Abstract
The Rulkov map model is an efficient model for reproducing different dynamics of the neurons. In specific neurons, the electrical activity is regulated by time-delayed self-feedback called autapse. This paper investigates how the dynamics of the Rulkov model change by considering the autaptic current. Both electrical and chemical autapses are considered, and bifurcation diagrams are plotted for different autapse gains and time delays. Consequently, various firing patterns of the model are illustrated. The results represent that the firing pattern is greatly dependent on the values of autapse parameters. Moreover, the average firing frequency is computed and it is shown that the enhanced firing activity is induced by the inhibitory autapse. The synchronous dynamics of coupled Rulkov maps in the presence of autapse is also studied. It is shown that the electrical autapse enhances synchronization in small time delays, while the enhancement is achieved by chemical autapse in any time delay. However, increasing the time delay reduces the synchronization region.
1. Introduction
The neuronal processes are regulated by the neurons’ electrical activities, which are influenced by many factors. The autapse is a kind of synapse connecting the axon and dendrites of the same neuron that can affect its response [1]. Two types of autapses, electrical and chemical, have been observed in different brain areas such as the neocortex and hippocampus [2, 3]. The dynamics of some continuous time neuron models, such as the three-dimensional Hindmarsh–Rose neuron, four-dimensional Filippov Hindmarsh–Rose, and Hodgkin–Huxley neuron [4–6], have been previously studied by considering the autapse. Moreover, some studies have introduced the memristive autapse since the electromagnetic effects can be considered using the memristor [7].
Apart from the neuron’s dynamics, the autapse plays an essential role in coupled neurons’ synchronized behavior [8]. Synchronization, which is defined as the coherence in the responses of connected neurons [9], is associated with several brain functions such as memory, attention, and learning [10, 11]. Hence, it has been the focus of neuroscientists for a long time [12–14]. The neurons’ synchronization has been studied by considering different synapses and autapses [15–17]. Ma et al. represented that synchronization can be induced in three coupled Hindmarsh–Rose neurons by appropriate selection of autapse gain and time delay [18]. Wu et al. reported synchronization transitions in the small-world network of autaptic Hodgkin–Huxley neurons [19]. They showed that the transitions for chemical autapse are more frequent than electrical autapse. Peng et al. represented that the synchronization in a small-world network of neurons relies on the autapse time delay; the synchronization in a small-world network of neurons relies nonmonotinically on the autapse time delay [20].
Although there is accuracy of continuous time neuron models, they often have high computational costs. This computational cost especially causes problems when studying the networks. Using the discrete time models is a solution to this problem. Therefore, many efforts have been devoted to proposing an appropriate discrete time neuron model [21–23]. Some researchers have also tried to modify these models by incorporating specific properties. One of the well-known discrete time neuron models is the Rulkov map with rich dynamics [24]. Wang and Cao analyzed the parameter space of the Rulkov map, and its different dynamics and firing regimes were obtained [25]. Bashkirtseva et al. studied the effects of noise on the dynamics of this map and showed that increasing the noise intensity can induce bursting [26]. Wang et al. investigated the impact of time delay on the dynamics of a nonchaotic Rulkov model [27]. In some studies, the effects of electromagnetic induction on the Rulkov model dynamics have been considered [28, 29]. In addition to the single neuron, dynamical behaviors of the coupled Rulkov map have been under consideration [30, 31]. Sun and Cao studied synchronizing two identical and nonidentical Rulkov maps [32]. They found that the electrically coupled chaotic Rulkov maps cannot reach complete synchronization.
In this paper, we propose the autaptic Rulkov neuron model. The dynamics of the Rulkov model are studied for each electrical and chemical autapse. For each case, the autapse gain and time delay are varied, bifurcation diagrams are plotted, and the firing patterns for different parameters are represented. Furthermore, the synchronization of two coupled autaptic Rulkov maps is investigated. The results show that chemical autapse enhances synchronization for any time delay. This enhancement is attained in the electrical autapse only in minor time delays.
