We previously highlighted that V1^R are spontaneously active at E12.5. Their response to a 2 s suprathreshold depolarizing current steps revealed four main patterns, depending on the recorded IN (Boeri et al., 2018): (1) single spiking (SS) V1^R that fires only 1–3 APs at the onset of the depolarizing pulse, (2) repetitive spiking (RS) V1^R, (3) mixed events (ME) V1^R that show an alternation of APs and PPs, or (4) V1^R that displays a long-lasting sodium-dependent PP (Figure 1A1–A4).

Figure 1

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We also uncovered a relationship between $I_{Nap}$ and the ability of embryonic V1^R to sustain repetitive firing (Boeri et al., 2018). However, the heterogeneous firing patterns of V1^R observed at E12.5 could not be fully explained by variations in $I_{Nap}$ (Boeri et al., 2018), suggesting the involvement of other voltage-gated channels in the control of the firing pattern of V1^R, in particular potassium channels, known to control firing and AP repolarization. Our voltage clamp protocol, performed in the presence of tetrodotoxin (TTX) (1 μM), did not disclose any inward rectifying current (hyperpolarizing voltage steps to –100 mV from $V_H$ = –20 mV, data not shown), but revealed two voltage-dependent outward potassium currents, a delayed rectifier current (${\displaystyle I_{Kdr}}$), and a transient potassium current (${\displaystyle I_A}$) in all embryonic V1^R, whatever the firing pattern; Figure 1B1–B4. These currents are known to control AP duration (${\displaystyle I_{Kdr}}$) or firing rate (${\displaystyle I_A}$), respectively (Coetzee et al., 1999). The activation threshold of $I_{Kdr}$ lied between –30 mV and –20 mV and the threshold of $I_A$ between –60 mV and –50 mV, (n = 27; N = 27 embryos) (Figure 1C1–C4). Removing external calcium had no effect on potassium current I/V curves (data not shown), suggesting that calcium-dependent potassium currents are not yet present at E12.5.

It was unlikely that the heterogeneity of V1^R firing patterns resulted from variations in the intensity of $I_A$. Indeed, its voltage-dependent inactivation (time constant: 23.3 ± 2.6 ms, n = 8; N = 8), which occurs during the depolarizing phase of an AP, makes it ineffective to control AP or PP durations. This was confirmed by our theoretical analysis (Figure 7—figure supplement 1). We thus focused our study on $I_{Kdr}$. At E12.5, PP V1^R had a significantly lower $G_{Kdr}$ (2.12 ± 0.44 nS, n = 6; N = 6) than SS V1^R (5.57 ± 0.56 nS, n = 9; N = 9) and RS V1^R (6.39 ± 0.83 nS, n = 7; N = 7) (Figure 1D). However, there was no significant difference in $G_{Kdr}$ between SS V1^R and RS V1^R at E12.5 (Figure 1D), which indicated that variations in $G_{Kdr}$ alone could not explain all the firing patterns observed at E12.5. Similarly, there was no significant difference in $G_{Nap}$ between RS V1^R (0.91 ± 0.21nS, n = 8; N = 8) and PP V1^R (1.24 ± 0.19 nS, n = 6; N = 6) at E12.5 (Figure 1E), indicating that variations in $G_{Nap}$ alone could not explain all the firing patterns of V1^R at E12.5 (Boeri et al., 2018). In contrast, $G_{Nap}$ measured in SS V1^R at E12.5 (0.21 ± 0.20 nS, n = 9; N = 9) were significantly lower compared to $G_{Nap}$ measured in RS V1^R and in PP V1^R at E12.5 (Figure 1E).

Mature neurons often display multiple stable firing patterns (O'Leary et al., 2013; Taylor et al., 2009; Alonso and Marder, 2019). This usually depends on the combination of several outward and inward voltage- or calcium-dependent conductances and on their spatial localization (O'Leary et al., 2013; Taylor et al., 2009; Alonso and Marder, 2019). In contrast, immature V1^R have a limited repertoire of voltage-dependent currents ($I_{Nat}$ and $I_{Nap}$, $I_{Kdr}$ and $I_A$) at E12.5, and we did not find any evidence of voltage-dependent calcium currents at this age (Boeri et al., 2018). Blocking $I_{Nap}$ prevented PP activity, PP-V1^R becoming unexcitable, and turned RS V1^R into SS V1^R (Boeri et al., 2018). Therefore, we hypothesized that the different firing patterns of V1^R observed at E12.5 were related to the $G_{Nap}$/$G_{Kdr}$ ratio only, with variations in the intensity of $I_A$ being unlikely to account for the heterogeneity of firing pattern. We found that this ratio was significantly lower for SS V1^R recorded at E12.5 ($G_{Nap}$/$G_{Kdr}$ = 0.043 ± 0.015, n = 9) compared to RS V1^R (0.154 ± 0.022, n = 8) and PP V1^R (0.66 ± 0.132, n = 6) (Figure 1F). We also found that the $G_{Nap}$/$G_{Kdr}$ ratio was significantly lower for RS V1^R compared to PP V1^R (Figure 1F).

Altogether, these results strongly suggest that, although the presence of $I_{Nap}$ is required for embryonic V1^R to fire repetitively or to generate PPs (Boeri et al., 2018), the heterogeneity of the firing pattern observed between E12.5 is not determined by $I_{Nap}$ per se but likely by the balance between $I_{Nap}$ and $I_{Kdr}$.

We previously showed that blocking $I_{Nap}$ with riluzole converted PP V1^R or RS V1^R into SS V1^R (Boeri et al., 2018). To confirm further that the balance between $G_{Nap}$ and $G_{Kdr}$ was the key factor in the heterogeneity of V1^R firing patterns, we assessed to what extent a given E12.5 SS V1^R cell could change its firing pattern when $I_{Kdr}$ was gradually blocked by 4-aminopiridine (4-AP). We found that $I_{Kdr}$ could be blocked by micromolar concentrations of 4-AP without affecting I_A (Figure 2—figure supplement 1). 4-AP, applied at concentrations ranging from 0.3 μM to 300 μM, specifically inhibited $I_{Kdr}$ with an IC₅₀ of 2.9 μM (Figure 2—figure supplement 1C1).

We then determined to what extent increasing the concentration of 4-AP modified the firing pattern of V1^R at E12.5. Applying 4-AP at concentrations ranging from 3 μM to 300 μM changed the firing pattern of SS V1^R (n = 10; N = 10) in a concentration-dependent manner (Figure 2A1–A3). In 50% of the recorded V1^R, increasing 4-AP concentrations successfully transformed SS V1^R into PP V1^R with the following sequence: SS → RS → ME → PP (Figure 2A1). In a second group of embryonic V1^R (25%), 4-AP application only evoked mixed activity, with the same sequence as aforementioned (SS → RS → ME) (data not shown). In the remaining SS V1^R (25%), increasing 4-AP concentration only led to sustained AP firing (Figure 2A2). Application of 300 μM 4-AP on RS V1^R at E12.5 evoked MEs or PPs (Figure 2—figure supplement 2). PPs and RS evoked in the presence of 300 μM 4-AP were fully blocked by 0.5–1 μM TTX, indicating that they were generated by voltage-gated Na⁺ channels (Figure 2B, C, Figure 2—figure supplement 2). It should be noted that the application of 300 μM of 4-AP induced a significant 30.5 ± 12.4% increase (p=0.0137; Wilcoxon test) of the input resistance (1.11 ± 0.08 GΩ versus 1.41 ± 0.12 GΩ; n = 11; N = 11).

