# Synchronization in Networks With Heterogeneous Adaptation Rules and Applications to Distance-Dependent Synaptic Plasticity

^{1}Institut für Mathematik, Technische Universität Berlin, Berlin, Germany^{2}Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

This work introduces a methodology for studying synchronization in adaptive networks with heterogeneous plasticity (adaptation) rules. As a paradigmatic model, we consider a network of adaptively coupled phase oscillators with distance-dependent adaptations. For this system, we extend the master stability function approach to adaptive networks with heterogeneous adaptation. Our method allows for separating the contributions of network structure, local node dynamics, and heterogeneous adaptation in determining synchronization. Utilizing our proposed methodology, we explain mechanisms leading to synchronization or desynchronization by enhanced long-range connections in nonlocally coupled ring networks and networks with Gaussian distance-dependent coupling weights equipped with a biologically motivated plasticity rule.

## 1 Introduction

In nature and technology, complex networks serve as a ubiquitous paradigm with a broad range of applications from physics, chemistry, biology, neuroscience, socioeconomic, and other systems [1]. Dynamical networks consist of interacting dynamical units, such as neurons or lasers. Collective behavior in dynamical networks has attracted much attention in recent decades. Depending on the network and the specific dynamical system, various synchronization patterns with increasing complexity were explored [2–5]. Even in simple models of coupled oscillators, patterns such as complete synchronization [6, 7], cluster synchronization [8–11], and various forms of partial synchronization have been found, such as frequency clusters [12], solitary [13–15], or chimera states [16–20]. In particular, synchronization is believed to play a crucial role in brain networks, for example, under normal conditions in the context of cognition and learning [21, 22], and under pathological conditions, such as Parkinson’s disease [23–25], epilepsy [26–29], tinnitus [30, 31], schizophrenia, to name a few [32].

The powerful methodology of master stability function [33] has been a milestone for the analysis of synchronization phenomena. This method allows for the separation of dynamic and structural features in dynamical networks. It greatly simplifies the problem by reducing the dimension and unifying the synchronization study for different networks. Since its introduction, the master stability approach has been extended and refined for various complex systems [34–42], and methods beyond the local stability analysis have been developed [43–47]. More recently, the master stability approach has been extended to another class of oscillator networks with high application potential, namely adaptive networks [48].

Adaptive networks are commonly used models for various systems from nature and technology [49–57]. A prominent example are neuronal networks with spike-timing dependent plasticity, in which the synaptic coupling between neurons changes depending on their relative spiking times [58–61]. There are a large number of studies investigating the dynamic properties induced by this form of synaptic plasticity [62]. However, analysis is usually limited to only one or two forms of spike timing-dependent plasticity within a neuronal population. On the other hand, experimental studies indicate that different forms of spike timing-dependent plasticity may be present within a neuronal population, where the form depends on the connection structure between the axons and dendrites [63]. Among all structural aspects, an important factor for the specific form of the plasticity rule is the distance between neurons [64–66]. More specifically, it has been found that the plasticity rule between proximal or distal neurons, respectively, can change from Hebbian-like to anti-Hebbian-like [67, 68].

This work introduces a methodology to study synchronization in adaptive networks with heterogeneous plasticity (adaptation) rules. As a paradigmatic system, we consider an adaptively coupled phase oscillator network [69–75], which is proven to be useful for predicting and describing phenomena occurring in more realistic and detailed models [76–79]. More specifically, in the spirit of the master stability function approach, we consider the synchronization problem as the interplay between network structure and a heterogeneous adaptation rule arising from distance- (or location-)dependent synaptic plasticity. For a given heterogeneous adaptation rule, our master stability function provides synchronization criteria for any coupling configuration. As illustrative examples, we consider a nonlocally coupled ring with biologically motivated plasticity rule, and a network with a Gaussian distance-dependent coupling weights. We explained such intriguing effects as synchronization or desynchronization by enhancement of long-distance links.

We introduce the model in Section 2. Building on findings from [48], we develop a master stability approach in Section 3 that takes a heterogeneous adaptation rule in account. In Section 4.1, we provide an approximation of the structural eigenvalues that determine the stability of the synchronous state. We then consider two different setups: a nonlocally coupled ring in Section 4.2 and a weighted network with Gaussian distance distribution of coupling weights in Section 4.3. Both systems are equipped with a biologically motivated plasticity rule. In Section 5, we summarize the results.

