 pulse, with strength 40 kV/cm, duration 10 ns, and linear ramps that define the 1 ns rise and fall times. Although terminology varies, a common designation is nsPEF (nanosecond pulsed electric field), indicating pulses with durations less than 1 ls. Such pulses have received considerable attention since 2001, when two papers appeared several months apart (Schoenbach et al. 2001;Müller et al. 2001). These pulses are close to the conditions studied by MD (Tieleman et al. 2003;Vernier et al. 2006b;Vernier and Ziegler 2007;Levine and Vernier 2010;Fernández et al. 2012;Romeo et al. 2013;Delemotte and Tarek 2012), and involve pore densities expected for supra-EP (Stewart et al. 2004;Gowrishankar et al. 2006;Esser et al. 2009). Potential applications include nonthermal cell death by apoptosis (Beebe, et al. 2002;Beebe, et al. 2003;Nuccitelli et al. 2006;Garon et al. 2007;Nuccitelli et al. 2014), and a variety of other intracellular effects. Even the introduction of bleomycin is reported for nsPEF (Silve et al. 2012).

Results and Discussion
Electrical and Poration Behavior Figure 2 shows examples of coupled electrical and poration behavior. Movement of small ions through pores and toward or away from the membrane is used to compute currents, fields, and transmembrane voltages, with the latter two quantities emphasized here. Charged solute transport contributes negligibly to electrical behavior. Accordingly, solute transport by electrodiffusion is computed using the electric fields determined solely by small, ubiquitous ions.

Transmembrane Voltage Temporal Behavior
Electroporation is characterized by electrical creation of transient aqueous pores, usually described by a rate equation that is strongly dependent on the transmembrane voltage, D/ m (Weaver and Chizmadzhev 1996;Neu and Krassowska 1999;Smith 2011;Smith et al. 2013). At one extreme, a pore can be created in about one nanosecond for D/ m % 1:2 V in a planar membrane patch (Vasilkoski et al. 2006), with similar times reported by MD studies using pure lipid bilayers (Gurtovenko and Vattulainen 2007). For longer times much smaller voltages are sufficient (Melikov et al. 2001). Cell models show behavior consistent with similar rate equations (DeBruin and Krassowska 1999a, b;Stewart et al. 2004;Krassowska and Filev 2007) and include the present model (Smith 2011;Smith et al. 2013). The transmembrane voltage, D/ m, is a key quantity for pore creation, and for dynamic pore behavior involving expansion and contraction, with the latter important to pores shrinking to a small enough size that they are destroyed by thermal fluctuations (Pastushenko et al. 1979;Levine and Vernier 2010;Smith et al. 2013). Figure 2a, g shows D/ m averaged over the cathodic and anodic quadrants for the two pulses, with D/ m (t) behavior that is quite different due to the very different durations and rise/fall times. Figure 2a shows the response to the conventional EP pulse (1.5 kV/cm, 100 ls). The display emphasizes the transient behavior at the beginning of the pulse (first 3 ls) and at the pulse end (from 99 to 101 ls). The membrane charges passively until t ¼ 0:7 ls; when the rise in D/ m is interrupted by REB (Benz et al. 1979;Barnett and Weaver 1991). This transmembrane voltage peak of 1.2 V is associated with a burst of pore creation. The accompanying large conductance increase interrupts the D/ m rise, and forces a change in direction such that D/ m quickly decreases, even though the applied field is still increasing. Within a microsecond after the REB peak the rate of D/ m decreases slowly, and reaches a quasi-plateau of about 0.5 V. Figure 2b, k shows the angular profiles of D/ m at the end of the maximum values of the two pulses. For the 100 ls pulse this is 99 ls, which is much longer than five charging times ðs m ¼ 370 ns). But the idealized 10 ns pulse peak value ends at 9 ns, which is much shorter than s m. This means that pore creation begins rapidly, at about 5 ns (Fig. 2j). At D/ m % 1:7 V (Fig. 2g) pores are made rapidly, even though the membrane never charges fully. Instead, a large local conductance exists when the pulse maximum value ends at 9 ns, and D/ m begins to relatively slowly decrease to zero over about 40 ns (Fig. 2g).

