**5. Regular and Chaotic neuronal dynamics**

The neuronal communication is known to be mediated by electrical pulses called spikes. Studies of various spiking patterns reveal nonlinear characteristics of slowfast neuronal dynamics. A considerable amount of information regarding neuronal activity has been obtained by studying the dynamics of spiking pattern [22]. The phenomenon of tonic firing, mixed mode (bursting and spiking) etc., are typical responses exhibited by an excitable neuron [23]. Cortical neurons have been reported to show tonic bursting wherein the neurons periodically switches between firing state and resting state. The mixed mode firing is observed in mammalian neocortex [24]. Spike generation in fact depends on the firing threshold and the stimulus intensity. In recent years, the perception regarding constancy of neuron's firing threshold has changed to dynamic [21]. In this work, we first briefly introduce the Fitzhugh-Nagumo model (FHN) that have been proposed for spike generation like well known Hodgkin-Huxley model. It is however to be noted that FHN model reproduce the experimental results less accurately. Our interest in FHN model emanates mainly due to its showing complex spiking pattern even though it is mathematically simple. The basic FHN model assumes the threshold to be constant. We also study the changes caused in the spiking pattern as a result of time varying threshold. This study assumes significance as such a model may throw insight into the model the dynamics of cortisol secretion from hypothalamus [25]. It is to be noted that the neuronal firings may take place at regular interval or randomly due to inherent mechanism or may be due to its interaction with the neighborhood neurons or result of exogenous stimulus [26–28].

#### **5.1 Basic dynamics of FHN model**

The FHN model describes the interaction between the voltage *v* across the axon membrane driven by input current *I* and the recovery variable *w*. The recovery

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

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

$$\begin{aligned} \frac{dv}{dt} &= av(\beta - v)(v - b\_0) - \sigma w + I\\\\ \frac{dw}{dt} &= \varepsilon (v - \delta w) \end{aligned} \tag{13}$$

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 for the system.

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 points *ve* ð Þ , *we* as:

$$E\_1 = (\mathbf{0}, \mathbf{0}),$$

$$E\_{2,3} = \frac{(\mathbf{1} + b\_0)}{2} \pm \frac{\sqrt{(\mathbf{1} - b\_0)^2 - \frac{4}{a\delta}}}{2}.$$

It may be noted that if

*Ri*,*j*ð Þ¼ *<sup>r</sup>* <sup>Θ</sup> *<sup>r</sup>* � <sup>∥</sup>**Xi** � **Xj**<sup>∥</sup> ; *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>n</sup>* (12)

where Θ is a Heavyside function, *r* is a small threshold distance between

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

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

A characteristic pattern emerges in RPs which characterizes a dynamical system.

The neuronal communication is known to be mediated by electrical pulses called spikes. Studies of various spiking patterns reveal nonlinear characteristics of slowfast neuronal dynamics. A considerable amount of information regarding neuronal activity has been obtained by studying the dynamics of spiking pattern [22]. The phenomenon of tonic firing, mixed mode (bursting and spiking) etc., are typical responses exhibited by an excitable neuron [23]. Cortical neurons have been reported to show tonic bursting wherein the neurons periodically switches between firing state and resting state. The mixed mode firing is observed in mammalian neocortex [24]. Spike generation in fact depends on the firing threshold and the stimulus intensity. In recent years, the perception regarding constancy of neuron's firing threshold has changed to dynamic [21]. In this work, we first briefly introduce the Fitzhugh-Nagumo model (FHN) that have been proposed for spike generation like well known Hodgkin-Huxley model. It is however to be noted that FHN model reproduce the experimental results less accurately. Our interest in FHN model emanates mainly due to its showing complex spiking pattern even though it is mathematically simple. The basic FHN model assumes the threshold to be constant. We also study the changes caused in the spiking pattern as a result of time varying threshold. This study assumes significance as such a model may throw insight into the model the dynamics of cortisol secretion from hypothalamus [25]. It is to be noted that the neuronal firings may take place at regular interval or randomly due to inherent mechanism or may be due to its interaction with the

neighboring points and ∥*:*∥ is an Euclidean norm.

**5. Regular and Chaotic neuronal dynamics**

neighborhood neurons or result of exogenous stimulus [26–28].

