Physiological and morphological characterization

We characterized TC neurons in slices of the rat VB thalamus, by combining whole-cell patch-clamp recordings, biocytin filling and 3D Neurolucida (MicroBrightField) reconstruction, along with anatomical localization in a reference atlas [19] (Fig 1).

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Fig 1. Simultaneous physiological and morphological characterization.

(A) View of a patched cell under optic microscope and anatomical localization of biocytin-filled neurons (insets). Letters D and E identify morphologies in a slice. (B) Voltage responses of two different thalamocortical (TC) neurons to a standardized battery of current stimuli. Each current amplitude was normalized by the threshold current of each neuron (e.g. 150% threshold, see Methods). Third row is a low-threshold burst response from a hyperpolarized holding potential, V_hold = −84 mV (burst mode), the other responses are elicited from a depolarized holding potential, V_hold = −64 mV (tonic mode). Two different holding currents (I_hold—tonic, I_hold—burst) are injected to obtain the desired V_hold. The vertical scale bar applies to all the traces, the first horizontal scale bar from the top refers to the first two rows, the second applies to the last four rows. (C) Analysis of adaptation index (AI) from recordings in tonic mode. Solid line is a non-parametric estimation of the distribution, dashed lines are two Gaussian distributions fitted to the data (see Methods). The vertical line indicates the cut-off value.

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Visual inspection of 50 reconstructed morphologies (24 from the VPL, 26 from the VPM nuclei) revealed variability in the number of principal dendritic trunks and their orientation, in agreement with previous anatomical studies [4]. The maximum radial extent of the dendrites ranged between 120 and 200 μm and they started to branch between 20 and 50 μm from the soma (S1 Fig). We then analyzed the morphologies with two methods in order to quantitively classify different morphological types. We used algebraic topology to extract the persistent homology of each morphology and to visualize the persistence barcode [20] (Fig 2A, see Methods). Each horizontal bar in the persistence barcode represents the start and end point of each dendritic component in terms of its radial distance from the soma. The barcodes of all the morphologies followed a semi-continuous distribution of decreasing length. To quantify the differences between the barcodes, we computed the pairwise distances of the persistence images (see Methods and S1 Fig). We found that they were in general small (<0.4, values expected to vary between 0 and 1). These findings indicate that the morphologies cannot be grouped in different classes based on the topology of their dendrites. Furthermore, we performed Sholl Analysis [21] to compare the complexity of the dendritic trees (Fig 2B). We observed that all the morphologies had dense dendritic branches, with a maximum number of 50–100 intersections between 50–80 μm from the soma. When comparing the Sholl profiles for each pair of neurons we could not find any statistically significant difference (S1C Fig). Considering the results of topological and Sholl analyses, we grouped all the morphologies in one morphological type (m-type) called thalamocortical (TC) m-type.

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Fig 2. Morphological properties.

(A) Renderings of 3D reconstructed TC neurons along with their persistence barcode. The persistence barcode is a topological description of the branching pattern of the neurons’ dendrites. Grey: soma and dendrites, blue: axon (only small sections available). (B) Sholl analysis of TC neuron dendrites. For each Sholl ring, the number of intersections is shown (mean ± standard deviation, N = 50). Each grey circle represents one morphology, colored lines correspond to the morphologies in A. See S1 Fig for further analysis.

