 Research
 Open Access
Wave Generation in Unidirectional Chains of Idealized Neural Oscillators
 Bastien Fernandez^{1} and
 Stanislav M. Mintchev^{2}Email author
https://doi.org/10.1186/s134080160037x
© Fernandez and Mintchev 2016
 Received: 4 December 2015
 Accepted: 29 March 2016
 Published: 8 April 2016
Abstract
We investigate the dynamics of unidirectional semiinfinite chains of typeI oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the fardownstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems. Here, we give a full mathematical proof of generation under uniform forcing in a simple piecewise affine setting for which the dynamics can be solved explicitly. In particular, our analysis proves existence, global stability, and robustness with respect to perturbations of the forcing, of families of waves with arbitrary period/wave number in some range, for every value of the parameters in the system.
Keywords
 Nonlinear waves
 Forced feedforward chains
 Coupled oscillators
 Type I neural oscillator
1 Introduction
Signal propagation in the form of waves is a ubiquitous feature of the functioning of neural networks. Waves transmitting electrical activity across neural structures have been observed in a large variety of situations, both in artificially grown cultures and in living brain tissues, see e.g. [1] and [2] for an instance of each case; many other examples can be found in the literature.
This experimental phenomenology has fostered numerous computational and analytical studies on theoretical models for wave propagation. To mention a single category of exact results, one can cite proofs of existence of waves with context dependent shape: fronts, pulses, periodic wave trains, etc., both in full voltage/conductance models and in firing rate models; see e.g. [3–5]. In parallel, numerical studies have investigated propagation features such as firing synchrony within cortical layers, and their dependence on dynamical ingredients: feedback, surrounding noise, or external stimulus; see for instance [6–9].
Our paper aims to develop a rigorous mathematical investigation of how the global dynamics of a (simple model of a) neural network may cause it to organize to a wave behavior, in spite of being forced by a rather unrelated signal. Given that the natural setting of neural ensembles typically features an external environment that is prone to providing an array of irregular stimuli, it seems that such forcing is in no way a priori tailored toward generating periodic patterns in layered ensembles. Nevertheless, recordings from tissues suggest that selforganization will often ensue despite this obvious and inherent mismatch.
In order to get insight into the generation of periodic traveling waves through ad hoc stimulus, we consider unidirectional chains of coupled oscillators. Inspired by the propagation of synfire waves through cortical layers [10], such systems can be regarded as basic phase variable models of feedforward networks featuring synchronized groups, in which each pool can be treated as a phase oscillator that repetitively alternates a refractory period with a firing burst. In addition, chains with unidirectional coupling as in Eq. (1) below are representative of some physiological systems, such as central pattern generators [11]. Also, acyclic chains of typeI oscillators have been used as simple examples for the analysis of network reliability [12].

\(\epsilon,\omega\in \mathbb {R}^{+}\). Up to a rescaling of time, we can always assume that \(\omega=1\).

Δ is the socalled phase response curve (PRC). Recall that a typeI oscillator is one for which the PRC is a nonnegative onehumped function [13].

δ mimics incoming stimuli from the preceding node and also takes the form of a unimodal function.

The first oscillator at site \(s=0\) evolves according to some forcing signal f (external stimulus), i.e. we have \(\theta_{0}(t)=f(t)\) for all \(t\in \mathbb {R}^{+}\). The forcing is assumed to be continuous, increasing, and periodic (viz. there exists \(\tau\in \mathbb {R}^{+}\) such that \(f(t+\tau)=f(t)+1\) for all \(t\in \mathbb {R}^{+}\)).
This behavior was somehow unanticipated because (1) does not reveal any crucial properties usually required in the proofs of wave existence, such as the monotonicity of the profile dynamics (analogous property to the maximum principle in parabolic PDEs); see e.g. [16–18] for lattice differential equations and [19–21] for discrete time recursions.^{2} In this context, proving the existence of waves remains unsolved and so is the stability problem, not to mention any justification of the generation phenomenon when forcing with ad hoc signal. Notice, however, that, by assuming the existence of waves and their local stability for the singlesite dynamics, a proof of stability for the whole chain has been obtained and applied to the design of numerical algorithms for the doubleprecision construction of wave shapes [15]. (Our stability proof here is inspired by this one.)
In order to get mathematical insights into wave generation under ad hoc forcing, here, we analyze simple piecewise affine systems for which the dynamics can be solved explicitly. This analysis can be viewed as an exploratory step in the endeavor of searching for full proofs in (more general) nonlinear systems. Hence, the functions Δ and δ are both assumed to be piecewise constant on the circle, taking on only the two distinct values of 1 (‘on’) or 0 (‘off’).
In this setting, our analysis shows that the numerical phenomenology can be mathematically confirmed. For all parameter values, we prove the existence of TW with arbitrary period in some interval, and their global stability with respect to initial perturbations in the phase space \(\mathbb {T}^{\mathbb {N}}\), not only when the forcing at \(s=0\) is chosen to be a TW shape but also for an open set of periodic signals with identical period. In addition, this open set is shown to contain uniform forcing \(f(t)=t/\tau\) provided that the coupling intensity ϵ is sufficiently small.
The paper is organized as follows. The next section contains the accurate definition of the initial value problem, the basic properties of the associated flow and the statements of the main results. The rest of the paper is devoted to proofs. In Sect. 3, we prove the existence of TW by establishing an explicit expression of their shape. We study the TW stability with respect to initial conditions in Sect. 4 by considering the associated stroboscopic dynamics, first for the first site, and then for the second site, from where the stability of the full chains is deduced. Finally, stability with respect to changes in forcing is shown in Sect. 5, as a byproduct of the arguments developed in the previous sections. Section 6 offers some concluding remarks.
2 Definitions, Basic Properties, and Main Results

