PMID-26621426 Causal Inference and Explaining Away in a Spiking Network
- Rubén 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.
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- N causes, one observation: ,
- -- 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 (see figure above)
- That is ... J looks like a correlation matrix!
- 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 [44].
- 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
- 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): , then
- 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.
- translates then to global inhibition, and to lowered reset voltage.
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- 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.
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