**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, *IC* (μA cm−2) associated with charging the membrane capacitance *C*(μF cm−2) and, the other, Iionic associated with the movement of specific types of ions across the membrane.

For the VOF model, we suggest that:

$$I\_c = CD^{\eta(\iota)}V,\tag{4}$$

where *V* is the membrane voltage ((mV), measured in volts), 0 < *η* (t) ≤ 1, and *D<sup>η</sup>*(*t*) is defined by Grünwald–Letinkov approximation for the VOF derivative as follows:

$$D^{\eta(t)}\mathbf{y}(t) = \lim\_{h \to 0} h^{-\eta(t)} \sum\_{i=0}^{\lfloor t \rfloor} (-\mathbf{l})^{\binom{\eta}{i}} \binom{\eta(t)}{i} \mathbf{y}\left(t - ih\right), \tag{5}$$

where [*t*] denotes the integer part of *t* and *h* is the step size.

Substituting (4) into (1), we obtain:

$$
\hbar \, CD^{\eta(t)} V = I\_{\text{cat}} - I\_{\text{ionic}}(t). \tag{6}
$$

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

$$D^{\eta(t)}p = \alpha\_\rho(V)(1-p) - \beta\_\rho(V)p, \qquad \qquad p = m, n, \hbar,\tag{7}$$

The conductance for each current component can be written in the form [5]:

$$\mathbf{G}\_{\rm Na} = \overline{\mathbf{g}}\_{\rm Na} m^3 h, \mathbf{G}\_{\rm Na} = \overline{\mathbf{g}}\_{\rm K} n^4, \mathbf{G}\_{\rm L} = \mathbf{g}\_{\rm L},\tag{8}$$

where *g* ¯ *Na* and *g* ¯ *<sup>K</sup>* are the maximal conductance of sodium and potassium, respectively, and *gL* is a constant. Substituting (2) and (8) into (6), we obtain:

$$\text{C}D^{\eta(t)}V = I\_{\text{arc}} - \overline{\mathbf{g}}\_{\text{Na}} \boldsymbol{m}^{3} \boldsymbol{h} \left( V - E\_{\text{Na}} \right) - \overline{\mathbf{g}}\_{\text{K}} \boldsymbol{n}^{4} \boldsymbol{h} \left( V - E\_{\text{K}} \right) - \mathbf{g}\_{\text{L}} \left( V - E\_{\text{L}} \right). \tag{9}$$

Eqs. (9) with (7) are a generalization of HHM [5], and it can be described as follows:

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

The variable *p* approaches the steady-state value [5]:

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

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, *IC* (μA cm−2) associated with charging the membrane capacitance *C*(μF cm−2) and, the other, Iionic associated with the

> ( ) , *<sup>t</sup> CI CD V* h

where *V* is the membrane voltage ((mV), measured in volts), 0 < *η* (t) ≤ 1, and *D<sup>η</sup>*(*t*)

*<sup>h</sup> <sup>h</sup> <sup>i</sup>*

® =

h

( ) ( ) ( ) [ ] <sup>0</sup> <sup>0</sup> li ( ) ( ) ( 1)m , *<sup>t</sup> t i <sup>t</sup>*


*D yt h y t ih <sup>i</sup>*

( ) ( .) *<sup>t</sup> CD V I I t ext ionic*

( ) ( )(1 ) ( ) , ,,, *<sup>t</sup> D p p p p V p p m nV*

 b

The conductance for each current component can be written in the form [5]:

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

*t*

h

by Grünwald–Letinkov approximation for the VOF derivative as follows:

h

= (4)

è ø <sup>å</sup> (5)

= - (6)


3 4 , ,, *G g mhG g n G g Na Na Na K L L* = == (8)

*<sup>K</sup>* are the maximal conductance of sodium and potassium, respectively, and

is defined

movement of specific types of ions across the membrane.

For the VOF model, we suggest that:

116 Numerical Simulation - From Brain Imaging to Turbulent Flows

h

Substituting (4) into (1), we obtain:

h

where *g* ¯ *Na* and *g* ¯ a

*gL* is a constant. Substituting (2) and (8) into (6), we obtain:

where [*t*] denotes the integer part of *t* and *h* is the step size.

variables.

