Appendix A Higher-Order Spike Interactions through Nonlinearity 301 point processes. Perhaps most important, verifying the duality between the two representations can provide valuable insight into the statistical limita- tions of a synaptic model. Therefore, these inner products (or their derived constructs) can be used to analyze neurophysiological data, and they hint at the underlying principles of information encoding from these perspectives. The inner product with the form given in Eq. (8.16) is statistically, com- pletely general but, in practice, its usefulness is limited by the current methods for estimation of the conditional intensity function. Although methods for point processes with memory have been recently proposed in the literature, currently effective intensity function estimators exist only for Poisson pro- cess. 9 In this regard, however, the inner products defined in Eqs. (8.18) and (8.20) show considerable promise. Through the use of a nonlinearity, these inner products circumvent these limitations and show sensitivity to spike time statistics, albeit with limited expressiveness. In our experiments using renewal point processes, both inner products clearly outperform the inner product in Eq. (8.17). It must be remarked that the inner product in Eq. (8.17) is in essence a continuous version of the widely used cross-correlation of binned