ComplexityInducedbyExternalStimulationsinaNeuralNetwork...
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Research Article ComplexityInducedbyExternalStimulationsinaNeuralNetwork System with Time Delay Bin Zhen , 1 Dingyi Zhang, 2 and Zigen Song 3 1 School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, China 2 College of Food and Technology, Shanghai Ocean University, Shanghai 201306, China 3 College of Information Technology, Shanghai Ocean University, Shanghai 201306, China Correspondence should be addressed to Zigen Song; [email protected] Received 8 July 2020; Accepted 12 August 2020; Published 7 September 2020 Academic Editor: Gonglin Yuan Copyright © 2020 Bin Zhen et al. is 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. Complexity and dynamical analysis in neural systems play an important role in the application of optimization problem and associative memory. In this paper, we establish a delayed neural system with external stimulations. e complex dynamical behaviors induced by external simulations are investigated employing theoretical analysis and numerical simulation. Firstly, we illustrate number of equilibria by the saddle-node bifurcation of nontrivial equilibria. It implies that the neural system has one/three equilibria for the external stimulation. en, analyzing characteristic equation to find Hopf bifurcation, we obtain the equilibrium’s stability and illustrate periodic activity induced by the external stimulations and time delay. e neural system exhibits a periodic activity with the increased delay. Further, the external stimulations can induce and suppress the periodic activity. e system dynamics can be transformed from quiescent state (i.e., the stable equilibrium) to periodic activity and then quiescent state with stimulation increasing. Finally, inspired by ubiquitous rhythm in living organisms, we introduce periodic stimulations with low frequency as rhythm activity from sensory organs and other regions. e neural system subjected by the periodic stimulations exhibits some interesting activities, such as periodic spiking, subthreshold oscillation, and bursting-like activity. Moreover, the subthreshold oscillation can switch its position with delay increasing. e neural system may employ time delay to realize Winner-Take-All functionality. 1. Introduction Imitating the properties of a biological nerve system to build artificial neural network model plays an important role in the fields of neural network applications. Complexity and dynamical analysis in neural systems are important re- quirements for the application of optimization problem and associative memory. In fact, many neural network systems, such as Hopfield neural system, Cohen–Grossberg neural system, and cellular neural system, have been applied in many research fields such as associative memory, secure communication, and signal processing. Actually, associative memory storage in neural networks is defined by the stable equilibrium points, periodic orbits, and even chaotic attractors. Neural system receives external stimulations from sensory organs and brain regions and then produces dif- ferent types of firing behaviors, such as periodic activity, spiking, bursting, and chaos to transfer and integrate neural information [1, 2]. To reveal firing mechanism and study neural dynamics, many neural system models have been constructed. e evolution of system is governed by the assumed dynamics of units and their interactions [3]. Time delay is an inevitable factor in signal transmission due to the finite propagation velocity and switching speed. Recently, delayed neural systems with few units have been proposed to get a deep understanding of the neural dynamics [4–6]. In this paper, we investigate a simplified delay neural system with external stimulations. e Wilson-Cowan (W-C) neural model with time delay [7] is chosen to describe dynamical behavior of individual neurons or neural function units. e W-C model is a set of differential equations that represent time evolution of neural units, using sigmoidal function to describe neural interaction. e W-C neural system has been used to address many problems in Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 5472351, 9 pages https://doi.org/10.1155/2020/5472351
Transcript of ComplexityInducedbyExternalStimulationsinaNeuralNetwork...
Research ArticleComplexity InducedbyExternal Stimulations in aNeuralNetworkSystem with Time Delay
Bin Zhen 1 Dingyi Zhang2 and Zigen Song 3
1School of Environment and Architecture University of Shanghai for Science and Technology Shanghai 200093 China2College of Food and Technology Shanghai Ocean University Shanghai 201306 China3College of Information Technology Shanghai Ocean University Shanghai 201306 China
Correspondence should be addressed to Zigen Song zigensong163com
Received 8 July 2020 Accepted 12 August 2020 Published 7 September 2020
Academic Editor Gonglin Yuan
Copyright copy 2020 Bin Zhen et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Complexity and dynamical analysis in neural systems play an important role in the application of optimization problem and associativememory In this paper we establish a delayed neural system with external stimulations e complex dynamical behaviors induced byexternal simulations are investigated employing theoretical analysis and numerical simulation Firstly we illustrate number ofequilibria by the saddle-node bifurcation of nontrivial equilibria It implies that the neural system has onethree equilibria for theexternal stimulation en analyzing characteristic equation to find Hopf bifurcation we obtain the equilibriumrsquos stability andillustrate periodic activity induced by the external stimulations and time delay e neural system exhibits a periodic activity with theincreased delay Further the external stimulations can induce and suppress the periodic activity e system dynamics can betransformed from quiescent state (ie the stable equilibrium) to periodic activity and then quiescent state with stimulation increasingFinally inspired by ubiquitous rhythm in living organisms we introduce periodic stimulations with low frequency as rhythm activityfrom sensory organs and other regions e neural system subjected by the periodic stimulations exhibits some interesting activitiessuch as periodic spiking subthreshold oscillation and bursting-like activity Moreover the subthreshold oscillation can switch itsposition with delay increasing e neural system may employ time delay to realize Winner-Take-All functionality
1 Introduction
Imitating the properties of a biological nerve system to buildartificial neural network model plays an important role inthe fields of neural network applications Complexity anddynamical analysis in neural systems are important re-quirements for the application of optimization problem andassociative memory In fact many neural network systemssuch as Hopfield neural system CohenndashGrossberg neuralsystem and cellular neural system have been applied inmany research fields such as associative memory securecommunication and signal processing Actually associativememory storage in neural networks is defined by the stableequilibrium points periodic orbits and even chaoticattractors Neural system receives external stimulations fromsensory organs and brain regions and then produces dif-ferent types of firing behaviors such as periodic activity
spiking bursting and chaos to transfer and integrate neuralinformation [1 2] To reveal firing mechanism and studyneural dynamics many neural system models have beenconstructed e evolution of system is governed by theassumed dynamics of units and their interactions [3] Timedelay is an inevitable factor in signal transmission due to thefinite propagation velocity and switching speed Recentlydelayed neural systems with few units have been proposed toget a deep understanding of the neural dynamics [4ndash6]
In this paper we investigate a simplified delay neuralsystem with external stimulations e Wilson-Cowan(W-C) neural model with time delay [7] is chosen to describedynamical behavior of individual neurons or neural functionunits e W-C model is a set of differential equations thatrepresent time evolution of neural units using sigmoidalfunction to describe neural interaction e W-C neuralsystem has been used to address many problems in
HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5472351 9 pageshttpsdoiorg10115520205472351
computational neuroscience [8] e mathematical char-acteristics of single and two-coupling W-C neural systemhave been studied extensively because of mathematicaltractability and rich dynamical properties [9 10] For ex-ample Borisyuk and Kirillov [11] found a coexistence regionwith equilibrium and periodic solution in WndashC model edynamics and bifurcations of two coupled WndashC oscillatorswith four different connection types were investigated [12]Maruyama et al [13] elucidated the chaos mechanism incoupled WndashC oscillators Further the modified WndashC os-cillator with delayed self-connection was proposed byMonteiro et al [14] Song and Xu have focused on the effectsof external inputs in delayed WndashC oscillator [15 16] Twocoupled WndashC models exhibited a chaotic oscillation forcedby periodic input [13] Moreover the neural system hadcoexistence of oscillatory and steady states e periodicstimulation transformed the level of activity to high or low[17 18] However to the best of our knowledge there is nopublished report on dynamical behavior of WndashC neuralmodel with time delay and external stimulations is is ourmotivation of the present research
e WndashC neural system with time delay and externalstimulations is described by the following delayed differ-ential equation
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2( 1113857
⎧⎨
⎩ (1)
where u1(t) and u2(t) denote neural activities at time t ci
(i 1 4) represent synaptic weights for inhibition(ci lt 0) and excitation (ci gt 0) τ is a transmission delay andI1 and I2 are external stimulations f(u) is neural activationfunction and chosen as f(u) 1(1 + eminusu) in this paper
e outline of this paper is as follows in Section 2 weillustrate number of equilibria by saddle-node bifurcation ofnontrivial equilibria e system exhibits onethree equi-libria for different values of external stimulations In Section3 we analyze the equilibriumrsquos stability and demonstrate aperiodic activity which can be induced by both externalstimulations and time delay e neural system exhibits aperiodic activity with delay increasing Further the externalstimulations can induce and suppress the periodic activitye system dynamics can be transformed from quiescentstate to periodic activity and then to quiescent state withincrease of the external stimulations In Section 4 we in-troduce periodic stimulations as rhythm activity fromsensory organs and another region e neural systemsubjected by the periodic stimulations exhibits periodicspiking subthreshold and bursting-like activities Conclu-sions are given in Section 5
2 Saddle-Node Bifurcation ofNontrivial Equilibrium
We start with equilibrium analysis by employing thedynamical bifurcation theory It is obvious that system (1)just has nontrivial equilibrium labeled as (u10 u20) be-cause the neural activation function f(u) 1(1 + eminusu) isnot an origin symmetry By letting u
1 0 and u
2 0 in
system (1) we have the nontrivial equilibrium (u10 u20)
satisfied with
u10 f c1u10 + c2u20 + I1( 1113857
u20 f c3u20 + c4u10 + I2( 11138571113896 (2)
e number and value of the nontrivial equilibrium justdepend on synaptic weights ci (i 1 4) and externalstimulations I1 and I2 Since time delay has no impact on thenumber and value of system equilibria we have rewrittensystem (1) as the following nondelayed system which is
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t) + I1( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t) + I2( 1113857
⎧⎨
⎩ (3)
By x1(t) u1(t) minus u10 x2(t) u2(t) minus u20 we obtainthe corresponding linearization system
x
1(t) minusx1(t) + c1Mx1(t) + c2Mx2(t)
x
2(t) minusx2(t) + c3Nx2(t) + c4Nx1(t)1113896 (4)
where
M e
minus m0
1 + eminusm0( 1113857
2 gt 0
N e
minusn0
1 + eminusn0( 1113857
2 gt 0
m0 c1u10 + c2u20 + I1
n0 c3u20 + c4u10 + I2
(5)
e corresponding characteristic equation is
λ + 1 minus c1M minusc2M
minusc4N λ + 1 minus c3N
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868 0 (6)
which is
λ2 + minusc1M minus c3N + 2( 1113857λ + c1c3 minus c2c4( 1113857MN minus c3N
minus c1M + 1 0(7)
In fact the critical value of equilibriumrsquos number cor-responds to a static bifurcation [19 20] So in Section 3 wewill describe the change of the number of equilibria bybifurcation analysis It follows from bifurcation theory thatsystem (1) has a static bifurcation at the nontrivial equi-librium (u10 u20) if the following equation is valid which is
c1c3 minus c2c4( 1113857MN minus c3N minus c1M + 1 0 (8)
It should be noted that M and N entirely depend onsynaptic weights ci (i 1 4) and external stimulationsI1 and I2 So equation (8) is a transcendental and compli-cated equation e static bifurcation point cannot be il-lustrated in theoretical expressions But for the fixed systemparameters the static bifurcation points can be obtained bynumerical computation Further we choose system pa-rameters as c1 minus3 c2 minus10 c3 minus3 and c4 minus10 Fordifferent values of the external stimulations I1 and I2 we can
2 Mathematical Problems in Engineering
show dynamic nullclines to demonstrate the types of bi-furcation of nontrivial equilibrium which is
I1 3u10 + 10u20 minus ln1
u10 minus 11113888 1113889
I2 3u20 + 10u10 + ln1
u20 minus 11113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(9)
e number of equilibria can be obtained by dynamicnullclines as shown in Figure 1 It follows from Figure 1(a)that the dynamic nullclines just have one intersection pointfor I1 8 and I2 5 which implies system (1) has just onenontrivial equilibrium However when external stimula-tions I1 and I2 are fixed as I1 8 I2 7 the dynamicnullclines have three intersection points with a nonsym-metrical pattern as shown in Figure 1(b) It implies thatsystem (1) exhibits a saddle-node bifurcation of nontrivialequilibrium Two new nontrivial equilibria generate withexternal stimulations increasing System (1) exhibits threeequilibria Similar results are obtained for system (1) whichis one equilibrium for I1 2 and I2 minus2 (Figure 1(c)) andthree equilibria for I1 2 and I2 2 (Figure 1(d))
3 Stability Analysis and Period Activity
In this section we will analyze the equilibriumrsquos stability andfind periodic activity induced by both external stimulationsand time delay To this end using x1 u1 minus u10 andx2 u2 minus u20 we firstly obtain a linearizing system at theequilibrium (u10 u20) which is
x
1(t) minusx1(t) + c1Mx1(t) + c2Mx2(t minus τ)
x
2(t) minusx2(t) + c3Nx2(t) + c4Nx1(t minus τ)1113896 (10)
e characteristic equation of system (10) is
λ2 + minusc1M minus c3N + 2( 1113857λ minus c2c4MNeminus2λτ
+ c1c3MN
minus c3N minus c1M + 1 0
(11)
It follows that the equilibrium of system (1) is locallystable when all eigenvalues of equation (11) present negativereal parts Supposing τ 0 in equation (11) produces
λ2 + minusc1M minus c3N + 2( 1113857λ + c1c3 minus c2c4( 1113857MN minus c3N minus c1M + 1 0
(12)
Using the Routh-Hurwitz criterion we obtain a neces-sary and sufficient condition to assure the equilibrium ofsystem (1) has local stability which is
c1M + c3Nlt 2
c1M + c3N minus c1c3MN + c2c4MNlt 11113896 (13)
With delay τ increasing the system equilibriummay loseits stability and evolve into a periodic activity To obtain thecritical values supposing λ iω (ωgt 0) is a pure imaginaryroot of the characteristic equation (11) we have
ω2+ minusc1M minus c3N + 2( 1113857iω minus c2c4MNe
minus2iωτ+ c1c3MN
minus c3N minus c1M + 1 0
(14)
Separating equation (14) into real and imaginary partsyields
1 minus c1M minus c3N + c1c3MN minus ω2minus c2c4MN cos(2ωτ) 0
2ω minus c1Mω minus c3Nω + c2c4MN sin(2ωτ) 0
⎧⎨
⎩
(15)
Eliminating τ from equation (15) one has
cos(2ωτ) 1 minus c1M minus c3N + c1c3MN minus ω2
c2c4MN
sin(2ωτ) minus2ω minus c1Mω minus c3Nω
c2c4MN
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(16)
By cos2(2ωτ) + sin2(2ωτ) 1 we have
L(ω) ω4+ p0ω
2+ q0 0 (17)
where
p0 2 minus 2c1M minus 2c3N + c21M
2+ c
23N
2
q0 1 minus c1M minus c3N + c1c3MN( 11138572
minus c2c4MN( 11138572
⎧⎨
⎩ (18)
