Analysis and Control of the Improved Denatured Morris-Lecar Neuron Model

Lakshmi N. Sridhar

Archivos de Ciencia e Investigación(ADCI)

ISSN: 3068-014X | DOI: 10.33140/ADCI

Researchers and authors can directly submit their manuscript online through this link Online Manuscript Submission.

Track Your Submission

Share this page:

Open Access Journals

Research Article - (2025) Volume 1, Issue 2

View PDF Download PDF

Analysis and Control of the Improved Denatured Morris-Lecar Neuron Model

Lakshmi N. Sridhar ^*

Chemical Engineering Department, University of Puerto Rico, USA

^*Corresponding Author: Lakshmi N. Sridhar, Chemical Engineering Department, University of Puerto Rico, USA

Received Date: Aug 06, 2025 / Accepted Date: Oct 23, 2025 / Published Date: Nov 04, 2025

Copyright: ©�2025 Lakshmi N. Sridhar. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Citation: Sridhar, L. N. (2025). Analysis and Control of the Improved Denatured Morris-Lecar Neuron Model. Arch Cienc Investig, 1(2), 01-08.

Abstract

The dynamics of neurons is very complex and nonlinear, and it is important to understand the nonlinearity and develop strategies to control mechanisms as effectively as possible. In this work, bifurcation analysis and multiobjective nonlinear model predictive control is performed on the Improved Denatured Morris-Lecar Neuron Model. Bifurcation analysis is a powerful mathematical tool used to deal with the nonlinear dynamics of any process. Several factors must be considered, and multiple objectives must be met simultaneously. The MATLAB program MATCONT was used to perform the bifurcation analysis. The MNLMPC calculations were performed using the optimization language PYOMO in conjunction with the state-of-the-art global optimization solvers IPOPT and BARON. The bifurcation analysis revealed the existence of a Hopf bifurcation point and a limit point. The MNLMC converged to the utopia solution. The Hopf bifurcation point, which causes an unwanted limit cycle, is eliminated using an activation factor involving the tanh function. The limit points (which cause multiple steady-state solutions from a singular point) are very beneficial because they enable the Multiobjective nonlinear model predictive control calculations to converge to the Utopia point (the best possible solution) in the model.

Keywords

Bifurcation, Optimization, Control, Neuron

Background

Levy et al investigated the high-frequency synchronization of neuronal activity in the subthalamic nucleus of Parkinsonian Pa- tients with Limb Tremor. Govaerts, and Sautois studied the onset and extinction of neural spiking using a numerical bifurcation approach. Duan et al performed a codimension-two bifurcation analysis on firing activities in the Chay neuron model. Tsumo- to et al studied the bifurcations in the Morris-Lecar neuron model. Duan et al performed a two-parameter bifurcation analysis of firing activities in the Chay neuronal model. Wang et al studied the response of Morris-Lecar neurons to various stimuli. Liu et al performed bifurcation analysis studies of a Morris–Lecar neuron model. Gonzalez-Miranda studied the pacemaker dynamics in the full Morris-Lecar model [1-9]. Li et al studied the dynamic behav- ior in firing rhythm transitions of neurons under electromagnetic radiation. Barry et al researched optical magnetic detection of sin- gle-neuron action potentials using quantum defects in diamond.

Lv and Ma showed the existence of multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Jia et al studied the dynamics of transitions from anti-phase to multiple in-phase synchronizations in inhibitory coupled bursting neurons. Et´em´e, et al investigated firing and synchronization modes in neural network under magnetic stimulation. Mondal et al performed bifurcation analysis of a modified excitable neuron model [12-14]. Xing et al researched bifurcations and excitability in the temperature-sensitive Morris–Lecar neuron. Rajagopal et al studied the effects of very low frequency electric fields and of magnetic fields on the local and network dynamics of an excitable medium on a modified Morris-Lecar neuron model. Yang et al in- vestigated the synchronization behaviors of coupled fractional-or- der neuronal networks under electromagnetic radiation [15-17]. Muni et al studied the dynamical effects of electromagnetic flux on Chialvo neuron map [18]. Fatoyinbo et al studied the influence of sodium inward current on the dynamical behaviour of modi- fied morris-lecar model [19]. Fatoyinbo et al performed numerical Bifurcation Analysis of the Improved Denatured Morris-Lecar Neuron Model [20]. In this work, bifurcation analysis and multi- objective nonlinear model predictive control is performed on the improved Denatured Morris-Lecar Neuron Model Fatoyinbo et al. The paper is organized as follows. First, the model equations are presented, followed by a discussion of the numerical techniques involving bifurcation analysis and multiobjective nonlinear model predictive control (MNLMPC). The results and discussion are then presented, followed by the conclusions.

Model Equations

Birfucation Analysis

The MATLAB software MATCONT is used to perform the bifurcation calculations. Bifurcation analysis deals with multiple steady-states and limit cycles. Multiple steady states occur because of the existence of branch and limit points. Hopf bifurcation points cause limit cycles. A commonly used MATLAB program that locates limit points, branch points, and Hopf bifurcation points is MATCONT [21-22]. This program detects Limit points (LP), branch points (BP), and Hopf bifurcation points(H) for an ODE system

For a limit point, there is only one tangent at the point of singularity. At this singular point, there is a single non-zero vector, y, where Jy = 0. This vector is of dimension n. Since there is only one tangent the vector y = ( y₁ , y₂ , y₃ , y₄ ,...yn ) must align with wˆ= (w₁, w₂ , w₃, w₄,...w_n ). Since

@ indicates the bialternate product while In is the n-square identity matrix. Hopf bifurcations cause limit cycles and should be eliminated because limit cycles make optimization and control tasks very difficult. More details can be found in Kuznetsov and Govaerts [22-24]. Hopf bifurcations cause limit cycles. The tanh activation function (where a control value u is replaced by) (u tanh u/ε) is used to eliminate spikes in the optimal control profiles explained with several examples how the activation factor involving the tanh function also eliminates the Hopf bifurcation points. This was because the tanh function increases the oscillation time period in the limit cycle [25-30].

Multi objective Nonlinear Model Predictive Control (MNLMPC)

The rigorous multiobjective nonlinear model predictive control (MNLMPC) method developed by Flores Tlacuahuaz et al (2012) [31] was used.

t_f being the final time value, and n the total number of objective variables and u the control parameter. The single objective optimal

Result and Discussion

Conclusion

Bifurcation analysis and multiobjective nonlinear control (MN-LMPC) studies on the Improved Denatured Morris-Lecar Neuron Model. The bifurcation analysis revealed the existence of Hopf bifurcation points and limit points. The Hopf bifurcation point, which causes an unwanted limit cycle, is eliminated using an acti- vation factor involving the tanh function. The limit points (which cause multiple steady-state solutions from a singular point) are very beneficial because they enable the Multiobjective nonlinear model predictive control calculations to converge to the Utopia point (the best possible solution) in the models. A combination of bifurcation analysis and Multiobjective Nonlinear Model Predic- tive Control(MNLMPC) for the Improved Denatured Morris-Le- car Neuron Model is the main contribution of this paper.