The FHN model describes the interaction between the voltage *v* across the axon membrane driven by input current *I* and the recovery variable *w*. The recovery

**5.1 Basic dynamics of FHN model**

**44**

diagram etc.

$$\left(\left(1 - b\_0\right)^2 - \frac{4}{a\delta} < 0,\tag{14}$$

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

$$h(\upsilon) = \upsilon(\mathbf{1} - \upsilon)(\upsilon - b\_0) \tag{15}$$

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

$$J = \begin{vmatrix} a h'(v) & -\mathbf{1} \\ \varepsilon & -\varepsilon \delta \end{vmatrix}.\tag{16}$$

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

$$\lambda\_{1,2} = \frac{-\left(\epsilon\delta - ab\_1\right) \pm \sqrt{\left(\epsilon\delta - ab\_1\right)^2 - 4\epsilon\left(\mathbf{1} - ab\_1\delta\right)}}{2} \tag{17}$$

where *b*<sup>1</sup> ¼ *h*<sup>0</sup> ð Þ *ve* .

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 and (3) if *αb*1*δ* ¼ 1 then the equilibrium point is stable (unstable) if *αb*<sup>1</sup> <*δε α*ð Þ *b*<sup>1</sup> >*δε* .

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 stable if

$$\operatorname{Re}\left[-\left(ab\_0 + \epsilon\delta\right) \pm \sqrt{\left(ab\_0 + \epsilon\delta\right)^2 - 4\epsilon\left(a\delta b\_0 + 1\right)}\right] < 0. \tag{18}$$

Based on Routh's criteria we may write equivalently

$$a b\_0 + \epsilon \delta > \qquad 0, \qquad a \delta b\_0 + \mathbf{1} > \mathbf{0}. \tag{19}$$

**5.2 FHN neuron in the presence of external periodic electrical stimulation**

ð Þ *d I* ¼ 6*:*8, ð Þ*e I* ¼ 7*:*5, ð Þ*f I* ¼ 9*:*5, ð Þ*g I* ¼ 11*:*75, ð Þ *h I* ¼ 12*:*42, *and i*ð Þ *I* ¼ 12*:*45*.*

external periodic stimulation, *I t*ð Þ, may be written as:

*Chaotic Dynamics and Complexity in Real and Physical Systems*

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

*dv*

**Figure 14.**

**47**

*dw*

i.e., i.e., *I*<sup>0</sup> ¼ 0*:*183, *ν* ¼ 0*:*1931, as shown in **Figure 17**.

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

Here, we take the external periodic stimulation as given by *I t*ðÞ¼ *<sup>I</sup>*<sup>0</sup>

Chaotic systems exhibit complexity and are sensitively dependent on initial condition of the system under investigation and also unpredictable. Chaos as a nonlinear phenomenon has attracted researchers from different disciplines e.g., physics, biology, ecology, neurobioscience etc. In this section, we investigate the effect of periodic electrical stimulation on the dynamics of an FHN system, Eq. (13). The basic equation that governs the dynamics of FHN system in the presence of

*Phase portrait of FHN system showing limit cycle for* 4*:*2 ≤*I* ≤12*:*45*. a*ð Þ *I* ¼ 4*:*23, ð Þ *b I* ¼ 4*:*8, ð Þ*c I* ¼ 5*:*5,

*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> *I t*ð Þ,

where *I*0, *ν* refers to the amplitude and frequency of the input stimulus. Further, we present the simulation results of the system, Eq. (20), by taking *α* ¼ 10, *β* ¼ 1, *b*<sup>0</sup> ¼ 0*:*10, *δ* ¼ 0*:*25, *ε* ¼ 1, and *σ* ¼ 1 and varying both *I*<sup>0</sup> and *ν*. The variation of both amplitude and frequency of the external periodic stimulus is found to result in the membrane potential *v* exhibiting regular or chaotic temporal behavior. The regular or periodic neuron spiking could be classified as *p* : *q* phase-locking, where *p* and *q* corresponds to the number of spikes and number of periodic stimuli per unit response period. For instance **Figures 15** and **16** illustrates respectively the response of the neuronal spiking corresponding to 1 : 1 and 1 : 2 phase locked rhythm.

