A new method to infer higher-order spike correlations from membrane potentials

Abstract

What is the role of higher-order spike correlations for neuronal information processing? Common data analysis methods to address this question are devised for the application to spike recordings from multiple single neurons. Here, we present a new method which evaluates the subthreshold membrane potential fluctuations of one neuron, and infers higher-order correlations among the neurons that constitute its presynaptic population. This has two important advantages: Very large populations of up to several thousands of neurons can be studied, and the spike sorting is obsolete. Moreover, this new approach truly emphasizes the functional aspects of higher-order statistics, since we infer exactly those correlations which are seen by a neuron. Our approach is to represent the subthreshold membrane potential fluctuations as presynaptic activity filtered with a fixed kernel, as it would be the case for a leaky integrator neuron model. This allows us to adapt the recently proposed method CuBIC (cumulant based inference of higher-order correlations from the population spike count; Staude et al., J Comput Neurosci 29(1-2):327-350, 2010c) with which the maximal order of correlation can be inferred. By numerical simulation we show that our new method is reasonably sensitive to weak higher-order correlations, and that only short stretches of membrane potential are required for their reliable inference. Finally, we demonstrate its remarkable robustness against violations of the simplifying assumptions made for its construction, and discuss how it can be employed to analyze in vivo intracellular recordings of membrane potentials.

Fig. 1

Relating subthreshold membrane potential fluctuations of a neuron and its presynaptic input spike trains. A neuron receives spikes which have been elicited by many presynaptic neurons (raster plot at top). While this activity is not observable, intracellular recordings allow to observe the subthreshold membrane potential fluctuations of a neuron (bottom). This signal contains information about the input spiking dynamics. For instance, coincident spikes give rise to deflections in the membrane potential. Thus, analysis of this signal may reveal higher-order spike correlations in the input population (arrow). Membrane potential trace has been recorded intracellularly in vivo in rat primary visual cortex

Fig. 2

Model and measurement. a Presynaptic spike activity. The sum activity is described as a compound Poisson process , where are independent Poisson processes with intensity . Each process represents the synchronized activity of n presynaptic neurons encoded by the same color. b Neuronal integration. The biological procedure is modeled as convolution with a fixed kernel . c Postsynaptic subthreshold activity. Filtering of the presynaptic spike activity yields a shot noise process with amplitude distribution . From the cumulants of this distribution the maximal order of correlation can be inferred via CuBICm (indicated by green arrow)

Fig. 3

Impact of correlated samples and associated correction. a Inferred maximal order of correlation in dependence of the filter time constant for a shot noise process with presynaptic activity consisting of independent Poisson processes with rate spikes/s each. Average results over 200 simulations for 50 s each are shown for CuBICm without correction for correlated samples (light blue) and CuBICm with correction (dark blue, dashed line). b Inferred for different sample intervals for the same shot noise processes as in (a) with . Simulation time has been adjusted to keep the sample size fixed. c Distribution of third cumulant of data in (a) with . Kernel density estimate of third sample cumulant (yellow) is shown in comparison with the mean distribution assumed under for CuBICm without correction (light blue) and CuBICm with correction (dark blue, dashed line). Additionally, dots indicate the cumulants of the data sets for which has been rejected by the use of CuBICm without correction (light red) and CuBICm with correction (dark red) which corresponds to 22.5 % and 3.5 % of all data sets, respectively. Inset presents same figure but with different axes limits. Error bars in a–b depict standard deviation. A significance level of has been used

Fig. 4

Subthreshold activity and corresponding estimated maximal order of correlation in dependence of various parameters. Presynaptic activity is mimicked as N Poisson processes, where a subpopulation of is correlated with pairwise correlation coefficient c. Each presynaptic neuron fires at rate . previous sectione performed for time T, and each data set filtered with an exponential kernel with time constant . Blue: Data set with default parameters . Green: Data set with default parameters . a, b Sample of 2 s of subthreshold activity S and its distribution for the whole data trace. Arrows indicate time stamps of higher-order events in the presynaptic spike activity. c Correlation structure. d, e, f Estimated maximal order of correlation averaged over 50 simulations. Error bars represent standard deviation. Black circles depict results for default parameter settings and triangle mark true maximal order of correlation

Fig. 5

Mean inferred maximal order of correlations for imprecise coincidences. Presynaptic activity as in Fig. 4b. All spike times of the presynaptic activity have been jittered according to a uniform distribution with support . a Same shot noise process as in Fig. 4b (blue) and the same data with jitter (red). b Mean estimated maximal order of correlation in dependence of time constant for various degrees of jitter (blue: , purple: , red: , orange: , yellow: ). Error bars depict one time standard deviation. Simulation time is 500 s

Fig. 6

Impact of non-Poissonian spiking on the inference of higher-order correlations. Average results for presynaptic processes with lognormal inter-spike intervals, , in relation to estimates, , for presynaptic Poisson processes are depicted. Non-Poissonian and Poisson processes have the same correlation structure. Lognormal processes with different coefficient of variations of their inter-spike interval distribution are considered. Statistics of presynaptic population are identical to the blue data set in Fig. 7a. An exponential filter kernel with amplitude and different time constants between 1 ms (black) and 50 ms (light blue) has been used. Simulation time is s

Fig. 7

Impact of misestimated kernel (parameters) on mean estimated maximal order of correlation. Three surrogate data sets have been analyzed. Purple: ms. Yellow: As purple but with . Blue: ms. The spike trains have been filtered with an exponential kernel and the resulting signal has been analyzed with an exponential kernel with true (or mean in e, respectively) amplitude a and time constant unless stated otherwise. has been averaged over 50 data sets per parameter setting. Error bars denote one time standard deviation. a Correlation structure of the three sample data sets. b The signal is considered as subthreshold membrane potential fluctuations where not the actual resting membrane potential but has been subtracted. is in units of the amplitude a of the filter kernel or postsynaptic potential, respectively. c Synaptic current as -synapse with time constant . d Time constant misestimated as . e Amplitude A misestimated as . f Presynaptic activity with lognormal distributed amplitude a for various coefficients of variation of the distribution. g Presynaptic activity consisting of an excitatory correlated population with spike rate per neuron and an additional inhibitory independent population of the same size with spike rate per neuron

Fig. 8

Inferred maximal order of correlation from the membrane potential fluctuations of a multi-compartment model of a reconstructed layer 5 pyramidal cell (Hay et al. 2011). Mean results of five simulations are shown for different estimated EPSP amplitudes and estimated resting membrane potential (color coded). The model has various active ionic currents where we removed the sodium channels to prevent the neuron from spiking. 10,000 (2,000) conductance based AMPA (GABA) synapses with rise time 0.2 ms (1 ms), decay time 1.7 ms (10 ms) and peak value 3 nS were randomly distributed, where all presynaptic neurons fired 1 (2.5) spikes/s. All spike trains were simulated as Poisson processes, which were independent of each other within the inhibitory population. a Uncorrelated excitatory population. b 200 neurons of the excitatory population are correlated with a pairwise correlation coefficient c = 0.2, and an order of correlation of 50. Black triangles mark maximal order of correlation. Yellow crosses mark the result which is obtained if and are closest to the true values. Simulation time was 100 s