**3. Neuron model with a band-pass filter property**

#### **3.1. Subthreshold resonance phenomenon**

As described in Section 2.2, neurons can generate action potentials depending on the strength of DC input stimuli. As shown in **Figure 6(a)**, for small DC inputs (dark blue, light blue, dashed red lines), action potentials are not generated by reason that membrane potentials do not exceed the threshold. However, if a larger input (red line) is given to a neuron, the membrane potential can exceed the threshold and as a result, an action potential is generated. On the contrary, when AC inputs are given to a neuron, outputs of a neuron are unlike the cases of DC inputs, apart from whether the membrane potential exceeds the threshold or not. We consider three AC inputs (blue, red, and green in **Figure 6(b)**), whose amplitudes are equal but their frequencies are different (*f*1<*f*2<*f*3). By using an AC input with frequency *f*1(blue), the membrane potential (blue line) is assumed to be obtained under the threshold level, that is, in a subthreshold level. If the input frequency increases from *f*1 to *f*2 (red), the membrane potential (red line) is still in a subthreshold level, but its amplitude becomes larger than that of frequency *f*<sup>1</sup> (blue line). However, if the input frequency further increases to *f*3 (green), the amplitude of the membrane potential (green line) reduces and becomes smaller than that of frequency *f*<sup>2</sup> (red line). Instead of AC inputs with a single frequency, let an AC input whose frequency increases with time be given to this neuron. This kind of AC is called a chirp current. Then, its membrane potential has the shape with an expanded center section as shown in **Figure 6(c)**, that is, the membrane potential takes the maximum at a specific frequency, however, remains at a subthreshold level. As its FFT shows, this neuron has a band-pass property, that is, frequency selectivity. These kinds of oscillatory phenomena in a subthreshold level are called the subthreshold resonance phenomena.

inactivation variables of *Na* channel, and *n*(*V,t*) is an activation variable of *K* channel. They

d( ,) ( ) (1 ( , )) ( ) ( , ), ( , , ) <sup>d</sup> *x x xV t V xV t V xV t x mhn <sup>t</sup>* = ×- - × =

 b

0.1 (25 ) ( ) , ( ) 4 exp( ), <sup>25</sup> <sup>18</sup> exp 1 <sup>10</sup>

<sup>1</sup> ( ) 0.07 exp , ( ) , <sup>20</sup> <sup>30</sup> exp 1 <sup>10</sup>

0.01 (10 ) ( ) , ( ) 0.125 exp . <sup>10</sup> <sup>80</sup> exp 1 <sup>10</sup>

**Figure 5(b)** shows one example of computer simulation results for the HH model. When a continuous DC input is given to the HH model, action potentials can be generated at certain interval, that is, with a refractory period. Regardless of the strength of inputs, action potentials have the same shape and size. All differential equations were solved by the fourth-order Rung-Kata method by using C++. Parameters used here were as *C* = 1μF/cm2

As described in Section 2.2, neurons can generate action potentials depending on the strength of DC input stimuli. As shown in **Figure 6(a)**, for small DC inputs (dark blue, light blue, dashed red lines), action potentials are not generated by reason that membrane potentials do not exceed the threshold. However, if a larger input (red line) is given to a neuron, the membrane potential can exceed the threshold and as a result, an action potential is generated. On the contrary, when AC inputs are given to a neuron, outputs of a neuron are unlike the cases of DC inputs, apart from whether the membrane potential exceeds the threshold or not. We consider three AC inputs (blue, red, and green in **Figure 6(b)**), whose amplitudes are equal but their frequencies are different (*f*1<*f*2<*f*3). By using an AC input with frequency *f*1(blue), the

 b- æ ö <sup>=</sup> = -ç ÷ æ ö - è ø ç ÷ - è ø

*V V V V*

 b

è ø æ ö - ç ÷ <sup>+</sup> è ø

 b- <sup>=</sup> = - æ ö - ç ÷ - è ø

*V V V V*

*m m*

*h h <sup>V</sup> V V*

*n m*

, *g* ¯

**3. Neuron model with a band-pass filter property**

*<sup>K</sup>* =0.3 mS/cm<sup>2</sup>

*V*

*V*

æ ö =- = ç ÷

(8)

*V*

, *ENa*= 115 mV, *EK* = −12 mV, *Iinp* = 10 mA/cm2

(9)

(10)

(11)

,

, and

gave the following empirical formula:

58 Numerical Simulation - From Brain Imaging to Turbulent Flows

a

a

a

, *g*

the resting potential = 0 mV.

¯ =0.3 mS/cm<sup>2</sup>

**3.1. Subthreshold resonance phenomenon**

*g* ¯

*<sup>L</sup>* =0.3 mS/cm<sup>2</sup>

a

**Figure 6.** Subthreshold resonance phenomena. (a) DC inputs with different amplitudes, (b) AC inputs with different frequencies, and (c) a chirp current input whose frequency increases as time increases.

