# Criticality or Supersymmetry Breaking?

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## Abstract

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## 1. Introduction

## 2. Basic Phase Diagram

#### 2.1. 1D Chain of Stochastic Neuron-Like Elements

#### 2.2. Stochastic Neurodynamics on Neuromorphic Hardware

#### Emulation Results

## 3. Neurodynamic Meaning of the Three Phases

#### Fine Structure

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DC | dynamical complexity |

FVF | flow vector field |

ND | neurodynamics |

SDE | stochastic differential equation |

SOC | self-organized criticality |

STS | supersymmetric theory of stochastics |

TS | topological supersymmetry |

## Appendix A. Key Elements of Sts

#### Appendix A.1. The Model

#### Appendix A.2. From SDE to Topological Field Theory

#### Appendix A.3. Operator Representation

#### Appendix A.4. Eigensystem

#### Appendix A.5. Dynamical Partition Function, Supersymmetry Breaking, and Chaos

#### Appendix A.6. The Phase Diagram

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**Figure 1.**(

**a**) The “border of chaos” as predicted by the supersymmetric theory of stochastics (STS). The thick black curve separates systems with unbroken (symmetric) and spontaneously broken (ordered) topological supersymmetry (TS). The vertical straight line separates models with integrable and non-integrable flow vector field (FVF) (laws of deterministic evolution). Accordingly, there are three phases in the weak noise limit: the phase of ordinary chaos (C, red) where TS is broken by the non-integrability of the FVF; the phase of noise-induced chaotic behavior (N, green) where TS is broken by condensation of noise-induced (anti)instantons; and the phase of ergodic dynamics or thermal equilibrium (T, white) with unbroken TS. In the deterministic limit, noise-induced antiinstantons disappear, and the N-phase collapses into the T-C boundary. The termination of the N-C boundary by a cross at a certain noise intensity indicates that this boundary is transition-like only at weaker noises, and at stronger noises, it must be smeared out into a crossover. (

**b**–

**d**) Three qualitatively different types of stochastic evolution operator (SEO) spectra with unbroken (

**b**) and broken (

**c**,

**d**) TS. Black dots are the ground states, which are the fastest growing eigenstates. Grey dots at the origin are supersymmetric states that are ground states only when TS is unbroken. In (

**d**), there are two equally good candidates for the status of the ground state, and the ground state is chosen between them by convention, similarly as is done in quantum theory. $\mathsf{\Gamma}$ is the real part of the ground state’s eigenvalue providing a lower bound to the topological entropy of the model (see the text).

**Figure 2.**(

**a**) Local potential function in Equation (2) and the process of tunneling from $x=0$ to $x=2\pi \sim 0$, which is a combination of an antiinstanton, $\overline{I}$, leading from $x=0$ to the unstable critical point $x=\pi -2{x}_{0}$, and an instanton, I, leading further to $x=2\pi $. (

**b**) In certain coordinates, the noise-induced tunneling process is spike-like, similar to that in real neurons. (

**c**) In 1D chain of neuron-like elements, the noise-induced antiinstanton becomes a process of the creation of a pair of solitons, a kink (K) and antikink ($\overline{K}$). Free solitons travel with constant velocity in opposite directions. The process of the creation of the $K-\overline{K}$ pair with subsequent propagation of solitons in different directions is this model’s counterpart of neuroavalanches. (

**d**) Qualitative representation (a contour plot for $x\left(tr\right)$) of the dynamics in the N-phase under the condition of the condensation of $\overline{I}$’s and instantons (I) and the processes of creation and annihilation of the K-$\overline{K}$ pairs.

**Figure 3.**Maximal stochastic Lyapunov exponent of the 1D chain of neuron-like elements. Crosses and open circles denote positive and negative values, respectively. The noise level range corresponds to the weak-noise regime in Figure 1. Positive Lyapunov exponents and their vanishing width in the deterministic limit confirm that the N-phase can indeed be identified as noise-induced chaos.

