Computational modeling of neurons: intensity-duration relationship of extracellular electrical stimulation for changes in intracellular calcium

Abstract

In many instances of extensive nerve damage, the injured nerve never adequately heals, leaving lack of nerve function. Electrical stimulation (ES) has been shown to increase the rate and orient the direction of neurite growth, and is a promising therapy. However, the mechanism in which ES affects neuronal growth is not understood, making it difficult to compare existing ES protocols or to design and optimize new protocols. We hypothesize that ES acts by elevating intracellular calcium concentration ([Ca²⁺]_i) via opening voltage-dependent Ca²⁺ channels (VDCCs). In this work, we have created a computer model to estimate the ES Ca²⁺ relationship. Using COMSOL Multiphysics, we modeled a small dorsal root ganglion (DRG) neuron that includes one Na⁺ channel, two K⁺ channels, and three VDCCs to estimate [Ca²⁺]_i in the soma and growth cone. As expected, the results show that an ES that generates action potentials (APs) can efficiently raise the [Ca²⁺]_i of neurons. More interestingly, our simulation results show that sub-AP ES can efficiently raise neuronal [Ca²⁺]_i and that specific high-voltage ES can preferentially raise [Ca²⁺]_i in the growth cone. The intensities and durations of ES on modeled growth cone calcium rise are consistent with directionality and orientation of growth cones experimentally shown by others. Finally, this model provides a basis to design experimental ES pulse parameters, including duration, intensity, pulse-train frequency, and pulse-train duration to efficiently raise [Ca²⁺]_i in neuronal somas or growth cones.

the peripheral nervous system is known for its remarkable ability to regenerate after injury, especially compared with the central nervous system. However, even in best surgical appositions of peripheral nerves, only about 10% of regenerating neurites reach their target organs (Zochodne 2012), which is generally sufficient for partial functional recovery. Proximal nerve injuries, such as those occurring at the sciatic notch, may take so long to regenerate that a distant end organ permanently loses function before the regenerating neuron can reach it. Regenerating neurites grow at ∼1 mm/day, and end organs and support cells start to lose their regenerative capacity after several months (Gordon et al. 2003). Improving neurite growth rate is therefore a primary challenge in the field of nerve regeneration (Brushart 2011).

Electrical stimulation (ES) has been shown to improve nerve regeneration in vivo and is a promising therapy (Al-Majed et al. 2000; English et al. 2007; Geremia et al. 2007; Singh et al. 2012). In vitro, ES has been shown to improve the rate of neurite growth (Adams et al. 2014; Graves et al. 2011; Macias et al. 2000; Wood and Willits 2006; Yan et al. 2014) and in some cases to orient neurite growth (Graves et al. 2011; Macias et al. 2000). However, it is currently poorly understood how ES improves the rate and affects the direction of neurite growth (Hronik-Tupaj and Kaplan 2012). To complicate matters, in most neuron ES experiments only one type of ES is used, such as DC, AC, or pulses of a specific duration and intensity. Recently, our group as well as others have hypothesized that the mechanism in which ES improves neurite growth is via the opening of voltage-dependent Ca²⁺ channels (VDCCs), leading to a subsequent rise in the concentration of intracellular Ca²⁺ ([Ca²⁺]_i) (Yan et al. 2014). Elevation of [Ca²⁺]_i may affect neuronal growth via signaling pathways modulated by Ca²⁺ sensors that include calmodulin, calcineurin, phospholipases, and protein kinases (Salazar et al. 2008). The effect of ES parameters, including intensity and duration of an electrical pulse, on increased [Ca²⁺]_i has not yet been examined.

Multiple publications request a neuronal [Ca²⁺]_i model to inform researchers of possible mechanisms of ES that may underlie nerve regeneration (Adams et al. 2014; Hronik-Tupaj and Kaplan 2012; Singh et al. 2012). The membrane potential and [Ca²⁺]_i of a neuron are difficult to resolve experimentally on the microsecond time scale during which extracellular electrical fields act. A computational model of the membrane potential and [Ca²⁺]_i of a neuron is a valuable tool for understanding and predicting how extracellular electrical fields may be increasing neuronal growth through an influx of [Ca²⁺]_i.

Most electrophysiology modeling papers use intracellular current injection to perturb membrane voltages, and with this technique, researchers have developed neuronal models that match the properties of live neurons (Herzog et al. 2001; Hodgkin and Huxley 1952; Vasylyev and Waxman 2012). However, because clinically applied ES is extracellular, an extracellular stimulation rather than a current injection model is more applicable. Decades ago, several studies used passive membrane properties to estimate the excitability of neurons on the basis of the geometry of spherical somas and cylindrical neurites (Norman 1972; Rattay 1986; Rubinstein and Spelman 1988). More recently, others have recognized the need for extracellular stimulation models of neurons with active membranes (Boinagrov et al. 2010, 2012; Rattay et al. 2012), and we have utilized these studies as a foundation for the simulations presented in this article. However, these previous studies focused on the role of ES in generating action potentials (APs), which is often the goal for neural prostheses, but did not model increases in [Ca²⁺]_i. To the best of our knowledge, the present results are the first to 1) look at the intensity-duration relationship of pulsed electrical fields (PEF) for raising [Ca²⁺]_i, 2) estimate the rise of [Ca²⁺]_i using the membrane properties of an unmyelinated sensory neuron, 3) compare the rise in [Ca²⁺]_i between the growth cone and soma of model neurons, and 4) analyze the model results to readily design optimal stimulation paradigms for increasing [Ca²⁺]_i while minimizing the energy (and likely damage) of the ES. Energy of stimulation may be a limiting factor for ES, because joule heating of tissue is directly proportional to energy of stimulation. Additionally, for implanted electrodes, the energy of stimulation will directly affect the battery life of the device.

In this study, we compare the anatomical regions of a neuron that have the most extreme responses induced by ES: 1) the growth cone tip of a neurite that has the greatest change in membrane potential from long duration ES, and 2) the soma that reacts the most rapidly to ES. Additionally, the growth cone tip is physiologically interesting given the region of growth and elongation where [Ca²⁺]_i control occurs (Lowery and Van Vactor 2009; Yao et al. 2006), and the soma of the neuron is integral for gene transcription and protein production. We show the results that support the three primary objectives for this study. First, the study is designed to assist in rationally designing new ES protocols. Second, the study compares ES already used by researchers for nerve regeneration. Last, the study is designed to better understand the mechanism through which ES is likely improving neuronal growth.

METHODS

Model Description

We created a neuron model based on a small dorsal root ganglion (DRG) neuron that gives rise to unmyelinated axons (Kreitler 2007). The diameters for small DRG neurons are described as 20–27 μm compared with ∼35 μm for medium or ∼48 μm for large DRG neurons (Scroggs and Fox 1992). Because regenerating axons are unmyelinated, this model also translates well to nerve regeneration. The active neuron membrane properties were modeled on mouse DRG neurons that include two Na⁺ channels and one K⁺ channel (Herzog et al. 2001). We added three VDCCs that are reported in rat DRG neurons (Scroggs and Fox 1992).

The representative shape of a “typical” small DRG neuron with a spherical soma and an attached cylindrical neurite is shown in Fig. 1A. The neuron soma diameter was set at the average small DRG neuron diameter of 24.1 μm (Scroggs and Fox 1992; Vasylyev and Waxman 2012), and the neurite diameter was set at the average unmyelinated axon diameter of 0.7 μm (Vasylyev and Waxman 2012). To simplify the model and reduce solving times, the DRG was divided into three geometries, shown in the boxes in Fig. 1A. Models 1 and 2 represent long neurites, and in these models we are only observing the [Ca²⁺]_i and generation of APs from the growth cone. Models 4 and 5 also include a long neurite, but in these models we are only observing the [Ca²⁺]_i and generation of APs from the soma. Whereas model 1 could be attached to model 5, and model 2 to model 4, in the simulations these models are isolated from one another and APs cannot propagate from one model to the other. All of the values and variables used in this simulation are shown in Table 1.

Fig. 1.

Modeling parameters for a dorsal root ganglion (DRG) neuron. A: a neuron is divided into 3 basic models (dashed boxes). B: models 1 and 2 represent the neuron growth cone that faces the negative (model 1) or positive (model 2) electrodes, causing depolarization or hyperpolarization, respectively. Model 3 represents the axotomized neuron soma with no extensions. Models 4 and 5 represent a neuron soma with a neurite attached. C: the neurite and soma were simplified into 1-dimensional models. The neurite was divided into 0.7-μm-diameter cylindrical segments, and intracellular Ca²⁺ concentration ([Ca²⁺]_i) was calculated in the growth cone with dimensions of 2-μm diameter by 5-μm length. The simulated neurite length was 2,000 μm, which is ∼10 length constants, and yields the same results as an infinitely long neurite. The simulated soma was divided into stacked truncated cones to estimate the volume and surface area of each segment. The representative image shows 10 segments, but the actual model was performed with ∼100 segments. The simulated soma with neurite has the same parameters as the soma with an additional infinitely long neurite attached at the equator of the soma and pointing directly toward the positive and negative electrodes.

