Research Article Nonautonomous Discrete Neuron Model with Multiple Periodic and Eventually Periodic Solutions

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1 Discrete Dyamics i Nature ad Society Volume 21, Article ID , 6 pages Research Article Noautoomous Discrete Neuro Model with Multiple Periodic ad Evetually Periodic Solutios Alexader N. Pisarchik, 1,2 Michael A. Radi, 3 ad Rya Vogt 3 1 Cetro de Ivestigacioes e Optica, Loma del Bosque 11, Lomas del Campestre, 371 Leó, GTO, Mexico 2 Ceter for Biomedical Techology, Techical Uiversity of Madrid, Campus de Motegacedo, Madrid, Spai 3 College of Sciece, Rochester Istitute of Techology, 1 Lomb Memorial Drive, Rochester, NY 14623, USA Correspodece should be addressed to Michael A. Radi; michael.radi@rit.edu Received 2 April 21; Accepted 2 May 21 Academic Editor: Lu Zhe Copyright 21 Alexader N. Pisarchik et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We itroduce a oautoomous discrete euro model based o the Rulkov map ad ivestigate its dyamics. Usig both the liear stability ad bifurcatio aalyses of the system of piecewise differece equatios, we determie dyamical bifurcatios ad parameter regios of steady-state ad periodic solutios. 1. Itroductio Piecewise differece equatios exhibit very rich dyamics because the lack of differetiability makes their solutios either evetually costat, evetually periodic of various periods, or evetually chaotic [1]. The 3 +1cojecture proposed by Lothar Collatz i 1937 ad the tet map were the first cosidered piecewise liear differece equatios [2, 3]. Later, piecewise differece equatios have bee used as mathematical models for various applicatios, icludig euros (for extesive review, see [4] ad refereces therei). I this paper, we focus o the Rulkov map as oe of the simplest systems to model euro dyamics []. The origial Rulkov map is the autoomous system which reproduces the spikig behavior similar to biological euros. Although this model does ot have real parameters, it is computatioally less costly tha europhysiological models, such as, the Hodgki-Huxley model [6], ad hece it ca be easily used for simulatio of a complex etwork of syaptically coupled euros. Beig a part of the complex eural etwork, a euroisotseparated;itsdyamicsisaffectedbyoscillatios of the eighborig euros through syapses. Therefore, the euro ca be cosidered as a oautoomous system. 2. Model The autoomous Rulkov map is the system of piecewise differece equatios that cosists of three compoets [, 7]: whe x, x +1 = +y 1 x, y +1 =y μ(x + 1)+μσ, =, 1,..., x +1 =+y, y +1 =y μ(x + 1)+μσ, =, 1,..., whe x +y ad x 1, ad whe x >+y or x 1 >. x +1 = 1, y +1 =y μ(x + 1)+μσ, =, 1,..., (1) (2) (3)

2 2 Discrete Dyamics i Nature ad Society To covert the autoomous Rulkov map equatios (1) (3) ito a oautoomous system, we itroduce two periodic parameters ad 1 as follows: = {, if is eve, { (4) { 1, if is odd. It is our goal to ivestigate mootoic, periodic, ad chaotic characters of solutios. We will start with a liear stability aalysis of equilibrium poits of the autoomous system equatios (1) (). 3. Liear Stability Aalysis of the Autoomous System We will aalyze the local stability of the equilibrium poits of the autoomous map liearizig each of the three compoets idividually. The first compoet is the followig system: x +1 = +y 1 x, y +1 =y μ(x + 1)+μσ, whe x. By settig x= 1 x + y, =, 1,..., y=y μ(x+1) +μσ, we get the followig equilibrium poit: Now, let The, x=σ 1, y=σ 1 f(x,y)= 1 x +y, g(x,y)=y μ(x+1) +μσ. f x (x, y) = (1 x) 2, f y (x, y) = 1, g x (x, y) = μ, g y (x, y) = 1. () (6) 2 σ. (7) So, we get the followig Jacobia matrix J 1 : J 1 =[ f x (x, y) f y (x, y) g x (x, y) g y (x, y) ]= [(1 (σ 1)) 2 1 ] [ ] = [(2 σ) 2 1 ]. [ ] (8) (9) (1) Thus, the eigevalues of J 1 are the roots of the followig characteristic polyomial: (λ ) 2 (λ 1) +μ (2 σ) =λ 2 λ[1 + (2 σ) 2 ]+[ (2 σ) 2 +μ]=. Hece, we see that λ 1 <1ad λ 2 <1ofJ 1 if ad oly if (11) 1 + (2 σ) 2 < (2 σ) 2 +μ+1 < 2 (12) ad if ad oly if (i) (ii) (iii) <(2 σ) 2 (1 μ), (13) σ = 2, (14) μ<1. (1) Next, we will aalyze the local stability of the equilibrium poits of the secod compoet of the autoomous system: x +1 =+y, y +1 =y μ(x + 1)+μσ, =, 1,..., whe x +y ad x 1. By settig x=+y, y=y μ(x+1) +μσ, we get the followig equilibrium poit: Now, let The, x=σ 1, y=σ 1. f(x,y)=+y, g(x,y)=y μ(x+1) +μσ. f x (x, y) =, f y (x, y) = 1, g x (x, y) = μ, g y (x, y) = 1. (16) (17) (18) (19) (2)

