Precise Time Superresolution by Event Correlation Microscopy

Abstract

Fluorescence imaging is often used to monitor dynamic cellular functions under conditions of very low light intensities to avoid photodamage to the cell and rapid photobleaching. Determination of the time of a fluorescence change relative to a rapid high time-resolution event, such as an action potential or pulse stimulation, is challenged by the low photon rate and the need to use imaging frame durations that limit the time resolution. To overcome these limitations, we developed a time superresolution method named event correlation microscopy that aligns repetitive events with respect to the high time-resolution events. We describe the algorithm of the method, its step response function, and a theoretical, computational, and experimental analysis of its precision, providing guidelines for camera exposure time settings depending on imaging signal properties and camera parameters for optimal time resolution. We also demonstrate the utility of the method to recover rapid nonstepwise kinetics by deconvolution fits. The event correlation microscopy method provides time superresolution beyond the photon rate limit and imaging frame duration with well-defined precision.

Introduction

Fluorescence microscopy has emerged as one of the key techniques in the field of cell biology and neuroscience. Much advancement for live cell imaging has come from the development of genetic tools to fluorescently label proteins in live cells with proteins such as green fluorescent protein (1), its derivatives, and other labels, including Förster resonance energy transfer (FRET) probes. Fluorescence imaging to monitor dynamic cellular functions is frequently performed at very low light intensities to avoid photodamage, rapid photobleaching, and excessive loading or overexpression of the fluorescent labels in the cells. To acquire useful signals from low photon emitters, imaging is performed with highly sensitive electron-multiplying charge-coupled device (EMCCD) or scientific complementary metal-oxide semiconductor (sCMOS) cameras; nevertheless, they require typical exposure times in the 100-ms range, although many biological processes, such as action potentials or synaptic transmission, occur on a timescale of 1 ms or less. To resolve the dynamics of rapid conformational changes, the dynamics of transmitter release, or action potential propagation, high time-resolution (HTR) information must be recovered. The low photon rate and long imaging frame duration impose limits on the time resolution of fluorescence changes relative to associated rapid events. This limitation is somewhat analogous to the spatial resolution diffraction limit (2) that is set by the point-spread function and the camera pixel size in conjunction with the image magnification in the image plane.

Single-molecule localization microscopy techniques such as stochastic optical reconstruction microscopy (3) and photoactivated localization microscopy (or fluorescence photoactivated localization microscopy) (4, 5) break the diffraction limit and achieve a spatial resolution down to tens of nanometers, which is smaller than the camera pixel size, by fitting a two-dimensional Gaussian to determine the center of the point-spread function (6). Analogous to the spatial superresolution single-molecule localization techniques, we developed a time superresolution technique named event correlation microscopy (ECOM) that breaks the time resolution limit given by the photon rate and imaging frame duration. The method takes advantage of the fact that the intensity in a certain region of interest reported by an imaging frame depends on the time at which the intensity change occurred. The method depends on the simultaneous recording of the imaging data and a related biological or physical HTR event such as a stimulation pulse, action potential, or amperometric spike of a quantal transmitter release event. When multiple events occurring at random times during an exposure are averaged, the time correlation between the physical events and the fluorescence change can be determined with subframe resolution. We previously applied ECOM to analyze small FRET changes associated with individual transmitter release events detected as an amperometric spike that serves as the HTR event (7, 8). Here, we provide a detailed theoretical, computational, and experimental analysis of the method and its precision in determination of the time of the intensity step with subimaging frame resolution.

As a computational method for image analysis, without extra demand of sophisticated hardware, ECOM can theoretically achieve imaging temporal resolution down to any precision, limited only by the signal/noise ratio (SNR) of the averaged imaging data. Analogous to the single-molecule localization precision (6), which beats the spatial resolution limit beyond pixel size, ECOM beats the time resolution limit beyond the imaging frame duration. Here, we describe the principle of ECOM and discuss in detail how the ECOM precision is limited by the SNR and other imaging factors. We also present its experimental application in biophysical studies and its extension to the deconvolution analysis of nonstepwise intensity changes.

Materials and Methods

Simulation of imaging data

To investigate the contribution of different experimental parameters, such as basal signal, amplitude of signal change, background noise, and camera exposure and readout time, to the precision of the step time determination using the ECOM method, Igor-based simulations were performed over a broad range of these parameters.

To test each parameter set, 1000 sweeps of intensity time courses were simulated by generating Poisson-distributed pseudorandom values of “basal signals” and “basal signal + amplitude of signal change.” Each sweep included 100 frames before and after the intensity step (the transition frame). The timing of each intensity step (t = 0) within the transition frame was set by drawing a random value from a uniform distribution within the frame time and occurred randomly relative to the frame start time. The intensities of the transition frame were obtained by starting the Poisson-based pseudorandom detection of signal photons at the selected intensity step times within the frame. The time course of each sweep was subsampled with 1-ms time points assigning the frame intensity value to all the points covering the corresponding frame. The time courses of the 1000 simulated sweeps were aligned at the time of the intensity step and averaged. The averaged time course was fitted by the ECOM step response function (Eq. 22) to determine t₀. The root mean square value of the residual noise obtained by subtracting the fitted curve from the averaged time course was used for the calculation of SNR. One hundred such data sets obtained by simulation of 1000 sweeps for the same parameter set were generated and fitted, providing 100 t₀ values such that the mean t₀ value and its SD (σ_t0) could be calculated. For the ideal camera, no extra noise sources were considered. But for the EMCCD and sCMOS cameras, additional Gaussian-distributed noise (Table 1) was added.

Table 1.

Typical Most Relevant Parameters for Ideal, EMCCD, and sCMOS Cameras

	Ideal	EMCCD Gain = 300	sCMOS Global Shutter
Noise factor F	1	1.4	1
Readout noise R (e−/pixel)	0	0.2	2.4
Readout gap time (ms)	0	1.8	0.3

Fluorescence imaging and data acquisition

To determine an unknown shutter delay, a drop of fluorescent beads (Fluoresbrite multifluorescent microspheres (0.20 μm; catalog no. 24050); Polysciences, Warrington, PA) was mixed with 400 μL H₂O on a cover glass placed on the stage of an inverted microscope (Olympus IX83 equipped with total internal reflection fluorescence (TIRF) excitation module IX2-RFAEVA-2 and Olympus UPlanSApo (NA 0.75; 20× epifluorescence objective); Shinjuku, Tokyo, Japan) followed by a removal of 200 μL from this volume and letting them dry. When the beads were dry, they were illuminated in epifluorescence mode by a 442-nm low-noise diode laser (MLL-III-442; CNI Optoelectronics, Changchun, China) via the TIRF adaptor (IX2-RFAEVA-2; Olympus) to use its built-in laser interlock shutter. The shutter was controlled manually by mouse clicks (using Prior Terminal V2.9) via the ProScan III automation controller (Prior Scientific) sending a transistor-transistor logic (TTL) pulse to a home-built trigger box operating the interlock shutter (Fig. 1 A). Fluorescence videos were acquired by an EMCCD camera (Andor iXon3; Andor Technology, Belfast, United Kingdom) using Solis software (Andor Technology). To record the precise times of the acquired images relative to the trigger pulse operating the shutter, the EMCCD camera fire output and the TTL pulse were recorded simultaneously with a second computer by a data acquisition board (PCIe-6321; National Instruments, Austin, TX).

Figure 1.

Schematic diagrams of experimental setup to determine the shutter opening delay (A) and kinetics of depolarization-induced [Ca²⁺] change (B) using ECOM. To see this figure in color, go online.

