Publications (255)800.35 Total impact
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ABSTRACT: In statistical data assimilation one evaluates the conditional expected values, conditioned on measurements, of interesting quantities on the path of a model through observation and prediction windows. This often requires working with very high dimensional integrals in the discrete time descriptions of the observations and model dynamics, which become functional integrals in the continuoustime limit. Two familiar methods for performing these integrals include (1) Monte Carlo calculations and (2) variational approximations using the method of Laplace plus perturbative corrections to the dominant contributions. We attend here to aspects of the Laplace approximation and develop an annealing method for locating the variational path satisfying the EulerLagrange equations that comprises the major contribution to the integrals. This begins with the identification of the minimum action path starting with a situation where the model dynamics is totally unresolved in state space, and the consistent minimum of the variational problem is known. We then proceed to slowly increase the model resolution, seeking to remain in the basin of the minimum action path, until a path that gives the dominant contribution to the integral is identified. After a discussion of some general issues, we give examples of the assimilation process for some simple, instructive models from the geophysical literature. Then we explore a slightly richer model of the same type with two distinct time scales. This is followed by a model characterizing the biophysics of individual neurons.  [Show abstract] [Hide abstract]
ABSTRACT: 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 9dimensional Lorenz96 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 Lorenz96 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).  [Show abstract] [Hide abstract]
ABSTRACT: Data assimilation transfers information from an observed system to a physically based model system with state variables x(t). The observations are typically noisy, the model has errors, and the initial state x(t0) is uncertain: the data assimilation is statistical. One can ask about expected values of functions ⟨G(X)⟩ on the path X = {x(t0), ..., x(tm)} of the model state through the observation window tn = {t0, ..., tm}. The conditional (on the measurements) probability distribution P(X) = exp[−A0(X)] determines these expected values. Variational methods using saddle points of the "action" A0(X), known as 4DVar (Talagrand and Courtier, 1987; Evensen, 2009), are utilized for estimating ⟨G(X)⟩. In a path integral formulation of statistical data assimilation, we consider variational approximations in a realization of the action where measurement errors and model errors are Gaussian. We (a) discuss an annealing method for locating the path X0 giving a consistent minimum of the action A0(X0), (b) consider the explicit role of the number of measurements at each tn in determining A0(X0), and (c) identify a parameter regime for the scale of model errors, which allows X0 to give a precise estimate of ⟨G(X0)⟩ with computable, small higherorder corrections.  [Show abstract] [Hide abstract]
ABSTRACT: Information in measurements of a nonlinear dynamical system can be transferred to a quantitative model of the observed system to establish its fixed parameters and unobserved state variables. After this learning period is complete, one may predict the model response to new forces and, when successful, these predictions will match additional observations. This adjustment process encounters problems when the model is nonlinear and chaotic because dynamical instability impedes the transfer of information from the data to the model when the number of measurements at each observation time is insufficient. We discuss the use of information in the waveform of the data, realized through a time delayed collection of measurements, to provide additional stability and accuracy to this search procedure. Several examples are explored, including a few familiar nonlinear dynamical systems and small networks of Colpitts oscillators.  [Show abstract] [Hide abstract]
ABSTRACT: Cardiac rhythm management devices provide therapies for both arrhythmias and resynchronization but not heart failure, which affects millions of patients worldwide. This paper reviews recent advances in biophysics and mathematical engineering that provide a novel technological platform for addressing heart disease and enabling beattobeat adaptation of cardiac pacing in response to physiological feedback. The technology consists of silicon hardware central pattern generators (hCPG) that may be trained to emulate accurately the dynamical response of biological central pattern generators (bCPG). We discuss the limitations of present CPGs and appraise the advantages of analogue over digital circuits for application in bioelectronic medicine. To test the system, we have focused on the cardiorespiratory oscillators in the medulla oblongata that modulate heart rate in phase with respiration to induce respiratory sinus arrhythmia (RSA). We describe here a novel, scalable hCPG comprising physiologically realistic (HodgkinHuxley type) neurones and synapses. Our hCPG comprises two neurones that antagonise each other to provide rhythmic motor drive to the vagus nerve to slow the heart. We show how recent advances in modelling allow the motor output to adapt to physiological feedback such as respiration. In rats, we report on the restoration of RSA using an hCPG that receives diaphragmatic electromyography input and use it to stimulate the vagus nerve at specific time points of the respiratory cycle to slow the heart rate. We have validated the adaptation of stimulation to alterations in respiratory rate. We demonstrate that the hCPG is tuneable in terms of the depth and timing of the RSA relative to respiratory phase. These pioneering studies will now permit an analysis of the physiological role of RSA as well as its any potential therapeutic use in cardiac disease.This article is protected by copyright. All rights reserved  [Show abstract] [Hide abstract]