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We used an adaptive stimulation protocol, called e-code, consisting of a battery of current stimuli (see Methods for details), where the stimulation amplitude was adapted to the excitability of different neurons. This standardized protocol has previously been used to build biophysically-accurate models of cortical electrical types (e-types) [16]. However, TC neurons from different thalamic nuclei and species fire action potentials in two distinct firing modes, namely tonic firing, when stimulated from a relatively depolarized membrane potential or low-threshold bursting, from a hyperpolarized membrane potential [5–8]. We thus extended the e-code to include two different holding currents. All the neurons recorded in this study displayed tonic and burst firing, when stimulated with the appropriate holding current (Fig 1B). Moreover, we were able to classify different e-types by considering the voltage traces recorded in tonic mode in response to step current injections (Fig 1B). The majority of the cells (59.3%) showed a non-adapting tonic discharge (continuous non-adapting low-threshold bursting, cNAD_ltb e-type) while others (40.7%) had higher adaptation rates (continuous adapting low-threshold bursting, cAD_ltb e-type), as reflected by the adaptation index (Fig 1C). We followed the Petilla convention [22] for naming the tonic firing discharge (cNAD or cAD), extending it to include “_ltb” for the low-threshold bursting property. In some rare examples, we noticed acceleration in the firing rate with decreasing inter-spike intervals (ISIs) towards the end of the stimulus. Similar adapting and accelerating responses have already been described in the VB thalamus of the cat [7]. We also observed stereotypical burst firing responses within the same cell, with variation of the number of spikes per burst in different cells, but the burst firing responses alone were insufficient to classify distinct e-types.

Constraining the models with experimental data

Multi-compartmental models comes with the need of tuning a large number of parameters [23], therefore we constrained the models as much as possible with experimental data. We first combined the morphology and the ionic currents models in the different morphological compartments (soma, dendrites and axon). Given that the reconstruction of the axon was limited, we replaced it with a stub representing the initial segment [16]. We used previously published ionic current models and selected those that best matched properties measured in rat TC neurons (see Methods). The kinetics parameters were not part of the free parameters of the models.

The distribution of the different ionic currents and their conductances in the dendrites of TC neurons is largely unknown. The current amplitudes of the fast sodium, persistent and transient (A-type) potassium currents were measured, but only up to 40–50 μm from the soma [24]. Indirect measures of burst properties [15] or Ca²⁺ imaging studies [25] suggest that the low-threshold calcium (T-type) channels are uniformly distributed in the somatodendritic compartments. We thus assumed different peak conductance in the soma, dendrites and axon for all the ionic currents, except for I_CaT, which had the same conductance value in the soma and dendrites. We then extracted the mean and standard deviation (STD) of different electrical features in order to capture the variability of firing responses from different cells of the same e-type [9,10,17] (Fig 3). We observed that some features extracted from tonic firing responses had distinct distributions between the cAD_ltb and cNAD_ltb e-types (Fig 3A).

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Fig 3. Histograms of electrical features.

Each vertical line represents the mean feature value for a cell. Tonic and burst refer to the holding voltage as in Fig 1. (A) Feature values extracted from recordings in tonic mode (N = 11 cAD_ltb cells, N = 16 cNAD_ltb cells). The features highlighted by a black box show different distributions for the cNAD_ltb and cAD_ltb electrical types (e-types) (p-value<0.05, two-sided Mann-Whitney U test with Bonferroni correction for multiple comparisons). Passive properties (V_rest, R_input) and spike shape features (AHP depth, AP amp., etc.) did not show clear differences between the two e-types. (B) Features measuring burst firing properties (N = 22 cells).

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For optimizing the models’ parameters, we chose features that quantified passive (input resistance, resting membrane potential), burst and tonic firing properties (number of spikes, inverse of inter-spike intervals, latency to first spike, adaptation index), action potentials shape (amplitude, half-width, depth of the fast after-hyperpolarization). We aimed at finding the minimal set of features that captured the most important properties in the two firing modes. This set was a trade-off between comprehensively describing the experimental data (i.e. extracting all possible features), which can lead to over-fitting and loss of generalizability, and a too small set that would miss some important characteristics. For the tonic firing responses, we used three stimulation amplitudes (150%, 200%, 250% of firing threshold) which have been shown to reproduce the complete input-output function of the neurons [16,17]. Responses to two hyperpolarizing steps of different amplitudes (−40% and −140% threshold) constrained the input resistance (conductance of the leak current) and the conductance of currents activated in hyperpolarization, for example the h-current, I_h (sag_amplitude feature). We included baseline voltage values to the optimization objectives to ensure that the models were in the right firing regime and spike count to penalize models that were firing in response to the holding currents. To constrain the low-threshold burst we used features (such as number of spikes) which are influenced by specific ionic currents, for example the low-threshold calcium current, I_CaT.