Videos were acquired with various exposure times (20, 50, 100, and 200 ms) in EMCCD frame transfer mode, with an electron-multiplying (EM) gain of 10, system readout rate of 10 MHz, pre-amplifier setting of 5.10, charge-coupled device (CCD) sensitivity of 12.54 e⁻/analog-to-digital count, and single-pixel readout noise of 52.54 e⁻. Because the EM gain is subject to aging, the actual camera gain value was experimentally calibrated from the analysis of fluorescent signals from individual pixels according to the equation gain = signal variance/(2 × signal), where the factor of 2 in the denominator comes from the EM gain, which effectively doubles the variance (9). Signal and variance were determined from the difference of counts between open and closed shutter intervals. The determined gain value of 0.67 was used to convert the signal intensity in camera counts to the detection rate of emitted photons (photons per second) over certain frame exposure times. With each exposure time, the beads were repeatedly excited 200 times, switching the excitation on and off by a manually operated trigger pulse such that the shutter opened in a random manner relative to the imaging frames.

The individual pixels within the video frames (215 × 240 pixels) exhibited a wide range of fluorescence intensities because of the uneven distribution of beads. This allowed us to treat the 200 11-frame videos from each individual pixel as 1 data set determining each pixel’s mean value of signal amplitude and noise SD.

The ECOM method was applied to determine experimentally the delay (t₀) between the TTL pulse, which triggered the shutter opening and the fluorescence signal, which indicated the actual time of shutter opening. Briefly, the imaging frames were desynchronized by subsampling the frame time (consisting of exposure time plus readout time) with sampling points corresponding to the sampling frequency of the HTR event (in this case, 1 ms), assigning the intensity value of a given frame to all points within the frame. This method allowed the alignment of the 200 signal time courses at the onsets of the corresponding electrical pulses (t = 0) and averaging the time courses. The averaged time course was fitted with the ECOM step response function (Eq. 22) in the same way as for the simulated signals, providing a t₀ value for each pixel. Mean t₀ values and their SDs σ_t0 were determined averaging t₀ values from pixels with similar SNR.

To verify the delay time t₀ obtained with the ECOM method, experiments were performed measuring the delay between the TTL trigger pulse and the fluorescence signal using a fast photodiode (Siemens SFH 21; Siemens, Munich, Germany).

Calcium measurements

Whole-cell patch-clamp measurements were combined with fura-2 fluorescence measurements to simultaneously monitor the calcium current I_Ca and fura-2 fluorescence signals in mouse embryonic chromaffin cells (E18), which were prepared and cultured as described (10). Recordings were performed at room temperature on the second day of culture on an Olympus IX83 inverted microscope equipped with an Olympus UAPON (NA 1.49; 100× objective).

Patch electrodes were fabricated from borosilicate glass capillaries. The pipette resistance in the bath was ∼2–3 MΩ when backfilled with intracellular solutions. The extracellular solution contained 145 mM NaCl, 2.8 mM KCl, 1 mM MgCl₂, 10 mM HEPES, 10 mM glucose, 2 mM CaCl₂ (pH 7.4). The intracellular solution contained 145 mM L-Glutamic acid, 145 mM CsOH, 8 mM NaCl, 1 mM MgCl₂, 2 mM Mg-ATP, 0.3 mM GTP, 10 mM HEPES (pH 7.3), supplemented with 0.1 mM fura-2 from 10 mM fura-2 stock solution freshly prepared immediately before the experiment. Once the whole-cell configuration was established, 0.1 mM fura-2 in the intracellular solution diffused into the cell. After ∼3 min of equilibration time, calcium currents were evoked by 10-ms depolarization pulses from nominally −70 to +10 mV (not corrected for liquid junction potentials) at ∼10-s interpulse intervals. The evoked I_Ca currents were digitized at 10 KHz, filtered at 2.9 KHz (8-pole Bessel) and acquired using a patchmaster-controlled EPC-9 amplifier (HEKA Elektronik) (Fig. 1 B).

Fura-2 was excited by a metal halide lamp (Lumen 220PRO; Prior) through excitation filters of 360/12 or 387/11 nm and a 409 nm dichroic (Fig. 1 B). Before and after a series of pulse stimulations, paired excitations at 360/387 nm were performed to obtain a ratiometric calibration of [Ca²⁺]_i. For each pulse stimulation recording, a 2000-frame image sequence was acquired using the EMCCD camera (Andor iXon3; EM gain 300) and its accompanying software Solis with a 100-ms exposure time and ∼1.7-ms readout interval. During these acquisitions, fura-2 was continuously excited at the more Ca²⁺-sensitive wavelength of 387 nm. To record the precise times of the acquired images relative to the pulse stimulation, the EMCCD camera fire output and the command voltage output of the EPC-9 were recorded simultaneously by a data acquisition board (PCIe-6321; National Instruments) (Fig. 1 B).

Fast line scans of fura-2 fluorescence were performed with an EM gain of 300 to image a small area of 5 lines in the cell with a 1-ms exposure time and 678-Hz frame rate.

Ethics statement

Mice were kept and maintained in the animal facility of the Max Planck Institute for Biophysical Chemistry in Göttingen. All procedures were approved by the Animal Care and Use Committee of the Max Planck Institute for Biophysical Chemistry in accordance with the German Animal Welfare Act in the version published on May 18, 2006 (Federal Law Gazette I, p. 1206, 1313), amended by article 4, section 90 of the Act of August, 7 2013 (Federal Law Gazette I, p. 3154).

ECOM analysis software

Imaging data were analyzed with the ECOM method using the pulse onset as the HTR event using the customized IgorPro routines (Wavemetrics). The IgorPro-based (Wavemetrics) ECOM software was developed to determine the timing of imaging signals (t₀) relative to the HTR events and is provided in Data S1 (ECOM_Software).

Results

Principle of the ECOM method

To illustrate the basic concept, we consider the opening of a shutter while imaging data are acquired. In this case, the shutter opening or the trigger pulse for shutter opening is an HTR event. For convenience, we begin assuming that the light intensity reaching the camera is zero while the shutter is closed and increases to a constant value I₀ when the shutter is open. When the shutter opens during an exposure, the light reaching the camera during this exposure depends on the time at which the intensity step occurs.

To derive the equation for the step response of the ECOM method, we consider image acquisition with exposure times t_e and gaps between the exposures for data readout with duration t_gap. We define the frame start time (t_start) as the time point that precedes the start of the exposure by t_gap/2 as illustrated in Fig. 2 A for an example, assuming t_e = 200 ms and t_gap = 20 ms. In this case, the duration of a frame is thus t_frame = t_e + t_gap = 220 ms.

Figure 2.

Definition of camera frames. (A) Imaging frames consist of exposure times (red bars; camera fire output: high) and readout gaps between the exposures (camera fire output: low). Camera frames are defined as having durations t_e + t_gap centered around the exposure periods (green). (B and C) An intensity step is assumed at t = 0, falling between two subsequent frames (B) or in the middle of an exposure (C) such that the frame receives half the light of a full exposure (I_F = I₀/2). To see this figure in color, go online.

We now consider an intensity step at time t = 0 from I = 0 to I = I₀ (Fig. 2, B and C, black traces), that is, in this synthetic data set, the intensity step is assumed to be synchronous with the shutter opening. If the frames are acquired such that the step falls in the gap between two frames, then the last frame preceding the step reports I = 0 and the next frame I = I₀ (Fig. 2 B, red bars). If the step occurs in the middle of an exposure, the respective frame reports an average intensity I = I₀/2 (Fig. 2 C, red bar). The average intensity of a frame I_F during which the intensity step occurs therefore depends on t_start relative to the time of the intensity step (t = 0). For the subsequent analysis method, we assign the intensity in a region of interest for a given exposure to the duration of the entire frame (Fig. 2, B and C, green traces). For averaging imaging data aligned with respect to the time of the intensity step, the intensity values of individual frames are converted to traces that consist of data points spaced by short time increments Δt such that each frame is split into n = t_frame/Δt points (e.g., 1 ms) (green traces).

As long as the actual exposure ends before the intensity step, which occurs by definition at t = 0, that is, for t_start between –(t_e + t_gap) and –(t_e + t_gap/2) (Fig. 3, A and B), the frame intensity is as follows.

$$I_F=0,\text{for}\mathrm{-}\left(t_e+t_{gap}\right)<t_{start}<-\left(t_e+t_{gap}/2\right).$$ (1)

Figure 3.

