Statistical Inference for Perturbed Stochastic Processes with Application to Neuroscience
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A latent internal process describes the state of some system, e.g. the membrane potential evaluation of a neuron, the price of a stock or the health status of a person. When this process reaches a predefined threshold, the process terminates and an observable event occurs, e.g. a neuron releases an electrical impulse, the stock is sold/bought or the person dies. Imagine an intervention, e.g., an input current, a speculation strategy or a medical treatment, is initiated to the process before the event occurs. How can we evaluate whether the intervention had an effect? How can we detect the type of stimulus applied only observing the events following the intervention? What can be said if both the time of the intervention and the type of stimulus are unknown? Imagine now that the intervention has an unknown intensity level. What is the highest decoding accuracy of the intensity level? How does this discrimination change if the system is observed for a longer time? Answering these questions is particularly difficult because the latent internal process describing the state of the system is perturbed, i.e. observed only on top of an indistinguishable background noise. From a mathematical point of view, the described problem is modeled by stochastic point processes obtained as hitting times of perturbed stochastic processes. A study of the decoding accuracy of the stimulus level based on either the first event after the intervention (assuming the change point to be either known or unknown) or the rate of events on a certain observation time window were performed, yielding counter-intuitive results, representing a novel manifestation of the noise-aided signal enhancement, which differs fundamentally from the usual kinds reported on, such as standard stochastic resonance. Our results are discussed in the framework information transfer in neural systems, but the same scenario can be found in many fields, e.g. reliability theory, social sciences, finance, biology or medicine.
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