scaling laws in cognitive science christopher kello cognitive and information sciences thanks to...
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Scaling Laws in Cognitive Science
Christopher KelloCognitive and Information Sciences
Thanks to NSF, DARPA, and the Keck Foundation
Reasons for FC in Cogsci• Intrinsic Fluctuations
• Critical Branching
• Lévy-like Foraging
• Continuous-Time Random Walks
= Disabled synapse = Unblamed synapses
= Enabled synapse = Blamed synapses
1. Choose a disabled synapse
2. If , enable with probability ρ
3. Set to
B
?
?
itiV ,
~B~B
1. Choose an enabled synapse
2. If , disable with probability ρ
3. Set to
BB
Spike triggers axonal & dendritic processes
~B
Sequ
ence
dPo
isson
Poiss
on+S
TDP
Source Reservoir
6.9 6.902 6.904 6.906 6.908 6.91
x 105
0
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Unit Time Interval
Sp
ike C
ou
nt
SequencePoissonPoisson+STDP
0 1 2 3 4 5
x 104Unit Time Interval
Sp
ike
Cou
nt
Sequence
Poisson
Poisson+STDP
CB on CB off
Intrinsic Fluctuations
• Neural activity is intrinsic and ever-present– Sleep, “wakeful rest”
• Behavioral activity also has intrinsic expressions– Postural sway, gait, any repetition
Lowen & Teich (1996), JASA
TN i
TN
TNTNTA
i
ii
2
21
TN i 1
Allan Factor Analyses Show Scaling Law Clustering
TTA
Intrinsic Fluctuations In Spike Trains
Intrinsic Fluctuations in LFPs
Beggs & Plenz (2003), J Neuroscience
Bursts of LFP Activity inRat Somatosensory Slice Preparations
Mazzoni et al. (2007), PLoS One
231 SSP
Burst Sizes Follow a 3/2 Inverse Scaling Law
Intrinsic Fluctuations in LFPs
Intact Leech Ganglia Dissociated Rat Hippocampus
Intrinsic Fluctuations in Speech
Am
pli
tud
e
Time
“Bucket” “Bucket” “Bucket” “Bucket”
12
9
3
6
3
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Trial Number
Power (dB)-40
-20
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Fre
qu
enc
y (K
Hz)
n n+1 n+2 n+3 n n+1 n+2 n+3
Buck Buck Buck Buck Ket Ket Ket Ket
Am
pli
tud
e
Time
“Bucket” “Bucket” “Bucket” “Bucket”
12
9
3
6
3
0
Trial Number
Power (dB)-40
-20
0
20
40
60
Fre
qu
enc
y (K
Hz)
n n+1 n+2 n+3 n n+1 n+2 n+3
Buck Buck Buck Buck Ket Ket Ket Ket
Intrinsic Fluctuations in Speech
0.15 kHz
6.05 kHz
13.15 kHz
Bucks KetsF
requ
ency
(K
Hz)
Inte
nsity
(st
anda
rdiz
ed)
Trial Number
0.15 kHz
6.05 kHz
13.15 kHz
Bucks KetsF
requ
ency
(K
Hz)
Inte
nsity
(st
anda
rdiz
ed)
Trial Number
Intrinsic Fluctuations in Speech
0.0 0.5 1.0 1.5 2.0
Alpha
0
30
60
90
120
150
Fre
qu
en
cy
M = 1.06SD = 0.26-0.85
Log f
Log
S(f)
S(f) ~ 1/fα
Scaling Laws in Brain and Behavior
• How can we model and simulate the pervasiveness of these scaling laws?
– Clustering in spike trains
– Burst distributions in local field potentials
– Fluctuations in repeated measures of behavior
Critical Branching• Critical branching is a critical point between
damped and runaway spike propagation
1~prepostc SN
1sub 1c 1super
Damped Runaway
pre
post
Spiking Network Model
PSPj,t : Ij,t = ωj
PSPk,t+τk
?, itiV
itiV ,
tjtt
titi IeVV i,
)'(',,
PSPk,t+τk
τk
Incoming PSP
Update Membrane(and floor at zero)
Crossed Threshold?(and not in refractory)
Reset Membrane
Outgoing PSPs forenabled synapses
ωkτk
ωk
LeakyIntegrate
&Fire
Neuron
Source
Sink
Rese
rvoir
Critical Branching Algorithm
= Disabled synapse = Unblamed synapses
= Enabled synapse = Blamed synapses
1. Choose a disabled synapse
2. If , enable with probability ρ
3. Set to
B
?
