Non-rectangular waveforms for neural stimulation with practical electrodes

Abstract

Historically the rectangular pulse waveform has been the choice for neural stimulation. The strength–duration curve is thus defined for rectangular pulses. Not much attention has been paid to alternative waveforms to determine if the pulse shape has an effect on the strength–duration relation. Similarly the charge injection capacity of neural electrodes has also been measured with rectangular pulses. In this study we questioned if non-rectangular waveforms can generate a stronger stimulation effect, when applied through practical electrodes, by minimizing the neural activation threshold and maximizing the charge injection capacity of the electrode. First, the activation threshold parameters were studied with seven different pulse shapes using computer simulations of a local membrane model. These waveforms were rectangular, linear increase and decrease, exponential increase and decrease, Gaussian, and sinusoidal. The chronaxie time was found to be longer with all the non-rectangular pulses and some provided more energy efficient stimulation than the rectangular waveform. Second, the charge injection capacity of titanium nitride microelectrodes was measured experimentally for the same waveforms. Linearly decreasing ramp provided the best charge injection for all pulse widths tested from 0.02 to 0.5 ms. Finally, the most efficient waveform that maximized the charge injection capacity of the electrode while providing the lowest threshold charge for neural activation was searched. Linear and exponential decrease, and Gaussian waveforms were found to be the most efficient pulse shapes.

Introduction

Electrical stimulation currently offers a method to improve the disrupted function in a number of neurological disorders and many new applications are on the horizon [1–6]. Historically the rectangular waveform has been the choice for the current or voltage pulses employed in neural stimulators. Current pulses are preferred over voltage pulses to eliminate variations in the stimulation threshold as a result of the changes in the electrode–tissue impedance. For rectangular stimuli, the amplitude and duration together determine the stimulus strength and therefore the volume of activation around the tip of the electrode. The relation between the amplitude and pulse duration was named strength–duration curve and it was formulated by Lapicque [7, 8] first who showed that the strength–duration curves for all excitable tissues had a similar form with different chronaxie times (see methods). Later Blair [9, 10] assumed that tissue can be modeled by an RC network and proposed an exponential function for the strength–duration curve. These two formulae have been found mostly sufficient to explain experimental data for intracellular and extracellular stimulation of neurons. Some reports recently criticized their ability to fit experimental data [11, 12].

The chronaxie time (C) and rheobase current (I_r) are the two parameters that determine the strength–duration curve. There is evidence that the chronaxie time for neural stimulation is a function of the stimulus waveform. Wessale et al quantitatively compared the threshold stimulus currents of rectangular and exponentially decaying waveforms, and found that the chronaxie time of the latter was about twice as long [13]. To our knowledge, there have been no other reports of how the strength–duration curve is affected by the stimulus pulse shape.

On the other hand, electrodes of micro scale are needed to localize the volume of activation in many neural prosthetic applications, particularly in the central nervous system. Small electrode sizes demand larger charge densities than traditional noble metal and capacitor electrodes can provide, such as platinum and tantalum oxide electrodes. Novel materials with higher charge injection capacities are being searched to reduce the size of stimulation electrodes [15–18].

The large interface capacitance of the electrode–electrolyte interface measured with slow cyclic voltammetry is not available at fast rates of current injection [14, 17]. Both capacitive and Faradaic-type interfaces, such as titanium nitride (TiN) and iridium oxide electrodes pass more charge at longer pulse widths of the rectangular stimulus waveform [16]. The differential behavior of the interface for short and long pulses led to the proposal of electrode–electrolyte interface models similar to that of a transmission line [15]. However, the effect of the stimulus waveform on the charge injection capacity of these novel electrode materials has not been investigated.

It leads from the above discussion that some non-rectangular pulse shapes may exist which can move the chronaxie time to longer pulse durations where the electrodes can handle more charge per unit surface area of the electrode. The optimal stimulus waveform should maximize the injectable charge through the electrode interface while keeping the activation threshold at a minimum.

In this report, the threshold charge for neural stimulation was investigated for seven different waveforms using a local mammalian nerve model [18]. The charge injection capacity of titanium nitride electrodes was also measured for the same waveforms experimentally. Considering these simulated and experimental data, the waveforms that minimized the threshold charge for activation and maximized the injectable charge were determined. Titanium nitride was chosen because of its popularity as an electrode material since it readily lends itself to reactive sputtering methods and provides significant charge injection rates. Preliminary results of this work were published in abstract form [19].