Table 1.

Glossary of values and variables used in the simulation

Description	Symbol	Value	Reference
Soma diameter	ds	24.1 μm	Vasylyev and Waxman 2012
Neurite diameter	dn	0.7 μm	Vasylyev and Waxman 2012
Membrane capacitance	Cm	1 μF/cm2	Herzog et al. 2001
Intracellular resistance	Ri	145.69 Ω·cm	Herzog et al. 2001 (modified)
Resting membrane potential	ur	−61.4 mV	Calculated from model
Resting membrane resistance	Rm	3,220 Ω·cm2	Calculated from model
TTX-sensitive Na+ channel maximum conductivity	GNas	105.4 mS/cm2	Manually adjusted from Herzog et al. 2001; Vasylyev and Waxman 2012
TTX-resistant Na+ channel maximum conductivity	GNar	2.3 mS/cm2	Manually adjusted from Herzog et al. 2001; Vasylyev and Waxman 2012
K+ channel maximum conductivity	GK	1.9 mS/cm2	Manually adjusted from Herzog et al. 2001; Vasylyev and Waxman 2012
L-type Ca2+ channel maximum conductivity	GCaL	1.5 mS/cm2	Manually adjusted from Fox et al. 1987; Vasylyev and Waxman 2012
N-type Ca2+ channel maximum conductivity	GCaN	1.2 mS/cm2	Manually adjusted from Fox et al. 1987; Vasylyev and Waxman 2012
T-type Ca2+ channel maximum conductivity	GCaT	1.3 mS/cm2	Manually adjusted from Fox et al. 1987; Vasylyev and Waxman 2012
Leak conductivity	GL	0.14 mS/cm2	Herzog et al. 2001
Na+ reversal potential	ENa	66.37 mV	Herzog et al. 2001 (modified)
K+ reversal potential	EK	−97.37 mV	Herzog et al. 2001 (modified)
Ca2+ reversal potential	ECa	46.97 mV	Benison et al. 2001 (modified)
Leak reversal potential	EL	−57.26 mV	Herzog et al. 2001 (modified)
Fraction of free intracellular Ca2+	fi	0.025	Benison et al. 2001
Equilibrium concentration of Ca2+	Caeq	100 nM	Benison et al. 2001
Time constant of intracellular Ca2+ stores	τstore	12.5 ms	Benison et al. 2001
Velocity of Ca2+ efflux pump	vpump	3.77 nmol/s/m2	Benison et al. 2001
Electrical field intensity	v	Variable, V/m
Intracellular voltage	Vi	Variable, mV
Extracellular voltage	Vo	Variable, mV
Location along axis of neuron	x	Variable, μm
Time	t	Variable, s

TTX, tetrodotoxin.

Models 1 and 2 simulate [Ca²⁺]_i in a growth cone of a neurite oriented toward the negative or positive electrode, respectively (Fig. 1B). Model 3 simulates the [Ca²⁺]_i in a soma that has no current flowing into or out of the soma through a neurite. Models 4 and 5 simulate [Ca²⁺]_i in a soma that has a neurite oriented toward the positive or negative electrode, respectively. The spectrum between models 3, 4, and 5 represents the neuron geometries we observed (Adams et al. 2014). Model 3 represents a neuron soma that expresses 1) no neurites, 2) a neurite growing perpendicular to the electrical field so that no current is drawn through it, and 3) equal growth toward the negative and positive electrodes so no net current is drawn into or out of the soma. Models 4 and 5 represent a neuron with either a single neurite or the majority of neurites oriented toward the positive or negative electrode, respectively, so that current is drawn either into (model 4) or out of (model 5) the neuron soma (Fig. 1B).

In living neurons, Ca²⁺ microdomains exist near the membrane where [Ca²⁺]_i rises rapidly and transiently via Ca²⁺ entry through VDCCs (Mehta and Zhang 2015). It is hypothesized that the density of VDCCs and possibly the downstream targets of Ca²⁺ (e.g., calmodulin) are at higher concentrations in these microdomains (Mehta and Zhang 2015). However, in our model we did not utilize the complexity of Ca²⁺ diffusion and Ca²⁺ microdomains, but instead modeled the total average [Ca²⁺]_i based on the volume of the neuron using a first-order Ca²⁺ buffering model (Benison et al. 2001). Therefore, our whole neuron averaged [Ca²⁺]_i from the model is comparable to the [Ca²⁺]_i reported by a Ca²⁺ indicator, such as fura 2, but underestimates the focal [Ca²⁺]_i in microdomains near the membrane.

In our model, we assumed that the electrical field is linear. This is nearly true in the microenvironment surrounding a neuron if the electrodes are more than several millimeters away from the neuron or if the neuron is directly between the electrodes that create the electrical field. Both of these conditions are true in our experimental stimulations (Adams et al. 2014).

Long Cylindrical Neurite

Unmyelinated axons range in diameter from 0.1 to 2 μm (Vasylyev and Waxman 2012), with the average diameter of regrowing motor neurites 1.9 μm and that of regrowing sensory neurites 1.3 μm (Brushart 2011). The average neurite diameter from a small DRG neuron is 0.7 μm (Vasylyev and Waxman 2012), and this value was used for all of the simulations presented in this report. All of the results for a 0.7-μm-diameter neurite can be intuitively scaled for other diameters by using the relationships discussed later.

Rarely are actual neurites perfectly aligned to the electrical fields. Unaligned neurites are polarized on the basis of the component of the applied electrical field that is in the direction of the neurite, calculated with the equation v_n = v·cos (θ), where v is the magnitude of the electrical field around the neuron and θ is the angle between the electrical field and the neurite, where 0° is aligned. For example, a neurite deviated 10° from the electrical field would experience 98.5% of the ES intensity, and a neurite deviated 45° from the electrical field would experience 71% of the ES intensity.

The length constant (λ) is the characteristic electrical length of a neurite. If current is injected into a passive neurite, the membrane polarization drops by 63% for every length constant distance away from the point of current injection. Length constant is directly proportional to the square root of the neurite's diameter (d_n), based on the cable theory equation (Eq. 1) (Rashbass and Ruston 1949):

$$\lambda =\sqrt{\frac{d_n\cdot R_m}{4\cdot R_t}}\mathrm{.}$$ (1)

The membrane potential within a cylindrical neurite varies such that the tip of the neurite is most polarized by an electrical field, and this polarization reduces exponentially to zero away from the tip based on the length constant of the neurite. The length constant of the neurite is ∼200 μm when determined using Eq. 1 and the values in Table 1. A neurite length of 2 mm was used in this simulation to approximate an infinitely long neurite. The cylindrical neurite was reduced to a one-dimensional model and divided into segments of logarithmically varying length, with the shortest segment of 50 nm at the growth cone (tip) and the longest segment of 20 μm at 2 mm from the growth cone. The surface area (SA) for each segment is the product of segment length and circumference (π·d_n, written as ∂SA in Eq. 2), and the volume (Vol) of each segment is the product of segment length and cross-sectional area (π·d_n²/4, written as ∂Vol in Eq. 2).

Spherical Soma

When the membrane conductance of a neuronal soma is negligible compared with the conductivity of the media, the electrical field around a spherical soma (v_s) distorts in such a way that the neuron experiences an electrical field 50% greater than that if there were no distortion (Gross et al. 1986) as shown by the equation v_s = 3/2·v.

In models 4 and 5, the neurite is simulated to be aligned with the electrical field, and to be infinitely long so that current flows through it at the same magnitude as the extracellular electrical field. As we described for neurites above, rarely are neurites perfectly in line with the electrical field. In these simulations we modeled the most extreme case of alignment with the electrical field. If a neurite was not aligned with the electrical field, the component of current drawn through the neurite would be less, and models 4 and 5 would behave more similarly to model 3. The neurite was modeled to attach to the equator of the soma (Fig. 1) so that the membrane potential at neurite origin is the average soma membrane potential.

The spherical soma of the neuron was divided into equally thick segments along the axis parallel to the direction of current, shown in Fig. 1C. In a three-dimensional passive membrane simulation (not shown), we found that at the onset of a pulse, all intracellular current moves parallel to this axis, and the membrane potential is equal around the entire circumference of each segment. The model was therefore reduced to a one-dimensional analysis. Each segment shown in Fig. 1C is a trapezoidal cone. The SA for each segment is the product of segment length and the circumference of the entire soma (π·d_s, written as ∂SA in Eq. 2), and the volume of each segment is the product of segment length and cross-sectional area [π(d_s²/4 − x²), written as ∂Vol in Eq. 2]. x is the distance along the axis of the soma and is zero at the center of the sphere. x is equal to the value of the radius at the top pole and the negative value of the radius at the bottom pole of the neuron soma in Fig. 1C.

