2.1.

Human Embryonic Kidney (HEK) Cell Culture

HEK293T cells were maintained at 37°C, 5% ${\mathrm{CO}}_2$ in Dulbecco’s Modified Eagle Medium (DMEM) supplemented with 10% fetal bovine serum, 1% GlutaMAX-I, penicillin ($100\mathrm{U}/\mathrm{mL}$), and streptomycin ($100\mathrm{mg}/\mathrm{mL}$). For maintaining or expanding the cell culture, we used a 35 mm tissue-culture-treated culture dish (CorningWare, Corning, New York, United States). For each imaging experiment, cells in one 35 mm dish were transiently transfected with the construct to be imaged using polyethyleneimine (PEI) in a 3:1 PEI-to-DNA mass ratio. For all the imaging experiments, cells were replated on glass-bottomed dishes (Cellvis, D35-14-1.5-N, Mountain View, California, United States) 36 h after transfection. Imaging was performed $\sim 6\mathrm{h}$ after replating. Before optical stimulation and imaging, the medium was replaced with extracellular (XC) buffer containing 125 mM NaCl, 2.5 mM KCl, 3 mM ${\mathrm{CaCl}}_2$, 1 mM ${\mathrm{MgCl}}_2$, 15 mM HEPES, and 30-mM glucose (pH 7.3). For BeRST1 experiments in HEK cells, cells were stained with $1\mu \text{M}$ BeRST1 for 30 min prior to 3× wash with XC buffer before imaging.

2.2.

Microscope and Illumination Control

All imaging was performed using the Luminos bi-directional microscopy control software²⁸ on a custom-built upright microscope equipped with 1P and 2P illumination paths and a shared emission path to a scientific complementary metal oxide semiconductor (sCMOS) camera (Hamamatsu, ORCA-Flash 4.0 v2, Bridgewater, New Jersey, United States). The 1P illumination path contained a 488 nm laser (Coherent OBIS, Saxonburg, Pennsylvania, United States), a 532 nm laser (Laserglow LLS-0532, North York, Canada), and a 635 nm laser (Coherent OBIS). The outputs of the 488 nm and 532 nm lasers were modulated using a multichannel acousto-optic tunable filter (Gooch & Housego PCAOM NI VIS driven by G&H MSD040-150, Ilminster, United Kingdom). The 635 nm laser was modulated using its analog modulation input and an external neutral density filter wheel (Thorlabs, Newton, New Jersey, United States). The 488 nm and 532 nm lasers were patterned via a digital micromirror display (Vialux V-7001 V-module, Chemnitz, Germany). To convert illumination intensity to power per cell, we approximated HEK cells as circles with a $10\mu \mathrm{m}$ diameter, e.g., an intensity of $1\mathrm{W}/{\mathrm{cm}}^2$ corresponded to $0.8\mu \mathrm{W}$ per cell.

The two-photon illumination path comprised an 80 MHz tunable femtosecond laser (InSight DeepSee, Spectra Physics, Milpitas, California, United States), an electro-optic modulator (ConOptics 350-80-02, Danbury, Connecticut, United States) and two galvo mirrors for steering (Cambridge Technology 6215H driven by 6671HP driver). Power calibration was performed with a Thorlabs PM400 power meter with a photodiode-based sensor (S170C) and a thermal sensor (S175C) for 1P and 2P illumination, respectively. Electrical stimuli and measurements were performed using a National Instruments 6063 PCIe DAQ. All imaging was performed with a 25× water immersion objective [Olympus XLPLN25XWMP2, Tokyo, Japan, 2 mm working distance, numerical aperture (NA) 1.05]. At each wavelength, the dispersion was adjusted to maximize the 2P fluorescence signal. The wavelengths for 2P excitation of QuasAr6a and BeRST1 were chosen to drive the $S_0$ to $S_2$ transition rather than the $S_0$ to $S_1$ transition because the $S_2$ transition was stronger.

2.3.

Measuring Voltage-Sensitive Fluorescence In Vitro

All imaging and electrophysiology experiments were performed in an XC buffer. Concurrent whole-cell patch clamp and fluorescence recordings were acquired on the microscope described above. Filamented glass micropipettes were pulled to a tip resistance of 5 to 8 MΩ and filled with an internal solution containing 125 mM potassium gluconate, 8 mM NaCl, 0.6 mM ${\mathrm{MgCl}}_2$, 0.1 mM ${\mathrm{CaCl}}_2$, 1 mM EGTA, 10 mM HEPES, 4 mM Mg-ATP, and 0.4 mM Na-GTP (pH 7.3), adjusted to 295 mOsm with sucrose. Whole-cell patch clamp recordings were performed with an Axopatch 200B amplifier (Molecular Devices, San Jose, California, United States). Fluorescence was recorded in response to a square wave from $-70$ to $+30\mathrm{mV}$.

2.4.

