Population density equations for stochastic processes with memory kernels
We present a method for solving population density equations (PDEs)–-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two diffe...
| Main Authors: | , |
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| Format: | Article |
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American Physical Society
2017
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| Online Access: | https://eprints.nottingham.ac.uk/43904/ |
| _version_ | 1848796793064652800 |
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| author | Lai, Yi Ming de Kamps, Marc |
| author_facet | Lai, Yi Ming de Kamps, Marc |
| author_sort | Lai, Yi Ming |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We present a method for solving population density equations (PDEs)–-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process. |
| first_indexed | 2025-11-14T19:53:37Z |
| format | Article |
| id | nottingham-43904 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:53:37Z |
| publishDate | 2017 |
| publisher | American Physical Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-439042020-05-04T18:51:03Z https://eprints.nottingham.ac.uk/43904/ Population density equations for stochastic processes with memory kernels Lai, Yi Ming de Kamps, Marc We present a method for solving population density equations (PDEs)–-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process. American Physical Society 2017-06-20 Article PeerReviewed Lai, Yi Ming and de Kamps, Marc (2017) Population density equations for stochastic processes with memory kernels. Physical Review E, 95 . 062125/1-062125/11. ISSN 2470-0053 https://journals.aps.org/pre/abstract/10.1103/PhysRevE.95.062125 doi:10.1103/PhysRevE.95.062125 doi:10.1103/PhysRevE.95.062125 |
| spellingShingle | Lai, Yi Ming de Kamps, Marc Population density equations for stochastic processes with memory kernels |
| title | Population density equations for stochastic processes with memory kernels |
| title_full | Population density equations for stochastic processes with memory kernels |
| title_fullStr | Population density equations for stochastic processes with memory kernels |
| title_full_unstemmed | Population density equations for stochastic processes with memory kernels |
| title_short | Population density equations for stochastic processes with memory kernels |
| title_sort | population density equations for stochastic processes with memory kernels |
| url | https://eprints.nottingham.ac.uk/43904/ https://eprints.nottingham.ac.uk/43904/ https://eprints.nottingham.ac.uk/43904/ |