Mode locking in a spatially extended neuron model: Active soma and compartmental tree

Svensson, C. M.; Svensson, Carl-Magnus; Coombes, S.

Abstract

Understanding the mode-locked response of excitable systems to periodic forcing has important applications in neuroscience. For example, it is known that spatially extended place cells in the hippocampus are driven by the theta rhythm to generate a code conveying information about spatial location. Thus, it is important to explore the role of neuronal dendrites in generating the response to periodic current injection. In this paper we pursue this using a compartmental model, with linear dynamics for each compartment, coupled to an active soma model that generates action potentials. By working with the piecewise linear McKean model for the soma we show how the response of the whole neuron model (soma and dendrites) can be written in closed form. We exploit this to construct a stroboscopic map describing the response of the spatially extended model to periodic forcing. A linear stability analysis of this map, together with a careful treatment of the nondifferentiability of the soma model, allows us to construct the Arnol'd tongue structure for 1:q states (one action potential for q cycles of forcing). Importantly we show how the presence of quasi-active membrane in the dendrites can influence the shape of tongues. Direct numerical simulations confirm our theory and further indicate that resonant dendritic membrane can enlarge the windows in parameter space for chaotic behavior. These simulations also show that the spatially extended neuron model responds differently to global as opposed to point forcing. In the former case spatio-temporal patterns of activity within an Arnol'd tongue are standing waves, whilst in the latter they are traveling waves. © 2009 World Scientific Publishing Company.

• Journal Article Type: Article
• Acceptance Date: Nov 21, 2008
• Online Publication Date: Jan 30, 2009
• Publication Date: Jan 30, 2009
• Deposit Date: Jan 28, 2009
• Publicly Available Date: Jan 30, 2009
• Journal: International Journal of Bifurcation and Chaos
• Print ISSN: 0218-1274
• Electronic ISSN: 0218-1274
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 8
• Pages: 2597-2607
• DOI: https://doi.org/10.1142/S0218127409024347
• Public URL: https://nottingham-repository.worktribe.com/output/1014907
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S0218127409024347
• Additional Information: Electronic version of an article to be published in International Journal of Bifurcation and Chaos © World Scientific Publishing Company. http://www.worldscinet.com/ijbc/