The response of the single FHN neuron to external periodic response could also be chaotic for certain values of the amplitude *I*<sup>0</sup> and frequency *ν* of driving stimulus

The observed dynamical transition from regular/periodic to chaotic of membrane potential *v* with increase in amplitude and frequency of external could be

(20)

<sup>2</sup>*πν* cos 2ð Þ *πν<sup>t</sup>* ,

Therefore if origin is the only fixed point of the system, then following [30], it is observed that the system has no limit cycle if *<sup>b</sup>*<sup>0</sup> � <sup>1</sup> 2 � �<sup>2</sup> <sup>þ</sup> <sup>3</sup> <sup>4</sup> � <sup>3</sup>*δε <sup>α</sup>* < 0. This result however assumes *αb*<sup>0</sup> þ *εδ*>0. In case *αb*<sup>0</sup> þ *εδ*<0, the origin becomes unstable and the system can be shown to exhibit one stable limit cycle. It is noted here that on varying the threshold parameter *b*0, the system may exhibit Andronov-Hopf bifurcation when *αb*<sup>0</sup> þ *εδ* ¼ 0 as per Eq. (18) and at this point the origin of the system becomes unstable causing a bifurcation to at least one stable limit cycle. It is to be noted that gaps exist in parameter space when origin is the only asymptotically fixed point and where limit cycles may exists. Interestingly, we numerically show the existence of bistable behavior [18, 30] in terms of occurrence of double cycle bifurcation by taking *ε* ¼ 0*:*015, *δ* ¼ 3*:*5, *α* ¼ 1*:*0 and allowing the threshold value *b*<sup>0</sup> <0 i.e., �0*:*044(**Figure 13**).

For the case of *I* 6¼ 0, the equilibrium points may be one, two or three and their stability may be analyzed following the foregoing analysis. Taking the parameter values: *a* ¼ 0*:*06, *b*<sup>0</sup> ¼ 0*:*50, *ε* ¼ 14, as in [29], the phase portrait were obtained using numerical integration of the system, Eq. (13), for different *I* values (**Figure 14**). The appearance of limit cycle behavior is due to supercritical Hopf bifurcation and as a consequence of loss of stability of the unique equilibrium point that exist for *I* <4*:*2 [29]. **Figure 14** also suggest that the amplitude of limit cycles first increases and subsequently decreases with increase in values of *I*. At around *I* � 12*:*45 the second bifurcation occurs and system is led to a stable equilibrium. [29] has provided a detailed discussion on the richness of various bifurcation event as *I* is varied.

**Figure 13.** *Phase portrait of FHN system showing bistability between limit cycle and stable fixed point.*

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

#### **Figure 14.**

*Re* �ð Þ� *αb*<sup>0</sup> þ *εδ*

Based on Routh's criteria we may write equivalently

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

observed that the system has no limit cycle if *<sup>b</sup>*<sup>0</sup> � <sup>1</sup>

*b*<sup>0</sup> <0 i.e., �0*:*044(**Figure 13**).

as *I* is varied.

**Figure 13.**

**46**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>α</sup>b*<sup>0</sup> <sup>þ</sup> *εδ* <sup>2</sup> � <sup>4</sup>*ε αδ* ð Þ *<sup>b</sup>*<sup>0</sup> <sup>þ</sup> <sup>1</sup>

*αb*<sup>0</sup> þ *εδ*> 0, *αδb*<sup>0</sup> þ 1>0*:* (19)

2 � �<sup>2</sup> <sup>þ</sup> <sup>3</sup>

<sup>4</sup> � <sup>3</sup>*δε*

< 0*:* (18)

*<sup>α</sup>* < 0. This result

� � q

Therefore if origin is the only fixed point of the system, then following [30], it is

For the case of *I* 6¼ 0, the equilibrium points may be one, two or three and their stability may be analyzed following the foregoing analysis. Taking the parameter values: *a* ¼ 0*:*06, *b*<sup>0</sup> ¼ 0*:*50, *ε* ¼ 14, as in [29], the phase portrait were obtained

using numerical integration of the system, Eq. (13), for different *I* values (**Figure 14**). The appearance of limit cycle behavior is due to supercritical Hopf bifurcation and as a consequence of loss of stability of the unique equilibrium point that exist for *I* <4*:*2 [29]. **Figure 14** also suggest that the amplitude of limit cycles first increases and subsequently decreases with increase in values of *I*. At around *I* � 12*:*45 the second bifurcation occurs and system is led to a stable equilibrium. [29] has provided a detailed discussion on the richness of various bifurcation event

