Chaotic spiking oscillators

Chaotic Spiking Oscillators (CSOs) are continuous-time autonomous circuits including an integrate-and-fire (IAF) switch and have the following properties.

* 1\. The IAF switch is the key element to generate chaos. The IAF is suitable for hardware implementation and chaotic dynamics can be confirmed experimentally.
* 2\. In a class of piecewise linear CSOs, return map is described using exact piecewise solution and chaos generation is guaranteed mathematically.
* 3\. The CSO can be a building block of pulse-coupled neural networks (PCNNs) that has rich synchronous phenomena. The PCNN has some potential applications including image processing.

Figure 1: Circuits with integrate-and-fire switching

## Contents

* 1 Integrate-and-Fire Switching in Circuits
* 2 A Simple Circuit
* 3 Chaotic dynamics
* 3.1 1D return map
* 3.2 Further development
* 4 A transistor-based realization
* 5 References
* 6 External Links
* 7 See Also

## [edit] Integrate-and-Fire Switching in Circuits

Chaotic circuits do not imply just a realization method of existing mathematical models but important real physical systems to investigate interesting nonlinear phenomena (Matsumoto, Chua and Komuro 1985; Maggio G. M., Feo O. D., Kennedy M. P. 1999; Ekwakil A. S., Kennedy M. P. 2001 ). A variety of nonlinear elements are used in existing chaotic circuits and the IAF switch is used in the CSO. In Fig. 1, \\(N\\) is a linear sub-circuit and \\(S\\) is the IAF switch. If the capacitor voltage \\(v_1\\) reaches a threshold voltage \\(V_T\ ,\\) \\(v_1\\) is reset to the base voltage \\(E\ ,\\) instantaneously. If the sub-circuit \\(N\\) consists of resistors and dependent sources, it can be replaced with the Thevenin equivalent sub-circuit and the circuit can exhibit periodic waveforms as shown in Fig. 1 (b). This periodic behavior is basic for an integrate-and-fire neuron model which can be a building block of PCNN (Mirollo and Strogatz 1990; Hopfield and Herz 1995). Applications of PCNNs include image processing (Campbell, Wang and Jayaprakash 1999), associative memory (Izhikevich 1999) and Spike-based communication (M. Maggio, Rulkov and Reggiani 2001).

If the sub-circuit \\(N\\) includes one memory element (inductor or capacitor) as shown in Fig.1 (c), the circuit can become a CSO. In this case the IAF switch causes vibrate-and-fire dynamics that relates deeply to resonant-and-fire neuron models (Izhikevich 2001).

Figure 2: Simple example of chaotic spiking oscillator

## [edit] A Simple Circuit

If \\(N\\) includes 1 or more memory elements, \\(v_1\\) can vibrate below the threshold and the IAF switch can cause chaotic behavior. Fig. 2 shows a simple example of the CSO. In the figure \\(-R\\) is a linear negative resistor. If the capacitor voltage \\(v\\) is below the threshold \\(V_T\ ,\\) the switch \\(S\\) is opened and the circuit dynamics is described by

\\( C\frac{dv_1}{dt} = i, \ \ L\frac{di}{dt} = - v_1 + R i, \ \ \mbox{ for } v_1 < V_T. \ \ \ (1) \\)

\\(v_1\\) is assumed to vibrate divergently. As \\(v_1\\) reaches \\(V_T\ ,\\) the comparator COMP triggers the monostable multivibrator MM to output an impulse \\(v_o\ .\\) The impulse \\(v_o\\) closes the IAF switch \\(S\\) and \\(v_1\\) is reset to the base \\(E\\) instantaneously holding the continuity property of \\(i\ :\\)

\\( ( v_1(t^+), i(t^+) ) = ( E, i(t) ), \quad \mbox{ if } v_1(t) = V_T. \ \ \ (2) \\)

Repeating IAF switching the circuit generate chaos. Using discrete elements such as op-amps, this circuit can be fabricated easily and chaotic behavior can be confirmed experimentally as shown in Fig. 3.

Figure 3: Chaotic attractor for C=22nF, L=300mH, \\(V_T\\)=1v, -R=-1.2k\\(\Omega\\) and E=-0.2V. i is in mA.

Note that there exist various CSOs with two memory elements: applying the IAF switch to some oscillator (e.g., Wien bridge oscillator ), we obtain a CSO.