## 2 Model

In this work, we study the synchronization on networks with adaptive coupling weights, where the adaptation (plasticity) rule depends on the distance between oscillators (neurons). We consider the model of adaptively coupled phase oscillators, which has proven to be useful for understanding dynamics in neuronal systems with spike timing-dependent plasticity [77, 79, 48]. The model reads as follows:

where *i*th oscillator, *j* to *i*, *ω* denotes the natural frequency of each oscillator, and *A* describing the network connectivity. The time scales of the “fast” phase oscillators and “slow” coupling weights are separated by the parameter ϵ, which we assume to be small *g* and

The main difference of system Eqs. 1, 2 from the models considered previously in the literature [40, 70, 71, 74, 82], is that the plasticity functions

A solution to Eqs. 1, 2 is called *phase-locked* if, for all *collective frequency**in-phase synchronous* or, short, *synchronous* state.

In the case of in-phase synchronous state, we can set

where we assume that the weighted row sum

In the following section, we show how the stability of the synchronous state is determined in a master-stability-like approach.

## 3 Master Stability Approach

In Section 2, we have introduced a general class of models and the synchronous state, that are considered throughout this paper. In this section, we derive a framework for the local stability analysis of the synchronous states. We note that the master stability approach for homogeneous adaptations

To describe the local stability, we introduce the variations

where *N*-dimensional vector containing the perturbations

The time-independent matrices *B* and *C* are

where

Note that due to the shift symmetry of Eqs. 1, 2, the Jacobian *J* in Eq. 5 is time independent. Therefore, the real parts of the *λ* of *J* are the Lyapunov exponents of the synchronous state and hence determine its local stability. In the following proposition, we exploit the fact that *J* contains a large diagonal block *J*.

Proposition 1. *Suppose**is an in-phase synchronous state of* Eqs. 1, 2*. Then its linear stability is determined by the* *-dimensional linear system*

*where* *and* *are as in* Eq. 5*and the**weighted Laplacian matrix* *possesses the following elements*

Proof. We remind that system Eq. 5 determines the spectrum (Lyapunov exponents) of the synchronous state. The Jacobian matrix in Eq. 5 is sparse with a large

This system has at least

with *R*. Here *K* is an

where *N* and *Q*. Note that the eigenvalues *J* in Eq. 5 in the limit of slow adaptation, i.e.,

Proposition 2. *Assume that**is diagonalizable, with**being the associated diagonal matrix and Q the corresponding transformation. Let**be an in-phase synchronous state of*Eqs. 1, 2*Then, the local stability of this state is determined by the solutions of N quadratic equations, which are given up to the first order in* ϵ *as*

*where**are the eigenvalues of**located on the diagonal of**and**are the corresponding diagonal elements of**If**and**commute, then*Eq. 11*is exact, and**are the eigenvalues of*

where we have used the transformation *Q* that brings

The latter equation is almost diagonal. The only off-diagonal components remain from *F* with entries

where *Q*. In this case, the diagonal elements *N*Eq. 11 determine the stability of the synchronous state. More precisely, the real parts of theses solutions determine the Lyapunov exponents. If *g*, the connectivity, and the adaptation structure. This dependence, however, is only encoded in the two complex parameters *ν*, we obtain two explicit stability conditions from Eq. 11: The synchronous state is locally stable (

These conditions agree with the black dashed lines in Figure 1 and are used subsequently to describe stability for certain network models.

**FIGURE 1**. The master stability function **(A–E)**. The dashed black line describes the border between regions corresponding to local stability and instability, respectively. Parameters: **(A)****(B)****(C)****(D)****(E)**

## 4 Synchronization on Networks With Distance-Dependent Plasticity

In the previous section, we established a generic analytic tool for studying stability of synchronous states. In this section, we focus on the application of the tool to certain network models. For the rest of the work, we restrict our attention to the following generalization of the Kuramoto-Sakaguchi system with distance-dependent synaptic plasticity

The plasticity function *h* depends on the phase difference

With this form of the adaptation function, we have a symmetric

where the distance dependence is encoded in the phase shift function

In Figure 2A, we illustrate the distance-dependent plasticity function Eqs. 18–20 for a network of

**FIGURE 2**. Panel **(A)** shows the plasticity function **(left)** correspond to the colors of the depicted plasticity function (right). Panel **(B)** displays the connectivity structure of a nonlocally coupled ring network with **(C)** displays the weighted connectivity structure of a network with **(left)** with distance-dependent Gaussian weight distribution **(right)**. Note that the colors of the links in the network (left) correspond to the colors of the bars in the weight distribution **(right)**.