Subthreshold resonance oscillations have been found in many excitatory and/or inhibitory neurons in the whole brain. Mauro et al. first reported a subthreshold resonance oscillation in squid giant axon [4]. Koch discussed these resonance oscillations in relation to the cable theory [5]. Since then, these resonance phenomena have been observed in many neurons in various regions of the brain, such as trigeminal root ganglion [6], inferior olive [7], and thalamus [8, 9]. These subthreshold resonance phenomena have been also reported in cortical neurons [10– 14], and in the 2000s, also in hippocampal neurons in CA1 [15, 16]. Although it is suspected that frequency selectivity of neurons should play an important role in behavioral or perceptual functions in animals, their practical roles have still been unclear. Recently, Narayanan and Johnston [17] reported that subthreshold resonance oscillations in hippocampal CA1 neurons are closely related to the long-term synaptic plasticity, which is currently considered as one of possible foundations of learning and memory [18, 19]. So, it is very interesting and attractive to study those resonance oscillatory features, in order to clarify the mechanisms of higher information processing functions in the brain, such as learning, short-term memory, or working memory.

As already described, the cell membrane is usually modeled by an RC circuit. However, if a chirp current is given to an RC circuit, a membrane potential shows only a property of lowpass filter shown in **Figure 7(a)**. On electrical circuit theory, resonance circuit must contain inductive elements, that is, inductance *L*. Indeed, if a chirp current is given to an RLC circuit, a membrane potential shows a band-pass property shown in **Figure 7(b)**. Having many neurons, the subthreshold resonance phenomena indicate that those neurons must have some kind of inductive factor. So exactly, what is a distinguishing major role of such inductive characteristics in the cell membrane? By advances of experimental technique, it has been reported that many kinds of voltage-dependent ion channels have an important role in subthreshold phenomena. Such ion channels involved in the subthreshold resonance phe‐ nomena are different from neuron to neuron that belongs to brain regions. Among them, slow non-inactivating *K*<sup>+</sup> channel (*Krs* channels) [10], hyperpolarization-activated cationic channel (*h* channel) [12], and persistent *Na*<sup>+</sup> channels (*NaP* channel) [13] are well known. In addition to these channels, voltage-dependent *Ca*2+ channels in neurons and/or dendritic spines [8] are also concerned in the subthreshold resonance oscillation.

**Figure 7.** Calculated membrane potential for a chirp current and its magnitude of FFT. (a) RC circuit and (b) RLC cir‐ cuit.

#### **3.2. Inductive property of voltage-dependent ion channels**

A hyperpolarization-activated cation channel (*h* channel) and a persistent sodium channel (*NaP* channel) are known to mediate the subthreshold resonance oscillation observed in entorhinal cortical neurons [12, 14]. In this section, we show how such voltage-dependent ion channels have inductive properties. We consider a compartment neuron model with h channel and *NaP* channel as shown in **Figure 8**. Its dynamics are expressed by the following conduc‐ tance-based equations [13]:

$$C\frac{\mathbf{d}V}{\mathbf{d}t} = -I\_L - I\_h - I\_{NaP} - I\_{inp},\tag{12}$$

**Figure 8.** A compartmental neuron model with h channel and NaP channel.

As already described, the cell membrane is usually modeled by an RC circuit. However, if a chirp current is given to an RC circuit, a membrane potential shows only a property of lowpass filter shown in **Figure 7(a)**. On electrical circuit theory, resonance circuit must contain inductive elements, that is, inductance *L*. Indeed, if a chirp current is given to an RLC circuit, a membrane potential shows a band-pass property shown in **Figure 7(b)**. Having many neurons, the subthreshold resonance phenomena indicate that those neurons must have some kind of inductive factor. So exactly, what is a distinguishing major role of such inductive characteristics in the cell membrane? By advances of experimental technique, it has been reported that many kinds of voltage-dependent ion channels have an important role in subthreshold phenomena. Such ion channels involved in the subthreshold resonance phe‐ nomena are different from neuron to neuron that belongs to brain regions. Among them, slow

to these channels, voltage-dependent *Ca*2+ channels in neurons and/or dendritic spines [8] are

**Figure 7.** Calculated membrane potential for a chirp current and its magnitude of FFT. (a) RC circuit and (b) RLC cir‐

A hyperpolarization-activated cation channel (*h* channel) and a persistent sodium channel (*NaP* channel) are known to mediate the subthreshold resonance oscillation observed in entorhinal cortical neurons [12, 14]. In this section, we show how such voltage-dependent ion channels have inductive properties. We consider a compartment neuron model with h channel and *NaP* channel as shown in **Figure 8**. Its dynamics are expressed by the following conduc‐

> <sup>d</sup> , <sup>d</sup> *L NaP <sup>h</sup> inp <sup>V</sup> C III I*

*<sup>t</sup>* =- - - - (12)

channel (*Krs* channels) [10], hyperpolarization-activated cationic channel

channels (*NaP* channel) [13] are well known. In addition

non-inactivating *K*<sup>+</sup>

cuit.

tance-based equations [13]:

(*h* channel) [12], and persistent *Na*<sup>+</sup>

60 Numerical Simulation - From Brain Imaging to Turbulent Flows

also concerned in the subthreshold resonance oscillation.

