# Partial and Entropic Information Decompositions of a Neuronal Modulatory Interaction

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Definitions

## 3. An Interaction Designed to Be Modulatory

- If the integrated RF input is extremely weak, then the value of the transfer function is close to zero.
- If the integrated CF input is extremely weak, then the value of the transfer function should be close to the integrated RF input.
- If the integrated RF and CF inputs have the same sign, then the absolute value of the transfer function should be greater than when based on the RF input alone. On the other hand, if the RF and CF inputs are of opposite sign then the absolute value of the transfer function should be less than when based on the RF input alone.
- The sign of the value of the transfer function is that of the integrated RF, so that the context cannot change the sign of the conditional mean of the output.

- M1:
- If the $RF$ signal is strong enough and the CF input is extremely weak then $I[Y;{X}_{1}|{X}_{2}]$ can have its maximum value, $I[Y;{X}_{1}]$ can be maximised and $I[Y;{X}_{2}|{X}_{1}]$ is close to zero. This shows that the RF input is sufficient, thus allowing the information in the $RF$ to be transmitted, and that the CF input is not necessary.
- M2:
- $I[Y;{X}_{2}|{X}_{1}]$ and $I[Y;{X}_{1}]$ are close to zero when the RF input is extremely weak no matter how strong the CF input. This shows that the RF input is necessary for information to be transmitted, and that the CF input is not sufficient to transmit the information in the RF input.
- M3:
- When ${s}_{1}<{s}_{2}$ and when the RF input is weak, $I[Y;{X}_{1}]$ and $I[Y;{X}_{1}|{X}_{2}]$ are both larger when the CF input is moderate than when the CF input is weak. Thus the CF input modulates the transmission of information about the RF input.

- (i)
- the modulatory input affects output only when the primary driving integrated RF input is non-zero but weak;
- (ii)
- that even when it does have an effect it has no effect on the sign of the conditional mean output, and
- (iii)
- that it can have those modulatory effects without the binary output transmitting any unique information about the modulator.

#### 3.1. Analysis Using Classical Shannon Measures

**Theorem**

**1.**

- (a)
- $I[Y;{X}_{1}|{X}_{2}]=h\left(w\right)-2\lambda h\left(u\right)-2\mu h\left(v\right);$
- (b)
- $I[Y;{X}_{2}|{X}_{1}]=h\left(z\right)-2\lambda h\left(u\right)-2\mu h\left(v\right);$
- (c)
- $I[Y;{X}_{1}]=1-h\left(z\right);$
- (d)
- $I[Y;{X}_{2}]=1-h\left(w\right);$
- (e)
- $I[Y;{X}_{1};{X}_{2}]=1-h\left(z\right)-h\left(w\right)+2\lambda h\left(u\right)+2\mu h\left(v\right);$
- (f)
- $I[Y;({X}_{1},{X}_{2})]=1-2\lambda h\left(u\right)-2\mu h\left(v\right),$

## 4. Information Decompositions

$\mathrm{Unq}{X}_{1}\equiv {I}_{unq}[Y;{X}_{1}|{X}_{2}]$ | denotes the unique information that ${X}_{1}$ conveys about Y; |

$\mathrm{Unq}{X}_{2}\equiv {I}_{unq}[Y;{X}_{2}|{X}_{1}]$ | is the unique information that ${X}_{2}$ conveys about Y; |

${\mathrm{Shar}}_{\mathrm{S}+\mathrm{M}}\equiv {I}_{shdS+M}[Y;({X}_{1},{X}_{2})]$ | gives the common (or redundant or shared) information that both ${X}_{1}$ and ${X}_{2}$ have about Y; |

$\mathrm{Syn}\equiv {I}_{syn}[Y;({X}_{1},{X}_{2})]$ | is the synergy or information that the joint variable $({X}_{1},{X}_{2})$ has about Y that cannot be obtained by observing ${X}_{1}$ and ${X}_{2}$ separately. |