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These results show that, in addition to $I_{Nap}$, $I_{Kdr}$ is also a major determinant of the firing pattern of embryonic V1^R. The above suggests that the firing patterns depend on a synergy between $I_{Nap}$ and $I_{Kdr}$ and that the different patterns can be ordered along the following sequence SS → RS → ME → PP when the ratio $G_{Nap}/G_{Kdr}$ is increased.

It was initially unclear whether these different firing patterns corresponded to well-separated classes within the E12.5 V1^R population or not. To address this question, we performed a hierarchical cluster analysis on 163 embryonic V1^R based on three quantitative parameters describing the firing pattern elicited by the depolarizing pulse: the mean duration of evoked APs or PPs measured at half-amplitude (mean ½Ad), the variability of the event duration during repetitive firing (coefficient of variation of ½Ad [CV ½Ad]), and the total duration of all events, expressed in percentage of the pulse duration (depolarizing duration ratio [ddr]) (Figure 3A insets). In view of the large dispersion of mean ½Ad and ddr values, cluster analysis was performed using the (decimal) logarithm of these two quantities (Sigworth and Sine, 1987). The analysis of the distribution of log mean ½Ad, CV ½Ad, and log ddr revealed multimodal histograms that could be fitted with several Gaussians (Figure 3—figure supplement 1A1–C1). Cluster analysis based on these three parameters showed that the most likely number of clusters was 5 (Figure 3A, B), as determined by the silhouette width measurement (Figure 3B). Two clearly separated embryonic V1^R groups with CV ½Ad = 0 stood out, as shown in the 3D plot in Figure 3C. The cluster with the largest ½Ad (mean ½Ad = 833.5 ± 89.99 ms) and the largest ddr (0.441 ± 0.044) contained all PP V1^R (n = 35; N = 29) (Figure 3C, D, Figure 3—figure supplement 1A2, C2). Similarly, the cluster with the shortest ½Ad (9.73 ± 0.66 ms) and the lowest ddr (0.0051 ± 0.0004) contained all SS V1^R (n = 46; N = 37) (Figure 3C, D, Figure 3—figure supplement 1A2, C2).

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Figure 3—source data 1

Numerical data used to perform cluster analysis shown in Figure 3.

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The three other clusters corresponded to V1^R with nonzero values of CV ½Ad (Figure 3C). A first cluster regrouping all RS V1^R (n = 69; N = 61) was characterized by smaller values of ½Ad (23.91 ± 1.43 ms), CV ½Ad (27.36 ± 1.64%), and ddr (0.11 ± 0.01) (Figure 3C–D, Figure 3—figure supplement 1A2, C2). The last two clusters corresponded to ME V1^R (Figure 3C, D). The smaller cluster, characterized by a larger CV ½Ad (170.9 ± 8.9%; n = 4; N = 4), displayed a mix of APs and short PPs, while the second cluster, with smaller CV ½Ad (87.61 ± 7.37%; n = 9; N = 9), displayed a mix of APs and long-lasting PPs (Figure 3D, Figure 3—figure supplement 1B2). Their ½Ad and ddr values were not significantly different (Figure 3—figure supplement 1A2, C2).

It must be noted that three embryonic V1^R (1.8%) were apparently misclassified since they were aggregated within the RS cluster although having zero CV ½Ad (Figure 3C, arrows). Examination of their firing pattern revealed that this was because they generated only two APs, although their ddr (0.16–0.2) and ½ Ad values (31.6–40.3 ms) were well in the range corresponding tto the RS cluster.

These different firing patterns of V1^R might reflect different states of neuronal development (Gao and Ziskind-Conhaim, 1998; Ramoa and McCormick, 1994; Belleau and Warren, 2000; Picken Bahrey and Moody, 2003). SS and/or PPs are generally believed to be the most immature forms of firing pattern, RS constituting the most mature form (Spitzer, 2006; Tong and McDearmid, 2012). If it were so, the firing patterns of embryonic V1^R would evolve during embryonic development from SS or PP to RS, this latter firing pattern becoming the only one in neonates (Perry et al., 2015) and at early postnatal stages (Bikoff et al., 2016). However, RS neurons already represent 41% of V1^R at E12.5. We therefore analyzed the development of firing patterns from E11.5, when V1^R terminate their migration and reach their final position (Alvarez et al., 2013), to E16.5. This developmental period covers a first phase of development (E11.5–E14.5), where lumbar spinal networks exhibit SNA, and a second phase (E14.5–E16.5), where locomotor-like activity emerges (Myers et al., 2005; Yvert et al., 2004; Allain et al., 2010; Branchereau et al., 2000). We first analyzed changes in the intrinsic properties (input capacitance $C_{in}$, input resistance $R_{in}$ = ${1/G}_{in}$, and spike voltage threshold) of V1^R. $C_{in}$ did not change significantly from E11.5 to E13.5 (Figure 4A1), remaining of the order of 12 pF, in agreement with our previous work (Boeri et al., 2018). However, it increased significantly at the transition between the two developmental periods (E13.5–E15.5) to reach about 23.5 pF at E15.5 (Figure 4A1). A similar developmental pattern was observed for $R_{in}$, which remained stable during the first phase from E11.5 to E14.5 ($R_{in}$ ≈ 1–1.2 GΩ) but decreased significantly after E14.5 to reach about 0.7 GΩ at E15.5 (Figure 4A2). Spike threshold also decreased significantly between the first and the second developmental phases, dropping from about –34 mV at E12.5 to about –41 mV at E16.5 (Figure 4A3). Interestingly, this developmental transition around E14.5 corresponds to the critical stage at which SNA gives way to a locomotor-like activity (Yvert et al., 2004; Allain et al., 2010; Branchereau et al., 2000) and rhythmic activity becomes dominated by glutamate release rather than acetylcholine release (Myers et al., 2005).

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This led us to hypothesize that this developmental transition could be also critical for the maturation of V1^R firing patterns. The distinct firing patterns observed at E12.5 were already present at E11.5 (Figure 4B1, C), but the percentage of RS V1^R strongly increased from E11.5 to E12.5, while the percentage of ME V1^R decreased significantly (Figure 4C). The heterogeneity of V1^R firing patterns then substantially diminished. PPs were no longer observed at E13.5 (Figure 4B2, C) and ME V1^R disappeared at E14.5 (Figure 4B3, C). Interestingly, the proportion of SS V1^R remained high from E13.5 to E15.5 and even slightly increased (91.23% at E14.5% and 93.33% at E15.5; Figure 4C). This trend was partially reversed at E16.5 as the percentage of RS V1^R increased at the expense of SS V1^R (67.86% SS V1^R and 32.34% RS V1^R; Figure 4B5, C). This decrease in repetitive firing capability after E13.5 was surprising in view of what is classically admitted on the developmental pattern of neuronal excitability (Moody and Bosma, 2005; Spitzer et al., 2000). Therefore, we verified that it did not reflect the death of some V1^R after E13.5. Our data did not reveal any activated caspase3 (aCaspase3) staining in V1^R (FoxD3 staining) at E14.5 (n = 10 SCs; N = 10) (Figure 5), in agreement with previous reports showing that developmental cell death of V1^R does not occur before birth (Prasad et al., 2008).

