Stabilizing Circadian Rhythms in Bipolar Disorder by Chaos Control Methods

Bipolar disorder (BD), which involves mood swings between mania and depression, is associated with multiple relapses during long-term treatment and high suicide and morbidity rates. In BD, the circadian rhythms, which are measured by daily mood scores and actigraphic records, are disturbed. Chronotherapy has emerged as a potential treatment for BD because it stabilizes the disturbed circadian rhythms and improves BD symptoms. Concrete treatments include light therapy and combination therapy (light therapy and drugs). However, some patients have difficulty adjusting to light therapy; inappropriate light and duration of treatment increase risks for inducing mixed states and the emergence of conditions, such as hypomania and autonomic hyperactivation. Therefore, it is important to devise methods for optimizing chronotherapy for BD. We aimed to develop feedback signals for the frontal cortex, which were based on the delayed feedback method as one of the chaos control methods, to stabilize the disturbed circadian rhythms of BD. Concrete procedures of this study are indicated as follows: first, circadian rhythms of BD are reproduced using the frontal cortex and hypothalamus neural system, which has been previously proposed. Second, the delayed feedback signal is developed by using bifurcation analysis. Third, the effect of delayed feedback signal is evaluated by index for complexity, and power spectrum under the condition with/without stochastic noise in feedback term. We found that application of the delayed feedback signal to the frontal cortical neural activity induces the periodic state of circadian rhythms from the disturbed complex and is feasible for treating BD. However, when increasing the influence of noise in feedback term, the stabilizing effect is diminished. In conclusion, we developed a stabilizing method for disturbed circadian rhythms of BD using the circadian neural systems. The present study highlights the potential usefulness of the chaos control method for treating BD.


