This requires the event file to be read out. The event file is read out using. Using the value and timestamp fields, we built a user-specified function that constructs a cfg. In this case, we take Alternatively you can download the trial function from the ftp server. The structure now contains the spike times in seconds spikeTrials. We have created three new fields in the spike structure, namely spikeTrials. Together, these three fields fully identify the structure of the spiketrain relative to the event trigger. The relationship between spikeTrials.
All spikes that fall in the same trial have the same value in spikeTrials. For every spike, we indicate in trial the spike was fired spikeTrials. Thus, spikeTrials. The spikeTrials. The first and second column of spikeTrials. For example,. Note that the end of the trial is variable because we defined our trials running until the first target or distractor change. The field spikeTrials. The advantage of the spike structure is that it is very memory efficient as compared to e. For many functions, e. PSTHs, raster-plots and cross-correlations, it is also the most natural format to perform computations.
Furthermore, the format makes it easy to associate certain data with single spikes, for example spike-triggered LFP spectra and waveform information. It is also possible to create only one trial. This is useful for two reasons. First of all, we explicitly convert timestamps to time. Secondly, we can correct for the fact the first recorded timestamp often does not start at zero for example, with Neuralynx data. In this case, the first recorded timestamp does correspond to zero. To this end, we run. For some analyses, it may be desired to have the data in binary format.
If fsample is too low compared to the spike firing rate, then the spike trains will not be binary as multiple spikes can fall into one bin, resulting in integer values larger than one to keep track of the number of spikes in one sample and the round-off errors will become larger. The structure data has the contents. Each dat. After the conversion, the waveform and timestamp information is lost. Note that these conversions are automatically performed in all the spike functions, such that data in both a spike or continuous raw representation can be entered.
If spike trains are governed by a Poisson process, then the statistics of the spike train can be fully described: the distribution of waiting times between subsequent spikes is exponential, and the distribution of spike counts is Poisson. However, neurons show various non-Poissonian behaviors, such as refractory periods, bursting, and rhythmicity. These behaviors may arise from intrinsic dynamics e. To investigate whether the recorded spike trains reveal such non-Poissonian history effects, we study the ISI distribution. We compute the isi histogram using. The field isih.
This gives two figures, one with a longer refractory period the narrow spiking cell; top , and one with a bursting pattern the broad spiking cell; bottom. We also read in an additional dataset consisting of an M-clust. This plot shows that after a burst, either a new burst follows, or a long waiting period on the order of a theta cycle ms.
Both spike-density functions and peri-stimulus time histograms are methods to compute the average firing rate at selected time points around event triggers. This is an important step to understand how neurons react to changes in external variables. The field psth. It is also possible but less computationally efficient to enter the binary spike trains that are stored in a continuous raw format.
The yellow lines in the raster plot indicate the trial borders. Also, multiple neurons are plotted with different colors. This can also be used to plot multiple conditions at the same time. We then run spike-density functions on the spike trains, to obtain spike density with rasters. The advantage of the spike-density function is that an estimate of the instantaneous firing rate or expected spike count can be obtained for every time-point, instead of larger bins as with the PSTH. To this end, do. One can compute noise correlations between units by doing. The cross-correlogram is one of the classic techniques to show rhythmic synchronization between different neurons e.
The auto-correlogram typically offers a more sensitive measure of the degree to which a single neuronal source displays rhythmic firing than the ISI distribution, especially if firing rates are high. For this analysis we select the unsorted multi-units from the same data-set, as they give more reliable cross-correlations. The observed cross-correlogram should always be compared against a cross-correlogram obtained by shuffling the trials. Cross-correlations between neurons can either arise because of common, time-locked fluctuations in the firing rate Brody et al.
These correlations are invariant to a change in the order of trials. If the observed features of the cross-correlogram that are not present in the shift-predictor cross-correlogram, then this indicates that they arise because of induced synchronous activity. Note that for the shift-predictor, it is required that the trials cover the full latency window that is specified by cfg. For example, if the first trial has a duration of 3 sec. Hence, cfg. For example, the computed cross-correlogram reveals strong zero-lag and alpha-band synchronization in the pre-stimulus period:.
