Existence and uniqueness of wave fronts in neuronal network with nonlocal post-synaptic axonal and delayed nonlocal feedback connections
© Zhang; licensee Springer 2013
Received: 28 February 2013
Accepted: 29 July 2013
Published: 14 August 2013
An integral-differential model equation, arising from neuronal networks with both axonal and delayed nonlocal feedback connections, is considered in this paper. The kernel functions in the feedback channel we study here include not only pure excitations but also lateral inhibition. For the kernel functions in the synaptic coupling, pure excitations, lateral inhibition, the lateral excitations and more general synaptic couplings (e.g., oscillating kernel functions) are considered. The main goal of this paper is the study of the existence and uniqueness of the traveling wave front solutions. The main method we applied is the speed index functions and principle of linear superposition.
which was proposed by Hutt  to understand the mechanism of the formation and propagation of activity patterns in neural networks. Here, represents the effective post-synaptic potential of the neuron population at location x and time t. The first term and the second one on the right side of equation (1.1) represent the synaptic input by axonal and feedback connections, respectively. The kernel functions and are introduced as probability density functions of connection, which may be negative at some points to allow for inhibitory behavior in coupling. The parameters α and β represent the synaptic strength of axonal and nonlocal feedback contributions, respectively. Both the intral-areal nonlocal axonal connections with a transmission delay and nonlocal feedback connections with a constant time delay τ are incorporated in this model equation, where c is the transmission speed for both excitatory and inhibitory connections. The transfer function H is always chosen to be the Heaviside step function: for all , , and for all . denotes the output firing rate of a neuron, which means that a neuron fires at its maximum rate when the potential exceeds a threshold, and does not fire otherwise. Here, θ is assumed to be the firing threshold for all neurons functions.
The first class consists of nonnegative kernel functions (pure excitation).
The second class consists of Mexican hat kernel functions (lateral inhibition), that is, on and on for some positive constants M and N.
The third class consists of upside down Mexican hat kernel functions (lateral excitation), that is, on and on , for two positive constants M and N.
where k and ρ are positive constants.
In , Magpantay and Zou studied the wave fronts of equation (1.1), in which the kernel function in the feedback channel is assumed to be nonnegative (pure excitation), and for the kernel function in the synaptic coupling, four types, including types (A), (B) and (C) and the pure inhibition type, were considered. Modeling of traveling phenomena in general neural systems necessitates the study for more general types of kernel functions. However, it is more complicated in practical application in neuronal network. We may meet the case that both excitation and inhibition happen at the same time. Thus, it is important to study equation (1.1) with more general kernel functions.
In the model equation (1.1), the parameters are always supposed to be positive and . To include the case , it is understood that when . In this paper, for the kernel functions mentioned below, we suppose that they all satisfy the condition (1.4). We say if satisfies condition (B).
Motivated by their exciting pioneering works [12–17], in this paper, we aim to study the existence and uniqueness of the wave front solutions of IDE (1.1) with more general kernel functions. The main idea in this paper is employing the speed index functions (the main idea in [12, 17] and other pioneering works) and the principle of linear superposition. It is easy to see that the kernel functions that were studied before are included in our study. For example, if the kernel functions satisfy (L4) and (R3) with (see Section 4), then they are upside down Mexican hat kernel functions; actually, if the kernel functions satisfy (L3) and (R5) with , then they are of case (A) in . We also prove the existence and uniqueness of the traveling wave solutions of this model equation with the general classes of kernel functions under less restrictive conditions.
The traveling wave solutions of an equation are the solutions of the form , where is the moving coordinate and μ is a constant, which represents the speed of the traveling wave. Generally speaking, there are two kinds of traveling waves that attracted much more research concerns, because they possess some important and practical meanings in a neural network, which are the traveling wave front and the traveling pulse. We mainly focus on the traveling wave front in this paper.
According to the property of the traveling wave front, we know that if is a traveling wave front of equation (1.1), then and exist, and , should be two different constant solutions of equation (1.1) or equation (2.3). However, it is easy to see that the constant solutions of equation (1.1) only could be 0 or , since or .