If not indicated differently, we consider the coupling structure given by

where

In the following section, we provide an approximation for the eigenvalues of

### 4.1 Approximation of the Eigenvalues for Large Systems With Circulant Structure

In the previous part, we have defined the plasticity functions

In this section, we briefly recall how one can derive the eigenvalues *L* is a circulant *k*th eigenvalue is explicitly given by

For the case of

with *A* is assumed to be symmetric, the eigenvalues of *N* that makes it harder to study the influence of other system properties, such as the coupling structure or the plasticity function. To remove this *N*-dependence, we consider the continuum limit

Due to the definition of *h* and the symmetry of

for any *k*. This explicit expression allows studying the distribution of the eigenvalues *h* and coupling structure *a*. Note that a similar expression as (23) can be analogously derived for the eigenvalues of

We note that

The results from Eqs. 23 and 24 are applied in the next sections to analyze different networks.

### 4.2 Synchronization on Nonlocally Coupled Ring Networks

In this section, we analyze the effect of long distance connections on the stability of synchronous states in nonlocally coupled ring networks. We consider the coupling structure given by

This means that any two oscillators are coupled if they are separated at most by the coupling range *P*. The coupling Eq. 25 defines a nonlocal ring structure with coupling range *p* to each side and two special limiting cases: local ring for *N* is even, else

In order to study the influence of the coupling range, we use the approximations for the eigenvalues

for the eigenvalues

for

In Figure 3A, we provide an error analysis of the approximations Eqs. 26 and 27 compared to the exact eigenvalues given by Eq. 22. As expected, the errors tend to zero as the number of oscillators increases. Additionally in Figures 3B,C, we display *k* depending on the relative coupling range *p*. We observe that *k*. This is due to given plasticity function Eqs. 18–20, for which the update is positive (or equal to zero) for all distances at

**FIGURE 3**. Panel **(A)** shows the errors *N* (number of oscillators). The relative coupling range is set to **(B)** and **(C)** show the approximated eigenvalues given by Eqs. 26, 27, respectively, depending on the relative coupling range *p* for different values of *k*.

It is important to note, that our choice of the circulant adaptation functions imply that the matrices

Combining the fact *k*. These limits are displayed in Figures 3B,C as black lines.

In Figure 4, we show different scenarios for the stability of the synchronous state depending on the phase lag parameter α and the coupling range *p*. Due to the necessary condition *p* larger than a critical value of the coupling range

**FIGURE 4**. Stability analysis of the synchronous state of system Eqs. 16, 17 with plasticity rule Eqs. 18–20 and coupling structure Eq. 25. Panels **(A–D)** show the function *α*, see Eq. 15, calculated with the approximations Eqs. 26, 27 depending on the relative coupling range *p*. In each panel, *k*. The gray shaded regions refer to unstable synchronous states. Panels (e,f,g,h) show the master stability function *α* with color code as in Figure 1. The crosses and dots correspond to two sets of eigenvalue pairs **(I–L)** show the synchronization error **(A, E, I)****(B, F, J)****(C, G, K)****(D, H, L)**

The situation changes for

We have shown that long distance interactions may stabilize or destabilize the synchronous state depending on the phase lag parameter α. In this section, all links have the same weight independent of the corresponding distance. In the next section, we analyze a network with a more realistic structure with a distance-dependent distribution of weights.

### 4.3 Synchronization on Isotropic and Homogeneous Network With Gaussian Distance Distribution

In the previous section, we used the prototypical example of a nonlocally coupled rings to study the effects of long-range interaction on synchronization. In this setup, however, all links are equally weighted. In realistic systems, in contrast, the number of links with a certain distance are distributed, see [67] for details. To incorporate this into our network model, we weight the links with respect to a distance distribution. Measurements suggest that the distance distribution can be estimated by a mean and a distribution width [67]. The Gaussian distributions is a paradigmatic distribution that allows for studying effects emanating from the mean and the distribution width. For the remainder of the section, we consider the link distance distribution given by a Gaussian distribution, and weight the links of the network connectivity structure *A* accordingly, i.e.

where *ξ* and *σ* are the mean value and the standard deviation, respectively. Note that the standard deviation characterizes the width of the distribution. For the numerical simulations, we normalize each row of *A* by

As we know from Eqs. 14 and 15, for *N* and unstable otherwise. In Figure 5A, we display *α*. As in the case of nonlocally coupled ring networks, with *ξ*.

