Correcting model error bias in estimations of neuronal dynamics from time series observations

Neuron models built from experimental data have successfully predicted observed voltage oscillations within and beyond training range. A tantalising prospect is the possibility of estimating the unobserved dynamics of ion channels, which is largely inaccessible to experiment, from membrane voltage recordings. The main roadblock here is our lack of knowledge of the equations governing biological neurons which forces us to rely on surrogate models and parameter estimates biassed by model error. Error correction algorithms are therefore needed to infer both observed and unobserved dynamics, and ultimately the actual parameters of a biological neuron. Here we use a recurrent neural network to correct the outputs of a surrogate Hodgkin-Huxley (HH) model. The reservoir-surrogate HH model hybrid was trained on the voltage oscillations of a reference HH model and its driving current waveform. Out of the six reservoir-surrogate model architectures investigated, we identify one that most accurately recovers the reference membrane voltage and ion channel dynamics. The reservoir was thus effective in correcting model error in an externally driven nonlinear oscillator and in reconstructing the dynamics of both observed and unobserved state variables from the reference model mimicking an actual neuron.