The average value and STD of each feature were used to calculate the feature errors (Fig 4C). Each error measured how much the features of the models deviated from the experimental mean, in units of the experimental STD. We used a multiobjective optimization approach (MOO), where each error was considered in parallel. To rank the resulting models after optimization, we considered model A better than model B if the maximum error of all the features of A was smaller than the maximum error of all the features of B.

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Fig 4. Models of different TC e-types and their fitting errors.

(A) Single neuron modelling pipeline. (B) Experimental and model voltage responses to a variety of stimuli pattern used during the optimization of cNAD_ltb and cAD_ltb e-types. (C) Feature errors of the models shown in (B) reported as deviation from the experimental mean. The models are compared with the mean of features shown in Fig 3. Note that the models shown in B are fitted in order to reproduce the mean firing properties, not only a specific experimental recording. See S2 Fig for a complete list of fitting errors.

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By applying this MOO procedure, we generated multiple models with distinct parameter combinations for each of the twenty-two free parameters (Fig 4B). The free parameters were allowed to vary between the upper and lower bounds shown in Fig 5B. The models reproduced well the key firing dynamics observed in the experimental recordings. They showed a low-threshold burst when stimulated from a hyperpolarized membrane potential, crowned by a variable number of sodium spikes (Fig 4B). In the tonic firing regime, they reproduced adapting and non-adapting firing discharges as observed in the two e-types. These results indicate that the ion channels included in the models were sufficient to reproduce the experimental firing properties and that different e-types in TC neurons could be generated by different ion channel densities.

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Fig 5. Diversity of model parameters and experimental variability.

(A) Example of model fitting errors (sum of all feature errors) during optimization. (B) Initial parameter ranges and diversity of solutions. Each vertical line represents the range for the parameters, when the horizontal lower bar is missing the bound is 0. The characters following”.” in the parameter name specifies the morphological compartment for the parameter (”s”: soma,”d”: dendrites,”a”: axon). Black circles: parameter values for one of the models in Fig 4, grey circles: parameter values of the models with all feature errors below 3 STD. (C) Features variability in the models and experiments. Blue crosses: feature errors of a sample of 10 models. Each grey circle is the z-scored feature value of one experimental cell, obtained from the feature values shown in Fig 3. The protocol names are shown in parenthesis and corresponds to the stimuli shown in Figs 1 and 4, tonic and burst refer to the holding current as in Fig 1.

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Model and experimental diversity

We found that different sets of parameter values reproduced the target firing behavior (Fig 5B). We further analyzed models that had all the feature errors below 3 STD. Models’ voltage responses reflected the characteristic firing properties of TC neurons (S3 Fig), indicating that the selected set of features and ion channels were sufficient to capture the two firing modes, in both the adapting and non-adapting e-types. The voltage traces from different models showed small differences in spike amplitude, firing frequency, and depth of the after-hyperpolarization, as reflected by the variability of features values (Fig 5C), arising from differences in ion channel densities between models.

Spike-shape related features (e.g. AP. amplitude) in the different models covered the space of the experimental variability, while for some features (e.g. input resistance, R_input), all models tended to cluster on one of the tails of the experimental distribution. R_input relates to the neuron passive properties and depends both on the number of channels open at rest (inverse of the leak conductance in the model) and the size of the cell. Given that all the models for a given e-type were constrained on a single morphology, this result is not surprising. Other features, such as sag amplitude were less variable in the models compared to experiments. We hypothesized that this depended on the variable stimulation amplitudes applied to different experimental cells, while all the models were stimulated with the same current amplitudes.

