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Article

A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model

by
Othman Abdullah Almatroud
1,
Viet-Thanh Pham
2,* and
Karthikeyan Rajagopal
3,4
1
Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia
2
Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City 70000, Vietnam
3
Center for Research, Easwari Engineering College, Chennai 600089, India
4
Center for Research, SRM Institute of Science and Technology-Ramapuram, Chennai 600089, India
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(19), 2970; https://doi.org/10.3390/math12192970
Submission received: 11 September 2024 / Revised: 22 September 2024 / Accepted: 23 September 2024 / Published: 25 September 2024
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)

Abstract

:
This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior within specific parameter ranges, while increased magnetic strength tends to maintain periodic dynamics. The emergence of various firing patterns, including periodic and chaotic spiking as well as square-wave and triangle-wave bursting is also evident. The modified model also demonstrates multistability across certain parameter ranges. Additionally, the dynamics of a network of these modified models are explored. This study shows that synchronization depends on the strength of the magnetic flux, with synchronization occurring at lower coupling strengths as the magnetic flux increases. The network patterns also reveal the formation of different chimera states, such as traveling and non-stationary chimera states.

1. Introduction

The study of mathematical neuronal models plays a crucial role in neuroscience, as these models help us understand the mechanisms behind neuronal firing and allow for the simulation of various firing patterns [1]. The foundation of many neuron models is the Hodgkin–Huxley (HH) model, which is based on the conductance of ion channels and explains how action potentials in neurons are propagated [2]. This model has been simplified over time, leading to the development of other models like the FitzHugh–Nagumo [3], Hindmarsh–Rose [4], and Morris–Lecar [5] models, which also generate action potential dynamics. The Morris–Lecar model, a two-dimensional differential equation, approximates the same three currents in the HH model and is particularly effective in modeling fast-spiking neurons, such as neocortical pyramidal neurons. Bifurcation analysis of the Morris–Lecar model has been explored in studies by Liu et al. [6] and Tsumoto et al. [7]. However, since two-dimensional models cannot produce the bursting firings typically seen in biological neurons with fast and slow variables, Izhikevich proposed a fast–slow 3D Morris–Lecar neuron model in 2000 [8]. Bao et al. [9] further analyzed chaotic bursting dynamics and multistability in this 3D autonomous Morris–Lecar model. In 2016, Hu et al. introduced an electronic circuit implementation of the ML model [10]. In addition, Shi and Wang [11] later examined the model’s firing patterns by incorporating fractional-order derivatives.
Recent research in neuroscience has increasingly focused on modifying traditional neuron models to enhance their dynamic responses, taking into account various external environmental factors [12,13,14]. For instance, Lv and Ma [15] demonstrated that the influence of electromagnetic induction can be incorporated into neuron models by integrating a memristor. They showed that fluctuations in membrane potential, caused by changes in ion concentrations, can also induce time-varying electric and magnetic fields. Their findings highlighted that the intensity of the magnetic field significantly impacts the model’s firing patterns. Building on this, various memristive neuron and neural network models have been developed [16,17,18,19,20]. Xu et al. [21] introduced a simplified Hodgkin–Huxley circuit featuring memristive sodium and potassium ion channels and explored its periodic and chaotic spiking behaviors. Chen et al. [22] developed an enhanced memristive FitzHugh–Nagumo (mFHN) model using a flux-controlled memristor with a sinusoidal mem-conductance function, revealing a rich array of hidden dynamics triggered by the memristor’s initial-offset boosting. Lin et al. [23] proposed a novel neuron model incorporating a new type of locally active, non-volatile memristor and examined its firing patterns and multistability.
In addition to the dynamics of individual neuron models, understanding the collective behavior of neuron groups is crucial [24]. Simplified and enhanced neuron models facilitate the study of neuronal population behaviors, such as synchronization and chimera states. Synchronization among a population of neurons is linked to various cognitive functions in the brain [25] and is also associated with several neurological disorders, including epileptic seizures [26]. Numerous studies have explored the characteristics of synchronization within neuronal networks [27,28,29,30]. Additionally, the phenomenon where coherent and incoherent groups of neurons coexist, known as a chimera state, is a distinctive feature of neuronal systems and is associated with conditions like Parkinson’s disease and schizophrenia [31,32]. The emergence of chimera states is influenced by factors such as synaptic connections, network structures, and coupling configurations [33,34,35]. Depending on the properties of the coherent and incoherent clusters, various types of chimera states can arise, including non-stationary chimera, traveling chimera, chimera death, etc. [36,37,38].
Recent research has introduced a novel memristor with a ReLU (Rectified Linear Unit) activation function featuring two-segment linearity, which is advantageous for enhancing neuron and neural network [39,40,41,42]. Since ReLU is a simple, piecewise linear function, memristors can quickly change states, speeding up calculations. By reducing the need for complicated digital processing to run ReLU, memristors help lower delays, making neural networks faster during training and prediction. Additionally, memristors can store and keep the neuron’s activation state (whether it is “on” or “off”), reducing the need for frequent memory access. Memristor-based ReLU setups also simplify circuit design by removing the need for extra digital parts, leading to more compact and efficient neuromorphic systems. Hence, it has been recently used in neuron and neural networks. For instance, Chen et al. [43] developed a ReLU-type HNN model that revealed a wide range of dynamic behaviors. In this study, we investigate the impact of the ReLU-type memristor on the dynamics of the Morris–Lecar neuron model. Using bifurcation and Lyapunov exponent diagrams, we identify regions of periodic and chaotic dynamics and analyze how each parameter influences these behaviors. This study demonstrates that various firing patterns, including periodic and chaotic spiking as well as different bursting behaviors, can be produced. Additionally, evidence of the system’s multistability is presented. Furthermore, we examine the effect of the memristor on the behavior of a network of coupled Morris–Lecar neurons, showing that synchronization is enhanced with increased magnetic induction. The network model also exhibits a range of chimera states.

