**4. Recurrence plot**

Natural system exhibits periodicities and also irregular cyclicities. Usually measures such as Lyapunov characteristic exponent (LCE), correlation dimension, Kolmogorov- Sinai (KS) entropy etc., have been used to characterize the complexity of observed nonlinear dynamical behavior of a system. But the analysis based on application of the foregoing tools inherently assumes the system to be noise free and stationary. An alternative framework based on the idea of recurrence plot was introduced in [19] for visualization of the dynamical behavior of a system in phase space and subsequently the formalism has been extended to quantify the recurrence plots to unravel the observed complexities i.e., regular, quasi-periodic, chaotic transition etc. For a discrete time series P with N data points such that

$$P: \{\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3, \dots, \mathfrak{x}\_N\},\tag{10}$$

where *xi*, *i* ¼ 1, 2, *::* … , *N* refers to observed values at time *t*1, *t*<sup>1</sup> þ Δ*t*, *:* … , *t*<sup>1</sup> þ *n*Δ*t*. If the system has true dimension *m* , a sequence of vectors may be constructed from the time series as:

$$\mathbf{X}\_{i} = \left(\mathbf{x}\_{i}, \mathbf{x}\_{i+\tau}, \mathbf{x}\_{i+2\tau}, \dots, \mathbf{x}\_{i+(m-1)\tau}\right); \ i = 1, 2, \dots, n; \ n = N - (m-1)\tau \tag{11}$$

where *τ* corresponds to time lag or delay and *m* - the embedding dimension of the phase space. By considering the distances in *m*- dimensional reconstructed points, we construct a recurrence plot (RP). In fact RP is an *n* � *n* symmetrical array where a *dot* is marked at a point ð Þ *i*, *j* if **Xi** is close to another point **Xj**. We may write

$$R\_{i,j}(r) = \Theta\left(r - \|\mathbf{X}\_i - \mathbf{X}\_j\|\right);\ i, j = 1, 2, \cdots, n \tag{12}$$

variable *w* is the result of mainly the reflecting outward potassium current (*K*þ) that results in hyperpolarization of the axon after each spike occurrence. We may

*dt* <sup>¼</sup> *<sup>α</sup>v*ð Þ *<sup>β</sup>* � *<sup>v</sup>* ð Þ� *<sup>v</sup>* � *<sup>b</sup>*<sup>0</sup> *<sup>σ</sup><sup>w</sup>* <sup>þ</sup> *<sup>I</sup>*

where *δ*>0 and the parameter *α*>0 scales the amplitude of the membrane potential *v*, and *ε* is used here to control the recovery variable *w* with respect to action potential *v*. The parameter *b*<sup>0</sup> i.e., 0ð Þ <*b*<sup>0</sup> < 1 , corresponds to the threshold value that controls the excitable behavior of the neuron. Also *β* and *σ* are constants

In our analysis of Eq. (13), we take *β* ¼ *σ* ¼ 1 and for the case of no external input current, *I* ¼ 0 the dynamical system (13) has three equilibrium points or fixed

*E*<sup>1</sup> ¼ ð Þ 0, 0 ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 � *b*<sup>0</sup>

*αδ* <sup>r</sup>

<sup>2</sup> � <sup>4</sup>

<sup>2</sup> *:*

*h v*ð Þ¼ *v*ð Þ 1 � *v* ð Þ *v* � *b*<sup>0</sup> (15)

*αδ* <sup>&</sup>lt;0, (14)

*:* (16)

�

<sup>2</sup> � <sup>4</sup>

ðÞ � *v* 1 *ε* �*εδ*

ð Þ *εδ* � *αb*<sup>1</sup>

Therefore (1) if *αb*1*δ*<1, the equilibrium point *ve* ð Þ , *we* is asymptotically stable if *αb*<sup>1</sup> <*εδ*, a repellor if *αb*<sup>1</sup> > *ε*, (2) if *αb*1*δ*>1 the equilibrium point is a saddle point

In case the parameters of the system are such that condition Eq. (14) holds then using Eq. (17) we find the equilibrium point *E*<sup>1</sup> i.e., origin, to be asymptotically

For the equilibrium or fixed points *ve* ð Þ , *we* , the eigenvalues *λ*1,2 of the Jacobian

q

and (3) if *αb*1*δ* ¼ 1 then the equilibrium point is stable (unstable) if

� � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � <sup>4</sup>*ε*ð Þ <sup>1</sup> � *<sup>α</sup>b*1*<sup>δ</sup>*

<sup>2</sup> (17)

ð Þ 1 � *b*<sup>0</sup>

*<sup>J</sup>* <sup>¼</sup> *<sup>α</sup>h*<sup>0</sup>

� � � � (13)

write the FHN model equation as [28–30]:

*DOI: http://dx.doi.org/10.5772/intechopen.96573*

for the system.

points *ve* ð Þ , *we* as:

It may be noted that if

Further defining

matrix are given by

where *b*<sup>1</sup> ¼ *h*<sup>0</sup>

*αb*<sup>1</sup> <*δε α*ð Þ *b*<sup>1</sup> >*δε* .

stable if

**45**

ð Þ *ve* .

*dv*

*Chaotic Dynamics and Complexity in Real and Physical Systems*

*dw*

*dt* <sup>¼</sup> *<sup>ε</sup>*ð Þ *<sup>v</sup>* � *<sup>δ</sup><sup>w</sup>*

*<sup>E</sup>*2,3 <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> 2

then the system possess *E*<sup>1</sup> as the only equilibrium point.

the Jacobian matrix *J* of the system may be written as:

*<sup>λ</sup>*1,2 <sup>¼</sup> �ð Þ� *εδ* � *<sup>α</sup>b*<sup>1</sup>

where Θ is a Heavyside function, *r* is a small threshold distance between neighboring points and ∥*:*∥ is an Euclidean norm.

A characteristic pattern emerges in RPs which characterizes a dynamical system. The method of RP is suitable for both stationary and non-stationary dynamical system. Since a trajectory may return to a point or close to it in phase space, the deterministic dynamical system shows recurrent behavior and RP therefore exhibits both horizontal and vertical lines. For a stochastic dynamical system, such lines in RP are of very small size and in fact appear by chance. Therefore the distribution of such points appear to be homogeneous. In case of periodic system the RP is filled with longer diagonal lines. Various measures that quantify RPs are mainly, *RR, DET, ENT, DIV, LAM,TT* which refers to density of recurrence points, determinism, divergence, entropy, laminarity and trapping time respectively. For a periodic system diagonal lines are longer which for chaotic system RP shows broken short lines. Recently [20] has provided a very useful description of applying RPs and recurrence quantification analysis to unravel the complex dynamics of general problem of three species interaction in ecology. In the following section, we extend the analysis of complexity to the problem associated with neuronal dynamics, an area of current interest in neuro-bio-science [21]. We however restrict ourselves to only RPs to supplement the analysis of complexity using phase portrait, bifurcation diagram etc.
