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 spike-trains 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 input-output 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.

Fig. 1

(A) Schematic representation of the decomposition of the MIMO model into an array of MISO modules. (B) Schematic detail of the structure of the MISO module comprising the cascade of the NVT and the TT operators.

Fig. 2

(A) Kernels of the second-order simulated MISO system with four inputs and one output: the first column shows the first-order kernels, the second column shows the second-order self-kernels, and the third column shows the cross-kernel between the first and the fourth input (see text). (B) The estimated kernels of this second-order model from the simulated data for the noise free case, along with ROC curves and theta values of in-sample and out-of-sample prediction (see text). The x axes in the first-order kernels and the x and y axes in the second-order kernels range from 0 to 1000 ms, while the ROC curves have in both axes values from 0 to 1.

Fig. 3

(A) The estimated model for the case of 50% spurious spikes in the inputs and output and (B) for the case of jitter in the spike location in the input–output data (see caption of Fig. 2 for detailed description of the layout and the axes of the plots). The results demonstrate the robustness of the proposed methodology in the presence of added spurious spikes and jitter.

Fig. 4

Schematic of the three behavioral tasks that the live rat is performing in each trial of the experiment: (I) right sample, (II) delay phase, (III) left nonmatch. (B) Schematic representation of the multielectrode array stimulating and recording arrangement in the CA3 and CA1 hippocampal regions of the behaving rat.

Fig. 5

Illustrative Case of a complete MIMO model for the left sample event. (A) ROC curve and theta estimate of each output, (B) first-order kernels, (C) second-order self-kernels and cross-kernels, (D) colormap of the meshes used for the second-order kernels, (E) schematic of the topology of the input–output neurons in the CA3 and the CA1, respectively, and the causal connections considered in the MIMO model, (F) the model prediction and the actual recorded activity of each output neurons during the left sample task). The x axes in the first-order kernels and the x and y axes in the second-order kernels range from 0 to 500 ms, while the ROC curves have in both axes, values from 0 to 1.

Fig. 6

Schematic of the topology of the input–output neurons in the CA3 and the CA1, respectively, and the causal connections considered in each model (left panels). The model predicted and the actual recorded activity of the output neurons during three behavioral tasks: (A) left nonmatch, (B) right sample, and (C) right nonmatch. The time axes in all the prediction plots are in seconds.