?
itiV ,
~B~B
1. Choose an enabled synapse
2. If , disable with probability ρ
3. Set to
BB
Spike triggers axonal & dendritic processes
~B
Critical Branching Tuning
0 1000 2000 3000 4000 5000 6000
Unit Time Interval X 10
Mea
n Lo
cal B
ranc
hing
Ra
tio
SequencePoissonPoisson+STDP
Tuning ON Tuning OFF
Allan Factor Results
100
101
102
103
100
101
102
Counting Time (T)
Alla
n F
acto
r A
(T)
Sequence
Poisson
Poisson+STDP
data5
data6
CB off
TN i
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TNTNTA
i
ii
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21
TN i 1
TTA
Se
qu
en
ce
dP
ois
so
nP
ois
so
n+
ST
DP
Source Reservoir
Neuronal Bursts
6.9 6.902 6.904 6.906 6.908 6.91
x 105
0
10
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Unit Time Interval
Spi
ke C
ount
SequencePoissonPoisson+STDP
Neuronal Avalanche Results
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10-8
10-6
10-4
10-2
100
Size
P(S
ize)
SequencePoissonPoisson+STDPdata4data5
1/f Noise in Simple Responses
Response Times Response Durations
10-4
10-3
10-2
10-1
100
10-1
100
101
102
Frequency
Po
we
r
Evenly Timed CuesRandomly Timed Cues
10-4
10-3
10-2
10-1
100
10-1
100
101
102
Frequency
Po
we
r
Evenly Timed CuesRandomly Timed Cues
Memory Capacity of Spike Dynamics
0 5 10 15 20 25 300.5
0.6
0.7
0.8
0.9
1
Time Lag
% C
orre
ct
BR ~ 1BR < 1 (~0.8)BR > 1 (~1.1)Random
0.7 0.8 0.9 1 1.10.68
0.69
0.7
0.71
0.72
0.73
0.74
Branching Ratio Bias
Mea
n %
Cor
rect
*Random
Critical Branching and FC
• The critical branching algorithm produces pervasive scaling laws in its activity.
FC might serve to:
– Analyze and better understand the algorithm
– Formalize the capacity for spike computation
– Refine and optimize the algorithm
Lévy-like Foraging𝑃 (𝑙 ) 𝑙−𝜇
1<𝜇<3
Animal Foraging
𝑃 (𝑡𝑖 ) (𝑡𝑖+1 )−𝜇
𝜇 2
Memory Foraging
𝑃 (𝑡𝑖 ) (𝑡𝑖+1 )−𝜇
𝜇 2
Lévy-like Visual Search
100
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Tile Size
Alla
n F
act
or V
aria
nce
Natural
Artificial
Natural
Artificial
Image
Eye
100
101
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103
10-6
10-5
10-4
10-3
10-2
10-1
100
Saccade Length
P(S
acca
de L
eng
th)
Natural
Artificial
Lévy-like Foraging Games
.05 .15 .25 .50
-2.2
-2.1
-2
-1.9
-1.8
-1.7Number of Resources Averaged
Resource Clustering
Slo
pe
Top 20 ScoresMiddle 20 ScoresBottom 20 Scores
25 50 100 150
-2.2
-2.1
-2
-1.9
-1.8
-1.7Degree of Clustering Averaged
Resource Quantity
Top 20 ScoresMiddle 20 ScoresBottom 20 Scores
“Optimizing” Search with Levy Walks• Lévy walks with μ ~ 2 are maximally efficient
under certain assumptions
• How can these results be generalized and applied to more challenging search problems?
Continuous-Time Random WalksIn general, the CTRW probability density obeys
Mean waiting time:
Jump length variance:
Human-Robot Search Teams
• Wait times correspond to times for vertical movements
• Tradeoff between sensor accuracy and scope
• Human-controlled and algorithm-controlled search agents in virtual environments
Conclusions
• Neural and behavioral activities generally exhibit scaling laws
• Fractional calculus is a mathematics suited to scaling law phenomena
• Therefore, cognitive mechanics may be usefully formalized as fractional order mechanics