Calcium-bound probe molecules and the emitted light signal.

The concentration of calcium-bound probe molecules can be used as a proxy for the concentration of calcium in the system. A simplified reaction scheme describing calcium binding is shown in Fig. 1b. As probe molecules bind to calcium quickly, we assume that during fast transients, the change in concentration of fluorophore bound to calcium is proportional to the calcium concentration ([ ]) times the concentration of free fluorophore at this location. This relationship holds as long as probe molecules are not saturated with calcium. Our optical probe, Oregon Green BAPTA-1, has a high affinity to calcium, due to its fast binding rate, , and slow unbinding rate, , leading to decay rates on the order of . As a result, the fluorescence response from a fast voltage transient, like that of a bAP or an EPSP, rises effectively instantaneously, and decays slowly [10], [11]. Thus the fluorescent signal provides a temporally filtered representation of the underlying intracellular calcium concentration. This complicates the interpretation of the recordings. Our goal will be to reconstruct the spatial and temporal structure of bound probe molecule dynamics. We therefore first describe how the emitted light intensity is related to the calcium concentration at a location within the cell.

The light intensity emitted at location at time , , is directly related to the concentrations of bound probe molecules bywhere and are the concentration of bound and free probe molecules respectively, at spatial location and time . The scalar represents the relative fluorescence of the bound fluorophore to free fluorophore ( for the dye used here [12]). The constants and represent the light intensity of free probe molecules and autofluorescence, respectively.

Assuming that probe molecules are conserved locally, i.e., , we obtain(1)where and . Therefore fluorescence, , is affinely related to the concentration of calcium-bound probe molecules when these are not saturated. To simplify our analysis we assume that Eq. (1) is deterministic and that all the noise in our model comes from the recording procedure, which will be described below. By defining the baseline fluorescence at location as , we have that .

The signal of interest.

The fast calcium signals in the tree, in our case, are due to a bAP that results in a transient increase in the concentration of bound probe molecules. High affinity calcium dyes, such as the Oregon Green BAPTA-1 (OGB-1) used here, bind calcium with a very fast onset and also have a much slower dissociation rate. Neglecting spatial correlations and diffusion terms, we can model the bound probe molecule dynamics as(2)Here is the time constant that determines the rate at which probe fluorescence molecules unbind from calcium. The term represents the times at which the sequence of bAPs reaches the location and can in this case be written as the sum of Dirac delta functions, where represents the time at which the -th bAP reaches location . We will generalize this assumption below. The -th bAP at location results in an instantaneous jump of size in bound probe molecule concentration, followed by an exponential decay to a baseline concentration at rate . For a high affinity dye a fast calcium influx leads to a transient with rapid onset. Moreover, from the reaction scheme of Fig. 1b the amplitude of this transient depends also on the baseline concentration of the optical probe at each location .

This baseline probe molecule concentration typically varies across the cell and is difficult to control a priori. For instance, probe molecule concentrations are typically higher in the apical dendrite and near the soma, resulting in the higher luminosity of these regions. This variation may be due to nonuniform concentration of basal calcium across the cell. Also contributing may be limitations of the experimental setup, such as non-uniform laser illumination across the microscope's field of view or non-uniform volume of fluorophore excitation, due to thin processes that extend beyond the focal volume. Moreover, recording sites are sometimes placed slightly off a dendrite. To account for such variability in baseline fluorescence, we assume that such non-uniformities are primarily due to the experimental limitations which affect the number of probe molecules being excited. As a result, and based on the dynamics of calcium binding, we expect that during fast transients , where is the binding rate of our optical probe. Implicit in this relation is the assumption that the concentration of probe molecules is away from saturation, and that the fast calcium transient is mainly due to the bAP. Therefore, Eq. (2) can be rewritten as(3)where represents (up to a multiplicative constant) the amplitude of the calcium transient at location and time .

Based on the above discussion, we make an important conclusion: The relative change in bound probe molecules is reflective of the transients in calcium concentration. Hence, by Eq. (1), we need to estimate relative changes in light intensity at a given location, as a proxy for transients in calcium concentration. Dividing Eq. (3) by we get(4)where . As discussed above cannot be easily estimated. However, from our measurements we have access to . To be able to compare the calcium concentration at different locations of the tree, we need that approximately for all the locations in the tree. This assumption is expected to hold if for all locations, which is expected since the autofluorescence signal is in general weak for high affinity calcium indicators. Therefore from now on we assume that the relative change in bound probe molecules is approximately proportional to the -transformed emitted light intensity, with a constant of proportionality that is the same across the tree:(5)Note that this resulting transformation is a standard heuristic used in the analysis of fluorescent recordings [9], [8].

