A study on dynamics and multiscale complexity of a neuro system
aa r X i v : . [ n li n . C D ] J a n A study on dynamics and multiscale complexity of a neuro system
Sanjay K. Palit a , Sayan Mukherjee b, ∗ a Basic Sciences and Humanities Department, Calcutta Institute of Engineering and Management, Kolkata, India b Department of Mathematics, Sivanath Sastri College, Kolkata, India
Abstract
We explore the chaotic dynamics and complexity of a neuro-system with respect to variable synaptic weightsin both noise free and noisy conditions. The chaotic dynamics of the system is investigated by bifurcationanalysis and 0 − Keywords:
Neuro dynamics, Power noise, 0 −
1. Introduction
An artificial neural network (ANN) is a mathe-matical model analogical with the biological struc-ture of a neuron, which consists of a cellular bodywith a dense centroid of activity called the nucleus,entering nerves that receive signals from other neu-rons called dendrites and the departing nerves thatcarry signals away from the neurons called axons [1].It is represented by a directed graph composed ofneurons as the nodes, nerves or synapses as the edgesand an algorithm describing the conduction of im-pulses through the network. The extent to which theinput of neuron i is driven by the output of the j neuron is characterized by its output and the synap-tic weight w ij . Positive value of the synaptic weight w ij indicates that the output of the neuron j excitesthe neuron i , while the negative value indicates theoutput of the neuron j inhibits the neuron i . If theoutput of the neuron j has no influence on the neuron i , then the synaptic weight w ij equals zero [2].The human neural system is very much complexand its complex dynamic evolutions [3] that lead tochaos have already been observed experimentally. Mostof the theoretical models of neural systems exhibitstable and cyclic behaviors, yet there also exists somemodels that illustrate the existence of chaos in neu- ∗ Corresponding author, Email: [email protected] ral networks. These models rely on complex archi-tectures or complex equations for both neuron andsynaptic dynamics to display chaos. Sometimes thequantities which exhibit chaotic evolutions in thesemodels have no direct physiological interpretations.In [4], chaos in neural networks appears for the evo-lution of the sum of the absolute values of the synap-tic weights of a network. A wide range of studies onsmall networks has been made by different investiga-tors. Glass et al. discussed a transition from steadystate through limit cycle to chaos for networks of sixor more neurons [5]. In [6], it has been demonstratedthat the onset of chaos in an eight neuron system andnumerically track down the transition from steadystate through limit cycles to chaos. In [7], differ-ent dynamical regimes has been reported, particu-larly the evidence of possibility of chaotic regimes inindividual neuron output activity. They have shownthe transition of the system from a stable to a chaoticregime as synaptic weight increases. In [8], authorshave shown a detailed numerical simulations on howthe stability of the system passes from stable state tochaotic state and also discussed some biological im-plications. They have also made an attempt to findthe parameters on which the stability of the systemdepends most sensitively.During the past few decades, complexity analysisof deterministic and stochastic systems has becomean integral part of nonlinear analysis. In all kinds
Preprint submitted to Elsevier February 2, 2021 f real world phenomena, some sort of uncertainty isalways being there. Obviously, for a stochastic phe-nomenon it is more than a deterministic phenomenon.This actually means that as the system becomes moreand more random, the amount of uncertainty grad-ually increases. This is measured by entropy, firstintroduced by C.E. Shannon [9]. More is the entropyvalue, more uncertainty is there in the correspondingphenomenon. The term complexity is used in thiscontext. In general complexity is positively corre-lated with entropy. Since the inception of Shannonentropy, several entropy measures have been devel-oped [10–14] and used widely in diverse domains ofresearch [15–17].After the introduction of the recurrence plots (RP)[18–20], few other measures of complexity [21–24] havebeen introduced. All of these measures