(A–C) Dependence of imaging frame intensity on relation between frame start time t_start and time of intensity step (set to t = 0). Exposure times and intensities are indicated in red, and imaging frames corresponding to exposure times are in green. (D) The averaged intensity trace (green line) and fit with Eq. 9 (dashed black line) are shown. To see this figure in color, go online.

For frames starting between $t_{start}=\mathrm{-}\left(t_e+t_{gap}/2\right)\text{and}{\mathit{t}}_{\mathit{start}}=\mathrm{-}{t}_{gap}/2$, I_F increases linearly from 0 to I₀ (Fig. 3) as follows.

$$I_F=I_0\times \left(1+\frac{t_{start}+t_{gap}/2}{t_e}\right)=I_0\times \left(\frac{t_e+t_{start}+t_{gap}/2}{t_e}\right),$$

$$\text{for}-\left(t_e+t_{gap}/2\right)<t_{start}<-t_{gap}/2.$$ (2)

For later frame start times, the intensity step occurs before the actual exposure starts such that the frame intensity is equal to I₀ as follows.

$$I_F=I_0,\text{for}-t_{gap}/2<t_{start}<0.$$ (3)

As shown in Fig. 3 C, the approach of ECOM is to align the time courses of individual imaging signals not based on synchronizing the frames from different image sequences, as is usually done, but rather by aligning the imaging data to the HTR event with subframe precision, which becomes possible because of the conversion to data points spaced by short time increments. After averaging the aligned time courses, the resulting averaged time course (Fig. 3 D, green line) exhibits a smooth increase that spans over two frame times.

The ECOM step response function

As described above, we assign the value I_F to the whole time interval of the respective frame, including the exposure time and half of the gap before and after the exposure (Figs. 2 and 3, green bars). Because the imaging frames are acquired independent of the time of the intensity step, all t_start values are equally likely. For averaging multiple events, image sequences with random t_start values are summed. The summed intensity at time t is thus obtained by integration.

$$I_{int}\left(t\right)=\underset{-\left(t_e+t_{gap}\right)}{\overset{t}{\int }}I\left(t_{start}\right)d{t}_{start}.$$ (4)

From t_start = −(t_e + t_gap) to t = −(t_e + t_gap/2), the frame intensity is zero (Eq. 1) giving

$$I_{int}\left(t\right)=\underset{-\left(t_e+t_{gap}\right)}{\overset{-\left(t_e+t_{gap}/2\right)}{\int }}I\left(t_{start}\right)d{t}_{start}=0.$$ (5)

From t = −(t_e + t_gap/2) to t = −t_gap/2, the integrated intensity is according to Eq. 2.

$$I_{\mathit{int}}\left(t\right)=\underset{-\left(t_e+\frac{t_{gap}}{2}\right)}{\overset{t}{\int }}{I}_0\times \left(\frac{t_e+t_{start}+t_{gap}/2}{t_e}\right)d{t}_{start}=I_0\times \frac{1}{2}\frac{{\left(t_e+t_{gap}/2+t\right)}^2}{t_e}.$$ (6)

The final value in this interval at t = −t_gap/2 is as follows.

$$I_{int}\left(-t_{gap}/2\right)=I_0\times \frac{1}{2}{t}_e.$$

For the subsequent interval from t = −t_gap/2 to t = 0, we get the additional contribution as follows.

$$I_{\mathit{int}}\left(t\right)=I_0\times \frac{1}{2}{t}_e+\underset{-\frac{t_{gap}}{2}}{\overset{t}{\int }}I\left(t_{start}\right)d{t}_{start}=I_0\times \frac{1}{2}{t}_e+\underset{-\frac{t_{gap}}{2}}{\overset{t}{\int }}{I}_{0}d{t}_{start},$$

$$I_{\mathit{int}}\left(t\right)=I_0\times \frac{1}{2}{t}_e+I_0\left(t+t_{gap}/2\right),\mathrm{or}$$

$$I_{\mathit{int}}\left(t\right)=I_0\left[\frac{1}{2}\left(t_e+t_{gap}\right)+t\right].$$ (7)

For the remaining period from t = 0 to t = t_e + t_gap, the time course of I_int continues accordingly for t = 0 to t = t_gap/2 as follows

$$I_{\mathit{int}}\left(t\right)=I_0\times \left[\frac{1}{2}\left(t_e+t_{gap}\right)+t\right],$$

and for t = t_gap/2 to t = t_e + t_gap/2 as

$$I_{\mathit{int}}\left(t\right)=I_0\times \left[1-\frac{1}{2}\frac{{\left(t_e+t_{gap}/2+t\right)}^2}{t_e}\right].$$ (8)

To account for a possible delay relative to a reference signal such that the intensity step occurs at t = t₀ rather than at t = 0 and changing from the integral to the average by dividing by the frame duration t_e + t_gap, we obtain the following for the average intensity I_Avg by combining Eqs. 5, 6, 7, and 8:

$$\begin{array}{cc}ift<t_0-t_e-\frac{t_{gap}}{2}:& I_{Avg}=0\end{array},$$

$$if{t}_0-t_e-\frac{t_{gap}}{2}<t<t_0-\frac{t_{gap}}{2}:I_{Avg}=I_0\times \frac{1}{2}\frac{{\left(t_e+t_{gap}/2+\left(t-t_0\right)\right)}^2}{\left(t_e+t_{gap}\right)\times t_e},$$

$$if{t}_0-\frac{t_{gap}}{2}<t<t_0+\frac{t_{gap}}{2}:I_{Avg}=I_0\times \left[\frac{1}{2}+\frac{t-t_0}{\left(t_e+t_{gap}\right)}\right],$$ (9)

$$if{t}_0+\frac{t_{gap}}{2}<t<t_0+t_e+\frac{t_{gap}}{2}:I_{Avg}=I_0\times \left[1-\frac{1}{2}\frac{{\left(t_e+t_{gap}/2-\left(t-t_0\right)\right)}^2}{\left(t_e+t_{gap}\right)\times t_e}\right],\text{and}$$

$$ift>t_0+t_e+\frac{t_{gap}}{2}:I_{Avg}=I_0.$$

For t₀ = 0, this is the step response function of the ECOM method, the shape of which depends on t_e and t_gap. For image acquisition, t_e and t_gap are known parameters, and Eq. 9 can thus be used to fit the time delay t₀ and signal amplitude I₀. As expected, Eq. 9 fits the averaged trace of the synthetic data of Fig. 3 perfectly (Fig. 3 D), giving t₀ = 0 (15.6 ± 16.4 μs) and I₀ = 1 (0.99986 ± 3e−05). It should be noted that I_Avg(t = 0) is exactly I₀/2.

Validation of the ECOM method

The ECOM method was previously validated by an experiment using a fast shutter while imaging data were recorded with t_e = 200 ms (t_gap ≈ 19 ms) and the light intensity simultaneously recorded by a fast photodiode (8). Using the photodiode signal as the reference HTR event, the fit of the resulting data averaged from 262 shutter openings yielded t₀ = −50 ± 80 μs, consistent with the HTR signal occurring at the same time as the intensity step. To test if ECOM properly determines nonzero time delay values, we performed imaging experiments using a shutter with unknown properties. For this test we used the TIRF laser interlock shutter of an Olympus IX2-RFAEVA-2 TIRF illuminator, which was controlled by a home-built trigger box (Fig. 1 A). In this experiment, the TTL trigger pulse was used as the HTR signal (Fig. 1 A).

The sample was prepared by applying fluorescent beads (see Materials and Methods) to the surface of a cover glass and letting them dry (Fig. 4 A). The apparent shadowing effect in this image is due to the oblique illumination via the TIRF illuminator. While a video was acquired (t_e = 100 ms, t_gap ≈ 1.8 ms), the shutter was opened and closed repeatedly, generating trigger pulses randomly with a manual mouse click. Shortly before the start of the camera acquisition, the simultaneous recording of the camera fire output and the trigger pulse controlling the shutter were started such that the time of the first camera frame and the timing of the trigger pulses (Fig. 4 B, blue) relative to the imaging frames (Fig. 4 B, black) were precisely known. The intensities of pixel 112,178 (Fig. 4 A, arrow) from this video are also shown (Fig. 4 B, red). As expected, the intensity is low when the shutter is closed (trigger pulse high) and high when the shutter is open (trigger pulse low).