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To determine whether $G_{Nap}$ and $G_{Kdr}$ also controlled the firing pattern of V1^R at E14.5 (see Figure 4B3, C), we assessed the presence of $I_{Nap}$ and $I_{Kdr}$ in SS V1^R at this embryonic age. Both $I_{Nap}$ and $I_{Kdr}$ were present in V1^R at E14.5 (Figure 6—figure supplement 1, Figure 6—figure supplement 2), whereas, as in V1^R at E12.5, no calcium-dependent potassium current was detected at this developmental age (not shown). In SS V1^R, $G_{Kdr}$ was significantly higher at E14.5 (11.11 ± 1.12 nS, n = 10; N = 10) than at E12.5 (Figure 1D). In contrast, $G_{Nap}$ was similar at E14.5 (0.13 ± 0.14 nS, n = 10; N = 10) and E12.5 (Figure 1E). We also found that the $G_{Nap}$/$G_{Kdr}$ ratio was significantly lower for SS V1^R recorded at E14.5 (0.012 ± 0.004, n = 10) compared to RS V1^R (0.154 ± 0.022, n = 8) and PP V1^R (0.66 ± 0.132, n = 6) recorded at E12.5 (Figure 1F).

We tested the effect of 4-AP in SS V1^R at E14.5. At this embryonic age, 300 μM 4-AP inhibited only 59.2% of $I_{Kdr}$. Increasing 4-AP concentration to 600 μM did not inhibit $I_{Kdr}$ significantly more (60.2%) (Figure 6—figure supplement 2), indicating that inhibition of $I_{Kdr}$ by 4-AP reached a plateau at around 300 μM. 600 μM 4-AP application had no significant effect on $I_A$ (Figure 6—figure supplement 2). The application of the maximal concentration of 4-AP tested (600 μM) converted SS V1^R (n = 13; N = 13) to PP V1^R (23.1%; Figure 6A1, B), RS V1^R (38.5%; Figure 6A2, B), or ME V1^R (38.4%; Figure 6B), as was observed at E12.5, thus indicating that the firing pattern of V1^R depends on the balance between $I_{Nap}$ and ${\displaystyle I_{Kdr}}$ also at E14.5. PP and RS recorded in the presence of 4-AP at E14.5 were fully blocked by 0.5–1 μM TTX, indicating that they were generated by voltage-gated sodium channels (Figure 6A1, A2), as observed at E12.5.

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As shown in Figure 7A for 26 cells, in which both $G_{Nap}$ and $G_{Kdr}$ were measured, the three largest clusters revealed by the hierarchical clustering analysis (SS, RS, and PP, which account together for the discharge of more than 95% of cells, see Figure 3) correspond to well-defined regions of the $G_{Nap}$ - $G_{Kdr}$ plane. SS is observed only when $G_{Nap}$ is smaller than 0.6 nS. For larger values of $G_{Nap}$, RS occurs when $G_{Kdr}$ is larger than 3.5 nS, and V1^R display PPs when $G_{Kdr}$ is smaller than 3.5 nS. ME (4.5% of the 163 cells used in the cluster analysis), where plateaus and spiking episodes alternate, are observed at the boundary of RS and PP clusters. This suggested to us that a conductance-based model incorporating only the leak current, $I_{Nat}$, $I_{Nap}$, and $I_{Kdr}$ (see Materials and methods), could account for most experimental observations, the observed zonation being explained in terms of bifurcations between the different stable states of the model. Therefore, we first investigated a simplified version of the model without $I_A$ and slow inactivation of $I_{Nap}$.

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A one-parameter bifurcation diagram of this 'basic' model is shown in Figure 7B for two values of $G_{Kdr}$ (2.5 nS and 10 nS) and a constant injected current $I$ = 20 pA. In both cases, the steady-state membrane voltage (stable or unstable) and the peak and trough voltages of stable and unstable periodic solutions are shown as a function of the maximal conductance $G_{Nap}$ of the $I_{Nap}$ current, all other parameters being kept constant. For $G_{Kdr}$ = 10 nS, the steady-state membrane voltage progressively increases (in gray) with $G_{Nap}$, but RS (in red, see voltage trace for $G_{Nap}$ = 1.2 nS) is not achieved until $G_{Nap}$ reaches point SN₁, where a saddle node (SN) bifurcation of limit cycles occurs. This fits with the experimental data, where a minimal value of $G_{Nap}$ is required for RS (see also Boeri et al., 2018), and is in agreement with the known role of $I_{Nap}$ in promoting repetitive discharge (Taddese and Bean, 2002; Kuo et al., 2006). Below SN_1, the model responds to the onset of a current pulse by firing only one spike before returning to quiescence (see voltage trace for $G_{Nap}$ = 0.2 nS) or a few spikes when close to SN₁ (not shown) before returning to quiescence. The quiescent state becomes unstable through a subcritical Hopf bifurcation (HB) at point HB₁, with bistability between quiescence and spiking occurring between SN₁ and HB₁ points. Repetitive firing persists when $G_{Nap}$ is increased further and eventually disappears at point SN₂. The firing rate does not increase much throughout the RS range (Figure 7—figure supplement 1C), remaining between 11.5 Hz (at SN₁) and 20.1 Hz (at SN₂). A stable plateau appears at point HB₂ through a subcritical HB. The model is bistable between HB₂ and SN₂, with plateau and large amplitude APs coexisting in this range.

The model behaves very differently when $G_{Kdr}$ is reduced to 2.5 nS (gray-blue curve in Figure 7B). It exhibits a unique stable fixed point whatever the value of $G_{Nap}$ is, and the transition from quiescence to plateau is gradual as $G_{Nap}$ is increased. No RS is ever observed. This indicates that the activity pattern is controlled not only by $G_{Nap}$ but also by $G_{Kdr}$. This is demonstrated further in Figure 7C, where $G_{Nap}$ was fixed at 1.2 nS while $G_{Kdr}$ was increased from 0 to 25 nS. The model exhibits a PP until $G_{Kdr}$ is increased past point the subcritical HB point HB₂, RS sets in before at point SN₂ via a SN of limit cycles bifurcation. When $G_{Kdr}$ is further increased, repetitive firing eventually disappears through a SN bifurcation of limit cycles at point SN₁, the quiescent state becomes stable through a subcritical HB at point HB₁, and bistability occurs between these two points. This behavior is in agreement with Figure 7A.

Since both conductances $G_{Nap}$ and $G_{Kdr}$ control the firing pattern of embryonic V1^R cells, we computed a two-parameter bifurcation diagram (Figure 7D), where the stability regions of the different possible activity states and the transition lines between them are plotted in the $G_{Nap}$ - $G_{Kdr}$ plane. The black curves correspond to the bifurcations HB₁ and HB₂ and delimit a region where only repetitive firing occurs. The red curves correspond to the SN bifurcations of periodic orbits associated with the transition from quiescence to firing (SN₁) and the transition from plateau to firing (SN₂). They encompass a region (shaded area) where RS can be achieved but may coexist with SS (between the HB₁ and SN₁ lines) or PP (in the narrow region between the HB₂ and SN₂ lines).

