The basic working principle of the family of CuBIC methods is to perform a sequence of statistical hypothesis tests, where the cumulants of measured data with higher-order correlations of unknown order are compared to the cumulants of a model with correlations only up to a certain order included (Staude et al.
2010b,
c and Section
2). Apparently, the reliability of such a testing procedure depends on how much the physiological characteristics of the measured system deviate from the assumed model and its parameters. Our model is based on the following set of assumptions: (i) lumped input spike trains are well characterized by a compound Poisson process, (ii) PSPs can be described by a fixed kernel function, (iii) all inputs to the neuron are integrated linearly and, (iv) all inputs are of the same sign, i.e. we have either only excitatory or only inhibitory PSPs. In addition, the resting membrane potential, the membrane time constant and the mean PSP amplitude must be known.
We found that CuBICm is remarkably robust against violations of model assumptions and misestimation of model parameters—even when the recording situation differs in several respects from the underlying model (see Section
5). As Eq. (
7) directly shows, a synchronous event of order
n makes a contribution to the
k-th cumulant of the membrane potential of order
\(n^{k}\). Our results suggest that all other perturbations considered here rather affect all cumulants equally, and thus hardly impair the inference of the maximal order of correlation. Here, we discuss the robustness of CuBICm when applying it to
in vivo intracellular recordings of membrane potentials, specifically with respect to the appropriateness of model assumptions and the estimation of model parameters.
6.1.1 Appropriateness of model assumptions
From the above mentioned assumptions, number (iv) appears most problematic. A well-regulated interplay between excitation and inhibition is a generic feature of many known biological networks, and only rarely will a cell in the brain get exclusively either excitatory or inhibitory input. However, as demonstrated in Section
5.3.2, CuBICm is stunningly robust against additional inhibitory input, if higher-order correlations within the pool of excitatory presynaptic neurons are to be measured. To which extent a correlation between excitatory and inhibitory inputs compromises the performance of CuBICm was not tested in our present study. However, such correlations have been experimentally demonstrated in the rat brain (Okun and Lampl
2008; Gentet et al.
2010) and the primate and mouse retina (Cafaro and Rieke
2010). Experimentally, problems associated with inhibition confounding the measurement of HOCs in excitatory inputs can be minimized with the help of pharmacological blockers of GABAergic receptors that act from the cytoplasmic face and can simply be added to the pipette solution (see e.g. Nelson et al.
1994; Lang and Par
1997). Another less invasive method to isolate excitatory input during the measurements would be to clamp the membrane potential to the reversal potential of inhibitory currents using the slow voltage clamp technique (Sutor et al.
2003), which annihilates slow deviations of the membrane potential from the prescribed clamping potential, but does not affect fast deflections like PSPs.
To what extent the assumption (iii) of linearity of PSP summation is justified in cells within an active network is still a matter of debate. Obviously, because of their cable properties (Hodgkin and Rushton
1946; Rall
1959), neurons without specific non-linear mechanisms will integrate in an essentially linear fashion (however, see Kuhn et al.
2004). Indeed, work in acute brain slices has experimentally demonstrated linear integration (Cash and Yuste
1999; Magee and Cook
2000), and even in the intact animal, summation of artificially evoked inputs has been shown to be linear (Léger et al.
2005; Jagadeesh et al.
1993). On the other hand, a rich repertoire of non-linear integration mechanisms has been described in dendritic regions of a number of neuron types, mostly in acute brain slices (Miyakawa et al.
1992; Amitai et al.
1993; Schiller et al.
2000; Larkum et al.
2009), and accordingly non-linear integration properties in single neurons have been described (Nettleton and Spain
2000; Yoshimura et al.
2000). To which extent these mechanisms are actually effective in the intact brain remains a matter of debate until today, and clear evidence for their functional relevance is still lacking. As soon as clear ideas evolve where and when non-linear summation effects play a role, appropriate compensation mechanisms should be included into CuBICm. Given the present uncertainty concerning non-linear summation in the intact brain, it does not seem appropriate to formulate the conditions for such a correction, however. Similarly, sub-threshold, voltage-dependent conductances like
\(\mathrm {I}_{\mathrm {h}}\) could be taken into consideration for an adapted version of CuBICm, but to date, most quantitative data for channel densities and overall conductances originate from slice work, and it is difficult to estimate their abundance
in vivo.
While linear summation might be, at present, the most reasonable model, assumption (ii) of all PSPs having the same fixed amplitude and decay time constant does definitely not apply to most biological recordings. Even in a simple cable model of a cell, filtering effects will lead to attenuation and slowdown of PSPs from distant synapses. Indeed, in certain cortical neuron types mechanisms for normalization of amplitude (Magee and Cook
2000) or time course (Williams and Stuart
2000) have been described. However, they seem to be specific to the corresponding cell type or even to certain dendritic regions (Williams and Stuart
2002). In general, PSP amplitudes vary strongly from synapse to synapse, and the distribution of amplitudes at one synapse sometimes shows a heavy tail (Berretta and Jones
1996; Lefort et al.
