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Dynamics of Two Coupled Neurons of Different Types of Excitability

Yasaman, Somayeh | 2019

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 53310 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Razvan, Mohammad Reza
  7. Abstract:
  8. Excitability is one of the most important characteristics of a neuron. In 1948, Hodgkin identified three different types of excitability of neurons. Excitability an all of its types can be observed in Hodgkin-Huxley model of neuronal dynamics (H-H model) as a four-dimensional system of differential equations and in at least two dimensional reductions of H-H type models. By applying the theory of dynamical systems (e.g. bifurcation theory), one can give a mathematical definition of excitability.Excitability of the neuron is equivalent to that the neuronal model is near a bifurcation through which the state of the system approaches to a stable limit cycle. In this thesis, a two-dimensional reduction of H-H type model is considered as a model for single neuron’s dynamics, then a neuron of type I excitability (equivalently, near ”saddle-node bifurcation on invariant circle”) couple a neuron of type II excitability (equivalently, near ”saddle-node bifurcation” or ”subcritical Hopf bifurcation”) through gap junction. Then, it is investigated that how the level of excitability of the neuron of type I excitability affects the dynamics of the coupled system when one of the coupling neurons is fixed and the other varies. By applying the theory of dynamical systems (e.g. bifurcation theory), the results showed that increasing the coupling strength give rise to complex dynamics in the coupled neurons. More precisely, although the neurons not intrinsically and individually burst, can show complex dynamics when they are coupled. The stated complexbehaviors can be bursting oscillations in one of the neurons, burst synchronization of the neurons and synchrony. Moreover, sequences of intermittency in the neurons, which is an abrupt and irregular transition between periodic oscillations and chaotic dynamics, have been observed.In addition, when a neuron of type I excitability (near saddle-node bifurcatio on invariant circle) couple the other one of type II excitability (near saddle-node bifurcation), there exist two different synchronous behaviors, one of them corresponds to 1-burst synchronization and the other corresponds to the synchronization of 1-bursting oscillations in one of the neuron and tonic spiking oscillations in the other. Moreover, when a neuron of type I excitability (near saddle-node bifurcation on invariant circle) couple the other one of type II excitability (near subcritical Hopf bifurcation), different mixed mode oscillations may be observed in the neuron of type II excitability (near subcritical Hopf bifurcation)
  9. Keywords:
  10. Synchronisation ; Burst Synchronization ; Mixed Mode Oscillation ; Coupled Neurons ; Bursting Emergence ; Excitability Type ; Intermittency Sequence

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