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Dieses Kapitel taucht in die faszinierende Welt der Neuroelektrophysiologie ein und zeichnet die Geschichte der Bioelektrizität von altägyptischen Wandmalereien bis hin zu den bahnbrechenden Arbeiten von Galvani und Volta nach. Es untersucht die elektrischen Eigenschaften neuronaler Membranen, einschließlich Widerstand, Kapazität und Zeitkonstante, die für das Verständnis der Erzeugung und Verbreitung elektrischer Signale durch Neuronen von entscheidender Bedeutung sind. Das Kapitel behandelt auch die Erzeugung von Ruhe- und Aktionspotenzialen, wobei die Rolle von Ionenkanälen und der Natrium-Kalium-Pumpe bei der Aufrechterhaltung des Ruhepotenzials hervorgehoben wird. Ein bedeutender Schwerpunkt liegt auf dem Hodgkin-Huxley-Modell, das eine quantitative Beschreibung der elektrischen Eigenschaften von Ionenkanälen in neuronalen Membranen liefert. Dieses Modell hat entscheidend dazu beigetragen, unser Verständnis neuronalen Verhaltens zu verbessern und den Weg für zahlreiche Anwendungen in der neurowissenschaftlichen Forschung geebnet. Darüber hinaus behandelt das Kapitel die Prinzipien der elektrischen Stimulation von Neuronen, einschließlich intra- und extrazellulärer Methoden, und die damit verbundenen Sicherheitsüberlegungen. Es werden die Unterschiede zwischen Spannungs- und Stromreizen sowie unipolaren und bipolaren Reizen und ihre jeweiligen Vor- und Nachteile diskutiert. Das Kapitel schließt mit einer Diskussion über die Sicherheit der elektrischen Stimulation, in der die Bedeutung der Verhinderung von Gewebeschäden und Elektrodenkorrosion betont wird. Insgesamt bietet dieses Kapitel einen umfassenden Überblick über die Neuroelektrophysiologie, von historischen Entdeckungen bis hin zu modernen Modellen und Anwendungen, was sie zu einer unschätzbaren Ressource für Fachleute auf diesem Gebiet macht.
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Abstract
This chapter describes the fundamentals of neuro-electrophysiology. The electrical properties of neuronal membranes can be simulated by equivalent circuits composed of resistance and capacitance. Ion concentration differences across the membrane, combined with varying membrane permeability to ions, form the basis of resting and action potentials. The chapter introduces quantitative membrane indices (including time constant, space constant, rheobase and chronaxie), membrane cable equations, and Hodgkin-Huxley computational model. It also covers electrical stimulation approaches for brain neurons, including intracellular versus extracellular modes, current versus voltage modes, and unipolar versus bipolar modes. Finally, it addresses stimulation safety rules to prevent tissue damage and electrode corrosion.
The nervous system plays a crucial role in regulating and controlling the activities of humans and other organisms. Its basic structural and functional unit is neuron. Electrical activity generated by neuronal membranes is essential for nervous system functions. The early understanding of bioelectricity and subsequent developments in related theories and technologies have laid the foundations for neuro-electrophysiology.
1.1 Brief History of Bioelectricity
The discovery of bioelectricity in fishes can date back to ancient times (Wickens 2015). Egyptian murals depicted Nile catfish capable of producing powerful electrical currents. People used these fish-generated currents to punish prisoners and treat ailments like headaches. However, without understanding the true nature of electricity and bioelectricity, they simply attributed these mysterious phenomena to ghostly forces.
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In 1791, Luigi Galvani (1737–1798), an anatomist and physiologist at the University of Bologna in Italy, published a paper on muscle contractions caused by touching metal arches to frog leg muscles or nerves (Piccolino 1998). Galvani pointed out that the muscle contractions were caused by electrical energy released from muscle cells. This discovery marked a significant milestone in electrophysiology. His finding generated immediate excitement in the academic community and caught the attention of prominent Italian physicist Alessandro Volta (1745–1827). However, Volta offered a different explanation. He argued that the current causing muscle contractions came not from the muscles themselves, but from chemical reactions between the different metals in the arches and the liquid on the muscles. These reactions created a potential difference to generate current through the muscles. This disagreement sparked a famous debate. Both Galvani and Volta pursued further research in two distinct directions, and eventually became pioneers in electrophysiology and electrical engineering (with Volta inventing chemical batteries), respectively. Their achievements were closely intertwined with the advances in electrical physics during that period. For more details about this scientific debate, refer to Sect. 12.2 History in the textbook “Introduction to Biomedical Engineering” (Enderle and Bronzino 2012).
Since Galvani's discovery over 200 years ago, electrophysiology has made significant progress, driven by the advances in electrical science and the inventions of measurement technologies and instruments. The field development has been marked by several key milestones. When Galvani first discovered bioelectricity, no instruments existed for measuring it directly. In 1827, the invention of ammeters enabled scientists to measure potential differences between injured and intact muscle tissues. By 1849, German physiologist Emil du Bois Reymond became another pioneer in electrophysiology by successfully recording resting and action potentials in nerves.
It took over 100 years for scientists to understand the mechanism of bioelectricity after its discovery. In 1902, German scientist Julius Bernstein proposed the cell membrane theory of bioelectricity. He suggested that a potential difference exists across cell membrane, related to the membrane permeability to potassium ions. This theory led to significant advances in electrophysiology. In the 1920s, American scientists Herbert Spencer Gasser and Joseph Erlanger made a breakthrough by using a cathode ray tube to observe and study the potential conduction on nerve fibers, earning them the 1944 Nobel Prize. In the 1940s and 1950s, British scientists Alan Lloyd Hodgkin and Andrew Fielding Huxley studied the conductivity changes of potassium and sodium ion channels in cell membranes by using voltage clamp technology with glass microelectrodes. They created a computational model for these ion channels. Additionally, Australian scientist John Carew Eccles discovered excitatory and inhibitory synaptic potentials. In 1963, Eccles, Hodgkin, and Huxley jointly received the Nobel Prize in Physiology or Medicine. Another major advancement came in 1976 when German scientists Erwin Neher and Bert Sakmann invented the patch clamp technique—a crucial method for studying membrane ion channels—leading to their 1991 Nobel Prize. These milestones showcase how science and technology mutually enhance each other to continuously build theoretical foundations and technical supports in the electrophysiology field. Today, extensive researches have deepened our understanding of cell membrane composition, structure, and electrical properties.
1.2 Electrical Properties and Membrane Potentials in Neurons
1.2.1 Electrical Properties of Neuronal Membrane
The neuronal membrane consists mainly of a lipid bilayer containing membrane proteins. It exhibits electrical properties of both resistance and capacitance.
1.
MembraneResistance
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The electrical resistivity of membrane lipid bilayer is extremely high, ranging from 1013 to 1015 Ω·cm. In contrast, the resistivity of the fluids on either side of the membrane—the intracellular fluid (cytosol) and extracellular fluid—is only 60–80 Ω·cm. This means that the membrane resistivity is more than ten orders of magnitude higher than the surrounding fluid. Additionally, the membrane resistance is not fixed, but varies with multiple factors, including changes in transmembrane potential and in intracellular and extracellular environments. When neurons shift between resting and excited states, their membrane resistance can span several orders of magnitude.
Similar to the definition of resistance in physics, membrane resistance (Rm) is defined as
Its unit is Ω. In this formula, ρm is electrical resistivity—the resistance per unit length across unit cross-sectional area—measured in Ω·cm. l is length and s is cross-sectional area. When calculating membrane resistance (or transmembrane resistance) for current flowing between the inside and outside of a neuron, the l represents membrane thickness while s represents membrane area. For example, in a slender neuronal axon, the l represents the thickness of axonal membrane and the s represents the membrane area of the axonal surrounding cylindrical surface. Beyond membrane resistance, other neuronal components also have resistance, calculated using the same formula. When calculating the axoplasmic resistance as current flows along the neuronal axon, the l represents the axon length and the s represents its cross-sectional area. Note that the definitions of length and cross-sectional area change based on the current path when calculating resistance in different neuronal components.
The reciprocal of membrane resistance Rm is the membrane conductivity Gm (or gm), measured in Siemens (S). Membrane permeability to charged ions is typically expressed by the ionic conductivity.
2.
MembraneCapacitance
The lipid bilayer in neuronal membrane has very low conductivity, making the membrane act like an insulator—similar to the dielectric layer in a capacitor. The ion-rich intracellular and extracellular fluids on both sides of the lipid bilayer act like the parallel plates of a capacitor. This structure of conductive fluids separated by the thin insulating membrane creates the membrane capacitance.
The capacitance of a parallel-plate capacitor is directly proportional to both the dielectric constant of insulating layer and the plate area, while inversely proportional to the distance between the plates. Membrane capacitance follows the same principle and is typically represented as capacitance per unit area, known as specific capacitance (\(\hat{C}_{m}\)), which is defined as
Here εm is the dielectric constant of membrane, and l is the thickness of membrane lipid bilayer. The εm of neuronal membrane is remarkably great at approximately 3–5, even greater than that of polypropylene plastic film used in common capacitors, which is only 2.25. The thickness of neuronal membranes is only 5–10 nm, resulting in a great membrane capacitance (\(\hat{C}_{m}\)) of about 1 µF/cm2. This \(\hat{C}_{m}\) is stable, varying hardly between resting and excited membrane states. Additionally, the \(\hat{C}_{m}\) values are similar across different neuron types.
Changes in transmembrane potential can generate charging or discharging currents across membrane capacitance. This capacitive current can interfere with the measurements of ionic currents flowing through membrane channels. Using voltage clamp to fix the membrane potential and eliminate the capacitive current can allow accurate measurements of the ionic currents. By using this technique, Hodgkin and Huxley created the famous mathematical membrane model describing ion channel dynamics, as detailed in Sect. 1.3 (note that transmembrane potential is simplified as membrane potential).
Membrane capacitance prevents any sudden changes in membrane potential. When subjected to an electrical stimulation with a sudden change in potential or current (e.g., a step waveform stimulation), the membrane potential always undergoes a transient state before reaching a steady state. The duration of this transient state can be described by a time constant.
3.
