Denoising Two-Photon Calcium Imaging Data

Two-photon calcium imaging is now an important tool for in vivo imaging of biological systems. By enabling neuronal population imaging with subcellular resolution, this modality offers an approach for gaining a fundamental understanding of brain anatomy and physiology. Proper analysis of calcium imaging data requires denoising, that is separating the signal from complex physiological noise. To analyze two-photon brain imaging data, we present a signal plus colored noise model in which the signal is represented as harmonic regression and the correlated noise is represented as an order autoregressive process. We provide an efficient cyclic descent algorithm to compute approximate maximum likelihood parameter estimates by combing a weighted least-squares procedure with the Burg algorithm. We use Akaike information criterion to guide selection of the harmonic regression and the autoregressive model orders. Our flexible yet parsimonious modeling approach reliably separates stimulus-evoked fluorescence response from background activity and noise, assesses goodness of fit, and estimates confidence intervals and signal-to-noise ratio. This refined separation leads to appreciably enhanced image contrast for individual cells including clear delineation of subcellular details and network activity. The application of our approach to in vivo imaging data recorded in the ferret primary visual cortex demonstrates that our method yields substantially denoised signal estimates. We also provide a general Volterra series framework for deriving this and other signal plus correlated noise models for imaging. This approach to analyzing two-photon calcium imaging data may be readily adapted to other computational biology problems which apply correlated noise models.National Institutes of Health (U.S.) (DP1 OD003646-01)National Institutes of Health (U.S.) (R01EB006385-01)National Institutes of Health (U.S.) (EY07023)National Institutes of Health (U.S.) (EY017098

Temporally Correlated Inputs to Leaky Integrate-and-Fire Models Can Reproduce Spiking Statistics of Cortical Neurons

There has been a controversy on whether the standard neuro-spiking models are consistent with the irregular spiking of cortical neurons. In a previous study, we proposed examining this consistency on the basis of the high order statistics of the inter-spike intervals, as represented by the coe#cient of variation and the skewness coe#cient. In that study we found that a leaky integrate-and-fire model incorporating the assumption of temporally uncorrelated inputs is not able to account for the spiking data recorded from a monkey prefrontal cortex. In the present paper, we attempt to revise the neuro-spiking model so as to make it consistent with the biological data. Here we consider the correlation coe#cient of consecutive inter-spike intervals, which was ignored in previous studies. Considering three statistical coe#cients, we conclude that the leaky integrate-and-fire model with temporally correlated inputs does account for the biological data. The correlation time scale of the inputs ..

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Probabilistic approach to non-local equations

The work focuses on probabilistic representation of solutions of non-local equations, where the considered non-local operators are either Caputo-type derivatives or trasformations of the Laplace operator via Bernstein functions. The first chapter is devoted to the introduction of the main tools, that is to say Bernstein functions, subordinators and their inverse processes, Caputo-type derivatives and Bochner subordination. In the second chapter we focus on theoretical results concerning probabilistic representation of non-local equations involving Caputo-type derivatives. First we study a general theory for abstract Cauchy problems involving such kind of derivatives, focusing also on the eigenfunctions of such Caputo-type derivatives and Gronwall-type inequalities. Then we exploit the link between time-changed birth-death processes and abstract Cauchy problems in suitable Banach sequence spaces, via a spectral decomposition approach. Then we consider how the Fokker-Planck equation of a non-Markov Gaussian process changes after applying a time-change via the inverse of a subordinator. Finally, we consider exit times of time-changed processes, their asymptotic problems and the link between their survival probability and solutions of non-local (in time) parabolic PDEs. The third chapter is devoted to applications of the previously presented theoretical results. In particular we focus on queueing theory and computational neuroscience. Moreover, we also exploit some simulation properties to work with such processes. In the fourth and last chapter we focus on non-local operators in space with two exemplary problems. The first one concerns the integral representation of Bernstein functions of the Laplace operator. In particular we prove asymptotic properties of the singular kernel of such integral representations depending on asymptotic properties of the Levy measure of the considered Bernstein function. The second problem deals with spectral properties of a Marchaud-type operator on the sphere. In particular, we prove an identity involving the first eigenvalue of the aforementioned operator and moments of the length of random segments in the unit ball