Hodgkin-Huxley Membrane Kinetics

The flow of current either through a spherical soma or through a cylindrical neurite was modeled using the cable theory equation and the Hodgkin-Huxley style ion channel currents in Eq. 2 (Hodgkin and Huxley 1952; Norman 1972):

where V_o(t) = v_s(t)·x for a spherical soma and V_o(t) = v_n(t)·x for a neurite. Additionally, m_s, h_s, m_r, h_r, s_r, n, m_L, m_N, h_N, m_T, and h_T are the individual channel gating variables, which vary depending on membrane potential and time, and are defined in the appendix.

The equations for the Na⁺ and K⁺ channels were defined by Herzog et al. (2001) for a small DRG neuron and include the definitions for m_s, h_s, m_r, h_r, s_r, and n (see appendix). The density of the Na⁺, K⁺, and leak channels defined by Herzog et al. (2001) were adjusted to match the resting membrane potential, AP threshold, and AP waveforms observed in small DRG neuron axons (Vasylyev and Waxman 2012). The density of K⁺ channels was scaled to 90% to bring the resting membrane potential to −61.5 mV. The density of TTX-resistant Na⁺ channels was scaled to 33% to bring the resting membrane resistance up to 6.8 kΩ·cm². The density of TTX-sensitive Na⁺ channels was scaled to 300% to 1) set the AP current threshold at 90 mA, 2) raise the AP overshoot from ∼0 to ∼25 mV, and 3) raise the AP conduction velocity from 50 to 200 mm/s.

The equations for L- and N-type VDCCs, including the definitions for m_L, m_N, and h_N, were defined by Benison et al. (2001) (see appendix). The equations from Benison et al. (2001) were increased by 5 mV to better match the Ca²⁺ current voltage responses reported by the Fox laboratory (Fox et al. 1987; Scroggs and Fox 1992). Equations for the T-type VDCC, including m_T, and h_T, were defined by Huguenard and McCormick (1992). The channel densities of the L-, N-, and T-type VDCCs in proportion to one another were matched to the magnitude of currents reported for each channel subtype during an AP (Fox et al. 1987). The total membrane concentration of VDCCs was scaled to 33% of the density reported by Fox et al. (1987) to match the Ca²⁺ current to the proportion of the total membrane current during an AP shown for small DRG axons (Vasylyev and Waxman 2012).

Channel kinetics were adjusted to represent in vivo temperature. The experimentally obtained kinetics measured at a temperature (T_m) of 21°C for the Na⁺ and K⁺ channel measurements and at 24°C for the VDCC measurements were adjusted to 37°C. The channel kinetics were adjusted using the Q₁₀ temperature coefficients of 2.5, 3.3, and 3 for Na⁺, K⁺, and Ca²⁺, respectively in the equation: IonFactor Q₁₀ = ${Q_{10}}^{(37°C-T_m)/10}$ (Russ and Siemen 1996; Wang et al. 1991).

Additionally, the reversal potentials (E_ion-measured) for Na⁺, K⁺, and leak channels (62.94, −92.34, and −54.3 mV, respectively) reported by Herzog et al. (2001) at 21°C and the reversal potential for VDCCs (45 mV) used by Benison et al. (2001) from experiments at 24°C were adjusted based on the Nernst equation: E_ion = E_ion-measured·(273K + 37°C)/(273K + T_m). The adjusted values are shown in Table 1. Finally, the resistivity of the intracellular cytosol reported at 200 Ω/cm at 21°C was adjusted to 37°C using a thermal coefficient of 2% increase in conductivity per degree Celsius (Bisen and Sharma 2013).

Ca²⁺ Modeling

To estimate the average [Ca²⁺]_i of our model neuron in response to extracellular ES, we used the first-order Ca²⁺ model developed by Benison et al. (2001). The model, shown in Eq. 3, incorporates 1) the free fraction of cytosolic Ca²⁺ (f_i), 2) a first-order model of intracellular Ca²⁺ stores with time constant τ_store, and 3) a membrane pump with velocity v_pump. The amount of [Ca²⁺]_i is elevated by influx of Ca²⁺ current through the L-, N-, and T-type VDCCs (I_CaL, I_CaN, and I_CaT), which is converted to moles of Ca²⁺ ions based on the Faraday constant and two charges per Ca²⁺ ion (2F) and averaged over the volume of the neuronal space (Vol_neuron). For the spherical soma, the volume is 1/6·π·d_s³. The growth cone was estimated as shown in Fig. 1C for a cylinder 2 μm in diameter and 5 μm in length (Vasylyev and Waxman 2012), so its volume was 5π μm³.

In Eq. 3, SA_neuron is the membrane surface area of the neuronal space. For the spherical soma, the surface area is ½·π·d_s². The surface area of a growth cone is dynamic and therefore difficult to estimate (Lowery and Van Vactor 2009). For the purposes of this Ca²⁺ model, the surface of the growth cone was a cylinder 2 μm in diameter and 5 μm long with a total area of 12π μm².

COMSOL Modeling Parameters

All five geometrical models shown in Fig. 1 were simulated with monophasic electrical pulses, for a total of five simulations. Monophasic electrical pulses were simulated as an electrical field that rose from zero to a defined intensity, remained at that intensity for a defined pulse duration, and returned to an electrical field of zero. The edges of the rectangular pulse were second order smoothed over 20 ns.

The monophasic pulses were simulated with 40 pulse intensities ranging from 10 V/m to ∼3 MV/m and 15 pulse durations ranging from 10 ns to 100 s, for a total of ∼600 conditions per monophasic simulation. In each simulation condition, the [Ca²⁺]_i peaked within 80 ms of the end of the pulse duration, so the simulation was continued for 300 ms after the end of each pulse. The finite element model was solved numerically using the built in “backward differentiation formula” in COMSOL Multiphysics. A total of ∼3,000 conditional simulations took ∼6 wk to run on an Intel i5 3.2 GHz personal computer (4 cores).

Passive Model

A passive model assumes that the resting membrane resistance of the neuron is permanently equal to the resting membrane resistance of the active membrane model (see Table 1) and computes about 1,000 times more quickly than the active model. The passive model was only used in results, Translating the Results to Neurons of Any Size, Using a Passive Model.

The geometry of the passive model of the neurite was cylindrical. The tip of the neurite represents the location where a growth cone occurs, and no current flowed through the end of the tip. The opposite end of the cylinder represented the point at which the neurite would attach to a soma and therefore was held at the average soma voltage. In this simplified model for a neurite, the voltage of the soma was not simulated to change, so the voltage at this end of the cylinder was fixed at the resting membrane potential and there was no restriction on current flow. The geometry of the passive model of the soma was that of a closed sphere. The geometry was segmented with the same methodology as the active membrane model with trapezoidal cones of equal thickness.

Both passive models were simulated with the pdepe(m,pdefun,icfun,bcfun,xmesh,tspan) function in MATLAB (The MathWorks).

Energy-per-Pulse Calculations

Using Ohm's law, the energy density of a pulse (J/cm³) is dependent on the pulse electrical field intensity (v, in V/cm), pulse duration (t_pulse, in s), and the tissue/media resistance (R_media, in Ω·cm) (Eq. 4, derived from Ohm's law). Like electrical field intensity, energy density is independent of electrode geometry. In the results, we estimated tissue/media resistance the same as intracellular resistance (R_i) from Table 1. To put energy density in perspective, a 4.184 J/cm³ pulse would raise the temperature of water-based media by about 1°C.

$$Pulse\hspace{0.17em}energy\hspace{0.17em}density=\frac{v^2\cdot t_{pulse}}{R_{media}}$$ (4)

Concerns with Soma Models for Pulses Shorter Than 10 μs

The Hodgkin-Huxley equations for gate variables of voltage-dependent ion channels are considered the best model of a neuron membrane (Hille 1992). The Hodgkin-Huxley style equations are still commonly used in published models to understand different neuron subtypes (Benison et al. 2001; Herzog et al. 2001; Huguenard and McCormick 1992; Wang et al. 1991), and as we have done in this article, to estimate the effect of ES on neurons (Boinagrov et al. 2010; Rattay et al. 2012; Sahin and Tie 2007; Wongsarnpigoon and Grill 2010).

The caveat with the Hodgkin-Huxley model.

