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Kadakia, N, Rey D, Ye J, Abarbanel HDI.  2017.  Symplectic structure of statistical variational data assimilation. Quarterly Journal of the Royal Meteorological Society. 143:756-771.   10.1002/qj.2962   Abstract

Data assimilation variational principles (4D-Var) exhibit a natural symplectic structure among the state variables x(t) and. x(t). We explore the implications of this structure in both Lagrangian coordinates {x(t), x(t)} andHamiltonian canonical coordinates {x(t), p(t)} through a numerical examination of the chaotic Lorenz 1996 model in ten dimensions. We find that there are a number of subtleties associated with discretization, boundary conditions, and symplecticity, suggesting differing approaches when working in the the Lagrangian versus the Hamiltonian description. We investigate these differences in detail, and accordingly develop a protocol for searching for optimal trajectories in a Hamiltonian space. We find that casting the problem into canonical coordinates can, in some situations, considerably improve the quality of predictions.

Kadakia, N, Armstrong E, Breen D, Morone U, Daou A, Margoliash D, Abarbanel HDI.  2016.  Nonlinear statistical data assimilation for HVCRA neurons in the avian song system. Biological Cybernetics. 110:417-434.   10.1007/s00422-016-0697-3   AbstractWebsite

With the goal of building a model of the HVC nucleus in the avian song system, we discuss in detail a model of HVCRA projection neurons comprised of a somatic compartment with fast Na+ and K+ currents and a dendritic compartment with slower Ca2+ dynamics. We show this model qualitatively exhibits many observed electrophysiological behaviors. We then show in numerical procedures how one can design and analyze feasible laboratory experiments that allow the estimation of all of the many parameters and unmeasured dynamical variables, given observations of the somatic voltage V-s(t) alone. A key to this procedure is to initially estimate the slow dynamics associated with Ca, blocking the fast Na and K variations, and then with the Ca parameters fixed estimate the fast Na and K dynamics. This separation of time scales provides a numerically robust method for completing the full neuron model, and the efficacy of the method is tested by prediction when observations are complete. The simulation provides a framework for the slice preparation experiments and illustrates the use of data assimilation methods for the design of those experiments.

Schumann-Bischoff, J, Parlitz U, Abarbanel HDI, Kostuk M, Rey D, Eldridge M, Luther S.  2015.  Basin structure of optimization based state and parameter estimation. Chaos. 25   10.1063/1.4920942   AbstractWebsite

Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here, we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses, and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series, the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e., the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment). (C) 2015 AIP Publishing LLC.

Meliza, CD, Kostuk M, Huang H, Nogaret A, Margoliash D, Abarbanel HDI.  2014.  Estimating parameters and predicting membrane voltages with conductance-based neuron models. Biological Cybernetics. 108:495-516.   10.1007/s00422-014-0615-5   AbstractWebsite

Recent results demonstrate techniques for fully quantitative, statistical inference of the dynamics of individual neurons under the Hodgkin-Huxley framework of voltage-gated conductances. Using a variational approximation, this approach has been successfully applied to simulated data from model neurons. Here, we use this method to analyze a population of real neurons recorded in a slice preparation of the zebra finch forebrain nucleus HVC. Our results demonstrate that using only 1,500 ms of voltage recorded while injecting a complex current waveform, we can estimate the values of 12 state variables and 72 parameters in a dynamical model, such that the model accurately predicts the responses of the neuron to novel injected currents. A less complex model produced consistently worse predictions, indicating that the additional currents contribute significantly to the dynamics of these neurons. Preliminary results indicate some differences in the channel complement of the models for different classes of HVC neurons, which accords with expectations from the biology. Whereas the model for each cell is incomplete (representing only the somatic compartment, and likely to be missing classes of channels that the real neurons possess), our approach opens the possibility to investigate in modeling the plausibility of additional classes of channels the cell might possess, thus improving the models over time. These results provide an important foundational basis for building biologically realistic network models, such as the one in HVC that contributes to the process of song production and developmental vocal learning in songbirds.