In-House AAV Packaging

AAV2/9 JEDI-2P vectors were packaged in-house based on a published protocol.²⁹ Briefly, $\sim 50\%$ to 70% confluent HEK293T cells grown in DMEM supplemented with 5% FBS were triple transfected with pHelper, pAAV ITRexpression, and pAAV Rep-Cap plasmids using acidified (pH 4) PEI (DNA-to-PEI ratio 1:3) in $\sim 1$ to 2 T175 flasks ($\sim 2\times {10}^7\text{cells}/\text{flask}$). The adeno-associated virus (AAV)-containing medium was harvested on day 3, and the AAV-containing medium and cells were harvested on day 5. For the second harvest, AAVs were released from the cells with citrate buffer (55 mM citric acid, 55 mM sodium citrate, and 800 mM NaCl, 3 mL per flask). The two harvests were then combined and precipitated with PEG/NaCl (5×, 40% PEG 8000 ($w/v$), 2.5 M NaCl, 4°C overnight). The low-titer virus was then purified with chloroform extraction [viral suspension and chloroform 1:1 ($v/v$)], aqueous two-phase partitioning [per 1 g of the AAV-containing supernatant, add 5 g of 20% $({\mathrm{NH}}_4{)}_2{\mathrm{SO}}_4$ solution and 1.5 g of 50% PEG 8000 solution], and iodixanol discontinuous gradient centrifugation (15%, 25%, 40%, and 54% iodixanol gradient prepared from OptiPrep] [60% ($w/v$) iodixanol, Axis-Shield PoC AS, Dundee, United Kingdom]. The purified AAV titer was determined via quantitative polymerase chain reaction (SYBR Green, primer for forward ITR: 5′-GGAACCCCTAGTGATGGAGTT-3′; primer for reverse ITR sequence 5′-CGGCCTCAGTGAGCGA-3′).

2.5.

In Vivo Imaging

All animal experiments were approved by the Institutional Animal Care and Use Committee of Harvard University. The cranial window surgery for in vivo imaging was based on previously published protocols.⁷ Briefly, an adult CD1 mouse was injected with a 50-nL viral mix in four sites in the whisker barrel cortex at 100, 200, 300, 400, and $500\mu \mathrm{m}$ below the tissue surface. The viral mix had a final concentration of $5\times {10}^{12}\mathrm{vg}/\mathrm{mL}$ pAAV-EF1a-DIO-JEDI-2P-Kv2.1motif and $1\times {10}^{11}$ CamKII-Cre in AAV 2/9. A cranial window and mounting plate were then installed over the injection sites. Two weeks after surgery and injection, a head-fixed CD1 mouse was imaged at 1.5% isoflurane with the dose adjusted to maintain a stable breathing rate. The mouse was kept on a heating pad (WPI ATC2000) to maintain a stable body temperature at 37°C, and its eyes were kept moist using ophthalmic eye ointment. 2P imaging was performed at a wavelength of 930 nm. The mouse was imaged for 2 h after which it recovered in less than 10 min.

Light for 1P imaging in vivo was patterned via a digital micromirror device to selectively illuminate the targeted cell body, as described above.

2.6.

Data Analysis

All analysis was performed in MATLAB. Cell fluorescence was measured from the regions of interest (ROIs) manually selected to lie on the cell membrane. 1P and 2P recordings from the same cell used identical analysis ROIs. Fluorescence from cell-adjacent illuminated areas was used to estimate the background.

To estimate shot noise for an ideal detector (as an upper bound of performance), we converted camera counts to incident photons by dividing by the camera quantum efficiency ($\mathrm{QE}=\sim 67\%$ at 525 nm) and multiplying by the conversion factor ($\mathrm{CF}=0.46$ photoelectrons/digital count).

3.1.

Shot Noise Constrains Functional Fluorescence Imaging

Shot noise imposes a fundamental limit on imaging performance. A source that generates, on average, $N$-detected photons will have fluctuations with standard deviation $\sqrt{N}$. To detect a 1-ms spike ($\mathrm{\Delta }F/F\sim 10\%$) with an SNR of 10, it requires determining fluorescence to 1% precision in 1 ms. We adopt 1% precision in 1 ms as a reasonable standard for a high-SNR voltage recording. Due to shot noise, this standard requires detecting at least ${10}^4\text{photons}/\mathrm{ms}$ or ${10}^7\text{photons}/\mathrm{s}$.

More generally, imaging an event of magnitude $\mathrm{\Delta }F/F=\beta$ with a given SNR requires determining fluorescence to a precision of $\beta /\mathrm{SNR}$. The photon flux ($\mathrm{\Gamma }$) required for a measurement rate ($f$), signal level ($\beta$), and SNR is

If one wishes only to detect spikes in pyramidal cells, one might tolerate $\mathrm{SNR}=3$ and $f=400\mathrm{Hz}$. A highly sensitive GEVI could give $\beta =0.2$ for a spike.⁶ Under these conditions, the minimum detection rate is $\mathrm{\Gamma }=9\times {10}^4\text{photons}/\mathrm{s}$. In comparison, for a typical calcium imaging scenario, $\mathrm{SNR}=10$, $\beta =1$, and $f=30\mathrm{Hz}$,³⁰ implying $\mathrm{\Gamma }=3000\text{photons}/\mathrm{s}$, which is 30-fold less than even low-SNR spike detection via voltage imaging. Thus, the brief duration of voltage spikes and the low fractional sensitivity of existing voltage indicators conspire to make voltage imaging a very photon-greedy technique.