Time Constant
Neuronal membrane at rest can be represented by an equivalent RC circuit with a resistor (Rm) and a capacitor (Cm) in parallel (Fig. 1.1A). As illustrated in Fig. 1.1B, after a glass pipette stimulation-electrode is inserted into the cell membrane, a square current is applied between this microelectrode and a grounded electrode outside the cell. When the duration of square current exceeds the charging time of the membrane capacitor, the step current changes in the rising and falling edges can cause the membrane potential to change exponentially over time toward a steady state (Fig. 1.1C), rather than a step potential change. This transient period can be characterized by a time constant.
Fig. 1.1
Equivalent circuit for quantitative analysis of membrane potential changes. A Equivalent RC circuit. B Schematic diagram showing membrane potential measurement during a square current injection into a cell. C Change in membrane potential ∆V(t) caused by step changes in membrane current Im
When a small-amplitude step current (Im) is applied across the membrane at time t = 0 (a subthreshold stimulus that cannot induce an action potential), the membrane potential begins to deviate from its resting value by ΔV(t). The applied Im divides into two components as it flows through the membrane: a capacitive current (IC) and a resistive current (IR) (Fig. 1.1A, C). According to the Kirchhoff's current law, we have
Here ΔV∞ = ImRm is the steady-state value of ΔV(t), representing the maximum change in membrane potential induced by the stimulus Im. The time constant τ equals RmCm. When t = τ, ΔV(t) = ΔV∞(1 − e−1) = 0.63ΔV∞. This means that after applying a step current, the change of membrane potential can reach 63% of its maximum value within one time constant. Conversely, by measuring how long it takes for the membrane potential to reach 63%ΔV∞, we can determine the membrane time constant τ. Obviously, the larger the membrane resistance and capacitance, the longer the time constant τ becomes, resulting in a longer transient period of membrane potential. In contrast, the membrane potential can approach its steady-state level more quickly. After a duration of 5τ, the membrane potential reaches over 99% of its steady-state value, indicating that the transient period approximately ends and the potential enters its steady-state. When the applied current is removed, the membrane potential undergoes a transient change in the opposite direction, immediately following the falling edge of the square current. The time constant τ remains the same.
Let Vm(t) represent the absolute value of membrane potential and Vrest be the resting potential, then
The direction of membrane potential change—whether depolarization or hyperpolarization—depends on the direction of transmembrane current (simplified as membrane current). When the current flows from inside to outside, it causes membrane depolarization (as shown in Fig. 1.1B), raising the negative resting potential. Conversely, current flowing from outside to inside results in hyperpolarization.
4.
Rheobase and Chronaxie
The RC circuit in Fig. 1.1A can only describe the membrane response to subthreshold stimulation when membrane resistance remains constant approximately. Once the change of membrane potential exceeds the activation threshold, an action potential (AP) is initiated through a series of ion channel opening and closing processes. During the processes, the membrane resistance changes significantly (refer to Sect. 1.3). Let's denote the AP threshold as Vth and the maximum potential change caused by the applied current Im as ΔV∞ = ImRm. Then, the minimum Im required to raise the membrane potential to the threshold Vth is known as rheobase Irh:
This current Irh must be maintained indefinitely (i.e. t = ∞) to raise the membrane potential to threshold Vth. When the capacitor Cm in the RC circuit is fully charged, its branch becomes open, and the applied current Irh flows entirely through the resistor Rm. TheRm correlates with cell size. For a spherical cell, its membrane resistance can be simplified as Rm = c/r2, where c is a constant and r is the cell radius. Based on Eq. (1.6), the rheobase Irh is directly proportional to the square of cell radius. Consequently, a larger cell requires a greater rheobase Irh to reach threshold potential due to its smaller resistance Rm. This principle extends to synaptic transmission—larger postsynaptic neurons require greater synaptic currents (integrated from more synaptic inputs) for activation. This is because current spreads across the entire membrane area. The current intensity per unit area, rather than total current, determines whether the membrane can be excited sufficiently to generate an AP. Therefore, a larger membrane area needs a proportionally larger current.
For a current applied as a brief pulse with width T rather than lasting infinitely long, based on Eq. (1.4), to reach the threshold Vth at the pulse end (i.e. t = T), the current intensity must be at least
As shown in Fig. 1.2, this equation describes how the threshold current Is,th changes with pulse width T, forming what is called the strength-duration curve, or I-T curve (Horch and Dhillon 2004). According to Eq. (1.7), when the current Is,th is twice the rheobase Irh (i.e., Is,th = 2Irh), the minimum pulse width (denoted as Tchr) required for the membrane potential to reach the threshold Vth is
Strength-duration curve of square current stimulation applied to a cell. Rheobase is defined as the minimum activation threshold for an infinite pulse duration. Chronaxie is the minimum pulse duration needed to activate a cell with a current twice the rheobase. The vertical axis (Is,th) is normalized to the rheobase Irh, while the horizontal axis (T) is normalized to the time constant τ. In short, let Irh = 1 and τ = 1
This Tchr, known as the chronaxie, is approximately 2/3 of the membrane time constant. For spherical cells, the membrane capacitance can be simplified as Cm = kr2, where k is a constant and r is the cell radius. The time constant τ = RmCm = (c/r2) (kr2) = ck. The membrane resistance Rm is inversely proportional to the membrane area (∝ 1/r2), while the membrane capacitance Cm is directly proportional to the membrane area (∝r2). Both the time constant τ and the chronaxie Tchr appear independent of cell size. However, for non-spherical neurons and other cells, τ and Tchr do show some size dependence, though their relations to cell size are much weaker than that of the rheobase Irh (Horch and Dhillon 2004). Additionally, c and k are not true constants—larger membrane areas typically have greater time constants. Thus, larger cells usually have greater τ values, while smaller cells have smaller ones.
Different neural tissues and neuronal structures (such as dendrites, cell bodies, and axons) have different membrane morphologies and features, resulting in distinct strength-duration curves and chronaxies. By using electrical pulses with an appropriate width based on various chronaxies, stimulations can selectively activate specific neural tissues or structures. For instance, the chronaxie of C-type axonal fibers in peripheral nerves exceeds 1 ms (Schneider et al. 2023), while neuronal axons in the central nervous system have chronaxies ranging only from 30 to 300 μs. Thicker myelinated axons have shorter chronaxies than thinner unmyelinated axons, making them more responsive to narrow pulses. Neuronal cell bodies (somata) can have a chronaxie as long as 1 ms (Ranck 1975; McIntyre and Grill 1999; Udupa and Chen 2015), and dendrites have a chronaxie about 10 times longer than axons (Stern et al. 2015). Therefore, narrow pulses (≤ 100 μs), commonly used in deep brain stimulation (DBS), can activate axons first due to their short chronaxies (Nowak and Bullier 1998; DiLorenzo and Bronzino 2008). Beyond neural tissues, other excitable tissues also have diverse chronaxies. For example, ordinary myocardial cells have a chronaxie of approximately 2 ms, while specialized myocardial cells with autonomy (His bundle) have a chronaxie of about 0.5 ms (Jastrzębski et al. 2019). Some smooth muscles have chronaxies reaching up to 10 ms. This explains why neural stimulations typically require pulse widths shorter than 2 ms, while muscle tissue stimulations may need pulses as long as 10 ms.
When the intensity of applied pulses is only slightly greater than the threshold, small changes in tissue excitability can prevent its activation. Therefore, in practical applications, stimulation intensity is typically set at least twice the threshold to ensure reliable effects.
5.
Space Constantand Cable Equation
When a cell membrane extends over a large spatial range, such as in a slender axon, the change of membrane potential at a stimulation site cannot maintain constant magnitude as it spreads throughout the entire membrane. Instead, the potential decreases exponentially with distance. This attenuation can be characterized by a space constant.
A slender cylindrical axon (unmyelinated) behaves like a cable (Fig. 1.3A), with its membrane serving as an insulating layer surrounding the axoplasm which has a much lower resistivity. When the tip of a glass pipette electrode is inserted into the axonal membrane and locally injects stimulation current into the axoplasm, part of the current can flow along the axoplasm bidirectionally toward both ends of the axon, known as axial current. Simultaneously, the current continuously penetrates the axonal membrane along its path, creating transmembrane radial current (indicated by brown arrows in Fig. 1.3A). Finally, the current converges at the extracellular electrode (e.g., a grounded electrode) to complete the current loop. As the current passes through the membrane at different sites along the axon, it alters the membrane potential there. The change magnitude of membrane potential varies with its distance from the stimulation site. The potential change is largest at the injection site and gradually decreases with distance, exhibiting an exponential decay (Fig. 1.3A, bottom curve). This pattern of potential distribution, formed by current diffusion within the axoplasm and across the membrane, is known as electrotonic potential.
Fig. 1.3
Creation of electrotonic potential along an axon and definition of space constant (A) and axonal equivalent circuit (B). For simplicity, the time variable t is omitted from the current and potential variables in (B)
Assume that a slender axon is evenly divided into identical small cylindrical segments with a length of Δx (Fig. 1.3B). The membrane of each segment is represented as an equivalent circuit with a resistor (Rm) and a capacitor (Cm) in parallel. The axoplasm resistance between two adjacent segments is Ri. The resistance of extracellular fluid is ignored due to its negligible value compared to the membrane and axoplasm resistances. Under these conditions, no potential gradient exists in the extracellular space, and the extracellular potential is set to 0. This axon model is known as core conductor model. From this model, the so called axon cable equation can be derived as below.
From the resistance formula Eq. (1.1), the axoplasm resistance Ri of cylindrical segment is
Here, ρi is the axoplasm resistivity, typically ranging from 50 to 250 Ω·cm (Koch 1999). Δx is the length of cylindrical segment, d is its diameter, and ri is the axoplasm resistance per unit length (measured in Ω/cm).
The membrane resistance Rm of cylindrical segment is
Here, ρm is the membrane resistivity, l is the membrane thickness, \(\hat{R}_{{\text{m}}}\) is the membrane resistance per unit area (in Ω·cm2), and rm is the membrane resistance per unit length (in Ω·cm).