2. Mathematical Model
The two-dimensional Rulkov map with the autapse current can be mathematically described as follows [24]:where . The variables are the fast and slow dynamical variables. The autapse current is a self-feedback current with a time delay that can be considered electrical or chemical. The electrical autapse can be mathematically described by the following equation [33]:
The chemical autapse can be defined as follows [33]:
The autapse coefficient is denoted by and is the time delay. The parameters of the chemical autapse are considered constant as and , so the chemical autapse is inhibitory. The Rulkov model has three parameters, namely, and , whose variation results in various firing patterns. The bifurcation diagrams of the model according to and for are shown in Figure 1 (first row). The corresponding Lyapunov exponent diagrams are shown in the second row of Figure 1. It can be observed that the model’s response bifurcates between different periodic and chaotic firings by varying or . In the following, the parameter is fixed at and the effects of are under consideration. The time series of the action potential of the neuron for different values are shown in Figure 2. The initial conditions are chosen randomly in the range [0 1]. For , the firing is chaotic and is a combination of square bursting and triangular bursting (Figure 2(a)). These firing patterns have been identified and analyzed in the previous studies [34, 35]. As increases to , the triangular bursting is disappeared (Figure 2(b)). By increasing , the chaotic bursting changes to chaotic firing, and the amplitude of the oscillations is also increased. Furthermore, the frequency of the oscillations is also increased. There are also some periodic windows in Figure 1(a). Figures 2(e) and 2(g) show a period-1 firing for and a period-2 firing for . Figures 2(f) and 2(h) illustrate two chaotic firings with higher amplitude and frequency for and , respectively.
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3. Dynamics of the Autaptic Rulkov Map
The firing patterns of the neuron model are impacted by considering the autapse. At first, the electrical autapse is investigated. To illustrate, the bifurcation diagrams by varying the autapse coefficient are plotted for different values in Figure 3. The autapse time delay is considered . For and , the dynamics remain chaotic by varying the autapse coefficient; however, the firing pattern is changed. For example, the time series of neurons for and and are represented in Figures 4(a)–4(c). It can be observed that for adding the autapse causes the elimination of triangular bursts (Figure 4(a)). For and , wider bursts can be seen. Figure 3(e) shows that the periodic firing in changes to chaotic by adding the autaptic current. However, some small periodic windows are observed. Figure 3(f) shows that for , the periodic windows are a bit larger. For according to Figure 3(g), the neuron has periodic firing until and then changes to chaotic. For , there is a periodic window in . The chaotic firing patterns and bursting of the model for and are shown in Figures 4(d)–4(f)).
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In Figures 3 and 4, the time delay was fixed at . To investigate the effects of the time delay, the bifurcation diagrams are plotted by varying for different time delays as in Figure 5. The value of is equal to and in parts (a)–(d), respectively. According to parts (a)–(b), for and , the time delay is influential only in low autapse coefficients. For , some periodic windows emerge when (Figure 5(c)). For , the time delay is more significant. By varying to , the period part in reduces to . As changes to or , some other periodic windows emerge whose region and width change according to the time delay value.
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Next, we investigate the effect of chemical autapse. The bifurcation diagrams for are plotted in Figure 6. Note that the figures are plotted for the range of with the bounded solution. Comparing Figures 3 and 6 shows that the bifurcation diagrams have more differences in different alpha values in the presence of chemical autapse. For , increasing leads to the decrement of the amplitude of oscillations, and the chaotic oscillation changes to periodic in (Figure 6(a)). Increasing to increases the coupling strength at which periodic dynamics emerge (Figure 6(b)). In , the amplitude of oscillations is decreased suddenly at (Figure 6(c)). In Figures 6(d)–6(h), small periodic windows are observed in small autapse gains. In and , the dynamics changes to periodic in and , and then instability emerges for larger values. Overall, the increasing of increases the amplitude of oscillations. Some different forms of time series of the model with chemical autapse are shown in Figure 7. It is observed that the model can exhibit triangular bursting (Figure 7(a)), combined triangular and square bursting (Figure 7(b)), plateau bursting (Figure 7(c)), different square bursting with different widths (Figures 7(d)–7(f)), and different periodic firings (Figures 7(g)–7(i)).
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One important point in the chemical synapses is the effect of the inhibitory and excitatory synapse on reducing or inducing the firing activity. In general, the inhibitory autapse can reduce the firing activity, and excitatory autapse induces the firing activity. However, in some studies, paradoxical roles that the inhibitory autapse induces enhanced the firing activity and excitatory autapse induces reduced the firing activity have been reported [36–41]. To obtain the effect of the introduced inhibitory autapse on the firing of the Rulkov map, the average firing frequency is computed. Figure 8 shows the average firing frequency of the Rulkov map for different values according to the gain of chemical autapse. It can be observed that except for , in general, the increment of the autapse gain leads to increasing in the average frequency. Hence, the paradoxical role is observed for the inhibitory autapse. For , the average frequency is first increased and then decreased by increasing the autapse gain.