**3.2. Inductive property of voltage-dependent ion channels**

where *V* is the membrane potential, *IL* is the leak current, *Ih* and *INaP* are the currents through *h* channel and *NaP* channel, respectively, and *Iinp* is the input current. The leak current *IL* is given by

$$I\_L = \overline{\mathbf{g}}\_L \cdot (V - E\_L). \tag{13}$$

where *g* ¯ *<sup>L</sup>* is the leak conductance and *EL* is the resting potential. Currents *Ih* and *INaP* are given as follows:

$$I\_h = \mathbf{g}\_h(V) \cdot (V - E\_h) = \overline{\mathbf{g}}\_h \cdot \langle 0.65 \ m\_{h'}(V) + 0.35 \ m\_{hs}(V) \rangle \cdot (V - E\_h). \tag{14}$$

$$I\_{NaP} = \mathbf{g}\_{NaP}(V) \cdot (V - E\_N) = \overline{\mathbf{g}}\_{NaP} \cdot m\_{NaP}(V) \cdot (V - E\_N). \tag{15}$$

where *gh*(*V*) and *gNaP*(*V*) are, respectively, the *h* channel conductance and the *NaP* channel conductance, *g* ¯ *<sup>h</sup>* and *g* ¯ are, respectively, the maximum amplitude of *gh*(*V*) and *gNaP*(*V*), and *Eh* and *EN* are the equilibrium potentials for *K*<sup>+</sup> through *h* channel and *Na*<sup>+</sup> for *NaP* channel, respectively. *mhf*(*V*) and *mhs*(*V*) are, respectively, the fast activation and slow activation variables of *h* channel, and *mNaP*(*V*) is an activation variable of *NaP* channel. They satisfy the following equations:

$$\frac{\text{dm}\_{\text{x}}}{\text{dt}} = \frac{1}{\tau\_{\text{x}}} \cdot (m\_{\text{x}o} - m\_{\text{x}}), \quad (\text{x} = hf, h\text{s}, NaP) \tag{16}$$

$$m\_{b\prime} = \frac{1}{1 + \exp\left(\frac{V + 79.2}{9.78}\right)}, \quad \tau\_{b\prime} = 1 + \frac{0.51}{\exp\left(\frac{V - 1.7}{10}\right) + \exp\left(-\frac{V + 340}{52}\right)},\tag{17}$$

$$m\_{h'} = \frac{1}{1 + \exp\left(\frac{V + 71.3}{7.9}\right)}, \quad \tau\_{ho} = 1 + \frac{5.6}{\exp\left(\frac{V - 1.7}{14}\right) + \exp\left(-\frac{V + 260}{43}\right)},\tag{18}$$

$$m\_{NaP\infty} = \frac{1}{1 + \exp\left(-\frac{V + 38}{6.5}\right)}, \quad \tau\_{NaP} = 0.15 \text{ ms}.\tag{19}$$

#### *3.2.1. Equivalent admittance (impedance) of h channel*

Let *V*\* be the equilibrium potential, *I*<sup>h</sup> \* be the *h*-current at *V*\*. From Eq. (14), *I*<sup>h</sup> \* satisfies the following relation:

$$I\_h "= \overline{\mathbf{g}}\_h \cdot \{ 0.65 \, m\_{h'}(V^\*) + 0.35 \, m\_{hs}(V^\*) \} \cdot (V^\* - E\_h). \tag{20}$$

When the membrane potential *V*(*t*) changes from *V*\* to *V*\* + δ*V*(*t*), where δ*V*(*t*) is a small variation of the membrane potential from *V*\*, the current *Ih*(*t*) also changes from *I*<sup>h</sup> \* to *I*h \* <sup>+</sup> <sup>δ</sup>*I*<sup>h</sup> \* (*t*), where δ*Ih*(*t*) is a small variation of *h* current caused by δ*V*(*t*). *I*<sup>h</sup> \* <sup>+</sup> <sup>δ</sup>*I*h(*t*) satisfies the following relation:

$$\mathcal{I}\_h \mathsf{I}\_h \mathsf{I}\_h(t) = \overline{\mathsf{g}}\_h \cdot \{0.65 \, m\_{h\overline{f}}(V^\ast + \delta V(t)) + 0.35 \, m\_{h\overline{f}}(V^\ast + \delta V(t)) \cdot (V^\ast + \delta V(t) - E\_h). \tag{21}$$

Let *mhf*(*V*\* + δ*V*) approximate by *mhf* (*V*\* ) + δ*mhf* and *mhs*(*V*\* + *δV*) by *mhs*(*V*\* ) + *δmhs* for a small variation δ*V*. Then, Eq. (21) can be expressed by the following equation:

$$\begin{split} I\_h \, ^\ast + \delta I\_h(t) &= \overline{\mathfrak{g}}\_h \cdot \{ 0.65 \, m\_{hf}(V^\ast) + 0.35 \, m\_{hs}(V^\ast) \} \cdot (V^\ast - E\_h) \\ &+ \overline{\mathfrak{g}}\_h \cdot \{ 0.65 \, \delta m\_{hf} + 0.35 \, \delta m\_{hs} \} \cdot (V^\ast - E\_h) \\ &+ \overline{\mathfrak{g}}\_h \cdot \{ 0.65 \, m\_{hf}(V^\ast) + 0.35 \, m\_{hs}(V^\ast) \} \cdot \delta V(t) \\ &+ \overline{\mathfrak{g}}\_h \cdot \{ 0.65 \, \delta m\_{hf} + 0.35 \, \delta m\_{hs} \} \cdot \delta V(t) .\end{split} \tag{22}$$