The membrane capacitance Cm of cylindrical segment is
Here, εm is the membrane dielectric constant, \(\hat{C}_{{\text{m}}}\) is the membrane capacitance per unit area (in μF/cm2), and cm is the membrane capacitance per unit length (in μF/cm).
As shown in Fig. 1.3B, set the extracellular potential Ve = 0. x is the axial distance from the stimulation site. Vi(x, t) is the intracellular potential (i.e., membrane potential) and im(x, t) is the membrane current, at the position x and time t. Then, we have
It is the linear cable equation, which has analytical solutions in two special cases.
The first case occurs when the membrane potential along the axon reaches a steady state with no further changes over time. For example, as shown in Fig. 1.3A, when a step current is applied at x = 0 and maintained constant long enough for the membrane potential to reach steady state, the potential at each site along the axon remains unchanged with time. As a result, the second term on the right side of Eq. (1.17) becomes 0, yielding the following second-order ordinary differential equation
The λ is known as space constant, representing the distance (x = λ) at which the membrane potential decreases to e−1, or 37% of the original potential V0 at the stimulation site (x = 0). Equation (1.20) shows that the λ is directly proportional to the square root of the axon diameter d. For an axon with diameter d = 4 μm, membrane resistance per unit area \(\hat{R}_{{\text{m}}}\) = 20,000 Ω·cm2 and axoplasm resistivity ρi = 200 Ω·cm, its space constant is λ = 1 mm. When the axon diameter decreases to 1 μm with other parameters remaining the same, the λ decreases to 0.5 mm. In describing membrane potential propagation through space, spatial distances are often expressed as dimensionless multiples of λ.
The second case for the analytical solution of Eq. (1.17) occurs when there is no potential difference within the entire axon. For example, inserting a bare metal wire axially through an axon interior can create a spatial clamp, equalizing the potential throughout the axoplasm. In this case, the left term of Eq. (1.17) becomes 0, resulting the following first-order ordinary differential equation
It is consistent with the time constant of a spherical cell membrane shown in Fig. 1.1B. Time variables can be converted to dimensionless values by dividing them by τ, yielding multiples of τ. The x-axis in Fig. 1.2, labeled “pulse width (T)”, shows this dimensionless scale in τ units.
By substituting Eqs. (1.20) and (1.23) into Eq. (1.17), the linear cable equation becomes
where Im(X, T) = λτIm(x, t) is the applied current waveform with spatial and temporal coordinates normalized by λ and τ, respectively. Here, Im(x, t) is the actual current waveform in original spatial and temporal coordinates (Koch 1999).
The above describes the basic electrical properties of cell membranes in subthreshold situations without inducing any action potential. They are known as passive properties that can be simulated using an equivalent RC circuit with constant parameters. Among them, the time constant describes the speed of membrane potential changing over time, which is determined by membrane resistance and capacitance. The space constant describes the attenuation of membrane potential over spatial distance, mainly determined by the membrane resistance and intracellular plasma resistance. Although the cable equation can describe passive potential propagation along axons, axonal fibers are essentially different from electric cables that are made of metal conductors and insulation layers for power transmission and information communication. Beyond passive properties, axons also possess active properties created by ion channels distributed in their membranes. When the membrane is stimulated to reach its activation threshold and triggered an action potential, the resulting changes in ion channels and membrane potential must be described using a time-varying equivalent circuit (introduced in Sect. 1.3) with so called active electrical properties. Before discussing those properties, let's first review the resting potential and action potential, along with their generation mechanisms.
1.2.2 Resting Potential and Action Potential
1.
RestingPotential
When an excitable cell is at rest and unstimulated, its intracellular potential is typically tens of millivolts lower than its extracellular potential, resulting in a resting potential. Different species of cells have distinct resting potentials. For example, brain neurons have a resting potential of around − 70 mV, while visual sensory cells (such as rod and cone cells) in the retina have a resting potential of approximately − 40 mV.
The resting potential is formed by ion concentration differences between the inner and outer fluids. For instance, the intracellular potassium (K+) concentration (~ 100 mM) is much higher than its extracellular concentration (~ 5 mM), resulting in an approximate 20-fold difference. In contrast, the extracellular sodium (Na+) concentration (~ 150 mM) is higher than its intracellular concentration (~ 15 mM), with a tenfold difference. The distribution of chloride ions is similar to that of sodium ions (Bear et al. 2016). Additionally, various membrane channels have different permeabilities to ions. At rest, the permeability of K+ channels in the neuronal membrane is much higher than that of other ion channels. Thus, driven by the great K+ concentration gradient across the membrane, K+ ions diffuse outward, carrying positive charges out. This results in accumulations of positive charges outside and negative charges inside the membrane. These opposite charges attract each other across the membrane, establishing a potential difference across the membrane. The force on K+ ions from this potential difference opposes the force from the concentration gradient. When these opposing forces reach equilibrium, K+ inflow equals outflow, resulting in no net movement of K+ ions across the membrane. At this equilibrium state, the potential difference that exactly balances an ionic concentration gradient is known as the transmembrane equilibrium potential for the ion, or simply equilibrium potential, termed as Eion, e.g., EK for K+.
The Eion of a specific ion, determined by its concentration gradient across the membrane, can be calculated using the Nernst equation
Here R is the gas constant of 8.314 J/(mol·K), T is the absolute temperature which equals Celsius temperature plus 273, F is the Faraday constant of 9.6485 × 104 C/mol, Zion is the number of charges carried by the ion, [ion]o is the extracellular ionic concentration, and [ion]i is the intracellular ion concentration. The equilibrium potential Eion is measured in volts (V).
According to the Nernst equation, the equilibrium potential of K+ at 37 ℃ is EK = − 80 mV. The resting potential of neurons measured in actual experiments is slightly higher, around − 70 mV. This difference occurs because at rest, while K+ channels have high permeability, the membrane also has some permeability to Na+, chloride (Cl−) and other ions. The final equilibrium potential resulting from multiple ions permeating simultaneously can be calculated by the Goldman equation, also known as the GHK equation
Here PK, PNa and PCl are the permeability coefficients of K+, Na+ and Cl− ions, respectively. [K+]o, [Na+]o and [Cl−]o are the extracellular concentrations of these three ions, while [K+]i, [Na+]i and [Cl−]i are their intracellular concentrations.
At rest, the permeability ratio of K+, Na+ and Cl− is PK:PNa:PCl = 1:0.02:0.45. Therefore, the final equilibrium potential of the membrane at 37 ℃ is Eion = − 67 mV. The resting potential can vary with the permeability ratio of ion channels, with a typical resting potential around − 70 mV. If the PK:PNa ratio decreases, the resting potential becomes less negative; conversely, if the ratio increases, the resting potential becomes more negative, tending toward -80 mV—the K+ equilibrium potential (EK).
During an AP, the opening and closing of Na+, K+ and other ion channels change dynamically, causing corresponding changes in the permeability ratio of these ions. For instance, when the membrane potential reaches the AP peak, the opening of Na+ channels leads to PK:PNa:PCl = 1:20:0.45. At this moment, the equilibrium potential at 37℃ is approximately 50 mV, which is close to the Na+ equilibrium potential (ENa).
Either at rest or during an AP, ion channels allow K+ and Na+ to flow along concentration gradients across the membrane to some extent. Won't these concentration gradients eventually disappear? Actually not! Sodium–potassium pumps in the membrane maintain these ion concentration gradients thereby maintaining the resting potential. Sodium–potassium pumps are integrated proteins embedded in the membrane that transport K+ and Na+ ions across the membrane against their concentration gradients by consuming energy from adenosine triphosphate (ATP). Each pump exchanges three internal Na+ for two external K+. The sodium–potassium pump protein has a molecular weight of approximately 2.75 × 105 Daltons (1 g = 6 × 1023 Da) and a size of approximately 6 × 8 nm. Neuronal membrane contains roughly 100–200 pumps/µm2, which means that a single neuron can have up to one million pumps. At peak speed, each pump can transport approximately 200 Na+ and 130 K+ ions per second.
In summary, ion concentrations are different between the inside and outside of a neuron. At rest, the neuronal membrane is highly permeable to K+ through K+ channels, resulting in K+ efflux that creates an equilibrium potential across the membrane. And, energy-consuming sodium–potassium pumps maintain the ion concentration gradients. Together, these mechanisms form the resting potential. The K+ concentration gradient plays a crucial role in generating the resting potential, which approximates the EK. The resting potential serves as the foundation for cell activity and bioelectricity generation. Significant changes in resting potential can seriously disrupt normal physiological functions and potentially cause fatal damage. To maintain stable resting potential, the extracellular K+ concentration must remain constant. In the brain, astrocyte glial cells create an effective K+ buffering system. These cells can rapidly absorb excess K+ nearby, then transport, diffuse and release the K+ ions to distant areas through extensive astrocytic networks. Once local neural activity causes an abnormal increase in [K+]o, the astrocytic buffer system can quickly adjust K+ distribution to protect neurons (Bear et al. 2016).
2.
ActionPotential
When a neuron receives excitatory synaptic inputs or is activated directly by an external stimulus to depolarize its membrane up to the activation threshold, an AP can be triggered. The AP generation follows this process: the initial depolarization causes voltage-gated Na+ channels to open, resulting in Na+ influx driven by the great Na+ concentration gradient across the membrane. This Na+ influx further accelerates membrane depolarization and opens more Na+ channels. Through this positive feedback, the regenerative Na+ influx rapidly raises the intracellular potential from negative to positive until it reaches a peak. This forms the AP rising phase, where the inside positive potential is called overshoot. The Na+ channels remain open briefly before entering an inactive state, causing a rapid decrease in Na+ influx. In the meantime, the membrane depolarization activates K+ channels, leading to a K+ efflux to form the AP falling phase—the repolarization phase—until the membrane potential approaches the equilibrium potential of K+ (EK). The end of falling phase can become more negative than the resting potential, creating a so called undershoot or after-hyperpolarization (AHP). The AHP then gradually disappears as the membrane potential returns to its resting potential (Fig. 1.4). During the repolarization, Na+ channels recover from their inactive state to closed state. Thus, Na+ channels cycle through three states: closed, open and inactive. During the inactive state, although the channels appear “open” due to membrane depolarization, their altered protein structure prevent Na+ from flowing through. Once membrane repolarization occurs, the Na+ channels are relieved of their inactivation and return to closed state.