Some important features of the diagram must be emphasized: (1) minimal values of both $G_{Nap}$ (to ensure sufficient excitability) and $G_{Kdr}$ (to ensure proper spike repolarization) are required for RS; (2) SS and PP can be clearly distinguished only when they are separated by a region of RS (see also Figure 7B for $G_{Kdr}$ = 10 nS), otherwise the transition is gradual (Figure 7B for $G_{Kdr}$ = 2.5 nS); and (3) only oblique lines with an intermediate slope cross the bifurcation curve and enter the RS region (e.g., see the red line in Figure 7D). This means that RS requires an appropriate balance between $I_{Nap}$ and $I_{Kdr}$. If the ratio $G_{Nap}$/$G_{Kdr}$ is too large (blue line) or too small (gray line), only PPs or SS will be observed at steady state. This is exactly what is observed in experiments, as shown by the cumulative distribution function of the ratio $G_{Nap}$/$G_{Kdr}$ for the different clusters of embryonic V1^R in Figure 7E (same cells as in Figure 7A). The ratio increases according to the sequence SS → RS → ME → PP, with an overlap of the distributions for SS V1^R and RS V1^R. Note also that the ratio for ME cells (around 0.25) corresponds to the transition between RS and PP (more on this below).

Embryonic V1^R cells display voltage fluctuations that may exceed 5 mV and are presumably due to channel noise. The relatively low number of sodium and potassium channels (of the order of a few thousands) led to voltage fluctuations in the stochastic version of our model comparable to those seen experimentally when the cell was quiescent (top voltage trace in Figure 7D) or when a voltage plateau occurred (bottom trace). Channel noise caused some jitter during RS (middle trace) and induced clearly visible variations in the amplitude of APs. However, RS proved to be very robust and was not disrupted by voltage fluctuations. Altogether, channel noise little alters the dynamics (compare the deterministic voltage traces in Figure 7B and the noisy traces in Figure 7D). This is likely because channel noise has a broad power spectrum and displays no resonance with the deterministic solutions of the model.

The one-parameter bifurcation diagram of the model was not substantially modified when we took $I_A$ into account, as shown in Figure 7—figure supplement 1. It just elicited a slight membrane hyperpolarization, an increase in the minimal value of $G_{Nap}$ required for firing, and a decrease of the firing frequency. The transition from repetitive firing to plateau was not affected because $I_A$ is then inactivated by depolarization.

The bifurcation diagram of Figure 7D accounts qualitatively for the physiological data on V1^R at E12.5 presented in Figure 7A, as shown in Figure 7F where the conductance data of Figure 7A were superimposed on it. However, one must beware of making a more quantitative comparison because the theoretical bifurcation diagram was established for a constant injected current of 20 pA, whereas the current injected in experiments data varied from neuron to neuron and ranged from 10 to 30 pA in the sample shown in Figure 7A. The position of bifurcation lines in the $G_{Nap}$ - $G_{Kdr}$ plane depends not only on the value of the injected current, but on the values chosen for the other parameters, which also vary from cell to cell but were kept at fixed values in the model (Ori et al., 2018). For instance, the diagrams were computed in Figure 7D, F for $G_{in}$ = 1 nS and $C_{in}$ = 13 pF, the median values of the input conductance and capacitance at E12.5, taking no account of the cell-to-cell variations of these quantities. Between E12.5 and E14.5, $C_{in}$, which provides an estimate of the cell size, increases by 38% in average, whereas $G_{in}$ is not significantly modified (see Figure 4). As illustrated in Figure 7G, the two-parameter bifurcation diagram is then shifted upward and rightward compared to Figure 7F because larger conductances are required to obtain the same firing pattern. The observed regression of excitability from E12.5 to E14.5–E15.5 (see Figure 4C) thus comes from a decrease in $G_{Nap}$ density (see presumable developmental trajectories indicated by arrows in Figure 7F) together with a shift of the RS region as cell size increases. As a result, all 10 cells shown in Figure 7G are deeply inside the SS region at E14.5.

It is less straightforward to explain on the basis of our model the experiments where 4-AP changed the firing pattern of SS V1^R (Figure 2). Indeed, the decrease of $G_{Kdr}$ (Figure 7—figure supplement 2), although it may exceed 70% at the higher concentrations of 4-AP we used, is not sufficient by itself to account for the change in the firing pattern of V1^R because data points in the SS cluster will not cross the bifurcation lines between SS and RS (SN₁) and between RS and PP (SN₂) when displaced downward in the $G_{Nap}$ - $G_{Kdr}$ plane. However, 4-AP at a 300 μM concentration also decreased $G_{in}$ (by 23% in average and up to 50% in some neurons), the rheobase current with it, and the current that was injected in cells during experiments was reduced accordingly. When we take into account this reduction of both $G_{in}$ and $I$, the two-parameter bifurcation diagram of the model remains qualitatively the same, but it is shifted leftwards and downwards in the $G_{Nap}$ - $G_{Kdr}$ plane (Figure 7—figure supplement 2). As a consequence, the bifurcation lines between SS and RS (SN1) and between RS and PP (SN₂) are then successively crossed when $G_{Kdr}$ is reduced, in accordance with experimental results.

Our basic model accounts for the firing pattern of 73% of the 163 cells used in the cluster analysis. However, bursting, under the form of recurring plateaus separated by brief repolarization episodes (see a typical trace in Figure 8A, left), was experimentally observed in half of PP V1^R (24 out of 46), and plateaus intertwined with spiking episodes were recorded in the 13 cells of the ME cluster (8% of the total sample, see Figure 8A, right, for a typical example). Recurrent plateaus indicate membrane bistability and require that the $I-V$ curve be S-shaped. This occurs when $G_{Nap}$ is large and $G_{Kdr}$ small (Figure 8B1, B2). However, our basic model lacks a mechanism for switching between quiescent state and plateau, even in this case. Channel noise might induce such transitions, but our numerical simulations showed that this is too infrequent to account for bursting (see voltage trace in Figure 8B1 where the plateau state is maintained despite channel noise).