In general based on conditions p0 lt 0 q0 gt 0 andp20 minus 4q0 gt 0 equation (17) may have at most two positive
roots ωi i 1 2 which are the frequencies of the periodicactivity
ω12
minusp0 plusmn
p20 minus 4q0
1113969
2
11139741113972
(19)
en equation (14) has the critical delayed values of theHopf bifurcation that is
τji
ϕi + 2jπωi
i 1 2 j 0 1 2 (20)
where ϕi isin [0 2π) and satisfies with
1 minus c1M minus c3N + c1c3MN minus ω2i minus c2c4MN cos ϕi 0
2ωi minus c1Mωi minus c3Nωi + c2c4MN sinϕi 0
⎧⎨
⎩
(21)
Define
τ0 min τ0i i 1 21113966 1113967 (22)
e Hopf bifurcation happens when the system eigen-values cross the imaginary axis with nonzero velocityDifferentiating λ with τ in (11) one has the crossing velocity
λprime(τ) 2c2c4MNλ
e2λτ
c1M + c3N minus 2λ minus 2( 1113857 minus 2c2c4MNτ (23)
Mathematical Problems in Engineering 3
By the Hopf bifurcation theory we obtain the followingconclusion with condition (11) If L(ω) 0 has at most twopositive roots ωi i 1 2 there is a critical value of delay τ0defined by (22) All eigenvalues have negative real parts forτ isin (0 τ0) where the dynamics of system (1) is asymptot-ically stable Furthermore system (1) has a Hopf bifurcationwith the critical delayed value τ τ0 when Re(λprime(τ))ne 0 Itimplies that the system has a periodic activity near theequilibrium (u10 u20) for τ isin (τ0 +infin)
For example we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 It follows thatL(ω) 0 has a positive rootω 06133eminimum delayof the Hopf bifurcation is τ0 12422 by equation (18) Alleigenvalues of equation (11) have negative real parts for thedelayed interval τ isin (0 12422) At this time the systemequilibrium is locally stable It follows from Figure 2(a) thatthe maximum real part of eigenvalues is Re(λ) minus00389 forthe fixed delay τ 1 e corresponding time history isillustrated in Figure 3(a) e trajectory evolves into thenontrivial equilibrium Furthermore system (1) undergoes aHopf bifurcation when τ passes through the critical delayed
value τ0 12422 e maximum real part of eigenvalueswill change its sign from negative to positive At least a rootof equation (9) has a positive part for τ isin (12422infin) asshown in Figure 2(b) for the fixed delay τ 2 e systemdynamic will lose its stability and enter a stable periodicactivity as shown in Figure 3(b) e real parts of systemeigenvalues with τ increasing are shown in Figure 4
On the other hand external stimulations can induce andsuppress the periodic activity in system (1) With increasingof the external stimulations the system dynamics can switchfrom quiescent state to periodic activity and then enter intothe quiescent state For example we increase externalstimulation I2 and fix other parameters as c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 2 and τ 2 It follows fromFigure 5(a) that the system trajectories evolve into a non-trivial equilibrium for I2 0 e equilibrium of system islocally stable With increasing of external stimulation toI2 2 the system dynamic loses its stability and enters into astable periodic activity as shown in Figure 5(b) It impliesthat the external stimulation induces a periodic activity insystem (1) Further the intensity of periodic activity is
0 05 1 15
0
05
1
15
u2
u1
E1
(a)
0 05 1 15
0
05
1
15
u2
u1
E1
E2
E3
(b)
ndash05 0 05 1
ndash05
0
05
1
u2
u1
E1
(c)
ndash05 0 05 1 15ndash05
0
1
05
15
u2
u1
E3
E1
E2
(d)
Figure 1 Intersection points of dynamic nullclines illustrate the saddle-node bifurcation of nontrivial equilibrium where (a) one nontrivialequilibrium for I1 8 and I2 5 (b) three equilibria for I1 8 and I2 7 (c) one equilibrium for I1 2 and I2 minus2 and (d) threeequilibria for I1 2 and I2 2 where the other parameters are c1 minus3 c2 minus10 c3 minus3 and c4 minus10
4 Mathematical Problems in Engineering
enhanced by external stimulation I2 It follows fromFigure 5(c) that the amplitude can reach to u1 05 forexternal stimulation I2 6 However when external stim-ulation is increased into I2 8 the periodic activity evolvesinto a quiescent state as shown in Figure 5(d) e systemtrajectories enter a nontrivial equilibrium again e systemequilibrium regains its stability e external stimulationssuppress the periodic activity in system (1)
4 Complex Activity Excited byPeriodic Stimulations
In the section above we have studied the equilibrium sta-bility and find a periodic activity where system (1) hasexternal stimulations I1 and I2 which are constant values Infact rhythm stimulation is ubiquitous in living organisms
So in this section we will analyze dynamical activity of theneural system encountered external stimulation with peri-odical rhythme periodic stimulations with low frequencyare introduced as rhythm activity from sensory organs andother regionse neural system considered in this section isdescribed by the following differential equations
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1 + A1 cos (vt)( 1113857( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2 + A2 sin (vt)( 1113857( 1113857
⎧⎨
⎩
(24)
where A1 and A2 are amplitudes of external stimulationswith periodical rhythm and v is frequency Physically insome experimental studies of living organisms periodicstimulations are used by injecting sinusoidal current intoneurons such as the squid giant axons [21] snail neurons[22] and lobster CPGs [23] Some complex activity of
ndash3 ndash2 ndash1 0ndash20
ndash10
0
10
20
Re (λ)
Im (λ
)
(a)
Re (λ)ndash15 ndash1 ndash05 0
ndash20
ndash10
0
10
20
Im (λ
)
(b)
Figure 2 Partial eigenvalues of the system equilibrium (u10 u20) with time delay τ increasing (a) τ 1 and (b) τ 2 for the fixed parametersc1 minus3 c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
0 100 200 300 400 5000
002
004
006
008
01
012
014
016
t
u1
(a)
0 100 200 300 400 5000
005
01
015
02
025
t
u1
(b)
Figure 3 Time histories of system (1) with the delay (a) τ 1 and (b) τ 2 for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2and I2 minus2
Mathematical Problems in Engineering 5
0 2 4 6 8 10ndash01
ndash005
0
005
01
τ
Re (λ
)
Figure 4 e real parts of system eigenvalues with τ increasing for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2 andI2 minus2
0 100 200 300 400 5000
005
01
015
02
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
06
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 5 Time histories of system (1) with external stimulation increasing (a) I2 0 (b) I2 2 (c) I2 6 and (d) I2 8 for the fixedparameters c1 minus3 c2 minus10 c3 minus3 c4 minus10 I1 2 and τ 2
6 Mathematical Problems in Engineering
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
computational neuroscience [8] e mathematical char-acteristics of single and two-coupling W-C neural systemhave been studied extensively because of mathematicaltractability and rich dynamical properties [9 10] For ex-ample Borisyuk and Kirillov [11] found a coexistence regionwith equilibrium and periodic solution in WndashC model edynamics and bifurcations of two coupled WndashC oscillatorswith four different connection types were investigated [12]Maruyama et al [13] elucidated the chaos mechanism incoupled WndashC oscillators Further the modified WndashC os-cillator with delayed self-connection was proposed byMonteiro et al [14] Song and Xu have focused on the effectsof external inputs in delayed WndashC oscillator [15 16] Twocoupled WndashC models exhibited a chaotic oscillation forcedby periodic input [13] Moreover the neural system hadcoexistence of oscillatory and steady states e periodicstimulation transformed the level of activity to high or low[17 18] However to the best of our knowledge there is nopublished report on dynamical behavior of WndashC neuralmodel with time delay and external stimulations is is ourmotivation of the present research
e WndashC neural system with time delay and externalstimulations is described by the following delayed differ-ential equation
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2( 1113857
⎧⎨
⎩ (1)
where u1(t) and u2(t) denote neural activities at time t ci
(i 1 4) represent synaptic weights for inhibition(ci lt 0) and excitation (ci gt 0) τ is a transmission delay andI1 and I2 are external stimulations f(u) is neural activationfunction and chosen as f(u) 1(1 + eminusu) in this paper
