**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].

$$C\_m \frac{dV\_m}{dt} = I\_{cat} - I\_{ionic} \,\,,\tag{1}$$

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where Cm is the membrane capacitance, Vm is the intracellular potential, *t* is the time, Iionic is the net ionic current flowing across the membrane, and Iext is the externally applied current [2].

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 *m*<sup>3</sup> *ħ*) [3].

The total ionic current is therefore

$$I\_{\rm ionic} = \overbrace{G\_{\rm Na}(V-E\_{\rm Na})}^{I\_{\rm Na}} + \overbrace{G\_{\rm K}(V-E\_{\rm K})}^{I\_{\rm K}} + \overbrace{G\_{\rm L}(V-E\_{\rm L})}^{I\_{\rm L}},\tag{2}$$

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 con‐ ductance [4, 5]. A probability *p* ranging from 0 to 1 can be defined for each gate, but since a large population of channels (and therefore gates) has to be considered, this leads to the following:

$$1 - p(t) = \prescript{\beta\_{\rho}}{}{p(t)}$$

where 1−p(t) is the fraction in non-permissive state, p(t) is the fraction in permissive state, and *αp* and *βp* are called rate functions.

Transitions between permissive and non-permissive states are assumed to obey the relation:

$$\frac{dp}{dt} = \alpha\_{\rho} \left( V \right) \left( 1 - p \right) - \beta\_{\rho} \left( V \right) p, \qquad \qquad p = m, n, \hbar,\tag{3}$$

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 propa‐ gated 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 disklike 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].

**Figure 1.** Neurons cells.

where Cm is the membrane capacitance, Vm is the intracellular potential, *t* is the time, Iionic is the net ionic current flowing across the membrane, and Iext is the externally applied current [2]. 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 transient Na+

( ) ( ) ( ), *Na K L I I I*

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

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 con‐ ductance [4, 5]. A probability *p* ranging from 0 to 1 can be defined for each gate, but since a large population of channels (and therefore gates) has to be considered, this leads to the

> 1 () () *<sup>p</sup> <sup>p</sup> pt pt* b - = a

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:

( )(1 ) ( ) , ,,, *p p dp V p V p p mn dt* = --

where m and n are the activation variables for the sodium and the potassium, respectively,

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 propa‐ gated 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

 b

*ionic Na Na K K L L I G V E GV E GV E* = - + -+ -

current with four activation gates (resulting in the term n4

6447448 64748 64748 (2)

and K+ channels, respectively. The current IL is

= h (3)

, where

current with three activation

voltage-gated persistent K+

an equilibrium potential.

*αp* and *βp* are called rate functions.

a

and represent the sodium de-inactivation.

other excitable cells [5–7].

following:

n is the activation variable for K+

The total ionic current is therefore

gates and one inactivation gate (the term *m*<sup>3</sup> *ħ*) [3].

114 Numerical Simulation - From Brain Imaging to Turbulent Flows

where INa and IK are the currents through Na+

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:

$$\begin{aligned} CD^\eta V &= I\_{\text{ext}} - \overline{\mathbf{g}}\_{\text{Na}} m^3 h \left( V - E\_{\text{Na}} \right) - \overline{\mathbf{g}}\_K n^4 h \left( V - E\_K \right) - \mathbf{g}\_L \left( V - E\_L \right), \\ D^\eta p &= \alpha\_p \left( V \right) \left( 1 - p \right) - \beta\_p \left( V \right) p, p = m, n, h, \end{aligned}$$

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.
