Generalized Integrate-and-Fire model

The GIF model discussed in this study [31, 37] is a leaky integrate-and-fire model augmented with a spike-triggered current η(t), a moving threshold γ(t) and the escape rate mechanism [38, 39] for stochastic spike emission (Fig 1A). This model is able to predict both the spiking activity and the subthreshold dynamics of individual neurons (Fig 1B), and it is flexible enough to capture the behavior of different neuronal cell types [37].

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Fig 1. The GIF model accurately predicts both the subthreshold and the spiking activity of cortical neurons.

(A) Block representation of the GIF model. The membrane acts as a low-pass filter κ(t) on the input current I(t) to produce the modeled potential V(t). The exponential nonlinearity (escape-rate) transforms this voltage into an instantaneous firing intensity λ(t), according to which spikes are generated. Each time a spike is emitted, both a current η(t) and a movement of the firing threshold γ(t) are triggered. (B) The GIF model accurately predicts the occurrence of individual spikes with millisecond precision. To evaluate the predictive power of the GIF model, the response of a L5 pyramidal neuron to a fluctuating input current (top, the horizontal dashed line represents 0 nA) has been recorded intracellularly (middle, black). The same protocol was repeated nine times to assess the reliability of the neural response (bottom, black raster). The GIF model (with parameters extracted using a different dataset) was able to accurately predict both the subthreshold (middle, red) and the spiking response (bottom, red raster) of the cell.

https://doi.org/10.1371/journal.pcbi.1004275.g001

In the model, the subthreshold membrane potential V(t) evolves according to the following differential equation: (1) where the parameters C, g_L and E_L define the passive properties of the neuron, I(t) is the input current and $\{{\widehat{t}}_j\}$ are the spike times. According to Eq 1, the passive properties of the membrane are described by an exponential filter $\kappa (t)=\frac{R}{{\tau }_{\mathrm{\text{m}}}}\text{exp}\left(\frac{-t}{{\tau }_{\mathrm{\text{m}}}}\right)$, with $R=g_{\mathrm{\text{L}}}^{-1}$ being the cell resistance and τ_m = RC being the membrane timescale (Fig 1A). Each time an action potential is fired, an intrinsic current with stereotypical shape η(t) is triggered. By convention, the spike-triggered current η(t) is hyperpolarizing when its amplitude is positive and depolarizing otherwise. Currents triggered by different spikes accumulate and produce spike-frequency adaptation, if η(t) > 0 (or facilitation, if η(t) < 0). The functional shape of η(t) varies among neuron types [37]. Consequently the time course of η(t) is not assumed a priori but is extracted from intracellular recordings. Each time a spike is emitted, the numerical integration is stopped during a short absolute refractory period T_ref and the membrane potential is reset to $V({\widehat{t}}_j+T_{\mathrm{\text{ref}}})=V_{\mathrm{\text{reset}}}$.

Spikes are produced stochastically according to a point process with conditional firing intensity λ(t∣V, V_T), which exponentially depends on the momentary difference between the membrane potential V(t) and the firing threshold V_T(t) [22, 39, 40]: (2) where λ₀ has units of s⁻¹, so that λ(t) is in Hz and ΔV defines the level of stochasticity. According to Eq 2, if ΔV ≠ 0, the probability of a spike to occur at a time $\widehat{t}\in [t;t+\Delta t]$ is given by: (3) In the limit ΔV → 0, the model becomes deterministic and action potentials are emitted at the precise moment when the membrane potential crosses the firing threshold. Importantly, the value of ΔV is extracted from experimental data.

Finally, the dynamics of the firing threshold V_T(t) is given by: (4) where $V_T^{*}$ is a constant and γ(t) describes the stereotypical time course of the firing threshold after the emission of an action potential. Since the contribution of different spikes accumulates, the moving threshold defined in Eq 4 constitutes an additional source of adaptation (or facilitation). Similar to η(t), the functional shape of γ(t) is not assumed a priori but is extracted from intracellular recordings. All model parameters are summarized in Table 1.

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Table 1. List of model parameters and symbols.

https://doi.org/10.1371/journal.pcbi.1004275.t001

GIF model parameter extraction

Given the intracellular voltage response V_data(t) evoked in vitro by a controlled input current I_tr(t), all of the GIF model parameters are extracted from experimental data (training set) using a three-step procedure (Fig 2) that we previously introduced [30, 31]. A detailed description of the fitting procedure can be found in the Materials and Methods section.

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Fig 2. Schematic representation of the procedure used for GIF model parameter extraction.

In Step 1 (first row), the experimental spike train S_data(t) is extracted from the voltage trace V_data(t) using a standard threshold-crossing method (left, dashed line). Parameters related to absolute refractoriness are extracted from the average spike shape (middle). In Step 2 (second row), given the injected current I_tr(t) and the recorded potential V_data, all the parameters θ_sub defining the dynamics of the subthreshold membrane potential (Eq 1) are extracted by performing a least-square multilinear regression on the membrane potential derivative ${\stackrel{.}{V}}_{\mathrm{\text{data}}}(t)$. Since Eq 1 does not describe the membrane potential dynamics during action potentials, all the data close to spikes are discarded. In Step 3 (third row), the subthreshold parameters θ_sub are first used to compute the subthreshold voltage of the model ${\widehat{V}}_{\mathrm{\text{model}}}(t)$. The parameters θ_th defining the dynamics of the firing threshold (left, dashed line) are then extracted by maximizing the probability (i.e., the log-likelihood) that the experimental spike train S_data(t) was produced by the model, given the subthreshold dynamics ${\widehat{V}}_{\mathrm{\text{model}}}(t)$.

https://doi.org/10.1371/journal.pcbi.1004275.g002

In Step 1 (Fig 2, Step 1), the experimental spike train $S_{\mathrm{\text{data}}}=\{{\widehat{t}}_j\}$ is first defined as the collection of instants ${\widehat{t}}_j$ at which V_data(t) crossed a certain threshold from below. The average spike shape V_STA(t) is then obtained by computing the spike-triggered average (STA) of V_data(t). Depending on the cell type (i.e., depending on the average spike shape), the absolute refractory period T_ref is fixed to a certain value and the reset potential is computed as V_reset = V_STA(T_ref). In the GIF model, a period of absolute refractoriness can alternatively be implemented by setting the first milliseconds of the spike-triggered threshold movement γ(t) to very large values. For this reason, as long as T_ref remains smaller than the shortest interspike interval (ISI) observed in the data, its precise value is not critical. A sensible choice is to set T_ref about twice the spike width at half maximum.

In Step 2 (Fig 2, Step 2), the first-order temporal derivative of the experimental voltage ${\stackrel{.}{V}}_{\mathrm{\text{data}}}(t)$ is estimated by finite differences and the parameters θ_sub = {C, g_L, E_L, η(t)} determining the membrane potential dynamics are extracted by fitting Eq 1 on ${\stackrel{.}{V}}_{\mathrm{\text{data}}}(t)$. This is done by exploiting the knowledge of the experimental voltage V_data(t) and the external input I_tr(t). To avoid a priori assumptions on the functional shape of the spike-triggered current, η(t) is expanded in a linear combination of rectangular basis functions. Consequently, optimal parameters minimizing the sum of squared errors between $\stackrel{.}{V}(t)$ and ${\stackrel{.}{V}}_{\mathrm{\text{data}}}(t)$ can be efficiently obtained by solving a multilinear regression problem [22] (cf. Eqs 17–18).

