Two-photon calcium imaging enables functional analysis of neuronal circuits by inferring action potential (AP) occurrence (“spike trains”) from cellular fluorescence signals. It remains unclear how experimental parameters such as signal-to-noise ratio (SNR) and acquisition rate affect spike inference and whether additional information about network structure can be extracted. Here we present a simulation framework for quantitatively assessing how well spike dynamics and network topology can be inferred from noisy calcium imaging data. For simulated AP-evoked calcium transients in neocortical pyramidal cells, we analyzed the quality of spike inference as a function of SNR and data acquisition rate using a recently introduced peeling algorithm. Given experimentally attainable values of SNR and acquisition rate, neural spike trains could be reconstructed accurately and with up to millisecond precision. We then applied statistical neuronal network models to explore how remaining uncertainties in spike inference affect estimates of network connectivity and topological features of network organization. We define the experimental conditions suitable for inferring whether the network has a scale-free structure and determine how well hub neurons can be identified. Our findings provide a benchmark for future calcium imaging studies that aim to reliably infer neuronal network properties.

One possible solution is to extract the dynamic coupling (or effective connectivity) structure based on the recorded spike trains. The effective coupling between two neurons quantifies how the spiking activity of one neuron enhances or reduces the spiking probability of the second one and should in general be directed and causal. After thresholding, such a coupling structure reduces to a binary, directed connectivity matrix in which a non-zero entry indicates a directed coupling between two neurons. The connectivity matrix defines a graph and its structural properties can be further quantified. Using this analysis scheme, the overall network activity can be summarized in a few quantitative parameters of the inferred network topology (Figure 1).The construction of the network graph proceeds therefore in three steps. First, we need to extract couplings from time-series observations. Several statistical methods have been proposed that extend beyond simple cross-correlation analysis. A particularly interesting statistical model is the generalized linear model (GLM; McCullagh and Nelder, 1989). GLMs allow the extraction of directed, i.e., causal coupling filters that are conditioned on the whole population response ( Figure 1B). This is in contrast to many other proposed schemes that are based only on pair-wise measures or very crude approximations to the overall population activity. IntroductIonIn recent years, techniques in systems neuroscience and electrophysiology to record from many neurons in parallel became widely available (see, e.g., Georgopoulos et al., 1986;Nicolelis et al., 1997Nicolelis et al., , 2003Santhanam et al., 2006;Jarosiewicz et al., 2008). While in the beginning, this was seen merely as a way to perform many single-neuron experiments simultaneously, the full potential of such parallel recordings was soon realized and led to a paradigm shift in the analysis from single-spike statistics (Perkel et al., 1967a;Brillinger, 1988;Bialek et al., 1991;Rieke et al., 1999;Gerstner and Kistler, 2002) toward the analysis of neural population activity and interactions between neurons. Early attempts focused on pair-wise cross-correlation analysis (Perkel et al., 1967b;Aertsen et al., 1989), but recent statistical models have tried to capture the global network activity (Iyengar, 2001;Brown et al., 2004;Truccolo et al., 2005;Schneidman et al., 2006;Pillow et al., 2008;Paninski et al., 2010) in an attempt to understand how neurons collectively encode and process information. The access to this high-dimensional type of data has also triggered a challenge for statistical data analysis and modeling (Kass et al., 2005): What is an adequate lowdimensional description of the structure in the spiking activity of neural populations? The simultaneous recording of the activity of many neurons poses challenges for multivariate data analysis. Here, we propose a general scheme of reconstruction of the functional network from spike train recordings. Effective, causal interactions are estimated by fitting generalized linear ...

Point process generalized linear models (PP-GLMs) provide an important statistical framework for modeling spiking activity in single-neurons and neuronal networks. Stochastic stability is essential when sampling from these models, as done in computational neuroscience to analyze statistical properties of neuronal dynamics and in neuro-engineering to implement closed-loop applications. Here we show, however, that despite passing common goodness-of-fit tests, PP-GLMs estimated from data are often unstable, leading to divergent firing rates. The inclusion of absolute refractory periods is not a satisfactory solution since the activity then typically settles into unphysiological rates. To address these issues, we derive a framework for determining the existence and stability of fixed points of the expected conditional intensity function (CIF) for general PP-GLMs. Specifically, in nonlinear Hawkes PP-GLMs, the CIF is expressed as a function of the previous spike history and exogenous inputs. We use a mean-field quasi-renewal (QR) approximation that decomposes spike history effects into the contribution of the last spike and an average of the CIF over all spike histories prior to the last spike. Fixed points for stationary rates are derived as self-consistent solutions of integral equations. Bifurcation analysis and the number of fixed points predict that the original models can show stable, divergent, and metastable (fragile) dynamics. For fragile models, fluctuations of the single-neuron dynamics predict expected divergence times after which rates approach unphysiologically high values. This metric can be used to estimate the probability of rates to remain physiological for given time periods, e.g., for simulation purposes. We demonstrate the use of the stability framework using simulated single-neuron examples and neurophysiological recordings. Finally, we show how to adapt PP-GLM estimation procedures to guarantee model stability. Overall, our results provide a stability framework for data-driven PP-GLMs and shed new light on the stochastic dynamics of state-of-the-art statistical models of neuronal spiking activity.

Calcium imaging has revolutionized systems neuroscience, providing the ability to image large neural populations with single-cell resolution. The resulting datasets are quite large (with scales of TB/hour in some cases), which has presented a barrier to routine open sharing of this data, slowing progress in reproducible research. State of the art methods for analyzing this data are based on nonnegative matrix factorization (NMF); these approaches solve a non-convex optimization problem, and are highly effective when good initializations are available, but can break down e.g. in low-SNR settings where common initialization approaches fail.Here we introduce an improved approach to compressing and denoising functional imaging data. The method is based on a spatially-localized penalized matrix decomposition (PMD) of the data to separate (low-dimensional) signal from (temporally-uncorrelated) noise. This approach can be applied in parallel on local spatial patches and is therefore highly scalable, does not impose nonnegativity constraints or require stringent identifiability assumptions (leading to significantly more robust results compared to NMF), and estimates all parameters directly from the data, so no hand-tuning is required. We have applied the method to a wide range of functional imaging data (including one-photon, two-photon, three-photon, widefield, somatic, axonal, dendritic, calcium, and voltage imaging datasets): in all cases, we observe ∼2-4x increases in SNR and compression rates of 20-300x with minimal visible loss of signal, with no adjustment of hyperparameters; this in turn facilitates the process of demixing the observed activity into contributions from individual neurons. We focus on two challenging applications: dendritic calcium imaging data and voltage imaging data in the context of optogenetic stimulation. In both cases, we show that our new approach leads to faster and much more robust extraction of activity from the video data.