**1.2. VOFHHM**

$$p^\*(V) = \frac{\alpha\_\rho}{\alpha\_\rho + \beta\_\rho}, \qquad \qquad p = m, n, \hbar,\tag{10}$$

when the membrane potential *V* remains constant.

#### **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 *ψ*(*h*). Also, the Grünwald–Letinkov definition will be used with *V*(*tk*)=*Vk*, *p*(*tk*), *p*(*tk*) = *pk*, and then, we claim

$$\sum\_{i=0}^{k-1} c\_i^{\eta(i)} p\_{k+1-i} = \alpha\_p V\_k \left(1 - p\_k \right) - \beta\_p V\_k p\_k, \qquad p = m, n, h,\tag{11}$$

$$\sum\_{l=0}^{k+l} \mathbf{c}\_{l}^{\eta(l)} V\_{k+l-l} = I\_{m} - \overline{\mathbf{g}}\_{\text{Na}} m\_{k-l}^{2} m\_{k} \hbar \left( V\_{k} - E\_{\text{Na}} \right) - \overline{\mathbf{g}}\_{\text{K}} n\_{k-l}^{3} n\_{k} \hbar \left( V\_{k} - E\_{\text{K}} \right) - \mathbf{g}\_{\text{L}} \left( V\_{k} - E\_{\text{L}} \right). \tag{12}$$

Doing some algebraic manipulation to Eqs. (11) and (12) yields the following equations:

$$\mathbf{p}\_{\mathbf{k}+1} = \frac{-\sum\_{i=1}^{\mathbf{k}+1} \mathbf{c}\_i^{\eta(t)} \mathbf{p}\_{\mathbf{k}+1-i} + \alpha\_p \mathbf{V}\_\mathbf{k} \left(\mathbf{l} - \mathbf{p}\_\mathbf{k}\right) - \beta\_p \mathbf{V}\_\mathbf{k} \mathbf{p}\_\mathbf{k}}{\mathbf{c}\_0^{\eta(t)}}, \qquad \mathbf{p} = \mathbf{m}, \mathbf{n}, \hbar, \tag{13}$$

$$V\_{k+1} = \frac{I\_{\rm ext} - \sum\_{i=1}^{k+1} \mathbf{c}\_i^{\eta(i)} V\_{k+1-i} - \overline{\mathbf{g}}\_{\rm Nu} m\_{k-1}^2 m\_k \hbar \left( V\_k - E\_{\rm Na} \right) - \overline{\mathbf{g}}\_{\rm x} n\_{k-1}^3 n\_k \hbar \left( V\_k - E\_{\rm x} \right) - \mathbf{g}\_L \left( V\_k - E\_L \right)}{C \mathbf{c}\_0^{\eta(i)}} \tag{14}$$

where *c*<sup>0</sup> *η*(*t*) =*ψ*(*h* ) −*η*(*t*) , *ψ*(*h* )=*e <sup>h</sup>* −1.

#### **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]:

$$\begin{aligned} \alpha\_{\pi} \left( \mathbf{V} \right) &= 0.01 \left( \mathbf{V} + 1 \right) \left[ \exp \left( \frac{V + 10}{10} \right) \right]^{-1}, \mathcal{J}\_{\pi} \left( V \right) = 0.125 \exp \left( \frac{V}{80} \right), \\\\ \alpha\_{\pi} \left( \mathbf{V} \right) &= 0.1 \left( \mathbf{V} + 25 \right) \left[ \exp \left( \frac{V + 25}{10} \right) \right]^{-1}, \mathcal{J}\_{\pi} \left( V \right) = 4 \exp \left( \frac{V}{18} \right), \\\\ \alpha\_{\hbar} \left( \mathbf{V} \right) &= 0.07 \exp \left( \frac{V}{20} \right), \mathcal{J}\_{\hbar} \left( V \right) = \left[ \exp \left( \frac{V + 30}{10} \right) \right]^{-1}, \end{aligned}$$

