Continuous Stopping Boundaries R Package Goldmann

Excitable Tissue and Bioelectric Signals

Sverre Grimnes , Ørjan G Martinsen , in Bioimpedance and Bioelectricity Basics (Third Edition), 2015

5.6 Problems

1.

Use the Nernst equation to find the potential difference across a cell membrane where the concentration of potassium on the outside is 5 times higher than on the inside. Should the calculation have been done with the Goldman equation?

2.

Calculate the change in intracellular potential if 2   million Na⁺ ions are transported out of the cell. The cell diameter is 10   μm, membrane capacitance is 1   μF/cm², ion charge is 1.6   ×   10⁻¹⁹ [coulomb], and q   =   UC.

3.

Find the difference between the Weiss and Lapique equations if t/τ  =   10.

4.

Define an action potential. Must it be measured with an intracellular microelectrode?

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BIOELECTRIC PHENOMENA

John Enderle PhD , in Introduction to Biomedical Engineering (Second Edition), 2005

Electromotive Force Properties

The three major ions K⁺, Na⁺, and Cl⁻ are differentially distributed across the cell membrane at rest and across the membrane through passive ion channels as illustrated in Figure 11.5. This separation of charge exists across the membrane and results in a voltage potential V _m as described by the Goldman Equation 11.33.

Across each ion-specific channel, a concentration gradient exists for each ion that creates an electromotive force, a force that drives that ion through the channel at a constant rate. The Nernst potential for that ion is the electrical potential difference across the channel and is easily modeled as a battery, as is illustrated in Figure 11.11 for K⁺. The same model is applied for Na⁺ and Cl⁻ with values equal to the Nernst potentials for each.

Figure 11.11. A battery is used to model the electromotive force for a K⁺ channel with a value equal to the K⁺ Nernst potential. The polarity of the battery is given with the ground on the outside of the membrane, in agreement with convention. From Table 11.1, note that the Nernst potential for K⁺ is negative, which reverses the polarity of the battery, driving K⁺ out of the cell.

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Physiological principles of electrical stimulation

Narendra Bhadra , in Implantable Neuroprostheses for Restoring Function, 2015

2.3.1 Passive electrical properties

Charge separation across the membrane results in a resting transmembrane potential V _m  = V _in  − V _out (where V _in and V _out are the respective potentials inside and outside the membrane). It has an average value of about −60   mV internally. Ions flow along their electrochemical gradients to an equilibrium potential (V _m) for any specific ionic species, given by

(2.1) $V_{\text{m}}=\frac{RT}{zF}\ln \frac{C_{\text{o}}}{C_{\text{i}}}$

where R  =   gas constant, T  =   absolute temperature, z  =   valence, F  =   Faraday's constant, C _o  =   ion concentration outside, and C _i  = ion concentration inside. A resting membrane potential is determined by a number of different ion species, and described by the Goldman equation ( Goldman, 1943), where the term within the logarithmic function is replaced by summations of individual external and internal concentrations multiplied by the valence. Intracellular concentrations of sodium ions are about 10   mM, with extracellular concentration of around 150   mM (equilibrium potential about +65   mV). For potassium, the concentrations are 140   mM intracellular and 5   mM extracellular (equilibrium potential about −95   mV). These differences account for ion leakage at resting potentials.

Biological tissues may be modeled as parallel resistor–capacitor combinations to analyze such potentials (Figure 2.2). With an applied constant voltage, a conduction current flows across this combination with some charge storage on the capacitance. With an alternating voltage (V ₀  cosωt, where V ₀ is the amplitude and ω is the angular frequency), there is a displacement current, I  =   −ωCV ₀  sinωt. The total current is a sum of conduction and displacement currents, which are 90° out of phase. Tissues can be characterized with a complex valued admittance (Y) for its ability to transmit current and a complex-valued impedance (Z) for its ability to restrict current.

Figure 2.2. (a) Stylized view of cell membrane formed by two layers of molecules that have polar or hydrophilic heads (dark circles) and nonpolar tails. (b) Passive membrane model with membrane capacitance C _m in parallel with a membrane resistance R _m. R _o and R _i are the resistances of the outer and inner cell compartment, respectively.

The bilayer lipid membrane has a capacitance (C _m) of about 1   μF/cm² of membrane surface due to the charge difference across it. This capacitance is modeled in parallel to a membrane resistance R _m across the membrane, in ohms/unit area. In axons, the longitudinal resistance of the internal cytoplasm is modeled as R _a, in ohms/unit length. A unit length of axon may be represented by a parallel resistance and capacitance, together with an intervening axonal resistance. Because of rapid spatial decay, changes in passive membrane potentials are only of use in short-distance communications as in some retinal rod cells and in some gap junctions.

When current is injected into the membrane, the transient voltage response is determined by the time constant, τ  = R _m  × C _m. Inward (positive) currents move the membrane potential to more negative values (hyperpolarization) and outward currents (negative) change it to less negative values (depolarization). Such responses are limited to values below activation thresholds that would generate an AP. Currents injected at one location on the axon exit through the membrane along its length, and the one-dimensional spatial potential profile is determined by a space constant λ  =   √(R _m/R _a). The voltage decreases by about 63% within a length λ. Beyond a threshold level, membrane depolarization can trigger an AP.

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Mechanisms of Ion Transport Across Cell Membranes and Epithelia

Luis Reuss , in Seldin and Giebisch's The Kidney (Fourth Edition), 2008

ELECTRODIFFUSION

Electrodiffusion is the main mechanism of passive transport of ions in homogeneous media, that is, bulk aqueous solution or relatively large water-filled pores. Electrodiffusive transmembrane ion transport is a mediated transport process, but it is better discussed at this point for continuity with diffusion. In large-diameter pores, electrodiffusion theory explains ion permeation very well. In ion channels, which have smaller diameter than pores and are highly selective, there are significant interactions between the ions and the permeation pathway. For this reason, simple electrodiffusion theory is not entirely applicable to ion channels, but is nevertheless a useful approximation. For ion transport across a membrane, two factors determine the flux: the chemical potential difference (difference in concentration across the membrane) and the electrical potential difference (membrane voltage). The net ion flux (J_i ) is given by the Nernst–Planck equation (see 44). If a constant electrical field is assumed in the membrane and other assumptions are made, the Nernst–Planck equation can be solved, yielding the Goldman-Hodgkin-Katz (GHK) equation (35, 45):

(10) $J_i=-P_i\frac{z_i{V}_{m}F}{RT}\left[\frac{C_i^0-C_i^i\text{exp}(V_{m}F/RT)}{1-\text{exp}(V_{m}F/RT)}\right]$

where R, T, z, and F have their usual meanings, P is permeability, V_m is membrane voltage, C is concentration, the subscript i denotes the ith ion and the superscripts i and o denote the two sides of the membrane.

