Discrete Dynamics in Nature and Society

Volume 2015, Article ID 836402, 12 pages

http://dx.doi.org/10.1155/2015/836402

## Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control

^{1}Key Laboratory of Biologic Resources Protection and Utilization, Hubei Minzu University, Enshi, Hubei 445000, China^{2}Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China^{3}College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China

Received 7 May 2015; Revised 14 August 2015; Accepted 30 August 2015

Academic Editor: Michael Radin

Copyright © 2015 Guangyao Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Spiking neuron models which exhibit rich dynamics are usually defined by hybrid dynamical systems. It is revealed that mathematical analysis of these models has important significance. Therefore, in this work, we provide a comprehensively qualitative analysis for a quadratic integrate-and-fire model by using the theories of hybrid dynamical system. Firstly, the exact impulsive and phase sets are defined according to the phase portraits of the proposed model, and then the Poincaré map is constructed. Furthermore, the conditions for the existence and stability of an order 1 periodic solution are provided. Moreover, the existence and nonexistence of an order periodic solution have been studied theoretically and numerically, and the results show that the system has periodic solutions with any period. Finally, some biological implications of the mathematical results are discussed.

#### 1. Introduction

To understand the working mechanism of the brain, it is necessary to combine experimental studies of nervous systems with numerical simulations of neuron models [1]. Therefore, a large number of differential equations have been proposed to model neurons in the neurocomputing community. However, there are two major impediments in computational neuroscience including the need for a computationally simple and biologically realistic model of the neuron [2]. To conquer these issues, FitzHugh and Nagumo introduced a new neuron model with cubic nonlinearity and two variables [3, 4]. Then this novel model was reduced to a class of nonlinear bidimensional spiking models with adaptation by many authors which are mathematically tractable, are efficiently implemented, and are able to reproduce many electrophysiological signatures [5–7]. These models often consist of two variables: the membrane potential of the neuron and the adaptation variable . In this paper, the leak and spike initiation currents function is assumed to be ; then the quadratic adaptive model can be described by the following ordinary differential equations:where represents the input intensity of the neuron, () is the characteristic time of the adaptation variable, and accounts for the interaction between the membrane potential and the adaptation variable. System (1) has been recently used by Izhikevich and Edelman in very large-scale simulations of neural networks [7]. In addition, the rich dynamics that system (1) presented have been investigated in detail by many authors [2, 8, 9].

In view of the spike and reset condition, a spike is emitted at time when the membrane potential reaches a threshold (or a cutoff value) . Then the membrane potential is reset to a constant value and the adaptation variable is updated to . Therefore, system (1) combining with the reset process can be written aswhere () describes the total amount of outward minus inward currents activated during the spike and is the reset membrane potential. System (2) is a hybrid dynamical system which is also known as a planar impulsive semidynamical system (ISDS) [10, 11]. Recently, Shlizerman and Holmes have studied system (2) and obtained explicit approximations of instantaneous firing rates for fixed values of the recovery variable, and then they used the averaging theorem to obtain asymptotic firing rates as a function of current and reset parameters, which provided explicit tools for the interpretation of different spiking patterns [8], whereas Touboul and Brette constructed an adaptation map to study the patterns of spikes numerically when the adaptation variable blows up [2].

In this paper, the global dynamical behaviors of system (2) will be studied theoretically. To this end, some useful definitions and lemmas of the planar ISDS will be presented in Section 2; then we will briefly review the rich dynamics of system (1). In Section 3, we present our main results. In particular, the impulsive and phase sets will be defined exactly in consideration of the phase portraits of model (1) in Section 3.1. In Section 3.2, the Poincaré map for impulsive point series defined in the exact phase set is obtained. In Section 3.3, the existence and stability of order 1 periodic solution will be addressed in detail. In Section 3.4, the existence of order periodic solutions will be studied theoretically and numerically. In the last section, Section 4, the related biological implications are discussed from a neurocomputational point of view.

#### 2. Preliminaries and Main Properties of System (1)

The generalized planar ISDS with state-dependent feedback control can be described as follows:where . We denote and for simplicity, and , , , and are continuous functions from into , and denotes the impulsive set. For each point , the map is defined: and is called an impulsive point of .

Let be the phase set (i.e., for any ), and . System (3) is generally known as a planar ISDS. We note that system (2) is an ISDS, where impulsive set is a closed subset of and continuous function . It follows that the phase set Without loss of generality, unless otherwise specified we assume the initial point .

In the following we briefly list some definitions related to ISDS, which are useful in this work.

Let or be a semidynamical system [10, 11], where is a metric space and is the set of all nonnegative reals. For any , the function is a continuous function such that for all , and for all and (denoted by ). The set is called the positive orbit of . For any set , let and , where and is the attainable set of at . Finally, we set . Before discussing the dynamical behavior of system (2), we need the following definitions and lemmas [12–16].