INTRODUCTION
Bipolar disorder (BD), which involves mood swings between mania and depression, is associated with multiple relapses during long-term treatment and high suicide and morbidity rates (Belmaker, 2004). Changes in the glutamatic acid and gamma-aminobutyric acid (GABA) neural pathways and abnormal cortical neural networks have been reported as the neural bases for BD (Sanacora et al., 2012;Schloesser et al., 2012). Moreover, Tobe et al. (2017) showed that abnormal phosphorylation of synaptic connections causes BD. Drugs, such as mood stabilizers (i.e., lithium carbonate and clozapine) and intramuscular neuroleptics are widely utilized for BD. Pharmacological mechanisms of these treatments have been elucidated in previous studies (Hirschfeld et al., 2003;López-Muñoz et al., 2006;Tobe et al., 2017). However, these treatments are also associated with several side effects, such as progressive renal failure and a narrow therapeutic index (Hirschfeld et al., 2003;López-Muñoz et al., 2006). Therefore, previous studies have focused on developing alternative and effective treatment methods for BD (Abreu and Bragança, 2015).
Recently, the clinical efficiency and feasibility of chronotherapy has been studied (Abreu and Bragança, 2015). In mood disorders, including BD, the circadian rhythms, which are measured by daily mood scores and actigraphic records, are disturbed (Yeragani et al., 2003;Glenn et al., 2006;Bonsall et al., 2011;Moore et al., 2014). Chronotherapy stabilizes the disturbed circadian rhythms and improves BD symptoms (Abreu and Bragança, 2015). Concrete treatments include light therapy and combination therapy (light therapy with drugs) (Leibenluft et al., 1995;Terman and Terman, 2005). However, it is difficult for some BD patients to adjust to light therapy; furthermore, inappropriate light and duration of treatment increase the risks of mixed states, hypomania, and autonomic hyperactivation (Terman and Terman, 2005;Sit et al., 2007;Abreu and Bragança, 2015).
The application of nonlinear modeling for drug treatments has also emerged as a potential therapy (Itik et  and Suzuki et al. (2010) proposed the concept of intermittent hormone therapy for preventing the growth of prostate cancer and its divergences; the model was based on the nonlinear control theory and utilized a prostate cancer model. Other control methods, such as optimized cancer immunotherapy and drug treatments involving human immunodeficiency virus have also been developed (Mhawej et al., 2010;Babaei and Salamci, 2015;Hamdache et al., 2016). That is, applying nonlinear control methods has opened a new avenue of medical treatments. Therefore, applying nonlinear control methods to chronotherapy for stabilizing the circadian rhythms of BD is promising for avoiding risks associated with chronotherapy. Hadaeghi et al. (2016) and Bayani et al. (2017) demonstrated that a mechanism of disturbed circadian rhythms in BD involves aperiodic daily neural activities, which is referred to as chaos-chaos intermittency. In the frontal cortex, chaoschaos intermittency perturbs the circadian pacemaker of the hypothalamus through model simulation of the neural system (which is comprised of the frontal cortex and hypothalamus). The accuracy of this model is high in comparison with physiological circadian rhythms of BD, and it can explain the relationship between abnormalities in cognitive function and the disturbance of circadian rhythms (Hadaeghi et al., 2013;Hadaeghi et al., 2016;Bayani et al., 2017;Hassanzadeh et al., 2017). Therefore, based on previous findings, it can be inferred that the method for shifting the chaos-chaos intermittency to a periodic state in frontal cortical activity stabilizes the circadian rhythms of BD and optimizes chronotherapy.
To stabilize and adjust chaotic behaviors, various kinds of chaos control methods have been proposed, such as the method established by Ott-Grebogi-Yorke (OGY) Ott et al. (1990), the delayed feedback method (Pyragas, 1992;Nakajima, 1997), H ∞ control (Jiang et al., 2005), and a reduced region of the orbit method Nobukawa et al., 2019b). These chaos control methods have been adopted and applied to many neural systems Zhou et al., 2008;Nobukawa et al., 2019c). Particularly, compared to the other chaos control methods, the delayed feedback method can stabilize the chaotic behaviors using a smaller number of parameters (Schöll and Schuster, 2008). Concretely, by applying the delayed feedback method at an appropriate strength, which is based on the previous system status before the target period, the chaotic behaviors may shift to periodic behaviors within the target period (Pyragas, 1992;Nakajima, 1997). Therefore, even under conditions where the detailed system dynamics cannot be comprehended (i.e., the actual neural systems), the delayed feedback method has high potential and feasibility.
Based on the delayed feedback system, we aimed to develop feedback signals for the frontal cortex that could be used to stabilize disturbed circadian rhythms associated with BD. Concrete procedures used in this study are as follows: first, circadian rhythms of BD were reproduced using bifurcation analysis and the frontal cortex and hypothalamus neural system, as proposed by Hadaeghi et al. (2016). Second, the delayed feedback signal was developed using the bifurcation diagram. Third, the effect of the delayed feedback signal was evaluated by index for complexity and power spectrum analysis under the condition with/without stochastic noise in feedback term.

