Dynamical estimation of neuron and network properties II: path integral Monte Carlo methods

Kostuk, Mark; Toth, Bryan; Meliza, C. Daniel; Margoliash, Daniel; Abarbanel, Henry D. I.

doi:10.1007/s00422-012-0487-5

Cited by 45 publications

(46 citation statements)

References 11 publications

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“…That exact formula is a path integral through model states encountered while observations are made, and in this article we have focused on the saddle path approximation to that integral representation of answers to data assimilation questions. More direct approaches to the evaluation of the desired path integral are in preparation (Kostuk et al 2011). …”

Section: Discussionmentioning

confidence: 99%

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Dynamical estimation of neuron and network properties I: variational methods

et al. 2011

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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 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.

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Section: Discussionmentioning

confidence: 99%

“…The direct evaluation of the path integral using Monte Carlo methods will be addressed using the same examples here in a companion paper (Kostuk et al 2011). …”

Section: Discussionmentioning

confidence: 99%

Dynamical estimation of neuron and network properties I: variational methods

et al. 2011

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“…Nonlinear extensions to DCM, incorporating quadratic terms, have been proposed as well recently [87]. State and parameter estimation has also been attempted in (noisy) nonlinear biophysical models [88; 89], but these approaches are usually computationally expensive, especially when based on numerical sampling [89], while at the same time pursuing objectives somewhat different from those targeted here (i.e., less focused on computational properties). A very recent article by Whiteway & Butts [90] discusses an approach closely related to the present one in that it also assumed piecewise linear latent states (or, ‘rectified linear units (ReLU)’).…”

Section: Discussionmentioning

confidence: 99%

A state space approach for piecewise-linear recurrent neural networks for identifying computational dynamics from neural measurements

Durstewitz

2017

PLoS Comput Biol

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The computational and cognitive properties of neural systems are often thought to be implemented in terms of their (stochastic) network dynamics. Hence, recovering the system dynamics from experimentally observed neuronal time series, like multiple single-unit recordings or neuroimaging data, is an important step toward understanding its computations. Ideally, one would not only seek a (lower-dimensional) state space representation of the dynamics, but would wish to have access to its statistical properties and their generative equations for in-depth analysis. Recurrent neural networks (RNNs) are a computationally powerful and dynamically universal formal framework which has been extensively studied from both the computational and the dynamical systems perspective. Here we develop a semi-analytical maximum-likelihood estimation scheme for piecewise-linear RNNs (PLRNNs) within the statistical framework of state space models, which accounts for noise in both the underlying latent dynamics and the observation process. The Expectation-Maximization algorithm is used to infer the latent state distribution, through a global Laplace approximation, and the PLRNN parameters iteratively. After validating the procedure on toy examples, and using inference through particle filters for comparison, the approach is applied to multiple single-unit recordings from the rodent anterior cingulate cortex (ACC) obtained during performance of a classical working memory task, delayed alternation. Models estimated from kernel-smoothed spike time data were able to capture the essential computational dynamics underlying task performance, including stimulus-selective delay activity. The estimated models were rarely multi-stable, however, but rather were tuned to exhibit slow dynamics in the vicinity of a bifurcation point. In summary, the present work advances a semi-analytical (thus reasonably fast) maximum-likelihood estimation framework for PLRNNs that may enable to recover relevant aspects of the nonlinear dynamics underlying observed neuronal time series, and directly link these to computational properties.

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“…Two generally useful approaches for evaluating this kind of high dimensional integral are Laplace's method [24,25] and the collection of techniques using Monte Carlo sampling [15,18,19,21,26].…”

Section: B the Goal Of Sdamentioning

confidence: 99%

Precision annealing Monte Carlo methods for statistical data assimilation and machine learning

Fang

Wong

Hao

et al. 2020

Phys. Rev. Research

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In statistical data assimilation (SDA) and supervised machine learning (ML), we wish to transfer information from observations to a model of the processes underlying those observations. For SDA, the model consists of a set of differential equations that describe the dynamics of a physical system. For ML, the model is usually constructed using other strategies. In this paper, we develop a systematic formulation based on Monte Carlo sampling to achieve such information transfer. Following the derivation of an appropriate target distribution, we present the formulation based on the standard Metropolis-Hasting (MH) procedure and the Hamiltonian Monte Carlo (HMC) method for performing the high dimensional integrals that appear. To the extensive literature on MH and HMC, we add (1) an annealing method using a hyperparameter that governs the precision of the model to identify and explore the highest probability regions of phase space dominating those integrals, and (2) a strategy for initializing the state space search. The efficacy of the proposed formulation is demonstrated using a nonlinear dynamical model with chaotic solutions widely used in geophysics.

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Dynamical estimation of neuron and network properties II: path integral Monte Carlo methods

Cited by 45 publications

References 11 publications

Dynamical estimation of neuron and network properties I: variational methods

Dynamical estimation of neuron and network properties I: variational methods

A state space approach for piecewise-linear recurrent neural networks for identifying computational dynamics from neural measurements

Precision annealing Monte Carlo methods for statistical data assimilation and machine learning

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