Inferring the Dynamics of Ionic Currents from Recursive Piecewise Data Assimilation of Approximate Neuron Models

We construct neuron models from data by transferring information from an observed time series to the state variables and parameters of Hodgkin-Huxley models. When the learning period completes, the model will predict additional observations and its parameters uniquely characterize the complement of ion channels. However, the assimilation of biological data, as opposed to model data, is complicated by the lack of knowledge of the true neuron equations. Reliance on guessed conductance models is plagued with multivalued parameter solutions. Here, we report on the distributions of parameters and currents predicted with intentionally erroneous models, overspecified models, and an approximate model fitting hippocampal neuron data. We introduce a recursive piecewise data assimilation algorithm that converges with near-perfect reliability when the model is known. When the model is unknown, we show model error introduces correlations between certain parameters. The ionic current waveforms reconstructed from these parameters are excellent predictors of true currents and carry a higher degree of confidence, greater than $95.5\%$, than underlying parameters, which is $53\%$. Unexpressed ionic currents are correctly filtered out even in the presence of mild model error. When the model is unknown, the covariance eigenvalues of parameter estimates are found to be a good gauge of model error. Our results suggest that biological information may be retrieved from data by focusing on current estimates rather than parameters.

Figure 1

RPDA algorithm. (a) Algorithm flowchart. (b) The RPDA algorithm synchronizes the membrane voltage variable $x_1$ to the time series data $V_{\text{mem}}$ in blocks of $M$ discrete time points. The algorithm reinjects $V_{\text{mem}}$ in the model at the beginning of each block (red circle). (c) Example of recursive convergence starting from $M_0=6$ and increasing the block size to $M=N$. This follows two “false starts” at $M_0=2$ and $M_0=4$.

Figure 2

Neuron oscillations and assimilation ranges. Membrane voltage (black trace) is generated by forward integration of the injected current protocol (blue trace) with the original RVLM model. The $w_1,\cdots ,w_{11}$ are the 11 assimilation windows used to generate the data in Table 1 from well-posed problems. Each window is 200 ms wide (yellow bands) and offset from the next by 40 ms. The top panel shows the 41 assimilation windows ${\omega }_1,\cdots ,{\omega }_{41}$ used to the generate the data in Table 2 from ill-posed problems. These windows are also 200 ms long but offset by 2 ms so that all encompass the same action potential labeled by an asterisk at 88 ms.

Figure 3

Covariance maps of parameters inferred with the RVLM, ErrM1, and ErrM2 models. (a) Eigenvalues of the covariance matrix of parameters estimated with erroneous models ErrM1 (blue trace) and ErrM2 (red trace). The eigenvalues of the well-posed RVLM model (black trace) are shown for reference. Covariance maps of the (b) RVLM, (c) ErrM1, and (d) ErrM2 parameters.

Figure 4

Comparison of ionic currents predicted by erroneous and exact models. (a) Current waveforms predicted by the RVLM model. The minimum and maximum in the Na and K traces give the range of variation of currents reconstructed from the 41 assimilation windows. For reference, the true Na and K current waveforms predicted by the original RVLM model are shown as the black dashed lines in (b)–(f). Current waveforms are predicted by erroneous models (b) ErrM1; (c) ErrM2, an overspecified RVLM model; (d) ErrM3, an overspecified erroneous model; and (e) ErrM4 and from noisy data (f) Noisy.

Figure 5

Uncertainty on currents and parameters estimated from a hippocampal CA1 neuron. (a) Membrane voltage oscillations (black trace) driven by injecting a calibrated current waveform (blue trace) into a hippocampal neuron (Wistar rat, CA1 neuron). Voltage oscillations were predicted by the hippocampal neuron model completed with one set of parameters. The asterisk labels the action potential whose ionic current waveforms we predict in (b). (b) Minimum and maximum current waveforms (Na and K) predicted by 71 CA1 models completed with the parameters from 71 assimilation windows. (c) Eigenvalues of the $70\times 70$ covariance matrix of estimated parameters.