Analysis of synaptic noise from intracellular recordings
As explained in detail in Section 2.1, synaptic background activity can be modeled by either biophysical models with detailed dendritic morphology and a large number (thousands) of synapses, or by simplified models using equivalent stochastic processes . Focusing on this second model, it can be shown that the steady-state probability density for the Vm can be obtained analytically (using the Fokker-Planck formalism; see refs. [2,3]; in collaboration with Michelle Rudolph, ICN). This expression provides a very good approximation of the statistical properties of subthreshold activity obtained in complex biophysical models where synaptic activity was simulated by thousands of synapses. The exact form of the distribution depends on the total synaptic conductances, their variances and decay time . This analytic approach was extended recently , leading to an "extended expression" matching the model for several orders of magnitude of parameter values. In the physiological range of parameters, the extended expression is the most accurate of the different analytic expressions and approximations available so far in the literature .
Because the steady-state Vm distribution is easily observable experimentally, this approach  provides a possible method to estimate the mean and variance of synaptic conductances in real neurons . In other words, from an approach that was initially relatively abstract (stochastic calculus), methods can be obtained to extract useful quantities from experimental data. This "VmD" method yields accurate estimates of the conductances and of their fluctuation levels from the sole knowledge of the Vm of the recorded neuron. This was shown in collaboration with Thierry Bal (ICN) using the dynamic-clamp technique, by which artificial conductances can be injected in real neurons. Using this technique, we showed that recreated artificial "up" states - based on measurements from the VmD method - are very close to the natural "up" states in cortical neurons  (Fig. 3).
Another approach to analyze synaptic noise is through power spectral analyses. With Michelle Rudolph, we have recently provided a theoretical analysis of the power spectral density (PSD) of both the conductance and the Vm of neurons subject to synaptic noise . This analysis showed that one can extract useful quantities about the decay kinetics of synaptic conductances, again from the sole knowledge of the Vm (see details in ). Together with the VmD method, this "Vm-PSD" analysis should provide very efficient tools to analyze network states, as seen from intracellular recordings in vivo.
In collaboration with Denis Paré (Rutgers University), these new approaches were applied to intracellular recordings in vivo during "active" states (as defined by desynchronized EEG activity, such as in aroused states) [7,11]. Application of the VmD and Vm-PSD methods revealed that active states are high-conductance states dominated by inhibitory conductances. This was confirmed using different methods, as well as using computational models. In such states, cortical neurons are in a stochastic integrative mode (see Section 2.1). In particular, the amplitude and timing of somatic EPSPs was nearly independent of the position of the synapses in dendrites, suggesting that EEG-activated states are compatible with coding paradigms involving the precise timing of synaptic events.
In addition to estimate conductances, other properties of network activity can also be extracted from synaptic noise in intracellular recordings [8,9,10]. Because cortical neurons are highly interconnected, their subthreshold Vm activity contains information about the activity of thousands of other neurons in the network. The VmD method  outlined above enables to extract the mean and variance of synaptic conductances from subthreshold Vm activity. It is easy to deduce that the mean conductances can be related to the mean firing rate of presynaptic neurons. In addition, extracting the variances of excitatory and inhibitory synaptic conductances can provide estimates of the mean temporal correlation - or level of synchrony - among thousands of neurons in the network . Thus, extracting conductance variances, which is the main originality of the VmD method, can provide estimates of the level of correlation among neurons in the network. Such a dependence on correlation was formalized recently using shot-noise stochastic processes .
Another property that can be extracted from the Vm activity is the optimal conductance pattern related to spikes, on average. We recently introduced a method to calculate the spike triggered average (STA) of conductance from the sole knowledge of Vm activity . We used a procedure based on maximum likelihood estimation to extract the conductances from the sole knowledge of the Vm activity of the recorded neuron. This leads to very useful tools to analyze the respective role and timing of excitation and inhibition in the spike selectivity of cortical neurons. The obtained method was then be tested in real cortical neurons using dynamic-clamp (experiments realized with Thierry Bal at ICN). In this case, known conductance variations are injected into the cell and the method can be tested by checking if the conductances extracted from the sole analysis of the Vm correspond to the conductances actually injected. The method was shown to provide excellent estimates .
The STA and VmD methods were later applied to analyze intracellular recordings from awake cats  (in collaboration with Igor Timofeev, Laval University). The main finding is that inhibitory conductances are generally larger than excitatory conductance, both during wakefulness, and during the ``up'' states of slow-wave sleep. Moreover, the variance of inhibitory conductance is also larger for inhibition. The STA of conductances also demonstrates that variations of inhibition mostly drive spiking activity in these states, suggesting that inhibitory processes are particularly determinant in aroused states.
We next generalized the VmD method to single Vm traces . This ``VmT'' method estimates conductance parameters using maximum likelihood criteria, under the assumption that synaptic conductances are described by Gaussian stochastic processes and are integrated by a passive leaky membrane. The method was tested using models and in guinea-pig visual cortex neurons in vitro using dynamic-clamp experiments. This VmT method holds promises for extracting conductances from single-trial measurements, which has a high potential for in vivo applications .
These approaches were reviewed in several review articles [13,14,15,16]. They were also be reviewed in a monograph that we published with Michelle Rudolph .
More recently, we focused on the estimation of excitatory and inhibitory conductances from single-trial recordings . This method is based on oversampling of the Vm, and is different from the VmT method described above. While the VmT estimates statistics of the conductances (mean and variance), the oversampling method provides an estimate of the full time course of the excitatory and inhibitory conductances. It was tested numerically using models of increasing complexity, as well as using controlled conductance injection in cortical neurons in vitro using the dynamic-clamp technique.