The recorded signal.

The signal, , is weak, and is measured using photomultiplier tubes (PMTs), sensitive light detectors which contain several amplifying stages. Briefly, an electron entering one stage will produce a random number of outgoing electrons. If is the number of incoming electrons at a stage, the number of outgoing electrons, , is random. The mean of is the gain of the stage. For instance, could be a sum of numbers drawn independently from a Poisson distribution with mean [13]. The output of an eight stage PMT is then an eight-fold composition of with itself. The noise in the output of the PMT can therefore be highly non-Gaussian, with a distribution that is determined by the strength of the signal . In particular, at lower light levels approximately follows an exponential distribution (data not shown). At higher light levels, the distribution is approximately Gaussian [14], [13]. The photomultiplier statistics can be fit with a gamma distribution with activity dependent parameters:(6)When we fit our data to this model we obtained (data not shown). Note however that the signal of interest is not itself but its relative change (see Eq. (5)). If is known, then the scaled random variable is distributed according to . In that case the variance of the relative change in light intensity remains approximately constant with small changes in relative to . The skewness and kurtosis of the distribution are given by and respectively. As becomes large, they become very small and the resulting distribution is approximately Gaussian [15]. This happens at locations with moderate or high baseline activity. As a result we first assume that the signal is sufficiently strong and the noise is Gaussian, so that for a Gaussian random variable . We relax this assumption subsequently. However, we always assume that the expected value of the measured voltage is proportional to the true light intensity .

A general statistical model for the spatiotemporal calcium signal.

Using Eq. (4) for the dynamics of relative bound probe molecules in the presence of incoming calcium spikes, we have that(7)where denotes the Heaviside function and denotes the number of spikes. If the time at which a bAP is initiated is known, then the only parameters that need to be estimated are the amplitudes . We can assume that a bAP propagates through the tree instantaneously, so that for all locations. Under this assumption amplitudes can be easily estimated using maximum likelihood techniques, with the added constraints that the inferred amplitudes are nonnegative. We will fully describe these techniques in a more general setup below.

More generally, the timing of the APs may not be known or not the same for all the locations in the tree and must be inferred from the bound probe molecule trace. This is a challenging problem in itself [6]. Moreover, the above approach neglects spatial correlations caused from the backpropapation and possible calcium diffusion mechanisms. As a result, the calcium profile at each location is treated independently and cannot be used to make predictions for neighboring, possibly non-imaged locations. Instead of analyzing the recorded signals from each location separately we would like to take advantage of the fact that the recording from one location may provide information about the signal at neighboring locations. We next describe a statistical approach that allows us to perform spatial and temporal smoothing concurrently.

To address the issue of spatial correlations, we model the relative concentration of bound probe molecules at location of the dendritic tree at time as a linear combination of spatial functions whose weights are allowed to change with time,(8)The spatial functions are bump functions which are smooth, nonnegative and local ( for further than some distance from the “center point” of the -th bump). These functions serve to spatially smooth the inferred signal, since cannot vary more quickly in space than ; by changing the number of bumps and their smoothness we can modify the overall spatial smoothness of . There are a number of possible interpolation approaches, and we chose to use B-splines [16]. Other choices include gaussian bumps or diffusion kernels [17] that preserve continuous differentiability at the tree branch points. However, the exact choice of basis functions did not considerably affect our results, as long as these functions are smooth and are placed approximately uniformly along the tree.

As the spline basis functions are typically defined on a line, here we use a suitable generalization to trees. Splines are defined on subdomains of the tree defined by the cell shape. This tree therefore needs to be partitioned. We chose to define the partition using the topological distance from the soma, i.e., the distance from the soma along the tree. Thus all points that lie within a given distance from the center of a spline function belong to the same subdomain. Picking this distance is important because a very large/small value can lead to significant under/over-fitting of the data. In the Results section where we apply our algorithm to experimental data, we propose a data-driven method for determining this parameter.

We discretize the cell into compartments so that the -dimensional state vector of normalized bound probe molecule concentration at each location on the cell can be written in matrix-vector form as(9)where is an matrix whose -th column is a vector obtained from the spatial discretization of , and is the column vector of the activation variables at time , and has length .

