1. Introduction

With the development of fluorescent calcium indicators and fast imaging technology, calcium imaging has enabled us to optically detect spikes within local neuronal population [1–3], which is essential for understanding how neural information is processed. In this approach, reconstruction methods read out spike patterns from calcium traces and error estimation methods provide quantitative evaluation of the reconstruction results. The reconstruction and its error estimation have proved to be an important issue in the neuronal information detection [4–6].

Various reconstruction methods have been developed to date and significant progress has been achieved [5, 7–14]. However, there is no error estimation method that evaluates the corresponding reconstruction results associated with unclear nonlinearities in the spike-calcium relationship [9, 15, 16], contamination of signals from other cellular parts [4], system noise [17], and the often relatively low temporal resolution of the calcium signal recording compared to the electrical signal [14], along with the lack of a strict mathematical model describing the relationship between the Ca²⁺ trace and spike firing (especially burst firing) [18, 19].

In this paper, we propose a relatively simple and effective reconstruction method. During the reconstruction process, the statistical data features corresponding to the reconstruction method are extracted and are then used to evaluate the results. Combined analog signals of several types, whose statistical data features are same as the actual trace, are regarded as having the same error ranges as the actual trace. This error estimation method uses analog data to approximate the experimental trace; thus, we are able to avoid searching for complex mathematical models to describe Ca²⁺ signals and burst activities. Moreover, this method only requires a small amount of a priori knowledge, such as noise models, Ca²⁺ transients to single spikes (i.e., the template), and basic types of burst firing. Evaluation results of real rates of 10 experimental Ca²⁺ traces within their estimated ranges confirmed this evaluation method to be effective.

2. Material and methods

2.1 Sample preparation and data acquisition

All animal studies were performed in compliance with protocols that had been approved by the Hubei Provincial Animal Care and Use Committee and conformed to the experimental guidelines of the Animal Experimentation Ethics Committee of Huazhong University of Science and Technology. Postnatal (P14−20) Wistar rats were used for the experiments. The animals were anesthetized with 1% sodium pentobarbital (50 mg/kg). After fast decapitation, the brain was rapidly dissected and placed in ice-cold oxygenated (95% O₂ and 5% CO₂) normal artificial cerebrospinal fluid (ASCF). Cortical slices were cut 350−400 µm thick with a vibratome (VTS1000, Leica) and maintained in an incubation chamber. For the imaging experiments, slices were transferred to a submersion-recording chamber and incubated in oxygenated ACSF at room temperature. To induce epileptiform discharging, a potassium channel blocker 4-aminopyridine and high potassium solution were applied to the bath at final concentrations of 100 μM and 10 mM, respectively [20]. To compare the reconstructed with the real spike train, neuronal Ca²⁺ transients and spike firing were simultaneously recorded. Spike firing is characterized by rapid voltage changes of the membrane potential, and was recorded with a patch-clamp amplifier (EPC-9, HEKA). The signals were sampled at 10 kHz by the amplifier system. The recording pipettes were routinely filled with electrode filling solution containing 100 µM Fluo-5F (calcium dye). The local neuron population was loaded sequentially with the calcium dye Fluo-5F through the recording pipettes [21], and the real voltage change and neuronal Ca²⁺ transients of the last patched neuron were recorded simultaneously.

In this paper, a custom-constructed fast two-photon fluorescence microscope system was used to image Ca²⁺ fluorescence transients. The system setup has been described by Lv et al [22]. Briefly, the mode-locked laser intensity (800 nm and ~100 fs pulse width at a 80 MHz repetitive frequency) was adjusted using an electric-optical modulator (EOM). After passing through the dispersion-compensation prism (tilted at 45°), the laser beam was x–y deflected by two orthogonal acousto-optic deflectors (AODs). The laser beam was then coupled to an upright microscope (BX61WI, Olympus) with a water immersion 40 × objective (Olympus) with a working distance of 3.3 mm and a numerical aperture of 0.80. The imaging process was controlled by custom-written software in LABVIEW. Relative fluorescence intensity fluctuation (ΔF/F) was used to indicate neuronal firing.