Filtering in space or time can increase the effective value of $N$ at a given pixel, at the cost of lower spatial or temporal resolution, but filtering does not change the $\sqrt{N}$ shot noise scaling. We discuss advanced analysis techniques below.

3.2.

1P Versus 2P Excitation of Commonly Used Voltage Indicators

We compared the brightness of commonly used voltage indicators in HEK293T (HEK) cells under alternating wide-field 1P illumination and 2P spiral illumination with an 80-MHz pulsed laser. Point-scanning 2P imaging typically uses a single-element detector such as a photomultiplier tube. We used a shared camera-based detection path for both 1P and 2P imaging to ensure equal photon detection efficiencies and to facilitate quantitative comparisons [Figs. 1(a)–1(c); Sec. 2: Methods]. For real-world 2P imaging, the choice of the detector can impact system performance via detector electronic noise and quantum efficiency. In our measurements, the per-pixel count rates were high enough that electronic noise contributed little to overall noise, compared with shot noise. To facilitate the translation of our results to other systems, we converted digital camera counts to photons incident on the camera using the known camera quantum efficiency and digitizer gain. This provides a “best-case” signal estimate for an ideal detector.

Fig. 1

Comparison of 1P and 2P brightness and sensitivity of fluorescent voltage indicators. (a) Diagram of the experiment. HEK cells were sequentially illuminated with wide-field 1P light in steps of increasing intensity and then by spiral scan 2P steps of increasing intensity. (b) Example HEK cell expressing the GEVI ASAP3. The 2P spiral scan pattern is shown in red, and the analysis ROI is shown in green. $\text{Scale bars}=5\mu \mathrm{m}$. (c) Example single-trial data for a cell expressing ASAP3. (d) Top: fluorescence in panel (c) as a function of 1P intensity with a linear fit. Bottom: fluorescence as a function of 2P power with quadratic fit. (e) Log–log plot of count rate versus optical power on the cell for seven voltage indicators. 1P data, filled symbols; 2P data, empty symbols. Error bars are standard error of the mean from at least $n=8$ cells. The excitation wavelengths used for 1P (2P) excitation of each of the reporters were ASAP3 488 nm (930 nm), BeRST1 635 nm (850 nm), JEDI-2P 488 nm (930 nm), QuasAr6a 635 nm (900 nm), ${\text{Voltron}2}_{525}$ 488 nm (930 nm), ${\text{Voltron}2}_{585}$ 594 nm (1100 nm), and ${\text{Voltron}2}_{669}$ 635 nm (1220 nm). A horizontal line is shown at $1.5\times {10}^7\text{counts}/\mathrm{s}$, equivalent on our camera to ${10}^7$ impinging photons/s. (f) Whole-cell patch clamp protocol for measuring voltage sensitivity under 1P and 2P excitation. (g) Average voltage responses of JEDI-2P and ${\text{Voltron}2}_{525}$ under 1P and 2P illumination. (h) Ratio of voltage contrast under 2P versus 1P illumination for JEDI-2P and ${\text{Voltron}2}_{525}$, $n=5$ cells per construct.

We compared a voltage-sensitive dye (BeRST1³¹), an opsin-derived GEVI imaged via intrinsic retinal fluorescence (QuasAr6a³²), a chemigenetic FRET-opsin GEVI (Voltron2³³), and two GEVIs that couple a voltage-sensitive phosphatase (VSP) to a circularly permuted partner fluorophore (ASAP3⁴ and JEDI-2P⁶).

As expected, the fluorescence scaled linearly with 1P illumination intensity and quadratically with 2P intensity [Fig. 1(d)]. For all but one indicator, 2P illumination required at least ${10}^4$-fold greater time-averaged power per cell to achieve comparable counts to 1P illumination [Fig. 1(e)]. The ratio of 2P-to-1P powers for QuasAr6a was only $\sim 300$ due to the requirement for high-intensity 1P illumination and selection rules that favor 2P over 1P excitation in opsins.³⁴ Consistent with prior reports,¹ we found that chemigenetic indicators were brighter under both 1P and 2P illumination than their purely protein-based counterparts, though our measurements did not assign the relative contributions of expression level versus per-molecule brightness.

The huge difference in optical power requirement between 1P and 2P (80 MHz) excitation is consistent with published reports: 2P imaging of JEDI-2P was reported at a power of 9 to $12\mathrm{mW}/\text{cell}$,⁶ whereas 1P imaging of similar GFP-based GEVIs is typically performed at 1 to $10\mathrm{W}/{\mathrm{cm}}^2$,^4,6 corresponding to 1 to $10\mu \mathrm{W}/\text{cell}$. Estimates based on tabulated 1P and 2P absorption coefficients³⁵ give a similar factor of $\sim {10}^4$ difference in power efficiency (see the Supplementary Material for calculation).