Under zero-current conditions, the GHK flux equation yields the membrane voltage as a function of the permeabilities and concentrations of all permeant ions. For the case of three monovalent permeant ions (Na⁺, K⁺, and Cl⁻), the equation (GHK voltage equation) is:

(11) $V_m=-\frac{RT}{F}\text{ln}\frac{P_{Na}[N{a}^{+}{]}_i+P_K[K^{+}{]}_i+P_{Cl}[C{l}^{-}{]}_o}{P_{Na}[N{a}^{+}{]}_o+P_K[K^{+}{]}_o+P_{Cl}[C{l}^{-}{]}_i}$

where the brackets denote concentrations. Note that if the fraction including permeability coefficients and ion concentrations is inverted, then the sign of the right side of the equation is also inverted. I prefer the notation given here because it gives the intracellular potential minus the extracellular potential, the convention used in electrophysiology. This also applies to the Nernst equation below.

Note that if only one ion is permeable, e.g., if P_Na and P_Cl are 0 in Eq. 11, then the membrane voltage becomes equal to the equilibrium potential for that ion, in this example K⁺. The equilibrium potential is given by the Nernst equation (76):

Under these conditions, the two compartments separated by the membrane are at a steady state (the amounts of K⁺ on each side remain constant with time) but also at equlibrium, which means that the net driving force on K⁺ is zero, and hence the unidirectional fluxes are equal (Fig. 3). In the case of cells, one frequently observes a steady-state K⁺ distribution without equilibrium: the net efflux through channels is exactly balanced by influx via the Na⁺,K⁺-ATPase.

FIGURE 3. Electrochemical equilibrium. A membrane permeable to K⁺ and impermeable to Cl⁻ separates two KCl solutions of concentrations, 10 mM and 100 mM, respectively. Because of the difference in concentrations, there is a chemical driving force for K⁺ and Cl⁻ fluxes from left to right. While the impermeant Cl⁻ cannot move, the permeant K⁺ moves across the membrane, and in doing so creates a difference in electrical potential across the membrane. The membrane becomes electrically charged by a tiny excess of K⁺ ions on the left and a tiny excess of Cl⁻ ions on the right. This difference in electrical potential (the transmembrane voltage) opposes further K⁺ flux and a state is reached at which the chemical driving force and the electrical driving force for K⁺ movement are equal and opposite. This condition, described by the Nernst equation (Eq. 12), is electrochemical equilibrium.

Another interesting point is that the Nernst equation indicates that if only one ion is permeable, then the membrane voltage is determined by the concentration ratio for the ion, not its absolute concentrations. In addition, the membrane voltage is independent of the absolute value of the ion permeability. As shown by the GHK voltage equation, in the case of a membrane permeable to more than one ion, the membrane voltage depends on the absolute concentrations and permeability coefficients of all permeant ions. The Nernst equation can be derived more directly from the definitions of electrochemical potential (Eq. 1) and equilibrium (Δ

= 0).

The most important points concerning electrodiffusion are that ion fluxes across membranes are determined by both permeability and driving force, and that the driving force has chemical and electrical components. Hence, these three elements must be known to predict the direction and magnitude of the flux. For example, knowledge of the K⁺ concentrations inside and outside a cell is insufficient to decide whether the ion is at equilibrium across the membrane or whether there is a passive driving force inwardly or outwardly directed. To establish this simple point, it is necessary to know the membrane voltage. However, knowledge of the electrochemical gradient is insufficient to predict the magnitude of the K⁺ flux expected for this gradient; the K⁺ permeability of the membrane must be known as well.

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Bioelectric Phenomena

John D. Enderle PhD , in Introduction to Biomedical Engineering (Third Edition), 2012

12.5.1 Electromotive, Resistive, and Capacitive Properties

Electromotive Force Properties

The three major ions, K ⁺, Na ⁺, and Cl ⁻, are differentially distributed across the cell membrane at rest using passive ion channels, as illustrated in Figure 12.5. This separation of charge exists across the membrane and results in a voltage potential V _m , as described by Eq. (12.33) (the Goldman equation).

Across each ion-specific channel, a concentration gradient exists for each ion that creates an electromotive force, a force that drives that ion through the channel at a constant rate. The Nernst potential for that ion is the electrical potential difference across the channel and is easily modeled as a battery, as shown in Figure 12.11 for K ⁺. The same model is applied for Na ⁺ and Cl ⁻, with values equal to the Nernst potentials for each.

Figure 12.11. A battery is used to model the electromotive force for a K ⁺ channel with a value equal to the K ⁺ Nernst potential. The polarity of the battery is given with the ground on the outside of the membrane in agreement with convention. From Table 12.1, note that the Nernst potential for K ⁺ is negative, which reverses the polarity of the battery, driving K ⁺ out of the cell.

Resistive Properties

In addition to the electromotive force, each channel also has resistance—that is, it resists the movement of ions through the channel. This is mainly due to collisions with the channel wall, where energy is given up as heat. The term conductance, G, measured in siemens (S), which is the ease with which the ions move through the membrane, is typically used to represent resistance. Since the conductances (channels) are in parallel, the total conductance is the total number of channels, N, times the conductance for each channel, G′:

$G=N\times G\prime$

It is usually more convenient to write the conductance as resistance $R=\frac{1}{G}$ , measured in ohms (Ω). An equivalent circuit for the channels for a single ion is now given as a resistor in series with a battery, as shown in Figure 12.12.

Figure 12.12. The equivalent circuit for N ion channels is a single resistor and battery.

Conductance is related to membrane permeability, but they are not interchangeable in a physiological sense. Conductance depends on the state of the membrane, varies with ion concentration, and is proportional to the flow of ions through a membrane. Permeability describes the state of the membrane for a particular ion. Consider the case in which there are no ions on either side of the membrane. No matter how many channels are open, G = 0 because there are no ions available to flow across the cell membrane (due to a potential difference). At the same time, ion permeability is constant and determined by the state of the membrane.

Equivalent Circuit for Three Ions

Each of the three ions $K^{+},N{a}^{+},\text{and}C{l}^{-}$ are represented by the same equivalent circuit, as shown in Figure 12.12, with Nernst potentials and appropriate resistances. Combining the three equivalent circuits into one circuit with the extracellular fluid and cytoplasm connected by short circuits completely describes a membrane at rest (Figure 12.13).

Figure 12.13. Model of the passive channels for a small area of nerve at rest, with each ion channel represented by a resistor in series with a battery.

