Activity of populations of sensory neurons carries stimulus information in both the temporal and the spatial dimensions. This poses the question of how to compactly represent all the information that the population codes carry across all these dimensions. Here, we developed an analytical method to factorize a large number of retinal ganglion cells' spike trains into a robust low-dimensional representation that captures efficiently both their spatial and temporal information. In particular, we extended previously used single-trial space-by-time tensor decomposition based on non-negative matrix factorization to efficiently discount pre-stimulus baseline activity. On data recorded from retinal ganglion cells with strong pre-stimulus baseline, we showed that in situations were the stimulus elicits a strong change in firing rate, our extensions yield a boost in stimulus decoding performance. Our results thus suggest that taking into account the baseline can be important for finding a compact information-rich representation of neural activity. Populations of neurons encode sensory stimuli across the time dimension (temporal variations), the space dimension (different neuron identities), or along combinations of both dimensions BID1 BID16 BID17 BID10 BID19 . Consequently, understanding the neural code requires characterizing the firing patterns along these dimensions and linking them to the stimuli BID0 BID9 BID17 BID18 BID12 . There are many methods for compactly representing neural activity along their most relevant dimensions. These methods include Principal Component Analysis (PCA), Independent Component Analysis (ICA) and Factor Analysis (FA) BID2 BID3 BID13 BID20 . Recently, a particularly promising tensor decomposition method was introduced that provides a compact representation of single trial neuronal activity into spatial and temporal dimensions and their combination in the given trial . The method is based on non-negative matrix factorization (NMF) BID14 BID6 BID21 which imposes non-negativity constraints on the extracted components leading to a parts-based, low dimensional, though flexible representation of the data, only assuming non-negativity of the model components. Though space-by-time NMF yielded robust decoding performance with a small number of parameters and good biological interpretability of its basis functions on data recorded from salamander retinal ganglion cells, the method does have a potential shortcoming: it cannot explicitly discount, and is partly confounded by, baseline activity that is not relevant for the neural response to a sensory stimulus. Although these non-negative tensor factorizations performed well on salamander retinal ganglion cells, which have almost non-existent spontaneous activity , it is not clear how well the method would perform on data with considerable spontaneous activity, which might require to explicitly correct for the pre-stimulus baseline.One way to reduce the baseline would be to subtract it from the stimulus-elicited response. This, however, would result in negative activities that cannot be modeled using a decomposition with full non-negativity constraints such as space-by-time NMF. In this study, we thus propose a variant of space-by-time NMF that discounts the baseline activity by subtracting the pre-stimulus baseline from each trial and then decomposes the baseline-corrected activity using a tri-factorization that finds non-negative spatial and temporal modules, and signed activation coefficients. We explored the benefits that this method provides on data recorded from mouse and pig retinal ganglion cells and showed that baseline-corrected space-by-time NMF improves decoding performance on data with non-negligible baselines and stimulus response changes. Here we introduced a novel computational approach to decompose single trial neural population spike trains into a small set of trial-invariant spatial and temporal firing patterns and into a set of activation coefficients that characterize single trials in terms of the identified patterns. To this end, we extended space-by-time non-negative matrix factorization to discount the neuronal pre-stimulus baseline activity. Subtraction of the baseline required the introduction of signed activation coefficients into the decomposition algorithm. This extension considerable widens the scope of applicability of the algorithm as it opens the possibility to decompose data that are inherently signed.Our method inherits many the advantages of the original space-by-time NMF decomposition such as yielding low-dimensional representations of neural activity that compactly carry stimulus information from both the spatial and temporal dimension. Using non-negativity constraints for the spatial and temporal modules, we could also retain the ability of space-by-time NMF to identify a partsbased representation of the concurrent spatial and temporal firing activity of the population. The factorization into space and time further still allows the quantification of the relative importance of these different dimensions on a trial-by-trial basis.Recently, introduced another tensor decomposition algorithm with the capacity to factorize signed data. Their algorithm differs from ours in that it introduces additional constraints for the spatial and temporal modules (cluster-NMF). Our algorithm, on the other hand, introduces no additional constraints, thereby facilitating the comparison with the original space-by-time NMF algorithm. In fact, our extension actually relaxes the non-negativity constraint for the activation coefficients without giving up the parts-based representation of the spatial and temporal modules. This made it possible to pinpoint the reason for the increase in performance as the introduction of the baseline-correction.While BC-SbT-NMF outperformed SbT-NMF overall on tasks with strong baseline activity, we also found that in a few cases, SbT-NMF performed better than BC-SbT-NMF. Previous studies showed that there is an effect of the baseline firing rate on the response BID5 BID8 . In these situations, the baseline might have an advantageous effect on the representation of neural responses and could lead to better decoding performance of SbT-NMF that we observed in some cases. One possibility to take this effect into account would be to devise a joint factorization-decoding framework that explicitly introduces the baseline into the optimization framework. While this is beyond the scope of the current work, we believe that development of such a framework is a promising direction for future research.In order to evaluate decoding performance, we applied LDA classification to the single trial activation coefficients to predict the stimulus identity and also to compare decoding performance of our baseline correction extension with the original space-by-time NMF decomposition. Specifically, we could show that our baseline-corrected version of space-by-time NMF increases decoding performance significantly when the difference between pre-stimulus baseline activity and stimulus-elicited rate was moderate to high. Importantly, this rate-change criterion makes it possible to select the best decomposition method (SbT-NMF vs. BC-SbT-NMF) following a simple data screening by means of the rate change. On our data, we obtained a relative difference in decoding performance on the order of 19.18% when picking the right method in this way and comparing to the inferior method.The requirement for such a rate change to perform well can be understood when considering the baseline-corrected activity. Without a substantial change from pre-stimulus to stimulus-elicited rate, most of the baseline-corrected activity will be close to zero. The Frobenius norm that is at the core of our objective function puts emphasis on high values and will be sensitive to outliers whenever most of the activity is close to zero. In this situation, our update rules are strongly affected by noise, thereby decreasing cross-validated decoding performance. In practical terms, this new method is expected to improve decoding performance when there is a large sensory-evoked response but the differences in responses across different sensory stimuli is of the order of spontaneous activity. In that case, the discounting of the spontaneous levels of firing would help to better discriminate among different stimuli based on neural responses. While the original space-by-time NMF algorithm could in principle identify spatial and temporal modules that fully account for the implicit baseline, the performance gain of our extension suggests that in practice, the original method cannot completely do so. Additional modules increases the model complexity and the number of parameters the method needs to fit which lowers decoding performance. The discount of the baseline provides an elegant way to avoid this unnecessary complication. 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000
We extended single-trial space-by-time tensor decomposition based on non-negative matrix factorization to efficiently discount pre-stimulus baseline activity that improves decoding performance on data with non-negligible baselines.
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