#### 4.1. The Ibroja PID

#### 4.2. The EID Using ${I}_{\mathrm{ccs}}$

## 5. Information Decomposition (ID) Spectra

#### 5.1. Definition and Illustrations

#### 5.2. Ibroja Spectra

#### 5.3. EID Spectra

#### 5.4. Contextual Modulation and Information Decompositions

- S1:
- If the $RF$ signal is strong enough, and the CF input is extremely weak, then both UnqX2 and Syn are close to zero, UnqX1 can have its maximum value, and the sum of UnqX1 and ${\mathrm{Shar}}_{\mathrm{S}+\mathrm{M}}$ can equal the total output entropy. This shows that the RF input is sufficient, thus allowing the information in the $RF$ to be transmitted, and that the CF input is not necessary.
- S2:
- All five partial information components are close to zero when the RF input is extremely weak no matter how strong the CF input. This shows that the RF input is necessary for information to be transmitted, and that the CF input is not sufficient to transmit the information in the RF input.
- S3:
- When ${s}_{1}<{s}_{2}$ and when the RF input is weak, then the sum of UnqX1 and Syn is larger when the CF input is moderate than it is when the CF input is weak. The same is true of the sum of UnqX1 and ${\mathrm{Shar}}_{\mathrm{S}+\mathrm{M}}$. Thus the CF input modulates the transmission of information about the RF input.

- S1’:
- When UnqX2 < 0, UnqX2 and Syn are approximately of the same magnitude, the sum of UnqX1 and Syn can have its maximum value, and the sum of UnqX1 and ${\mathrm{Shar}}_{\mathrm{S}+\mathrm{M}}$ can equal the total output entropy.
- S2’:
- If at least one component is negative, then we can set the left-hand sides of (18)–(21) to zero and use the rule that the sum of the magnitudes of the negative components is approximately equal to the sum of the magnitudes of the positive components. If in any of (18)–(21) there is no negative term then all terms on the right-hand side are close to zero.

#### 5.5. Comparison of PID and EID

## 6. Analysis of the Transfer Functions Using the Ibroja PID over a Wide Range of Input Strengths

**Theorem**

**2.**

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

- (a)
- When transfer function ${T}_{M}$ is employed then
- (i)
- Syn $=I(Y;{X}_{2}|{X}_{1})=h\left({z}_{M}\right)-2\lambda h\left({u}_{M}\right)-2\mu h\left({v}_{M}\right)$;
- (ii)
- ${Shar}_{S+M}=I(Y;{X}_{2})=1-h\left({w}_{M}\right)$;
- (iii)
- Unq${X}_{1}=I(Y;{X}_{1}|{X}_{2})-I(Y;{X}_{2}|{X}_{1})=h\left({w}_{M}\right)-h\left({z}_{M}\right),\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}$ Unq${X}_{2}=0$.

- (b)
- When the transfer function ${T}_{A}$ is used and ${s}_{1}={s}_{2}$ then
- (i)
- Syn $=I(Y;{X}_{2}|{X}_{1})=h\left({z}_{A}\right)-2\lambda h\left({u}_{A}\right)-2\mu ;$
- (ii)
- ${Shar}_{S+M}=I(Y;{X}_{1})=1-h\left({z}_{A}\right);$
- (iii)
- Unq${X}_{1}$ = Unq${X}_{2}=0;$

- (c)
- When the transfer function ${T}_{A}$ is used and ${s}_{1}<{s}_{2}$ then
- (i)
- Syn $=I(Y;{X}_{1}|{X}_{2})=h\left({w}_{A}\right)-2\lambda h\left({u}_{A}\right)-2\mu h\left({v}_{A}\right);$
- (ii)
- ${Shar}_{S+M}=I(Y;{X}_{1})=1-h\left({z}_{A}\right);$
- (iii)
- Unq${X}_{2}=I(Y;{X}_{2}|{X}_{1})-I(Y;{X}_{1}|{X}_{2})=h\left({z}_{A}\right)-h\left({w}_{A}\right)\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}$ Unq${X}_{1}=0.$

- (d)
- When the transfer function ${T}_{A}$ is used and ${s}_{1}>{s}_{2}$ then
- (i)
- Syn $=I(Y;{X}_{2}|{X}_{1})=h\left({z}_{A}\right)-2\lambda h\left({u}_{A}\right)-2\mu h\left({v}_{A}\right)$;
- (ii)
- ${Shar}_{S+M}=I(Y;{X}_{2})=1-h\left({w}_{A}\right)$;
- (iii)
- Unq${X}_{1}=I(Y;{X}_{1}|{X}_{2})-I(Y;{X}_{2}|{X}_{1})=h\left({w}_{A}\right)-h\left({z}_{A}\right),\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}$ Unq${X}_{2}=0$.