Cross-correlations are computed over the complete trial period. We have shown how to perform several common spike train analyses. A number of methods estimate spiking probabilities or instantaneous spiking rates 24 , 25 , 26 , 27 , 28 , 29 , 30 , This approach has advantages when used on data that clearly lack single-spike resolution or when it is important to assess the uncertainty of the estimation. However, when it comes to investigating temporal coding and causal network relations, estimating the optimal time series of the spikes themselves, as do MLspike, Peeling and others, can be advantageous.
From the practical point of view, it should also be noted that dealing with a single spike train has the advantage of being able to use—essentially as-is —the large thesaurus of standard methods available today for spike train analysis. Furthermore, the ongoing progress in both fluorescent marker- and imaging technology is likely to make robust and precise single-spike estimation increasingly accessible Conversely, both approaches can be used to investigating rate coding or average responses.
Interestingly, the MCMC algorithm 26 , 27 returns actual sample spike trains that can be used, for example, to investigate network properties based on spike times. At the same time, MCMC returns many such spike trains, sampled according to the posterior probability distribution, thus allowing both the estimation of spiking probabilities and an indication of the level of estimation uncertainty. Similarly, we have adapted MLspike so it can return, upon user choice, the spiking probability distribution, or a set of spike trains sampled according to the posterior distribution, rather than the unique MAP spike train see Methods and Supplementary Fig.
Caveats apply, however, to the interpretation of such sample spike trains. Those can result, for example, from a mismatch between the used response model and the data. Such a situation can be seen in Fig. The resulting low variability would make the user overly confident with respect to the quality of the reconstruction.
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In terms of algorithm and implementation, MLspike implements a Viterbi algorithm to estimate a MAP spike train from calcium signals, for the first time to our knowledge. Additional novelties include the representation of probability distributions as discretized onto a grid, as in histogram filters, so they can easily be spline-interpolated over the whole state space as opposed to, for example, particle representations as in ref.
For additional details, including simplifications that further increase computation speed, see Methods. The need to further improve the response model may increase the number of its dimensions and with it the dimensionality of the grid onto which probabilities are represented. This would result in a prohibitively large number of points at which the probability has to be calculated. Our autocalibration procedure is somewhat more ad hoc.
Other, more well-defined, methods maximize the likelihood of the fluorescence signals 24 , 25 , 26 , 27 , 28 , 29 , but such optimization is computationally more expensive. This is not the case of our autocalibration, which uses such information by allowing only a range of values for certain parameters for example, A , and even clamps some others to fixed values such as those governing nonlinearities, which are particularly difficult to estimate.
Such a priori can prove advantageous in situations with noisy or little data, or when only few isolated spikes are available. Ideally, autocalibration methods would be able to combine such information with that provided by the data itself. The open source code of our method is available as Supplementary Software and includes introductory demos.
It qualitatively but simply and intuitively illustrates how to adjust the few parameters of MLspike in the rare cases that the default values should be inadequate for a quantitative and systematic study of parameter dependencies, see Supplementary Figs 1—4. Our novel method makes it possible to optimally exploit the capabilities of current hardware. Warranting more accurate spike train extraction from larger sets of cells than so-far is a step forward in the investigation of local network properties, such as temporal coding at very high SNR and correlations at the single-spike level or, at lower SNR, at the level of changes in spiking rate.
It also extends the applicability of two-photon imaging to investigating more densely connected networks than so-far, improvements in the determination of functional connectivity for a quantitative analysis see ref. Recent progress in waveform shaping 43 , 44 that corrects for scattering-induced deformations should also allow a significant extension of the volume accessible for imaging, into depth in particular. Recently, more general approaches have been proposed, aimed to jointly infer regions of interest and spikes 23 , 27 , 45 , Although the strength of these methods resides in exploiting the full spatio-temporal structure of the problem of spike inference in calcium imaging and in offering an unbiased approach for ROI determination, they have the disadvantage of requiring that the full two-dimensional 2D or 3D data are available, which is not the case in random-access scanning.
Indeed, there, one scans only the points of interest—albeit at 3D and at much higher speeds, for instance using AOD-technology Nevertheless, MLspike could straightforwardly be added to the list of available spike estimation algorithms even in algorithms of these kind 27 , thus increasing their data processing power. Finally, we have shown that it is straightforward to modify our method to include different response models—here, to account for the specific nonlinearities of GECIs. Similarly, our method could be easily adapted to event detection in other noisy signals, such as the fluorescence of new voltage probes 47 or even intracellular patch- and sharp-electrode recordings of super- and sub-threshold neuronal activity.
