1 Introduction

Networks of coupled oscillators receive large amounts of interest across various disciplines, due to their relevance to for instance neural networks [6] and power systems [21]. In the context of neuromorphic computing, synchronization plays a key role in designing novel and power-efficient technologies. Understanding synchronization in oscillator networks has, for example, aided in designing these systems in a way that lets them naturally solve optimization problems [2, 10, 13, 41] or tasks like image recognition [1] and gait pattern classification [38]. Hardware implementations of neural networks are of great interest [30] due to their power eficiency. The combination of neural networks with memristors [11, 12] exhibits remarkable properties such as fault tolerance [30] and event-triggered synchronization [34]. The design of such networks is an active area of research [22, 35, 57]. It is hence desirable to understand synchronization on a level relatively close to the hardware, for instance in terms of electrical parameters [43]. Necessary synchronization conditions have been approached for systems of linearly coupled oscillators via the Master stability function method [44], while other approaches leading to sufficient conditions are for instance based on contraction theory [3,4,5, 9, 27, 28, 36, 40], QUAD-conditions [15, 17, 18] or semi-passivity [46, 47, 49]. The approaches based on semi-passivity usually construct suitable (quadratic but also non-smooth) Lyapunov functions [14, 20, 56]. Most of these conditions do not allow an immediate physical interpretation and are sometimes hard to apply to actual physical systems. In one way or another, all these methods compare a one-sided Lipschitz condition to a suitable measure of algebraic connectivity of the coupling graph. We will discuss this in more detail in Sect. 5.2.

The FitzHugh-Nagumo oscillator (FHNO) is a relevant oscillator for neuromorphic applications [42] because it is technically realizable [45] and shows biologically plausible behavior [39]. Its key ingredient is the nonlinear conductance, which we view as being characterized by its (i, u)-curve in this paper. We derive a sufficient synchronization criterion for identical diffusively coupled FHNOs that is phrased in terms of conditions on the coupling network and the (general) (i, u)-curve such that our condition also applies to other models such as the van-der-Pol oscillator [25, 48]. We consider diffusive coupling with time-varying, state-dependent and locally adaptive coupling strength, where our main application consists of FHNOs coupled by ideal memristors.

In the setting of phase-oscillators, the setting of adaptive coupling is already well-studied, but still an active area of research. In order to not rely on the validity of a phase-approximation, we study the full oscillator dynamics and therefore only name a few references concerning synchronization of phase-oscillators [23, 29, 54].

Similar to [14] we use an approach based on semi-passivity but we choose a quadratic Lyapunov function. The major novelty of our approach consists of the generality of the coupling strength’s time evolution, but also in the way we bound its time derivative, which is reminiscent of the strategy in [17] but without the need for the individual oscillators’ vector fields to be Lipschitz or to satisfy a QUAD-condition. We exploit that the nonlinear conductance is semi-passive in the following sense: The dissipated power at the conductance is always positive for large enough magnitudes of applied voltage and negative differential conductance occurs only for voltages with magnitude smaller than some possibly large threshold. Our sufficient criterion essentially states that for all times the algebraic connectivity of the coupling network needs to be larger than the slope of this linear term for the oscillators to synchronize. The Lyapunov-function used includes the power dissipated by the coupling network as a term, so that this power tends to zero if our sufficient condition is met. Specialized to the standard unitless FitzHugh-Nagumo-model (FHN-model) and static diffusive coupling, it turns out that our criterion is sharper than that of [14] based on semi-passivity and also that of [31] which uses a direct Lyapunov approach. For static diffusive coupling our criterion coincides with [3, cor. 4.1] and [27, thm. 30], whose derivation is based on contraction theory and [56],which is based on Lyapunov’s method. We note that most of these sources obtain their result as a special case of more general considerations and we will discuss in Sect. 5.2 the reason why our bound coincides with that of [3, 27], and [56].

An advantage of our criterion is that it spells out in terms of the ideal circuit’s parameters, which makes it easily applicable to electronical models of the FHNO such as the ones presented in [45]. Its application only requires knowledge of the nonlinear conductance’s (i, u)-curve and the coupling network.

In summary, we shed some light on a physically-interpretable sufficient synchronization condition of identical diffusively coupled FHNOs that is more specialized w.r.t. the oscillators but as sharp as the condition in [3] and has three distinct advantages:

  1. 1.

    It gives practitioners an easy way of ensuring the occurrence of synchronization in dependence on the system’s nonlinearity.

  2. 2.

    It has a physical interpretation that is also embedded into the associated mathematical analysis, namely the minimization of dissipated power.

  3. 3.

    It also applies to locally adaptive, state-dependent couplings including memristors or nonlinear couplings

The paper is structured as follows. In Sect. 2 we first fix our notation and describe the class of considered couplings. Afterwards we recapitulate the FitzHugh-Nagumo oscillator and describe a generalization of it, discuss the electrical coupling, and derive a compact unitless description. In Sect. 3, we derive an explicit sufficient condition for synchronization. As an application, we spell out this condition for FHNOs coupled by ideal memristors and static linear conductances in Sect. 4. We demonstrate its correctness on an example in Sect. 5, where we also discuss how our result for static diffusive coupling is related to the existing literature. Finally, Sect. 6 summarizes the main contributions of this work and gives an outlook on further research in this area. All proofs to lemmas and propositions stated in the main text can be found in the appendix.

Notation: Throughout, vector and matrix objects are typeset with bold symbols. For instance, we denote by \({\mathbf {0}}\) the zero-vector of a given vector space. Given the amount of necessary notation, we included a nomenclature.

This work is a completely revised and substantially extended version of the earlier preprint "On the Electrical Modeling and Synchronization of Diffusively Coupled FitzHugh-Nagumo Oscillators", available at https://doi.org/10.36227/techrxiv.21679532.v1.

2 Diffusively coupled oscillators of FitzHugh-Nagumo type

2.1 Interconnected input-affine systems

Consider the input-affine dynamical system

$$\begin{aligned} \dot{{\varvec{z}}} = {\varvec{f}}({\varvec{z}})+ {\varvec{B}}({\varvec{z}}){\varvec{x}} , \quad {\varvec{y}} = {\varvec{h}}({\varvec{z}}) , \quad {\varvec{z}}(t_{0}) = {\varvec{z}}_{0}, \end{aligned}$$
(1)

with (Lipschitz) continuous functions \({\varvec{f}}:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\), \({{\varvec{B}}}:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^{n\times k}\), \({\varvec{h}}:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^k\). We refer to the objects as the state vector \({\varvec{z}}\in {\mathbb {R}}^{n}\), input vector \({\varvec{x}}\in {\mathbb {R}}^{k}\), and output vector \({\varvec{y}}\in {\mathbb {R}}^{k}\), where we have dropped the time argument for the sake of brevity. The input vector \({\varvec{x}}\) is assumed to be a continuous and bounded function of time.

We obtain an isolated system by considering N systems of type (1) interconnected by a generalized diffusiveFootnote 1 coupling:

$$\begin{aligned} \dot{{\varvec{z}}}_{\mu }&= {\varvec{f}}({\varvec{z}}_{\mu }) + {{\varvec{B}}}({\varvec{z}}_{\mu }){\varvec{x}}_{\mu } , \end{aligned}$$
(2a)
$$\begin{aligned} {\varvec{y}}_{\mu }&= {\varvec{h}}({\varvec{z}}_{\mu }) , \end{aligned}$$
(2b)
$$\begin{aligned} {\varvec{x}}_{\mu }&= -\sum _{\nu =1}^{N} a_{\mu \nu }(t,{\varvec{z}},c_{\mu \nu }) \left[ {\varvec{y}}_{\mu }-{\varvec{y}}_{\nu }\right] , \end{aligned}$$
(2c)
$$\begin{aligned} \dot{c}_{\mu \nu }&= k_{\mu \nu }\left( {\varvec{y}}_{\mu },{\varvec{y}}_{\nu },t,c_{\mu \nu }\right) , \end{aligned}$$
(2d)

with state-variables \({\varvec{z}}_{\mu }\), inputs \({\varvec{x}}_{\mu }\) and outputs \({\varvec{y}}_{\mu }\) for \(\mu =1,\dots ,N\), where \({\varvec{z}}= \left[\begin{array}{ccc} {\varvec{z}}_1^T,\dots ,{\varvec{z}}_N^{\text {T}} \end{array}\right] ^{\text {T}}\) denotes the stacked state vectors. The coupling weights \(a_{\mu \nu }=a_{\nu \mu } \ge 0\) are nonnegative and symmetric for all arguments. The dependence of the coupling \(a_{\mu \nu }\) on the state vector’s of all subsystems is a very general case and for most applications dependence on \({\varvec{z}}_\mu ,{\varvec{z}}_\nu\) suffices as most coupling mechanisms are local in nature. Since our synchronization condition’s proof also works for the more general case, we decided to work in this setting. We assume that the outputs and inputs are of the same dimension k so that the difference \({\varvec{y}}_{\mu }-{\varvec{y}}_{\nu }\) makes sense. The coupling weights are allowed to be functions of time t as well as the systems’ state variables and can furthermore be locally adaptive in the sense that \(a_{\mu \nu }\) depends on a state variable \(c_{\mu \nu }\) that evolves over time. We call this a locally adaptive coupling because the evolution law of \(a_{\mu \nu }\) depends only on time, \(c_{\mu \nu }\) and the outputs \({\varvec{y}}_{\mu }\) and \({\varvec{y}}_{\nu }\).

