# Contextual Modulation in Mammalian Neocortex is Asymmetric

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Notation and Definitions

#### 2.2. Partial Information Decompositions

#### 2.3. Transfer Functions

#### 2.4. Different Signal-Strength Scenarios

- CM1:
- The drive, R, is sufficient for the output to transmit information about the input, so context, C, is not necessary.
- CM2:
- The drive, R, is necessary for the output to transmit information about the input, so context, C, is not sufficient.
- CM3:
- The output can transmit unique information about the drive, R, but little or no unique information or misinformation about the context, C.
- CM4:
- The context can strengthen the transmission of information about R when R is weak.

#### 2.5. Bivariate Gaussian Mixture Model (BGM)

#### 2.6. Single Bivariate Gaussian Model (SBG)

#### 2.7. First Simulation, with Inputs Generated Using the BGM Model

#### 2.8. The Second Simulation, with Inputs Generated Using the SBG Model

## 3. The Results

#### 3.1. First Simulation, with Inputs Generated Using the BGM Model

#### 3.1.1. Distinctive Information Transmission Properties of Contextual Modulation

#### 3.1.2. Comparison of Five Different Forms of Information Decomposition

#### 3.2. Second Simulation, with Inputs Generated Using the SBG Model

#### 3.2.1. Distinctive Information Transmission Properties of Contextual Modulation

#### 3.2.2. Comparison of Five Different Forms of Information Decomposition

#### 3.3. Partial Information Decomposition of Binarized Action Potential Data from a Detailed Multi-Compartment Model of A Neuron

## 4. Conclusions and Discussion

#### 4.1. Contextual Modulation Contrasts with the Arithmetic Operators

#### 4.2. There are Various Forms of Contextual Modulation

#### 4.3. Is Coordinate Transformation an Example of Contextual Modulation?

#### 4.4. Is the Contrast Between ‘Modulation’ and ‘Drive’ Adequately Defined?

#### 4.5. What Forms of Modulation Occur in Neocortex?

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Mathematical Symbols | |

Y | A discrete random variable representing the binary output |

y | A realization of the random variable Y |

R | A continuous random variable representing the receptive field input |

r | A realization of the random variable R |

C | A continuous random variable representing the contextual field input |

c | A realization of the random variable C |

$p(\xb7),p(\xb7,\xb7)$ | A univariate and bivariate probability mass functions, e.g., $p\left(u\right),p(u,v)$ |

$f(\xb7,\xb7),{f}_{i}(\xb7,\xb7)$ | Bivariate probability density functions, as in $\left(19\right)$ |

${\mu}_{i},\mathsf{\Sigma}$ | Mean vector and covariance matrix, as in $\left(19\right)$ |

${\pi}_{i}$ | The mixing proportions in the bivariate Gaussian mixture model, as in $\left(19\right)$ |

${X}_{1},{X}_{2}$ | Continuous random variables following various Gaussian probability models |

${x}_{1},{x}_{2}$ | The realized values of random variables ${X}_{1},{X}_{2}$ |

${s}_{1},{s}_{2}$ | The signal strengths used to compute r and c |

$T(r,c)$ | The general form of transfer function, given receptive field input r and contextual field input c, as in (1). |

${T}_{F}(r,c)$ |

Classical Information Terms | |

$H\left(Y\right)$ | The Shannon entropy of the random variable Y |

$H{\left(Y\right)}_{\mathrm{res}}$ | The residual entropy of the output random variable Y, which is equal to $H\left(Y\right|R,C)$, the entropy in Y that is not shared with R or C. |

$H(\xb7)$ | The Shannon entropy where the random variable is evident in the text |

$H(\xb7,\xb7)$ | The Shannon entropy of a bivariate random vector, e.g., $H(Y,R)$ |

$H(Y,R,C)$ | The Shannon entropy of the trivariate random vector $(Y,R,C)$ |

$I(\xb7;\xb7)$ | The mutual information shared between two random variables, e.g., $I(Y;R)$ |