**Theorem**

**5.**

## 7. Analysis of the Transfer Functions Using EID over a Wide Range of Input Strengths

## 8. Applications of ID Measures to Psychophysical Data

## 9. Conclusions and Discussion

#### 9.1. Implications of These Findings for Conceptions of ‘Modulation’ in the Cognitive and Neurosciences

#### 9.2. Comparisons between PID and EID

#### 9.3. Using EID and PID to Analyze and Interpret Psychophysical Data

#### 9.4. Using ID Spectra to Analyze and Interpret Empirical Data in General

#### 9.5. Modulatory Regulation of Activity as a Crucial and Non-Trivial Aspect of Information Processing

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Preliminary Results

## Appendix B. Proof of Theorem 1

## Appendix C. Proof of Theorem 2

## Appendix D. Proof of Corollary 1

## Appendix E. Proof of Corollary 2

## Appendix F. Proof of Theorem 3

## Appendix G. Proof of Theorem 4

## Appendix H. Proof of Theorem 5

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**Figure 1.**A local processor with binary receptive field (RF) input ${X}_{1},$ contextual field (CF) input ${X}_{2}$ and output Y. The weights on the connections from the RF and CF inputs into the output unit are ${s}_{1},{s}_{2},$ which represent the strengths given to the input signals. The integrated RF input, r, and the integrated CF input, c, are passed through a transfer function T and a logistic nonlinearity within the output unit to produce the conditional output probability, $\theta $, as well as the output conditional mean, m.

**Figure 2.**Classical Shannon measures (in bits), based on additive and modulatory transfer functions, and a correlation between inputs of 0.78.

**Figure 3.**Classical Shannon measures (in bits), based on additive and modulatory transfer functions, and a zero correlation between inputs.

**Figure 4.**Partial information decomposition (PID) and entropic information decomposition (EID) spectra (in bits), based on additive (A) and modulatory (M) transfer functions for four combinations of signal strengths: 1. (${s}_{1}=10.0,{s}_{2}=0.05$), 2. (${s}_{1}=0.05,{s}_{2}=10.0$), 3. (${s}_{1}=1.0,{s}_{2}=0.05$), 4. (${s}_{1}=1.0,{s}_{2}=5.0$), and two values of the correlation between inputs: 0.78 and zero.

**Figure 5.**Ibroja surfaces, based on additive and modulatory transfer functions, and a correlation between inputs of 0.78.

**Figure 6.**Ibroja surfaces, based on additive and modulatory transfer functions, and zero correlation between inputs.

**Figure 7.**EID surfaces, based on additive and modulatory transfer functions, and a correlation between inputs of $0.78$.

**Figure 8.**EID surfaces, based on additive and modulatory transfer functions, and a zero correlation between inputs.

**Figure 9.**Examples of gabor patch stimuli used in the psychophysical experiment. In all conditions, the task was to detect the presence of a centrally presented target gabor.

**Figure 10.**Partial information decomposition (PID) and EID spectra (in bits) calculated for subject 10 (S10), subject 18 (S18) and the whole group of subjects (G) in the contrast detection experiment calculated at threshold (AT) and over threshold (OT).

**Table 1.**Estimated accuracy, with estimated standard error, for each combination of the three conditions and the absence or presence of flankers.

No Target | At Threshold | Over Threshold | |
---|---|---|---|

Without Flankers | 0.9096 (0.0273) | 0.8797 (0.0289) | 0.9824 (0.0037) |

With Flankers | 0.9629 (0.0150) | 0.3766 (0.0532) | 0.9849 (0.0039) |

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**MDPI and ACS Style**

Kay, J.W.; Ince, R.A.A.; Dering, B.; Phillips, W.A. Partial and Entropic Information Decompositions of a Neuronal Modulatory Interaction. *Entropy* **2017**, *19*, 560.
https://doi.org/10.3390/e19110560

**AMA Style**

Kay JW, Ince RAA, Dering B, Phillips WA. Partial and Entropic Information Decompositions of a Neuronal Modulatory Interaction. *Entropy*. 2017; 19(11):560.
https://doi.org/10.3390/e19110560

**Chicago/Turabian Style**

Kay, Jim W., Robin A. A. Ince, Benjamin Dering, and William A. Phillips. 2017. "Partial and Entropic Information Decompositions of a Neuronal Modulatory Interaction" *Entropy* 19, no. 11: 560.
https://doi.org/10.3390/e19110560