2. Model

The Morris–Lecar model is a simplified yet powerful framework for modeling neuron dynamics. It strikes a balance between computational simplicity and biological accuracy, making it ideal for simulating excitable and oscillatory neuronal behaviors, bistability, and different firing patterns in neurons. Its ability to exhibit both Type I and Type II excitability and its capacity for phase plane analysis make it a versatile tool for studying neural circuits and behavior. This neuron model is described by the following equations:
C d V d t = g C a M V C a V + g K W V K V + g L V L V + I 0 , d W d t = τ W W W ,
where M and W are the opening probabilities of Ca++ and K+ channels at steady state, and τ W denotes the opening rate constant of K+ channels, which are defined as follows:
M V = 0.5 + 0.5 tan h V V 1 V 2 , W V = 0.5 + 0.5 tan h V V 3 V 4 , τ W V = σ 1 cos h [ 0.5 ( V V 3 ) V 4 ] .
The variables V and W denote the membrane voltage and the activation of delayed K+ current. The maximal conductances associated with the three transmembranar currents are shown by g C a , g K , and g L , and the steady-state Nernst potentials of calcium ions, potassium ions, and leak channels are shown by V C a , V K , and V L . The input current is denoted by I 0 , and C is the capacitance of the membrane. The parameters of the model can be set to C = 1 , V C a = 1 ,   V L = 0.5 ,   g C a = 1.2 ,   g K = 2 ,   g L = 0.5 ,   V 1 = 0.01 ,   V 2 = 0.15 , V 3 = 0.1 , a n d   V 4 = 0.05 , σ = 3 . Note that V K and I 0 are chosen as the varying parameters for creating abundant firing patterns.
It has been demonstrated that a memristor can be employed to characterize the magnetic induction flow triggered by the neuron’s membrane potential. The magnetic induction flow of the memristor can act as a slow modulation current introduced into the ion channel. A voltage-controlled generic memristor can be introduced and defined to characterize the magnetic induction flow. A simple activation function is the ReLU function defined as R e L U ( x ) = m a x ( 0 , x ) . The ReLU-type memristor has been frequently used in neuron model recently. The memristor with ReLU function as a memductance function can be described as
I M = W ϕ V M = a b     R e L U ϕ V M , ϕ ˙ = e ( V M + V 0 ) ,
where I M and V M are the memristor’s current and voltage and ϕ is the magnetic flux. Therefore, the three-dimensional memristive Morris–Lecar model can be defined as follows:
C d V d t = g C a M V C a V + g K W V K V + g L V L V + I 0 k a b   R e L U ϕ V , d W d t = τ W W W , d ϕ d t = e ( V + V 0 ) ,
where k is the strength of the magnetic induction. The other parameters are set to a = 1 , b = 0.2 , e = 0.1 , and V 0 = 0.2 [44]. The effect of the magnetic induction through the defined memristor on the dynamics of the ML model is studied in the next section.

