Dealing with Unknown Unknowns: Identification and Selection of Minimal Sensing for Fractional Dynamics with Unknown Inputs

Abstract

This paper focuses on analysis and design of timevarying complex networks having fractional order dynamics. These systems are key in modeling the complex dynamical processes arising in several natural and man made systems.Notably, examples include neurophysiological signals such as electroencephalogram (EEG) that captures the variation in potential fields, and blood oxygenation level dependent (BOLD) signal, which serves as a proxy for neuronal activity. Notwithstanding, the complex networks originated by locally measuring EEG and BOLD are often treated as isolated networks and do not capture the dependency from external stimuli, e.g., originated in subcortical structures such as the thalamus and the brain stem. Therefore, we propose a paradigm-shift towards the analysis of such complex networks under unknown unknowns (i.e., excitations). Consequently, the main contributions of the present paper are threefold: (i) we present an alternating scheme that enables to determine the best estimate of the model parameters and unknown stimuli; (ii) we provide necessary and sufficient conditions to ensure that it is possible to retrieve the state and unknown stimuli; and (iii) upon these conditions we determine a small subset of variables that need to be measured to ensure that both state and input can be recovered, while establishing sub-optimality guarantees with respect to the smallest possible subset. Finally, we present several pedagogical examples of the main results using real data collected from an EEG wearable device.

A dynamical systems framework to uncover the drivers of large-scale cortical activity

“…The regularization parameter λ was obtained experimentally (λ = 0.4) while assessing the total mean squared error of the one-step ahead prediction capabilities in a testing subset. Notice that the change of the objective is crucial to ensure that the unknown inputs are close to the real external input to the different ROIs -see more details in (50). Therefore, future work should examine the robustness of our results and test the sensitivity of our observations to different methodological choices, including the dimensions of the input matrix, the size of the system identification window, and the value of the regularization parameter.…”

“…Moreover, the inability of models such as DCM to capture signal variations beyond those caused by the external inputs makes the connectivity estimation highly dependent on the assumed number and form of the inputs (94). In this work, we treat the exogenous inputs to the cortex as unknown in the model, and we simultaneously estimate the internal system parameters and unknown excitations leveraging recent developments in linear systems theory (50). To the best of the authors' knowledge, this is the first use of joint-estimation for an LTI system and its unknown inputs in the context to BOLD dynamics, which allows us to uncover the spatiotemporal structure of the drivers of cortical activity and provides new insights on how the brain responds to the requirements of the ongoing task.…”

Controlling Network Dynamics

“…Under the classical linear time-invariant dynamics, the evolution of the system's states is determined by the current states, namely it is Markovian time dependent. In this issue, Cao et al [28] investigate the controllability of the discrete-time fractional-order linear dynamical networks [45], where the system's non-Markovian time properties are incorporated with longterm memory. They present the trade-offs between the MDNS and the time to control based on the concept of actuation spectrum [46].…”

A Separation Principle for Discrete-Time Fractional-Order Dynamical Systems and Its Implications to Closed-Loop Neurotechnology

“…The data used for these experiments are from subject 11 from the CHB-MIT Scalp EEG database [45]. To achieve this identification, we leveraged the tools developed in [46], which led us to…”