2.2 Reconstruction of neuronal spiking with local differential method (LDM)

The differential algorithm is a simple and classical method to rebuild spike trains, but it is incredibly sensitive to noise [12, 23]. We improved this method by only analyzing the differential signals that corresponded to the rising segment of the Ca²⁺ transient (named local differential method, LDM). The procedure of LDM, as shown in Fig. 1, is described as follows:

 

Fig. 1 Schematic depiction of neuronal spiking reconstruction using local differential signal. (A) Using 100-Hz and 10-Hz low-pass filters to smooth the original Ca²⁺ trace. A1 is an enlarged segment of the filtered signals. As shown in A1, the time interval of spiking is estimated from the 10-Hz low-pass filtered signal. (B) Applying LDM to 100-Hz low-pass filtered signal to generate the differential signal. B1 is an enlarged segment of the differential signal. B1 shows the extraction of peak width and peak timing of the differential signal. (C) Identifying neuronal spikes from the extracted peak widths. When the value of a peak width drops into the interval of spike identification, its corresponding peak timing is regarded as the neuronal spike timing. (D) Reconstruction of neuronal spike train from the identified peak timings. In the case where the neuronal electrical signal is recorded and the real spike timing is known, the error time of reconstruction can be calculated by comparison of the real and the reconstructed spike timings. Note that the circle indicated by the red arrow in (C) represents two reconstructed spikes, and the time difference between these two reconstructed spikes and the real spikes is indicated by the arrows in (D).

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In process (1), two low-pass filters with 10 Hz and 100 Hz were used (the number of zeroes and poles were set to 3 and 10) for smoothing the Ca²⁺ trace (Fig. 1(A)). To estimate the time interval of spiking, the Ca²⁺ trace was first smoothed with an extreme low-pass filter (10 Hz), which resulted in a single rising phase that could correspond to burst spiking or a single spike of a neuron. We then extracted the rise duration by setting the threshold of the rise amplitude of the 10-Hz-filtered Ca²⁺ trace. We let t₀ and t₁ denote the start and end time of the rise duration, respectively. As shown in Fig. 1(A1) and 1(B1), the times of the first and last spikes in the burst are later than t₀ and t₁, respectively. The time interval [t₀, t₁] cannot contain all spike timings. Time interval [t₀ + 50 ms, t₁ + Δt] was used to ascertain the interval of neuronal spiking, as indicated by the green arrows in Fig. 1(A1). t₁ + Δt is the time corresponded to the signal value that equaled 2/3 times the peak value of t₁ in the 10-Hz-filtered signal. As shown in Fig. 1(A1) and 1(B1), the last spike timings of neuronal burst are later than t₁. Correspondingly, the time interval [t₀, t₁] cannot contain all spike timings. In addition, the first spike time is later than t₀, and interval [t₀, t₀ + 50 ms] corresponds to noise signal with a 100 Hz filter. So we use the time interval [t₀ + 50 ms, t₁ + Δt] to be the estimated time interval of neuronal spiking for achieving that the estimated time interval is small and contain as many spike timings as possible. The selections of 50 ms and Δt, experimental determined, makes the estimated time intervals contain all most spike timings (results not shown).

In process (2), a differential operation was performed on the 100-Hz-filtered Ca²⁺ trace segments that were matched to the time intervals derived from the above process in (1a), and for the other signal segments, the differential values were set to zero. Importantly, this was only performed on the 100-Hz-filtered trace segments and not the whole Ca²⁺ trace, and is referred as LDM. From the locally differential signal, the peak timing whose value was more than the given threshold (twice the standard deviation of the differential of the Ca²⁺ trace (noise)) and the corresponding peak width were extracted, as shown in Fig. 1(B1).