Example Problem 12.4

Find V _m for the frog skeletal muscle (Table 12.2) if the Cl ⁻ channels are ignored. Use $R_K=1.7\text{k}\Omega$ and $R_{Na}=15.67\text{k}\Omega .$

Solution

The following diagram depicts the membrane circuit with mesh current I, current I _Na through the sodium channel, and current I _K through the potassium channel. Current I is found using mesh analysis:

$E_{Na}+I{R}_{Na}+I{R}_K-E_K=0$

and then solving for I,

$I=\frac{E_K-E_{Na}}{R_{Na}+R_K}=\frac{(-105-56)\times {10}^{-3}}{\left(15.67+1.7\right)\times {10}^3}=-9.27\mu A$

yields

$V_m=E_{Na}+I{R}_{Na}=-89mV$

Notice that $I=-I_{Na}$ and $I=-I_K,$ or $I_{Na}=I_K$ as expected. Physiologically, this implies that the inward Na ⁺ current is exactly balanced by the outward bound K ⁺ current.

Example Problem 12.5

Find Vm for the frog skeletal muscle if $R_{Cl}=3.125\text{k}\Omega$ .

Solution

To solve, first find a Thevenin's equivalent circuit for the circuit in Example Problem 12.4:

$V_{TH}=V_m=-89\text{mV}$

and

$R_{TH}=\frac{R_{Na}\times R_K}{R_{Na}+R_K}=1.534\text{k}\Omega$

The Thevenin's equivalent circuit is shown in the following figure.

Since $E_{Cl}=V_{TH}$ according to Table 12.2, no current flows. This is the actual situation in most nerve cells. The membrane potential is determined by the relative conductances and Nernst potentials for K ⁺ and Na ⁺. The Nernst potentials are maintained by the Na-K active pump that maintains the concentration gradient. Cl ⁻ is usually passively distributed across the membrane.

The Na-K Pump

As shown in Example Problem 12.4 and Section 12.4, there is a steady flow of K ⁺ ions out of the cell and Na ⁺ ions into the cell even when the membrane is at the resting potential. Left unchecked, this would drive E _K and E _Na toward 0. To prevent this, current generators depicting the Na-K pump are used that are equal to and the opposite of the passive currents and incorporated into the model, as shown in Figure 12.14.

Figure 12.14. Circuit model of the three passive channels for a small area of the nerve at rest with each ion channel represented by a resistor in series with a battery. The Na-K active pump is modeled as two current sources within the shaded box.

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Mathematical Modeling of Cellular Electric Activity

Riccardo Sacco , ... Aurelio Giancarlo Mauri , in A Comprehensive Physically Based Approach to Modeling in Bioengineering and Life Sciences, 2019

17.4 The Hodgkin–Huxley Model

In a series of historical papers [149,150] that earned them the Nobel Prize in Medicine in 1963, Hodgkin and Huxley proposed and analyzed a differential algebraic mathematical model for transmembrane ion current conduction in the squid giant axon. The original form of the Hodgkin–Huxley model falls into the category of whole-cell models but the formulation can be used also as a local ordinary differential equation model, as described in Chapter 18.

The Hodgkin–Huxley model is characterized by the following main properties:

•

the capacitance $C_m$ is linear;

•

the conductance $G_m$ is a nonlinear function of ${\psi }_m$ .

Fig. 17.8 gives a schematic illustration of the electric equivalent circuit of the Hodgkin–Huxley whole-cell model. Three ion species, Na⁺, K⁺, and Cl⁻, are included in the biophysical picture of ion current conduction in the squid giant axon. Experimental evidence indicates that the principal ion currents are due to Na⁺ and K⁺ ions. The remaining ion currents are conventionally associated with the Cl⁻ ion. Since this current is small compared to the principal currents, it is usually referred to as leakage current (see [175,148,107]).

Figure 17.8. Equivalent electric representation of the Hodgkin–Huxley model.

The symbol $c_m$ is the membrane specific capacitance (unit: $\mathrm{F}{\mathrm{m}}^{-\mathrm{2}}$ ), defined as

(17.35) $c_m:=\frac{C_m}{{\mathcal{S}}_{cell}}.$

The symbols $J_{\mathrm{Na}}^{tm}$ and $J_{\mathrm{K}}^{tm}$ denote the current densities of sodium and potassium ions, whereas $J_{\mathrm{L}}^{tm}$ is the leakage current density. The symbol $v_m$ is related to the membrane potential ${\psi }_m$ by the following relation:

(17.36) $v_m:={\psi }_m-E_{c,m},$

where $E_{c,m}$ is the membrane resting potential (i.e., the equilibrium potential of the membrane), computed using Eq. (17.24b). The numerical evaluation of $E_{c,m}$ is described in detail in Example 17.2.

The variable $v_m$ biophysically represents the displacement of the membrane potential from the resting state of the membrane, which corresponds to the situation of thermodynamic equilibrium introduced in Definition 17.1 in the case of a single ion species. Correspondingly, the quantities $v_{\mathrm{\text{K}}}$ , $v_{\mathrm{\text{Na}}}$ , and $v_L$ represent the deviations of the Nernst potentials of potassium, sodium, and chloride from the membrane resting potential; they are treated in [150] as model fitting parameters (so-called ''adjusted potentials''). All potentials $v_m$ , $v_{\mathrm{\text{K}}}$ , $v_{\mathrm{\text{Na}}}$ , and $v_{\mathrm{\text{L}}}$ are expressed in mV, and, precisely, the shifted reversal potentials are set equal to

$v_{\mathrm{\text{K}}}=-12\text{mV},v_{\mathrm{\text{Na}}}=115\text{mV},v_{\mathrm{\text{L}}}=10.613\text{mV}.$

The specific membrane capacitance $c_m$ is equal to $1\mathrm{\mu }\text{F\hspace{0.17em}c}{\mathrm{m}}^{-\mathrm{2}}$ .