Methods

Computer simulations

Several models have been developed to study the basic mechanisms underlying the generation and propagation of action potentials [20–24]. In this study, simulation of the membrane dynamics was based on one of the mammalian nerve models available. The CRRSS model was developed by Chiu et al [23] and modified by Sweeney et al [18] to set the model temperature to 37 °C. The CRRSS model was chosen because of its ability to fire at fast rates (up to 2 kHz) that are observed in some parts of the mammalian nervous system thus making it suitable for functional electrical stimulation applications [25]. This model is similar to the Hodgkin-Huxley model [20] in terms of the gate variables except for the absence of the potassium channel. All simulations were conducted in Matlab^®. Model variables were generated recursively using a time resolution of 0.1 μs.

Seven stimulus waveforms shown in table 1 were tested while the pulse width was varied from 0.001 to 0.5 ms. The computer algorithm searched for the activation threshold at each pulse width by increasing the stimulus strength (K) in small steps. Activation was decided when the action potential peak crossed the zero line. The stimulus charge and energy were computed for this threshold stimulus.

Table 1.

Current pulse waveforms tested in this study. 0 < t < τ for all. K is the stimulus strength factor for all.

1		Rectangular (Rect)	K[u(t) − u(t − τ)]
2		Linear increase (LinInc)	Kt
3		Linear decrease (LinDec)	K(τ − t)
4		Exponential increase (ExpInc)	$K({\text{e}}^{-\frac{5(\tau -t)}{\tau }})$
5		Exponential decrease (ExpDec)	$K({\text{e}}^{-\frac{5t}{\tau }})$
6		Gaussian (Gauss)	$\frac{K}{(\tau /5)\sqrt{2\pi }}{\text{e}}^{-{(\frac{t-\frac{\tau }{2}}{\sqrt{2}\frac{\tau }{5}})}^2}$
7		Sinusoidal (Sin)	$K\text{sin}(\frac{\pi t}{\tau })$

The threshold charge (Q_th) was calculated as the integral of the stimulus waveform (equation (1)) and the threshold energy (E_th) as the integral of the stimulus waveform squared (equation (2)). Definitions of rheobase current (I_r) and chronaxie time (C) were adopted from the Lapicque's expression for strength–duration curve (equation (3) [7, 8]):

$$Q_{\text{th}}(\tau )={\int }_0^{\tau }{I}_{\text{th}}(t)\text{d}t$$ (1)

$$E_{\text{th}}(\tau )\propto {\int }_0^{\tau }{I}_{\text{th}}{\left(t\right)}^2\text{d}t$$ (2)

$$K_{\text{th}}(\tau )=I_r\left(1+\frac{C}{\tau }\right),$$ (3)

where τ is the pulse duration, and I_th(t) is the current pulse waveform at threshold strength (K_th). The stimulus strength (K_th) is a scalar for each waveform as defined in table 1. The electrode impedance is omitted from the energy equation (equation (2)) for simplicity since the impedance appears as a scalar for all waveforms.

Charge injection capacity (CIC) measurements

Silicon substrate electrodes were provided by the Center of Neural Communication Technology at University of Michigan. The details of electrode fabrication were reported earlier by this group [16]. Briefly, metal contacts were formed by sputtering 30 nm of titanium followed by 150 nm of iridium. The final titanium nitride (TiN) layer above the contacts was formed by sputtering titanium in a 50% nitrogen/50% argon atmosphere to a thickness of about 1 μm. In this paper, three TiN contacts with areas of 177 μm² were studied. The electrodes were placed in a phosphate buffered normal saline (pH = 7.4) at room temperature and the bias voltage was set to zero with respect to a large Ag/AgCl reference electrode. For measurements of CIC, cathodic current pulses were applied at a slow repetition rate (2.5 Hz) to allow recovery between pulses. The pulse duration was set to 20, 40, 60, 80, 100, 200, 300 and 500 μs. The selected stimulus waveforms shown in table 1 were generated in LabVIEW and outputted through a PCI 6071 data acquisition board (both from National Inst.). The computer generated voltage waveform was converted into a current pulse using a custom designed circuit to ensure a fast rise time (<0.5 μs) and thereby allowing an accurate measurement of the access voltage with a step function. The access voltage was subtracted from the applied voltage to find the electrode back voltage. The back voltage and the electrode current were sampled into the computer at 1 MHz. The spike triggered averaging method was employed to reduce the noise in the current signal.