Hodgkin and Huxley defined gates using α and β equations that can be converted to gating time constants by the equation τ_x = 1/(α_x + β_x). The physiological range of membrane voltages spans from the lowest reversal potential to the highest, which are −97 mV for K⁺ channels and +67 mV for Na⁺ channels in our model (see Table 1). Hodgkin and Huxley fit the α and β equations to experimentally measured gate time constants within the physiological range of membrane voltages; two of the gates are shown in Fig. 2A (Hodgkin and Huxley 1952). The caveat arises from the time constant equations that asymptote to zero at extremely hyperpolarized membrane values: −200 mV for the Na⁺ h gate and −500 mV for the K⁺ n gate, shown in Fig. 2A. Although the ability of a channel's gate to open in microseconds may be possible (Hallermann et al. 2005; Shapovalov and Lester 2004; Sigg et al. 2003), we are not aware of experimental evidence showing that the bulk stochastic opening and closing time constants of any of the gates approach zero (i.e., instantaneous opening/closing) at extremely hyperpolarized membrane potentials. Furthermore, the original Hodgkin-Huxley equations that asymptote to zero at extremely hyperpolarizing potentials have been adapted to newer characterized voltage-dependent channels, including the two Na⁺ channels and one of the Ca²⁺ channels presented in our model.

Fig. 2.

Channel kinetics based on Hodgkin-Huxley equations. A: Hodgkin and Huxley (1952) measured the time constants of Na⁺ and K⁺ channels of a squid giant axon (open circles) and fit the data with model equations (lines). As the equations are extrapolated to extremely hyperpolarized potentials, the equations approach zero, with no experimental support. B: our modification to the N-type voltage-dependent Ca²⁺ channel (VDCC) adjusts the negative asymptote of the time constant from zero to a value equal to the positive asymptote. This modified gate is only used in the “modified-gate model” shown in Fig. 5E.

Why the caveat has more recently become a problem.

Hodgkin and Huxley may have only intended for their membrane model to be used within the range of physiological membrane voltages, wherein the caveat does not exist. However, more recently, the Hodgkin-Huxley model has been used to simulate extracellular ES of neurons. Extracellular ES can depolarize and hyperpolarize parts of a membrane far beyond the physiological range. For a more extreme example, Scarlett et al. (2009) used 60-ns electrical pulses to stimulate Jurkat cells and found the depolarization threshold for Ca²⁺ entry to be ∼10 MV/m. With the use of Eq. 12, this ES condition translates to roughly 8,600-mV depolarization of the negative electrode-facing membrane and roughly 8,600-mV hyperpolarization of the positive electrode-facing membrane. This 8,600-mV hyperpolarization is well below −150 to −200 mV, where most channels are defined. Therefore, to accurately model how neurons react to short, high-intensity fields such as these, the time constants of ion channels now need to be defined at extreme values, not just the physiological range.

How the caveat affects our model.

In our model, the caveat of ion channel gating time constants that approach zero only becomes an issue at membrane potentials less than about −200 mV. Monophasic pulse parameters that cause hyperpolarization below −200 mV are marked with diagonal hatching in Figs. 4 and 5. Model 1 is only depolarized and is not affected by the caveat. Model 2 reaches membrane potentials less than −200 mV from pulses greater than ∼1,000 V/m. However, the most efficient Ca²⁺ entry in this model occurs with a 30-ms and 100 V/m pulse, which falls outside the caveat area. In summary, models 1 and 2 are generally unaffected by the caveat. However, models 3, 4, and 5 all reach membrane potentials of −200 mV with pulse intensities greater than ∼10,000 V/m. All supra-AP pulses less than 10 μs in duration are in the caveat area in these three models (see Fig. 4, C–E).

Fig. 4.

Ca²⁺ per energy efficiency of monophasic pulses (A–E) for 5 models. In each of the graphs (A–E), Ca²⁺ efficiency is plotted as a function of pulse intensity and pulse duration. The color scale graphically represents the [Ca²⁺]_i per pulse energy for growth cones (A and B), for the soma alone (C), and for the somas with neurites (D and E). The line in the middle of each graph is the AP threshold for each model (from Fig. 3). The stimulation parameters for monophasic pulses that cause hyperpolarization beyond −200 mV, where the model is not well defined, are filled with black diagonal hash lines. The pulses marked by numbers and symbols represent either special conditions (1, 2, and 9) or the most efficient pulses for that model (all others: 3, *, 4, 5, 6, 7, +, and 8). The numbered pulses are further described in Figs. 5 and 6. Note: pulse parameters that are not tested are shown as the white space surrounding the color scale [Ca²⁺]_i per pulse energy for each model.

Fig. 5.

The rise in [Ca²⁺]_i in response to monophasic pulses (A–D, F) is shown for all five models. In each of the graphs (A–F), the rise in [Ca²⁺]_i is plotted as a function of pulse intensity and pulse duration. The color scale graphically represents the rise in [Ca²⁺]_i for growth cones (A and B), for the soma alone (C), and for the somas with neurites (D and F). The line in the middle of each graph is the AP threshold for each model (from Fig. 3). The stimulation parameters for monophasic pulses that cause hyperpolarization beyond −200 mV, where the model is not well defined, are filled with black diagonal hash lines. The numbers 1–9 on the monophasic graphs depict the parameter values for the representative pulses shown in Fig. 6. E: the modified-gate model has altered gate time constants that do not asymptote to zero (from Fig. 2). The modified-gate model was simulated for pulse durations ranging from 10 ns to 1 ms (inset within dashed line) and is shown overlaid on the original model shown in C. Note: pulse parameters that are not tested are shown as the white space surrounding the color scale [Ca²⁺]_i per pulse energy for each model.

Modified-gate model.

To see how the caveat affects the efficient rise in [Ca²⁺]_i for pulses shorter than 10 μs in models 3, 4, and 5, we created a “modified-gate model” to compare the original model. We are not proposing a correction to the membrane channel gating equations at extremely hyperpolarized membranes, because, to the best of our knowledge, the time constants of channels in this range are not experimentally known, and we do not have the equipment to measure such possible time constants. The modified-gate model removes the zero-time constant from the gating variables by changing the original α equations (in appendix) of the TTX-sensitive Na⁺ h gate (Eq. 5), the TTX-resistant Na⁺ s gate (Eq. 6), and N-type VDCC h gate (Eq. 7). The gate variables were modified so that the negative asymptote is equal to the positive asymptote. As an example, the modification of the time constant of h gate of the N-type VDCC is shown in Fig. 2B.

$${\alpha }_{h_{s}c}=NaFactor{Q}_{10}\cdot \frac{3}{1+e^{\left(\frac{u+197.6}{20.33}\right)}}$$ (5)

$${\alpha }_{S_{r}c}=NaFactor{Q}_{10}\cdot \frac{0.0005}{1+e^{\left(\frac{u+95.6}{12}\right)}}$$ (6)

$${\alpha }_{h_{N}c}=CaFactor{Q}_{10}\cdot \frac{0.03}{1+e^{\left(\frac{u+67.2}{10}\right)}}$$ (7)

This corrected membrane model was used for the geometry of model 3 for a neuron soma and was tested for pulse durations less than 1 ms where the zero-time constant problem occurred, as shown in Fig. 5E.

We only use the modified-gate model in Fig. 5E; we use the original model throughout the rest of this article. We do not know whether the modified-gate model is a more accurate representation of the ion channels in our neuron model. The sole purpose of the modified-gate model for this report is to show that efficient rise in [Ca²⁺]_i seen in the original model for pulses shorter than 10 μs is solely due to the caveat and is nonexistent in the modified-gate model (see Fig. 5E). We do not focus on pulses shorter than 10 μs in results or discussion because 1) the efficient [Ca²⁺]_i increase from pulses shorter than 10 μs is solely due to the caveat, 2) we do not know whether the caveat represents the activity of a real neuron, and 3) the minimum pulse duration of many stimulators such as the HEKA EPC 10 used by many is 10 μs.

RESULTS

Intensity-Duration Relationship for AP Threshold for Monophasic Pulses

AP threshold.

The AP threshold for the neurite (models 1 and 2, Fig. 3) is defined such that ES above this threshold caused at least one AP to propagate down the length of the neurite. The threshold for the soma (models 3–5, Fig. 3) is defined as a pulse that caused the average intracellular voltage of the soma to surpass −20 mV. When looking at pulse intensities at the AP threshold, we found that pulses that caused a peak depolarization above −20 mV led to a propagated AP, whereas pulses that led to a peak depolarization below −20 mV led to a failed AP because K⁺ channels opened before the AP propagated. To put the intensity and duration of a pulse in perspective, equal pulse energy lines are shown in Fig. 3.

Fig. 3.

Action potential (AP) thresholds are shown for each of the 5 neuronal models for monophasic stimulation. Pulse energy is plotted as a function of the pulse duration. All parameters with equal pulse energy are shown as dashed diagonal lines. The threshold to initiate an AP for each model is shown as a solid gray or black line, as indicated. For ease of distinguishing the AP thresholds for each model, a unique symbol was added to each line as shown in the key.

Comparing all five models.

