Controllability analysis and design for underactuated stochastic neurocontrol
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Neuroengineering has advanced tremendously over the past decade, but for sensory prosthetics and similar applications, it remains an extraordinary challenge to access neurons at the single cell resolution of most sensory encoding theories. In particular, if each neuron is “tuned” to particular stimulus features, then eliciting a target percept requires activating only neurons tuned to that percept and not others. However, most available technology is underactuated, with orders of magnitude fewer independent control inputs than neural degrees of freedom, possibly limiting its effectiveness given the inherent trade-off of resolution with network size. Here I analyze controllability for pairs of neurons receiving a common input. In particular, I extend previous work on the deterministic control problem to include stochastic membrane dynamics, treating both cases as a bifurcation problem in the noise parameter. I determine controllable regions in parameter space using a combination of mathematical analysis and numerical solution of stochastic differential and Fokker-Planck equations. I explain how boundaries between these regions change with noise level, and connect to the dynamical mechanisms by which controllability is lost. I show that in stochastic systems, in contrast to deterministic systems, expanding the allowable input space to include exponential ramps expands the parameter range over which neuron pairs are controllable. I also describe an alternative controllability definition using only mean spike times, as compared to the probability distribution of spiking within prespecified time intervals. These results could guide future control strategies in the development of sensory neuroprosthetics and other neurocontrol application.