We describe the coupled subsystems in terms of a weighted undirected graph \({\mathcal {G}}=\left( {\mathcal {V}},{\mathcal {E}}\right)\) with \(\vert {\mathcal {V}} \vert =N\) nodes \({\mathcal {V}}\), one for each subsystem, and \({N_{{\mathcal {E}}}} =\vert {\mathcal {E}} \vert\) edges of respective weights \(a_{\mu \nu }\) depending on the variables t, \({\varvec{z}}\) and \(c_{\mu \nu }\). For each unoriented edge \(\{\mu ,\nu \}\) select an orientation, denoted by \((\mu ,\nu )\) if the edge is oriented from \(\mu\) to \(\nu\) or \((\nu ,\mu )\) otherwise. Let \({{\varvec{N}}}\in {\mathbb {Z}}^{N\times {N_{{\mathcal {E}}}} }\) be the incidence matrix of the resulting directed graph \({\mathcal {G}}\) with the elements

$$\begin{aligned} n_{\mu e}= {\left\{ \begin{array}{ll} +1 &{} \text {if } e=(\mu ,\nu )\text { for some }\nu \in {\mathcal {V}}\\ -1 &{} \text {if } e=(\nu ,\mu )\text { for some }\nu \in {\mathcal {V}}\\ 0 &{} \text {otherwise.} \end{array}\right. } . \end{aligned}$$
(3)

We collect the edge weights in the diagonal matrix \({{\varvec{D}}}\in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} \times {N_{{\mathcal {E}}}} }\), i.e., for the edge \(e=\{\mu ,\nu \}\) one has \(D_{ee}=a_{\mu \nu }\). We define the Laplacian matrix as

$$\begin{aligned} {{\varvec{\varGamma }}}= {{\varvec{N}}} {{\varvec{D}}} {{\varvec{N}}}^{\text {T}} . \end{aligned}$$
(4)

It is a standard fact of algebraic graph theory [8] that \({{\varvec{\varGamma }}}\) can alternatively be defined as

$$\begin{aligned} \Gamma _{\mu \mu }= \sum _{\nu \ne \mu }a_{\mu \nu } \quad \text {and} \quad \Gamma _{\mu \nu }= -a_{\mu \nu } , \quad \text {for} \quad \mu \ne \nu , \end{aligned}$$
(5)

which coincides with (4). Throughout this paper we sort the eigenvalues of \({{\varvec{\Gamma }}}\) in ascending order, i.e.,

$$\begin{aligned} 0=\lambda _{1}\{ {{\varvec{\Gamma }}}\} \le \lambda _2\{ {{\varvec{\Gamma }}}\} \le \dots \le \lambda _n \{ {{\varvec{\Gamma }}}\}, \end{aligned}$$
(6)

where the first inequality is strict if and only if the coupling graph is connected. Note that the eigenvalues of \({{\varvec{\Gamma }}}\) are always real because \({{\varvec{\Gamma }}}^{\text {T}}= {{\varvec{\Gamma }}}\) and nonnegative because \({{\varvec{\Gamma }}}\) is diagonally dominant by (5). For later use, we define the two (stacked) input and output vectors

$$\begin{aligned} {\varvec{x}}&= \left[ \!\begin{array}{ccc} {\varvec{x}}_1^T,\dots ,{\varvec{x}}_N^{\text {T}} \end{array}\!\right] ^{\text {T}} \quad \text {and} \quad {\varvec{y}}= \left[ \!\begin{array}{ccc} {\varvec{y}}_1^T,\dots ,{\varvec{y}}_N^{\text {T}} \end{array}\!\right] ^{\text {T}} , \end{aligned}$$

respectively, which are related according to (2c) by

$$\begin{aligned} {\varvec{x}} = - {{\varvec{L}}}{\varvec{y}} \quad \text {with} \quad {{\varvec{L}}}= {{\varvec{\Gamma }}}\otimes {\mathbf {1}}_k , \end{aligned}$$
(7)

where \({\mathbf {1}}_k\) denotes the unit matrix of dimension k. Note that \({{\varvec{L}}}\) is positive semi-definite because \({{\varvec{\Gamma }}}\) is. The \(\mu\)-th input \({\varvec{x}}_{\mu }\) is then obtained as

$$\begin{aligned} {\varvec{x}}_{\mu }= - {{\varvec{e}}_{\mu}^{\text {T}}} {{\varvec{L}}}{\varvec{y}}, \quad \text{where} \quad {\varvec{e}}_{\mu}= e_{\mu}\otimes {{{\mathbbm{1}}}_k}, \end{aligned}$$

with \(e_\mu\) being the \(\mu\)-th unit vector in \({\mathbb {R}}^N\) and \({{\mathbbm{1}}_k} \in {{\mathbb{R}}^k}\) the vector of ones.

Assume that the N subsystems in (2) are identical and therefore of the same dimension n for all \(\mu =1,\dots ,N\). If the initial values of the subsystems are also identical, then the solution to the coupled system (2) is that the state vectors \({\varvec{z}}_{\mu }\) of the subsystems are equal to each other at all times, i.e., \({\varvec{z}}_{\mu }(t)={\varvec{z}}_{\nu }(t)\) and such that \({\varvec{z}}_{\mu }(t)\) is a solution to the subsystem (1) with zero-input and an appropriate initial value. We call such solutions synchronous.

Definition 1

(Synchronization manifold) The (partial) synchronization manifold to the system (2) with identical subsystems of dimension n is defined as

$$\begin{aligned} {\mathcal {S}} {:}{=}\left\{ {\varvec{z}} \in {\mathbb {R}}^{nN}\ \vert \ {\varvec{z}}_{\mu }={\varvec{z}}_{\nu },\ \ \forall \ \mu ,\nu =1,\dots ,N \right\} . \end{aligned}$$

Synchronous solutions are automatically contained in the synchronization manifold. This definition of synchrony does not include clustered synchronization or phase-locked solutions with phase differences other than 0. We call this a partial synchronization manifold, because only the state variables \({\varvec{z}}\) synchronize whereas the couplings’ state variables \({\varvec{c}}\) may not.

2.2 FitzHugh-Nagumo oscillator

Fig. 1
figure 1

Equivalent circuit of a FitzHugh-Nagumo oscillator

Full size image

The FitzHugh-Nagumo oscillator (FHNO) is a technically realizable [45] and biologically plausible neuronal oscillator [39]. In this section, we briefly recapitulate the electrical model depicted in Fig. 1 which will be used to describe the FHNO throughout this paper. We will show that this model satisfies the definition of a strictly semi-passive system as we need this property to apply prop. 13 later. The differential equations associated with the circuit can be deduced from Kirchhoff’s laws and the constitutive relations of the circuit elements:

$$\begin{aligned} C {\dot{u}}_{C}&= i_{C} = j-i_{G}(u_{C}) - i_{L} + i ,&u_{C}(t_{0})&= u_{C,0} , \end{aligned}$$
(8a)
$$\begin{aligned} L {\dot{i}}_{L}&= u_{L} = u_{C} - R i_{L} ,&i_{L}(t_{0})&= i_{L,0} . \end{aligned}$$
(8b)

Here, \(u_{C}\) and \(i_{L}\) are state-space quantities corresponding to a capacitor voltage and an inductor current, respectively, while \(u_{C,0}\) and \(i_{L,0}\) denote their initial values at the starting time \(t_{0}\), respectively. The current i denotes an external excitation current, while the current j represents a (bias) supply current. The electrical parameters C, L, R denote a capacitance, an inductance, and a resistance, respectively. Lastly, \(i_{G}: {\mathbb {R}} \mapsto {\mathbb {R}}\) is a nonlinear conductance function, which has been realized by a tunnel diode in the past [25, 39]. In the following, we work with the cubic nonlinearity

$$\begin{aligned} i_{G}(u_{C}) = G_{0}\left[ \frac{u_{C}^{3}}{3U^{2}_{0}} -u_{C}\right] , \end{aligned}$$
(9)

where the conductance \(G_{0}\) and voltage \(U_{0}\) are normalization constants. Even though we will stick to the above nonlinear conductance function in our examples, our results hold for more general functions. We will refer to this more general case as an oscillator of FitzHugh-Nagumo-type.

The FHNO can be written as a system of type (1), i.e., an input affine system with one-dimensional input given by the current i and one-dimensional output given by the voltage \(u_C\). We introduce the following quantities in order to recast (8) in terms of unitless variables:

$$\begin{aligned} \omega _0&= \frac{1}{\sqrt{LC} },\quad Z_0 = \sqrt{\frac{L}{C}},\quad I_0 =G_0U_0, \end{aligned}$$
(10a)
$$\begin{aligned} z_1&=\frac{u_C}{Z_{0} I_0 } ,\quad \ \, z_2 = \frac{i_L}{I_0}, \qquad \, \tau =\omega _0 t, \end{aligned}$$
(10b)
$$\begin{aligned} \iota&=\frac{j}{I_0},\quad \beta = \frac{R}{Z_0}, \quad f_{G}(z_1)=\frac{1}{I_0}i_G\left( Z_0I_0z_1\right) , \end{aligned}$$
(10c)

where \(G_0\), \(U_0\) are the parameters of (9) which transforms to

$$\begin{aligned} f_{G}(z_1)= \alpha \left[ \frac{\alpha ^2}{3}z_1^3 -z_1\right] ,\quad \alpha = G_0 Z_0. \end{aligned}$$
(10d)

In terms of the unitless parameters of (10) and in dependence on \(\tau\), (8) can be written as

$$\begin{aligned} {{\varvec{z}}}^\prime&= {\varvec{f}}({\varvec{z}})+ {\varvec{B}}({\varvec{z}}){\varvec{x}} ,&{\varvec{z}}(\tau _{0}) = {\varvec{z}}_{0},\ \text {with} \end{aligned}$$
(11a)
$$\begin{aligned} {\varvec{z}}&= \left[ \begin{array}{cc} z_1&z_2 \end{array}\right] ^{T} ,\quad {\varvec{x}}=[{i}/I_0],&{\varvec{y}}=[z_1] ,\ \text {and}\end{aligned}$$
(11b)
$$\begin{aligned} {\varvec{f}}({\varvec{z}})&=\left[ \begin{array}{c}-f_{G}\left( z_{1}\right) - z_2 +\iota \\ z_{1}-\beta z_{2} \end{array}\right] , \ {}&{{\varvec{B}}}({\varvec{z}})=\left[ \begin{array}{c}1 \\ 0 \end{array}\right] , \end{aligned}$$
(11c)

where we denote \({{\varvec{z}}}^\prime =\frac{\text {d}}{\text {d}\tau }{\varvec{z}}\).

Lemma 2

The system (11) is strictly semi-passiveFootnote 2 and admits a radially unbounded storage function for all functions \(f_{G}: {\mathbb {R}} \mapsto {\mathbb {R}}\) satisfying

$$\begin{aligned} \lim _{z_1\rightarrow \pm \infty } f_{G}\left( z_1\right) = \pm \infty , \end{aligned}$$

and all \(\iota \in {\mathbb {R}}\). Here, the function \(\rho\) from definition 12 can be chosen as

$$\begin{aligned} \rho (\Vert {\varvec{z}} \Vert ) = c_0 \Vert {\varvec{z}} \Vert \quad \text {for} \quad c_0> 0 . \end{aligned}$$

The physical meaning behind lemma 2 is that the system (8) is semi-passive as long as the nonlinear conductance’s behavior is eventually passive in the following sense: For voltages with magnitude larger than some finite threshold the conductance always dissipates power; power injection can only occur for voltage magnitudes below that threshold. The characterization of the FHNO as a semi-passive system provided by lemma 2 allows us to apply prop. 13 to networks of such oscillators coupled by a generalized diffusive coupling mechanism as in (2).

