoaldo Grotta-Ragazzo
    • 1
  • Coraci P. Malta
    • 2
    Email author
  • Khashayar Pakdaman
    • 3
  1. 1.Instituto de Matemática e Estatística, Universidade de São PauloSão PauloBrazil
  2. 2.Instituto de Física, Universidade de São PauloSão PauloBrazil
  3. 3.Institut Jacques Monod, Univ Paris DiderotParisFrance

Section editors and affiliations

  • John Milton
    • 1
  1. 1.W.M. Keck Science CenterThe Claremont CollegesClaremontUSA