Figure 4.

For a Figure360 author presentation of this figure, see https://doi.org/10.1016/j.bpj.2019.03.034 SNR affects the precision of ECOM in experimental study. (A) A sample with dried fluorescent beads, providing a fluorescence image with pixels having widely varying intensities, was used to obtain variable signals in parallel. The image was generated by averaging the first 4 frames acquired after each of the 200 shutter openings (800 frames total). The intensity scale (right) indicates the pixel brightness and includes the camera dark level of ∼88 counts. The white arrow points to the pixel analyzed in (B) and (C). (B) The initial part of recording with t_e = 100 ms showing camera fire output (black), trigger pulses (blue), and frame intensity trace (red) from the pixel marked by white arrow in (A) is shown. (C) The intensity time course of the same pixel analyzed with the ECOM method (N = 200 sweeps; black trace) and fit with the ECOM step response function (superimposed red line) is shown. The point at t = 11.3 ms at which 50% of the maximal intensity is reached is marked by a gray line. (D) The measurement of shutter delay using a fast photodiode (13.9 ± 0.1 ms) is shown. (E) The shutter delay values obtained from analysis of each individual pixel using ECOM from averaging the 200 sweeps plotted versus the SNR of the corresponding traces (black dots) is shown. The mean of all values is 13.7 ± 0.2 ms (red line). The mean t₀ values in individual bins of 1000 SNR values each and their SDs (green) are shown. (F) The SDs from (E) (green bars) plotted versus the SNR are shown. The line is not a fit but was calculated using Eq. 28. To see this figure in color, go online.

Averaging the data from 200 shutter openings using the trigger pulse high-to-low transition used as the HTR reference signal provides the time course shown in Fig. 4 C (black trace). Fitting this time course with Eq. 9 plus nonzero baselin (Fig. 4 C, red trace) reproduces the time course well and gives a value for t₀ = 11.3 ± 2.2 ms for the delay between the trigger pulse and shutter opening. To verify that the ECOM determination of the delay is correct, the intensity change was recorded using a fast photodiode together with the trigger signal (Fig. 4 D), indicating a delay between the trigger pulse and shutter opening of 13.9 ± 0.1 ms, not significantly different from the ECOM result obtained for a single pixel.

The precision of ECOM

The time delay t₀ between the HRT event and the fluorescence change is obtained by fitting Eq. 22 (Figs. 3 D and 4 C). The accuracy of this method will obviously depend on the SNR of the averaged intensity change, which was ∼56 in the example of Fig. 4 C. To evaluate experimentally how the precision of the t₀ determination depends on the SNR, all of the 51,600 pixels in the video were analyzed individually, averaging 200 shutter openings. Depending on the brightness of the pixel, these averaged intensity traces yielded SNRs ranging from 0 to 230. Fig. 4 E shows the t₀ values obtained for the individual pixels as a function of SNR (black dots) in the range defined by the figure axes. The average t₀ = 13.7 ± 0.2 ms from all data points as indicated by the horizontal red line is in excellent agreement with the 13.9 ± 0.1-ms delay obtained with the fast photodiode (Fig. 4 D). As expected, the scatter decreases with increasing SNR. To determine the relation between the precision of the t₀ values (σ_t0) and SNR, data points were binned according to SNR such that each bin contained the values from 1000 pixels in ascending ranges of SNR and the mean t₀, as well as its SD were determined separately for each bin (Fig. 4 E, green symbols with error bars). For SNR >5, σ_t0 is inversely proportional to SNR (Fig. 4 F). The solid line shows the relation σ_t0 = 124 ms/SNR, which gives the ±2.2-ms error estimate for the SNR of the trace in Fig. 4 C.

Theoretical analysis of step time determination

We begin the theoretical analysis by considering the simple case of a stepwise intensity change from intensity I = 0 to I = I₀ at the time t = t₀ relative to an HTR event at t = 0. As in the example of Fig. 4, this change could come from a shutter opening after a trigger pulse or from an instantaneous change in a fluorescent molecule from nonfluorescent to fluorescent. However, because the photon emission rate is not infinitely high, photons will be emitted and a fraction of them detected stochastically at a certain rate in a Poisson process starting at t = t₀ such that the precise time of the change that initiates the emission of photons is not exactly known. The first detected photon will be an indicator of this time and the higher the photon emission rate, the more precise will the measurement indicate the time of the change of interest.

For a Poisson process, the time intervals Δt between individual photon detections are exponentially distributed with the probability density function as follows.

$$F\left(\mathit{\Delta }t,\lambda \right)=\lambda \times e^{-\lambda \times \mathit{\Delta }t},$$ (10)

where λ is the average rate of photon detections. Eq. 10 gives the distribution of the time delays of arrival of the first photon after the step change at t = t₀ such that there will be an exponentially distributed waiting time until the first signal photon is detected with a mean value as follows.

$$\langle \mathit{\Delta }{t}_S\rangle =\frac{1}{\lambda },$$ (11)

and a variance

$${\sigma }_{\mathit{\Delta }ts}^2=\frac{1}{{\lambda }^2}.$$ (12)

We use again the example for which we have a trigger pulse at time t = 0, which is followed by the shutter opening at t = t₀ such that the average waiting time until arrival of the first signal photon after the trigger pulse is as follows.

$$\langle \mathit{\Delta }{t}_S\rangle =\langle t_0\rangle +\frac{1}{\mathit{\lambda }}.$$ (13)

An estimate of t₀ can be obtained from the mean of the waiting times and the average photon detection rate λ after shutter opening as follows.

$$\langle t_0\rangle =\langle \mathit{\Delta }{t}_S\rangle -\frac{1}{\lambda },$$ (14)

with a variance

$${\sigma }_{t0}^2={\sigma }_{\mathit{\Delta }ts}^2=\frac{1}{{\lambda }^2}.$$ (15)

This estimate, however, assumes that the average photon detection rate is precisely known, which is generally not justified, particularly for low photon detection rates. If we continue measuring photons after the shutter opening for a time period t_S, the inverse detection rate is obtained from the number of detected signal photons n_S as follows.

$$\frac{1}{\lambda }=\frac{t_S}{n_S},$$ (16)

with a variance

$${\sigma }_{1/\lambda }^2=\frac{1}{{\lambda }^2}\left(1+\frac{1}{n_s}\right).$$ (17)

Therefore, the estimate of t₀ in a single measurement has the variance

$${\sigma }_{t0}^2=\frac{1}{{\lambda }^2}\left(1+\frac{1}{n_S}\right).$$ (18)

However, because n_S will have to be considerably greater than one to obtain a reasonable estimate of λ, the term 1/n_S can be neglected.

An additional uncertainty in determination of the waiting time at which a photon is actually detected comes from the finite duration of the imaging frames, which produces an additional variance of a top-hat distribution with a size t_frame because it is not known at what time during the frame the photon was detected,

$${\sigma }_{frame}^2=\frac{t_{frame}^2}{12}.$$ (19)

Averaging N trials, the total variance will be as follows.

$${\sigma }_{t0}^2=\frac{\frac{1}{{\lambda }^2}+\frac{t_{frame}^2}{12}}{N}.$$ (20)

As long as t_frame does not exceed 1/λ, the term t_frame²/12 in Eq. 20 is negligible, increasing SD_t0 less than 4% such that the standard error (SE) of the mean t₀ value is approximately as follows.

$${\sigma }_{t0}=\frac{1}{\sqrt{N}\times \lambda }.$$ (21)

Eq. 21 assumes, of course, that there are no false photon counts, which is, in general, not justified. We will show below that the limit of Eq. 21 can be broken using the ECOM method in the absence as well as in the presence of background counts.