By subtracting Eq. (20) from Eq. (22) and dropping the higher-order variation terms than the second, which appeared on the right-hand side of Eq. (22), the following equation is obtained:

$$
\delta I\_{\hbar} \approx \overline{\mathcal{g}}\_{\hbar} \cdot \{0.65 \,\delta m\_{\hbar \prime} + 0.35 \,\delta m\_{\hbar \prime}\} \cdot \{V^\* - E\_{\hbar}\} + \overline{\mathcal{g}}\_{\hbar} \cdot \{0.65 \, m\_{\hbar \prime \prime} + 0.35 \, m\_{\hbar \prime \prime}\} \cdot \delta V \,. \tag{23}
$$

As an activation variable *mhf*(*V*) satisfies Eq. (16), m*hf*(*V*\* ) and m*hf*(*V*\* + *δV*) must satisfy the following equations:

<sup>1</sup> 5.6 , 1 , 71.3 1.7 <sup>260</sup> 1 exp exp exp 7.9 <sup>14</sup> <sup>43</sup>

<sup>1</sup> , 0.15 ms. <sup>38</sup> 1 exp 6.5

When the membrane potential *V*(*t*) changes from *V*\* to *V*\* + δ*V*(*t*), where δ*V*(*t*) is a small variation of the membrane potential from *V*\*, the current *Ih*(*t*) also changes from *I*<sup>h</sup>

t

be the *h*-current at *V*\*. From Eq. (14), *I*<sup>h</sup>

\* {0.65 ( ) 0.35 ( )} ( ). \* \*\* *h h hf hs <sup>h</sup> I g mV mV V E* =× + × - (20)

dd

+ *δV*) by *mhs*(*V*\*

 = + æ ö æ öæ ö + -+ <sup>+</sup> ç ÷ ç ÷ç ÷ + - è ø è øè ø

(18)

(19)

\* to

(22)

\*

\* <sup>+</sup> <sup>δ</sup>*I*h(*t*) satisfies the

(21)

d

(23)

) + *δmhs* for a small

satisfies the

*mhf hs V VV* <sup>=</sup> t

*mNaP NaP <sup>V</sup>*

\*

(*t*), where δ*Ih*(*t*) is a small variation of *h* current caused by δ*V*(*t*). *I*<sup>h</sup>

d

variation δ*V*. Then, Eq. (21) can be expressed by the following equation:

*h hf hs h hf hs*

+× + ×

+× + × - +× + ×

d

d

 d

\* ( ) {0.65 ( ( )) 0.35 ( ( )) } ( ( ) ). \* \*\* *h h h hf hf <sup>h</sup> I I t g m V Vt m V Vt V Vt E* <sup>+</sup>

) + δ*mhf* and *mhs*(*V*\*

\*

d

=× + + + × + -

\* \* \*\*

*I It g m V m V V E g m m VE g m V m V Vt*

{0.65 0.35 } ( ).

*g m m Vt*

{0.65 0.35 } ( ) {0.65 ( ) 0.35 ( ) } ( )

 d

 dd

\* \* {0.65 0.35 } ( ) {0.65 0.35 } . \* *h h hf hs h h hf hs*

*I g m m VE g m m V* »× + × - +× + ×

*h h h hf hs h h hf hs h*

+ =× + × -

\* \*

By subtracting Eq. (20) from Eq. (22) and dropping the higher-order variation terms than the second, which appeared on the right-hand side of Eq. (22), the following equation is obtained:

( ) {0.65 ( ) 0.35 ( ) } ( )

= =

 æ ö + + -ç ÷ è ø

¥

62 Numerical Simulation - From Brain Imaging to Turbulent Flows

*3.2.1. Equivalent admittance (impedance) of h channel*

+ δ*V*) approximate by *mhf* (*V*\*

d

d

Let *V*\* be the equilibrium potential, *I*<sup>h</sup>

following relation:

following relation:

d

Let *mhf*(*V*\*

d

*I*h \* <sup>+</sup> <sup>δ</sup>*I*<sup>h</sup> \*

$$\frac{\mathrm{d}m\_{h'}(V^{\ast})}{\mathrm{d}t} = \frac{1}{\tau\_{h'}(V^{\ast})} \cdot \left\{ m\_{h'^{\ast}}(V^{\ast}) - m\_{h'} \right\},\tag{24}$$

$$\frac{dm\_{\mathbb{M}}\left(V^{\ast} + \delta V\right)}{dt} = \frac{1}{\tau\_{\mathbb{M}}\left(V^{\ast} + \delta V\right)} \cdot \left\{m\_{\mathbb{M}^{\otimes n}}\left(V^{\ast} + \delta V\right) - m\_{\mathbb{M}}\left(V^{\ast} + \delta V\right)\right\}.\tag{25}$$

where left terms of Eqs. (24) and (25), d*m*(*V* \* ) / *dt* and d*m*(*V* \* + δ*V* )/ d*t*, represent the quantity d*m* / d*t* evaluated at *V*\* and *V*\* + δ*V*, respectively. By approximating also *τ* (*V* \* + δ*V* ) by *τ* (*V* \* ) + δ*τhf*, where δ*τ* is a small variation caused by δ*V*, Eq. (25) is written as follows:

$$\begin{split} \frac{d[m\_{\boldsymbol{\eta}\prime}(V^{\star}) + \delta m\_{\boldsymbol{\eta}\prime}]}{dt} & \approx \frac{1}{\tau\_{\boldsymbol{\eta}\prime}(V^{\star}) + \delta \tau\_{\boldsymbol{\eta}\prime}} \cdot \{m\_{\boldsymbol{\eta}\prime\ast}(V^{\star}) + \delta m\_{\boldsymbol{\eta}\prime\alpha} - \{m\_{\boldsymbol{\eta}\prime}(V^{\star}) + \delta m\_{\boldsymbol{\eta}\prime}\}\} \\ & \approx \frac{1}{\tau\_{\boldsymbol{\eta}\prime}(V^{\star})} [1 - \frac{\delta \tau\_{\boldsymbol{\eta}\prime}}{\tau\_{\boldsymbol{\eta}\prime}(V^{\star})} + \dots] \cdot \{m\_{\boldsymbol{\eta}\prime\alpha}(V^{\star}) + \delta m\_{\boldsymbol{\eta}\prime\alpha} - \{m\_{\boldsymbol{\eta}\prime\prime}(V^{\star}) + \delta m\_{\boldsymbol{\eta}\prime}\}]. \end{split} \tag{26}$$

By dropping the higher-order variation terms than the second and subtracting Eq. (24) from Eq. (26), the following equation is obtained:

$$\frac{d\delta m\_{h\circ}}{dt} \approx \frac{1}{\tau\_{h\circ}(V^{\ast})} \cdot \{\delta m\_{h\circ o} - \delta m\_{h\circ}\}.\tag{27}$$

Furthermore, as a small variation δ*m∞* may be approximately expressed by <sup>d</sup>*m∞*(*<sup>V</sup>* \* ) / d*V* ⋅δ*V* , Eq. (27) becomes as follows:

$$\frac{d\delta m\_{\boldsymbol{\mu}^{\star}}}{dt} \approx \frac{1}{\tau\_{\boldsymbol{\mu}^{\star}}(V^{\star})} \cdot \left\{ \frac{d m\_{\boldsymbol{\mu}^{\star}\boldsymbol{\alpha}}(V^{\star})}{dV} \cdot \delta V - \delta m\_{\boldsymbol{\mu}^{\star}} \right\}.\tag{28}$$

Using the differential operator *p* instead of time derivative (d/d*t*), Eq. (28) can be written as follows:

$$
\left(\mathbf{p} + \frac{1}{\tau\_{\mathbf{h}\circ}(V^{\star})}\right) \cdot \delta m\_{\mathbf{h}\circ} = \frac{1}{\tau\_{\mathbf{h}\circ}(V^{\star})} \cdot \left(\frac{d m\_{\mathbf{h}\circ\circ}(V^{\star})}{dV}\right) \cdot \delta V. \tag{29}
$$

From this relation, a small variation δ*mhf* is definitely expressed by δ*V* as follows:

$$\delta m\_{\mathbf{\dot{w}}} = \frac{1}{\frac{\tau\_{\mathbf{w'}}(V^{\mathbf{w}})}{\mathbf{p} + \frac{1}{\tau\_{\mathbf{w'}}(V^{\mathbf{w}})}} \cdot \delta V} \cdot \delta V. \tag{30}$$

Notice that d*mhf*<sup>∞</sup> (*V*\* )/d*V* in the numerator of the right-hand side can be directly calculated from Eq. (17), that is, it is given by

$$\frac{\text{dm}\_{hf\,\infty}(V^{\ast})}{\text{d}V} = \frac{1}{9.78} \cdot \frac{\exp\left(\frac{V^{\ast} + 79.2}{9.78}\right)}{\left\{1 + \exp\left(\frac{V^{\ast} + 79.2}{9.78}\right)\right\}^{2}}.\tag{31}$$

Exactly in the same way, a small variation δ*mhs* is expressed by δ*V* as follows:

$$
\delta m\_{hs} = \frac{\frac{1}{\tau\_{hs}(V^{\ast})} \cdot \frac{\mathbf{d}m\_{hs}(V^{\ast})}{\mathbf{d}V}}{\mathbf{p} + \frac{1}{\tau\_{hs}(V^{\ast})}} \cdot \delta V. \tag{32}
$$

By substituting Eqs. (30) and (32) into Eq. (23), δ*Ih* is finally expressed by δ*V*(*t*) as follows:

$$\begin{split} \frac{\delta I\_h}{\delta V} &\approx \overline{\mathbf{g}}\_h \cdot \langle 0.65 \ m\_{\mathbf{y}^\*} + 0.35 \ m\_{\mathbf{h}^\*} \rangle \\ &+ \overline{\mathbf{g}}\_h \cdot \left[ 0.65 \frac{1}{\frac{\tau\_{\mathbf{y}^\*}(V^\mathbf{v})}{p + \frac{1}{\tau\_{\mathbf{y}^\*}(V^\mathbf{v})}}} + 0.35 \frac{1}{\frac{\tau\_{\mathbf{x}\_h}(V^\mathbf{v})}{p + \frac{1}{\tau\_{\mathbf{x}\_h}(V^\mathbf{v})}}} \frac{\operatorname{dm}\_{\text{hxc}}(V^\mathbf{v})}{\mathbf{d}V} \right] \cdot (V^\mathbf{v} - E\_h) . \end{split} \tag{33}$$