e outline of this paper is as follows in Section 2 weillustrate number of equilibria by saddle-node bifurcation ofnontrivial equilibria e system exhibits onethree equi-libria for different values of external stimulations In Section3 we analyze the equilibriumrsquos stability and demonstrate aperiodic activity which can be induced by both externalstimulations and time delay e neural system exhibits aperiodic activity with delay increasing Further the externalstimulations can induce and suppress the periodic activitye system dynamics can be transformed from quiescentstate to periodic activity and then to quiescent state withincrease of the external stimulations In Section 4 we in-troduce periodic stimulations as rhythm activity fromsensory organs and another region e neural systemsubjected by the periodic stimulations exhibits periodicspiking subthreshold and bursting-like activities Conclu-sions are given in Section 5
2 Saddle-Node Bifurcation ofNontrivial Equilibrium
We start with equilibrium analysis by employing thedynamical bifurcation theory It is obvious that system (1)just has nontrivial equilibrium labeled as (u10 u20) be-cause the neural activation function f(u) 1(1 + eminusu) isnot an origin symmetry By letting u
1 0 and u
2 0 in
system (1) we have the nontrivial equilibrium (u10 u20)
satisfied with
u10 f c1u10 + c2u20 + I1( 1113857
u20 f c3u20 + c4u10 + I2( 11138571113896 (2)
e number and value of the nontrivial equilibrium justdepend on synaptic weights ci (i 1 4) and externalstimulations I1 and I2 Since time delay has no impact on thenumber and value of system equilibria we have rewrittensystem (1) as the following nondelayed system which is
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t) + I1( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t) + I2( 1113857
⎧⎨
⎩ (3)
By x1(t) u1(t) minus u10 x2(t) u2(t) minus u20 we obtainthe corresponding linearization system
x
1(t) minusx1(t) + c1Mx1(t) + c2Mx2(t)
x
2(t) minusx2(t) + c3Nx2(t) + c4Nx1(t)1113896 (4)
where
M e
minus m0
1 + eminusm0( 1113857
2 gt 0
N e
minusn0
1 + eminusn0( 1113857
2 gt 0
m0 c1u10 + c2u20 + I1
n0 c3u20 + c4u10 + I2
(5)
e corresponding characteristic equation is
λ + 1 minus c1M minusc2M
minusc4N λ + 1 minus c3N
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868 0 (6)
which is
λ2 + minusc1M minus c3N + 2( 1113857λ + c1c3 minus c2c4( 1113857MN minus c3N
minus c1M + 1 0(7)
In fact the critical value of equilibriumrsquos number cor-responds to a static bifurcation [19 20] So in Section 3 wewill describe the change of the number of equilibria bybifurcation analysis It follows from bifurcation theory thatsystem (1) has a static bifurcation at the nontrivial equi-librium (u10 u20) if the following equation is valid which is
c1c3 minus c2c4( 1113857MN minus c3N minus c1M + 1 0 (8)
It should be noted that M and N entirely depend onsynaptic weights ci (i 1 4) and external stimulationsI1 and I2 So equation (8) is a transcendental and compli-cated equation e static bifurcation point cannot be il-lustrated in theoretical expressions But for the fixed systemparameters the static bifurcation points can be obtained bynumerical computation Further we choose system pa-rameters as c1 minus3 c2 minus10 c3 minus3 and c4 minus10 Fordifferent values of the external stimulations I1 and I2 we can
2 Mathematical Problems in Engineering
show dynamic nullclines to demonstrate the types of bi-furcation of nontrivial equilibrium which is
I1 3u10 + 10u20 minus ln1
u10 minus 11113888 1113889
I2 3u20 + 10u10 + ln1
u20 minus 11113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(9)
e number of equilibria can be obtained by dynamicnullclines as shown in Figure 1 It follows from Figure 1(a)that the dynamic nullclines just have one intersection pointfor I1 8 and I2 5 which implies system (1) has just onenontrivial equilibrium However when external stimula-tions I1 and I2 are fixed as I1 8 I2 7 the dynamicnullclines have three intersection points with a nonsym-metrical pattern as shown in Figure 1(b) It implies thatsystem (1) exhibits a saddle-node bifurcation of nontrivialequilibrium Two new nontrivial equilibria generate withexternal stimulations increasing System (1) exhibits threeequilibria Similar results are obtained for system (1) whichis one equilibrium for I1 2 and I2 minus2 (Figure 1(c)) andthree equilibria for I1 2 and I2 2 (Figure 1(d))
3 Stability Analysis and Period Activity
In this section we will analyze the equilibriumrsquos stability andfind periodic activity induced by both external stimulationsand time delay To this end using x1 u1 minus u10 andx2 u2 minus u20 we firstly obtain a linearizing system at theequilibrium (u10 u20) which is
x
1(t) minusx1(t) + c1Mx1(t) + c2Mx2(t minus τ)
x
2(t) minusx2(t) + c3Nx2(t) + c4Nx1(t minus τ)1113896 (10)
e characteristic equation of system (10) is
λ2 + minusc1M minus c3N + 2( 1113857λ minus c2c4MNeminus2λτ
+ c1c3MN
minus c3N minus c1M + 1 0
(11)
It follows that the equilibrium of system (1) is locallystable when all eigenvalues of equation (11) present negativereal parts Supposing τ 0 in equation (11) produces
λ2 + minusc1M minus c3N + 2( 1113857λ + c1c3 minus c2c4( 1113857MN minus c3N minus c1M + 1 0
(12)
Using the Routh-Hurwitz criterion we obtain a neces-sary and sufficient condition to assure the equilibrium ofsystem (1) has local stability which is
c1M + c3Nlt 2
c1M + c3N minus c1c3MN + c2c4MNlt 11113896 (13)
With delay τ increasing the system equilibriummay loseits stability and evolve into a periodic activity To obtain thecritical values supposing λ iω (ωgt 0) is a pure imaginaryroot of the characteristic equation (11) we have
ω2+ minusc1M minus c3N + 2( 1113857iω minus c2c4MNe
minus2iωτ+ c1c3MN
minus c3N minus c1M + 1 0
(14)
Separating equation (14) into real and imaginary partsyields
1 minus c1M minus c3N + c1c3MN minus ω2minus c2c4MN cos(2ωτ) 0
2ω minus c1Mω minus c3Nω + c2c4MN sin(2ωτ) 0
⎧⎨
⎩
(15)
Eliminating τ from equation (15) one has
cos(2ωτ) 1 minus c1M minus c3N + c1c3MN minus ω2
c2c4MN
sin(2ωτ) minus2ω minus c1Mω minus c3Nω
c2c4MN
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(16)
By cos2(2ωτ) + sin2(2ωτ) 1 we have
L(ω) ω4+ p0ω
2+ q0 0 (17)
where
p0 2 minus 2c1M minus 2c3N + c21M
2+ c
23N
2
q0 1 minus c1M minus c3N + c1c3MN( 11138572
minus c2c4MN( 11138572
⎧⎨
⎩ (18)
In general based on conditions p0 lt 0 q0 gt 0 andp20 minus 4q0 gt 0 equation (17) may have at most two positive
roots ωi i 1 2 which are the frequencies of the periodicactivity
ω12
minusp0 plusmn
p20 minus 4q0
1113969
2
11139741113972
(19)
en equation (14) has the critical delayed values of theHopf bifurcation that is
τji
ϕi + 2jπωi
i 1 2 j 0 1 2 (20)
where ϕi isin [0 2π) and satisfies with
1 minus c1M minus c3N + c1c3MN minus ω2i minus c2c4MN cos ϕi 0
2ωi minus c1Mωi minus c3Nωi + c2c4MN sinϕi 0
⎧⎨
⎩
(21)
Define
τ0 min τ0i i 1 21113966 1113967 (22)
e Hopf bifurcation happens when the system eigen-values cross the imaginary axis with nonzero velocityDifferentiating λ with τ in (11) one has the crossing velocity
λprime(τ) 2c2c4MNλ
e2λτ
c1M + c3N minus 2λ minus 2( 1113857 minus 2c2c4MNτ (23)
Mathematical Problems in Engineering 3
By the Hopf bifurcation theory we obtain the followingconclusion with condition (11) If L(ω) 0 has at most twopositive roots ωi i 1 2 there is a critical value of delay τ0defined by (22) All eigenvalues have negative real parts forτ isin (0 τ0) where the dynamics of system (1) is asymptot-ically stable Furthermore system (1) has a Hopf bifurcationwith the critical delayed value τ τ0 when Re(λprime(τ))ne 0 Itimplies that the system has a periodic activity near theequilibrium (u10 u20) for τ isin (τ0 +infin)
For example we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 It follows thatL(ω) 0 has a positive rootω 06133eminimum delayof the Hopf bifurcation is τ0 12422 by equation (18) Alleigenvalues of equation (11) have negative real parts for thedelayed interval τ isin (0 12422) At this time the systemequilibrium is locally stable It follows from Figure 2(a) thatthe maximum real part of eigenvalues is Re(λ) minus00389 forthe fixed delay τ 1 e corresponding time history isillustrated in Figure 3(a) e trajectory evolves into thenontrivial equilibrium Furthermore system (1) undergoes aHopf bifurcation when τ passes through the critical delayed