Some other features were systematically above or below the experimental values in both e-types. We suggest that this depend on the exact dynamics of some specific ion channels. For example, the amplitudes of the first and second spikes in the burst tended to be similar or above and below the experimental values, respectively. This can depend on the specific activation/inactivation properties of some ionic currents, for example the transient sodium current (I_NaT) and delayed potassium current (I_Kd). During the rising phase of the low-threshold spike, I_NaT in the model is readily activated and generated a first spike with higher amplitude, but is not repolarized enough by the activation of I_Kd. At higher potentials, reached towards the peak of the low-threshold spike, the availability of I_NaT and other depolarizing currents seemed reduced and generated a spike with smaller amplitude. Sensitivity analysis confirmed that I_NaT and I_Kd had an influence on the amplitude of the first and second spike in the burst. Furthermore, these two currents operate together with currents that generate the burst, such as the low-threshold calcium current (I_CaT) and the I_h in shaping the amplitude of the second spike in the burst. Interestingly, the models also tended to have lower instantaneous frequency of the first two spikes in the burst (Inv. 1^st ISI) and this feature had similar sensitivity (but of opposite signs) to the amplitude of the second spike in the burst.

Another possible explanation is the lack of some ionic currents in the model, for example some specific subtype of potassium channels that promote higher firing rates (Kv3.1 and Kv3.3). While neurons of the thalamic reticular nucleus are known to express this channel subunit [26], the expression in TC neurons has not been confirmed yet. The dynamics of I_Kd could also explain why the after-hyperpolarization (AHP depth) tended to be smaller in the models compared to the experimental values. AHP depth is also influenced by other ionic currents, such as high-threshold calcium current (I_CaL), calcium-activated potassium current (I_SK) and the intracellular calcium dynamics. The number of action potentials (Num. of APs) in different conditions (No stim, I_hold) ensured that the models did not spike in the absence of a stimulus or in response to the holding current. For this reason, all the experimental and model feature values in Fig 5C are equal to 0.

We examined the diversity of the parameter values with respect to the initial parameter range (Fig 5B). Most of the optimized parameter values spanned intervals larger than one order of magnitude. On the other hand, some parameter values were restricted to one order of magnitude, for example the permeability of the low-threshold calcium current P_CaT. This result is in agreement with experiments showing a minimum value of I_CaT is critical to generate burst activity and this critical value is reached only at a certain postnatal age [27]. The value of P_CaT was constrained by features measuring burst activity (such as number of spikes, frequency, etc.).

Assessment of model generalization

We used different stimuli for model fitting (current steps) and for generalization assessment (current ramps and noise). We simulated the experimental ramp currents, by stimulating the models with the appropriate holding currents for the two firing modes and a linearly increasing current. We first compared visually the model responses with the experimental recordings (Fig 6A). In burst mode, the models reproduced the different behaviors observed experimentally: absence of a burst, small low-threshold spike, burst (S4A Fig). Moreover, the latency of burst generation substantially overlapped with the experimental one. However, a small fraction of models (1.2%) generate repetitive burst that we have never observed in the experimental recordings (S4B Fig). These models were quantitatively rejected by considering the number of spikes and the inter-spike intervals. In tonic mode, the latency to first spike, the voltage threshold, the shape of the subsequent action potentials and the increase in firing frequency were comparable with the experimental recordings (Fig 6A). In addition, we quantified the generalization error to ramp stimuli (Fig 6C), by considering the latency to first spike, firing frequency increase over time (tonic mode) or number of spikes (burst mode).

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Fig 6. Model generalization.

(A) Responses to a ramp current injection in burst mode (left) and tonic mode (right). (B) Responses to a noise current generated according to an Ornstein-Uhlenbeck process and scaled based on the excitability of the different experimental cells and models (see Methods). (C) Generalization errors for all the models that passed the generalization test (all generalization errors <3 STD). (D) Proportion of models that passed the generalization test (see S4 Fig for examples of models that failed this test).