$i(\xb7;\xb7)$ | The local mutual information shared between realizations of two random variables, e.g., $i(u;v)$ |

$I(Y;R,C)$ | The mutual information shared between the random variable Y and the random vector $(R,C)$ |

$I(Y;R|C)$ | The conditional mutual information shared between the random variables Y and R but not shared with C |

$I(Y;C|R)$ | The mutual information shared between the random variables Y and C but not shared with R |

$II(Y;R;C)$ | The interaction information – a measure involving synergy and shared information which involves all three random variables $Y,R,C$. |

Partial Information Decomposition | |

PID | Partial Information Decomposition, with components UnqR, UnqC, Shd and Syn, defined in Section 2.2 |

${I}_{\mathrm{broja}}$ | The PID developed by Bertschinger et al. [49] |

${I}_{\mathrm{min}}$ | The PID developed by Williams and Beer [30] |

${I}_{\mathrm{proj}}$ | The PID developed by Harder et al. [48] |

${I}_{\mathrm{ccs}}$ | The PID developed by Ince [51] |

${I}_{\mathrm{dep}}$ | The PID developed by James et al. [52] |

Other Acronyms | |

M1–M4 | Four modulatory transfer functions, with ${M}_{i}$ with transfer function ${T}_{{M}_{i}}(i=1,2,3,4)$, as defined in Section 2.3 |

A, S, P, D | Four arithmetic transfer function, with e.g., A with transfer function ${T}_{A}$, as defined in Section 2.3 |

S1–S4 | Four signal-strength scenarios, as defined in Section 2.4 |

CM1–CM4 | Four key properties of contextual modulation, as defined in Section 2.4 |

BGM | Bivariate Gaussian Mixture Model, as defined in Section 2.5 |

SBG | Single bivariate Gaussian Model, as defined in Section 2.6 |

AP | Action potential |

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**Figure 2.**

**Williams-Beer diagram.**A Williams-Beer diagram showing the decomposition of the joint mutual information $I(Y;R,C)$ between the output Y and the two inputs R and C into a sum of the four partial information components, Shd, UnqR, UnqC and Syn, as defined in the text.

**Figure 3.**Probability density plots of $({X}_{1},{X}_{2})$ in the bivariate Gaussian model for two different values of the correlation, d.

**Figure 4.**Normalized spectra for the ${I}_{\mathrm{broja}}$, ${I}_{\mathrm{dep}}$ and ${I}_{\mathrm{ccs}}$ PIDs, for each of the scenarios: S1 $({s}_{1}=10.0,{s}_{2}=0.05)$, S2 $({s}_{1}=0.05,{s}_{2}=10.0)$, S3 $({s}_{1}=1.00,{s}_{2}=0.05)$, S4 $({s}_{1}=1.0,{s}_{2}=5.0)$. The data were generated from the bivariate Gaussian mixture model. The correlation between inputs, R and C, is 0.8. The modulatory transfer functions, M1, M2, M3, M4, and the arithmetic transfer functions, A, S, P, D, are defined in Section 2.3.

**Figure 5.**Normalized spectra for the ${I}_{\mathrm{broja}}$, ${I}_{\mathrm{dep}}$ and ${I}_{\mathrm{ccs}}$ PIDs, for each of the scenarios: S1 $({s}_{1}=10.0,{s}_{2}=0.05)$, S2 $({s}_{1}=0.05,{s}_{2}=10.0)$, S3 $({s}_{1}=1.00,{s}_{2}=0.05)$, S4 $({s}_{1}=1.0,{s}_{2}=5.0)$. The data were generated from the bivariate Gaussian mixture model. The correlation between inputs, R and C, is 0.2. The modulatory transfer functions, M1, M2, M3, M4, and the arithmetic transfer functions, A, S, P, D, are defined in Section 2.3.

