Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Abstract

Relaxation oscillators may exhibit small amplitude oscillations (SAOs) in addition to the typical large amplitude oscillations (LAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. We investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear (PWL) relaxation oscillators where global feedback acting on the rate of change of the activator (fast variable) involves the inhibitor (slow variable). We also investigate of these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry breaking global feedback effects.

1 Introduction

Several chemical, biochemical and biological systems exhibit oscillatory temporal patterns when they are far from equilibrium [1–7]. These phenomena are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Point (single) oscillators require at least one variable (activator) that favors both changes in its own production via autocatalytic effects and the production of a second variable (inhibitor). Inhibitors oppose changes in the activator on a slower time scale. Activators and inhibitors represent different state variables in different systems. Examples are the chemical compounds in the Belousov-Zhabotinsky (BZ) reaction [8,9], the substrate and product in product-activated glycolytic oscillations [4,10], the activator and repressor in genetic oscillators, and the voltage and the recovery variabls of the ionic currents in neurons [5].

In many biologically realistic systems the time scales between activators and inhibitors are well separated, and the resulting oscillations are of relaxation type [2,5]. These are captured by the prototypical van der Pol (VDP) model for a triode circuit [11] and the FitzHugh-Nagumo (FHN) tunnel-diode model for nerve cells [12,13]; and, also, by more detailed models as the Oregonator for the BZ reaction [14–16], the Morris-Lecar model for neuronal oscillations [17] the modified versions of the Selkov model for glycolytic oscillations [18–21] and genetic oscillators [22].

The complexity of individual relaxation oscillators results from the combined effect of two distinct inherent properties: (i) the presence of characteristic types of nonlinearities (typically cubic-like) and (ii) the time scale separation between the participating variables referred to above. In addition to the typical large amplitude oscillations (LAOs) of relaxation type, relaxation oscillators may exhibit small amplitude oscillations (SAOs) with an amplitude difference of roughly an order of magnitude as well as abrupt transitions between them (canard phenomenon) as a control parameter changes through a critical range (exponentially small in the parameter defining the slow time scale) [23–29]. Individual 2D relaxation oscillators may display either SAOs or LAOs, but not both. Higher dimensional relaxation oscillators may exhibit mixed-mode oscillations (MMOs) [30,31], where LAOs are interspersed with SAOs. This creates additional effective time scales.

In addition to the individual oscillators’ intrinsic feedback effects, oscillatory networks have feedback effects that result from the network connectivity and its interaction with the intrinsic properties of the individual oscillators. One such type of feedback is generated by global coupling where each oscillator in the network is affected by the dynamics of the rest through one or more of the participating variables. The effects of global coupling in shaping the network oscillatory patterns has been studied in a variety of systems both experimentally and theoretically. These include oscillatory chemical reactions [32–37], electrochemical oscillators [38–48], laser arrays [49], catalytic reactions [50], salt-water oscillators [51], metabolic oscillators and cellular dynamics [20,52,53], cardiac oscillators [54,55], coupling through quorum sensing [56–60], circadian oscillators [61–63], neuronal networks [5,64–69] and image processing [65,70].

Globally coupled networks of 2D relaxation oscillators have been shown to generate oscillatory cluster patterns [20,32–35,38,39,64,71–75]. Each cluster consists of synchronized in-phase identical oscillators. Oscillators in different clusters differ in at least one of their attributes (e.g., frequency, amplitude and phase). A typical example are the antiphase and, more generally, the phase-locked oscillatory cluster patterns. In a phase-locked two-cluster pattern the two oscillators exhibit LAOs. In some cases, they may also exhibit MMOs, which typically reflect the effects of the network connectivity (e.g., inhibition transiently pushing the activator down or terminating an oscillation before it reaches high enough values), but may also reflect the interaction between the connectivity and the intrinsic canard structure [76] of the individual oscillators [72,73].

A more complex type of pattern that emerges in these globally coupled networks are localized oscillations, where one cluster exhibits LAO or MMOs and the other shows no oscillations or SAOs [32–36,72,73]. Each oscillator in the network is monostable (it can display SAOs or LAOs but not both). The break of symmetry into patterns where each oscillator is in a different amplitude regime requires some type of network heterogeneity such as different cluster sizes or the same cluster size with different global feedback intensities in each cluster. In previous work we showed that the generation of these patterns involves the interplay of the network connectivity and the intrinsic properties of the individual oscillators, particularly their ability to exhibit the canard phenomenon and canard-like SAOs (generated in a supercritical Hopf bifurcation). However, the dynamic mechanisms that give rise to localized oscillatory patterns in networks of relaxation oscillators and how these patterns depend on the properties of the participating oscillators is not fully understood.

The goal of this paper is to address these issues in the context of globally coupled networks where the global feedback acting on the rate of change of the activator involves the inhibitor [32–36,72,73]. An additional goal is to understand how these patterns are affected by the presence of a diffusive type of coupling. Since, in contrast to global inhibition, diffusion tends to synchronize oscillators, their interplay generates a competition between the two opposing effects. We use a cluster reduction of dimensions argument [35] and assume the system is divided into two clusters with the same or different sizes. The effects of the cluster size on the dynamics of this two-cluster networks are absorbed into the global feedback parameter coding for the intensity. Different cluster sizes result in an effective heterogeneous connectivity.

To capture the intrinsic dynamics we use a piecewise-linear (PWL) relaxation oscillator model of FitzHugh-Nagumo (FHN) type, which is an extension of the one we used in [77] to investigate the mechanisms of generation of the canard phenomenon. PWL models can be explicitly analyzed using linear tools of dynamical systems and matching “pieces of solutions” corresponding to consecutive linear regimes. PWL models have been used in a variety of fields as caricature of nonlinear models to provide insights into the dynamics of smooth nonlinear models either to investigate the dynamics of individual nodes or networks [78–104].

As in [77], the activator (v) nullcline we use is cubic-like and has four linear pieces (Fig. 1, red curve). The inhibitor (w) nullcline is sigmoid-like and has three linear pieces (Fig. 1, green curve). The canard phenomenon requires the presence of the two linear pieces in the middle branch of the v-nullcline, but a linear w-nullcline is enough. However, localization in models having a linear w-nullcline is more difficult to obtain and is less robust than in models having sigmoid-like w-nullclines.

Figure 1: Cubic- and sigmoid-like piecewise linear v- and w- nullclines for system (1).