As δ*I* (*t*)/ δ*V* (*t*) represents admittance, Eq. (33) shows an equivalent admittance of *h* channel for a small variation δ*V* and its admittance can be expressed by parallel coupling circuits of one conductance and two admittances. That is, the first term of Eq. (33) represents a conduc‐ tance of *h* channel, which is expressed by the inverse of a pure resistance *Rh*; the second term is an admittance of a fast activation variable *Yhf*, which can be expressed by the inverse of series coupling of an inductance *Lhf* and a resistance *Rhf*, that is, *Yhf* = 1/(*Rhf* + *p* ⋅ *Lhf*); and the third term is an admittance element of a slow activation variable *Yhs*, which is also expressed by the inverse of series coupling of an inductance *Lhs* and a resistance *Rhs*, that is, *Yhs* = 1/(*Rhs*+ *p* ⋅ *Lhs*). **Figure 9** shows an equivalent RLC circuit of *h* channel, where *Rh*, *Rhf*, *Rhf*, *Rhs*, and *Lhs* are given as follows:

$$R\_h \text{ - } \frac{1}{\overline{\mathbf{g}}\_h \cdot (0.65 \ m\_{h'^\*} + 0.35 \ m\_{h^\*})},\tag{34}$$

$$R\_{Lf} = \frac{1}{0.65 \text{ } \overline{\text{g}}\_h \cdot \frac{d \text{m}\_{h'o}(V^\*)}{dV} \cdot (V^\* - E\_h)} \quad L\_f = \tau\_{hf} \cdot R\_{Lf'} \\ \text{(35)}$$

$$R\_{hs} = \frac{1}{0.35 \text{ g}\_h \cdot \frac{\text{dm}\_{hs}(V^\bullet)}{\text{d}V} \cdot (V^\bullet - E\_h)}, \quad L\_{hs} = \tau\_{hs} \cdot R\_{hs} \,. \tag{36}$$

**Figure 9.** An equivalent RLC circuit for *h* channel.

1 1 ( \*) . ( \*) ( \*)

1 d ( \*)

*hf*

( \*) d . <sup>1</sup>

*m V*

¥

( \*)

*V*

*hf*

t

exp d ( \*) <sup>1</sup> 9.78 . d 9.78 \* 79.2 1 exp 9.78

è ø = ×

1 d ( \*)

*hs*

*m V*

¥

( \*)

*V*

( \*) d δ . <sup>1</sup>

t

*V V m V*

× = × *p* +

*hs*

By substituting Eqs. (30) and (32) into Eq. (23), δ*Ih* is finally expressed by δ*V*(*t*) as follows:

1 d ( \*) 1 d ( \*)

*hf hs*

*h h*

*V V V V g V E*

ì ü ï ï × × + × í ý + × ï ï + + î þ *p p*

( \*) d ( \*) d 0.65 0.35 ( ). <sup>1</sup>

As δ*I* (*t*)/ δ*V* (*t*) represents admittance, Eq. (33) shows an equivalent admittance of *h* channel for a small variation δ*V* and its admittance can be expressed by parallel coupling circuits of one conductance and two admittances. That is, the first term of Eq. (33) represents a conduc‐

t

*hf hs*

*m V m V*

*V V*

Exactly in the same way, a small variation δ*mhs* is expressed by δ*V* as follows:

*hs*

t

*hs*

\* \*

1 (V\*)

*hf hs*

¥

*hf hs*

t

{0.65 0.35 } <sup>δ</sup>

*gm m <sup>V</sup>*

»× +

t

*h*

*I*

d

*h*

*V V m V*

× = × *p* +

*hf hf hf*

*hf*

t

*hf*

d

*hf*

*m V*

¥

Notice that d*mhf*<sup>∞</sup> (*V*\*

Eq. (17), that is, it is given by

d

t

64 Numerical Simulation - From Brain Imaging to Turbulent Flows

*V V dV*

 t

From this relation, a small variation δ*mhf* is definitely expressed by δ*V* as follows:

¥ æ ö æ ö ç ÷ + ×= × × ç ÷ è ø è ø

*hf*

*dm V m V*

> d

)/d*V* in the numerator of the right-hand side can be directly calculated from

\* 79.2

d

æ ö <sup>+</sup> ç ÷

*V*

ì ü æ ö <sup>+</sup> í ý <sup>+</sup> ç ÷ î þ è ø

2

\*

( \*)

¥

*V*

 t

*p* (29)

 d

(30)

(31)

(32)

(33)

#### *3.2.2. Equivalent admittance (impedance) of NaP channel*

As in the case of with *h* channel, let *V*\* be the equilibrium potential and *I* \* be the *NaP* current at *V*\*. From Eq. (15), *I* \* satisfies the following relation:

$$
\overline{\mathbf{g}}\_{\text{NaP}}^{\*} = \overline{\mathbf{g}}\_{\text{NaP}} \cdot \boldsymbol{m}\_{\text{NaP}}(\boldsymbol{V}^{\*}) \cdot (\boldsymbol{V}^{\*} - \boldsymbol{E}\_{N}).\tag{37}
$$

