Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks

The eigenvalue spectrum of the matrix of directed weights defining a neural network model is informative of several stability and dynamical properties of network activity. Existing results for eigenspectra of sparse asymmetric random matrices neglect spatial or other constraints in determining entries in these matrices, and so are of partial applicability to cortical-like architectures. Here we examine a parameterized class of networks that are defined by sparse connectivity, with connection weighting modulated by physical proximity (i.e., asymmetric Euclidean random matrices), modular network partitioning, and functional specificity within the excitatory population. We present a set of analytical constraints that apply to the eigenvalue spectra of associated weight matrices, highlighting the relationship between connectivity rules and classes of network dynamics.

Figure 1

Empirical and analytical eigenvalue distributions are closely matched. (a) Analytical predictions of ${\lambda }_b$ (curve) match empirical observations (dots) in a sparse network without community structure with $N=500,h_E=10\%,h_I=50\%,w_I=-1$, and $f_I=20\%$. (b) Analytical predictions of ${\lambda }_Q$ (curve) match empirical observations (dots) in a full network (i.e., $h_E,h_I=100\%)$ with $N=500,w_E=1,w_I=-10,f_I=20\%$, and with two subnetwork partitions $\left(M=2\right)$. (c) The distribution of empirical eigenvalues (black circles) matches the analytical bounds (gray circles and markers) in a sparse network $(N=1000,h_E=10\%$, and $h_I=50\%)$ with many subnetwork partitions $\left(M=20\right)$ and with $w_E=2,w_I=-12,f_I=20\%$. (d) Analytical predictions of the maximum real eigenvalue $max\mathrm{Re}\left(\lambda \right)$ (curves) correspond well to the empirical maximum real eigenvalue, in a sparse network $\left(N=5000,h_E=10\%,h_I=\{10\%,20\%,80\%\}\right)$ containing two subnetwork partitions $\left(M=2\right)$ and with $w_E=2,w_I=-12,f_I=20\%$, and $h_E=10\%$.

Figure 2

(a) Analytical distributions of ${\Sigma }_L{S}_{i\ne j},\lceil {\Sigma }_L{S}_{i\ne j}\rceil$, and $\lfloor {\Sigma }_L{S}_{i\ne j}\rfloor$ (curves) closely match empirical measurements over 2000 network instances (shaded histograms). Network parameters: $N=1200,L=1199,{\kappa }_s=\frac{1}{8},D=5$. (b) Analytical predictions of eigenspectra bounds $\mathrm{E}\lceil {\lambda }_{+}\rceil$ and $\mathrm{E}\lfloor {\lambda }_{-}\rfloor$ (curves) and empirical maximum and minimum eigenvalues (dots) for networks with $r=0,h_I,h_E=1,f_I=20\%,N=6000,D=2,w_E=2$, and $w_I=8$.

Figure 3

Analytical and approximate distributions of ${\Delta }_D^2$. (a–c) Analytical distributions (black) compared against empirical distributions for 5000 variates (shaded histograms). These distributions are exact; note the perfect overlap between empirical and analytical distributions. (d–f) A Rice distribution (black) gives a close approximation to the empirical distribution of ${\Delta }_D^2$ (shaded histograms) for $D=4–10$. Values of $D$ are as indicated on the figure.

Figure 4

Plots of extremal weight component estimates ${\widehat{w}}_{*}$ used to calculate $\mathrm{E}\lceil {\lambda }_{-}\rceil$ (left column) and $\mathrm{E}\lfloor {\lambda }_{+}\rfloor$ (middle column). At right are shown analytical estimates for $\mathrm{E}\lceil {\lambda }_{+}\rceil$ and $\mathrm{E}\lfloor {\lambda }_{-}\rfloor$ (gray curves), as well as the empirical ${\lambda }_{+}$ and ${\lambda }_{-}$ (black dots) computed numerically for many instances of $W$. For all panels $r=0,h_I,h_E=1,N=6000,w_E=2,w_I=8$, and $f_I=20\%$. Values for $D$: (a) 1, (b) 5, (c) 10. A comparison between predicted and empirical ${\lambda }_{+}$ and ${\lambda }_{-}$ for $D=2$ is shown in Fig. 2.