2.3 Coupled FHN-oscillators

Fig. 2
figure 2

Schematic representation of an example network consisting of three coupled FHNOs \({\mathcal {N}}_1,\dots ,{\mathcal {N}}_3\) with the output voltages \(u_\mu\) and input currents \(i_\mu\). A memristor symbol is used to indicate the adaptive coupling in (14). At every coupling we have the interaction current \(j_{\mu \nu }=W_{\mu \nu }(t,v_{\mu \nu },c_{\mu \nu })v_{\mu \nu }\). The ODEs governing this example are (17) with the choice of \({{\varvec{W}}}\) as depicted above

Full size image

Now that we have characterized the FHNOs as strictly semi-passive systems, we discuss the generalized diffusive coupling network used to connect the oscillators, where we use the graph-theoretical language developed in Sect. 2.1. Every pair of adjacent oscillators is coupled by an adaptive coupling as depicted in Fig. 2. Hence, to every undirected edge \(\{\mu ,\nu \}\in {\mathcal {E}}\) there is a (positive) conductance \(W_{\mu \nu }\) which we allow to depend on time t, the voltage \(v_{\mu \nu }\) across the edge \(\{\mu ,\nu \}\) and an edge variable \(c_{\mu \nu }\). We also associate a locally Lipschitz-continuous function \(k_{\mu \nu }:{\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) to each edge that describes the time-evolution of \(c_{\mu \nu }\):

$$\begin{aligned} \dot{c}_{\mu \nu }= k_{\mu \nu }\left( t,v_{\mu \nu },c_{\mu \nu }\right) \ . \end{aligned}$$
(12)

We call such a time evolution for \(W_{\mu \nu }\) locally adaptive because it is driven by local information, namely the edge voltage \(v_{\mu \nu }\) and the edge variable \(c_{\mu \nu }\).

We denote by \({\varvec{u}} = [u_{1},u_{2},\dots ,u_{N}]^{\text {T}}\) and \({\varvec{i}} = [i_{1},i_{2},\dots ,i_{N}]^{\text {T}}\) the port quantities of the interaction ports and introduce the vectors \({\varvec{v}} \in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} }\), \({\varvec{j}} \in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} }\) and \({\varvec{c}} \in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} }\) containing the voltage differences \(v_{\mu \nu }\), the interaction currents \(j_{\mu \nu }\), and edge variables \(c_{\mu \nu }\) respectively . Using the graph’s incidence matrix \({{\varvec{N}}}\in {\mathbb {Z}}^{N\times {N_{{\mathcal {E}}}} }\) the Kirchhoff equations governing the coupling network spell out as follows:

$$\begin{aligned} {\varvec{i}} = - {{\varvec{N}}} {\varvec{j}} , \quad {\varvec{v}} = {{\varvec{N}}}^{\text {T}} {\varvec{u}} . \end{aligned}$$
(13a)

Collecting the edge weights in a diagonal matrix denoted by \({{\varvec{W}}}_{\text {d}} \in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} \times {N_{{\mathcal {E}}}} }\) we can simultaneously state Ohm’s law for every coupling conductance as:

$$\begin{aligned} {\varvec{j}} = {{\varvec{W}}}_{\text {d}} {\varvec{v}} ,\ \text {with } {{\varvec{W}}}_{\text {d}} = {{\varvec{W}}}_{\text {d}} \left( t,{\varvec{v}},{\varvec{c}}\right) . \end{aligned}$$
(13b)

Combination of (13b) with (13a) yields the specialized version of (7):

$$\begin{aligned} {\varvec{i}} = - {{\varvec{W}}} {\varvec{u}} , \quad \text {with} \quad {{\varvec{W}}} = {{\varvec{W}}} \left( t,{\varvec{v}},{\varvec{c}}\right) = {{\varvec{N}}} {{\varvec{W}}}_{\text {d}} {{\varvec{N}}}^{\text {T}} . \end{aligned}$$
(14)

The current directions are chosen such that a negative sign emerges in the coupling formula (14). Note that the conductance needs not be linear despite the appearance of (14). The dependence of \({{\varvec{W}}}\) on the voltage differences \({\varvec{v}}\) allows for nonlinear conductances where the shape of (14) only guarantees that a zero voltage difference results in a zero current.

In order to state the coupled version of (11) we define the vectors of normalized capacitor voltages and normalized inductor currents and normalized supply currents

$${\varvec{\zeta }}_{1} = \left[ z_{\mu ,1} \right] _{\mu =1}^{N}, \quad \varvec{{\zeta }}_{2} = \left[ z_{\mu,2} \right] _{\mu =1}^{N},\quad {\varvec{\iota}}\in {\mathbb {R}}^N ,$$
(15)

as well as a parameter matrix \(\varvec{{\beta }}\in {\mathbb {R}}^{N \times N}\) and the unitless analog to \({{\varvec{W}}}\):

$$\begin{aligned} \varvec{{\beta }}= \beta {\mathbf {1}},\quad {{\varvec{\Gamma }}}\left( \tau ,{\varvec{z}}, {\varvec{c}}\right) = Z_0 {{\varvec{W}}} \left( t,{\varvec{v}},{\varvec{c}}\right) , \end{aligned}$$
(16a)

with \(t=\omega _0^{-1} \tau\), \({\varvec{v}}=Z_0I_0 {{\varvec{N}}}^{\text {T}}\varvec{ {\zeta }}_{1}\), where \(\beta\) is from (10) and \({\mathbf {1}}\) denotes the unit matrix. Furthermore, we define vector-valued functions

$$\begin{aligned} {\varvec{f}}_{G}: {\mathbb {R}}^{N} \rightarrow {\mathbb {R}}^{N},\quad \text {and} \quad \tilde{{\varvec{k}}} :{\mathbb {R}}\times {\mathbb {R}}^{N}\times {\mathbb {R}}^{ {N_{{\mathcal {E}}}} }\rightarrow {\mathbb {R}}^{ {N_{{\mathcal {E}}}} }, \end{aligned}$$
(16b)

where the \(\mu\)-th element of \({\varvec{f}}_{G}\) is \(f_{G}\) evaluated at \(z_{\mu ,1}\), while the \(\mu \nu\)-th element of \(\tilde{{\varvec{k}}}\) is \(\omega _0^{-1}k_{\mu \nu }\) evaluated at \(t=\omega _0^{-1}\tau\), \(v_{\mu \nu }=Z_0I_0[z_{\mu ,1}-z_{\nu ,1}]\), and \(c_{\mu \nu }\) for \(\{\mu ,\nu \}\) ranging over the edges \({\mathcal {E}}\) of the coupling graph \({\mathcal {G}}\). Using these definitions, we obtain the ODE describing the coupled FHN-type oscillators w.r.t. a generalized diffusive coupling parametrized by normal time \(\tau\):

$$\begin{aligned} \varvec{ {\zeta }}_{1}^{\prime }&= \varvec{\iota }-{\varvec{f}}_{G}\left( \varvec{ {\zeta }}_{1} \right) -\varvec{ {\zeta }}_{2} - {{\varvec{\Gamma }}}\varvec{ {\zeta }}_{1} \end{aligned}$$
(17a)
$$\begin{aligned} \varvec{ {\zeta }}_{2}^{\prime }&= \varvec{ {\zeta }}_{1} -\varvec{ {\beta }}\varvec{ {\zeta }}_{2} \end{aligned}$$
(17b)
$$\begin{aligned} {{\varvec{c}}}^\prime&= \tilde{{\varvec{k}}} \left( \tau ,\varvec{ {\zeta }}_{1} , {\varvec{c}} \right) . \end{aligned}$$
(17c)

This is an instance of an ODE of type (2) in unitless form. The example network sketched in Fig. 2 has three oscillators so that \(\varvec{ {\zeta }}_{1} ,\varvec{ {\zeta }}_{2} \in {\mathbb {R}}^3\) and three adaptive couplings \(W_{12},W_{13},W_{23}\) that are functions of \(\tau ,\varvec{ {\zeta }}_{1}\) and \({\varvec{c}}\). Its coupling matrix \({{\varvec{\Gamma }}}\) is

$$\begin{aligned} {{\varvec{\Gamma }}}= Z_0\begin{bmatrix} W_{12}+W_{13} &{} -W_{12} &{} -W_{13}\\ -W_{12} &{} W_{12}+W_{23} &{} -W_{23} \\ -W_{13} &{} -W_{23} &{} W_{13} +W_{23} \end{bmatrix}. \end{aligned}$$
(18)

3 Main result

We first state a sufficient condition for the synchronization of N FHNOs described in (11) with a coupling as in (17). We will prove this result throughout the rest of this section. In accordance with def. 1, we say that the states \({\varvec{z}}_1,\dots ,{\varvec{z}}_N\) of N oscillators synchronize if \(\lim _{t\rightarrow \infty } {\mathop {\left\| {{\varvec{z}}_\mu -{\varvec{z}}_{\nu }}\right\| }\limits }=0\) for all \(\mu ,\nu =1,\dots ,N\), which excludes clustered synchronization.

Theorem 3

Consider N identical, diffusively coupled FitzHugh-Nagumo-type oscillators (17). Let \({\mathcal {G}}\) denote the weighted undirected graph with time-variant and locally adaptive weights associated to the coupling network and let \({{\varvec{\Gamma }}}(\tau ,{\varvec{z}},{\varvec{c}})\) denote the associated Laplacian matrix of \({\mathcal {G}}\). The states \({\varvec{z}}_1,\dots ,{\varvec{z}}_N\) of the oscillators synchronize if the following conditions hold:

  1. 1.

    the coupling graph \({\mathcal {G}}\) is connected,

  2. 2.

    the normalized nonlinear conductance function \(f_{G}:{\mathbb {R}}\rightarrow {\mathbb {R}}\) satisfies \(\lim _{z_1\rightarrow \pm \infty }f_{G}\left( z_1\right) =\pm \infty\) and \(-\frac{\text {d} f_{G}}{\text {d}z_1}\) admits a global maximum \(K{:}=\max _{z_1\in {\mathbb {R}}} \left\{ -\frac{\text {d} f_{G}}{\text {d}z_1}\right\}\);

  3. 3.

    the algebraic connectivity \(\lambda _{2}\{ {{\varvec{\Gamma }}}(\tau ,{\varvec{z}},{\varvec{c}})\}\), i.e., the smallest nonzero eigenvalue of \({{\varvec{\Gamma }}}(\tau ,{\varvec{z}},{\varvec{c}})\), satisfies

    $$\begin{aligned} \lambda _{2} \{ {{\varvec{\Gamma }}}(\tau ,{\varvec{z}},{\varvec{c}}) \} >&\max \{0,K\}\quad \forall \ (\tau ,{\varvec{z}},{\varvec{c}}); \end{aligned}$$
  4. 4.

    The coupling’s evolution law \(\tilde{{\varvec{k}}}\) in (17c) is such that a solution to (17c) exist for all \(\tau \ge \tau _0\) regardless of the inputs \({\varvec{x}}\).

The initial conditions of the system do not matter in the above theorem. As a consequence the synchronization manifold is globally asymptotically stable and the individual oscillators all converge to the same state.

