Learning and Adaptation Strategies in an Obstacle - Avoidance Task

Performed in Monkeys

CALTECH

Biology Division – Andersen Lab

Elizabeth B. TorresRichard Andersen

Motivation: What is encoded in the PRR region of the Posterior Parietal Cortex of

the Monkey (Macaca Mulatta)

Cohen & Andersen (2002) Nat Rev Neurosci 3

Lewis & Van Essen (2000) J Comp Neuroll 428

Goal? or

Hand path? Posture ?

Experimental Design:

Obstacle Avoidance, 2 very different handpath solutions ??

Handpaths constrained to a plane

Handpaths constrained to a plane

Results

1 – Subutilize 3D space

Results

1 - Subutilize 3D space

2 - Adaptation

Obstacles

No Obstacles - Aftereffect

No Obstacle - Deadapted

Results

1 - Subutilize 3D space

2 – Adaptation

3 – Speed Independence (during learning)

Learning Period Speed Independence

As the subject learns,

More consistent, shorter motions,

approach bell-shaped speed profiles

From geometric (local) strategy

(decoupled from speed) to

Kinetic-based (global) optimization

(eventually smooth, ballistic motion)

Kalaska’s experim loads effect on PD

M1 A5 Dorsal

The Straight-line Path of a Curved World

Metric Tensor ijg

Generalized Pythagorean Theorem (curved world)

2 2 211 1 12 1 2 22 2ds g dx g dx dx g dx

Euclidean Case (flat world)

2 2 21 2

ij ij

ds dx dx

g

1 nij

1

1 np

n

b

X Y g u ua a

b

, ,, , ,

Inner Product (norm)

1 0

flat world

0 1

11 1

1

curved n

n nn

g x g x

g x g x

f

mR

q

nR

x

XQ

rR

r f q

TARGET WORLD

ij

gPOSTURE WORLD

kl

g ?

tkl ij

g J g J Pullback the Metric of X into Q

1f

targetx 1 targetf x

Local Isometric Imbedding

The gradient flow generates geodesics paths (“straight-line”

paths of a space whose curvature is task-dependent,

because we have optimized with respect to a geometry

dictated by the norm/cost the task dictates:

i.e. dictated by the TARGET !!!

Given this, What norm could we optimize in order to

approximate these solution paths in hand space?, i.e. to

capture the geometry (curvature) of task space and that of

the underlying parameter space?

Via Point

Temporally, speed-based

Spatially-based

3 62 2ViaPoint target

1 21 4

i i i ii i

r x f q x f q

11

1

1traversedd D

e

22

1

1traversedd D

e

D1D2

Target

ViaPoint

Init Hand

Norm in this TASK Space

-100 -50 50 100

0.2

0.4

0.6

0.8

1

-100 -50 50 100

0.2

0.4

0.6

0.8

1

1. Obstacles Weight such that first priority is Via Point

2. No Obstacle (Deadaptation residual aftereffects) More weight to Main Target, Via Point is not as important

3. No Obstacle Straight-line Paths 0 weight for Via Point, 1 for Main Target

Solving the Task

D1D2

Target

ViaPoint

Init Hand

Apply Method to Data Paths

curved 'sijg

flat 'sijg

Simulations

Future Work

• Neural Recordings

• Neural Systems Identification

Acknowledgements

Sloan-Swartz Foundation

Richard Andersen

All members of the Andersen Lab for their immense help and incredible patience while teaching me

Nobody asked questions related to this, but I had included the following 2 slides here in case someone wanted to know more about the model implementation of the theory in general

Equation describing the autonomous flow of geometric motion

1 target,qdq G r f q x t

t

q m n n mf qm m n nG J G J

target f qrr f q x

f q q

,

target targetn 2

i ii 1

r f q x x f q ,

Compatibility Condition q Q q Q W p

nq Q R : M

nQ RnQ R

1 1 1q q q W q W :

nq Q R : M

|

|

i1 1

j

i1 1

j

qq q Q Q J q q

q

qq q Q Q J q q

q

:

:

ijg rsg

r st1 1

ij rs rs i jrs

q qg J q q g J q q g

q q

2 i j r sij rs

ij rsds g dq dq g dq dq

1 1 1q q q W q W :

1q 1q