Data Assimilation Methods for Neuronal State and Parameter Estimation

Low-dimensional models of single neurons: a review

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as subthreshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type.

Parameter and coupling estimation in small networks of Izhikevich's neurons

Nowadays, experimental techniques allow scientists to have access to large amounts of data. In order to obtain reliable information from the complex systems that produce these data, appropriate analysis tools are needed. The Kalman filter is a frequently used technique to infer, assuming a model of the system, the parameters of the model from uncertain observations. A well-known implementation of the Kalman filter, the unscented Kalman filter (UKF), was recently shown to be able to infer the connectivity of a set of coupled chaotic oscillators. In this work, we test whether the UKF can also reconstruct the connectivity of small groups of coupled neurons when their links are either electrical or chemical synapses. In particular, we consider Izhikevich neurons and aim to infer which neurons influence each other, considering simulated spike trains as the experimental observations used by the UKF. First, we verify that the UKF can recover the parameters of a single neuron, even when the parameters vary in time. Second, we analyze small neural ensembles and demonstrate that the UKF allows inferring the connectivity between the neurons, even for heterogeneous, directed, and temporally evolving networks. Our results show that time-dependent parameter and coupling estimation is possible in this nonlinearly coupled system.

Adaptive unscented Kalman filter for neuronal state and parameter estimation

Data assimilation techniques for state and parameter estimation are frequently applied in the context of computational neuroscience. In this work, we show how an adaptive variant of the unscented Kalman filter (UKF) performs on the tracking of a conductance-based neuron model. Unlike standard recursive filter implementations, the robust adaptive unscented Kalman filter (RAUKF) jointly estimates the states and parameters of the neuronal model while adjusting noise covariance matrices online based on innovation and residual information. We benchmark the adaptive filter's performance against existing nonlinear Kalman filters and explore the sensitivity of the filter parameters to the system being modelled. To evaluate the robustness of the proposed solution, we simulate practical settings that challenge tracking performance, such as a model mismatch and measurement faults. Compared to standard variants of the Kalman filter the adaptive variant implemented here is more accurate and robust to faults.

A data-assimilation approach to predict population dynamics during epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT) is a biological process that plays a central role in embryonic development, tissue regeneration, and cancer metastasis. Transforming growth factor-β (TGFβ) is a potent inducer of this cellular transition, comprising transitions from an epithelial state to partial or hybrid EMT state(s), to a mesenchymal state. Recent experimental studies have shown that, within a population of epithelial cells, heterogeneous phenotypical profiles arise in response to different time- and TGFβ dose-dependent stimuli. This offers a challenge for computational models, as most model parameters are generally obtained to represent typical cell responses, not necessarily specific responses nor to capture population variability. In this study, we applied a data-assimilation approach that combines limited noisy observations with predictions from a computational model, paired with parameter estimation. Synthetic experiments mimic the biological heterogeneity in cell states that is observed in epithelial cell populations by generating a large population of model parameter sets. Analysis of the parameters for virtual epithelial cells with biologically significant characteristics (e.g., EMT prone or resistant) illustrates that these sub-populations have identifiable critical model parameters. We perform a series of in silico experiments in which a forecasting system reconstructs the EMT dynamics of each virtual cell within a heterogeneous population exposed to time-dependent exogenous TGFβ dose and either an EMT-suppressing or EMT-promoting perturbation. We find that estimating population-specific critical parameters significantly improved the prediction accuracy of cell responses. Thus, with appropriate protocol design, we demonstrate that a data-assimilation approach successfully reconstructs and predicts the dynamics of a heterogeneous virtual epithelial cell population in the presence of physiological model error and parameter uncertainty.