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Although conductance-based models can be fit by using step and ramp currents, these stimuli are different from synaptic inputs, which can be simulated by injecting noisy currents. To test the response to such network-like input, we used a noisy current varying accordingly to an Ornstein-Uhlenbeck (OU) process [28] to compare models’ responses with the experimental data. Each experimentally recorded cell was stimulated with the same OU input, scaled by a factor w. Experimentally, w was calculated by evaluating the responses to previous stimuli. We developed a similar approach to generate the noise stimuli in silico (see Methods). The noise current was injected on top of the holding currents used during the optimization. We found that the models reproduced well the subthreshold potential, spike times and the distribution of single spikes and bursts (Fig 6B). Moreover, we quantitatively evaluated the generalization to the noise stimulus by extracting features (e.g. number of spikes) and comparing them with the experimental mean.

We computed generalization errors for each model, which were calculated similarly to the optimization errors (Fig 6C). We considered a model acceptable after generalization if it had all generalization errors <3 STD and we found that the majority of the models (>90%) passed the generalization test (Fig 6D).

Sensitivity of electrical features to small parameter perturbations

We assessed the robustness of the models to small changes in their parameter values. To that end, we varied each parameter at a time by a small amount (± 2.5% of the optimized value) and computed the values of the features. A sensitivity value of 2 between parameter p and feature y means that a 3% change in p caused a 6% change in f. We ranked the parameters from the most to the least influential and the features from the most sensitive to the least sensitive.

Some features resulted to be more sensitive to parameter changes, both in term of magnitude of the sensitivity and number of parameters (e.g. adaptation index, inverse of inter-spike intervals, ISIs, AHP depth). Most of these features describe the model firing pattern, which depend more on the interplay between the different ionic currents than on the specific activation/inactivation dynamics. Conversely, spike shape-related features were less sensitive to parameter changes (e.g. AP half-width, AP amp.) because they depend more on specific ionic current dynamics (e.g. I_Kd, I_L, I_NaT,). Some features were very weakly influenced by small parameter changes, e.g. baseline voltage, which depend more on the holding current amplitude, than on the model parameters (Fig 7A).

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Fig 7. Local sensitivity analysis.

(A) Sensitivity of the feature values to small changes to the parameter values for the cNAD_ltb model in Fig 4. Sensitivities (Δy/Δp) are color coded as a heat map. Features are ranked from the most to the least sensitive and parameters are ranked from the most to the least influential. The last three rows are features that ensure that the models were not firing without input or in the response to the holding current. Small changes to the parameter values are not expected to make the model firing and thus the sensitivity of these features is 0. (B) Same sensitivity values as in (A), with features and parameters clustered by similar sensitivity and influences.

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The conductance of the leak current g_leak emerged as the most influential parameter (Fig 7A). An increase in g_leak caused a decrease in firing frequency (inverse of ISIs) in both the tonic and burst firing modes. These results are easy to interpret when considering Ohm’s law: increasing g_leak means decreasing the input resistance of the model, so that for the same input current the voltage response becomes smaller. The second most influential parameter was the conductance of the persistent sodium current g_NaP in the dendrites, which increased the tonic firing rate as expected from a depolarizing current. Interestingly, g_NaP had an effect on the late phase of the low-threshold burst (inverse last ISI—burst), suggesting that the low-threshold burst is initiated by the activation of I_T and modulated by I_NaP. An increase in the permeability of the low-threshold calcium current P_CaT, known to be one of the main currents underlying low threshold bursting, enhanced burst firing responses (it increased the inverse of ISIs, i.e. the firing frequency) and had effects on some of the tonic features. Increasing the somatic permeability of the high threshold calcium current P_CaL decreased the tonic firing rate, despite being a depolarizing current. Increasing P_CaL means higher Ca²⁺ influx and higher activation of the Ca^2+-activated potassium current (I_SK). The parameter g_SK had indeed a similar effect on the features and thus clustered together with parameters regulating the intracellular calcium dynamics γ_Ca and τ_Ca (Fig 7B). Sag amplitude, that is known to depend on the activity of I_h, was mainly influenced by change in g_leak, P_CaT and g_h. In summary, each parameter influenced at least one feature. These results indicate that the model ability to generate tonic and burst firing is robust to small changes in parameter values and that all the parameters were constrained during the optimization by one or more features.

