{1517} revision 1 modified: 07-22-2020 18:13 gmt

PMID-26621426 Causal Inference and Explaining Away in a Spiking Network

  • RubĂ©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: μ=Σ i=1 Nu ir i+ε \mu = \Sigma_{i=1}^{N} u_i r_i + \epsilon ,
    • r i0r_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 TUβI J = -U^TU - \beta I (see figure above)
    • That is ... J looks like a correlation matrix!
    • UU 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 p(μ,r)exp(L(μ,r))Π i=1 NH(r i) 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 TUW = -U^T U , h=U Tμ h = U^T \mu then E(r)=0.5r TWrr Th+α1 Tr+0.5βr TrE(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.
    • α\alpha translates then to global inhibition, and β\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.