Browsing by Author "Rubchinsky, Leonid"

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    A-Optimal Subsampling For Big Data General Estimating Equations
    (2019-08) Cheung, Chung Ching; Peng, Hanxiang; Rubchinsky, Leonid; Boukai, Benzion; Lin, Guang; Al Hasan, Mohammad
    A significant hurdle for analyzing big data is the lack of effective technology and statistical inference methods. A popular approach for analyzing data with large sample is subsampling. Many subsampling probabilities have been introduced in literature (Ma, \emph{et al.}, 2015) for linear model. In this dissertation, we focus on generalized estimating equations (GEE) with big data and derive the asymptotic normality for the estimator without resampling and estimator with resampling. We also give the asymptotic representation of the bias of estimator without resampling and estimator with resampling. we show that bias becomes significant when the data is of high-dimensional. We also present a novel subsampling method called A-optimal which is derived by minimizing the trace of some dispersion matrices (Peng and Tan, 2018). We derive the asymptotic normality of the estimator based on A-optimal subsampling methods. We conduct extensive simulations on large sample data with high dimension to evaluate the performance of our proposed methods using MSE as a criterion. High dimensional data are further investigated and we show through simulations that minimizing the asymptotic variance does not imply minimizing the MSE as bias not negligible. We apply our proposed subsampling method to analyze a real data set, gas sensor data which has more than four millions data points. In both simulations and real data analysis, our A-optimal method outperform the traditional uniform subsampling method.
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    IUPUI Center for Mathematical Biosciences
    (Office of the Vice Chancellor for Research, 2010-04-09) Boukai, Benzion; Chin, Ray; Dziubek, Andrea; Fokin, Vladimir; Ghosh, Samiran; Kuznetsov, Alexey; Li, Fang; Li, Jiliang; Rader, Andrew; Rubchinsky, Leonid; Sarkar, Jyotirmoy; Guidoboni, Giovanna; Worth, Robert; Zhu, Luoding
    At-Large Mission: “to serve as an umbrella center for spearheading research and programmatic activities in the general bio-mathematics area” • promote and facilitate faculty excellence in mathematical and Computational research in the biosciences; • provide a mechanism and an environment that fosters collaborative research activities across the mathematical sciences and the life and health sciences schools at IUPUI— specifically with the IUSOM; • provide foundations and resources for further strategic development in targeted areas of mathematical and computational biosciences research; and • create greater opportunities and increase competitiveness in seeking and procuring extramural funding.
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    Mathematical Models of Basal Ganglia Dynamics
    (2013-07-12) Dovzhenok, Andrey A.; Rubchinsky, Leonid; Kuznetsov, Alexey; Its, Alexander R.; Worth, Robert; Mukhin, Evgeny
    Physical and biological phenomena that involve oscillations on multiple time scales attract attention of mathematicians because resulting equations include a small parameter that allows for decomposing a three- or higher-dimensional dynamical system into fast/slow subsystems of lower dimensionality and analyzing them independently using geometric singular perturbation theory and other techniques. However, in most life sciences applications observed dynamics is extremely complex, no small parameter exists and this approach fails. Nevertheless, it is still desirable to gain insight into behavior of these mathematical models using the only viable alternative – ad hoc computational analysis. Current dissertation is devoted to this latter approach. Neural networks in the region of the brain called basal ganglia (BG) are capable of producing rich activity patterns. For example, burst firing, i.e. a train of action potentials followed by a period of quiescence in neurons of the subthalamic nucleus (STN) in BG was shown to be related to involuntary shaking of limbs in Parkinson’s disease called tremor. The origin of tremor remains unknown; however, a few hypotheses of tremor-generation were proposed recently. The first project of this dissertation examines the BG-thalamo-cortical loop hypothesis for tremor generation by building physiologically-relevant mathematical model of tremor-related circuits with negative delayed feedback. The dynamics of the model is explored under variation of connection strength and delay parameters in the feedback loop using computational methods and data analysis techniques. The model is shown to qualitatively reproduce the transition from irregular physiological activity to pathological synchronous dynamics with varying parameters that are affected in Parkinson’s disease. Thus, the proposed model provides an