We then analyzed which features depended similarly on parameter changes, as they may add superfluous degrees of freedom during parameters search. Fig 7B shows the same sensitivities as in Fig 7A, clustered by their similarities (see Methods). Features clustered together if they were sensitive to similar parameter combinations and parameters clustered based on their similar influence on the features. Not surprisingly, the same tonic features measured at different level of current stimulation clustered together (e.g. AP amplitude and half-width, AHP depth, latency of the first ISI) and tonic firing features belonged to a cluster that was different from burst features. Some features measured in tonic mode (such as AP half-width and AP amp.) clustered together because they depended mainly on the dynamics of I_NaT and I_Kd: increasing the conductance of I_NaT increased the amplitude of the APs and decreased its duration. This was also true for the amplitude of the 1^st AP in the burst. Features measured in burst mode had similar sensitivities because they depend on currents that are active at relatively hyperpolarized potential (such I_H and I_CaT).

Preservation of model firing properties with different morphologies

We optimized the parameters for the adapting and non-adapting e-models in combination with two different experimental morphologies selected at random and then tested them with the other 48 morphologies. Considering that morphologies could not be classified in different m-types based on topological analysis of their dendrites and that TC neurons have been shown to be electrically compact [15], we expected the electrical behavior to be conserved when changing morphology. Nonetheless, different neurons vary in their input resistance R_input and rheobase current I_thr due to variation in the surface area. Variation in R_input and I_thr made the current amplitude applied during the optimization inadequate to generate the appropriate voltage trajectories. We thus devised an algorithm to search for the holding current to obtain the target holding voltage (for example −64 mV or −84 mV for tonic and burst firing, respectively) and I_thr from the desired holding voltage. The different e-model/morphology combinations (me-combinations) were evaluated by computing the same feature errors calculated during optimization (Fig 8A). For each morphology, we selected the e-model that generated the smallest maximum error. We chose the value of 3 STD as the threshold to define which me-combinations were acceptable [29], yielding 48 acceptable me-combinations out of the 48 tested (Fig 8A). All me-combinations reproduced burst and tonic firing (Fig 8B).

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Fig 8. Model generalization to different experimental morphologies.

(A) Feature errors from the best electrical models (e-models) showed in Fig 4 applied to 50 different TC cell morphologies. Each morphology is represented with a different color. None of the e-models/morphology combinations were rejected, because all the feature errors were < 3 STD (dashed line). (B) Example of voltage responses from two accepted e-model/morphology combinations.

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Given that the generalization of the electrical models to the other 48 morphologies worked well, we can conclude that the morphological properties of the modeled neurons are very similar, at least for properties that have an impact on the electrical models (e.g. surface area, diameters of the compartments).

Experimental procedures

Experimental data were collected 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.

All the experiments were conducted on coronal or horizontal brain slices (300 μm thick- ness) from the right hemisphere of male and female juvenile (P14-18) Wistar Han rats. The region of interest was identified using the Paxinos and Watson rat brain atlas [19]. After decapitation, brains were quickly dissected and sliced (HR2 vibratome, Sigmann Elektronik, Germany) in ice-cold standard ACSF (in mM: NaCl 125.0, KCl 2.50, MgCl₂ 1.00, NaH₂PO₄ 1.25, CaCl₂ 2.00, D-(+)-Glucose 50.00, NaHCO₃ 50.00; pH 7.40, aerated with 95% O₂ / 5% CO₂). Recordings of thalamocortical neurons in the VB complex were performed at 34°C in standard ACSF with an Axon Instruments Axopatch 200B Amplifier (Molecular Devices, USA) using 5–7 MΩ borosilicate pipettes, containing (in mM): K⁺-gluconate 110.00, KCl 10.00, ATP-Mg²⁺ 4.00, Na₂-phosphocreatine 10.00, GTP-Na⁺ 0.30, HEPES 10.00, biocytin 13.00; pH adjusted to 7.20 with KOH, osmolarity 270–300 mOsm. Cells were visualized using infrared differential interference contrast video microscopy (VX55 camera, Till Photonics, Germany and BX51WI microscope, Olympus, Japan).

