scaling laws in cognitive science
DESCRIPTION
Scaling Laws in Cognitive Science. Christopher Kello Cognitive and Information Sciences Thanks to NSF, DARPA, and the Keck Foundation. Background and Disclaimer. Cognitive Mechanics…. Fractional Order Mechanics?. Reasons for FC in Cogsci. Intrinsic Fluctuations Critical Branching - PowerPoint PPT PresentationTRANSCRIPT
Scaling Laws in Cognitive Science
Christopher KelloCognitive and Information Sciences
Thanks to NSF, DARPA, and the Keck Foundation
Background and Disclaimer
Cognitive Mechanics…
Fractional Order Mechanics?
Reasons for FC in Cogsci• Intrinsic Fluctuations
• Critical Branching
• Lévy-like Foraging
• Continuous-Time Random Walks
= Disabled synapse = Unblamed synapses
= Enabled synapse = Blamed synapses
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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 6.91x 10
5
0
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Unit Time Interval
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ke C
ount
SequencePoissonPoisson+STDP
0 1 2 3 4 5x 104Unit Time Interval
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ke C
ount
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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
Ampl
itude
Time
“Bucket” “Bucket” “Bucket” “Bucket”
12
9
3
6
3
0
Trial Number
Power (dB)-40-20
0204060
Freq
uenc
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
Ampl
itude
Time
“Bucket” “Bucket” “Bucket” “Bucket”
12
9
3
6
3
0
Trial Number
Power (dB)-40-20
0204060
Freq
uenc
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 KetsFr
eque
ncy
(KH
z)In
tens
ity (s
tand
ardi
zed)
Trial Number
0.15 kHz
6.05 kHz
13.15 kHz
Bucks KetsFr
eque
ncy
(KH
z)In
tens
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tand
ardi
zed)
Trial Number
Intrinsic Fluctuations in Speech
0.0 0.5 1.0 1.5 2.0Alpha
0
30
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Freq
uenc
y
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
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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
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Critical Branching Algorithm
= Disabled synapse = Unblamed synapses
= Enabled synapse = Blamed synapses
1. Choose a disabled synapse2. If , enable with probability ρ3. Set to
B
?
?
itiV ,
~B~B
1. Choose an enabled synapse2. If , disable with probability ρ3. Set to
BB
Spike triggers axonal & dendritic processes
~B
Critical Branching Tuning
0 1000 2000 3000 4000 5000 6000Unit Time Interval X 10
Mea
n Lo
cal B
ranc
hing
Rat
io
SequencePoissonPoisson+STDP
Tuning ON Tuning OFF
Spike TrainsSe
quen
ced
Poiss
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isson
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Source Reservoir
Allan Factor Results
100 101 102 103
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Counting Time (T)
Alla
n Fa
ctor
A(T
)
SequencePoissonPoisson+STDP
data5data6
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TTA
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ence
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Neuronal Bursts
6.9 6.902 6.904 6.906 6.908 6.91x 105
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Neuronal Avalanche Results
100 101 102 103 104 10510-8
10-6
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Size
P(S
ize)
SequencePoissonPoisson+STDPdata4data5
Simple Response Series
Predictable Cues Unpredictable Cues
Spik
e Co
unt
Sour
ceRe
serv
oir
Time
1/f Noise in Simple Responses
Response Times Response Durations
10-4 10-3 10-2 10-1 10010-1
100
101
102
Frequency
Pow
er
Evenly Timed CuesRandomly Timed Cues
10-4 10-3 10-2 10-1 10010-1
100
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Frequency
Pow
er
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−𝜇1<𝜇<3
Animal Foraging
𝑃 (𝑡𝑖 ) (𝑡𝑖+1 )−𝜇
𝜇 2
Memory Foraging
𝑃 (𝑡𝑖 ) (𝑡𝑖+1 )−𝜇
𝜇 2
Lévy-like Visual Search
Lévy-like Visual Search
100 101 102 103 104 105100
101
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Tile Size
Alla
n Fa
ctor
Var
ianc
e
NaturalArtificialNaturalArtificial
Image
Eye
100 101 102 10310-6
10-5
10-4
10-3
10-2
10-1
100
Saccade Length
P(S
acca
de L
engt
h)
NaturalArtificial
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
Collaborators
• Gregory Anderson• Brandon Beltz• Bryan Kerster• Jeff Rodny• Janelle Szary
• Marty Mayberry• Theo Rhodes
• John Beggs• Stefano Carpin• YangQuan Chen• Jay Holden• Guy Van Orden