Parameter estimation of neuron models using in-vitro and in-vivo electrophysiological data

Abstract

Spiking neuron models can accurately predict the response of neurons to somatically injected currents if the model parameters are carefully tuned. Predicting the response of in-vivo neurons responding to natural stimuli presents a far more challenging modeling problem. In this study, an algorithm is presented for parameter estimation of spiking neuron models. The algorithm is a hybrid evolutionary algorithm which uses a spike train metric as a fitness function. We apply this to parameter discovery in modeling two experimental data sets with spiking neurons; in-vitro current injection responses from a regular spiking pyramidal neuron are modeled using spiking neurons and in-vivo extracellular auditory data is modeled using a two stage model consisting of a stimulus filter and spiking neuron model.

Keywords: auditory neurons; evolutionary algorithms; parameter estimation; spike train metrics; spiking neurons.

Figure 1

Schematic of the flow of the STRF-Neuron cascade model optimization algorithm. The input stimulus, s(τ, ω), is a spectro-temporal representation of a sound. It is convolved with a STRF, h, which forms the input to an aEIF neuron. The predicted spike train, $\tilde{u}$, is then compared against the training data and validation data spike trains using the van Rossum distance, d(u, $\tilde{u}$). The population results on the training data are then used to evolve either the population of STRFs or the population of aEIF parameter sets, P, independently. This is done in tandem and in real simulations, a regime of alternating between 50 iterations of evolving each population was used. The result against the validation data is used for tracking convergence and for convergence criteria; the algorithm can be stopped after a maximum number of specified iterations or when d(u, $\tilde{u}$) < ϵ then we have a sufficiently accurate solution.

Figure 2

An illustration of the effect of filter width on van Rossum metric; (A) Shows two spike trains, one in black, the other in blue, (B) Shows the functions obtained by filtering the spike trains with a causal exponential with a time constant of τ = 10 ms, and (C) Shows the same trains filtered with time constant of 50 ms, 5 × τ.

Figure 3

Sample of data from the in-vitro L5 pyrmadial neuron experiment. (A) Shows 2 s of current input which was injected during the experiments. (B) Is the corresponding voltage trace from one of the 13 experimental trials.

Figure 4

Illustration of stimulus and response from an example in-vivo cell. (A) Shows the raw sound waveform of 2 s of Zebra Finch song, (B) shows a spectrographic representation of the sound, the y axis indicating the frequency in kHz, and (C) shows the peri-stimulus time histogram of the response of the cell.

Figure 5

Convergence of the van Rossum distance and coincidence factor in model finding runs. These simulations used synthetic model data as targets. The convergence was measured and averaged over 20 identical runs. The algorithm used the van Rossum distance, d, between the experimental and model spike trains, u and $\tilde{u}$, as the fitness function. The van Rossum distance is normalized by its initial value. (A) shows the average performance of the algorithm. (B) shows the convergence behavior of the coincidence factor with different timescale choice in three cases; using a long timescale (T/2), a short timescale (1/f) and a varying timescale, decreasing from an initial value of T/2 down to 1/f. The dotted lines show the standard deviation from the mean (solid) lines.

Figure 6

(A) Shows the average number of iterations to reach and not again fall below 90% of the maximum coincidence factor achieved on that run plotted with the maximum achieved coincidence factors for each neuron. We chose to plot the data in this way because the coincidence factor does not increase monotonically with decreasing van Rossum distance. (B) Shows the evolution of average population best coincidence factor averaged across trials for each model used for modeling the in-vitro data set. Note that the coincidence factor does not strictly increase because it is not the fitness function, but rather, the van Rossum distance is. The dotted lines indicate the standard deviation from the mean (solid) lines.

Figure 7

Performance of the full model on 110 cell data sets plotted against measures of the reliability of the data under two metrics. (A) Shows the best coincidence factor plotted against the intrinsic reliability for each cell. (B) Shows the best van Rossum distance plotted against the cluster size—the average inter-trial van Rossum distance of the experimental data set. ⁺refer to the individual data points; each ⁺corresponds to an individual cell in the cohort of cells studied.

Figure 8

Comparison of model predictions and normalized reverse correlation predictions on the validation set under three metrics; (A) the coincidence factor and (B) the van Rossum distance. The average values of the coincidence factor scaled by the intrinsic reliability were Γ = 0.76 ± 0.08 for the STRF-aEIF models and Γ = 0.61 ± 0.05 for the STRF-poisson model. The STRF-aEIF model had an average better coincidence factor in 70.6% of cells. The average van Rossum distances were 1.39 ± 0.05 for the STRF-aEIF model and 1.50 ± 0.03 for the STRF-Poisson model with the STRF-aEIF performing better in 59% of cases. ⁺refer to the individual data points; each ⁺corresponds to an individual cell in the cohort of cells studied.

Figure 9

Histograms and scatter plots of aEIF parameters obtained from the STRF-aEIF model optimization runs. The ratios on the x-axis in panels (B,C) were chosen as they were presented as criterion for classifying the firing type of aEIF neurons in Naud et al. (2008). The scatter plots show the ratio of threshold to reset V_T/Vr, the ratio of time constants τ_m/τ_w and (A) shows the slope factor of the exponential term, Δ_T, in the aEIF model. (D–F) show sample scatter plots of some of the parameters plotted against each other which were plotted to see if there may be any correlations present.

Figure 10

Raster plot showing 2 s of spike trains in response to Zebra Finch conspecific song. The first spike train, in red with the arrow next to it, is a spike train generated by the cascade model. The 12 black spike trains are individual trials from experimental recordings The cell is “pupi01414_10,” taken from the CRCNS aa-2 data set (Theunissen et al., 2011).