Long-range interaction effects on coupled excitable nodes: traveling waves and chimera state.

In this paper, analytical and numerical studies of the influence of the long-range interaction parameter on the excitability threshold in a ring of FitzHugh-Nagumo (FHN) system are investigated. The long-range interaction is introduced to the network to model regulation of the Gap junctions or hemichannels activity at the connexins level, which provides links between pre-synaptic and post-synaptic neurons. Results show that the long-range coupling enhances the range of the threshold parameter. We also investigate the long-range effects on the network dynamics, which induces enlargement of the oscillatory zone before the excitable regime. When considering bidirectional coupling, the long-range interaction induces traveling patterns such as traveling waves, while when considering unidirectional coupling, the long-range interaction induces multi-chimera states. We also studied the difference between the dynamics of coupled oscillators and coupled excitable neurons. We found that, for the coupled system, the oscillation period decreases with the increasing of the coupling parameter. For the same values of the coupling parameter, the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics. The analytical approximation shows good agreement with the numerical results.

Introduction

In neural circuits, network operations and communication are supported by action potentials [1], [2]. The mechanism of generation of action potentials is a key component in the regulation of information flow [3] and it is initiated at the gap junction (GJ) level. For this reason, GJs have been regarded as an important component within the neuronal networks that underlie synchronous neuronal activity and field potential oscillations [4]. Soon after GJs were identified as structural and functional components of intercellular junctions, attempts were made to discover their protein makeup. Results showed that GJs are formed by the interaction of 2 hemichannels (HCs), one contributed by each of the adjacent cells [5], [6]. Each HC is composed of connexins in vertebrates or innexins in invertebrates [7], [8]. Gap junctions of identical HCs are known as homo-typic GJs, while those of different HCs are known as hetero-typic GJs [9], [10]. A human presents more than 20 different connexins species [11], [12]. They are all named after their predicted molecular weight, expressed in kilodaltons (e.g., connexing43 call Cx43, has a molecular weight of 43 kilodaltons). Gap junctions permit the intercellular diffusion of ions, metabolites, and cytoplasmic messenger molecules smaller than 1 kilodalton. This short route connection serves as a powerful cell function coordinator in complex tissues like the heart and brain. But HC activity can be regulated by phosphorylation [13], [14], [15], redox potential [16], [17], [18], [19], pH [20], [21], and other conditions [22]. Exposition to low extracellular Ca2+ concentration [22], mechanical stress [23], or positive membrane voltages [22] has been observed to increase the open probability of different Cx HCs. This regulation affects the membrane fluidity of the GJs [24].

However, studies regarding the transmission of the information taking into account the regulation of HC activity still lack, the same as the studies on the influence of this regulation on the chimera state [25], [26], [27], [28], [29], [30], [31]. Chimera state represents a peculiar collective behavior of coupled identical oscillators characterized by the coexistence of coherent or synchronized and incoherent or unsynchronized groups. After its discovery in 2002 by Kuramoto and Battogtokh [32], the chimera state has been investigated in several neuronal processes, and some particular patterns have been found such as multi-chimera [25], [33], [34] and traveling chimera [25], [35], [36], [37], [38], [39].

In this work, we will study the effects of the regulation of HC activity on the electrical current at the connexins level of electrical synapses.

We hypothesized that a specific long-range parameter represents the regulation parameter of HC activity on the communication in the brain areas. It characterizes the coupling between cells at different sites. The variation of the long-range parameter can be related to the variation of the temperature or of the fluidity of the milieu, or redox potential, etc. In recent years, many studies on the effect of long-range interactions have been carried out on different approaches [40], [41]. It has been shown that long-range interactions have effects on calcium-wave propagation [42] or on chimera patterns [43], [44]. We will also study the difference between the dynamics of coupled oscillators [45] and coupled excitable neurons.