Applying the KCL to the circuit of Fig. 17.8 with $J^{tot}=0$ , we obtain the following nonlinear system of ordinary differential equations and algebraic equations, which is referred to as the Hodgkin–Huxley model:

(17.37a) $c_m\frac{d{v}_m}{dt}=-(J_{\mathrm{\text{K}}}^{tm}+{J_{\mathrm{\text{Na}}}}^{tm}+J_L^{tm}),$

(17.37b) $J_{\mathrm{\text{K}}}^{tm}=n^4{g}_{\mathrm{\text{K}}}(v_m-v_{\mathrm{\text{K}}}),$

(17.37c) $J_{\mathrm{\text{Na}}}^{tm}=h{m}^3{g}_{\mathrm{\text{Na}}}(v_m-v_{\mathrm{\text{Na}}}),$

(17.37d) $J_L^{tm}=g_{\mathrm{\text{L}}}(v_m-v_{\mathrm{\text{L}}}),$

(17.37e) $\frac{\partial m}{\partial t}={\alpha }_m(1-m)-{\beta }_{m}m,$

(17.37f) $\frac{\partial n}{\partial t}={\alpha }_n(1-n)-{\beta }_{n}n,$

(17.37g) $\frac{\partial h}{\partial t}={\alpha }_h(1-h)-{\beta }_{h}h,$

where the time rates ${\alpha }_j$ and ${\beta }_j$ , $j=m,n,h$ , are positive coefficients that nonlinearly depend on the potential $v_m$ as specified below. The dimensionless functions of time n, m, and h are the so-called gating variables. This terminology is motivated by the fact that they describe the opening state of the channel. Their introduction, and, more importantly, their phenomenological characterization, is the main contribution of the Hodgkin–Huxley theory and permits the simulation of the propagation of an action potential, as shown below in the numerical examples. The variables m, n, and h typically vary between 0 and 1, and, at each point of the membrane, are governed by the ordinary differential equations (17.37e)–(17.37g), where the time rates ${\alpha }_s$ and ${\beta }_s$ , $s=m,n,h$ , are experimentally determined functions expressed in $\mathrm{m}{\mathrm{s}}^{-\mathrm{1}}$ . Hodgkin and Huxley [150] used the following expressions for them:

(17.38a) ${\alpha }_m(v_m)=\mathcal{B}((25-v_m)/10),$

(17.38b) ${\beta }_m(v_m)=4\exp (-v_m/18),$

(17.38c) ${\alpha }_n(v_m)=0.1\mathcal{B}((10-v_m)/10),$

(17.38d) ${\beta }_n(v_m)=0.125\exp (-v_m/80),$

(17.38e) ${\alpha }_h(v_m)=0.07\exp (-v_m/20),$

(17.38f) ${\beta }_h(v_m)=\mathcal{C}((30-v_m)/10),$

where $\mathcal{B}(t):=t/(e^t-1)$ and $\mathcal{C}(t):=1/(e^t+1)$ . The constant specific conductances $g_{\mathrm{\text{K}}}$ , $g_{\mathrm{\text{Na}}}$ , and $g_{\mathrm{\text{L}}}$ are expressed in $\mathrm{mS\hspace{0.17em}c}{\mathrm{m}}^{-\mathrm{2}}$ and are set equal to

$g_{\mathrm{\text{K}}}=36\mathrm{mS\hspace{0.17em}c}{\mathrm{m}}^{-\mathrm{2}},g_{\mathrm{\text{Na}}}=120\mathrm{mS\hspace{0.17em}c}{\mathrm{m}}^{-\mathrm{2}},g_{\mathrm{\text{L}}}=0.3\mathrm{mS\hspace{0.17em}c}{\mathrm{m}}^{-\mathrm{2}}.$

Example 17.2 Thermodynamic equilibrium for the squid giant axon

In this example we apply the Goldman equation (17.24b) to determine the equilibrium potential of the squid giant axon that was originally studied by Hodgkin and Huxley in [149] and [150]. Since three ion species are flowing across the squid axon cell membrane, Na⁺, K⁺, and Cl⁻, we have $M_{ion}=3$ , with $M^{+}=2$ and $M^{-}=1$ , and Eq. (17.24b) becomes

(17.39a) $E_{c,M_{ion}}=V_{th}\ln \left(\frac{P_{\mathrm{\text{K}}}{C}_{\mathrm{\text{K}}}^{(out)}+P_{\mathrm{\text{Na}}}{C}_{\mathrm{\text{Na}}}^{(out)}+P_{\mathrm{\text{Cl}}}{C}_{\mathrm{\text{Cl}}}^{(in)}}{P_{\mathrm{\text{K}}}{C}_{\mathrm{\text{K}}}^{(in)}+P_{\mathrm{\text{Na}}}{C_{\mathrm{\text{Na}}}}^{(in)}+P_{\mathrm{\text{Cl}}}{C}_{\mathrm{\text{Cl}}}^{(out)}}\right).$

Experimental data reveal that the permeabilities verify the following ratio (see [107]):

(17.39b) $P_{\mathrm{\text{K}}}:P_{\mathrm{\text{Na}}}:P_{\mathrm{\text{Cl}}}=1:0.03:0.1,$

so that, dividing numerator and denominator in the argument of the logarithm in (17.39a) by $P_{\mathrm{\text{K}}}$ and using (17.39b), we get

(17.39c) $E_{c,M_{ion}}=V_{th}\ln \left(\frac{1C_{\mathrm{\text{K}}}^{(out)}+0.03C_{\mathrm{\text{Na}}}^{(out)}+0.1C_{\mathrm{\text{Cl}}}^{(in)}}{1C_{\mathrm{\text{K}}}^{(in)}+0.03C_{\mathrm{\text{Na}}}^{(in)}+0.1C_{\mathrm{\text{Cl}}}^{(out)}}\right).$

Setting $\mathrm{\Theta }=300\mathrm{K}$ and replacing in (17.39c) the values of the intracellular and extracellular concentrations listed in Table 17.1, we obtain

(17.39d) $E_{c,M_{ion}}=-72.705\text{mV}.$

To evaluate the role played by the various ions in determining membrane electrochemical equilibrium, we compute the resting potential associated with the sole potassium ion, obtaining

(17.39e) $E_{c,\mathrm{\text{K}}}=V_{th}\ln \left(\frac{C_{\mathrm{\text{K}}}^{(out)}}{C_{\mathrm{\text{K}}}^{(in)}}\right)=-95.910\text{mV}.$

Comparing (17.39e) with (17.39d), we see that:

1.

the equilibrium of the squid giant axon is dominated by the potassium ion;

2.

the chlorine ion has the effect of depolarizing the cell because it brings the resting potential closer to zero.

Table 17.1. Values of intracellular and extracellular ion concentrations in the study of membrane equilibrium of the squid giant axon.

