{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/spike-train-cumulants-for-linear-nonlinear","title":"Spike Train Cumulants for Linear-Nonlinear Poisson Cascade Models","arxiv_id":"2001.05057","date":"2020-01-14","proceeding":null,"authors":[],"abstract":"Spiking activity in cortical networks is nonlinear in nature. The\nlinear-nonlinear cascade model, some versions of which are also known as\npoint-process generalized linear model, can efficiently capture the nonlinear\ndynamics exhibited by such networks. Of particular interest in such models are\ntheoretical predictions of spike train statistics. However, due to the\nmoment-closure problem, approximations are inevitable. We suggest here a series\nexpansion that explains how higher-order moments couple to lower-order ones.\nOur approach makes predictions in terms of certain integrals, the so-called\nloop integrals. In previous studies these integrals have been evaluated\nnumerically, but numerical instabilities are sometimes encountered rendering\nthe results unreliable. Analytic solutions are presented here to overcome this\nproblem, and to arrive at more robust evaluations. We were able to deduce these\nanalytic solutions by switching to Fourier space and making use of complex\nanalysis, specifically Cauchy's residue theorem. We formalized the loop\nintegrals and explicitly solved them for specific response functions. To\nquantify the importance of these corrections for spike train cumulants, we\nnumerically simulated spiking networks and compared their sample statistics to\nour theoretical predictions. Our results demonstrate that the magnitude of the\nnonlinear corrections depends on the working point of the nonlinear network\ndynamics, and that it is related to the eigenvalues of the mean-field stability\nmatrix. For our example, the corrections for the firing rates are in the range\nbetween 4 % and 21 % on average. Precise and robust predictions of spike train\nstatistics accounting for nonlinear effects are, for example, highly relevant\nfor theories involving spike-timing dependent plasticity (STDP).","url_abs":"http://arxiv.org/abs/2001.05057v1","url_pdf":"http://arxiv.org/pdf/2001.05057v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"spike-train-cumulants-for-linear-nonlinear","repo_url":"https://github.com/gocker/PoissonGLMCumulants","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}