**2.1. Electrical circuit model of the cell membrane**

Neurons play a key role in almost all brain functions. Fundamental function of neurons is to generate action potentials when they received sufficient stimuli from the environment. Once action potentials are generated, they are transmitted to other neurons so as to communicate information from one neuron to another. There exist many types of neurons in the brain, such as pyramidal neurons in the hippocampus and neocortex (**Figure 1(a)**), motor neurons in motor cortex (**Figure 1(b)**), or Purkinje cells in the cerebellum (**Figure 1(c)**) [1]. Although their shapes are different, they have basically the same structure. As shown in **Figure 2(a)**, a neuron is composed of three parts, that is, the soma (cell body) where action potentials are generated, the dendrite that receives inputs from other neurons, and the axon along which action potentials are transmitted to axon terminals. One thing especially worth mentioning, the dendrite of a neuron has hundreds to thousands of spines, on which axon terminals of other neurons connect. This junction is called a synapse, through which information are transmitted from one neuron to another (**Figure 2(b)**). Actually, there exist two kinds of synapses, one of which is an electric synapse and the other is a chemical synapse [1]. The former is a junction where neurons are directly contacted each other and information are electrically conducted from one neuron to another. This junction is also called a gap junction. On the other hand, the latter one is a junction with a cleft, called a synaptic cleft, into which neurochemical transmit‐ ters are released from the axon terminal and they bind to receptors on the spine head. Electrical synapses are found at the sites that require the fastest possible response, such as nociceptive reflex, whereas chemical synapses are found in almost all neurons of the brain. **Figure 2(b)** shows an example of a chemical synapse. Both synapses have a very important role in signal processing between neurons.

takes the central role in advanced information processing, such as visual, auditory, speech or language faculties, motion control, recognition, emotion, and so on. According to advances of experiment and computer technology, the research of brain science or neuroscience has been made not only in the fields of medicine, biology, biochemistry, pharmacology, and psycholo‐

The present-day computers have outstanding processing capacity. For example, they can find the data that satisfy some requirements among huge quantities of data (database) or can calculate over five trillion figures of pi(π). Therefore, many people are inclined to think that our brain will be able to be replaced by computer in near future. Surely, computers excel at processing of digitized data and processing by following a standard algorithm. However, it can hardly execute processing, such as recognition of ambiguity figures (such as illusionism) or inference based on imperfect information, which our brain can instantaneously carry out. Reason for this comes from differences in ways of information processing of the computer and our brain. The current computers, called von Neumann computer, are grounded in sequential processing by using central processing units (CPUs) and memory storages, while on the other hand, our brain bases on parallel and distributed processing through neural networks whose

There exist tens of billions of neurons in our brain, which build neural networks in complicated arrangement. All information processing in the brain is accomplished by neural activity in the form of neural oscillations that cause cortical oscillations (delta, theta, alpha, beta, or gamma oscillation). In order to clarify the mechanisms of advanced information processing in the brain, such as learning and memory, it is necessary to understand functions and features of neurons and neural networks. Although the current progress in experiment technology and measuring system is remarkable, only experiments by themselves cannot uncover the behavior of only a single neuron, because even a single neuron has complex biophysical characteristics and never stops growth. Computational neuroscience is a research field which fills up such a deficiency in experiments. By modeling the essential features of a neuron or a neural network at multiple spatial-temporal scales, we can capture and analyze the fundamental properties of a neuron or a neural network by computer simulation. Moreover, we can even offer some suggestions to experimental study by taking into account the probable results obtained from

Neurons play a key role in almost all brain functions. Fundamental function of neurons is to generate action potentials when they received sufficient stimuli from the environment. Once action potentials are generated, they are transmitted to other neurons so as to communicate information from one neuron to another. There exist many types of neurons in the brain, such as pyramidal neurons in the hippocampus and neocortex (**Figure 1(a)**), motor neurons in motor cortex (**Figure 1(b)**), or Purkinje cells in the cerebellum (**Figure 1(c)**) [1]. Although their shapes

gy but also in the field of engineering.

52 Numerical Simulation - From Brain Imaging to Turbulent Flows

components are neurons.

the simulation.

**2. Neuron model with a low-pass filter property**

**2.1. Electrical circuit model of the cell membrane**

**Figure 1.** Various types of neurons. (a) Pyramidal neuron (cortex), (b) motor neuron (spinal cord),and (c) Purkinje cell (cerebellum).