We then measured the voltage sensitivity of one representative from each GEVI family, comparing 1P and 2P illumination. Using whole-cell voltage clamp in HEK cells [Fig. 1(f)], we found that the contrast ($\mathrm{\Delta }F/F$ per 100 mV) of the VSP-based JEDI-2P was similar for 1P and 2P illumination. The opsin-based chemigenetic indicator ${\text{Voltron}2}_{525}$ showed voltage sensitivity under 1P but not 2P illumination [Figs. 1(g) and 1(h)]. Loss of voltage sensitivity under 2P illumination of FRET-Opsin GEVIs has been reported previously.³⁶ We recently determined the photophysical mechanism underlying this effect³⁷ but did not pursue 2P imaging of FRET-Opsin GEVIs here.

3.3.

Testing the Dependence of 1P and 2P Signals as a Function of Depth

To characterize the depth dependence of 1P and 2P voltage imaging in brain tissue, we sparsely expressed soma-localized JEDI-2P in the mouse cortex. Chien et al.³⁸ previously showed that for voltage imaging in brain tissue, restricting 1P illumination to the soma led to an approximately eightfold improvement in signal-to-background ratio compared with wide-field illumination, by minimizing the background from off-target illumination. We thus compared the soma-targeted 1P imaging and raster-scanned 2P imaging at several depths [Figs. 2(a) and 2(b); Sec. 2: Methods].

Fig. 2

Depth-dependent 1P and 2P signals in the brain. (a) Experimental protocol. In a mouse expressing JEDI-2P, raster-scanned 2P ($\lambda =920\mathrm{nm}$) and DMD-patterned 1P imaging ($\lambda =488\mathrm{nm}$) were alternately applied to neurons at different depths. 1P illumination patterned to cell-free regions was used to estimate the background signal. (b) Example 1P and 2P images of cells at three depths. $\text{Scale bars}=5\mu \mathrm{m}$. (c) Estimated signal-to-background ratio for $n=43$ neurons under patterned 1P illumination. (d) Mean count rate from the membranes of $n=43$ neurons under 2P illumination. The color indicates excitation power. The inset shows the cell at $473\mu \mathrm{m}$ depth (boxed on the graph). $\text{Scale bar}=5\mu \mathrm{m}$.

With 1P illumination, JEDI-2P-expressing cells were resolvable down to $d\sim 200\mu \mathrm{m}$. The challenge for 1P imaging at greater depths was not shot noise but rather a decrease in signal-to-background ratio. At depths $>200\mu \mathrm{m}$, cells were not distinguishable from the background by 1P imaging with patterned 488 nm excitation [Fig. 2(c)].

With 2P illumination, cells were resolvable to $d=473\mu \mathrm{m}$, the greatest depth we tested [Fig. 2(b)]. As the cell depth increased, we increased the 2P laser power to maintain a sufficient count rate to resolve the cells, up to $P_0=140\mathrm{mW}$ at $d=473\mu \mathrm{m}$.

3.4.

Thermal Limits to 2P Excitation Power in Brain Tissue

The heating caused by 2P illumination can transiently perturb neural function and at high levels can damage tissue. Most ion channel gating properties have a $Q_{10}$ (i.e., ratio of rates at temperatures separated by 10°C) between 1 and 3.³⁹ Changes of 1°C can cause changes in neuronal firing rates.⁴⁰ In rodents, brain temperature may fluctuate under physiological conditions by up to 4°C,⁴¹ and implantation of a glass imaging window may lead to some local brain cooling, partially canceling the effect of laser heating. Podgorski and Ranganathan⁴² found lasting damage after continuous illumination of a $1{\mathrm{mm}}^2$ scan at 250 mW, corresponding to a steady-state temperature change of $\sim 5°\mathrm{C}$.

The relation between laser power and heating depends on the scan area, scan pattern, and measurement duty cycle. For a $1\text{-}{\mathrm{mm}}^2$ square scan pattern, Podgorski and Ranganathan found steady-state temperature coefficients between 0.012 and $0.02°\mathrm{C}/\mathrm{mW}$ at wavelengths from 800 to 1040 nm, equivalent to a temperature rise of $<2°\mathrm{C}$ at 100 mW illumination. They simulated the dependence of temperature rise on the scan area and found a weak dependence. Their results predict a maximum temperature coefficient of $0.03°\mathrm{C}/\mathrm{mW}$ for a square scan of side length $20\mu \mathrm{m}$ (representing a single neuronal soma). The precise value of this coefficient depends both on the brain region and the wavelength.⁴³ Finally, Podgorski and Ranganathan showed that reducing the illumination duty cycle to 10 s on and 20 s off allowed brighter illumination to be used during “on” periods while still keeping time-averaged heating beneath the damage threshold. Some experiments may permit low-duty cycle imaging, whereas others may not. Hereafter, we use 200 mW as a reasonable upper bound on the power, acknowledging that this limit may be up to approximately twofold higher (or lower) depending on many experimental details.

3.5.