Knowlton, C, Meliza CD, Margoliash D, Abarbanel HDI.  2014.  Dynamical estimation of neuron and network properties III: network analysis using neuron spike times. Biological Cybernetics. 108:261-273.   10.1007/s00422-014-0601-y   AbstractWebsite

Estimating the behavior of a network of neurons requires accurate models of the individual neurons along with accurate characterizations of the connections among them. Whereas for a single cell, measurements of the intracellular voltage are technically feasible and sufficient to characterize a useful model of its behavior, making sufficient numbers of simultaneous intracellular measurements to characterize even small networks is infeasible. This paper builds on prior work on single neurons to explore whether knowledge of the time of spiking of neurons in a network, once the nodes (neurons) have been characterized biophysically, can provide enough information to usefully constrain the functional architecture of the network: the existence of synaptic links among neurons and their strength. Using standardized voltage and synaptic gating variable waveforms associated with a spike, we demonstrate that the functional architecture of a small network of model neurons can be established.

Rey, D, Eldridge M, Kostuk M, Abarbanel HDI, Schumann-Bischoff J, Parlitz U.  2014.  Accurate state and parameter estimation in nonlinear systems with sparse observations. Physics Letters A. 378:869-873.   10.1016/j.physleta.2014.01.027   AbstractWebsite

Transferring information from observations to models of complex systems may meet impediments when the number of observations at any observation time is not sufficient. This is especially so when chaotic behavior is expressed. We show how to use time-delay embedding, familiar from nonlinear dynamics, to provide the information required to obtain accurate state and parameter estimates. Good estimates of parameters and unobserved states are necessary for good predictions of the future state of a model system. This method may be critical in allowing the understanding of prediction in complex systems as varied as nervous systems and weather prediction where insufficient measurements are typical. (C) 2014 Elsevier B.V. All rights reserved.

Kostuk, M, Toth BA, Meliza CD, Margoliash D, Abarbanel HDI.  2012.  Dynamical estimation of neuron and network properties II: path integral Monte Carlo methods. Biological Cybernetics. 106:155-167.   10.1007/s00422-012-0487-5   AbstractWebsite

Hodgkin-Huxley (HH) models of neuronal membrane dynamics consist of a set of nonlinear differential equations that describe the time-varying conductance of various ion channels. Using observations of voltage alone we show how to estimate the unknown parameters and unobserved state variables of an HH model in the expected circumstance that the measurements are noisy, the model has errors, and the state of the neuron is not known when observations commence. The joint probability distribution of the observed membrane voltage and the unobserved state variables and parameters of these models is a path integral through the model state space. The solution to this integral allows estimation of the parameters and thus a characterization of many biological properties of interest, including channel complement and density, that give rise to a neuron's electrophysiological behavior. This paper describes a method for directly evaluating the path integral using a Monte Carlo numerical approach. This provides estimates not only of the expected values of model parameters but also of their posterior uncertainty. Using test data simulated from neuronal models comprising several common channels, we show that short (< 50 ms) intracellular recordings from neurons stimulated with a complex time-varying current yield accurate and precise estimates of the model parameters as well as accurate predictions of the future behavior of the neuron. We also show that this method is robust to errors in model specification, supporting model development for biological preparations in which the channel expression and other biophysical properties of the neurons are not fully known.

Quinn, JC, Abarbanel HDI.  2011.  Data assimilation using a GPU accelerated path integral Monte Carlo approach. Journal of Computational Physics. 230:8168-8178.   10.1016/   AbstractWebsite

The answers to data assimilation questions can be expressed as path integrals over all possible state and parameter histories. We show how these path integrals can be evaluated numerically using a Markov Chain Monte Carlo method designed to run in parallel on a graphics processing unit (GPU). We demonstrate the application of the method to an example with a transmembrane voltage time series of a simulated neuron as an input, and using a Hodgkin-Huxley neuron model. By taking advantage of GPU computing, we gain a parallel speedup factor of up to about 300, compared to an equivalent serial computation on a CPU, with performance increasing as the length of the observation time used for data assimilation increases. (C) 2011 Elsevier Inc. All rights reserved.

