An advanced approach for the electrical responses of discrete fractional-order biophysical neural network models and their dynamical responses

Abstract

The multiple activities of neurons frequently generate several spiking-bursting variations observed within the neurological mechanism. We show that a discrete fractional-order activated nerve cell framework incorporating a Caputo-type fractional difference operator can be used to investigate the impacts of complex interactions on the surge-empowering capabilities noticed within our findings. The relevance of this expansion is based on the model's structure as well as the commensurate and incommensurate fractional-orders, which take kernel and inherited characteristics into account. We begin by providing data regarding the fluctuations in electronic operations using the fractional exponent. We investigate two-dimensional Morris-Lecar neuronal cell frameworks via spiked and saturated attributes, as well as mixed-mode oscillations and mixed-mode bursting oscillations of a decoupled fractional-order neuronal cell. The investigation proceeds by using a three-dimensional slow-fast Morris-Lecar simulation within the fractional context. The proposed method determines a method for describing multiple parallels within fractional and integer-order behaviour. We examine distinctive attribute environments where inactive status develops in detached neural networks using stability and bifurcation assessment. We demonstrate features that are in accordance with the analysis's findings. The Erdös-Rényi connection of asynchronization transformed neural networks (periodic and actionable) is subsequently assembled and paired via membranes that are under pressure. It is capable of generating multifaceted launching processes in which dormant neural networks begin to come under scrutiny. Additionally, we demonstrated that boosting connections can cause classification synchronization, allowing network devices to activate in conjunction in the future. We construct a reduced-order simulation constructed around clustering synchronisation that may represent the operations that comprise the whole system. Our findings indicate the influence of fractional-order is dependent on connections between neurons and the system's stored evidence. Moreover, the processes capture the consequences of fractional derivatives on surge regularity modification and enhance delays that happen across numerous time frames in neural processing.

Figure 1

According to group i and ii, bifurcation cases of the two-dimensional Morris–Lecar neural networks. (a) $\Im$ as a bifurcation factor: ($\Im =90,\vartheta =0.97$) and ($\Im =40,\vartheta =0.1$) indicate the presence of Hopf and saddle component type plots in the framework (18), respectively. The blue and red lines represent the structure’s balance and imbalance branches, respectively. (b) reflects the appearance of an unsteady limit process.

Figure 2

Phase depictions (which incorporates nullclines) in the $(u_1,u_2)$-plane for the two-dimensional Morris–Lecar neural networks system having discrete fractional-order commensurate technique: (a) $\vartheta =1$, (b) $\vartheta =0.97$, (c) $\vartheta =0.92$, (d) $\vartheta =0.89$, (e) $\vartheta =0.85$, (f) $\vartheta =0.82$, (g) $\vartheta =0.79$, (h) $\vartheta =0.75$, (i) $\vartheta =0.72$, (j) $\vartheta =0.70.$.

Figure 3

Time analysis of group i and ii of discrete fractional-order Morris–Lecar neural networks (18) for various fractional-orders (a-d) $\vartheta =1,0.97,0.95,0.93,0.90$ having $\Im =43;$ (e–h) $\vartheta =1,0.97,0.95,0.93,0.90$ having $\Im =50$ including the specification of group i and ii; (i–l) $\vartheta =1,0.97,0.95,0.93,0.90$ having $\Im =90$ including the specification of group iii.

Figure 4

Bifurcation plot as a single panel and time dependent plots of slow-swift active three-dimensional discrete fractional-order Morris–Lecar neural network model (19) for various fractional-orders, $\vartheta =1,0.95,0.90,0.85,0.80,0.75,0.72,0.70$ with model specifications group i, ii and iii.

Figure 5

Time-analysis plots interactions of an independently associated ensemble of group i and ii discrete fractional-order two-dimensional Morris–Lecar neural network (18) with various fractional-orders; (a-d) Group i: ${\vartheta }_1=\dots ={\vartheta }_{65}=1$ and ${\mathrm{\Phi }}_{66}=\dots ={\mathrm{\Phi }}_{100}=0.80$ having ${\mathcal{W}}_{ϵ}=0.0005,0.005,0.05,1.$ (e–h) Group ii: ${\vartheta }_1=\dots ={\vartheta }_{65}=0.95$ and ${\mathrm{\Phi }}_{66}=\dots ={\mathrm{\Phi }}_{100}=0.80$ having ${\mathcal{W}}_{ϵ}=0.0005,0.007,0.05,1.$ (i-l) Group iii: ${\vartheta }_1=\dots ={\vartheta }_{65}=0.84$ and ${\mathrm{\Phi }}_{66}=\dots ={\mathrm{\Phi }}_{100}=0.72$ having ${\mathcal{W}}_{ϵ}=0.0005,0.001,0.05,1.$ We selected these two structures from two samples to display the time indications. The interval for the assessment of the particular components identified via a blue region is selected from a subclass of components in inactive states (when ${\mathcal{W}}_{ϵ}=0$). The red indicator is drawn from points that were maintained in jumping assertions in the dearth of interaction.

Figure 6

Nature of neural network reactions depending on association abilities in a reduction in order discrete fractional-order three-dimensional Morris–Lecar system (19) for activated cells of group i and group ii via multiple fractional-orders. (a–d) Group i: $(\vartheta ,\mathrm{\Phi },\gamma )=(1,0.95,0.85)$ and ${\mathcal{W}}_{ϵ}=0.0005,0.005,0.05,1,$ respectively. (e–h) Group ii: $(\vartheta ,\mathrm{\Phi },\gamma )=(0.95,0.85,0.80)$ and ${\mathcal{W}}_{ϵ}=0.0005,0.007,0.05,1.$ (i–l) Group iii: $(\vartheta ,\mathrm{\Phi },\gamma )=(0.90,0.73,0.65)$ and ${\mathcal{W}}_{ϵ}=0.0005,0.007,0.05,1$.