value τ0 12422 e maximum real part of eigenvalueswill change its sign from negative to positive At least a rootof equation (9) has a positive part for τ isin (12422infin) asshown in Figure 2(b) for the fixed delay τ 2 e systemdynamic will lose its stability and enter a stable periodicactivity as shown in Figure 3(b) e real parts of systemeigenvalues with τ increasing are shown in Figure 4
On the other hand external stimulations can induce andsuppress the periodic activity in system (1) With increasingof the external stimulations the system dynamics can switchfrom quiescent state to periodic activity and then enter intothe quiescent state For example we increase externalstimulation I2 and fix other parameters as c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 2 and τ 2 It follows fromFigure 5(a) that the system trajectories evolve into a non-trivial equilibrium for I2 0 e equilibrium of system islocally stable With increasing of external stimulation toI2 2 the system dynamic loses its stability and enters into astable periodic activity as shown in Figure 5(b) It impliesthat the external stimulation induces a periodic activity insystem (1) Further the intensity of periodic activity is
0 05 1 15
0
05
1
15
u2
u1
E1
(a)
0 05 1 15
0
05
1
15
u2
u1
E1
E2
E3
(b)
ndash05 0 05 1
ndash05
0
05
1
u2
u1
E1
(c)
ndash05 0 05 1 15ndash05
0
1
05
15
u2
u1
E3
E1
E2
(d)
Figure 1 Intersection points of dynamic nullclines illustrate the saddle-node bifurcation of nontrivial equilibrium where (a) one nontrivialequilibrium for I1 8 and I2 5 (b) three equilibria for I1 8 and I2 7 (c) one equilibrium for I1 2 and I2 minus2 and (d) threeequilibria for I1 2 and I2 2 where the other parameters are c1 minus3 c2 minus10 c3 minus3 and c4 minus10
4 Mathematical Problems in Engineering
enhanced by external stimulation I2 It follows fromFigure 5(c) that the amplitude can reach to u1 05 forexternal stimulation I2 6 However when external stim-ulation is increased into I2 8 the periodic activity evolvesinto a quiescent state as shown in Figure 5(d) e systemtrajectories enter a nontrivial equilibrium again e systemequilibrium regains its stability e external stimulationssuppress the periodic activity in system (1)
4 Complex Activity Excited byPeriodic Stimulations
In the section above we have studied the equilibrium sta-bility and find a periodic activity where system (1) hasexternal stimulations I1 and I2 which are constant values Infact rhythm stimulation is ubiquitous in living organisms
So in this section we will analyze dynamical activity of theneural system encountered external stimulation with peri-odical rhythme periodic stimulations with low frequencyare introduced as rhythm activity from sensory organs andother regionse neural system considered in this section isdescribed by the following differential equations
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1 + A1 cos (vt)( 1113857( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2 + A2 sin (vt)( 1113857( 1113857
⎧⎨
⎩
(24)
where A1 and A2 are amplitudes of external stimulationswith periodical rhythm and v is frequency Physically insome experimental studies of living organisms periodicstimulations are used by injecting sinusoidal current intoneurons such as the squid giant axons [21] snail neurons[22] and lobster CPGs [23] Some complex activity of
ndash3 ndash2 ndash1 0ndash20
ndash10
0
10
20
Re (λ)
Im (λ
)
(a)
Re (λ)ndash15 ndash1 ndash05 0
ndash20
ndash10
0
10
20
Im (λ
)
(b)
Figure 2 Partial eigenvalues of the system equilibrium (u10 u20) with time delay τ increasing (a) τ 1 and (b) τ 2 for the fixed parametersc1 minus3 c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
0 100 200 300 400 5000
002
004
006
008
01
012
014
016
t
u1
(a)
0 100 200 300 400 5000
005
01
015
02
025
t
u1
(b)
Figure 3 Time histories of system (1) with the delay (a) τ 1 and (b) τ 2 for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2and I2 minus2
Mathematical Problems in Engineering 5
0 2 4 6 8 10ndash01
ndash005
0
005
01
τ
Re (λ
)
Figure 4 e real parts of system eigenvalues with τ increasing for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2 andI2 minus2
0 100 200 300 400 5000
005
01
015
02
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
06
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 5 Time histories of system (1) with external stimulation increasing (a) I2 0 (b) I2 2 (c) I2 6 and (d) I2 8 for the fixedparameters c1 minus3 c2 minus10 c3 minus3 c4 minus10 I1 2 and τ 2
6 Mathematical Problems in Engineering
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
show dynamic nullclines to demonstrate the types of bi-furcation of nontrivial equilibrium which is
I1 3u10 + 10u20 minus ln1
u10 minus 11113888 1113889
I2 3u20 + 10u10 + ln1
u20 minus 11113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(9)
e number of equilibria can be obtained by dynamicnullclines as shown in Figure 1 It follows from Figure 1(a)that the dynamic nullclines just have one intersection pointfor I1 8 and I2 5 which implies system (1) has just onenontrivial equilibrium However when external stimula-tions I1 and I2 are fixed as I1 8 I2 7 the dynamicnullclines have three intersection points with a nonsym-metrical pattern as shown in Figure 1(b) It implies thatsystem (1) exhibits a saddle-node bifurcation of nontrivialequilibrium Two new nontrivial equilibria generate withexternal stimulations increasing System (1) exhibits threeequilibria Similar results are obtained for system (1) whichis one equilibrium for I1 2 and I2 minus2 (Figure 1(c)) andthree equilibria for I1 2 and I2 2 (Figure 1(d))
3 Stability Analysis and Period Activity
In this section we will analyze the equilibriumrsquos stability andfind periodic activity induced by both external stimulationsand time delay To this end using x1 u1 minus u10 andx2 u2 minus u20 we firstly obtain a linearizing system at theequilibrium (u10 u20) which is
x
1(t) minusx1(t) + c1Mx1(t) + c2Mx2(t minus τ)
x
2(t) minusx2(t) + c3Nx2(t) + c4Nx1(t minus τ)1113896 (10)
e characteristic equation of system (10) is
λ2 + minusc1M minus c3N + 2( 1113857λ minus c2c4MNeminus2λτ
+ c1c3MN
minus c3N minus c1M + 1 0
(11)
It follows that the equilibrium of system (1) is locallystable when all eigenvalues of equation (11) present negativereal parts Supposing τ 0 in equation (11) produces
λ2 + minusc1M minus c3N + 2( 1113857λ + c1c3 minus c2c4( 1113857MN minus c3N minus c1M + 1 0
(12)
Using the Routh-Hurwitz criterion we obtain a neces-sary and sufficient condition to assure the equilibrium ofsystem (1) has local stability which is
c1M + c3Nlt 2
c1M + c3N minus c1c3MN + c2c4MNlt 11113896 (13)
With delay τ increasing the system equilibriummay loseits stability and evolve into a periodic activity To obtain thecritical values supposing λ iω (ωgt 0) is a pure imaginaryroot of the characteristic equation (11) we have
ω2+ minusc1M minus c3N + 2( 1113857iω minus c2c4MNe
minus2iωτ+ c1c3MN
minus c3N minus c1M + 1 0
(14)
Separating equation (14) into real and imaginary partsyields
1 minus c1M minus c3N + c1c3MN minus ω2minus c2c4MN cos(2ωτ) 0
2ω minus c1Mω minus c3Nω + c2c4MN sin(2ωτ) 0
⎧⎨
⎩
(15)
Eliminating τ from equation (15) one has
cos(2ωτ) 1 minus c1M minus c3N + c1c3MN minus ω2
c2c4MN
sin(2ωτ) minus2ω minus c1Mω minus c3Nω
c2c4MN
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(16)
By cos2(2ωτ) + sin2(2ωτ) 1 we have
L(ω) ω4+ p0ω
2+ q0 0 (17)
where
p0 2 minus 2c1M minus 2c3N + c21M
2+ c
23N
2
q0 1 minus c1M minus c3N + c1c3MN( 11138572
minus c2c4MN( 11138572
⎧⎨
⎩ (18)
In general based on conditions p0 lt 0 q0 gt 0 andp20 minus 4q0 gt 0 equation (17) may have at most two positive
roots ωi i 1 2 which are the frequencies of the periodicactivity
ω12
minusp0 plusmn
p20 minus 4q0
1113969
2
11139741113972
(19)
en equation (14) has the critical delayed values of theHopf bifurcation that is
τji
ϕi + 2jπωi
i 1 2 j 0 1 2 (20)
where ϕi isin [0 2π) and satisfies with
1 minus c1M minus c3N + c1c3MN minus ω2i minus c2c4MN cos ϕi 0
2ωi minus c1Mωi minus c3Nωi + c2c4MN sinϕi 0
⎧⎨
⎩
(21)
Define
τ0 min τ0i i 1 21113966 1113967 (22)