Fig. 1.4
Typical waveform of a neuronal action potential with its defined phases
Neuronal action potentials typically exhibit an “all-or-none” feature. A subthreshold stimulus is too weak to trigger an AP, but once the stimulus exceeds the activation threshold, both the waveform and amplitude of the triggered AP are independent of stimulus intensity. Through Na+ channel opening regeneration, the membrane automatically reaches peak potential before returning to resting potential. A sustained suprathreshold stimulus with a higher intensity can only increase the AP occurrence rate without significantly altering their waveforms. However, AP waveforms may change under certain special circumstances. Additionally, different neuron species have distinct AP waveforms, as do different parts of neuronal structures, such as the soma and axon (Bean 2007).
AP generation produces a refractory period consisting of two phases: an absolute refractory period followed by a relative one. The refractory period occurs because inactive Na+ channels prevent Na+ from flowing into the cell. During the absolute refractory period, no new AP can be triggered, regardless of stimulus strength. As the membrane gradually repolarizes, Na+ channels begin to recover. This recovery phase is called the relative refractory period, during which an AP can be triggered, but only by a stimulus significantly stronger than its normal threshold.
The above provides only a qualitative description of AP, while the classic Hodgkin-Huxley model (HH model), named after its creators, offers a quantitative description as introduced below.
1.3 Modeling Membrane Ion Channels (HH Model)
In the 1950s, British scientists Hodgkin and Huxley created a computational model to describe the electrical properties of K+ and Na+ channels in axon membranes, long before the physical and chemical structures of these ion channels were understood. Using a voltage clamp technique, they measured membrane currents flowing through ion channels at different membrane potentials. Their electrophysiological experiments used giant squid axons with a diameter about 0.5 mm (Hodgkin et al. 1952). Through studying conductance changes in ion channels during AP generation, they created the classic HH model. This groundbreaking work earned them the 1963 Nobel Prize in Physiology and Medicine. This section will present their experimental data and mathematical equations.
1.3.1 Voltage-Clamp Experiments on Squid Giant Axons
During an AP, the voltage-gated Na+ and K+ channels in the membrane rapidly change their states, causing changes in ion channel conductance. As Na+ and K+ ions flow through these channels, they result in a rapid change in membrane potential, which always generates a charging or discharging current through membrane capacitor. Consequently, the measured membrane current includes currents flowing through both ion channels and the membrane capacitor. Measuring pure ion channel currents is challenging under this situation.
Figure 1.5 illustrates an equivalent circuit describing the active electrical properties of membrane during AP period. Compared to the RC circuit that describes the passive electrical properties during subthreshold stimulation (Fig. 1.1A), this circuit adds two conductance variables for Na+ and K+ channels (gNa and gK), along with three ion equilibrium potentials (ENa, EK and EL). Now, the total current (Im) flowing through the membrane is
Simplified equivalent circuit for the active properties of squid giant axon membrane. Variables gNa, gK and gL represent the conductance of Na+, K+ and other ion channels (the leakage conductance), respectively. ENa, EK and EL represent the equilibrium potentials of Na+, K+ and other ions, respectively
Bild vergrößern
Here Vm is the membrane potential, INa, IK and IL are the currents through Na+, K+ and other ion channels, respectively, and Cm is the membrane capacitance.
Using a voltage-clamp technique, the membrane potential can be clamped at a preset level and kept constant for a sufficient duration, resulting in \(\frac{{{\text{d}}V_{{\text{m}}} }}{{{\text{d}}t}} = 0\). Under this condition, \(I_{{\text{m}}} = I_{{{\text{Na}}}} + I_{{\text{K}}} + I_{{\text{L}}}\), the pure current through ion channels can be measured after eliminating the capacitive current \(C_{{\text{m}}} \frac{{{\text{d}}V_{{\text{m}}} }}{{{\text{d}}t}}\). Since ion channels are purely resistive, we have \(I_{{{\text{Na}}}} = g_{{{\text{Na}}}} (V_{{\text{m}}} - E_{{{\text{Na}}}})\), \(I_{{\text{K}}} = g_{{\text{K}}} (V_{{\text{m}}} - E_{{\text{K}}})\) and \(I_{{\text{L}}} = g_{{\text{L}}} (V_{{\text{m}}} - E_{{\text{L}}})\). By clamping the membrane potential (Vm) at different depolarization levels respectively and measuring the correspondingINa and IK, the conductances of Na+ and K+ channels (gNa and gK) can be determined. These conductances are functions of time (t) and membrane potential (Vm). The other parameters in the equivalent circuit (Fig. 1.5), including the ionic equilibrium potentials ENa, EK and EL, the membrane capacitance Cm, and the leakage conductance gL, can be treated as constants. After obtaining the experiment gNa and gK data at different t and Vm values, mathematical equations can be developed for simulating membrane dynamics. To collect the necessary data, Hodgkin and Huxley conducted experiments on squid giant axons.
Figure 1.6 illustrates their voltage-clamp experiment setup, which included both intracellular space-clamp and transmembrane-clamp. The space-clamp was achieved by inserting a thin bare metal wire axially inside the axon to keep an isoelectric potential along its entire length. The transmembrane-clamp was achieved using a feedback circuit formed as following: the tip of a glass microelectrode was inserted inside the axon to measure the membrane potential with another electrode placed outside. The measured potential was amplified by a high-gain operational amplifier to obtain the membrane potential Vm. Then, through a feedback amplifier, Vm was compared with a preset clamping voltage—a step potential from a signal generator. WhenVm differed from the clamping voltage, the feedback amplifier injected a positive or negative current into the axon to drive Vm toward the clamping voltage and maintained it. The output current of feedback amplifier was measured as the membrane current Im. In this way, under the situation of both space-clamp and transmembrane-clamp, the potential within the entire axon was clamped at different preset voltages one by one. When the membrane potential remained at constant for a sufficient duration, the current through the membrane capacitor became zero (IC = 0). Thus, the measured Im represented the total current flowing through the ion channels (Eq. (1.29) and Fig. 1.5).
Fig. 1.6
Schematic diagram of the voltage-clamp experiment on a squid giant axon
Define the direction of membrane current from inside to outside as positive. As shown in Fig. 1.7, when the applied clamping voltage was small and caused only a subthreshold reaction on the membrane, an initial IC appeared before the membrane current Im settled at a constant low level—the leakage current IL (Fig. 1.7A). When the applied clamping voltage was large enough to cause a super-threshold depolarization, Im changed with an initial IC followed by a rapid inward Na+ current (INa) and then reversed to an outward K+ current (IK) (Fig. 1.7B). Today, channel blockers, such as tetraethylamine (TEA) for K+ channels and tetrodotoxin (TTX) for Na+ channels, can readily separate the two ion currents, IK andINa. However, in the 1950s, the nature of ion channels was unknown, let along channel blockers. Hodgkin and Huxley devised an elegant method to remove Na+ from the external solution, thereby eliminating INa from the measured Im. ThisIm then contained only IK and IL. By subtracting this Im from the Im measured with normal external solution containing Na+, they estimated INa (Hodgkin and Huxley 1952a). With IL obtained from subthreshold measurement (Fig. 1.7A), they then determined IK.
Fig. 1.7
Membrane currents measured during voltage-clamp at subthreshold (A) and super-threshold (B) depolarization levels
Using the determined IK data and clamping voltage Vm values, the experimental result of K+ channel conductance (gK) can be determined by the formula \(g_{{\text{K}}} = \frac{{I_{{\text{K}}} }}{{V_{{\text{m}}} - E_{{\text{K}}} }}\). Figure 1.8 shows the gK data at three different clamping voltages—depolarization potentials (denoted on the top). The depolarization potential was the difference between the clamped Vm and the resting potential, denoted by V. When the Vm was clamped at a specific depolarization level, a series of gK measurements was determined over time (t), shown by small circles in the plots. The original paper by the two scientists included more depolarization levels than shown in Fig. 1.8. The smooth curves were simulation data calculated by the HH model, closely matching the experimental data.
Fig. 1.8
Changes in K+ channel conductance (gK) over time (t) at three different levels of membrane depolarization. The small circles represent experimental data, while the smooth curves are the simulation results of HH model
Here \({\overline{\text{g}}}_{{\text{K}}}\) is a constant representing the maximum value of K+ conductance (gK). n represents the probability of K+ channel opening (0 ≤ n ≤ 1), also known as K+ channel gating variable. The variables \(\alpha_{n}\) and \(\beta_{n}\) are rate constants (measured in s−1) describing the ion channel transitions between open and closed states at different membrane potentials:
Thus, Eq. (1.31) is the first-order kinetic equation describing how n varies with both time and membrane potential. At rest (V = 0), n has a resting value of \(n_{0} = \frac{{\alpha_{{n_{0} }} }}{{\alpha_{{n_{0} }} + \beta_{{n_{0} }} }}\). With the boundary condition of \(n = n_{0}\) att = 0, the solution of Eq. (1.31) is
where \(g_{{{\text{K}}0}}\) is the initial conductance value at t = 0. \(g_{{{\text{K}}\infty }}\) is the steady-state conductance value that finally stabilizes after sufficient time. The values of \(g_{{{\text{K}}0}}\), \(g_{{{\text{K}}\infty }}\) and \({\overline{\text{g}}}_{{\text{K}}}\) can be read from the experimental data at different depolarization levels (Fig. 1.8). Choose an appropriate value for \(\tau_{n}\) to optimize the fit between the gK curve calculated by Eq. (1.36) and the experimental data. Then, use the following three formulas to calculate series of \(n_{\infty }\), \(\alpha_{n}\) and \(\beta_{n}\) values at different depolarization levels (shown by the small circles in Fig. 1.9):
Parameters of K+ channel model versus membrane potential, including the steady-state value of gating variable \(n_{\infty }\) (A), time constant \(\tau_{n}\) (B), and rate constants \(\alpha_{n}\) (C) and \(\beta_{n}\) (D). In each plot, the abscissa V is the depolarization potential relative to resting potential. A positive V value indicates depolarization, while a negativeV value indicates hyperpolarization. The small circles are the values determined from experiment measurements, while the smooth curves are the simulation values from the HH model
The smooth curves in Fig. 1.9 were calculated using Eqs. (1.33), (1.34), (1.40) and (1.41). They show how the variables \(n_{\infty }\), \(\tau_{n}\), \(\alpha_{n}\) and \(\beta_{n}\) in the conductance model of K+ channels vary with the depolarization potential V.
