The execution of reaching movements involves the coordinated activity of multiple brain regions that relate variously to the desired target and a path of arm states to achieve that target. These arm states may represent positions, velocities, torques, or other quantities. Estimation has been previously applied to neural activity in reconstructing the target separately from the path. However, the target and path are not independent. Because arm movements are limited by finite muscle contractility, knowledge of the target constrains the path of states that leads to the target. In this letter, we derive and illustrate a state equation to capture this basic dependency between target and path. The solution is described for discrete-time linear systems and gaussian increments with known target arrival time. The resulting analysis enables the use of estimation to study how brain regions that relate variously to target and path together specify a trajectory. The corresponding reconstruction procedure may also be useful in brain-driven prosthetic devices to generate control signals for goal-directed movements.

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