Large-scale neural recordings have established that the transformation of sensory stimuli into motor outputs relies on low-dimensional dynamics at the population level, while individual neurons exhibit complex selectivity. Understanding how low-dimensional computations on mixed, distributed representations emerge from the structure of the recurrent connectivity and inputs to cortical networks is a major challenge. Here, we study a class of recurrent network models in which the connectivity is a sum of a random part and a minimal, low-dimensional structure. We show that, in such networks, the dynamics are low dimensional and can be directly inferred from connectivity using a geometrical approach. We exploit this understanding to determine minimal connectivity required to implement specific computations and find that the dynamical range and computational capacity quickly increase with the dimensionality of the connectivity structure. This framework produces testable experimental predictions for the relationship between connectivity, low-dimensional dynamics, and computational features of recorded neurons.

Asynchronous activity in balanced networks of excitatory and inhibitory neurons is believed to constitute the primary medium for the propagation and transformation of information in the neocortex. Here we show that an unstructured, sparsely connected network of model spiking neurons can display two fundamentally different types of asynchronous activity that imply vastly different computational properties. For weak synaptic couplings, the network at rest is in the well-studied asynchronous state, in which individual neurons fire irregularly at constant rates. In this state, an external input leads to a highly redundant response of different neurons that favors information transmission but hinders more complex computations. For strong couplings, we find that the network at rest displays rich internal dynamics, in which the firing rates of individual neurons fluctuate strongly in time and across neurons. In this regime, the internal dynamics interact with incoming stimuli to provide a substrate for complex information processing and learning.