Information Flow in Neural Circuits? Praveen Venkatesh with - - PowerPoint PPT Presentation
How else can we define Information Flow in Neural Circuits? Praveen Venkatesh with Sanghamitra Dutta and Pulkit Grover Dept. of Electrical & Computer Engineering Carnegie Mellon University How should we define Information Flow in Neural
“How should we define Information Flow in Neural Circuits?”, ISIT 2019 “Information Flow in Computational Systems”, IEEE Trans. IT 2020 (https://praveenv253.github.io/publications)
with Sanghamitra Dutta and Pulkit Grover
Carnegie Mellon University
Theorists Neuroscientists Clinicians Labmates & Colleagues Advisor
Pulkit Grover (CMU) Marlene Behrmann (CMU) Rob Kass (CMU) Mark Richardson (MGH/Harvard) Vasily Kokkinos (MGH/Hvd) Bobak Nazer (BU) Venkatesh Saligrama (BU) Sanghamitra Dutta (CMU) Aditya Gangrade (BU) Haewon Jeong (CMU) Ashwati Krishnan (CMU) Alireza Chamanzar (CMU)
Fellowships
Todd Coleman (UCSD)
Grants
Research
Our main insight: controversy exists because “information flow” is not formally defined
(Venkatesh & Grover SfN’15, Allerton’15) (Venkatesh & Grover, Cosyne’19) (Venkatesh, Dutta & Grover, IEEE Trans IT ’20)
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(Almeida et al., Cortex, 2013)
Region A Region C Region B “Stimulus” Goal: Find a definition for information flow, so that we can track info paths
brain regions
stimulus
6 𝑁: “Message” = stimulus 𝑌(𝐹𝑢): Transmission on edge 𝐹𝑢 at time 𝑢
the edge 𝐹0 at 𝑢 = 0 𝑌(𝐹1) = 𝑔
𝐵1 𝑌 𝐹0 , …
measured
arrives at and only at 𝑢 = 0
(Thompson, 1980: VLSI) (Ahlswede et al., 2000: Network Info Theory) (Peters et al., 2016: Causality)
𝐵0 𝐶0 𝐷0
𝑢 = 0
𝐵1 𝐶1 𝐷1
𝑢 = 1
𝐵2 𝐶2 𝐷2
𝑢 = 2 𝑁 𝑔(𝑁) 𝑌(𝐹1) 𝑌(𝐹0)
Region A Region C Region B Stimulus
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the edge 𝐹0 at 𝑢 = 0 𝑌(𝐹1) = 𝑔
𝐵1 𝑌 𝐹0 , …
measured
arrives at and only at 𝑢 = 0
(Thompson, 1980: VLSI) (Ahlswede et al., 2000: Network Info Theory) (Peters et al., 2016: Causality)
𝐵0 𝐵1 𝐵2 𝐶0 𝐶1 𝐶2
𝑁 𝑔(𝑁)
𝐷0 𝐷1 𝐷2
𝑌(𝐹0) 𝑌(𝐹1) 𝑢 = 0 𝑢 = 1 𝑢 = 2
Goal: Define info flow + track info path
𝑁, 𝑎 ~ iid Ber(1/2)
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𝑁 ⊕ 𝑎 = 𝑁 xor 𝑎 𝐽 𝑁; 𝑁 ⊕ 𝑎 = 0 Why not a simpler definition like: 𝐽 𝑁; 𝑌 𝐹𝑢 > 0
?
Definition [𝑁-Information Flow]: (ISIT ’19, Trans IT ‘20) We say that an edge 𝐹𝑢 has 𝑁-information flow if ∃ ℰ𝑢
′ ⊆ ℰ𝑢 s.t. 𝐽 𝑁; 𝑌 𝐹𝑢 | 𝑌(ℰ𝑢 ′) > 0.
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Using the above definition: 𝐽 𝑁; 𝑁 ⊕ 𝑎 𝑎 > 0 Conditioning on 𝑎 reveals the dependence between 𝑁 ⊕ 𝑎 and 𝑁: Definition [𝑁-Information Flow]: (ISIT ‘19, Trans IT ‘20) We say that an edge 𝐹𝑢 has 𝑁-information flow if ∃ ℰ𝑢
′ ⊆ ℰ𝑢 s.t. 𝐽 𝑁; 𝑌 𝐹𝑢 | 𝑌(ℰ𝑢 ′) > 0.
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Definition [𝑁-Information Flow]: (ISIT ’19, Trans IT ‘20) We say that an edge 𝐹𝑢 has 𝑁-information flow if ∃ ℰ𝑢
′ ⊆ ℰ𝑢 s.t. 𝐽 𝑁; 𝑌 𝐹𝑢 | 𝑌(ℰ𝑢 ′) > 0.