Theoretical precision of step time determination by fitting the ECOM function

For a theoretical estimate of the resulting precision of the t₀ determination, we consider the least-squares fitting approach based on Eq. 9, which provides the fitted values Y(t_0, B, A, t_i), where A and B are the signal amplitude and baseline value, respectively.

$$ift<t_0-t_e-\frac{t_{gap}}{2}:Y=B,$$

$$if{t}_0-t_e-\frac{t_{gap}}{2}<t<t_0-\frac{t_{gap}}{2}:Y=A\times \frac{1}{2}\frac{{\left(t_e+t_{gap}/2+\left(t-t_0\right)\right)}^2}{\left(t_e+t_{gap}\right)\times t_e}+B,$$

$$if{t}_0-\frac{t_{gap}}{2}<t<t_0+\frac{t_{gap}}{2}:Y=A\times \left[\frac{1}{2}+\frac{t-t_0}{\left(t_e+t_{gap}\right)}\right]+B,$$ (22)

$$if{t}_0+\frac{t_{gap}}{2}<t<t_0+t_e+\frac{t_{gap}}{2}:Y=A\times \left[1-\frac{1}{2}\frac{{\left(t_e+t_{gap}/2-\left(t-t_0\right)\right)}^2}{\left(t_e+t_{gap}\right)\times t_e}\right]+B,\text{and}$$

$$ift>t_0+t_e+\frac{t_{gap}}{2}:Y=A+B.$$

The parameter t₀ is fitted by minimizing the sum of the squared deviations between the fitted values and the data points y_i according to the following:

$$\frac{1}{2}\frac{\partial }{\partial t_0}{\sum }_{i=1}^n{\left[y_i-Y\left(t_0,B,A,t_i\right)\right]}^2=0\text{or}$$

$${\sum }_{i=1}^n\left[y_i-Y\left(t_0,B,A,t_i\right)\right]\frac{\partial Y\left(t_0\text{,}B\text{,}A\text{,}{t}_i\right)}{\partial t_0}=0.$$ (23)

Linearizing $\mathrm{Y}\left(t_0,B,A,t_i\right)$ for small deviations from the true value of t₀ as follows.

$$Y\left(t_0,B,A,t_i\right)=Y\left(\widetilde{t_0},\tilde{B},\tilde{A},t_i\right)+\frac{\partial Y\left(t_0\text{,}B\text{,}A\text{,}{t}_i)\right)}{\partial t_0}\Delta t_0,$$

yields

$${\sum }_{i=1}^n\left[y_i-Y\left(\widetilde{t_0},\tilde{B},\tilde{A},{\mathit{t}}_i\right)-\frac{\mathit{\partial }Y\left(t_0\text{,}\mathit{B}\text{,}\mathit{A}\text{,}{\mathit{t}}_{\mathit{i}}\right)}{\mathit{\partial }{t}_0}\mathit{\Delta }{t}_0\right]\frac{\mathit{\partial }Y\left(t_0\text{,}\mathit{B}\text{,}\mathit{A}\text{,}{\mathit{t}}_{\mathit{i}}\right)}{\mathit{\partial }{t}_0}=0\text{and}$$

$$\mathit{\Delta }{t}_0=\frac{{\sum }_{i=1}^n\frac{\mathit{\partial }Y\left(t_0\text{,}\mathit{B}\text{,}\mathit{A}\text{,}{\mathit{t}}_{\mathit{i}}\right)}{\mathit{\partial }{t}_0}\cdot \left[y_i-Y\left(\widetilde{t_0}\text{,}\tilde{\mathit{B}}\text{,}\tilde{\mathit{A}}\text{,}{\mathit{t}}_{\mathit{i}}\right)\right]}{{\sum }_{i=1}^n\left(\frac{\mathit{\partial }Y\left(t_0\text{,}\mathit{B}\text{,}\mathit{A}\text{,}{\mathit{t}}_{\mathit{i}}\right)}{\mathit{\partial }{t}_0}\right)}=0,$$ (24)

where the parameters denoted with the tilde are the closest estimates.

The variance of t₀ is as follows.

$${\sigma }_{t0}^2={\sum }_{i=1}^n{\left(\frac{\mathit{\partial \Delta }{t}_0}{\mathit{\partial }{y}_i}\right)}^2{\sigma }_{yi}^2=\frac{1}{{\sum }_{i=1}^n{\left(\frac{\mathit{\partial }Y\left(t_0\text{,}\mathit{B}\text{,}\mathit{A}\text{,}{\mathit{t}}_{\mathit{i}}\right)}{\mathit{\partial }{t}_0}\right)}^2/{\sigma }_{yi}^2}.$$ (25)

If the noise of the traces is dominated by constant background noise, σ_yi² can be approximated by the constant background variance σ_B² giving the following:

$${\sigma }_{t0}^2=\frac{{\sigma }_B^2}{{{\sum }_{i=1}^n\left(\frac{\mathit{\partial }Y\left(t_0,B,A,t_i\right)}{\mathit{\partial }{t}_0}\right)}^2}.$$

In the absence of background noise, the variance comes only from the detected signal photons, and the variance of the averaged trace increases from zero at t<(t₀−t_e−t_gap/2) to the mean number of signal photons per frame multiplied with a possible noise factor F from camera gain that must be included for EMCCD or intensified CCD cameras (F² × λ × t_e) at t>(t₀ + t_e + t_gap/2). In this case, the average variance is well approximated by the mean value σ_S² = (F² × λ × t_e)/2N, where λ is the signal photon rate.

The standard error of t₀ is thus as follows.

$${\sigma }_{t0}\approx \sqrt{\frac{{\sigma }_B^2+F^2\times \mathit{\lambda }/2\times t_e/N}{{\sum }_{i=1}^n{\left(\frac{\mathit{\partial }Y\left(t_0,B,A,t_i\right)}{\left(\mathit{\partial }{t}_0\right)}\right)}^2}}.$$ (26)

An analytical expression can be obtained by converting the sum in the denominator to an integral.

$${\sum _{i=1}^n\left(\frac{\mathit{\partial }Y\left(t_0\text{,}\mathit{B}\text{,}\mathit{A}\text{,}{\mathit{t}}_{\mathit{i}}\right)}{\mathit{\partial }{t}_0}\right)}^2\approx {\int }_{t_0-t_e-t_{gap}/2}^{t_0+t_e+t_{gap}/2}\frac{dY\left(t_0,B,A,t\right)}{d{t}_0}dt.$$

By integrating the squared derivatives over the corresponding time regions of Eq. 22, we obtain the result as follows.

$${\sigma }_{to}\approx t_{frame}\times \frac{\sqrt{{\sigma }_B^2+F^2\times \mathit{\lambda }/\mathit{2}\times {\mathit{t}}_{\mathit{e}}/N}}{A}\sqrt{\frac{3}{2}\times \left(\frac{1}{1+\raisebox{1ex}{$\mathit{\gamma }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)},$$ (27)

where γ is the gap fraction of the acquired frames γ = t_gap/t_frame, with t_frame = (t_e + t_gap). It should be noted that in this equation, signal amplitude A, gap fraction γ, and the background variance will also depend on t_e. Eq. 27 may also be expressed as a function of SNR as follows.

$${\sigma }_{t0}=\sqrt{\left(\frac{3}{2+\mathit{\gamma }}\right)\times \frac{t_{frame}}{SNR}},$$ (28)

keeping in mind that SNR also depends on t_e. For the example of Fig. 4 with t_frame = 101.8 ms and γ∼0.018, σ_t0 = 124 ms/SNR is obtained, matching the experimental data (Fig. 4 F).

