Correcting observation model error in data assimilation.

Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications, the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models and on a single-column radiative transfer model with multicloud parameterization.

Tracking intracellular dynamics through extracellular measurements.

Extracellular recordings of neuronal cells are frequently a part of in vitro and in vivo experimental studies as a means of monitoring network-level dynamics. Their connections to intracellular dynamics are not well understood. Single-unit recordings are a more direct way to measure intracellular dynamics, but are typically difficult and expensive. On the other hand, simple differential equations models exist for single neurons. In this article, we apply a recent advance in data assimilation theory, designed to correct bias in general observation functions, toward the reconstruction of model-based intracellular dynamics from extracellular recordings.

Detection of Bladder Contractions From the Activity of the External Urethral Sphincter in Rats Using Sparse Regression.

Bladder overactivity and incontinence and dysfunction can be mitigated by electrical stimulation of the pudendal nerve applied at the onset of a bladder contraction. Thus, it is important to predict accurately both bladder pressure and the onset of bladder contractions. We propose a novel method for prediction of bladder pressure using a time-dependent spectrogram representation of external urethral sphincter electromyographic (EUS EMG) activity and a least absolute shrinkage and selection operator regression model. There was a statistically significant improvement in prediction of bladder pressure compared with methods based on the firing rate of EUS EMG activity. This approach enabled prediction of the onset of bladder contractions with 91% specificity and 96% sensitivity and may be suitable for closed-loop control of bladder continence.

Forecasting and Uncertainty Quantification Using a Hybrid of Mechanistic and Non-mechanistic Models for an Age-Structured Population Model.

In this paper, we present a new method for the prediction and uncertainty quantification of data-driven multivariate systems. Traditionally, either mechanistic or non-mechanistic modeling methodologies have been used for prediction; however, it is uncommon for the two to be incorporated together. We compare the forecast accuracy of mechanistic modeling, using Bayesian inference, a non-mechanistic modeling approach based on state space reconstruction, and a novel hybrid methodology composed of the two for an age-structured population data set. The data come from cannibalistic flour beetles, in which it is observed that the adults preying on the eggs and pupae result in non-equilibrium population dynamics. Uncertainty quantification methods for the hybrid models are outlined and illustrated for these data. We perform an analysis of the results from Bayesian inference for the mechanistic model and hybrid models to suggest reasons why hybrid modeling methodology may enable more accurate forecasts of multivariate systems than traditional approaches.

Adaptive filtering for hidden node detection and tracking in networks.

The identification of network connectivity from noisy time series is of great interest in the study of network dynamics. This connectivity estimation problem becomes more complicated when we consider the possibility of hidden nodes within the network. These hidden nodes act as unknown drivers on our network and their presence can lead to the identification of false connections, resulting in incorrect network inference. Detecting the parts of the network they are acting on is thus critical. Here, we propose a novel method for hidden node detection based on an adaptive filtering framework with specific application to neuronal networks. We consider the hidden node as a problem of missing variables when model fitting and show that the estimated system noise covariance provided by the adaptive filter can be used to localize the influence of the hidden nodes and distinguish the effects of different hidden nodes. Additionally, we show that the sequential nature of our algorithm allows for tracking changes in the hidden node influence over time.

Hybrid modeling and prediction of dynamical systems.

Scientific analysis often relies on the ability to make accurate predictions of a system's dynamics. Mechanistic models, parameterized by a number of unknown parameters, are often used for this purpose. Accurate estimation of the model state and parameters prior to prediction is necessary, but may be complicated by issues such as noisy data and uncertainty in parameters and initial conditions. At the other end of the spectrum exist nonparametric methods, which rely solely on data to build their predictions. While these nonparametric methods do not require a model of the system, their performance is strongly influenced by the amount and noisiness of the data. In this article, we consider a hybrid approach to modeling and prediction which merges recent advancements in nonparametric analysis with standard parametric methods. The general idea is to replace a subset of a mechanistic model's equations with their corresponding nonparametric representations, resulting in a hybrid modeling and prediction scheme. Overall, we find that this hybrid approach allows for more robust parameter estimation and improved short-term prediction in situations where there is a large uncertainty in model parameters. We demonstrate these advantages in the classical Lorenz-63 chaotic system and in networks of Hindmarsh-Rose neurons before application to experimentally collected structured population data.

Time-Dependent Increase in Network Response to Stimulation.

