arXiv subjects

Sarthak Chatterjee

Publications and source records attributed to Sarthak Chatterjee.

10 recordsLinked to original sources

arXiv2021

Fractional cyber-neural systems -- a brief survey

Neurotechnology has made great strides in the last 20 years. However, we still have a long way to go to commercialize many of these technologies as we lack a unified framework to study cyber-neural systems (CNS) that bring the hardware, software, and the neural system together. Dynamical systems play a key role in developing these technologies as they capture different aspects of the brain and provide insight into their function. Converging evidence suggests that fractional-order dynamical systems are advantageous in modeling neural systems because of their compact representation and accuracy in capturing the long-range memory exhibited in neural behavior. In this brief survey, we provide an overview of fractional CNS that entails fractional-order systems in the context of CNS. In particular, we introduce basic definitions required for the analysis and synthesis of fractional CNS, encompassing system identification, state estimation, and closed-loop control. Additionally, we provide an illustration of some applications in the context of CNS and draw some possible future research directions. Ultimately, advancements in these three areas will be critical in developing the next generation of CNS, which will, ultimately, improve people's quality of life.

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arXiv2021

On Learning Discrete-Time Fractional-Order Dynamical Systems

Discrete-time fractional-order dynamical systems (DT-FODS) have found innumerable applications in the context of modeling spatiotemporal behaviors associated with long-term memory. Applications include neurophysiological signals such as electroencephalogram (EEG) and electrocorticogram (ECoG). Although learning the spatiotemporal parameters of DT-FODS is not a new problem, when dealing with neurophysiological signals we need to guarantee performance standards. Therefore, we need to understand the trade-offs between sample complexity and estimation accuracy of the system parameters. Simply speaking, we need to address the question of how many measurements we need to collect to identify the system parameters up to an uncertainty level. In this paper, we address the problem of identifying the spatial and temporal parameters of DT-FODS. The main result is the first result on non-asymptotic finite-sample complexity guarantees of identifying DT-FODS. Finally, we provide evidence of the efficacy of our method in the context of forecasting real-life intracranial EEG time series collected from patients undergoing epileptic seizures.

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arXiv2021

Discrete-Time Fractional-Order Dynamical Networks Minimum-Energy State Estimation

Fractional-order dynamical networks are increasingly being used to model and describe processes demonstrating long-term memory or complex interlaced dependencies amongst the spatial and temporal components of a wide variety of dynamical networks. Notable examples include networked control systems or neurophysiological networks which are created using electroencephalographic (EEG) or blood-oxygen-level-dependent (BOLD) data. As a result, the estimation of the states of fractional-order dynamical networks poses an important problem. To this effect, this paper addresses the problem of minimum-energy state estimation for discrete-time fractional-order dynamical networks (DT-FODN), where the state and output equations are affected by an additive noise that is considered to be deterministic, bounded, and unknown. Specifically, we derive the corresponding estimator and show that the resulting estimation error is exponentially input-to-state stable with respect to the disturbances and to a signal that is decreasing with the increase of the accuracy of the adopted approximation model. An illustrative example shows the effectiveness of the proposed method on real-world neurophysiological networks.

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arXiv2019

Fractional-Order Model Predictive Control for Neurophysiological Cyber-Physical Systems: A Case Study using Transcranial Magnetic Stimulation

Fractional-order dynamical systems are used to describe processes that exhibit temporal long-term memory and power-law dependence of trajectories. There has been evidence that complex neurophysiological signals like electroencephalogram (EEG) can be modeled by fractional-order systems. In this work, we propose a model-based approach for closed-loop Transcranial Magnetic Stimulation (TMS) to regulate brain activity through EEG data. More precisely, we propose a model predictive control (MPC) approach with an underlying fractional-order system (FOS) predictive model. Furthermore, MPC offers, by design, an additional layer of robustness to compensate for system-model mismatch, which the more traditional strategies lack. To establish the potential of our framework, we focus on epileptic seizure mitigation by computational simulation of our proposed strategy upon seizure-like events. We conclude by empirically analyzing the effectiveness of our method, and compare it with event-triggered open-loop strategies.

