Calculating event-triggered average synaptic conductances from the membrane potential 

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Test of the accuracy of the method using numerical simulations

To test this method, we first considered numerical simulations of the IF model in four different situations. We distinguished high-conductance states, where the total conductance is dominated by inhibition, from low-conductance states, where both synaptic conductances are of comparablemagnitude. We also varied the standard deviations of the conductances such that for both high- and low-conductance  states we have the cases si > se as well as se > si. The results are summarized in Fig. 3.1, where the STA traces of excitatory and inhibitory conductances recorded from simulations are compared to the most likely (equivalent to the average) conductance traces obtained from solving Eq. (3.10). In general, the plots demonstrate a very good agreement. To quantify our results, we investigated the effect of the statistics as well as of the broadness of the conductance distributions on the quality of the estimation. The latter is crucial, because the derivation of the most likely conductance time course allows for negative conductances, whereas in the simulations negative conductances lead to numerical instabilities, and conductances are bound to positive values. We thus expect an increasing error with increasing ratio SD/mean of the conductance distributions. We estimated the root-mean-square (RMS) of the difference between the recorded and the estimated conductance STAs. The results,  summarized in Fig. 3.2, are as expected. Increasing the number of spikes enhances the match between theory and simulation (Fig. 3.2A shows the RMS deviation for excitation, 3.2B for inhibition) up to the point where the effect of negative conductances becomes dominant. In this example, where the ratio SD/mean was fixed at 0.1, the RMS deviation enters a plateau at about 7000 spikes. The plateau values can also be recovered from the neighboring plots ( i.e., the RMS deviations at SD/mean = 0.1 in Fig. 3.2 C and D correspond to the plateau values in A and B). On the other hand, a broadening of the conductance distribution yields a higher  deviation between simulation and estimation. However, at SD/mean = 0.5, the RMS deviation is still as low as ∼ 2% of the mean conductance for excitation and ∼ 4% for inhibition.

Test of the method in real neurons

We also tested the method on voltage STAs obtained from dynamic-clamp recordings of guinea-pig cortical neurons in slices. In real neurons, a problem is the strong influence of spike-related voltage-dependent (presumably sodium) conductances on the voltage time course. Since we maximize the global probability of  ge(t) and gi(t), the voltage in the vicinity of the spike has an influence on the estimated conductances at all times. As a consequence, without removing the effect of sodium, the estimation fails (see Fig. 3.4). Fortunately, it is rather simple to correct for this effect by excluding the last 1–2 ms before the spike from the analysis. The corrected comparison between the recorded and the estimated conductance traces is shown in Fig. 3.5.
Finally, to check for the applicability of this method to in vivo recordings, we assessed the sensitivity of the estimates with respect to the different parameters by varying the values describing passive properties and synaptic activity. We assume that the total conductance can be constrained by input resistance measurements, and that time constants of the synaptic currents can be estimated by power spectral analyses (Destexhe and Rudolph 2004). This leaves gL, C, ge0, se and si as the main parameters. The impact of these parameters on STA conductance estimates is shown in Fig. 3.6. Varying these parameters within ±50% of their nominal value led to various degrees of error in the STA estimates. The dominant effect of a variation in the mean conductances is a shift in the estimated STAs, whereas a variation in the SDs changes the curvature just before the spike.
To address this point further, we fitted the estimated conductance STAs with an exponential function: fs(t) = Gs(1+Kse t−t0 Ts ).