*Phase portrait of FHN system showing bistability between limit cycle and stable fixed point.*

however assumes *αb*<sup>0</sup> þ *εδ*>0. In case *αb*<sup>0</sup> þ *εδ*<0, the origin becomes unstable and the system can be shown to exhibit one stable limit cycle. It is noted here that on varying the threshold parameter *b*0, the system may exhibit Andronov-Hopf bifurcation when *αb*<sup>0</sup> þ *εδ* ¼ 0 as per Eq. (18) and at this point the origin of the system becomes unstable causing a bifurcation to at least one stable limit cycle. It is to be noted that gaps exist in parameter space when origin is the only asymptotically fixed point and where limit cycles may exists. Interestingly, we numerically show the existence of bistable behavior [18, 30] in terms of occurrence of double cycle bifurcation by taking *ε* ¼ 0*:*015, *δ* ¼ 3*:*5, *α* ¼ 1*:*0 and allowing the threshold value

*Phase portrait of FHN system showing limit cycle for* 4*:*2 ≤*I* ≤12*:*45*. a*ð Þ *I* ¼ 4*:*23, ð Þ *b I* ¼ 4*:*8, ð Þ*c I* ¼ 5*:*5, ð Þ *d I* ¼ 6*:*8, ð Þ*e I* ¼ 7*:*5, ð Þ*f I* ¼ 9*:*5, ð Þ*g I* ¼ 11*:*75, ð Þ *h I* ¼ 12*:*42, *and i*ð Þ *I* ¼ 12*:*45*.*

#### **5.2 FHN neuron in the presence of external periodic electrical stimulation**

Chaotic systems exhibit complexity and are sensitively dependent on initial condition of the system under investigation and also unpredictable. Chaos as a nonlinear phenomenon has attracted researchers from different disciplines e.g., physics, biology, ecology, neurobioscience etc. In this section, we investigate the effect of periodic electrical stimulation on the dynamics of an FHN system, Eq. (13).

The basic equation that governs the dynamics of FHN system in the presence of external periodic stimulation, *I t*ð Þ, may be written as:

$$\begin{split} \frac{dv}{dt} &= a v(\beta - v)(v - b\_0) - \sigma w + I(t), \\\\ \frac{dw}{dt} &= \varepsilon (v - \delta w). \end{split} \tag{20}$$

Here, we take the external periodic stimulation as given by *I t*ðÞ¼ *<sup>I</sup>*<sup>0</sup> <sup>2</sup>*πν* cos 2ð Þ *πν<sup>t</sup>* , where *I*0, *ν* refers to the amplitude and frequency of the input stimulus. Further, we present the simulation results of the system, Eq. (20), by taking *α* ¼ 10, *β* ¼ 1, *b*<sup>0</sup> ¼ 0*:*10, *δ* ¼ 0*:*25, *ε* ¼ 1, and *σ* ¼ 1 and varying both *I*<sup>0</sup> and *ν*. The variation of both amplitude and frequency of the external periodic stimulus is found to result in the membrane potential *v* exhibiting regular or chaotic temporal behavior. The regular or periodic neuron spiking could be classified as *p* : *q* phase-locking, where *p* and *q* corresponds to the number of spikes and number of periodic stimuli per unit response period. For instance **Figures 15** and **16** illustrates respectively the response of the neuronal spiking corresponding to 1 : 1 and 1 : 2 phase locked rhythm.

The response of the single FHN neuron to external periodic response could also be chaotic for certain values of the amplitude *I*<sup>0</sup> and frequency *ν* of driving stimulus i.e., i.e., *I*<sup>0</sup> ¼ 0*:*183, *ν* ¼ 0*:*1931, as shown in **Figure 17**.