**Figure 4.**(Top row) Filled contour plots with the horizontal and vertical axes being respectively the time, t, and the spatial coordinate, r, of simulated $x\left(rt\right)$ at a low noise level ($\mathsf{\Theta}=0.05$) and three values of $\alpha =(0.95,0.99,1.05)$ from the three dynamical phases as seen from Figure 3. Dynamics in the N-phase (middle) is clearly dominated by anti-instantonic processes (cf., Figure 2d). The middle and bottom rows are respectively $f\left(t\right)=sinx\left({r}_{0}t\right)$ at some position ${r}_{0}$ as a function of time and the corresponding power spectrum. The data reveal the qualitative difference between dynamics in the three major phases and have the same features as the data from the emulated dynamics using neuromorphic hardware in Figure 5b.

**Figure 5.**(

**a**)The phase diagram of emulated neurodynamics (ND) using the Spikey neuromorphic chip on the plane of the noise intensity and the firing threshold. The insets present the power spectra of the membrane potential of a neuron and are centered at the points in the phase diagram corresponding to the parameters used for the emulation. The scale of the insets is the same everywhere. The results show that in the deterministic limit, the N-phase collapses onto a sharp transition between the T-phase and C-phase (vertical dashed line at around −60.0 mV), as predicted by the STS picture of the N-phase dynamics. (

**b**) Three typical power-spectra (top) with their corresponding membrane potential recordings (bottom). The thermal equilibrium phase (“coma”-like), the noise-induced chaotic phase (“conscious”-like), and the regular chaotic phase (“seizure”-like) are featured, respectively, by no membrane potential dynamics with a sharp decrease at low frequency on the power spectra (left column), an avalanche-like membrane potential with 1/f noise-like spectra (middle column), and a non-stop firing pertinent to the seizure-like collective neuronal behavior with 1/f noise-like spectra superimposed by equidistant peaks (right column) representing periodic dynamics. This power-spectrum is also compared to the one generated by an isolated neuron (lower red curve). As can be seen, the spectrum of the isolated neuron exhibits 1/f noise, which is thus a signature of the collective neuronal dynamical behavior in the network.

**Figure 6.**The integral intensity of dynamics defined as ${\int \left|f\left(\omega \right)\right|}^{2}d\omega $, where $f\left(\omega \right)$ is the Fourier component of the membrane potential of a neuron. Even though this characteristic cannot be viewed as the “chaotic” order parameter, it does experience a relatively sudden jump and a smeared plateau at, respectively, the T-N and the N-C transition lines of Figure 5.

**Figure 7.**The phase diagram of emulated ND using the Spikey chip at a fixed noise intensity of 10 arb. units in Figure 5 and for various values of the firing threshold and repolarization time. As in Figure 5, the insets show the power spectra at the corresponding values of the parameters. As expected, the N-phase, featured by the $1/f$-type spectra, is sandwiched between the T- and the C-phases. The results show that the position of the N-phase shifts to the right with the increase of the repolarization time, thus indicating that the effect of the increase of the repolarization time is similar to the increase of the firing threshold.

**Figure 8.**A hypothetical phase diagram of the ND with the rudimentary structure as displayed in Figure 7 and with an additional fine-structure that resolves the larger scale collective dynamical phenomena (see the discussion in the text). The boundaries between the qualitatively different subphases are presented as dashed curves. The “awakeness” phase (the thicker green filled rectangle-like area) must be a subphase of the N-phase. The two closed red arrows qualitatively represent the phase space trajectory during an anesthesiological cycle.

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## Share and Cite

**MDPI and ACS Style**

Ovchinnikov, I.V.; Li, W.; Sun, Y.; Hudson, A.E.; Meier, K.; Schwartz, R.N.; Wang, K.L.
Criticality or Supersymmetry Breaking? *Symmetry* **2020**, *12*, 805.
https://doi.org/10.3390/sym12050805

**AMA Style**

Ovchinnikov IV, Li W, Sun Y, Hudson AE, Meier K, Schwartz RN, Wang KL.
Criticality or Supersymmetry Breaking? *Symmetry*. 2020; 12(5):805.
https://doi.org/10.3390/sym12050805

**Chicago/Turabian Style**

Ovchinnikov, Igor V., Wenyuan Li, Yuquan Sun, Andrew E. Hudson, Karlheinz Meier, Robert N. Schwartz, and Kang L. Wang.
2020. "Criticality or Supersymmetry Breaking?" *Symmetry* 12, no. 5: 805.
https://doi.org/10.3390/sym12050805