The v-nullcline f(v) (red) is given by (2). We used the following parameter values: η = 0.3, v_c = 0.3 The w-nullcline g(v; λ) (green) is given by (3). We used the following parameter values for the two superimposed w-nullclines: α = 4 (solid-green), α = 2 (dashed-green), λ = 0.3 (solid-green), λ = 0.21 (dashed-green), β_L = −0.2, β_R = 0.2. The arrows indicate the effects of increasing values of λ and α. Increasing (decreasing) λ displaces the w-nullcline to the right (left), while increasing (decreasing) α increases (decreases) the slope of the w-nullcline.

An advantage of using PWL models for this study is that they provide a way of understanding how the intrinsic properties of the individual oscillators affect the network dynamics in terms of the different linear portions of the PWL nullclines whose properties are easily captured by their slopes and end-points. The scenarios we explore in this paper make heavy use of this property. For example, by increasing the values of β_L and β_R in Fig. 1 the w-nullcline becomes “more linear” in the region of the phase-plane where the oscillations occur (around the four branches of the cubic-like v-nullcine). Along this paper we will compare two such scenarios where the w-nullcline is sigmoid-like (as in Fig. 1) and linear-like (relatively large values of both β_L and β_R).

The localized patterns as well as the other types of MMO patterns analyzed in this paper can be a desired or an undesired result of the network activity. For memory devices and working memory [105–108], localized patterns allow for the effective representation of information in the LAO components. In contrast, the presence of localized oscillations may disrupt the communication between neurons [5] and the effective pulsatile secretion of insulin when controlled by glycolytic oscillators or other oscillatory systems (e.g., calcium) [20,109–111] (but see [112]). Our results will contribute to understand the mechanisms underlying the generation of these patterns and how to control or prevent them when necessary.

2 Methods

2.1 Piecewise linear models of FitzHugh-Nagumo type

We consider the following piecewise linear (PWL) models of FitzHugh-Nagumo (FHN) type where the prime sign represent the derivative with respect to the variable t and the functions f and g are PWL cubic- and sigmoid-like functions (see Fig. 1) given, respectively, by and

The PWL cubic-like function f (Fig. 1, red) has a minimum at (0,0) and a maximum at (1,1). As in the smooth case, this choice ensures that large amplitude oscillations are 𝒪(1) [77]. The parameter η governs the slopes of the two middle branches L₂ and L₃. The slope of L₃ also depends on the parameter v_c (v-coordinate of the point joining L₂ and L₃). The slopes of both the left (L₁) and right (L₄) branches are equal to −1.

The PWL sigmoid function g (Fig. 1, green) has three branches. The two horizontal branches S₁ and S₃ are below and above the minimum and maximum of f, respectively. The middle branch S₂ joins these two horizontal branches. The parameter λ controls the displacement of g to the right (lda > 0) or the left (λ < 0). The parameter α controls the slope of the middle branch S₂, which increases with increasing values of α. In the limit of β_L, β_R → ∞, the PWL system is the one used in [77] where g is a linear function.

2.2 Linear regimes and virtual fixed-points

The dynamics of a PWL model of the form (1)-(3) can be divided into four linear regimes R_k (k = 1,…, 4), corresponding to the four linear pieces L_k of the cubic-like PWL function f(v) (Fig. 2). The initial conditions in each regime are equal to the values of the variables v and w at the end of the previous regime where the trajectory has evolved.

Figure 2: Linear regimes and actual/virtual fixed-point for system (1).

The v-nullcline (red) is as in Fig. 1. For the w-nullcline (green) we used α = 2, λ = 0.3 β_L = −0.05 and β_R = 0.05. The superimposed dashed-green w-nullcline is linear (extension the linear piece S₂). The virtual fixed-points for each regime (blue dots) are the intersection between the extensions of the corresponding linear pieces and the w-nullcline. The stable virtual fixed-point for R₂ coincides with the actual fixed-point.

In each linear regime the dynamics are organized around a virtual fixed-point (Fig. 2), which results from the intersection between the w-nullcline (green line) and the corresponding linear piece (red line) or its extension beyond the boundaries of this regime (dashed-red line). In the latter case the virtual fixed-points do not coincide with the actual fixed-points, and are located outside the corresponding regime, but still play an important role in determining the dynamics in that regime. The trajectories in a given regime never reach the purely virtual stable fixed-points (outside the regime), but their presence provides information about the trajectory’s direction of motion. More specifically, within the boundaries of each regime trajectories evolve according to the linear dynamics defined in that regime as if the dynamics were globally linear, and they “do not feel” that the “rules” governing their evolution will change at a future time when the trajectory moves to a different regime. We refer the reader to [77] for more details.

2.3 Networks of PWL oscillators with global inhibitory feedback

We consider networks of PWL oscillators of FHN type of the form (1) globally coupled through the inhibitor variable (w) for k = 1,…, N, where N is the total number of oscillators in the network, γ ≥ 0 is the global feedback parameter and

2.4 Cluster reduction of dimensions and heterogeneous coupling

Following previous work [35,36,72,73] we assume the network is divided into two clusters where all oscillators in each cluster are identical and have identical dynamics, while oscillators in different clusters may have different dynamics. Accordingly, for a two-cluster network, where σ₁ and σ₂ (σ₁ + σ₂ = 1) are the fractions of oscillators in each cluster. Alternatively, the global coupling term (6) can be also interpreted as consisting of clusters with the same fraction of oscillators each, but heterogeneous connectivity.

System (4) with (6) can be written as for k, j = 1, 2 with j ≠ k.

The zero-level surfaces (“higher-dimensional nullclines”) for the k^th oscillator are given by and respectively.

Eq. (8) describes a two-dimensional surface having the shape of the first term in the right hand side of N_v,k(v_k, 0; γ). For γ > 0, we view the nullsurface (8) as the v-nullcline for the individual (uncoupled) oscillator N_v,k (v, 0; 0), flattened by the effect of the denominator and forced by the second oscillator via the variable w_j(t). When there is no ambiguity, we refer to the autonomous part N_v,k(v_k, 0; γ) in (8) as the v-nullcline for the oscillator O_k. The oscillations in the latter “raise” and “lower” this v-nullcline following the dynamics of w_j and therefore affect the evolution of the trajectories in the phase-plane diagrams.

2.5 Diffusive coupling between clusters

System (7) with an added diffusion term reads for k, j = 1,2 with j ≠ k, where D_v is the diffusion coefficient.