e Hopf bifurcation happens when the system eigen-values cross the imaginary axis with nonzero velocityDifferentiating λ with τ in (11) one has the crossing velocity
λprime(τ) 2c2c4MNλ
e2λτ
c1M + c3N minus 2λ minus 2( 1113857 minus 2c2c4MNτ (23)
Mathematical Problems in Engineering 3
By the Hopf bifurcation theory we obtain the followingconclusion with condition (11) If L(ω) 0 has at most twopositive roots ωi i 1 2 there is a critical value of delay τ0defined by (22) All eigenvalues have negative real parts forτ isin (0 τ0) where the dynamics of system (1) is asymptot-ically stable Furthermore system (1) has a Hopf bifurcationwith the critical delayed value τ τ0 when Re(λprime(τ))ne 0 Itimplies that the system has a periodic activity near theequilibrium (u10 u20) for τ isin (τ0 +infin)
For example we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 It follows thatL(ω) 0 has a positive rootω 06133eminimum delayof the Hopf bifurcation is τ0 12422 by equation (18) Alleigenvalues of equation (11) have negative real parts for thedelayed interval τ isin (0 12422) At this time the systemequilibrium is locally stable It follows from Figure 2(a) thatthe maximum real part of eigenvalues is Re(λ) minus00389 forthe fixed delay τ 1 e corresponding time history isillustrated in Figure 3(a) e trajectory evolves into thenontrivial equilibrium Furthermore system (1) undergoes aHopf bifurcation when τ passes through the critical delayed
value τ0 12422 e maximum real part of eigenvalueswill change its sign from negative to positive At least a rootof equation (9) has a positive part for τ isin (12422infin) asshown in Figure 2(b) for the fixed delay τ 2 e systemdynamic will lose its stability and enter a stable periodicactivity as shown in Figure 3(b) e real parts of systemeigenvalues with τ increasing are shown in Figure 4
On the other hand external stimulations can induce andsuppress the periodic activity in system (1) With increasingof the external stimulations the system dynamics can switchfrom quiescent state to periodic activity and then enter intothe quiescent state For example we increase externalstimulation I2 and fix other parameters as c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 2 and τ 2 It follows fromFigure 5(a) that the system trajectories evolve into a non-trivial equilibrium for I2 0 e equilibrium of system islocally stable With increasing of external stimulation toI2 2 the system dynamic loses its stability and enters into astable periodic activity as shown in Figure 5(b) It impliesthat the external stimulation induces a periodic activity insystem (1) Further the intensity of periodic activity is
0 05 1 15
0
05
1
15
u2
u1
E1
(a)
0 05 1 15
0
05
1
15
u2
u1
E1
E2
E3
(b)
ndash05 0 05 1
ndash05
0
05
1
u2
u1
E1
(c)
ndash05 0 05 1 15ndash05
0
1
05
15
u2
u1
E3
E1
E2
(d)
Figure 1 Intersection points of dynamic nullclines illustrate the saddle-node bifurcation of nontrivial equilibrium where (a) one nontrivialequilibrium for I1 8 and I2 5 (b) three equilibria for I1 8 and I2 7 (c) one equilibrium for I1 2 and I2 minus2 and (d) threeequilibria for I1 2 and I2 2 where the other parameters are c1 minus3 c2 minus10 c3 minus3 and c4 minus10
4 Mathematical Problems in Engineering
enhanced by external stimulation I2 It follows fromFigure 5(c) that the amplitude can reach to u1 05 forexternal stimulation I2 6 However when external stim-ulation is increased into I2 8 the periodic activity evolvesinto a quiescent state as shown in Figure 5(d) e systemtrajectories enter a nontrivial equilibrium again e systemequilibrium regains its stability e external stimulationssuppress the periodic activity in system (1)
4 Complex Activity Excited byPeriodic Stimulations
In the section above we have studied the equilibrium sta-bility and find a periodic activity where system (1) hasexternal stimulations I1 and I2 which are constant values Infact rhythm stimulation is ubiquitous in living organisms
So in this section we will analyze dynamical activity of theneural system encountered external stimulation with peri-odical rhythme periodic stimulations with low frequencyare introduced as rhythm activity from sensory organs andother regionse neural system considered in this section isdescribed by the following differential equations
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1 + A1 cos (vt)( 1113857( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2 + A2 sin (vt)( 1113857( 1113857
⎧⎨
⎩
(24)
where A1 and A2 are amplitudes of external stimulationswith periodical rhythm and v is frequency Physically insome experimental studies of living organisms periodicstimulations are used by injecting sinusoidal current intoneurons such as the squid giant axons [21] snail neurons[22] and lobster CPGs [23] Some complex activity of
ndash3 ndash2 ndash1 0ndash20
ndash10
0
10
20
Re (λ)
Im (λ
)
(a)
Re (λ)ndash15 ndash1 ndash05 0
ndash20
ndash10
0
10
20
Im (λ
)
(b)
Figure 2 Partial eigenvalues of the system equilibrium (u10 u20) with time delay τ increasing (a) τ 1 and (b) τ 2 for the fixed parametersc1 minus3 c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
0 100 200 300 400 5000
002
004
006
008
01
012
014
016
t
u1
(a)
0 100 200 300 400 5000
005
01
015
02
025
t
u1
(b)
Figure 3 Time histories of system (1) with the delay (a) τ 1 and (b) τ 2 for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2and I2 minus2
Mathematical Problems in Engineering 5
0 2 4 6 8 10ndash01
ndash005
0
005
01
τ
Re (λ
)
Figure 4 e real parts of system eigenvalues with τ increasing for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2 andI2 minus2
0 100 200 300 400 5000
005
01
015
02
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
06
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 5 Time histories of system (1) with external stimulation increasing (a) I2 0 (b) I2 2 (c) I2 6 and (d) I2 8 for the fixedparameters c1 minus3 c2 minus10 c3 minus3 c4 minus10 I1 2 and τ 2
6 Mathematical Problems in Engineering
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
By the Hopf bifurcation theory we obtain the followingconclusion with condition (11) If L(ω) 0 has at most twopositive roots ωi i 1 2 there is a critical value of delay τ0defined by (22) All eigenvalues have negative real parts forτ isin (0 τ0) where the dynamics of system (1) is asymptot-ically stable Furthermore system (1) has a Hopf bifurcationwith the critical delayed value τ τ0 when Re(λprime(τ))ne 0 Itimplies that the system has a periodic activity near theequilibrium (u10 u20) for τ isin (τ0 +infin)
For example we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 It follows thatL(ω) 0 has a positive rootω 06133eminimum delayof the Hopf bifurcation is τ0 12422 by equation (18) Alleigenvalues of equation (11) have negative real parts for thedelayed interval τ isin (0 12422) At this time the systemequilibrium is locally stable It follows from Figure 2(a) thatthe maximum real part of eigenvalues is Re(λ) minus00389 forthe fixed delay τ 1 e corresponding time history isillustrated in Figure 3(a) e trajectory evolves into thenontrivial equilibrium Furthermore system (1) undergoes aHopf bifurcation when τ passes through the critical delayed
value τ0 12422 e maximum real part of eigenvalueswill change its sign from negative to positive At least a rootof equation (9) has a positive part for τ isin (12422infin) asshown in Figure 2(b) for the fixed delay τ 2 e systemdynamic will lose its stability and enter a stable periodicactivity as shown in Figure 3(b) e real parts of systemeigenvalues with τ increasing are shown in Figure 4
On the other hand external stimulations can induce andsuppress the periodic activity in system (1) With increasingof the external stimulations the system dynamics can switchfrom quiescent state to periodic activity and then enter intothe quiescent state For example we increase externalstimulation I2 and fix other parameters as c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 2 and τ 2 It follows fromFigure 5(a) that the system trajectories evolve into a non-trivial equilibrium for I2 0 e equilibrium of system islocally stable With increasing of external stimulation toI2 2 the system dynamic loses its stability and enters into astable periodic activity as shown in Figure 5(b) It impliesthat the external stimulation induces a periodic activity insystem (1) Further the intensity of periodic activity is