Model parameter	Value	Unit
$C_{\mathrm{\text{K}}}^{(in)}$	400	[mM]
$C_{\mathrm{\text{K}}}^{(out)}$	10	[mM]
$C_{\mathrm{\text{Na}}}^{(in)}$	50	[mM]
$C_{\mathrm{\text{Na}}}^{(out)}$	460	[mM]
$C_{\mathrm{\text{Cl}}}^{(in)}$	40	[mM]
$C_{\mathrm{\text{Cl}}}^{(out)}$	540	[mM]

Example 17.3 Action potential simulation using the Hodgkin–Huxley model

To conclude this section we apply the Hodgkin–Huxley model (17.37) to the simulation of the elicitation of an action potential in an excitable cell and its subsequent propagation along a nerve fiber. The action potential occurs in response to the external stimulus (a voltage or a current stress) that triggers the excitable cell. Let us consider the case of a current density stimulus $J_{out}\ne 0$ . Applying the KCL to the circuit in Fig. 17.8 with an external current source, Eq. (17.37) becomes

(17.40) $c_m\frac{d{v}_m}{dt}=-(J_{\mathrm{\text{K}}}^{tm}+J_{\mathrm{\text{Na}}}^{tm}+J_L^{tm})+J_{out},$

where $J_{\mathrm{\text{K}}}^{tm}$ , $J_{\mathrm{\text{Na}}}^{tm}$ , and $J_L^{tm}$ are given in (17.37). Assuming a cell radius of $5\cdot {10}^{-3}\text{cm}$ and an external stimulus of $J_{out}=2\cdot {10}^{-6}\mathrm{\mu }\mathrm{A\hspace{0.17em}c}{\mathrm{m}}^{-\mathrm{2}}$ , we obtain the results illustrated in Figs. 17.9 and 17.10.

Figure 17.9. Simulation of the elicitation of an action potential in an excitable cell using the Hodgkin–Huxley model. Left panel: action potential. Right panel: gating variables.

Figure 17.10. Left panel: channel ion conductances. Right panel: ion current densities predicted by the Hodgkin–Huxley model.

Fig. 17.9 shows the action potential and gating variables predicted by the Hodgkin–Huxley equation system (17.37): the fast rise of the potential is determined by the opening of the Na⁺ channels controlled by the gating variable m (solid line in the right panel), whereas the slow ramp down is driven by the K⁺ ions whose conduction is determined by the gating variable n (dashed line in the right panel) till the K⁺ Nernst potential is reached. It is worth noting that the ion channel conductance in the Hodgkin–Huxley model is not constant but is determined by the solution of the gating variable system, as illustrated in Fig. 17.10 (left panel). Fig. 17.10 (right panel) shows the current density $J_{\mathrm{\text{K}}}^{tm}$ associated with potassium (solid line) and the current density $J_{\mathrm{\text{Na}}}^{tm}$ associated with sodium (dashed line): they have opposite signs, corresponding to the fact that Na⁺ flows out of the cell whereas K⁺ flows into the cell. As anticipated, simulation results indicate that the leakage current density $J_{\mathrm{\text{L}}}^{tm}$ associated with Cl⁻ (dash-dotted line) is far smaller than the other two principal current densities.

A complete mathematical analysis of the ordinary differential equation system (17.37) goes beyond the scope of this section and can be found in [175] and [107]. We refer to Section 28.3.1 for the illustration of the Matlab script that implements the Hodgkin–Huxley model.

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Finite Element Approximations of Initial Value/Boundary Value Problems of Advection–Diffusion–Reaction Type

Riccardo Sacco , ... Aurelio Giancarlo Mauri , in A Comprehensive Physically Based Approach to Modeling in Bioengineering and Life Sciences, 2019

24.2 Simulation of the Propagation of an Action Potential Along an Axon

In this section we numerically investigate the time-dependent GFEM illustrated in Section 24.1 applied to the simulation of the propagation of an action potential along an ideal axon of length $L=10\text{cm}$ and radius $a=2.5\text{mm}$ . The computational procedure consists of the use of (i) the functional iteration techniques illustrated in Section 21.5.1; (ii) the time discretization methods illustrated in the present chapter, and (iii) the mixed GFEM illustrated in Chapter 23.

24.2.1 Model Data

The axon membrane thickness is $t_m=10\text{nm}$ , the load output resistance $R_{out}$ is set equal to +∞, so that the terminal of the axon at $x=L$ is, from the electrical standpoint, an open circuit and no electric current is allowed to flow out of it. (See Table 24.1.)

Table 24.1. Values of model parameters in the simulation of the propagation of an action potential along an axon using the cable equation model. The temperature of the system under investigation is Θ = 298.15 K. Transmembrane conduction current is assumed to be driven by potassium, sodium, and chloride ions.

Model parameter	Description	Value
${\epsilon }_m^r$	Dielectric relative constant	80 [⋅]
t 0, t 1, t 2, t 3	Time characteristic of the axon potential	0,5,7.5,10 ms
T fin	Final simulation time	15 ms
R out	Load output resistance	+∞ Ω
a	Radius of the ideal axon	2.5 ⋅ 10−3 m
L	Length of the ideal axon	0.1 m
t m	Membrane thickness of the ideal axon	10−8 m
c m	Unit area capacitance of the ideal axon	10−2 F m−2
V 1, V 2	Electric characteristic of the axon potential	28.73, −91.27 mV

The resting potential of the cell can be computed by using the Goldman equation (17.24b) in which the ion permeabilities $P_{\mathrm{\text{K}}}$ , $P_{\mathrm{\text{Na}}}$ , and $P_{\mathrm{\text{Cl}}}$ satisfy relation (17.39b). Setting $T=300\mathrm{K}$ and using the values of the intracellular and extracellular concentrations listed in Table 24.2, we obtain

Table 24.2. Values of intracellular and extracellular ion concentrations in the study of propagation of an action potential along an axon in the human body (cf. [107, Table 1.1].)

Model parameter	Description	Value
$C_{\mathrm{\text{K}}}^{(in)}$	K+ concentration in the intracellular region	150 mM
$C_{\mathrm{\text{Na}}}^{(in)}$	Na+ concentration in the intracellular region	50 mM
$C_{\mathrm{\text{Cl}}}^{(in)}$	Cl− concentration in the intracellular region	10 mM
$C_{\mathrm{\text{K}}}^{(out)}$	K+ concentration in the extracellular region	5 mM
$C_{\mathrm{\text{Na}}}^{(out)}$	Na+ concentration in the extracellular region	460 mM
$C_{\mathrm{\text{Cl}}}^{(out)}$	Cl− concentration in the extracellular region	125 mM