Toth, BA, Kostuk M, Meliza CD, Margoliash D, Abarbanel HDI.  2011.  Dynamical estimation of neuron and network properties I: variational methods. Biological Cybernetics. 105:217-237.   10.1007/s00422-011-0459-1   AbstractWebsite

We present a method for using measurements of membrane voltage in individual neurons to estimate the parameters and states of the voltage-gated ion channels underlying the dynamics of the neuron's behavior. Short injections of a complex time-varying current provide sufficient data to determine the reversal potentials, maximal conductances, and kinetic parameters of a diverse range of channels, representing tens of unknown parameters and many gating variables in a model of the neuron's behavior. These estimates are used to predict the response of the model at times beyond the observation window. This method of data assimilation extends to the general problem of determining model parameters and unobserved state variables from a sparse set of observations, and may be applicable to networks of neurons. We describe an exact formulation of the tasks in nonlinear data assimilation when one has noisy data, errors in the models, and incomplete information about the state of the system when observations commence. This is a high dimensional integral along the path of the model state through the observation window. In this article, a stationary path approximation to this integral, using a variational method, is described and tested employing data generated using neuronal models comprising several common channels with Hodgkin-Huxley dynamics. These numerical experiments reveal a number of practical considerations in designing stimulus currents and in determining model consistency. The tools explored here are computationally efficient and have paths to parallelization that should allow large individual neuron and network problems to be addressed.

Quinn, JC, Abarbanel HDI.  2010.  State and parameter estimation using Monte Carlo evaluation of path integrals. Quarterly Journal of the Royal Meteorological Society. 136:1855-1867.   10.1002/qj.690   AbstractWebsite

The process of transferring information from observations of a dynamical system to estimate the fixed parameters and unobserved states of a system model can be formulated as the evaluation of a discrete-time path integral in model state space. The observations serve as a guiding 'potential' working with the dynamical rules of the model to direct system orbits in state space. The path-integral representation permits direct numerical evaluation of the conditional mean path through the state space as well as conditional moments about this mean. Using a Monte Carlo method for selecting paths through state space, we show how these moments can be evaluated and demonstrate in an interesting model system the explicit influence of the role of transfer of information from the observations. We address the question of how many observations are required to estimate the unobserved state variables, and we examine the assumptions of Gaussianity of the underlying conditional probability. Copyright (C) 2010 Royal Meteorological Society

Abarbanel, HDI, Creveling DR, Farsian R, Kostuk M.  2009.  Dynamical state and parameter estimation. SIAM Journal on Applied Dynamical Systems. 8:1341-1381.   10.1137/090749761   AbstractWebsite

We discuss the problem of determining unknown fixed parameters and unobserved state variables in nonlinear models of a dynamical system using observed time series data from that system. In dynamical terms this requires synchronization of the experimental data with time series output from a model. If the model and the experimental system are chaotic, the synchronization manifold, where the data time series is equal to the model time series, may be unstable. If this occurs, then small perturbations in parameters or state variables can lead to large excursions near the synchronization manifold and produce a very complex surface in any estimation metric for those quantities. Coupling the experimental information to the model dynamics can lead to a stabilization of this manifold by reducing a positive conditional Lyapunov exponent (CLE) to a negative value. An approach called dynamical parameter estimation (DPE) addresses these instabilities and regularizes them, allowing for smooth surfaces in the space of parameters and initial conditions. DPE acts as an observer in the control systems sense, and because the control is systematically removed through an optimization process, it acts as an estimator of the unknown model parameters for the desired physical model without external control. Examples are given from several systems including an electronic oscillator, a neuron model, and a very simple geophysical model. In networks and larger dynamical models one may encounter many positive CLEs, and we investigate a general approach for estimating fixed model parameters and unobserved system states in this situation.