In process (3), we classified the extracted peak widths into three groups that corresponded to noise, single-spiking, and bursting activity (Fig. 1(C)). We selected two threshold values, denoted by b₁ and b₂, for distinguishing the three groups. When peak width was less than b₁, there was no spike corresponding to the peak. When peak width was greater than b₁ and smaller than b₂, there was one single spike at the peak time. When peak width was greater than b₂, it represented a special burst-firing pattern where two spikes fired with an interval of about 10 ms, so that their wave forms were not completely separated. Here, b₁ was determined by the statistical information we obtained from the peak widths of the noise trace (more than 95% of the peak widths from noise were less than b₁), and b₂ was set to about 1.3 times the peak width of the differential of the template.

In process (4), the reconstructed spike timing refers to the series of peak timing whose corresponding peak width was more than the threshold value b₁. From the reconstructed spike timings (t_i, i = 1,2,..,n), the spike train (m(t) = 1 when t = t_i, i = 1,2,...,n and m(t) = 0, otherwise) can be generated. Note that, if the corresponding electrical signal was also recorded, that is, the real spike timings were known, we can measure the difference between the reconstructed and real spike timings, as shown in Fig. 1(D).

2.3 Estimating the error range of the reconstructed results

To estimate the error range of neuronal spiking reconstruction, we generated a simulated Ca²⁺ trace whose peak-width distribution was as close as possible to that of the evaluated Ca²⁺ trace, and we then used the reconstruction of the simulated Ca²⁺ trace for assessing the reliability of the reconstruction results from the evaluated trace. The procedures for generation of analog signals, as shown in Fig. 2, are described as follows:

 

Fig. 2 Schematic depiction of signal simulation close to the experimental trace. (A) Extraction of peak-width distribution curve. (A1) a 100-Hz low-pass filtered Ca²⁺ trace; (A2) the locally differential Ca²⁺ trace; (A3) the distribution of the peak widths extracted from the signals in (A2). (B) Generating the four types of simulated signals and calculating the peak-width distribution curves from the simulated signals. (C) Using the linear combination of peak-width distribution curves of the simulated signals to approach the peak width distribution of the evaluated trace in (A3) and obtaining the optimal coefficients. The locally differential signal can be decomposed into the four types of simulated signals, and the portions of this decomposing correspond to the optimal coefficients.

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In process (5), for the given 100-Hz-filtered Ca²⁺ trace (Fig. 2(A1)), we obtained its corresponding differential signals by using LDM (Fig. 2(A2)). From the local differential signals, we extracted the peak widths, and their distribution curve was generated (Fig. 2(A3)).

In process (6), based on the fact that (1) a Ca²⁺ trace without noise could be modeled as the convolution of a given template (corresponding to single spiking) and a 0-1 sequence (spike train) [4, 5, 18], and (2) the noise of the Ca²⁺ trace was close to Gaussian white noise [17, 24], the simulated Ca²⁺ trace could be generated by the model written as the following equation.

Here, Template is an average of Ca²⁺ transients to single spike (Fig. 2(B)), s(t) indicates a 0-1 sequence (Fig. 2(B)), Gnoise(t) is defined as Gaussian white noise, filter() is a 100-Hz low pass filter, and the ratio of signal-to-noise (SNR) is the ratio of amplitudes of the template to the standard deviation of noise. The template is extracted from the experimental Ca²⁺ trace shown as Fig. 2(A1), and the noise is the original Ca²⁺ trace of the time intervals, which are the complement of the estimated time interval of spiking. Considering that peak-width distribution of the experimental Ca²⁺ trace contained the information of noise, single-spiking, and burst firing, the corresponding four types of analog signals shown in Fig. 2(B) were generated using Eq. (1), and respectively determined by s(t), in which all elements were assigned 0 (noise); only one element was assigned 1 (single-spiking), only two elements were assigned 1 with interval time equal to 10 ms (burst I), and equal to 20 ms (burst II). Burst I and II correspond to situations where the two Ca²⁺ traces behave totally transient and partly separated, respectively. Note that in our analysis, we deleted the first wave of burst II due to the fact that its rising duration was the same as that of the wave of single-spiking, and the second wave of burst II was multiplied by a scale from the normal distribution with a mean value of 0.75 and standard deviation of 0.11 for approaching the diversity of waves in the experimental signal. We performed the differential operation on these analog signals, extracted the peak widths from the differential signals with the same procedure applied to the experimental signal, and computed the corresponding distribution of peak widths, denoted by the distribution curve (Fig. 2(B)).