were foundto be more effective even than the Lyapunov expo-nent for the determination of the divergence behav-ior of dynamical systems. In RP, various structuresprovide different information regarding the nature ofphase space. Diagonal lines describe parallel move-ments, while trapping situation/ laminar states aredescribed by vertical/horizontal lines. Presence ofonly diagonal lines with equal/unequal time span in-dicates periodicity/quasi-periodicity of the phase space.Chaotic regime can be understood from rectangularlike structure consists of diagonal lines with someisolated points and vertical lines. All of these ba-sic features of the phase space can be characterizedby Recurrence period density (RPD). The idea ofRPD is based on recurrent time between the recur-rent points. Shannon entropy of recurrence times iscalled Recurrence plot density entropy (RPDE) [31],which is found to be very effective to calculate thedegree of complexity of the phase space. However,a multiscale approach [25–30] of the RPDE has notbeen explored so far, which is expected to reflect thedynamical characteristics of complex systems moreaccurately.In this article, the dynamics of the three neu-ron systems [8] has been further investigated in noisefree, noise induced and music perturbed condition tolook after the dynamical changes of the system. Thedynamics is quantified by single and two parameterbifurcation diagrams followed by 0 − − − K c ) of MSD servesas a measure to quantify the dynamics of a system ora time series. For chaotic and regular dynamics, K c comes close to 1 and 0 respectively. The main advan-tage of 0 − − − K c and bifurcation analysis. Inorder to know the long term characteristics of the sys-tems, multiscale RPDE is proposed, which stronglycorrelates with K c . Finally, this multiscale RPDE isused to explore the changes in complexity of the neurosystems in noise and music perturbed condition.
2. Dynamics of three neurons
Let x , x , x respectively denotes the output ac-tivity of the three neurons 1 , ,
3. The weights of thesynaptic connections from neuron 2 to 1 and neuron2 to 1 are denoted by w and w respectively. Thecorresponding schematic diagram is given in Fig.1.With each neuron, there is associated a non-negative
12 3 (a) (b)
Figure 1: (a) represents three connected neurons-1 , ,
3. Arrowindicates the direction of the output generated by the neuron.(b) represents schematic diagram of a three neuron network. x , x , x indicates output of the respective excited neurons 1,2 and 3. bounded (bounded by 0 ,
1) sigmoidal response func-tion given by f i ( s ) = (1 + e − β i ( s − θ i ) ) − , i = 1 , , β i , θ i respectively denotes the slope and thethreshold of the response function for the neuron i .The equations of control for this sequence of eventswith the response function f i is described by Das et.al [8]. The corresponding noise induced system isgiven by dx dt = f ( w x + w x ) − α x + Kφ ( ξ ( t )) , (1) dx dt = f ( x ) − α x ,dx dt = f ( x ) − α x , where where α , α , α are the respective decay rates,assumed to be constant. K is the noise strength and φ ( ξ ) is the Gaussian white noise. For the entire simu-lation, we choose α = 0 . , α = 0 . , and α = 0 . − test In this section, we investigate the dynamics of (1)with individual as well as combined effect of w and w . The investigation is done in both noisy and noisefree conditions. In this section, we investigate thedynamics of (1) with individual as well as combinedeffect of w and w . The investigation is done inboth noisy and noise free conditions. w and w We first investigate the bifurcation scenario of (1)with the changes of w , w . Fig.2a, b shows thecorresponding bifurcation diagrams for K = 0 with w ∈ [0 . , . , w = 5 . w ∈ [4 , . , w = 1respectively. Fig.2a shows single/double and multi-ple periods for w ≤ . , w > .
75 respectively.However, the multi-periodicity is lost for w > . . , .
1] and[0 . , . \ [0 . , .
1] respectively. On the other hand,the system (1) shows periodic/quasi-periodic behav-ior for w < .
75 but becomes multi-periodic withthe increase of w as evident from Fig.2b. Similaranalysis has been done with K = 0 .
05. The cor-responding bifurcation diagrams are given by Fig.2eand f respectively. It is seen from Fig.2e and f that thesystem always possesses multiple periods for w ∈ [0 . , . w = 5 . w ∈ [4 , .