Intracellular recordings in awake and naturally sleeping animals

Intracellular recordings were performed in cat association cortex (areas 5, 7 and 21 of parietal cortex) using a technique described in detail previously (seeMethods in Steriade et al. 2001; Timofeev et al. 2001). Briefly, a chamber specially designed for chronic intracellular recordings, local field potential (LFP), electromyogram (EMG) and electrooculogram (EOG) electrodes as well as head fixation screws, were implanted under anesthesia. Electrodes for LFP recordings were inserted in cortical depth and their tips were located at a depth of 0.8-1.0 mm. The animals were gradually habituated to the recording setup and the head-fixed position. After 3-4 days of training the cats started to display normal sleep-waking cycles and could move their limbs.
Intracellular recordings were performed under current-clamp using standard borosilicate glass micropipettes 1.5/0.84 mm (OD/ID) filled with 2.5 mM potassium acetate. Initially, pipettes were pulled to a resistance of 120-150 MW and under visual guidance their tips were broken to obtain a resistance of 50-70 MW.
The pipettes descended in the recording chamber. After insertion in the brain the resistance of some of the pipettes further decreased. The intracellular electrodes with resistance lower than 30 MW were discarded because intracellular recordings obtained with these electrodes often revealed signs of deterioration appearing as unstable membrane potential, which becomes progressively depolarized, and broadening of action potential duration. Typically the recording sessions lasted up to 3 hours. Only recordings showing stable Vm (no drift) and overshooting action potentials were considered here. To correctly estimate themembrane potential, we recorded the potential drop during penetration of the recorded neuron, and after pulling out of the neuron. If the difference in the estimation of recorded membrane potential at the beginning and at the end of recording exceeded 2 mV, these data were discarded. A total of 118 neurons were recorded during the wake/sleep cycle, including 96 presumed excitatory cells and 22 presumed inhibitory interneurons (see Results).
To accumulate sufficient statistics for the analysis (see below), recordings needed to last typically more than 30 min, and be conducted using several levels of injected DCs. A total of 15 neurons were obtained satisfying these criteria and were used for conductance analysis. In some cases, the recording was long enough to span across several states of vigilance (quiet wakefulness, slow-wave sleep and REM sleep). Experiments were conducted in agreement with ethics guidelines of the Canadian Council on Animal Care and were approved by the committee for animal care of Laval University.

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Analysis and computational models

The basis of the analysis was to match the Vm of the intracellularly-recorded neurons to a stochastic model of synaptic background activity. This model was used for estimating synaptic conductances, to perform spike-triggered averages of synaptic conductance variations related to spikes, as well as for simulating membrane dynamics based on those estimates. The methods related to these different approaches are described below.
The basis of the analysis was a fluctuating conductance model, which approximates the highly complex synaptic activity as a random process, using global synaptic conductances described by the following stochastic equation (Destexhe et al. 2001): dge(t) dt = − 1 te [ge(t)−ge0]+ s 2s2e te xe(t) .

Intracellular recordings in awake and naturally sleeping animals

Intracellular recordings of cortical neurons were performed in parietal cortex of awake and naturally-sleeping cats (seeMaterials andMethods). Intracellular recordings were done simultaneously with the local field potential (LFP), electromyogram (EMG) and electrooculogram (EOG) to identify behavioral states. With pipettes filled with K+-Acetate (KAc), we recorded and electrophysiologically identified activities of 96 presumed excitatory neurons during the waking state. Of them, 47 neurons revealed a regular-spiking (RS) firing pattern, with significant spike-frequency adaptation in response to depolarizing current pulses, and spike width of 0.69 ± 0.20 ms (range 0.4-1.5 ms). The Vm of RS neurons varied in a range between -56 mV and -76 mV (mean -64.0 ± 5.9 mV). 26 of these RS neurons were “wake-active” cells, in which the firing was sustained all through the wake state, as described previously (Matsumara et al. 1988; Baranyi et al. 1993; Steriade et al. 2001; Timofeev et al. 2001). In these “wake-active” neurons (Fig. 4.1A), the Vm was depolarized (around -65 mV) and showed high-amplitude fluctuations and sustained irregular firing (3.1 Hz on average; range 1 to 11 Hz) during wakefulness. During slow-wave sleep, all neurons always showed up and down states in the Vm activity in phase with the slow-waves (Fig. 4.1A, SWS), as described previously (Steriade et al. 2001).
Almost half of the RS neurons recorded (21 out of 47), were “wake-silent” cells, which systematically ceased firing during periods of quiet wakefulness (cf. Fig. 4.1B). Note that during the transition from slow-wave sleep to waking, these wake-silent neurons continued to fire for 10-60 s, and after that period, their Vm hyperpolarized by several mV and they stopped to fire action potentials as long as the animal remained in the state of quiet wakefulness. Figure 4.1B illustrates one example of a wake-silent cell which upon awakening had a Vm of -53.0 ± 4.9 mV and fired with frequency 10.1±7.9 Hz for about 30 s. Thereafter, the Vm hyperpolarized to -62.5 ± 2.6 mV and the same neuron stopped firing. The hyperpolarization during waking state was not due to K+ load because on 2 occasions, we were able to obtain intracellular recordings from wake-silent neurons during waking state that was preceded and followed by other states of vigilance (see Supplementary  Fig. 1). In this case, the recorded neuron was relatively depolarized and fired  action potentials during REM sleep. Upon awakening, this neuron was hyperpolarized by about 10 mV and stopped firing. After 3 min of waking state the animal went to slow-wave sleep state and the same neuron was depolarized and started to fire action potentials. On one occasion (not shown) we recorded extracellularly spikes from two units. One of the units stopped firing during waking state lasting about 10 min while another unit continued to emit action potentials. This observation suggests that it is a particular set of neurons and not local networks that stop firing during quiet wakefulness. The mean firing rates for RS neurons were 6.1 ± 6.7 Hz (silent neurons included; 10.1 ± 5.6 Hz with silent neurons excluded). No wake-silent cells were observed for other neuronal classes than RS cells, and all together, wake-silent neurons represented about 25% of the totalm number of recorded cells in the wake state. This large proportion of wake-silent neurons constitutes a first hint for an important role for inhibitory conductances during waking.