The injected current amplitude was increased slowly with the help of a potentiometer in the voltage/current converter circuitry until the electrode back voltage reached a peak of −1.2 V. This value was reported as the negative limit of the water window for TiN electrodes [17]. This procedure was repeated at each pulse width and the maximum current amplitudes were recorded. The CIC was calculated as the area under the current waveform and plotted against the pulse width. The ratio of the measured charge injection capacity of the electrode over the simulated threshold charge from the nerve model was defined as the goal function to be maximized. This goal function was used to compare the waveforms quantitatively in terms of their stimulation effect. A larger value of the goal function implies that the selected waveform can activate the neurons with the smallest area of the stimulating electrode required. This in turn would allow smallest contact designs for the same stimulation effect in the tissue.

Results

Model variables

Response to a long stimulus pulse (0.2 ms) was studied at the threshold current amplitude to observe the effect of stimulus waveform on the model variables. The membrane voltage, and the gate variables m and h (figure 1) were plotted for each waveform. A sluggish response was observed in the membrane voltage and the m gate following the offset of the stimulus in cases of ExpInc and LinInc. The same effect was very small for the Rect, Gauss and Sin waveforms. A full scale action potential was generated even before the end of the pulse for ExpDec and LinDec stimuli. However, more importantly the trajectory of the membrane voltage and the gate variables were substantially different during the course of the stimulus for each one of the waveforms.

Figure 1.

Comparison of membrane dynamics for various stimulus waveforms for a 0.2 ms long pulse. (A) membrane voltage, (B) m gate and (C) h gate of the membrane model. All stimulus intensities are at threshold level. The threshold stimulus was decided when the peak of the membrane voltage reached zero as the stimulus strength was being increased in small steps.

Chronaxie times

From Lapicque's expression (equation (3)), the chronaxie time could be determined as the point where the energy was minimum (equation (2)). The chronaxie times were listed in table 2 for each stimulus waveform. The Rect waveform had the shortest chronaxie while the exponential waveforms had the longest.

Table 2.

Chronaxie, threshold charge (Q_th), and threshold energy (E_th) values for each pulse waveform considered. Q_th and E_th are measured at the chronaxie point. Minimum values are in bold.

Stimulus waveform	Chronaxie (μs)	Qth (pC cm−2)	Eth (nJ cm−2)
Rectangular	59.5	126	268
Linear increase	85	129	260
Linear decrease	90	136	274
Exp. increase	191	144	276
Exp. decrease	223	160	293
Gaussian	106	132	236
Sinusoidal	85	129	240

Charge as a function of pulse width

Threshold charges are plotted in figure 2 as a function of the pulse width. At short pulse widths, the charge for activation converges approximately to 85 pC cm⁻² in this local model for all waveforms. At longer durations, ExInc and ExpDec require the lowest amount of charge whereas the Rect calls for the highest. The LinDec crosses the charge curves for Sin and LinInc at approximately 0.18 ms and 0.21 ms, respectively. The threshold charges at the chronaxie times are given in table 2. Although Rect requires the highest charge at any given pulse width, it has the lowest charge threshold when each waveform is evaluated at its own chronaxie time.

Figure 2.

Threshold charge versus pulse width for all seven stimulus waveforms.

Energy as a function of pulse width

Threshold energy is plotted in figure 3 for all waveforms. Each curve has a minimum at the chronaxie time of the corresponding waveform. The minimum energy varies as a function of the stimulus type as listed in table 2. For pulse durations less than 40 μs, the Rect stimulus requires the lowest energy. Between 40–81 μs, Sin is the most efficient stimulus waveform with a minimum less than that of the Rect. From 81 μs to 228 μs, the Gauss is the most energy efficient waveform before ExpInc takes over. The waveforms that have lower minima than Rect are LinInc, Gauss and Sin (table 2).

Figure 3.

Threshold energy versus pulse width for all seven stimulus waveforms. Energy was calculated using equation (2) and the current pulse density that it takes to stimulate the local model. Color coding of the waveforms is the same as in figure 2.

