Abstract
The effects of time delays on the nonlinear dynamics of neural networks are investigated. A decomposition method is utilized to derive modal equations that allow one to analyze the dynamics around synchronous states. The D-subdivision method is used to study the stability of equilibria while the stability of periodic orbits is investigated using Floquet theory. These methods are applied to a system of delay coupled Hodgkin-Huxley neurons to map out stable and unstable synchronous states. It is shown that for sufficiently strong coupling there exist delay ranges where stable equilibria coexist with stable oscillations which allow neural systems to respond to different environmental stimuli with different spatiotemporal patterns.
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References
Cepeda-Gomez, R., Olgac, N.: An exact method for the stability analysis of linear consensus protocols with time delay. IEEE Transactions on Automatic Control 56(7), 1734–1740 (2011)
Coombes, S.: Neuronal networks with gap junctions: a study of piecewise linear planar neuron models. SIAM Journal on Applied Dynamical Systems 7(3), 1101–1129 (2008)
D’Huys, O., Fischer, I., Danckaert, J., Vicente, R.: Role of delay for the symmetry in the dynamics of networks. Physical Review E 83(4), 046223 (2011)
Ermentrout, G.B., Terman, D.H.: Mathematical Foundations of Neuroscience. In: Interdisciplinary Applied Mathematics., vol. 35, Springer (2010)
Flunkert, V., Yanchuk, S., Dahms, T., Schöll, E.: Synchronizing distant nodes: a universal classification of networks. Physical Review Letters 105(25), 254101 (2010)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 117(4), 500–544 (1952)
Insperger, T., Stépán, G.: Semi-discretization for Time-delay Systems: Stability and Engineering Applications. Applied Mathematical Sciences, vol. 178. Springer (2011)
Kantner, M., Yanchuk, S.: Bifurcation analysis of delay-induced patterns in a ring of Hodgkin-Huxley neurons. Philosophical Transactions of the Royal Society A (to appear, 2013)
Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control 49(9), 1520–1533 (2004)
Orosz, G.: Decomposing the dynamics of delayed networks: equilibria and rhythmic patterns in neural systems. In: Sipahi, R. (ed.) 10th IFAC Workshop on Time Delay Systems, pp. 173–178. IFAC (2012)
Orosz, G.: Decomposition of nonlinear delayed networks around cluster states with applications to neuro-dynamics. SIAM Journal on Applied Dynamcal Systems (submitted, 2013)
Orosz, G., Moehlis, J., Ashwin, P.: Designing the dynamics of globally coupled oscillators. Progress of Theoretical Physics 122(3), 611–630 (2009)
Orosz, G., Moehlis, J., Murray, R.M.: Controlling biological networks by time-delayed signals. Philosophical Transactions of the Royal Society A 368(1911), 439–454 (2010)
Roose, D., Szalai, R.: Continuation and bifurcation analysis of delay differential equations. In: Krauskopf, B., Osinga, H.M., Galan-Vioque, J. (eds.) Numerical Continuation Methods for Dynamical Systems, Understanding Complex Systems, pp. 359–399. Springer (2007)
Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Pitman Research Notes in Mathematics, vol. 210. Longman (1989)
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Orosz, G. (2014). Decomposing the Dynamics of Delayed Hodgkin-Huxley Neurons. In: Vyhlídal, T., Lafay, JF., Sipahi, R. (eds) Delay Systems. Advances in Delays and Dynamics, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-01695-5_25
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DOI: https://doi.org/10.1007/978-3-319-01695-5_25
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