Dynamics Of An Inertial Neuron Model With Delay-dependent Parameters

Over the past decades, many researchers have focused on the study of the dynamics of neural network models with delay. However, these studies typically assume that parameters are delay-independent. Memory performance of the biological neuron usually depends on time history, its memory intensity is usually lower and lower as time is gradually far away from the current time. Therefore, in neural network modeling, it is necessary to consider the parameters with delay-dependent.Some biological neurons can be modeled as having the inertia, such as the hair cells in thin film of some animals. In addition, inertial term was seen as the factor of affecting the stability of the neural model and generating chaos. Therefore, based on the aforementioned studies, this paper adds inertial term in the model of Ref.[17], to consider some dynamical behaviors of an inertial neuron model with delay-dependent parameters.This paper presents a detailed analysis on the dynamical behaviors of an inertial neuron model with delay-dependent parameters, containing the linear stability analysis, an analysis on the steady state bifurcation. On Hopf bifurcation including the direction and stability of bifurcation periodic solutions. Consider the impact of inertia and delay on the behaviors of a model with delay-dependent parameters.This letter considers the inertial neuron model as follows: Where x(t) denotes the neuron response,βdenotes damping, a>0 denotes the range of the continuous variable, b is a memory function, can be considered as a measure of the inhibitory influence from the past history. In this letter we assume that b(Ï„)(>0) is a decreasing function ofÏ„.F:[0,+∞)â†'[0,+∞) is a continuous delay kernel function.If the kernel function is a Dirac function of the form F(s)=δ(s-Ï„),Ï„> 0, the characteristic equation of the linearization of Eq.(1) at equilibrium is infinite-dimension, it is difficult to determine the characteristic roots. Therefore, it need to decide the crossing direction of the characteristic roots through the imaginary axis by the critia of stability switches.For determining the stability of the system at the equilibrium, according to Ref.[16], we shall show that the stability of a given steady state is simply determined by the graphs of some functions ofÏ„which can be expressed explicitly and thus can be easily depicted by some popular software, and give the formula of the crossing direction of the characteristic roots through the imaginary axis. It shows that system will undergo the finite number of stability switches with an increase of time delay. The system finally will be stable or unstable. And a moderate large delay could also stabilize the system. The analysis is demonstrated by numerical simulation, it shows that the dynamics of the neuron model with delay-dependent parameters is quite different from that the systems with delay-independent parameters only. Furthermore, it shows the necessarily of considering delay-dependent parameters. For the analysis on Hopf bifurcation, the method is the normal form theory and the center manifold theorem[28]. This method is the most common one of simplifying the delay system. Moreover, the direction and the stability of the bifurcating periodic solutions are obtained. It shows that the increasing delay could cause Hopf bifurcation and periodical solutions.Therefore, not only the kernel function effects the dynamics but also memory function, choosing the appropriate memory function also changes the stability of the neuron system. This provides a promising scheme to achieve special motivation for artificial representations of neural networks or to control dynamics of non-linear delayed systems.