the forcing signal f is assumed to be a Lipschitzcontinuous, τperiodic^{4} (\(\tau>0\)), and increasing function with slope (wherever defined) at least 1,^{5} and satisfying \(f(0)=0\),

\(\{\vartheta_{s}\}_{s\in \mathbb {N}}\in \mathbb {T}^{\mathbb {N}}\) is an arbitrary initial configuration,

\(\epsilon>0\) and \(a_{0},a_{1}\in(0,1)\) are arbitrary parameters.
Solutions of Eq. (2) are in general denoted by \(\{\theta _{s}^{(f)}(t)\}_{s\in \mathbb {N}}\) but the notation \(\{\theta_{s}^{(f)}(\vartheta _{1},\ldots,\vartheta_{s},t)\}_{s\in \mathbb {N}}\) is also employed when the dependence on initial condition needs to be explicitly mentioned.
2.1 Basic Properties
The solutions of Eq. (2) have a series of basic properties which we present and succinctly argue in a rather informal way. These facts can be formally established by explicitly solving the dynamics. The details are left to the reader.
2.1.1 Existence of the Flow
Given any forcing signal f and any initial condition \(\{\vartheta_{s}\}_{s\in \mathbb {N}}\), for every \(s\in \mathbb {N}\), there exists a unique function \(t\mapsto\theta_{s}^{(f)}(t)\) which satisfies Eq. (2) for all \(t\in \mathbb {R}^{+}\). This function \(\theta_{s}^{(f)}\) is continuous, increasing, and piecewise affine with alternating slope in \(\{1,1+\epsilon\}\). Moreover, each piece of slope 1 must have length ≥ \(1a_{1}\).
These facts readily follow from solving the dynamics inductively down the chain. Assuming that \(t\mapsto\theta_{s}^{(f)}(t)\) is given for some \(s\in \mathbb {N}\) (or considering the forcing term f if \(s=0\)), the slope of the first piece of \(t\mapsto\theta_{s+1}^{(f)}(t)\) only depends on the relative position of \(\vartheta_{s+1}\) with respect to \(a_{1} \ (\mathrm{mod}\ 1)\) and of \(\vartheta_{s}\) with respect to \(a_{0} \ (\mathrm{mod}\ 1)\). The length of this piece depends on its slope, on \(\vartheta_{s+1}\) and on the smallest \(t>0\) such that \(\theta _{s}(t)\in\{0,a_{0}\} \ (\mathrm{mod}\ 1)\); this infimum time has to be positive. By induction, this process generates the whole function \(\theta_{s+1}^{(f)}\) by using the location of \(\theta_{s}\) and \(\theta _{s1}\) at the end of each piece, and the next time when \(\theta_{s}\in \{0,a_{0}\} \ (\mathrm{mod}\ 1)\).
2.1.2 Continuous Dependence on Inputs
Endow \(\mathbb {T}^{\mathbb {Z}^{+}}\) with a pointwise topology and, given \(T>0\), endow continuous and monotonic functions of \(t\in[0,T]\) with a uniform topology and norm \(\\cdot\ \). For every \(s\in \mathbb {N}\) and \(T>0\), the quantity \(\\theta _{s}^{(f)}_{[0,T]}\\) continuously depends both on the forcing signal \(f_{[0,T]}\) and on the initial condition \(\{\vartheta_{s}\}_{s\in \mathbb {N}}\).
Indeed, if two forcing signals f and g are close, then the lower bound on their derivatives implies that the respective times in \([0,T]\) at which they reach \(\{0,a_{0}\} \ (\mathrm{mod}\ 1)\) must be close. If, in addition, the initial conditions \(\vartheta_{1}\) and \(\xi_{1}\) are close, then the trajectories \(t\mapsto\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(t\mapsto\theta_{1}^{(g)}(\xi_{1},t)\) alternate their slopes at close times; hence \(\\theta_{1}^{(f)}(\vartheta_{1},\cdot )_{[0,T]}\theta_{1}^{(g)}(\xi_{1},\cdot)_{[0,T]}\\) must be small. Since the slopes are at least 1, the respective times at which \(\theta _{1}^{(f)}(\vartheta_{1},\cdot)_{[0,T]}\) and \(\theta_{1}^{(g)}(\xi _{1},\cdot)_{[0,T]}\) reach \(\{0,a_{0}\} \ (\mathrm{mod}\ 1)\) are close. By repeating the argument, we conclude that \(\\theta _{2}^{(f)}(\vartheta_{1},\vartheta_{2},\cdot)_{[0,T]}\theta_{2}^{(g)}(\xi _{1},\xi_{2},\cdot)_{[0,T]}\\) must be small when \(\vartheta_{2}\) and \(\xi_{2}\) are sufficiently close. Then the result for an arbitrary \(s\in \mathbb {N}\) follows by induction.
2.1.3 Semigroup Property
As suggested above, a large part of the analysis consists in focusing on the onedimensional dynamics of the first oscillator \(\theta_{1}\) forced by the stimulus f, prior to extending the results to subsequent sites. Indeed, the dynamics of the oscillator s can be regarded as a forced system with input signal \(\theta_{s1}\).