All of the AP thresholds in Fig. 3 asymptote to a slope of zero for long pulses and to the well-established log-log slope of −1 observed in vitro and in vivo for pulses shorter than the electrical time constants (∼2.58 ms for the neurite, ∼176 ns for the soma) (Boinagrov et al. 2012; Lapicque 1907). In an in vitro culture of neurons, the lowest AP threshold between the models present is the overall AP threshold for the neurons.

In Fig. 3, the hyperpolarized neurite (model 2) experiences APs at the lowest voltage for pulses longer than 100 ms in duration. For pulses with durations between 10 μs and 100 ms, the depolarized neurite (model 1) has the lowest AP threshold. All of the soma models (models 3–5) equally show the most sensitivity to pulses shorter than 10 μs, due to the caveat in the Hodgkin-Huxley-style equations.

[Ca²⁺]_i Rise-Per-Pulse Energy Ratio for Monophasic Pulses

Use in designing ES protocols.

All of the contour plots in Fig. 4 display the rise in [Ca²⁺]_i for a single pulse divided by the energy of that pulse. We term this ratio “Ca²⁺ efficiency.” Figure 4, A and B, shows the Ca²⁺ efficiency for ES of growth cones, and Fig. 4, C–E, shows the Ca²⁺ efficiency of somas. The growth cones have approximately one order of magnitude greater maximum Ca²⁺ efficiency (shown in the color bar legend in Fig. 4), due to the increased electrical sensitivity of neurites over somas (shown in Fig. 3), and the roughly three-fold greater rise in [Ca²⁺]_i in response to a pulse (shown in Fig. 4), because of the larger surface area-to-volume ratio of the growth cone versus the soma that is used in Eq. 3. We propose that Ca²⁺ efficiency is the most helpful method for designing many ES protocols because an optimally efficient set of pulse parameters can be used in pulse trains to achieve a desired total rise in [Ca²⁺]_i.

Ca²⁺ efficiency for depolarizing monophasic pulses in neurites.

The most Ca²⁺-efficient monophasic pulse parameters for model 1 are shown in Fig. 4A. The most efficient supra-AP pulse (Fig. 4A, pulse 3) can more generally be described as a pulse duration approximately equal to the neurite time constant and a pulse intensity just above the AP threshold. This result is consistent with analysis by Lapicque (1907) showing that a pulse duration equal to chronaxie generates an AP using the least energy, where chronaxie is about 70% of the membrane time constant (Rattay et al. 2012) and neurite time constant is about 80% of the membrane time constant (Eq. 10). The efficient sub-AP pulse (Fig. 4A, asterisk) is efficient because the pulse energy is very low, but the total [Ca²⁺]_i elevation is much less than 1 nM (dark blue, Fig. 5), so this pulse may not be physiologically effective in neurons.

Ca²⁺ efficiency for hyperpolarizing monophasic pulses in neurites.

The most Ca²⁺-efficient monophasic pulse for model 2 is shown in Fig. 4B (pulse 5). Even though pulses in model 2 are hyperpolarizing, the most efficient pulse (Fig. 4B, pulse 5) has a comparable efficiency to the depolarizing pulse shown in model 1 (Fig. 4A, pulse 3). Interestingly, the pulse (Fig. 4B, pulse 5) is below all AP thresholds in Fig. 3, suggesting that substantial and efficient [Ca²⁺]_i rise is possible in neurons without causing any APs.

Ca²⁺ efficiency for monophasic pulses in somas.

For pulses longer than 10 μs, the most efficient monophasic pulse to model 3 is shown in Fig. 4C (pulse 6). Sub-AP pulses do not cause an efficient rise in [Ca²⁺]_i in model 3. Models 4 and 5 show similarities to models 1 and 2, respectively, due to the direction of current flow from the attached neurite in each of the models (red arrows in schematized neurons, Fig. 4). With the depolarizing current of the attached neurite in model 4, the most Ca²⁺ efficient supra-AP pulse is shown in Fig. 4D (pulse 7). Like the pulse indicated by the asterisk in model 1, the pulse in model 4 also has an efficient but low [Ca²⁺]_i increase (Fig. 4D, plus sign). The most efficient pulse for model 5 is shown in Fig. 4E (pulse 8). Model 5 had some sub-AP [Ca²⁺]_i increase, shown in Fig. 4E (pulse 9), though it was one-third the efficiency of the supra-AP pulse (Fig. 4E, pulse 8).

Peak [Ca²⁺]_i Rise for Monophasic Pulses

Use in comparing ES protocols.

All of the contour plots in Fig. 5 show the peak [Ca²⁺]_i in each neuronal model after a single monophasic pulse. In consideration of a mechanism of nerve regeneration that is based on increases in [Ca²⁺]_i, researchers can utilize the ES parameters from previously accomplished studies to compare to these results and can utilize the present results to hypothesize the outcomes of newly designed ES protocols (Fig. 5).

Peak [Ca²⁺]_i rise in neurites.

For depolarizing pulses in model 1 (Fig. 5A), sub-AP pulses cause no more than a 3 nM rise in [Ca²⁺]_i per pulse (dark and light blue contours in Fig. 5A), whereas most supra-AP pulses cause a 30–100 nM rise in [Ca²⁺]_i (orange contours in Fig. 5A). The two exceptions to this are a short-duration, high-intensity region with no [Ca²⁺]_i elevation (Fig. 5A, pulse 1) and a long-duration, high-intensity region with >100 nM [Ca²⁺]_i elevation (Fig. 5A, pulse 2).

For hyperpolarizing pulses in model 2 (Fig. 5B), [Ca²⁺]_i elevation is directly related to pulse intensity at all pulse durations and is less dependent on an AP. For the most extreme comparison, supra-AP pulses longer than 1 s in model 2 can cause an increase in [Ca²⁺]_i as low as ∼10 nM (yellow contour in Fig. 5B), and a sub-AP pulse 30 ms in duration can achieve an ∼100 nM rise (orange contour in Fig. 5B).

Peak [Ca²⁺]_i rise in somas.

For models 3–5 (Fig. 5, C–E), pulses shorter than 100 μs are essentially dependent on an AP for an increase in [Ca²⁺]_i such that sub-AP pulses cause less than a 0.3 nM rise, and most supra-AP pulses cause a 10–30 nM rise. For pulses longer than 100 μs, models 3 and 4 remain mostly dependent on an AP for an increase in [Ca²⁺]_i. Model 5 exhibits a Ca²⁺ gradient that is directly proportional to field intensity that permits supra-AP and sub-AP increases in [Ca²⁺]_i, similarly to model 2. Of special note, model 3 (Fig. 5C) exhibits a region at ∼1 ms and ∼8,000 V/m where an AP is not generated. This is because the depolarized part of the soma membrane is near the Na⁺ reversal potential during the duration of the pulse. This causes the neuron to fail to depolarize enough to initiate an AP (called the “stimulation gap” by Boinagrov et al. 2010). In Fig. 5C, the small black oval indicated by the arrow is the stimulation gap. The region above the stimulation gap is the region of ES where the Na⁺ current reversal, in addition to increased K⁺ efflux, causes a hyperpolarization of the average soma membrane potential. This leads to the K⁺ channels closing and a rebound AP after the pulse.

Mechanism for [Ca²⁺]_i Rise for Monophasic Pulses

By exploring the mechanism in which ES increases [Ca²⁺]_i via VDCCs, an ES protocol can be better designed for certain objectives such as raising [Ca²⁺]_i preferentially in a growth cone or causing a sub-AP [Ca²⁺]_i rise. Figure 6 displays an example pulse for each of the pulses numbered in Figs. 4 and 5. Each trace shows the membrane potential, VDCC currents, and [Ca²⁺]_i during and after a pulse. In the neurite models (models 1 and 2), the membrane potential and the VDCC currents are measured at the tip of the growth cone, and the [Ca²⁺]_i is measured as the average within the growth cone. In the soma models (models 3–5), the membrane potential is measured at the equatorial midline and is equal to the intracellular average. The VDCC currents are the average of the currents measured at the hyperpolarized and depolarized poles, and the [Ca²⁺]_i is measured as the average within the soma. The membrane potential regulates VDCC currents, and in turn, in this model VDCC currents set [Ca²⁺]_i.

Fig. 6.

Representative currents and [Ca²⁺]_i elevations from 9 representative pulses. A–I: the 9 most representative and efficient (or special) stimulation pulses are shown from Figs. 4 and 5. For A–I, the top set of graphs depicts how the electrical pulse increases [Ca²⁺]_i and membrane potential, and the lower set of graphs depicts how the VDCC currents change with the electrical pulse. The 9 pulses (gold) shown above the graphs are labeled with their stimulation parameters. [Ca²⁺]_i is shown as the red trace and right-hand scale, and membrane potential is shown as the black trace and left-hand scale. The bottom set of graphs show the Ca²⁺ current passing through each of the L-type (blue), N-type (gray), and T-type (green) VDCCs. VDCC current magnitudes are based on the scale bars shown in each panel and begin and end at zero in every graph.