**Figure 6.**Normalized spectra for the ${I}_{\mathrm{broja}}$, ${I}_{\mathrm{dep}}$ and ${I}_{\mathrm{ccs}}$ PIDs, for each of the scenarios: S1 $({s}_{1}=10.0,{s}_{2}=0.05)$, S2 $({s}_{1}=0.05,{s}_{2}=10.0)$, S3 $({s}_{1}=1.00,{s}_{2}=0.05)$, S4 $({s}_{1}=1.0,{s}_{2}=5.0)$. The data were generated from the single bivariate Gaussian model. The correlation between inputs, R and C, is 0.8. The modulatory transfer functions, M1, M2, M3, M4, and the arithmetic transfer functions, A, S, P, D, are defined in Section 2.3.

**Figure 7.**Normalized spectra for the ${I}_{\mathrm{broja}}$, ${I}_{\mathrm{dep}}$ and ${I}_{\mathrm{ccs}}$ PIDs, for each of the scenarios: S1 $({s}_{1}=10.0,{s}_{2}=0.05)$, S2 $({s}_{1}=0.05,{s}_{2}=10.0)$, S3 $({s}_{1}=1.00,{s}_{2}=0.05)$, S4 $({s}_{1}=1.0,{s}_{2}=5.0)$. The data were generated from the single bivariate Gaussian model. The correlation between inputs, R and C, is 0.2. The modulatory transfer functions, M1, M2, M3, M4, and the arithmetic transfer functions, A, S, P, D, are defined in Section 2.3.

**Figure 8.**Partial information decompositions are shown (the unit is bit) of Shai et al.’s binarized action potential data in relation to the numbers of basal and apical inputs to the detailed multi-compartmental model of a layer 5 pyramidal cell. All five PID methods were used, with the results from ${I}_{\mathrm{min}}$ and ${I}_{\mathrm{proj}}$ being virtually identical to those obtained using the ${I}_{\mathrm{broja}}$ method, so their results are not shown.

S1 | Scenario 1: Strong R, with C near 0 | (${s}_{1}=10.0,{s}_{2}=0.05$) |

S2 | Scenario 2: R near zero, with strong C | (${s}_{1}=0.05,{s}_{2}=10.0$) |

S3 | Scenario 3: Weak R, with C near 0 | (${s}_{1}=1.00,{s}_{2}=0.05$) |

S4 | Scenario 4: Weak R, with moderate C | (${s}_{1}=1.00,{s}_{2}=5.00$) |

**Table 2.**Some estimated classical Shannon information measures, given to two significant figures. The unit is bit.

H(Y) | $\mathit{I}(\mathit{Y};\mathit{R})$ | $\mathit{I}(\mathit{Y};\mathit{C})$ | $\mathit{I}(\mathit{Y};\mathit{R}|\mathit{C})$ | $\mathit{I}(\mathit{Y};\mathit{C}|\mathit{R})$ | $\mathit{I}(\mathit{Y};\mathit{R},\mathit{C})$ | $\mathit{II}(\mathit{Y};\mathit{R};\mathit{C})$ |
---|---|---|---|---|---|---|

0.88 | 0.28 | 0.16 | 0.43 | 0.31 | 0.59 | 0.16 |

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Kay, J.W.; Phillips, W.A.
Contextual Modulation in Mammalian Neocortex is Asymmetric. *Symmetry* **2020**, *12*, 815.
https://doi.org/10.3390/sym12050815

**AMA Style**

Kay JW, Phillips WA.
Contextual Modulation in Mammalian Neocortex is Asymmetric. *Symmetry*. 2020; 12(5):815.
https://doi.org/10.3390/sym12050815

**Chicago/Turabian Style**

Kay, Jim W., and William A. Phillips.
2020. "Contextual Modulation in Mammalian Neocortex is Asymmetric" *Symmetry* 12, no. 5: 815.
https://doi.org/10.3390/sym12050815