Validating σ_t0 predictions by simulations

The treatment we present here is based on the rate of signal and background photon detection and thus does not depend on the specific camera quantum efficiency. We also do not normalize to camera pixel area and analyze first the noise in time traces obtained from individual pixels as in the example of Fig. 4. However, the noise generated by signal and background photons, as well as readout noise and the gap fraction depend on the specific camera type that is used (Table 1). Background noise can arise from background photons, background thermal dark current generated in the camera, or from camera readout noise. The background variance after averaging N traces is approximately as follows.

$${\sigma }_B^2=\frac{F^2\times \left(b+d\right)\times t_e+R^2}{N},$$

where t_e is the exposure time, b is the rate of background photon counts, d is the camera dark current, F the noise factor introduced by camera gain, and R is the camera readout noise. Substituting the signal amplitude A by (λ × t_e), σ_t0 can be expressed as follows.

$${\sigma }_{t0}\approx t_{frame}\times \frac{\sqrt{\left[F^2\times \left(\mathrm{b}+\mathrm{d+}\mathit{\lambda }/2\right)\times t_e+R^2\right]/N}}{\mathit{\lambda }{t}_e}\sqrt{\frac{3}{2}\times \left(\frac{1}{1+\raisebox{1ex}{$\mathit{\gamma }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}.$$ (29)

Eq. 29 is a useful tool to predict the precision of timing σ_t0 and to optimize the exposure time. One way to validate this equation is by performing corresponding simulations. Simulations can cover a broad range of the different parameters that affect σ_t0, specifically background and signal photon rates, exposure time, readout time, and readout noise. The results of the simulations provide detailed profiles of the correlations between these parameters and ECOM accuracy, which are difficult to obtain experimentally in an equally systematic way.

Idealized camera parameters

For ideal conditions, the pixel records signal photons and background photons without any other noise contributions and with zero gap time (i.e., t_frame = t_e). In this case, the precision of t₀ is as follows.

$${\sigma }_{t0}\approx \sqrt{\frac{3}{2}}\times \frac{\sqrt{\left(b+\mathit{\lambda }/2\right)\times t_e/N}}{\mathit{\lambda }}.$$ (30)

This relation is shown as solid black lines for averages of 1000 sweeps assuming λ = 10 photons/s either in the absence of background (Fig. 5 A) b = 0 or in the presence of b = 100 photons/s background (Fig. 5 B) as a function of exposure time t_e. In the absence of background, σ_t0 is ∼5.5 ms for 400-ms exposure times and approaches zero as the exposure time approaches zero. However, with the reduced exposure time, SNR also goes to zero (black dashed lines), indicating that there will be practical limitations. To determine which σ_t0 values can actually be realized in the analysis, we performed simulations of photon detection in a Poisson process with these parameters. For each parameter set, 1000 sweeps were simulated with the intensity step occurring at t = 0, randomly distributed relative to the frame start times, averaged as illustrated in Fig. 3 and fitted using Eq. 22. For each parameter set, this was repeated 100 times such that 100 determinations of t₀ were performed, providing the mean t₀ as well as its SD. The resulting data points (solid black circles) are in good agreement with the predictions of Eq. 30 down to an SNR of ∼6–7 (Fig. 5, arrows). Whereas fits at a higher SNR are generally very good (Fig. 5 C, t_e = 50 ms, fitted t₀ = −7 ms), at a lower SNR, many fits fail (Fig. 5 D, t_e = 20 ms, fitted t₀ = −18 ms) giving large σ_t0 values (Fig. 5 B, black dots at t_e = 10 ms and t_e = 20 ms), indicating that a robust determination of σ_t0 requires an SNR of ∼7, which, in the absence of background, requires t_e >2 ms and provides a σ_t0 of ∼0.4 ms. In the presence of a 100 photons/s background, an SNR of ∼6 requires a t_e of ∼50 ms and provides a σ_t0 of ∼8 ms.

Figure 5.

ECOM precision calculated using Eq. 30 (solid lines) for the ideal (black), EMCCD (red), and sCMOS (green) camera. Dependence of SNR on exposure time is indicated by dashed lines. Simulation results are indicated as solid circles for σ_t0 and open circles for SNR based on 100 simulations of 1000 sweeps averages for each data point. (A) 10 signal photons/s starting at t₀ with no background and (B) 10 signal photons/s with 100 continuous background photons/s are shown. (C–F) Example fits are shown: (C) ideal camera t_e = 50 ms (background = 100 photons/s), (D) ideal camera t_e = 20 ms (background = 100 photons/s), (E) EMCCD t_e = 20 ms (no background), and (F) sCMOS t_e = 20 ms (no background). To see this figure in color, go online.

Comparison of typical camera parameters

For current imaging setups, we consider EMCCD cameras in frame transfer mode and sCMOS cameras in global shutter mode. The parameters are given in Table 1, and we, again, compare image acquisition, assuming λ = 10 photons/s with either b = 0 or b = 100 photons/s. Quantum efficiency was set to 1 in all simulations because, here, we are only concerned with precision based on detected photons. As expected, the EMCCD (red lines) performs close to the ideal camera. However, in the absence of background, σ_t0 has a minimal value of ∼1.5 ms at ∼6-ms exposure such that σ_t0 is rather insensitive to a considerable increase in t_e (Fig. 5 A). SNR at ∼7, requires an ∼7-ms exposure time with σ_t0 at ∼1.5 ms in the absence of background. In the presence of 100 photons/s background (Fig. 5 B) an SNR >7 requires ∼80-ms exposure time providing a σ_t0 at ∼16 ms, which is double the value for the ideal camera and is largely due to the noise factor that doubles the variance generated by the background photons. The results shown in Fig. 5 A may be compared to the precision obtained by averaging the precise waiting times for arrival of the first photon (Eq. 21), which for an average of 1000 trials provides a precision of σ_t0 = 3.2 ms, whereas EMCCD imaging with 10-ms exposure time provides a σ_t0 of ∼1.5 ms.

The sCMOS camera considered here has a noise factor of 1 but has a median 2.4 photons root mean square readout noise. It should be noted that for sCMOS cameras, the pixel readout noise varies between pixels, and the precision of the ECOM result will vary accordingly between pixels as predicted by Eq. 30. In the absence of background, SNR >7 requires ∼50-ms exposure time providing a σ_t0 of ∼9.5 ms. However, increasing the exposure time increases SNR almost linearly, and σ_t0 is thus not much increased with strongly increased t_e (Fig. 5 A, green lines). In the presence of 100 photons/s background, SNR >6 is obtained at a 70-ms exposure time and provides a σ_t0 of ∼14 ms, which slightly better than the EMCCD because of the lower noise factor.

As for the ideal camera, simulations were performed for the two camera types and analyzed using the ECOM method. The data points are in good agreement with the theoretical predictions for SNR >7. For t_e = 20 ms in the absence of background, the SNR of the EMCCD is ∼13 and reliable fits are obtained (Fig. 5 E, fitted t₀ = 1 ms), whereas the SNR of the sCMOS is only 2.7 and fits frequently fail (Fig. 5 F, fitted t₀ = 41 ms), giving rise to the large σ_t0 values at t_e = 10 and t_e = 20 ms (Fig. 5 A, green dots).

Compiling a large number of data sets simulated for the EMCCD with different SNR in the averaged traces and sorting them according to the exposure time (Fig. 6 A, data points) shows the reciprocal dependence of σ_t0 on SNR expected from Eq. 28 (Fig. 6 A, solid lines). As shown above, the ECOM method functions reliably with any exposure time as long as the SNR is >7. When the SNR falls below 7, ECOM exhibits poor precision because the fits frequently fail. The required minimal SNR of ∼7 is thus not specific for the signal and background intensities specified in the analysis of Fig. 5 but, in general, holds independent of the specific signal and background intensities. The determining quantities are only t_e and the SNR of the fitted averaged trace. The SNR depends on all the experimental parameters, including camera type, background signal, signal intensity (the amplitude of intensity change), frame exposure time, and number of sweeps that are averaged. It should be noted that the equations and corresponding results for ECOM precision are applicable for any values of t_e and t_gap and not restricted to specific types of instrumentation. They may be applied also to line scan imaging or other imaging modalities with very fast image acquisition rates.

Figure 6.