3. The Magnetic Induction Effect on the ML Model Dynamics

One of the key parameters affecting the system’s dynamics is V K , where changes can result in chaotic or periodic bursting and spiking. The bifurcation diagram of the model as a function of V K for various magnetic induction strengths is illustrated in Figure 1. Below each bifurcation diagram, the corresponding maximum Lyapunov exponent is displayed. In part (a), with k = 0.5 , the oscillation period increases as V K rises from 360 , followed by a small chaotic region in 205 < V K < 203 . For k = 3 (part (b)), the oscillation period also increases at lower V K values, with two chaotic regions appearing in 227 < V K < 223.3 and 207.1 < V K < 201.4 . As k increases to 5 (part (c)), the bifurcation pattern remains similar, but a larger chaotic region emerges in 221.7 < V K < 213.3 . At higher k values, the chaotic region disappears, as seen in part (d) for k = 8 . Comparing parts (a) through (d) reveals that increasing k leads to a decrease in the ISI value.
Another key parameter influencing the model is the current I 0 . The bifurcation diagrams based on I 0 , along with the corresponding Maximum Lyapunov Exponent (MLE) diagrams, are displayed in Figure 2. These diagrams reveal that bifurcations occur at very small values of I 0 . In part (a), where k = 2 , a period-4 behavior transitions into chaos at I 0 = 0.008 because of a crisis event. This is followed by a periodic window resulting from period halving, and the chaotic region eventually ends at I 0 = 0.0135 . For k = 5 (part (b)), the system exhibits chaotic dynamics in the range 0 < I 0 < 0.0095 , which then shifts to periodic behavior through period halving. Figure 3 illustrates the bifurcation diagrams with respect to magnetic strength for I 0 = 0 (a) and I 0 = 0.01 (b). It is evident that chaotic firing occurs over a larger range of k when I 0 = 0.01 . To further demonstrate, Figure 4 showcases various periodic and chaotic firing patterns of the model.
The model’s dynamical behavior in two-dimensional planes is illustrated in Figure 5. The left column displays the oscillation periods, while the right column presents the maximum Lyapunov exponent. In the bifurcation diagrams, blue indicates dynamics with low periods, while purple, orange, pink, and green regions correspond to periodic dynamics with higher periods. In part (a), it is observed that as the value of k increases, the oscillation period decreases. The chaotic region, depicted in yellow, is very narrow. Part (b) reveals that for higher current values, the dynamics remain periodic. At lower current values, high-period dynamics and chaotic behaviors are evident. However, increasing k also causes the dynamics to transition to periodic behavior.
It is noteworthy that the ReLU-based memristor causes the ML model to exhibit multistability within a small range of parameters. The multistable dynamics have also been previously shown in memristive ML models [45]. Two examples of multistable dynamics are illustrated in Figure 6. In part (a), with parameters set to k = 0.6 , V K = 220 , and I 0 = 0 , a period-1 spiking behavior coexists with a period-9 bursting dynamic. In part (b), when the k value is adjusted to k = 6.5 , it is observed that either period-2 or period-4 dynamics can emerge, depending on the initial conditions.

4. The Magnetic Induction Effect on the Synchronization

This section explores the impact of magnetic induction on the synchronization of the introduced ML neurons when they are coupled. A network of N diffusively coupled memristive ML neurons is defined as follows:
C d V i d t = g C a M V C a V i + g K W i V K V i + g L V L V i + I k a b   R e L U ϕ i V i                       + ϵ   j = 1 N G i j ( V j V i ) , d W i d t = τ W W W i , d ϕ i d t = e ( V i + V 0 ) ,
where ϵ denotes the coupling strength and G represents the network structure which is a ring structure with two nearest-neighbor coupling. To characterize synchronization, the synchronization error is calculated as follows:
E = 1 N T t = 0 T j = 2 N ( V j t V 1 t ) 2 .
Zero synchronization error indicates synchronization. The synchronization error according to the coupling strength and the magnetic induction strength is shown in Figure 7a. The parameters are set to V K = 220 and I 0 = 0 . The dark blue color in the two-dimensional plane shows zero error. It can be observed that as the magnetic induction strength increases, the coupled ML neurons synchronize in lower coupling strength. For better illustration, 1D synchronization errors for k = 3 , 5 , and 8 are presented in Figure 7b.
The network’s patterns before achieving complete synchronization are also examined. The spatiotemporal patterns reveal that varying parameters can lead to the formation of different chimera states. In Figure 8, part (a), with k = 8 , ϵ = 1.4 , and part (b), with k = 4 , ϵ = 0.5 , there are both synchronous and asynchronous groups, where the asynchronous cluster moves regularly over time, resulting in a traveling chimera pattern. In part (c), although the asynchronous cluster moves in time, its movement is imperfect, leading to an imperfect traveling chimera for k = 1 , ϵ = 0.5 . In part (d), with k = 1 , ϵ = 1 , the positions of the asynchronous cluster change over time, creating a non-stationary chimera pattern.

