The stability analysis of a neural field model with small delay

Abstract

Keywords: Neural Field Model; Routh-Hurwitz Criterion; Stability Analysis

Introduction

Because of the complexity of biological systems in neuroscience, we use mathematical models to understand their general behavior and express the fundamental principles. Two fundamental types of cells in the brain are neurons and glia. Interconnection of neurons is important for living being. Three main parts in connection between neurons are receiving signals from dendrites, sending signals along the axon and creating the response for the signal transmitted. The whole process is based on an electrochemical mechanism [3]. Details about membrane dynamics of a single neuron are introduced in the work of Hodgkin and Huxley [9] by a nonlinear dynamical system. Later, when the case at which neurons are grouped, some researches are made and mathematical models giving the relation on the average pre-synaptic firing rates and the average post-synaptic membrane potentials of neural populations are written. In modelling

the brain activity and nonlinear dynamics of populations of neurons, the neural field models are developed by propounding parameters and making estimations on experimental data. For the applications of models of neural field theory in neuroscience, the studies of brain activity, sleep cycles, epilepsy and Alzheimer’s disease may be given [3]. For further reading on the neural field models, one can refer to fundamental studies made by Amari [1] and Wilson and Cowan [21]. In the study of Wilson and Cowan, we see excitatory and inhibitory populations of neurons. These neural field models demonstrate the activity of neurons and consist of integro-differential equations. In brain, neurons have a significant role in receiving and transmiting signals. In communications of neurons, the time for releasing the neurotransmitter and for a signal passing through the axon, is introduced to the model by a delay term. Refining and developing the analysis on behavior of solutions of these models needs some analytical approaches or numerical methods. There are some substantial researches

*Corresponding author. *E-mail address: berrak.ozgur@idu.edu.tr This paper was recommended for publication in revised form by Editor in Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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on the stability of the model and the effect of the delay term [2,4-6,8,17-20]. From a dynamical systems point of view, the investigation of the roots of the characteristic equation for the neural field model made by the Routh-Hurwitz criterion or D-curves method can be seen in [11-15]. The details for these methods can be found in [7,10,16]. In this study we are interested in the stability of a neural field model for two neuron populations with small delay. First, the model is introduced. In next section, the stability switches in terms of delay term and system parameters are given. The results obtained are given in summary in the last part of this section. The conclusion of this study is given in the last section.

Here we consider the effect of the small delay ε in the functions U1(x,t) and U2(x,t) of the model written for two . We assume the propneuron populations on agation delay as τ(x,y) = τ. The synaptic connectivity functions generally describe how neurons in the Jth population at position y influence the neurons in the ith population at position x. As a special case for the model, we consider the case in which synaptic connectivity functions as J11(x,y) = J22(x,y) = 0. An illustrative explanation of the model considered here is given below in Figure 1. This is an adapted version of the figure given in [3] showing the relation of neurons in population I and II. The main idea of this figure is based on the work by Wilson and Cowan [21].

The Model The neural field equations for p neural populations on the space Ω ⊂ Rd modelling the dynamics of mean membran potential [18-20] is given below

Stability Analysis If we write the first order Taylor series expansion for the functions U1(x, t − ε) and U2(x, t − ε) then we get

In this neural field model, the functions Vi(t,r) represent the synaptic inputs for a large group of neurons at position x and time t. The synaptic connectivity is given by the functions Ji,j(r,r-). These functions are π periodic even functions. Hence the linearized system near (0,0) in terms of the functions U1(x,t) and U2(x,t) for the synaptic inputs is the following:

We are looking for the solutions in the form U1(x,t) = u1(t)eikx = c1eλteikx and U2(x,t) = u2(t)eikx = c2eλteikx . Substituting them we get the following

For the simplicity we take K1 = σ1s1, K2 = σ2s1 and Figure 1. This figure is an illustration of the mathematical model studied. This is an adapted version of the figure given in [3] showing the relation of neurons in population I and II. The main idea of this figure is based on the work by Wilson and Cowan [21]. Wilson and Cowan [21], with permission from Springer.].