In Step 3 (Fig 2, Step 3), the parameters estimated so far are first used to compute the subthreshold membrane potential of the model ${\widehat{V}}_{\mathrm{\text{model}}}(t)$. For that, Eq 1 is numerically solved by enforcing adaptation currents η(t) at all the observed spike times $\{{\widehat{t}}_j\}$. Given ${\widehat{V}}_{\mathrm{\text{model}}}(t)$, the parameters ${\theta }_{\mathrm{\text{th}}}=\{V_T^{*},\Delta V,\gamma (t)\}$ defining the firing threshold dynamics (cf. Eqs 2–4) are then extracted by maximizing the probability (i.e., the log-likelihood) of the experimental spike train S_data(t) being produced by the GIF model (cf. Eqs 20–21). Similar to η(t), the spike-triggered threshold movement is extracted by expanding γ(t) in a linear combination of rectangular basis functions. Since the parameters λ₀ and $V_T^{*}$ are redundant, λ₀ is fixed to 1 Hz. With the exponential function in Eq 2, the log-likelihood to maximize is guaranteed to be a concave function of θ_th[41] and the optimization problem can be solved using standard gradient ascent techniques. The method used in this last step closely resembles the standard GLM fitting procedure [35, 36]. However, here, by exploiting the information contained in the subthreshold dynamics of the membrane potential, the maximum likelihood approach is specifically used to infer the dynamics of the firing threshold. In contrast to GLMs, the GIF model can consequently disentangle adaptation processes mediated by intrinsic currents and threshold movements.

GIF model validation

To obtain a high-throughput pipeline for GIF model parameter extraction, the method described in the previous section has to be complemented with a validation protocol designed to automatically detect and discard trials in which the fitting procedure fails. Good spiking neuron models should be able to accurately predict the occurrence of individual action potentials with millisecond precision [6]. To take into account the stochastic nature of single neurons [42], we designed a validation protocol based on the measurement of the model performance in predicting spike emission probability. After the acquisition of the training dataset used for parameter extraction, a new set of recordings (test dataset) is performed in which single neurons are stimulated repetitively with a test current I_test(t). The resulting set of experimental spike trains is then compared against a set of spike trains predicted by repetitive simulations of the GIF model. To obtain a quantitative measure of the model’s predictive power, the similarity $M_d^{*}$ [34] between the two sets of spike trains is computed (Materials and Methods). $M_d^{*}$ takes values between 0 and 1, where $M_d^{*}=0$ indicates that the model is unable to predict any of the experimental spikes and $M_d^{*}=1$ indicates a perfect match. Importantly, $M_d^{*}$ avoids the small-sample bias known to occur when measuring the similarity between small groups of spike trains as well as the deterministic bias known to favor noise-free models [34].

Testing GIF model parameter extraction and validation on artificial data

To estimate the amount of data required to perform GIF model parameter extraction, we first tested our fitting procedure on an artificial training set generated by simulating the response of a GIF model to a fluctuating current I(t). The choice of reference parameters (Fig 3A–3D, black) was based on previous results [31]. In particular, both the spike-triggered current η(t) and the threshold movement γ(t) were defined as a linear combination of K = 26 log-spaced rectangular basis functions approximating a power-law decay over 5 seconds [31, 43]. Overall, the reference model had 59 parameters: 31 were related to the subthreshold dynamics and 28 to the firing threshold.

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Fig 3. Validation of the procedure used for GIF model parameter extraction.

(A)-(D) GIF model parameters used to generate the artificial data (black) recovered using a training set of T_tr = 10 s (gray) and T_tr = 100 s (red). Error bars and shaded areas represent one standard deviation obtained using five different data sets. In case of perfect agreement, black lines, gray lines and shaded areas are not visible. (A) Membrane filter κ(t). Inset: membrane timescale τ_m = C/g_L. (B) Spike-triggered current η(t). (C) Spike-triggered movement of the firing threshold γ(t). (D) Reversal potential (E_L, top left); cell resistance ($R=g_{\mathrm{\text{L}}}^{-1}$, top right); threshold baseline ($V_{\mathrm{\text{T}}}^{*}$, bottom left) and threshold sharpness (ΔV, bottom right). (E) Estimation error ϵ_param on model parameters (upper panel), performance on spike-timing prediction $M_d^{*}$ (middle panel) and computing time required for parameter extraction (lower panel) as a function of the training set size T_tr. Gray areas indicate one standard deviation across different artificial datasets generated using the same reference parameters. Gray and red arrows indicate the performance obtained with a training set of 10 s and 100 s, respectively. (F) Reliability of the validation procedure as a function of the number of repetitions n_test and the duration T_test of the test current. For different values of n_test and T_test, $M_d^{*}$ was computed 1000 times using different test currents. Consistent with the result that $M_d^{*}$ corrects the small-sample bias, the mean value of $M_d^{*}=0.998$ (dashed line) obtained across repetitions of different test currents did not depend on n_test and T_test. The continuous lines represent the 0.25-quantiles of the $M_d^{*}$ distribution obtained with n_test = {3,6,9,12,15} (from dark to light gray) and indicate that the reliability of the measure increases with n_test and T_test.

https://doi.org/10.1371/journal.pcbi.1004275.g003

The input current I(t) used to build the artificial training set was generated at ΔT ⁻¹ = 20 kHz by numerically solving the stochastic differential equation $\tau \stackrel{.}{I}=-I+I_0+\sqrt{2\tau }\sigma (t)\xi (t)$ in discrete time (5) where ξ(t) is a Gaussian white-noise process generated by independently sampling from a Normal distribution 𝓝(0,1), τ = 3 ms is the characteristic timescale on which the input fluctuates, I₀ defines the mean input and σ(t) is the time-dependent standard deviation of I(t). Ornstein-Uhlenbeck processes (i.e. stationary filtered Gaussian processes) have been extensively used to model the input current received in vivo at the soma of neocortical neurons [44]. Here, we relaxed the assumption of stationarity by modulating the variance of the input with a periodic oscillation [43] given by: (6) where σ₀ and Δσ are constants and f = 0.2 Hz is the modulation frequency. An input current with non-stationary statistics drives the neurons through different regimes producing broad ISI distributions that better constrain the fit of adaptation processes. The input parameters I₀, σ₀ and Δσ were adjusted to generate an artificial training set in which the GIF model emitted spikes at an average firing rate of 10 Hz oscillating over 5 seconds between around 7 and 13 Hz.

The fitting procedure illustrated in Fig 2 was then applied to recover the reference parameters of the GIF model used to generate the artificial dataset (Fig 3A–3D, black). To estimate the amount of data required to guarantee a high degree of accuracy, this operation was repeated several times by varying the size of the training set T_tr (i.e., the duration of the input current I(t)). Fig 3A–3D shows a comparison between the reference parameters and the results obtained by fitting a training set of T_tr = 10 seconds (gray) and T_tr = 100 seconds (red). Overall, we found that 100 seconds were sufficient to accurately recover the reference parameters. To quantify the accuracy of the fit, we computed the mean error ϵ_param on model parameters as a function of T_tr (see Materials and Methods). The results indicate that the minimum amount of data required for accurate parameter extraction is 30–40 seconds. In particular, we found that 100 seconds were sufficient to limit the error to ϵ_param < 2.0% (Fig 3E, top). The great accuracy with which the fitted model was able to predict the spiking activity of the reference model ($M_d^{*}=0.998$) confirmed the goodness of this fit (Fig 3E, middle). To achieve high-throughput and perform parameter extraction on the fly, it is crucial to minimize the computing time (CPU time) required for the fit. We measured the CPU time as a function of the training set duration T_tr (Fig 3E, bottom) and we found that accurate parameter extraction from a training set of T_tr = 100 seconds requires around 60 seconds of computing. We concluded that GIF model parameter extraction is suitable for high-throughput.