3 Discrete Dyamics i Nature ad Society 3 So, we get the followig Jacobia matrix J 2 : J 2 =[ f x (x, y) f y (x, y) g x (x, y) g y (x, y) ]=[ 1 ]. (21) y f(μ) = (2 σ) 2 (1 μ); μ 1 σ is costat ad σ 2 Thus, the eigevalues of J 2 are the roots of the followig characteristic polyomial: (, (2 σ) 2 ) (λ )(λ 1) +μ=λ 2 λ+μ=. (22) Hece, we see that λ 1 <1ad λ 2 <1ofJ 2 if ad oly if 1 < 1 +μ<2 (23) ad if ad oly if Covergece regio (1/2, (3/4)(2 σ) 2 ) Stable periodicity Periodicity ad bifurcatio μ<1. (24) (, ) (1/4, ) (1, ) μ Fially, we will aalyze the local stability of the equilibrium poits of the third compoet of the autoomous system: x +1 = 1, y +1 =y μ(x + 1)+μσ, =, 1,..., whe x >+y or x 1 >. By settig x= 1, y=y μ(x+1) +μσ, we get the followig equilibrium poit: Now, let The, x= 1, y=r. f(x,y)=+y, g(x,y)=y μ(x+1) +μσ. f x (x, y) =, f y (x, y) =, g x (x, y) = μ, g y (x, y) = 1. So, we get the followig Jacobia matrix J 3 : (2) (26) (27) (28) (29) J 2 =[ f x (x, y) f y (x, y) g x (x, y) g y (x, y) ]=[ ]. (3) Thus, the eigevalues of J 3 are λ 1 = adλ 2 =. Figure 1: Bifurcatio diagram with μ as a cotrol parameter for σ= Stability of the Noautoomous System From the liearized stability aalysis, we decompose the oautoomous system ito six compoets ad apply the liearized stability aalysis o each oe as previously doe from which we obtai the followig stability coditios: (1) stability coditios: (2 σ) 2 (1 μ), μ 1 4, (31) (2) stable periodic coditios: 1 (2 σ) 2 (1 μ), 1 <μ<1. (32) 4 These coditios result i two bifurcatio diagrams show i Figures 1 ad 2. Bylettigσ=1becostat,wegetthe straight lie which bouds differet stability regios. Next, by lettig μ=.2 be costat, we get the parabola show i Figure 2, which bouds differet stability regios. From the liear stability aalysis of the autoomous systems ad from the two bifurcatio diagrams show i Figures 1 ad 2, we obtai the followig stability coditios: (i) stability coditios: max {, 1 } (2 σ) 2 (1 μ), μ 1 4, (33) (ii) stable periodic coditios: max {, 1 } (2 σ) 2 (1 μ), (iii) istability ad bifurcatios 1 <μ<1, (34) 4 mi {, 1 }>(2 σ) 2 (1 μ). (3)

