Data-driven Koopman operator approach for computational neuroscience

Marrouch, Natasza; Sławińska, Joanna; Giannakis, Dimitrios; Read, Heather L.

doi:10.1007/s10472-019-09666-2

Cited by 26 publications

(19 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These fields include systems biology, neuroscience, and chemokinetics, among others. These are all areas where KOT has, if at all, only begun to be applied 15,18,59 , and largely remains seen as an exotic method. Second, to our knowledge, TCA has been seen as an approach for representing existing data.…”

Section: Discussionmentioning

confidence: 99%

“…One common goal of these works has been to highlight to researchers in fields that are familiar with PCA but not KOT, such as neuroscience where dimensionality reduction techniques are seen as essential tools, that KMD can give similar and, in certain scenarios, superior mode representations of the data as PCA. Despite this, to date KOT remains a niché and not widely adopted tool in neuroscience 15,18 and other such communities.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Koopman Mode Decomposition and Tensor Component Analysis

Redman

2021

Preprint

View full text Add to dashboard Cite

Koopman mode decomposition and tensor component analysis are two tools that decompose high dimensional data sets into low dimensional modes that capture relevant features and/or dynamics. Despite their similar goal, the two methods are largely used by distinct scientific communities. For the first time, examine the two together and show that, under a certain (reasonable) condition on the data, the theoretical decomposition given by tensor component analysis is the same as that given by Koopman mode decomposition. This realization provides possibilities for new algorithmic approaches to both Koopman mode decomposition and tensor component analysis, new insight into what is being captured by the tensor component analysis modes, and a "bridge" with which the two communities can more effectively communicate.Koopman operator theory (KOT) has emerged as a powerful and general framework with which to understand nonlinear dynamical systems in a data-driven manner. Despite its many strengths, there are numerous scientific fields, such as neuroscience and biology, where it remains under employed. As part of this stems from KOT being formulated mathematically in a distinct way from existing, commonly used methods, such as principal component analysis (PCA), there is a need for bridging KOT with such methods. Here, for the first time, we investigate how a major KOT analysis method differs from tensor component analysis (TCA), an extension of PCA that has quickly become adopted by some of those same fields hesitant on KOT. We show that, in certain scenarios, TCA and KOT, in theory, give the same decomposition of data. This not only makes strides in establishing a bridge between TCA and KOT, but also provides new insight into TCA and new possible algorithms for implementing KOT and TCA.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Koopman Mode Decomposition and Tensor Component Analysis

Redman

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, isostable coordinates are at present limited to situations where a model is available. Obtaining the Koopman operator from data is being researched [77][78][79], which could make possible to recover isostables from data as there is a strong connection between the Koopman operator and isostables [37]. A method to directly obtain isostables from data has very recently been suggested [80].…”

Section: Future Directionsmentioning

confidence: 99%

Optimizing deep brain stimulation based on isostable amplitude in essential tremor patient models

et al. 2021

View full text Add to dashboard Cite

Objective. Deep brain stimulation is a treatment for medically refractory essential tremor. To improve the therapy, closed-loop approaches are designed to deliver stimulation according to the system's state, which is constantly monitored by recording a pathological signal associated with symptoms (e.g. brain signal or limb tremor). Since the space of possible closed-loop stimulation strategies is vast and cannot be fully explored experimentally, how to stimulate according to the state should be informed by modeling. A typical modeling goal is to design a stimulation strategy that aims to maximally reduce the Hilbert amplitude of the pathological signal in order to minimize symptoms. Isostables provide a notion of amplitude related to convergence time to the attractor, which can be beneficial in model-based control problems. However, how isostable and Hilbert amplitudes compare when optimizing the amplitude response to stimulation in models constrained by data is unknown. Approach. We formulate a simple closed-loop stimulation strategy based on models previously fitted to phase-locked deep brain stimulation data from essential tremor patients. We compare the performance of this strategy in suppressing oscillatory power when based on Hilbert amplitude and when based on isostable amplitude. We also compare performance to phase-locked stimulation and open-loop high-frequency stimulation. Main results. For our closed-loop phase space stimulation strategy, stimulation based on isostable amplitude is significantly more effective than stimulation based on Hilbert amplitude when amplitude field computation time is limited to minutes. Performance is similar when there are no constraints, however constraints on computation time are expected in clinical applications. Even when computation time is limited to minutes, closed-loop phase space stimulation based on isostable amplitude is advantageous compared to phase-locked stimulation, and is more efficient than high-frequency stimulation. Significance. Our results suggest a potential benefit to using isostable amplitude more broadly for model-based optimization of stimulation in neurological disorders.

show abstract

“…Bernard Koopman introduced the Koopman operator in [7] and this seminal work became a popular tool with [1] and several other works followed where Koopman operator theory is used in the system identification [3,8], for control design [9,10,11], sensor fusion [12] and analysis of spectrally conjugate systems [13], finding observability gramians or observers [14,15,16], study of chaotic systems [17,18], data-driven information transfer [19,20,21], data-driven based causal inference in dynamical systems [22], in data-driven classification of power system stability [23], and in power system coherency identification, power system stability monitoring [24,25,26,25,27], dynamic state estimation in power networks [28], fault, cyber-attack localization in cyber-physical systems [29,30], computational neuroscience applications [31] and data assimilation for climate forecasting [32].…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Operator Theoretic Methods for Global Phase Space Learning

Nandanoori

Sinha

Yeung

2020

2020 American Control Conference (ACC)

View full text Add to dashboard Cite

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system.We defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator.

show abstract

Data-driven Koopman operator approach for computational neuroscience

Cited by 26 publications

References 57 publications

On Koopman Mode Decomposition and Tensor Component Analysis

On Koopman Mode Decomposition and Tensor Component Analysis

Optimizing deep brain stimulation based on isostable amplitude in essential tremor patient models

Data-Driven Operator Theoretic Methods for Global Phase Space Learning

Contact Info

Product

Resources

About