Dependence of ECOM precision on the signal/noise ratio (SNR) from simulations and on signal photon rates from experimental data. The data points are grouped according to different exposure times indicated by colors (10 ms: gray; 20 ms: black; 50 ms: red; 100 ms: green; 200 ms: blue). (A) ECOM precision (SD of fitted t₀ values) versus SNR is shown. Precision agrees with theoretical values from Eq. 28 for a SNR above ∼7 (vertical arrow). (B) ECOM precision versus signal photon rates from the experiment of Fig. 4 is shown. Data points are grouped according to frame exposure times. Solid lines indicate theoretical values for zero background from Eq. 29, and dashed lines indicate SNRs of averaged traces obtained with the corresponding exposure times. ECOM fits provide reliable values for SNRs >5–10 (vertical arrows). To see this figure in color, go online.

Fig. 6 A shows the dependence of σ_t0 on t_e and SNR. However, SNR also depends obviously on t_e. How does one select a suitable exposure time to achieve the optimal ECOM precision in a given experimental situation, that is, for a given signal amplitude? To address this question, we analyzed the imaging data of the fluorescent beads (Fig. 4 A) acquired with different exposure times (20–200 ms) as a function of signal photon detection rate (Fig. 6 B).

The σ_t0 values (Fig. 6 B) are the SDs obtained from averaging t₀ values from 100 pixels with similar signal amplitudes. The data points are in close agreement with the theoretical predictions from Eq. 29 (solid lines) for SNR > 5–10. The SNRs (right-hand scale) for the different exposure times are shown as dashed lines. It should be noted that the SDs provided by the IGOR curve fitting routine output for the fit parameters are much smaller than the actual SDs because of the resampling of the measured frame intensity to a large number of points (N_framepoints), which are not independent data points. The SEs of t₀ provided by the fit program must thus be multiplied by a factor sqrt(N_framepoints) to obtain the appropriate error estimate. For lower SNRs, σ_t0 values are much higher, as expected from the results of Figs. 5 and 6 A. It should be noted that even for relatively high signal photon rates (>500 photons/s), the impacts of different exposure times on the σ_t0 values are very limited, as indicated by the data points as well as the theoretical curves. For signals with 500 photons/s, ECOM precision is improved less than twofold when exposure time is decreased from 200 to 20 ms. On the other hand, for low photon rates (<100 photons/s), the theoretical lines are practically indistinguishable, with longer exposure times producing much more robust and reliable fits because of higher SNR with a precision that is not compromised by increased exposure times. For an ∼20 photons/s signal, the 200-ms exposure gives a σ_t0 at ∼30 ms a better precision than any of the shorter exposure times (Fig. 6 B). These results suggest that exposure times in the 100–200-ms range should be a safe choice to achieve good ECOM precision for both strong and weak signals, especially when the signal strength is not a priori known.

The time course of cellular [Ca²⁺] rise after brief depolarization

In chromaffin cells, the intracellular Ca²⁺ concentration ([Ca²⁺]_i) changes on the millisecond timescale in response to depolarization, triggering catecholamine release from dense core vesicles. To study the time course of [Ca²⁺]_i using ECOM, mouse chromaffin cells were loaded with fura-2 in the whole-cell patch-clamp configuration and stimulated by 10-ms depolarization pulses at ∼10-s interpulse intervals. The resultant I_Ca was recorded, and a sequence of 2000 fura-2 fluorescence images of the cell was simultaneously acquired with 100-ms exposure time (EM gain = 300) (Fig. 7). The camera fire output was recorded together with the depolarization pulse for subsequent ECOM analysis. To track the time course of [Ca²⁺]_i continuously, fura-2 was only excited at the more Ca²⁺-sensitive wavelength of 387 nm during the stimulation process (Fig. S1). Paired excitations of 360/390 nm were performed before and after the pulse sequence to obtain a ratiometric calibration of total fluorescence and [Ca²⁺]_i.

Figure 7.

Determination of [Ca²⁺]_i kinetics using ECOM. (A) 10-ms depolarizing pulse (black) and the averaged Ca²⁺ current trace (green) are shown. (B) The averaged time course of whole-cell fluorescence (t_e = 100 ms) from a total of 89 pulses averaged using the ECOM method (black trace), fitted using the ECOM Eq. 22 (red dashed line; t₀ = 16.8 ms) is shown. The top orange trace shows the fit residual. An average of 89 fast line scan recordings (five lines; t_e∼1 ms; 678 Hz) for comparison (noisy blue trace) is shown. I_Ca and imaging data were aligned at the onset of the depolarizing pulses (t = 0) before averaging. (C) The step response (dashed, left scale) and impulse response normalized to unity integral (solid line, right scale) for ECOM analysis with t_e = 100 ms and t_gap = 1.8 ms are shown. (D) The deconvolution fit of the ECOM intensity data using a single exponential rise and decay (red dashed line) with residual (orange trace) is shown. The deconvolved intensity time course is shown as a smooth red line, and a single exponential fit of the fast line scan data is shown as a smooth blue line. To see this figure in color, go online.

The Ca²⁺ currents (Fig. 7 A, green trace) and imaging data from trials with I_Ca >10 pA were aligned at the onsets of the depolarization pulses and averaged. The averaged ECOM fluorescence trace (Fig. 7 B, black trace) shows a large decrease of fluorescence intensity with SNR = 74. Fitting the time course with the ECOM step response function (Eq. 22) from −200 to +190 ms, where the fluorescence signal had its minimum (Fig. 7 B, red dashed line), revealed a 16.8 ± 1.7-ms delay between the pulse onset and the fluorescence intensity decrease.

The result of the ECOM analysis was compared to the time course obtained by fast line scans of the fura-2 fluorescence in a small region of the cell measuring the time course of the fura-2 fluorescence change induced by the depolarization pulse with HTR (five lines, t_e ≈ 1 ms; frame rate ∼678 Hz; Fig. S2). Averaging 89 trials, a rapid fluorescence change was obtained (Fig. 7 B, noisy blue trace), which showed a rapid change consistent with the ∼17-ms delay obtained by ECOM analysis based on 100-ms exposure time. The time course of the ECOM trace (black) is much slower than that obtained from the fast line scan data because the long exposure time acts as a low-pass filter with the step response function given in Eq. 9.

The time course obtained by ECOM analysis represents the convolution of the actual averaged intensity time course with the ECOM impulse response function (Fig. 7 C), which is the derivative of the step response function (Eq. 9). In principle, it is possible to perform a numerical deconvolution, but these inverse problems suffer from instabilities when applied to noisy data. To determine if ECOM can provide reasonable estimates of the time course of nonstepwise intensity changes, we performed a deconvolution based on the assumption that the fluorescence intensity change follows a single exponential time course. The time course of the intensity change was fitted (Fig. 7 D, red dashed line) such that its convolution with the ECOM impulse response function fits the ECOM data. The deconvolved time course of the Fura-2 fluorescence change (Fig. 7 D, smooth red line) revealed a 20.2 ± 0.6-ms time constant for the fura-2 intensity change, which is very close to the 17.1 ± 0.9-ms time constant obtained by a single exponential fit of the fast line scan data (Fig. 7 D, smooth blue line).

ECOM analysis of rapid transients

In many imaging applications, fluorescence changes involve rapid transients with a rise and decay on the millisecond timescale. Examples are the glutamate “sniffers” (11) or genetically encoded voltage indicators (12, 13), which are typically also noisy and require signal averaging. To determine if ECOM can provide reasonable estimates of the timing and kinetics of such changes, we performed simulations for EMCCD imaging with t_e = 15 ms and t_gap = 1.8 ms, which are realistic parameters for wide-field imaging. Because of the short frame duration, the subsampling was performed with 100 μs rather than 1-ms dwell time. Transients were simulated assuming a linear rise, reaching a peak signal photon rate of 2000 photons/s after 1 ms, followed by an exponential decay with a 3-ms (cyan), 10-ms (green), or 30-ms (red) time constant (Fig. 8 A). In each simulation, 36 transients were averaged. To achieve a SNR of ∼20 of the averaged ECOM signals, a background intensity of 200, 800, and 2000 photons/s was assumed for the respective decay time constants. When these signals were fitted, with baseline, signal start time t₀, signal amplitude, and decay time constant τ_decay as free fitting parameters, these parameters were correctly determined by the fit and the assumed input signals were correctly recovered (Fig. 8 B). Fitted kinetic parameters were t₀ = 0.5 ± 1.8 ms, τ_decay = 2.8 ± 2.0 ms; t₀ = −0.66 ± 1.0 ms, τ_decay = 9.9 ± 1.7 ms; and t₀ = 0.90 ± 0.85 ms, τ_decay = 31.0 ± 2.6 ms for the transients with 3-, 10-, and 30-ms decay time, respectively.

