Travelling waves in a neural field model with refractoriness
At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and i...
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| Format: | Article |
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Springer Verlag
2014
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| Online Access: | https://eprints.nottingham.ac.uk/2670/ |
| _version_ | 1848790845305651200 |
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| author | Meijer, Hil G.E. Coombes, Stephen |
| author_facet | Meijer, Hil G.E. Coombes, Stephen |
| author_sort | Meijer, Hil G.E. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit
of an original prescription by Wilson and Cowan (Biophys J 12:1–24, 1972 ), using a term that multiplies the usual spatial convolution with a moving time average of local
activity over some refractory time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts.We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both
an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic
travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically. |
| first_indexed | 2025-11-14T18:19:05Z |
| format | Article |
| id | nottingham-2670 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:19:05Z |
| publishDate | 2014 |
| publisher | Springer Verlag |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-26702020-05-04T16:43:46Z https://eprints.nottingham.ac.uk/2670/ Travelling waves in a neural field model with refractoriness Meijer, Hil G.E. Coombes, Stephen At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit of an original prescription by Wilson and Cowan (Biophys J 12:1–24, 1972 ), using a term that multiplies the usual spatial convolution with a moving time average of local activity over some refractory time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts.We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically. Springer Verlag 2014-04-01 Article PeerReviewed Meijer, Hil G.E. and Coombes, Stephen (2014) Travelling waves in a neural field model with refractoriness. Journal of Mathematical Biology, 68 (5). pp. 1249-1268. ISSN 1432-1416 neural field models; travelling waves; refractoriness; delay differential equations http://link.springer.com/article/10.1007%2Fs00285-013-0670-x doi:10.1007/s00285-013-0670-x doi:10.1007/s00285-013-0670-x |
| spellingShingle | neural field models; travelling waves; refractoriness; delay differential equations Meijer, Hil G.E. Coombes, Stephen Travelling waves in a neural field model with refractoriness |
| title | Travelling waves in a neural field model with refractoriness |
| title_full | Travelling waves in a neural field model with refractoriness |
| title_fullStr | Travelling waves in a neural field model with refractoriness |
| title_full_unstemmed | Travelling waves in a neural field model with refractoriness |
| title_short | Travelling waves in a neural field model with refractoriness |
| title_sort | travelling waves in a neural field model with refractoriness |
| topic | neural field models; travelling waves; refractoriness; delay differential equations |
| url | https://eprints.nottingham.ac.uk/2670/ https://eprints.nottingham.ac.uk/2670/ https://eprints.nottingham.ac.uk/2670/ |