3.1 Preparations

We begin with providing a candidate for a (weak) quadratic Lyapunov function, to which we show that it is decreasing along the solutions of (17) under some assumptions on the coupling graph and the nonlinearity \(f_{G}\). Its decrease along the solutions of (17) will be key to deduce synchronization, because its 0-locus coincides with the synchronization manifold. To this end we denote by \({{\mathbbm{1}}^{\perp}_{N}}\) the subspace of vectors perpendicular to \({{\mathbbm{1}}_{N}}\), and introduce the orthogonal projection matrix

$$\begin{aligned} {{\varvec{P}}}={\mathbf {1}}_{N}-\frac{1}{N} {{\mathbbm {1}}_N} {{{\mathbbm{1}}}_{N}^{T}} \end{aligned}$$
(19)

to \({{\mathbbm{1}}^{\perp}_{N}}\). One has \({{\varvec{P}}}{{{\mathbbm{1}}}_N}={\mathbf {0}}\) and \({{\varvec{P}}}{\varvec{v}}={\varvec{v}}\) for all \({\varvec{v}} \in {{{\mathbbm{1}}}^{\perp}_{N}}\). Recall that the stacked state vectors are denoted by \({\varvec{z}}=\begin{bmatrix} {\varvec{z}}_1^T&\dots&{\varvec{z}}_{N}^T \end{bmatrix}^T\) and that \({\varvec{z}}\in {\mathbb {R}}^{2N}\) given that each oscillator has a two-dimensional state variable \({\varvec{z}}_{\mu }\). The following computations will be easier if one works with the stacked normalized voltages and currents \(\varvec{ {\zeta }}_{1}\) and \(\varvec{ {\zeta }}_{2}\) defined in (15) instead. The two are related by a permutation matrix denoted by

$$\begin{aligned} {{\varvec{\varPi }}}\in {\mathbb {R}}^{2N\times 2N}:\qquad {{\varvec{\varPi }}}{\varvec{z}}= \begin{bmatrix} \varvec{ {\zeta }}_{1} \\ \varvec{ {\zeta }}_{2} \end{bmatrix}. \end{aligned}$$
(20)

Lemma 4

Let the graph \({\mathcal {G}}\) describing the coupling network be connected and assume that the normalized supply currents \(\varvec{\iota }\) are identical, i.e., \(\varvec{\iota}=\iota {\mathbbm{1}}\) in (2), where \({{\mathbbm{1}}}\) denotes the vector of ones. Define \(V:{\mathbb {R}}^{2N}\rightarrow {\mathbb {R}}\) as

$$\begin{aligned} V\left( {\varvec{z}}\right) = \frac{1}{2}{\varvec{z}}^{\text {T}} {{\varvec{M}}}{\varvec{z}} , \quad \text {with} \quad {{\varvec{M}}} = {{\varvec{\Pi }}}^{\text {T}} \begin{bmatrix} {{\varvec{P}}}&{} 0\\ 0 &{} {{\varvec{P}}}\end{bmatrix} {{\varvec{\Pi }}}. \end{aligned}$$
(21)

Then \(V({\varvec{z}})=0\) if and only if \(\varvec{{\zeta}}_{1} = {\bar{z}_{1}} {\mathbbm{1}}\) and \(\varvec{{\zeta }}_{2} = {\bar{z}_{2}} {\mathbbm{1}}\) for \(\bar{z}_{1},\bar{z}_{2}\in {\mathbb {R}}\), i.e., \(V({\varvec{z}})=0\) \(\Leftrightarrow\) \({\varvec{z}}\in {\mathcal {S}}\), where \({\mathcal {S}}\) is the synchronization manifold to (17) as in definition 1.

Suppose there exists a monotonically increasing function \(f_{\text {m}}:{\mathbb {R}} \rightarrow {\mathbb {R}}\) and a constant \(K_{\text {l}}\ge 0\) such that,

$$\begin{aligned} f_{G}(z_1) = f_{\text {m}}(z_1) - K_{\text {l}}z_{1} \, \quad \text {for all} \quad z_{1} \in {\mathbb {R}} . \end{aligned}$$
(22)

In this case, if \(\lambda _{2}\{ {{\varvec{\Gamma }}}(\tau ,{\varvec{z}},{\varvec{c}})\}\ge K_{\text {l}}\) for all \((\tau ,{\varvec{z}},{\varvec{c}})\) then \({V}^\prime ({\varvec{z}}) \le 0\) along the solutions of (17) , where \(\lambda _{2}\{ {{\varvec{\Gamma }}}\}\) refers to the second-smallest eigenvalue of \({{\varvec{\Gamma }}}\).

The condition (22) on the normalized conductance function requires some clarification concerning when it holds. This is the goal of the next lemma.

Lemma 5

Let \(f_{G}: {\mathbb {R}} \rightarrow {\mathbb {R}}\) be a (at least once) differentiable function such that

$$\begin{aligned} \lim _{z_1\rightarrow \pm \infty } f_{G}(z_1)&= \pm \infty . \end{aligned}$$

Furthermore, let its derivative with respect to \(z_1\) be a function that is bounded from below and set

$$\begin{aligned} K{:}{=}\max _{z_1\in {\mathbb {R}}}\left\{ -\frac{\text {d}}{\text {d}z_1} f_{G}(z_1)\right\} \ . \end{aligned}$$

Then \(f_{G}\) satisfies the conditions of lemma 4 with \(K_{\text {l}}= K+ \epsilon\) for \(\epsilon >0\) arbitrarily small, and is such that (11) is semi-passive with unbounded storage function.

3.2 Proof of Theorem 3

Now, we will exploit the above results in order to show that the synchronization manifold is globally asymptotically stable under the conditions of theorem 3.

The strategy is as follows: We show that the bound on \({V}^\prime\) obtained in lemma 4 can be spelled out in terms of a vector \(\hat{{\varvec{z}}}\), defined below in (23), that is perpendicular to the kernel of \({ {{\varvec{M}}}}\) from (21). We then show that \({\varvec{z}} \in {\mathcal {S}}\) if and only if \(\hat{{\varvec{z}}}=0\). We spell out the bound on \({V}^\prime\) in terms of the norm of \(\hat{{\varvec{z}}}\) which we use to bound its integral. Afterwards we use a variation of Barbalat’s lemma (provided in the appendix, see lemmas 15 and 16) to conclude

$$\begin{aligned} \lim _{t\rightarrow \infty } \hat{{\varvec{z}}}(t) = {\mathbf {0}}, \end{aligned}$$

which implies that \({\varvec{z}} \in {\mathcal {S}}\). In order to apply Barbalat’s lemma we must show that the solutions are uniformly continuous and it is there where we will use semi-passivity of the coupled system.

Under the conditions of theorem 3, the conditions of lemma 4 are satisfied according to lemma 5. We had seen that then

$$\begin{aligned} {V}^\prime ({\varvec{z}})&\le - \varvec{ {\zeta }}_{1} ^{\text {T}} \left[ {{\varvec{P}}} {{\varvec{\Gamma }}}- K_{\text {l}} {{\varvec{P}}}\right] \varvec{ {\zeta }}_{1} - {\beta } \varvec{ {\zeta }}_{2} ^{\text {T}} {{\varvec{P}}}\varvec{ {\zeta }}_{2} . \end{aligned}$$

Lemma 4 characterizes the synchronization manifold as the linear subspace consisting of those \({\varvec{z}}\in {\mathbb {R}}^{2N}\) for which \(\varvec{{\zeta}}_{1} = {\bar{z}_{1}} {{\mathbbm{1}}}\) and \(\varvec{{\zeta }}_{2} = {\bar{z}_{2}} {{\mathbbm{1}}}\). In terms of \({{\varvec{P}}}\) and the permutation matrix \({{\varvec{\Pi }}}\) from (20) one has that

$$\begin{aligned} {\mathcal {S}}=\ker \left\{ \begin{bmatrix} {{\varvec{P}}}&{} 0\\ 0 &{} {{\varvec{P}}}\end{bmatrix} {{\varvec{\Pi }}}\right\} . \end{aligned}$$

Since \({\mathcal {S}}\) is a linear subspace of \({\mathbb {R}}^{2N}\) it is always possible to split up the state vector \({\varvec{z}}\) into a part \({\varvec{z}}_{\text {s}}\) in \({\mathcal {S}}\) and a part \(\hat{{\varvec{z}}}\) orthogonal to it. By the above this split is facilitated by constant projections:

$$\begin{aligned} {\varvec{z}}_{\text {s}} (\tau )&= {{\varvec{\Pi }}}^{-1} \begin{bmatrix} {{\varvec{P}}}-{\mathbf {1}}&{} 0\\ 0 &{} {{\varvec{P}}}-{\mathbf {1}}\end{bmatrix} {{\varvec{\Pi }}}{\varvec{z}}(\tau ), \end{aligned}$$
(23a)
$$\begin{aligned} \hat{{\varvec{z}}} (\tau )&= {{\varvec{\Pi }}}^{-1} \begin{bmatrix} {{\varvec{P}}}&{} 0\\ 0 &{} {{\varvec{P}}}\end{bmatrix} {{\varvec{\Pi }}}{\varvec{z}}(\tau ). \end{aligned}$$
(23b)

In terms of \(\varvec{ {\zeta }}_{1}\) and \(\varvec{ {\zeta }}_{2}\) this looks a bit simpler:

$$\left[\begin{array}{ll} \varvec{ {\zeta }}_{1,\text {s}} (\tau )\\ \varvec{ {\zeta }}_{2,\text {s} } (\tau) \end{array} \right] = \left[\begin{array}{cc} {{\varvec{P}}}-{\mathbf {1}}& 0\\ 0 & {{\varvec{P}}}-{\mathbf {1}}\end{array} \right] \left[\begin{array}{ll} \varvec{ {\zeta }}_{1} (\tau )\\ \varvec{ {\zeta }}_{2} (\tau) \end{array} \right] =\left[\begin{array}{ll} \bar{z}_{1}(\tau ) {{{\mathbbm {1}}}}_N \\ \bar{z}_{2}(\tau ) {{{\mathbbm {1}}}}_N \end{array} \right],$$
(24a)
$$\begin{aligned} \hspace{-10pt}\begin{bmatrix} \hat{\varvec{ {\zeta }}}_{1} (\tau )\\ \hat{\varvec{ {\zeta }}}_{2} (\tau ) \end{bmatrix}&= \begin{bmatrix} {{\varvec{P}}}&{} 0\\ 0 &{} {{\varvec{P}}}\end{bmatrix} \begin{bmatrix} \varvec{ {\zeta }}_{1} (\tau )\\ \varvec{ {\zeta }}_{2} (\tau ) \end{bmatrix}, \end{aligned}$$
(24b)

where

$$\begin{aligned} \bar{z}_{1}(\tau )=\frac{1}{N}\sum _{\mu =1}^{N} z_{\mu ,1}(\tau )\,\ \text {and}\,\ \bar{z}_{2}(\tau )=\frac{1}{N}\sum _{\mu =1}^{N} z_{\mu ,2}(\tau ) \end{aligned}$$
(24c)

are the averages of the state variables taken over the ensemble of N oscillators. One has that \({\varvec{z}}\in {\mathcal {S}}\) if and only if \(\hat{{\varvec{z}}}=0\) which is equivalent to \(\hat{\varvec{ {\zeta }}}_{1} =\hat{\varvec{ {\zeta }}}_{2} =0\).