Approaches to Parameter Estimation from Model Neurons and Biological Neurons

Model optimization in neuroscience has focused on inferring intracellular parameters from time series observations of the membrane voltage and calcium concentrations. These parameters constitute the fingerprints of ion channel subtypes and may identify ion channel mutations from observed changes in electrical activity. A central question in neuroscience is whether computational methods may obtain ion channel parameters with sufficient consistency and accuracy to provide new information on the underlying biology. Finding single-valued solutions in particular, remains an outstanding theoretical challenge. This note reviews recent progress in the field. It first covers well-posed problems and describes the conditions that the model and data need to meet to warrant the recovery of all the original parameters—even in the presence of noise. The main challenge is model error, which reflects our lack of knowledge of exact equations. We report on strategies that have been partially successful at inferring the parameters of rodent and songbird neurons, when model error is sufficiently small for accurate predictions to be made irrespective of stimulation.

The brainstem connectome database

Connectivity data of the nervous system and subdivisions, such as the brainstem, cerebral cortex and subcortical nuclei, are necessary to understand connectional structures, predict effects of connectional disorders and simulate network dynamics. For that purpose, a database was built and analyzed which comprises all known directed and weighted connections within the rat brainstem. A longterm metastudy of original research publications describing tract tracing results form the foundation of the brainstem connectome (BC) database which can be analyzed directly in the framework neuroVIISAS. The BC database can be accessed directly by connectivity tables, a web-based tool and the framework. Analysis of global and local network properties, a motif analysis, and a community analysis of the brainstem connectome provides insight into its network organization. For example, we found that BC is a scale-free network with a small-world connectivity. The Louvain modularity and weighted stochastic block matching resulted in partially matching of functions and connectivity. BC modeling was performed to demonstrate signal propagation through the somatosensory pathway which is affected in Multiple sclerosis.

Measurement(s)	brainstem
Technology Type(s)	tract tracing metastudy
Factor Type(s)	brain region
Sample Characteristic - Organism	Rattus rattus
Sample Characteristic - Environment	Experimental setup
Sample Characteristic - Location	Germany

Parameter Estimation in the Age of Degeneracy and Unidentifiability

Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.

The E3 ubiquitin ligase adaptor Tango10 links the core circadian clock to neuropeptide and behavioral rhythms

Circadian transcriptional timekeepers in pacemaker neurons drive profound daily rhythms in sleep and wake. Here we reveal a molecular pathway that links core transcriptional oscillators to neuronal and behavioral rhythms. Using two independent genetic screens, we identified mutants of Transport and Golgi organization 10 (Tango10) with poor behavioral rhythmicity. Tango10 expression in pacemaker neurons expressing the neuropeptide PIGMENT-DISPERSING FACTOR (PDF) is required for robust rhythms. Loss of Tango10 results in elevated PDF accumulation in nerve terminals even in mutants lacking a functional core clock. TANGO10 protein itself is rhythmically expressed in PDF terminals. Mass spectrometry of TANGO10 complexes reveals interactions with the E3 ubiquitin ligase CULLIN 3 (CUL3). CUL3 depletion phenocopies Tango10 mutant effects on PDF even in the absence of the core clock gene timeless Patch clamp electrophysiology in Tango10 mutant neurons demonstrates elevated spontaneous firing potentially due to reduced voltage-gated Shaker-like potassium currents. We propose that Tango10/Cul3 transduces molecular oscillations from the core clock to neuropeptide release important for behavioral rhythms.

Robust data assimilation with noise: Applications to cardiac dynamics

Reconstructions of excitation patterns in cardiac tissue must contend with uncertainties due to model error, observation error, and hidden state variables. The accuracy of these state reconstructions may be improved by efforts to account for each of these sources of uncertainty, in particular, through the incorporation of uncertainty in model specification and model dynamics. To this end, we introduce stochastic modeling methods in the context of ensemble-based data assimilation and state reconstruction for cardiac dynamics in one- and three-dimensional cardiac systems. We propose two classes of methods, one following the canonical stochastic differential equation formalism, and another perturbing the ensemble evolution in the parameter space of the model, which are further characterized according to the details of the models used in the ensemble. The stochastic methods are applied to a simple model of cardiac dynamics with fast-slow time-scale separation, which permits tuning the form of effective stochastic assimilation schemes based on a similar separation of dynamical time scales. We find that the selection of slow or fast time scales in the formulation of stochastic forcing terms can be understood analogously to existing ensemble inflation techniques for accounting for finite-size effects in ensemble Kalman filter methods; however, like existing inflation methods, care must be taken in choosing relevant parameters to avoid over-driving the data assimilation process. In particular, we find that a combination of stochastic processes-analogously to the combination of additive and multiplicative inflation methods-yields improvements to the assimilation error and ensemble spread over these classical methods.