This way of adding diffusion is somehow artificial and does not reflect the diffusive effects in the original system nor is it derived from it. However, its inclusion helps understand the competitive effects of global inhibition and diffusion.

Equation (8) is extended to

For D_v > 0 the v-nullcline N_v,k (v_k, 0,0; γ, D_v) is linearly modified by the term D_v v_k. In contrast to global coupling, this effect is not homogeneous for all values of v_k, but is dependent on its sign. For positive values of v_k the v-nullcline is flattened, while for negative values of v_k the v-nullcline is sharpened. The oscillations in v_j “raise” and “lower” this v-nullcline following its dynamics. In order for the linear piece L₂ to remain positive for D_v > 0, we will restrict D_v < η.

2.6 Dynamics of the linear regimes

The dynamics of system (1)-(3) in each linear regime are governed by a system of the form centered at the fixed-point (v̅,w̅) (virtual or actual) corresponding to each linear regime. In (12) η represents the generic slope of each linear piece of f(v) and not necessarily the slope of the linear piece L₂. The coordinates of the fixed-point for each regime, where f(v) is described by η (v − v̂) + ŵ, are given by

Note that we are using the same notation for the translated system (12) and the original system.

The case κ = 1 corresponds to the uncoupled system, while the case κ = 1 + σγ corresponds to the autonomous part of the globally coupled system (7). The effects of D_v are included in the parameter η.

The eigenvalues for each fixed-point are given by

The fixed-points for linear regime (12) are stable if η < ∊ and unstable if η > ∊. They are foci if and nodes otherwise. Since α ≥ 0 and κ > 0, saddles are possible only for α = 0. We refer the reader to [77] for a more detailed discussion for the case κ = 1. The global feedback parameter γ > 0 affects both the location of the fixed-points and the eigenvalues. For large enough values of γ a node can transition into a focus.

2.7 Numerical simulation

The numerical solutions were computed using the modified Euler method (Runge-Kutta, order 2) [113] with a time step Δt = 0.1 ms (or smaller values of Δt when necessary) in MATLAB (The Mathworks, Natick, MA).

3 Results

3.1 The canard phenomenon for PWL models of FHN type revisited

In a two-dimensional relaxation oscillator, the canard phenomenon refers to the abrupt transition between small amplitude oscillations (SAOs) and large amplitude oscillations (LAOs) as a control parameter crosses a very small critical range which is exponentially small in the parameter defining the slow time scale [23–29]. We identify this critical range with a critical value for the control parameter (e.g., λ_c if the control parameter is λ). If the Hopf bifurcation underlying the creation of the small amplitude limit cycles is supercritical, then the SAOs are stable. In contrast, if the Hopf bifurcation is subcritical, then the small amplitude limit cycle is unstable. The relaxation-type LAOs are always stable. In the subcritical case, there is bistability between a fixed-point and the LAOs for a range of values of the control parameter.

The canard phenomenon for PWL models of FHN type with a linear w-nullcline has been described in [77,98] and has been throughly analyzed in [77]. Here we briefly describe it in the context of the PWL models of FHN type with sigmoid-like PWL w-nullclines using the parameter λ as the control parameter. Our results are presented in Fig. 3 for two representative types of w-nullclines. The first type (β_L = β_R = 0.05 is truly sigmoid-like in the region of the phase-plane where the oscillations occur (Figs. 3-a and -b). The second type (β_L = β_R = 1) is linear in that region (Fig. 3-b). In this paper we refer to them as sigmoid- and linear-like w-nullclines, respectively. The primary difference between them is the way they affect the effective time scales in the region of the phase-plane where SAOs and the transition from SAOs to LAOs occur.

Figure 3: Dynamics of system (1) for representative parameter values.

The v-nullcline (red) is as in Fig. 1. For the w-nullcline we used the following parameter values: α = 2, β_L = β_R = 0.05 (a, b), β_L = β_R = 0.5 (c).

For the SAOs to be generated (Figs. 3-a1 and -c1) the limit cycle must cross either the linear piece L₂ or the first portion of the linear piece L₃ of the v-nullcline. Otherwise (Figs. 3-a2 and -c2) the limit cycle trajectory moves into the linear regime R₄ and the system displays LAOs. For a trajectory arriving in R₂ to be able to cross L₂ or the first portion of L₃ the actual fixed-point in R₂ must be a focus (see eq. 15 for κ = 1). In addition, the initial amplitude of the trajectory in R₂ (the distance between the actual fixed-point and the initial point in R₂) must be small enough so that the trajectory reaches the v-nullcline before reaching the region of fast motion that would cause it to move towards the right branch. For the parameter values in Fig. 3-a, | η + ∊ | = 0.4 and in the linear regime R₂ and therefore the actual fixed-point is a focus. However, as ∊ decreases this inequality may no longer hold. For example, for the parameters in Fig. 3-b, | η + ∊ | = 31 and ) and therefore the actual fixed-point is a node and, as a consequence, the system is no longer able to exhibit the canard phenomenon.

The canard critical value λ_c is affected by the vector field away from the local vicinity of the small amplitude limit cycle (compare Figs. 3-a and -c). The fact that the horizontal branches of the w-nullcline in Fig. 3-c are further away from the v-nullcline than in Fig. 3-a (β_L and β_R are larger in Fig. 3-c than in Fig. 3-a) causes the oscillation frequency to increase and λ_c to decrease.

3.2 The canard phenomenon induced by global feedback

Here we follow previous work [35,36,73] and focus on the dynamics of the one-cluster globally coupled system (7) with σ₁ = 1 (σ₂ = 0). This is not likely to be a realistic situation, but it provides information about how the global feedback affects the oscillatory dynamics and the occurrence of the canard phenomenon in an autonomous system. The results of this section will be helpful in understanding the dynamics of the autonomous part of the two-cluster systems discussed below in this paper.

From (15) with κ = 1 + γ, increasing values of γ (all other parameters fixed) can cause the fixed-point to transition from a node to a focus. In addition, from (14), increasing values of γ change the location of the fixed-point. Therefore, the global feedback parameter γ can act as the control parameter that induces the canard phenomenon for fixed-values of λ.

The top panels in Fig. 4 show curves of the oscillation amplitude versus γ for representative parameter values and α = 4. The corresponding bottom panels show the phase-plane diagrams for specific values of γ (for the blue curves in the top panels). The parameter values in Figs. 4-a and -b are the same, except for β_L and β_R that are smaller in Fig. 4-a than in Fig. 4-b (the w-nullcline is sigmoid-like in panel a and linear-like in panel b).

