 m8ta
 {1517} hide / / print ref: -2015 tags: spiking neural networks causality inference demixing date: 07-22-2020 18:13 gmt revision:1  [head] RubÃ©n Moreno-Bote & Jan Drugowitsch Use linear non-negative mixing plus nose to generate a series of sensory stimuli. Pass these through a one-layer spiking or non-spiking neural network with adaptive global inhibition and adaptive reset voltage to solve this quadratic programming problem with non-negative constraints. N causes, one observation: $\mu = \Sigma_{i=1}^{N} u_i r_i + \epsilon$ , $r_i \geq 0$ -- causes can be present or not present, but not negative. cause coefficients drawn from a truncated (positive only) Gaussian. linear spiking network with symmetric weight matrix $J = -U^TU - \beta I$ (see figure above) That is ... J looks like a correlation matrix! $U$ is M x N; columns are the mixing vectors. U is known beforehand and not learned That said, as a quasi-correlation matrix, it might not be so hard to learn. See ref . Can solve this problem by minimizing the negative log-posterior function: $$L(\mu, r) = \frac{1}{2}(\mu - Ur)^T(\mu - Ur) + \alpha1^Tr + \frac{\beta}{2}r^Tr$$ That is, want to maximize the joint probability of the data and observations given the probabilistic model $p(\mu, r) \propto exp(-L(\mu, r)) \Pi_{i=1}^{N} H(r_i)$ First term quadratically penalizes difference between prediction and measurement. second term, alpha is a L1 regularization term, and third term w beta is a L2 regularization. The negative log-likelihood is then converted to an energy function (linear algebra): $W = -U^T U$ , $h = U^T \mu$ then $E(r) = 0.5 r^T W r - r^T h + \alpha 1^T r + 0.5 \beta r^T r$ This is where they get the weight matrix J or W. If the vectors U are linearly independent, then it is negative semidefinite. The dynamics of individual neurons w/ global inhibition and variable reset voltage serves to minimize this energy -- hence, solve the problem. (They gloss over this derivation in the main text). Next, show that a spike-based network can similarly 'relax' or descent the objective gradient to arrive at the quadratic programming solution. Network is N leaky integrate and fire neurons, with variable synaptic integration kernels. $\alpha$ translates then to global inhibition, and $\beta$ to lowered reset voltage. Yes, it can solve the problem .. and do so in the presence of firing noise in a finite period of time .. but a little bit meh, because the problem is not that hard, and there is no learning in the network. {1430} hide / / print ref: -2017 tags: calcium imaging seeded iterative demixing light field microscopy mouse cortex hippocampus date: 02-13-2019 22:44 gmt revision:1  [head] Tobias NÃ¶bauer, Oliver Skocek, Alejandro J PernÃ­a-Andrade, Lukas Weilguny, Francisca MartÃ­nez Traub, Maxim I Molodtsov & Alipasha Vaziri Cell-scale imaging at video rates of hundreds of GCaMP6 labeled neurons with light-field imaging followed by computationally-efficient deconvolution and iterative demixing based on non-negative factorization in space and time.  Utilized a hybrid light-field and 2p microscope, but didn't use the latter to inform the SID algorithm. Algorithm: Remove motion artifacts Time iteration: Compute the standard deviation versus time (subtract mean over time, measure standard deviance) Deconvolve standard deviation image using Richardson-Lucy algo, with non-negativity, sparsity constraints, and a simulated PSF. Yields hotspots of activity, putative neurons. These neuron lcoations are convolved with the PSF, thereby estimating its ballistic image on the LFM. This is converted to a binary mask of pixels which contribute information to the activity of a given neuron, a 'footprint' Form a matrix of these footprints, p * n, $S_0$ (p pixels, n neurons) Also get the corresponding image data $Y$ , p * t, (t time) Solve: minimize over T $|| Y - ST||_2$ subject to $T \geq 0$ That is, find a non-negative matrix of temporal components $T$ which predicts data $Y$ from masks $S$ . Space iteration: Start with the masks again, $S$ , find all sets $O^k$ of spatially overlapping components $s_i$ (e.g. where footprints overlap) Extract the corresponding data columns $t_i$ of T (from temporal step above) from $O^k$ to yield $T^k$ . Each column corresponds to temporal data corresponding to the spatial overlap sets. (additively?) Also get the data matrix $Y^k$ that is image data in the overlapping regions in the same way. Minimize over $S^k$ $|| Y^k - S^k T^k||_2$ Subject to $S^k >= 0$ That is, solve over the footprints $S^k$ to best predict the data from the corresponding temporal components $T^k$ . They also impose spatial constraints on this non-negative least squares problem (not explained). This process repeats. allegedly 1000x better than existing deconvolution / blind source segmentation algorithms, such as those used in CaImAn