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A simple integrate-and-fire system and various super-stable periodic orbits

This paper studies super-stable periodic orbits and related phenomena in a simple switched dynamical system. The system repeats integrate-and-fire behavior between a constant threshold and a periodic base signal. The base signal consists of two components: the fundamental and higher frequency triangular signals. First, we demonstrate “chaos + chaos = order” such that the system exhibits chaos if the base signal is either of the two components whereas the system exhibits a super-stable periodic orbit if the base signal consists of both components. This phenomenon is confirmed experimentally and is explained theoretically. Second, we extract two key parameters and show that the system can exhibit a variety of super-stable periodic orbits. Applying a mapping procedure, existence regions of basic super-stable periodic orbits are calculated precisely.


Introduction
The bifurcating neuron (BN) is a simple switched dynamical system inspired by spiking neuron models [1][2][3][4][5][6]. Repeating integrate-and-fire behavior between a constant threshold and a periodic base signal, the BN can exhibit various chaotic/periodic phenomena. On the other hand, integrate-andfire neuron models have been applied to many engineering systems including associative memories [2], image processors [7], and communication systems [4,8]. In the spiking neuron models (incl. BNs), analysis of the chaotic/periodic phenomena is important not only as basic study of nonlinear dynamics but also for the engineering applications.
This paper studies dynamics of a class of BNs whose periodic base signal is given by addition of two triangular signals. The first signal is period T , the second signal is period T/3, and they are synchronized. In order to analyze the dynamics precisely, we derive one-dimensional return map (Rmap). The Rmap is piecewise linear and can be described exactly. For convenience of the analysis, we extract two key parameters from many original parameters. We then present two main results.
First, we demonstrate an interesting phenomenon as the following. If the base signal is either the first or the second signal only, the BN exhibits chaos [9]. If the base signal is given by addition of the first and the second signals, the BN exhibits a super-stable periodic orbit (SSPO) such that a set of initial states fall rapidly into the periodic orbit. Such a phenomenon is referred to as "chaos + chaos = order (CCO)". Presenting a simple test circuit, the CCO is confirmed experimentally. Using the Rmap, the CCO is explained theoretically. Such phenomena have been observed in coupled BNs [5,6]. However, the coupled BNs and their Rmap are complicated and analysis of the CCO is not sufficient.
Second, we analyze a variety of SSPOs in a space of the two key parameters. The analysis of the SSPOs is integrated into the analysis of the Rmap including four segments of slope zero. Using the Rmap, existence regions of basic SSPOs can be calculated precisely. The existence regions distribute in complicated shapes in the parameter space. Such complicated existence regions have not been presented in previous works [4][5][6]. Preliminary results along these lines can be found in our conference paper [10]. When v reaches a constant threshold V T , the comparator outputs a spike. The spike closes a switch SW and v is reset to the base signal B(t) with period T . Repeating this integrate-and-fire behavior, the circuit outputs a spike-train Y (t). For simplicity, we ignore inner resistors, i.e., r 1 → ∞ and r 2 → 0. In this case, the circuit dynamics is described by

Circuit model and typical phenomena
where ±E denotes output voltages of the comparator. The base signal B(t) consists of three components: the first component E 1 (t) with period T , the second component E 3 (t) with period T/3, and the dc component E 0 ≥ 0. The first and second components are triangular signals and are assumed to be synchronized.
In this paper, we consider three cases: The base signal in Case 3 is given by addition of base signals in Cases 1 and 2. We have performed laboratory experiments and Fig. 3 shows typical phenomena. In the experiments, the current source with its inner resistor r 1 is implemented by an equivalent circuit of a voltage source E with its inner resistor r 1 . The parameter I in Eq. (1) is approximated by I ≈ E/(r 1 − r 1 e −1 ). In the figure, we can see that the BN exhibits chaotic waveforms in Cases 1 and 2; and that the BN exhibits periodic waveform in Case 3. That is, adding two base signals each of which causes chaotic waveforms, the BN exhibits periodic waveforms. We refer to such phenomena as "chaos + chaos = order" (CCO, [10]). In the following sections, we consider the CCO and various periodic phenomena 1 .