Figure 4: The canard phenomenon induced by global inhibitory feedback in bulk oscillatory systems.

The solid-red v- nullcline corresponds to the actual values of γ used in each panel. The v-nullcline for γ = 0 (dashed-red) is as in Fig. 1 and is presented for reference. For the w-nullcline we used the following parameter values: (a) α = 4, ∊ = 0.1, λ = 0.7, β_L = β_R = 0.05. (b) α = 4, ∊ = 0.1, λ = 0.7, β_L = β_R = 1. (c) α = 4, ∈ = 0.01, λ = 0.08, β_L = β_R = 1.

For ∊ = 0.1 and the sigmoid-like w-nullcline in Fig. 4-a the canard phenomenon is induced by γ. The transition is more pronounced for lower values of λ. As γ increases, the v-nullcline flattens (Figs. 4-a2 and -a3) and for γ_c the limit cycle trajectory is able to cross L₃, thus generating SAOs (Figs. 4-a3), instead of moving towards R₄ to generate LAOs (Figs. 4-a2).

For ∊ = 0.1 and the linear-like w-nullcline in Fig. 4-b the system fails to exhibit the canard phenomenon as γ increases. The effective time scale separation in the vicinity of the minimum of the v-nullcline is smaller than in Fig. 4-a because of the absence of the horizontal piece of the w-nullcline (compare panels a1 and a2 with panels b1 and b2), and therefore the limit cycle trajectories are more rounded in Figs. 4-b than in Figs. 4-a. This causes the limit cycle trajectory to move further away from L₂ and L₃ in Fig. 4-b2 than in Fig. 4-a2. As a result, the v-nullcline is able to flatten significantly before the limit cycle trajectory is able to cross the middle branch, and therefore the oscillations’ amplitude decreases gradually instead of abruptly. For lower values of λ (red and green curves in Fig. 4-b) the transition from LAOs to SAOs is faster and the final amplitude smaller than for λ = 0.7, but still this transition is not abrupt

A decrease in ∈ for the same parameter values as in Fig. 4-b (including the same linear-like w-nullcline) restores the ability of γ to induce the canard phenomenon (Fig. 4-c). The decrease in ∊ compensates for the lack of the horizontal pieces of the w-nullcline, thus maintaining similar levels of the time scale separation in the vicinity of the minimum of the v-nullcline.

As we discussed in the previous section, for ∊ = 0.01 and α = 2 the uncoupled oscillator (γ = 0) fails to exhibit the canard phenomenon. However, the canard phenomenon can be induced by γ (not shown) with similar properties as for α = 4. The values of γ_c increase with λ and, in contrast to the α = 4 case, they are both significantly larger for α = 2 than for α = 4. Also, the range of values of γ_c spanned by λ is significantly larger for α = 2 than for α = 4.

3.3 Canard and non-canard (standard) SAOs for two interacting oscillators forcing one another

As discussed above, the interaction between two oscillators due to global coupling can be thought of as the oscillators forcing one another through the last term in the first equation in (7). If the product σ_k γ (k = 1,2) is large enough, then the autonomous part of N_v,k (8) can be in an SAO regime. This means that if w_j (j = 1, 2 with j ≠ k) would be artificially made equal to zero, then the oscillator O_k would exhibit the type of canard-like SAOs discussed in the previous section. However, since w_j is not necessarily equal to zero or very small, but also oscillates, then N_v,k raises and shifts down from their baseline location in an oscillatory fashion. This may interfere with the canard SAOs to create the more complex patterns that we discuss in the following sections.

Among these patterns there are the MMOs (LAOs interspersed with SAOs) shown in Fig. 5-a (blue curve). This MMO pattern consists of two types of SAOs. The ones along the ascending phase and the ones on the more shallow phase (Fig. 5-a2). The former correspond to the portion of the trajectory evolving along the left branch of N_v,1 (Fig. 5-c) as they respond to the motion of N_v,1 following the forcing exerted by O₂ (Fig. 5-c). The second group corresponds to the trajectories moving around the minimum of N_v,1 as they are able to cross the linear piece L₁ to create SAOs. We refer to them as canard-like SAOs. The canard-like and standard SAOs are created by different mechanisms. The standard SAOs in v₁ (Fig. 5-a, blue) primarily respond to the oscillatory input from w₂ (Fig. 5-b, red). During the ascending phase, w₁ is decreasing, therefore the oscillations in v₂ and w₂ are intrinsically generated by a canard-like mechanism (Fig. 5-d) that does not require oscillations in the input. The canard-like SAOs are created by the canard-like mechanism described above. Note that although v₁ receives an oscillatory input from w₂, the oscillations in w₂ during the shallow phase have a smaller amplitude than during the ascending phase, indicating that they are less important in the generation of the SAOs in O₁.

Figure 5: Canard and non-canard (standard) SAOs in two-cluster networks.

(a) Curves of v₁ and v₂ vs. t. (b) Curves of w₁ and w₂ vs. t. (c,d) Phase-plane diagrams. The gray-dashed curves represent to the vnullcline for the uncoupled system (γ = 0). The blue- and red-dashed curves represent to the v-nullclines for the autonomous part of the globally coupled system. The green-dashed curves represent the w-nullclines. The solid blue and red curves represent the trajectories of the globally coupled system for the oscillators O₁ and O₂ respectively. The right panels are magnifications of the left ones. We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.08, σ₁ = 0.2, σ₂ = 0.8, γ − 5 and β_L = β_R = 1.

3.4 Localized, mixed-mode, phase-locked and SAO network oscillatory patterns

In the next sections we examine the consequences of the global feedback’s ability to induce the canard phenomenon in a single-cluster oscillator (σ₁ = 1 and σ₂ = 0) discussed above for two-cluster network dynamics (σ₁ < 1 and σ₂ > 0). We focus on the case σ₁ =0.2 (σ₂ = 0.8) as a representative case of heterogeneous clusters. As we briefly discuss below, homogeneous clusters (σ₁ = σ₂ = 0.5) produce relatively simple network patterns.

From (7), the autonomous part of each oscillator is affected by both the cluster size (σ_k) and γ. In the absence of the forcing exerted by the other oscillator (w_j), the canard phenomenon would be induced by increasing values of both σ_k and γ [35,36]. The global feedback parameter critical value for the autonomous part of each oscillatory cluster is given by where γ_c is the global feedback parameter critical value for the single-cluster oscillator discussed above (e.g., γ_c = 0.32 in Fig. 4-a and γ_c = 0.38 in Fig. 4-c).