0 05 1 15
0
05
1
15
u2
u1
E1
(a)
0 05 1 15
0
05
1
15
u2
u1
E1
E2
E3
(b)
ndash05 0 05 1
ndash05
0
05
1
u2
u1
E1
(c)
ndash05 0 05 1 15ndash05
0
1
05
15
u2
u1
E3
E1
E2
(d)
Figure 1 Intersection points of dynamic nullclines illustrate the saddle-node bifurcation of nontrivial equilibrium where (a) one nontrivialequilibrium for I1 8 and I2 5 (b) three equilibria for I1 8 and I2 7 (c) one equilibrium for I1 2 and I2 minus2 and (d) threeequilibria for I1 2 and I2 2 where the other parameters are c1 minus3 c2 minus10 c3 minus3 and c4 minus10
4 Mathematical Problems in Engineering
enhanced by external stimulation I2 It follows fromFigure 5(c) that the amplitude can reach to u1 05 forexternal stimulation I2 6 However when external stim-ulation is increased into I2 8 the periodic activity evolvesinto a quiescent state as shown in Figure 5(d) e systemtrajectories enter a nontrivial equilibrium again e systemequilibrium regains its stability e external stimulationssuppress the periodic activity in system (1)
4 Complex Activity Excited byPeriodic Stimulations
In the section above we have studied the equilibrium sta-bility and find a periodic activity where system (1) hasexternal stimulations I1 and I2 which are constant values Infact rhythm stimulation is ubiquitous in living organisms
So in this section we will analyze dynamical activity of theneural system encountered external stimulation with peri-odical rhythme periodic stimulations with low frequencyare introduced as rhythm activity from sensory organs andother regionse neural system considered in this section isdescribed by the following differential equations
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1 + A1 cos (vt)( 1113857( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2 + A2 sin (vt)( 1113857( 1113857
⎧⎨
⎩
(24)
where A1 and A2 are amplitudes of external stimulationswith periodical rhythm and v is frequency Physically insome experimental studies of living organisms periodicstimulations are used by injecting sinusoidal current intoneurons such as the squid giant axons [21] snail neurons[22] and lobster CPGs [23] Some complex activity of
ndash3 ndash2 ndash1 0ndash20
ndash10
0
10
20
Re (λ)
Im (λ
)
(a)
Re (λ)ndash15 ndash1 ndash05 0
ndash20
ndash10
0
10
20
Im (λ
)
(b)
Figure 2 Partial eigenvalues of the system equilibrium (u10 u20) with time delay τ increasing (a) τ 1 and (b) τ 2 for the fixed parametersc1 minus3 c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
0 100 200 300 400 5000
002
004
006
008
01
012
014
016
t
u1
(a)
0 100 200 300 400 5000
005
01
015
02
025
t
u1
(b)
Figure 3 Time histories of system (1) with the delay (a) τ 1 and (b) τ 2 for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2and I2 minus2
Mathematical Problems in Engineering 5
0 2 4 6 8 10ndash01
ndash005
0
005
01
τ
Re (λ
)
Figure 4 e real parts of system eigenvalues with τ increasing for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2 andI2 minus2
0 100 200 300 400 5000
005
01
015
02
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
06
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 5 Time histories of system (1) with external stimulation increasing (a) I2 0 (b) I2 2 (c) I2 6 and (d) I2 8 for the fixedparameters c1 minus3 c2 minus10 c3 minus3 c4 minus10 I1 2 and τ 2
6 Mathematical Problems in Engineering
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
enhanced by external stimulation I2 It follows fromFigure 5(c) that the amplitude can reach to u1 05 forexternal stimulation I2 6 However when external stim-ulation is increased into I2 8 the periodic activity evolvesinto a quiescent state as shown in Figure 5(d) e systemtrajectories enter a nontrivial equilibrium again e systemequilibrium regains its stability e external stimulationssuppress the periodic activity in system (1)
4 Complex Activity Excited byPeriodic Stimulations
In the section above we have studied the equilibrium sta-bility and find a periodic activity where system (1) hasexternal stimulations I1 and I2 which are constant values Infact rhythm stimulation is ubiquitous in living organisms
So in this section we will analyze dynamical activity of theneural system encountered external stimulation with peri-odical rhythme periodic stimulations with low frequencyare introduced as rhythm activity from sensory organs andother regionse neural system considered in this section isdescribed by the following differential equations
u
1(t) minusu1(t) + f c1u1(t) + c2u2(t minus τ) + I1 + A1 cos (vt)( 1113857( 1113857
u
2(t) minusu2(t) + f c3u2(t) + c4u1(t minus τ) + I2 + A2 sin (vt)( 1113857( 1113857
⎧⎨
⎩
(24)
where A1 and A2 are amplitudes of external stimulationswith periodical rhythm and v is frequency Physically insome experimental studies of living organisms periodicstimulations are used by injecting sinusoidal current intoneurons such as the squid giant axons [21] snail neurons[22] and lobster CPGs [23] Some complex activity of
ndash3 ndash2 ndash1 0ndash20
ndash10
0
10
20
Re (λ)
Im (λ
)
(a)
Re (λ)ndash15 ndash1 ndash05 0
ndash20
ndash10
0
10
20
Im (λ
)
(b)
Figure 2 Partial eigenvalues of the system equilibrium (u10 u20) with time delay τ increasing (a) τ 1 and (b) τ 2 for the fixed parametersc1 minus3 c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
0 100 200 300 400 5000
002
004
006
008
01
012
014
016
t
u1
(a)
0 100 200 300 400 5000
005
01
015
02
025
t
u1
(b)
Figure 3 Time histories of system (1) with the delay (a) τ 1 and (b) τ 2 for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2and I2 minus2
Mathematical Problems in Engineering 5
0 2 4 6 8 10ndash01
ndash005
0
005
01
τ
Re (λ
)
Figure 4 e real parts of system eigenvalues with τ increasing for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2 andI2 minus2
0 100 200 300 400 5000
005
01
015
02
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
06
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 5 Time histories of system (1) with external stimulation increasing (a) I2 0 (b) I2 2 (c) I2 6 and (d) I2 8 for the fixedparameters c1 minus3 c2 minus10 c3 minus3 c4 minus10 I1 2 and τ 2
6 Mathematical Problems in Engineering
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
0 2 4 6 8 10ndash01
ndash005
0
005
01
τ
Re (λ
)
Figure 4 e real parts of system eigenvalues with τ increasing for the fixed parameters c1 minus3 c2 minus10 c3 3 c4 5 I1 2 andI2 minus2
0 100 200 300 400 5000
005
01
015
02
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
06
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 5 Time histories of system (1) with external stimulation increasing (a) I2 0 (b) I2 2 (c) I2 6 and (d) I2 8 for the fixedparameters c1 minus3 c2 minus10 c3 minus3 c4 minus10 I1 2 and τ 2
6 Mathematical Problems in Engineering
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
neurons such as periodic spiking subthreshold burstingand even chaos behaviors are illustrated for external stim-ulations with periodical rhythm
It follows from equilibrium analysis in Section 2 thatneural system (1) exhibits one equilibrium and threeequilibria for different parameter values Further with timedelay and external stimulations the neural system illustratesa periodic activity from a quiescent state and even regainsthe quiescent state So in Section 5 we exhibit dynamicactivity of neural system (24) with two cases One is c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 for system (1)having one equilibrium and the other is c1 minus3 c2 minus10c3 minus3 c4 minus10 I1 8 and I2 7 for three equilibriaFurther by choosing A1 01 A2 01 and v 3 as theperiodic stimulations we illustrate the effect of time delayand external stimulation on system activitye results showthat the neural system subjected by periodic stimulationsexhibits some interesting activities such as periodic spikingsubthreshold oscillation and bursting-like activity
Firstly we choose system parameters as c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2 where system(1) has one equilibrium e neural activities of neural