The aim of the present work is embedded in the following twofold questions, considering [46] in which the electrical coupling between neurons has been elaborated, and [42] in which long-range interaction has been defined:

- Firstly, what are the effects of long-range coupling on the generation of action potentials in the electrically coupled excited neurons? Furthermore, how do these action potentials differ from the ones of coupled oscillators?

- Secondly, how does long-range interaction influences the chimera state?

The paper is organized as follows: Section 2 introduces the mathematical model of long-range interaction. Section 3 describes the mathematical model of the threshold parameter of electrical coupling, considering long-range interaction and exploiting the FitzHugh-Nagumo (FHN) neuron model used for most of the studies in the literature. Section 4 and Section 5 study the effects of the long-range coupling on the bidirectional coupling and the unidirectional coupling respectively. In the concluding section, we summarize our main results, present possible applications, and discuss open problems.

Model description

While electrical synapses are known to be the target of the neuromodulatory transmitters, increasing evidence indicates that they are profoundly influenced by the activity of the networks in which they are embedded. It is in this order that the HC activity of transmission of the information could be affected. We use a long-range coupling to model this influence. The model of the long-range coupling between neurons j and k is given by the following relation [42], [47]:

In this relation, represents the system's coupling parameter, and r defines the range of interaction. This r can represent the parameter that regulates the HC activity during the interaction. According to the nature of this regulation parameter, the long-range coupling could be optimal for the proper choice of the range r. For example, the human brain's suitable temperature is taken very near 37 °C [20]. Far from this value, it does not work conveniently.

The absolute value represents the measure of the distance between neurons j and k.

However, Eq. (1) is more adapted for to ensure that the long-range would be equal to zero when . To manage even the case of we will consider in this work the following model of long-range:

The model of Eq. (2) is similar to that of Eq. (1) when .

The graphical model of the long-range coupling is given by (Fig. 1) representing its variation with the range of interaction r and the distance . It shows that the coupling between cells decreases gradually as the distance increases. For the same HC activity, the near neighbors' long-range coupling parameter is greater than that of far neighbors. This description characterizes that as the signals propagate in the milieu, the number of receptors in the considered cells decreases.

Figure 1

Influence of the distance between neurons |k − j| on the long-range coupling σ for σ0 = 1. The long-range coupling decreases with the distance between neurons.

Mathematical model

Network model

In order to properly predict certain biological processes that are associated with the neuronal behavior, R. FitzHugh and J. Nagumo suggested the FitzHugh-Nagumo (FHN) model [2], [48]. It is an example of a relaxation model.

The mathematical form of the individual system describing the process of activation and deactivation dynamics of a spiking neuron is given by the following set of ordinary differential equations:

Here x and y represent the activator (the membrane potential) and inhibitor (slow variable for current) variables. b is the small parameter characterizing the time scale separation of fast activator and slow inhibitor. In the present paper, we fix it at . The system exhibits limit cycle relaxation oscillations for the proper choice of the excitability threshold a. Otherwise, it exhibits either oscillatory behavior for or excitable behavior for .

In the following parts, we assume identical elements in the excitable regime. We consider a network with a ring of non-locally coupled topology. In the network, we let N nodes be uniformly distributed on the ring and let each node be connected with its nearest neighbors. Each node represents a neuron where the interaction between two neurons can only be implemented through the connected connexins. The fluidity at the level of connexins is represented by the long-range coupling given below:

Neuronal networks are often structured in topologies where strong interconnections between different neurons are found within a range R, but much fewer connections exist at longer distances [49], [50]. We approximate this feature by coupling with a strength within the R nearest neighbors from both sides. 2R is the total number of connections in the network for node j. Another important feature of Eq. (4) is that it contains not only direct and couplings, but also cross-couplings between the membrane potential (x) and the slow variable for current (y). For the sake of simplicity, a rotational coupling matrix depending on a single parameter ϕ has been defined [49], [51] to model this feature as:

In this section, we address the question of the influence of long-range coupling on the excitability threshold a by taking the parameter ϕ near to as suggested in [49]. For , ,