Neural System Composed of the Frontal Cortex and Hypothalamus
The pathology of BD involves multiple complex neural pathways  Figure 1 shows an overview of this neural system.
The daily neural activity of the frontal cortex x(n) (n 1, 2, . . .) is controlled by the competition of excitatory and inhibitory neural populations: x(n + 1) B tanh(w 2 x(n)) − A tanh(w 1 x(n)). (1) Here, w 1 , w 2 the synaptic weights of input to the inhibitory neural population and excitatory neural population are determined, respectively. A, B correspond with the synaptic weights of output from the inhibitory neural population and excitatory neural population, respectively. The output from the frontal cortex to the hypothalamus is defined as the temporal variation from the periodic state with period p: The dynamics of the circadian pacemaker in the hypothalamus are represented by a two-dimensional mapbased model (Pavlov et al., 2011): where y and z represent the fast and slow variable at time step (k 1, 2, . . .), and μ refers to the length of the circadian cycle. The function of ϵ is defined by a Heaviside step function, with a threshold yth: ϵ(z) H(z − yth). The time scale n of Eq. (1) is the daily scale, which corresponds with the period from peak to peak of z(k) in Eq. (4). The function f is a nonlinear function, as follows: where u is given by the external stimulus I ext , and an internal parameter β: u β + I ext . z s exhibits the peak value of oscillation of z(k). The other parameter, g in Eq. (4) is a parameter for the rising shape of the z(k) from bottom state: Here, A normal , B normal , g opt represent the optimal value of inhibitory and excitatory neurotransmitters, and the optimal value of g, respectively. Δ is given by the connection w 4 from the frontal cortex to the circadian pacemaker of the hypothalamus, Δ D(w 4 e(n)) where D is a function of tansig. The output of the hypothalamus is a function of the slow-state variable z that is regulated by input from the frontal cortex e(n): where M is a function of tansig. The set of parameters used in this study was determined based on previous research by Hadaeghi   October 2020 | Volume 6 | Article 562929 t th 0.01,z s 1.3,y p −0.8,α 3.2,β −2.578,I ext 0.001,A normal 13.0,B normal 5.82. The period between peak to peak of output in A A normal ,B B normal is defined as one day in Eqs (3)−(9). Hadaeghi et al. (2016) demonstrated that healthy periodic circadian rhythms and disturbed circadian rhythms associated with BD are produced by period-p state in the periodic window and chaos-chaos intermittency state, respectively, in the frontal cortical neural activity, as demonstrated by Eq. (1) (Hadaeghi et al., 2016). In this study, we developed the feedback signal to shift the chaos-chaos intermittency of x(n) the periodic-p state using the delayed feedback method (Pyragas, 1992;Nakajima, 1997) for inducing the stable circadian rhythm. In the delayed feedback method, the feedback signal-which is based on the difference between the current state and the previous state before p-period-stabilizes an unstable periodic-p orbit embedded in a chaotic attractor (Pyragas, 1992;Nakajima, 1997). The daily neural activity of the frontal cortex x(n) is controlled by delayed feedback signals, as follows:

Controlling Frontal Cortical Neural Activity by Delayed Feedback Signals
Here, in the delayed feedback method, the target period is not only the period-p, but also involves p/m(m 1, 2, . . .).
In the actual treatment, the estimation of frontal cortical activity and the applying of stimulus involve the measurement error and background noise. To evaluate the influence of stochastic noise in the feedback term, we considered the daily neural activity of the frontal cortex x(n) controlled by delayed feedback signals involving stochastic noise Dξ(n) where D and ξ(n) indicate noise strength and Gaussian white noise (mean: 0, standard deviation: 1) given by

Power Spectrum
A power spectrum analysis was performed to evaluate the periodicity of circadian rhythm given by (k). We calculated the power spectrum density (PSD) (dB·day) for output(k) using a fast Fourier transform. A Hanning window was applied to this time-series.

Multiscale Entropy
The approximate entropy was used to evaluate the disturbance of circadian rhythms (Yeragani et al., 2003;Glenn et al., 2006). This approximate entropy was extended to the sample entropy (SampEn) and multiscale entropy (MSE) by modifying the data length dependency and robustness of outliers (Costa et al., 2002). In this study, along with PSD, we used MSE to evaluate the complexity with time scale dependency in the timeseries of output(k).
Against stochastic variable {x 1 , x 2 , . . . , x N }, a SampEn is defined by where C m (r) indicates the probability to satisfy with Here, x m i is m a dimensional vector given by In MSE analysis, against coarse-grained series of {x 1 , x 2 , . . . , x N } with the scale factor τ(τ 1, 2, . . .): SampEn h τ (r, m) is calculated. By the dependency of h τ (r, m) with a scale factor τ, we evaluated the characteristic of complexity in the time-series of output (k). In this study, we set m 2, r 0.2 (Costa et al., 2002).

