Abstract
In this chapter, some real-life model problems that can be formulated as ordinary differential equations (ODEs) are introduced and numerically studied. These models are the variable-order fractional Hodgkin–Huxley model of neuronal excitation (VOFHHM) and other models with the variable-order fractional (VOF) time delay, such as the 4-year life cycle of a population of lemmings model, the enzyme kinetics with an inhibitor molecule model, and the Chen system model. A class of numerical methods is used to study the above-mentioned models such as non-standard finite difference (NSFD) and Adams-Bashforth-Moulton (ABM) methods. Numerical test examples are presented.
Keywords
- Ordinary differential equations
- Variable and fractional order real-life mathematical models
- Adams-Bashforth-Moulton method
- Non-standard finite difference method
- Error analysis in numerical computations
1. Numerical simulations for VOFHHM of neuronal excitation
1.1. Introduction
Hodgkin and Huxley proposed the famous Hodgkin–Huxley equations which quantitatively describe the generation of action potential of squid giant axon [1].
where Cm is the membrane capacitance, Vm is the intracellular potential,
The ionic current described in Hodgkin and Huxley’s giant squid axon can be divided into three different cases: sodium current, potassium current, and leakage current, for example, voltage-gated persistent K+ current with four activation gates (resulting in the term n4, where n is the activation variable for K+), voltage-gated transient Na+ current with three activation gates and one inactivation gate (the term
The total ionic current is therefore
where INa and IK are the currents through Na+ and K+ channels, respectively. The current IL is the leak current and denotes all the residue currents through a cell membrane other than Na+ and K+ currents. G denotes conductance value for each individual ionic component, and E is an equilibrium potential.
Hodgkin and Huxley defined the probability for a gate that can be found in its permissive state to depend on the membrane potential, therefore incorporating the voltage-dependent conductance [4, 5]. A probability
where 1−p(t) is the fraction in non-permissive state, p(t) is the fraction in permissive state, and
Transitions between permissive and non-permissive states are assumed to obey the relation:
where m and n are the activation variables for the sodium and the potassium, respectively, and represent the sodium de-inactivation.
In the literature, Alan Lloyd Hodgkin and Andrew Huxley authored a succession of five papers describing the nonlinear ODEs that model how action potentials can be initiated and propagated through an axon [1–5]. HHM can be considered one of the most successful mathematical models in quantity lively describing biological phenomena, and it is the method which can be used in deriving the model of a squid that is directly applicable to many kinds of neurons and other excitable cells [5–7].
Action potentials occur in excitable cells, including neurons, muscle cells, and endocrine cells. It is known that the brain is a complicated network of a tremendous number of neurons. A neuron has a very special shape, which is much different from usual sphere-shaped or disk-like cells [8]. A soma is the main body of neuron from which a long cable called an axon is extended. Neurons transmit and exchange electric signals called action potentials or spikes. Then, the electric signals or information is transmitted in the direction from a dendrite to an axon, see Figure 1 [8].
Doi et al. [8] have briefly explained the framework of the Hodgkin–Huxley formalism to model the action potential generation of neuron sand of other excitable cells. In 2012, Siciliano [9] has studied different numerical methods to solve the HHM. Six different numerical methods are first introduced and compared using a simple and arbitrary ODE. In 2014, Sweilam and Nagy [6] have presented numerical method for solving fractional Hodgkin–Huxley model (FHHM), described as follows:
where 0 < η ≤ 1.
In this work, the above-described model in [6] will be extended to VOFHHM, which is more general than fractional model.
The main aim of studying the VOFHHM is to show the behavior of the action potential when the derivative expressed as the VOF in order to explain the extent of this impact on the gating variables.
1.2. VOFHHM
It is well known that HHM is based on the parallel thought of a simple circuit with batteries, resistors, and capacitors [9]. Current can be carried through the circuit as ions passing through the membrane (resistors) or by charging the capacitors of the membrane. In this circuit, current flow across the membrane has two major components: one,
For the VOF model, we suggest that:
where
where [
Substituting (4) into (1), we obtain:
Transitions between permissive and non-permissive states are assumed to obey the relation:
The conductance for each current component can be written in the form [5]:
where
Eqs. (9) with (7) are a generalization of HHM [5], and it can be described as follows:
The variable
when the membrane potential
1.3. Discretizations of the model using NSFD method
The aim of this part is to introduce numerical discretization for Eqs. (7) and (9). The NSFD method [10–17] is used where the step size h in the FDM is replaced by a function
Doing some algebraic manipulation to Eqs. (11) and (12) yields the following equations:
where
1.4. Numerical experiments
VOFHHM can be summarized neatly into four separate variable-order fractional ODEs with some supporting functions. These equations are described in Eqs. (7) and (9), where the rate functions are listed below [5]:
and
Number 1 means that the action potential begins to fire, the sodium channels are now open, and an influx of sodium occurs.