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Theorem [Existence of Information Paths]: If the transmissions of an “output” node 𝑊
𝑢 𝑝𝑞 depend on 𝑁, then there
exists a path from 𝑊
𝑗𝑞 to 𝑊 𝑢 𝑝𝑞, every edge of which has 𝑁-info flow. (Venkatesh, Dutta & Grover, ISIT 2019; Trans. IEEE 2020)
𝐵0 𝐵1 𝐵2 𝐶0 𝐶1 𝐶2
𝑁 𝑔 𝑁 𝑌(𝐹0) 𝑌(𝐹1) 𝑊
𝑗𝑞
𝑊
𝑢 𝑝𝑞
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An edge 𝐹𝑢 has 𝑁-info flow if ∃ ℰ𝑢
′ ⊆ ℰ𝑢 s.t. 𝐽 𝑁; 𝑌 𝐹𝑢 | 𝑌(ℰ𝑢 ′) > 0.
𝑁-info flows out of a node ⇒ 𝑁-info flows into the node 𝐽 𝑁; 𝑁 ⊕ 𝑎 𝑎 > 0 𝐽 𝑁; 𝑎 𝑁 ⊕ 𝑎 > 0 𝐷1 is an orphan 𝑎 has 𝑁-info flow!
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Algorithm: Prune edges that do not lead to 𝑊
𝑗𝑞 using Depth First Search (Venkatesh, Dutta & Grover, IEEE Trans. IT 2020; to appear)
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Algorithm: Prune edges that do not lead to 𝑊
𝑗𝑞 using Depth First Search (Venkatesh, Dutta & Grover, IEEE Trans. IT 2020; to appear)
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Algorithm: Prune edges that do not lead to 𝑊
𝑗𝑞 using Depth First Search (Venkatesh, Dutta & Grover, IEEE Trans. IT 2020; to appear)
Remove orphans such as 𝐷1 Hopefully prune out edges like 𝑎, which are not “computed” from 𝑁
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𝐽 𝑁; 𝑁 ⊕ 𝑎 = 0 𝐽 𝑁; 𝑁 ⊕ 𝑎 | [𝑁, 𝑎] = 0 The only information path from 𝐵0 to 𝐶3 is through an edge carrying 𝑎! The orphan removed by pruning transmits 𝑁 ⊕ 𝑎! (𝐵1, 𝐵2) has no 𝑁-info flow:
𝑁, 𝑎 ~ Ber(1/2) 𝑁 ⊥ 𝑎
𝐵2 is an 𝑁-information Orphan
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We disliked orphans for two reasons: 1. No “conservation” of info flow at an orphan 2. 𝑎 is not “computed” from 𝑁 Is 𝑎 really all that different from 𝑁 ⊕ 𝑎 in these examples? To differentiate, we need to go beyond the joint distribution! 𝑞 𝑛, 𝑨, 𝑧 = 1 4 𝕁 𝑧 = 𝑛 ⊕ 𝑨 = 1 4 𝕁 𝑧 ⊕ 𝑨 = 𝑛 Using 𝑍 for 𝑁 ⊕ 𝑎, Joint distribution is symmetric in 𝑁 ⊕ 𝑎 and 𝑎!
𝑁, 𝑎 ~ Ber(1/2) 𝑁 ⊥ 𝑎
19 𝑌0 𝑌1 𝑌4 𝑌2 𝑌5 𝑌3 Directed Acyclic Graph with functional relationships
𝑌𝑗 = 𝑔
𝑗 𝑄𝑏𝑌𝑗, 𝑋 𝑗
𝐵0 𝐶0 𝐵1 𝐶1 𝐵2 𝐶2
𝑁 𝑔(𝑁) 𝑌1 𝑌2 𝑌3 𝑌3 = 𝑔
𝐵1 𝑌1, 𝑌2, 𝑋 𝐵1
The Computational System is also a Structural Causal Model Structural Causal Model
(Peters, Janzing & Schölkopf, “Elements of Causal Inference”, 2016)
Computational System Model
(Venkatesh, Dutta & Grover, ISIT 2019; Trans. IT 2020)
𝑞 𝑌5 ? 𝑌0 ≔ 𝑦
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For a particular realization of all RVs, What would have happened (to downstream variables) If 𝑁 had taken a different value? (keeping all other sources of randomness fixed) = 0 = 1 = 0 = 1 = 1 = 1 = 0 1 1 1 We can now differentiate: 𝑁 ⊕ 𝑎 is CCI’d by 𝑁, but 𝑎 is not!
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= 0 = 1 = 0 = 1 = 1 = 1 = 0 1 1 1 We can now differentiate: 𝑁 ⊕ 𝑎 is CCI’d by 𝑁, but 𝑎 is not!