For the onedimensional forced system, letting \(R^{T}\theta(t)=\theta (t+T)\) for all \(t\in \mathbb {R}^{+}\) denotes the time translations, we shall especially rely on the ‘semigroup’ property of the flow, viz. if \(\theta_{1}^{(f)}(\vartheta_{1},\cdot)\) is a solution with initial condition \(\vartheta_{1}\) and forcing f, then, for every \(T\in \mathbb {R}^{+}\), \(R^{T}\theta_{1}^{(R^{T}f)}(\theta_{1}^{(f)}(\vartheta_{1},T),\cdot)\) is a solution with initial condition \(\theta_{1}^{(f)}(\vartheta_{1},T)\) and forcing \(R^{T}f\).
2.1.4 Monotonicity Failure
2.2 Main Results
As we shall see below, TW exist that are asymptotically stable, not only with respect to perturbations of the initial phases \(\{\vartheta _{s}\}\), but more importantly, also with respect to changes in the forcing signal. For convenience, we first separately state existence and uniqueness.
Theorem 2.1
(Existence.) For every ϵ, \(a_{0}\) and \(a_{1}\), there exists a (nonempty) interval \(I_{\epsilon,a_{0},a_{1}}\), and for every \(\tau\in I_{\epsilon,a_{0},a_{1}}\), there exist a τperiodic forcing signal f and \(\alpha\in(0,\tau)\) such that \(\{f(t+\alpha s)\}_{s\in \mathbb {N},t\in \mathbb {R}^{+}}\) is a TW.
 (C)
f is a piecewise affine forcing signal whose restriction \(f_{(0,\tau)}\) has slope \(1+\epsilon\) only on a subinterval whose left boundary is α.
For the proof, see Sect. 3, in particular Corollary 3.5. Notice that constraint (C) has no intrinsic interest other than unambiguously identifying appropriate TW shapes for the stability statement. To identify stable waves matters because the system (2) also possesses neutral and unstable traveling waves.
For stability, we shall use notions that are appropriate to forced systems, and adapted to our setting. In particular, since the information flow is unidirectional here, it is natural to only require that perturbations relax in pointwise topology, rather than in uniform topology. Therefore, we shall consider the dynamics on arbitrary finite collections of sites which, without loss of generality, can be chosen to be the first s sites, for an arbitrary s. Moreover, there is no need for local stability considerations here because we shall be concerned with TW for which the basin of attraction is as large as it can get from a topological viewpoint.
Theorem 2.2
There exists a (nonempty) subinterval \(I'_{\epsilon,a_{0},a_{1}}\subset I_{\epsilon,a_{0},a_{1}}\) such that for every \(\tau\in I'_{\epsilon ,a_{0},a_{1}}\), the TW \((f,\alpha)\) determined by constraint (C) is globally asymptotically stable.
See Fig. 3 for an illustration of this result. Theorem 2.2 is proved in Sect. 4; see especially the concluding statement Corollary 4.5. In addition, for initial conditions not satisfying the stability condition, our proof shows that the first coordinate s for which this condition fails asymptotically approaches an unstable periodic solution.
Theorem 2.3
In addition, for ϵ small enough (depending on \(a_{0}\), \(a_{1}\)), there exists a (nonempty) subinterval \(I''_{\epsilon,a_{0},a_{1}}\subset I'_{\epsilon,a_{0},a_{1}}\) such that, for every \(\tau\in I''_{\epsilon ,a_{0},a_{1}}\), the neighborhood U contains the uniform forcing \(g(t)=\frac{t}{\tau}\), for all \(t\in \mathbb {R}^{+}\).
Theorem 2.3 is established in Sect. 5 and implies in particular robustness with respect to forcing perturbations.
Corollary 2.4
For every \(\tau\in I'_{\epsilon,a_{0},a_{1}}\), the TW determined by the constraint (C) is robust with respect to perturbations of the forcing.
3 Existence of Traveling Wave Solutions
Since we assume \(f(0)=0\), the forcing signal/TW shape f can be entirely characterized by the partition of \([0,\tau]\) into intervals where the slope is constant. Without loss of generality, we can also assume that \(\alpha\in(0,\tau)\).^{6} Under these assumptions, there can only be four cases depending on the initial location \(f(\alpha)\) of \(\theta_{1}\) (above or below \(a_{1}\)) and the location \(f(\tau\alpha)\) of \(\theta_{0}\) (above or below \(a_{0}\)) when \(\theta_{1}\) reaches 1. By examining each case, one easily constructs the desired partition.
Lemma 3.1
 (a):

