On the relationship between equilibria and dynamics in large, random neuronal networks

Abstract

We investigate the equilibria of a random model network exhibiting extensive chaos. In this regime, a large number of equilibria is present. They are all saddles with low-dimensional unstable manifolds. Surprisingly, despite network's connectivity being completely random, the equilibria are strongly correlated and, as a result, they occupy a very small region in the phase space. The attractor is inside this region. This geometry explains why the collective states sampled by the dynamics are dominated by correlation effects and, hence, why the chaotic dynamics in these models can be described by a fractionally-small number of collective modes.

Figures (3)

• Figure 1: (a) $\Sigma^*_q$ (full) and $\Sigma^*_a$ (dashed) as a function of $\sigma_w$. Dots with errorbars (SD) represent numerical estimates ($n=50$ sample networks for each value of $\sigma_w$ and $N$). The vertical line (dashed blue) indicates the critical point $\sigma_w=\sqrt{2}$. (b) Instability index averaged over all the equilibria as a function of $\sigma_w$, according to the quenched (full) and annealed (dashed) theory. Dots and vertical line as in (a). (c) $\Sigma_q$ as a function of $\sigma_w$ and $f$. The full line denotes the location of $\Sigma^*_q$, the dashed one that of $\Sigma^*_a$. Vertical dashed line as in (a). (d) Spectral distribution of $\tilde{\lambda}\equiv\left(\sigma_w\sqrt{f}\right)^{-1}\left(1+\lambda\right)$, where $\lambda$ is a non-trivial eigenvalue of $\mathbf{J}_{\mathbf{w}}\left(\mathbf{x}^a\right)$ and $\mathbf{x}^a$ is an equilibrium with a fraction $f^a=f$ of active neurons. Main panel: cumulative distribution function of $\lvert\tilde{\lambda}\rvert$; Inset: full eigenspectrum. Black: theory; Colors: $3$ equilibria randomly chosen from $3$ different sample networks.
• Figure 2: (a) Cosine similarity (CS) between network's equilibria and their centroid as a function of $f$. Full line: theory; Red dots: estimates from the equilibria numerically found (shaded area is $\pm$SD across networks); Gray line: CS between random equilibria and their centroid. Parameters: $\sigma_w=2.5$ and $N=200$. (b) CS between centroids at different $f_1$ and $f_2$, both in the same network (above diagonal), or in different networks (below diagonal). Parameters as in (a). (c) Cartoon illustrating the spatial organization of actual and random equilibria.
• Figure 3: (a) Cosine similarity between $\boldsymbol{\tilde{\phi}}^t$ and $\langle\boldsymbol{\tilde{\phi}}\rangle$ (black) and between $\boldsymbol{\tilde{\phi}}^t$ and $\boldsymbol{\bar{\phi}}$ (orange), when all vectors are computed in the same sample network ($\sigma_w=2.5$, $N=200$). For comparison, we also plot the cosine similarity between $\boldsymbol{\tilde{\phi}}^t$ and $\boldsymbol{\bar{\phi}}$, when these vectors are computed in different sample networks (gray). (b) $Q_\mathrm{D}$ as a function of $\sigma_w$. Red dots with errorbars (SD) represent numerical estimates ($n=50$ samples with $N=5000$). The gray area represents the range of $Q(f)$ over the corresponding equilibria. (c) $q_\mathrm{D}$ (full) and $q_\mathrm{eq}$ (dashed) as a function of $\sigma_w$. Red dots as in (b).