**3.3. Properties of voltage responses to oscillatory current inputs**

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

model and its equivalent RLC circuit is described as follows:

68 Numerical Simulation - From Brain Imaging to Turbulent Flows

, *g* ¯

model (**Figure 8**) and (b) its equivalent RLC circuit (**Figure 11**).

simulations; *C* = 1.5 μF/cm2

*EN* = 55 mV, *EK* = −90 mV, *Eh* = −20mV.

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

> sin ( ( ) ) , ( ) 2 , *inp inp <sup>t</sup> I A t t id t f <sup>T</sup>* = ×+ = w

> > *<sup>L</sup>* =0.15 mS/cm<sup>2</sup>

 wp

, *EL* = −65 mV, *g*

¯

NaP =0.5 mS/cm<sup>2</sup>

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

**Figure 12.** Membrane potential and the magnitude and the phase of its FFT. Simulation result for (a) a compartment

**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

(47)

, *g* ¯

*<sup>h</sup>* =1.5 mS/cm<sup>2</sup>

,

A neuron can generate action potentials whenever its membrane potential exceeds the threshold except during a refractory period. If the membrane potential of a neuron stays in a subthreshold level, a neuron cannot generate action potentials. However, as described in the previous section, many neurons in the brain have the subthreshold resonance properties. This fact indicates that a neuron may be able to generate an action potential when AC inputs whose frequencies are close to the resonance frequency of a neuron are given, because the resulting membrane potential for that input has the potential to exceed the threshold by the effects of the subthreshold resonance property. Actually, this fact has been observed in neurons of the brain. For example, Hutcheon et al. studied subthreshold voltage responses to AC inputs in neurons from the sensorimotor cortex of rats [11]. **Figure 13** shows one of their results. Cases 1–3 show effects of the input amplitude: (Case 1) if a chirp current with a small amplitude is given to a neuron, the membrane potential does not exceed the threshold and no action potentials are generated; (Case 2) if its input amplitude increases a little bit, the membrane potential becomes larger but it still stays in a subthreshold level and no action potentials are generated; and (Case 3) if its amplitude increases more, the membrane potential exceeds for AC inputs with frequencies close to a resonance frequency of a neuron. As a result, action potentials are generated around that frequency. Cases 4 and 5 show effects of a DC bias input: (Case 4) if a DC-bias in the input current increases a little bit from 0μA/cm2 , the membrane potential cannot exceed the threshold and no action potentials are generated and (Case 5) if the value of a DC bias is raised more, the membrane potential can exceed the threshold and action potentials are generated.

**Figure 13.** Experimental results: Subthreshold voltage responses for a chirp current input with different amplitudes and DC bias currents observed in sensory cortex of rats [11].

We consider here a compartment neuron model with *h* channel and *NaP* channel described in the previous section, into which the HH model is incorporated in order to generate action potentials. **Figure 14** shows its integrated neuron model, and the following equation is obtained:

$$\text{C}\frac{dV}{dt} = -I\_L - I\_h - I\_{NaP} - I\_{Na} - I\_K + I\_{imp} \,. \tag{48}$$

**Figure 14.** A compartment model with *h* channel and *NaP* channel into which the Hodgkin-Huxley model (the HHmodel) is incorporated.

Eq. (48) is an extension form of Eq. (12), into which two currents *IN* and *IK* of the HH model are added. Detail dynamics of *IL*, *Ih*, and *INaP* are given by Eqs. (13)–(15) in Section 3.2. Dynamics of *INa* and *IK* are also given by Eqs. (6) and (7) in Section 2.2. In addition to parameter values shown in the previous section, the following values were used in simulations: *g* ¯ Na =52 mS/cm<sup>2</sup> , *g* ¯ *<sup>K</sup>* =52 mS/cm<sup>2</sup> , *EN* = 55 mV, and *EK* = −90 mV.

**Figure 15(a)** shows the membrane potential for a chirp current input with *Ainp* = 2.5 μA/cm2 and *id* = 0 μA/cm2 . For this input, the membrane potential cannot exceed the threshold, and no action potentials are generated. However, if the amplitude of AC input (*Ainp*) increases, the situation changes. **Figure 15(b)** shows the membrane potentials for a chirp current input whose amplitude increases to 3.3 μA/cm2 . In this case, the membrane potential exceeds the threshold for the input frequency close to the resonance frequency of this neuron model. As shown in **Figure 12**, it is 13 Hz in this neuron model. On the other hand, **Figure 15(c)** shows the membrane potential for a chirp input with *Ainp* = 2.5 μA/cm2 and increased *id* = 1 μA/cm2 . Almost the same response as **Figure 12(b)** is obtained. However, resulting membrane potentials in **Figures 13** and **15** are not completely identical, because neurons used in experiments by Hutcheon et al. and a neuron model used in this simulations are different. However, simulation results follow a similar tendency as experimental results by Hutcheon et al.[11].