Charge injection capacity of TiN electrodes

Charge injection capacity (CIC) of the TiN microelectrodes was measured experimentally as a function of pulse width (figure 4). The CIC increased several-fold from 0.02 ms to 0.5 ms for all waveforms. The LinDec allowed the maximum charge injection for all pulse durations. The ExpInc stimulus was the lowest followed by LinInc. The remaining four waveforms were very close to each other in terms of their CIC for all pulse widths.

Figure 4.

Charge injection capacity of the titanium nitride microelectrodes tested for the seven current pulse waveforms shown in table 1. Surface area of the electrodes was 177 μm². The voltage was limited at −1.2 V and the bias was zero. The mean CIC at 0.5 ms was 950 ± 100, 1057 ± 93, 740 ± 77, 928 ± 80, 538 ± 49, 906 ± 107, 932 ± 93 pC (n = 3 for all) for Rect, LinDec, LinInc, ExpDec, ExpInc, Gauss, and Sin waveforms, respectively.

The goal function: stimulation effect

The CIC divided by the activation threshold was used as a measure to compare the stimulus waveforms quantitatively. Because each parameter was obtained in different platforms the ratio could only be used for comparison as a unitless variable. This measure is plotted in figure 5 at the pulse width values where the CIC was measured with the TiN electrodes. For three out of seven waveforms, including the Rect, the maximum did not occur within the pulse width range studied. For the remaining four traces, the maximum was captured within the range. Three out of these four traces, LinDec, ExpDec, and Gauss, had maxima that were higher than that of the best value of the Rect stimulus measured at 0.2 ms. Thus, some of the non-rectangular waveforms considered here proved to be more efficient for the delivery of the stimulus current through TiN electrodes. This implies that the stimulation effect would be stronger for the maximum charge that can be delivered through an electrode with a given surface area. This in turn should result in more effective use of the electrode area and allow smaller electrode designs.

Figure 5.

The goal function. The CIC values measured in figure 4 were divided by the threshold charges simulated in figure 2 for each waveform. The vertical scale is in arbitrary units.

Discussion

This study shows that there is a pulse width which maximizes the stimulation effect by optimizing the trade-off between the strength-duration curve for neural activation and the CIC capacity of the electrode (figure 5). This optimum pulse width varies as a function of the stimulus waveform used. Although not demonstrated here, the optimum point will evidently be different for various materials also since the charge injection versus pulse width curve will be different. In our tests, both TiN and iridium oxide (not presented) electrodes showed increases of several-fold in the CIC as the pulse width was varied from 0.02 to 0.5 ms. A steeper charge injection curve for the electrode would move the peak of the goal function in figure 5 towards longer pulse widths. Another important conclusion that follows from figure 5 is that a non-rectangular stimulus waveform can generate a larger peak in the goal function and thus allow the design of smaller electrodes for the same stimulation effect. The exponentially decreasing pulse generated the goal function with the largest peak.

The rectangular shape was not the best choice for the CIC measurements either. A linearly decreasing ramp provided the largest CIC for the TiN electrodes that were tested. In general, it seems that linearly or exponentially increasing pulse shapes are not favorable to obtain the best CICs from the electrodes (figure 4). These waveforms did not produce very large peaks in the goal function either, as seen in figure 5.

In computer simulations, each stimulus waveform generated a unique pattern of membrane dynamics during action potential initiation (first 0.2 ms in figure 1). The differential effects were more evident for longer pulse durations. These simulations raise the question if there exists an optimal pulse shape that matches the membrane dynamics perfectly and thus minimizes the threshold energy. The total energy that is injected into the tissue may not be significant in neural prostheses with only a few channels of stimulation compared to the energy consumed in the control electronics. But, the energy per pulse may become an important parameter in applications where the number of channels exceeds hundreds such as in visual prostheses.

Our simulations confirm the previous report of Wessale et al [13] that the chronaxie time is stimulus waveform dependent. Chronaxie is useful as a measure to differentiate excitable cells in terms of their activation speed. It was historically defined using rectangular stimuli where the rate of current injection is constant for the duration of the pulse. A non-rectangular waveform implies a non-uniform current injection rate. It needs to be investigated if the chronaxie times with these alternative waveforms can provide a better scheme for classification of neurons found in different parts of the nervous system. Another relevant question regarding functional neural stimulation is whether the fiber size selectivity can be improved using non-rectangular stimulus shapes.

