Simons Center for Systems Biology - Seminar

Abstract: In many experimental situations, large populations of simultaneously recorded neurons (up to 1 million) have been found to exhibit ‘low-dimensional’ dynamics, in the sense that the population dynamics can be effectively described by a number of modes significantly smaller than the number of recorded neurons. It has been suggested that this requires some structural property of the synaptic connectivity matrix; in particular that the connectivity matrix is low-rank (i.e., it has rapidly decreasing singular values). However, while low-rank connectivity is a sufficient condition for low-dimensional dynamics, it is unclear whether it is also a necessary condition and, hence, whether low-rank connectivity can be legitimately inferred from the observation of low-dimensional dynamics. Here, we show that low-dimensional dynamics occurs in neuronal networks with unstructured (i.e., random, full-rank) synaptic connectivity and provide a simple geometrical explanation of why this is so. Using replica theory, we analytically characterize the number, stability and phase-space geometry of the network's equilibria. In the chaotic regime, exhibited by the network for a suitable choice of the parameters, there are exponentially many equilibria, all unstable with small instability indexes (the fractional dimension of their unstable manifolds). Remarkably, despite the absence of structure in the synaptic connectivity, the equilibria are strongly correlated and, therefore, occupy a small region of the phase space; the chaotic attractor is contained within 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. Furthermore, the number of unstable directions of a point on the attractor is upper-bounded by the instability index of the ‘most unstable’ equilibrium, which is fractionally small compared to the total number of neurons. This results in low dimensionality, as estimated from the Lyapunov spectrum. These results show that low-dimensional dynamics can be a purely emergent phenomenon, with no structural counterpart; low-rank connectivity is not a necessary condition for low-dimensional activity. Our results add to the theoretical tools that can be used to analyze large neuronal populations and provide an alternative interpretation of the observed low-dimensional dynamics.