Eq. (4) is reduced to:

This yields the following approximation when considering only the node j in the network: with

The system exhibits either oscillatory behavior for or excitable behavior for . The bifurcation points are obtained for each value of and by:

The maximum value of a for each value of R is obtained for:

The analytical result has revealed that by increasing R, the interval of values of a allowing oscillatory behavior is reduced (Fig. 2a), and its maximum is translated to the right way on r−line as presented in Fig. 2b. But whatever values of R and , the bifurcation point tends to 1 when r tends to 0 or when r tends to 1. By taking r values near to 0 or near to 1 the slopes at the right side in Fig. 3a and b decrease and are closed to .

Figure 2

Variation of the excitability threshold a, when σ0 = 5, (a) with r, for R = 10 (black dashed line), R = 15 (blue dashed line), R = 20 (red dashed line), R = 65 (green dashed line), (b) with R for r = 0.8.

Figure 3

Variation of the excitability threshold a, (a) with r for R = 10 and σ0 = 5, (b) with σ0 for R = 10 and r = 0.8.

For the numerical simulation, we have considered a ring of neurons in which a signal can be reproduced after neurons and thus the boundary conditions are taken to be cyclic and defined as and .

Under random initials conditions and to be able to compare the analytical approximations' results (Fig. 3a) with those of numerical calculations (Fig. 4), we considered the following parameters' values: , , , and . The maximum value of a is analytically equal to , obtained for . From Fig. 4a, when r tends to 0 or to 1, a is greater than the bifurcation value at this level, and the nodes are in an excitable state. By taking suitable values of r, values of a are less than the bifurcation value as seen in (Fig. 4b, c, d), so the nodes are in the oscillatory state after transition time from excitable state. The duration of transition decreases when r tends to its value corresponding to the bifurcation maximum value. The analytical approximation (Fig. 3a) is then in good agreement with the numerical interpretation (Fig. 4).

Figure 4

Influence of the long-range coupling on the appearing of oscillations at the excitable state when a = 1.27, (a) for r = 0.1 or for r = 0.98, (b) for r = 0.72, (c) for r = 0.8, (d) for r = 0.85.

Period of network of neurons

This part aims to compare the oscillation period of a non-coupled oscillator and coupled oscillators for with the oscillation period of coupled excitable neurons for .

Considering a single oscillator (Fig. 5), or coupled system (Fig. 6) for , the local dynamics are then periodically oscillating. The trajectories are linked to the nullcline at the right branch (firing phase, from to ) and at the left branch (refractory phase, from to ). The cycle approximated by four points A, B, C, and D corresponds to one period T. The approximation of the period for a single neuron when is given by the following equation [52], [53]: Where denotes the period of single oscillator.

Figure 5

(a) Phase portrait (trajectories in red line and nullclines in blue dashed line) of a single FitzHugh-Nagumo system and (b) the corresponding time series of activator x(t) for a = 0.1.

Figure 6

(a) Phase portrait (trajectories in red line and nullclines in blue dashed line) of coupled oscillatory FitzHugh-Nagumo neurons and (b) the corresponding time series of activator x(t) for a = .35.

In the case of coupled oscillators when , the approximation of the period is given by the following equation [45]:

Where the parameter σ denotes the coupling strength, is the period of coupled oscillators. The period decreases with increasing of σ [45].

In the case of excitable dynamics, by example for , for a single neuron, the neuron does not oscillate, it stays constant at the value of a as shown in Fig. 4a. By coupling excitable neurons, it appears oscillation as shown in Fig. 7. However, contrarily for the precedents cases, the blue trajectory is not linked to the nullcline at the right branch (firing phase) and the left branch (refractory phase). The slow phases of the trajectory are separated from the nullcline by a certain gap. Since it is the coupling of excitable neurons, the trajectory starts to the value of , and performs a transient phase before adopting the permanent trajectory Fig. 8. The cycle approximated by four points A, B, C, and D corresponds to one period T.