Finally, it should be noted that the strength–duration curves reported here are based on computer simulations of a local membrane model. The differential behavior of the membrane parameters for each stimulus waveform may not extrapolate to real nerves as predicted by our simulations. These results should be verified with in vivo experiments. Stimulus waveforms other than those studied here should also be tested for potentially better results.

Conclusions

Computer simulations of our local nerve membrane model show that the strength-duration relation and hence the chronaxie time varies as a function of the stimulus waveform. Some non-rectangular stimulus pulses provide more energy efficient neural stimulation than the rectangular pulses at optimal pulse widths. The CIC of titanium nitride microelectrodes is also stimulus waveform dependent. Linearly decreasing ramp provided the best charge injection for the pulse width range of 0.02–0.5 ms. Linear and exponential decrease, and Gaussian waveforms were most efficient in that they required the smallest electrode surface area or else generated the strongest stimulation effect for unit electrode area.

Appendix. Mammalian nerve axon model

Figure A.1 and the following equations show a local model for mammalian nerves developed by Chiu et al [23] and modified by Sweeney et al [18] for temperature.

The rate of change in the membrane potential V is given by

$$\frac{\text{d}V}{\text{d}t}=\frac{i_{\text{st}}-i_{\text{Na}}-i_{\text{L}}}{C_m},$$ (A.1)

where i_st is the stimulus current density, i_Na and i_L are current densities (μA cm⁻²) of sodium and leakage channels respectively, and are formulated as

$$i_{\text{Na}}=\overline{g_{\text{Na}}}{m}^{2}h(V-E_{\text{Na}}),$$ (A.2)

$$i_{\text{L}}=\overline{g_{\text{L}}}(V-E_{\text{L}}).$$ (A.3)

The gating variable m, and the corresponding opening and closing rate coefficients (α_m and β_m)are given as

$$m(t)=m_0-[(m_0-m_{\infty })(1-{\text{e}}^{-t/{\tau }_m})],$$ (A.4)

$$m_{\infty }=\frac{{\alpha }_m}{{\alpha }_m+{\beta }_m},$$ (A.5)

$${\alpha }_m=k\frac{126+0.363V}{1+{\text{e}}^{-\frac{V+49}{5.3}}},$$ (A.6)

$${\beta }_m=k\frac{{\alpha }_m}{{\text{e}}^{\frac{V+56.2}{4.17}}},$$ (A.7)

$${\tau }_m=\frac{1}{{\alpha }_m+{\beta }_m}.$$ (A.8)

Similarly, the equations for h, α_h and β_h are

$$h(t)=h_0-[(h_0-h_{\infty })(1-{\text{e}}^{-t/{\tau }_h})],$$ (A.9)

$$h_{\infty }=\frac{{\alpha }_h}{{\alpha }_h+{\beta }_h},$$ (A.10)

Figure A.1.

A local model for mammalian nerves developed by Chui et al [23] and modified by Sweeney et al [18] for temperature.

Table A.1.

Description of the terms used in the equations.

$\overline{g_{\text{Na}}}$	Maximum conductance of sodium channels	1445 mmho cm−2
$\overline{g_{\text{L}}}$	Maximum conductance of leakage channels	128 mmho cm−2
ENa	Equilibrium potential of sodium channels	35.64 mV
EL	Equilibrium potential of leakage channels	−80.01 mV
Cm	Membrane capacitance	2.5 μF cm−2
V0	Resting membrance potential	−80 mV

$${\beta }_h=k\frac{15.6}{1+{\text{e}}^{-\frac{V+56}{10}}},$$ (A.11)

$${\alpha }_h=k\frac{{\beta }_h}{{\text{e}}^{\frac{V+74.5}{5}}},$$ (A.12)

$${\tau }_h=\frac{1}{{\alpha }_h+{\beta }_h}.$$ (A.13)

The resting membrane potential (V₀) was set to −80 mV.

At higher temperatures the membrane kinetics becomes considerably faster, therefore an acceleration factor k was included in the rate coefficients (αs and βs) to account for the temperature effect [23]. k depends on a special constant Q₁₀ by the equation

$$k=Q_{10}^{\frac{T-T_0}{10}},$$ (A.14)

where Q₁₀ = 3, as suggested by Hodgkin and Huxley, T₀ is the experimental temperature in the CRRSS model (14 °C), and T is the temperature of the simulation.