5] with w =1. Since bifurcation analysis is done only for finding-‘period route to chaos’, the above analysis can onlyindicate that the dynamics of the noise-induced sys-tem (1) has a higher tendency of producing chaoticlike structures for a wider range of parameter valuesthan the same in noise-free condition.To investigate regular (periodic/quasi-periodic) andchaotic behavior of the system (1), we have used 0 − x ( j ) , j = 1 , , .., N is translated by p c ( n ) = n X j =1 x ( j ) cos( jc ) , q c ( n ) = n X j =1 x ( j ) sin( jc ) , (2)where c ∈ (0 , π ) and n = 1 , , .., N .The diffusive and non-diffusive behavior of p c and q c is then investigated by measuring mean square dis-placement (MSD) M c [32, 33] given by, M c = lim N →∞ N N X j =1 [ p c ( j + n ) − p c ( j )] +[ q c ( j + n ) − q c ( j )] , (3)where n << N . The limiting value of M c is assuredonly for n ≤ n cut , where n cut << N . For the practicalpurpose, n cut = N reveals good result [32, 33]. In or-der to investigate the behavior of M c , the asymptoticgrowth K c of M C is calculated by K c = lim n →∞ log M c ( n )log n . (4)3he value of K c close to 1 and 0 indicates chaotic andregular dynamics respectively [32, 33].For numerical simulation, we have considered x -components of (1). Fig.2c, d represents the fluctua-tion of K c with K = 0 under the variables w (keep-ing fixed w = 5 .
2) and w (keeping fixed w = 1)respectively. It can be observed from the Fig.2c that Figure 2: (a), (b) respectively represents the bifurcation dia-grams of the Neuro system (1) in noise free condition ( K = 0)for varying synaptic weights w ∈ [0 . , . , w ∈ [4 , . w and w vs. K c graphs with fixed w = 5 . w = 1 respectively. (e), (f) respectively represents thebifurcation diagrams for the same range of parameter values of w , w in noisy condition with noise strength K = 0 .
05. (g),(h) respectively represents w and w vs. K c graphs withfixed w = 5 . w = 1 respectively in noisy condition. K c is close to 0 and 1 for w ∈ [0 . , . ∪ (1 . , . w ∈ [0 . , . ∪ [0 . , .
1] respectively. Onthe other hand, it can be observed from Fig.2d that K c is close to 0 for w < .
63 and w ∈ [4 . , . K c comes close to 1 for w ∈ (6 . , . K c can quantify the chaotic aswell as the non-chaotic regime of (1) for the vari-able synaptic weights w , w respectively. Similarinvestigation is done with K = 0 .
05. The corre-sponding fluctuations are shown in Fig.2g and h re-spectively. From the figures, it can be observed thatthe respective values of K c are close to 1 and henceindicates chaos for w ∈ [0 . , . , w = 5 . w ∈ [4 , . , w = 1. Therefore, inclusion of whitenoise with a small strength can enhance the chaos in a certain range of parameter space. As chaotic dy-namics is a signature of complex phenomenon in asystem, it assures greater paradigm of complex dy-namics exists in noise-induced system compared tothe same in noise-free condition. w and w We first investigate two parameter bifurcation ofthe system (1) with K = 0 , .
05. The correspond-ing diagrams are shown in Fig.3a,d respectively. It w w w w Lag w -200 0 200 L a g w -2000200 w w w w Lag w -200 0 200 L a g w -2000200 (e)(d)(a) (b) (c)(f) Figure 3: (a), (d) respectively represents the 2D bifurcationdiagrams and contour diagram representing K c values for theNeuro-system (1) in noise free condition ( K = 0) with vary-ing synaptic weights w ∈ [0 . , . , w ∈ [4 , . K = 0 .
05. (c),(f) represent the 2D cross correlationdiagram of (a),(b) and (d), (e) respectively. The associate colorbars indicate values of the cross-correlation. can be observed from Fig.3a that the system exhibitsmultiple periods (3 or more) in the region [0 . , × [4 . , . − [0 . , . × [4 . , . − [0 . , × [4 . , . . , × [4 . , . K = 0 .
