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Conceived and designed the experiments: GU SJS. Performed the experiments: GU. Analyzed the data: GU. Wrote the paper: GU SJS.

The authors have declared that no competing interests exist.

Observability of a dynamical system requires an understanding of its state—the collective values of its variables. However, existing techniques are too limited to measure all but a small fraction of the physical variables and parameters of neuronal networks. We constructed models of the biophysical properties of neuronal membrane, synaptic, and microenvironment dynamics, and incorporated them into a model-based predictor-controller framework from modern control theory. We demonstrate that it is now possible to meaningfully estimate the dynamics of small neuronal networks using as few as a single measured variable. Specifically, we assimilate noisy membrane potential measurements from individual hippocampal neurons to reconstruct the dynamics of networks of these cells, their extracellular microenvironment, and the activities of different neuronal types during seizures. We use reconstruction to account for unmeasured parts of the neuronal system, relating micro-domain metabolic processes to cellular excitability, and validate the reconstruction of cellular dynamical interactions against actual measurements. Data assimilation, the fusing of measurement with computational models, has significant potential to improve the way we observe and understand brain dynamics.

To understand a complex system such as the weather or the brain, one needs an exhaustive detailing of the system variables and parameters. But such systems are vastly undersampled from existing technology. The alternative is to employ realistic computational models of the system dynamics to reconstruct the unobserved features. This model based state estimation is referred to as data assimilation. Modern robotics use data assimilation as the recursive predictive strategy that underlies the autonomous control performance of aerospace and terrestrial applications. We here adapt such data assimilation techniques to a computational model of the interplay of excitatory and inhibitory neurons during epileptic seizures. We show that incorporating lower scale metabolic models of potassium dynamics is essential for accuracy. We apply our strategy using data from simultaneous dual intracellular impalements of inhibitory and excitatory neurons. Our findings are, to our knowledge, the first validation of such data assimilation in neuronal dynamics.

A universal dilemma in understanding the brain is that it is complex, multiscale, nonlinear in space and time, and we never have more than partial experimental access to its dynamics. To better understand its function one not only needs to encompass the complexity and nonlinearity, but also estimate the unmeasured variables and parameters of brain dynamics. A parallel comparison can be drawn in weather forecasting

The most prominent of the model based predictor-controller strategies is the Kalman filter (KF)

Our hypothesis is that seizures arise from a complex nonlinear interaction between specific excitatory and inhibitory neuronal sub-types

It has recently been shown that the interrelated dynamics of

Brain dynamics emerge from within a system of apparently unique complexity among the natural systems we observe. Even as multivariable sensing technology steadily improves, the near infinite dimensionality of the complex spatial extent of brain networks will require reconstruction through modeling. Since at present, our technical capabilities restrict us to only one or two variables at a restricted number of sites (such as voltage or calcium), computational models become the “lens” through which we must consider viewing all brain measurements

As a first example of assimilating neural data we used intracellular voltage data from a spiking pyramidal cell (PC) from the Cornu Ammonis region 1 (CA1) of rat hippocampus. Using only the noisy membrane potential measurement,

In (A) we show measured (red) and estimated (black) voltage,

Root mean squared error for measured and estimated

Model inadequacy is an issue of intense research in the data assimilation community – no model does exactly what nature does. To deal with inadequate models, researchers in areas such as meteorology have developed various strategies to account for the inaccuracies in the models for weather forecasting

(A) measured (red), and estimated (black) voltage,

Despite decades of effort neuroscientists lack a unifying dynamical principle for epilepsy. An incomplete knowledge of the neural interactions during seizures makes the quest for unifying principles especially difficult

(A) Measured

In

(A) membrane potential,

Considering the slow time scale of seizure evolution (period of more than 100 seconds in our experiments), we test the importance of slow variables such as ion concentrations for seizure tracking. As shown in

Observed (A) and estimated (B) membrane potential using the model with ion concentrations dynamics. In (C) we show estimated membrane potential using the model without ion concentrations dynamics.