and

*g* ¯ Na =120mS / cm<sup>2</sup> , *g* ¯ <sup>K</sup> =36mS / cm<sup>2</sup> , *gL* =0.3mS / cm<sup>2</sup> , *ENa* =120mV, *EK* = −12mV, *EL* =10.6mV, with the initial conditions *V*(0) = *V*0, and *p*(0) = *p*0 = *p*\* (*V*0). The behavior of the neuron can be simulated for different initial values of *V*. **Figure 2** shows an action potential simulated by the solver ode45, i.e., fourth-order Runge–Kutta method, and NSFD method at *η*(*t*) = 1, of the variable-order Hodgkin–Huxley equations for zero Iext, and *V*0 = −40 mV. Also, in this figure, all parts of the action potential are presented including the rapid upraise, downfall, and unexcitable phase, according to the number sequence as follows:


**Figure 2.** An action potential created by using ode45 and NSFD method at *η* (t) = 1.

**1.4. Numerical experiments**

118 Numerical Simulation - From Brain Imaging to Turbulent Flows

functions are listed below [5]:

a

a

and

Na =120mS / cm<sup>2</sup>

, *g* ¯

*g* ¯ a

<sup>K</sup> =36mS / cm<sup>2</sup>

with the initial conditions *V*(0) = *V*0, and *p*(0) = *p*0 = *p*\*

unexcitable phase, according to the number sequence as follows:

open, and an influx of sodium occurs.

sodium is allowed to enter.

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

<sup>10</sup> V 0.01 V 1 , 0.125 , <sup>10</sup> <sup>80</sup> *n n*

<sup>25</sup> V 0.1 V 25 ,4, <sup>10</sup> <sup>18</sup> *m m*

<sup>30</sup> V 0.07 , , <sup>20</sup> <sup>10</sup> *V V*

 b*exp V exp*

æö æ ö é ù <sup>+</sup> = = ç÷ ç ÷ ê ú èø è ø ë û h h

simulated for different initial values of *V*. **Figure 2** shows an action potential simulated by the solver ode45, i.e., fourth-order Runge–Kutta method, and NSFD method at *η*(*t*) = 1, of the variable-order Hodgkin–Huxley equations for zero Iext, and *V*0 = −40 mV. Also, in this figure, all parts of the action potential are presented including the rapid upraise, downfall, and

**i.** Number 1 means that the action potential begins to fire, the sodium channels are now

**ii.** 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.

**iii.** Number 3 means that the sodium channels have now become refractory and no more

**iv.** Number 4 means that the sodium channel is still in its refractory stage and only the

to leave the cell, the membrane potential moves toward the resting value.

**v.** Number 5 means that the potassium channels close and the sodium channels begin

to leave the refractory phase and reset to its resting phase.

potassium ions are passing through the membrane. As the potassium ions continue

é ù æ ö <sup>+</sup> æ ö = + <sup>=</sup> ê ú ç ÷ ç ÷ ë û è ø è ø

1

 b *exp V exp* é ù æ ö <sup>+</sup> æ ö = + <sup>=</sup> ê ú ç ÷ ç ÷ ë û è ø è ø

1

 b *exp V exp* -

*V V*

*V V*

1


, *ENa* =120mV, *EK* = −12mV, *EL* =10.6mV,

(*V*0). The behavior of the neuron can be

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

, *gL* =0.3mS / cm<sup>2</sup>

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.

**Figure 3.** Plot of the different gating variables using ode45 (left) and NSFD method (right) at *η* (t) = 1.

**Figure 4.** The approximated action potential (left) and the gating variables (right) using the NSFD method at (*t*) = *0.99* – (*0.01*/*100*)*t*.