In process (7), we used a linear combination of the four types of distribution curves, denoted by f₁, f_2, f_3, f₄, respectively, to approach the distribution curve of the evaluated Ca²⁺ trace denoted by s under the condition that all linear parameters should be more than zero, and ||.|| be L₂-norm.

By solving the optimization problem (2) [25], we can obtain the optimal coefficients. Correspondingly, the linear combination of four types of simulated signals (S₁, S₂, S₃, S₄; shown in Fig. 2(B)) with the obtained coefficients can be regarded as an approximation of the experimental signal in Fig. 2(A2).

After obtaining the analog signals used to approach the experimental signal, we can estimate the error range of the reconstructed results. As described previously, the analog signals are the linear combination of four types of simulated signals with the coefficients, denoted by a_i (i = 1, 2, 3, 4), obtained by solving the optimization problem (2).

For the given type of analog Ca²⁺ signal associated with the known electrical signal (Fig. 2(B)), 1000 pieces of signals are generated for error estimation. Each piece of signal has a same length and a same sampling frequency with the evaluated Ca²⁺ trace. And then spike train of each piece can be reconstructed with the same procedure applied in the evaluated trace. The corresponding false-negative ratio (FNR) and false-positive ratio (FPR) are generated by comparing the real and reconstructed spike train. Here, FPR is defined as the ratio of the number of falsely detected spikes to the number of totally detected spikes [26, 27]. FNR is defined as the ratio of the number of omitted spikes to the number of real spikes. A suspected spike was regarded as falsely detected spike satisfying that time difference between the suspected spike and the true spike was more than 10 ms, and regarded as real spikes, otherwise. We analyze these 1000 pieces of signals and calculate their FNRs and FPRs. The mean μ and standard deviation σ of the calculated FPRs are used to construct the confidence interval [μ(FPRs)-2σ(FPRs), μ(FPRs)-2σ(FPRs)] of FPR of the spiking reconstruction from the given type of analog signal. Similarly, the confidence interval [μ(FNRs) + 2σ(FNRs), μ(FNRs) + 2σ(FNRs)] of FNR are also obtained. These two confidence intervals are regarded as error ranges of spiking reconstruction from the given type of analog signal. We use a linear combination of four types of analog Ca²⁺ signals with parameters a_i (i = 1, 2, 3, 4) to approach the evaluated trace. Correspondingly, with slight modification, the same combination of error ranges from analog signals can be regarded as an approximate error range approximating that from the evaluated trace. In detail, the confidence intervals of FNR and FPR of spiking reconstruction from the evaluated trace are estimated as the following Eq. (3) and (4) respectively.

In Eq. (3), FNRs is the false negative ratios of each type analog trace as described above, and i is the indicator of the type of analog signal. In Eq. (4), FPRs is the false positive ratios of each type analog trace, i has the same meaning as that in Eq. (3), and b₁ is the boundary of noise and single spike type, that is, if the peak width was greater than b₁, there was one single spike firing. s_b1 and s_b1+1 are the numbers of extracted spike events with a width of b₁ and b₁ + 1, respectively, and s_total is the total number of spikes extracted from the evaluated trace. From Eq. (4), we can see that the actual peak-width function was taken into account in the upper boundary estimation of FPR mainly based upon two considerations. On one hand, the probability is very low when the spikes with peak width far more than b₁ are falsely detected, and thus detection of spikes with peak widths at the vicinity of b1 is the main cause of false detection. On the other hand, when using the combination of peak-width curves to approach the real curve, it is difficult to fit well at the vicinity of b₁ due to their small values, that is, the fitting values at the vicinity of b₁ cannot well represent the real values closely related to FPR.