Elevation in [Ca²⁺]_i only occurs during membrane depolarization.

Pulses applied to models 1 and 4 are depolarizing (pulses 1, 2, 3, and 7 in Figs. 4–6), and pulses applied to models 2 and 5 are hyperpolarizing (pulses 4, 5, 8, and 9 in Figs. 4–6). The depolarizing pulses cause Ca²⁺ flux when the membrane potential depolarizes above −20 mV, within milliseconds of the onset of a pulse (Fig. 6). The hyperpolarizing pulses do not cause Ca²⁺ flux during the pulse, but rather after the pulse ends as the membrane potential rebound causes a depolarization.

Action potentials cause L- and N-type VDCC current.

Only the supra-AP pulses (pulses 1, 2, 3, 4, 6, 7, and 8 in Figs. 4–6) exhibit a substantial L- and N-type VDCC current (Fig. 6). The depolarization above −20 mV during the AP causes an ∼3-ms surge of L- and N-type VDCC current in pulses 3, 4, 6, 7, and 8. In pulse 1, the membrane potential is above the Ca²⁺ reversal potential during the majority of the AP, causing Ca²⁺ to flow out of, rather than into, the neuron during the AP.

Hyperpolarizing pulses cause T-type VDCC current.

Only the hyperpolarizing pulses (pulses 4, 5, 8, and 9 in Figs. 4–6) exhibit T-type VDCC current (Fig. 6). The hyperpolarized membrane potential causes the h gate of the T-type VDCCs to open, allowing Ca²⁺ entry upon rebound depolarization. Sub-AP hyperpolarizing pulses (pulses 5 and 9) only exhibit T-type VDCC current because they lack an AP. Supra-AP hyperpolarizing pulses (pulses 4 and 8) exhibit L, N, and T-type VDCC current.

Specific pulse for continuous [Ca²⁺]_i rise.

When an electrical field greater than ∼300 V/m is applied to model 1 for more than 10 ms, the membrane potential becomes continually depolarized at a potential greater than −20 mV following the initial AP. Pulse 2 in Fig. 6 shows that after the initial AP, the membrane potential is continually held at ∼0 mV. Such a depolarized membrane potential will permit the L-type VDCCs to remain in an open state. Based on the simulation, Ca²⁺ would continually enter the growth cone of the neuron until the pulse ended. In pulse 2, the [Ca²⁺]_i rises at ∼15 nM/ms, and the total rise in [Ca²⁺]_i is directly proportional to the length of the pulse.

Translating the Results to Neurons of Any Size, Using a Passive Model

The active membrane simulation results described are for an average small DRG soma (24.1-μm diameter) and neurite (0.7-μm diameter). However, the active membrane simulation can be translated (intuitively and mathematically) to neurons of other sizes. The time constant of the passive membrane is calculated using the cable theory equation: τ_m = R_m·C_m (Rall 1969). The results of the passive models were fit with Eqs. 8–13 below.

Effect of neurite diameter.

For a neurite of length l and length constant λ (from Eq. 1) with the component of electrical field in the direction of the neurite (v_n; from methods, Long cylindrical neurite), the polarization voltage of the tip of a neurite (V_tip) increases with time (t) toward the steady-state polarization voltage (V_tip-ss) with a half-order exponential time constant (τ_n), shown in Eqs. 8–10. With the use of Eq. 10, the time constant for an infinitely long neurite with parameters from Table 1 is 2.58 ms. Shape variable 2 (s2), which controls the tightness of the elbow in the time-dependent rise of the tip voltage, is defined by the equation s2(l) = 1 − 0.1443·ln {1 − exp [−l/(1.577λ)]}.

$$V_{tip}\left(t,l,\lambda ,v_n\right)=V_{tip-ss}\left(l,\lambda ,v_n\right){\left[1-e^{-{\left(\frac{t}{{\tau }_n\left(l,\lambda \right)}\right)}^{S2}}\right]}^{\frac{1}{2\cdot S2}}$$ (8)

$$V_{tip-ss}\left(l,\lambda ,v_n\right)=v_n\cdot \lambda {\left[1-e^{-{\left(\frac{l}{\lambda }\right)}^{S1}}\right]}^{\frac{1}{S1}}$$ (9)

where s1 (shape variable 1) = 1.68, and

$${\tau }_n\left(l,\lambda \right)=0.8\cdot {\tau }_m\cdot {\left(1-e^{-{\left(\frac{l}{\lambda }\right)}^{S1}}\right)}^{\frac{2}{S1}}$$ (10)

Equation 8 can be simplified for neurites several times longer than the length constant or shorter than a fraction of the length constant, as well as for time constants longer and shorter than the membrane time constant. A long-duration pulse applied to a long neurite simplifies to V_tip-ss(t > τ, l > λ, λ, v_n) ≈ v_n·λ, so the polarization is proportional to the square root of diameter. A long-duration pulse applied to a short neurite simplifies to V_tip-ss(t > τ, l < λ, λ, v_n) ≈ v_n·l. For a short-duration pulse applied to any length neurite, the peak growth cone polarization is proportional to the square root of diameter (Eq. 11):

$$V_{tip}\left(t<\tau ,l,v_n,\lambda \right)=v_n\cdot \lambda \cdot \sqrt{\frac{t}{0.8\cdot {\tau }_m}}\mathrm{.}$$ (11)

Therefore, to adjust the results of models 1 and 2 in Figs. 3–5 for a long neurite of a diameter different from 0.7 μm, the actual applied ES intensity can be multiplied by $\sqrt{d_{\text{n-actual}}\hspace{0.17em}/\hspace{0.17em}0.7\mu \text{m}}$, and this adjusted voltage should be used for comparison with voltages in Figs. 3–5. For example, if an electrical field of 1,000 V/m stimulation was applied to a neurite of 2.8 μm (4 times greater than 0.7 μm), then the expected values for AP generation and [Ca²⁺]_i elevation should be obtained by using pulse intensities of 2,000 V/m in Figs. 3 and 5.

Effect of soma diameter.

A spherical soma is geometrically defined only by its diameter and does not have a variable length-to-diameter aspect ratio like a neurite. The steady-state voltage of the pole (V_pole-ss) of the soma is linearly proportional to half of the diameter of the soma (d_s) and is proportional to 3/2 of the voltage field (Eq. 12). The time constant (τ_s) of the soma is also linearly proportional to the diameter of the soma: τ_s(R_i, C_m, d_s) = R_i·C_m·(d_s/2). The time constant of a spherical soma with parameters from Table 1 is 176 ns.

$$V_{pole-ss}\left(d_s,v\right)=\frac{3}{2}\cdot v\cdot \frac{d_s}{2}$$ (12)

When an electrical field is applied, the soma begins to hyperpolarize and depolarize at the poles facing the positive electrode and negative electrodes, respectively. The polarization of both poles follow the same time course defined by the first-order exponential function in Eq. 13:

$$V_{pole}\left(t,d_s,v,R_i,C_m\right)=V_{pole-ss}\left(d_s,v\right)\left[1-e^{-\left(\frac{t}{\tau \left(R_i,C_m,d_s\right)}\right)}\right]\mathrm{.}$$ (13)

Thus a neuron with membrane properties from Table 1 would have a time constant of 72 ns for a 10-μm-diameter soma or 291 ns for a 40-μm-diameter soma. These time constants are orders of magnitude shorter than the time constants of the ion channels, and their variation does not affect the outcome of a pulse longer than 1 μs. Therefore, the practical outcome of variation in a neuron soma diameter is the proportional effect on the magnitude of hyperpolarization and depolarization of the poles. To adjust the results of model 3 in Figs. 3–5 for a soma with a diameter other than 24.1 μm, the actual applied ES intensity can be multiplied by d_s-actual/24.1 μm, and this adjusted voltage should be used for comparison with voltages in Figs. 3–5. For example, if an electrical field of 1,000 V/m was applied to a neuron with a soma that had a 50% greater diameter (36.3 μm), then the expected values for AP generation and [Ca²⁺]_i elevation should be obtained by using pulse intensities of 1,500 V/m in Figs. 3 and 5C.

DISCUSSION

Experimental Validation

This model has been validated by measuring the [Ca²⁺]_i of dissociated DRG neurons suspended in a three-dimensional collagen scaffold in vitro. [Ca²⁺]_i was measured using the intracellular fluorescent Ca²⁺ indicator fura 2-AM. The neurons were stimulated with electrical pulses from 10 μs to 100 ms in duration and from 70 to 70,000 V/m in intensity. The experimental validation was initiated only after the computational model was complete, so the computational model was not tuned to any experimental data from our laboratory. Because of resolution limits of [Ca²⁺]_i in neurites, the [Ca²⁺]_i was only calculated for the neuron somas. The imaged somas are best represented by the spectrum of the geometries of models 3–5. The computational estimation of the AP threshold for models 3–5 (Fig. 3) was within a factor of 2 of the threshold for ES intensity to cause an experimental increase in [Ca²⁺]_i. The methodology, detailed results, and discussion of these experiments are beyond the scope of this article and are anticipated in a forthcoming manuscript.