We assume the same temporal dynamics as in Eq. (2), but now for the weight variables , which are assumed to evolve according to(10)Because of linearity, has the same interpretation as the time constant that determines the rate at which probe fluorescence molecules unbind from calcium, and have similar interpretations as before. After discretizing time, and introducing , Eq. (10) can be approximated by(11)In this discretized representation when , and otherwise. In the general case, where the timing of the spikes is unknown, we replace the product of with a nonnegative random variable .(12)This model is a spatiotemporal generalization of that discussed in [6].

There are two essential differences between the model described here by Eqs. (8)–(12) and the motivating model described in Eq. (7): estimation of spike times, and incorporation of spatial correlations. In the present model the input, , is not assumed to be a sum of delta functions with unknown weights occurring at known times. Moreover, as we will see below, the inference will be done at once for the whole tree and not individually and independently for each location. Thus, this statistical approach is more general since the algorithm is capable of finding spike times, and more biologically accurate, since it allows for spatial correlations for nearby compartments.

We use a Bayesian approach, and start by specifying a prior distribution on , and hence construct a prior distribution on the hidden weights (and consequently on (t)). For example, if we know that calcium influx is small at certain times or at certain locations , then we can choose the prior variance and mean of the corresponding to be small. Local correlations in may also be incorporated to increase the smoothness of the state vector . The prior on can also be interpreted as an autoregressive prior on which serves to temporally smooth the inferred signal, on a scale set by the time constant . This time constant could also depend on the spatial variable (modeling different buffer constants at different locations in the tree) without significantly changing the presentation below.

Now taking into account the dynamics of probe molecules (see Eqs. (1),(4) and (5)), the emitted light intensity can be written as(13)where is a diagonal matrix where the non-zero entries denote the probe molecule baseline concentration, is a column vector of ones with length and is a scalar constant, related to the proportionality constant in Eq. (5), and is assumed to be constant throughout the tree. Note that our signal of interest and the emitted light intensity are related through an affine transformation.

We have access to through spatially localized noisy measurements. In each measurement, we image a single compartment of the tree and at each time , such measurements are performed. Let be a matrix where each row has one entry equal to one, corresponding to the compartment being imaged, and the rest equal to zero. Assuming additive noise, the measurements can be expressed as(14)The noise is assumed to be spatiotemporally uncorrelated, although this could be generalized to the case that the covariance matrix of has a local structure. Note that , the number of measurements per time step, can easily change with time. The matrix can represent more general linear measurements. Initially we assume noise is additive for notational simplicity. We show below that more general forms of noise can be modeled within this framework.

After discretizing time, Eqs. (8) and Eq. (14), correspond to the state-space model(15)where and we abbreviated (with dimensions ) and . This completes the specification of the model.

We emphasize again that Eq. (14) provides a biophysically inspired statistical model. As earlier, we aim to approximate a signal proportional to the bound calcium concentration across the dendritic tree. Note that certain parameters in the model such as the time constant do have direct physical interpretations

Computing and optimizing the log-posterior.

We would like to compute the maximum a-posteriori (MAP) estimate of the relative spatiotemporal concentration of calcium-bound probe molecules , given the sequence of observations . Here and denote the vectors with the underlying calcium concentrations and measured light intensity at all times, and have lengths , respectively. From Eq. (9) it follows thatThus we will focus on finding the MAP estimate of the activation profile , where is a vector of length defined similarly to and . Using Bayes rule and the state-space nature of the model we can compute the log-posterior directly [18], [19]:(16)where the constant does not depend on the and thus can be ignored. The log-posterior , can be maximized efficiently if this function is concave in . Since is a sum of terms of the form and , it is sufficient that these observation and transition densities be log-concave, i.e., that and be concave in and respectively. This means that other log-concave noise distributions can be used in our framework, not just additive Gaussian noise.

Additionally, since the calcium concentration is nonnegative, we have to ensure that all the activation variables, as well as the bumps are nonnegative. These additional constraints can be expressed , where the inequalities are in vector form, i.e., they have to hold for all the entries. Therefore our optimization problem can be written as(17)

In the Supporting Information we describe the technical details of a fast method for the solution of Eq. (17). Briefly, we construct a log-barrier method for the incorporation of the non-negativity constraints [20]. In general, barrier methods are iterative procedures that converge within only a few iterations, but at cost per iteration in time and memory requirement. The so called “big O” notation for the time cost is used to denote that every iteration requires a number of operations that scales linearly with the length of the experiment and cubically with the size of the imaged tree . We see that for large trees this cost can be prohibitive. However, by exploiting the tree-structure and the spatially localized nature of the measurements, we show that the cost per iteration can be dropped to in terms of both time and space, making our method applicable to arbitrary large dendritic trees. Details can be found in the Supporting Information.