In vitro neuronal cultures have become a popular method with which to probe network-level neuronal dynamics and phenomena in controlled laboratory settings. One of the key dynamics of interest in these in vitro studies has been the extent to which cultured networks display properties indicative of learning. Here we demonstrate the effects of a high frequency electrical stimulation signal in training cultured networks of cortical neurons. Networks receiving this training signal displayed a time-dependent increase in the response to a low frequency probing stimulation, particularly in the time window of 20-50 ms after stimulation. This increase was found to be statistically significant as compared to control networks that did not receive training. The timing of this increase suggests potentiation of synaptic mechanisms. To further investigate this possibility, we leveraged the powerful Cox statistical connectivity method as previously investigated by our group. This method was used to identify and track changes in network connectivity strength.

Predicting chaotic time series with a partial model.

Methods for forecasting time series are a critical aspect of the understanding and control of complex networks. When the model of the network is unknown, nonparametric methods for prediction have been developed, based on concepts of attractor reconstruction pioneered by Takens and others. In this Rapid Communication we consider how to make use of a subset of the system equations, if they are known, to improve the predictive capability of forecasting methods. A counterintuitive implication of the results is that knowledge of the evolution equation of even one variable, if known, can improve forecasting of all variables. The method is illustrated on data from the Lorenz attractor and from a small network with chaotic dynamics.

Dynamic steering of in vitro cortical neurons using field stimulation.

Neurological disorders are often characterized by abnormal neuronal activity. In the case of epilepsy, this can manifest itself in the form of uncontrolled synchronous activity often in the form of bursting. Pattern steering is the ability to apply stimulation to a network that effectively changes its dynamical firing pattern. In an epileptic network, the stimulation would be used to move the seizing network from its abnormal state to a normal state. This idea is explored here in cultured networks of cortical neurons plated on microelectrode arrays. Stimulation was applied to the bath resulting in an electric field generated throughout the network. This field was verified as sub-threshold in strength using a finite element model simulation. Stimulated networks showed a significant suppression in the number of bursts and increase in the interburst interval as compared to control networks. This observed burst suppression suggests that the sub-threshold stimulating field moved networks from a state of high frequency bursting to a state of low frequency bursting.

Real-time tracking of neuronal network structure using data assimilation.

A nonlinear data assimilation technique is applied to determine and track effective connections between ensembles of cultured spinal cord neurons measured with multielectrode arrays. The method is statistical, depending only on confidence intervals, and requiring no form of arbitrary thresholding. In addition, the method updates connection strengths sequentially, enabling real-time tracking of nonstationary networks. The ensemble Kalman filter is used with a generic spiking neuron model to estimate connection strengths as well as other system parameters to deal with model mismatch. The method is validated on noisy synthetic data from Hodgkin-Huxley model neurons before being used to find network connections in the neural culture recordings.

Differential responses to ω-agatoxin IVA in murine frontal cortex and spinal cord derived neuronal networks.

ω-Agatoxin-IVA is a well known P/Q-type Ca(2+) channel blocker and has been shown to affect presynaptic Ca(2+) currents as well postsynaptic potentials. P/Q-type voltage gated Ca(2+) channels play a vital role in presynaptic neurotransmitter release and thus play a role in action potential generation. Monitoring spontaneous activity of neuronal networks on microelectrode arrays (MEAs) provides an important tool for examining this neurotoxin. Changes in extracellular action potentials are readily observed and are dependent on synaptic function. Given the efficacy of murine frontal cortex and spinal cord networks to detect neuroactive substances, we investigated the effects of ω-agatoxin on spontaneous action potential firing within these networks. We found that networks derived from spinal cord are more sensitive to the toxin than those from frontal cortex; a concentration of only 10nM produced statistically significant effects on activity from spinal cord networks whereas 50 nM was required to alter activity in frontal cortex networks. Furthermore, the effects of the toxin on frontal cortex are more complex as unit specific responses were observed. These manifested as either a decrease or increase in action potential firing rate which could be statistically separated as unique clusters. Administration of bicuculline, a GABAA inhibitor, isolated a single response to ω-agatoxin, which was characterized by a reduction in network activity. These data support the notion that the two clusters detected with ω-agatoxin exposure represent differential responses from excitatory and inhibitory neuronal populations.

Detecting connectivity changes in neuronal networks.

We develop a method from semiparametric statistics (Cox, 1972) for the purpose of tracking links and connection strengths over time in a neuronal network from spike train data. We consider application of the method as implemented in Masud and Borisyuk (2011), and evaluate its use on data generated independently of the Cox model hypothesis, in particular from a spiking model of Izhikevich in four different dynamical regimes. Then, we show how the Cox method can be used to determine statistically significant changes in network connectivity over time. Our methodology is demonstrated using spike trains from multi-electrode array measurements of networks of cultured mammalian spinal cord cells.