A second time-consuming procedure that has to be analyzed is the validation protocol. To quantify the predictive power of the fitted model, the reference model was stimulated with repetitive injections of a test current I_test(t) generated according to Eqs 5–6. To estimate the number of repetitions n_test and the duration T_test of the test current required to obtain a reliable estimate of the model predictive power, the similarity measure $M_d^{*}$ was computed multiple times using different values of n_test and T_test (Fig 3F). On average, the value of $M_d^{*}$ was independent of both the input current duration and the number of repetitions, confirming that the spike-train metrics $M_d^{*}$ successfully eliminates the small sample bias [34]. We measured the variability of $M_d^{*}$ across validation procedures performed with different realizations of I_test(t) and found that the reliability of $M_d^{*}$ increased with both the number of repetitions n_test and the duration of the test current T_test (Fig 3F). Spike-triggered processes can last for several seconds [31, 43]. This sets a constraint on the minimal duration of both the test current I_test(t) and the interstimulus interval. By taking into account these constraints, we concluded that, while respecting high-throughput constraints, a validation protocol based on nine injections of a 10-second current guarantees a reliable estimation of the model’s predictive power (Fig 3F).

A protocol for automated high-throughput single-neuron characterization

Based on the results reported in the previous section, we designed a protocol for the fit and the validation of GIF models on in vitro intracellular recordings (Fig 4). The protocol is conceptually divided in two phases. In the first part, a training set is acquired by recording the single-neuron response to a fluctuating input I_tr(t) lasting for T_tr = 100 seconds and generated according to Eqs 5–6. These data are then used for parameter extraction. In the second part of the protocol, nine repetitive injections of a new 10-second current I_test(t) are performed with an interstimulus interval of 10 seconds, so as to allow the cell to recover. These data (test set) are then used to quantify the predictive power of the GIF model with the spike-train similarity measure $M_d^{*}$. Since all the computations required for parameter extraction and model validation can be performed on the fly, the whole protocol requires 5 minutes and is suitable for high-throughput.

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Fig 4. Schematic representation of the protocol for high-throughput single-neuron characterization.

To characterize the properties of the electrode required for AEC, the experimental protocol starts with the injection of a short subthreshold current. While the filtering properties of the patch clamp are estimated (AEC box, left), the training dataset is collected. After training set collection, the raw data are preprocessed with AEC (AEC box, right). Then, in parallel with GIF model parameter extraction and successive spike timing prediction, the test dataset is collected by injecting nine repetitions of the same time-dependent current. Finally, after complete acquisition of the test set, the similarity measure $M_{\mathrm{\text{d}}}^{*}$ between the observed and the predicted spike trains is computed. Overall, GIF model parameter extraction and validation requires around five minutes.

https://doi.org/10.1371/journal.pcbi.1004275.g004

Current-clamp experiments in which the same electrode is used both for stimulating and recording from single neurons are biased due to the voltage drop across the electrode [32]. To remove this bias, intracellular recordings are preprocessed using a technique called Active Electrode Compensation (AEC, refs. [32, 33], see Materials and Methods). To perform AEC, the filtering properties of the electrode have to be estimated. For that, an additional 10-second subthreshold current injection is performed before the acquisition of the training set (Fig 4).

Testing the high-throughput protocol on in silico recordings

A different class of models used to describe the electrical activity of individual neurons includes the so called multi-compartment conductance-based models (or detailed biophysical models). In contrast to point-neuron models, detailed biophysical models account for the intricate morphology of both dendritic and axonal arborizations and explicitly describe the dynamics of a large variety of ion channels mediating active currents. Both aspects are likely to play a role in single-neuron information processing [45, 46]. A detailed biophysical model (DBM) has recently been proposed that captures several features of L5b thick-tufted pyramidal neurons [14]. In particular, this model includes active dendrites and describes the interactions between Na⁺-spiking at the soma, back-propagating action potentials and Ca²⁺-spikes generated at the distal apical dendrites.

To validate our procedure for high-throughput single-neuron characterization, the protocol described in Fig 4 was tested in silico by simulating the DBM response to a set of current injections (Fig 5A, see Materials and Methods). The input parameters were calibrated to obtain an average firing rate of 10 Hz with slow rate fluctuations between 7 and 13 Hz. Moreover, to model stochastic spike emission, a source of noise was introduced by corrupting the input current with some additive white-noise (see Materials and Methods). Capturing the DBM spiking response to dendritic injections goes beyond the scope of this study. Since we are ultimately interested in automatic somatic patching, all in silico experiments were preformed by delivering the current at the somatic compartment (Fig 5A). DBM somatic recordings were then used to perform GIF model parameter extraction (Fig 5B–5D). Compared with previous results from in vitro recordings in L5 pyramidal neurons [30, 31], the membrane filter κ(t) was characterized by a relatively short timescale (τ_m = 6.7 ms, s.d. 0.1 ms, Fig 5B). GIF model parameter extraction also revealed the presence of a long-lasting adaptation current (Fig 5C) as well as a long-lasting spike-triggered movement of the firing threshold (Fig 5D). Consistent with the tendency of L5b pyramidal neurons to produce bursts of action potentials (ref. [14] and Fig 5G), the activation of the spike-triggered current was not instantaneous.

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Fig 5. Testing GIF model parameter extraction on in silico recordings from a detailed biophysical model: optimal GIF model parameters.

(A) Reconstructed morphology of the detailed biophysical model (DBM, ref. [14]) used to validate the protocol for high-throughput single-neuron characterization. The recording site is indicated by the red pipette. (B)-(D) GIF model parameters extracted from in silico recordings obtained by simulating the DBM response to a somatic current injection. The filters obtained by averaging the parameters extracted from five independent training sets of T_tr = 100 s each are shown in red. Gray areas indicate one standard deviation. (B) Membrane filter κ(t). Inset: comparison between the membrane timescale extracted using a GIF model and the timescale of the GLM linear filter (c.f., exponential fit of κ_GLM(t) in panel E). Each couple of open circles indicates the timescale extracted from a specific training set. Bar plots represent the mean and one standard deviation across training sets (τ_m = 6.7 ms, s.d. 0.1 ms, GIF; τ_m = 8.9 ms, s.d. 1.3 ms, GLM). (C) Spike-triggered current η(t). Inset: zoom on the first 400 ms. (D) Spike-triggered movement of the firing threshold γ(t). (E)-(F) GLM parameters extracted from the same in silico recordings used to fit the GIF model. Average filters are shown in blue. Gray areas indicate one standard deviation across training sets. (E) Linear filter κ_GLM(t) (blue) and exponential fit (dashed black). For comparison, a rescaled version of the membrane filter κ(t) is shown in red. Inset: same data displayed on semi-logarithmic scales. (F) Spike-history filter h_GLM(t). For comparison, a rescaled version of the GIF model effective filter h(t) (Eq 7) is shown in red. Inset: same data displayed on double-logarithmic scales. (G) ISI distributions computed using the test set data (black line and gray area) the GIF model prediction (red) and the GLM prediction (blue).