5. Conclusions

This paper introduced a new neuron model that builds on the Morris–Lecar neuron model by incorporating a ReLU-based memristor. The ReLU-based memristor, characterized by a memductance function based on the maximum function, is a simple memristor that has recently been applied in various neuron models. In this study, we explored the impact of this memristor on the Morris–Lecar neuron model. We first analyzed the dynamics by varying model parameters and the strength of magnetic flow. Adjusting the parameter V K (the Nernst potential of potassium ions) revealed that as V K increases, the oscillation period increases, leading to bursts with multiple spikes and eventually creating a small chaotic region. However, the dynamics were also influenced by the magnetic flow strength, where high magnetic strengths led to the disappearance of chaos. Examining the effect of input current showed that bifurcations occur at very low current values, and as the current increases, the dynamics remain periodic. Visualizing the model’s attractor and time series demonstrated its ability to exhibit periodic and chaotic spiking, as well as square-wave and triangle-wave bursts. Our investigations also revealed the emergence of multistability. In the next phase, we studied the patterns within a network of memristive ML neurons. To do this, we organized a simple ring network and explored its synchronization and the patterns observed before synchronization was achieved. We found that synchronization could be attained with less coupling strength as the magnetic induction strength increased. Additionally, interactions among memristive ML neurons led to the emergence of traveling, imperfect traveling, and non-stationary chimera states before synchronization was reached.

Author Contributions

Conceptualization, O.A.A.; investigation, O.A.A.; methodology, K.R.; project administration, O.A.A.; resources, V.-T.P.; validation, V.-T.P.; writing—original draft, V.-T.P.; writing—review and editing, K.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research has been funded by the Scientific Research Deanship at the University of Ha’il, Saudi Arabia, through project number <<RG-23 087>>.