05 increases the number of peri-ods of the Neuro-system than the same with K = 0.The chaotic and non-chaotic region is then classi-fied by using 0 − w , w . The contour diagram in Fig.3b and erepresent the variation of K c values with respect to w , w respectively. The K c values in Fig.3b indi-cates that the system is chaotic in the range [0 . , × [4 . , . − [0 . , . × [4 . , . − [0 . , × [4 . , . . , × [4 . , .
2] as evidentfrom Fig.3e. Therefore, the white noise even with a4inimal strength has a strong influence on the sys-tem and it makes the system chaotic irrespective ofthe synaptic weights w , w . To check whether ornot the bifurcation analysis and 0 − w , w . This are given by Fig.3c and f for noisefree and noise induced condition respectively. Boththe 2D correlation diagrams show strong correlationbetween the two parameters bifurcation and 0 − K = 0)and noise induced condition ( k = 0 . x x x x x x x x x x x x (a) (b) (c)(d) (f)(e) Figure 4: (a), (b), (c) respectively represents the 2D projectionof the phase space of the neuro system of for different combina-tion of synaptic weights w = 0 . , w = 5 . w = 1 , w =4 . w = 1 , w = 5 . K = 0). (d), (e),(f) respectively represent the similar diagrams in noisy condi-tion ( K = 0 .
3. Multiscale complexity in noise-induced neuro-system
Recurrence in a n -dimensional phase space X = { ( ~x i ) : ~x i ∈ ℜ n , i = 1 , , ..., N } , indicates the close-ness of its points. Two points x i , x j ∈ X, i = 1 , , ..., N are considered close i.e. recurrent if k ~x i − ~x j k < ǫ .The corresponding recurrent matrix is defined as R i,j = Θ( ǫ − k ~x i − ~x j k ) , i = 1 , , ..., N, (5)where Θ is the Heaviside function, k . k is the Eu-clidean norm of the phase space, and ǫ denotes theradius of the neighborhood. The symbols ‘1’ (blackdots) and ‘0’ (white dots) are used to represent therecurrent and non-recurrent points respectively. Re-current time denoted by T k is computed as the num-ber of non-recurrent points or white lines between tworecurrent points x i , x j in the RP R i,j . Formally, re-current time for a pair of recurrent points x i , x j ∈ R i,j is defined as T k = ( i − j ). Thus, T corresponds tothe least recurrent time, T corresponds to the nextand so on. A series of recurrent time interval n ( T k )for all points in R i,j is obtained as the number ofoccurrence of T k . RPD denoted by P ( T k ) is definedas the probability of n ( T k ) among the sample space { n ( T k ) } . This is given by (8). P ( T k ) = n ( T k ) P T max k =1 n ( T k ) , (6)where T max = max { T k } . RPD can quantify the com-plexity of the phase space. However, it can not mea-sure the order of complexity. This is done by a RPDbased entropy called Normalized Recurrence perioddensity entropy (NRPDE). Recurrence periodic en-tropy (RPDE) of the reconstructed phase space, wherethe points are independently identically distributed isdefined by utilizing the concept of Shannon entropy[9]. Thus, RPDE is given by H = − T max X k =1 P ( T k ) log P ( T k ) . (7)Since T max varies with sampling time, a normaliza-tion of RPDE is necessary. The normalized RPDE(NRPDE) is defined as H norm = − ( logT max ) − T max X k =1 P ( T k ) log P ( T k ) . (8)5ere log( T max ) is equal to the entropy of a purelyrandom variable, given bylog( T max ) = − T max X k =1 P ( T k ) log P ( T k ) , where P ( T k ) ∼ T max .To measure the order of complexity more accu-rately, MNRPDE is defined by utilizing the MAVmultiscaling technique [41] on the NRPDE H norm asfollows:For the time series x (defined as above), the mul-tiscale time series, denoted by { z ( s ) j } N − s +1 j =1 is definedas z ( s ) j = 1 s j + s − X i = j x i (9)For each scale s , we can define the multiscale NRPDE H ( s ) norm by Eq.(10). The mean of { H ( s ) norm } s s =1 is thendefined by < H norm > = 1 s s X s =1 H ( s ) norm , (10)where < . > represents statistical average.In the following section, we verify the effectivenessof < H norm > by measuring the dynamical complex-ity of (1). To measure the dynamical complexity, we havefirst investigated the multi-scaling behavior of (1) us-ing H ( s ) norm with the scale s = 1 , , ..,
8. This is givenby Fig.5. Fig.5a, c show the fluctuations of H ( s ) norm forfixed ( w , w ) = (0 . , . , (1 , . , (1 . , .