\(f(\alpha)< a_{1}\) and \(f(\tau\alpha)\geq a_{0}\). Then there exists \(\sigma\in(0,\tau)\) such that we have for the TW coordinate at site \(s=1\)$$\frac{d\theta_{1}^{(f)}}{dt}= \textstyle\begin{cases} 1+\epsilon&\textit{if } 0< t< \sigma, \\ 1&\textit{if } \sigma< t< \tau. \end{cases} $$
 (b):

\(f(\alpha)\geq a_{1}\) and \(f(\tau\alpha)< a_{0}\). Then there exists \(\sigma\in(\tau\alpha,\tau)\) such that we have$$\frac{d\theta_{1}^{(f)}}{dt}= \textstyle\begin{cases} 1+\epsilon&\textit{if } \tau\alpha< t< \sigma, \\ 1&\textit{if } 0< t< \tau\alpha\textit{ or if } \sigma< t\leq\tau. \end{cases} $$
 (c):

\(f(\alpha)\geq a_{1}\) and \(f(\tau\alpha)\geq a_{0}\). Then \(\frac{d\theta_{1}^{(f)}}{dt}=1\) for all \(t\in[0,\tau]\).
 (c′):

\(f(\alpha)< a_{1}\) and \(f(\tau\alpha)< a_{0}\). Then there exist \(\sigma\in(0,\tau\alpha)\) and \(\nu\in(\tau\alpha ,\tau)\) such that we have$$\frac{d\theta_{1}^{(f)}}{dt}= \textstyle\begin{cases} 1+\epsilon&\textit{if } 0\leq t< \sigma \textit{ or if } \tau\alpha < t< \nu, \\ 1&\textit{if } \sigma< t< \tau\alpha \textit{ or if } \nu< t< \tau. \end{cases} $$
Proof
 (a):

The assumption \(f(\alpha)< a_{1}\) implies that \(\theta_{1}\) immediately increases with speed \(1+\epsilon\) when t crosses 0, and until either it reaches \(a_{1}\) or \(\theta_{0}\) reaches \(a_{0}\). Then \(\theta_{1}\) must increase with speed 1 until (at least) time \(\tau \alpha\) when it reaches 1. The inequality \(f(\tau\alpha)\geq a_{0}\) implies that \(\theta_{0}\) is above \(a_{0}\) for all \(t>\tau\alpha\); hence \(\theta_{1}\) continues to grow at rate 1 until \(\theta_{1}\) reaches 1.
 (c′):

Assuming a first phase as before, if otherwise \(f(\tau \alpha)< a_{0}\), then \(\theta_{1}\) increases again with speed \(1+\epsilon \) after it has crossed 1, and until either \(\theta_{0}\) reaches \(a_{0}\) or \(\theta_{1}\) reaches \(1+a_{1}\). The latter case is impossible because it would imply that \(\theta_{1}\) increases by more than 1 over the fundamental time interval. This uniquely determines case (c′) and we have \(f(\sigma\alpha)=a_{1}\) and \(f(\nu)=a_{0}\).
 (c):

If \(f(\alpha)\geq a_{1}\), then \(\theta_{1}\) cannot have speed 1 before it reaches 1. Since we assume that \(\theta_{0}\) is already larger than \(a_{0}\) when this happens, it follows that \(\theta_{1}\) can never have speed \(1+\epsilon\).
 (b):

