Nonlinear Modeling of Causal Interrelationships in Neuronal Ensembles

Abstract

The increasing availability of multiunit recordings gives new urgency to the need for effective analysis of "multidimensional" time-series data that are derived from the recorded activity of neuronal ensembles in the form of multiple sequences of action potentials-treated mathematically as point-processes and computationally as spike-trains. Whether in conditions of spontaneous activity or under conditions of external stimulation, the objective is the identification and quantification of possible causal links among the neurons generating the observed binary signals. A multiple-input/ multiple-output (MIMO) modeling methodology is presented that can be used to quantify the neuronal dynamics of causal interrelationships in neuronal ensembles using spike-train data recorded from individual neurons. These causal interrelationships are modeled as transformations of spiketrains recorded from a set of neurons designated as the "inputs" into spike-trains recorded from another set of neurons designated as the "outputs." The MIMO model is composed of a set of multiinput/single-output (MISO) modules, one for each output. Each module is the cascade of a MISO Volterra model and a threshold operator generating the output spikes. The Laguerre expansion approach is used to estimate the Volterra kernels of each MISO module from the respective inputoutput data using the least-squares method. The predictive performance of the model is evaluated with the use of the receiver operating characteristic (ROC) curve, from which the optimum threshold is also selected. The Mann-Whitney statistic is used to select the significant inputs for each output by examining the statistical significance of improvements in the predictive accuracy of the model when the respective inputs is included. Illustrative examples are presented for a simulated system and for an actual application using multiunit data recordings from the hippocampus of a behaving rat.

Relationships between spike-free local field potentials and spike timing in human temporal cortex

“…This method is described in detail in Marmarelis (2004) and Zanos et al (2008). Briefly, to compute up to second-order Volterra kernels for our continuous input-binary output model, the filtered versions of the LFP were used as the input and the simultaneously recorded spike trains (binary sequences), on the same or on a nearby electrode, were used as the output.…”

“…Selection of the optimal alpha parameter of the Laguerre functions is based on the optimal in-sample model prediction. Considerable advantages in terms of estimation accuracy, input-output data requirements, and model complexity result from this significant reduction in the number of estimated values and model compactness (Marmarelis 1993;Zanos et al 2008). Finally, a threshold is applied on the output of the kernel subsystem through a triggering threshold function that provides the predicted binary output r(n).…”

“…To establish the statistical significance of each estimated Volterra model (as the quantitative representation of a causal link between the inputs and the output) in the presence of noise or other sources of stochastic variability in the data, we utilized the Mann-Whitney two-sample statistic (MWS) (Zanos et al 2008) that relates to the area under the curve (AUC) value of the ROC curve and can be used to test statistically whether a specific model is better than another as a binary predictor. The MWS represents the probability that a randomly selected sample X i from the intermediate variable, u p (n), that corresponds to zero output will be less than (or equal to) a randomly selected sample Y i from the values of u p (n) that correspond to spikes in the output.…”

Dynamic nonlinear modeling of interactions between neuronal ensembles using principal dynamic modes

“…The obtained results to date have provided evidence of the validity and efficacy of this MIMO modeling approach in the rat hippocampus [5][6][7][8]. Notwithstanding the success of these efforts, the challenge remains with regard to the practical ''scaling up'' of this approach to large populations of neurons.…”

“…We have participated in this collective effort by exploring the use of parsimonious forms of Volterra-type models which offer the requisite flexibility to represent actual neural function with adequate fidelity but have been viewed heretofore as prohibitively cumbersome. Our results to date have offered considerable promise in achieving reasonable parsimony and predictive capability for the dynamic nonlinear MIMO models that are required for satisfactory representation of actual neural systems [1,[4][5][6][7][8].…”