See also our Supplementary Note 1 and the two demos in the code for guidance in using MLspike. Surgical procedures. Body temperature was monitored and maintained at A metal chamber was attached with dental cement to the exposed skull above the primary somatosensory cortex 2. A 3-mm-wide craniotomy was opened and the dura mater was carefully removed. The latter was applied such as to leave a narrow rostro-caudal gap along the most lateral side of the chamber, in order to allow access to the micropipette used for dye injection or for electrical recordings.
V1 was localized first anatomically 0. The rest of the surgical procedure was as described for rats. To awaken the mice from anaesthesia for the imaging, they were given a mixture of nexodal, reventor and flumazenil 1. Slice preparation. GIN mice PP24 anaesthetized with isoflurane were decapitated; their brain was rapidly removed from the skull and placed in ice-cold artificial cerebrospinal fluid ACSF.
A cranial window was implanted 2 weeks after the injection over the injection site as described in Surgical procedures section. Scanning was performed with 6-mm-large scanning mirrors mounted on galvanometers Cambridge Technology. In both setups, fluorescent light was separated from excitation light using custom-ordered dichroic filters and collected by a GaAsP photomultiplier PMT for the green calcium fluorescence and a multi-alkali-PMT for the red sulforhodamine fluorescence.
In the anaesthetized rat experiments, activity was recorded in the absence of a stimulus. For imaging and stimulation in anaesthetized mice, see ref. In the experiments performed on awake head-restrained mouse, a visual stimulus was delivered during data acquisition, in form of drifting gratings spatial frequency: 0. For details see ref. Electrophysiological recordings. After the preselection of neurons showing activity based on the bolus-loaded OGB1-AM, cell-attached in vivo or patch in vitro recordings were started on visually targeted neurons using borosilicate microelectrodes 6.
When patching, the dye also served to check membrane integrity. Electrical recordings were made Multiclamp B, Digidata, Molecular Devices simultaneously with imaging.
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Table 1 summarizes type, origin and amount of the recorded data. Our model equations for OGB1 use equations given in refs 24 , 32 , 49 and reparameterize them so as to decrease the total number of parameters and use final parameters whose effects on the final dynamics are more intuitive. F 0 and F max the fluorescence levels at rest and when the dye is saturated, respectively. K d the dissociation constant of the dye. We can now replace the measure equation 2 with.
We also introduce, instead of the fix baseline F 0 , a drifting baseline B t. This yields the model equations:. A major advantage of the reparameterization is to reduce the total number of parameters, which had redundant effects on the original model dynamics. In the case of GECIs, three different models were assessed. These three models are compared in Supplementary Fig.
The first and largest difference between genetically engineered and organic calcium sensors is the supra-linear behaviour of the fluorescence response function to calcium. In the first model, we followed 15 , that is, fitted this function with a cubic polynomial:.
In the second model, the measure function was thus replaced by.
The slower rise time is due to a slower calcium binding to the indicator, and the supralinear behaviour is due to the cooperative binding of more than one calcium ions to one indicator protein The full kinetics of the binding process should be taken into account then:. Thus the evolution of calcium and bound indicator concentrations must be dissociated in two distinct terms, while the fluorescence measure 2 is replaced by. These new equations introduce a significant number of new parameters.
We also introduce two new parameters: is the normalized level of baseline calcium concentration and is the binding time constant when calcium is at baseline. We detail here this discretization and the full derivations of probability distributions p x t x t-1 and p y t x t in the case of the simpler physiological model. The model equations become:. Note that the first line of the equation is a simplification for the more rigorous but complicate formula. It is the a priori probability of the hidden state, in absence of any measurement.
In practice, we used a uniform distribution for both c 1 and B 1. Regarding c 1 , indeed we found that when the true spiking rate was not known, a uniform probability was better than a distribution determined mathematically based on the value of a priori spiking rate, because if that value was not correct, errors were increased. If the true spiking rate is known however, the following a priori can be used: one can observe that c 1 is a weighted sum of Poisson random variables:.
Its probability distribution can thus not only be computed exactly with iterative convolutions but is also well-approximated with a truncated normal distribution:. Let T be the number of time instants. This approach relies on the following recursion of maximizations:.