Designing an ES Protocol

Objective for elevation of [Ca²⁺]_i.

The mechanism for how ES causes an improvement in neuronal regrowth is not known. However, one proposed mechanism is through an elevation of intracellular Ca²⁺. In the present study, we explored one step in this potential Ca²⁺ mechanism for ES improvement of neurite growth. A subsequent step involved in the Ca²⁺ hypothesis of regenerative regrowth involves the downstream activation of genes and protein expression that was not considered in this work but that must somehow be involved in the control of regrowth. Intracellular pathways controlled by [Ca²⁺]_i are cooperative pathways as described in the detailed summary by Salazar et al. (2008). A cooperative pathway is most efficient when the pulses in [Ca²⁺]_i are at a level close to the dissociation constant of the Ca²⁺ sensor controlling the pathway. Such sensors would include phospholipase Cδ, protein kinase Cβ, calmodulin, calmodulin-dependent protein kinase II, and calcineurin. The dissociation constants of these proteins range from 0.4 to 35 μM. Although we do not know the particular Ca²⁺ sensors targeted for neurite regrowth, for the purposes of this discussion we will assume that the target Ca²⁺ sensor(s) have a dissociation constant of ∼1 μM in the DRG soma.

Pulse parameters.

In our defined DRG model, there are generally four types of efficient ES pulses: a supra-AP stimulation for neurite-only models, a supra-AP stimulation for all models, a sub-AP stimulation for neurite-only models, and a high-voltage, long-duration stimulation of model 1. The supra-AP stimulations for neurites are shown in the four models with neurite (models 1, 2, 4, and 5) that are efficiently stimulated by a supra-AP pulse with parameters of ∼10 ms in duration and ∼200 V/m in intensity (pulses 3, 4, 7, and 8 in Fig. 4). The supra-AP stimulation for the soma without a neurite (model 3) was most efficiently stimulated by a 300-μs and 3,000 V/m pulse (pulse 6 in Fig. 4). Only model 5 was inefficiently stimulated by the 300-μs and 3,000 V/m pulse. To be conservative, the pulse intensity could be increased to 6,000 V/m to cause APs directly in all models (Fig. 3). Only the four models with neurites (models 1, 2, 4, and 5) show any efficient sub-AP rise in [Ca²⁺]_i, and a pulse of ∼30 ms and ∼60 V/m generalizes the efficient sub-AP pulses (pulses 5 and 9 in Fig. 4). ES of greater than ∼300 V/m applied for more than 10 ms to model 1 causes L-type VDCCs to open for a continuous Ca²⁺ influx and increase during the pulse (pulse 2 in Fig. 6). Although this ES parameter causes an AP in all of the models that have neurites, this particular ES is distinct because it raises [Ca²⁺]_i in model 1 substantially higher than the other models. Additionally, this pulse parameter is the only pulse of reasonable efficiency that can cause >700 nM rise from a single pulse.

The choice of a pulse type is largely dependent on the regeneration scenario and the ultimate goal for using electrical stimulation. For the most efficient [Ca²⁺]_i elevation in growth cones and somas with neurites, supra-AP ES for neurite-only models is most suitable. If many neurons are lacking neurites in the regeneration scenario, then the supra-AP ES for all models may be a more suitable choice, or, to potentially cause directional neurite regrowth, either sub-AP ES or high-voltage, long-duration ES may be preferable. Furthermore, we do not know if the rate of [Ca²⁺]_i rise is a significant variable for regeneration. To achieve rates of [Ca²⁺]_i rise higher than the most efficient pulses, higher energy pulses are necessary.

Pulse train.

The upper limit of stimulation frequency is dependent on multiple factors including the duration of a pulse, the refractory period of the neuron for supra-AP pulses (∼15 ms for our model neuron), and the time it takes for the full [Ca²⁺]_i rise to occur (3–50 ms in Fig. 6). Based on these estimations, short depolarizing pulses would have an upper limit of stimulation of ∼50 Hz, and short hyperpolarizing pulses would have an upper limit of stimulation of ∼20 Hz for supra-AP ES and ∼10 Hz for sub-AP ES. If the supra-AP ES for neurite models (10 ms and 200 V/m) is chosen to elevate [Ca²⁺]_i in somas from ∼100 nM resting to 1 μM, this ES parameter would have to be applied in a pulse train of 90 pulses, because each pulse only increases [Ca²⁺]_i by ∼10 nM, conservatively (pulses 7 and 8 in Fig. 5). At 20 Hz, this pulse train would raise the [Ca²⁺]_i by ∼ 900 nM in 4.5 s. From experimentation in our laboratory using the kinetically delayed indicator fura 2-AM, embryonic chicken DRG neurons were observed to have an [Ca²⁺]_i time constant on the order of ∼14 s (not shown; unpublished observations) that roughly followed first-order dynamics. Furthermore, with the use of fura 2-AM, adult rat DRG were reported to have an ∼10-s [Ca²⁺]_i time constant (Duflo et al. 2004). Approximating first-order dynamics, pulse trains shorter than this time constant of 10 s should have an additive increase in [Ca²⁺]_i from each pulse, but pulse trains longer than the [Ca²⁺]_i time constant would reach an asymptotic total rise in [Ca²⁺]_i with continued stimulation. For our example, each 4.5-s pulse train can be applied every 20–40 s, allowing [Ca²⁺]_i to return to baseline between pulse trains, thereby maximizing [Ca²⁺]_i efficiency of cooperativity while reducing energy used by stimulation.

Hypotheses for ES Causing Directional Growth

We hypothesized that ES may cause directional growth through two mechanisms: 1) by causing APs in neurites already aligned with a field or 2) by causing preferential rise in [Ca²⁺]_i in neurites facing either the positive or negative electrode.

AP in aligned neurites.

The first mechanism for aligned directional growth would occur by causing APs in neurites that are facing toward the positive or negative electrode. In this scenario, somas with neurites aligned will experience more APs that would permit a preferential growth of neurites that are already aligned toward the positive or negative electrode. Patel and Poo (1984) found that 5-ms pulses greater than 125 V/m had preferential growth toward the negative electrode. This result is consistent with our model that estimates that negative electrode-facing neurites have an AP threshold of ∼125 V/m for 5-ms pulses (Fig. 5A) and that positive electrode-facing neurites have an AP threshold closer to ∼500 V/m for 5-ms pulses (Fig. 5B).

Preferential [Ca²⁺]_i rise in growth cones of aligned neurites.

The second mechanism for aligned directional growth would occur by elevating the [Ca²⁺]_i preferentially in neurites facing toward one electrode, using either sub-AP stimulation or high voltage (pulse 5 in Fig. 5) or long-duration ES from model 1 (pulse 1 in Fig. 5). This scenario would preferentially cause [Ca²⁺]_i in the growth cones of a neurite already facing toward the electrode, even if there are other growth cones attached to the same neuron facing in other directions. Because sub-AP stimulation only elevates [Ca²⁺]_i on the positive electrode-facing part of the neuronal membrane, this type of ES has greater potential for turning growth cones toward an electrode. Additionally, high-voltage, long-duration pulses that initiate a single propagated AP throughout the neuron would continuously elevate [Ca²⁺]_i in the negative electrode-facing growth cone. Patel and Poo (1982) found that neurites grow more quickly toward the negative electrode of a DC electrical field if the field is greater than 250 V/m, which is consistent with our model for high-voltage, long-duration stimulation.

Sub-AP DC Stimulation

Sub-AP DC stimulation has been shown to increase neuronal growth rate (Adams et al. 2014; Cork et al. 1994; Wood and Willits 2006), but the mechanism for this form of ES is not well explained by our simulation models. In our present models, sub-AP depolarizing DC only elevates [Ca²⁺]_i during the immediate milliseconds following the onset of the DC stimulation and actually depresses Ca²⁺ levels during the remainder of the DC stimulation. In fact, our model shows that DC stimulation only causes a substantial rise in [Ca²⁺]_i when the hyperpolarizing DC stimulation ends due to the rebound depolarization. One possible explanation for the increased growth from sub-AP DC stimulation is that in some experiments, the DC is occasionally interrupted due to circuit characteristics or increases in resistance in the electrode path. This interruption would effectively cause a series of long-duration, sub-AP pulses that could substantially raise [Ca²⁺]_i of neurites facing the positive electrode (shown in Fig. 5B) each time the DC is interrupted for at least a few milliseconds. A second possible explanation is that DC acts through a mechanism that our model does not explore, such as intracellular current or spontaneous neuronal activity.

Intracellular current.