https://doi.org/10.1371/journal.pcbi.1004275.g005

According to cable theory [47], the large number of dendritic branches explicitly modeled in the DBM, is expected to manifest itself in a membrane filter κ(t) decaying over multiple timescales. To verify the accuracy of the single-exponential assumption and to compare the GIF model performance against a reference model, we also used the in silico recordings to fit a GLM [35, 36], (Fig 5E and 5F). In the GLM, the linear filter κ_GLM(t) acting on the input current is not assumed a priori to be an exponential function and its shape is extracted from experimental data using a non-parametric method (see Materials and Methods). We found that the GLM filter κ_GLM(s) and the membrane filer κ(t) of the GIF model were in good agreement (Fig 5E), suggesting that complex dendritic morphologies weakly affect temporal integration at the somatic compartment. Further quantitative evidence was provided by fitting κ_GLM(t) with a single exponential function and comparing the resulting timescale against τ_m (Fig 5B, inset). The GLM spike-history filter h_GLM(t) extracted from in silico recordings (Fig 5F) was also in good agreement with the effective adaptation filter h(t) of the GIF model [31, 37]: (7) This result confirmed that h_GLM(t) combines, but cannot disentangle, the effects of the adaptation current η(t) and the movement of the firing threshold γ(t). In contrast to GIF models, GLMs do not model absolute refractoriness with a dead time followed by a voltage reset. This explains why, during the first milliseconds, h_GLM(t) is much larger than h(t) (Fig 5F). Finally, consistent with previous results that in L5 pyramidal neurons spike-frequency adaptation occurs on multiple timescales [31, 43], we noticed that both h(t) and h_GLM(t) were approximatively linear on double logarithmic scales (Fig 5F, inset).

The predictive power of both the GIF model and the GLM was then assessed on a test set obtained by simulating the DBM response to nine repetitive injections of a new 10-second current (Fig 6A). Both models achieved a similar performance and were able to predict around 80% of the spikes emitted by the DBM (temporal precision Δ = 4 ms; $M_d^{*}$ = 0.80, s.d. 0.01, GIF; $M_d^{*}$ = 0.79, s.d. 0.01, GLM; Fig 6B). Compared to the GLM, the GIF model presented two advantages. First, the GIF model, but not the GLM, explicitly modeled the dynamics of the membrane potential and could therefore predict the DBM subthreshold voltage with an average root mean squared error (RMSE) of 3.4 mV, s.d. 0.03 mV (variance explained ϵ_V = 74.3 %, s.d. 1.1%; Fig 6C). Second, the time required to perform parameter extraction was faster for the GIF model than for the GLM (T_CPU = 86 s, GIF; T_CPU = 143 s, GLM).

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Fig 6. Testing GIF model parameter extraction on in silico recordings from a detailed biophysical model: GIF model validation.

(A) Fraction of the input current I_test(t) (top, gray) used for model validation; typical DBM response evoked by a single current injection (middle, black); DBM spiking activity in response to nine repetitive injections of the same input (bottom, black raster); PSTH constructed by averaging the nine spike trains smoothed with a rectangular window of 500 ms (bottom, black line). GIF model and GLM predictions are shown in red and blue, respectively. Dashed black lines represent 0 nA (top) and 0 Hz (bottom). (B)-(D) Performance comparison between GIF model (red) and GLM (blue) in predicting the DBM activity. Parameter extraction and model validation were repeated five times using different datasets. Each couple of open circles indicates the performance obtained by both models on a specific dataset. Bar plots indicate the mean and one standard deviation across repetitions. (B) Spike-timing prediction as quantified by $M_d^{*}$ with precision Δ = 4 ms. (C) Mean prediction error ϵ_V on subthreshold membrane potential fluctuations. The GLM does not explicitly model the subthreshold membrane potential dynamics and is therefore not applicable (N/A). (D) GIF model spike-timing prediction ($M_d^{*}$, with precision Δ = 4 ms) as a function of the training set size used for parameter extraction. Increasing the duration of the training set from 100 s to 120 s does not improve the GIF model predictive power ($M_d^{*}$ = 0.80, s.d. 0.01, T_tr = 100 s; $M_d^{*}$ = 0.80, s.d. 0.01, T_tr = 120 s; n = 10, paired Student t-test, t₄ = 0.05, p = 0.97; n.s. > 0.05).

https://doi.org/10.1371/journal.pcbi.1004275.g006

Repeating the entire protocol by varying the duration of I_tr(t) confirmed that a training set of T_tr = 100 s was sufficient to ensure convergence of the fitting procedure (Fig 6D). Overall, these results suggest that, despite their simplicity, modern point-neuron models are capable of predicting most of the spikes emitted by a detailed biophysical model in response to complex somatic current injections.

Testing the high-throughput protocol on in vitro patch-clamp recordings

To confirm the results reported in the previous section, the protocol for high-throughput single-neuron characterization was further tested using standard current-clamp in vitro recordings from L5 pyramidal neurons (see Materials and Methods). At the beginning of the experiment, the input current was calibrated to obtain an average firing rate of 10 Hz with amplitude fluctuations between 7 and 13 Hz. For that, we set Δσ = 0.5, I₀ = σ₀ and adjusted I₀ in order to obtain an average firing rate of around 10 Hz. While this simple approach works well for L5 pyramidal neurons, different cell types might require a more involved calibration protocol in which I₀ ≠ σ₀. In these cases, an alternative solution consist of: i) temporarily setting I₀ = 0 pA and looking for two values of σ₀, denoted ${\sigma }_0^{\mathrm{\text{min}}}$ and ${\sigma }_0^{\mathrm{\text{max}}}$, giving rise to subthreshold voltage fluctuations σ_V of desired magnitudes (e.g., ${\sigma }_{\mathrm{\text{V}}}^{\mathrm{\text{min}}}\approx 3$ mV and ${\sigma }_{\mathrm{\text{V}}}^{\mathrm{\text{max}}}\approx 7$ mV); ii) set ${\sigma }_0=({\sigma }_0^{\mathrm{\text{min}}}+{\sigma }_0^{\mathrm{\text{max}}})/2$ and $\Delta \sigma =({\sigma }_0^{\mathrm{\text{max}}}-{\sigma }_0^{\mathrm{\text{min}}})/2{\sigma }_0$; iii) adjust I₀ to obtain an average firing rate of around 10 Hz.

