Inferring Dynamic Delays Using a Sample Path Causal Measure

Understanding dynamic relationships between random processes is important in many applications, such as neuroscience. The interactive relationship between multiple recorded neural processes is a challenge to discover, particularly when the delays in these interactions are dynamic. Although informationtheoretic measures such as directed information have been adopted to understand propagation delays in neuroscience applications, they are insufficient to capture time-varying delays. We introduce a sample path causal measure to help infer the directed delay between two random processes that considers the relative entropy between two predictors, dynamically changing as needed, where one has more (delayed) information than another. We develop a procedure to determine dynamic delays by identifying phase transitions in these relative entropies as a function of delay order and time. We apply this procedure to a simulated constant delay case and a time-varying delay case using a sequential probability approach to dynamically track the evolving statistics. We demonstrate with simulations how the method identifies the delay in both the constant and dynamic situations.