Abstract
The understanding of the activity of neurons in the brain has been modeled as nonlinear systems using mathematical modeling for decades. Nonlinearity in brain dynamics is complex structure to do mathematically but computational techniques make this area of research quite interesting and easy to study the dynamics. With advancement of new technology, mathematical and computational studies are more preferable to understand the behavior of neurons in a single cell to global cognitive process. In the present study, the impacts of different externally applied currents on the behavior of neurons in a simple BVP model (Bonhoeffer-Vander Pol Model) are analyzed thoroughly. The results of BVP model are similar to the characteristics of neurons shown by the Hodgkin-Huxley Model. In the BPV model, when system is stable, neurons are in resting-state. Unlike Hodgkin-Huxley model which follows all-or-none law, the BVP model does not follow this all-or-none rule. In the BVP model, there is an intermediate phase where no spike forms, but when sufficiently large input applied then spikes emerge. On applying constant current in BVP model, system is stable while it exhibits oscillatory behavior when current is applied externally above threshold value of it. If sinusoidal, continuous wavelet, and har wavelet form of external applied currents are injected then continuous firing emerges which have several interesting dynamics. Numerical simulations have been performed to understand the bifurcation analysis of the BVP model. Oneparameter and two-parameter bifurcation diagrams have been drawn in which threshold current values are discussed.
Highlights
The brain is a major organ that is responsible for thought, feelings, learning, and memory, etc
In the phase-space analysis, with initial condition, the solutions of the differential equations are given by trajectories or paths [2] which may be useful to understand the dynamics of neurons
When constant external current is applied to the system 1, the different behavior of membrane potential and relation between membrane potential and recovery variable has been observed for the different values of the current
Summary
The brain is a major organ that is responsible for thought, feelings, learning, and memory, etc. Neuron activity is due to a change in the membrane potential of the cell [2]. FitzHugh extracted minimum informations from Hodgkin-Huxley equations and had formed a model named Bonhoeffer-Vander Pol Model (BVP Model). This is a simple model in two dimensions with three parameters which gives maximum information as similar to the Hodgkin-Huxley model [6]. Using the numerical simulations technique, the sensitivity of parameters and dynamical behavior of the BVP model are analyzed and discussed here. We have applied wavelet-current(see 2.2.4 & 2.2.5) to system 1 to observe the behavior of the system and pattern of the relation between membrane potential and recovery variable. The results have been be concluded in discussion section (see 3)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