(24.2) ${\psi }_{eq}=-71.27\text{mV}.$

24.2.2 Action Potential

The input voltage stimulus ${\bar{V}}_{in}$ is the piecewise step function illustrated in Fig. 24.3. This function is a schematic representation of the action potential which develops in an excitable cell such as a neuron or a cardiac myocyte. The applied voltage is equal to the resting potential of the cell ${\psi }_{eq}$ from time $t=t_0=0\text{ms}$ until time $t_1=5\text{ms}$ . In these conditions, the intracellular potential is lower than the extracellular potential so that the "−" pole is the intracellular site and the "+" pole is the extracellular site. At time $t=t_1$ , the applied voltage is suddenly raised to the value $V_1={\psi }_{eq}+100\text{mV}=+28.73\text{mV}$ and is kept equal to $V_1$ until time $t_2=7.5\text{ms}$ . In these conditions, a change of polarity takes place because the intracellular potential is higher than the extracellular potential so that the "+" pole is the intracellular site and the "−" pole is the extracellular site. This change of polarity is the result of a flow of a positive ion charge, such as Na⁺, that was attracted to the cytoplasm by the "+" pole in the resting condition. This phase of the action potential is referred to as depolarization (see [107, Chapter 1]). At time $t=t_2$ , the voltage stimulus is again suddenly changed to the value $V_2={\psi }_{eq}-20\text{mV}=-91.27\text{mV}$ and is kept equal to $V_2$ until time $t_3=10\text{ms}$ . In these conditions, another change of polarity takes place because the intracellular potential is more negative than the resting potential so that the "−" pole is the intracellular site and the "+" pole is the extracellular site. This change of polarity is the result of a flow of a negative ion charge, such as Cl⁻, that was attracted to the cytoplasm by the "+" pole in the depolarization condition. This phase of the action potential is referred to as hyperpolarization (see [107, Chapter 1]). At time $t=t_3$ , the voltage stimulus is again suddenly changed to the value of the resting potential ${\psi }_{eq}$ and is kept equal to ${\psi }_{eq}$ until the final time $t=T_{fin}=15\text{ms}$ .

Figure 24.3. Input voltage stimulus. The piecewise constant function shown in the figure schematically represents an action potential. The jump between the resting potential to the positive value V ₁ = 28.73 mV corresponds to the depolarization phase. The jump from V ₁ to the value V ₂ = −91.27 mV corresponds to the hyperpolarization phase.

24.2.3 Overview

In Chapter 17 we illustrated two different models to characterize the transmembrane current, namely, the linear resistor formulation (17.44) and the nonlinear Goldman–Hodgkin–Katz (GHK) formulation (17.51). In the following paragraphs, we utilize the functional iteration (21.16) and its time and spatial discretization to compare the solutions obtained using the two models. The numerical implementation of the computational algorithm is performed by the Matlab script 28.49. The Matlab function 29.28 implements the solution of the problem using the linear resistor formulation, whereas the Matlab function 29.29 implements the solution of the problem using the GHK formulation. In both functions, the time discretization of (18.2) is performed using the backward Euler method illustrated in Section 3.5.2.1, whereas the spatial discretization is performed using the mixed GFEM illustrated in Chapter 23. A uniform time step $\mathrm{\Delta }t=(T_{end}-t_0)/250=6\cdot {10}^{-5}\mathrm{s}$ and a uniform spatial discretization parameter $h=L/100=0.1\text{cm}$ are used in the numerical computations. The Matlab function 29.27 computes the voltage stimulus graphically represented in Fig. 24.3.

Simulation With the Linear Resistor Model In this paragraph we perform the numerical simulation of the propagation of the action potential elicited by the voltage stimulus of Fig. 24.3, by representing the axon transmembrane current density through the linear resistor model (17.44). We consider three values of the longitudinal resistivity, and we set ${\rho }_{in}={\rho }_{in}^{SW}$ (case (a)), ${\rho }_{in}={10}^{+3}{\rho }_{in}^{SW}$ (case (b)), and ${\rho }_{in}={10}^{-3}{\rho }_{in}^{SW}$ (case (c)). To use the linear resistor model, we need to determine the ion specific conductance $g_{\alpha }$ (unit: $\mathrm{S}{\mathrm{m}}^{-\mathrm{2}}$ ), $\alpha \in \{\mathrm{K},\mathrm{Na},\mathrm{Cl}\}$ . With this aim, we use the definition of local ion conductivity (13.75d) in the following average form:

(24.3) ${\sigma }_{\alpha }=q|z_{\alpha }|{\mu }_{\alpha }^{el}\frac{n_{\alpha }(0)+n_{\alpha }(t_m)}{2t_m}=q|z_{\alpha }{|}^2{D}_{\alpha }\frac{n_{\alpha }^{in}+n_{\alpha }^{out}}{2t_m},\alpha \in \{\mathrm{K},\mathrm{Na},\mathrm{Cl}\},$

where $n_{\alpha }^{in}$ and $n_{\alpha }^{out}$ are the intra and extracellular number densities of the αth ionic species and Einstein's relation (13.69) is used to express the ion electric mobility ${\mu }_{\alpha }^{el}$ in favor of the ion diffusion coefficient $D_{\alpha }$ . Results are shown in Figs. 24.4, 24.5, and 24.6.

Figure 24.4. Simulation of axon potential evolution using the cable equation model (18.2) supplied by the linear resistor model for the transmembrane current. In this simulation the axoplasm resistivity ρ _in is set equal to ${\rho }_{in}^{SW}=2\cdot {10}^{-1}\mathrm{\Omega }\mathrm{m}$ , corresponding to an axoplasm conductivity ${\sigma }_{in}=5\mathrm{S}{\mathrm{m}}^{-\mathrm{1}}$ . Left panel: view from x =L. Right panel: view from x = 0.

Figure 24.5. Simulation of axon potential evolution using the cable equation model (18.2) supplied by the linear resistor model for the transmembrane current. In this simulation the axoplasm resistivity ρ _in is set equal to ${10}^3{\rho }_{in}^{SW}=2\cdot {10}^2\mathrm{\Omega }\mathrm{m}$ , corresponding to an axoplasm conductivity ${\sigma }_{in}=5\cdot {10}^{-3}\mathrm{S}{\mathrm{m}}^{-\mathrm{1}}$ . Left panel: view from x =L. Right panel: view from x = 0.

Figure 24.6. Simulation of axon potential evolution using the cable equation model (18.2) supplied by the linear resistor model for the transmembrane current. In this simulation the axoplasm resistivity ρ _in is set equal to ${10}^{-3}{\rho }_{in}^{SW}=2\cdot {10}^{-4}\mathrm{\Omega }\mathrm{m}$ , corresponding to an axoplasm conductivity ${\sigma }_{in}=5\cdot {10}^3\mathrm{S}{\mathrm{m}}^{-\mathrm{1}}$ . For each t ∈ [0,T _fin ], the spatial distribution of the membrane potential along the axon is constant because of the elevated longitudinal conductivity.