As \({{\varvec{\Gamma }}}\) is the (symmetric) Laplacian matrix to a connected graph its only eigenvector with eigenvalue 0 is \({{{\mathbbm{1}}}_N}\). Even though \({{\varvec{\Gamma }}}\) may vary over (normal) time, its kernel does not change but is always spanned by \({{{\mathbbm{1}}}_N}\). Hence, \(\varvec{ {\zeta }}_{1,\text {s}} (\tau )\) and \(\varvec{ {\zeta }}_{2,\text {s} } (\tau )\) defined above are always contained in the kernel of \({{\varvec{\Gamma }}}\). Furthermore, \({\varvec{z}}_{s}\) and \(\hat{{\varvec{z}}}\) are given by linear combinations of the entries of \({\varvec{z}}\) with constant coefficients as can be seen from (23). We can now express the bound on \({V}^\prime\) via \(\hat{{\varvec{z}}}\) and exploit that \(\hat{{\varvec{z}}}\) can be decomposed as a function of the eigenvectors of \({{\varvec{\Gamma }}}\) excluding the eigenvector corresponding to the eigenvalue zero. Here, we have that

$$\begin{aligned} {V}^\prime ({\varvec{z}})&\le - \varvec{ {\zeta }}_{1} ^{\text {T}} \left[ {{\varvec{P}}} {{\varvec{\Gamma }}}- K_{\text {l}} {{\varvec{P}}}\right] \varvec{ {\zeta }}_{1} - {\beta } \varvec{ {\zeta }}_{2} ^{\text {T}} {{\varvec{P}}}\varvec{ {\zeta }}_{2} \\&\le -\left[ \lambda _{2}\left\{ {{\varvec{\Gamma }}}\right\} -K_{\text {l}}\right] {\mathop {\left\| {\hat{\varvec{ {\zeta }}}_{1} }\right\| }\limits }^2 -{\beta } {\mathop {\left\| {\hat{\varvec{ {\zeta }}}_{2} }\right\| }\limits }^2 . \end{aligned}$$

Hence, by our theorem’s assumption

$$\begin{aligned} \lambda _{2} \{{{\varvec{\Gamma }}}(\tau ,{\varvec{z}},{\varvec{c}}) \} > K_{\text {l}}, \quad K_{\text {l}}= \max \{0,K\}, \quad \forall \,(\tau ,{\varvec{z}},{\varvec{c}}), \end{aligned}$$

there exists a constant \(c_{0}>0\) such that,

$$\begin{aligned} {V}^\prime ({\varvec{z}})\le -c_{0} {\mathop {\left\| {\hat{{\varvec{z}}}}\right\| }\limits }^2, \end{aligned}$$

and from this, we obtain via integration that

$$\begin{aligned} V({\varvec{z}}(\tau ))-V({\varvec{z}}_{0})&\le -c_{0}\int _{0}^{\tau } {\mathop {\left\| {\hat{{\varvec{z}}}(s)}\right\| }\limits }^2 \text {d}s . \end{aligned}$$
(25)

As \(V\ge 0\), the integral is bounded from above by \(c_{0}^{-1}V({\varvec{z}}_{0})\) and from below by 0 for all \(\tau \in {\mathbb {R}}\).

According to prop. 13 the solutions of the diffusively coupled system (17) are bounded and as its right hand side is continuous we conclude that \({{\varvec{z}}}^\prime\) is bounded. This implies that \({\varvec{z}}\) is uniformly continuous (u.c.) and as a consequence, \({\mathop {\left\| {\hat{{\varvec{z}}}}\right\| }\limits }:{\mathbb {R}}\rightarrow {\mathbb {R}}\), \(\tau \mapsto {\mathop {\left\| {\hat{{\varvec{z}}}(\tau )}\right\| }\limits }\) is u.c. because \(\hat{{\varvec{z}}}= {{\varvec{P}}}{\varvec{z}}\) and linear maps are u.c. It now follows from lemma 16 that

$$\begin{aligned} \lim _{\tau \rightarrow \infty } {\mathop {\left\| {\hat{{\varvec{z}}}(\tau )}\right\| }\limits } \rightarrow 0 \quad \Rightarrow \quad \lim _{\tau \rightarrow \infty } {\varvec{z}}(\tau ) \in {\mathcal {S}}. \end{aligned}$$

We conclude that \({\mathcal {S}}\) is globally asymptotically stable and therefore proved theorem 3.

4 Application to resistive and memristive coupling

We would like to consider some applications of theorem 3. We consider ideal voltage-controlled memristors as described in [50] as coupling elements, containing a simplification to purely resistive coupling as a special case. As we consider explicit circuit elements we will spell out the conditions of theorem 3 in terms of the circuit elements’ parameters. The ideal memristors we consider are governed by the following ODE in input-state-output form, where the variables below are parametrized w.r.t. time t and carry units with the exception of the memristor’s state \(c\):

$$\begin{aligned} \dot{c}&=Q_{0}^{-1}\,i,\quad i=W(c) u,\quad c\in [0,1], \end{aligned}$$
(26a)
$$\begin{aligned} W(c)&=W_{0}+c\left[ W_{1}-W_{0}\right] . \end{aligned}$$
(26b)

Here, \(W(c)\) is the memductance with lower bound \(W_0\) and upper bound \(W_1\) as the state variable \(c\) is restricted to the interval [0, 1]. In the above description, the input u is the voltage across the device, while the current i is both the output as well as the quantity driving the evolution of the memristor’s state. Concerning simulations of this particular model we refer to the literature (cp. for instance [50]). We consider the case where each edge \(\{\mu ,\nu \}\in {\mathcal {E}}\) of the coupling graph \({\mathcal {G}}\) is realized by a memristor of the above type and such that the constants \(Q_0\), \(W_0\) and \(W_1\) may depend on the edge \(\{\mu ,\nu \}\), indicated by replacing W with \(W_{\mu \nu }\) and \(Q_0\) with \(Q_{\mu \nu ,0}\), etc., which we also collect in a diagonal matrix \({{\varvec{Q}}}_{\text {d}}\in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} \times {N_{{\mathcal {E}}}} }\). Solutions to (26) exist for all times regardless of the input u as the memristor’s state variable is restrictedFootnote 3 to the interval [0, 1]. We collect the memductances \(W_{\mu \nu }(c_{\mu \nu })\) in a diagonal matrix \({{\varvec{W}}}_{\text {d}}({\varvec{c}})\in {\mathbb {R}}^{ {N_{{\mathcal {E}}}} \times {N_{{\mathcal {E}}}} }\) and define the coupling matrix as in (14) to be

$$\begin{aligned} {{\varvec{W}}} = {{\varvec{W}}} ({\varvec{c}})= {{\varvec{N}}} {{\varvec{W}}}_{\text {d}}({\varvec{c}}) {{\varvec{N}}}^T . \end{aligned}$$
(27)

We would like to provide the full set of ODEs for identical FHNOs coupled by ideal memristors of type (26) in circuit quantities instead of the normalized ones used in (17). We denote by \({\varvec{u}}_{C} \in {\mathbb {R}}^{N}\) the vector of capacitor voltages, \({\varvec{i}}_{L} \in {\mathbb {R}}^{N}\) the vector of inductor currents, \({\varvec{j}}= {{{\mathbbm {1}}}}_N\) the vector of supply currents, and \({{\varvec{R}}}_{0} = R{\mathbf {1}}_N\), \({{\varvec{C}}} = C {\mathbf {1}}_N\), \({{\varvec{L}}} = L {\mathbf {1}}_N\) diagonal matrices carrying the resistances, capacitances and inductances respectively that occur in the FHNO circuit model. The ODE corresponding to (17) is then given by (cp. (11) and (10) concerning the translation between the two models)

$$\begin{aligned} {{\varvec{C}}}\dot{{\varvec{u}}}_{C}&= {\varvec{j}} - {\varvec{i}}_{G}({\varvec{u}}_{C}) - {\varvec{i}}_{L} - {{\varvec{W}}} ({\varvec{c}}){\varvec{u}}_{C} , \end{aligned}$$
(28a)
$$\begin{aligned} {{\varvec{L}}}\dot{{\varvec{i}}}_{L}&= {\varvec{u}}_{C} - {{\varvec{R}}}_{0} {\varvec{i}}_{L} , \end{aligned}$$
(28b)
$$\begin{aligned} \dot{{\varvec{c}}}&= {{\varvec{Q}}}_{\text {d}}^{-1} {{\varvec{W}}}_{\text {d}}({\varvec{c}}) {{\varvec{N}}}^{\text {T}}{\varvec{u}}_{C}, \end{aligned}$$
(28c)

with \({{\varvec{W}}} ({\varvec{c}})\) as in (26) and (27) and where \({\varvec{i}}_{G}\) denotes the vectorized form of \(i_G\) analogous to \({\varvec{f}}_{G}\) and \(f_{G}\).

In order to apply theorem 3 one needs to find a lower bound to \(\lambda _2\{ {{\varvec{W}}} ({\varvec{c}}(t))\}\) for all \(t\ge t_0\). One certainly has

$$\begin{aligned} \lambda _2\{ {{\varvec{W}}} ({\varvec{c}}(t))\} \ge \lambda _{2,\text {min}}{:}{=}\min _{{\varvec{c}}\in [0,1]^ {N_{{\mathcal {E}}}} }\{\lambda _2\{ {{\varvec{W}}} ({\varvec{c}})\}\}. \end{aligned}$$
(29)

Since the eigenvalues of Laplacian matrices are monotonic functions of the edge-weights [8], this minimum can be computed by setting each edge weight equal to the minimal memductance \(W_{\mu \nu ,0}\) which is assumed at \(c_{\mu \nu }=0\). Thus,

$$\begin{aligned} \lambda _{2,\text {min}}=\lambda _2\{ {{\varvec{W}}} \left( {\mathbf {0}}\right) \}\ . \end{aligned}$$
(30)

We arrive at the following corollary to theorem 3:

Corollary 6

Consider the N identical FHNOs coupled diffusively by ideal memristors in (28). Let \({\mathcal {G}}\) denote the weighted undirected graph with locally adaptive weights associated to the coupling network and let \({{\varvec{W}}} ({\varvec{c}})\) denote the associated Laplacian matrix of \({\mathcal {G}}\) described in (27). The states of the oscillators synchronize if the following conditions hold:

  1. 1.

    the coupling graph \({\mathcal {G}}\) is connected,

  2. 2.

    the nonlinear conductance function \(i_{G}:{\mathbb {R}}\rightarrow {\mathbb {R}}\) from (8) satisfies \(\lim _{u\rightarrow \pm \infty }i_{G}(u)=\pm \infty\) and \(-\frac{\text {d} i_{G}}{\text {d}u}\) admits a global maximum \(\Delta G{:}{=}\max _{u\in {\mathbb {R}}} \left\{ - \frac{\text {d} i_{G}}{\text {d}u}(u)\right\}\),

  3. 3.

    the lower bound \(\lambda _{2,\text {min}}=\lambda _2\{ {{\varvec{W}}} \left( {\mathbf {0}}\right) \}\) for the algebraic connectivity \(\lambda _{2}\{ {{\varvec{W}}} ({\varvec{c}}(t))\}\), i.e. the smallest nonzero eigenvalue of \({{\varvec{W}}} ({\varvec{c}}(t))\), satisfies

    $$\begin{aligned} \lambda _{2,\text {min}} >&\max \{0,\Delta G\}. \end{aligned}$$

For the special case of static coupling described by a constant conductance matrix \({{\varvec{W}}}\), one replaces \(\lambda _{2,\text {min}}\) with the algebraic connectivity \(\lambda _{2}\{ {{\varvec{W}}} \}\).