One may wonder why the variable n was raised to the fourth power in the conductance equation \(g_{{\text{K}}} = {\overline{\text{g}}}_{{\text{K}}} n^{4}\). Actually, Hodgkin and Huxley tried different powers in their fitting process. They found that although higher powers yielded better fits between simulation results and experimental data, the fourth power provided sufficient accuracy. Using higher powers would have only added unnecessary computational complexity.
Totally, the K+ channel model consists of the following four equations
Similarly, the Na+ channel model can be determined. First, Na+ currents (\(I_{{{\text{Na}}}}\)) were measured at different depolarization potentials during voltage-clamp experiments. Then, using the applied clamping voltages and the measured \(I_{{{\text{Na}}}}\) data, the experimental data of Na+ conductance \(\left( {g_{{{\text{Na}}}} = \frac{{I_{{{\text{Na}}}} }}{{V_{{\text{m}}} - {\text{E}}_{{{\text{Na}}}} }}} \right)\) were determined, as shown by the small circles in Fig. 1.10.
Fig. 1.10
Changes in Na+ channel conductance (gNa) over time at three different levels of membrane depolarization. The small circles represent experimental data, while the smooth curves are the simulation results of HH model
Unlike K+ channels which have only two states (open and closed), Na+ channels have an additional inactive state. Although voltage-gated Na+ channels can be activated by a superthreshold depolarization, they can remain open briefly before automatically entering an inactive state. The inactive Na+ channels can return to their closed state—becoming ready for reactivation—only after membrane repolarization occurs. To describe how Na+ channel conductance changes during activation and inactivation, two gating variables, m and h, were used in the conductance equation
Here \({\overline{\text{g}}}_{{{\text{Na}}}}\) is a constant representing the maximum value of Na+ conductance (gNa). The two pairs of rate constants α and β are functions of membrane potential, since the Na+ channel is voltage-gated.
The two scientists proposed Eq. (1.42) based on the following hypothesis: Na+ channel activation is determined by three independent activation particles (M) and one inactivation particle (H). A Na+ channel opens only when all three M particles and one H particle locate in specific positions on the membrane. The probability of the channel opening is then represented by \(m^{3} h\), where the variable m represents the probability of an M particle being in the open position (0 ≤ m ≤ 1), and the variable h represents the probability of an H particle being in the open position (0 ≤ h ≤ 1). Thus, \(1 - m\) and \(1 - h\) represent the probabilities of the two particles being in the closed position, respectively. Similar to the K+ channel model, variables \(\alpha_{m}\), \(\beta_{m}\) and \(\alpha_{h}\), \(\beta_{h}\) represent the rate constants describing the particle transitions between open and closed states:
Equations (1.43) and (1.44) describe the first-order kinetics of \(m\) and \(h\) over time (t) and membrane potential (V). With the boundary conditions \(m = m_{0}\) and \(h = h_{0}\) at t = 0, the solutions of these equations are
Here we need to determine the functions of \(\alpha_{m}\), \(\beta_{m}\) and \(\alpha_{h}\), \(\beta_{h}\) varying with the membrane potential. Since the Na+ channel conductance at rest is very small, the \(m_{0}\) becomes negligible compared to the \(m_{\infty }\) under large depolarization. In addition, the inactivation of Na+ channel is approximately complete. Thus, \(h_{\infty }\) is negligible compared to \(h_{0}\). Under these conditions, substituting Eqs. (1.45) and (1.46) into Eq. (1.42) yields
where \(g^{\prime}_{{{\text{Na}}}} = {\overline{\text{g}}}_{{{\text{Na}}}} m_{\infty }^{3} h_{0}\). Choose appropriate values for \(\tau_{m}\) and \(\tau_{h}\) to reach the \(g_{{{\text{Na}}}}\) curve that best fits the experimental data at different depolarization levels. Then, use the following equations to calculate series of \(\alpha\) and \(\beta\) values at different depolarization potentials
Finally, using these calculated \(\alpha\) and \(\beta\) values, the following equations for the four rate constants can be determined by mathematical fitting:
Here V is the displacement of membrane potential from its resting potential. VNa, VK and VL are the displacements in equilibrium potentials of Na+, K+ and other ions from the resting potential. All these potentials are measured in mV. The membrane current Im is measured in µA/cm2, conductance in mS/cm2, capacitance in µF/cm2, time in ms, and all rate constants \(\alpha\) and \(\beta\) in ms−1. In addition, the \(\alpha\) and \(\beta\) values are temperature-dependent. The equations above were derived at 6.3℃. At a temperatures of T ℃, the \(\alpha\) and \(\beta\) values should be adjusted by a multiplication factor \(\varphi = 3^{{\frac{T - 6.3}{{10}}}}\) (Hodgkin and Huxley 1952a).
1.3.3 Example Simulations of the HH Model
The HH model has no analytical solution but numerical ones. Using a computer program, we can solve the model equations and perform simple simulations. For example, using the MATLAB program provided in the appendix at the end of this book, we can calculate the curves of the three pairs of \(\alpha\) and \(\beta\) over V values (Fig. 1.11A, B), and time constants and steady-state values of the gating variables (m, h and n) for Na+ and K+ channels against V values (Fig. 1.11C, D). The abscissa (V) in these plots is the difference between the membrane potential and its resting potential, where positive values represent depolarization and negative values represent hyperpolarization.
Among the time constants of three gating variables (Fig. 1.11C), τm of Na+ channel activation is the smallest, much smaller than both τh of Na+ channel inactivation and τn of K+ channel activation. This indicates that Na+ channels open very rapidly. Additionally, as membrane depolarization increases, both activation variables m∞ and n∞ increase monotonically (Fig. 1.11D). However, m∞ rises more rapidly than n∞, indicating that Na+ channels can be activated by smaller depolarization displacements. Nevertheless, with increased membrane depolarization, the inactivation variable h∞ of Na+ channels decreases, thereby reducing Na+ channel opening. Under excessive depolarization, complete inactivation occurs with a very smallh∞, resulting in a depolarization block that prevents action potential generation. For example, applying a prolonged positive step current into a neuron can cause sustained depolarization leading to this block. On the other hand, when the membrane is excessively hyperpolarized, the activation variable m∞ can decrease to a very small value (Fig. 1.11D), causing a hyperpolarization block. This block occurs through deep closure of Na+ channels, distinct from depolarization block caused by Na+ channel inactivation (Bhunia et al. 2015).
Fig. 1.11
Simulated curves of rate constants (A and B), time constants (C) and steady-state values (D) for Na+ and K+ channel gating variables in the HH model. The four plots are the outputs of the MATLAB script “constants_variables.m” provided in the appendix. The βm curve is truncated in (B) to better display the βn and βh curves. Gaps in the curves occur where corresponding equations have zero denominators. The abscissa V represents the depolarization potential relative to resting potential
Assume no axial current within the axon, as shown in the space-clamp experiment in Fig. 1.6. Under this condition, the net current across the membrane remains zero unless an external current is applied. In other words, the Im in Eq. (1.55) represents an applied stimulation current. Using the MATLAB program in the appendix, we can simulate neuronal responses to step (or square) current stimulations (Im) with various intensities and directions. The Im is represented by the variable Iapp in the program. As shown in Fig. 1.12, with t = 0 marking the start of stimulation, the changes in membrane potential are shown in the left column, while the changes in gating variables n, m and h of Na+ and K+ channels are shown in the right column. These changes induced by various Im stimulations can be explained based on the properties of both ion channels shown in Fig. 1.11.
Fig. 1.12
Changes in the membrane potential (left column) and gating variables (right column) induced by different current stimulations—positive step currents with different intensities starting at time zero (A–E), a negative wide pulse (F) and a narrow biphasic pulse (G). These curves were generated by the MATLAB script “HH_simulation.m” provided in the appendix. You can produce different simulation results by adjusting the waveform and amplitude of the applied current Im (represented by the variable Iapp in the script). The Iapp settings are detailed in the script comments
The resting potential is set to −70 mV. Figure 1.12A–E show the simulation results with a positive step currentIm (i.e., an outward current) starting at t = 0, with an intensity ranging from 2 to 150 μA/cm2. At a subthreshold small current of Im = 2 µA/cm2 (Fig. 1.12A), only small membrane potential fluctuations between − 70 and − 65 mV are induced in the initial period. Then, the membrane potential stabilizes at − 68.5 mV, maintaining a depolarization of about 1.5 mV without triggering an action potential (AP). The changes in the gating variables of ion channels are also very small. The activation variable m of Na+ channels increases slightly from 0.053 to 0.063, while its inactivation variable h decreases from 0.60 to 0.54. The activation variable n of K+ channels increases slightly from 0.32 to 0.34. These changes follow the curves shown in Fig. 1.11D, slightly increasing the conductance of both types of channels. However, the inward Na+ flow cannot exceed the outward K+ flow. The two opposite ionic flows reach an equilibrium at the new membrane potential of −68.5 mV, slightly above the original resting potential of −70 mV.
Increasing the step current to Im = 5.5 µA/cm2 initially induces an AP (Fig. 1.12B). Afterwards, the applied current generates only small sustained depolarization at about − 66 mV. This depolarization reduces the inactivation variable h of Na+ channels. Although conductance increases in both types of channels, the Na+ flow still cannot overcome the K+ flow, preventing continuous AP generation. When the applied current increases further to Im = 6.5 µA/cm2 (Fig. 1.12C), the activation variable m of Na+ channels periodically reaches its peak (~ 1.0), causing a sharp increase in Na+ conductance. The Na+ flow exceeds the K+ flow, inducing periodic APs. Increasing the applied current to Im = 20 µA/cm2 enhances the AP occurrence rate (Fig. 1.12D). However, when the current is set to Im = 150 µA/cm2 (Fig. 1.12E), APs cannot occur continuously, because the membrane cannot repolarize sufficiently to allow inactive Na+ channels to recover after the first AP. The inactivation variable h remains too low to generate new AP. Eventually, the membrane potential only oscillates slightly around − 48 mV, resulting in a sustained depolarization of ~ 22 mV. This depolarization keeps the h at a low level while enhancing the activation variables m and n significantly higher than their resting values (see Fig. 1.11D).
