Abstract
We investigate a minimal network model consisting of a 2D linear (non-oscillatory) resonator and a 1D linear cell, mutually inhibited with piecewise-linear graded synapses. We demonstrate that this network can produce oscillations in certain parameter regimes and the corresponding limit gradually transition from regular oscillations (of non-relaxation type) to relaxation oscillations as the levels of mutual inhibition increase.
This work was partially supported by the National Science Foundation grant DMS-1608077 (HGR), the NJIT Faculty Seed Grant 211278 (HGR) and the Universidad Nacional del Sur Grant PGI 24/L096. HGR is grateful to the Courant Institute of Mathematical Sciences at New York University and the Centre de Recerca Matemàtica, Barcelona. The authors are grateful to Antoni Guillamon for useful comments on an earlier version of this manuscript.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Bel, H.G. Rotstein, Membrane potential resonance in non-oscillatory neurons interacts with synaptic connectivity to produce network oscillations. In preparation (2018)
R.L. Burden, J.D. Faires, Numerical Analysis (PWS Publishing Company, Boston, 1980)
Y. Chen, X. Li, H.G. Rotstein, F. Nadim, Membrane potential resonance frequency directly influences network frequency through electrical coupling. J. Neurophysiol. 116(4), 1554–1563 (2016)
B. Hutcheon, Y. Yarom, Resonance, oscillations and the intrinsic frequency preferences of neurons. Trends Neurosci. 23(5), 216–222 (2000)
M.J.E. Richardson, N. Brunel, V. Hakim, From subthreshold to firing-rate resonance. J. Neurophysiol. 89, 2538–2554 (2003)
H.G. Rotstein, Frequency preference response to oscillatory inputs in two-dimensional neural models: a geometric approach to subthreshold amplitude and phase resonance. J. Math. Neurosci. 4, 11 (2014)
H.G. Rotstein, F. Nadim, Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents. J. Comput. Neurosci. 37, 9–28 (2014)
R.A. Tikidji-Hamburyan, J.J. Martínez, J.A. White, C. Canavier, Resonant interneurons can increase robustness of gamma oscillations. J. Neurosci. 35, 15682–15695 (2015)
R. Zemankovics, S. Káli, O. Paulsen, T.F. Freund, N. Hájos, Differences in subthreshold resonance of hippocampal pyramidal cells and interneurons: the role of \(h\)-current and passive membrane characteristics. J. Physiol. 588, 2109–2132 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Bel, A., Rotstein, H.G. (2019). Resonance-Based Mechanisms of Generation of Relaxation Oscillations in Networks of Non-oscillatory Neurons. In: Korobeinikov, A., Caubergh, M., Lázaro, T., Sardanyés, J. (eds) Extended Abstracts Spring 2018. Trends in Mathematics(), vol 11. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-25261-8_24
Download citation
DOI: https://doi.org/10.1007/978-3-030-25261-8_24
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-25260-1
Online ISBN: 978-3-030-25261-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)