Doubly Periodic Traveling Waves in a Cellular Neural Network with Linear Reaction
© Copyright © 2009 2009
Received: 4 June 2009
Accepted: 13 October 2009
Published: 1 December 2009
Szekeley observed that the dynamic pattern of the locomotion of salamanders can be explained by periodic vector sequences generated by logical neural networks. Such sequences can mathematically be described by "doubly periodic traveling waves" and therefore it is of interest to propose dynamic models that may produce such waves. One such dynamic network model is built here based on reaction-diffusion principles and a complete discussion is given for the existence of doubly periodic waves as outputs. Since there are 2 parameters in our model and 4 a priori unknown parameters involved in our search of solutions, our results are nontrivial. The reaction term in our model is a linear function and hence our results can also be interpreted as existence criteria for solutions of a nontrivial linear problem depending on 6 parameters.
Szekely in  studied the locomotion of salamanders and showed that a bipolar neural network may generate dynamic rhythms that mimic the "sequential" contraction and relaxation of four muscle pools that govern the movements of these animals. What is interesting is that we may explain the correct sequential rhythm by means of the transition of state values of four different (artificial) neurons and the sequential rhythm can be explained in terms of an -periodic vector sequence and subsequently in terms of a "doubly periodic traveling wave solution" of the dynamic bipolar cellular neural network.
Similar dynamic (locomotive) patterns can be observed in many animal behaviors and therefore we need not repeat the same description in . Instead, we may use "simplified" snorkeling or walking patterns to motivate our study here. When snorkeling, we need to float on water with our faces downward, stretch out our arms forward, and expand our legs backward. Then our legs must move alternatively. More precisely, one leg kicks downward and another moves upward alternatively.
Let and be two neuron pools controlling our right and left legs, respectively, so that our leg moves upward if the state value of the corresponding neuron pool is and downward if the state value of the corresponding neuron pool is Let and be the state values of and during the time stage where Then the movements of our legs in terms of will form a -periodic sequential pattern
To this end, in , we build a (nonlogical) neural network and showed the exact conditions such doubly periodic traveling wave solutions may or may not be generated by it. The network in  has a linear "diffusion part" and a nonlinear "reaction part." However, the reaction part consists of a quadratic polynomial so that the investigation is reduced to a linear and homogeneous problem. It is therefore of great interests to build networks with general polynomials as reaction terms. This job is carried out in two stages. The first stage results in the present paper and we consider linear functions as our reaction functions. In a subsequent paper, as a report of the second stage investigation, we consider polynomials with more general form (see the statement after (2.11)).
2. The Model
We briefly recall the diffusion-reaction network in . In the following, we set and For any we also use [ ] to denote the greatest integer part of Suppose that are neuron pools, where placed (in a counterclockwise manner) on the vertices of a regular polygon such that each neuron pool has exactly two neighbors, and where For the sake of convenience, we have set and to reflect the fact that these neuron pools are placed on the vertices of a regular polygon. For the same reason, we define for any and let each be the state value of the th unit in the time period During the time period , if the value of the th unit is higher than , we assume that "information" will flow from the th unit to its neighbor. The subsequent change of the state value of the th unit is , and it is reasonable to postulate that it is proportional to the difference , say, , where is a proportionality constant. Similarly, information is assumed to flow from the -unit to the th unit if . Thus, it is reasonable that the total effect is
If we now assume further that a control or reaction mechanism is imposed, a slightly more complicated nonhomogeneous model such as the following
where is the greatest common divisor of and For this reason, we will pay attention to the condition that Formally, given any and with a real double sequence is called a traveling wave with velocity if
Next, recall that a positive integer is called a period of a sequence if for all . Furthermore, if is the least among all periods of a sequence then is said to be -periodic. It is clear that if a sequence is periodic, then the least number of all its (positive) periods exists. It is easy to see the following relation between the least period and a period of a periodic sequence.
We may extend the above concept of periodic sequences to double sequences. Suppose that is a real double sequence. If such that for all and then is called a spatial period of Similarly, if such that for all and then is called a temporal period of . Furthermore, if is the least among all spatial periods of , then is called spatial -periodic, and if is the least among all temporal periods of then is called temporal -periodic.