When the membrane potential *V*(*t*) changes from *V*\* to *V*\* + *δV*(*t*), the current *INaP* (*t*) also changes from *I*NaP \* to *I*NaP \* <sup>+</sup> <sup>δ</sup>*I*NaP(*t*), where δ*INaP*(*t*) is a small variation of *NaP* current caused by δ*V*(*t*). Then, *I*NaP \* <sup>+</sup> <sup>δ</sup>*I*NaP(*t*) satisfies the following relation:

$$
\delta I\_{\rm NaP}^\* + \delta I\_{\rm NaP}(t) = \overline{\mathbf{g}}\_{\rm NaP} \cdot m\_{\rm NaP}(V\* + \delta V(t)) \cdot (V\* + \delta V(t) - E\_N). \tag{38}
$$

Let *mNaP*(*V*\* + δ*V*) approximate by *mNaP*(*V*\*) + δ*mNaP* for a small variation δ*V*. Then, Eq. (38) can be expressed by the following equation:

$$\begin{split} I\_{\textit{NaP}}^{\*} + \delta I\_{\textit{NaP}}(t) &= \overline{\sf{g}}\_{\textit{NaP}} \cdot m\_{\textit{NaP}}(V\*) \cdot (V\* - E\_{\textit{N}}) + \overline{\sf{g}}\_{\textit{NaP}} \cdot \delta m\_{\textit{NaP}} \cdot (V\* - E\_{\textit{N}}) \\ &+ \overline{\sf{g}}\_{\textit{NaP}} \cdot m\_{\textit{NaP}}(V\*) \cdot \delta V(t) + \overline{\sf{g}}\_{\textit{NaP}} \cdot \delta m\_{\textit{NaP}} \cdot \delta V(t). \end{split} \tag{39}$$

By subtracting Eq. (37) from Eq. (39) and dropping the higher-order variation terms than the second, which appeared on the right-hand side of Eq. (39), the following equation is obtained:

$$
\delta I\_{NaP} \approx \overline{\mathbf{g}}\_{NaP} \cdot \delta m\_{\text{NaP}} \cdot (V^\* - E\_N) + \overline{\mathbf{g}}\_{NaP} \cdot m\_{\text{NaP}}^\bullet \cdot \delta V. \tag{40}
$$

By following the same procedure from Eq. (24) to Eq. (26) except that *τNaP* is constant (0.15 ms), a small variation δ*mNaP* can be expressed as follows:

$$\frac{\mathbf{d}\,\delta m\_{\mathrm{NaP}}}{\mathrm{d}t} \approx \frac{1}{0.15} \cdot \left\{ \delta m\_{\mathrm{NaP\,c}} - \gamma m\_{\mathrm{NaP}} \right\}.\tag{41}$$

By approximating a small variation δ*mNaP*∞ by [d*mNaP*∞ (*V*\*)/d*V*] ⋅ *δV*, Eq. (41) becomes

$$\frac{\mathbf{d}\,\delta m\_{\mathrm{NaP}}}{\mathrm{d}t} \approx \frac{1}{0.15} \left\{ \frac{\mathrm{d}\,\delta m\_{\mathrm{NaP}c}(V^{\ast})}{\mathrm{d}V} \cdot \delta V - \delta m\_{\mathrm{NaP}} \right\}.\tag{42}$$

By using the differential operator *p*, a small variation δ*mNaP* can be expressed by δ*V* as follows:

$$
\delta m\_{NaP} = \frac{\frac{1}{0.15} \cdot \frac{\text{d}m\_{NaPe}(V^\*)}{\text{d}V}}{\mathbf{p} + \frac{1}{0.15}} \cdot \delta V. \tag{43}
$$

By substituting Eq. (43) into Eq. (40), δ*INaP* is finally expressed by δ*V*(*t*) as follows:

$$\frac{\delta I\_{NaP}}{\delta V} \approx \overline{\mathbf{g}}\_{NaP} \cdot m\_{NaP}(V^{\sf a}) + \overline{\mathbf{g}}\_{NaP} \cdot \frac{\frac{1}{0.15} \cdot \frac{\mathbf{d}m\_{NaP}(V^{\sf a})}{\mathbf{d}V}}{\mathbf{p} + \frac{1}{0.15}} \cdot (V^{\sf a} - E\_N). \tag{44}$$

Eq. (44) shows an equivalent admittance of *NaP* channel for a small variation δ*V*, and it can be expressed by parallel coupling circuits of one conductance and one admittance. That is, the first term of Eq. (44) represents a conductance of *NaP* channel, which is expressed by the inverse of a pure resistance *RNaP*, and the second term is an admittance of an activation variable *YNaP*, which is expressed by the inverse of series coupling of an inductance *LP* and a resistance *RP*, that is, *YNaP* = 1/(*RP*+ *p* ⋅ *LP*). **Figure 10** shows an equivalent RLC circuit of *NaP* channel, where *RNaP*, *RP*, and *LP* are given as follows:

$$R\_{NaP} \sim \frac{1}{\overline{\mathbf{g}}\_{NaP} \cdot m\_{NaP}(V^\*)},\tag{45}$$

$$R\_P = \frac{1}{\frac{1}{0.15}\overline{\mathfrak{g}}\_{NaP} \cdot \frac{\text{d}m\_{NaPo}(V^{\bullet})}{\text{d}V} \cdot (V^{\bullet} - E\_N)}, \quad L\_P = 0.15 \cdot R\_P,\tag{46}$$