The intracellular electrical field in the center of a neurite remains equal to the extracellular electrical field during the entirety of the electrical pulse. Cations and anions within a neurite would move toward the negative and positive electrodes, respectively. It is difficult to hypothesize the effect of such ionic movements within the neurites, but the ionic movements could affect transport of ionically charged proteins as they are transported along the length of the neurite. Relative ion concentrations at different sections of the neuron may also be altered due to the ionic movements.

Spontaneous neuronal activity.

Spontaneous alterations in the membrane potential of a neuron occur and can cause APs (Scarlett et al. 2009). A sub-AP DC field will change the excitability of parts of the neuron and will change the rise in [Ca²⁺]_i due to opening of VDCCs. For example, if a spontaneous AP were to reach a growth cone already hyperpolarized by sub-AP DC, then the growth cone would have a higher rise in [Ca²⁺]_i than a growth cone that is unaffected by DC. This hyperpolarization would leave some of the Na⁺ and Ca²⁺ channel gates in a more open state.

Both the interrupted DC explanation and the spontaneous neuronal activity explanation permit us to hypothesize that sub-AP DC would cause increased [Ca²⁺]_i in neurites facing the positive electrode (model 2), which is the opposite of high-intensity DC that increases [Ca²⁺]_i preferentially in model 1. Consistent with these hypotheses, Cork et al. (1994) found that neurites grow more quickly toward positive electrodes when low intensity DC fields between 30 and 100 V/m were applied to neurons.

Additional Considerations

A common goal of an ES protocol is to reduce electrolysis, which is dependent on electrode properties (for review see Merrill et al. 2005) but is generally achieved in practice by using biphasic pulses of shorter duration. In addition, whereas this model used rectangular pulses to make the values of pulse intensity and duration clear, the use of ramp, exponential, sinusoidal, and even Gaussian-shaped pulses may be 10–20% more efficient than rectangular pulses at generating APs (Sahin and Tie 2007; Wongsarnpigoon and Grill 2010). Finally, some neurons and axons have membrane time constants ∼0.2–1 ms (Rattay et al. 2012; Sahin and Tie 2007; Wongsarnpigoon and Grill 2010) shorter than the 3.22-ms membrane time constant in this model, so optimal pulse durations for generating APs will be reduced accordingly.

Conclusions

In conclusion, ES can efficiently elevate [Ca²⁺]_i in all geometrical models of neurons by generating APs. In models where ES hyperpolarizes the growth cone or soma of a neuron, sub-AP stimulation can also efficiently elevate [Ca²⁺]_i to levels comparable to supra-AP ES. Specific pulse parameters including those for sub-AP stimulation may be able to guide neurites by causing preferential [Ca²⁺]_i elevation only in neurites facing toward either the positive or negative electrode. This model provides a basis for choosing actual ES pulse durations, pulse intensities, pulse-train frequencies, and pulse-train durations for efficiently raising [Ca²⁺]_i in neuronal growth cones or somas.

GRANTS

This work was supported in part by National Institutes of Health (NIH) Grant GM008306-24 (to R. D. Adams), the Margaret F. Donovan Endowed Chair for Women in Engineering (to R. K. Willits), and NIH Grant 1R03EB015048 (to A. B. Harkins).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

R.D.A., R.K.W., and A.B.H. conception and design of research; R.D.A. performed experiments; R.D.A. analyzed data; R.D.A. and A.B.H. interpreted results of experiments; R.D.A. and A.B.H. prepared figures; R.D.A. and A.B.H. drafted manuscript; R.D.A., R.K.W., and A.B.H. edited and revised manuscript; R.D.A., R.K.W., and A.B.H. approved final version of manuscript.

APPENDIX

Gate variables n, m_T, and h_T are governed by a first-order differential equation with the form of

$$\frac{d{x}_i}{dt}=\frac{x_{i_{\infty }}-x_i}{{\tau }_{x_i}},$$

where x_i refers to any of the gate variables just listed. The x_i∞ and τ equations for all of the listed gates are governed by the membrane potential u = V_i − V_o in millivolts, and the τ equations are in milliseconds:

$${\tau }_n=\frac{1}{KFactor{Q}_{10}}\cdot \left[\frac{1}{\frac{0.001265\cdot \left(u+14.273\right)}{1-e\left(\frac{u+14.273}{-10}\right)}+0.125\cdot e^{\left(\frac{u+55}{-2.5}\right)}}+1\right]$$

$$n_{\infty }=\frac{1}{1+e^{\left(\frac{u+14.62}{-18.38}\right)}}$$

$${\tau }_{m_T}=\frac{2}{CaFactor{Q}_{10}}\cdot \left[\frac{1}{e^{\left(\frac{u+132}{-16.7}\right)}+e^{\left(\frac{u+16.8}{18.2}\right)}}+0.612\right]$$

$$m_{T_{\infty }}=\frac{1}{1+e^{\left(\frac{u+57}{-6.2}\right)}}$$

$${\tau }_{h_T}=\left\{\begin{array}{cc}\frac{1}{CaFactor{Q}_{10}}\cdot e^{\left(\frac{-200+467}{66.6}\right)}& u<-200\\ \frac{1}{CaFactor{Q}_{10}}\cdot e^{\left(\frac{u+467}{66.6}\right)}& -200\le u\le -80\\ \frac{1}{CaFactor{Q}_{10}}\cdot \left[e^{\left(\frac{u+22}{-10.5}+28\right)}\right]& -80<u\le 200\\ \frac{1}{CaFactor{Q}_{10}}\cdot \left[e^{\left(\frac{200+22}{-10.5}+28\right)}\right]& u>200\end{array}\right\}$$

$$h_{T_{\infty }}=\frac{1}{1+e^{\left(\frac{u+81}{4}\right)}}$$

Gate variables m_s, h_s, m_r, h_r, s_r, m_L, m_N, and h_N are governed by a first-order differential equation with the form of

$$\frac{d{x}_i}{dt}={\alpha }_{x_i}\left(1-x_i\right)-{\beta }_{x_i}-x_i,$$

where x_i refers to any of the gate variables just listed. The α and β equations are in ms⁻¹ for all of the listed gates and are governed by the membrane potential u = V_i − V_o in millivolts:

$${\alpha }_{m_s}=NaFactor{Q}_{10}\cdot \frac{11.49}{1+e^{\left(\frac{u+8.58}{-8.47}\right)}}$$

$${\beta }_{m_s}=NaFactor{Q}_{10}\cdot \frac{11.49}{1+e^{\left(\frac{u+67.2}{27.8}\right)}}$$

$${\alpha }_{h_s}=NaFactor{Q}_{10}\cdot 0.0658\cdot e^{\left(\frac{u+120}{-20.33}\right)}$$

$${\beta }_{h_s}=NaFactor{Q}_{10}\cdot \frac{3}{1+e^{\left(\frac{u-6.8}{-12.998}\right)}}$$

$${\alpha }_{m_r}=NaFactor{Q}_{10}\cdot \frac{1.032}{1+e^{\left(\frac{u+6.99}{-14.8712}\right)}}$$

$${\beta }_{m_r}=NaFactor{Q}_{10}\cdot \frac{5.79}{1+e^{\left(\frac{u+130.4}{22.9}\right)}}$$

$${\alpha }_{h_r}=NaFactor{Q}_{10}\cdot \frac{0.06435}{1+e^{\left(\frac{u+73.26415}{3.71928}\right)}}$$

$${\beta }_{h_r}=NaFactor{Q}_{10}\cdot \frac{0.13496}{1+e^{\left(\frac{u+10.27853}{-9.09334}\right)}}$$

$${\alpha }_{s_r}=NaFactor{Q}_{10}\cdot 0.00000016\cdot e^{\left(\frac{u}{-12}\right)}$$

$${\beta }_{s_r}=NaFactor{Q}_{10}\cdot \frac{0.0005}{1+e^{\left(\frac{u+32}{-23}\right)}}$$

$${\alpha }_{m_L}=CaFactor{Q}_{10}\cdot \frac{0.061\cdot \left(u-8\right)}{1-e^{\left(\frac{u-8}{-12.5}\right)}}$$

$${\beta }_{m_L}=CaFactor{Q}_{10}\cdot 0.058\cdot e^{\left(\frac{u-15}{-15}\right)}$$

$${\alpha }_{m_N}=CaFactor{Q}_{10}\cdot \frac{0.1\cdot \left(u-25\right)}{1-e^{\left(\frac{u-25}{-10}\right)}}$$

$${\beta }_{m_N}=CaFactor{Q}_{10}\cdot 0.4\cdot e^{\left(\frac{u+20}{-18}\right)}$$

$${\alpha }_{h_N}=CaFactor{Q}_{10}\cdot 0.003\cdot e^{\left(\frac{u+45}{-10}\right)}$$

$${\beta }_{h_N}=CaFactor{Q}_{10}\cdot \frac{0.03}{1+e^{\left(\frac{u+12}{-17}\right)}}$$