# Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Network Model

#### 2.2. Statistical Properties of the Network Model

#### 2.2.1. Probability Distribution of the Bifurcation Points

#### 2.2.2. Mean Multistability Diagram

#### 2.2.3. Occurrence Probability of the Stationary States for a Given Combination of Stimuli

#### 2.2.4. Occurrence Probability of the Stationary States Regardless of the Stimuli

#### 2.3. The Special Case of Multi-Population Networks Composed of Statistically Homogeneous Populations

#### 2.4. Large-Network Limit

#### 2.5. Numerical Simulations

## 3. Results

## 4. Discussion

#### 4.1. Progress with Respect to Previous Work on Bifurcation Analysis

#### 4.2. Limitations of Our Approach

#### 4.3. Directions for Future Work on the Statistical Mechanics of Networks with Quenched Disorder

#### 4.4. Possible Implications of This Work for Circuit Neuroscience

## Supplementary Materials

**File “Statistics.py”:**this Python script calculates semi-analytically the cumulative distribution function of the bifurcation points, the mean multistability diagram, and the occurrence probability of the stationary states (for fixed stimuli and regardless of them) for a network with quenched disorder, according to the methods described in Section 2.2. Moreover, the script compares these results with the numerical evaluation of the corresponding quantities, obtained through the Monte Carlo approach described in Section 2.5.

**File “Permanent.py”:**this Python script calculates the permanent of block matrices with homogeneous blocks by means of Equation (20), and compares the computational time of this formula with that of the Balasubramanian-Bax-Franklin-Glynn algorithm.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Probability distribution of the synaptic weights. This figure shows the probability distribution of the synaptic weights ${J}_{i,j}$ (see Equation (2)), in the specific case of the small-size network model described in Section 3. (

**A**) Graph of the connection probability matrix $\mathcal{P}$ reported in Table 1. An arrow from the vertex j to the vertex i represents a connection probability ${\mathcal{P}}_{i,j}>0$. (

**B**) Wigner semicircle distribution of some variables ${W}_{i,j}$, according to Equation (25) and the values of the parameters $\mathfrak{C}$ and $\mathfrak{R}$ reported in Table 1.

**Figure 2.**Examples of cumulative distribution functions of the bifurcation points. This figure reports the cumulative distribution functions of the bifurcation points ${\Lambda}_{E}$ and ${\Lambda}_{I}$ of the firing-rate state 1110 (panels (

**A**) and (

**B**), respectively), in the case of the small-size network described in Section 3. The red curves represent the semi-analytical functions, calculated through Equation (14), while the blue dots represent the numerical functions, computed over 5000 network realizations as described in Section 2.5. Similar results can be derived for all the other bifurcation points of the network, if desired.

**Figure 3.**Mean multistability diagram. This figure reports the mean multistability diagram of the small-size network described in Section 3, see Equation (25) and Table 1. The diagram shows how the degree of multistability of the network, namely the number of stationary solutions, depends on average on the external currents ${I}_{E}$ and ${I}_{I}$. Each color represents a different degree of multistability $\mathcal{M}$ (e.g., blue = tristability). (

**A**–

**C**) Numerical multistability diagrams, obtained through Monte Carlo simulations as described in Section 2.5, for an increasing number of network realizations (5, 50 and 5000). (

**D**) Semi-analytical multistability diagram, obtained through the techniques described in Section 2.2.2. Please note that by increasing the number of network realizations, the numerical multistability diagram converges to the semi-analytical one.

**Figure 4.**Occurrence probability of the firing-rate states. This figure reports the occurrence probability of the ${2}^{N}=16$ firing-rate states of the network described in Section 3 (see Equation (25) and Table 1), from the state 0000 to the state 1111. (

**A**) Occurrence probability of the firing-rate states for fixed stimuli, obtained for ${\widehat{I}}_{E}=0$ and ${\widehat{I}}_{I}=4$. The red bars represent the occurrence probability calculated semi-analytically through the method described in Section 2.2.3. The blue bars represent the same probability, evaluated numerically by a Monte Carlo simulation, as explained in Section 2.5; (

**B**) Occurrence probability of the firing-rate states, regardless of the stimuli (red bars calculated according to the approach of Section 2.2.4, blue bars computed again through a Monte Carlo simulation). In both panels, we computed the blue bars over 5000 network realizations. Please note that the occurrence probabilities are not normalized over the set of ${2}^{N}$ firing-rate states, see text.