Notice that function (2.6) we obtained is just a solution of equation (2.5), but not necessarily a solution of equation (2.1), that is to say that it is not necessarily a traveling wave front of equation (1.1). Function (2.6) could be a traveling wave front of equation (1.1) only if it satisfies , when , and when . Based on the discussion above, we study the existence and uniqueness of the wave front solution of IDE (1.1) in the following two sections under some conditions by proving the existence of a unique wave speed μ such that function (2.6) with this unique μ satisfies , when , and when .
3 Existence and uniqueness of the wave solution of IDE (1.1) with
Similar to the analysis given in Section 2, we know that function (3.3) could be a solution of equation (3.1) if it satisfies the phase conditions ; when and when .
which is a continuous function with μ.
3.1 Existence and uniqueness of the wave speed
then it is easy to see that for any kernel function of type (A), since β, c and τ are all positive parameters. Thus, for any θ on interval , there exists , such that , i.e., (3.5) holds when , which follows directly from the intermediate value theorem for continuous function.
For type (A), i.e., , it is obvious that and if and only if on , since on ℝ. So is a nonincreasing function with μ. Then the uniqueness of the zero to the equation follows from the property of . If the uniqueness of the zero to the equation does not hold, then there exists an interval such that when ; when and when . Then, we know that when , which means that on , and thus, when , which contradicts with the conclusion above that when . Consequently, is the unique zero of the equation if the kernel function is of type (A).
For type (B), i.e., , we have the following lemma.
Lemma 3.1 Suppose that the kernel function is of type (B), that is, on and on , and , then there exists a unique such that and if .
Consequently, for any , function (3.6) is decreasing with μ on interval , and then , which can be applied to get the uniqueness of the zero of the equation when . We summarize the discussion above in the following theorem.
Theorem 3.1 (Existence and uniqueness of the wave speed)
if the kernel function is of type (A), then for any , there exists a unique , such that ;
if the kernel functions is of type (B), then for any , there exists a unique , such that , where and is uniquely determined by .
3.2 Existence and uniqueness of the wave
We prove that function (3.3) could satisfy when and when , when and the kernel function is of type (A) or type (B), i.e., nonnegative function or Mexican hat function. In the rest of this subsection, we write μ for for simplicity.
Then has the same sign with . Obviously, when is of type (A). Thus, function is a nondecreasing function, since when is of type (A). Especially, we know that when is of type (A). Otherwise, there should be when , and thus, we get from (3.5), which is contradictory to the assumption. So when and when when the kernel function is of type (A), since and .
is nonincreasing on and nondecreasing on .
, and there exists such that on and on .
changes its sign at most once on , that is to say, on , or there exists , such that on and on .
From the discussion above, we know that there are two cases that may happen to if the kernel function is of type (B).
Case (1) on , on .
Case (2) on , on and on .
By a simple computation, we get and . If the first case holds, then is nonincreasing on , nondecreasing on . Note that , and . Consequently, when and when if case (1) holds. However, for case (2), it is easy to see that is also nonincreasing on , nondecreasing on and nonincreasing on . Because and , when and when if the case (2) holds. From the discussion above, we know that when and when for any kernel function of type (B). Consequently, we get the following theorem on the existence and uniqueness of wave solution to equation (3.1).
Theorem 3.2 (Existence and uniqueness of the wave front)
is of type (A), ;
is of type (B), , , where .
where , and the wave speed is uniquely determined by the wave speed equation (3.5).
4 Existence and uniqueness of the wave solution of IDE (1.1)
In this section, we will study the existence of the traveling wave fronts to this equation (1.1) with general types of kernel functions, which includes the cases that have been studied before. To be more precise, we will show the existence of traveling wave front to this equation when the kernel function in feedback channel is of nonnegative function or Mexican hat function and the kernel function in synaptic coupling is of some general types of functions. Not only three typical types of kernel functions but also oscillatory kernel functions within certain range of model parameters in synaptic coupling are considered. First, we give some assumptions for the kernel functions . In addition to the assumptions (1.4) for kernel functions, we also assume that the kernel function in satisfies one of the following conditions.
(L1) in .
where and , .
(L4) when and when , where .
Assume that the kernel function in satisfies one of the following conditions.
(R1) for all .
(R4) satisfies that there exist such that when and when .