4 4 Discrete Dyamics i Nature ad Society y f(σ) = (2 σ) 2 (1 μ); σ ad σ 2 μ is costat ad μ 1 1 (, 4(1 μ)) Periodicity ad bifurcatio x[] Stability (2, ) Stability σ Figure 4: Periodic orbit satisfies coditio (i). Figure 2: Bifurcatio diagram with σ as a cotrol parameter for μ= x[] x[] Figure : Periodic orbit satisfies coditio (i). Figure 3: Evetually steady periodic cycle satisfies coditio (i).. Time Series To illustrate the map dyamics, we will preset some graphical examples for various parameters i differet regios of the stability diagrams show i Figures 1 ad 2. Example 1. I this example, we assume that coditio (i) metioed above is satisfied, where μ=.2, σ=1, =.7, ad 1 =.7. The, we obtai a evetually steady periodic cycle i Figure 3. Weseethatithiscasethesolutiobecomesevetually periodic. Example 2. I this example, we assume that (ii) is satisfied, where μ=., σ=1, =.4, ad 1 =.3. This gives us the periodic orbit show i Figure 4. Now,weobservethatevethoughweareithestability regio, the solutio is evetually periodic istead of beig evetually costat as we have complex eigevalues whe μ>1/4. Example 3. I this example, we assume that (ii) is satisfied, where μ=.7, σ=1, =.2, ad 1 =.1, which gives us the graph i Figure. Now, we observe that the periodic character of the solutios chages as we have complex eigevalues whe μ> 1/4. Example 4. I this example, we assume that (ii) fails by lettig μ=.7, σ=.1, = 1., ad 1 =.7 which gives us the graph i Figure 6. Notice that this is evetually periodic orbit with a differet period compared to Example 4 due to the fact that the stability coditios fail.

5 Discrete Dyamics i Nature ad Society 1 1 x[] x[] Figure 6: Chaotic orbit satisfies coditio (ii). Figure 8: Evetually periodic orbit satisfies coditio (ii). 6. Coclusios ad Future Works x[] Figure 7: Ubouded solutios satisfy coditio (ii). Example. I this example, we will assume that (ii) fails by lettig μ=1, σ=1, = 4, ad 1 = 4.1whichgivesusthe graph i Figure 7. Notice that the periodicity is quite substatially differet compared to the previous examples due to the fact that the stability coditios fail. Example 6. I this example, we will assume that (ii) fails by lettig μ=1, σ=1, =.4, ad 1 =.whichgivesusthe graph i Figure 8. Notice that the periodicity is quite substatially differet compared to the previous examples due to the fact that the stability coditios fail. Furthermore, we observe that whe μ > 1/4, the periodic character of the solutios chages, ustable periodic orbits appear, ad chaotic behavior appears as well. O the base of a autoomous Rulkov map, we desiged a oautoomous discrete euro model. We demostrated steady state, periodic, ad chaotic character of solutios ad performed a liear stability aalysis of the equilibrium poits of both the autoomous ad oautoomous Rulkov maps. Our future goal is to geeralize the results whe { } = is periodic with period p 3. I particular, it would be iterestig to compare similarities ad differeces that will arise with the results of this paper whe { } = is periodic with period 2. Furthermore, it would be of paramout iterest to itroduce the periodic parameter {σ } = to icrease the iteractio betwee the euros of the model ad make the model much more accurate ad, moreover, to show how the iteractio betwee the terms of { } = ad {σ } = determies the stability of solutios, periods of solutios, ad boudedess of solutios. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmet ThisstudywassupportedbytheBBVA-UPMIsaacPeral BioTech Program. Refereces [1] V. C. Carmoa, E. Freire, E. Poce, ad F. Torres, O simplifyig ad classifyig piecewise-liear systems, IEEE Trasactios o Circuits ad Systems I: Fudametal Theory ad Applicatios, vol.49,o.,pp.69 62,22. [2] D. L. Johso ad C. D. Maddux, Logo: A Retrospective, HaworthPress,NewYork,NY,USA,1997. [3] J. Gleich, Chaos: The Amazig Sciece of the Upredictable, Vitage Books, Lodo, UK, 1998.

6 6 Discrete Dyamics i Nature ad Society [4] B. Ibarz, J. M. Casado, adm. A. F. Sajuá, Map-based models i euroal dyamics, Physics Reports, vol. 1, o. 1-2, pp. 1 74, 211. [] N. F. Rulkov, Modelig of spikig-burstig eural behavior usig two-dimesioal map, Physical Review E, vol. 6, o. 4, Article ID 41922, 22. [6] A. L. Hodgki ad A. F. Huxley, A quatitative descriptio of membrae curret ad its applicatio to coductio ad excitatio i erve, The Joural of Physiology, vol. 117, o. 4, pp. 44, 192. [7] J. M. Sausedo-Solorio ad A. Pisarchik, Sychroizatio of map-based euros with memory ad syaptic delay, Physics Letters A,vol.378,o.3-31,pp ,214.

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