**Figures 4**–**8** show the approximated action potential and the gating variables using the NSFD method obtained for different values of (*t*), where *η*(*t*) = 0.99 − (0.01/100)*t, η*(*t*) = 0.87 − (0.01/100)*t, η*(*t*) =0.79 − (0.001)*t*, and *η*(*t*) = 0.67 − (0.001)*t*, respectively. These figures show an example of a numerically solved solution of the Hodgkin–Huxley equations. **Figures 4**–**8** (left) are a waveform of the membrane potential. A pulsatile input applied at a time t = 5 ms ((ms) time measured in second) induces an action potential. **Figures 4**–**7** (right) show the dynamics of all three gating variables m, n, and h. Sodium activation, m, changes much more rapidly than either h or n. **Figure 9** (left) describes the relationship between the variables n and h, where n is the activation variable of potassium channel expressed in a dark color, while h is inactivation variable of sodium channel expressed in a light color. The action potential, n, be in stillness stage and begins to enter the cell until it reaches the highest activity, unlike the action potential, h, at the top of its activity. Step by step, the action potential, h, decreases its activities and continues to leave the cell and moves toward the resting value. At a certain moment in time, the intersection between them are occurred, where n increased while h decreased. **Figure 9** (right) describes the relationship between the variables m and h, where m is the activation variable of potassium channel expressed in a dark color, while h is inactivation variable of sodium channel expressed in a light color. The dynamics variables m and h have the same dynamics similar to n and h, where m increased while h decreased. **Figure 10** describes the dynamics of gating variables n and m in 3D. The variables n and m are the activation variables of K+ and Na+ ionic channels, respectively. They are expressed in a dark color and a light color, respectively. They are in stillness stage and begin to enter the cell until they reach the highest activity, where they are in the case of increasing.

**Figure 5.** The approximated action potential (left) and the gating variables (right) using the NSFD method at (*t*) = *0.87* – (*0.01*/*100*)*t*.

**Figure 6.** The approximated action potential (left), and the gating variables (right) using the NSFD method at (*t*) = *0.79* – (*0.001*)*t*.

Numerical Simulations of Some Real-Life Problems Governed by ODEs http://dx.doi.org/10.5772/63958 121

(0.01/100)*t, η*(*t*) =0.79 − (0.001)*t*, and *η*(*t*) = 0.67 − (0.001)*t*, respectively. These figures show an example of a numerically solved solution of the Hodgkin–Huxley equations. **Figures 4**–**8** (left) are a waveform of the membrane potential. A pulsatile input applied at a time t = 5 ms ((ms) time measured in second) induces an action potential. **Figures 4**–**7** (right) show the dynamics of all three gating variables m, n, and h. Sodium activation, m, changes much more rapidly than either h or n. **Figure 9** (left) describes the relationship between the variables n and h, where n is the activation variable of potassium channel expressed in a dark color, while h is inactivation variable of sodium channel expressed in a light color. The action potential, n, be in stillness stage and begins to enter the cell until it reaches the highest activity, unlike the action potential, h, at the top of its activity. Step by step, the action potential, h, decreases its activities and continues to leave the cell and moves toward the resting value. At a certain moment in time, the intersection between them are occurred, where n increased while h decreased. **Figure 9** (right) describes the relationship between the variables m and h, where m is the activation variable of potassium channel expressed in a dark color, while h is inactivation variable of sodium channel expressed in a light color. The dynamics variables m and h have the same dynamics similar to n and h, where m increased while h decreased. **Figure 10** describes the dynamics of gating variables n and m in 3D. The variables n and m are the

color and a light color, respectively. They are in stillness stage and begin to enter the cell until

**Figure 5.** The approximated action potential (left) and the gating variables (right) using the NSFD method at (*t*) = *0.87* –

**Figure 6.** The approximated action potential (left), and the gating variables (right) using the NSFD method at (*t*) = *0.79*

ionic channels, respectively. They are expressed in a dark

activation variables of K+

(*0.01*/*100*)*t*.

– (*0.001*)*t*.

and Na+

120 Numerical Simulation - From Brain Imaging to Turbulent Flows

they reach the highest activity, where they are in the case of increasing.

**Figure 7.** The approximated action potential (left) and the gating variables (right) using the NSFD method at (*t*) = *0.67* – (*0.001*)*t*.

**Figure 8.** An action potential created from the NSFD method using different values of *η* (t).

**Figure 9.** The waveforms of the gating variables n and h (left), and m and h (right) in 3D.

**Figure 10.** The waveforms of the gating variables n and m in 3D.