Assuming a first phase as in (c), if \(f(\tau\alpha )< a_{0}\), then \(\theta_{1}\) accelerates after time \(\tau\alpha\) and until either \(\theta_{0}\) reaches \(a_{0}\) or \(\theta_{1}\) reaches \(1+a_{1}\). After that, \(\theta_{1}\) has speed 1 until either \(\theta_{0}\) reaches 1 or \(\theta_{1}\) reaches 2. Again the latter case is impossible because total increase over one period is at most 1. □
For our purpose here, it is enough to focus on case (a). (Indeed, Lemma 4.1 below shows that these are the only possibly asymptotically stable waves.) In this case, Lemma 3.1 indicates that the TW shape is completely determined by the numbers α, σ, and τ. Our next statement claims that σ and τ are actually entirely determined by the phase shift α, and, therefore, so is the TW shape.
Lemma 3.2
For every choice of the parameters ϵ, \(a_{0}\), \(a_{1}\) and every phase shift \(\alpha\in \mathbb {R}^{+}\), there exists at most one TW solution in case (a).
One can actually prove that a similar statement holds in cases (b), (c), and (c′). More importantly, this statement paves the way to uniqueness as stated in Theorem 2.1.
Proof of Lemma 3.2
Claim 3.3
For every choice of parameters ϵ, \(a_{0}\), \(a_{1}\) and every phase shift \(\alpha\in \mathbb {R}^{+}\), the equation \(\varTheta_{\alpha,t_{0}}(t_{0})=a_{0}\) (resp. \(\varTheta_{\alpha,t_{1}}(t_{1}+\alpha)=a_{1}\)) has a unique positive solution denoted \(t_{0}\) (resp. \(t_{1}\)). Moreover, we have \(\sigma=\min\{ t_{0},t_{1}\}\).
Notice that the quantity σ in this statement satisfies the inequalities \(0\leq\sigma\leq 1\epsilon\sigma=\tau\).
Proof of Claim 3.3
Now, σ is the first time in \([0,\tau]\) when \(\theta_{1}\) adopts speed 1. As argued in the proof of Lemma 3.1, this time corresponds to the smallest of times when \(\theta_{1}\) reaches \(a_{1}\) or \(\theta_{0}\) reaches \(a_{0}\); viz. \(\sigma=\min\{t_{0},t_{1}\}\) and the claim is proved. □
□
To conclude about existence of waves, it remains to investigate the conditions on α such that a TW shape \(\varTheta_{\alpha,\sigma }\) satisfies the conditions of Lemma 3.2. The result is given in the next statement.
Lemma 3.4
The constraints here define a nonempty interval of α for every possible choice of parameters ϵ, \(a_{0}\), \(a_{1}\). By Lemma 3.2, let \(I_{\epsilon,a_{0},a_{1}}\) be the corresponding interval of forcing periods τ. We have proved the following statement.
Corollary 3.5
Equation (2) has a unique traveling wave solution in case (a), for every forcing period in \(I_{\epsilon,a_{0},a_{1}}\).
Proof of Lemma 3.4
We need to check the conditions \(0< f(\alpha)< a_{1}\) and \(f(\tau\alpha)\geq a_{0}\) of Lemma 3.1 for a shape \(\varTheta_{\alpha,\sigma}\) as in the proof of Lemma 3.2.
First, we have \(f(\alpha)=\varTheta_{\alpha,\sigma}(\alpha)=\alpha\). So the condition \(0< f(\alpha)< a_{1}\) is equivalent to \(0<\alpha< a_{1}\).
In the first case, the inequality \(f(\tau\alpha)\geq a_{0}\) is equivalent to \(\tau\alpha\geq t_{0}=\sigma\). However, we know from the proof of Lemma 3.2 that we must have \(\tau\alpha >\sigma\); hence there is nothing to prove.
4 Stability Analysis
4.1 Local Stability of Stroboscopic Map Fixed Points
In this section, we study the local stability of the fixed point \(f(\alpha)\) of the stroboscopic map \(F_{f}\) associated with a TW shape. A first statement, respectively, identifies the stable, unstable, and neutral cases according to the decomposition in Lemma 3.1.
Lemma 4.1

the fixed point \(f(\alpha)\) is locally asymptotically stable if in case (a) with \(f(\sigma)< a_{0}\) when \(\theta _{1}^{(f)}(\sigma)=a_{1}\);

it is unstable if in case (b) with \(\theta _{1}^{(f)}(\sigma)<1+a_{1}\) when \(f(\sigma)=a_{0}\);

it is neutral in any other case.
Proof
We want to evaluate the behavior of the difference \(F_{f}^{n}(f(\alpha))F_{f}^{n}(\xi_{1})\), where \(\xi_{1}\in \mathbb {T}\) is a small perturbation of the initial condition \(\vartheta_{1}=f(\alpha)\).

both \(\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(\theta _{1}^{(f)}(\xi_{1},t)\) have speed \(1+\epsilon\) for \(t\in(0,\sigma)\);

\(\theta_{1}^{(f)}(\vartheta_{1},t)\) has speed 1 and \(\theta_{1}^{(f)}(\xi_{1},t)\) has speed \(1+\epsilon\) for \(t\in(\sigma ,\sigma')\);

both \(\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(\theta _{1}^{(f)}(\xi_{1},t)\) have speed 1 for \(t\in(\sigma',\tau)\).

both \(\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(\theta _{1}^{(f)}(\xi_{1},t)\) have speed 1 for \(t\in(0,\tau\alpha)\) (NB: \(\tau\alpha=1f(\alpha)\));

\(\theta_{1}^{(f)}(\vartheta_{1},t)\) has speed \(1+\epsilon\) and \(\theta_{1}^{(f)}(\xi_{1},t)\) has speed 1 for \(t\in (\tau\alpha,1\xi_{1})\);

both \(\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(\theta _{1}^{(f)}(\xi_{1},t)\) have speed \(1+\epsilon\) for \(t\in(1\xi_{1},\sigma)\);

both \(\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(\theta _{1}^{(f)}(\xi_{1},t)\) have speed 1 for \(t\in(\sigma,\tau)\).
Neutral case. The analysis is similar. One shows that the total duration when the perturbed trajectory has speed \(1+\epsilon\) is identical to that of the TW; hence \(F_{f}(f(\alpha))F_{f}(\xi _{1})=f(\alpha)\xi_{1}\). □
In the proof of Lemma 3.4 above, we have identified the condition \(\alpha>\alpha_{0}\) (see Eq. (4)) as sufficient to ensure being in case (a) with \(f(\sigma)< a_{0}\) when \(\theta _{1}^{(f)}(\sigma)=a_{1}\). By crosschecking this constraint with the one in the statement of that lemma (and using also Lemma 3.2, i.e. that the TW is entirely determined by the parameters ϵ, \(a_{0}\), \(a_{1}\) and its phase shift α), we get the following conclusion.
Corollary 4.2
For every choice of parameters ϵ, \(a_{0}\), \(a_{1}\), there exists an interval \(I'_{\epsilon,a_{0},a_{1}}\subset I_{\epsilon,a_{0},a_{1}}\) such that for every \(\tau\in I'_{\epsilon,a_{0},a_{1}}\), there exists a τperiodic TW shape f and a phase shift \(\alpha\in(0,\tau)\) such that \(f(\alpha)\) is a locally asymptotically stable fixed point of \(F_{f}\).
Proof
If \(a_{1}\leq a_{0}\), then the left hand side of the inequality is equal to 0, while the right hand side always remains nonnegative; hence the inequality holds.