Membrane potentials were sampled at 10 kHz using an ITC-18 digitizing board (InstruTECH, USA) controlled by custom-written software operating within IGOR Pro (Wavemetrics, USA). Voltage signals were low-pass filtered (Bessel, 10 kHz) and corrected after acquisition for the liquid junction potential (LJP) of −14 mV. Only cells with a series resistance <25 MΩ were used.

After reaching the whole-cell configuration, a battery of current stimuli was injected into the cells and repeated 2–4 times (e-code). During the entire protocol, we defined offset currents in order to keep the cell at −50 mV (tonic firing) or −70 mV (burst firing) before LJP correction and applied them during the entire protocol. The step and ramp currents were injected with a delay of 250 ms in the experiment. In the models, the stimuli were injected with a delay of 800 ms, to allow for the decay of transients due to initialization. Each stimulus was normalized to the rheobase current of each cell, calculated on-line as the current that elicited one spike (stimulus TestAmp, duration 1350 ms). The stimuli used in the experiments, for fitting and testing the models were:

• IDRest: current step of 1350 ms, injected at different amplitude levels in 25% increments (range 50–300% threshold). IDRest was renamed to Step in the model.
• IDThresh: current step with duration of 270 ms, 4% increments (range 50–130%).
• IV: hyperpolarizing and depolarizing steps of 3000 ms injected in 20% increments (range −140–60%).
• SponNoHold: the first 10 seconds of this stimulus was used to calculate the resting membrane potential. No holding or stimulation currents were applied.
• SponHold: the first 10 seconds of this stimulus was used to calculate the holding current applied to keep the cells at the target potential.
• PosCheops: ramps of current from 0 to 300% and from 300 to 0% having progressively shorter durations (4000 ms, 2000 ms, 1250 ms). To test the models in tonic mode we used the first increasing ramp in the stimulus, while we used the last one in the bursting firing mode. We chose the last one because the biological cells were more likely to generate a burst, while they were silent during the first two ramps.
• NOISEOU3: the original wave was scaled and offset for each cell based on the spike frequency in response to the IDRest protocol. The scaling factor w was extracted from the frequency-current curve and corresponded to the current value that made the cell fire at 7.5 Hz.

Neurons that were completely stained and those with high contrast were reconstructed in 3D and corrected for shrinkage as previously described [16]. Reconstruction used the Neurolucida system (MicroBrightField). The location of the stained cells was defined by overlaying the stained slice and applying manually an affine transformation to the Paxinos and Watson’s rat atlas [19].

Electrical features extraction

Electrical features were extracted using the Electrophys Feature Extraction Library (eFEL) [42]. We calculated the adaptation index (AI) from recordings in tonic mode (Step 200% threshold) and classified TC VB neurons into adapting (AI> = 0.029) and non-adapting (AI<0.029) electrical types. AI was calculated using the eFEL feature adaptation_index2 and corresponded to the average of the difference between two consecutive inter-spike intervals (ISI) normalized by their sum. The cut-off value was calculated after fitting a Gaussian mixture model to the bimodal data, using available routines for R [43,44]. In order to group data from different cells and generate population features, we normalized all the stimuli by the rheobase current I_thr of each cell. To calculate I_thr, we used IDRest and IDThresh and selected the minimal amplitude that evoked a single spike. Along with the voltage features, we extracted mean holding and threshold current values for all the experimental stimuli. Description of the features and the details on their calculation are available on-line [42]. Current stimuli applied during the optimization and generalization were directly obtained from the experimental values or automatically calculated by following the experimental procedures (e.g. noise stimulus).