For σ₁ = σ₂ = 0.5, γ_c,1 = γ_{c, 2} and therefore both oscillators would simultaneously be either in the LAO or SAO regime (Fig. 6) with no intermediate types of patterns. However, due to the forcing effects that the oscillators exerts on each other the values of γ at which these abrupt transitions occur are larger than the ones predicted by (16). For example, for the parameter values in Fig. 6-a, γ_c,1 = γ_c,2 ~ 0.76 (see Fig. 4-c) and the transition occurs at γ ~ 5.21. For another example, for the parameter values in Fig. 6-b, γ_c,1 = γ_c,2 ~ 0.72 (not shown) and the transition occurs at γ ~ 0.99.

Figure 6: Abrupt transition between antiphase LAO and SAO patterns in two-cluster networks for representative values of γ.

(a) Parameter values are as in Fig. 4-c. (b) Parameter values are as in Fig. 4-c, except for β_L = 0.05 and β_R = 0.05 (same as in Figs. 4-a and -b). We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.08, σ₁ = σ₂ = 0.5, β_L = β_R = 1 (a) and β_L = β_R = 0.05 (b).

For σ₁ ≠ σ₂ it is be possible for one oscillator (σ₁ < 0.5) to be in the LAO, while the other (σ₂ > 0.5) is in the SAO, thus generating localized patterns (described in more detail below). However, as discussed above, the forcing effects that the oscillators exert on each other is expected to generate more complex dynamics that disrupt this scenario, and it is not a priori clear whether and under what conditions these localized patterns exist. For this to happen, the forcing effects should not interfere with the canard phenomenon for each oscillator. A richer repertoire of intermediate patterns that are not “purely LAO” or “purely SAO” are expected to result from the complex interactions between oscillators as it happens for other systems [72,73].

We have identified various types of network patterns for different parameter regimes.

• Phase-locked LAO patterns (e.g., Figs. 7-a1 and -a2) correspond to both oscillators in the LAO regime. All other parameters fixed, the phase-difference between the two oscillators depends on the relative cluster sizes. For σ₁ = σ₂ = 0.5 the patterns are antiphase (Fig. 6). The underlying mechanisms are qualitatively similar to these described in [72,73] involving the standard SAOs discussed above, and will not be discussed further in the context of this paper.

• Mixed-mode oscillatory (MMO) patterns (e.g., Fig. 7-a3) correspond to either one or both oscillators exhibiting MMOs.

• Localized patterns (e.g., Figs. 7-a5 and -a6 and Figs. 7-b2 and -b3) correspond to one oscillator exhibiting LAOs or MMOs, while the other exhibits exclusively SAOs. From (16), the oscillator with the larger cluster size is the one expected be in the SAO regime.

• LAO localized patterns (e.g., Figs. 10-a3 and -a3) correspond to the two oscillators exhibiting LAOs or MMOs, but the number of LAOs per cycle is different between the two oscillators. The typical situation is one oscillator exhibiting one LAO per cycle, while the other exhibits a burst of LAOs.

• SAO patterns correspond to both oscillators exhibiting SAOs that may or may not be synchronized in phase or have the same amplitude.

Figure 7: Localization in a two-cluster network for representative values of γ.

(a) Parameter values are as in Fig. 4-c. (b) Parameter values are as in Fig. 4-c, except for β_L = 0.05 and β_R = 0.05 (same as in Figs. 4-a and -b). We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.08, σ₁ = 0.2, σ₂ = 0.8, β_L = β_R = 1 (a) and β_L = β_R = 0.05 (b).

In addition, we have identified various irregular patterns that emerge mostly as transition patterns between these mentioned above. We will not analyze these patterns in this paper.

3.5 From phase-locked to localized patterns through network MMOs in the PWL model with a linear-like w-nullcline

Fig. 7-a shows various representative two-cluster patterns for the same parameter values as in Fig. 4-c. The global feedback critical values are γ_c,1 ~ 1.9 and γ_c,2 ~ 0.475. The corresponding phase-plane diagrams are presented in Fig. 8-a.

Figure 8: Localization in a two-cluster network for representative values of γ.

Phase-plane diagrams for the parameter values is Fig. 7. The gray-dashed curves represent to the vnullcline for the uncoupled system (γ = 0). The blue- and red-dashed curves represent to the v-nullclines for the autonomous part of the globally coupled system. The green-dashed curves represent the w-nullclines. The solid blue and red curves represent the trajectories of the globally coupled system for the oscillators O₁ and O₂ respectively. We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.08, σ₁ = 0.2, σ₂ = 0.8, β_L = β_R = 1 (a) and β_L = β_R = 0.05 (b).

For low values of γ (Fig. 7-a1 and -a2) the system exhibits phase-locked LAO patterns. The duty cycle is smaller for the larger cluster (oscillator O₂) since its nullcline is flatter (Fig. 8-a2). The relative size of the (smaller to larger) duty cycles for the two oscillators O₁ and O₂ decreases with increasing values of γ.

As γ increases above these values the system transitions to MMO patterns (Fig. 7-a3 and -a4). The SAOs for O₂ in Fig. 7-a3 are canard-like (Fig. 8-a3) (the limit cycle trajectories cross the linear piece L₂ or at most the early portion of L₃). The last SAO in each cycle for O₁ is also canard-like. They all occur as both w₁ and w₂ are very small so their forcing effects are almost negligible. In contrast, the first SAOs in each cycle are standard (not canard-like) and reflect the motion of N_v,1 in response to the dynamics of O₂ as explained in Section 3.3.

During the active phase of O₁, O₂ is almost silent (constant). When O₁ jumps down, w₁ decreases and N_v,2 raises, thus releasing O₂. Because N_v,2 is flatter than N_v,1, O₂ completes the cycle just before O₁, and for some time they are both silent (w₁ ~ 0 and w₂ ~ 0). Fig. 7-a4 corresponds to a slightly higher value of γ. This causes the first O₂ oscillation to transition to a SAO. As this happens O₁ is moving along the left branch of N_v,1 and continuing to release O₂ from inhibition. As a result, the second O₂ oscillation is a LAO.

For larger values of γ the system transitions to localized patterns (Fig. 7-a5 and -a6) where the smaller cluster (O₁) exhibits MMOs and larger cluster (O₂) exhibits canard-like SAOs (Fig. 8-a5 and -a6). The SAOs displayed by O₁ are a combination of canard- and standard SAOs. (as described above) in response to the dynamics of O₂. Note that the transition to localized patterns requires a much larger value of γ than the one predicted by γ_c,1 and γ_c,2.