Since the same patch-clamp electrode was used to simultaneously stimulate and record from single neurons, the acquired signal V_rec(t) is a biased version of the real membrane potential V_data(t) [32, 33]. This bias is due to the voltage drop V_e(t) across the patch-clamp electrode and was removed using a technique called Active Electrode Compensation (AEC, see Materials and Methods and Fig 7A). In AEC [32, 33], the electrode is modeled as an arbitrarily complex linear filter κ_e(t) estimated at the beginning of the experiment from the optimal linear filter κ_opt(t) between a 10-second subthreshold current I_sub(t) and the recorded response V_sub(t) (Fig 7B). For all subsequent injections, we estimated the voltage drop across the electrode V_e(t) by convolving the input current with the electrode filter κ_e(t) (Fig 7C). We finally recovered the membrane potential V_data(t) by subtracting V_e(t) from the recorded signal V_rec(t) (Fig 7A and 7D): (8)

According to our high-throughput protocol, the training set was compensated only after its complete acquisition. With this strategy, the time-consuming procedure required to estimate the electrode filter can be performed during the acquisition of the training set (see Fig 4), limiting the total duration of the protocol. Consistent with previous results [31], we found that the electrode filter κ_e(t) decayed on a very rapid timescale τ_e = 0.54 ms, s.d. 0.11 ms (Fig 7C). Consequently, AEC acted on the raw data as a low-pass filter with a cutoff frequency of around 2 kHz.

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Fig 7. Data preprocessing: Active Electrode Compensation.

(A) Schematic representation of the Active Electrode Compensation technique used to correct for the experimental bias known to occur when the same patch-clamp electrode is used to simultaneously inject and record from a single neuron. The artifactual voltage V_e(t) across the pipette is estimated by filtering the input current I(t) with the electrode filter κ_e(t). The intracellular membrane potential V_data(t) if finally obtained by subtracting the artifactual voltage V_e(t) from the recorded signal V_rec(t). (B) Typical optimal linear filter κ_opt(t) between the subthreshold input current I_sub(t) and the recorded signal V_sub(t). To estimate the electrode filter, an exponential fit is performed on the tail of κ_opt(t) (dashed black). Inset: Magnification of the y-axis illustrating the good accuracy of the exponential fit (dashed black) on the tail of the optimal linear filter κ_opt(t) (red). (C) Typical electrode filter κ_e(t) obtained by subtracting the exponential fit from the optimal linear filter κ_opt(t) (see panel B). Since in vitro recordings were performed with the standard bridge compensation technique, the electrode filter κ_e(t) is characterized by a strong initial negative peak. The characteristic timescale of the electrode filter τ_e was measured by performing an exponential fit (dashed black) on κ_e(t). Inset: distribution of the electrode timescales τ_e measured in ten different recordings included in this study. (D) Comparison between recorded signal V_rec(t) (black) and membrane potential V_data(t) (red) obtained after AEC. Inset: zoom indicating that AEC operates as a low-pass filter by removing high-frequency components from the acquired signal. Scale bars: 30 ms, 5 mV.

https://doi.org/10.1371/journal.pcbi.1004275.g007

After AEC, the in vitro recordings acquired from ten different L5 pyramidal neurons (Fig 8A) were used to perform GIF model parameter extraction (Fig 8B–8E). All of the extracted parameters were consistent with the ones previously obtained by fitting the GIF model on in vitro recordings from L5 pyramidal neurons responding to a mean-modulated input [31]. The parameters describing the passive properties of the membrane (i.e., the resting membrane potential E_L, the membrane timescale τ_m and the input resistance R) revealed the presence of cell-to-cell variability (Fig 8B and 8E) and were on average consistent with previous results obtained using standard characterization protocols based on current-step injections [48]. Our characterization approach further showed that, in L5 Pyr neurons, spike-frequency adaptation is mediated by a long-lasting adaptation current featuring a power-law decay (Fig 8C; see also ref. [31]), which possibly results from the combined action of multiple channels operating on different timescales [49]. In standard protocols for single-neuron characterization, spike-triggered currents are generally assessed indirectly by measuring the spike after-hyperpolarization (AHP) induced by an artificial current pulse designed to evoke one or more action potentials (see, e.g., refs. [49, 50]). Importantly, with our characterization method the magnitude and the time-course of adaptation currents mediating AHPs can be measured simultaneously along with the other GIF model parameters, while neurons are processing in vivo-like inputs. Finally, our characterization protocol showed the presence of large firing threshold movements triggered by the emission of action potentials and lasting for several hundreds of milliseconds (Fig 8D). The dynamical properties of the firing threshold have been previously shown to be cell-type specific [30, 51] and functionally relevant [52], but are generally not considered in standard characterization protocols.

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Fig 8. Testing the protocol for high-throughput single-neuron characterization on in vitro patch-clamp recordings: optimal GIF model parameters.

(A) Staining of a biocytin-filled L5 pyramidal neuron included in this study. (B)-(E) GIF model parameters extracted from ten L5 pyramidal neurons. Average filters are shown in red. Gray lines show the results from individual neurons. (B) Membrane filter κ(t). Inset: comparison between the characteristic timescale of κ(t) and the slow timescale τ_slow of κ_GLM(t) (see panel F). Each couple of open circles indicates the parameters measured in a single neuron. Bar plots indicate the mean and one standard deviation across neurons. (C) Spike-triggered current η(t) displayed on double-logarithmic scales. (D) Spike-triggered movement of the firing threshold γ(t) displayed on double-logarithmic scales. (E) Histograms of GIF model parameters extracted from ten L5 pyramidal neurons. From left to right: reversal potential, E_L; membrane timescale, τ_m = C/g_L; cell resistance, $R=g_{\mathrm{\text{L}}}^{-1}$; firing threshold baseline, $V_{\mathrm{\text{T}}}^{*}$; firing threshold sharpness, ΔV. (F)-(H) GLM parameters extracted from ten L5 pyramidal neurons. Average filters are shown in blue. Gray areas indicate one standard deviation across neurons. (F) GLM linear filter κ_GLM(t) (blue). For comparison, a rescaled version of the GIF filter κ(t) is shown in red. Inset: same data shown on semi-logarithmic scales. To quantify the slow τ_slow (panel B, inset) and the fast τ_fast timescale of κ_GLM(t), we performed a double exponential fit (not shown for clarity) of κ_GLM(t). (G) GLM spike-history filter h_GLM(t) (blue). For comparison, a rescaled version of the effective GIF adaptation filer h(t) (c.f., Eq 7) is shown in red. (H) Distribution of the GLM parameter E₀ extracted from ten L5 pyramidal neurons.

https://doi.org/10.1371/journal.pcbi.1004275.g008

To allow for a comparison, we also used the in vitro recordings to perform GLM parameter extraction (Fig 8F–8H, see Materials and Methods). Confirming the results reported in the previous section, the effective spike-history filter h(t) of the GIF model obtained by combining the spike-triggered current η(t) and threshold movement γ(t) was in good agreement with the GLM spike-history filter h_GLM(t) (Fig 8G). The linear filters κ_GLM(t) and κ(t) were also in good agreement (Fig 8B and 8F, τ_m = 20.9 ms, s.d. 6.5 ms GIF; τ_slow = 22.5 ms, s.d. 3.0 ms GLM). However, the large values observed in the first two bins of κ_GLM(t) indicated the presence of a second rapid component (τ_fast = 1.9 ms, s.d. 0.5 ms), which is neglected in the GIF model (Fig 8F, inset).

We tested the predictive power of both the GIF model and the GLM on a new set of recordings (test set) in which a test current I_test(t) was repetitively injected (Fig 9A). In terms of mere spike-timing prediction, the GIF model and the GLM achieved similar results ($M_d^{*}=0.79\pm 0.04$, GIF; $M_d^{*}=0.81\pm 0.04$, GLM; Fig 9B). Moreover, the GIF model, but not the GLM, could predict the subthreshold response of real neurons with an RMSE of 3.6 mV, s.d. 0.5 mV (variance explained ϵ_V = 80.1 %, s.d. 4.1%; Fig 9C). These results indicate that the difference observed between the linear filters κ(t) and κ_GLM(t) does not have a major impact on the predictive power of the models.