**Figure 5.**Speed test for the analytical formula of the permanent. This figure shows the mean computational times $\left(\right)$ and $\left(\right)$ (see text) required for calculating the permanent of homogeneous block matrices $\mathcal{B}$ by means of an Intel

^{®}Core™ i5-5300U CPU clocked at 2.30 GHz with 16 GB RAM. We chose the entries of the matrices $\mathcal{B}$ to be independent random numbers $0\le {B}_{\lambda ,\mu}<0.3$ with two decimal digits, generated from a uniform probability distribution (see the Python script “Permanent.py”), while the remaining parameters of $\mathcal{B}$ are shown in Table 2. The average times $\left(\right)$ and $\left(\right)$ are calculated over 100 repetitions of the matrices. (

**A**) Mean computational times $\left(\right)$ (blue line) and $\left(\right)$ (red line) in seconds, as a function of $\mathcal{N}$. The green line represents the mean speed gain $\left(\right)$ of Equation (20) over the BBFG algorithm. (

**B**) Mean computational times and speed gain as a function of $\mathfrak{X}$.

**Figure 6.**Large-size limit of a statistically homogeneous two-population network. This figure shows the probability distribution of the bifurcation points in a random network composed of two statistically homogeneous populations (one excitatory and one inhibitory) with Laplace-distributed weights (see Equation (26)), in the large-size limit. The parameters of the network are reported in Table 3. In particular, note that in this figure we compute the probability density of the bifurcation points for every firing-rate state $\mathit{\nu}$ that is composed of ${\gamma}_{E,1}=240$ (respectively ${\gamma}_{I,1}=80$) active neurons in the excitatory (respectively inhibitory) population. The analytical curves (see the green and red solid lines) are obtained from Equations (22) and (23), while the numerical probability densities (magenta and blue dots) are calculated over 100,000 network realizations, as described in Section 2.5. (

**A**) Analytical and numerical probability distributions of the the bifurcation points ${\Lambda}_{E}$ and ${\Xi}_{E}$; (

**B**) Probability distributions of ${\Lambda}_{I}$ and ${\Xi}_{I}$.

**Table 1.**

**An example of network parameters**. This table contains the values of the parameters of the small-size network that we study in Section 3 (see Equation (25) and Figure 1). The symbol × in the matrices $\mathfrak{C}$ and $\mathfrak{R}$ means that the statistics of the stationary states and of the bifurcation points are not affected by those parameters, since the corresponding synaptic connections are absent (${\mathcal{P}}_{i,j}=0$).

$\mathcal{P}=\left(\right)open="["\; close="]">\begin{array}{cccc}0& 0.5& 1& 0.6\\ 0.4& 0.5& 0.1& 1\\ 0.5& 0.7& 0.3& 0.8\\ 0& 1& 0.9& 0\end{array}$ | $\mathit{\theta}=\left(\right)open="["\; close="]">\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}$ | |||

$\mathfrak{C}=\left(\right)open="["\; close="]">\begin{array}{cccc}\times & 4& -3& -10\\ 6& 5& -2& -4\\ 3& 4& -6& -7\\ \times & 2& -5& \times \end{array}$ | $\mathfrak{R}=\left(\right)open="["\; close="]">\begin{array}{cccc}\times & 4& 2& 3\\ 5& 3& 2& 3\\ 3& 4& 5& 6\\ \times & 2& 4& \times \end{array}$ |

**Table 2.**Set of parameters used for generating Figure 5.

Panel A | |||
---|---|---|---|

$\mathcal{N}=10,\phantom{\rule{0.222222em}{0ex}}12,\cdots ,\phantom{\rule{0.222222em}{0ex}}22$ | $\mathfrak{X}=3$ | ${X}_{0}=3$ | ${Y}_{0}=8$ |

$\mathfrak{Y}=2$ | ${X}_{1}=5$ | ${Y}_{1}=\mathcal{N}-8$ | |

${X}_{2}=\mathcal{N}-8$ | |||

Panel B | |||

$\mathcal{N}=16$ | $\mathfrak{X}=1,\phantom{\rule{0.222222em}{0ex}}2,\phantom{\rule{0.222222em}{0ex}}4,\phantom{\rule{0.222222em}{0ex}}8,\phantom{\rule{0.222222em}{0ex}}16$ | ${X}_{\lambda}=\frac{\mathcal{N}}{\mathfrak{X}}$ | ${Y}_{0}=7$ |

$\mathfrak{Y}=2$ | $\forall \lambda \in \left(\right)open="\{"\; close="\}">0,\cdots ,\mathfrak{X}-1$ | ${Y}_{1}=9$ |

**Table 3.**Set of parameters used for generating Figure 6.

${N}_{E}=640$ | ${\gamma}_{E,1}=240$ | ${\vartheta}_{E}=3$ | ${\mu}_{EE}=11$ | ${\sigma}_{EE}=0.8$ | ${P}_{EE}=0.7$ |

${N}_{I}=160$ | ${\gamma}_{I,1}=80$ | ${\vartheta}_{I}=0$ | ${\mu}_{EI}=-8$ | ${\sigma}_{EI}=0.6$ | ${P}_{EI}=0.9$ |

${\mu}_{IE}=5$ | ${\sigma}_{IE}=0.7$ | ${P}_{IE}=1.0$ | |||

${\mu}_{II}=-10$ | ${\sigma}_{II}=0.9$ | ${P}_{II}=0.8$ |

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Fasoli, D.; Panzeri, S.
Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder. *Entropy* **2019**, *21*, 630.
https://doi.org/10.3390/e21070630

**AMA Style**

Fasoli D, Panzeri S.
Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder. *Entropy*. 2019; 21(7):630.
https://doi.org/10.3390/e21070630

**Chicago/Turabian Style**

Fasoli, Diego, and Stefano Panzeri.
2019. "Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder" *Entropy* 21, no. 7: 630.
https://doi.org/10.3390/e21070630