Quasilinearization-Based Controllability Analysis Of Neuronal Rate Networks

Recent interest has developed around the problem of assaying the controllability of networks in the brain. The analysis of such networks is highly nontrivial, owing to their overwhelming complexity. Thus, any controllability analysis must tradeoff against model complexity/explanatory power, and analysis tractability. Here, we consider a class of neuronal network models with nearly linear dynamics, whose primary complication arises due to a sigmoidal nonlinearity in the neuronal coupling. Exploiting the equivalence between the controllability gramian and the steady state covariance matrix of a linear system under white noise, we develop an approximate controllability analysis based on the method of stochastic linearization (quasilinearization). We show that for this relatively simple system, the quasilinear approach generates a significantly better characterization of controllability as compared with a Jacobian linearization. Our results provide a new tool for assessing controllability of networks with sigmoidal interactions, and, moreover, highlight the potential inaccuracy of linear characterizations of networks with even relatively mild nonlinearities.