2) in bothnoise free and noisy conditions respectively, while Fig.5b,d represent the similar graphs for fixed ( w , w ) =(1 , . , (1 , . H ( s ) norm gives different values for different scales.Thus, the mean value of H ( s ) norm is expected to reflectthe degree of complexity of the neuro system prop-erly. Fig.5e and f respectively shows the variation of < H norm > over variable w , w in both noise freeand noisy conditions. It can be seen from the figuresthat the degree of complexity increases for the neurosystem in noisy condition with respect to both theparameters. This correlates with the earlier results H no r m ( s ) H no r m ( s ) H no r m ( s ) H no r m ( s ) w < H no r m > w < H no r m > (b)(f)(d)(a)(c)(e) Figure 5: (a), (b) respectively represents the graph of MNR-PDE − H ( s ) norm for some fixed value of the synaptic weights( w , w ) = (0 . , . , (1 , . , (1 . , .
2) and ( w , w ) =(1 , . , (1 , .
1) in noise free condition ( K = 0). (c), (d) respec-tively represents the similar graphs in noisy condition ( K =0 . < H norm > for varying w ∈ [0 . , . w = 5 . w ∈ [4 , .
5] with a fixed w = 1. RP is constructedfrom the attractor reconstructed from x component of the so-lution vector with embedding dimension 3 and time-delay 10. of bifurcation analysis and 0 − < H norm > under the com-bined effect of ( w , w ) ∈ [0 . , × [4 . , .
2] in bothnoise free and noisy conditions. The correspondingmatrix plots are given in Fig.6a and c respectively.Comparing these plots with the same in Fig.3b ande, it can be observed that both of < H norm > and K c plots are almost similar for same set of parametervalues of w , w in noise free and noisy conditions.The correlation between them has also been inves-tigated. Fig.6b and d represents respective 2D cor-relation contour, which establishes almost correlatedpatterns between < H norm > and K c .
4. Application on the music perturbed neurosystem
In this section, we investigate chaotic dynamicsand complexity of the system (1) under an effect ofmusic signal. For the numerical experiment, we haveconsidered an instrumental music signal
M u ( t ) withpower S ( f ) = f α . Fig.7a shows corresponding f vs. S ( f ) graph. From the figure, it can be observed thatthe slope α of the line representing the mean trend6 w w Lag w -100 0 100 200 L a g w -1000100200 w w Lag w -100 0 100 200 L a g w -1000100200 (b)(d)(c)(a) Figure 6: (a), (c) respectively represents the contour plots of − < H norm > for varying synaptic weights w ∈ [0 . , , w ∈ [4 . , .
2] in noise free ( K = 0) and noise induced ( K = 0 . < H norm > plot with twoparameter 0 − of S ( f ) is approximately 2. So α = 2. The musicperturbed system of (1) is given by dx dt = f ( w x + w x ) − α x + K M u ( t ) , (11) dx dt = f x − α x ,dx dt = f x − α x . where K denotes the strength of the music.Fig.7b shows the attractors of the neuro system(1) with K = 0 (blue) and the music perturbed neurosystem (11) (red) with w = 1 , w = 5 . K = 0 .
