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. 2016 Sep 8:6:32749.
doi: 10.1038/srep32749.

Automatic Construction of Predictive Neuron Models through Large Scale Assimilation of Electrophysiological Data

Alain Nogaret  1 , C Daniel Meliza  2 , Daniel Margoliash  3 , Henry D I Abarbanel  4   5
Affiliations

Automatic Construction of Predictive Neuron Models through Large Scale Assimilation of Electrophysiological Data

Alain Nogaret et al. Sci Rep. .

Abstract

We report on the construction of neuron models by assimilating electrophysiological data with large-scale constrained nonlinear optimization. The method implements interior point line parameter search to determine parameters from the responses to intracellular current injections of zebra finch HVC neurons. We incorporated these parameters into a nine ionic channel conductance model to obtain completed models which we then use to predict the state of the neuron under arbitrary current stimulation. Each model was validated by successfully predicting the dynamics of the membrane potential induced by 20-50 different current protocols. The dispersion of parameters extracted from different assimilation windows was studied. Differences in constraints from current protocols, stochastic variability in neuron output, and noise behave as a residual temperature which broadens the global minimum of the objective function to an ellipsoid domain whose principal axes follow an exponentially decaying distribution. The maximum likelihood expectation of extracted parameters was found to provide an excellent approximation of the global minimum and yields highly consistent kinetics for both neurons studied. Large scale assimilation absorbs the intrinsic variability of electrophysiological data over wide assimilation windows. It builds models in an automatic manner treating all data as equal quantities and requiring minimal additional insight.

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Figures

Figure 1
Figure 1. Assimilation-prediction method.
The injected current waveform formula image and resultant time series membrane voltage formula image of the epoch to assimilate (Epoch 0) are input into the nonlinear optimization filter (IPOPT). These provide the equality constraints. The user specifies the inequality constraints by choosing the upper and lower boundaries of the parameter search intervals, pL and pU. IPOPT outputs the state variables x*(t) solution of the minimization problem at each point of the assimilation window, together with the parameter solutions p*. The extracted parameters p* are inserted in the model equations to construct the completed model. The state of the neuron xpredict(t) is predicted by integrating the current protocol formula image forward from initial conditions x*(0) using a fifth order, adaptive step size, Runge-Kutta solver (RK5). The model is validated by comparing the predicted membrane voltage formula image with the voltage formula image recorded in Epoch e.
Figure 2
Figure 2. Membrane voltage oscillations used to extract the model parameters of two HVC neurons (Epoch 0).
The experimental voltage formula image (black line) was recorded under current clamp stimulation by current waveform formula image (blue line). N1 is a putative RA-projecting neuron (top panel) and N2 is a X-projecting neuron (bottom panel). The assimilation window used to extract the parameters spans the interval [0–1600 ms] for N1 and [190 ms–1090 ms] for N2. The membrane voltage solution of the constrained optimization problem is V *(t) (green line). The membrane voltage predicted by integrating the experimental current waveform is formula image (red line). Details of the oscillations of N2 are plotted in supplementary Figure S7.
Figure 3
Figure 3. Validation of completed models by comparing predicted membrane voltages (red lines) with experimentally measured voltages (black lines).
The current protocols in Epochs 1–4 (blue lines) include a broad range of waveforms and timescales which are distinct from those used to assimilate data. Each current protocol was integrated from the origin onwards to obtain the predicted voltage (red line). The same set of parameters was used to predict all Epochs of N1 (resp. N2).
Figure 4
Figure 4. Experimental and theoretical voltage output by Neuron 1 under stimulation by longer current steps.
A depolarizing current step is applied between 100 ms and 300 ms. Its amplitude increases from 50 pA (Epoch 5) to 130 pA (Epoch 9) in steps of 20 pA. A constant hyperpolarizing current of −80 pA is applied between 500 ms and 700 ms in all epochs. The predicted voltage (right column) is obtained by integrating the 5 current protocols with the completed model of N1.
Figure 5
Figure 5. Effect of the choice of initial conditions on model predictions.
The initial values of state variables at the beginning of forward integration were obtained by two methods: data assimilation (green and violet lines) and by assuming steady state conditions at the beginning of the epoch (red line). Assimilation of the first 20 ms of data yielded initial conditions at t = 0 (green line) and at t = 20 ms (violet line). Panel A shows the differences in neuron oscillations induced by different initial conditions over the first 300 ms. These differences however vanish with time. Panel B shows that all predictions become identical near the end of the epoch, irrespective of initial conditions.
Figure 6
Figure 6
(a) Covariance matrix of the random vector formula image of N1. The dashed arrows indicate the sloppy parameters with the largest off-diagonal matrix elements. (b) Spectrum of eigenvalues of the covariance matrix of N1 (black squares) and N2 (red circles) which measure the length of the principal axes of the data misfit ellipsoid (inset). (c) Standard deviations of activation and inactivation voltage thresholds and recovery times of neuron N1 (square symbols) and N2 (circles). (d) Maximum likelihood expectation values of the activation and inactivation thresholds of N1 (dark squares) and N2 (red circles). The error bars are the standard deviations. (e) Stationary values (m∞, h∞) and time constants (τm, τh) of the activation and inactivation variables of NaT, NaP, K1 and K2 ion currents for neurons N1 and N2. These were calculated using the maximum likelihood expectation of parameters.
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