Pyramidal cells and interneurons in the hippocampus reside in different layers with different cell densities. To investigate whether there exist significant differences in the microenvironment surrounding these two cell types we assimilated membrane potential data from OLM interneurons in the hippocampus and reconstructed

Membrane potential measured (red) by whole cell recording from OLM interneurons during spontaneous seizures (A). In (B–D) we show membrane potential,

Since the interaction of neurons determines network patterns of activity, it is within such interactions that we seek unifying principles for epilepsy. To demonstrate that the UKF framework can be utilized to study cellular interactions, we reconstructed the dynamics of one cell type by assimilating the measured data from another cell type in the network. In

Measured (A, red) and estimated (B, black)

There has been intense interest in the neuroscience communities in bringing control-theoretical tools to bear on neuronal encoding and decoding problems

Our conjecture is that the parallels with numerical meteorology are deep. By the turn of the 20th century, it was apparent that the lack of periodicities in weather limited forecasts based on previous state (autoregressive) statistical models, and that integrating the actual equations of motion of the atmosphere would be required. Infeasible initially, the turning point came when integrating such models gave ‘first approximations that bore a recognizable resemblance to the actual motions’

Our findings suggest that an analogous use of biophysical models of neuronal processes using the recursive predictive strategies employed in meteorological data assimilation is now feasible. We are presently exploring such application in frameworks for model-based data assimilation and control of Parkinson's disease

In conclusion, we have demonstrated the feasibility for data assimilation within neuronal networks using detailed biophysical models. In particular, we demonstrated that estimating the neuronal microenvironment and neuronal interactions can be performed by embedding our improving biophysical neuronal models within a model based state estimation framework. This approach can provide a more complete understanding of otherwise incompletely observed neuronal dynamics during normal and pathological brain function.

We used two-compartmental models for the pyramidal cells and interneurons: a cellular compartment and the surrounding extracellular microenvironment. The membrane potentials of both cells were modeled by Hodgkin-Huxley equations containing sodium, potassium, calcium-gated potassium (after-hyperpolarization), and leak currents. For the network model, the two cell types are coupled synaptically and through diffusion of potassium ions in the extracellular space. A schematic of the model is shown in

Potassium is released to the extracellular space and is pumped back to the cell by the ATP-dependent

The membrane potential

Parameter | Value | Description |

Membrane capacitance | ||

Conductance of Sodium Current | ||

Conductance of potassium current | ||

Conductance of afterhyperpolarization current | ||

Conductance of potassium leak current | ||

Conductance of sodium leak current | ||

Conductance of chloride leak current | ||

Time constant of gating variables | ||

Conductance of calcium current | ||

Reversal potential of calcium | ||

Ratio of intracellular to extracellular volume of the cell | ||

Maximum pump strength | ||

Maximum strength of glial uptake | ||

Diffusion constant of extracellular |
||

Extracellular chloride concentration | ||

Intracellular chloride concentration |

The rate equations for the gating variables are

The current equations were augmented with dynamic variables representing the intra- and extracellular ion concentrations (

Given the potassium ion currents

We consider a spherical cell with a radius of

To complete the description of

The intracellular and extracellular

The intracellular

The reversal potentials for

Equation (7) binds the ion concentrations dynamics to the Hodgkin-Huxley equations (1, 2).

The pyramidal cells and OLM interneurons are coupled both synaptically and through extracellular

Where the superscripts

In the case of coupled pyramidal cells and interneurons, the rate equation for

To estimate and track the dynamics of the neuronal networks, we applied a nonlinear ensemble version of the Kalman filter, the unscented Kalman filter (UKF)

Given a function

Applying one step of the dynamics

In our simulations, the state

An iteration of the filter is performed in the following three steps (see

Next, we estimate the cross covariance between

The

We calculate the AIC measure for the two models used in

All simulations were carried out using MATLAB on 2

Estimates of remaining variables for the INs shown in ^{2+}^{+}^{+}

(0.40 MB TIF)

Estimates of remaining variables for the PCs shown in ^{2+}

(0.30 MB TIF)

Estimates of synaptic variables for PCs and INs shown in _{p}_{p}_{i}_{i}_{i}_{i}_{i}_{i}

(0.21 MB TIF)

We extend our heartfelt thanks to Jokubas Ziburkus for his constructive comments on the manuscript and generously providing us with access to the experimental data.