**Figure 2.** (a) Schematic neuron (Structure of neuron). (b) Synaptic connection at the synapse.

Surfaces of a neuron are covered with the cell membrane, which separates the interior of cell from the exterior environment. The cell membrane is composed of protein, lipid, and carbo‐ hydrate [1]. As shown in **Figure 3(a)**, it is composed of two layers of phospholipid molecules, each of which has a hydrophilic head (circle) and hydrophobic tail (two waved lined), and both hydrophobic tails face each other inside the cell membrane. This structure is called lipid bilayer. Furthermore, the concentration of the extracellular ions, such as *Na*<sup>+</sup> , *Cl*- , or *Ca*2+, is higher than the intracellular one. Contrarily, the concentration of intracellular ion, such as *K*<sup>+</sup> , is higher than the extracellular one. In addition, many types of ion channels (protein) are penetrating the cell membrane. Those ion channels are normally closed. If neurochemical transmitters released from the presynaptic axon terminal bind to receptors of the correspond‐ ing ion channels on the spine head, those ion channels are activated and open. Subsequently, specific ion flow occurs according to their ionic gradients. At the resting state, those channels are closed and no ionic flows occur except for small leakage.

**Figure 3.** Cell membrane. (a) Cross section of a cell membrane lipid bilayer and (b) equivalent RC circuit model of cell membrane.

Based on the above properties, the cell membrane has the following electrical properties:


With these points in mind, the cell membrane can be modeled by an equivalent RC circuit, which is shown in **Figure 3(b)**. Once an RC circuit is obtained, we can obtain its dynamics by using Ohm's law, Kirchhoff's law, or other knowledge of electrical circuit theory. From **Figure 3(b)**, the following equation is obtained:

$$\mathcal{C}\frac{dV}{dt} = -\frac{1}{\mathcal{R}\_L} \cdot (V - E\_L) - \frac{1}{\mathcal{R}\_i} \cdot (V - E\_i) + I\_{imp'} \tag{1}$$

where *V* is the membrane potential of the cell membrane, *C* is the membrane capacitance, *R*<sup>L</sup> is the leakage resistance, *EL* is the reversal potential, *Ri* is the flowability of ion *i*(*i* = *Na*<sup>+</sup> , *K*<sup>+</sup> , *Cl*<sup>−</sup> , or *Ca*2+), *E*<sup>i</sup> is the corresponding ionic equilibrium potential, and *I*inp is the specific input current given to the cell membrane. As a neuron is covered with the cell membrane, a synapse or a soma can be also expressed by using an RC circuit. Accordingly, we can study the synaptic properties or neuronal characteristics by using computer simulations.

### **2.2. Generation of action potentials (Hodgkin-Huxley model)**

Surfaces of a neuron are covered with the cell membrane, which separates the interior of cell from the exterior environment. The cell membrane is composed of protein, lipid, and carbo‐ hydrate [1]. As shown in **Figure 3(a)**, it is composed of two layers of phospholipid molecules, each of which has a hydrophilic head (circle) and hydrophobic tail (two waved lined), and both hydrophobic tails face each other inside the cell membrane. This structure is called lipid

higher than the intracellular one. Contrarily, the concentration of intracellular ion, such as *K*<sup>+</sup>

is higher than the extracellular one. In addition, many types of ion channels (protein) are penetrating the cell membrane. Those ion channels are normally closed. If neurochemical transmitters released from the presynaptic axon terminal bind to receptors of the correspond‐ ing ion channels on the spine head, those ion channels are activated and open. Subsequently, specific ion flow occurs according to their ionic gradients. At the resting state, those channels

**Figure 3.** Cell membrane. (a) Cross section of a cell membrane lipid bilayer and (b) equivalent RC circuit model of cell

**a.** It is lipid bilayer, that is, it is composed of two parallel plates. Thus, the cell membrane

**b.** The difference between intracellular and extracellular ion concentrations corresponds to

With these points in mind, the cell membrane can be modeled by an equivalent RC circuit, which is shown in **Figure 3(b)**. Once an RC circuit is obtained, we can obtain its dynamics by

or *Ca*2+).

**c.** The ionic flowability of opening ion channels is thought of as "resistance" *R*<sup>i</sup>

, *K*<sup>+</sup> , *Cl*<sup>−</sup>

Based on the above properties, the cell membrane has the following electrical properties:

, *Cl*-

, or *Ca*2+, is

or "conduc‐

, or *Ca*2+ through ion channels are "current," *INa*, *IK*,

,

bilayer. Furthermore, the concentration of the extracellular ions, such as *Na*<sup>+</sup>

are closed and no ionic flows occur except for small leakage.

54 Numerical Simulation - From Brain Imaging to Turbulent Flows

has characteristics similar to "capacitance," *C*.

(*i* = *Na*<sup>+</sup>

, *K*<sup>+</sup> , *Cl*<sup>−</sup>

membrane.

"power source," *Ei*

.

**d.** The corresponding flows of *Na*<sup>+</sup>

tance" 1/*Ri*

*ICl*, or *ICa*.