Data Availability Statement

No new data were created or analyzed in this study.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The bifurcation diagram of the ReLU-based memristor ML model for I 0 = 0 , according to V K . The corresponding MLE diagram is shown below the bifurcation diagram. (a) k = 0.5 . (b) k = 3 . (c) k = 5 . (d) k = 8 .
Figure 1. The bifurcation diagram of the ReLU-based memristor ML model for I 0 = 0 , according to V K . The corresponding MLE diagram is shown below the bifurcation diagram. (a) k = 0.5 . (b) k = 3 . (c) k = 5 . (d) k = 8 .
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Figure 2. The bifurcation diagram of the ReLU-based memristor ML model for V k = 220 , according to I 0 . The corresponding MLE diagram is shown below the bifurcation diagram. (a) k = 2 . (b) k = 5 .
Figure 2. The bifurcation diagram of the ReLU-based memristor ML model for V k = 220 , according to I 0 . The corresponding MLE diagram is shown below the bifurcation diagram. (a) k = 2 . (b) k = 5 .
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Figure 3. The bifurcation diagram of the ReLU-based memristor ML model for V k = 220 , according to k . The corresponding MLE diagram is shown below the bifurcation diagram. (a) I 0 = 0 . (b) I 0 = 0.01 .
Figure 3. The bifurcation diagram of the ReLU-based memristor ML model for V k = 220 , according to k . The corresponding MLE diagram is shown below the bifurcation diagram. (a) I 0 = 0 . (b) I 0 = 0.01 .
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Figure 4. The phase space of the ReLU-based memristor ML model in different parameters of k and V K and I 0 = 0 . The time series of each attractor is shown in its subset. (a) k = 0.5 ,   V K = 400 . (b) k = 0.5 , V K = 300 . (c) k = 0.5 ,   V K = 220 . (d) k = 0.5 ,   V K = 200 . (e) k = 3 , V K = 200 . (f) k = 3 , V K = 220 . (g) k = 5 , V K = 300 . (h) k = 5 ,   V K = 220 .
Figure 4. The phase space of the ReLU-based memristor ML model in different parameters of k and V K and I 0 = 0 . The time series of each attractor is shown in its subset. (a) k = 0.5 ,   V K = 400 . (b) k = 0.5 , V K = 300 . (c) k = 0.5 ,   V K = 220 . (d) k = 0.5 ,   V K = 200 . (e) k = 3 , V K = 200 . (f) k = 3 , V K = 220 . (g) k = 5 , V K = 300 . (h) k = 5 ,   V K = 220 .
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Figure 5. Dynamical map of the ReLU-based memristor ML model in two-dimensional planes of ( V K , k ) in parts (a,b) and ( I 0 , k ) in parts (c,d). The period of oscillation is shown in the left column and the maximum Lyapunov exponent is shown in the right column, where the yellow color shows the chaotic region.
Figure 5. Dynamical map of the ReLU-based memristor ML model in two-dimensional planes of ( V K , k ) in parts (a,b) and ( I 0 , k ) in parts (c,d). The period of oscillation is shown in the left column and the maximum Lyapunov exponent is shown in the right column, where the yellow color shows the chaotic region.
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Figure 6. Multistability emerges in the ReLU-based memristor ML model. (a) The parameters are k = 0.6 , V K = 220 , and I 0 = 0 and the initial conditions of the orange and red firings are [ 0.16 , 0 , 187.11 ] and 0.19 , 0 , 2.97 , respectively. (b) The parameters are k = 6.5 , V K = 220 , and I 0 = 0 and the initial conditions of the orange and red firings are [ 0.22 , 0 , 4.8 ] and [ 0.24 , 0 , 4.78 ] , respectively.
Figure 6. Multistability emerges in the ReLU-based memristor ML model. (a) The parameters are k = 0.6 , V K = 220 , and I 0 = 0 and the initial conditions of the orange and red firings are [ 0.16 , 0 , 187.11 ] and 0.19 , 0 , 2.97 , respectively. (b) The parameters are k = 6.5 , V K = 220 , and I 0 = 0 and the initial conditions of the orange and red firings are [ 0.22 , 0 , 4.8 ] and [ 0.24 , 0 , 4.78 ] , respectively.
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Figure 7. (a) The synchronization error ( E ) of the network of memristive ML neurons (Equation (5)) according to the coupling strength ( ϵ ) and the magnetic induction strength ( k ). (b) One-dimensional synchronization error according to the coupling strength ( ϵ ) for k = 3 , 5 , and 8 .
Figure 7. (a) The synchronization error ( E ) of the network of memristive ML neurons (Equation (5)) according to the coupling strength ( ϵ ) and the magnetic induction strength ( k ). (b) One-dimensional synchronization error according to the coupling strength ( ϵ ) for k = 3 , 5 , and 8 .
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Figure 8. Temporal evolution of the network of memristive ML neurons (Equation (5)). (a) For k = 8 and ϵ = 1.4 , a traveling chimera is formed. (b) For k = 4 and ϵ = 0.5 , a traveling chimera is formed. (c) For k = 1 and ϵ = 0.5 , an imperfect traveling chimera is formed. (d) For k = 1 and ϵ = 1 , a non-stationary chimera is formed.
Figure 8. Temporal evolution of the network of memristive ML neurons (Equation (5)). (a) For k = 8 and ϵ = 1.4 , a traveling chimera is formed. (b) For k = 4 and ϵ = 0.5 , a traveling chimera is formed. (c) For k = 1 and ϵ = 0.5 , an imperfect traveling chimera is formed. (d) For k = 1 and ϵ = 1 , a non-stationary chimera is formed.
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Almatroud, O.A.; Pham, V.-T.; Rajagopal, K. A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model. Mathematics 2024, 12, 2970. https://doi.org/10.3390/math12192970

AMA Style

Almatroud OA, Pham V-T, Rajagopal K. A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model. Mathematics. 2024; 12(19):2970. https://doi.org/10.3390/math12192970

Chicago/Turabian Style

Almatroud, Othman Abdullah, Viet-Thanh Pham, and Karthikeyan Rajagopal. 2024. "A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model" Mathematics 12, no. 19: 2970. https://doi.org/10.3390/math12192970

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