Estimating Measurable Cells as a Function of Depth

Shot noise places an upper bound, $N_{\text{cells}}^{2P}$, on the number of cells that can be measured simultaneously via 2P illumination with a given illumination power and SNR. Under a protocol that sequentially visits single cells, $N_{\text{cells}}^{2P}$ depends on both the brightness and voltage sensitivity (see the Supplementary Material for derivation),

Here, $A$ is the brightness coefficient derived from the HEK cell experiments [Eq. (S1) in the Supplementary Material], $\tau$ is the integration time, $P$ is the laser power at the focus, $\beta =\mathrm{\Delta }F/F$ per spike, $\varphi$ is the fraction of the scan that intersects with the cell membrane (i.e., the imaging duty cycle), and SNR is the target ratio of the spike amplitude to shot noise. The value of $A$ is specific to the laser repetition rate, pulse width, focal parameters, and detection optics. We discuss the effects of varying these parameters below. For an analysis that includes the effect of light scattering on depth-dependent collection efficiency, see Ref. 15.

The parameter $\varphi$ approaches 1 for perfectly membrane-targeted illumination. To estimate $\varphi$ for a raster scan over a bounding box around a single cell body, we examined 2P images of pyramidal cells with membrane-targeted fluorescent tags in cortex layer 2/3. In these images, the membrane-labeled area fraction within the bounding box was ${\varphi }_{\mathrm{bb}}=0.18\pm 0.07$ (mean ± std, $n=10$ cells). For a raster scan over multiple sparsely expressing cells, $\varphi$ is lower than ${\varphi }_{\mathrm{bb}}$ by a factor of the sparsity. The low values of $\varphi$ for raster-scanned 2P imaging are a consequence of the membrane-localized signal and highlight the importance of membrane-targeted illumination. However, precise targeting of the illumination to the membrane increases sensitivity to motion artifacts. The ULoVE technique brackets the membrane with pairs of spots and thereby mitigates the effect of small motions, at the cost of less-than-optimal membrane targeting of the spots.⁴

Equation (2) can be used to predict the scaling of $N_{\text{cells}}^{2P}$ as a function of depth, $d$, for 2P voltage imaging. The power at a laser focus decays exponentially with $d$, with an extinction length, $l_e$, in brain tissue. At $\lambda =920\mathrm{nm}$, $l_{\mathrm{e}}$ is between 112 (Refs. 15 and 44) and $155\mu \mathrm{m}$.⁴⁵ Due to the quadratic dependence of the 2P signal on focal intensity, the 2P signal decays with a length constant of $l_{\mathrm{e}}/2$. Substituting $P=P_0{e}^{-d/l_e}$ into Eq. (2) implies a decay in $N_{\text{cells}}^{2P}$ by a factor of 10 for every 130 to $180\mu \mathrm{m}$ increase in $d$ [Fig. 3(a)], assuming constant power $P_0$ into the tissue.

Fig. 3

Scaling of 2P voltage measurements with depth and with GEVI properties. (a) Predicted number of simultaneously measurable cells as a function of depth, based on brightness derived from HEK cells expressing JEDI-2P (Fig. 1). We assumed a spike contrast of $\beta =0.2$; target SNRs of 3 (green dotted), 5 (blue dashed), or 10 (red solid) in an integration time $\tau =1\mathrm{ms}$; a total power $P_0=200\mathrm{mW}$, targeting fraction $\varphi =1$, scattering length $l_{\mathrm{e}}=112\mu \mathrm{m}$,⁴⁴ and a detector with perfect quantum efficiency. (b) Predicted number of simultaneously measurable cells at $\mathrm{SNR}=10$ and 200 mW power for each experimentally measured single-cell count rate reported in Fig. 2(d). (c) Number of simultaneously measurable cells under 2P illumination at a depth of $500\mu \mathrm{m}$, assuming brightness and contrast improvements of future GEVIs, target SNR of 10, and all other parameters as in panel (a).

We applied Eq. (2) to the instantaneous count rates measured from the cell membranes [Fig. 2(d)] to estimate $N_{\text{cells}}^{2P}$ for $\varphi \sim 1$, i.e., a perfectly membrane-targeted annular scan pattern. We converted the digital camera counts to collected photon rates to provide an upper performance bound for a perfect detector. Modern back-illuminated sCMOS cameras have detection efficiencies that approach 100%. We assumed $\beta =0.2$, based on the reported spike response of JEDI-2P⁶ and a target shot noise-limited SNR of 10 in a 1 kHz bandwidth, and extrapolated the count rates to an input power of $P_0=200\mathrm{mW}$. The estimated number of measurable cells decreased quickly at $d>200\mu \mathrm{m}$ and dropped below three at $d>470\mu \mathrm{m}$ and input power 200 mW [Fig. 3(b)]. These results are similar to the prediction from the simple model using the 2P count rates from our HEK cell experiments [Fig. 3(a), blue line].

The palette of available voltage indicators is continually improving.⁴⁶ We therefore considered the scaling of $N_{\text{cells}}^{2P}$ with changes in brightness and spike height ($\beta$). $N_{\text{cells}}^{2P}$ depends linearly on $A$ and quadratically on $\beta$ [Fig. 3(c)]. An order-of-magnitude increase in molecular brightness coupled with a 2.5-fold increase in $\beta$ compared with JEDI-2P could enable high SNR measurement of $>8$ cells at depths up to $500\mu \mathrm{m}$ using the optical configuration we considered above.