In this section, we give one model that can generate an action potential, which is the basic function of a neuron. When a dendritic spine receives stimuli from an axon terminal of other neuron, the membrane potential of a spine head changes depending on that stimulus. Those potential changes are transmitted to the soma (strictly speaking, the axon hillock in the neighborhood of the soma) through dendrites and integrated there. If the accumulated potential of the soma exceeds the threshold, an action potential is generated. Generated action potentials are transmitted to axon terminals along the axon. Based on this knowledge, McCulloch and Pitts expressed a neuron as a product-sum threshold element in 1943 [2]. Their model is a formal neuron model, called McCulloch-Pitts model, and is shown in **Figure 4**.

**Figure 4.** Formal neuron model (The McCulloch-Pitts model).

The McCulloch-Pitts model is expressed as follows:

$$\mathbf{w}(t) = \sum\_{l=1}^{N} \mathbf{w}\_l(t) \cdot \mathbf{x}\_l(t), \tag{2}$$

$$y(t) = \Phi(u(t)) = \begin{cases} 1 & \colon \ u(t) \ge \theta \\ 0 & \colon \ u(t) < \theta \end{cases} \tag{3}$$

where *xi* (*t*) is an input from *i*th neuron, *wi* (*t*) is a weight from a neuron *i, u*(*t*) is a state (potential) of a neuron, *y*(*t*) is its output, and *θ* is a threshold. In this model, if a state *u*(*t*) exceeds a threshold *θ*, output 1 is send to other neurons. Notice that, however, McCulloch-Pitts model does not consider a refractory period, during which neurons cannot or find it hard to generate the next action potential.

As the McCulloch-Pitts model was a very easy model for engineers to understand the mechanism of generation of action potentials, many engineers have applied this model to study basic neuronal behaviors. The most prominent example is the application to the perceptron, which was known as one of the powerful tools for some kinds of pattern rec‐ ognition problems. Although there exist many variations of the McCulloch-Pitts model, one of them uses a sigmoid function instead of a step function expressed by Eq. (3). This kind of model is applied to the back propagation algorithm and recently the deep learn‐ ing method, because a sigmoid function is a differentiable function. However, the practi‐ cal neurons are not so simple as the McCulloch-Pitts model and the back propagation algorithm. Therefore, more profound considerations were necessary to describe complicat‐ ed neuronal behaviors.

In 1952, Hodgkin and Huxley developed one mathematical model that explains the generation of an action potential (impulse or spike) based on physiological experiments for a squid giant axon [3]. As described in the previous section, the extracellular concentration of *Na*<sup>+</sup> is higher than the intracellular one, and the intracellular concentration *K*<sup>+</sup> is higher than the extracellular one, and the cell membrane has both *Na*<sup>+</sup> permeable channel (*Na* channel) and *K*<sup>+</sup> permeable channel (*K* channel). Hodgkin and Huxley found that both *Na* and *K* channels are voltagedependently activated, that is, the activation and inactivation of these channels are affected by the membrane potential of the cell membrane. They also elucidated that action potentials are generated by increased or decreased activation and inactivation of *Na* channel and increased or decreased activation of *K* channel. Based on the results of physiological experiments for a squid giant axon, they showed that an action potential is generated whenever the cell mem‐ brane is depolarized over the threshold. They proposed a schematic electrical circuit model that can explain the mechanism for generation of action potentials, called the Hodgkin-Huxley model (HH model).

**Figure 5.** The Hodgkin-Huxley model. (a) Conductance-based electrical circuit of the Hodgkin-Huxley model and (b) simulation result.

**Figure 5(a)** shows the HH model and its dynamics is given as follows:

1 ( ) ( ) ( ), *N*

1 : () ( ) ( ( )) , 0 : ()

*i ut w t x t* =

*yt ut u t*

=F = í

(*t*) is an input from *i*th neuron, *wi*

56 Numerical Simulation - From Brain Imaging to Turbulent Flows

where *xi*

the next action potential.

ed neuronal behaviors.

model (HH model).

