Linear Network Models of the Oculomotor Integrators

The oculomotor system uses integrators to transform velocity signals to control eye position. We model such an integrator as a linear network with a marginal mode separated from stable modes by a large spectral gap. The model neurons carry a superposition of position and velocity signals, just as observed in single unit recordings from hindbrain oculomotor areas. The single unit properties of tonic activity, position sensitivity, velocity sensitivity, and position susceptibility are shaped by feedforward input and local feedback. Single unit response can also be given a geometric interpretation in terms of a line attractor in the state space of the network. Unlike nonlinear models, the linear model predicts uniformity of linear response along the line attractor. Both the transverse relaxation time under normal conditions and the integrator time constant after lesion or inactivation suggest that local feedback in the integrator is mediated by synapses with persistence times on the order of 100 ms. Control theoretic models of the oculomotor system rely on integrators to transform velocity signals into eye position signals. While there has been great progress in localizing these integrators to hindbrain areas, the neural basis of integration is still obscure. Robinson and colleagues have advanced a series of linear network models that are constructed from neurons with short persistence times, but can integrate over long time scales by virtue of their local feedback connectionss1, 2, 3, 4]. Because they found a diversity of integrator networks, robust predictions could only be made on the basis of characteristics common to many networks, rather than any particular one. This required an understanding of network function, which seemed impossible given the uninter-pretability of single unit behavior: \the behavior of hidden units becomes more and more divorced from intuitive behavior as their tasks become more and more complicated""5]. The present work solves Robinson's problems of nonuniqueness and uninter-pretability by precisely characterizing the set of all linear network integrators.