# Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed

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## Abstract

**:**

## 1. Introduction

## 2. Birhythmic Dynamics in 3-Dimensions

## 3. Electronic Physical Realization

#### 3.1. Nonlinear Activation Function Circuit

#### 3.2. Schematics of Electronic Birhythmic RNN

`-tanh`”, where terminals IOP1 and IOP2 are the input and output voltage to each hyperbolic tangent unit, respectively. Furthermore, two analog multiplication chips can be used to introduce the phase flow speed dependence on $z\left(t\right)$ in Equations (8) and (9). Thus, the entire system can be constructed entirely from simple analog components.

`stim_out`represents the output of the nonlinear stimulator circuit and will be discussed in Section 4. The schematic of the electronic realization of Equations (8) and (9) is shown in Figure 5, where the state variables, $x\left(t\right)$ and $y\left(t\right)$, are represented by the voltages across capacitors C1 and C2, respectively. The “z” input to analog multipliers, M1 and M2, is taken from the integrator output labeled “z” in Figure 4.

#### 3.3. Circuit Construction and Experimental Results

## 4. Nonlinear Stimulator Circuit Design and Discussion

`stim_out`in Figure 4, where the output to the stimulator circuit will be fed in and summed with the current value of $\dot{z}$ in the system. By extending the system properly in this way, we can prevent straightforward stimulation patterns (i.e., constant, random, periodic, etc.) from inducing attractor transitions.

`stim_out`, to either increase or decrease. Furthermore, the amount by which the signal can change should exist on a continuum, rather than outputting a voltage from a finite set of values. The final requirement we will enforce for such a circuit will be that a stimulation pulse delivered at a random time should have equal probability of increasing or decreasing the output signal. This last requirement ensures that if one stimulates randomly or continuously, the expected value of

`stim_out`averaged across all time will be zero as time approaches infinity.

`Stimulus`.

`stim_out`.

`stim_out`node in Figure 8 and apply constant stimulation. All of the trials successfully traveled over the separating unstable manifold at $z=0$ as depicted by the red and blue curves in Figure 9, representing the trials initialized on the slow and fast limit cycles, respectively. An example of the behavior for all system coordinates and the stimulator circuit output is shown in Figure 12 for both trajectories initialized on the slow and fast limit cycles.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Circuit Component Values

#### Appendix A.1. Component Values for Figure 4

#### Appendix A.2. Component Values for Figure 5

#### Appendix A.3. Component Values for Figure 8

## Appendix B. Hyperbolic Tangent Implementation

**Figure A1.**Input/output relation of three physically realized hyperbolic tangent circuits, interpolated through 21 points, compared with the analytic hyperbolic tangent function.

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**Figure 1.**Planar Limit Cycle with 2D continuous-time gated recurrent unit (ct-GRU) depicted in phase space: The red dot indicates an unstable fixed point at the origin unstable, while orange and pink lines represent the x and y nullclines, respectively. Purple lines indicate various trajectories of the hidden state. Direction of the flow is determined by the black arrows, where the colormap underlying the figure depicts the magnitude of the velocity of the flow in log scale.

**Figure 2.**Birhythmicy in 3-dimensions: (

**A**): light blue manifold on the $x-y$ plane separates the basins of attraction of the upper and lower limit cycles. Trajectories are colored either dark blue or purple, depending on which basin of attraction they are initialized in. Red dots indicate fixed points, and black arrows depict the direction of flow. (

**B**,

**C**): x, y, and z components of trajectories initialized in the basins of attractions for the top and bottom limit cycles, respectively. Solid colored lines indicate $x\left(t\right)$, dashed lines indicate $y\left(t\right)$, and black lines indicate $z\left(t\right)$.

**Figure 3.**Electronic circuit realization of the hyperbolic tangent function, as implemented in Reference [30]. ${V}_{in}$ and ${V}_{o}$ represent the input and output signals, respectively.

**Figure 4.**Circuit schematic of $\dot{z}$ for the birhythmic system. The system block labeled

`-tanh`represents the circuit depicted in Figure 3, where

`I01`and

`I02`correspond to ${V}_{in}$ and ${V}_{o}$, respectively. The terminal labeled

`stim_out`represents the output to the stimulator circuit, as discussed in Section 4.

**Figure 5.**Circuit schematic of $\dot{x}$ and $\dot{y}$ for the birhythmic system. The system blocks labeled

`-tanh1`and

`-tanh2`represent the circuit depicted in Figure 3, where

`I01`and

`I02`correspond to ${V}_{in}$ and ${V}_{o}$, respectively, for both blocks. The two multiplier chips, M1 and M2, are assumed to operate with unity gain.

**Figure 6.**Physical birhythmic circuit constructed on a breadboard. Blue boxes represent hyperbolic tangent units. The magenta box indicates the subsection of the circuit generating $\dot{z}$, and the green box indicates the analog multipliers.

**Figure 7.**Experimental recordings of the birhythmic circuit: (

**A**,

**C**): x (yellow), y (blue), and z (pink) with respect to time of trajectories within the basin of attraction of the fast and slow limit cycles, respectively. (

**B**,

**D**): Projection of the corresponding trajectories in (

**A**,

**C**) onto the x-y plane, respectively.

**Figure 8.**Schematic for nonlinear stimulator circuit, with input labeled as Stimulus. The output, labeled

`stim_out`, is fed into the terminal with the same name presented in the circuit diagram shown in Figure 4. The two multiplier chips, M1 and M2, are assumed to operate with unity gain, and the x and y terminals are fed into the equivalently named terminals depicted in Figure 5.

**Figure 9.**A time window of $z\left(t\right)$ for all experimental trials: Red and blue trajectories demonstrate resultant behavior from stimulation patterns designed to transition states from the slow and fast limit cycles, respectively. Turquoise trajectories depict trials of constant stimulation, and orange trajectories show trials of random stimulation.

**Figure 10.**x (yellow), y (blue), z (pink), and stim_out (green) with respect to time of trajectories within the basin of attraction of the slow (

**A**) and fast (

**B**) limit cycles, under constant stimulation. Note that this stimulation regime does not successfully transition between the two attracting states in either direction.

**Figure 11.**x (yellow), y (blue), z (pink), and stim_out (green) with respect to time of trajectories within the basin of attraction of the slow (

**A**) and fast (

**B**) limit cycles, with random stimulation. Note that this stimulation regime does not successfully transition between the two attracting states in either direction.

**Figure 12.**Examples of stimulation patterns capable of inducing transition between states: x (yellow), y (blue), z (pink), and stim_out (green) with respect to time of trajectories initialized within the basin of attraction of the slow (

**A**) and fast (

**B**) limit cycles. As $z\left(t\right)$ changes with stimulation so does the frequency of oscillation. As such, the time window to stimulate shifts continuously.

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Jordan, I.D.; Park, I.M. Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed. *Entropy* **2020**, *22*, 537.
https://doi.org/10.3390/e22050537

**AMA Style**

Jordan ID, Park IM. Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed. *Entropy*. 2020; 22(5):537.
https://doi.org/10.3390/e22050537

**Chicago/Turabian Style**

Jordan, Ian D., and Il Memming Park. 2020. "Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed" *Entropy* 22, no. 5: 537.
https://doi.org/10.3390/e22050537