ABSTRACT: Recent results demonstrate techniques for fully quantitative, statistical inference of the dynamics of individual neurons under the HodgkinHuxley framework of voltagegated 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.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the dynamics of a conductancebased neuron model coupled to a model of intracellular calcium uptake and release by the endoplasmic reticulum. The intracellular calcium dynamics occur on a time scale that is orders of magnitude slower than voltage spiking behavior. Coupling these mechanisms sets the stage for the appearance of chaotic dynamics, which we observe within certain ranges of model parameter values. We then explore the question of whether one can, using observed voltage data alone, estimate the states and parameters of the voltage plus calcium (V+Ca) dynamics model. We find the answer is negative. Indeed, we show that voltage plus another observed quantity must be known to allow the estimation to be accurate. We show that observing both the voltage time course V(t) and the intracellular Ca time course will permit accurate estimation, and from the estimated model state, accurate prediction after observations are completed. This sets the stage for how one will be able to use a more detailed model of V+Ca dynamics in neuron activity in the analysis of experimental data on individual neurons as well as functional networks in which the nodes (neurons) have these biophysical properties.  [Show abstract] [Hide abstract]
ABSTRACT: Utilizing the information in observations of a complex system to make accurate predictions through a quantitative model when observations are completed at time $T$, requires an accurate estimate of the full state of the model at time $T$. When the number of measurements $L$ at each observation time within the observation window is larger than a sufficient minimum value $L_s$, the impediments in the estimation procedure are removed. As the number of available observations is typically such that $L \ll L_s$, additional information from the observations must be presented to the model. We show how, using the time delays of the measurements at each observation time, one can augment the information transferred from the data to the model, removing the impediments to accurate estimation and permitting dependable prediction. We do this in a core geophysical fluid dynamics model, the shallow water equations, at the heart of numerical weather prediction. The method is quite general, however, and can be utilized in the analysis of a broad spectrum of complex systems where measurements are sparse. When the model of the complex system has errors, the method still enables accurate estimation of the state of the model and thus evaluation of the model errors in a manner separated from uncertainties in the data assimilation procedure.  [Show abstract] [Hide abstract]
ABSTRACT: Utilizing the information in observations of a complex system to make accurate predictions through a quantitative model when observations are completed at time $T$, requires an accurate estimate of the full state of the model at time $T$. When the number of measurements $L$ at each observation time within the observation window is larger than a sufficient minimum value $L_s$, the impediments in the estimation procedure are removed. As the number of available observations is typically such that $L \ll L_s$, additional information from the observations must be presented to the model. We show how, using the time delays of the measurements at each observation time, one can augment the information transferred from the data to the model, removing the impediments to accurate estimation and permitting dependable prediction. We do this in a core geophysical fluid dynamics model, the shallow water equations, at the heart of numerical weather prediction. The method is quite general, however, and can be utilized in the analysis of a broad spectrum of complex systems where measurements are sparse. When the model of the complex system has errors, the method still enables accurate estimation of the state of the model and thus evaluation of the model errors in a manner separated from uncertainties in the data assimilation procedure. 
Article: Dynamical estimation of neuron and network properties III: Network analysis using neuron spike times
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ABSTRACT: 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.  [Show abstract] [Hide abstract]
ABSTRACT: 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 timedelay 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.  [Show abstract] [Hide abstract]
ABSTRACT: The authors consider statistical ensemble data assimilation for a onelayer shallowwater equation in a twin experiment: data are generated by an N x N enstrophyconserving grid integration scheme along with an Ekman vertical velocity at the bottom of an Ekman layer driving the flow and Rayleigh and eddy viscosity dissipation damping the flow. Data are generated for N = 16 and the chaotic flow that results is analyzed. This analysis is performed in a pathintegral formulation of the data assimilation problem. These path integrals are estimated by a Monte Carlo method using a Metropolis Hastings algorithm. The authors' concentration is on the number of measurements Lc that must be assimilated by the model to allow accurate estimation of the full state of the model at the end of an observation window. It is found that for this shallowwater flow approximately 70% of the full set of state variables must be observed to accomplish either goal. The number of required observations is determined by examining the number needed to synchronize the observed data Lc and the model output when L data streams are assimilated by the model. Synchronization occurs when L >= Lc and the correct selection of which Lc data are observed is made. If the number of observations is too small, so synchronization does not occur, or the selection of observations does not lead to synchronization of the data with the model output, state estimates during and at the end of the observation window and predictions beyond the observation window are inaccurate. 