By computer simulations, effects of the amplitude of AC input, its frequency, and a DC-bias current on the membrane potential were studied [20]. Some results are shown here. **Figure 16** shows the membrane potentials for the AC inputs with a single frequency, setting *Ainp* = 3.3 μA/cm2 . **Figure 16(a)** shows the results for 2 Hz input frequency. In this case, no action potentials are generated, because the membrane potential cannot exceed the threshold at all.

We consider here a compartment neuron model with *h* channel and *NaP* channel described in the previous section, into which the HH model is incorporated in order to generate action potentials. **Figure 14** shows its integrated neuron model, and the following equation is

> =- - - - - + . *L NaP Na K <sup>h</sup> inp dV C III I I I*

**Figure 14.** A compartment model with *h* channel and *NaP* channel into which the Hodgkin-Huxley model (the HH-

Eq. (48) is an extension form of Eq. (12), into which two currents *IN* and *IK* of the HH model are added. Detail dynamics of *IL*, *Ih*, and *INaP* are given by Eqs. (13)–(15) in Section 3.2. Dynamics of *INa* and *IK* are also given by Eqs. (6) and (7) in Section 2.2. In addition to parameter values shown in the previous section, the following values were used in simulations:

, *EN* = 55 mV, and *EK* = −90 mV.

**Figure 15(a)** shows the membrane potential for a chirp current input with *Ainp* = 2.5 μA/cm2

action potentials are generated. However, if the amplitude of AC input (*Ainp*) increases, the situation changes. **Figure 15(b)** shows the membrane potentials for a chirp current input whose

for the input frequency close to the resonance frequency of this neuron model. As shown in **Figure 12**, it is 13 Hz in this neuron model. On the other hand, **Figure 15(c)** shows the membrane

response as **Figure 12(b)** is obtained. However, resulting membrane potentials in **Figures 13** and **15** are not completely identical, because neurons used in experiments by Hutcheon et al. and a neuron model used in this simulations are different. However, simulation results follow

By computer simulations, effects of the amplitude of AC input, its frequency, and a DC-bias current on the membrane potential were studied [20]. Some results are shown here. **Figure 16** shows the membrane potentials for the AC inputs with a single frequency, setting *Ainp* = 3.3

potentials are generated, because the membrane potential cannot exceed the threshold at all.

. **Figure 16(a)** shows the results for 2 Hz input frequency. In this case, no action

. For this input, the membrane potential cannot exceed the threshold, and no

. In this case, the membrane potential exceeds the threshold

. Almost the same

and increased *id* = 1 μA/cm2

*dt* (48)

obtained:

model) is incorporated.

Na =52 mS/cm<sup>2</sup>

and *id* = 0 μA/cm2

μA/cm2

, *g* ¯

amplitude increases to 3.3 μA/cm2

*<sup>K</sup>* =52 mS/cm<sup>2</sup>

70 Numerical Simulation - From Brain Imaging to Turbulent Flows

potential for a chirp input with *Ainp* = 2.5 μA/cm2

a similar tendency as experimental results by Hutcheon et al.[11].

*g* ¯

**Figure 15.** Simulated membrane potential for a chirp current input with different amplitude and DC bias current. (a) Response for *Ainp* = 2.5 μA/cm2 and *id* = 0 μA/cm2 . (b) Response for *Ainp* = 3.3 μA/cm2 and *id* = 0 μA/cm2 . (c) Response for *Ainp* = 2.5 μA/cm2 and *id* = 1 μA/cm2 .

**Figure 16.** Simulated membrane potential for an AC input with a single frequency (*A*inp is fixed to 2.5 μA/cm2 ). (a) Re‐ sponse for *id* = 0 μA/cm2 and *f* = 2 Hz. (b) Response for *id* = 0 μA/cm2 and *f* = 13 Hz. (c) Response for *id* = 0 μA/cm2 and *f* = 30 Hz. (d) Response for *id* = 1 μA/cm2 and *f* = 13 Hz. (e) Response for *id* = 1 μA/cm2 and *f* = 30 Hz.

**Figure 16(b)** shows that action potentials are generated, because the input frequency is 13 Hz, which is close to the resonance frequency of this neuron model. Thus, the maximum amplitude of membrane potentials exceeds the threshold by the effect of the subthreshold resonance property. In the case of **Figure 16(c)**, for 30 Hz input frequency, no action potentials are generated, because the membrane potential is less than the threshold. On the other hand, if a DC bias current input increases to 1μA/cm2 , as **Figure 16(d)** shows, more action potentials are generated for 13 Hz input frequency input than the case of no DC bias input. This indicates the effect of a DC bias current on generation of action potentials. Comparing **Figure 16(b)** and **(d)**, it is clarified that the more spikes are observed than the case of no DC bias current input. For the case of *id*= 1 μA/cm2 ,a neuron model can also generate an action potential for 30Hz input frequency, as shown in **Figure 16(e)**. As no action potential is generated for the case of *id*= 0 μA/cm2 , this is also caused by the effect of a DC bias current input.