3.6.

Effect of GEVI Kinetics

Some GEVIs have response times that are slow compared with the duration of a spike. On the one hand, this blunts the amplitude of the spike response; on the other hand, it permits one to average for longer to detect whether a spike has occurred (assuming that the interval between spikes remains long compared with the recovery time of the GEVI). Here, we analyze this tradeoff.

Consider a GEVI subjected to a voltage step that induces a steady-state change in fluorescence, $\frac{\mathrm{\Delta }F}{F}=M$. Assume that the GEVI responds to a voltage step of duration $t$ with exponential response time constants ${\tau }_{\text{on}}$ and ${\tau }_{\text{off}}$ [Figs. 4(a) and 4(b)]. We define $R$ as the area under the curve of $\mathrm{\Delta }F/F$ versus $t$ (see the Supplementary Material for derivation)

Fig. 4

Effect of reporter kinetics on signal. (a) Response of a reporter to a 1-ms voltage pulse. An exponential rise with time constant ${\tau }_{\text{on}}$ reaches a maximum of $\beta$ followed by an exponential decay with time constant ${\tau }_{\text{off}}$. The signal comprises the total area under both response phases (cross-hatched). (b) Area under the curve in panel (a) as a function of ${\tau }_{\text{on}}$ and ${\tau }_{\text{off}}$, keeping constant pulse duration (1 ms) and steady-state voltage sensitivity (${\beta }_{\mathrm{ss}}=0.2$). Increasing ${\tau }_{\text{off}}$ allows longer integration time, whereas increasing ${\tau }_{\text{on}}$ truncates the response. (c) Effects on SNR of changing ${\tau }_{\text{on}}$ or ${\tau }_{\text{off}}$ while keeping the other fixed. The fixed parameter is shown in the legend, and variable 1 is indicated by the $x$-axis. At large ${\tau }_{\text{on}}$, $\mathrm{SNR}\sim 1/{\tau }_{\text{on}}$. At large ${\tau }_{\text{off}}$, $\mathrm{SNR}\sim {\tau }_{\text{off}}^{1/2}$.

Equation (3) assumes the collection of the entire tail of the decay and thus provides an upper bound to the signal. For a GEVI with a symmetric upstroke and downstroke (${\tau }_{\text{on}}={\tau }_{\text{off}}$), $R=M$ $t$. For a finite integration time equal to $t+{\tau }_{\text{off}}$, the relation of SNR and $R$ is (see the Supplementary Material for derivation)

A large ${\tau }_{\text{on}}$ will decrease the magnitude of the response $\beta$ to a short electrical spike, decreasing instantaneous SNR proportionally. A large ${\tau }_{\text{off}}$, however, increases the duration of the impulse response, increasing the duration of the signal that can be integrated and leading to an increase in SNR proportional to ${\tau }_{\text{off}}^{1/2}$ [Eq. (S7) in the Supplementary Material, Fig. 4(c)]. When ${\tau }_{\text{on}}$ and ${\tau }_{\text{off}}$ can be independently chosen, ${\tau }_{\text{on}}$ should be minimized (with diminished effect once ${\tau }_{\text{on}}$ is below the spike width) and ${\tau }_{\text{off}}$ maximized (while remaining short compared with the interspike interval). Often, these time constants are biophysically related. If ${\tau }_{\text{on}}\approx {\tau }_{\text{off}}=\tau$, then $\mathrm{SNR}\sim {\tau }^{-1/2}$ [see Eq. (S7) in the Supplementary Material]. That is, faster GEVIs are better than slower ones in terms of shot noise-limited SNR, all else being equal.

3.7.

Effect of Optical Parameters on 2P Fluorescence

Advances in 2P voltage imaging typically have two aims: (1) increasing the number of cells, $N$, which are sampled with a high enough revisit rate to capture all spikes and (2) increasing fluorescence per cell to improve the shot noise-limited SNR. In many cases, these two aims are in tension.

3.7.1.

Changing numerical aperture

We distinguish between the numerical aperture of excitation (${\mathrm{NA}}_e$) and of collection (${\mathrm{NA}}_c$). Often, ${\mathrm{NA}}_c$ is set by the objective NA, whereas ${\mathrm{NA}}_e$ may be lower as a result of underfilling the objective back aperture. The photon detection efficiency (PDE) scales as $\mathrm{PDE}\sim {\mathrm{NA}}_c^2$. The effective ${\mathrm{NA}}_c$ may be increased beyond the objective NA by collecting high-angle fluorescence photons with either reflective⁴⁷ or fiber-optic⁴⁸ auxiliary collectors. The effect of ${\mathrm{NA}}_e$ on the 2P signal depends on the sample geometry. Within the Gaussian beam approximation [Fig. 5(a)], the width of the focus scales as