Number 2 means that the membrane becomes more positive at this point due to sodium entering the cell. Now, the potassium channels open and leave the cell.
Number 3 means that the sodium channels have now become refractory and no more sodium is allowed to enter.
Number 4 means that the sodium channel is still in its refractory stage and only the potassium ions are passing through the membrane. As the potassium ions continue to leave the cell, the membrane potential moves toward the resting value.
Number 5 means that the potassium channels close and the sodium channels begin to leave the refractory phase and reset to its resting phase.
Finally, the potassium returns to its resting value where it can wait another action potential (for more details see [8, 9]). Moreover, in Figure 3, the gating variables are correctly constrained between 0 and 1 as expected using ode45 and NSFD method, respectively.
Figures 4–8 show the approximated action potential and the gating variables using the NSFD method obtained for different values of (
2. Real-life models governed by nonlinear delay differential equations (DDEs)
2.1. Introduction
DDEs are differential equations in which the derivatives of some unknown functions at present time are dependent on the values of the functions at previous times. In real-world systems, delay is very often encountered in many practical systems, such as control systems [18], lasers, traffic models [19], metal cutting, epidemiology, neuroscience, population dynamics [20], and chemical kinetics [21]. Recent theoretical and computational advancements in DDEs reveal that DDEs are capable of generating rich and plausible dynamics with realistic parameter values. Naturally, occurrence of complex dynamics is often generated by well-formulated DDE models. This is simply due to the fact that a DDE operates on an infinite-dimensional space consisting of continuous functions that accommodate high-dimensional dynamics.
For example, the Lotka–Volterra predator prey model [22] with crowding effect does not produce sustainable oscillatory solutions that describe population cycles. However, the Nicholson’s blowflies model [23] can generate rich and complex dynamics. Delayed fractional differential equations (DFDEs) are correspondingly used to describe dynamical systems [23]. In recent years, DFDEs begin to arouse the attention of many researchers [7, 19, 20, 25–27]. Simulating these equations is an important technique in the research, and accordingly, finding effective numerical methods for the DFDEs is a necessary process. Several methods based on Caputo or Riemann–Liouville definitions [28] have been proposed and analyzed. For instance, based on the predictor–corrector scheme, Diethelm et al. introduced the ABM algorithm [29–31] and mean while some error analysis was presented to improve the numerical accuracy. In 2011, Bhalekar and Daftardar-Gejji [25] have been extended the ABM algorithm to solve DDEs of fractional order and presented numerical illustrations to demonstrate utility of the method. In 2012, Ma et al. [32] have presented the numerical solution of a VOF financial system which is calculated by using the ABM method.
This section presents class of numerical method for solving the variable-order fractional nonlinear delay differential equations (VOFDDEs). The main aim of this part is to study VOFDDEs numerically.
2.2. ABM method for the VOFDDEs
In this section, the ABM algorithm has been extended to study the VOFDDEs, where the derivative is defined in the Caputo VOF sense. Special attention is given to prove the error estimate of the proposed method. Numerical test examples are presented to demonstrate utility of the method. Chaotic behaviors are observed in variable-order one-dimensional delayed systems.
In the following, we apply the ABM predictor–corrector method to implement the numerical solution of VOFDDEs.
Let us consider the following VOF system:
where
Also, consider a uniform grid {
such that
and note that
Applying the integral
on both sides of (15) and using (16), we claim to:
Further, the integral in Eq. (20) is evaluated using product trapezoidal quadrature formula [29–31, 33]. Then, we have the following corrector formula:
or
where
0 ≤
Due to nonlinearity off, Eq. (21) cannot be solved explicitly for
or
where
2.3. Error analysis of the algorithm
In this subsection, we aim to introduce the following lemma, which will be used in the proof of main theorem.
Lemma 2.1. [31] (a) Let
(b) Let
Proof. To prove statement (a), by construction of the product rectangle formula, we find that the quadrature error has the representation [31]
Apply the mean value theorem of differential calculus to the second factor of the integrand on the right-hand side of the above equation
Here, the term in parentheses is simply the remainder of the standard rectangle quadrature formula, applied to the function
Similarly, we can prove (b).