## [edit] Chaotic dynamics

We assume that Equation (1) has unstable complex characteristic root \\(\delta \omega \pm j \omega\ :\\)

\\( \omega^2=\frac{1}{LC}-\left(\frac{R}{2L}\right)^2>0, \;\; \delta = \frac{R}{2 \omega L}>0\. \ \ \ (3) \\)

In this case \\(v_1\\) can vibrate divergently below the threshold \\(V_T\ .\\) The divergent vibration and the IAF switching correspond to stretching and folding mechanisms, respectively, which are fundamental for chaos generation. Using the following dimensionless variables and parameters\\[ \tau=\omega t,\ q=\frac{E}{V_T}, \ x=\frac{v_1}{V_T} \ y=-\frac{\delta}{V_T}v+\frac{1}{\omega C V_T}i, \ \ \ (4) \\]

Equations (1) and (2) are transformed into the following.

\\( \begin{array}{c} \left( \begin{array}{c} \dot{x}\\\ \dot{y} \end{array} \right) = \left( \begin{array}{cc} \delta & 1\\\ -1 & \delta \end{array} \right) \left( \begin{array}{c} x\\\ y \end{array} \right) \mbox{ for } x < 1 \ \ \ (5)\\\ \\\ \mbox{SW: } (x(\tau^+), y(\tau^+))=(q, y(\tau)-\delta(1-q)) \mbox{ if } x_1(\tau)=1 \end{array} \\)

where \\(\dot{x} \equiv dx/d\tau\\) and \\(p=\delta\ .\\) This dimensionless equation is characterized by two parameters \\(\delta\\) and \\(q\\) which can be controlled by \\(-R\\) and \\(E\ ,\\) respectively. This equation can reproduce chaotic attractor as shown in Fig. 4.

Figure 4: Chaotic attractor of dimensionless equation for \\(\delta=0.05\\) and \\(q=-0.2\ .\\)

Note that the case \\(p \ne \delta\\) governs wider class of CSOs (Mitsubori and Saito 2000).

### [edit] 1D return map

The circuit dynamics can be analyzed using 1-D return map. Some objects of the map are shown in Fig. 5: the domain of the map \\(L_d=\\{(x,y)\;|\;x=0,\; y\ge0\\}\ ,\\) the threshold line \\(L_T=\\{(x,y)|\;x=1 \\}\ ,\\) and the base line \\(L_q=\\{(x,y)|\;x=q \\}\ .\\) Let a point on these objects be represented by their \\(y\\)-coordinate. Also let \\(D\\) be a point on \\(L_d\\) such that a trajectory started from \\(D\\) touches \\(L_T\\) within half period.

For the case \\(q>0\\) let a trajectory start from a point \\(y_0\\) on \\(L_d\\) at \\(\tau=0\ .\\) If \\(0<y_0<D\ ,\\) the trajectory return to \\(L_d\\) at \\(\tau=2\pi\\) without reaching \\(L_T\ .\\) If \\(D \le y_0\ ,\\) the trajectory hits the threshold \\(L_T\\) and is reset to the base \\(L_q\ .\\) Then the trajectory re-starts from \\(L_q\\) and returns to \\(L_d\ .\\) Since any trajectory started from \\(y_0\\) on \\(L\\) must return to \\(L_d\ ,\\) a 1-D return map can be defined\\[ y_1 = f(y_0), \ f: L_d \rightarrow L_d \ \ \ (6) \\]

where \\(y_1\\) is the return point on \\(L_d\ .\\) That is, the circuit dynamics can be integrated into the iteration \\(y_{n+1} = f(y_n)\\) as shown in Fig. 5.

Figure 5: Return Map. The orbit enters eventially into the interval \\([a, b]\\) and behaves chaotically.

Using exact piecewise solution of Equation (3), the return map can be described and chaos generation can be guaranteed theoretically (Nakano and Saito 2002) in the sense of ergodic and positive Lyapunov exponent (Lasota and Mackey). The theoretical results can be extended for the case \\(q<0\\) and other CSO examples (Mitsubori and Saito 2000).

### [edit] Further development

The CSO can be developed into a variety of interesting systems:

* 1\. If periodic spike-train input is applied, the CSO exhibits rich synchronous/asynchronous phenomena (Miyach, Nakano and Saito 2003)

* 2\. If the sub-circuit has two or more memory elements, the IAF switch can cause hyperchaos and rich bifurcation phenomena (Takahashi, Nakano and Saito 2005)

* 3\. The PCNN of CSO can exhibits rich periodic/chaotic synchronous phenomena.