$$w_0\propto \frac{1}{{\mathrm{NA}}_e}.$$

Fig. 5

Scaling of fluorescence with numerical aperture, sample geometry, and excitation modality. (a) Geometry of a Gaussian beam, showing the width ($W_0\sim 1/\mathrm{NA}$) and waist ($b\sim {1/\mathrm{NA}}^2$). (b) Scaling of total 2P fluorescence as a function of excitation ${\mathrm{NA}}_e$ for different sample geometries. All slender dimensions are assumed to be $\ll W_0$, and all extended dimensions are assumed to be $\gg b$. In a planar raster scan, the fraction of time that a subwavelength structure is excited, $\varphi$, depends on the focus width and hence the ${\mathrm{NA}}_e$. In cases (iii), (v), and (vi), we assume that the object is perfectly in focus, i.e., in the axial plane where focus size is minimum. The collection efficiency for all geometries depends on the collection solid angle $\sim N{A}_c^2$. To calculate the total signal for targeted illumination, multiply the first and third lines; for a raster scan, multiply all three lines. (c) Total fluorescence evoked by the intersection of a laser focus and a spherical membrane, $10\mu \mathrm{m}$ diameter. We compared 1P and 2P excitations with equal ${\mathrm{NA}}_c$ and ${\mathrm{NA}}_e$, with powers adjusted to match per-molecule excitation rates at the focus at NA = 1.2. The much smaller 2P focal volume led to a 5.3-fold smaller maximum fluorescence at the highest NA (and even greater discrepancy at lower NA) and a 3-fold greater sensitivity to misalignment compared with 1P excitation.

The intensity, $I$, at the focus scales inversely with the cross-sectional area (so $I\propto {\mathrm{NA}}_e^2$), and the rate of 2P excitation per molecule, $E$, scales with the intensity squared. Hence,

$$\mathrm{E}\propto {\mathrm{NA}}_e^4.$$

The axial extent of the beam waist scales as

$$\mathrm{b}\propto \frac{1}{{\mathrm{NA}}_e^2}.$$

The total signal from a volume element depends on the number of fluorophores excited. In a three-dimensional bulk solution—an approximation of 2P ${\mathrm{Ca}}^{2+}$ imaging when the beam waist is significantly smaller than a single cell—the volume scales approximately as $V\sim w_0^2\mathrm{b}\propto \frac{1}{{\mathrm{NA}}_e^4}$. The total fluorescence emission ${\mathrm{\Gamma }}_{2P}$ scales as $V\times E$. Hence, in bulk solution, ${\mathrm{\Gamma }}_{2P}\propto {\mathrm{NA}}_e^0$, so that the total collected fluorescence, $F$, scales only with ${\mathrm{NA}}_c$ as $F\sim {\mathrm{NA}}_c^2$.

In contrast, for 2P voltage imaging, the fluorescence rate from the sample scales with ${\mathrm{NA}}_e$ raised to a power between 1 and 3, depending on the orientation and geometry of the membranes in the focal spot [Fig. 5(b), first row]. This different scaling arises because the fluorophores are arranged in a 2D membrane instead of a 3D volume. The scaling of membrane excitation argues strongly for maximizing the ${\mathrm{NA}}_e$ for 2P voltage imaging. On the other hand, a smaller excitation spot leads to (a) a higher rate of photobleaching and possibly photodamage and (b) greater sensitivity to misalignment between the focus and the sample, e.g., from sample motion [Fig. 5(c)]. The optimal ${\mathrm{NA}}_e$ for 2P voltage imaging in vivo likely involves a sample-dependent balance of these considerations.

3.7.3.

Effect of nonlinear saturation

Maximizing focal intensity is only beneficial up to a point. Saturation of the 2P excitation typically occurs at diffraction-limited pulse energies of $\sim 1\mathrm{nJ}$¹⁴ at the focus, and nonlinear local photodamage may occur at similar pulse energies.^55–57 For a fixed pulse duration and spot size, this limit is independent of $f$. We define the effective repetition rate, $f_{\mathrm{eff}}=f\times N_{\text{spots}}$. The optimal $f_{\mathrm{eff}}$ is the repetition rate at which the focal pulse energy density reaches but does not exceed the saturation or damage limit [Fig. 6(a)]. However, if emitted photons are collected in a single-element detector (as opposed to a camera), then the interval between laser pulses should also be several-fold larger than the electronic excited state lifetime (typically 2 to 4 ns) to avoid crosstalk between successive pulses. This constraint limits the pulse rate in the sample to $<\sim 150\mathrm{MHz}$.

