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Temporal learning in animals: Empirical tests of two mathematical models of timing * Armando Machado University of Minho, Portugal Université Charles-de-Gaulle, Lille 3
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Temporal learning in animals:
Empirical tests of two mathematical models of timing
*Armando Machado
University of Minho, Portugal
Université Charles-de-Gaulle, Lille 3
The Skinner boxThe microscope of learning psychologist
• Two models of timing
• Scalar Expectancy Theory (SET)
• Learning to Time (LeT)
• Critical tests: SET vs LeT
• Temporal Bisection
• Double Bisection
• Successes and Failures: what have we learned?
• The role of context
• Errorless learning and mediating behavior
The pigeon’s sense of time
T
Extinction Reinforc. Time in Interval (s)0 100 200 300
0
10
20
30
40
50
0 10 20 300
20
40
60
80
0 25 50 75 1000
20
40
60
80
Res
pons
es p
er M
inut
eKilleen et al. (1978)
FI-30 s
FI-100 s
FI-300 s
The scalar property
30 s
3,000 s
0.0 0.2 0.4 0.6 0.8
Proportion in Interval
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Res
pons
e R
ate
/ Ter
min
al R
ate
300 s
Dews (1970)
Retrospective TimingThe Bisection Experiment
Red
Retrospective TimingThe Bisection Experiment
Green
Retrospective TimingThe Bisection Experiment
TRAINING
Red
Green
TESTING
4-16
2-8
1-4
Church & Deluty (1977)
3-12
0 1 2 3 4 5
Relative Stimulus Duration
0.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("Sh
ort"
)Fetterman &
Killeen (1991)
2 4 6 8 10 12 14
Stimulus Duration (s)
0.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("Sh
ort"
)
Catania (1970)
0 1 2 3 4 5
Stimulus Duration (s)
0.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("Sh
ort"
)
Pacemaker
Accumulator
Memory S
Choose
Comparator
Memory LChoose
Scalar Expectancy Theory (SET)
n*
T
T0
n*
T
ST0
n*
T
S LT0
SET and temporal bisection
n*
T
S Lt
XS
XL
Xt
SET and temporal bisection
Similarity , SS t
t
XX X
X
SET and temporal bisection
Similarity , SS t
t
XX X
X
( | ) S t
t L
X XP t P
XR
X
Similarity , tt L
L
XX X
X
Stimulus duration (s)1 2 3 4
Pro
babi
lity
("S
hort
")
0.0
0.5
1.0
Learning to Time (LeT)
BehavioralStatesX(t,n)
Response StrengthsR(t), G(t)
Signal
AssociativeConnections
Wn
Behavioral States
0 1 2
Deterministic
interpretation
Stochastic
interpretation
Behavioral States
0 1 2
( ,0) ( ,0)
( , ) ( , 1) ( , )
1 if 0(0, )
0 if 0
dX t X t
dt
dX t n X t n X t n
dt
nX n
n
Machado (1997)
n=3n=2
n=1
...
0 5 10 15 20 25
Time (s)30
0.0
0.1
0.2
0.3
0.4
Act
ivat
ion
X(t
,n)
exp( )( )( , )
!
nt tX n t
n
( , ) 1 , if reinforcement
( , ) , if extinction
n n
n n
WR X S n WR
WR X L n WR
Associative Connections
nWR
( , ) 1 , if reinforcement
( , ) , if extinction
n n
n n
WG X L n WG
WG X S n WG
Associative Connections
nWG
( ) ( , ) nn
G t X t n WG
Response Strength
( ) ( , ) nn
R t X t n WR
Learning to Time (LeT)
1
( | )1 exp ( ) ( )
P tR t G t
R
4-16
2-8
1-4
Church & Deluty (1977)
3-12
0 1 2 3 4 5
Relative Stimulus Duration
0.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("Sh
ort"
)Fetterman &
Killeen (1991)
2 4 6 8 10 12 14
Stimulus Duration (s)
0.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("Sh
ort"
)
Catania (1970)
0 1 2 3 4 5
Stimulus Duration (s)
0.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("Sh
ort"
)
LeT and temporal bisection
For the bisection task, SET and LeT make similar predictions.
We need a new procedure to contrast the two models
The Double Bisection task
Double BisectionExperiment 1
TRAINING
Green
Red
Trial Type 1
Violet
Blue
Trial Type 2
TESTING
Green
Signal Durations Choice keys
Blue
vs
TRAINING
Green
Red
Trial Type 1
Violet
Blue
Trial Type 2
TESTING
Green
Signal Durations Choice keys
Blue
vs
Pacemaker
Accumulator
m
Response 1
Comparator
M
Response 2
M M
SET
LeTBehavioral
States
Responses
Associativeconnections
1 s Red
4 s Green
4 s Blue
16 s Violet
PredictionsA context effect?
1 2 84 160.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
("G
reen
")
LeT
SET
Duration of test stimulus (s)
Machado & Keen (1999)
1 s Red
4 s Green
4 s Blue
16 s Violet
DataYes, a context effect!