Figure 8

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To explain recurrent plateaus during a constant current pulse, we have to incorporate in our model an additional slow dynamical process. Therefore, we took into account the slow inactivation of $I_{Nap}$ that is observed in experiments. $I_{Kdr}$ also inactivates slowly but over times that are much longer than the timescale of bursting, which is why we did not take its slow inactivation into account. The one-parameter bifurcation diagram of the basic model without slow inactivation of $I_{Nap}$ is shown in Figure 8C for $G_{Kdr}$ = 5 nS and an injected current reduced to 10 pA (as compared to 20 pA in the previous section), so as to allow for bistability (see Figure 8B2). The $G_{Nap}-V$ curve is then S-shaped, as shown in Figure 8B1, with a bistability region for $G_{Nap}$ between 1.36 and 1.85 nS. This is in contrast with Figure 7B where the $G_{Nap}-V$ curve was monotonic. Adding the slow (de)inactivation of $I_{Nap}$ then causes periodic transitions between up (plateau) and down (quiescent) states, as illustrated by the top voltage trace on the right of Figure 8C, and the model displayed a stable limit cycle (shown in black in the bifurcation diagram on the left of Figure 8C). This mechanism is known as pseudo-plateau or plateau-like bursting (a.k.a. fold-subcritical HB bursting) (Teka et al., 2011). In contrast with square wave bursting (Bertram et al., 1995; Izhikevich, 2000a; Borisyuk and Rinzel, 2005; Rinzel, 1985), where the up state is a stable limit cycle arising from a supercritical Hopf bifurcation (Stern et al., 2008; Osinga and Tsaneva-Atanasova, 2010; Osinga and Tsaneva-Atanasova, 2010; Osinga et al., 2012), the up state here is a stable fixed point (which coexists with an unstable limit cycle). This is why one does not observe bursts of APs separated by quiescent periods as, for instance, observed in postnatal CA1 pyramidal cells (Golomb et al., 2006) and in neurons of neonatal pre-Bötzinger Complex (Del Negro et al., 2002; Rybak et al., 2004), but recurrent plateaus. The duration of the plateaus and repolarization episodes depends on the values of $G_{Nap}$ and $G_{Kdr}$. A voltage-independent time constant ${\displaystyle {\tau }_s}$= 2 ms leads to up and down states of comparable durations (see top-left voltage trace in Figure 8C). In agreement with the bifurcation diagram of Figure 8C, the persistent sodium current inactivates during plateaus (phase 1, see top-right trace in Figure 8C) and de-inactivates during quiescent episodes (phase 3, see top-right trace). Transitions from the up state to the down state occurs when inactivation is maximal (phase 2) and transition from the down state to the up state when it is minimal (phase 4). Adding channel noise preserves bursting but introduces substantial randomness in the duration of plateaus and repolarization episodes (bottom-left voltage trace in Figure 8C). Moreover, it substantially decreases the duration of both plateaus and quiescent episodes by making transition between the two states easier (compare the top and bottom voltage traces on the left, both computed for ${\displaystyle {\tau }_s=2ms}$).

Increasing $G_{Nap}$ (or decreasing $G_{Kdr}$) makes plateaus much longer than quiescent episodes (see bottom-right voltage trace in Figure 8C). This again points out to the fact that the ratio of these two conductances is an important control parameter. We also noted that adding the $I_A$ current lengthened the quiescent episodes (Figure 7—figure supplement 1).

Slow inactivation of $I_{Nap}$ also provides an explanation for mixed patterns, where plateaus alternate with spiking episodes (Figure 8A, right). They take place in our model near the transition between RS and PP, as in experiments (see Figure 8A). Slow inactivation can lead to elliptic bursting, notably when the bifurcation HB is subcritical (Izhikevich, 2000b; Su et al., 2004), which is the case here (Figure 8D). The model then displays a stable limit cycle with alternating plateaus and spiking episodes, arising from crossing the bifurcation points HB₂ and SN₂ back and forth (see bifurcation diagram in Figure 8D and top voltage trace). We note that sufficient de-inactivation of $I_{Nap}$ for triggering a new plateau (phase 1 in the bottom trace of Figure 8D) may be difficult to achieve during spiking episodes because voltage oscillates over a large range, which tends to average out the variations of the inactivation level. If de-inactivation is not sufficient, the model keeps on spiking repetitively without returning to the plateau state. This is what occurs for cells well within the RS region, far away from the RS-PP transition. It also probably explains why it was difficult in many recorded cells to elicit plateaus by increasing the injected current, activation of $I_{Nap}$ induced by the larger current being balanced by the increased inactivation.

Altogether, our study shows that a model incorporating the slow inactivation of $I_{Nap}$ accounts for all the firing patterns displayed by cells from the PP and ME clusters.

It is generally assumed that, during early development, newborn neurons cannot sustain repetitive firing (Pineda and Ribera, 2010; Spitzer et al., 2000). Later on, neurons progressively acquire the ability to fire repetitively, APs become sharper, and neurons eventually reach their mature firing pattern due to the progressive appearance of a panoply of voltage-gated channels with different kinetics (Moody and Bosma, 2005; Pineda and Ribera, 2010; Spitzer et al., 2000). Our results challenge the general view that SS is a more primitive form of excitability (Pineda and Ribera, 2010). Indeed, we show that repetitive firing and PPs dominated at early stages (E11.5–E12.5), while SS was prevailing only later (E13.5–E16.5).

The different V1^R firing patterns observed at E11.5–E12.5 might reflect variability in the maturation level between V1^R at a given developmental stage, as suggested for developing MNs (Vinay et al., 2000; Durand et al., 2015). However, this is unlikely since V1^R transiently lose their ability to sustain tonic firing or PP after E13.5. The heterogeneous discharge patterns of V1^R observed before E13.5 contrast with the unique firing pattern of V1^R at postnatal age (Bikoff et al., 2016). Accordingly, the transient functional heterogeneity of V1^R rather reflects an early initial developmental stage (E11.5–E13.5) of intrinsic excitability.

The physiological meaning of the transient functional involution of V1^R that follows, after E12.5, is puzzling. To our knowledge, such a phenomenon was never described in vertebrates during CNS development. So far, a functional involution was described only for inner hair cells between E16 and P12 (Marcotti et al., 2003a; Marcotti et al., 2003b) and cultured oligodendrocytes (Sontheimer et al., 1989), and it was irreversible. Because most V1^R cannot sustain tonic firing after E12.5, it is likely that their participation to SNA is limited to the developmental period before other GABAergic IN subtypes mature and start to produce GABA and glycine (Allain et al., 2004). Interestingly, embryonic V1^R begin to recover their capability to sustain tonic firing when locomotor-like activity emerges (Myers et al., 2005; Yvert et al., 2004), a few days before they form their recurrent synaptic loop with MNs (around E18.5 in the mouse embryos; Sapir et al., 2004). One possible function of the transient involution between E12.5 and E15.5 could be to regulate the growth of V1^R axons toward their targets. It is indeed known that low calcium fluctuations within growth cones are required for axon growth while high calcium fluctuations stop axon growth and promote growth cone differentiation (Henley and Poo, 2004).

Blockade of $I_{Nap}$ leads to single spiking (Boeri et al., 2018), which emphasizes the importance of this current for the occurrence of repetitive firing and plateau potentials in V1^R at early developmental stages. But these neurons can also switch from one firing pattern to another, when $G_{Kdr}$ is decreased by 4-AP, which emphasizes the importance of $I_{Kdr}$. We found that the main determinant of embryonic V1^R firing pattern is the balance between $G_{Nap}$ and $G_{Kdr}$.

A Hodgkin–Huxley-type model incorporating a persistent sodium current $I_{Nap}$ provided a parsimonious explanation of all five firing patterns recorded in the V1^R population at E12.5. It provided a mathematical interpretation for the clustering of embryonic V1^R shown by the hierarchical analysis and accounted for the effect of 4-AP and riluzole (Boeri et al., 2018) on the discharge. Remarkably, it highlighted how a simple mechanism involving only the two opposing currents $I_{Nap}$ and $I_{Kdr}$, could produce functional diversity in a population of developing neurons. The model explained why minimal $G_{Nap}$ and $G_{Kdr}$ are required for firing, and how a synergy between $G_{Nap}$ and $G_{Kdr}$ controls the firing pattern and accounts for the zonation of the $G_{Nap}-G_{Kdr}$ plane that is observed experimentally.