Estimation of the model parameters.

Our model has several parameters that have to be estimated before we apply our smoothing algorithm. These are the baseline probe molecules concentration, the time constant of calcium unbinding and the noise statistics. Below we describe simple heuristic methods to estimate the model parameters. In practice more sophisticated and powerful methods can be developed such as Monte Carlo estimation methods (e.g. via the Expectation Maximization algorithm) [25]. However, we saw that the results of our algorithm are qualitatively robust with respect to the various parameter values. We also discuss how we chose the appropriate spline matrix for each subtree and the prior on the transient amplitudes.

Baseline probe molecule concentration.

The baseline fluorophore concentration was estimated by taking the time average of the measurements over the first on all the imaged locations. This corresponds to the maximum likelihood estimate of the baseline activity both under Gaussian and under gamma measurement noise.

Time constant estimation.

Next, the decay constant of the recorded signal, , through its discretized version , was estimated: First the measured calcium traces were passed through a low-pass, moving average filter (with width) to eliminate the high frequency noise effects (shot noise). After that, the discretized time constant was estimated by fitting a first order autoregressive model to the filtered data which was first normalized to have zero baseline.(22)where is the filtered measured calcium trace, is the baseline concentration at location , and is the number of recording locations. was chosen , i.e., shortly after the last bAP. This estimator gave , or . The time constant was estimated separately for each subtree and the estimates were approximately equal.

Spline matrix and amplitude prior.

The state transition (amplitude) prior was considered to be an exponential distribution, as discussed in the Methods section. There a marked point process with rate , mean amplitude , and discretized in time at resolution can be approximated by a log-concave exponential distribution with parameter . Therefore we need to determine the amplitude of the hidden states at each location and the rate which is time varying. Note that in general the mean amplitude can be time varying, but we approximate it as constant with time. The B-spline matrix and the amplitude of the bump were estimated via an alternating procedure as follows: The basis functions were third order polynomials whose centers were placed uniformly on the tree, and the only parameter required for determining the splines was the topological distance (i.e., number of compartments), between two neighboring center points. The choice of this distance is important since a large distance can lead to a less flexible model, whereas a small one can cause over-fitting. We picked the maximum distance that allows the model to fit the estimated amplitude of the calcium bumps. Choosing a smaller distance gives more degrees of freedom to the system and can potentially lead to overfitting. Note that the prior amplitude pertains to the hidden state vectors and thus cannot be readily estimated from the data. We first obtained a crude estimate of the amplitude of the normalized calcium transient as(23)i.e., the maximum difference after the first bAP, measured in the filtered traces. Then to determine the amplitude for the hidden states and the B-spline matrix we proceeded as follows: For any topological distance between two neighboring spline function centers, a different basis matrix was constructed and the prior amplitude vector on the state variables was given by the solution of the quadratic program(24)This provided the non-negative amplitude parameters of the exponential priors that can best approximate (the estimated calcium transients) for the given topological distance . If this approximation had a relative error larger than 0.1, we decreased the topological distance and repeated the same procedure. The actual used topological distance , the splines matrix , and amplitude prior parameters were picked as(25)i.e., we picked the maximum distance, and hence the smaller number of hidden variables, that allow for a satisfactory interpolation of the estimated calcium transients from the hidden state amplitude prior parameters. The rate of bumps was time varying and equal for all the states, given by(26)The rate starts with a low value at the first since we do not expect any spikes there. Then, it is set at the value 10 since we expect 10 bAPs during the one-second excitation window. After that, the rate decays smoothly back to the initial value. As we explain further below, this smooth decay prevents the algorithm from inferring a non-existent spike at due to the temporal discontinuity of the rate. Note that with this model, the exact spike times are considered unknown and are also to be estimated by the algorithm.

Noise statistics.

Finally, the noise likelihood at each measurement location was assumed to be independent and Gaussian with zero mean, and variance estimated directly from the measurement traces. To find the noise statistics we subtracted the filtered traces from the -normalized data and then fitted a Gaussian distribution on the residual. Although this is a simplified approximation, in most of the cases this residual could be fit satisfactorily with a zero mean Gaussian distribution (data not shown).