*Definition 1. *An ISDS consists of a continuous semidynamical system together with a nonempty closed subset (or impulsive set) of and a continuous function such that the following property holds:(i)No point is a limit point of ,(ii) is a closed subset of .

Throughout the paper, we denote the points of discontinuity of by and call an impulsive point of .

We define a function from into the extended positive reals as follows: let ; if we set ; otherwise and we set , where for but .

*Definition 2. *A trajectory in is said to be periodic of period and order if there exist nonnegative integers and such that is the smallest integer for which and .

For simplification, we denote a periodic trajectory of period and order by an order periodic solution. An order periodic solution is called an order limit cycle if it is isolated. For more details of the concepts and properties of continuous dynamical systems and impulsive dynamical systems, see [12, 15, 17].

Lemma 3 (Analogue of Poincaré Criterion [10, 11]). *The -periodic solution of systemis orbitally asymptotically stable and enjoys the property of asymptotic phase if the Floquet multiplier satisfies the condition , where with and , , , , , , , and are calculated at the point and . Here is a sufficiently smooth function such that , and is the time of the th jump.*

Since system (1) has been investigated by many scholars, bifurcations and phase portraits were also addressed in detail [2, 8, 9]. Here we briefly review the dynamics of system (1) which are useful in this study. The two isoclines of system (1) are denoted by and , where , . If we fix parameters and , choosing as a variable, then the following results can be obtained easily.

Lemma 4 (see [2, 8, 9]). *(1) For , there is no equilibrium in system (1). A saddle-node bifurcation curve is defined by , and there is a unique equilibrium for system (1).**(2) For , there are two equilibria for system (1); here , . An Andronov-Hopf bifurcation line is defined by . A homoclinic bifurcation occurs on a curve approximated by . Moreover, the curves , , and meet with common tangents at the codimension two Bogdanov-Takens bifurcation point .*

*Remark 5. *From Lemma 4, model (1) could present six different qualitative dynamics when choosing as a variable including the following:(i)if , then the homoclinic orbit disappears and the stable and unstable manifolds separate to create an unbounded trapping region;(ii)if , then there exists a homoclinic orbit;(iii)if , then there exists an unstable limit cycle;(iv)if , then an Andronov-Hopf bifurcation occurs;(v)if , then is an unstable equilibrium point and is a saddle;(vi)if , then ( and coincide) is a saddle-node point; if , then there is no fixed point for system (1). Moreover, case (i) to case (v) are classified as the type 2 neurons, while case (vi) is corresponding to the type 1 neurons (for details see [9]).

In order to address the global dynamics of system (2), we consider case () at which there exists a homoclinic orbit based on the qualitative behaviors of ODE model (1). In this case the system has two fixed points and , and the solution initiating from the inside of the homoclinic orbit may not reach the threshold under certain conditions (see more details later), which means that the impulsive effects may not happen. This indicates that system (2) may exhibit rich dynamics due to different values of the threshold and the dynamics presented in this case could be more complicated than other cases. Therefore, if we can investigate the global dynamics of this case in detail by using the theories of the ISDS and analytic techniques, then by using similar methods the dynamics of the remaining cases could be studied.

#### 3. Mathematical Analysis and Main Results

##### 3.1. Impulsive Set and Phase Set

In order to investigate the existence of order periodic solutions, the Poincaré map is constructed first. Nevertheless, it is essential to know that the exact conditions under which the solution of system (2) starting from is free from impulsive effects, that is to say, the more exact phase set , should be provided. Moreover, the impulsive set defined in Section 2 is the maximum interval for the vertical coordinates. Therefore, the part of which the solution of system (2) can not reach will be removed and then the exact domains of impulsive set can be obtained. From now on, unless otherwise specified we assume that throughout the paper.

Based on different positions of we consider the following three cases: ; ;

The homoclinic cycle is denoted by , and we let be the horizontal component of the small intersection point of the homoclinic cycle with the line (denoted by ; see Figure 1). For case , intersects with the right branch of the homoclinic cycle at two points and we denote the lower point by . According to the relations among , , and , there may be three cases for , , or . If , then there exists a trajectory such that the line tangents to the curve at the point ; here . Clearly, the curve also intersects the line at two points and we denote the lower point by . Therefore, if , then the impulsive set is defined by and the phase set is defined by ; here . If so, any trajectory of system (2) initiating from the interior of the curve either cannot reach the impulsive set or reaches the interval and then maps to the phase set after one time impulsive effect. If , then the line intersects the left branch of homoclinic cycle at two points, denoted by and . Thus, the impulsive set can be defined as and the phase set is defined by ; here . Moreover, the trajectories initiating from with will be free from the impulsive effects and will tend to the stable equilibrium . If , then by using the same methods as subcase the impulsive set is defined by , and the phase set is defined by .