Heterogeneous firing rate response of mouse layer V pyramidal neurons in the fluctuation-driven regime

Abstract

Key points: We recreated in vitro the fluctuation-driven regime observed at the soma during asynchronous network activity in vivo and we studied the firing rate response as a function of the properties of the membrane potential fluctuations. We provide a simple analytical template that captures the firing response of both pyramidal neurons and various theoretical models. We found a strong heterogeneity in the firing rate response of layer V pyramidal neurons: in particular, individual neurons differ not only in their mean excitability level, but also in their sensitivity to fluctuations. Theoretical modelling suggest that this observed heterogeneity might arise from various expression levels of the following biophysical properties: sodium inactivation, density of sodium channels and spike frequency adaptation.

Abstract: Characterizing the input-output properties of neocortical neurons is of crucial importance for understanding the properties emerging at the network level. In the regime of low-rate irregular firing (such as in the awake state), determining those properties for neocortical cells remains, however, both experimentally and theoretically challenging. Here, we studied this problem using a combination of theoretical modelling and in vitro experiments. We first identified, theoretically, three somatic variables that describe the dynamical state at the soma in this fluctuation-driven regime: the mean, standard deviation and time constant of the membrane potential fluctuations. Next, we characterized the firing rate response of individual layer V pyramidal cells in this three-dimensional space by means of perforated-patch recordings and dynamic clamp in the visual cortex of juvenile mice in vitro. We found that individual neurons strongly differ not only in terms of their excitability, but also, and unexpectedly, in their sensitivities to fluctuations. Finally, using theoretical modelling, we attempted to reproduce these results. The model predicts that heterogeneous levels of biophysical properties such as sodium inactivation, sharpness of sodium activation and spike frequency adaptation account for the observed diversity of firing rate responses. Because the firing rate response will determine population rate dynamics during asynchronous neocortical activity, our results show that cortical populations are functionally strongly inhomogeneous in young mouse visual cortex, which should have important consequences on the strategies of cortical computation at early stages of sensory processing.

Figure 1. Investigating somatic computation in the fluctuation‐driven regime

A, schematic illustration on a layer V pyramidal cell in cat V1 (Contreras et al. 1997) together with theoretical distribution of inhibitory (red) and excitatory (green) synaptic inputs. Synaptic and dendritic integration of presynaptic spike trains (two sample spike trains in upper right) produce membrane potential fluctuations at the soma as recorded intracellularly in current clamp. B, a sample trace of membrane potential integrating inputs from all inhibitory and excitatory synapses (black trace). We characterize those fluctuations by a mean ${\mu }_V$ (dotted horizontal line), a standard deviation ${\sigma }_V$ of sub‐threshold oscillations (grey background) and global autocorrelation time ${\tau }_V$ (determined from the normalized autocorrelation function in the inset; see Methods). The properties of those fluctuations determine the spiking probability of the neuron (three spikes visible in the membrane potential trace).

Figure 2

The analytical template (eqns (18) and (9)) can capture the firing rate response of various theoretical models Shown for the leaky integrate‐and‐fire model (LIF) with V _thre = −47 mV (kept for all following models), the exponential integrate‐and‐fire model (EIF) with $k_a$ = 2 mV, the LIF model with spike‐frequency adaptation only (sfaLIF) with b = 20pA, the inactivating leaky integrate‐and‐fire model (iLIF) with $a_i$ = 0.6 and the iAdExp model that combines all the previously mentioned mechanisms with $k_a$ = 2 mV, b = 6pA and $a_i$ = 0.6. A, response of the models to a current step. Continuous line: response to depolarizing current step, dashed line: response to hyperpolarizing current step. For the iLIF and iAdExp models, we show in red the dynamics of the threshold $\theta \left(t\right)$. B, firing rate response in the $\left({\mu }_V,{\sigma }_V\right)$ space. Various colours indicate various levels of the global autocorrelation ratio ${\tau }_V^N={\tau }_V/{\tau }_{\mathrm{m}}^0$. C, projections along the standard deviation ${\sigma }_V$ axis for different mean polarization levels ${\mu }_V$. Data (points) and fitted analytical template (thick transparent lines) are shown. Note the shifts in the scanned $\left({\mu }_V,{\sigma }_V,{\tau }_V^N\right)$ domain to reach a comparable firing range despite a reduced excitability (see main text). D, phenomenological threshold $V_{\mathrm{thre}}^{\mathrm{eff}}$ that leads to the fitted firing rate response; the coefficients of the linear functions can be found in Table 1.