In the general case where drifts are estimated, m t x t is a function defined over a 2D space the set of all possible values for c t ,B t , and can thus be easily encoded into a 2D array by using appropriate sampling values for c t and B t. This way of encoding probabilities is the basis of histogram filters ref. This iterative calculation of the conditional probabilities m t x t is illustrated in Fig. But the full best trajectories x t ,…. To store in memory the conditional probability m t x t , the state space needs to be discretized. Approximating to a value on the grid could lead to important estimation errors, unless the discretization grid is extremely dense, implying unreasonable calculation times and memory usage.
The maximization can be performed successively over different state variables, thanks to the independence of B t and c t evolutions:. With this respect, we found that a spline interpolation resulted on average in less error than a linear interpolation when coarsening the grid discretization to a point leading to estimation errors.
This quadratic interpolation is also obtained by a matrix multiplication, with the matrix being precomputed. In our more detailed physiological model used for GECIs, we have introduced an additional state variable, p t , the normalized concentration of indicator bound to calcium. More specifically, we write. During the final forward sweep, the estimated x t values are not restricted to lie on the discretization grid either.
Taken together, these techniques allow minimizing computation time by keeping the discretization grid relatively coarse typically, we use calcium values and baseline values, but these number can in most cases be reduced to 30 without generating estimation errors , and by limiting the maximization search to a small number of tested values. The algorithm can be modified to return spike probabilities in each time bin instead of a unique spike train, or a set of spike trains sampled according to the posterior probability. Then arbitrary number of spike trains can be generated: they are initiated by drawing x 1 according to.
As for earlier MAP estimations, it is noteworthy that the abovementioned probability updates for one step in time can be decomposed into two sub-computations. For example, we have. Accurate estimations require accurately setting the six model parameters for details on how each of them influences estimation quality, see Supplementary Methods.
It would be tempting to estimate both the spike train and the parameters altogether by maximizing the likelihood. Indeed, our model considers fluorescence signals as the sum of calcium-related signals possibly modulated by the baseline drifts and of a white noise with s. The highest frequencies were also eliminated in this calculation, because the noise present in our data is actually not purely white see the spectra in Fig.
This resulted in a more equilibrated number of misses and false detections. As shown in Fig. Noise obviously increases the variability, but it is possible to obtain histograms of transient amplitudes that show several peaks corresponding to different numbers of spikes. Next, the amplitude of single spike transients is best estimated from isolated calcium transients of moderate amplitude.
At this point, a histogram of all event amplitudes is constructed Supplementary Fig. It is first smoothed, yielding x 1. Thereafter, peaks are enhanced by dividing x 1 by a low-passed version of itself, x 2. A first estimate of A is chosen as the value that maximizes x 3 the green star in Supplementary Fig. As a final note, several parts of the autocalibration algorithm being based on somewhat intuitive heuristics, large room for ameliorations is expected, notably through a more rigorous formulation.
Other parameters currently not autocalibrated. We thus expect autoestimation to be successful only at very high SNR. A summary of details on simulations and estimations shown in this study, such as parameter values, settings and so on, is provided in our Supplementary Table 1. Simulated spike trains generally consisted in Poissonian trains Figs 2a,b and 3a—c , and Supplementary Figs 1—4. Finally, in the case of OGB, we also performed the estimations using fixed parameter values from ref. Peeling has the ability to model transients with two such exponentials but does not model dye saturation effect see also next section.
When Peeling was run using the same approximation with a single exponential rather than two it performed even worse Peeling algorithm. The Peeling algorithm 10 , similarly to MLspike, returns a unique estimated spike trains that accounts for the recorded fluorescence signal. It requires a certain number of physiological and algorithmic parameters to be set. Regarding algorithmic parameters, preliminary testing of the algorithm on simulated and real data allowed us to determine which parameters could be kept fixed to their default value, and which are needed to be tuned depending on the quality of the data.
Two other parameters, slidwinsiz and maxbaseslope , had to be tuned according to the level of baseline drifts in the signals. In all simulations these parameters were optimized independently for each conditions Supplementary Figs 2 and 4 , while on real data they were assigned fixed values found heuristically. Peeling has an option for performing nonlinear estimations that account for dye saturation; however, this option resulted in poor baseline drift estimations, even after we edited and improved the code, therefore all Peeling estimations even on real data were rather performed using the linear model.