*i i*

*u t*

ìï ³

of a neuron, *y*(*t*) is its output, and *θ* is a threshold. In this model, if a state *u*(*t*) exceeds a threshold *θ*, output 1 is send to other neurons. Notice that, however, McCulloch-Pitts model does not consider a refractory period, during which neurons cannot or find it hard to generate

As the McCulloch-Pitts model was a very easy model for engineers to understand the mechanism of generation of action potentials, many engineers have applied this model to study basic neuronal behaviors. The most prominent example is the application to the perceptron, which was known as one of the powerful tools for some kinds of pattern rec‐ ognition problems. Although there exist many variations of the McCulloch-Pitts model, one of them uses a sigmoid function instead of a step function expressed by Eq. (3). This kind of model is applied to the back propagation algorithm and recently the deep learn‐ ing method, because a sigmoid function is a differentiable function. However, the practi‐ cal neurons are not so simple as the McCulloch-Pitts model and the back propagation algorithm. Therefore, more profound considerations were necessary to describe complicat‐

In 1952, Hodgkin and Huxley developed one mathematical model that explains the generation of an action potential (impulse or spike) based on physiological experiments for a squid giant

channel (*K* channel). Hodgkin and Huxley found that both *Na* and *K* channels are voltagedependently activated, that is, the activation and inactivation of these channels are affected by the membrane potential of the cell membrane. They also elucidated that action potentials are generated by increased or decreased activation and inactivation of *Na* channel and increased or decreased activation of *K* channel. Based on the results of physiological experiments for a squid giant axon, they showed that an action potential is generated whenever the cell mem‐ brane is depolarized over the threshold. They proposed a schematic electrical circuit model that can explain the mechanism for generation of action potentials, called the Hodgkin-Huxley

axon [3]. As described in the previous section, the extracellular concentration of *Na*<sup>+</sup>

one, and the cell membrane has both *Na*<sup>+</sup> permeable channel (*Na* channel) and *K*<sup>+</sup>

than the intracellular one, and the intracellular concentration *K*<sup>+</sup>

q

q

= × <sup>å</sup> (2)

ï < î (3)

(*t*) is a weight from a neuron *i, u*(*t*) is a state (potential)

is higher

permeable

is higher than the extracellular

$$I\_{imp} = C\frac{dV}{dt} + I\_L + I\_{Na} + I\_{K'} \tag{4}$$

where *V* is the membrane potential of the cell, *C* is the capacitance, *IL* is the leak current, *INa* and *IK* are currents through *Na* channel and *K* channel, respectively, and *Iinp* is the input current. They proposed the empirical formulae, which appropriately indicate activation and inactiva‐ tion properties of *Na* channel and activation properties of *K* channel, that is, the change of ionic permeability of *Na*<sup>+</sup> and *K*<sup>+</sup> . Currents *IL*, *INa*, and *IK* are given as follows [3]:

$$I\_L = \overline{\mathbf{g}}\_L \cdot (V - E\_L),\tag{5}$$

$$I\_{Na} = \overline{\mathcal{g}}\_{Na} \cdot m(V, t)^3 \cdot h(V, t) \cdot (V - E\_{\mathbb{N}a})\_{\prime} \tag{6}$$

$$I\_K = \overline{\mathbf{g}}\_K \cdot \mathbf{n}(V, t)^4 \cdot (V - E\_\kappa), \tag{7}$$

where *g* ¯ <sup>L</sup> is the leakage conductance, *g* ¯ and *<sup>g</sup>* ¯ <sup>K</sup> are the amplitude of *Na* channel conductance and *K* channel conductance, respectively, *EL* is the resting potential, and *ENa* and *EK* are equilibrium potentials of *Na* channel and K *channel*. *m*(*V, t*) and *h*(*V, t*) are activation and inactivation variables of *Na* channel, and *n*(*V,t*) is an activation variable of *K* channel. They gave the following empirical formula:

$$\frac{d\mathbf{x}(V,t)}{dt} = \alpha\_{\mathbf{x}}(V) \cdot (\mathbf{l} - \mathbf{x}(V,t)) - \beta\_{\mathbf{x}}(V) \cdot \mathbf{x}(V,t), \quad (\mathbf{x} = m, h, n) \tag{8}$$

$$\alpha\_m(V) = \frac{0.1\left(25 - V\right)}{\exp\left(\frac{25 - V}{10}\right) - 1}, \quad \beta\_m(V) = 4\exp(-\frac{V}{18}),\tag{9}$$

$$
\alpha\_h(V) = 0.07 \exp\left(-\frac{V}{20}\right), \qquad \beta\_h(V) = \frac{1}{\exp\left(\frac{30 - V}{10}\right) + 1}, \tag{10}
$$

$$\alpha\_n(V) = \frac{0.01\,(10-V)}{\exp\left(\frac{10-V}{10}\right) - 1}, \quad \beta\_m(V) = 0.125 \exp\left(-\frac{V}{80}\right). \tag{11}$$

**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 , *g* ¯ *<sup>L</sup>* =0.3 mS/cm<sup>2</sup> , *g* ¯ =0.3 mS/cm<sup>2</sup> , *g* ¯ *<sup>K</sup>* =0.3 mS/cm<sup>2</sup> , *ENa*= 115 mV, *EK* = −12 mV, *Iinp* = 10 mA/cm2 , and the resting potential = 0 mV.