Article: Dynamical estimation of neuron and network properties II: Path integral Monte Carlo methods
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ABSTRACT: HodgkinHuxley (HH) models of neuronal membrane dynamics consist of a set of nonlinear differential equations that describe the timevarying 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 timevarying 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.  [Show abstract] [Hide abstract]
ABSTRACT: Neuroscientists often propose detailed computational models to probe the properties of the neural systems they study. With the advent of neuromorphic engineering, there is an increasing number of hardware electronic analogs of biological neural systems being proposed as well. However, for both biological and hardware systems, it is often difficult to estimate the parameters of the model so that they are meaningful to the experimental system under study, especially when these models involve a large number of states and parameters that cannot be simultaneously measured. We have developed a procedure to solve this problem in the context of interacting neural populations using a recently developed dynamic state and parameter estimation (DSPE) technique. This technique uses synchronization as a tool for dynamically coupling experimentally measured data to its corresponding model to determine its parameters and internal state variables. Typically experimental data are obtained from the biological neural system and the model is simulated in software; here we show that this technique is also efficient in validating proposed network models for neuromorphic spikebased very largescale integration (VLSI) chips and that it is able to systematically extract network parameters such as synaptic weights, time constants, and other variables that are not accessible by direct observation. Our results suggest that this method can become a very useful tool for modelbased identification and configuration of neuromorphic multichip VLSI systems.  [Show abstract] [Hide abstract]
ABSTRACT: We present a method for using measurements of membrane voltage in individual neurons to estimate the parameters and states of the voltagegated ion channels underlying the dynamics of the neuron's behavior. Short injections of a complex timevarying 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 [Formula: see text] 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 HodgkinHuxley 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.  [Show abstract] [Hide abstract]
ABSTRACT: 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 HodgkinHuxley 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.  [Show abstract] [Hide abstract]
ABSTRACT: In using data assimilation to import information from observations to estimate parameters and state variables of a model, one must assume a distribution for the noise in the measurements and in the model errors. Using the path integral formulation of data assimilation~ cite{abar2009}, we introduce the idea of self consistency of the distribution of stochastic model errors: the distribution of model errors from the path integral with observed data should be consistent with the assumption made in formulating the the path integral. The path integral setting for data assimilation is discussed to provide the setting for the consistency test. Using two examples drawn from the 1996 Lorenz model, for $D = 100$ and for $D = 20$ we show how one can test for this inconsistency with essential no additional effort than that expended in extracting answers to interesting questions from data assimilation itself. \end{abstract}  [Show abstract] [Hide abstract]
ABSTRACT: 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 discretetime 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 pathintegral 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 © 2010 Royal Meteorological Society  [Show abstract] [Hide abstract]
ABSTRACT: Throughout the brain, neurons encode information in fundamental units of spikes. Each spike represents the combined thresholding of synaptic inputs and intrinsic neuronal dynamics. Here, we address a basic question of spike train formation: how do perithreshold synaptic inputs perturb the output of a spiking neuron? We recorded from single entorhinal principal cells in vitro and drove them to spike steadily at ∼5 Hz (theta range) with direct current injection, then used a dynamicclamp to superimpose strong excitatory conductance inputs at varying rates. Neurons spiked most reliably when the input rate matched the intrinsic neuronal firing rate. We also found a striking tendency of neurons to preserve their rates and coefficients of variation, independently of input rates. As mechanisms for this rate maintenance, we show that the efficacy of the conductance inputs varied with the relationship of input rate to neuronal firing rate, and with the arrival time of the input within the natural period. Using a novel method of spike classification, we developed a minimal Markov model that reproduced the measured statistics of the output spike trains and thus allowed us to identify and compare contributions to the rate maintenance and resonance. We suggest that the strength of rate maintenance may be used as a new categorization scheme for neuronal response and note that individual intrinsic spiking mechanisms may play a significant role in forming the rhythmic spike trains of activated neurons; in the entorhinal cortex, individual pacemakers may dominate production of the regional theta rhythm.
Publication Stats
12k  Citations  
800.35  Total Impact Points  
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Institutions

19702015

University of California, San Diego
 • Department of Physics
 • Marine Physical Laboratory (MPL)
 • Institute for Nonlinear Science (INLS)
 • Scripps Institution of Oceanography (SIO)
San Diego, California, United States


20082012

The Scripps Research Institute
لا هویا, California, United States


2004

Universidad Autónoma de Madrid
Madrid, Madrid, Spain


19922003

CSU Mentor
Long Beach, California, United States


19872001

National University (California)
San Diego, California, United States


1996

University of California, Santa Barbara
 Department of Physics
Santa Barbara, California, United States


19751983

University of California, Berkeley
 • Lawrence Berkeley Laboratory
 • Department of Physics
Berkeley, California, United States


19751978

Fermi National Accelerator Laboratory (Fermilab)
Батавия, Illinois, United States


1976

Stanford University
Palo Alto, California, United States


1974

California Institute of Technology
Pasadena, California, United States


1972

Weizmann Institute of Science
Israel


19651972

Princeton University
 Department of Physics
Princeton, New Jersey, United States