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arXiv2019

Analysis of a Generalized Expectation-Maximization Algorithm for Gaussian Mixture Models: A Control Systems Perspective

The Expectation-Maximization (EM) algorithm is one of the most popular methods used to solve the problem of parametric distribution-based clustering in unsupervised learning. In this paper, we propose to analyze a generalized EM (GEM) algorithm in the context of Gaussian mixture models, where the maximization step in the EM is replaced by an increasing step. We show that this GEM algorithm can be understood as a linear time-invariant (LTI) system with a feedback nonlinearity. Therefore, we explore some of its convergence properties by leveraging tools from robust control theory. Lastly, we explain how the proposed GEM can be designed, and present a pedagogical example to understand the advantages of the proposed approach.

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arXiv2019

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

Closed-loop neurotechnology requires the capability to predict the state evolution and its regulation under (possibly) partial measurements. There is evidence that neurophysiological dynamics can be modeled by fractional-order dynamical systems. Therefore, we propose to establish a separation principle for discrete-time fractional-order dynamical systems, which are inherently nonlinear and are able to capture spatiotemporal relations that exhibit non-Markovian properties. The separation principle states that the problems of controller and state estimator design can be done independently of each other while ensuring proper estimation and control in closed-loop setups. Lastly, we illustrate, as proof-of-concept, the application of the separation principle when designing controllers and estimators for these classes of systems in the context of neurophysiological data. In particular, we rely on real data to derive the models used to assess and regulate the evolution of closed-loop neurotechnologies based on electroencephalographic data.

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arXiv2018

Convergence of the Expectation-Maximization Algorithm Through Discrete-Time Lyapunov Stability Theory

In this paper, we propose a dynamical systems perspective of the Expectation-Maximization (EM) algorithm. More precisely, we can analyze the EM algorithm as a nonlinear state-space dynamical system. The EM algorithm is widely adopted for data clustering and density estimation in statistics, control systems, and machine learning. This algorithm belongs to a large class of iterative algorithms known as proximal point methods. In particular, we re-interpret limit points of the EM algorithm and other local maximizers of the likelihood function it seeks to optimize as equilibria in its dynamical system representation. Furthermore, we propose to assess its convergence as asymptotic stability in the sense of Lyapunov. As a consequence, we proceed by leveraging recent results regarding discrete-time Lyapunov stability theory in order to establish asymptotic stability (and thus, convergence) in the dynamical system representation of the EM algorithm.

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arXiv2018

Dealing with State Estimation in Fractional-Order Systems under Artifacts

Fractional-order dynamical systems are used to describe processes that exhibit long-term memory with power-law dependence. Notable examples include complex neurophysiological signals such as electroencephalogram (EEG) and blood-oxygen-level dependent (BOLD) signals. When analyzing different neurophysiological signals and other signals with different origin (for example, biological systems), we often find the presence of artifacts, that is, recorded activity that is due to external causes and does not have its origins in the system of interest. In this paper, we consider the problem of estimating the states of a discrete-time fractional-order dynamical system when there are artifacts present in some of the sensor measurements. Specifically, we provide necessary and sufficient conditions that ensure we can retrieve the system states even in the presence of artifacts. We provide a state estimation algorithm that can estimate the states of the system in the presence of artifacts. Finally, we present illustrative examples of our main results using real EEG data.

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arXiv2017

Cooperative Curve Tracking in Two Dimensions Without Explicit Estimation of the Field Gradient

We design a control law for two agents to successfully track a level curve in the plane without explicitly estimating the field gradient. The velocity of each agent is decomposed along two mutually perpendicular directions, and separate control laws are designed along each direction. We prove that the formation center will converge to the neighborhood of the level curve with the desired level value. The algorithm is tested on some test functions used in optimization problems in the presence of noise. Our results indicate that in spite of the control law being simple and gradient-free, we are able to successfully track noisy planar level curves fast and with a high degree of accuracy.

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arXiv2013

Stacked Patch Antenna With Cross Slot Electronic Band Gap Structure

A cross slotted electronic band gap (EBG) with stacked rectangular patches shorted with a shorting pin is proposed in this paper. The study is being done on how the various parameters are varied by changing the probe feed location. The design is constructed by using stacking of patches, shorting pin and cross slotted EBG to form an optimized antenna design with antenna efficiency of approximately 99.06%. The radiation patterns are given at 2.586 GHz which can be used for wireless communications.

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