3.6 Abrupt transition between phase-locked LAO and localized patterns for the PWL model with a sigmoid-like w-nullcline

Fig. 7-b shows various representative two-cluster patterns for the same parameter values as in Fig. 7-a, but a sigmoid-like w-nullcline instead of a linear-like one as in Fig. 7-a. The corresponding phase-plane diagrams are presented in Fig. 8-b. The global feedback critical values are γ_c,1 ~ 1.8 and γ_c,2 ~ 0.45.

In contrast to the case discussed in the previous section, the transition from phase-locked SAO patterns (Fig. 7-b1) to localized patterns (Fig. 7-b2) is abrupt and occurs for a value of γ slightly higher than γ_c,2. This is the result of the stronger time scale separation imposed by the sigmoid-like w-nullcline, particularly in the regions of the phase-plane where the left and right branches of the v-nullcline are located (Fig. 8-b).

When O₁ jumps up, it causes N_v,2 to shift down, thus inhibiting O₂. For the parameter values in Fig. 7-b1 (phase-locked SAO patterns), the trajectory for O₂ is above the minimum of N_v,2 and it continues to move down along N_v,2. After O₁ jumps down and begins to move down along N_v,1, decreasing the forcing exerted on O₂, this is released from inhibition and the trajectory moves through R₂ without crossing L₂, thus jumping up.

For the parameter values in Fig. 7-b2 (localized patterns) the trajectory for O₂ is almost at the minimum of N_v,2 when O₁ jumps up. The trajectory for O₂ first displays a small non-canard SAO, which is the result of O₁ causing N_v,2 to move down, and then two canard SAOs after O₁ jumps down and moves down along N_v,1. The larger value of γ increases the ability of the trajectory for O₂ to generate canard-like SAOs by crossing N_v,2 without jumping up.

The two models considered in this and the previous sections differ in the distances (β_L and β_R) between the horizontal pieces (S₁ and S₃) of the w-nullcline and the v-nullcline. To determine which one of β_L or β_R has a stronger effect in creating the abrupt transitions between the phase-locked LAO and localized patterns described in this section we looked at models with mixed values of these parameters. We found that for β_L = 1 and β_R = 0.05 the system behaves as in Fig. 7-a, while for β_L = 0.05 and β_R = 1 the system behaves as in Fig. 7-b. This confirms that the increase in the effective time scale separation created by the left horizontal piece of the sigmoid-like w-nullcline is key for the results discussed above (and in the next section).

3.7 The oscillatory frequency of the PWL model with sigmoid- and linear-like w-nullclines has different monotonic dependencies with γ

Comparison between the localized patterns in Figs. 7-a (panels a5 and a6) and -b (panels b2 and b3) shows that the LAO frequency of the oscillator O₁ decreases with increasing values of γ for the linear-like w-nullcline (Figs. 7-a5 and a6), while it increases with increasing values of γ for the sigmoid-like w-nullcline (Figs. 7-b2 and b3).

The underlying mechanisms in both cases involve the presence of canard-like SAOs. In Fig. 7-a5, O₁ jumps up right after reaching the minimum of N_v,1 (Fig. 8-a5). In Fig. 7-a6, O₁ engages in canard-like SAOs after reaching the minimum of N_v,1 (Fig. 8-a6), thus increasing the LAO period. This is the result of the forcing exerted by O₂ and lower time scale separation for the linear-like w-nullcline in Fig. 7-a as compared to the sigmoid-like w-nullcline in Figs. 7-b.

For the sigmoid-like w-nullcline (Figs. 7-b2 and -b3), the number of SAOs per cycle also increases as γ increases. However, O₁ jumps up upon reaching the minimum of N_v,1. Also, more importantly, the number of cycles per unit of time increases with γ because the active phase of O₁ significantly decreases with increasing values of γ. This is the result of the flattening of the the v-nullcline as γ increases and the fact that O₁ jumps down near the maximum of the baseline N_v,1.

3.8 The localized patterns persist for lower values of α for the PWL model with a sigmoid-like w-nullcline, but not for a linear-like w-nullcline

From our previous discussion about the effects of decreasing values of α on the ability of λ and γ to induce the canard phenomenon in the uncoupled and coupled systems, respectively, it is not a priori clear whether the localized patterns found in the previous section for α = 4 will persist when we decrease α. In Fig. 9 we present our results for the same parameter values as in Fig. 7 for α = 2 (instead of α = 4). For the uncoupled system and α = 2 (and all other parameters as in Fig. 9), the PWL model fails to exhibit the canard phenomenon as λ changes. For the one-cluster system, in contrast, changes in γ are able induce the canard phenomenon, although for significantly larger values of γ_c than for α = 4.

Figure 9: Localized and non-localized patterns in a two-cluster network for representative values of γ.

We used the following parameter values: α = 2, ∊ = 0.01, λ = 0.08, σ₁ = 0.2, σ₂ = 0.8, β_L = β_R = 1 (a) and β_L = β_R = 0.05 (b).

Our results in Fig. 9-b show that for α = 2 and sigmoid-like w-nullclines the ability of the PWL model to exhibit abrupt transitions between phase-locked and localized patterns persist with similar properties as for α = 4, but the abrupt transition occurs for much higher values of γ. In contrast, for linear-like w-nullclines the PWL model fails to produce localized patterns (Fig. 9-a). There is an abrupt transition from the LAO patterns in Fig. 9-a4 to the SAO patterns in Fig. 9.

There are additional differences between the patterns in Figs. 9-a and 7-a such as the occurrence of two LAOs per cycle for α = 2, which we did not observe for α = 4.

3.9 The localized patterns are robust to changes in λ for the PWL model with a sigmoid-like w-nullcline, but not for a linear-like w-nullcline

Increasing values of λ increase the global feedback critical value γ_c (Fig. 4-c), and therefore it increases both γ_c,1 and γ_c,2, and is expected to increase the values of γ at the transition to localized patterns (if they exist) are present. If the values of γ are to high, then the v-nullcline flattens before the canard phenomenon can be induced by γ as for the case illustrated in Fig. 4-b3. Therefore, it is not clear a priori that the transitions observed for λ = 0.08 in Figs. 7 and 8 persist for larger values of λ. To address this issue we used the same parameter values as in these figures, but with λ = 0.4 (instead of λ = 0.08). Our results are presented in Fig. 10.

