לוח שנה משולב

Since the 19th century neuroscientists have pondered the question of how the brain maintains a spatially accurate visual signal despite a constantly moving eye. Hering said to measure where the eye is in the orbit. Helmholtz said to adjust the visual representation by the dimensions of an upcoming movement. Both were right. Visual responses in the lateral intraparietal area (LIP) are modulated by eye position, and target position in supraretinnal coordinates can be calculated from this modulation. The eye position signals for this modulation come from the representation of eye position in somatosensory cortex. This eye position signal is, however, too slow to be accurate within 150 ms after a saccade. However, immediately before a saccade neurons in LIP respond to stimuli that will be brought into their receptive fields by an impending saccade. The signal that remaps the receptive field arises from a corollary discharge of the motor command, and a computational model shows that this remapping can be effected by a wave of activity in the cortex that propagates from the cell driven by the stimulus before the saccade to the cell in whose receptive field the stimulus will lie after the saccade. Thus spatial accuracy is effected by two systems, a relatively inaccurate, fast system using corollary discharge, and a slower proprioceptive system that more accurately measure the position of the eye in the orbit.

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by systems of differential equations. These systems describe the interrelationships between the variables involved, and depend in a complicated way on unknown quantities (e.g., initial values, constants or time dependent parameters). Most often, the researcher would like to execute important tasks such as testing the validity of a model, analyzing its qualitative behavior or predicting future states of the system. In order to execute these tasks, one usually needs to estimate the unknown quantities of the system from real data. However, in the case of differential equations, the inverse problem of parameter estimation is considered as the bottleneck in modeling dynamical systems and estimating parameters based on observed noisy state variables has a relatively sparse statistical literature.

In this talk we focus on the fairly general and often applied class of systems of ordinary differential equations linear in (functions of) the parameters. For such systems we first characterize a necessary and sufficient condition for identifiability of parameters. Then we present a novel estimation approach and support it by a general statistical theory that enables the development of estimators tailored for a variety of experimental scenarios. In particular, we present estimators corresponding to some common experimental setups and discuss their statistical properties. Simulation studies and application of the method to real data will be discussed.