When the applied current changes direction from outward to inward, the stimulation can hyperpolarize membrane. Although hyperpolarization typically cannot induce an AP directly, it may trigger one when the stimulation is withdrawn, provided the current has sufficient intensity and duration (Fig. 1.12F). This occurs because Na+ channels at rest are partially inactivated (e.g., h0 = 0.60 < 1). Hyperpolarization decreases the activation variable m, but simultaneously increases the inactivation variable h (Fig. 1.12F, right), thereby reducing inactivation. Once the hyperpolarization ends and membrane potential rises, them increase is much faster than the h decrease due to their different time constants (see Fig. 1.11C). Thus, the increased m combined with the still-high h leads to a rapid increase in Na+ channel conductance (∝ m3h). Meanwhile, the recovery of K+ channel activation (n), which has decreased during hyperpolarization, is slow, resulting in a slower increase in K+ channel conductance (∝ n4) than that of Na+ channels. As a result, the increased Na+ inflow generates an AP. This phenomenon is known as “anodic break excitation” or “rebound excitation” (Durand 2000). The AP latency in this case is long, as shown in the left of Fig. 1.12F. The latency between the AP peak and the end of negative current stimulation (20 ms width) is about 5.0 ms. In comparison, with positive stimulations (Fig. 1.12B–E), the latencies between the first AP peak and the current start point (t = 0) are only 3.0, 2.7, 1.5 and 0.6 ms. The greater the positive current, the shorter the latency becomes.
When a biphasic current pulse is applied with a first positive phase followed by a second negative phase (Fig. 1.12G), the negative phase cannot counteract the effect of positive phase to prevent AP generation. When the positive phase has sufficient intensity and duration, an AP can still generate even if the negative phase occurs before its full generation. In the example shown in Fig. 1.12G, the negative phase spans from t = 0.75 to 1.5 ms, occurring earlier than the AP peak at 2.1 ms. However, the threshold current intensity required for a biphasic pulse is higher than that of a single-phase positive pulse.
These simulation results show that membrane depolarization can generate APs. However, depolarization is not always beneficial for AP generation. In some cases, it can actually prevent APs. When membrane maintains a certain level of depolarization, even at a subthreshold level, the depolarization can increase the threshold of AP initiation due to increased Na+ channel inactivation (see Fig. 1.11D). Continuous excessive depolarization can prevent AP generations (see Fig. 1.12E). In contrast, membrane hyperpolarization can reduce Na+ channel inactivation and lower the threshold of AP initiation. Once this established hyperpolarization is removed, an AP can occur as the membrane potential rises toward its resting potential (Fig. 1.12F).
Note that the description above refers to intracellular stimulation, where the stimulation current flowing from inside to outside is defined as positive (Fig. 1.1B). Under this definition, a positive current depolarizes the cell membrane, while a negative current hyperpolarizes it. This differs from extracellular stimulation, where a negative pulse depolarizes the cell membrane. Please refer to Sect. 1.4.1 for detail differences between intracellular and extracellular stimulations.
The simulation results shown in Fig. 1.12 were obtained by solving the HH model equations using the MATLAB program in the appendix. The parameter settings were: Cm = 1 μF/cm2, \(\overline{g}_{{\text{K}}}\) = 36 mS/cm2, \(\overline{g}_{{{\text{Na}}}}\) = 120 mS/cm2, \(\overline{g}_{{\text{L}}}\) = 0.3 mS/cm2, VK = − 12 mV, VNa = 115 mV, and VL = 10.6 mV. With a resting potential of − 70 mV, all potentials (V, VK, VNa and VL) represented the differences of corresponding membrane potentials from the resting potential. The membrane potentials displayed in the left column of Fig. 1.12 were calculated as (V − 70 mV). The initial gating variables (resting values) were n0 = 0.3177, m0 = 0.0529 and h0 = 0.5961.
The HH model enables simulation and exploration of neuronal responses to various external stimulations. While the nonlinear characteristics of ion channels in neuronal membranes make it difficult to intuitively predict these responses, computational simulations can demonstrate complex responses and reveal their underlying mechanisms. In addition to the HH model, other computational models have emerged to simulate the behaviors of various neurons and neural networks. Some of them simplify calculations to address the high computation demands of the HH model, particularly for complex neural networks containing many neurons. However, the HH model remains a classic and irreplaceable foundation for other models. Nevertheless, the HH model was originally based on squid giant axons. Different neuron types possess distinct ion channels in their membranes, including various subtypes of Na+ and K+ channels with different properties. Additional ion channels, such as calcium (Ca2+) channels, may also exist on membranes. Therefore, when applying the model, parameters and equations must be adjusted according to specific simulation objects to achieve results that better align with actual physiological situations.
The original literature on the HH model consists of five consecutive papers published in the Journal of Physiology in 1952 (Hodgkin et al. 1952; Hodgkin and Huxley 1952a, b, c, d). This journal has since published commemorative articles every decade (Schwiening 2012; Catterall 2012; Catacuzzeno and Franciolini 2022). Many books and publications cover the HH model introduction and applications, including teaching materials to help beginners (Hopper et al. 2022).
1.4 Electrical Stimulation of Neurons
Neurons can respond to various external stimuli to generate APs or to alter their firing activity. While other stimulations—such as heat, mechanical force and magnetic field—can activate neurons by depolarizing their membranes to reach the AP threshold, electrical stimulation is the most common one. It is widely used in both research and therapeutic applications. In electrophysiological research, for example, it can help revel AP mechanisms, investigate synaptic excitations and inhibitions, measure nerve fiber conduction velocity, as well as explore pathways, structures, and functions of nervous systems. In clinical applications, electrical stimulation has been used in diagnosing neuromuscular junction disorders, treating sensory deficits and motor paralysis, controlling pain and epileptic seizures, and modulating brain disorders. Both intracellular (Fig. 1.1B) and extracellular electrical stimulation methods are used for research purposes. However, clinical applications requiring simultaneous activation of many neurons must use extracellular stimulation. It was also the method used in the in-vivo rat experiments described in Part II of this book. Additionally, electrical stimulation can be delivered through various forms, including voltage or current waveforms and monopolar or bipolar modes. Stimulation safety is a crucial prerequisite for both research and clinical applications. These aspects are discussed in the following sections.
1.4.1 Intracellular and Extracellular Stimulations
To activate neurons, external electrical stimulation can be applied either intracellularly or extracellularly. In the squid axon experiment shown in Fig. 1.6 and the model simulations in Fig. 1.12, intracellular stimulations were used. By delivering an outward positive current through a stimulation electrode (anode) placed inside a cell, we can depolarize the cell membrane to generate APs. Additionally, through rebound excitation, an inward negative current can also induce an AP (Fig. 1.12F). However, negative current is rarely used for intracellular stimulation because its activation efficiency is much lower, requiring longer duration. A key limitation of intracellular stimulation is that each electrode can only act on one cell at a time, making it challenging to activate multiple neurons simultaneously. Moreover, maintaining long-term stimulation of even a single cell is difficult. Current technology restricts intracellular manipulations to microelectrodes made of thin glass pipettes, which cannot maintain stability for long periods. This limitation is particularly pronounced in in-vivo neural electrical stimulation. As a result, intracellular neuronal stimulation is confined to scientific research rather than clinical applications. This technique is used primarily in isolated neural tissues (such as brain slices) or cultured neurons, and less frequently in live animal experiments.
In contrast, extracellular electrical stimulation can activate many neurons around the electrode simultaneously. Its clinical applications include deep brain stimulation (DBS), vagus nerve stimulation (VNS), spinal cord stimulation (SCS), cochlear implants (CI), and visual prostheses, among many others. Unlike intracellular stimulation, extracellular stimulation achieves higher activation efficiency when its working electrode functions as a cathode delivering negative current. Figure 1.13A illustrates how an axon is activated extracellularly by the exposed tip of a wire electrode. The tip, also called a contact, acts as a point source to deliver stimulation current. When a negative pulse is applied, it produces an outward current on the axonal membrane directly beneath the point source, causing membrane depolarization. (Let's call this membrane area the central site.) If the pulse is strong enough to make the depolarization reach the activation threshold, an AP occurs at the central site. Simultaneously, this pulse produces inward current on the membranes on both flank sides as the current completes its electrical loop. These inward currents hyperpolarize the flank membranes. The AP initiated at the central site must overcome the flank hyperpolarization before propagating along the axon bilaterally. The propagation toward the axon terminals is called orthodromic propagation, aligning with the normal physiological direction, while the propagation toward the soma is called antidromic.
Fig. 1.13
Schematic diagrams illustrating axonal responses to extracellular pulse stimulations. A Depolarization and hyperpolarization sites on the axonal membrane under a negative pulse stimulation. B Three distinct response zones (within axon fibers) at different distances from the stimulation point source, shown in the cross-sectional view through the point source. C and D Corresponding diagrams for a positive pulse stimulation
The magnitude of membrane depolarization and hyperpolarization produced by a negative pulse depends on the distance between the stimulation point source and the membranes—the shorter this distance, the greater the change in membrane potential. As shown in the cross-sectional view of axon fibers in Fig. 1.13B, where a point source is surrounded by axons perpendicular to the figure plane, three distinct zones emerge. In “Zone I”, closest to the point source, a sufficiently strong negative pulse can produce substantial hyperpolarization on the flank membranes, preventing the outward propagation of centrally initiated AP (Durand 2000; van de Steene et al. 2020). As distance increases, in "Zone II” (the activatable zone), the flank hyperpolarization weakens, allowing APs to propagate outward successfully. In “Zone III”, the most distant zone, the depolarization produced by the negative pulse cannot reach the activation threshold, thus no APs generate. Consequently, for axons with similar characteristics, effective activation that can propagate outward occurs only on axons with an appropriate distance from the stimulation site. Axons too close to or too far from the stimulation site may not generate effective activation. Even at the same distance, different types of axons may respond differently to stimulation.
