In this study we present a new method to model the nonlinear interactions resulting from the co-activity of two pathways that converge on a common population of cells, the granule cells of the hippocampus. The hippocampus is the brain area that underlies higher cognitive functions, such as learning and memory. It receives neuronal inputs from multiple brain regions that are involved in processing different sensory information and converge to a common population of postsynaptic cells. The dentate gyrus is the first region of the hippocampus which receives and integrates sensory information (e.g., visual, auditory, and olfactory) via the perforant path. The perforant path is composed of two distinct information pathways, the Lateral Perforant Path (LPP) and the Medial Perforant Path (MPP). Both pathways converge in a common population of neurons, the granule cells.

We present a general methodological approach (termed the Poisson-Volterra modeling approach) that yields non-parametric models capable of characterizing the short-term nonlinear dynamic interactions between two pathways. The proposed third order Poisson-Volterra model can provide a compact quantitative representation of the dentate gyrus based on experimentally available datasets. It is mathematically rigorous and scalable with predictive capabilities. The predictive accuracy of the proposed model is quantitatively evaluated using the normalized mean squared error (NMSE). This model has the ability to represent the combined activities of all known and unknown characteristics of neuronal interactions without explicit knowledge of the underlying neuronal mechanisms. As additional aspects of neuronal mechanisms and functions are introduced it can be extended to incorporate increased complexity. The Volterra self and cross kernels are the quantitative descriptors of the system's input/output relationships that fully capture and quantify the nonlinear interactions associated with the neuronal functional properties. The self-kernels capture the nonlinear dynamic interactions within one pathway. The cross-kernels capture the nonlinear dynamic interactions between the two pathways.

The descriptor widely used in the literature to characterize the two pathways has been the Paired Impulse Function (PIF). The measured PIF is defined as the ratio of the amplitude of the 2nd pulse over the 1st pulse resulting from stimulation with paired pulses (PP) with a given inter-pulse interval. PIF values greater than one indicate PP facilitation, while PIF values less than one indicate PP depression. In the context of the proposed third-order Poisson-Volterra modeling approach, the estimated self-kernels was used to compute the PIF equivalent to the one measured in the PP experiments. The value of the computed PÎF using the Poisson-Volterra modeling approach is an estimate of the measured PIF for the corresponding pathway. The computed PÎF will be a close approximation of the measured PIF if, and only if, the employed 3rd-order Poisson-Volterra model is an adequate model for the subject system. To examine the adequacy of the employed model, we compared the measured PIFs with the computed PÎFs for all experimental preparations, and quantified the degree of agreement by computing their correlation.

The Poisson-Volterra model presented here was validated with datasets recorded from in-vitro hippocampal slice preparations. Standard electrophysiological methods were used to prepare 350-400 µm thick slices and record from them. The electrophysiological activity of the dentate gyrus was measured experimentally using multi-electrode arrays that simultaneously stimulate and record the activity in the hippocampal slices. The multi-electrode arrays were designed and fabricated so that the spatial distribution of electrode sites be appropriately arranged to cover the subregions of the dentate gyrus. Stimulation electrodes were chosen in the outer-molecular layer and in the mid-molecular layer to activate the LPP and MPP, respectively.

The results of this study show that the proposed Poisson-Volterra model has excellent predictive capabilities. The kernels exhibit consistent waveforms across all experiments and are distinctive for each pathway, uniquely identifying the LPP and the MPP neuronal transformations. The NMSE was calculated both with and without including the cross- kernels in the third order Poisson-Volterra model. When only the self-kernels were included, the value of the NMSE was 14.39±3.86%. When both the self-kernels and the cross-kernels were included the NMSE dropped to 5.88±2.51%. Application of the paired t-test indicated that the inclusion of the cross- kernels resulted in a statistically significant reduction of the NMSE (p<0.01), which implies that significant dynamic interactions exist between the two pathways (thus the cross-kernels should be included in the model).

This model also exhibited better predictive capabilities than the predictions provided by the measured PIF of the widely used PP approach. The superior predictive capability of the Poisson-Volterra model is probably due to the fact that this model takes into consideration the entire history of the stimulus impulses (possibly including multiple impulses within the memory epoch of the system),whereas the PIF is based only on the current and one previous stimulus impulse. Therefore, the general behavior of these synapses cannot be extrapolated from PP responses and the associated analysis, because the system was shown to possess significant third-order dynamics that involve interactions among triplets of impulses. Consequently, although the third order Poisson-Volterra model was able to reproduce the PIF results obtained from PP data, the reverse is not true (i.e. the response of the system to triplets of impulses can be predicted by the third order Poisson-Volterra model but not from the PP analysis). This was further corroborated by the finding that the computed PÎF using a third order Poisson-Volterra model was in good agreement with the measured PIF. In this regard, note that the response observed during PP experiments includes the third-order kernels that exist within the system. This finding also indicates that there is no need to employ higher order (e.g., fourth order) models for this system.

In conclusion, our results indicate that the third order model has predictive capabilities of the nonlinear interactions between the two information pathways and indicate that this approach can be generalized to complex interactions between distinct inputs to the same set of cells.

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