Transmembrane Voltage Angular Profiles
Flattening of the transmembrane voltage profile around a cell is a classic indicator of EP, usually shown as the angular profile of D/ m. Flattening is observed experimentally (Kinosita et al. 1988;Frey et al. 2006;Flickinger et al. 2010), and is also exhibited by cell membrane models and has been found in cell model responses for both spherical (DeBruin and Krassowska 1999a; Smith and Weaver 2008;Talele et al. 2010;Sadik et al. 2013), and cylindrical membranes (Stewart et al. 2004;Gowrishankar et al. 2006;Smith et al. 2013). Figure 2b, h shows the model's angular response for the two pulses. In addition to profile flattening around the poles, the local peaks in Fig.  2b indicate transition regions that separate porated (shaded regions) from unporated membrane (DeBruin and Krassowska 1999a; Krassowska and Filev 2007;Esser et al. 2010).

Pore Densities
The model's poration response is shown as pore density plots (Fig. 2c, i) for both pulses, and shows where pores are present, independent of their size at the time indicated.
Here the pore density (number of pores per square lm) is indicated by white bands superposed on the cell membrane. The width of the white bands is proportional to pore density. Figure 2c shows two thin white bands that occupy about a third of the membrane, with 2:7 Â 10 4 pores (Fig. 3a-d) created by a 1.5 kV/cm, 100 ls pulse. The time is the end of the pulse peak, 99 ls, when pores have evolved to form two subpopulations (Fig. 3c). Similarly, Fig. 2i shows two broad white bands that occupy more of the membrane, with 2:1 Â 10 6 pores ( Fig. 3f, h) created by a 40 kV/cm, 10 ns pulse. The time is also at the end of the pulse peak, 9 ns. For this much larger and shorter pulse, there is very little pore expansion ( Fig. 3f-h). All the pores are small, but there are about 100-fold more pores.

Total Pore Number
The total number of pores in the cell membrane, N, is another partial indicator of EP. A general expectation is that more pores provide greater transport capability for small ions that dominate electrical behavior and for small solutes. According to the model, NðtÞ rapidly increases during a pulse, but after a pulse NðtÞ decreases exponentially with a 4 s pore lifetime (recovery or resealing time). This general behavior is a consequence of the importance of electrical interactions during a pulse. However, membrane depolarization results for most pulses, resulting in  Fig. 3 Pore histograms show the distribution of pore radii, Nðr p Þ, at various times in response to the 1.5 kV/cm, 100 ls pulse (a-d) and the 40 kV/cm, 10 ns pulse (e-h). The maximum pore radius r max ¼ 60 nm, but the display has been truncated to 40 nm in consideration of the maximum pore radii observed D/ m % 0 post pulse, with pore destruction proceeding spontaneously, due to local fluctuations. This is consistent with MD simulations, but MD lifetimes are much shorter, for reasons not yet understood.
For the conventional EP pulse, Fig. 2d shows the initial increase in total pores, which occurs in less than a microsecond, well before the pulse has reached its peak value. On a much longer time scale Fig. 2e shows N both during the pulse and for five pore lifetimes (20 s total) of stochastic pore decay post pulse. During essentially all of the post pulse decay, there are only thermalized pore distributions (Fig. 3d, h). The shape of the thermalized distribution is narrow, collapsed to only small pores. This narrow distribution is on average retained, but the magnitude, NðtÞ, of the population decays exponentially (Fig. 2e,  k).
The response to the nsPEF pulse ( Fig. 2j) also shows a rapid initial increase in total pores. In this case, the pulse ends (at 9 ns) before REB occurs, but already about two million pores have appeared, all small ( Fig. 3f-h). Both the small pores and their large number are features of supra-EP (Stewart et al. 2004;Smith and Weaver 2008;Joshi and Schoenbach 2010), which is also predicted to porate many organelles. The longer time scale result again shows a sharp increase in N, followed by the same exponential decay (Fig. 2k), but the magnitude of N is two orders of magnitude larger.

Membrane Conductance