Prospective engineering applications include flexible image processing and spike-based communications (Nakano and Saito 2004).

## [edit] A transistor-based realization

Figure 6: Experimental recording of a spiking signal for \\(L_1=15\mu\\)H, \\(L_2=68\mu\\)H, \\(L_3=150\mu\\)H, \\(C=470\\)pF, \\(R\approx210\Omega\\) and \\(V=5\\)V.

Among the several possible electronic realizations of chaotic spiking oscillators, one can find neurally-inspired circuits wherein electrical variables have clear physiological counterparts (Indiveri et al., 2011) as well as more compact circuits where this is not the case, and which more closely exploit the physical characteristics of semiconductor devices, such as bipolar junction transistors, to realize spiking dynamics. A particularly simple example of the latter was recently discovered in a study wherein a large number of atypical transistor circuits were derived from a random search (Minati et al., 2017). As shown in Fig. 6, this circuit comprises a network of three inductors and one capacitor, two discrete transistors, and one resistor connected in series to a DC voltage source which acts as the main control parameter. It generates trains of spikes having nearly-quantized amplitude and irregular spacing.

To some extent, the dynamics of this physical circuit can be captured by representing each transistor with exponential current sources and associated junction capacitances, yielding

\\( \dot{v}_1=\left((V-v_1)/R-i_1-i_a\left(e^{v_1/kV_T}-1\right)\right)/C_1\\\ \dot{v}_2=\left(i_1-i_2-i_b/\beta\right)/C_2\\\ \dot{v}_3=\left(i_2-i_3+i_b/\beta+i_b\left(e^{(v_4-v_3)/kV_T}-1\right)\right)/C_3\\\ \dot{v}_4=\left(i_3-i_b\left(e^{(v_4-v_3)/kV_T}-1\right)-i_a/\beta\right)/C_4\\\ \dot{i}_1=(v_1-v_2)/L_1\\\ \dot{i}_2=(v_2-v_3)/L_2\\\ \dot{i}_3=(v_3-v_4)/L_3 \\)

where \\(i_a=I_se^{v_4/V_T}\\) and \\(i_b=I_se^{(v_2-v_3)/V_T}\\), \\(L_1=15\mu\textrm{H}\\), \\(L_2=68\mu\textrm{H}\\), \\(L_3=150\mu\textrm{H}\\), \\(R=40\Omega\\), \\(V=5\textrm{V}\\), \\(V_T=0.025\textrm{V}\\), \\(C_1=C_2=C_4=0.2\textrm{pF}\\), \\(C_3=470\textrm{pF}\\), \\(\beta=75\\), \\(I_s=0.5\textrm{fA}\\) and \\(k=20\\).

While inherently less suitable for neat mathematical tractation, physical realizations of this kind find relevance in the construction of experimental networks, for which implementation using operational amplifiers, switching elements and multipliers generally leads to considerably larger circuits.

## [edit] References

* Campbell, S. R.; Wang, D. L. and Jayaprakash, C. (1999). Synchrony and Desynchrony in Integrate-and-Fire Oscillators. Neural Computation 11 (7): 1595-1619. doi:10.1162/089976699300016160.

* Elwakil, A. and Kennedy, M. (2001). Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 48 (3): 289-307. doi:10.1109/81.915386.

* Hopfield, J. J. and Herz, A. V. (1995). Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire neurons.. Proceedings of the National Academy of Sciences 92 (15): 6655-6662. doi:10.1073/pnas.92.15.6655.

* Indiveri, G.; Linares-Barranco, B.; Hamilton, T.J.; van Schaik, A.; Etienne-Cummings, R.; Delbruck, T.; Liu, S.C.; Dudek, P.;  et al. (2011). Neuromorphic silicon neuron circuits. Frontiers in Neuroscience 5: 73. doi:10.3389/fnins.2011.00073.

* Izhikevich, E. M. (1999). Weakly pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory. IEEE Transactions on Neural Networks 10 (3): 508-526. doi:10.1109/72.761708.

* Izhikevich, E. M. (2001). Resonate-and-fire neurons. Neural Networks 14 (6-7): 883-894. doi:10.1016/S0893-6080(01)00078-8.

* Lasota A., Mackey M. C. (1994) Chaos, Fractals, and Noise - Second Edition, Springer-Verlag.

* Matsumoto, T.; Chua, L. and Komuro, M. (1985). The double scroll. IEEE Transactions on Circuits and Systems 32 (8): 797-818. doi:10.1109/TCS.1985.1085791.

