# Natural Brain-Inspired Intelligence for Non-Gaussian and Nonlinear Environments with Finite Memory
## 1 Introduction  
The cognitive dynamic system (CDS) is inspired by the human brain neuroscience model that is built on the principles of cognition based on perception-action cycle (PAC), memory, attention, intelligence, and language [1; 2; 3]. The basic model of a CDS is provided in Figure 1. Also, Figure 1 shows three main subsystems of CDS: (i) perceptor for sensing the environment, (ii) the feedback channel for sending internal rewards, and (iii) the executive for performing actions on the environment. The CDS cycles through these subsystems which are known as the perception-action cycle. More details about CDS and PAC can be found in [2]. The CDS can be considered as enhanced artificial intelligence (AI). A CDS creates internal rewards and uses it to take some actions, while AI takes actions based on the rewards [2]. The CDS was proposed as an alternative to artificial intelligence in most AI applications [2; 3]. In [2, 3], the CDS was used for cognitive radar applications to provide optimal target detection and intelligent signal processing. In [2, 3, 4], the CDS was applied for cognitive radio applications, such as dynamic spectrum management in wireless communication networks [4]. The CDS was discussed for theory and applications of cognitive control [3, 5]. In [6], the general concept of CDS was introduced for risk control in physical systems. Specific applications of CDS for risk control were presented for vehicular radar transmit waveform selection [7], mitigating the cyber-attack in a smart grid [8], detecting the cyber-attack in smart grid [9], and to mitigate vehicle to vehicle (V2V) jamming [10]. Also, CDS was applied as the brain of complex network [11]. A prospective block diagram of a CDS for smart homes was presented in [12]. In [13], the application of CDS for cybersecurity applications was discussed. To date, the algorithms of CDS proposed in literature are for linear and Gaussian environments (LGE) [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Here, LGE means that the outputs of the environment linearly depend on the inputs and they have a Gaussian distribution. Therefore, the typical CDS in [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] uses the Kalman filter. Usually, the Kalman filter, can be applied to linear environments only. The simplified version of [PERSON] theory is used in [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] for a Gaussian environment for calculating the entropic state at time \(n\). A typical CDS can use the entropic state at time \(n\) as the internal reward. However, [PERSON]'s entropy equation cannot be simplified for non-Gaussian environment. Therefore, algorithms proposed in the literature for a typical CDS cannot be applied for non-Gaussian and nonlinear environments (NGNLE). Here, NGNLE means that the outputs of the environment are not linearly dependent on the input and do not have a Gaussian distribution. We should mention here that many AI applications [14, 15, 16, 17, 18, 19, 20] are for NGNLEs. For example, most data obtained or measured from health conditions, education, and social sciences are often not normally distributed [14]. Some examples of non-Gaussian distribution for health conditions, education, and social sciences are prostate cancer modeling [15, 16], psychometrics [17], and labor income [18], respectively. The same holds true for a long-haul fiber-optic link as a NGNLE [19, 20]. The computational complexity of CDS algorithms in [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] is high in a complex environment. Also, in an NGNLE, the complexity of CDS algorithms in [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] is higher than those for a LGE. In this paper, we develop a CDS for NGNLE with simple, straightforward, and faster PAC algorithms. In [20, 21, 22], the CDS was proposed for smart fiber optic communication systems as an example of smart systems using cognitive decision-making (CDM). Here, a CDS is used to control the quality of service in long-haul fiber-optic links.