Figure 10: Localized and non-localized patterns in a two-cluster network for representative values of γ.

We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.4, σ₁ = 0.2, σ₂ = 0.8, β_L = β_R = 1 (a) and β_L = β_R = 0.05 (b).

The model with a sigmoid-like w-nullcline (Fig. 10-b) shows an abrupt transition between phase-locked LAOs to localized patterns with similar properties as for λ = 0.08 (Fig. 7-b). In contrast, the patterns displayed for the model with a linear-like w-nullcline (Fig. 10-a) differ from these for λ = 0.08. Importantly, for λ = 0.4 the model does not exhibit localized patterns. Other differences include the presence of in-phase patterns (Fig. 10-a1) and LAO localized patterns (Figs. 10-a3, -a4 and -a5) where the number of LAOs for O₂ per cycle increases with increasing values of γ. There is an abrupt transition between these patterns and the ones in Fig. 10-a6.

3.10 Localized patterns are more robust for the sigmoid-like w-nullcline than for the linear-like w-nullcline for higher values of ∊

In Figs. 11-a and -b we show the different types of patterns for ∊ = 0.1 and the parameter values in Figs. 4-a and -b, respectively. In both cases, for low enough values of γ the system shows in-phase patterns (Fig. 11-a1 and -b1), consistently previous findings for the smooth FHN model [72].

Figure 11: Localization in a two-cluster network for representative values of γ.

We used the following parameter values: α = 4, ∊ = 0.1, λ = 0.7, σ₁ = 0.2, σ₂ = 0.8, β_L = β_R = 1 (a) and β_L = β_R = 0.05 (b).

As γ increases, the patterns in the PWL model with a linear-like w-nullcline transition to the complex type of patterns shown in Fig. 11-a2 and then to the synchronized in-phase patterns shown in Fig. 11-a3. The phase-plane diagrams for these patterns (not shown) are qualitatively similar to the ones obtained for the single-cluster case (Fig. 4-b1), which does not exhibit the canard phenomenon as γ increases. The absence of localization for the two-cluster system is associated to this lack of ability of the single-cluster system to exhibit the canard phenomenon as γ increases.

In contrast, for the PWL model with a sigmoid-like w-nullcline, as γ increases the patterns transition to the localized patterns shown in Figs. 11-b2 and -b3. The larger and smaller SAOs in Figs. 11-b2 and -b3 correspond to the limit cycle trajectory crossing the linear pieces L₃ and L₂, respectively. In Fig. 11-b3 the limit cycle trajectory crosses L₃ at a lower height than in Fig. 11-b2. A significant difference between these localized patterns and the ones for ∊ = 0.01 (Figs. 7-b and 10-b) is that in the latter the SAOs are interrupted during LAOs, while in the former SAOs and LAOs may occur simultaneously.

While localization does not occur for λ = 0.7 in the PWL model with a linear-like w-nullcline, it may be restored for lower values of λ (Fig. 11-c3). For these parameter values the system also shows antiphase patterns (Fig. 11-b2) for lower values of γ.

3.11 The canard phenomenon can be induced by the diffusion autonomous component

In the next sections we investigate the combined effect of global coupling and diffusion. Here we first look at the effects of the diffusion coefficient D_v on the dynamics of the autonomous part of system (10). This is not a realistic situation, but, as for the effects of γ on the one-cluster systems discussed in Section 3.2, it provides information about the dynamics of the autonomous part of each oscillator.

Increasing values of D_v decrease the slopes (η) of the linear pieces L₂ and L₃. From (15) this can cause the transition of the actual fixed-point in R₂ from a node to a focus, therefore favoring the occurrence of the canard phenomenon. This is illustrated in Fig. 12 for the same parameter values as in Fig. 4 and γ = 0. (The baseline v-nullclines for D_v =0 in panels a, b and c, are as in Figs. 4-a1, -b1, and -c1, respectively.)

Figure 12: The canard phenomenon induced by diffusion in bulk oscillatory systems.

Parameter values are as in Fig. 4 The baseline v-nullcline for γ = 0 (dashed-red) is as in Fig. 1 and is presented for reference. The solid-red v-nullcline corresponds to the actual values of γ used in each panel. For the w-nullcline we used the following parameter values: (a) α = 4, ∊ = 0.1, λ = 0.7, β_L = −0.05 and β_R = 0.05. (b) α = 4, ∊ = 0.1, λ = 0.7, β_L = −1 and β_R = 1. (c) α = 4, ∊ = 0.01, λ = 0.08, β_L = −1 and β_R = 1.

3.12 Interplay of diffusion and global feedback for equal-size clusters: in-phase synchronization and the canard phenomenon

Global feedback and diffusion have opposite effects. While global feedback favors the generation of phase-locked clusters (Fig. 6), diffusion favors in-phase synchronization (Fig. 13-a). Here and in the next section we investigate the patterns that result from the interplay of global coupling and diffusion. For visualization purposes, in Figs. 13 and 14 we present the patterns for the globally coupled system in the absence of diffusion to the left of the dashed-gray line. In a separate set of simulations we have checked that the patterns for both γ > 0 and D_v > 0 (right of the dashed-gray line) remain unchanged when global feedback and diffusion are activated simultaneously (not shown).

Figure 13: Interplay of global coupling and diffusion in a two-cluster network for representative values of γ.

We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.08, σ₁ = 0.5, σ₂ = 0.5, β_L = β_R = 0.05. The dashed-gray line indicates the time at which diffusion was activated.

Figure 14: Interplay of global coupling and diffusion in a two-cluster network for representative values of γ.

We used the following parameter values: α = 4, ∊ = 0.01, λ = 0.4 (a), λ = 0.08 (b), σ₁ = 0.2, σ₂ = 0.8, β_L = β_R = 0.05. The dashed-gray line indicates the time at which diffusion was activated.

For relative low D_v/γ ratios, the system shows antiphase patterns (Fig. 13). As this ratio increases, the two oscillators synchronize in-phase (Fig. 13-c). In-phase patterns are also obtained for γ = 0.3 and D_v = 0.15 (not shown) for which D_v/γ = 0.5. For larger values of γ, but similar D_v/γ ratios, the two oscillators exhibit in-phase SAOs. The increase in D_v does not always cause the transition from LAOs to SAOs since once the two oscillators synchronize in-phase they behave as a single cluster and the diffusive effects are negligible. Therefore, the transition from LAOs to SAOs in these cases depends on whether whether γ > γ_c or not. For example, for γ = 0.3 and values of D_v larger than the one in Fig. 13-c the patterns remain in the LAO regime in contrast to the patterns in Fig. 13-d).