Obviously, the kernel functions, which satisfy one of the conditions R i () in the interval and one of the conditions L j () in the interval , can form a very general class of functions, which includes all the classes of the kernel functions that appeared in previous works. In , we proved the existence of the traveling wave front of the IDE (1.1) when with the kernel function satisfying one of the assumptions L i () on and one of the assumptions R j () on . We cite the main results as follows.
Next, we give the following lemma to prove the existence of traveling wave front to equation (1.1) when the kernel function is of nonnegative function or Mexican hat function and the kernel function satisfy the assumptions as in Theorem 4.1 within a certain range of model parameters.
If the kernel function satisfies one of the assumptions L i () on and one of R j () on , then is monotonic decreasing on .
If the kernel function satisfies the assumptions L4 on and one of R j () on and , then is monotonic decreasing on .
- (2)If satisfies the assumptions L4 on and one of R j () on , then
Consequently, if , and thus, when . So, is monotonic decreasing on . □
Theorem 4.2 (Existence and uniqueness of the wave front)
which is (4.5). Note that the solution of equation (4.6) is not necessary; the traveling wave solution of the model equation (1.1), unless satisfies the phase conditions that , when and when . In the following, we prove that there exists a unique wave speed such that (4.4) with is the unique wave front solution of equation (1.1) satisfying the required conditions in this theorem.
It is easy to see that and , and is continuous with μ. Consequently, for any , there exists , such that . From the proofs of Theorem 3.1 and Theorem 4.1, we know that if the kernel functions and satisfy the conditions of this theorem, and are decreasing with μ, so the function is also decreasing with μ. Consequently, we get the wave speed, which is uniquely determined by (4.5). Let and , then , and .
where is uniquely determined by (4.5). □
For the case that the kernel functions satisfying L4 on and R j () on , the parameters α and β and the kernel function satisfying the same conditions as those in Theorem 4.2, we can get the existence of the wave speed by the same process. But the monotonic property of the function can not be guaranteed, neither the function , so the uniqueness of the wave speed cannot be obtained in the same way as Theorem 4.2. However, we can prove the uniqueness of the wave speed when .
Theorem 4.3 Suppose the kernel function satisfies L4 on and one of R j () on , the parameters α and β and the kernel function satisfy the same conditions as those in Theorem 4.2. Then for any , equation (1.1) has a unique traveling wave front solution satisfying the same phase and boundary conditions as in Theorem 4.2.
Proof By the same process as the proof of Theorem 4.2, we know that for any , there exists , such that . We show that there would be a contradiction if there existed such that , . Actually, , since is decreasing with μ on , and thus, . However, according to Lemma 4.1, we know that the function should be decreasing on . We get a contradiction. Consequently, for any , there exists a unique , such that , and . Just as the proof of Theorem 4.2, we can get the conclusion. □
5 Discussion and conclusion
The existence and uniqueness of the traveling wave fronts of the general integral differential model equation (1.1) arising from neuronal networks with both axonal and delayed nonlocal connections are investigated in this paper. Besides the three class of typical kernel functions, more general kernel functions are considered. Some known results are amended. In Theorem 4.1 of , Magpantay and Zou showed that for of type (C) and , if there exists a unique wave speed , then there exists a solution to equation (1.1) satisfying the phase and boundary conditions. It is easy to see that if satisfies (L4) and (R3) with , then is of type (C). From the results in Theorem 4.3, we know that the wave speed is unique, and thus, the unique traveling solution to equation (1.1) exists when .
We have obtained the existence and uniqueness of the wave front solution to IDE (1.1) with a very general kernel function in the first nonlinear term on the right side, but the kernel function is restricted to type (A) or type (B). However, at present, if (4.1) is still a wave front solution to (1.1) when the kernel function of type (C) is still open.
This work is supported by the Nature Science Foundation of China (No. 11101371). This work was motivated during the period that I was visiting the Department of Mathematics of Lehigh University. I would like to acknowledge the kind invitation from the Department of Mathematics of Lehigh University. Especially, I wish to express my sincere appreciation to Professor Linghai Zhang for his patient constructive instructions and discussions. I also would like to dedicate this paper to Prof. Jibin Li for his 70th birthday. I would like to express my gratitude to anonymous reviewers, whose comments helped to improve the presentation of the paper.
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