Case (a). Fig. 24.4 shows the axon membrane potential ${\psi }_m(x,t)$ for $x\in [0,L]$ and for $t\in [0,T_{end}]$ , computed by solving (21.16) with the linear resistor model for the axon transmembrane current density. The value of the axoplasm resistivity ${\rho }_{in}$ is set equal to ${\rho }_{in}^{SW}=2\cdot {10}^{-1}\mathrm{\Omega }\mathrm{m}$ , the resistivity of salt water at $20{}^{\circ }\mathrm{C}$ . This value is very close to the data shown in [83], where the experimentally measured resistivity of the extruded squid axon turned out to be equal to $(1.4\pm 0.2){\rho }_{in}^{SW}$ . Results indicate the following:

•

At $t=0$ , the membrane potential undergoes a fast increase in the neighborhood of $x=0$ and reaches a constant value close to $0\text{mV}$ along the axon length because of a flux of positive charge inside the axon.

•

As t increases, the spatial distribution of the membrane potential remains uniform but with a smaller value compared to $t=0$ because of a flux of part of the accumulated positive charge out of the intracellular axon region.

•

At $t=t_1$ , the membrane potential rises abruptly at $x=0$ and then monotonically decreases along the axon length, passing from positive values to negative values, because of a positive accumulated charge inside the axon near $x=0$ and a negative accumulated charge for larger values of x. The membrane potential exhibits an opposite behavior at $t=t_2$ .

•

At $t=t_3$ , the membrane potential returns abruptly to its resting value, which is less negative than $V_2$ . This induces an accumulation of positive charge inside the intracellular region of the axon and the membrane potential becomes uniform along the axon length, reaching a value of about $-55\text{mV}$ .

•

For larger values of t, the accumulated positive charge flows out of the axon, thereby giving rise to more negative values of the membrane potential. After a certain time instant, all the accumulated positive charge has flown out of the axon, so that the membrane potential becomes flat along the whole axon length and equal to the boundary condition at $x=0$ .

•

For all $t\in [0,T_{fin}]$ , the spatial distribution of the membrane potential becomes flat at the end of the axon because of the no-flux boundary condition at $x=L$ .

Case (b). Fig. 24.5 shows the axon membrane potential computed by solving (21.16) with the linear resistor model and with an axoplasm resistivity increased by a factor of 10³ with respect to the previous case. Results indicate that the membrane potential undergoes a practically linear decrease with respect to time for each $x\in [0,L]$ . This behavior is due to the elevated value of the longitudinal axon resistivity which considerably slows down the transmembrane flow of positive and negative charge for each $x\in [0,L]$ . We also note that the axon is far from resting conditions at $t=T_{fin}$ because the membrane potential is about $-50\text{mV}$ . Resting conditions would be reached only for a significantly larger value of the observational time $T_{fin}$ .

Case (c). Fig. 24.6 shows the axon membrane potential ${\psi }_m(x,t)$ for $x\in [0,L]$ and for $t\in [0,T_{end}]$ in the case where the axoplasm resistivity is reduced by a factor of 10³, so that we have ${\rho }_{in}={10}^{-8}{\rho }_{in}^{SW}=2\cdot {10}^{-9}\mathrm{\Omega }\mathrm{m}$ . Results indicate that for each value of $t\in [0,T_{end}]$ the spatial distribution of the membrane potential along the axon is constant because the elevated longitudinal conductivity gives rise to an almost instantaneous transport of the value of the Dirichlet boundary condition ${\psi }_m(0,t)={\bar{V}}_{in}(t)$ .

Simulation With the GHK Model In this paragraph we perform the numerical simulation of the propagation of the action potential elicited by the voltage stimulus of Fig. 24.3, by representing the axon transmembrane current density through the GHK model (17.51). We consider the same conditions as in Paragraph 'Simulation With the Linear Resistor Model', that is, we set ${\rho }_{in}={\rho }_{in}^{SW}$ (case (a)), ${\rho }_{in}={10}^{+3}{\rho }_{in}^{SW}$ (case (b)), and ${\rho }_{in}={10}^{-3}{\rho }_{in}^{SW}$ (case (c)). Computations show that four iterations were needed to satisfy the convergence check (21.17) with a tolerance $\mathtt{toll}={10}^{-5}$ . Results are shown in Figs. 24.7, 24.8, and 24.9.

Figure 24.7. Simulation of axon potential evolution using the cable equation model (18.2) supplied by the GHK model for the transmembrane current. In this simulation the axoplasm resistivity ρ _in is set equal to ${\rho }_{in}^{SW}=2\cdot {10}^{-1}\mathrm{\Omega }\mathrm{m}$ , corresponding to an axoplasm conductivity ${\sigma }_{in}=5\mathrm{S}{\mathrm{m}}^{-\mathrm{1}}$ . Left panel: view from x =L. Right panel: view from x = 0.

Figure 24.8. Simulation of axon potential evolution using the cable equation model (18.2) supplied by the GHK model for the transmembrane current. In this simulation the axoplasm resistivity ρ _in is set equal to ${10}^3{\rho }_{in}^{SW}=2\cdot {10}^2\mathrm{\Omega }\mathrm{m}$ , corresponding to an axoplasm conductivity ${\sigma }_{in}=5\cdot {10}^{-3}\mathrm{S}{\mathrm{m}}^{-\mathrm{1}}$ . Left panel: view from x =L. Right panel: view from x = 0.

Figure 24.9. Simulation of axon potential evolution using the cable equation model (18.2) supplied by the GHK model for the transmembrane current. In this simulation the axoplasm resistivity ρ _in is set equal to ${10}^{-3}{\rho }_{in}^{SW}=2\cdot {10}^{-4}\mathrm{\Omega }\mathrm{m}$ , corresponding to an axoplasm conductivity ${\sigma }_{in}=5\cdot {10}^3\mathrm{S}{\mathrm{m}}^{-\mathrm{1}}$ . For each t ∈ [0,T _fin ], the spatial distribution of the membrane potential along the axon is constant because of the elevated longitudinal conductivity.

Model predictions in case (c) cannot be distinguished from the predictions given by the linear resistor model in the same conditions. Model predictions in cases (a) and (b) are qualitatively similar to the corresponding simulations with the linear resistor model.

In case (a), we see that:

•

the increase of membrane potential at $t=0$ in the neighborhood of $x=0$ is less sharp than predicted by the linear resistor model;

•

the membrane potential at $x=L$ is less negative than predicted by the linear resistor model;

•

the membrane potential distribution at $t=T_{fin}$ is not as flat as predicted by the linear resistor model. This indicates that, according to the GHK model, the axon has not reached resting conditions as in the case of the linear resistor model. In particular, the value of the membrane potential predicted by the GHK model at $x=L$ is less negative than that predicted by the linear resistor model.