3. Results

We subsequently processed real experimental Ca²⁺ traces acquired using fast calcium imaging as described above (section 2.1). The spike train was reconstructed by LDM, and then evaluated using the data characteristics closing method with the results shown in Fig. 3. Figure 3(A) shows a demonstration of spike reconstruction; the top trace is the electrical signal obtained from whole-cell patch-clamp recording; the middle trace is the corresponding calcium trace from a neuron labelled with Fluo-5F sampled at 1000 Hz with a fast two-photon system; and the bottom trace is the reconstructed spike train from LDM. By comparing the electrical signal with the reconstructed spike train, we can easily observe that the reconstructed spike train is very similar to the real electrical signals. Indeed, there are 301 spikes in this 300-s calcium trace, and the real FPR and FNR are 7% and 2%, respectively (the 4th triangle in Fig. 3(B)). The algorithm was applied to other 9 samples of experimental data with durations of 300 s and sampling frequency of 1000 Hz. Reconstruction and estimation results are listed in Fig. 3(B). This figure demonstrates that most FPRs and FNRs were less than 10%, and that these rates were indeed within the range of estimated FPRs and FNRs. The 7% FPR of data set 4 and the 9% FNR of data set 6 exceeded the estimated range of 0−4% and 2−6%. This may be because long-duration spiking of one neuron led to difficulties of the convolution model in Eq. (1) in estimating the Ca²⁺ trace.

 

Fig. 3 Evaluation accuracy of reconstructed results of single neurons calcium traces from LDM. (A) A 300-s experimental trace, including electrical signal from whole-cell recording (top), the corresponding calcium fluorescence trace sampled at 1000 Hz (middle), and the reconstructed spike train from LDM (bottom). In this trace, there are 301 spikes in this 300-s calcium trace, and the real FPR and FNR are 7% and 2%, respectively. (B) Real and estimated error rangesof 10 samples of experimental data. The triangles represent the real FPR and FNR, and red vertical solid lines represent the estimated false-positive ranges (eFPR) and estimated false-negative ranges (eFNR). In this figure, most real FPRs and FNRs are less than 10%, and these real rates were indeed within the ranges of eFPRs and eFNRs.

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We next compared the reconstruction efficiency of LDM to that of current popular methods, such as deconvolution [5] and nonnegative deconvolution methods [14]. The same data as in Fig. 3(B) were processed by the deconvolution and nonnegative deconvolution methods, and the results are shown in Fig. 4. The figure illustrates that LDM exhibits an outstanding performance in both FPR and FNR. The deconvolution method, similar to the differential algorithm, is suitable for analyzing Ca²⁺ trace with high SNR. However, when SNR of Ca²⁺ trace was reduced, the effective reconstruction is depreciated. Nonnegative deconvolution method has a strong immunity to noise; however, it needs to be further developed for the reconstruction of burst I, that is, when two spikes fire with 10-ms interval. Thus, the nonnegative deconvolution performs well in same data sets, but deteriorates in analyzing data sets 7–10, which contain a certain amount of burst I. Based on these finding, we can conclude that the LDM is more robust to burst firing and demonstrates better reconstruction results than deconvolution and nonnegative convolution methods in analyzing these data sets.

 

Fig. 4 Comparison of reconstruction results of the same calcium traces through local differential, deconvolution and nonnegative deconvolution methods. (A) and (B) show FPR and eFPR, and FNR and eFNR, respectively. The same set data as in Fig. 3(B) was processed with deconvolution and nonnegative deconvolution methods. As previously pointed out, these calcium traces were sampled at 1000 Hz recorded with a fast two-photon system. The triangles, rhombi and squares in the figure denote FPR and FNR from LDM, deconvolution and nonnegative deconvolution methods, respectively. The LDM exhibits an excellent performance in both FPR and FNR across all 10 traces, while the performance of the two deconvolution methods is highly variable.

