Complexity and dynamics of partially symmetric random neural networks

Abstract

Neural circuits exhibit structured connectivity, including an overrepresentation of reciprocal connections between neuron pairs. Despite important advances, a full understanding of how such partial symmetry in connectivity shapes neural dynamics remains elusive. Here we ask how correlations between reciprocal connections in a random, recurrent neural network affect phase-space complexity, defined as the exponential proliferation rate (with network size) of the number of fixed points that accompanies the transition to chaotic dynamics. We find a striking pattern: partial anti-symmetry strongly amplifies complexity, while partial symmetry suppresses it. These opposing trends closely track changes in other measures of dynamical behavior, such as dimensionality, Lyapunov exponents, and transient path length, supporting the view that fixed-point structure is a key determinant of network dynamics. Thus, positive reciprocal correlations favor low-dimensional, slowly varying activity, whereas negative correlations promote high-dimensional, rapidly fluctuating chaotic activity. These results yield testable predictions about the link between connection reciprocity, neural dynamics and function.

Figures (6)

• Figure 1: Topological complexity, $c,$ given in \ref{['eq:TC']} as a function of the symmetry parameter, $\tau,$ for $g_{\mathrm{eff}}=1.05$ (solid blue line). The dashed red line shows $\max(0,c)$, the contribution of $c$ to the scaling of the expected number of fixed points $\mathbb E[A_N]$ in the limit $N \rightarrow\infty$.
• Figure 2: (a) The distributions of eigenvalues of the connectivity matrix $\mathbf{W}$ for $\tau=-0.5$ (green), $\tau=0$ (magenta), and $\tau=0.5$ (blue) with effective gain $g_{\text{eff}}=1.2$, and $N=4000$. The dashed red line corresponds to $\mathrm{Re}\,\lambda=1$. (b) Three representative solutions of the corresponding system given by \ref{['eq:nn']}. The colors of the trajectories match the colors of the connectivity matrices in panel (a).
• Figure 3: (a) The maximal Lyapunov exponent as a function of $\tau$. (b) Participation dimension as a function of $\tau$. The data for these plots were obtained from $200$ realizations of the connectivity matrix ($N=2000$, $g_{\mathrm{eff}}=1.4$); gray areas indicate standard errors.
• Figure 4: (a) The time-averaged standard deviation of the activity $[\sigma]$ as a function of $\tau$. Inset shows the noise standard deviation $[\sigma_{\mathrm{n}}]$ (blue) and coupling strength $g$ (red). (b) The mean sensitivity $[S']$ (averaged over network size) as a function of $\tau$. In both panels, each data point is obtained by averaging over $200$ realizations of the connectivity matrix with $N=2000$ and $g_{\mathrm{eff}}=1.4$; shaded regions denote standard errors (mostly smaller than marker size). Square brackets $[\cdots]$ denote time averages over the respective dynamics of each realization.
• Figure 5: (a) Median path length $\mathcal{L}$ as a function of $\tau$ for $N=700$; shading indicates the interquartile range (spanning first to third quartiles). (b) Average path length $\mathcal{L}$ versus $N$ for different values of $\tau\in\{0,0.2,0.4,0.6,0.8,1\}$, ordered top to bottom as $\tau$ increases: $\tau=0$ (solid blue, top) to $\tau=1$ (pink, bottom), shown by the diagonal arrow. In both panels, $g_{\mathrm{eff}}=1.2$.