Fig. 6

Optimal temporal and spatial splitting. (a) At fixed total power, the per-spot pulse energy is inversely proportional to the effective repetition rate, $f_{\mathrm{eff}}=f\times N_{\text{spots}}$. SNR is increased by increasing focal pulse energy up to the threshold (1-nJ horizontal line shown). Therefore, the optimal effective repetition rate lies at the intersection of the iso-power line with the threshold (circled in red). At a nonzero depth (dotted lines, $l_e$ = attenuation length), a lower repetition rate is needed to produce the same focal pulse energy. (b) For a single diffraction-limited focal spot, a total power of 200 mW, and a pulse energy threshold of 1 nJ, the optimal $f_{\mathrm{eff}}$ goes beneath 80 MHz (horizonal line) at $\sim 100\mu \mathrm{m}$ depth. Decreasing the threshold focal energy (e.g., to 0.5 nJ) increases the optimal repetition rate at a given depth. (c) For a fixed $f_{\mathrm{eff}}$, the focal pulse energy decays exponentially with depth. At $f_{\mathrm{eff}}=250\mathrm{MHz}$, a 1-nJ pulse is not achievable at any depth. At $f_{\mathrm{eff}}=40\mathrm{MHz}$, the pulse energy crosses the 1 nJ threshold at $\sim 200\mu \mathrm{m}$ depth.

Simultaneously approaching the 2P saturation energy at the focus ($\sim 1\mathrm{nJ}$) and the maximum power into the sample ($\sim 200\mathrm{mW}$) implies a depth-dependent optimum value of $f_{\mathrm{eff}}$ [Fig. 6(b)]. For example, for an 80 MHz laser illuminating the brain with 200 mW, a single focal spot will exceed the 1 nJ limit at the brain surface. Splitting the focus into three diffraction-limited spots ($f_{\mathrm{eff}}=240\mathrm{MHz}$) allows each spot to remain beneath the 1-nJ limit while using the full 200-mW budget, though time-resolved fluorescence detection at this frequency would induce crosstalk between successive pulses. Meanwhile, at a depth of 5 $l_e$, where $l_e$ is the exponential attenuation length, optimal $f_{\mathrm{eff}}\approx 1$ to 10 MHz.¹⁴ A 2.5 MHz fiber-based soliton laser gave a 26-fold greater average signal than an 80 MHz Ti-to-Sa laser of equal time–average power at depths up to $680\mu \mathrm{m}$ in the live mouse brain.¹⁴ In summary, the $f_{\mathrm{eff}}$ should be tuned to the power limit, focal energy threshold, and imaging depth for each experiment.

The effective repetition rate must also be high enough to visit each measurement point at least once per measurement cycle. For example, an 800 kHz laser could visit 800 points at a 1 kHz revisit rate (assuming a suitable scanner existed) and would provide 100-fold higher time–average count rate (and 10-fold higher shot noise–limited SNR) than the same laser power delivered at 80 MHz, assuming a subsaturation pulse energy. Mechanical and acousto-optical scanners have finite slew rates, which can limit the number of cells measurable within 1 ms. When this limit is below the limit set by optical SNR, beam splitting may increase the number of measurable cells.

3.8.

Can Advanced Analysis Techniques Overcome the Shot Noise Limit?

Consider the goal of detecting whether a spike occurred (hypothesis $H^{(1)}$) or did not occur ($H^{(0)}$) during a measurement time $\tau$. The mean number of detected photons in the case of a spike is $n_1$, and that in the absence of a spike is $n_0$. The probability distributions for the number of detected photons in the two cases are then given by Poisson distributions with means $n_1$ and $n_0$ respectively

If the number of photons collected in either scenario is not large and the contrast $\beta =(n_1-n_0)/n_0$ is also small, then the two probability distributions overlap: a given set of detected photons could have been produced by either the presence or absence of a spike. In such cases, no analysis algorithm can unambiguously determine whether a spike occurred; at best, one can determine the relative probabilities of the two hypotheses. This argument is analyzed in detail in Ref. 12.

Voltage signals corresponding to spikes are typically correlated across multiple pixels and sometimes across frames (depending on the frame rate and spike duration). As the photon shot noise is statistically independent among all pairs of pixels, the relative contribution of shot noise can be diminished by weighted averaging across pixels and possibly across frames. If the expected number of photon detections at pixel $i$ is $⟨n_i⟩$ and a filter assigns weight $a_i$ to the pixel, then the expected signal is $S=\sum _i{a}_i⟨n_i⟩$, and the variance in this quantity due to photon shot noise is ${\sigma }_S^2=\sum _i{a}_i^2⟨n_i⟩$. The art of voltage imaging analysis comprises determining the $a_i$ to maximize the difference between $p(S|H^{(1)})$ and $p(S|H^{(0)})$.

Determination of the weights $a_i$ can be via simple manual or activity-based selections of regions of interest or via optimal detection algorithms, e.g., as in Ref. 61. When signals from multiple sources overlap, a variety of unmixing algorithms are useful.^62–66 It is possible even to apply a filter during image acquisition to reduce the data burden.⁶⁷ Recently introduced machine learning algorithms^24,25 can help determine the optimal weighting of pixel signals. Noise reduction and signal extraction algorithms can play an important role in voltage imaging data analysis.

None of these techniques, however, overcomes the fundamental uncertainty that different voltages can give identical photon distributions at the detector. It is straightforward to simulate voltage imaging datasets with realistic shot noise (as well as other noise sources), where ground truth is known. Analysis methods should be validated against simulated data, and false-positive and false-negative spike detection rates should be quantified. Claims that denoising methods can “overcome fundamental limits”²³ of shot noise are misleading.