With sufficient intensity, a positive pulse can also activate axons extracellularly. Figure 1.13C shows that while a positive pulse generates hyperpolarization at the axonal membrane directly beneath the point source, depolarizations occur on both flank sides under the action of current loop. The depolarizations can produce APs that propagate outward. Since there is no hyperpolarization barrier in their propagation pathway, axons in the zone close to the point source can be activated, while axons outside this zone cannot, due to subthreshold depolarization at a distance (Fig. 1.13D). With a same pulse intensity, the activatable zone of a positive pulse is much smaller than that of a negative pulse. Therefore, in extracellular stimulation, negative pulses are typically much more efficient than positive pulses. However, under special circumstances, such as during sustained high-frequency pulse stimulation, the axonal membrane can change, altering the activation efficiency of positive and negative pulses. For more details, please refer to Chap. 7 in this book.
For extracellular pulse stimulation, it is crucial to select an appropriate pulse width based on the chronaxie of target cell membrane (refer to Fig. 1.2). For example, narrow pulses with about 100 μs width can effectively activate axonal fibers of brain neurons, especially myelinated ones, but cannot activate smooth muscle cells with a chronaxie of several milliseconds (Horch and Dhillon 2004). Different neuronal structures—axons, dendrites, and samata (cell bodies)—have distinct membrane properties. Among them, axonal membranes have the shortest chronaxie and are most readily activated by narrow pulses (Ranck 1975; Nowak and Bullier 1998; Buzsáki 2006; Brocker and Grill 2013). Dendrites contain numerous synapses that receive excitatory and inhibitory inputs to generate postsynaptic potentials. Their membranes generally have few voltage-gated Na+ channel capable of generating APs. Therefore, dendrite potentials spread primarily as passive electrotonic potential, decaying with distance (see Fig. 1.3A). In contrast, soma and axon membranes are rich in voltage-gated Na+ channels. The regenerative response of Na+ channels can generate an “all-or-none” AP that propagates along the axon without attenuation. Due to its shorter chronaxie, an axon is more sensitive to narrow pulses commonly used in electrical neuromodulations than a soma. Therefore, narrow pulse stimulation in the brain can typically first induce APs in axons, which then propagate bilaterally outward (see Fig. 1.13). In myelinated axons, the multi-layer myelin sheath increases resistance and reduces capacitance. Their space constant is typically twice the internode length between Ranvier's nodes. These allow APs to jump between bare nodes, significantly accelerating AP propagation speed.
Extracellular stimulation of peripheral nerves has long shown that thicker axons require lower current intensity for activation than thinner ones (Gilbert et al. 2023). This occurs because thick axons have lower axial resistance and larger axial current proportional to the square of their diameter. Their larger diameters also result in longer space constants. In myelinated axon fibers, thicker axons feature longer internode lengths and space constants, which enhance excitation conduction. Through graded stimulation intensities, axons of different thicknesses can be selectively activated—lower intensities activate thicker axons, while higher intensities gradually recruit thinner ones. Axon activation also depends on other factors, such as the distance from stimulation site, as well as the relationship between axon direction and stimulating current flow.
The in-vivo rat experiments described in this book employed extracellular electrical stimulation. Unless noted otherwise, the following contents focuse specifically on extracellular stimulation.
1.4.2 Voltage and Current Stimulations
In extracellular stimulation, both voltage and current waveforms can be used. During pulse stimulation, voltage waveforms maintain constant voltage while current waveforms maintain constant current. The efficiency of electrical stimulation on excitable tissues mainly depends on the current flowing through the target tissue. For brain tissue specifically, neuronal responses to stimuli depend on the current flowing through the neuronal membrane—the amount of injected charges per unit time (Montgomery 2014).
When applying electrical stimulation, the stimulation current flows through several parts in the circuit loop: instruments (including a stimulator) that generate stimulation waveforms, electrodes that deliver stimulation, interfaces between electrodes and biological tissues, and the stimulated object—the biological tissues. In voltage stimulation, despite constant output voltage from the stimulator, impedance changes in any parts of the circuit can alter the current flowing through the stimulated object (such as neurons), thereby affecting the stimulation efficiency. These impedance changes can arise from electrode polarization, inflammation and glial proliferation around electrodes, and other factors during sustained stimulations (Lempka et al. 2010; Cheung et al. 2013). In current stimulation, however, the stimulator automatically adjusts its output voltage to compensate for impedance changes. This adjustment maintains the output current at the preset level, ensuring the stimulated neurons receive a steady current. However, if abnormal changes occur in the stimulation circuit, the stimulator may fail to maintain its preset constant current.
Nevertheless, voltage stimulation is commonly used when the impedance of stimulation circuit remains stable. To monitor impedance changes during stimulation, a resistor (like 10 kΩ) can be connected in series with the stimulation electrode. Measuring the voltage across the resistor allows detection of current and impedance fluctuations in the circuit (Kim et al. 2012).
In animal experiments, current stimulation is typically used to avoid the influence of circuit impedance changes. Clinical applications, however, present a more complex situation. The complexity of current-mode stimulators makes them difficult to manufacture as implantable devices. Additionally, voltage stimulation is generally safer than current stimulation, since the latter may apply excessive voltage to the human body during impedance accidents. For these reasons, early implantable products for neural stimulation therapies mainly used voltage-mode. Over time, these products evolved to offer both stimulation modes. Some of the most recent products have even provided current stimulation exclusively (Kern et al. 2020; Gilbert et al. 2023). Theoretically, current stimulation should improve efficiency during unstable impedance periods, such as the postoperative adaptation period. However, its clinical applications still require further verification.
1.4.3 Unipolar and Bipolar Stimulations
Based on the arrangement of cathodic and anodic electrodes, extracellular stimulations can be applied in two modes: unipolar or bipolar.
In unipolar stimulation, the working electrode is placed in the target area, while the other electrode—called independent or ground electrode—is placed far away and serves only to complete the current loop without contributing to stimulation effect. Figure 1.13 illustrates an ideal unipolar stimulation—point source stimulation. To depolarize the cell membrane to its activation threshold, the stimulation must produce a sufficient current density (current per unit area). With a same applied current intensity, an electrode contact with a larger surface area produces a lower current density. Therefore, in neural stimulation, the exposed tip (the contact) of working electrode is typically small to generate sufficient current density. This electrode usually delivers a negative voltage or current to depolarize surrounding cell membranes. Its structural design must also consider both potential tissue damage during implantation and the desired shape and size of its action area. The independent electrode usually has a much larger surface area to prevent depolarizing nearby cells.
The effect of unipolar stimulation decays rapidly with distance, focusing its action near the working electrode. However, the broader spread of stimulation current between the two distant electrodes may activate cells in non-target areas and cause side effects. Nevertheless, its broad action range offers an advantage: therapeutic effects can be achieved even with less precise electrode positioning.
Unipolar stimulation is widely used in clinical applications. For example, for the implant DBS electrodes with multiple contacts, it can be achieved by designating one contact as the working electrode while using the stimulator outer shell (buried under the clavicle) as the independent electrode (Wang et al. 2021). Unipolar stimulation is also common in non-invasive treatments, such as using skin surface electrodes to activate superficial nerves and muscle tissues.
In bipolar stimulation, both electrodes are placed in or near the target area. The two electrode contacts are small and positioned close to each other. This stimulation can be created by using two thin wires with exposed tips only about 0.5 mm apart, or by using the inner and outer poles of a concentric bipolar electrode. The contacts can also be separated by several millimeters. Which contact serves as the working electrode depends on factors such as its position relative to the target, the polarity of the stimulator output it is connected to, and its contact size. For optimal efficiency, the working electrode should be positioned closer to the target area, connected to the stimulator anodic output, and have a smaller contact surface area. Bipolar stimulation offers a key advantage: it can confine the stimulation action to the target area and immediate surroundings, thereby achieving higher efficiency than unipolar stimulation. Additionally, during simultaneous neural signal recording, bipolar stimulation produces smaller stimulation artifacts (see Sect. 4.5 for details). However, it has a disadvantage: when cathodic and anodic contacts are very close, their respective depolarization and hyperpolarization effects may interfere with each other, reducing stimulation efficiency.
1.4.4 Electrical Stimulation Safety
The safety of electrical stimulation involves two key aspects: preventing tissue damage and avoiding electrode corrosion. In clinical applications, both implanted electrodes and applied voltages/currents must meet these safety requirements. Long-term safety must also be maintained during continuous use. Similarly, safety is crucial in animal experiments to prevent tissue damage and electrode corrosion from affecting experimental results. Even when metal electrodes show no corrosion in in-vitro tests, sustained current application through implanted electrodes in a living body may still damage both biological tissues and electrodes (Merrill et al. 2005).
Electrical neural stimulation in clinical applications typically uses metal electrodes. When implanted, the electrodes form an electrode–electrolyte interface with the surrounding tissue fluid—extracellular fluid (ECF). This ECF is an electrolyte solution rich in various ions, including Na+, K+, Cl−, Ca2+ and etc. In metal electrodes and their connected electrical circuits, electrons carry charges to form current. In ECF, however, ions carry charges to form current. These two types of currents convert to each other at the electrode–electrolyte interface. This conversion of charge carriers—from electrons in the metal electrode to ions in the electrolyte—occurs mainly through two mechanisms: non-Faradaic reaction and Faradaic reaction (Merrill et al. 2005; Gilbert et al. 2023). The non-Faradaic reaction is capacitive and reversible. It does not involve real electron transfer from electrode to ECF. Instead, it only redistributes charges at the interface—much like charging a capacitor. As electrons accumulate on the electrode side, they attract positively charged ions from the ECF side.