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Fig 9. Testing the protocol for high-throughput single-neuron characterization on in vitro patch-clamp recordings: GIF model validation.

(A) Input current I_test(t) (top, gray) used for model validation, typical L5 pyramidal neuron response evoked by a single current injection (middle, black) spiking activity observed in response to nine repetitive injections of the same input (bottom, black raster) and PSTH constructed by averaging the nine spike trains within rectangular windows of 500 ms (bottom, black line). GIF model and GLM predictions are shown in red and blue, respectively. (B)-(D) Summary data for the performance of the GIF model and the GLM in predicting the responses of ten L5 pyramidal neurons. Each couple of open circles indicates the performance on an individual cell. Error plots indicate the mean and one standard deviation across neurons. (B) Spike-timing prediction as quantified by $M_d^{*}$ with precision Δ = 4 ms. (C) Mean prediction error ϵ_V on subthreshold membrane potential fluctuations. The GLM does not explicitly model the subthreshold membrane potential dynamics and is therefore not applicable (N/A). (D) GIF model spike-timing prediction ($M_d^{*}$, with precision Δ = 4 ms) as a function of the training set size used for parameter extraction. Increasing the duration of the training set from 100 s to 120 s does not improve the GIF model predictive power ($M_d^{*}$ = 0.79, s.d. 0.04, T_tr = 100 s; $M_d^{*}$ = 0.79, s.d. 0.04, T_tr = 120 s; n = 10, paired Student t-test, t₉ = 0.25, p = 0.8; n.s. > 0.05).

https://doi.org/10.1371/journal.pcbi.1004275.g009

Finally, comparing the predictive power of different GIF models with parameters extracted from five training sets of different durations (T_tr = 10, 30, 60, 100 and 120 s; Fig 9D) confirmed that 100 seconds of intracellular recordings are sufficient to accurately constrain the GIF model parameters. We conclude that our protocol for GIF model parameter extraction is suitable for high-throughput single-neuron characterization.

Ethics Statement

All procedures in this study were conducted in conformity with the Swiss Welfare Act and the Swiss National Institutional Guidelines on Animal Experimentation for the ethical use of animals. The Swiss Cantonal Veterinary Office approved the project following an ethical review by the State Committee for Animal Experimentation.

Electrophysiological recordings

In vitro electrophysiological recordings were performed on 300 μm thick parasagittal acute slices from the right hemispheres of male P13–15 C57Bl/6J mouse brains, which were quickly dissected and sliced (HR2 vibratome, Sigmann Elektronik, Germany) in ice-cold artificial cerebrospinal fluid (ACSF) (in mM: NaCl 124, KCl 2.5, MgCl₂ 10, NaH₂PO₄ 1.25, CaCl₂ 0.5, D-(+)-Glucose 25, NaHC0₃ 25; pH 7.3, s.d. 0.1, aerated with 95% O₂, 5% CO₂), followed by a 15 minute incubation at 34°C in standard ACSF (in mM: NaCl 124, KCl 2.5, MgCl₂ 1, NaH₂PO₄ 1.25, CaCl₂ 2, D-(+)-Glucose 25, NaHC0₃ 25; pH 7.4, aerated with 95% O₂, 5% CO₂), equally used as bath solution. Cells were visualized using infrared differential interference contrast video microscopy (VX55 camera, Till Photonics, Germany and BX51WI microscope, Olympus, Japan). Somatic whole-cell current clamp recordings of layer 5 pyramidal cells in the primary somatosensory cortex were performed at 32, s.d. 1°C with an Axon Multiclamp 700B Amplifier (Molecular Devices, USA) using 6.5–7.5 MΩ borosilicate pipettes, containing (in mM): K⁺–gluconate 110, KCl 10, ATP–Mg²⁺ 4, ${\mathrm{\text{Na}}}_2^{+}$–phosphocreatine 10, GTP–Na⁺ 0.3, HEPES 10, biocytin 5 mg/ml; pH 7.3, 300 mOsm). To ensure intact axonal and dendritic arborisation, recordings were conducted in slices cut parallel to the apical dendrites.

Data were acquired at ΔT ⁻¹ = 20 kHz using an ITC-18 digitising board (InstruTECH, USA) controlled by a custom-written software module operating within IGOR Pro (Wavemetrics, USA). Voltage signals were low-pass filtered (Bessel, 10 kHz) and not corrected for the liquid junction potential. Only cells with an access resistance < 25MΩ (20.2, s.d. 3.2 MΩ, n = 10), which was compensated throughout the recording, and a drift in the resting membrane potential < 2.5 mV (1.2 mV, s.d. 0.8 mV, n = 10) between the start and the end of the recording were retained for further analysis.

In silico recordings: multi-compartmental model simulations

In silico recordings were performed by simulating a multi-compartmental model of aN L5b pyramidal neuron [14]. The model was obtained from Model DB (accession number 139653) and all simulations were performed in Neuron [68]. Similar to the in vitro experiments, input currents I(t) were generated according to Eqs 5–6 (with sampling frequency ΔT ⁻¹ = 20 kHz) and were delivered at the somatic compartment. To obtain an average firing rate fluctuating between 7 and 13 Hz, the input parameters were set to I₀ = 520 pA, σ₀ = 320 pA and Δσ = 0.5. To reproduce spike timing variability between responses to repetitive injections of the same current I(t), a source of noise was included in the model by adding a zero-mean white-noise signal ξ_w.n.(t) to I(t). In order to capture the autocorrelation function between spike trains recorded in vitro in response to different repetitions of the test set current, the magnitude of the noise was set to $\sqrt{\langle {\xi }_{\text{w}.\text{n}.}{\left(t\right)}^2\rangle }=$ 160 pA. The same amount of noise was also used to generate the training dataset. GIF model and GLM parameter extraction were performed by treating the noise current ξ_w.n.(t) as being unknown.

Data preprocessing: Active Electrode Compensation

All the in vitro recordings included in this study were preprocessed using AEC [32] according to the following four-step procedure [31, 33].

Step 1: Shortly before the acquisition of the training dataset (see Fig 4), we recorded the intracellular response V_sub(t) evoked by the injection of a short subthreshold current I_sub(t). The input was generated according to Eq 5 with parameters I₀ = 0 pA, σ(t) = 75 pA and τ = 3 ms and evoked small-amplitude subthreshold fluctuations around the resting potential. With this parameter choice, the standard deviation of V_sub(t) was around 2–3 mV.

Step 2: We then estimated the optimal linear filter κ_opt(t) between the subthreshold input I_sub(t) and the recorded signal V_sub(t) (Fig 7B). To reduce computing time, κ_opt(t) was defined over a finite interval [0,200ms] as (11) with {f^(m)(t)} being a set of M = 202 rectangular basis functions of linearly increasing width. The parameters b = [b₁, …, b_M] determining the shape of κ_opt(t) were then estimated by solving the following multilinear regression: (12) where, using the discrete-time notation x_t = x(t · ΔT) and by removing the subscripts sub for clarity, V is a vector whose t-th element is given by the membrane potential V_t = V(tΔT) and Z is a matrix made of vectors $z_t^T$ defined as: (13) with I_t = I(tΔT).