**FIGURE 5**. Stability analysis of the synchronous state of system Eqs. 16, 17 with plasticity rule Eqs. 18–20 and coupling structure Eq. 28. Panels **(A, C)** show the minimum over all *ξ* and the standard deviation *σ* of the weight distribution. The minima are displayed in color code. Panels **(B, D)** show the boundaries between stable and unstable regions in **(A)****(C)****(B, C)**

An opposite scenario is shown in Figure 5C for *α*, the synchronous state is stable for almost all values of *σ* and *ξ*, see Figure 5D. Only in cases of distribution sharply peaked at long distances, i.e., *ξ* close to *σ* close to 0, the synchronous state is unstable. This effect could not be found in networks with nonlocally coupled rings, see Section 4.2.

## 5 Conclusion

In summary, we have investigated the phenomenon of synchronization on adaptive networks with heterogeneous plasticity rules. In particular, we have modeled systems with distance-dependent plasticity as they have been found in neuronal networks experimentally [64–67] as well as computational models [68]. For the realization, we have used a ring-like network architecture and associated the distance of two nodes with the distance of their placement on the ring.

In Section 3, we have developed a generalized master stability approach for phase oscillator models that are adaptively coupled and where each link has its own adaptation rule (plasticity). By using an explicit splitting of the time scales between fast dynamics of the phase oscillators and slow dynamics of the link weights, we have established an explicit stability condition for the synchronous state. More precisely, we found that the stability is governed by the coupling function and the eigenvalues of two structure matrices. These structure matrices

In Section 4, we applied the novel technique to a system of adaptively coupled oscillators with distance-dependent plasticity. Here, we have used a ring-like network structure to study the impact of long- and short-distance connections on the stability of synchronization. For this purpose we introduced an approximation of the eigenvalues for the structure matrices in Section 4.1. This approximation allows for a comprehensive analysis of the stability as a function of various system parameters. Moreover, it enables us to identify critical eigenvalues that govern the stability of the synchronous state. In Sections 4.2 and 4.3, we have brought together all methodological findings and applied them to systems with a nonlocally coupled ring structure and with a Gaussian distribution of link weights. The latter structure accounts for the fact that in realistic neuronal populations the number of links with different distances are not uniformly distributed [67]. We found that long-distance connections can stabilize or destabilize the synchronous state, depending on the coupling function between the oscillators. A remarkable fact with respect to neuronal applications relates to the destabilization scenario. Here we observed that the destabilization can be attributed to the pronounced change of the plasticity rule from Hebbian to anti-Hebbian. For more realistic connectivity structures, we found that weight distributions of the connectivity structure with sharp peaks at long distances lead to destabilization for a wide range of the coupling function.

All in all, in this article, we have provided a general framework to study the emergence of synchronization in neuronal system with a heterogeneous plasticity rule. The developed methodology is not limited to distance-dependent types of plasticity and can also be used for non-symmetric setups. For the latter case, we have provided the necessary analytical result. In this work, we have restricted our attention to the case of phase oscillators, but the methods can be extended to more realistic neuron models by using techniques established, for example, in [48]. Moreover, techniques are available that allow for further generalization toward systems with slightly different local dynamics at each node [87]. On the one hand, the master stability approach offers a great tool to study the stability of the synchronous state depending on the networks structure. On the other hand, this approach allows for characterizing the network structures that are, in some sense, optimal for synchronization [88, 89]. In this regard, it remains an open question as to how plasticity optimizes the synchronizability of the network in a self-organized way. In addition, recent studies have shown that there is a great interest in synchronization phenomena to understand diseases such as Parkinson’s disease [90–92] or epilepsy [29, 93] for the development of proper therapeutic treatments. We believe that our work provides an important step toward understanding synchronization under realistic conditions.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author Contributions

RB designed the study and did the numerical simulations. RB and SY developed the analytical results. Both authors contributed to the preparation of the manuscript. Both authors read and approved the final manuscript.

## Funding

This work was supported by the German Research Foundation DFG, Project Nos. 411803875 and 440145547, and the Open Access Publication Fund of TU Berlin.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: synaptic plasticity, adaptive networks, phase oscillator, synchronization, distance-dependent synaptic plasticity, nonlocally coupled rings, master stability approach

Citation: Berner R and Yanchuk S (2021) Synchronization in Networks With Heterogeneous Adaptation Rules and Applications to Distance-Dependent Synaptic Plasticity. *Front. Appl. Math. Stat.* 7:714978. doi: 10.3389/fams.2021.714978

Received: 26 May 2021; Accepted: 21 June 2021;

Published: 15 July 2021.

Edited by:

Jun Ma, Lanzhou University of Technology, ChinaCopyright © 2021 Berner and Yanchuk. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Rico Berner, rico.berner@physik.tu-berlin.de