The quantity \(\Delta G\) in the above corollary can be interpreted as the maximal negative differential conductance in the FHNO’s circuit realization. It is striking that the only parameters in the above sufficient synchronization condition are \(\Delta G\) and the minimal possible connectivity \(\lambda _{2,\text {min}}\) while the other parameters of the FHNO play no role at all. In particular the initial values of the oscillators and the memristors do not matter. This is not in contradiction to the behavior that memristively coupled oscillators are known to exhibit from experiments, because our criterion is only met if the memristors’ high-Ohmic states and the network topology are still conductive enough to allow for synchronization. Hence, synchronization patterns that depend on the initial values can only occur if condition (iii) of Corollary 6 is not satisfied, under the assumption that the others are.

We note that for most applications, indiscriminate synchronization of neural oscillators with memristive coupling is undesirable. In this case the memristors need to be chosen such that the minimal possible connectivity \(\lambda _{2,\text {min}}\) does not exceed \(\Delta G\). Then, condition (iii) is violated and one can hope to observe the interesting phenomena that memristive networks are known for, so that Corollary 6 should be interpreted as a No-Go-theorem that tells us what kind of memristors, whose parameters enter \(\lambda _{2,\text {min}}\), fit to the chosen FHNOs, whose relevant parameter is \(\Delta G\). In view of the simulation results below one should in fact choose the memristors such that \(\lambda _{2,\text {min}}\) is orders of magnitude smaller than \(\Delta G\).

For synchronized states no current flows through the coupling network and therefore synchronization coincides with minimization of power dissipated by the coupling network. The projection \({{\varvec{P}}}\) used in the Lyapunov function (21) is in fact the Laplacian matrix to the unweighted complete graph. For static coupling, one can define an alternative Lyapunov function to (28) where the electrical counterpart to \({{\varvec{\Gamma }}}\) is replaced by the coupling matrix \({{\varvec{W}}}\). Explicitly, one can show that the storage function

$$\begin{aligned} S\left( {\varvec{u}}_{C},{\varvec{i}}_{L}\right) = {\varvec{u}}_{C}^{\text {T}} {{\varvec{W}}} {\varvec{u}}_{C} + Z_{0}^{2} \,{\varvec{i}}_{L}^{\text {T}} {{\varvec{W}}} {\varvec{i}}_{L} \end{aligned}$$
(31)

is a weak Lyapunov function to (28) leading to the same result as cor. 6 (for static coupling). The first term is in fact the power dissipated by the coupling network. This is another way to observe the implication "synchronization implies minimization of dissipated power in the coupling network" by having the Lyapunov function dominate the dissipated power.

If one chooses identical memristors, then \(\lambda _{2,\text {min}}=W_0\, \lambda _2( {{\varvec{\Gamma }}}_0)\), where \({{\varvec{\Gamma }}}_0\) describes the unweighted Laplacian matrix to the network. The algebraic connectivity \(\lambda _2( {{\varvec{\Gamma }}}_0)\) is bounded from above both by the vertex- as well as the edge-connectivity [24], defined as the minimal number of vertices (together with the edges connected to them) or edges respectively that one has to remove to render the graph disconnected. This is again bounded from above by the minimal degree of the vertices, i.e., one finds the vertex with the least neighbors and uses this as a bound. Of course, this can be much too conservative as it is possible to construct graphs with edge-connectivity equal to 1 but such that every neighbor has at least \(N_0\) edges (just take two complete graphs on \(N_0\) vertices each and join them by a single edge); but nonetheless this gives an estimate if one is unwilling to compute \(\lambda _2( {{\varvec{\Gamma }}}_0)\) directly.

5 Simulation results and discussion

5.1 Simulation results

In the first part of this section, we describe a practical guide towards the application of the derived synchronization condition. We use the ODE in terms of circuit parameters (28) and the criterion from cor. 6, once for memristive couplings and once for purely resistive couplings.

Table 1 FitzHugh-Nagumo oscillator parameters
Full size table

In order to compare the sharpness of our synchronization condition to the one of [3] and its predecessor [14] in Sect. 5.2, we simulate the same example as the latter, which is depicted in Fig. 3. Here, every vertex represents a FHNO as depicted in Fig. 1, whereas every edge represents the memristive and resistive interconnections from Sect. 4, which are chosen to be identical. The circuit parameters used within our simulations are given in Table 1.

Fig. 3
figure 3

Graph abstraction of the emulated example. Each vertex represents a FitzHugh-Nagumo oscillator. The edges represent the memristive interconnections depicted in Fig. 2 with identical coupling elements

Full size image
Fig. 4
figure 4

The nonlinear conductance function (9) and its negative derivative w.r.t. u

Full size image
Fig. 5
figure 5

Left: The plot shows the synchronization error (34) over time for 100 identical networks with different initializations. Right: 100 randomly chosen initializations of the system with coupling topology as in Fig. 3, where \(\mu\) labels the oscillators

Full size image

The nonlinear conductance function \(i_{G}\) is the one given in (9) with \(U_0\) and \(G_0\) as in table 1.

In order to apply our synchronization condition, we must first calculate the negative derivative of \(i_{G}\), which is bounded from above and has the maximum \(\Delta G=G_0= 100 \, \upmu \text {S}\), see Fig. 4. In a practical scenario, one must measure the (i, u)-curve of the nonlinear conductance, calculate the negative derivative numerically, and find its global maximum. According to cor. 6, the next step is to verify, whether for all times and memductance states, the connectivity of the graph is greater than \(\Delta G\).

The connectivity of the unweighted graph is given by

$$\begin{aligned} \lambda _{2}\{ {{\varvec{\Gamma}}}_0\} = 1, \quad \text {with} \quad {{\varvec{\Gamma}}}_0 = {\mathbf{{diag}}} ({{\varvec{A}}}_0 {\mathbbm{1}}) - {{\varvec{A}}}_0, \end{aligned}$$

where \({{\varvec{\Gamma }}}_0\) denotes the unweighted graph’s Laplacian and \({{\varvec{A}}}_0\) is the unweighted adjacency matrix. For uniform static coupling of strength \(G_{\text {c}}\), the weighted graph’s Laplacian is given by \({{\varvec{W}}} =G_{\text {c}} {{\varvec{\Gamma }}}_0\) and so \(\lambda _{2}\{ {{\varvec{W}}} \}=G_{\text {c}}\) in this case. For identical memductances as coupling elements with high-ohmic state \(W_0\) one has by (30) that

$$\begin{aligned} \lambda _{2,\text {min}}= {{\varvec{W}}} ({\mathbf {0}})=W_0\lambda _2\left\{ {{\varvec{\Gamma }}}_0\right\} =W_0. \end{aligned}$$
(32)

Hence, \({{\varvec{W}}}\) satisfies the inequality of corollary 6 if

$$\begin{aligned} G_{\text {c}}> \Delta G = 100 \, \upmu \text {S},\quad W_{0} > \Delta G = 100 \, \upmu \text {S} . \end{aligned}$$
(33)

We stress that it is not necessary to pick the coupling weights uniformly. Any coupling graph \({{\varvec{W}}}\) with \(\lambda _{2}\{ {{\varvec{W}}} ({\mathbf {0}})\}>100 \, \upmu \text {S}\) works here.

We display in Fig. 5 the results of a simulation for 100 copies of the system with memristive coupling for the case of \(W_0=100.01 \, \upmu \text {S}\) just above the boundary of our condition. The memristors are initialized identically in the high-ohmic state to give the system the "slowest" start possible. The oscillators themselves are initialized randomly such that \(i_{L,0}=0\,\text {A}\) and \(u_{C,0}\) is distributed uniformly in \([-400,400]\ \text {mV}\), where \(400\ \text {mV}\) is the maximal amplitude of the uncoupled FHNO’s stable limit cycle. The quantity plotted is

$$\begin{aligned} e(t)=\frac{1}{e_{\text {max}}}\sum _{\mu =2}^{N} \left[ u_{C,\mu }(t)-u_{C,1}(t)\right] ^2, \end{aligned}$$
(34)

with \(e_{\text {max}}\) chosen s.t. \(\max _te(t)=1\). This serves as a measure for the synchronization error although it neglects the second state variable \(i_L\) but has the advantage that it is also a measure for the power distributed by the coupling network. We observe that eventually all FHNOs synchronize but that the time required can vary. With the chosen parameters the period of a single oscillator is about \(2\,\text {ms}\) so that synchronization occurs within at most 10 oscillation cycles for the chosen range of initial values. We also observe that the power dissipated by the coupling network is not monotonically decreasing but exhibits a damped oscillation in magnitude with a similar period as the FHNO.

Fig. 6
figure 6

The synchronization error (34) over time for static coupling and a wide range of conductances. The y-axis starts at the boundary \(G_{\text {c}}=100\ \upmu \text {S}\) of the sufficient condition. Synchronization is achieved for coupling conductances orders of magnitude lower than required but with reduced and varying speed

Full size image

We have also tested the distance between our sufficient synchronization condition from the (unknown) necessary one by running a series of simulations for different (uniform) static coupling conductances on the graph depicted in Fig. 3 with the parameters in Table 1. The results are displayed in Fig. 6, where the boundary of our condition \(G_{\text {c}}=100 \, \upmu \text {S}\) is placed at the very top of the scale. We observe that synchronization is (eventually) achieved for conductances three orders of magnitude smaller than the criterion requires. However, we also observe that the time at which synchronization is achieved increases drastically from a few oscillation cycles to more than 1000.

While we did not observe a large variance in synchronization speed in the memristive scenario, we did so in the static case for situations where the criterion was not met. We think that this suggests that the gap between sufficient and necessary conditions for synchronization needs additional exploration. We also think that our sufficient condition should be further refined and augmented by criteria that guarantee a certain synchronization speed, potentially in dependence of the initial values, as it may be unacceptable to have a required synchronization time of a factor more than a 1000 time larger than the system’s time scale defined by the FHNOs period.

5.2 Comparison with the literature

To the best of our knowledge there exists no sufficient synchronization condition for memristively coupled FHNOs beyond the situation of two memristively coupled oscillators [32] although the phenomenon has been studied numerically on numerous occasions for two or more oscillators not necessarily of FHN-type [26, 33, 37, 53]. While there exist sufficient conditions towards systems coupled by a time-variant coupling matrix [3, 27], the time-variance there is not allowed to be depend on the oscillators’ state or additional state variables which excludes both nonlinear coupling elements as well as locally adaptive ones. A few types of diffusive, locally adaptive coupling have been studied in [15, 16], where the coupling strength is only allowed to be increasing so that our general form of admissible coupling and its evolution law also adds to the existing literature. For purely static, linear coupling the picture is more complicated, which is why we discuss this topic to more extent below and summarize our discussion in table 2. Given the vast amount of literature on the topic we can of course only include a selection of publications. We begin with comparing the sharpness of our synchronization condition for static diffusive coupling to the one presented in [14] as this is also based on semi-passivity, albeit working with a non-smooth Lyapunov function.We also compare our criterion briefly to the more recent but not as strong as possible result of [31] . Afterwards we analyze why our result coincides with the bound obtained from contraction theory more recently [3, 27]. We also relate it to QUAD-conditions [17] and conclude with a direct approach based on Lyapunov’s method [56]. Although [56] was published almost 30 years ago, it still sets the bar of the sharpest sufficient condition; it is reached but not outperformed by most of the other works, including the specialization of our own result to the static linear case.