**Figure 10.** An equivalent RLC circuit for *NaP* channel.

When the membrane potential *V*(*t*) changes from *V*\* to *V*\* + *δV*(*t*), the current *INaP* (*t*) also changes

d

Let *mNaP*(*V*\* + δ*V*) approximate by *mNaP*(*V*\*) + δ*mNaP* for a small variation δ*V*. Then, Eq. (38) can

() ( ) ( ) (

*g m V Vt g m Vt*

By subtracting Eq. (37) from Eq. (39) and dropping the higher-order variation terms than the second, which appeared on the right-hand side of Eq. (39), the following equation is obtained:

By following the same procedure from Eq. (24) to Eq. (26) except that *τNaP* is constant (0.15 ms),

*NaP NaP*

 g

d d

d 1 { }. d 0.15

d

*<sup>m</sup> m m <sup>t</sup>*

By approximating a small variation δ*mNaP*∞ by [d*mNaP*∞ (*V*\*)/d*V*] ⋅ *δV*, Eq. (41) becomes

*m mV V m*

By using the differential operator *p*, a small variation δ*mNaP* can be expressed by δ*V* as follows:

1 d ( \*)

*NaP*

*m V*

¥ × = × *p* +

0.15

0.15 d <sup>δ</sup> δ . <sup>1</sup>

By substituting Eq. (43) into Eq. (40), δ*INaP* is finally expressed by δ*V*(*t*) as follows:

*V m V*

*NaP NaP NaP NaP N NaP NaP N NaP NaP NaP NaP*

*I I tg mV V E g m V E*

\* + = × \* × \*- + × × \*-

d

( ) () ( ).

 dd

\* δ \* δ( ) δ . *NaP NaP NaP N NaP NaP I g m VE g m V* » × ×- + × × (40)

»× - <sup>µ</sup> (41)

*NaP*

<sup>µ</sup> ì ü \* » í ý ×- × î þ (42)

(43)

( ) ( ( )) ( ( ) ). *NaP NaP NaP NaP <sup>N</sup> I I t g m V Vt V Vt E*

\* <sup>+</sup> <sup>δ</sup>*I*NaP(*t*), where δ*INaP*(*t*) is a small variation of *NaP* current caused by δ*V*(*t*).

 d\* + = × \*+ × \*+ - (38)

> d

+ × \*× + × × (39)

from *I*NaP

Then, *I*NaP

\* to *I*NaP

\* <sup>+</sup> <sup>δ</sup>*I*NaP(*t*) satisfies the following relation:

d

66 Numerical Simulation - From Brain Imaging to Turbulent Flows

be expressed by the following equation:

d

a small variation δ*mNaP* can be expressed as follows:

d

*NaP*

d 1 d () d 0.15 d *NaP NaP*

 d

*t V*

*NaP*

d

**Figure 11.** An equivalent RLC circuit for a compartment model with *h* channel and *NaP* channel.

#### *3.2.3. Equivalent RLC circuit of a neuron with h channel and NaP channel*

By combining the results of Sections 3.2.1 and 3.2.2, an equivalent RLC circuit for a neuron model with *h* channel and *NaP* channel is obtained. **Figure 11** shows its equivalent RLC circuit. In this section, we show some simulation results for a compartment neuron model (**Figure 8**) and its equivalent RLC circuit(**Figure 11**). A chirp current given to both a compartment neuron model and its equivalent RLC circuit is described as follows:

$$I\_{imp} = A\_{imp} \sin\left(\alpha(t) \cdot t\right) + id, \quad \alpha(t) = 2\pi f \frac{t}{T},\tag{47}$$

where the angular frequency *ω*(*t*) increases from 0 to 2*πf* over the period [0, *T*]. *id* is a DC bias current, which is set to zero in this subsection. The following parameter values were used in simulations; *C* = 1.5 μF/cm2 , *g* ¯ *<sup>L</sup>* =0.15 mS/cm<sup>2</sup> , *EL* = −65 mV, *g* ¯ NaP =0.5 mS/cm<sup>2</sup> , *g* ¯ *<sup>h</sup>* =1.5 mS/cm<sup>2</sup> , *EN* = 55 mV, *EK* = −90 mV, *Eh* = −20mV.

**Figure 12.** Membrane potential and the magnitude and the phase of its FFT. Simulation result for (a) a compartment model (**Figure 8**) and (b) its equivalent RLC circuit (**Figure 11**).

**Figure 12(a)** shows one simulation result for a compartment model with *h* channel and *NaP* channel. The membrane potential *V* and the amplitude and the phase of its FFT are shown. As the magnitude of FFT shows, this neuron model has a band-pass property. **Figure 12(b)** shows the simulation result for its equivalent RLC circuit. Comparing **Figure 12(a)** and **(b)**, the membrane potentials and their FFTs are exactly similar. This fact indicates that the derived RLC circuit represents almost the same properties of a compartment neuron model within a small variation of the membrane potential. In other words, voltage-dependent *h* channel and *NaP* channel may surely have inductive properties and contribute to the subthreshold resonance phenomena.