Morphology analysis

Reconstructed morphologies were analyzed to objectively identify different morphological types. The Sholl profiles of each pair of cells was statistically tested by using k-samples Anderson-Darling statistics. This test was preferred to the most common Kolmogorov-Smirnov test, because it does not assume that the samples are drawn from a continuous distribution. The different Sholl profiles are indeed an analysis of the intersections with discrete spheres.

To compare the topological description of each morphology we transformed the persistence barcodes into persistence images and calculated their distances as in [20]. Briefly, we converted the persistence barcode, which encodes the start and end radial distances of a branch in the neuronal tree, into a persistence diagram. In the persistence diagram, each bar of the barcode is converted into a point in a 2D space, where the X and Y coordinates are the start and end radial distances of each bar. The persistence diagram was then converted into a persistence image by applying a Gaussian kernel. We used the library NeuroM [45] to perform Sholl and morphometrics analyses.

Ionic currents models

We used Hodgkin-Huxley types of ionic current models, starting from kinetics equations already available in the neuroscientific literature. Along with kinetics of the ionic currents, we stored information on the experimental conditions, such as temperature and LJP, by using the software NeuroCurator [46]. Whenever the data was available, we compared simulated voltage-clamp experiments to experimental data from juvenile rats. Ionic currents I_i were defined as functions of the membrane potential v, its maximal conductance density g_i and the constant value of the reversal potential E_i: m_ion and h_ion represent activation and inactivation probability (varying between 0 and 1), with integer exponents x and y. Each probability varied according to: where n_∞(v) is a function of voltage that represents the steady-state activation/inactivation function (normally fitted with a Boltzmann curve) and τ_n(v) is a voltage-dependent time constant. Exceptions to this formalism are ionic currents that do not inactivate (y = 0) and ionic currents with (in)activation processes mediated by two or more time constants. Calcium currents (I_CaT and I_CaL) were modeled according to the Goldman-Hodgkin-Katz constant field equation and had permeability values instead of conductance [47].

Simulation and parameters optimization

NEURON 7.5 software was used for simulation [56]. We used NEURON variable time step method for all simulations. For the sake of spatial discretization, each section was divided into segments of 40 μm length. The following global parameters were set during optimization and generalisation: initial simulation voltage (−79 mV), simulation temperature (34°C), specific membrane capacitance (1 μF/cm²), specific intracellular resistivity 100 Ωcm for all the sections, equilibrium potentials for sodium and potassium were 50 mV and −90 mV, respectively.

BluePyOpt [18] with Indicator Based Evolutionary Algorithm (IBEA) were used to fit the models to the experimental data. Each optimization run was repeated with three different random seeds and evaluated 100 individuals for 100 generations. The evaluation of these 300 individuals for 100 generations was parallelized using the iPython ipyparallel package and took between 21 and 52 h on 48 CPU cores (Intel Xeon 2.60 GHz) on a computing cluster. Each optimization run typically resulted in tens or hundreds of unique acceptable solutions, defined as models having all feature errors below 3 STD from the experimental mean.

Sensitivity analysis

We performed a sensitivity analysis of an optimization solution by varying one parameter value (p_m) at a time and calculating the electrical features from the voltage traces (y⁺ and y^-). We defined the sensitivity as the ratio between the normalized feature change and the parameter change, which for smooth functions approximates a partial derivative [57,58]. The features changes were normalized by the optimized feature value. For small changes of parameter values, we assumed that the features depend linearly on its parameters. We could thus linearize the relationship between the features and the parameters around an optimized parameter set and calculate the derivatives. The derivatives were calculated with a central difference scheme [57].

We collected the derivatives (sensitivities) in the N X M Jacobian matrix, with N representing the number of features and M the number of parameters.

To rank parameters and features we computed their relative importance by calculating their norms (the square root of the summed squared values) from the Jacobian columns and rows, respectively. To cluster parameters based on similar influences on the features and to cluster features that were similarly dependent on the parameters, we used angles between columns (or rows) to compute distances D between parameters (or features):

Features where thus considered similar if they depended in a similar manner on the parameters, independent of sign or magnitude.