Using Data Assimilation for Quantitative Electroencephalography Analysis

We propose a method based on the ensemble Kalman filter (EnKF) together with quantitative electroencephalogram (QEEG) coherence and power spectrum analysis for evaluating changes in brain activity associated with cognitive processes. Such analysis framework has been widely used in the context of data assimilation (DA) in areas such as geosciences, meteorology, and aerospace. However, the use of this approach is less common in neurosciences. In our case, EnKF highlights the spectral contribution of brain signals that are more likely (according to their coherence analysis) to be related to the cognitive process of interest. The power enhancement, due to the cognitive activity, is later validated in the power spectrum analysis by comparing through statistical tests relevant frequency content in two datasets in which assessing the development of cognitive abilities is of interest: the process of getting concentrated and of learning a new skill. Our results show that our DA-based methodology can highlight important frequency characteristics of the electroencephalogram (EEG) data that have been related to different cognitive processes. Hence, our proposal has the potential to understand of neurocognitive phenomena that is tracked through QEEG.

Cell Fate Forecasting: A Data-Assimilation Approach to Predict Epithelial-Mesenchymal Transition

Epithelial-mesenchymal transition (EMT) is a fundamental biological process that plays a central role in embryonic development, tissue regeneration, and cancer metastasis. Transforming growth factor-β (TGFβ) is a potent inducer of this cellular transition, which is composed of transitions from an epithelial state to intermediate or partial EMT state(s) to a mesenchymal state. Using computational models to predict cell state transitions in a specific experiment is inherently difficult for reasons including model parameter uncertainty and error associated with experimental observations. In this study, we demonstrate that a data-assimilation approach using an ensemble Kalman filter, which combines limited noisy observations with predictions from a computational model of TGFβ-induced EMT, can reconstruct the cell state and predict the timing of state transitions. We used our approach in proof-of-concept “synthetic” in silico experiments, in which experimental observations were produced from a known computational model with the addition of noise. We mimic parameter uncertainty in in vitro experiments by incorporating model error that shifts the TGFβ doses associated with the state transitions and reproduces experimentally observed variability in cell state by either shifting a single parameter or generating “populations” of model parameters. We performed synthetic experiments for a wide range of TGFβ doses, investigating different cell steady-state conditions, and conducted parameter studies varying properties of the data-assimilation approach including the time interval between observations and incorporating multiplicative inflation, a technique to compensate for underestimation of the model uncertainty and mitigate the influence of model error. We find that cell state can be successfully reconstructed and the future cell state predicted in synthetic experiments, even in the setting of model error, when experimental observations are performed at a sufficiently short time interval and incorporate multiplicative inflation. Our study demonstrates the feasibility and utility of a data-assimilation approach to forecasting the fate of cells undergoing EMT.

Special Issue from the 2017 International Conference on Mathematical Neuroscience

The ongoing acquisition of large and multifaceted data sets in neuroscience requires new mathematical tools for quantitatively grounding these experimental findings. Since 2015, the International Conference on Mathematical Neuroscience (ICMNS) has provided a forum for researchers to discuss current mathematical innovations emerging in neuroscience. This special issue assembles current research and tutorials that were presented at the 2017 ICMNS held in Boulder, Colorado from May 30 to June 2. Topics discussed at the meeting include correlation analysis of network activity, information theory for plastic synapses, combinatorics for attractor neural networks, and novel data assimilation methods for neuroscience-all of which are represented in this special issue.