Extracting non-linear integrate-and-fire models from experimental data using dynamic I-V curves

Abstract

The dynamic I-V curve method was recently introduced for the efficient experimental generation of reduced neuron models. The method extracts the response properties of a neuron while it is subject to a naturalistic stimulus that mimics in vivo-like fluctuating synaptic drive. The resulting history-dependent, transmembrane current is then projected onto a one-dimensional current-voltage relation that provides the basis for a tractable non-linear integrate-and-fire model. An attractive feature of the method is that it can be used in spike-triggered mode to quantify the distinct patterns of post-spike refractoriness seen in different classes of cortical neuron. The method is first illustrated using a conductance-based model and is then applied experimentally to generate reduced models of cortical layer-5 pyramidal cells and interneurons, in injected-current and injected- conductance protocols. The resulting low-dimensional neuron models-of the refractory exponential integrate-and-fire type-provide highly accurate predictions for spike-times. The method therefore provides a useful tool for the construction of tractable models and rapid experimental classification of cortical neurons.

Fig. 1

Summary of the dynamic I–V curve method and its application to the Wang-Buszáki model. a The derivative of the membrane voltage (top graph), multiplied by the cellular capacitance, is subtracted from the injected current (middle graph) to yield the intrinsic membrane current I _ion (bottom graph). b The intrinsic membrane current I _ion is plotted against the membrane voltage (black symbols). The dynamic I–V curve (red symbols) is obtained by averaging I _ion in small voltage bins. Error bars represent the standard deviation. c Measuring the cellular capacitance. At a fixed subthreshold voltage, the dynamic membrane current has a variance that depends on the value of C used in the calculation; the correct value of C corresponds to the point of minimal variance. d Relating dynamic I–V curves and non-linear integrate-and-fire models. The function (symbols) is plotted as a function of voltage, together with the EIF model fit (red line). Inset: semi-log plot of F(V) with leak current subtracted, showing a nearly exponential run-up. e Spike-triggered dynamic I–V curves. The functions F(V) measured in small time slices after each spike (symbols) are plotted together with the the EIF fit (green) and the pre-spike I–V curve as a reference (red). At early times it is clearly seen that both the conductance and the spike threshold are significantly increased. f Dynamics of the EIF model parameters during the refractory period. The parameters obtained from the fits of the I–V curves in e are plotted as a function of the time since the last spike (symbols) and fitted with exponential functions (green). g Comparison of the prediction of the refractory EIF (rEIF) model (green) with a voltage trace of the Wang-Buzsáki model (black) shows excellent agreement, with 96% of the spikes correctly predicted by the EIF model within a 5ms window

Fig. 2

Application of the dynamic I–V method to layer-5 pyramidal cells. a The intrinsic membrane current is plotted against the membrane voltage (black symbols). The dynamic I–V curve (red) is clearly seen to comprise a linear component in the subthreshold region followed by a sharp activation in the region of spike initiation. Inset: Examination of the variance of I _ion near the resting potential in the absence (black) or presence (red) of injected current suggest that the majority of the variance comes from intrinsic noise. b The function is plotted here (symbols) together with the EIF model fit (red). Inset: The exponential rise of the spike generating current is shown in a semi-log plot of F(V) with the leak currents subtracted. c Histograms of the EIF model parameters for a sample (N = 12) of pyramidal cells, showing considerable heterogeneity in the response properties of neurons in this population. d The cellular capacitance calculated with our optimization method (see text) is compared to the result of the standard current-pulse protocol, showing a good agreement between the two methods. e Spike-triggered dynamic I–V curves. The I–V curves measured in small time slices after a spike are plotted together with the EIF fit (green) and the pre-spike I–V curve as a reference (red). f Post-spike dynamics of the EIF model parameters (symbols) together with the fits to an exponential model. While conductance and spike threshold could be accurately fitted with a single exponential, the variation in the equilibrium potential E _L required two exponential components for a good fit. The spike width Δ _T did not vary significantly for these cells. g Comparison of the prediction of the rEIF model (green) with experimental data shows good agreement in the subthreshold region and in the prediction of spike times. h Summary of the performance of the rEIF model for the 12 cells investigated. Left: Prediction of the firing rate. Top right: Histogram of the performance measure. Bottom right: Voltage distribution for the rEIF model (green) and the experimental data (black). The figure is adapted from (Badel et al. 2008)

Fig. 3

GABAergic cortical interneuronmodels derived using the dynamic I–V methodology. a The function for a fast-spiking interneuron is plotted (symbols) together with the EIF model fit (red). Inset: The exponential rise of the spike generating current is shown in a semi-log plot of F(V) with the leak currents subtracted. b Distribution of the EIF model parameters for a sample (N = 6) of interneurons. The histograms overlap significantly with those of pyramidal cells shown in Fig 2. c Post-spike dynamics of the EIF model parameters (symbols) together with the fits to an exponential model. In the case of cortical interneurons all parameters could be fitted satisfactorily with a single exponential. Note that the transient, post-spike increase in the spike onset V _T (∼4mV) is smaller in this fast-spiking interneuron than that for pyramidals (∼15mV-see Fig. 2). d Comparison of the prediction of the rEIF model (green) with experimental data shows close agreement in the subthreshold region and in the prediction of spike times. e Summary of the performance of the rEIF model for the 6 cells investigated. Top: Prediction of the firing rate. Bottom: Histogram of the performance measure

Fig. 4

Application of the dynamic I–V method with conductance injection. a The function is plotted (symbols) together with the fit (red) to the function (14). Inset: Two exponential components are clearly seen in a semi-log plot of F(V) with the leak currents subtracted. b Post-spike dynamics of the EIF model parameters (symbols) together with the fits to an exponential model. The time-dependent resting potential shows a clear biphasic response as was seen for layer-5 pyramidals in the current-injection protocol. c Comparison of the prediction of the rEIF model (green)with experimental data again shows good agreement in the subthreshold region and in the prediction of spike times