the inequality \(\frac{a_{1}a_{0}}{1+\epsilon}\leq a_{1}(1+\epsilon)a_{0}\) is equivalent to \((2+\epsilon)a_{0}\leq a_{1}\),

the condition \(a_{1}\leq\max \{(1+\epsilon )(1a_{0})\epsilon a_{1},\frac{1a_{0}+\epsilon(1a_{1})}{1+\epsilon} \}\) simplifies to \(a_{1}\leq1\min \{a_{0},\frac{a_{0}+\epsilon a_{1}}{1+\epsilon} \}\) and we obviously have \(\min \{ a_{0},\frac{a_{0}+\epsilon a_{1}}{1+\epsilon} \}=a_{0}\) when \(a_{1}>a_{0}\).

If \((2+\epsilon)a_{0}\leq a_{1}\) then \((2+\epsilon )a_{0}+\epsilon a_{1}\leq1+\epsilon\) and the statement holds provided that \(a_{1}(1+\epsilon)a_{0}\leq(1+\epsilon)(1a_{0})\epsilon a_{1}\), which is true.

If \((2+\epsilon)a_{0}+\epsilon a_{1}> 1+\epsilon\) then \((2+\epsilon)a_{0}> a_{1}\) and the inequality to check iswhich holds true.$$\frac{a_{1}a_{0}}{1+\epsilon }\leq\frac{1a_{0}+\epsilon(1a_{1})}{1+\epsilon}, $$