3.13 Interplay of diffusion and global feedback for clusters with different size: Diffusion-induced localized and MMO patterns

In Fig. 14-a1 we illustrate the diffusion-induced localized patterns for the same parameter values as in Fig. 10-b and a relatively low D_v/γ ratio. In the absence of diffusion, the system exhibits phase-locked LAOs and localization is induced by increasing values of γ (Fig. 10-b). Similar patterns were obtained for other values of γ and low D_v/γ ratios.

As D_v increases, different types of MMO patterns emerge (Fig. 14-a2 to -a5), which combine the two competing effects of global coupling and diffusion. These patterns include in-phase MMO patterns with different ratios of SAOs and LAOs per cycle (e.g., Fig. 14-a2 and -a5) and MMO patterns where the LAOs in both oscillators are phase-locked (e.g., Fig. 14-a3 and -a4). As D_v increases further, the system exhibits in-phase SAOs.

In Fig. 14-b we show some representative patterns for low D_v/γ ratios. Fig. 14-b1 shows a transition from localized to in-phase MMO patterns (similar to this in Fig. 14 -a5) that combine the features of both oscillators when D_v = 0. The in-phase MMOs have a lower LAO frequency than the localized pattern for D_v = 0. Fig. 14-b2 also shows a transition between localized and in-phase MMOs. However, these MMOs have less SAOs per cycle and a higher LAO frequency than for D_v = 0. Finally, in Fig. 14-b3 there is a transition between two types of localized patterns with different ratios of SAOs per cycle and a lower frequency. In all cases, there is relatively abrupt transition between these patterns and SAO patterns, often not synchronized in-phase (not shown). These transitions sometimes involve irregular patterns for very small ranges of D_v

4 Discussion

Localized patterns in oscillatory networks where one oscillator (or cluster) exhibits LAOs or MMOs, while the other exhibits SAOs have been observed both experimentally and theoretically [32–36,72,73, 114–116]. In previous work we have established that these type of localized patterns can be obtained in networks of relaxation oscillators such as the FHN model and the Oregonator where the individual oscillators exhibit the supercritical canard phenomenon. In these networks, localized patterns required the presence of heterogeneity in the cluster distribution, which effectively creates heterogeneity in the inter-cluster connectivity. One important aspect of these networks is that the individual oscillators are monostable (they exhibit either LAOs or SAOs, but not both). The break of symmetry in the oscillation amplitude regime between the two (or more) clusters is a network phenomenon. However, how and under what conditions do localized patterns emerge as the result of the interaction between the network connectivity and the intrinsic properties of the individual oscillators (e.g., the canard phenomenon) was not fully understood.

In this paper we set out to address these issues in the context of PWL model of FHN type where the v-nullcline is cubic-like and the w-nullcline is either sigmoid- or linear-like. This model belongs to the set of minimal models that are able to produce localized patterns. Oscillatory patterns in globally coupled models have also been studied using the so called phase oscillators [37,75,117–122]. In these models, each oscillator is described solely by its phase and the effects of the interaction of oscillators on their amplitude is neglected by assuming weak coupling. These models are successful in capturing the phase-lock cluster patterns where the two oscillators are in the same amplitude regime, but they fail to capture the generation of the more complex patterns that involve more than one oscillatory amplitude regime and transitions between both.

In order to identify the principles that govern how the interplay of the intrinsic properties of the individual oscillators and the network connectivity interact to produce the localized patterns, we have considered a number of representative scenarios which include qualitatively different types of w-nullclines (sigmoid- and linear-like) and different parameter values that control the slope of the w-nullcline (α), its displacement with respect to the v-nullcline (λ), and the time scale separation between the participating variables (∊).

Our results show that the presence of the supercritical canard phenomenon in the individual oscillatory clusters is a necessary ingredient to produce localized patterns, but it is not sufficient (e.g., Figs. 9-a and 10-a). Localized patterns require a specific tuning between the various model parameters and the shape of the w-nullcline. The robustness of these patterns is strongly dependent on the shape of the w-nullcline. Models with a sigmoid-like w-nullcline produced more robust localized patterns than models with linear-like w-nullclines (e.g., Fig. 7) as well as abrupt transitions between phase-locked and localized patterns that were absent in models with linear-like w-nullclines.

An additional goal of this study was to understand the effects of the interplay between the two competing types of coupling: global inhibition and diffusion. Global inhibition tends to create clusters. Diffusion is local and tends to cause oscillators to synchronized in-phase. Indeed, when the two clusters have equal size and the oscillators are initially in the LAO regime, the addition of diffusion cause them to synchronize in-phase either in the LAO or SAO regimes depending on the D_v/γ ratio. However, when the cluster sizes were different, the addition of diffusion induced localized or MMO network patterns that were either synchronized in-phase or not depending also on the D_v/γ ratio. Even when the resulting patterns are synchronized in-phase, they do not resemble the patterns in the absence of diffusion.

We emphasize that the diffusive type of coupling we used in this paper is not realistic and does not reflect the diffusive effects between oscillators in each cluster in the original system. The question of how oscillators in each clusters are held together and how the different cluster sizes are generated as the result of the interplay of global coupling and diffusion remains open.

In this paper we have considered a specific type of global coupling motivated by previous work. Other studies have considered global feedback from the activator variable onto itself, rather than from the inhibitor onto the activator [69,123–126]. More research is needed to establish if and under what conditions localized patterns are possible in these networks and, if they exist, to characterize the similarities and differences between the patterns generated by the two types of global feedback.

An alternative scenario to the one we present here would involve the presence of bistability in the individual oscillators [108]. In this case, the role of the network connectivity would be to separate the oscillators into clusters by causing each oscillator to choose between the stationary solutions of the individual oscillators. This will require the presence of bistability between two oscillatory regimes. Alternatively, the localized solutions would involve one oscillatory and one silent cluster.

Network patterns can be generated by various mechanisms. On one extreme, these patterns can be imposed by the network connectivity with little or no participation of the individual oscillators. Our results highlight the richness of the patterns generated by the interplay of the network connectivity and the intrinsic properties of the individual oscillators.

5 Acknowledgments

This work was partially supported by the National Science Foundation grant DMS-1313861 (HGR)