Impact of external applied currents in BVP model

¹ DST-CIMS, Institute of Science, Banaras Hindu University, Varanasi, India
² Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India

^* e-mail: avinita.gautam2@bhu.ac.in
^** e-mail: anupam240@gmail.com

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.