Circadian Rhythms in the Neural System Are Composed of the Frontal Cortex and Hypothalamus
We have demonstrated that the system behavior in the neural system is composed of the frontal cortex and hypothalamus. Figure 2A shows the bifurcation diagram of the frontal neural activity x(n) as a function of synaptic weight from the inhibitory neural population A. With an increasing A value, x(n) exhibits the period doubling bifurcation and enters a chaotic state Aa8.1.  Figure 2B shows typical examples of the frontal neural activity x(n) given by Eq. (1) and the circadian rhythms output(k) given by Eq. (8) in HC and BD patients. In correspondence with HC behavior, x(n) exhibits the periodic-4 state where this parameter set is located at the periodic window in Figure 2A. The circadian rhythms are not disturbed because the temporal variation of x(n): e(n) becomes zero. In A 15.0 correspondence with BD behavior, x(n) exhibits chaos-chaos intermittency; therefore, e(n) > 0 is applied to the circadian pacemaker in the hypothalamus. This perturbation leads to intermittent states, with smaller peaks of output(k) and a shortened period from peak to peak. Consequently, circadian rhythms are disturbed, which is Frontiers in Applied Mathematics and Statistics | www.frontiersin.org October 2020 | Volume 6 | Article 562929

Stabilizing Disturbed Circadian Rhythms Using the Delayed Feedback Method
To stabilize the disturbed circadian rhythms of BD (A 15.0), the delayed feedback signal was applied to the frontal neural activity using Eq. (10). Figure 4A shows the bifurcation diagram of the frontal neural activity x(n), which is given by Eq. (10) as a function of K. In Ka0.3, the orbit exhibits the periodic state. In Figure 4B, the typical examples of the time-series of frontal neural activity x(n) and circadian rhythms output(k) are shown for the cases with and without feedback signals. In K 0 case, x(t) shows chaos-chaos intermittency; due to this perturbation, output(k) exhibits irregular behaviors. On the other hand, in K 0.5 case, x(t) converged its periodic state 65 days later. Consequently, output(k) also exhibited the periodic state. This stabilized effect was evaluated using MSE analysis. The dependence of SampEn as a function of temporal scale is shown in Figure 5A. It is confirmed that, with feedback signals (K 0.5), SampEn in the scale ((7 days) is lower, compared to the case without feedback signal (K 0). Along with it, the power spectrum analysis showed that under the feedback signal (K 0.5), the PSD in lower and higher frequency component around peaks ( ≈ 1 [1/day]) decreased more than the case without feedback signal (K 0). That is, the delayed feedback signal stabilizes the disturbed circadian rhythms.

Stabilizing Disturbed Circadian Rhythms in the Case With Delayed Feedback Signals Involving Stochastic Noise
Assuming the estimation of frontal cortical activity and the applying of stimulus involve the measurement error and background noise, we evaluated the influence of stochastic noise in the feedback term given by Eq. (11) to stabilizing disturbed circadian rhythms. Figure 6A showed that the result of MSE analysis in circadian rhythms output(k) given by Eq.

