Fractional-order systems in biological applications: estimating causal relations in a system with inner connectivity using fractional moments

This chapter aims to expand the application of fractional-order cumulants to determine the types of connection and the directions of causalities' interactions among neurons. Fractional-order signal processing methods can provide a more in-depth interpretation of dynamical systems than their conventional integer counterparts. Therefore, in this chapter, a new neuroscientific usage for the method is introduced. The directionality of the excitatory and inhibitory connections is found using fractional cumulants. The efficiency of the proposed approach is evaluated by using the Hindmarsh–Rose model time series. To identify the type (excitatory or inhibitory) and direction of a connection, a data-driven model based on clustering and a deep learning model (convolutional neural network) are used. Based on the results, excitatory/inhibitory connections are better detectable using fractional cumulants than using the conventional Granger causality method when other kinds of connections such as magnetic and electrical connections exist among the two regions.