Understanding perspectives for mixed mode oscillations of the fractional neural network approaches to the analysis of neurophysiological data from the perspective of the observability of complex networks

Saima Rashid; Ilyas Ali; Sobia Sultana; Zeemal Zia; S K Elagan

doi:10.1016/j.heliyon.2024.e40659

Understanding perspectives for mixed mode oscillations of the fractional neural network approaches to the analysis of neurophysiological data from the perspective of the observability of complex networks

Heliyon. 2024 Nov 26;10(23):e40659. doi: 10.1016/j.heliyon.2024.e40659. eCollection 2024 Dec 15.

Authors

Saima Rashid^{1

2}, Ilyas Ali³, Sobia Sultana⁴, Zeemal Zia¹, S K Elagan⁵

Affiliations

¹ Department of Mathematics, Government College University, Faisalabad 38000, Pakistan.
² Department of Computer Science and Mathematics, Lebanese American University, 11022801, Beirut, Lebanon.
³ Department of Basic Sciences and Humanities, University of Engineering and Technology Lahore, Faisalabad Campus, Faisalabad, Pakistan.
⁴ Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia.
⁵ Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia.

Abstract

Using discrete fractional calculus, a wide variety of physiological phenomena with various time scales have been productively investigated. In order to comprehend the intricate dynamics and activity of neuronal processing, we investigate the behavior of a slow-fast FitzHugh-Rinzel (FH-R) simulation neuron that is driven by physiological considerations via the Caputo fractional difference scheme. Taking into account the discrete fractional commensurate and incommensurate mechanisms, we speculate on the numerical representations of various excitabilities and persistent activation reactions brought about by the administered stimulation. Furthermore, the outcomes concentrate on the variability of several time scales, encompassing mixed-mode oscillations and mixed-mode bursting oscillations formed by the canard occurrence. It is confirmed that the fast-analyzing component, which was isolated within this framework with the slow-fast evaluation process, is bistable, and the criterion for bistability is added as well. The architecture appears to be bistable based on this. The pertinent factors for examining time evolution, Poincaré maps, the bifurcation configuration of the system and chaos illustrations involve the inserted power stimulation using commensurate and incommensurate fractional-order values. We investigate the canards adjacent to the folded platforms using the folded node hypothesis. Additionally, we are employing mixed-mode oscillations to illustrate the homoclinic bifurcation and the resulting chaotic trajectory. Also, we determine our research results by computing the Lyapunov spectra as an expression of time in conjunction with the dominating factor ℑ to demonstrate the chaotic behavior in a particular domain. Besides that, we estimate intricacy employing the sample entropy (Sp-En) approach and $C_{0}$ complexity. The emergence of chaos within the hypothesized discrete fractional FH-R system is verified using the $0 - 1$ criterion. Ultimately, we examine the prospective implications of mixed-mode oscillations in neuroscience and draw the inference that our observed outcomes could potentially be of great relevance. As a result, the predicted intricacy decreases while applying it to non-horizontal significant models. Finally, the simulation's characteristic phases, canards and mixed model oscillations are achieved statistically with the assistance of varying fractional orders.

Keywords: Caputo fractional difference operator; Data driven analysis; FitzHugh–Rinzel model; Geometric desingularization dynamics; Homoclinic bifurcation; Mixed-mode oscillations; Statistical test.