Fig 1.
Generative autoregessive model for calcium dynamics.
Spike train s gets filtered to produce calcium trace c; here we used p = 2 as order of the AR process. Added noise yields the observed fluorescence y.
Fig 2.
Illustration of OASIS for an AR(1) process (see S2 Video).
Red lines depict true spike times. The shaded background shows how the time points are gathered in pools. The pool currently under consideration is indicated by the blue crosses. A constraint violation is encountered for the second time step (A) leading to backtracking and merging (B). The algorithm proceeds moving forward (C-E) until the next violation occurs (E) and triggers backtracking and merging (F-G) as long as constraints are violated. When the most recent spike time has been reached (G) the algorithm proceeds forward again (H). The process continues until the end of the series has been reached (I). The solution is obtained and pools span the inter-spike-intervals.
Fig 3.
Optimizing sparsity parameter λ and AR coefficient .
(A) Running the active set method, with conservatively small estimate , yields an initial denoised estimate (blue) of the data (gray) roughly capturing the truth (red). We also report the correlation between the deconvolved estimate and true spike train as a direct measure for the accuracy of spike train inference. (B) Updating sparsity parameter λ according to Eq (18) such that RSS = σ2 T (left) shifts the current estimate downward (right, blue). (C) Running the active set method enforces the constraints again and is fast due to warm-starting. (D) Updating
by minimizing the polynomial function RSS(
) and (E) running the warm-started active set method completes one iteration, which yields already a decent fit. (F) A few more iterations improve the solution further. The obtained estimate (blue) is hardly distinguishable from the one obtained with known true γ (yellow dashed trace, plotted in addition to the traces in A-E, is on top of blue solid line). Note that determining
based on the autocovariance (additionally plotted purple trace) yields a crude solution that even misses spikes (at 24.6 s and 46.5 s).
Fig 4.
OASIS produces the same high quality results as convex solvers at least an order of magnitude faster.
(A) Raw and inferred traces for simulated AR(1) data, (B) simulated AR(2) and (C) real data from [36] fitted with an AR(2) model. OASIS solves Eq (3) exactly for AR(1) and just approximately for AR(2) processes, nevertheless well extracting spikes. (D) Computation time for simulated AR(1) data with given λ (blue circles, Eq 3) or inference with hard noise constraint (green x, Eq 15). GUROBI failed on the noise constrained problem. The inset shows the same data in logarithmic scale. (E) Computation time for simulated AR(2) data. (F) Normalized computation time of OASIS for simulated AR(1) data with given λ (blue circles, Eq 3) and inference with hard noise constraint on full data (green x, Eq 15) or small initial batch followed by processing in online mode (orange crosses).
Table 1.
Cost and quality of spike inference with parameter optimization.
Fig 5.
Thresholding can improve the accuracy of spike inference.
(A) Inferred trace using L1 penalty (L1, blue) and the thresholded OASIS (Thresh., green). The data (gray) are simulated with AR(1) model. (B) Inferred spiking activity. (C) The detected events using thresholded OASIS depend on the selection of smin. The ground truth is shown in red. (D,E,F), same as (A,B,C), but the data are simulated with AR(2).
Fig 6.
Varied lag in the online estimator.
(A,B) Performance of spike inference as function of lag for various noise levels (i.e., inverse SNR) without (A) and with positive threshold smin (B). We used correlation of the inferred spike train as similarity measure and compared to ground truth as well as to the optimally recoverable activity when the lag is unlimited as in offline processing. (C) Inferred trace with positive threshold smin for increasing lag using the data depicted in Fig 4A with high noise level (σ = 0.3). The gray lines indicate the true spike times.