* Maggio, G.; De Feo, O. and Kennedy, M. (1999). Nonlinear analysis of the Colpitts oscillator and applications to design. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 46 (9): 1118-1130. doi:10.1109/81.788813.

* Maggio, G.; Rulkov, N. and Reggiani, L. (2001). Pseudo-chaotic time hopping for UWB impulse radio. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 48 (12): 1424-1435. doi:10.1109/TCSI.2001.972849.

* Minati, L.; Frasca, M.; Oświȩcimka, P.; Faes, L. and Drożdż, S. (2017). Atypical transistor-based chaotic oscillators: Design, realization, and diversity. Chaos 27: 073113. doi:10.1063/1.4994815.

* Mirollo, R. E. and Strogatz, S. H. (1990). Synchronization of Pulse-Coupled Biological Oscillators. SIAM Journal on Applied Mathematics 50 (6): 1645-1662. doi:10.1137/0150098.

* Mirollo, R. E. and Strogatz, S. H. (1990). Synchronization of Pulse-Coupled Biological Oscillators. SIAM Journal on Applied Mathematics 50 (6): 1645-1662. doi:10.1137/0150098.

* Saito, T. and Mitsubori, K. (2000). Mutually pulse-coupled chaotic circuits by using dependent switched capacitors. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47 (10): 1469-1478. doi:10.1109/81.886977.

* Miyachi, K.; Nakano, H. and Saito, T. (2003). Response of a simple dependent switched capacitor circuit to a pulse-train input. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 50 (9): 1180-1187. doi:10.1109/TCSI.2003.816335.

* Nakano, H. and Saito, T. (2002). Basic dynamics from a pulse-coupled network of autonomous integrate-and-fire chaotic circuits. IEEE Transactions on Neural Networks 13 (1): 92-100. doi:10.1109/72.977276.

* Nakano, H. and Saito, T. (2004). Grouping Synchronization in a Pulse-Coupled Network of Chaotic Spiking Oscillators. IEEE Transactions on Neural Networks 15 (5): 1018-1026. doi:10.1109/TNN.2004.832807.

* Takahashi, Y.; Nakano, H. and Saito, T. (2005). Hyperchaotic spiking oscillators with periodic pulse-train input. IEEE Transactions on Circuits and Systems II: Express Briefs 52 (6): 344-348. doi:10.1109/TCSII.2005.849002.

Internal references

* Milnor, J. (2006). Attractor. Scholarpedia 1 (11): 1815. doi:10.4249/scholarpedia.1815.
* Guckenheimer, J. (2007). Bifurcation. Scholarpedia 2 (6): 1517. doi:10.4249/scholarpedia.1517.
* Meiss, J. (2007). Dynamical systems. Scholarpedia 2 (2): 1629. doi:10.4249/scholarpedia.1629.
* Letellier, C. and Rossler, O. (2007). Hyperchaos. Scholarpedia 2 (8): 1936. doi:10.4249/scholarpedia.1936.
* Wang, D. (2006). LEGION: locally excitatory globally inhibitory oscillator networks. Scholarpedia 1 (9): 1620. doi:10.4249/scholarpedia.1620.
* Moehlis, J.; Josic, K. and Shea-Brown, E. (2006). Periodic orbit. Scholarpedia 1 (7): 1358. doi:10.4249/scholarpedia.1358.
* Letellier, C. and Rossler, O. (2006). Rossler attractor. Scholarpedia 1 (10): 1721. doi:10.4249/scholarpedia.1721.
* Destexhe, A. (2007). Spike-and-wave oscillations. Scholarpedia 2 (2): 1402. doi:10.4249/scholarpedia.1402.
* Holmes, P. and Shea-Brown, E. (2006). Stability. Scholarpedia 1 (10): 1838. doi:10.4249/scholarpedia.1838.
* Pikovsky, A. and Rosenblum, M. (2007). Synchronization. Scholarpedia 2 (12): 1459. doi:10.4249/scholarpedia.1459.
* Hoppensteadt, F. (2006). Voltage-controlled oscillations in neurons. Scholarpedia 1 (11): 1599. doi:10.4249/scholarpedia.1599.

## [edit] External Links

* Toshi Saito

## [edit] See Also

Voltage-Controlled Oscillations in Neurons, LEGION: Locally Excitatory Globally Inhibitory Oscillator Networks, Bifurcation, Rossler Attractor, Spike-and-Wave Oscillations