Finally, if \((2+\epsilon)a_{0}> a_{1}\) and \((2+\epsilon )a_{0}+\epsilon a_{1}\leq1+\epsilon\), then the inequality to verify iswhich is equivalent to \(a_{1}a_{0}\leq(1+\epsilon) (1+\epsilon (1a_{1})(1+\epsilon)a_{0} )\). However, the inequality \((2+\epsilon )a_{0}+\epsilon a_{1}\leq1+\epsilon\) implies \((1+\epsilon) (1+\epsilon(1a_{1})(1+\epsilon)a_{0} )\geq(1+\epsilon)a_{0}\) and we have \((1+\epsilon)a_{0}\geq a_{1}a_{0}\) because of \((2+\epsilon)a_{0}> a_{1}\). The proof of the corollary is complete. □$$\frac{a_{1}a_{0}}{1+\epsilon }\leq(1+\epsilon) (1a_{0})\epsilon a_{1}, $$
4.2 Global Stability of Fixed Points
Once local stability has been established, a careful computation of \(F_{f}\) allows one to show that the fixed points are actually globally stable.
Proposition 4.3
Proof of Proposition 4.3
We are going to prove that, for the TW in case (a) with α simultaneously satisfying \(\alpha>\alpha_{0}\) and the conditions of Lemma 3.4, the restriction of \(F_{f}\) to the interval \([0,1)\) consists of four affine pieces (each piece being defined on an interval): two pieces are rigid rotations and they are interspersed by one contracting and one expanding piece.
Since \(F_{f}\) is a lift of an endomorphism of the circle that preserves the orientation, and since it has a locally stable fixed point \(f(\alpha)\) (Corollary 4.2), the graph of the contracting piece must intersect the main diagonal of \(\mathbb {R}^{2}\). As a consequence, the graphs of the following and preceding neutral pieces cannot intersect this line. By continuity and periodicity, the graph of the remaining expanding piece must intersect this line as well. Let \(\vartheta_{\mathrm{unst}}\in[0,1)\) be this unique unstable fixed point. The proposition immediately follows.
 (a)
either the last rapid phase with speed \(1+\epsilon\) (when \(\theta_{1}\geq n\)) stops when \(t=t_{0}\); this occurs iff \(t_{0}< n(\frac {a_{1}}{1+\epsilon}+1a_{1})+\frac{a_{1}}{1+\epsilon}\);
 (b)
or its stops when \(\theta_{1}\) reaches \(n+a_{1}\);^{9} this occurs when \(n(\frac{a_{1}}{1+\epsilon }+1a_{1})+\frac{a_{1}}{1+\epsilon}\leq t_{0}<(n+1)(\frac{a_{1}}{1+\epsilon }+1a_{1})\).
Throughout the proof, we shall frequently make use of the time \(t_{a,\vartheta_{1}}\) when the coordinate \(\theta_{1}\) reaches the value a, i.e. \(\theta _{1}^{(f)}(\vartheta_{1},t_{a,\vartheta_{1}})=a\). Here \(0\leq\vartheta _{1}\leq a\) are arbitrary.
Case (a). By continuous and monotonic dependence on the initial conditions, every trajectory starting initially with \(\vartheta _{1}\) sufficiently close to 0 (and >0) will not only experience the same number \(n+1\) of rapid phases with speed \(1+\epsilon\), but the last rapid phase (the unique rapid phase if \(n=0\)) will also stop at \(t=t_{0}\).
If \(n=0\), the same conclusion immediately follows from the fact that the lengths of the rapid phases are simply given by \(t_{0}\).
 (a1)
either \(\theta_{1}^{(f)}(a_{1},t_{0})\leq n+1\),
 (a2)
or \(\theta_{1}^{(f)}(a_{1},t_{0})> n+1\).
Let \(\vartheta_{\ast}\) be such that \(\theta_{1}^{(f)}(\vartheta_{\ast},t_{0})=n+1\). The dichotomy (a1) vs. (a2) is equivalent to \(a_{1}\leq\vartheta_{\ast}\) vs. \(a_{1}>\vartheta_{\ast}\). In the latter case, the same conclusion as before applies to the interval \((\vartheta_{\max},\vartheta_{\ast})\).
If \(n=0\), that \(F_{f}\) is a rigid rotation on \((a_{1},\vartheta_{\ast})\) immediately follows from the fact that there is no rapid phase at all. In case (a2), the same conclusion applies to \((\vartheta_{\ast},a_{1})\).
Case (b). The arguments are similar. For \(\vartheta_{1}>0\) in the neighborhood of 0, the length of the last rapid phase is now independent of \(\vartheta_{1}\) (\(n=1\)) and the length of the first rapid phase decreases when \(\vartheta_{1}\) increases. Hence \(F_{f}\) is contraction on this first interval. Its upper boundary is given by \(\min\{a_{1},\vartheta_{\ast}\}\), where, as before, \(\vartheta_{\ast}\) is defined by \(\theta_{1}(\vartheta_{\ast},t_{0})=n+1\).
For \(\vartheta_{1}\in(a_{1},\vartheta_{\ast})\) (or \(\vartheta_{1}\in (\vartheta_{\ast},a_{1})\) depending on the case), the cumulated duration of the accelerated phases does not depend on \(\vartheta_{1}\) (in particular because the last rapid phase stops when the coordinate reaches \(n+a_{1}\)).
In the case \(a_{1}<\vartheta_{\ast}\), for \(\vartheta_{1}>\vartheta_{\ast}\) the trajectory has an extra final rapid phase when compared to \(\vartheta_{\ast}\) and this holds for all \(\vartheta_{1}\leq\vartheta _{\dagger}\) where \(\vartheta_{\dagger}\) is defined by \(\theta_{1}(\vartheta _{\dagger},t_{0})=n+1+a_{1}\).^{11} Finally, for \(\vartheta_{1}\in (\vartheta_{\dagger},1)\), \(F_{f}\) is a rigid rotation. The case \(\vartheta _{\ast}\leq a_{1}\) can be treated similarly. □
4.3 Global Stability of TW: Proof of Theorem 2.2
Proposition 4.4
Proof of Proposition 4.4
 (P)
We have \(\lim_{n\to+\infty}\F_{R^{n\tau}\theta _{1}^{(f)}(\vartheta_{1},\cdot)}F_{R^{\alpha}f}\=0\) where the uniform norm \(\\cdot\\) is extended to functions on \(\mathbb {T}\).
To complete the proof, it remains to establish the property (9). Notice first that, as n gets large, the times at which the forcing signals \(R^{n\tau}\theta_{1}^{(f)}(\vartheta_{1},\cdot)\) and \(R^{\alpha}f\), respectively, cross the levels 0, \(a_{0}\) and 1 come close together. As a consequence, the restriction of \(F_{R^{n\tau }\theta_{1}^{(f)}(\vartheta_{1},\cdot)}\) to the interval I, since it is close to \(F_{R^{\alpha}f}_{\mathrm{I}}\) by the property (P), must consist of an expanding piece, possibly preceded and/or followed by a rigid rotation.