2009). As was demonstrated in Section
5.3.1, CuBICm works well in similar scenarios. In principle, an adaptation of CuBICm for randomly distributed amplitudes would be feasible. Methodologically this would be similar to the adaptation for non-stationary presynaptic spike activity, which is outlined below. As we also demonstrated in Section
5.2.2, the shape of the PSP does not need to be exactly matched, but an approximation also yields good results.
While the compound Poisson process (i) is a flexible model, it does not capture all features of spiking in biological neurons. We investigated the robustness of CuBICm for non-Poissonian spiking in Section
5.1.2. In fact, our results are in line with our previous findings, where we employed the method empirical de-Poissonization (Ehm et al.
2007; Reimer et al.
2012). It infers higher-order correlations from the population spike count, again assuming the CPP model. We therefore expect that, similarly, the degree of misestimation of
\(\xi \) by CuBICm does not only depend on the time constant, but also on the detailed spike statistics like firing rate, inter-spike interval distribution, spiking irregularity, and population size. In particular, the results will be most reliable for large populations of sparsely firing neurons—a parameter regime reported for the neocortex (see Barth and Poulet
2012, for a review). Especially in networks which are engaged in the processing of sensory information, spike rates are often strongly fluctuating on a short time scale. The component processes
\(Y_{n}(t)\) of the compound Poisson process are, however, stationary in the CPP model (see Section
2.1.1), which restricts the type of time variation of spike rates in single neurons. A simple example would be that half of the population spikes only within the first half of the observation interval, while the remaining neurons are silent in this period and fire only in the second half (see Staude et al.
2010b, for less obvious examples). In order to capture also scenarios like rapidly co-fluctuating firing rates of all neurons (Staude et al.
2010b), we extended the CPP model and adjusted CuBIC accordingly. The same approach is also applicable to CuBICm. Briefly, the component processes
\(Y_{n}(t)\) are conceived as doubly stochastic Poisson processes with a common (but random) rate profile. The cumulants of the population spike count, or postsynaptic subthreshold activity, respectively, are obtained by the law of total cumulance. In doing so, only assuming a parametric family of distributions for the rate fluctuations, and not a specific rate profile, the inference of higher-order correlations for non-stationary processes is made possible.
6.1.2 Estimation of model parameters
Among the three parameters which have to be estimated and inserted into the model for proper analysis, the mean PSP amplitude is the most problematic issue. As is shown in Section
5.2.4, an underestimation of the mean PSP amplitude by 50 % can lead to a substantial overestimate of correlation order. The opposite effect occurs for an overestimated PSP size, but it is much less pronounced. This asymmetric dependency of CuBICm suggests that the latter scenario (assumed PSPs bigger than in reality) leads to a conservative use of the method. The order of correlation will in this case, if at all, be slightly underestimated. Estimating PSP amplitudes in individual cells within intact networks is, in fact, not really feasible based on experimental methods available to date. The only reliable measurements of PSP amplitudes come from recordings in acute brain slices, where spontaneous activity is low and individual presynaptic cells can be stimulated repeatedly (either by paired recordings or via light-induced activation). Data from such experiments demonstrate a wide range of amplitudes which strongly depend on age, species, presynaptic/postsynaptic cell type and brain region (for a review see Thomson and Lamy
2007). Most PSP amplitude distributions reported in recent years have their peak below 1 mV, and this number seems to be a good approximation even if inputs from different layers or different cell populations are considered (Schnepel et al.
2011). However, neurons
in vivo receive the spiking activity of thousands of other neurons which can reduce the PSP amplitude drastically (Kuhn et al.
2004). Thus, estimates should be adapted accordingly (Kumar et al.
2008).
To get a reasonable estimate of the resting membrane potential is, compared to the mean PSP amplitude, much easier in practice, since the only requirement is to block all synaptic inputs for a limited period of time. In the intact animal, this could be achieved either by local application of ion channel or receptor blockers, or, alternatively, by administration of anesthetics which generate pronounced up/down states (Steriade et al.
1993; Mahon et al.
2001). Such pharmacological manipulations can only be performed after the recordings intended for the assessment of higher-order correlations, and the choice of manipulating agent depends on the details of the experimental procedure. In any case, an over- or under-estimation of the resting membrane potential of a few millivolts would not be too detrimental for the estimation of the degree of higher-order correlations, as we demonstrated in Section
5.2.1. However, the resting membrane potential during intracellular recordings can depend on the pipette solution, and drifting offset potentials may lead to a considerably erroneous read-out, so one of the above mentioned pharmacological interventions seem advisable wherever possible.
In general, the PSP decay time constant should (as far as dendritic filtering effects are neglected or covered by working with distributions of time constants rather than a single fixed value) conform to the membrane time constant. This value can, even in
in vivo intracellular recordings, be easily assessed by brief current pulse injections (Waters and Helmchen
2006; Léger et al.
2005). Moreover, CuBICm is hardly affected by misestimated membrane time constants (see Section
5.2.3).