05. It is observed that the dynamical pat-tern of both the attractors are almost similar. Toquantify this, we measure distance d ij = k x i − y j k for different windows W s with w = 1 , w = 5 . x i , y i ( i, j = 1 , , ..., N ) respectively denotes the i, j th point on the attractors of neuro systems (1)( K = 0) and (11). The windows are defined by W s = { ( d i,j ) M s × M s : M s ≤ N } . Fig.7c, d, e showthree such window matrix plots as sample illustra-tions. It can be observed that d ij ∈ [0 , .
4] for all i, j in each case. As d ij indicates dispersion betweenthe trajectories of (1) ( K = 0) and (11), its corre-sponding windows reflect changes between the respec-tive attractors. We define a ratio R = ¯ W s ¯ W s − , where¯ W s = N P M s i =1 P M s j =1 d ij ( d ij ∈ W s and M s ≤ N ). We call R by ratio of mean distance (RMD). Naturally, R ≈ K = 0) and (11) does not vary over time. Fig.7fshows the values of R (RMD) over s = 1 , , ..,
8. Itis observed that the R ≈ s and hence provesthat system (1) ( K = 0) and (11) have the similartrajectory movements with w = 1 , w = 5 . x x f S (f) Window index R M D Time
100 200 300 400 500 T i m e Time
100 200 300 400 500 T i m e Time
100 200 300 400 500 T i m e (a) (b) (c)(d) (f)(e) Figure 7: (a) represents the graph of power spectral density ofthe music signal with respect to variable frequencies. (b) rep-resents the joint attractors of the neuro system (1) (blue) andthe corresponding music perturbed system (11) (red). (c), (d),(e) represent three samples of sub distance matrix plots. Theassociate color bars represents values of d ij between the points( x i , y i ). (f) represents the graph of RMD i ( R i ) for differentwindow index i . The distance matrix ( d ij ) N × N thus obtainedis then subdivided into m = [ N ] sub matrices, each of size500. Keeping fixed w = 1 , w = 5 .
2, we furtherinvestigated the same dispersion between the trajec-tories over K ∈ [0 , . | − R | vs. K graph is shown in Fig.8a. From the figure, itcan be observed that values of | − R | ≈ K ∈ [0 . , . | − R | = 0 for K =0 . , .
05. It implies R = 1, i.e; almost similarphase spaces can be obtained for the systems (1) and(11) at K = 0 . , .
05 with w = 1 , w = 5 . | − R | is calculated over the re-gion ( w , w ) ∈ [0 . , × [4 . , . | − R | ≤ .
006 for all( w , w ) ∈ [0 . , × [4 . , .
2] with fixed K = 0 . K = 0) and (11) pos-sess almost similar phase spaces with the changes in( w , w ) ∈ [0 . , × [4 . , .
2] (for fixed K = 0 . K | - R | | - R | -3 w w -3 (a) (b) Figure 8: (a) represents | − R | vs. K ∈ [0 . .
1] graph for thesystem (11) with w = 1 , w = 5 .
2. (b) represent surface of | − R | over the region ( w , w ) ∈ [0 . , × [4 . , .
2] with fixed K = 0 .
05 for the same system. ics and complexity in the dynamics of (11) underthe variation of ( w , w ) with fixed K = 0 . − K c with ( w , w ) ∈ [0 . , × [4 . , .
2] (for fixed K = 0 . K c ≈
1. It verifies existence of chaotic dynam-ics in (11). Further, complexity is measured by cal-culating < H norm > over same ( w , w ) ∈ [0 . , × [4 . , .
2] with fixed K = 0 .
05. Fig.9b shows corre-sponding matrix plot. From Fig.9a and b, similarpatterns can be observed between the respective fluc-tuation in K c and < H norm > . To confirm the sim-ilarity, we have done a 2D cross-correlation analysis.The cross-correlation contour is given in Fig.9c. FromFig.9c, it can be investigated that cross-correlation isalmost equal to 1 at ( Lag w , Lag w ) = (0 , K c and < H norm > under the variation of ( w , w ) ∈ [0 . , × [4 . , . K = 0 . K = 0) and noise induced system (1)( K = 0 . K = 0) and (11)cannot be classified from this study.To classify the changes, we have considered twohypotheses: H / A : A /CaseI = A /CaseIIH / A : A /CaseI = A /CaseIIH / B : B /CaseI = B /CaseIIH / B : B /CaseI = B /CaseII, where A , B denotes the event for K c and < H norm > respectively. A /CaseI and A /CaseII stands for the w w w w Lag w L a g w (a) (b) (c) Figure 9: (a) K c vs. ( w , w ) ∈ [0 . , × [4 . , .