In seeking solutions of (2.2) that satisfy (2.5) and (2.6), in view of Lemma 2.1, there is no loss of generality to assume that the numbers and are the least spatial and the least temporal periods of the sought solution. Therefore, from here onward, we will seek such doubly-periodic traveling wave solutions of (2.2). More precisely, given any function and with in this paper, we will mainly be concerned with the traveling wave solutions of (2.2) with velocity which are also spatial -periodic and temporal -periodic. For convenience, we call such solutions -periodic traveling wave solutions of (2.2) with velocity
where are real numbers, and is a real parameter. In , the trivial polynomial and the quadratic polynomial are considered. In this paper, we will consider the linear case, namely,
Since the trivial polynomial is considered in , we may avoid the case where A further simplification of (2.11) is possible in view of the following translation invariance.
Therefore, from now on, we assume in (2.2) that
As for the traveling wave solutions, we also have the following reflection invariance result (a direct verification is easy and can be found in ).
Lemma 2.3 (cf. proof of [2, Theorem 3]).
3. Basic Facts
Second, we set
It is known (see, e.g., ) that for any the eigenvalues of are and the eigenvector corresponding to is
Suppose that case (a) holds. Take
The other cases (b)–(e) can be proved in similar manners and hence their proofs are skipped.
In other words, if then is -periodic. Next, suppose that is, there exists some such that and Note that and hence we also have Since and we have Taking then we have Hence is a period of and That is, is not -periodic. In conclusion, if is -periodic, then
The above can be used, as we will see later, to determine the spatial periods of some special double sequences.
To see (i), suppose that and is a nonzero vector Note that the fact that with implies By Lemma 3.1(i), is a period of if, and only if, and . By Lemma 3.1(i)again, and if, and only if, is a period of Hence the least period of is the same as and is spatial -periodic if, and only if, is -periodic. Note that is a period of By Lemma 2.1 and Lemma 3.1(i), we have is -periodic if and only if or for any with
The assertions (ii) and (iii) can be proved in similar manners. The proof is complete.
That is, This is contrary to our assumption. That is, if are both even, then we have for some and By similar arguments, in case where are both odd or where is even and is odd, we also have for some and In summary, if for all then is odd and is even.
The assertion (ii) is proved in a manner similar to that of (i). The proof is complete.
4. Necessary Conditions
Then we have
where is either the zero vector or an eigenvector of corresponding to the eigenvalue and is either a zero vector or an eigenvector of corresponding to the eigenvalue 1. Suppose that is the zero vector, or, is not an eigenvalue of Then must be an eigenvalue of and must be an eigenvector corresponding to the eigenvalue 1; otherwise, and this is impossible. Thus, is a temporal period of This is contrary to being the least among all periods and In conclusion, has eigenvalue and where is an eigenvector of corresponding to the eigenvalue and is either a zero vector or an eigenvector of corresponding to the eigenvalue Since are all distinct eigenvalues of there exists some such that
To see (iv), recall the result in (ii). We have
Then is a temporal period of and this is contrary to Thus Since the fact that is odd implies This leads to a contradiction. So we must have that is even and Note that and implies for some odd integer The proof is complete.
5. Existence Criteria
Now we turn to our main problem. First of all, let with and satisfy (2.16). If with and if (4.4) has a -periodic traveling wave solution of (4.4) with velocity by Lemmas 4.2 and 4.3, must be For this reason, we just need to consider five mutually exclusive and exhaustive cases: (i) (ii) and (iii) and (iv) and and (v) and
By Lemma 4.1, we also know that is not invertible and is a nonzero vector in Note that are all distinct eigenvalues of with corresponding eigenspaces respectively. Since and is not invertible, we have for some Hence and it is clear that
where such that is a nonzero vector. If we see that must be since It is clear that Suppose and recall that the extension of is -periodic. By Lemma 3.1(ii),the extension of is -periodic if and only if .
