Lior Segev

Ranit Aharonov

Alon Keinan

Isaac Meilijson

www.math.tau.ac.il/~ruppin

Localization of Function in Localization of Function in Neurocontrollers Neurocontrollers

Localization of Function

– How does one ``understand’’ neural information processing?

– A classical, good point to start with is localization of function(s) in neurocontrollers

– A good model to start with is Evolutionary Autonomous Agents (EAAs)

– Scope of analysis method may be more general

Evolved neurocontrollersEvolved neurocontrollers

Talk Overview

• The basic Functional Contribution Analysis (FCA)

• Localization of Subtasks• Synaptic Analysis• High-dimensional FCA• Informational Lesioning• Playing games in the brain, or “My fair lady”.

The basic FCA

• A multi-lesion approach: learning about normal, intact functioning via lesion ``perturbations’’

• Given are a set of neurocontroller lesions and the agent’s corresponding performance levels

• Assign ``importance’’ levels to the different units of the neurocontroller?

• The FCA: Find such assginments that maximize performance prediction of unseen lesions

Lesioning

C1

C2

C3

C4

C5

C6

p = f(c1+c3+c4+c6)~

argmin = Σ(p-p)2

{f,c}

12N

~

The Functional Contribution Algorithm (FCA)

f

module

c

module

optimal

f and c

training set

min(p-p)2~

The performance prediction function

(m . c)

P

Single Lesions vs. FCA

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0.1

0.15

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0.25

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0.35

S10 (General)

S10 (Grazing)

S10 (Exploration)

S22 (General)

SP10 (General)

Single Lesions

FCA

Generalization – an Adaptive Lesion Selection algorithm

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No. of configurations

MS

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AdaptiveRandom

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30 50 70

Task Comparison

Grazing

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Exploration

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General task

The Contribution Matrix – Localization and Specification

Task

Neuron

1 2 P

1 C11 C12 C1P

2 C21 C22 C2P

3 C31 C32 C3P

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N CN1 CN2 CNP

Synaptic AnalysisSynaptic Analysis

Network BackboneBy weights

By contributions

High-dimensional FCA

• The inherent limitations of basic FCA (e.g., paradoxical lesioning)

• Compound Elements

• Order (dimension) of compound elements

• An efficient High-D algorithm for compound element selection

Complexity of Task Localization

00.0020.0040.0060.008

0.010.0120.0140.0160.018

0.02

10 30 50 70 90 110

No. of elements

MSE

1D-FCA

2D-FCA

3D-FCA

4D-FCA

Types of 2D Interactions

• Paradoxical Interactions – element 1 is advantageous only if element 2 is intact

• Inverse Paradoxical interactions – element 1 is advantageous only if element 2 is lesioned

• All significant 2D compound elements belong to either type (there can be others..)

Informational Lesioning Method (ILM)

• The paradox of the lesioning paradigm• The dependence on the lesioning method• Controlled lesioning – approaching the limit of

intact behavior• Implement a lesion as a channel whose input

is the firing of the intact element and output is the firing of the lesioned element (given an input).

• Quantify the lesioning level as an inverse function of the Mutual Information between the input and output of the channel

ILM – In summary:

• Increased localization precision

• Portraying a spectrum of short-to-long term functional effects of system units

• Approaching the limit CVs of the intact state, in the ILM lesioning family

• Does such a limit exist more generally? Is the beauty inherently in the of the beholder?

Where Game Theory meets Brain Research..

• “George said: You know, we are on a wrong track altogether. We must not think of the things we could do with, but only of the things that we can’t do without.”

[Three men in a boat: to say nothing of the dog!, by Jerome K. Jerome, chapter 3]

FCA and the Shapley Value

• The Shapley value (SH): A famed, unique solution of cost allocation in a game theory axiomatic system

• Many functioning networks (including our EAA neurocontrollers) can be addressed within this framework

• An alternative formulation of the FCA is equivalent to the SH (even though the starting standpoints and motivations are different).

Ongoing FCA Research

• Optimal Lesioning ?

• Relation to SH and more efficient algorithms (sampling, high-D..).

• Generalization to PPR

• Application to neuroscience data (reverse inactivation, TMS, fMRI).

• Application to gene networks?

•The contribution values can be efficiently determined using the simple FCA.• More complex networks require higher dimensional FCA descriptions. •The minimal dimension of the FCA may provide an interesting measure of functional complexity.• The importance of being lesioned (in the “right” way..) – ILM and beyond.• Even if the brain is not “a society of minds”, it can be analyzed with the aids of fundamental tools from game theory.•www.math.tau.ac.il/~ruppin – papers (and code)

SummarySummary

Network backbone: 2D interactions