Stochastic activity in low-rank recurrent neural networks
Fig 5
Stochastic activity in rank-two recurrent neural networks.
A. Rank-two connectivity. B. Eigenvalues of the covariance matrix that are different than the reference value . As connectivity is rank-two, four eigenvalues are perturbed; we sort them in ascending order. Violin plots show the distribution of perturbed eigenvalues for different values of the parameters
and
. Note that, for all sets of parameters, two eigenvalues are increased and two decreased with respect to
. C. Dimensionality as a function of
and
. The black dashed lines indicate the parameter values for which dynamics become unstable. The tiny black square indicates the parameter values that are used for simulations in F–G. In both B and
, we keep the values of
and
fixed to zero (see Methods 7). D–E. Same as for B–C, but for a different parametrization, where we keep
and
fixed to zero. F–G. Example of a simulated network, parameters indicated in C. In F: covariance spectrum. In G: overlap between four selected principal components (the strongest and the weakest) estimated from simulated activity and the theoretically-estimated covariance eigenvectors (left) and the connectivity vectors (right). Overlaps are quantified via Eq 4. The theoretical expressions for this case are reported in Methods 7.