Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models
The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to...
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| Format: | Article |
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2008
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| Online Access: | https://eprints.nottingham.ac.uk/884/ |
| _version_ | 1848790497062027264 |
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| author | Coombes, Stephen |
| author_facet | Coombes, Stephen |
| author_sort | Coombes, Stephen |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits. |
| first_indexed | 2025-11-14T18:13:33Z |
| format | Article |
| id | nottingham-884 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:13:33Z |
| publishDate | 2008 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-8842020-05-04T16:27:43Z https://eprints.nottingham.ac.uk/884/ Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models Coombes, Stephen The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits. 2008-06-04 Article PeerReviewed Coombes, Stephen (2008) Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models. SIAM Journal on Applied Dynamical Systems . (In Press) http://epubs.siam.org/SIADS/ |
| spellingShingle | Coombes, Stephen Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models |
| title | Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models |
| title_full | Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models |
| title_fullStr | Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models |
| title_full_unstemmed | Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models |
| title_short | Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models |
| title_sort | neuronal networks with gap junctions: a study of piece-wise linear planar neuron models |
| url | https://eprints.nottingham.ac.uk/884/ https://eprints.nottingham.ac.uk/884/ |