Table of contents :

1 Introduction 
1.1 Preface
1.2 Computational neuroscience
1.2.1 Models with a single state variable
1.2.2 Models with multiple state variables
1.3 The point-conductance model
1.4 The dynamic-clamp technique
1.4.1 Overview
1.4.2 The AEC-method
1.5 Spike-triggered averages
1.6 Outline of the thesis
2 Inhibitory conductance dynamics in cortical neurons during activated states
2.1 Abstract
2.2 Introduction
2.3 Spike-triggered averages during activated states
2.4 Discussion
2.5 References
3 Calculating event-triggered average synaptic conductances from the membrane potential 
3.1 Abstract
3.2 Introduction
3.3 Material and Methods
3.3.1 Models
3.3.2 In vitro experiments
3.4 Results
3.4.1 Method to extract conductance STA
3.4.2 Test of the accuracy of the method using numerical simulations
3.4.3 Test of the method in real neurons
3.5 Discussion
3.6 Acknowledgments
3.7 References
4 Inhibition determines membrane potential dynamics and controls action potential generation in awake and sleeping cat cortex 
4.1 Abstract
4.2 Introduction
4.3 Materials and Methods
4.3.1 Intracellular recordings in awake and naturally sleeping animals
4.3.2 Analysis and computational models
4.4 Results
4.4.1 Intracellular recordings in awake and naturally sleeping animals
4.4.2 Synaptic conductances in wakefulness and natural sleep .
4.4.3 Conductance time course during up and down state transitions
4.4.4 Dynamics of spike initiation during activated states .
4.5 Discussion
4.5.1 Supplementary Information
4.6 References
5 Characterizing synaptic conductance fluctuations in cortical neurons and their influence on spike generation 
5.1 Abstract
5.2 Introduction
5.3 Methods
5.3.1 Computational methods
5.3.2 Biological preparation
5.3.3 Electrophysiology
5.3.4 Data analysis
5.4 Results
5.4.1 The VmD method for extracting synaptic conductance parameters
5.4.2 Estimating time constants from Vm power spectral density
5.4.3 Estimating spike-triggering conductance configurations .
5.4.4 Estimating spike-triggered averages of synaptic conductances from the Vm
5.5 Discussion
5.6 References
6 Which model best captures the spiking response of cortical neurons to excitatory inputs? 
6.1 Abstract
6.2 Introduction
6.3 Materials and Methods
6.3.1 In vitro experiments
6.3.2 Models
6.3.3 Integrate-and-fire models
6.3.4 The 2–state–variable models
6.3.5 The Hodgkin-Huxley model
6.3.6 The protocol
6.3.7 The optimisation
6.4 Results
6.5 Discussion
6.6 References
7 General Conclusions 
7.1 Summary
7.2 Outlook
A Estimating conductance parameters fromthemembrane potential time course 
A.1 Synopsis
A.2 The Method
A.3 Application to model data
A.4 Application to in vitro data
A.5 Discussion


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