system (24) subjected by periodic stimulations are illustratedin Figure 6 for the different values of time delay It followsfrom Figure 6(a) that system (24) exhibits a subthresholdoscillation for small delay τ 01 Further the amplitude ofsubthreshold oscillation decreases with time delay increas-ing e subthreshold oscillation degenerates into a quies-cent state as shown in Figure 6(b) for τ 1 On the otherhand with delay further increasing the quiescent state canbe excited and the subthreshold oscillation enters into aperiodic spiking as shown in Figure 6(c) for τ 3 At lastwhen time delay is chosen as τ 10 the periodic spikingtransforms into a bursting-like activity as shown inFigure 6(d)
For case 2 system parameters are fixed as c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8 where neuralsystem (1) exhibits three equilibria by saddle-node bifur-cation of nontrivial equilibrium e activities of neuralsystem (24) subjected by periodic stimulations illustrate thetopdown subthreshold oscillation and periodic spiking asshown in Figure 7 It follows from Figure 7(a) that the neuralsystem (24) exhibits a top subthreshold oscillation because ofthe top equilibrium in system (1) for delay τ 15 With
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(a)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(b)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(c)
0 100 200 300 400 5000
01
02
03
04
05
t
u1
(d)
Figure 6 Time histories of system (24) with delay increasing (a) τ 01 (b) τ 1 (c) τ 3 and (d) τ 10 for the fixed parameters c1 minus3c2 minus10 c3 3 c4 5 I1 2 and I2 minus2
Mathematical Problems in Engineering 7
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
delay increasing slightly the top oscillation changes itsposition and enters into the down subthreshold oscillationas shown in Figure 7(b) for time delay τ 16 e neuralsystem having multiple equilibria may employ time delay torealize Winner-Take-All functionality Further a periodicactivity will be excited by increasing delay as shown inFigure 7(c) for τ 6 e oscillatory intensity of neuralactivity is stronger than the spiking illustrated in case 1 eoscillation surrounds all top and down quiescent statesFinally the periodic spiking transforms into a bursting-likebehavior as shown in Figure 7(d) for time delay τ 15
5 Conclusion
Complexity and dynamical analysis in neural systems playan important role in the application of optimization problemand associative memory In this paper we considered adelayed neural system with contentperiodic externalstimulations e results show that content stimulations caninduce and suppress a periodic activity e neural systemexhibits a periodic activity with delay increasing Furtherthe system dynamics can be changed from quiescent state to
periodic activity and then enter into the quiescent statewith stimulation increasing Additionally in view of theubiquitous rhythm in living organisms we introduce theperiodic stimulations with low frequency as the rhythmactivity e results show that the neural system subjectedby periodic stimulations exhibits some interesting activi-ties such as the periodic spiking subthreshold oscillationand bursting-like ones Further with delay increasingslightly the subthreshold oscillation can change its positionfrom top to down e neural system having multipleequilibria may employ time delay to realize Winner-Take-All functionality
Data Availability
All data models and code generated or used during thestudy are provided within the article
Conflicts of Interest
e authors declared that they have no conflicts of interestregarding this work
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(a)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(b)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(c)
0 100 200 300 400 5000
02
04
06
08
1
t
u1
(d)
Figure 7 Time histories of system (20) with delay varying (a) τ 15 (b) τ 16 (c) τ 6 and (d) τ 15 for the fixed parameters c1 minus3c2 minus10 c3 minus3 c4 minus10 I1 8 and I2 8
8 Mathematical Problems in Engineering
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
Acknowledgments
is work was supported by the National Science Foun-dation of China (grant nos 11672177 and 11672185)
References
[1] E M Izhikevich ldquoNeural excitability spiking and burstingrdquoInternational Journal of Bifurcation and Chaos vol 10 no 6pp 1171ndash1266 2000
[2] M Yao and R Wang ldquoNeurodynamic analysis of merkel cell-neurite complex transduction mechanism during tactilesensingrdquo Cognitive Neurodynamics vol 13 no 3pp 293ndash302 2019
[3] Z Song B Zhen and D Hu ldquoMultiple bifurcations andcoexistence in an inertial two-neuron system with multipledelaysrdquo Cognitive Neurodynamics vol 14 no 3 pp 359ndash3742020
[4] X Liao K W Wong and Z Wu ldquoBifurcation analysis on atwo-neuron system with distributed delaysrdquo Physica DNonlinear Phenomena vol 149 no 1-2 pp 123ndash141 2001
[5] Z Song J Xu and B Zhen ldquoMultitype activity coexistence inan inertial two-neuron system with multiple delaysrdquo Inter-national Journal of Bifurcation and Chaos vol 25 no 13p 1530040 2015
[6] S Sen S N Daimi K Watanabe K Takahashi et al ldquoSwitchor stay automatic classification of internal mental states inbistable perceptionrdquo Cognitive Neurodynamics vol 14 no 1pp 95ndash113 2020
[7] H R Wilson and J D Cowan ldquoA mathematical theory of thefunctional dynamics of cortical and thalamic nervous tissuerdquoKybernetik vol 13 no 2 pp 55ndash80 1973
[8] A Destexhe and T J Sejnowski ldquoe Wilson-Cowan model36 years laterrdquo Biological Cybernetics vol 101 no 1 pp 1-22009
[9] T Ueta and G Chen ldquoOn synchronization and control ofcoupled Wilson-cowan neural oscillatorsrdquo InternationalJournal of Bifurcation and Chaos vol 13 no 1 pp 163ndash1752003
[10] J Harris and B Ermentrout ldquoBifurcations in the Wilson--cowan equations with nonsmooth firing raterdquo SIAM Journalon Applied Dynamical Systems vol 14 no 1 pp 43ndash72 2015
[11] R M Borisyuk and A B Kirillov ldquoBifurcation analysis of aneural network modelrdquo Biological Cybernetics vol 66 no 4pp 319ndash325 1992
[12] G N Borisyuk R M Borisyuk A I Khibnik and D RooseldquoDynamics and bifurcations of two coupled neural oscillatorswith different connection typesrdquo Bulletin of MathematicalBiology vol 57 no 6 pp 809ndash840 1995
[13] Y Maruyama Y Kakimoto and O Araki ldquoAnalysis ofchaotic oscillations induced in two coupled Wilson-Cowanmodelsrdquo Biological Cybernetics vol 108 no 3 pp 355ndash3632014
[14] L H A Monteiro A P Filho J G Chaui-Berlinck andJ R C Piqueira ldquoOscillation death in a two-neuron networkwith delay in a self-connectionrdquo Journal of Biological Systemsvol 15 no 1 pp 49ndash61 2007
[15] Z Song and J Xu ldquoBursting near bautin bifurcation in aneural network with delay couplingrdquo International Journal ofNeural Systems vol 19 no 5 pp 359ndash373 2009
[16] Z Song and J Xu ldquoCodimension-two bursting analysis in thedelayed neural system with external stimulationsrdquo NonlinearDynamics vol 67 no 1 pp 309ndash328 2012
[17] B D Noonburg and B Pollina ldquoA periodically forced Wil-sonndashCowan systemrdquo SIAM Journal on Applied DynamicalSystems vol 63 no 5 pp 1585ndash1603 2003
[18] R Decker and V W Noonburg ldquoA periodically forcedWilson-cowan systemwithmultiple attractorsrdquo SIAM Journalon Mathematical Analysis vol 44 no 2 pp 887ndash905 2012
[19] Y A Kuznetsov Elements of Applied Bifurcation 5eorySpringer New York NY USA 1995
[20] A Mondal R K Upadhyay J Ma B K Yadav S K Sharmaand A Mondal ldquoBifurcation analysis and diverse firing ac-tivities of a modified excitable neuron modelrdquo CognitiveNeurodynamics vol 13 no 4 pp 393ndash407 2019
[21] D T Kaplan J R Clay T Manning L Glass et al ldquoSub-threshold dynamics in periodically stimulated squid giantaxonsrdquo Physical Review Letters vol 76 no 21 pp 4074ndash40771996
[22] S Chillemi M Barbi and A D Garbo ldquoDynamics of theneural discharge in snail neuronsrdquo Biosystems vol 40 no 1-2pp 21ndash28 1997
[23] A Szucs R C Elson M I Rabinovich H D Abarbanel andA I Selverston ldquoNonlinear behavior of sinusoidally forcedpyloric pacemaker neuronsrdquo Journal of Neurophysiologyvol 85 no 4 pp 1623ndash1638 2001
Mathematical Problems in Engineering 9