DISCUSSION AND CONCLUSION
In this study, we reproduced the periodic and disturbed circadian rhythms that corresponded with neural system (frontal cortex and hypothalamus) behaviors of HC and BD patients, which was proposed by Hadaeghi et al. (2016). Furthermore, MSE analysis and power spectrum analysis revealed that the disturbed circadian rhythms of BD exhibited higher complexity than those associated with HC. A disturbed circadian rhythm corresponded with BD; the delayed feedback signal was applied to the frontal cortical neural activity following the delayed feedback method. As a result, the feedback signal induced a periodic circadian rhythm from the disturbed rhythm.
In order to determine the feedback signals in the actual treatment of BD, we must consider the method used to estimate the daily frontal cortical activity. In this estimation, its accuracy strongly affects the ability to stabilize the disturbed circadian rhythm (see Figure 6); therefore, a highly accurate estimation method is needed. As the candidates for the estimation method, Mitsukura and her colleagues developed a method with portable, single-channel electroencephalogram (EEG) devices;  According to the delayed feedback method (Pyragas, 1992;Nakajima, 1997), the feedback signals can also be determined by measuring the observation variables and analyzing their effects on the target dynamics. Therefore, the frontal cortical feedback signals can be developed by physiological and psychological signals, such as actigraphic records, body temperature, and daily mood states, which reflect circadian rhythms. However, based on the construct of the neural systems of the frontal cortex and hypothalamus, as proposed by Hadaeghi et al. (2016), the daily variation of cortical neural activity reflects the disturbance of circadian rhythms. That is, it can be assumed that, in many cases, the observation variable regarding circadian rhythms might include historical frontal cortical activity. In the original delayed feedback methods, the historical effects of observation variables were not considered (Pyragas, 1992;Nakajima, 1997). Therefore, further modifications of the delayed feedback method are necessary for this control.
The actual signals corresponding to the feedback signals in the treatment must be considered. Light therapy and combination therapy (light and drugs) control the melatonin secretion which affects circadian rhythms (Leibenluft et al., 1995;Terman and Terman, 2005). Therefore, the strength and duration of light and the amount of drugs needed to realize the target concentration of melatonin in the blood can correspond with feedback signals. Optimizing chronotherapy based on the delayed feedback method may reduce the risk for inducing mixed states, hypomania and autonomic hyperactivation (Terman and Terman, 2005;Sit et al., 2007;Abreu and Bragança, 2015).
This study has some limitations that should be addressed. Recent studies showed that the network structures, especially topological features, are strongly related with the temporal behavior of neural activity and its functions (Kawai et   . However, the frontal cortical network, which was dealt with in this study, is described by the neural population model with only one variable for cortical neural activity. As long as this population model is used, the cortical network structures cannot be implemented. Therefore, future directions for study, should use a neural network model with higher physiological validity, such as a spiking neural network. We plan to construct the frontal cortical neural network including brain network structures; by using the network model with high physiological validity, the effectiveness for our proposed treatment can be further evaluated. Another limitation lies in the delayed feedback method, where the inherent unstable periodic orbit becomes stabilize (Pyragas, 1992;Nakajima, 1997). Therefore, if this inherent unstable periodic orbit is inconsistent with periodic orbit in HC condition, the stabilized orbit is different compared to one in HC condition. Therefore, in our simulation, the frontal cortical activity exhibited the period-2 state and higher amplitude than HC. However, from the viewpoint of actual treatment, the objective is that the pathological disturbed circadian rhythm is changed to a healthy circadian rhythm. Therefore, the target period must be set to healthy period. To realize this purpose, the other chaos control methods that can specify a more detailed target parameter, such as OGY method (Ott et al., 1990), H ∞ control (Jiang et al., 2005), and the method utilizing synchronization mechanism (Doho et al., 2020) must be evaluated and compared with the delayed feedback method. We plan to deal with these points in future works. To determine the possible clinical application of this proposed method, the vital reaction regarding the melatonin secretion against light stimulus and drugs must be modeled to determine the strength and duration of light and the amount of drugs needed. Therefore, future studies should implement these vital reactions when modeling for circadian neural systems. Developing feedback signals based on the observation variables of frontal cortical activity is important because actigraphic records and body temperature are easier to obtain, compared to EEG measurements. Furthermore, in addition to BD, abnormal circadian rhythm appears in various chronic-degenerative diseases, caused by the pathological network conditions (Fossion et al., 2017;Fossion et al., 2018). Therefore, focusing on bio-signals that reflect these pathological network conditions' efficient feedback signals to stabilize it might be designed.
In this study, we developed a method for stabilizing disturbed circadian rhythm in the circadian neural system, which are associated with BD. Although several limitations remain, this method highlights the potential usefulness of the chaos control method for treating BD.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.