In contrast, Faradaic reaction involves electrochemical processes, transferring electrode electrons to ECF ions to form reactants. This reaction can be reversible providing the electrode delivers opposite charges (reversing the current direction) before the reactants disperse. However, if the reactants diffuse away and cannot be recovered, the electrode gradually corrodes and damages the surrounding tissue. At the anode, the electrochemical reaction is oxidation, while at the cathode, it is reduction. For example, water can be reduced to form hydroxide ions and hydrogen gas, or oxidized to produce hydrogen ions and oxygen gas. Thus, during electrical stimulation, tissue and electrode damages may manifest as bubble formation in the surrounding tissues. Thus, an appearance of bubbles around stimulation electrodes during an experiment may indicate damages and require immediate inspection.
In extracellular stimulation, negative pulses activate nerve tissue most efficiently. However, sustained applying negative pulses, especially at high-frequencies, can cause local charge accumulations, leading to irreversible electrochemical reactions and tissue damage. To prevent this, clinical applications typically use biphasic pulses—a negative phase immediately followed by a charge-balanced positive phase. The positive charges of subsequent phase quickly neutralize the negative charges of preceding phase, preventing irreversible reactions. Once the negative phase depolarizes the cell membrane and activate Na+ channels, their regenerative response continues despite the following positive phase. Therefore, the positive phase cannot fully counteract the effect of negative phase, and an action potential can still be triggered. However, the positive phase may weaken the activation of negative phase. To minimize this impact, the positive phase can be slightly delayed or delivered with reduced amplitude and increased width. Various waveforms for the balanced positive phase have been proposed (Merrill et al. 2005).
Additionally, limiting the stimulation current below the Faradaic reaction threshold can ensure safety by restricting the stimulation process to non-Faradaic reactions only. Previous studies have shown that when the charge density delivered by the electrode is kept sufficiently low, oxidation–reduction reactions become negligible. The recommended charge density limitation ranges from 15 to 65 μC/cm2 per pulse. This serves only as a guideline, as it can vary with electrode material, shape, size, and stimulation waveform, etc (Shannon 1992; Cogan et al. 2016).
For current pulse stimulation, the charge Q conveyed by each pulse equals the integral of the pulse current I(t), i.e., the product of the constant current I and the pulse width PW:
$$Q = \int {I(t){\text{d}}t} = I \times {\text{PW}}$$
(1.56)
The charge density D is the charge Q per unit area of the electrode contact surface, calculated by dividing Q by the surface area A:
$$D = \frac{Q}{A}$$
(1.57)
Shannon (1992) proposed using the following k value to determine the current limitation delivered by an electrode:
$$k = \lg D + \lg Q$$
(1.58)
where D is measured in μC/cm2/pulse-phase, and Q in μC/pulse-phase. A stimulus is considered safe when k falls below 1.5–2.0. Higher values may cause tissue damage (Shannon 1992; Cogan et al. 2016; Gilbert et al. 2023). This guideline serves as a reference only and should not be treated as an absolute rule.
1.5 Summary
This chapter introduces characteristics of neuronal membranes. The membrane electrical properties can be modeled using an equivalent circuit composed of resistance and capacitance. During subthreshold changes, both membrane resistance and capacitance remain constant. The time constant can describe how fast the membrane potential changes after applying a step current stimulation. The rheobase is the minimum current required to trigger an AP, while the chronaxie is the minimum pulse duration required to trigger an AP at twice the rheobase current. The space constant and cable equations describe how the subthreshold changes in membrane potential spread along a neuron.
Ion concentration differences between the inside and outside of neurons, along with varying membrane permeabilities to different ions, create the foundations of resting potential and AP. During an AP, voltage-gated Na+ and K+ channels manifest nonlinear changes in their conductivity against time and membrane potential. These changes can be simulated by Hodgkin-Huxley (HH) model, which consists of differential equations and nonlinear algebraic equations that require computer programs to solve. This foundational model can simulate responses to stimulations in both individual neurons and neural networks.
Electrical stimulation can activate neurons through both intracellular and extracellular methods. Extracellular stimulation—widely used in clinical applications and in-vivo animal experiments—typically employs square pulse waveforms with either constant current or constant voltage. The pulse width is determined by the characteristics of target tissue. Among neuronal structures, the axonal membrane has the shortest chronaxie, which makes it most responsive to narrow pulses. Extracellular stimulation can be applied in unipolar or bipolar manners, depending on specific requirements. Safety is a prerequisite in electrical stimulation applications—excessive current density must be avoided to prevent tissue damage and electrode corrosion.
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Bean BP (2007) The action potential in mammalian central neurons. Nat Rev Neurosci 8(6):451–465CrossRef
Bear MF, Connors BW, Paradiso MA (2016) Neuroscience: exploring the brain (4th ed.). Wolters Kluwer
Bhunia S, Majerus S, Sawan M (2015) Implantable biomedical microsystems: design principles and applications. Elsevier, Amsterdam
Brocker DT, Grill WM (2013) Principles of electrical stimulation of neural tissue. In: Handbook of clinical neurology, vol 116. Elsevier, Amsterdam, pp 3–18
Buzsáki G (2006) Rhythms of the brain. Oxford University Press, OxfordCrossRef
Catacuzzeno L, Franciolini F (2022) The 70-year search for the voltage-sensing mechanism of ion channels. J Physiol 600(14):3227–3247CrossRef
Catterall WA (2012) Voltage-gated sodium channels at 60: structure, function and pathophysiology. J Physiol 590(11):2577–2589CrossRef
Cheung T, Nuño M, Hoffman M et al (2013) Longitudinal impedance variability in patients with chronically implanted DBS devices. Brain Stimul 6(5):746–751CrossRef
Cogan SF, Ludwig KA, Welle CG et al (2016) Tissue damage thresholds during therapeutic electrical stimulation. J Neural Eng 13(2):021001CrossRef
DiLorenzo DJ, Bronzino JD (2008) Neuroengineering. CRC Press, Taylor & Francis Group, Boca Raton
Durand DM. 2000. Electrical stimulation of excitable tissue. In: Bronzino JD (ed) The biomedical engineering handbook. Springer, Berlin
Gilbert Z, Mason X, Sebastian R et al (2023) A review of neurophysiological effects and efficiency of waveform parameters in deep brain stimulation. Clin Neurophysiol 152:93–111CrossRef
Hodgkin AL, Huxley AF, Katz B (1952) Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J Physiol 116(4):424–448CrossRef
Hodgkin AL, Huxley AF (1952a) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544CrossRef
Hodgkin AL, Huxley AF (1952b) Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J Physiol 116(4):449–472CrossRef
Hodgkin AL, Huxley AF (1952c) The components of membrane conductance in the giant axon of Loligo. J Physiol 116(4):473–496CrossRef
Hodgkin AL, Huxley AF (1952d) The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. J Physiol 116(4):497–506CrossRef
Hopper AJ, Beswick-Jones H, Brown AM (2022) A color-coded graphical guide to the Hodgkin and Huxley papers. Adv Physiol Educ 46(4):580–592CrossRef
Horch KW, Dhillon GS. 2004. Neuroprosthetics: theory and practice. World Scientific Publishing Company.
Jastrzębski M, Moskal P, Bednarek A et al (2019) His bundle has a shorter chronaxie than does the adjacent ventricular myocardium: implications for pacemaker programming. Heart Rhythm 16(12):1808–1816CrossRef
Kern DS, Fasano A, Thompson JA et al (2020) Constant current versus constant voltage: clinical evidence supporting a fundamental difference in the modalities. Stereotact Funct Neurosurg 99(2):171–175CrossRef
Kim E, Owen B, Holmes WR et al (2012) Decreased afferent excitability contributes to synaptic depression during high-frequency stimulation in hippocampal area CA1. J Neurophysiol 108(7):1965–1976CrossRef
Koch C (1999) Biophysics of computation: Information processing in single neurons. Oxford University Press, UK
Lempka SF, Johnson MD, Miocinovic S et al (2010) Current-controlled deep brain stimulation reduces in vivo voltage fluctuations observed during voltage-controlled stimulation. Clin Neurophysiol 121(12):2128–2133CrossRef
McIntyre CC, Grill WM (1999) Excitation of central nervous system neurons by nonuniform electric fields. Biophys J 76(2):878–888CrossRef
Merrill DR, Bikson M, Jefferys JGR (2005) Electrical stimulation of excitable tissue: design of efficacious and safe protocols. J Neurosci Methods 141(2):171–198CrossRef
Montgomery EB (2014) Intraoperative neurophysiological monitoring for deep brain stimulation: principles, practice, and cases. Oxford University PressCrossRef
Nowak LG, Bullier J (1998) Axons, but not cell bodies, are activated by electrical stimulation in cortical gray matter. Exp Brain Res 118(4):489–500CrossRef
Piccolino M (1998) Animal electricity and the birth of electrophysiology: the legacy of Luigi Galvani. Brain Res Bull 46(5):381–407CrossRef
Ranck JB (1975) Which elements are excited in electrical stimulation of mammalian central nervous system: a review. Brain Res 98(3):417–440CrossRef
Schneider T, Filip J, Soares S et al (2023) Optimized electrical stimulation of c-nociceptors in humans based on the chronaxie of porcine C-fibers. J Pain 24(6):957–969CrossRef
Schwiening CJ (2012) A brief historical perspective: Hodgkin and Huxley. J Physiol 590(11):2571–2575CrossRef
Shannon RV (1992) A model of safe levels for electrical stimulation. IEEE Trans Biomed Eng 39(4):424–426CrossRef
Stern S, Agudelo-Toro A, Rotem A et al (2015) Chronaxie measurements in patterned neuronal cultures from rat hippocampus. PLoS ONE 10(7):e0132577CrossRef
Udupa K, Chen R (2015) The mechanisms of action of deep brain stimulation and ideas for the future development. Prog Neurobiol 133:27–49CrossRef
van de Steene T, Tanghe E, Tarnaud T et al (2020) Sensitivity study of neuronal excitation and cathodal blocking thresholds of myelinated axons for percutaneous auricular vagus nerve stimulation. IEEE Trans Biomed Eng 67(12):3276–3287CrossRef
Wang Z, Feng Z, Yang G et al (2021) Advances in the development of new stimulation paradigms of deep brain stimulation. Prog Biochem Biophys 48(3):263–274 (in Chinese)
Wickens AP (2015) A history of the brain: from stone age surgery to modern neuroscience. Psychology Press