The following unitless version of the FitzHugh-Nagumo model is used in [4]:

$$\begin{aligned} {\dot{x}}&= -\left[ \frac{x^3}{3}-x\right] +I + u , \end{aligned}$$
(35a)
$$\begin{aligned} {\dot{y}}&= \varepsilon [x + a - b y] , \end{aligned}$$
(35b)

where \(b,\varepsilon ,a>0\) and I are constants and u denotes the input. We remark that while our model (11) allows more general nonlinearities in x than (10d) as well as a dependence of \({\dot{x}}\) on y, it only treats the case \(a=0\). The relevant ingredient for our criterion is the nonlinearity

$$\begin{aligned} f_{G}(z_1) = \frac{1}{3} z_1^3 - z_1 \end{aligned}$$
(36)

in (11). Since the maximum of \(-\frac{\text {d}}{\text {d}z_1}f_{G}\) evaluates to 1 (the slope of \(f_{G}\) is extremal at the origin), we have the condition \(\lambda _{2}\{ {{\varvec{\Gamma }}}\} > 1\). The condition presented in [14] spells out as

$$\begin{aligned} \lambda _2\{ {{\varvec{\Gamma }}}\}\ge 1+\varepsilon +\frac{\beta _1^2}{3}, \end{aligned}$$
(37)

where \(\beta _1\) is the bound on x resulting from the fact that the trajectories are ultimately bounded. According to [14] common values (of the parameters relevant for the criterion) for biologically plausible firing behavior of the FHN-oscillators are \(\varepsilon \approx \frac{1}{12}\) and \(\beta _1\approx 2\) which would require the connectivity to fulfill \(\lambda _{2}\{ {{\varvec{\Gamma }}}\} > 2.41\). But even without explicit values we observe that our condition is sharper than (37) which is always strictly greater than 1.

The bound \(\lambda _{2}\{ {{\varvec{\Gamma }}}\} > 1\) for the above FHN-model (35) is the sharpest available sufficient condition for synchronization and has also been obtained in [3, cor. 4.1] with the methods of contraction theory, in [27, thm. 30] as part of the generalization of contraction theory to so-called semi-contracting systems, and in [56] by a direct application of Lyapunov’s method.

[31] considers static diffusive coupling of FitzHugh-Nagumo and van-der-Pol oscillators and their result is weaker than ours and that of [3, 27, 56] and ours, the reason being that they do not consider the coupling graph as a weighted graph. As a consequence, their criterion compares the smallest coupling weight to the quotient of the unweighted graph’s algebraic connectivity \(\lambda _2\) divided by a quantity that correspond to our \(\Delta G\). This provides a weaker result and has some peculiar properties: Given a network that meets their criterion, one could always add a new edge with small enough coupling to produce a situation in which their criterion is not met. However, the algebraic connectivity of a weighted graph is monotone in the sense that adding new edges only increases connectivity. We think that a synchronization criterion that uses the algebraic connectivity should reflect this fact.

The general sufficient condition of [3] is spelled out in terms of the log-matrix norm \(\mu _{2,P}\) and it is also obtained for more general log-matrix (semi-)norms in [27]. According to [27] (this is the phrasing of [9, thm. 5.19]) the diffusively coupled system

$$\begin{aligned} \dot{{\varvec{x}}}_{\mu }={\varvec{f}}\left( t,{\varvec{x}}_{\mu }\right) -\sum _{\nu }a_{\mu \nu }\left[ {\varvec{x}}_{\mu }-{\varvec{x}}_{\nu }\right] ,\ \mu \in \left\{ 1,\dots ,N\right\} . \end{aligned}$$
(38)

synchronizes if there exist \(p\in \left[ 1,\infty \right]\), positive definite \({{\varvec{Q}}}\in {\mathbb {R}}^{n\times n}\) and \(\varepsilon >0\) such that for every \((t,{\varvec{x}})\in {\mathbb {R}}_{\ge 0}\times {\mathbb {R}}^{n}\) one has

$$\begin{aligned} \mu _{p, {{\varvec{Q}}}}\left( D{\varvec{f}}(t,{\varvec{x}})\right) \le \lambda _{2}\left\{ {{\varvec{L}}}\right\} -\varepsilon , \end{aligned}$$
(39)

where \({{\varvec{L}}}\) is the Laplacian matrix to the weighted undirected graph with adjacency matrix \({{\varvec{A}}}\) and \(\mu _{p, {{\varvec{Q}}}}\) denotes the weighted log-matrix-norm. The only one needed explicitly in the following is \(\mu _{2, {{\varvec{Q}}}}( {{\varvec{M}}})=\lambda _{\text {max}}\left( {{\varvec{Q}}}\frac{ {{\varvec{M}}}+ {{\varvec{M}}}^{\text {T}}}{2} {{\varvec{Q}}}^{-1} \right)\). Now by the Demidovic-lemma ( [19], cp. [9, lem. 3.1]) the Jacobian \(D{\varvec{f}}\) satisfies \(\mu _{2, {{\varvec{Q}}}^{\frac{1}{2}}}\left( D{\varvec{f}}(x)\right) \le C\) for some positive-definite matrix \({{\varvec{Q}}}\) if and only if \({\varvec{f}}\) satisfies the one-sided Lipschitz condition

$$\begin{aligned} \left[ {\varvec{f}}({\varvec{x}})-{\varvec{f}}({\varvec{y}})\right] ^{\text {T}} {{\varvec{Q}}}\left[ {\varvec{x}}-{\varvec{y}}\right] \le C\left\| {\varvec{x}}-{\varvec{y}}\right\| _{2, {{\varvec{Q}}}^{\frac{1}{2}}}^2 \end{aligned}$$
(40)

for all \({\varvec{x}},{\varvec{y}}\in {\mathbb {R}}^n\), where \(\left\| {\varvec{x}}\right\| _{2, {{\varvec{Q}}}}{:}{=}\left\| {{\varvec{Q}}}{\varvec{x}}\right\| _{2}\), so that for symmetric \({{\varvec{Q}}}\) one has \({\mathop {\left\| {{\varvec{x}}-{\varvec{y}}}\right\| }\limits }_{2, {{\varvec{Q}}}^{\frac{1}{2}}}^2=\left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{Q}}}^{\frac{1}{2}} {{\varvec{Q}}}^{\frac{1}{2}}\left[ {\varvec{x}}-{\varvec{y}}\right] =\left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{Q}}}\left[ {\varvec{x}}-{\varvec{y}}\right]\). One can check that the nonlinear function \(f_{G}\) satisfies the one-sided Lipschitz condition

$$\begin{aligned} -\left[ f_{G}(x)-f_{G}(y) \right] \left[ x-y\right] \le K_{\text {l}}\left[ x-y\right] ^2, \end{aligned}$$
(41)

where \(K_{\text {l}}\) is as in (22), i.e., such that \(f_{\text {m}}(z_1)=f_{G}(z_1)+K_{\text {l}}z_1\) is strictly monotonic increasing. Due to the choice of coordinate transformation used to describe the unitless model, this translates to the following condition on \({\varvec{f}}\):

$$\begin{aligned} \left[ {\varvec{f}}({\varvec{x}})-{\varvec{f}}({\varvec{y}})\right] ^{\text {T}}\left[ {\varvec{x}}-{\varvec{y}}\right] \le K_{\text {l}}\left\| {\varvec{x}}-{\varvec{y}}\right\| _{2}^2. \end{aligned}$$
(42)

Therefore, our condition \(\lambda _2\{ {{\varvec{\Gamma }}}\}>\max \{0,K\}\) with \(K=\max _{z_1\in {\mathbb {R}}}\left\{ -\frac{\text {d}f_{G}}{\text {d}z_1}(z_1)\right\}\) implies that (39) is satisfied due to the equivalence of one-sided Lipschitz-conditions and bounds on log-matrix-norms established by the Demidovic-lemma.

QUAD-conditions, which have been used for instance in [15,16,17,18], are a variation of one-sided Lipschitz-conditions:

Definition 7

Let \({\varvec{f}}:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}\) be a vector field, let \({{\varvec{\Delta }}}\) be a diagonal matrix and \(\omega >0\) a real number. One says that \({\varvec{f}}\) is \(QUAD( {{\varvec{\Delta }}},\omega )\) if it satisfies the inequality

$$\begin{aligned}{}[{\varvec{x}}-{\varvec{y}}]^{\text {T}}[{\varvec{f}}({\varvec{x}})-{\varvec{f}}({\varvec{y}})]&\le [{\varvec{x}}-{\varvec{y}}]^{\text {T}} {{\varvec{\Delta }}}[{\varvec{x}}-{\varvec{y}}]\\&\quad -\omega [{\varvec{x}}-{\varvec{y}}]^{\text {T}}[{\varvec{x}}-{\varvec{y}}]. \end{aligned}$$

There is an intimate connection between QUAD-conditions and contraction theory summarized in [17]. Denote by \({{\varvec{\Lambda }}}_0=- {{\varvec{\Gamma }}}\) the negative Laplacian to (38), let \({\varvec{f}}\) be \(QUAD( {{\varvec{\Delta }}}_0,\omega )\) for some \(\omega >0\) and introduce the following matrices:

$$\begin{aligned} {{\varvec{\Lambda }}}{:}= {{\varvec{\Lambda }}}_{0}\otimes {\mathbf {1}}_{n},\ {{\varvec{\Pi }}}{:}{=} {{\varvec{P}}}_{N}\otimes {\mathbf {1}}_{n},\ {{\varvec{\Delta }}}{:}{=}{\mathbf {1}}_{N}\otimes {{\varvec{\Delta }}}_{0}, \end{aligned}$$
(43)

where \({{\varvec{P}}}_N\) is the orthogonal projection to the complement of \({{{\mathbbm{1}}}}_N\). According to [18, thm. 2]) the network of oscillators (38) synchronizes if the matrix \(\left[ \varvec{{\Pi \Delta }+{\Pi \Lambda }}\right]\) is negative semi-definite. The negative semi-definiteness of the matrix \(\left[ \varvec{{\Pi \Delta }+{\Pi \Lambda }}\right]\) can be characterized by an inequality between the maximal eigenvalue of \({{\varvec{\Delta }}}_{0}\) and the algebraic connectivity of \({{\varvec{\Gamma }}}\) via a standard argument. To our surprise we could not find this in the literature so we record a proof for the reader’s convenience.

