functional equations

Neural networks in the replica-mean field limits

In this talk, we propose to decipher the activity of neural networks via a “multiply and conquer” approach. This approach considers limit networks made of infinitely many replicas with the same basic neural structure. The key point is that these so-called replica-mean-field networks are in fact simplified, tractable versions of neural networks that retain important features of the finite network structure of interest. The finite size of neuronal populations and synaptic interactions is a core determinant of neural dynamics, being responsible for non-zero correlation in the spiking activity and for finite transition rates between metastable neural states. Theoretically, we develop our replica framework by expanding on ideas from the theory of communication networks rather than from statistical physics to establish Poissonian mean-field limits for spiking networks. Computationally, we leverage our original replica approach to characterize the stationary spiking activity of various network models via reduction to tractable functional equations. We conclude by discussing perspectives about how to use our replica framework to probe nontrivial regimes of spiking correlations and transition rates between metastable neural states.