Conversely, suppose ; there exists some such that and when . Let satisfy (5.1). By the definition of it is clear that is temporal -periodic and is a spatial period of . Suppose and then we have that The fact that is not a zero vector implies By Lemma 3.1(iii), we have that is -periodic. By (5.1), it is clear that is spatial -periodic. Suppose Since by Lemma 3.1(ii), we have is -periodic. By (5.1) again, it is also clear that is spatial -periodic. In conclusion, we have that is spatial -periodic, that is,
To see (ii), note that from the second part of the proof in (i), it is easy to see that any satisfying (5.1) is a -periodic traveling wave solution of (4.4) with velocity Also, by the first part of the proof in (i), the converse is also true. The proof is complete.
Now we consider the case In this case, and are specific integers. Hence it is relatively easy to find the -periodic traveling wave solutions of (4.4) with velocity for any satisfying (2.16). Depending on the parity of we have two results.
For (ii), let and is odd. By direct computation, we have and are eigenvalues of with corresponding eigenvectors and respectively. Suppose is a -periodic traveling wave solution with velocity By Lemma 4.3(ii), we have
For (iii), suppose that is odd, and Then we have that is an eigenvalue of with corresponding eigenvector and another eigenvalue Suppose that is a -periodic traveling wave solution with velocity By Lemma 4.3(ii), we have
The assertion (iv) is proved by the same method used in (ii).
Here the facts in Lemma 3.2 will be used to check the spatial period of a double sequence Furthermore, when is odd, the conclusions in Lemma 3.3 will be used to check whether a double sequence is a traveling wave.
(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is even, for some odd integer and there exists some such that and either (a) or (b) is odd and for any with one has either or ;
If and is even, by Lemma 3.3(i), does not satisfy (5.31). This leads to a contradiction. If and for some with by Lemma 3.2(i), we see that is not spatial -periodic. This leads to a contradiction again. In conclusion, we have that satisfies condition (a) or (b).
For the converse, suppose that is even, for some odd integer and for some We further suppose that satisfies (a) and let be defined by (5.20). Recall that and are eigenspaces of corresponding to the eigenvalues and respectively. Hence by direct computation, we have that is a solution of (4.4). Since , we also have that is temporal -periodic. Since we have for any with By (i) and (iii) of Lemma 3.2, it is easy to check that is spatial -periodic. The fact implies that is odd. From (i) and (ii) of Lemma 3.3, we have that
where and is a nonzero vector. Since is a spatial period of we have that is of the form (5.20). Now we just need to show that if then we have . Suppose to the contrary that is a zero vector, and By Lemma 3.2 (iii), is not spatial -periodic. This leads to a contradiction. The converse has been shown in the second part of the proof of (i).
(i)(4.4) has a - periodic traveling wave solution with velocity if, and only if, is odd, for some odd integer and for some such that either (a) and or (b) with or (c) with and for any with one has either or ;
(ii)furthermore, if satisfies condition (i)–(a) above, every such solution is of the form (5.20), and the converse is true; while if satisfies condition (i)–(b) above, every such solution is of the form (5.22), and the converse is true.
6. Concluding Remarks and Examples
Recall that one of our main concerns is whether mathematical models can be built that supports doubly periodic traveling patterns (with a priori unknown velocities and periodicities). In the previous discussions, we have found necessary and sufficient conditions for the existence of traveling waves with arbitrarily given least spatial periods and least temporal periods and traveling speeds. Therefore, we may now answer our original question as follows. Suppose that we are given the parameters and where with and the reaction-diffusion network:
( )Suppose (i) where is even, with or (ii) where is even, with odd and even and for any with one has either or Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which satisfy (2.16), is odd, and for some odd integer
( )Suppose (i) where is odd, with or (ii) where is odd, and for any with one has either or Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which satisfy (2.16), is even, and .
Finally, we provide some examples to illustrate the conclusions in the previous sections.
By direct computation,
By direct computation, we have
Secondly, let where Recall (6.15), we see that for all By our assumption, it is easy to check that is even and for some odd integer . We also have and note that because of By Theorem 5.7(i), (6.13) has doubly periodic traveling wave solutions. By Theorem 5.7(ii), any solution is of the form
for and In Figure 4, we take and for illustration.
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