Bifurcations underlying different excitability transitions modulated by excitatory and inhibitory memristor and chemical autapses

Abstract

Memristor is an emerging modulation factor to mimic neuronal synapse, and the switch or transition between neuronal excitability classes is an important topic in nonlinear dynamics and neurophysiology. In the present investigation on the Morris-Lecar model, the memristive autapse is identified to induce excitability transitions different from those of the chemical one. The inhibitory and excitatory chemical autapses induce excitability switch from class I to II and from class II to I, respectively. However, the inhibitory and excitatory memristive autapses induce the transition from class II to I and from class I to II, respectively. Furthermore, comprehensive bifurcation mechanisms underlying the transitions are acquired. The class I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC), and class II excitability to the saddle-node (SN) bifurcation and Hopf bifurcation. Type I spiking corresponds to the SNIC or big homoclinic (BHom) orbit bifurcations, and type II spiking to the Fold Limit Cycle (FLC) bifurcation or supercritical Hopf bifurcation. Among various codimension-2 bifurcations, the saddle-node homoclinic orbit bifurcation, which is related to the SNIC, BHom, and SN bifurcations, is responsible for the excitability transition. A nameless degenerate bifurcation associated with the BHom and FLC bifurcations, is responsible for the spiking transition. In addition, the nullcline shapes present the geometric mechanism for the transitions. The obtained results present different roles of memristive and chemical autapses on modulating neuronal electrical activities and the underlying dynamical mechanisms, which are helpful for the design and development of novel memristive synapses with excitatory or inhibitory effects.