Taking into account the slow inactivation of $I_{Nap}$ to the model allowed us to explain the bursting patterns displayed by cells of the PP and ME clusters. We showed, in particular, that smooth repetitive plateaus could be explained by a pseudo-plateau bursting mechanism (Teka et al., 2011; Osinga and Tsaneva-Atanasova, 2010). Such bursting scenario has been previously studied in models of endocrine cells (Stern et al., 2008; Tsaneva-Atanasova et al., 2010; Tagliavini et al., 2016) and adult neurons (Oster et al., 2015), but rarely observed in experiments (Chevalier et al., 2016). It contrasts with the more common square wave bursting at firing onset, that is, alternating bursts of APs and quiescent episodes, on which most studies of bursting focused (Golomb et al., 2006; Del Negro et al., 2002; Rybak et al., 2004). Our model can also display such square wave bursting, but this occurs for physiologically unrealistic parameter values, so we did not dwell on that bursting mode that we never observed in embryonic V1^R. The model also provides a mathematical explanation for mixed events, where bursts of APs alternate with plateau episodes. It is due to an elliptic bursting scenario at the RS-PP transition, a firing range that the aforementioned studies did not examine. This further emphasizes the capacity of our simple model to account for a wide diversity of firing patterns.

Pseudo-plateau bursting has also been observed in the embryonic pre-Bötzinger network (Chevalier et al., 2016). However, it is produced there by the calcium-activated nonselective cationic current $I_{CAN}$, while $I_{Nap}$ leads to square wave bursting. Pseudo-plateau bursting, displayed by half of the cells at E16.5, largely disappears at E18.5 because of the change in the balance between $I_{CAN}$ and $I_{Nap}$ during embryonic maturation (Chevalier et al., 2016). Such a scenario cannot account for the variety of discharge patterns observed in embryonic V1^R at the E11.5–12.5 stage of development. Our theoretical analysis and experimental data clearly indicate that the interplay between two opposing currents is necessary to explain all the firing patterns of V1^R. Our model is of course not restricted to embryonic V1^R, but may also apply to any electrically compact cell, the firing activity of which is dominated by $I_{Nap}$ and delayed rectifier potassium currents. This is the case of many classes of embryonic cells in mammals at an early stage of their development. It can also apply to the axon initial segment, where $G_{Nap}$ and $G_{Kdr}$ are known to play the major role in the occurrence of repetitive firing (Kole and Stuart, 2012).

Altogether, our experimental and theoretical results provide a global view of the developmental trajectories of embryonic V1^R (see Figure 7F, G). At E12.5, conductances of embryonic V1^R are widely spread in the $G_{Nap}-G_{Kdr}$ plane, which explains the heterogeneity of their firing patterns. This likely results from the random and uncorrelated expression of sodium and potassium channels from cell to cell at this early stage. Between E12.5 and E14.5–15.5, cell size increases, and $G_{Kdr}$ with it, while the density of sodium channels decreases (see Figures 1 and 4). The functional involution displayed by V1^R between E12.5 and E15.5 thus mainly results from a decrease of $G_{Nap}$ coordinated with an increase of $G_{Kdr}$. How these synergistic processes are controlled during this developmental period remains an open issue.

It is important to note that the presence of $I_{Nap}$ is required for the functional diversity of V1^R. Indeed, in the absence of $I_{Nap}$, V1^R lose their ability to generate plateau potentials or to fire repetitively. More generally, when the diversity of voltage-gated channels is limited, as observed in embryonic neurons (Moody and Bosma, 2005), changes in the balance between $I_{Kdr}$ and non inactivating inward currents can modify the firing pattern. This can be achieved not only by $I_{Nap}$, but also by other slowly or non-inactivating inward conductances, such as $I_{CAN}$ (Chevalier et al., 2016). Our work also clearly indicates that a change in the firing pattern can only occur if a change in inward conductances cannot be counterbalanced by a corresponding change in outward conductances. This implies that there is no homeostatic regulation of channel density to ensure the robustness of V1^R excitability during its early development, contrarily to the mature CNS (O'Leary et al., 2013). In addition, the poor repertoire of voltage-gated channels at this developmental stage precludes channel degeneracy, which is also known to ensure the robustness of excitability in mature neurons (O'Leary et al., 2013).

In conclusion, our study shows that there is no universal pattern of development in embryonic neurons, and it demonstrates that a simple general mechanism involving only two slowly inactivating voltage-gated channels with opposite effects is sufficient to produce a wide variety of firing patterns in immature neurons having a limited repertoire of voltage-gated channels.

Experiments were performed in accordance with European Community guiding principles on the care and use of animals (86/609/CEE, CE Off J no. L358, 18 December 1986), French decree no. 97/748 of 19 October 1987 (J Off République Française, 20 October 1987, pp. 12245–12248). All procedures were carried out in accordance with the local ethics committee of local universities and recommendations from the CNRS. We used Gad1^GFP knock-in mice to visualize putative GABAergic INs (Tamamaki et al., 2003), as in our previous study (Boeri et al., 2018). To obtain E12.5–E16.5 Gad1^GFP embryos, 8–12-week-old wild-type Swiss female mice were crossed with Gad1^GFP Swiss male mice.

Isolated mouse SCs from 420 embryos were used in this work and obtained as previously described (Delpy et al., 2008; Scain et al., 2010). Briefly, pregnant mice were anesthetized by intramuscular injection of a mix of ketamine and xylazine and sacrificed using a lethal dose of CO₂ after embryos of either sex were removed. Whole SCs were isolated from eGFP-positive embryos and maintained in an artificial cerebrospinal fluid (ACSF) containing 135 mM NaCl, 25 mM NaHCO₃, 1 mM NaH₂PO₄, 3 mM KCl, 11 mM glucose, 2 mM CaCl₂, and 1 mM MgCl₂ (307 mOsm/kg H₂O), continuously bubbled with a 95% O₂-5% CO₂ gas mixture.

In the lumbar SC of Gad1^GFP mouse embryos, eGFP neurons were detected using 488 nm UV light. They were localized in the ventrolateral marginal zone between the motor columns and the ventral funiculi (Stam et al., 2012). Embryonic V1^R identity was confirmed by the expression of the forkhead transcription factor Foxd3 (Boeri et al., 2018).

The isolated SC was placed in a recording chamber and was continuously perfused (2 ml/min) at room temperature (RT) (22–26°C) with oxygenated ACSF. Whole-cell patch-clamp recordings of lumbar spinal embryonic V1^R were carried out under direct visualization using an infrared-sensitive CCD video camera. Whole-cell patch-clamp electrodes with a resistance of 4–7 MΩ were pulled from thick-wall borosilicate glass using a P-97 horizontal puller (Sutter Instrument Co., USA). They were filled with a solution containing (in mM): 96.4 K methanesulfonate, 33.6 KCl, 4 MgCl₂, 4 Na₂ATP, 0.3 Na₃GTP, 10 EGTA, and 10 HEPES (pH 7.2; 290 mOsm/kg-H₂O). This intracellular solution led to an equilibrium potential of chloride ions, $E_{Cl}$, of about –30 mV, close to the physiological values measured at E12.5 in spinal MNs (Delpy et al., 2008). The junction potential (6.6 mV) was systematically corrected offline.

Signals were recorded using Multiclamp 700B amplifiers (Molecular Devices, USA). Data were low-pass filtered (2 kHz), digitized (20 kHz) online using Digidata 1440A or 1550B interfaces, and acquired using pCLAMP 10.5 software (Molecular Devices, USA). Analyses were performed offline using pCLAMP 10.5 software packages (Molecular Devices; RRID:SCR_014284) and Axograph 1.7.2 (AxoGraph; RRID:SCR_002798).

