Recent rapid developments in the field of neurotechnology and neurostimulation techniques have enabled millions of people all over the world to lead increasingly normal lives in spite of suffering from neurodegenerative disorders. With technology enabling the growth of such techniques progressing at a never-before-seen pace, it becomes important to understand the myriad issues arising from the interactions between such technologies and humans. Although the concepts of fractional calculus (i.e., differentiation and integration to an arbitrary order) have been around for centuries, only recently has fractional calculus-based modeling and analysis of systems found great success. For instance, it has allowed us to explain phenomena ranging from diffusion to viscoelasticity to the complex spatiotemporal relationships existing between the components of a network of agents. A unifying theme among systems where fractional-order based modeling has found success is the presence of long-term memory with non-exponential power-law dependence of system trajectories. In this dissertation, we focus on recent trends in applying fractional-order based system modeling, analysis, and control of neurophysiological signals like electroencephalogram (EEG) and intracranial EEG (iEEG) in the context of a systems and control framework in order to develop sustainable approaches for future neurotechnologies.
To this effect, we are interested in system identification, state estimation, and closed-loop control of fractional-order dynamical systems. We first consider the problem of least-squares system identification in discrete-time linear fractional-order dynamical systems assuming that the temporal components of the aforementioned systems do not change over time. With the advent of data-driven methods dictating the performance of most cyber-physical systems in the real world, it becomes a matter of great interest to characterize the finite-sample complexity properties of the above methods, i.e., the number of input-output samples needed for robust system identification. The working of our approach is then demonstrated on real-world EEG data.
We then consider the problem of resilient forecasting of the states of a discrete-time fractional-order dynamical system when the sensor measurements are corrupted by adversarial noise and also when the noise associated with the dynamics and measurement processes does not have any particular stochastic characterization, but are, instead, deterministic but unknown. Neurophysiological signals such as EEG often exhibit the presence of artifacts, which can be seen as (passive) adversarial noise often associated with external events. Furthermore, restricting the disturbances associated with the state evolution and measurement processes to be Gaussian, white, and having known variances often proves to be unrealistic. Therefore, we first propose necessary and sufficient conditions that enable us to guarantee when resilient state estimation can be performed even in the presence of the said artifacts and provide a state estimation algorithm using the techniques of compressive sensing. Next, we detail the construction of a minimum-energy estimator for discrete-time fractional-order dynamical systems, one that produces an estimate of the system state that is ``most consistent'' with the dynamics and measurement updates of the system. Pedagogical examples, as well as results using real-world EEG data, are presented to show the efficacy of both approaches.
Finally, we turn our attention to the problem of closed-loop design of controllers and estimators for discrete-time fractional-order dynamical systems. There is concrete evidence of closed-loop neurotechnologies based on fractional-order based modeling greatly improving the quality-of-life of patients suffering from neurodegenerative disorders such as dementia, Alzheimer's disease, or Parkinson's disease. Such neurotechnologies require the efficient design of closed-loop controllers and estimators to predict state evolution and regulation under (possibly) partial measurements. We propose a separation principle for deterministic discrete-time fractional-order dynamical systems, which guarantees the independent design of closed-loop controllers and estimators for such systems, which, in turn, enables us to design model-based feedback mechanisms that can stabilize these highly dynamic processes. We also illustrate the application of the separation principle in designing controllers and estimators for these systems in the context of real neurophysiological data. We then elucidate the development of a model predictive control (MPC) framework for discrete-time fractional-order dynamical systems. This framework is then utilized in order to develop real-time prediction and control capabilities that aid the development of closed-loop neurostimulation strategies. We finally demonstrate the usefulness of our framework in suppressing the overall duration and strength of epileptic seizures using a host of models currently used by the neuroscience community to simulate seizures as well as real-world data from subjects undergoing seizures.;