Now, let us consider that
where
Theorem 1. Suppose the solution y(t) ∈ C2 [0, T], of the Eqs. (15) and (16), satisfies the following two conditions:
with some γ1, γ2 ≥ 0, and
where
Proof. We will use the mathematical induction to prove the result. Suppose that the conclusion is true for j = 0, 1,…, n, then we have to prove that the inequality also holds for j = n + 1. To do this, we first consider the error of the predictor
So
and from
we have
Since
we have
2.4. Numerical examples
We present the purpose of this subsection to show that the proposed scheme designed provides good approximations for VOFDDEs. Throughout this subsection, we discuss four examples and their numerical solutions.
Example 2.1. Consider a VOFDDE:
In Figures 11 and 1(a, b) show the solutions
2.5. The four-year life cycle of a population of lemmings model
Lemmings are small rodents, usually found in or near the Arctic, in tundra biomes. It used to be that every three to 4 years, there were massive numbers of lemmings in the mountains and then the next year it was gone. Therefore, it has been an interesting thing to try to find out why. Are there factors that affect the lemming population such as climate, temperature, precipitation, and the like, that is, what we refer to as the lemming cycle. The modern scientific study of lemmings started with work carried out by the Norwegian Professor of Zoology Robert Collett, who at the end of the nineteenth century gathered a great deal of information about lemmings. To try to understand why lemmings fluctuate both regularly and extensively is indeed an important problem in ecology [34]. In [27], Tavernini has solved a model of the 4-year life cycle of a population of lemmings in an integer order. In [25], Bhalekar and Daftardar-Gejji have studied the model in a fractional order. In the following example [27], we study the extension of the lemming model:
Example 2.2. Consider the variable-order version of the four-year life cycle of a population of lemmings
Figure 16 shows the solutions
2.6. The enzyme kinetics with an inhibitor molecule model
An enzyme inhibitor is a molecule that binds to an enzyme and decreases its activity or completely inhibits the enzyme catalytic activity. It is well known that all these inhibitors follow the same rule to interplay in enzyme reaction. Furthermore, there are many factors that affect enzyme’s activity, such as temperature and pH. Many drug molecules are enzyme inhibitors, so their discovery and improvement are an active area of research in biochemistry and pharmacology to protect enzyme from any change. Therefore, studying the enzyme kinetics and structure–function relationship is vital to understand the kinetics of enzyme inhibition that in turn is fundamental to the modern design of pharmaceuticals in industries [35]. The frequency conversion mechanism in enzymatic feedback systems has been investigated with computer simulations in 1984 by Okamoto and Hayashi [36]. In [25], Bhalekar and Daftardar-Gejji have been study the enzyme kinetics with an inhibitor molecule model in a fractional order. In the following, we will study the model in [25] in general form where the derivative is given in a VOF.
Example 2.3. Consider the variable-order version of four-dimensional enzyme kinetics with an inhibitor molecule
Figure 23 shows the solutions
2.7. The Chen system model
The atmosphere is a layer of gases surrounding the planet Earth that has, at each altitude above each point of the Earth’s surface, a density, a pressure, a temperature, etc., and all these vary over time. It is of course unthinkable to know all this infinite number of data and to understand something about it. Approximations must be made. Edward Norton Lorenz (1917–2008) is a pioneer of chaos theory. He introduced the strange attractor notion and coined the term butterfly effect. In 1963 [37], he developed a simplified mathematical model for atmospheric convection when he studied the atmospheric convection. His model of the atmosphere was reduced to only three numbers x, y, and z and the evolution of the atmosphere to a tiny equation, where each point x, y, and z in space symbolizes a state of the atmosphere and the evolution follows a vector field. Later, several dynamical systems exhibiting chaos have been presented in various branches of science [24]. For example, in 1999, Chen and Ueta found another simple three-dimensional autonomous system, which is not topologically equivalent to Lorenz’s system and which has a chaotic attractor [38]. In [39], Chen proved that “The Chen system is a special case of the Lorenz system.” In 2004, Li and Peng [40] have presented Chen system with a fractional order. In 2012, Bhalekar et al. [41] proposed the fractional-order Chen system with time delay. In the following example, we will extend Chen system to a VOF, which can be more general.
Example 2.4. Consider the generalization of the delay fractional-order version of the Chen system [3] which involves the variable order:
on the interval [
At =
At
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