In voltage-clamp mode, voltage-dependent K⁺ currents ($I_{Kv}$) were elicited in the presence of 1 μM TTX (Alomone Labs, Cat# T550, CAS No.: 18660-81-6) by 500 ms depolarizing voltage steps (10 mV increments, 10 s interval) after a prepulse of 300 ms at $V_H$ = –100 mV. To isolate $I_{Kdr}$, voltage steps were applied after a 300 ms prepulse at $V_H$ = –30 mV that inactivated the low threshold transient potassium current $I_A$. $I_A$ was then obtained by subtracting offline $I_{Kdr}$ from the total potassium current $I_{Kv}$. Capacitance and leak current were subtracted using online P/4 protocol provided by pCLAMP 10.5.

In current-clamp mode, V1^R discharge was elicited using 2 s depolarizing current steps (from 0 to ≈ 50 pA in 5–10 pA increments, depending on the input resistance of the cell) with an 8 s interval to ensure that the membrane potential returned to $V_H$. When a cell generated a sustained discharge, the intensity of the depolarizing pulse was reduced to the minimal value compatible with repetitive firing.

$I_{Nap}$ was measured in voltage-clamp mode using a 70 mV/s depolarizing voltage ramp (Huang and Trussell, 2008). This speed was slow enough to preclude substantial contamination by the inactivating transient current and fast enough to avoid substantial inactivation of $I_{Nap}$. Subtraction of the current evoked by the voltage ramp in the presence of 1 μM TTX from the control voltage ramp-evoked current revealed $I_{Nap}$.

During patch-clamp recordings, bath application of TTX (1 µM, Alomone Labs, Cat# T550, CAS No.: 18660-81-6) or 4-AP (Sigma-Aldrich Cat# T550, CAS No.: 18660-81-6) was done using 0.5 mm diameter quartz tubing positioned, under direct visual control, 50 µm away from the recording area. The quartz tubing was connected to six solenoid valves linked with six reservoirs via a manifold. Solutions were gravity-fed into the quartz tubing. Their application was controlled using a VC-8 valve controller (Warner Instruments, USA).

4-AP was used to block $I_{Kdr}$. To determine the concentration–response curve, $I-V$ curves of $I_{Kdr}$ for different concentrations of 4-AP (0.3–300 μM) were compared to the control curve obtained in the absence of 4-AP. The percentage of inhibition for a given concentration was calculated by dividing the peak intensity of $I_{Kdr}$ by the peak value obtained in control condition. The obtained normalized concentration–response curves were fitted using the Hill equation:

where [4-AP] is the 4-AP concentration, $I_{min}$ is the residual current (in percentage of the peak $I_{Kdr}$), ${100-I}_{min}$ is the maximal inhibition achieved for saturating concentration of 4-AP, ${IC}_{50}$ is the 4-AP concentration producing half of the maximal inhibition, and $n_H$ is the Hill coefficient. Curve fitting was performed using KaleidaGraph 4.5 (Synergy Software, USA).

E14.5 embryos were collected from pregnant females. Once dissected out of their yolk sac, SCs were dissected and immediately immersion-fixed in phosphate buffer (PB 0.1 M) containing 4% paraformaldehyde (PFA; freshly prepared in PB, pH 7.4) for 1 hr at 4°C. Whole SCs were then rinsed out in 0.12 M PB at 4°C, thawed at RT, washed in PBS, incubated in NH₄Cl (50 mM), diluted in PBS for 20 min, and then permeabilized for 30 min in a blocking solution (10% goat serum in PBS) with 0.2% Triton X-100. They were incubated for 48 hr at 4°C in the presence of the following primary antibodies: guinea pig anti-FoxD3 (1:5000, gift from Carmen Birchmeier and Thomas Müller of the Max Delbrück Center for Molecular Medicine in Berlin) and rabbit anti-cleaved Caspase-3 (1:1000, Cell Signaling Technology Cat# 9661, RRID:AB_2341188). SCs were then washed in PBS and incubated for 2 hr at RT with secondary fluorescent antibodies (goat anti-rabbit-conjugated 649; donkey anti-guinea pig-conjugated Alexa Fluor 405 [1:1000, ThermoFisher]) diluted in 0.2% Triton X-100 blocking solution. After washing in PBS, SCs were dried and mounted in Mowiol medium (Millipore, Molsheim, France). Preparations were then imaged using a Leica SP5 confocal microscope. Immunostaining was observed using a 40× oil-immersion objective with a numerical aperture of 1.25, as well as with a 63× oil-immersion objective with a numerical aperture of 1.32. Serial optical sections were obtained with a Z-step of 1 µm (40×) and 0.2–0.3 µm (63×). Images (1024 × 1024; 12-bit color scale) were stored using Leica software LAS-AF and analyzed using ImageJ 1.5 (N.I.H., USA, RRID:SCR_003070) and Adobe Photoshop CS6 (Adobe, USA, RRID:SCR_014199) software.

To classify the firing patterns of embryonic V1^R, we performed a hierarchical cluster analysis on a population of 163 cells. Each cell was characterized by three quantitative measures of its firing pattern (see legend of Figure 5). After normalizing these quantities to zero mean and unit variance, we performed a hierarchical cluster analysis using the hclust function in R 3.3.2 software (Cran project; https://cran.r-project.org/; RRID:SCR_001905) that implements the complete linkage method. The intercluster distance was defined as the maximum Euclidean distance between the points of two clusters, and, at each step of the process, the two closest clusters were merged into a single one, thus constructing progressively a dendrogram. Clusters were then displayed in data space using the dendromat function in the R package ‘squash’ dedicated to color-based visualization of multivariate data. The best clustering was determined using the silhouette measure of clustering consistency (Rousseeuw, 1987). The silhouette of a data point, based on the comparison of its distance to other points in the same cluster and to points in the closest cluster, ranges from −1 to 1. A value near 1 indicates that the point is well assigned to its cluster, a value near 0 indicates that it is close to the decision boundary between two neighboring clusters, and negative values may indicate incorrect assignment to the cluster. This allowed us to identify an optimal number k of clusters by maximizing the overall average silhouette over a range of possible values for k (Rousseeuw, 1987) using the silhouette function in the R package ‘cluster’.

To understand the relationship between the voltage-dependent membrane conductances and the firing patterns of embryonic V1^R, we relied on a single-compartment conductance-based model that included the leak current, the transient and persistent components of the sodium current, $I_{Nat}$ and $I_{Nap}$, a delayed rectifier potassium current $I_{Kdr}$, and the inactivating potassium current $I_A$ revealed by experiments. Voltage evolution then followed the equation

where $C_{in}$ is the input capacitance; $G_{in}$ the input conductance; $G_{Nat}$, $G_{Nap}$, $G_{Kdr}$, and $G_A$ the maximal conductances of the aforementioned currents; $m,m_p$, ${\displaystyle n}$, and $m_A$ their activation variables; $h$ the inactivation variable of $I_{Nat}$; $s$ the slow inactivation variable of $I_{Nap}$; and ${\displaystyle h_A}$ the inactivation variable of $I_A$. $V_r$ is the baseline potential imposed by ad hoc current injection in current-clamp experiments; $E_{Na}$ and $E_K$ are the Nernst potentials of sodium and potassium ions, and $I$ the injected current. All gating variables satisfied equations of the form

where the (in)activation curves were modeled by a sigmoid function of the form

with $k_x$ being positive for activation and negative for inactivation. The time constant ${\tau }_x$ was voltage-independent except for the inactivation variables $h$ and $s$. The activation variable $m_A$ of $I_A$ was assumed to follow instantaneously voltage changes.