Figure 7

(a) Phase portrait (trajectories in red line and nullclines in blue dashed line) of coupled excitable FitzHugh-Nagumo neurons and (b) the corresponding time series of activator x(t) for a = 1.35.

Figure 8

Snapshot of coupled excitable FitzHugh-Nagumo neurons for a = 1.35.

In the following, we consider the excited coupled FHN system as described by Eq. (4). Its phase portrait and the corresponding time series of activator plotted in Fig. 7 are different from those of single neurons plotted in Fig. 5 and those of coupled oscillators plotted in Fig. 6. Nevertheless, the snapshot of each neuron of the system (Fig. 8) shows that each node spends the most significant part of its time in the firing () and the refractory () phases.

Similar to [45], [52], [53], we employ an analytic approximation for the period of neurons defined by Eq. (7) when . We have: with defined by Eq. (8).

One of the differences between coupled oscillators' dynamics and that of coupled excited neurons resides at the time series of activator level. For the first case (Fig. 6b), oscillations are produced between ±2, that is why to express the period, authors of [45], for integration, used ±2 as the maximum and the minimum.

For the second case (Fig. 7b), the maximum and the minimum are respectively greater than 2 and less than −2. We can integrate Eq. (13) analytically from to defined by and . α is the distance between the firing phase of the trajectory of uncoupled oscillators and coupled excitable neurons. It increases with increasing a, for the chosen value , . Let consider as a time of the firing phase and the time of the refractory phase:

, or and , or

The time duration of one cycle of a coupled excitable neuron when we neglect the fast motions and is then given by : where denotes the period of coupled excited neurons.

The analytical approximations of the period for an FHN system in the case of coupled oscillators (blue line) and of coupled excitable neurons (red line) are plotted in Fig. 9 considering the coupling parameter () constant. It comes out from Fig. 9 and from [45], [52], [53] that for the same value of , the oscillation period of a single oscillator is greater than the oscillation period of coupled ones . For the coupled system (oscillatory or excitable states), the oscillation period decreases when σ increases. For the same values of σ the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics . This analytical approximation is in good agreement with the numerical results from Figs. 5b, 6b, 7b.

Figure 9

Analytical approximations of the period for an FHN system. (a) Oscillatory dynamics for a = 0.35 (blue line), (b) excitable dynamics for a = 1.35 (red line).

It is important to notice that, the amplitude of the time series of activator of coupled excitable neurons increases as a increases far from ±1.

The analytical approximations of the period of excitable coupled FHN in term of the long-range parameter r are plotted in Fig. 10. The oscillation period firstly decreases for the range of r from to 0.8 which corresponds to the interval where the coupling parameter increases. Secondly, from until , the oscillation period increases which corresponds to the interval where the coupling parameter decreases.

Figure 10

Analytical approximations of the period of excitable coupled FHN in term of the long range parameter r for a = 1.35.

However, as in [52], [53], the transition from the left to the right branch or from the right to the left branch is not directly made. To account for the delay of transition, authors of [52], [53] modified the period of Eq. (11) by introducing a small parameter e as follows:

In the light of the above, our expression of the period becomes:

In Fig. 11, we compare the analytical results with numerical simulations of the excitable coupled FHN system. For , and for , the analytical approximation (Solid red line) is in good agreement with the numerical results (black circles).

Figure 11

Oscillation period of excitable coupled FHN in term of the coupling parameter for a = 1.35. Period obtained from numerical simulations (black circles), period obtained from analytical approximation of Eq. (16) (solid red line), period obtained from analytical approximation of Eq. (18) with the parameter e = 0.22 (solid black line).

Effect of the range of interactions on the bidirectional coupling

In this section, we study the influence of the long-range parameter in the system described by Eq. (4). Elements are in excitable dynamics and parameters are set as , and .