Figure 3. Extracting the mean properties of a single neuron response: excitability and sensitivities to the variables of the fluctuation‐driven regime

A, illustration for the LIF model. From the fitting procedure we obtain an analytical description of the firing rate response (centre plot). We focus the analysis on the domain $\mathcal{D}$ of the low‐rate fluctuation‐driven regime (see main text); its extent is delimited by the white square in the bottom and top‐right insets. The mean phenomenological threshold in the $\mathcal{D}$ domain quantifies the excitability (top left: large dashed line). Then for each variable (top right: ${\tau }_V^N$, bottom left: ${\sigma }_V$, bottom right: ${\mu }_V$), we show the projections of the firing response along this dimension for different combinations of the two other variables within the $\mathcal{D}$ domain (dotted lines). The mean derivative (represented by arc angle and large dashed line) with respect to the variable in x‐axis over different combinations of the remaining variables quantifies the mean sensitivity to this variable. B, excitabilities for the LIF, EIF, sfaLIF and iLIF models (parameters as in Fig. 2). C, mean sensitivities to ${\mu }_V$. D, mean sensitivities to ${\sigma }_V$. E, mean sensitivities to ${\tau }_V^N$ for the four models.

Figure 4. The exploration of a physiologically relevant space in layer 5 pyramidal neurons of juvenile mice

B to E, an example of a single cell. A, a typical layer 5 pyramidal neuron in the primary visual cortex of juvenile mice. Picture from additional experiments: marking with 2% Biocytin (Sigma Aldrich) in whole cell configuration. B, after diffusion of the perforant molecule toward the patch of membrane, a step voltage clamp protocol estimates the quality of the seal and perforation (see details in Methods). C, a step current clamp protocol estimate of the passive membrane properties. Those properties are used by the stimulation protocols to constrain the V _m fluctuations (see Methods). D, throughout the recording, we monitor the cellular properties: the resting membrane potential E _L, the membrane resistance R _m and the variations of the firing rate with respect to the stationary behaviour (see details in the main text and in Methods). The smoothed data (red curve) shows the global trend; it removes the measurement error due to the short sampling time for R _m and E _L (see E), for CV_ν, it removes the effect of the intrinsic spiking irregularity. For the $\mathrm{C}{\mathrm{V}}_{\nu }$curve we have added the standard deviation (mean ± SD in red) for comparison with a stationary Poisson process (mean ± SD in black). E, sample of the membrane potential V _m, the injected current I and the total conductance ${\mu }_G$ at the beginning of the recording (left, t = 17.7 min, blue star in D) and after 1 h (right, t = 71.8 min, red star in D). Within an episode, we scan one combination of the $\left({\mu }_V,{\sigma }_V,{\tau }_V^N\right)$ variables. For example, the middle episode corresponds to the most depolarized level ${\mu }_V$ (hence the lower spike amplitude due to sodium inactivation) with the fastest fluctuations ${\tau }_V^N$ (i.e. a high input conductance μ, also shunting the spikes; see the strong opposite current) and an intermediate variance ${\sigma }_V$. In between the two first episodes, one can see a rest period (to monitor E _L) followed by a current pulse (to monitor R _m).

Figure 5. Characterization of the firing rate response of the recorded neocortical pyramidal neurons

A, four examples of the firing rate response of single neurons, showing data (diamonds, error bars indicate variability estimated as the standard deviation from responses to multiple trials where available) and fitted template function (continuous line); the cells are indexed from 1 to 4 to identify them in the heterogeneity analysis (Fig. 6). B, for 21 neurons scanned with at least of 70 different combinations of input statistics, we split the dataset into two and investigated the similarity of the coefficients between the two subsets. The relatively high and significant (P < 0.05, Pearson correlation) correlation coefficients between characterizations in the first and second datasets indicate a robust characterization of the firing rate response.

Figure 6. Heterogeneity and underlying structure of the firing response of neocortical cells

A, histogram of data from recorded cells showing the mean excitabilities and sensitivities to the variables of the fluctuations. The dashed coloured lines show the values of theoretical models for comparison. B, scatter plot of the mean excitability and sensitivities to the variables of the fluctuation‐driven regime; the cells shown in Fig. 5 are highlighted with larger markers. C, principal component analysis; the inset show the vector coordinates of the two first components.

Figure 7. Variations in the expression of biophysical mechanisms explain the observed cellular heterogeneity in their firing rate response

A, increasing the threshold V _thre of the LIF model. Note that this only affects the excitability and negligibly the sensitivities to ${\mu }_V$, ${\sigma }_V$ and ${\tau }_V^N$. B, decreasing the sharpness of the sodium activation curve in the EIF model, $k_a$ = 0 mV corresponds to the LIF model, $k_a$ = 3.7 mV corresponds to a very smooth activation. Note the strong impact on the sensitivity to ${\tau }_V^N$. C, increasing spike frequency adaptation in the sfaLIF model, $b=$0 mV corresponds to the LIF model, $b=$35pA corresponds to a strongly adapting model. Note the concomitant variations of the sensitivities to ${\mu }_V$ and ${\sigma }_V$. D, increasing sodium inactivation in the iLIF model, $a_i$ = 0 corresponds to the LIF model, $a_i$ = 0.7 corresponds to a strongly inactivating model. Note the strong impact on the increase in sensitivity to ${\sigma }_V$ and ${\tau }_V^N$. E, histogram of the data from the n = 30 neurons for the four characteristics of the firing rate responses.