On our OGB data set, we used the ability of Peeling to model calcium transients with two exponentials; values from ref. Finally, to take into account the finite risetime in the case of GEGIs, for the precise temporal quantifications in Fig. We compared our real data estimations also to three algorithms published by the Paninsky group 24 , 26 , These algorithms have in common that they estimate model parameters directly from the data, either in a direct or iterative fashion, thus requiring no or little parameter tuning.
They all return an estimated spiking rate or spiking probability; up to a scaling factor in the case of CD at each time point of the original fluorescence signal, but the MCMC algorithm does this by generating a number of spike trains theoretically sampled from the posterior distribution that can be directly used, for example, for error quantification. Their underlying dynamic models are simpler than the one used for MLspike, as they do not include dye saturation for CD and MCMC SMC does include it , and, more importantly, do not include baseline fluorescence fluctuations SMC includes noise in the calcium evolution that can account for part but unfortunately not all of spike-unrelated fluctuations in the signals.
On our data sets, we observed that the lack of baseline fluctuations in the models could lead to important errors, for example with large inaccurate spiking activity being estimated where the baseline was higher. We therefore improved the estimations by detrending the signals before applying the algorithms: this increased estimation accuracies of the three algorithm; we also tried high-pass filtering the signals having noticed that signals are high-pass filtered in ref.
MCMC did not require such a correction because its estimations were run with an autoregressive model of order 2, which takes into account the finite rise time. A specific advantage of our MLspike implementation is that autocalibration can be performed globally on many trials recorded from the same neuron. This in fact improved overall estimation accuracy only very slightly, with improvements for some neurons but deteriorations for others: probably in the latter case neurons mismatcheing with the model for example, baseline fluctuations in some trials were misleading the global parameter estimation, therefore decreasing estimation accuracy in other trials.
If, as opposed to Peeling, we did not need to set parameters for the estimations, we did change a few default algorithmic parameters to increase the robustness of estimations at the expense of speed. Namely, the number of EM iterations for SMC was increased to 6; numbers of burn-in and used samples for MCMC were both increased to for example, sample spike trains were generated, and only the last were kept. Finally, because of their probabilistic nature the SMC and MCMC algorithms yield slightly different results when being repeated on the same data; to ensure repeatability; we thus reinitialized the random number generator previously to each estimation.
Error rate. Once spike trains have been estimated, they need to be compared with the real simulated or electrically recorded spikes. We used the F 1 -score to define an ER, defined as the harmonic mean between sensitivity and precision 51 :. We consider a given spike detection correct when it matches a real spike with a temporal precision better than 0. The estimated and real spikes were matched by computing distances using a simple metric over spike trains 52 that assigns costs to spike insertions, deletions and shifts, and is calculated using a dynamic programming algorithm.
Noise level. We quantified the noise level in the real data by taking the RMS of the difference between the measured fluorescence signals and those predicted by the electrically recorded spikes using the calibrated parameter values. Before computing this RMS however, the signals were filtered between 0. Then, this RMS was normalized by a quantification of the signal amplitude. In the case of the simulations or of the OGB data, using parameter A for this quantification led to satisfying properties of the noise level.
However in the case of GECIs, noise levels calculated that way could become very high due to weak responses to single spikes while, at the same time, leading to underestimating the strong responses to bursts. Calibration of the PMTs in order to estimate the photonic contribution to the noise. It was even possible to determine which part of this noise corresponded to photonic noise by an independent calibration of the PMTs, where we measured photonic noise corresponding to different signal levels and at different PMT voltages.
Indeed, the variance of the photonic noise is proportional to the number of photons collected by the PMT: if s is a signal whose noise is purely photonic, we note as N the corresponding average number of photons collected per time bin and a the gain of the PMT:. Therefore the gain can be estimated as. However, this is true only when the variance in the signal is only due to photonic noise. Even when imaging steady signals from fluorescent beads, we cannot estimate a in this manner because their signals will always contain system noise as well, which is non-negligible compared with photonic noise.
The variance of is now due purely to photonic noise, which we note:. Therefore, we have. Then for any new signal s acquired at the same PMT voltage at a given frame rate f , the contribution of photonic noise to the total noise RMS is , and using the same argument as above of the flat spectrum of the photonic noise, its contribution inside a specific frequency band [ f 1 f 2 ] is. All other data are available from the authors upon request.
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A finite rate of innovation algorithm for fast and accurate spike detection from two-photon calcium imaging.
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