Machado & Keen (1999)
1 2 84 160.0
0.2
0.4
0.6
0.8
1.0P
roba
bilit
y ("
Gre
en")
Duration of test stimulus (s)
1 s Red
4 s Green
4 s Blue
16 s Violet
Double BisectionExperiment 2
Group 8
1 s Red
4 s Green
4 s Blue
8 s Violet
1 s Red
4 s Green
4 s Blue
16 s Violet
Group 16
0 4 8 12
Stimulus duration (s)
160.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
(G
reen
)
Group 16
Group 8
Double BisectionExperiment 2
SET LeT
Group 8, Group 16
0 4 8 12
Stimulus duration (s)
160.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
(G
reen
)
Group 8: 1-4 and 4-8
Group 16: 1-4 and 4-16
SET vs LeT vs Data
Group 8 Group 16
Duration of test stimulus (s)
0 4 8 12 16
Pro
babi
lity
( G
reen
)
0.0
0.2
0.4
0.6
0.8
1.0
Duration of test stimulus (s)
0 4 8 12 16P
roba
bilit
y ( G
reen
)
0.0
0.2
0.4
0.6
0.8
1.0
LeT: Predictions vs Data
0 4 8 12
Stimulus duration (s)
160.0
0.2
0.4
0.6
0.8
1.0
Pro
babi
lity
(G
reen
)
4 vs 164 vs 8
Stimulus duration (s)
0 4 8 12 16
Pro
babi
lity
( G
reen
)
0.0
0.2
0.4
0.6
0.8
1.0
Double BisectionExperiment 3
1 s Red 4 s Green
4 s Blue 16 s Violet
1 s Green
16 s Blue
1 s Blue
16 s Green
Group “Consistent”
Group “Inconsistent”
How fast will the new discrimination be learned?
LeT: Predictions vs dataGroup “Consistent”
Stimulus duration (s)0 4 8 12 16
Pro
babi
lity
Gre
en
0.0
0.2
0.4
0.6
0.8
1.0
Stimulus duration (s)0 4 8 12 16
Pro
babi
lity
Gre
en
0.0
0.2
0.4
0.6
0.8
1.0
Day 1Day 2Day 10
1 s Red 4 s Green
4 s Blue 16 s Violet
1 s Blue
16 s Green
LeT: Predictions vs dataGroup “Inconsistent”
1 s Red 4 s Green
4 s Blue 16 s Violet
1 s Green
16 s Blue
Day 2
Stimulus duration (s)0 4 8 12 16
Pro
babi
lity
Gre
en
0.0
0.2
0.4
0.6
0.8
1.0
Stimulus duration (s)0 4 8 12 16
Pro
babi
lity
Gre
en0.0
0.2
0.4
0.6
0.8
1.0
Double BisectionExperiment 3
1 s Red 4 s Green
4 s Blue 16 s Violet
1 s Green
16 s Blue
1 s Blue
16 s Green
1 s Red 4 s Green
4 s Blue 16 s Violet
Group “Consistent”
Group “Inconsistent”
Which discrimination(s) will be disrupted?
Sessions0 2 4 6 8 10
Pro
babi
lity
cor
rect
0.0
0.2
0.4
0.6
0.8
1.0
Sessions0 2 4 6 8 10
Pro
babi
lity
cor
rect
0.0
0.2
0.4
0.6
0.8
1.0
LeT: Predictions vs dataGroup “Consistent”
1 s Red 4 s Green
4 s Blue 16 s Violet
1 s Blue
16 s Green
LeT: Predictions vs dataGroup “Inconsistent”
Sessions0 2 4 6 8 10
Pro
babi
lity
cor
rect
0.0
0.2
0.4
0.6
0.8
1.0
Sessions0 2 4 6 8 10
Pro
babi
lity
cor
rect
0.0
0.2
0.4
0.6
0.8
1.0
1 s Red 4 s Green
4 s Blue 16 s Violet
1 s Green
16 s Blue
Conclusions• SET and its memories
– Failure of independence from alternatives
• LeT and the role of context– Which key to peck after a signal– Which key not to peck after a signal
Current work• LeT and the role of context
– Errorless (temporal) learning?
• Behavioral patterns– Structuring and restructuring of patterns
during the time signal.– Some videos of mediational(?), collateral(?)
behavior.
Behavioral patterns
• P689– 1s: approaching keys– 4s R-G and 4s B-Y: pecking– 16 s: approaching keys, pecking, pausing, pecking
• P65– 16sf: pecking and wing flapping;– 16s: pecking and wing flapping ERROR;– 38: pecking and wing flapping;
• P68– 34 & 54: wall pecking and head-in-the-feeder
Collaborators
• Paulo Rodrigues• Marco Vasconcelos• Paulo Pata• Joana Arantes• Antonio Fidalgo• Ana Paula Leite• Paula Magalhães
• Richard Keen• Ozlem Cevik• Paulo Guilhardi
The full picture
SET LeT
Machado & Keen (1999)
Duration of test stimulus (s) Duration of test stimulus (s)
1 2 8 164
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty (R
or G
)
1 2 8 164
0.0
0.2
0.4
0.6
0.8
1.0
1 s Red
4 s Green
4 s Blue
16 s Violet
Machado & Keen (1999)
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty (G
)
1 2 4 8 16
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty (R
)
1 2 4 8 16
Duration of test stimulus (s)
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty (G
)
1 2 4 8 16
Duration of test stimulus (s)
1 2 4 8 160.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty (R
)