Similar considerations apply for case (b). In particular, we see that:

•

for each $x\in [0,L]$ the decay in time of the membrane potential is significantly smaller than predicted by the linear resistor model;

•

the resulting membrane potential at $x=L$ predicted by the GHK model is much less negative than in the case of the linear resistor model, so that a much larger observational time is needed to reach resting conditions in the axon.

24.2.4 Comparison and Concluding Remarks

The same considerations developed in Section 17.7.4.3 can be applied to discuss the quality and reliability of model predictions obtained in Paragraphs 'Simulation With the Linear Resistor Model' and 'Simulation With the GHK Model' using the linear resistor model and the GHK model, respectively. In particular, the accuracy of the computed action potential profiles may be warranted through the following two approaches:

•

an error analysis against an exact solution, and/or

•

comparison with experimental data, whenever available.

As far as the first approach, we refer to [223], whereas for the second approach we refer, e.g., to [317].

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Electrochemical Materials and Engineering • Sensors and Biosensors

Alexander R. Harris , Gordon G. Wallace , in Current Opinion in Electrochemistry, 2019

Introduction

Cells interact with their environment by sensing chemical, electrical, optical, thermal and mechanical stimuli. They can then act as a transducer, inducing a response in other nearby cells. Electrically excitable cells, including neurons and muscle cells, are of particular interest as they are involved in information transfer around the body, movement and cognition. These cells create an electrical potential by developing a charge separation across their membranes. Typically, the extracellular fluid contains high sodium and low potassium concentration, whereas the intracellular fluid has low sodium and high potassium concentration. The movement of ions across the membrane is controlled by ion pumps and ion channels, resulting in a resting membrane potential of approximately −40 to −80 mV. The membrane potential is modelled by the Goldman equation, which relates ion concentrations and their permeability across the membrane [1]. The current passing through the membrane is modelled by the Hodgkin–Huxley equation [2] and more recent derivatives [3]. Stimulation of a neuron results in a rise in membrane potential; if the threshold potential is reached, then an action potential is initiated. An action potential is a rapid rise or depolarisation of the membrane potential to around 40 mV and subsequent hyperpolarisation close to −90 mV before returning to the resting potential. The action potential propagates along the nerve fibre at approximately 1 m s⁻¹ in unmyelinated nerves and up to 120 m s⁻¹ in myelinated nerves. This high-speed action potential propagation and the complex neural networks that form between cells allow greater information processing and storage than would be possible if only chemical diffusion was used.

The ability to electrically stimulate excitable cells using an external power source was first discovered by Luigi Galvani in the 1780s when he applied charge generated by static electricity to induce movement in a dead frog's leg. In 1800, Alessandro Volta connected two metal rods to a battery and then placed them into his ears. He described the sensation as a shock within his head, followed by the sound of boiling thick soup [4]. Subsequently, the use of electrodes to the interface with electrically excitable cells has grown to enable an understanding of cell behaviour and the development of novel biomedical devices [5].

Electrodes can be used to record or stimulate electrically excitable cells. Recording neural behaviour can be used to control prosthetic devices for patients who have suffered trauma or locked-in syndrome. For instance, an electrode placed into the motor cortex could enable a patient to move a robotic arm or a cursor on a computer. Electrical stimulation can provide sensory cues to patients, such as the cochlear implant or bionic eye. Functional electrical stimulation can be used for numerous conditions including assisting with muscle contraction to aid in walking after spinal cord injury or for bladder control. Other uses of electrical stimulation include reduction of chronic pain and cardiac pacemakers. Closed-loop devices are now being developed [6,7]; by recording brain activity to control neural stimulation for alleviating epileptic seizures, battery life can be improved and side effects reduced over constant stimulating devices [8]. And recently, electrical stimulation of paralysed patients has enabled them to walk again [9].

Depending on the intended target, electrodes can be placed into different tissues. The central nervous system can be interfaced with electrodes placed on the scalp (electroencephalography), placed on the surface of the cortex (electrocorticography) or penetrated through the brain (deep brain stimulation or cortical implant). Electrodes placed on peripheral nerves may wrap around (cuff electrode), lie across (longitudinal intrafascicular electrode) or penetrate through (transverse intrafascicular multichannel electrode) the nerve fibres [10] or interface with specific sensory neurons. Electrodes can also be placed onto muscle tissue or on the overlying skin (electromyography) [11]. To design appropriate electrodes for medical devices, the electrode geometry and position must be tailored to the target tissue. Electrodes that are placed on the skin or surface of tissue are less invasive, reducing surgical complications. Conversely, electrodes placed closer to the target cells enable more localised recording and stimulation, improving the signal-to-noise ratio, reducing power usage and side effects from off-target stimulation. Recently, transcranial electrical stimulation from multiple electrodes at frequencies too high to induce neural firing but with differences in frequency that were in the dynamic range of neural firing was performed [12]. The electric field interference pattern enabled steerable neural stimulation, suggesting noninvasive, localised stimulation may be possible.

Simple models of current flow between an electrode and target cells assume tissue is a uniform medium and the electrodes are point sources. However, there is a large inhomogeneity in resistivity and permittivity in tissue due to variations in the cell type and structure and the associated variations in ionic strength and cell membrane capacitance. This leads to severe attenuation of high-frequency components in tissue, so measurement of electrical activity at large electrode-target cell distances is limited to low frequencies (local field potentials) [13]. For neural stimulation, an applied charge must lead to a sufficient change in membrane potential to induce an action potential. Cellular composition will affect the electric field, and numerous factors such as the specific neurite alignment, neurite size, cell and ion channel type and number will impact on the cell response [14]. Furthermore, after electrode implantation, an immune response to the foreign object can occur, encapsulating the electrode in fibrotic tissue [15]. This increases the distance between the electrode and target tissue. Efforts to reduce this immune response include reducing the device size [16], stiffness [17] and hardness [18], optimising the material biocompatibility [19] and applying anti-inflammatory drugs [20,21] or growth factors [22] to encourage growth of cells towards the electrode.

These devices typically operate in a 2-electrode configuration, so current that flows between the working and ground electrode must be shaped to pass through the target cells. Current steering can be achieved by varying the location of the stimulating and ground electrodes. For instance, a 42-channel, 600-μm diameter platinum hexagonal electrode array placed in the suprachoroidal space of a cat could be stimulated through different electrode configurations [23]. By passing current through the 6 electrodes of a hexagon, a virtual central electrode could be created, and by adjusting the current flow, the position of the centroid could be shifted. Modification of electrode geometry can also be used to improve the charge transfer efficiency to the target neurons [24].

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