2] graph with K = 0 .
CaseI indicates correlation betweenthe system (1) with K = 0 and the same with K =0 .
05. Similarly,
CaseII indicates the same betweenthe systems (1) with K = 0 and (11). In order to findthe correlation, we calculate cross-correlation (CR)at zero lag for each w = ω ∈ [4 . , .
2] under thevariation of w ∈ [0 . , K c and < H norm > respectively with w = ω ∈ [4 . , . , w ∈ [0 . , ≥ .
95 for
CaseII . Onthe other hand, the same CR ≤ .
56 for
CaseI . Itindicates weak and strong correlation for the
CaseI and
CaseII respectively. C r o rr e l a t i on w C r o rr e l a t i on w
100 1 (a) (b)
Figure 10: (a) represents correlation values for K c in CaseI (in red color) and
CaseII (in violet color) at each w = ω ∈ [4 . , .
2] under the variation w ∈ [0 . , < H norm > in the aforesaid cases at each w = ω ∈ [4 . , .
2] over w ∈ [0 . , ω ∈ [4 . , . Further, two sample t -test confirms that both H / A and H / B are true with p ( < . CaseII thanthe same in
CaseI with w = ω ∈ [4 . , . , w ∈ [0 . , K = 0) are highly correlatedwith the music perturbed system (11) compared tothe noise induced system (1) (with K = 0).
5. Conclusions
In this article, the dynamics and complexity ofa neuro system both have been studied under noisefree, noisy and music perturbed conditions. To inves-tigate complex dynamics, bifurcation analysis is doneonly for noise free and noise induced systems. Theresults indicate that larger number of multi-periodsexist in the noise induced system compared to thesame in noise free condition, whatever may be thevariation in both synaptic weights. Further, 0 − < H norm > shows a strong cor-relation with K c in both noise free and noisy condi-tions. So, < H norm > can reflect the complex na-ture of neuro dynamics properly. The neuro systemis then perturbed with an instrumental music. It hasbeen observed that the dynamics of the music per-turbed system has a close similarity with the originalneuro system. Since music has a soothing effect onhuman feeling and mood, the inclusion of music signalwith the neuro system keeps the dynamics almost un-changed. To investigate this, distances between everypair of points on the attractors of the respective origi-nal and music perturbed neuro system are computed.Based on these distance window based ratio RMD isthen defined which clearly establishes the similaritybetween the dynamics. Fluctuation of both K c and < H norm > are finally investigated for a certain rangeof parameter values w and w . Both of them reflectthe actual changes in the dynamics of the noise free,noise induced and music perturbed neuro systems.In fact, it assures similarity between the dynamics ofthe original (noise free) and music perturbed neurosystems, while they show dissimilarity in the dynam-ics of the original and noise induced neuro systems.Finally two samples t -test hypothesis confirms thatalmost similar dynamics can be obtained in the caseof music perturbed dynamics compared to the noisyneuro system. Thus, our newly proposed measure < H norm > can properly interpret the complexity ofthe neuro dynamics in noise free, noisy and music per-turbed conditions. Since the values of < H norm > of the original and music perturbed neuro systems arefound to be almost same for variable synaptic weights w , w and an optimal music strength K = 0 . < H norm > also reflects the soothing effect of musicon the neuro system. The present study also revealsthat the soothing effect of music will be destroyed if K < .
05 as | − R | highly deviates from 0 in thisrange. However, | − R | shows a mixed trend for K > .
05 and thus it needs further investigation onhow the neuro system reacts on music perturbationin this case. This is definitely a future scope of thepresent research.
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