Moreover, the length of these (putative) rigid rotation pieces must be bounded above by a number that depends only on γ and on \(\ F_{R^{n\tau}\theta_{1}^{(f)}(\vartheta_{1},\cdot)}F_{R^{\alpha}f}\\), and which vanishes as \(n\to+\infty\). (Indeed, if otherwise, the length(s) of the rigid rotation piece(s) remained bounded below by a positive number, we would have a contradiction with the property (P) above, because the distance between \(F_{R^{n\tau}\theta _{1}^{(f)}(\vartheta_{1},\cdot)}(\vartheta_{2})\) and \(F_{R^{\alpha}f}(\vartheta_{2})\) would remain bounded below by a positive number for \(\vartheta_{2}\) close to the boundaries of I.) This proves the existence of \(\mathrm{I}'\subset\mathrm{I}\) on which all \(F_{R^{n\tau }\theta_{1}}\) for n sufficiently large must be expanding.
In addition, by taking n even larger if necessary (so that \(\ F_{R^{n\tau}\theta_{1}^{(f)}(\vartheta_{1},\cdot)}F_{R^{\alpha}f}\\) is even smaller), we can make sure that the expanding piece of \(F_{R^{n\tau}\theta_{1}^{(f)}(\vartheta_{1},\cdot)}\) over \(\mathrm{I}'\) intersects the diagonal, since the limit map \(F_{R^{\alpha}f}\) does so. The first claim of property (9) then easily follows.
The second claim that the expanding slope must be bounded below by \(\gamma'\) can be proved using a similar contradiction argument to above. The proof of the proposition is complete. □
Corollary 4.5
5 Generation of TW: Proof of Theorem 2.3
Recall that for \(\tau\in I'_{\epsilon,a_{0},a_{1}}\) we have \(f(\sigma )< a_{0}\) for the associated TW where \(\sigma=\frac{a_{1}\alpha }{1+\epsilon}\) is such that \(f(\alpha+\sigma)=a_{1}\).
6 Concluding Remarks
For simple feedforward chains of typeI oscillators, our analysis proved that periodic wave trains can be generated from arbitrary initial condition, even when the root node is forced using an unrelated signal. Moreover, these stable waves exist for an open (parameterdependent) interval of wave number and period.
The existence of globally attracting waves for arbitrary wave number in some range is reminiscent of the inertiafree dynamics of tilted Frenkel–Kontorova chains, which constitute coupled oscillator models for spatially modulated structures in solidstate physics [23]. There are, however, essential differences between the two situations. Instead of a unidirectional interaction, the coupling is of bidirectional type in Frenkel–Kontorova chains and involves left and right neighbors. More importantly, the overall dynamics there is monotonic and, as mentioned in the introduction, this property is critical for the proof of the existence and stability of waves [24].
Finally, we notice that the results on asymptotic stability and on stability with respect to changes in forcing are based on hyperbolic properties of the stroboscopic dynamics. Accordingly, we believe that, using continuation methods, these results and, more generally, results on generation of traveling waves in unidirectional chains of typeI oscillators can be established in a rigorous mathematical way, in more general models with smooth PRC and stimulus nonlinearities. This will be the subject of future studies.
The wave period is neither necessarily equal to τ, nor to \(1/\omega\); see [15] for an illustration.
In the end of Sect. 2.1, we show that monotonicity with respect to ‘pointwise’ ordering on sequences in \(\mathbb {R}^{\mathbb {Z}^{+}}\) fails in this system.
Notations. \(\mathbb {Z}^{+}=\{0,1,2,\ldots\}\), \(\mathbb {N}=\{1,2,\ldots\}\) and \(\lfloor\cdot\rfloor \) denotes the floor function (recall that \(0\leq x\lfloor x\rfloor<1\) for all \(x\in \mathbb {R}\)). ϑ and \(\vartheta_{s}\) denote elements in \(\mathbb {T}\) whereas θ and \(\theta_{s}\) represent functions of the real positive variable with values in \(\mathbb {T}\).
We refer to the \(\mathbb {T}\)valued f as τperiodic provided that \(f(t+\tau)=f(t)+1\) for all \(t\in \mathbb {R}^{+}\).
In order to ensure continuous dependence of solutions of Eq. (2) on forcing, we actually only need that the forcing slope be bounded below by a positive number, the same number for all forcing. The choice 1 for the bound here is consistent with the minimal rate at which solutions can grow. Typically, f is thought of as being piecewise affine or even simply affine.
That \(\theta_{1}^{(f)}(0)=f(\alpha)=\alpha\) is a consequence of the fact that \(f(\tau\alpha)\geq a_{0}\)—see figures in cases (a) and (c).
Indeed, we obviously have \(F_{f}(x+1)=F_{f}(x)+1\) for all x, from Eq. (2). Moreover, the continuous dependence of \(\theta _{1}^{(f)}(\vartheta_{1},\tau)\) on \(\vartheta_{1}\) implies that \(F_{f}\) must be continuous. As for monotonicity, by contradiction, if we had \(\vartheta_{1}< \xi_{1}\) and \(F_{f}(\vartheta_{1})>F_{f}(\xi_{1})\), then the corresponding trajectories \(t\mapsto\theta_{1}^{(f)}(\vartheta_{1},t)\) and \(t\mapsto\theta_{1}^{(f)}(\xi_{1},t)\) would have crossed for some \(t\in(0,\tau)\). This is impossible by uniqueness of the vector field acting on this coordinate.
In particular, this shows that \((\vartheta_{\ast},1)\) is the fourth and last interval to consider in this analysis. There are no more acceleration patterns and \(F_{f}\) indeed decomposes over exactly four intervals.
The existence of \(\vartheta_{\dagger}\) is granted from the fact that \(\theta_{1}(1,t_{0})> n+1+a_{1}\), which follows from \(\theta_{1}(0,t_{0})>n+a_{1}\).
Notes
Declarations
Acknowledgements
The work of BF was supported by EU Marie Curie fellowship PIOFGA2009235741 and by CNRS PEPS Physique Théorique et ses interfaces.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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