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Similar to the electrical autapse, the next step is to examine the time delay impact. For and , increasing the time delay changes the periodic dynamics in large values to chaotic (Figures 9(a) and 9(b)). On the other hand, for and , increasing the time delay increases the periodic regions (Figures 9(c) and 9(d)).
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4. Synchronization of Coupled Autaptic Rulkov Maps
In this section, we investigate the synchronous behavior of coupled Rulkov maps. Two autaptic Rulkov maps are considered to be connected through inner linking function differences as follows:where is the coupling strength. The complete synchronization of two neurons is examined under different autapse gains and time delays. The synchronization error is computed bywhere is the total number of samples. Moreover, the phase synchronization of the neurons is evaluated bywherein shows the phase of th neuron, which is here defined bywhere is the initiation of th spike of th neuron. The phase synchronization of the neurons is identified by .
Figure 10 shows the synchronization error (first row) and parameter (second row) of coupled neurons with electrical autapse. The time delay is set at and in the left, middle, and right columns. The dark blue color region in the first row and the white region in the second row represent complete synchronization and phase synchronization, respectively. For , increasing the autapse gain decreases the required coupling strength for synchronization. While for and , the autapse gain has a negligible effect. Overall, by increasing time delay, the synchronization region reduces. Moreover, it can be observed that for the neurons are not completely synchronous but phase synchronized. The synchronous firing pattern of the neurons differs in different parameter values. Two examples of the synchronous firing patterns are shown in the first row of Figure 11.
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The synchronization of coupled neurons with chemical autapse is illustrated in Figure 12. The time delay in the left, middle, and right columns is and , respectively. It is observed that for all cases, increasing the autapse gain enhances synchronization by decreasing the needed coupling strength. However, as the time delay increases, the synchronous region is reduced. Similar to electrical autapse, as exceeds 0.4, the neurons have only phase synchronization. The same as the electrical autapse, the synchronous firing pattern in this case depend on the value of coupling strength and autapse gain. For example, two synchronous time series of neurons with chemical autapse are shown in the second row of Figure 11.
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5. Conclusion
This paper proposed a modified neuron map by introducing the electrical and chemical autapse to the Rulkov neuron model. The Rulkov model is a simple map-based model which can represent various neurons’ behaviors. This paper aimed to investigate the effect of autapse on the dynamics of the model in different parameters. Hence, the bifurcation diagrams of the autaptic models were obtained for different autapse gains and time delays for each electrical and chemical autapse. The time series of the model in different parameters were shown. It was observed that increasing the autapse gain can change the dynamics of the neuron from chaotic to periodic and vice versa. Therefore, multiple firing patterns could emerge. Also, the time delay had a significant impact on the dynamics of the neuron. Computing the average firing frequency according to the chemical autapse gain revealed that the firing frequency increases by increasing the autapse gain. Therefore, the inhibitory autapse induces the enhanced firing activity.
After investigating the dynamical behavior of the Rulkov model with autapse, its synchronization behavior was under consideration. Two autaptic Rulkov maps were studied by inner linking function difference coupling. It was found that in the presence of electrical autapse, the autapse can enhance synchronization when there is slight time delay. In larger time delays, the effect of electrical autapse was insignificant. For the chemical autapse, increasing the autapse gain decreased the required coupling strength for synchronization in all time delays. However, the synchronization region was reduced by increasing the time delay.
A summarization of some recent related papers is presented in Table 1. In the previous studies, different neuron models have been considered and the effects of autapse on their dynamics have been explored. The importance of the present study compared to others is that here a discrete time neuron model has been adopted, while the other works are all based on the continuous time models. In this paper, the effect of autapse on the firing patterns of the well-known Rulkov map was reported for the first time.
The dynamical mechanism for bursting modulated by autapse and the change in the bursting patterns, which is obtainable by the fast-slow variable dissection method [42, 43], can be investigated in the future works. Moreover, the energy flow and the power consumption [44, 45] needed for the emergence of different patterns is another important point that can be considered in the forthcoming study.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by the Center for Nonlinear Systems, Chennai Institute of Technology, India, vide funding number CIT/CNS/2023/RP/003.