Step 3: An exponential function f(t;a₁, a₂) = a₁exp(−t/a₂) was then fitted to the tail of κ_opt(t) by minimizing the error $E(a_1,a_2)={\int }_{T_{\mathrm{\text{min}}}}^{\infty }{({\kappa }_{\mathrm{\text{opt}}}(s)-f(s;a_1,a_2))}^{2}ds$ (Fig 7B). In AEC, the electrode is assumed to operate on fast timescales (< 1 ms) and the slow decay in κ_opt(t) is attributed to the cell. For this reason the fit was performed with T_min = 5 ms, and the electrode filter was estimated as (14) with ${\widehat{a}}_1$ and ${\widehat{a}}_2$ being the optimal parameters minimizing E(a₁, a₂). To improve accuracy, Steps 2 and 3 were repeated 15 times by resampling from the available data and the final electrode filter used for AEC was obtained by averaging the results across repetitions (Fig 7C). Alternatively, the electrode filter κ_e(t) can be extracted from κ_opt(t) by considering that also the net current flowing through the cell membrane is affected by the electrode properties (see ref. [32]).

Step 4: Finally, for all subsequent current-clamp injections, the membrane potential V_data(t) was estimated as follows (Fig 7A and 7D): (15) where I_ext(t) is the injected current, V_rec(t) is the recorded signal and the convolution integral on the right-hand side of Eq 15 approximates the voltage drop V_e(t) across the electrode.

Expanding κ_opt(t) in rectangular basis functions drastically reduces the computing time required in Step 2. Overall, Steps 1–3 were performed in around 62 seconds and can in principle be executed while the training set is being acquired. Step 4 requires less than 1 second and can be performed after training set collection without compromising high-throughput (Fig 4). Since in our protocol model validation only relies on spike-timing prediction, AEC only has to be applied to the training dataset. Here, in order to asses the prediction error on the subthreshold membrane potential, we also performed AEC on all test set recordings. In order to evaluate the quality of the recordings, our protocol for high-throughput single-neuron characterization (Fig 4) could in principle be extended by repeating Steps 1–3 after complete acquisition of the test set. These additional data could indeed be used to verify whether the electrode properties (i.e., the access resistance, the electrode timescale and, more generally, the electrode filter κ_e(t)) were satisfactory and sufficiently stable during the experiment (see ref. [31]).

GIF model parameter extraction

Given the intracellular membrane potential V_data(t) measured at a sampling frequency ΔT ⁻¹ in response to a known input current I_tr(t), as well as the spike times $\{{\widehat{t}}_j\}$ defined as instants at which V_data(t) crosses 0 mV from below, all the GIF model parameters are extracted following a three-step procedure [31, 37] (Fig 2).

Step 1: The absolute refractory period T_ref is fixed to an arbitrary value and the voltage reset is estimated by the average membrane potential recorded T_ref milliseconds after a spike $V_{\text{reset}}={\langle V_{\text{data}}({\widehat{t}}_j+T_{\text{ref}})\rangle }_j$. Since absolute refractoriness can be captured by a spike-triggered movement of the firing threshold, the particular choice of T_ref is not crucial and the only constraint is given by the shortest interspike interval in the dataset. Here, for L5 pyramidal neurons, we set the refractory period to T_ref = 4 ms.

Step 2: The parameters determining the subthreshold dynamics of the membrane potential are extracted. To allow convex optimization, the spike-triggered current η(t) is expanded as a linear combination of basis functions [22]: (16) where {η_k} is a set of parameters controlling the time course of η(t). The parameters ${\theta }_{\mathrm{\text{sub}}}^T=C^{-1}·[g_{\mathrm{\text{L}}},E_{\mathrm{\text{L}}}{g}_{\mathrm{\text{L}}},{\eta }_1,...,{\eta }_K,1]$ are then extracted by minimizing the sum of squared errors between the observed voltage derivative ${\stackrel{.}{V}}_{\mathrm{\text{data}}}$ and that of the model (i.e., Eq 1). Since all subthreshold parameters θ_sub act linearly on the observables, this optimization problem can be efficiently solved by computing the following multilinear regression [12, 22]: (17) where X is a matrix whose rows are given by the vectors (18) and ${\stackrel{.}{V}}_{\mathrm{\text{data}}}$ is a column-vector containing the voltage first-order derivative estimated by finite differences ${\stackrel{.}{V}}_{\mathrm{\text{data}}}(t)=(V_{\mathrm{\text{data}}}(t+\Delta T)-V_{\mathrm{\text{data}}}(t))/\Delta T$. Since the GIF model does not capture the subthreshold dynamics during spike initiation, all the data points close to action potentials $\{t\mid t\in [{\widehat{t}}_j-5\mathrm{\text{ms}};{\widehat{t}}_j+T_{\mathrm{\text{ref}}}]\}$ are excluded from the regression.

In principle, the residuals of this multilinear regression could provide some additional information about the quality of the recordings and could therefore be used for online quality control. This is true especially in cases where a large number of experiments are repeated in similar cell types and under similar conditions. In such a situation, one can indeed evaluate the results with respect to the distribution of errors obtained in previous experiments. Although the magnitude of high-frequency noise in patch-clamp recordings is generally low, this metric might however depend on the experimental sampling frequency. Overall, to limit the impact of high-frequency noise, it is generally safer to assess the model performance by computing the residuals on V_data (c.f., Eq 27), rather than on ${\stackrel{.}{V}}_{\mathrm{\text{data}}}$.

Step 3: The parameters defining the dynamics of the firing threshold are extracted. To determine the functional shape of the spike-triggered movement of the firing threshold, we first expand γ(t) as a sum of basis functions: (19)

Given the parameters obtained in the first two steps and the spike times observed in the experiment, the subthreshold membrane potential ${\widehat{V}}_{\mathrm{\text{model}}}(t)$ is then computed by numerical integration of Eq 1. Without loss of flexibility, the parameter λ₀ is fixed to 1 Hz and all threshold parameters ${\theta }_{\mathrm{\text{th}}}^T=\Delta V^{-1}·[1,V_{\mathrm{\text{T}}}^{*},{\gamma }_1,...,{\gamma }_P]$ are finally extracted by maximizing the log-likelihood of the experimental spike-train [35, 36, 69]: (20) where $\Omega =\{t\mid t\notin [{\widehat{t}}_j,{\widehat{t}}_j+T_{\mathrm{\text{ref}}}]\}$ is a set that excludes all the data points falling in the absolute refractory periods and the vectors $y_t^T$ are defined as (21)

With the exponential function in Eq 2, the log-likelihood to maximize is a convex function of θ_th[41] and both its gradient and Hessian can be computed analytically. Consequently, the optimization problem of Eq 20 can be efficiently solved using the Newton-Raphson (gradient ascent) method. Alternatively, Step 3 can be performed using the recorded potential V_data(t) instead of ${\widehat{V}}_{\mathrm{\text{model}}}(t)$ in Eq 21. Since small inaccuracies in Step 2 can be compensated in Step 3, performing the fit using ${\widehat{V}}_{\mathrm{\text{model}}}(t)$ generally improves spike-timing prediction.