The observed dynamical transition from regular/periodic to chaotic of membrane potential *v* with increase in amplitude and frequency of external could be

**Figure 15.**

1 : 1 *phase locking rhythm of spiking neuron. (a) Time series of membrane potential with I*<sup>0</sup> ¼ 0*:*1, *ν* ¼ 0*:*05*. (b) v* � *w phase portrait with same parameters as in (a).*

**Figure 17.**

**Figure 18.**

*rhythm.* **49**

*(b) v* � *w phase portrait with same parameters as in (a).*

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

1 : 1 *phase locking rhythm of spiking neuron. (a) Time series of membrane potential with I*<sup>0</sup> ¼ 0*:*183, *ν* ¼ 0*:*1931*.*

*RP of of the membrane potential with (a)* 1 : 1 *phase locking rhythm, (b)* 1 : 2 *phase locking and (c) chaotic*

**Figure 16.**

1 : 1 *phase locking rhythm of spiking neuron. (a) Time series of membrane potential with I*<sup>0</sup> ¼ 0*:*1, *ν* ¼ 0*:*1015*. (b) v* � *w phase portrait with same parameters as in (a).*

further seen by constructing the RPs. The construction of RP, discussed earlier in section 4, involves the reconstruction of phase space using a time series of a dynamical variable, say the membrane potential *v*, based on the information regarding the delay parameter *τ* and the embedding dimension *m*. The delay parameter *τ* for the time series of *v* could be obtained using the method of mutual information (MI) [31] and the embedding dimension *m* may be determined using the algorithm of [32, 33]. The time series of **Figures 15**–**17** for the membrane potential *v* have been used to construct the RP shown in **Figure 18**. The change in spiking patterns caused by external periodic stimulation from regular to chaotic is well indicated in RP of almost equally spaced diagonal lines to irregularly occurring broken diagonal lines of varying length.

#### **5.3 FHN neuron with time varying threshold**

The dynamics of cortisol secretion from hypothalamus could be modeled using FHN system with time varying threshold [25]. Complexities of spike dynamics of FHN neuron has been earlier investigated in [34] incorporating the time varying

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

1 : 1 *phase locking rhythm of spiking neuron. (a) Time series of membrane potential with I*<sup>0</sup> ¼ 0*:*183, *ν* ¼ 0*:*1931*. (b) v* � *w phase portrait with same parameters as in (a).*

**Figure 18.** *RP of of the membrane potential with (a)* 1 : 1 *phase locking rhythm, (b)* 1 : 2 *phase locking and (c) chaotic*

*rhythm.*

further seen by constructing the RPs. The construction of RP, discussed earlier in section 4, involves the reconstruction of phase space using a time series of a dynamical variable, say the membrane potential *v*, based on the information regarding the delay parameter *τ* and the embedding dimension *m*. The delay parameter *τ* for the time series of *v* could be obtained using the method of mutual information (MI) [31] and the embedding dimension *m* may be determined using the algorithm of [32, 33]. The time series of **Figures 15**–**17** for the membrane potential *v* have been used to construct the RP shown in **Figure 18**. The change in spiking patterns caused by external periodic stimulation from regular to chaotic is well indicated in RP of almost equally spaced diagonal lines to irregularly occurring

1 : 1 *phase locking rhythm of spiking neuron. (a) Time series of membrane potential with I*<sup>0</sup> ¼ 0*:*1, *ν* ¼ 0*:*1015*. (b) v* � *w phase portrait with same parameters as in (a).*

1 : 1 *phase locking rhythm of spiking neuron. (a) Time series of membrane potential with I*<sup>0</sup> ¼ 0*:*1, *ν* ¼ 0*:*05*.*

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The dynamics of cortisol secretion from hypothalamus could be modeled using FHN system with time varying threshold [25]. Complexities of spike dynamics of FHN neuron has been earlier investigated in [34] incorporating the time varying

broken diagonal lines of varying length.

**Figure 15.**

**Figure 16.**

**48**

*(b) v* � *w phase portrait with same parameters as in (a).*

**5.3 FHN neuron with time varying threshold**

threshold. Models of observed tonic firing and spike bursting were simulated by considering both periodic and noisy form of the threshold variation. The effect of a mixed mode threshold on the spiking FHN system was also investigated. Here we present results of bifurcation analysis of different states of neuronal firing of FHN neuron by considering a discrete form of the system [34, 35]. Following [34], the discrete form of the FHN system may be written as:

$$\boldsymbol{\nu}\_{n+1} = \boldsymbol{\nu}\_n + \Delta \boldsymbol{a} [-\boldsymbol{\nu}\_n (\boldsymbol{v}\_n - \mathbf{1})(\boldsymbol{v}\_n - \boldsymbol{b}\_n) - \boldsymbol{w} + I] \tag{21}$$