Proposition 8

The matrix \(\left[ \varvec{{\Pi \Delta }+{\Pi \Lambda }}\right]\) is negative semi-definite if and only if the largest eigenvalue \(\lambda _{\text {max}}\left\{ {{\varvec{\Delta }}}_{0}\right\}\) of \({{\varvec{\Delta }}}_{0}\) and the algebraic connectivity \(\lambda _{2}\left\{ {{\varvec{\Gamma }}}\right\}\) satisfy \(\lambda _{\text {max}}\left\{ {{\varvec{\Delta }}}_{0}\right\} \le \lambda _{2}\left\{ {{\varvec{\Gamma }}}\right\}\).

In order to compare our synchronization condition to the one of [18, thm. 2] based on QUAD-conditions we need to derive QUAD-estimates for the FHNO in dependence of the parameters. We derive the following estimate in the appendix:

Lemma 9

Consider the FitzHugh-Nagumo oscillator in the form

$$\begin{aligned} \begin{bmatrix}{\dot{x}}\\ {\dot{y}} \end{bmatrix}={\varvec{f}}\left( \begin{bmatrix}x\\ y \end{bmatrix}\right) =\begin{bmatrix}-ax^{3}+bx-c_{0}y+j\\ c_{1}x-c_{2}y \end{bmatrix} \end{aligned}$$
(44)

with \(a,b,c_{0},c_{1},c_{2}\ge 0\) and constant current injection j. Then \({\varvec{f}}\) is QUAD for \({{\varvec{\Delta }}},\omega\) such that \(\Delta _{11}-\omega \ge b+\frac{\left| c_{1}-c_{0}\right| }{2}\) and \(\Delta _{22}-\omega \ge -c_{2}+\frac{\left| c_{1}-c_{0}\right| }{2}\).

In the case of static diffusive coupling one can now compare the criteria obtained from QUAD-conditions and ours. As \(\omega\) can be chosen arbitrarily small one has that f is \(QUAD(\Delta ,\omega )\) for all \(\Delta\) such that \(\Delta _{11}>b+\frac{\left| c_{1}-c_{0}\right| }{2}\) and \(\Delta _{22}>-c_{2}+\frac{\left| c_{1}-c_{0}\right| }{2}\) and as all parameters are nonnegative \(\Delta _{11}\ge \Delta _{22}\). By the use of Proposition 8 this results in the sufficient condition

$$\begin{aligned} b+\frac{\left| c_{1}-c_{0}\right| }{2}>\lambda _{2}\left( {{\varvec{\Gamma }}}\right) . \end{aligned}$$
(45)

Our condition \(K=\max _{z_1\in {\mathbb {R}}}\left( -\frac{\text {d}f_{G}(z_1)}{\text {d}z_1}\right) <\lambda _{2}\left( {{\varvec{\Gamma }}}\right)\) spells out in terms of the above parameters as \(b<\lambda _{2}\left\{ {{\varvec{\Gamma }}}\right\}\) since the slope of the conductance function is extremal at the origin. Hence, our condition appears to be sharper in this case, because it does not depend on the cross terms proportional to \(c_{0},c_{1}\) that occur in the above inequalities. However, we had also seen that a coordinate transformation can ensure \(c_1=c_0\) so that the two in fact coincide if the ODE is set up appropriately.

Lastly, we would like to present one of the sufficient synchronization conditions from Wu and Chua [56] dating back to 1995. In this work the authors provide a rather general and powerful framework to study synchronization of identical oscillators subject to static linear coupling with particular emphasis on allowing different classes of coupling matrices. Their results are therefore divided according to those classes of coupling matrices and we only cite the case that applies to the system (38):

Assume there exist a positive definite matrix \({{\varvec{V}}}\), a diagonal matrix \({{\varvec{T}}}=\text {diag}(t_1,\dots ,t_n)\) and a continuous, nondecreasing function \(c:{\mathbb {R}}\rightarrow {\mathbb {R}}\) satisfying \(c(0)=0\) and \(c(s)>0\) for all \(s\ne 0\). If [56, eq. 17] holds, i.e., if

$$\begin{aligned} \left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{V}}}\left[ {\varvec{f}}({\varvec{x}},t)-{\varvec{f}}({\varvec{y}},t) - {{\varvec{T}}}\left[ {\varvec{x}}-{\varvec{y}}\right] \right] \end{aligned}$$
(46a)
$$\begin{aligned} \le -c\left( \left\| {\varvec{x}}-{\varvec{y}} \right\| \right) \, \end{aligned}$$
(46b)

for all \({\varvec{x}},{\varvec{y}}\in {\mathbb {R}}^n\) and t then (38) synchronizes if \(\lambda _{2}\{ {{\varvec{L}}}\}>t_i\) for all i (This is [56, thm. 7, cor. 3]). One observes that (46) is a generalized form of QUAD-condition as it is more flexible due to the matrix \({{\varvec{V}}}\). Based on the previous example one sees that this provides a more powerful criterion than [18, thm. 2] based on QUAD-conditions as one has a more tractable method to sharpen the bound by variation of \({{\varvec{V}}}\). One can check that (11) satisfies (46) for

$$\begin{aligned} {{\varvec{V}}}={\mathbf {1}}\ \text {and }\ {{\varvec{T}}}=\text {diag}(t_1,0) \end{aligned}$$
(47)

with \(t_1\ge K\). Thus, for \(t_1=K\) one recovers the condition \(\lambda _2\{ {{\varvec{L}}}\}>K\) from thm. 3 for the static case which coincided with [3, cor. 4.1]. This is again no coincidence, as (46) and the one-sided Lipschitz-condition (40) are similar and at least for diagonal matrices lead to equivalent results. To see this, spell out (46) with, for simplicity, diagonal \({{\varvec{V}}}= {{\varvec{Q}}}\):

$$\begin{aligned} \left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{Q}}}\left[ {\varvec{f}}({\varvec{x}},t)-{\varvec{f}}({\varvec{y}},t)\right] \\ \le \left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{Q}}} {{\varvec{T}}}\left[ {\varvec{x}}-{\varvec{y}}\right] -c\left( \left\| {\varvec{x}}-{\varvec{y}} \right\| \right) \\ \le \left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{Q}}} {{\varvec{T}}}\left[ {\varvec{x}}-{\varvec{y}}\right] \\ \le \max _{i}\{t_i\} \left\| {\varvec{x}}-{\varvec{y}}\right\| _{2, {{\varvec{Q}}}^{\frac{1}{2}}}^2, \end{aligned}$$

which is a one-sided Lipschitz-condition (40) with \(C=t_{\text {max}}=\max _{i}\{t_i\}\).

Conversely assume that (40) holds for diagonal, positive definite \({{\varvec{Q}}}\) and \(C>0\), then

$$\begin{aligned} \left[ {\varvec{f}}({\varvec{x}})-{\varvec{f}}({\varvec{y}})\right] ^{\text {T}} {{\varvec{Q}}}\left[ {\varvec{x}}-{\varvec{y}}\right] \le C\left\| {\varvec{x}}-{\varvec{y}}\right\| _{2, {{\varvec{Q}}}^{\frac{1}{2}}}^2 \end{aligned}$$
(48)
$$\begin{aligned} = C\left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{Q}}}\left[ {\varvec{x}}-{\varvec{y}}\right] \le \left[ {\varvec{x}}-{\varvec{y}}\right] ^{\text {T}} {{\varvec{T}}}_0 {{\varvec{Q}}}\left[ {\varvec{x}}-{\varvec{y}}\right] \end{aligned}$$
(49)

for \({{\varvec{T}}}_0=\text {diag}(t_1,\dots ,t_n)\) such that \(t_i\ge C\). Hence, (46) holds for all \({{\varvec{T}}}= {{\varvec{T}}}_0+\varepsilon {\mathbf {1}}\) with \(\varepsilon >0\) as \(-\varepsilon \Vert {\varvec{x}}-{\varvec{y}}\Vert ^2\) provides a function \(-c\left( \Vert {\varvec{x}}-{\varvec{y}}\Vert \right)\) with the required properties. The sharpest choice consists of \(t_i=C\) so that \(t_{\text {max}}=C\) which is the relevant part of \({{\varvec{T}}}\) towards the synchronization condition. In conclusion, although (46) and (40) are not fully equivalent due the higher flexibility in the matrix \({{\varvec{T}}}\) they lead to the same synchronization condition, because only the maximal eigenvalue of \({{\varvec{T}}}\) matters.

It is fascinating that the sharpness of sufficient synchronization conditions for oscillators coupled by static linear coupling has not increased over the last 30 years. We wonder, if this is only due to the similarities of the used methods, or if there is something deeper to be learned here.

Table 2 A comparison of the different approaches to synchronization mentioned, which for the static case all compare the algebraic connectivity \(\lambda _2\{ {{\varvec{L}}}\}\) to some other parameter that is set to 1 in the column "main bound"
Full size table

6 Conclusion and outlook

This work is dedicated to the derivation of a sufficient synchronization condition for a network of (identical) diffusively coupled FitzHugh-Nagumo oscillators with time-variant, state-dependent and locally adaptive coupling, which among others includes coupling by memristors and/or nonlinear conductances. We started by briefly reviewing the necessary theory of dissipative and semi-passive systems and provided a description of uncoupled and diffusively coupled FitzHugh-Nagumo-type oscillators by ordinary differential equations fitting in this framework. Then, we provided a Lyapunov function candidate V and derived conditions on the oscillators’ nonlinear conductance relative to the connectivity of the network for \({\dot{V}}\) to be decreasing along the solutions of the system. Our condition on the nonlinear conductance function \(i_G\) is that it admits a modification by a linear term \(i_\text {m}(u)=i_G(u)+\Delta G\,u\) such that \(i_\text {m}\) is strictly monotonically increasing. We then showed that the oscillators synchronize globally, i.e., from any initial state, if the connectivity \(\lambda _2\{ {{\varvec{W}}} \}\) of the network exceeds the slope \(\Delta G\) of this modification for all times. To examine the validity and sharpness of this condition, we conducted a few numerical experiments. Finally, we showed that our synchronization condition specialized to linear static coupling is as sharp as those from the literature [3, 27, 56] but also easy to use for practitioners who need these conditions to be given in terms of parameters of the ideal circuit. Towards application to real-world systems it is an open question to which extent our result applies to noisy systems. One would expect that small magnitudes of noise do not affect the system’s behavior but a quantitative answer requires a different kind of analysis. Towards this one would have to replace the deterministic ODEs by stochastic ones and study the system as a stochastic process.

In future research, we would like to sharpen our synchronization condition and to loosen the condition on \(\lambda _2\{ {{\varvec{W}}} \}\) for state-dependent and adaptive coupling. In the current form the condition does not take into account that only a minor part of phase space is reached by a trajectory but the criterion considers the minimum of \(\lambda _2\{ {{\varvec{W}}} \}\) over all times and phase space. We would therefore like to spell out this conditions with respect to the initial values in the sense that we want to study how the minimum over a given trajectory evaluates and depends on the initial value. We also plan to investigate if our approach can be adapted to the case of heterogeneous oscillators and clustered synchronization.