In order to avoid numerical problems resulting from the inappropriate choice of the initial condition θ_th,0 used in Step 3, it is convenient to first find the optimal threshold parameters ${\widehat{V}}_{\mathrm{\text{T}}}^{*}$ and $\widehat{\Delta V}$ of a reduced GIF model in which the firing threshold is constant (i.e., $V_{\mathrm{\text{T}}}=V_{\mathrm{\text{T}}}^{*}$). For that, a first gradient ascent is performed with initial conditions ΔV₀ = 50 mV and $V_{T,0}^{*}=-\Delta V_0·\text{log}\bar{r}$, where $\bar{r}$ denotes the average firing rate in the data. Then, a second gradient ascent is performed on the log-likelihood of the full model using ${\theta }_{\mathrm{\text{th,0}}}={\widehat{\Delta V}}^{-1}·[1,{\widehat{V}}_{\mathrm{\text{T}}}^{*},0,...,0]$ as initial condition.

Generalized Linear Model

All GIF model performance included in this study are compared against the ones of a standard GLM [35, 36]. In the GLM, spikes are emitted stochastically according to the following conditional intensity (22) with λ₀ = 1 Hz. In the GLM, the linear filter κ_GLM(t) is not assumed to be exponential but is extracted from experimental data through linear expansion in rectangular basis functions. Moreover, the GLM accounts for spike-history effects with a unique filter h_GLM(t). The GLM also differs from the GIF model because it has neither an absolute refractory period nor an explicit reset after the emission of a spike. To obtain a fair comparison between the two models, the filter h_GLM(t) was expanded using the same basis functions as used for γ(t) in the GIF and the number of basis functions used for κ_GLM(t) was such that, in total, the two models had the same number of parameters. Given the input current I(t) and the observed spike train S_data(t), GLM parameters θ_GLM = {E₀, κ_GLM(t), h_GLM(t)} were extracted with standard methods [35, 36] by maximizing the model log-likelihood L(θ_GLM) = logp(S_data∣I, θ_GLM) using ${\theta }_{\mathrm{\text{GLM,0}}}=\{\text{log}\bar{r},0,0\}$ as initial condition. Importantly, the GLM fitting procedure does not exploit the information available in the subthreshold membrane potential fluctuations and all of its parameters are extracted using the maximum likelihood approach. This explains why fitting a GLM requires more CPU time than fitting a GIF model.

Similarity measure $M_d^{*}$ between sets of spike trains

To quantify the model performance in predicting spikes, we used the normalized, bias-corrected metrics $M_d^{*}$ [34]. $M_d^{*}$ relies on a measure of the distance between the experimental and the predicted spike-emission probability, which are in turn inferred from the responses to a limited number of repetitive current injections. Importantly, $M_d^{*}$ resolves the small sample bias known to affect most of the similarity measures when the number of available spike trains is small. Also, in contrast to previous measures based on naive pairwise comparisons (e.g., the Γ coincidence factor used in ref. [70]), $M_d^{*}$ does not suffer from the so-called deterministic bias known to favor noise-free models and is therefore well suited for the evaluation of stochastic spiking models [34]. Moreover, in contrast to many other correlation-based measures, $M_d^{*}$ is sensitive to the accuracy with which both the shape and the amplitude of the spike probability is predicted.

Given a small set of experimental spiketrains $S_i^{\mathrm{\text{(d)}}}={\sum }_f\delta (t-{\widehat{t}}_f)$ recorded in response to N_d repetitive injections i = 1, …, N_d of the same input current I_test(t), as well as a large set of spike trains $S_j^{\mathrm{\text{(m)}}}={\sum }_f\delta (t-{\widehat{t}}_f)$ predicted by N_m repetitive simulations j = 1, …, N_m of a stochastic model, the similarity $M_d^{*}$ between the two sets of spike trains is defined as [34]: (23) where $S_i^{\mathrm{\text{(d)}}}$ denotes the i-th experimental spike-train, ${\nu }_{\mathrm{\text{d}}}=\frac{1}{N_d}{\sum }_{i=1}^{N_{\mathrm{\text{d}}}}{S}_i^{\mathrm{\text{(d)}}}$ is the average experimental response across trials (that is,i.e., the experimental peri-stimulus-time histogram (PSTH) computed with infinitesimally small bins), ${\nu }_{\mathrm{\text{m}}}=\frac{1}{N_m}{\sum }_{i=1}^{N_{\mathrm{\text{m}}}}{S}_i^{\mathrm{\text{(m)}}}$ is the average model response and ⟨ν_m, ν_m⟩ represents its norm. Due to high-throughput requirements and experimental constraints, only a small number N_d of experimental spike-trains are available. For this reason, the norm of ν_d must be computed using an unbiased estimator (cf. first term in the denominator of Eq 23). Finally, the brackets ⟨·, ·⟩ denote the inner product used to quantify the distance between two spike trains [34]: (24) where K(s, s′) is a two-dimensional kernel defining the degree of coincidence between two spikes occurred at times s and s′.

While different windows K(s, s′) may be used, the Kistler coincidence kernel K(s, s′) = δ(s′) · Θ(s+Δ) · Θ(−s+Δ) was chosen with Δ = 4 ms as in refs.[30, 31]. With this particular choice, the inner product ⟨S_i, S_j⟩ equals the number of spikes in S_i that fell ±Δ ms apart to one of the spikes in S_j and, consequently, $M_d^{*}$ becomes: (25) with n_dm being the average number of coincident spikes between data (d) and model (m), n_mm being the average number of coincident spikes computed across N_m = 500 repetitions generated by the model and $n_{\mathrm{\text{dd}}}^{*}$ being the bias-corrected average number of coincident spikes between different experimental spike trains (i.e., the number of coincident spikes between experimental spike trains $S_i^{(d)}$ and $S_j^{(d)}$ averaged across (i, j) ∈ [1, N_d] × [1, N_d] with i ≠ j, see Eq 23).

Performance evaluation

Prediction error ϵ_V on the subthreshold response. For each repetition i in the test set, we computed the coefficient of determination $R_i^2$ between the experimental membrane potential $V_i^{\mathrm{\text{(data)}}}(t)$ and the GIF model prediction ${\widehat{V}}_i^{\mathrm{\text{(model)}}}(t)$ (obtained by solving Eq 1 and enforcing the spikes to occur at the same time as in the experiment): (26)

The prediction error ϵ_V on the subthreshold response was then obtained by averaging the results from each repetition: (27) ϵ_V takes values between 0 and 1 and can be interpreted as the fraction of variance of the subthreshold membrane potential fluctuations that the model was able to predict. The parameters T_test and n_test denote the duration and the number of repetitions in the test set, respectively.

Prediction error ϵ_param on the GIF model parameters. The mean error ϵ_param on the parameters θ extracted from artificial data is defined as (28) where $\Delta {\theta }_i=\mid {\theta }_i-{\widehat{\theta }}_i\mid$ is the L1-error between the estimated parameter ${\widehat{\theta }}_i$ and the reference parameter θ_i (used to generate the artificial data). Overall, ϵ_param measures the absolute percentage error averaged across model parameters.

All the CPU times reported in this study were obtained using an IntelCore i7 CPU920 @ 2.67GHz with 24 GB RAM. Both GLM and GIF model parameters were extracted using custom-written Matlab procedures.