Defining info flow using CCI:
e.g. 𝑌 𝐹𝑢 = 𝕁 𝑁 ≥ 0
for some values of 𝑨 ∈ 𝒶 e.g. 𝑌 𝐹𝑢 = 𝑁 ⋅ 𝑎
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Definition [𝑁-Counterfactual Causal Influence]: We say that an edge 𝐹𝑢 is counterfactually causally influenced by 𝑁 if ∃ 𝑛, 𝑛′ ∈ ℳ and 𝑥 ∈ 𝒳 s.t. 𝑌 𝐹𝑢
𝑛,𝑥 ≠ 𝑌 𝐹𝑢 𝑛′,𝑥
𝑁-information flow 𝑁-counterfactual causal influence
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Theorem [𝑁-CCI has no orphans]: If an outgoing edge 𝐹𝑢 of a node 𝑊
𝑢 is 𝑁-CCI’d, then there exists some
incoming edge 𝐹𝑢−1 of 𝑊
𝑢 that is also 𝑁-CCI’d.
Theorem [𝑁-CCI guarantees info paths]: If the transmissions of an “output” node 𝑊
𝑢 𝑝𝑞 depend on 𝑁, then there
is a path from 𝑊
𝑗𝑞 to 𝑊 𝑢 𝑝𝑞 such that every edge of this path is 𝑁-CCI’d.
Definition [𝑁-Counterfactual Causal Influence]: We say that an edge 𝐹𝑢 is counterfactually causally influenced by 𝑁 if ∃ 𝑛, 𝑛′ ∈ ℳ and 𝑥 ∈ 𝒳 s.t. 𝑌 𝑛,𝑥 𝐹𝑢 ≠ 𝑌 𝑛′,𝑥 𝐹𝑢
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𝑞 𝑛, 𝑨, 𝑧 = 1 4 𝕁 𝑧 = 𝑛 ⊕ 𝑨 = 1 4 𝕁 𝑧 ⊕ 𝑨 = 𝑛
(using 𝑍 for 𝑁 ⊕ 𝑎)
Joint distribution is symmetric in 𝑁 ⊕ 𝑎 and 𝑎!
𝑁, 𝑎 ~ Ber(1/2) 𝑁 ⊥ 𝑎
variables to be held fixed while 𝑁 is changed
to make interventions
estimate We would like observational measures. But…
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Can we find an alternative definition which does not suffer from the pruning counterexample? Definition [Modified 𝑁-Information Flow]: We say that an edge 𝐹𝑢 has modified 𝑁-information flow if ∃ ℰ𝑢
′ = 𝐹𝑢 𝑗 ⊆ ℰ𝑢 and functions 𝑔 𝑗 s.t. 𝐽 𝑁; 𝑌 𝐹𝑢 | {𝑔 𝑗 ∘ 𝑌 𝐹𝑢 𝑗 } > 0.
𝑁
𝐵0 𝐶0 𝐵1 𝐶1 𝐵2 𝐶2
𝑊
𝑗𝑞
𝑊
𝑢 𝑝𝑞
𝐵3 𝐶3
𝑁 𝑁 𝑁 ⊕ 𝑎 [𝑁, 𝑎] 𝑁 ⊕ 𝑎 𝑁 𝑎 𝑎 𝑎 = 𝐽 𝑁; 𝑁 ⊕ 𝑎 𝑎 > 0 ∃ 𝑔: 𝑁, 𝑎 ↦ 𝑎 s.t. 𝐽 𝑁; 𝑁 ⊕ 𝑎 𝑔 𝑁, 𝑎
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All definitions have their shortcomings: 1. 𝑁-CCI identifies 𝑁 ⊕ 𝑎2 (but it cannot be decoded) 2. Modified 𝑁-info flow identifies 𝑎1 (but it cannot help decode 𝑁) 3. 𝑁-CCI ⇔ 𝑁-info flow 𝑁-CCI ⇔ Modified 𝑁-info flow 4. Modified 𝑁-info flow ⇒ 𝑁-info flow 𝑁-info flow 𝑁-CCI Modified 𝑁-info flow
Information Flow & PID
(Venkatesh, Dutta and Grover, ISIT 2019; IEEE Trans. IT 2020, to appear)
Neuroengineering: Fundamental Limits and Algorithms
(Experimental validation: Robinson, Venkatesh et al., Sci. Rep. 2017) (Fundamental limits: Venkatesh and Grover, ISIT 2017) (with Improvements by Xu, Coleman in this ISIT!) (Theory: Grover and Venkatesh, Proc. IEEE 2016) (Clinical use case: Chaman Zar, …, Venkatesh et al., IEEE Trans. BME 2018)
AI and Fairness
(Dutta, Venkatesh, Mardziel, Datta and Grover, AAAI 2020)