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At last, we applied the LDM to reconstruct spike trains from calcium traces of a neuronal population, and used the data characteristics closing method to evaluate the reconstruction effect, for which the results are shown in Fig. 5. Based on the calcium traces in the figure, the three neurons fire frequently, and there are burst activities in this period. From comparison of the real electrical signal with the reconstructed spike train from the corresponding calcium trace, the real FPR and FNR are both 9%. Using the data characteristics closing method, the estimated error ranges of FPR and FNR are 1–21% and 6–14%, respectively. The real rates are also within the estimated error ranges. The eFPR and eFNR of cell2 and cell3, shown in Fig. 5(C), demonstrate that when the real electrical signal is not available, we cannot only reconstruct the spike train, but we can also evaluate the reconstruction effect. It is important to note that a reasonable evaluation of reconstructed spike trains is highly valuable for the quantification of research based on the reconstructed neuronal activity. Through comparison of reconstruction results of neuron population in Fig. 5(C) to that of single neurons in Fig. 3(B), we can find that the reconstructed results of calcium traces from neuronal population are worse than that of single neurons. One probable cause may be the lower sampling frequency in this case; the other important cause may be the deteriorated state of neuron population. In the process of labeling, the local neuron population was loaded with the calcium dye Fluo-5F with the recording pipettes sequentially, and the real voltage change and neuronal Ca²⁺ transients of the last patched neuron were recorded simultaneously. Thus, the previously labeled neurons are whole-cell recorded first, and then the pipettes are withdrawn [21]. These manipulations could potentially harm the neurons and resting calcium levels could subsequently rise following that, which in turn would influence reconstructed results.

 

Fig. 5 Reconstructed results and evaluation of calcium traces of neuronal population. (A) The fluorescence image shows three neurons labeled sequentially with Fluo-5F with recording pipettes. The boxes indicate the recorded regions. (B) 300-s calcium traces from these three neurons, sampled at 666 Hz with (B1) representing the magnified signal within the black box in (B). The colors of the traces corresponded to the same colors as the boxes in (A). (C) The real electrical voltage trace of cell1 and reconstructed spike trains of cell1 to cell3 from LDM are shown. In the right panel, the real FPR and FNR of cell1, and eFPRs and eFNRs of cells1–3 are listed.

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4. Discussion and conclusion

In this paper, we propose a method to estimate the error range of the reconstructed neuronal spiking from a local differential algorithm. The results from 10 Ca²⁺ traces confirmed that this error estimation was effective and accurate. This error estimation method also was used to analyze the Ca²⁺ traces from a neuronal population, and provides a quantitative approach to estimate the reliability of inferring the activity of a population of neurons from their Ca²⁺ traces.

Despite the advances in reconstruction of neuronal spiking from Ca²⁺ traces [5, 9, 15, 19], only a few methods are available to estimate of the error range of the neuronal spiking reconstruction, mainly due to the lack of a rigorous mathematical model to quantitatively describe the relationship between calcium transients and bursting electrical activity. To overcome this problem, we employed statistical methods to generate an analog signal similar to the real neural signal, and then used the error range of reconstructed spiking from the analog signal to approximate that of the real signal. This strategy avoids building a complicated mathematical model to describe the spike-calcium relationship. Indeed, only peak-width distributions were used to distinguish spike events, and thus similar distribution curves correspond to similar reconstruction results, even in the light of differences between these signals. Using this assessment strategy with only a few similar characteristics proved to be reasonable, because not all features are needed to reconstruct electrical signals. In addition, after the operations that extract the template are completed, and the SNR of the real signal is estimated, error estimation can be performed to reconstruct results from the LDM, meaning that this error estimation can be carried out easily.