Figure 8.

(A) Simulated transient changes starting at t = 0 with a linear rise reaching a peak of 2000 signal photons/s after 1 ms, followed by an exponential decay with 3-ms (cyan), 10-ms (green), or 30-ms (red) decay time constant. Each transient was simulated averaging 36 signals. Background intensities were 200, 800, and 2000 photons/s for the 3-, 10-, and 30-ms decay time constants, respectively, to obtain an SNR of ∼20 for each averaged signal. ECOM fits of signal start and decay time (black smooth lines) recovered the correct input signal time courses (B). To see this figure in color, go online.

Discussion

Fluorescence imaging of dynamic cellular processes has become an essential tool in cell biology and neuroscience. Many cellular events are very rapid and require time resolution of a millisecond or less. In particular, determination of the time of fluorescence changes relative to rapid events such as pulse stimulation, electrophysiological, or electrochemical signals with HTR is of considerable importance. Such imaging experiments are frequently performed at very low intensity, which means that the rate of photon detection is low and camera exposure times are long such that signal averaging is used to obtain an acceptable SNR. We developed the ECOM method that aligns and averages the imaging data relative to an HTR event with a precision that is not limited by camera exposure time and applied it to analyze small FRET changes associated with individual transmitter release events detected as an amperometric spike that serves as the HTR event (7, 8). Here, we provide a detailed theoretical, computational, and experimental analysis of the method and its precision in determination of the time of the intensity step with subimaging frame resolution.

When the sampling frequency is insufficient to resolve the time course of a signal of interest, one way to improve time resolution is to perform repetitive measurements in a way in which the data are only sampled once after the trigger with a variable time interval between trigger and sampling, which can be much shorter than the sampling interval. This method was originally applied using the sampling oscilloscope (14). In cellular imaging experiments, a 350-ns pulsed laser was used to generate snapshots sampling fura-2 fluorescence images at various times after different individual stimuli during the exposure of a cooled CCD camera (15). Using this method, the time course of sarcomeric calcium gradients was visualized in frog skeletal muscle fibers with a time resolution of a few milliseconds. Another possibility is to sample repetitive sweeps at a constant (low) rate but shift the sampling times relative to the trigger in increments smaller than the sampling interval such that in repetitive measurements, sample points are spaced at improved resolution. Such signal reconstructions typically involve averaging the repetitive scans. When the time shifts are randomly distributed in the “random equivalent time sampling” method, the sampled values are binned into sampling subintervals according to their actual sampling time.

The basic principle involving the averaging of time-shifted image sequences has been applied in a few studies to obtain time superresolution better than that given by the imaging frame rate (8, 16, 17, 18). It was used in high-speed two-photon imaging of Ca²⁺ transients (16) and wide-field fluorescence imaging of Ca²⁺ and voltage-sensitive dye (18) associated with back-propagating action potentials, using the somatic action potential recorded by a whole-cell patch pipette as the reference HTR event. Alignment of confocal imaging intensity data indicating assembly and disassembly of actin patches has also been performed with subframe precision by interpolating data points from acquisitions with low frame rates and fitting their time shift for optimal overlap and subsequent averaging without the a priori knowledge of a related HTR event (17).

In random equivalent time sampling, the signal is typically sampled over a very short time period, which differs from typical camera-based image acquisitions for which the signal is integrated over most of the acquisition period to maximize the number of detected photons. The significance of timing fluorescence intensity changes with subframe resolution has been recognized for signal averaging (8, 16, 17, 18) as well as for aligning mathematical models with experimentally measured Ca²⁺ transients (19). The ECOM method takes advantage of the fact that for a rapid stepwise intensity change, the intensity reported by the imaging frame during which this change occurs depends on the time of the change during the exposure in a predictable way. For a single step, the intensity reported by the transition frame thus indicates the time of the transition with subframe resolution.

We derived the exact ECOM step response function for averaged intensity traces in which intensity steps occur randomly during the exposure based on the known camera exposure times and gap times between subsequent exposures. Experimental and simulated data measuring repetitive step changes are exactly fitted by this equation. We provide a rigorous analysis of the precision of the method. We show how the ECOM precision depends on the SNR of the averaged signal as well as exposure and gap times and provide equations that correctly predict the optimal choice of camera exposure time to achieve the optimal timing precision for given signal and background photon fluxes. It should be noted that the precision depends on the SNR of the averaged intensity traces. A given SNR may be achieved by averaging a small number of traces with low noise or a large number of traces with high noise. The analysis was applied here to camera-based imaging but is also applicable to a wide range of other signal detection mechanisms that involve integration of the signal over sampling intervals with significant duration. It is also applicable to confocal or multiphoton imaging in which the equations then apply to the integration time and repetition rate for individual spots that determine exposure and gap time.

We also show that the time course of nonstepwise intensity changes may be analyzed by deconvolution based on the ECOM impulse response function. The deconvolution approach further increases the breadth of potential ECOM applications, making ECOM analysis superior to alternative approaches that do not explicitly take into account the transfer function of camera exposure and readout time. The time course determined by deconvolution of the averaged trace is, of course, the average time course of the intensity change. In the presence of variability between individual events in terms of amplitude, time delay, or rise/fall time constants, this variability is contained in the average time course and as such also reflected in the ECOM analysis result.

We applied the ECOM analysis using 100-ms exposure times to determine the time course of the [Ca²⁺]_i rise in response to 10-ms depolarization pulses in chromaffin cells dialyzed with fura-2, revealing a delay of ∼17 ms relative to the pulse onset. This result was consistent with the time course obtained by fast line scan imaging through the center of the cell (Fig. 7 B). The dynamics of [Ca²⁺]_i near release sites after a short depolarization pulse to evoke exocytosis has been previously estimated in chromaffin cells by combining measurements of secretion by amperometry and capacitance measurement (20). Measurements of the average fluorescence changes in the entire cell involve not only rapid Ca²⁺ influx but also Ca²⁺ diffusion and the kinetics of Ca²⁺ binding to fura-2. During the stimulation pulse and shortly afterward, pronounced [Ca²⁺] gradients exist, showing partial rings or hotspots immediately beneath the plasma membrane, which dissipate mostly by diffusion of mobile buffers (21, 22, 23, 24, 25, 26). The apparent effective Ca²⁺ diffusion coefficient in the cell is ∼40 μm² s⁻¹ (22), which leads to equilibration of [Ca²⁺]_i over tens to hundreds of milliseconds (21, 26). Initially, the fura-2 at the [Ca²⁺] hotspots will thus be saturated, explaining the delay of the average [Ca²⁺]_i rise reported by fura-2. A continued [Ca²⁺] rise for ∼20 ms after the end of a 20-ms pulse was previously reported (21). Assuming a single exponential time course for the [Ca²⁺]_i rise, ECOM deconvolution analysis also revealed a ∼20-ms rise time, which is very close to the time constant obtained by a single exponential fit of fast line scan data obtained in a separate set of measurements (Fig. 7 D). As a wide-field imaging technique, ECOM analysis may be applied to individual pixels or to arbitrary regions of interest (see also ECOM_Software in Data S1) and is thus suitable to achieve temporal superresolution of multiple spatially distinct signals such as subcellular [Ca²⁺] transients.

Author Contributions

Q.F. performed research, developed software, analyzed data, designed research, and wrote the article. Y.Z. performed research, analyzed data, designed research, and wrote the article. M.L. designed research, performed research, developed software, analyzed data, and wrote the article.

Editor: Catherine Galbraith.