explanation for the basal ganglia-thalamo-cortical loop mechanism of tremor generation. Besides tremor-related bursting activity BG structures in Parkinson’s disease also show increased synchronized activity in the beta-band (10-30Hz) that ultimately causes other parkinsonian symptoms like slowness of movement, rigidity etc. Suppression of excessively synchronous beta-band oscillatory activity is believed to suppress hypokinetic motor symptoms in Parkinson’s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson’s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronized dynamics used to study the delayed feedback stimulation. The second project of this dissertation explores the action of delayed feedback stimulation on partially synchronous oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces the fine temporal structure of the synchronous dynamics observed experimentally. Modeling results suggest that delayed feedback DBS in Parkinson’s disease may boost rather than suppresses synchronization and is therefore unlikely to be clinically successful. Single neuron dynamics may also have important physiological meaning. For instance, bistability – coexistence of two stable solutions observed experimentally in many neurons is thought to be involved in some short-term memory tasks. Bistability that occurs at the depolarization block, i.e. a silent depolarized state a neuron enters with excessive excitatory input was proposed to play a role in improving robustness of oscillations in pacemaker-type neurons. The third project of this dissertation studies what parameters control bistability at the depolarization block in the three-dimensional conductance-based neuronal model by comparing the reduced dopaminergic neuron model to the Hodgkin-Huxley model of the squid giant axon. Bifurcation analysis and parameter variations revealed that bistability is mainly characterized by the inactivation of the Na+ current, while the activation characteristics of the Na+ and the delayed rectifier K+ currents do not account for the difference in bistability in the two models.
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    Modeling Temporal Patterns of Neural Synchronization: Synaptic Plasticity and Stochastic Mechanisms
    (2020-08) Zirkle, Joel; Rubchinsky, Leonid; Kuznetsov, Alexey; Arciero, Julia; Barber, Jared
    Neural synchrony in the brain at rest is usually variable and intermittent, thus intervals of predominantly synchronized activity are interrupted by intervals of desynchronized activity. Prior studies suggested that this temporal structure of the weakly synchronous activity might be functionally significant: many short desynchronizations may be functionally different from few long desynchronizations, even if the average synchrony level is the same. In this thesis, we use computational neuroscience methods to investigate the effects of (i) spike-timing dependent plasticity (STDP) and (ii) noise on the temporal patterns of synchronization in a simple model. The model is composed of two conductance-based neurons connected via excitatory unidirectional synapses. In (i) these excitatory synapses are made plastic, in (ii) two different types of noise implementation to model the stochasticity of membrane ion channels is considered. The plasticity results are taken from our recently published article, while the noise results are currently being compiled into a manuscript. The dynamics of this network is subjected to the time-series analysis methods used in prior experimental studies. We provide numerical evidence that both STDP and channel noise can alter the synchronized dynamics in the network in several ways. This depends on the time scale that plasticity acts on and the intensity of the noise. However, in general, the action of STDP and noise in the simple network considered here is to promote dynamics with short desynchronizations (i.e. dynamics reminiscent of that observed in experimental studies) over dynamics with longer desynchronizations.
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    Modeling the Origin of Parkinsonian Tremor
    (Office of the Vice Chancellor for Research, 2011-04-08) Dovzhenok, Andrei; Rubchinsky, Leonid
    Even though much is known about the biophysics, anatomy and physiology of basal ganglia networks, the cellular and network basis of parkinsonian tremor remains an open question. Multiple experimental data suggest that the physiological origin of parkinsonian tremor is different from the physiological origin of other parkinsonian motor symptoms. However, the exact origin of the tremor genesis in Parkinson’s disease remains unknown. A large body of experimental evidence supports the hypothesis, that the tremor arises due to pathological interaction of potentially oscillatory cells within the loop formed by basal ganglia and thalamocortical circuits. We suggest a model of this circuitry, which helps to clarify this potential mechanism of tremor genesis.