**Figure 19b** wherein windows of quasi-periodic behavior that sets sets in say at *δ* � 0*:*68 for *I* ¼ 0 (**Figure 20a**) make a transition to chaos as shown in **Figure 20b**. In the following section, we study the complex dynamics in economics, an area actively pursued by researchers, by introducing another measure called 'multi-scale

In mainstream economics, economic dynamics has assume great importance in recent years in view of the availability of market and other data. Economic dynamics has therefore influenced both micro- and macroeconomics. Therefore lot of research output has poured in explaining irregular micro-economic fluctuation, erratic business cycles, irregular growth and aperiodic behavior of economic data etc. Nonlinear systems provides an alternative simple and deterministic framework that easily can explain aperiodic or chaotic behaviors of various financial systems. One of the important features of nonlinear system is that the irregular/chaotic behavior supports an endogenous mechanism for the observed complexity in economic time series. As a result nonlinear dynamic framework has been applied to

In the present work we revisit the synthetic chaotic financial model discussed in

[45, 46] which is based on interest rate, investment demand and price index as dynamical variables. We numerically explored and analyze the complexity of the model using the multiscale entropy (MPE) frame work. In this section, we briefly describe the chaotic financial model and its basic characteristics. We also outlines

We consider a dynamic finance model composed of three coupled first order differential equation. This model describes the temporal evolution of the state variables viz. the interest rate *X*, the investment demand *Y* and the price index *Z*. The

*dt* <sup>¼</sup> *<sup>Z</sup>* <sup>þ</sup> ð Þ *<sup>Y</sup>* � *<sup>a</sup> <sup>X</sup>*,

,

(23)

*dt* <sup>¼</sup> <sup>1</sup> � *bY* � *<sup>X</sup>*<sup>2</sup>

*dt* ¼ �*<sup>X</sup>* � *cZ:*

investment and elasticity of demand of the commercial markets. First equation appears, representing the changes in *X*, as a result of contradiction in the investment market and structural adjustment from the goods prices. Second equation representing the changes in *Y* appears due to proportionality to the rate of investment and also to an inversion of the cost of investment and interest rate. The third equation emerges due to contradiction between supply and demand in commercial

Eq. (23) has been numerically integrated using fourth-order Runge-Kutta method to obtain the time series of the dynamic variables *X*, *Y* and *Z*, shown in **Figure 21** with *a* ¼ 3*:*0, *b* ¼ 0*:*1, *c* ¼ 1*:*0 and initial condition ð Þ¼ *X*0, *Y*0, *Z*<sup>0</sup>

Here *a*, *b* and *c* are positive constants and represent the saving amount, cost per

the procedure of MPE for analyzing the complexity of the finance model.

permutation entropy' by considering a nonlinear financial model.

*Chaotic Dynamics and Complexity in Real and Physical Systems*

economic modeling and several examples are available in [36–46].

*dX*

*dY*

*dZ*

markets which is influenced by interest rates.

ð Þ 2, 3, 2 . Similar choice of the parameter were made in [45].

**6. Chaotic dynamics in finance model**

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

**6.1 Chaotic financial model**

model is described as [39]:

**51**

$$
\omega\_{n+1} = \omega\_n + \Delta(v\_n - \delta w\_n) \tag{22}
$$

where Δ refers to the integral step size and is treated here as a bifurcation parameter.

In case of mixed mode threshold variation, the membrane potential *v* exhibits a complex behavior as shown in bifurcation diagram (**Figure 19a**). It is readily observed that the temporal behavior is chaotic in the region 0*:*42≤ Δ ≤0*:*68. Thereafter windows of regular and chaotic regimes are observe till Δ ¼ 0*:*8 for *I* ¼ 0. A slight increase in *I* changes the dynamics to a more complex behavior as shown in

**Figure 19.** *Bifurcation of membrane potential. (a) I = 0; (b) I = 1.*

**Figure 19b** wherein windows of quasi-periodic behavior that sets sets in say at *δ* � 0*:*68 for *I* ¼ 0 (**Figure 20a**) make a transition to chaos as shown in **Figure 20b**.

In the following section, we study the complex dynamics in economics, an area actively pursued by researchers, by introducing another measure called 'multi-scale permutation entropy' by considering a nonlinear financial model.