We used LDM to reconstruct neuronal spiking from the Ca²⁺ trace, mainly based on the following two facts. On the one hand, compared with the classic differential method that uses the peak value of the differential signal to identify neuronal spiking, our LDM vastly enhances the immunity to noise. First, LDM employs an extreme low-pass filter on the Ca²⁺ trace to extract the time intervals of neuronal spiking, and only analyzes Ca²⁺ traces within the extracted intervals and regards Ca²⁺ traces of the remaining intervals as noise. This operation eliminates the interference of most parts of noise on neuronal spiking reconstruction. Second, besides the peak threshold used in the differential method, LDM introduces the clustering of peak widths to identify neuronal spiking, which increases the robustness to noise. In fact, LDM can effectively analyze Ca²⁺ trace with SNR 3.6 (both FNR and FPR less than 10%), and the common differential method only can analyze Ca²⁺ trace with SNR 5.2 (data not shown). On the other hand, the LDM provided the statistical characteristics of the peak width of the distribution curve. Thus, by comparing the distribution curve method, a simulation signal similar to the actual signal was easily obtained and could be used to estimate the reconstruction error.

It is well known that the estimation of reconstruction results is more challenging than the reconstruction itself, especially for experimental signals. Indeed, many methods have been proposed for neuronal spiking reconstruction from Ca²⁺ traces, however, few methods succeed in providing an error estimation range of the reconstruction results. The core of our error estimation is that we only focus on the features of experimental signal that are close to the reconstruction, and generate the simulated signals with the selected features similar to the experimental signal. The effectiveness of this strategy has been demonstrated in this study. Essentially, this strategy can be applied to other biological signals that satisfy the convolution model, and can be further extended to other methods in which some signal features are used for reconstruction. We believe that this strategy is generally useful for the field of measuring reconstruction errors.

In this paper, we recognize spike activities only of pyramidal neurons from brain slices indicated with calcium dye Fluo-5F, and estimate the error ranges of reconstruction results only from LDM. In practice, there are many calcium dyes and genetic indicators available imaging, many kinds of neurons in the nervous system, and many reconstruction methods available for determine spiking rate using fluorescence imaging. To different calcium dyes and genetic indicators, this strategy would be work if only the dye has a fast rising dynamic when using the differential method. To some different neurons, this strategy would work, because generally the firing rate of neurons is no more than several tens of Hz, but the reconstruction and error estimation would still challenges when some neurons fire at a frequency of more than 100 Hz, such as fast-firing neurons. To the other reconstruction methods, this strategy should be adapted theoretically if some suitable features of experimental signal can be extracted, the proper analog signals can be easily generated. In practice, the application of this strategy in some methods may be difficult due to the implicit signal features close to the reconstruction.

As pointed out previously, our estimation of reconstruction is suitable for Ca²⁺ traces, which can be described by the convolution model, that is, a linear model. For sparse spiking or short durations of burst spiking, the convolution model can well describe the Ca²⁺ trace, and many studies are based on the convolution model for neuronal spike reconstruction. However, for burst spiking with long durations, the Ca²⁺ trace contains a certain amount of nonlinear signals that cannot be modeled with the convolution method. In this case, the simulated signals will deviate from the true signals, which makes it difficult to approximate the feature of the experimental signal by that of the simulated signal. To deal with this case, more information about the relationship between neuronal spiking and its corresponding Ca²⁺ trace is required to refine the reconstruction and error estimation method.

The development of imaging and labeling techniques has enabled us to acquire calcium signals of neuronal populations at finer time scales [28, 29] and larger spatial scales [28, 30, 31]. Moreover, many methods have been proposed for reconstructing the electrical activity of neurons based on optical signals. This progress clearly indicates that the electrical activity of a neuronal population can be inferred. Our method provides a quantitative assessment of the reliability of the inferred electrical activity of a neuronal population, which is beneficial for affirming the communication between different neurons.