On a more theoretical point of view, we showed that many statistical properties of neuronal membranes subject to noisy synaptic inputs can be estimated analytically . Using the theory of Poisson Point Processes transformations, the statistics of such systems can be obtained analytically, even when the input is non-stationary and of multiplicative type. This leads to effective approximations for the time evolution of Vm distributions and simple method to estimate the pre-synaptic rate from a small number of Vm traces. This work opens the perspective of obtaining analytic access to important statistical properties of conductance-based neuronal models.
 Destexhe, A., Rudolph, M., Fellous, J-M. and Sejnowski, T.J. Fluctuating synaptic conductances recreate in-vivo-like activity in neocortical neurons. Neuroscience 107: 13-24, 2001 (see abstract)
 Rudolph, M. and Destexhe, A. Characterization of subthreshold voltage fluctuations in neuronal membranes. Neural Computation 15: 2577-2618, 2003 (see abstract)
 Rudolph, M. and Destexhe, A. An extended analytic expression for the membrane potential distribution of conductance-based synaptic noise. Neural Computation 17: 2301-2315, 2005 (see abstract)
 Rudolph, M. and Destexhe, A. On the use of analytic expressions for the voltage distribution to analyze intracellular recordings. Neural Computation 18: 2917-2922, 2006 (see abstract)
 Rudolph, M., Piwkowska, Z., Badoual, M., Bal., T. and Destexhe, A. A method to estimate synaptic conductances from membrane potential fluctuations. Journal of Neurophysiology 91: 2884-2896, 2004 (see abstract)
 Destexhe, A. and Rudolph, M. Extracting information from the power spectrum of synaptic noise. Journal of Computational Neuroscience 17: 327-345, 2004 (see abstract)
 Rudolph, M., Pelletier, J-G., Paré, D. and Destexhe, A. Characterization of synaptic conductances and integrative properties during electrically-induced EEG-activated states in neocortical neurons in vivo. Journal of Neurophysiology 94: 2805-2821, 2005 (see abstract)
 Rudolph, M. and Destexhe, A. Inferring network activity from synaptic noise. Journal of Physiology (Paris) 98: 452-466, 2004 (see abstract)
 Rudolph, M. and Destexhe, A. A multichannel shot noise approach to describe synaptic background activity in neurons. European Physical Journal B52: 125-132, 2006 (see abstract)
 Pospischil, M., Piwkowska, Z., Rudolph, M., Bal, T. and Destexhe, A. Calculating event-triggered average synaptic conductances from the membrane potential. Journal of Neurophysiology 97: 2544-2552, 2007 (see abstract)
 Rudolph, M., Pospischil, M., Timofeev, I. and Destexhe, A. Inhibition determines membrane potential dynamics and controls action potential generation in awake and sleeping cat cortex. Journal of Neuroscience 27: 5280-5290, 2007 (see abstract)
 Pospischil, M., Piwkowska, Z., Bal, T. and Destexhe, A. Extracting synaptic conductances from single membrane potential traces. Neuroscience 158: 545-552, 2009 (see abstract)
 Piwkowska, Z., Pospischil, M., Brette, R., Sliwa, J., Rudolph-Lilith, M., Bal, T. and Destexhe, A. Characterizing synaptic conductance fluctuations in cortical neurons and their influence on spike generation. J. Neurosci. Methods 169: 302-322, 2008 (see abstract)
 Destexhe, A. Inhibitory ``noise''. Frontiers in Cellular Neuroscience 4: 9, 2010 (see abstract)
 Destexhe, A and Rudolph-Lilith, M. Synaptic ``noise'': Experiments, computational consequences and methods to analyze experimental data. In: Stochastic Methods in Neuroscience, Edited by Lord G. and Laing C., Oxford University Press, Oxford UK, pp.~242-271, 2010 (see abstract)
 Destexhe, A. 20 years of ``noise'' - Contributions of computational neuroscience to the exploration the effect of background activity on central neurons. In: 20 years of Computational Neuroscience, Edited by Bower J, Springer, New York, pp. 167-186, 2013 (see abstract)
 Destexhe, A. and Rudolph, M. Neuronal Noise. Springer, New York, 2012.
 Bedard, C., Behuret, S., Deleuze, C., Bal, T. and Destexhe, A. Oversampling method to extract excitatory and inhibitory conductances from single-trial membrane potential recordings. J. Neurosci. Meth. 210: 3-14, 2012 (see abstract)
 Brigham, M. and Destexhe, A. Non-stationary filtered shot noise processes and applications to neuronal membranes. Phys Rev E 91: 062102, 2015 (see abstract)
Figure 3: Method to estimate synaptic conductances and test of the method using dynamic-clamp. A. Scheme of the method and test of this method. Top left: intracellular recording during active states (up-states) obtained in vitro, using two DC current levels (Iext1, Iext2). Top right: membrane potential (Vm) distributions are calculated (gray) and the conductances are estimated by fitting an analytic expression of the Vm distribution (continuous lines). Lower right: histograms of the mean and variance of excitatory and inhibitory conductances obtained by this procedure (gray). Lower left: injection of stochastic conductances in dynamic-clamp based on this estimation. This injection recreates artificial active states in the same neuron. B. Example of natural and artificial active states in the same neuron as in A. The procedure recreates states very similar to natural states, as shown by the Vm distribution and discharge variability. Modified from Rudolph et al., J. Neurophysiol., 2004.
Department of Integrative and Computational Neuroscience (ICN),