Fig. 12 is made of two columns for (a), and (b). In the first row, the space-time plot of the node variables is depicted, and their corresponding snapshot at two different time (red color) and (blue color) are figured out in the second row. The last row presents the times series of the membrane potential at the nodes (red color) and (black color). We can see that nodes are organized into spatial traveling groups of elements, interrupted by regions of oscillating elements that travel from the last to the first node. As the moving fronts on the panels indicate, the velocity of the oscillating regions around the ring increases with r.

Figure 12

Traveling wave by increasing the long-range r. (a) For r = 0.53 and (b) for r = 0.54.

The time series confirm the traveling behavior for two different times. Since incoherent domains are moving from the right side to the left, the node oscillates before the node . Even looking at the snapshot panels of the node variables at two different times, (red color) and (blue color), the incoherent behaviors in red are at the right side of the incoherent behaviors in blue. That is with respect to the time of the snapshot. The increasing of r facilitates the generation of new incoherent regions, and the number of spikes that form the spatial profile of the wave increases. It is to notice that those incoherent domains travel around the ring using a constant velocity [26], [31], as clearly indicated by the corresponding space-time plot.

For greater values of r, the area of traveling coherent regions decreases while the number of traveling waves increases. This traveling patterns behavior is illustrated by the Fig. 13. The area of the coherent regions in Fig. 13b for , is smaller than the area of the coherent regions in Fig. 13a for .

Figure 13

Traveling patterns generated by increasing the long-range r. Panels present the space-time plot of the node variables x and their corresponding snapshot. (a) For r = 0.55, (b) for r = 0.56.

We found in Fig. 13 the traveling patterns in which the proportion of coherent domain is reduced when r increases. It is on this base that the coherent domains disappear; from , the whole system becomes completely unsynchronized. It starts to form the oblique stripes, and another typical pattern of traveling waves appears as shown in Fig. 14.

Figure 14

Traveling action potential for r = 0.7. (a) The space-time plot of the node variables x. (b) The times series of membrane potential x.

We conclude from the above results that long-range coupling takes a key role in the formation of different traveling patterns of Eq. (4).

Effect of the range of interactions on the unidirectional coupling

In this section, we study the influence of the long-range parameter in the system unidirectionally coupled described by Eq. (19). Elements are in excitable dynamics and parameters are set to , and .

Fig. 15 has two columns for (left side), and (right side). In the first row, the space-time plot of the node variables is presented, and their corresponding snapshot at the time is depicted in the second row. The last row presents the times series of membrane potential . We can see that nodes are organized into coherent groups of elements, separated by regions of incoherent elements. This behavior of the coexistence of multiple synchronized and unsynchronized regions is well known as the multi-chimera states.

Figure 15

Multi-chimera states. (a) For r = 0.53. (b) for r = 0.54.

Conclusion

The main objective of this work was to investigate the dynamic of excitable coupled FitzHugh-Nagumo oscillators in a ring for which we have introduced the long-range interactions. We have reported how the long-range interactions can modify the excitability threshold a in an electrical coupled excitable neurons for each coupled parameter . For an appropriate choice of the rotational coupling matrix, the position of the maximum of a for each coupled neuron j on the depends only on the number of neurons R which are coupled to j. The above results are in accordance with the results of the (Fig. 4) presenting the influence of the long-range coupling on the appearing of oscillations at the excitable state. We also study the difference between the dynamics of coupled oscillators and coupled excitable neurons. We found that, for the coupled system (oscillatory or excitable states), the oscillation period decreases with the increasing of σ. For the same values of σ the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics . The analytical approximation is in good agreement with the numerical results. We also demonstrated that the combination of the parameter ϕ and the long-range interaction r can produce different traveling patterns such as traveling waves in the case of bidirectional coupling and multi-chimera in the case of unidirectional coupling.

These findings strongly suggest that the long-range coupling applied to the neurons' activities could play a critical role in the dynamics of cerebral activities.

Declarations

Author contribution statement

Guy Blondeau Soh: Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Robert Tchitnga: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Paul Woafo: Conceived and designed the experiments; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

No data was used for the research described in the article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.