The effect of channel noise was investigated with a stochastic realization of the model, where channels kinetics were described by Markov-type models, assuming a unitary channel conductance of 10 pS for all channels.

Most model parameters were chosen on the basis of experimental measurements performed in the present study or already reported (Boeri et al., 2018). Parameters that could not be constrained from our experimental data were chosen from an experimentally realistic range of values. ${V_r\mathrm{}}_{}$ was set at –60 mV as in experiments (see Table 1). $C_{in}$ (average 13.15 pF, 50% between 11.9 and 15.1 pF, only 18 cells out of 246 in the first quartile below 7.2 pF or in the fourth quartile above 19 pF) and $G_{in}$ (50% of cells between 0.71 and 1.18 nS, only 7 out of 242 with input conductance above 2 nS) were not spread much in the cells recorded at E12.5, which showed that most embryonic V1^R were of comparable size. Interestingly, $C_{in}$ and $G_{in}$ were not correlated, which indicated that the input conductance was determined by the density of leak channels rather than by the sheer size of the cell. Moreover, no correlation was observed between the passive properties and the firing pattern (Boeri et al., 2018). Therefore, we always set $G_{in}$ and $C_{in}$ to 1 nS and 13 pF in the model (except in Figure 6—figure supplement 2), close to the experimental medians (0.96 nS and 13.15 pF, respectively). The membrane time constant $C_{in}/G_{in}$ was then equal to 13 ms, which was also close to the experimental median (13.9 ms, N = 241).

$E_{Na}$ was set to 60 mV (see Boeri et al., 2018). The activation curve of $I_{Nap}$ was obtained by fitting experimental data, leading to an average mid-activation of –36 mV and an average steepness of 9.5 mV. The experimentally measured values of $G_{Nap}$ were in the range 0–2.2 nS. We assumed that the activation curve of $I_{Nat}$ was shifted rightward by 10 mV in comparison to $I_{Nap}$. No experimental data was available for the inactivation of $I_{Nat}$. We chose a mid-inactivation voltage $V_h$ = –45 mV and a steepness $k_h$ = –5 mV. We also assumed that the activation time constant of both $I_{Nat}$ and $I_{Nap}$ was 1.5 ms, and that the inactivation time constant was voltage-dependent: ${\tau }_h\left(V\right)=16.5-13.5\tanh \left((V+20)/15\right)$, decreasing by one order of magnitude (from 30 ms down to 3 ms) with the voltage increase. This enabled us to account for the shape of the APs recorded in experiments, showing a slow rise time and rather long duration. The conductance $G_{Nat}$ was not measured experimentally. When choosing a reasonable value of 20 nS for $G_{Nat}$, the model behaved very much as recorded embryonic V1^R: with similar current threshold (typically 10–20 pA) and stable plateau potentials obtained for the largest values of $G_{Nap}$.

When taking into account slow inactivation of $I_{Nap}$ (see Figure 8), we chose $V_s$ = –30 mV for the mid-inactivation voltage and set the steepness $k_s$ at –5 mV (as for the inactivation of $I_{Nat}$). For simplicity, we assumed that the inactivation time constant was voltage-independent and set at 2 s.

$E_K$ was set to the experimentally measured value of –96 mV (Boeri et al., 2018). The activation parameters of $I_{Kdr}$ were obtained by fitting the experimental data: $V_n$ = –20 mV, $k_n$ = 15 mV, ${\tau }_n$ = 10 ms, and an activation exponent of 3. The activation and inactivation properties of $I_A$ were also chosen based on experimental measurements. Accordingly, ${\displaystyle V_{m_A}}$ = –30 mV, ${\displaystyle k_{m_A}}$ = –12 mV, ${\displaystyle V_{h_A}}$ = –70 mV, ${k_h}_A$ = –7 mV, and ${\displaystyle {\tau }_{h_A}}$ = 23 ms. When $I_A$ was taken into account, we assumed that $G_A=G_{Kdr}$, consistently with experimental data (see Figure 6—figure supplement 1).

We integrated numerically the deterministic model using the freeware XPPAUT 8.0 (University of Pittsburgh; Pennsylvania; USA; RRID:SCR_001996) (Ermentrout, 2002) and a standard fourth-order Runge–Kutta algorithm. XPPAUT was also used to compute one-parameter and two-parameter bifurcation diagrams. The stochastic version of the model was also implemented in XPPAUT and computed with a Gillespie’s algorithm (Gillespie, 1976).

To investigate the dynamics of the model with slow inactivation of $I_{Nap}$, we relied on numerical simulations together with fast/slow dynamics analysis (Witelski and Bowen, 2015). In this approach, one distinguishes slow dynamical variables (here only s) and fast dynamical variables. Slow variables vary little at the time scale of fast variables and may therefore be considered as constant parameters of the fast dynamics in first approximation. In contrast, slow variables are essentially sensitive to the time average of the fast variables, much more than to their instantaneous values. This separation of time scales allows one to conduct a phase plane analysis of the full dynamics.

Samples sizes (n) were determined based on previous experience. The number of embryos (N) is indicated in the main text and figure captions. No power analysis was employed, but sample sizes are comparable to those typically used in the field. All values were expressed as mean with standard error of mean (SEM). Statistical significance was assessed by non-parametric Kruskal–Wallis test with Dunn’s post hoc test for multiple comparisons, Mann–Whitney test for unpaired data or Wilcoxon matched pairs test for paired data using GraphPad Prism 7.0e Software (USA). Significant changes in the proportions of firing patterns with age were assessed by chi-square test for large sample and by Fisher's exact test for small sample using GraphPad Prism 7.0e Software (GraphPad Software; RRID:SCR_002798). Significance was determined as *p<0.05, **p<0.01, or ***p<0.001. The exact p values are mentioned in the result section or in the figure captions.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Susanne Bolte, Jean-François Gilles, and France Lam for assistance with confocal imaging (IBPS imaging facility) and IBPS rodent facility team for animal care and production. We thank University Paris Descartes for hosting Yulia Timofeeva as an invited professor. This work was supported by INSERM, CNRS, Sorbonne Université (Paris), Université de Bordeaux, Université Paris Descartes, and Fondation pour la Recherche Médicale.

Animal experimentation: Experiments were performed in accordance with European Community guiding principles on the care and use of animals (86/609/CEE, CE Off J no. L358, 18 December 1986), French decree no. 97/748 of October 19, 1987 (Journal Officiel République Française, 20 October 1987, pp. 12245-12248). All procedures were carried out in accordance with the local ethics committee of local Universities and recommendations from the CNRS. pregnant mice were anesthetized by intramuscular injection of a mix of ketamine and xylazine and sacrificed using a lethal dose of CO2 after embryos of either sex were removed. Every effort was made to minimize suffering.

© 2021, Boeri et al.

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