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Rearranging the system for u1(t) and u2(t) and considering the coefficient determinant, we get the characteristic values λ satisfying the following equation

Separating the real and imaginary parts, we get the following two equations

The Case: Absence of Delay Term For the stability analysis, we can make use of the RouthHurwitz criterion [7]. First we analyze the stability considering the absence of the propagation delay term τ. In this case the equation (6) turns into

(7) If the coefficients of λ and λ2 and the constant term are positive, then all the roots have negative real parts. Let . If

then we may conclude that all roots have negative real parts. According to the model, the conditions l1 > 0 and l2 > 0 are satisfied. Hence the conditions on the stability given above may be summarized to show the relations among parameters of the system in the following theorem. Theorem 1 : Consider the system (2) and its characteristic equation (7) in case of no delay term exists. Hence, if the conditions , and are satisfied then the system is stable. Proof : Considering the equation (7) and investigating its roots according to the Routh-Hurwitz criteria, if the conditions given are hold then all roots have negative real parts. Hence the system is stable. The Case: Existence of Delay Term In this section, as a general case, we investigate the stability considering the propagation delay term. For this aim, we consider the imaginary roots λ = iσ for the equation. After converting the characteristic equation into a polynomial form, we make an analysis for the roots whether they have positive real parts or not. Writing λ = iσ in (6) then we have the following equation

Now our aim is to apply the Routh-Hurwitz criterion for the stability analysis. Hence we add the squares of both sides of these equations. Then we have

We will make the analysis using a polynomial including the characteristic values. Hence we make the substitution µ = σ2, then we get the following polynomial equation making the leading coefficient 1, (8) Existence of a single positive root for the equation (8) leads the unstability of the system. Hence we may conclude this case by the following theorem. Theorem 2 : Consider the system (2) and its characteristic equation (8) in case of delay term exists. If

is satisfied then the system is unstable. Proof : According to the Routh-Hurwitz criteria, since the leading coefficient of (8) is positive, if the condition

is satisfied then there is a single positive root. Hence the unstability of the system occurs. The General Stability and Unstability Cases In this section we give the conclusions made by two theorems in the preceding sections. The following corollaries include the changes in the stability of the system in terms of the effects of the system parameters. Corollary 1 : Consider the system (2) with small delay. Preserving the stability conditions given in Theorem 1, the following two cases occur.

1. If K1K2F1F2 < 0 and l1 l2 < |K1K2F1F2| are satisfied then

there is a positive root. Hence the system becomes unstable.

2. If K1K2F1F2 < 0 but l1 l2 < |K1K2F1F2| is satisfied or the

condition K1K2F1F2 > 0 holds, then the system is still stable. Proof : Considering the system (2) with delay term, we have the characteristic equation (8). Since the leading coefficient is positive, if the constant term is negative then there

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exists a positive root. If the conditions K1K2F1F2 < 0 and l1 l2 < |K1K2F1F2| are satisfied then (l1 l2 )2 - (K1K2F1F2)2 < 0 and hence there is a positive root. This implies the unstability of the given system. Otherwise, no positive root exists. Corollary 2 : Consider the system (2). Under the condition l1 l2 - K1K2F1F2 < 0, the system is always unstable. Proof : For the system (2) with or without delay, the characteristic equation has a positive root under given condition. Hence the system is unstable.

Conclusion

In this study, a special case of the neural field model for two populations including a small delay is addressed. Investigation of the characteristic equation and using the Routh-Hurwitz criterion, we identify the connections between the parameters and the newly added delay term for the stability of the system. The powerful and practical Routh-Hurwitz criterion captures some practical results that the small delay term ε does not have an effect on the stability of the model considered when the propagation delay τ exists.

Acknowledgements

The authors would like to thank the reviewers for their valuable and helpful comments.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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