Complexity and dynamics of partially symmetric random neural networks.

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.