Dimensionless equations and return map
In order to analyze the dynamics of BN, we derive a dimensionless equation and return map (Rmap). First, we introduce the following dimensionless variables and parameters.
Here, we note that Eqs. (5) to (7) are characterized by 6 parameters a 1 > 0, b 1 > 0, a 3 > 0, b 3 > 0, b 0 ≥ 0, and s > 0. Since it is extremely hard to consider effects of all the 6 parameters, we reduce the number of parameters. That is, we focus on the case a 3 ≡ a, b 3 = a, a 1 = a − 1, b 1 = a + 1, s = 1 (8) In this case, Eqs. (5) to (7) depend on the two key parameters In order to analyze the dynamics of BN, we derive the spike-position map and Rmap. Let τ n denote the n-th spike position. Since τ n+1 is determined by τ n , we can define the spike-position map F from positive reals to itself.
Let θ n = τ n mod 1 denote the n-th spike phase. We can define the Rmap f of spike-phases from unit interval [0, 1) to itself.

Super-stable periodic orbits
We analyze the CCO and various super-stable periodic orbits (SSPOs). First, we introduce several concepts. Let Df 1 (θ), Df 2 (θ), and Df 3 (θ) denote slope of Rmaps f 1 , f 2 , and f 3 , respectively. If an Rmap f is expanding, |Df | > 1, then the Rmap exhibits chaos [5]. A point p ∈ I is said to be a periodic point with period k if an orbit started from p returns to itself after k iterations: f k (p) = p and f l (p) = p for 0 < l < k where f k is the k-fold composition of f . A sequence of periodic points {f (p), · · · , f k (p)} is said to be a periodic orbit. A periodic point p is said to be super-stable if Df (p) = 0. A sequence of super-stable periodic points is said to be a super-stable periodic orbit (SSPO) 2 .
Referring to Eqs. (12) and (13), the slope Df 1 is a or −a in Case 1 and the slope Df 2 is 1 + a or 1 − a in Case 2. Hence, if 2 < a < ∞ then the Rmap is expanding and exhibits chaos [5] in Cases 1  Here, we consider SSPOs in Case 3 for two key parameters a and b 0 . As a and b 0 vary, the Rmap can exhibit a variety of SSPOs. Figure 7 shows several SSPOs as b 0 varies. Some of SSPOs can co-exist and Rmap exhibits either SSPO depending on initial condition. Figure 8 shows several examples of co-existing SSPOs. Since all the SSPOs must pass through either of the four intervals S 1 to S 4 on which the slope is 0, the four initial points f (S 1 ) to f (S 4 ) are sufficient to consider the SSPOs. That is, for some values of parameters a and b 0 , the Rmap exhibits SSPO with period k if f k (θ 0 ) = θ 0 for That is, we can calculate existence regions of SSPOs precisely as the following.
Step 1: Give a value of key parameters (a, b 0 ).
Step 2: Set an initial point on S i , i = 1 ∼ 4.
Step 3: If an orbit returns to S i after k iterations, then the Rmap exhibits a SSPO with period k. Figures 9 to 12 show existence regions of SSPOs with period 1 to 4. In the figures, four kinds of SSPOs passing through S 1 to S 4 are classified by four colors: red, blue, pink, and orange, respectively. As period increases, shape of existence regions becomes complex. Even in the case of period 4, the

Conclusions
Basic dynamics of a class of BNs has been considered and two main results are presented in this paper. First, presenting a simple test circuit, the CCO is confirmed experimentally. Using the Rmap, the generation of CCO is explained theoretically. Second, a variety of SSPOs are demonstrated from the BNs in case 3. Extracting two key parameters and calculating slopes of Rmap, existence regions of SSPOs with period 1 to 4 are calculated precisely. The existence regions are complicated shape in the parameter space.
Future problems include analysis of bifurcation phenomena for the SSPOs and engineering applications of the SSPOs such as encoders in communication systems [8]. It should be noted that SSPOs with longer period are hard to be observed in actual circuits because zero-slope (Df 3 = 0) is impossible in a noisy hardware. Careful laboratory experiments are required.