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Dynamic Instabilitiesin
NeuroscienceBard ErmentroutOctober 2004
The Visions of Shamans:
The Vision of Shamans – p.1/35
How does an animal switch gaits?
Trot
Walk
GallopTransverse
LH RHRF LF
LH−RH LF−RF
LH LF RH RF
The Vision of Shamans – p.2/35
Dynamic Instabilities
Transitions from one state to anothergoverned by nonlinear equations
Mathematics uncovers common features
What are the basic principles for patternformation?
The Vision of Shamans – p.5/35
Dynamic Instabilities
Transitions from one state to anothergoverned by nonlinear equations
Mathematics uncovers common features
What are the basic principles for patternformation?
The Vision of Shamans – p.5/35
Meanings?
What do the geometric signs in UpperPaleolithic Art mean?
Compare to other modern culturesSan (bushman) rock paintingsShoshonean Coso tribeTukano tribes in BrazilAustralian aboriginal tribes
The Vision of Shamans – p.8/35
Meanings?
What do the geometric signs in UpperPaleolithic Art mean?
Compare to other modern culturesSan (bushman) rock paintingsShoshonean Coso tribeTukano tribes in BrazilAustralian aboriginal tribes
The Vision of Shamans – p.8/35
A Hypothesis
Lewis-Williams, Hedges, and other anthropologists suggestnonrepresentational paleolithic art inspired by shamanic visions
Psycho-active substances, eg datura (jimson weed), peyote, andyaje’ common in ceremonies
with flickering fire, chantingleads to altered states
The Vision of Shamans – p.10/35
Huichol shamanism
Huichol yarn painting depictsthe hunt for peyote
Huichol rug designs inspiredby visions
The Vision of Shamans – p.11/35
Entoptic phenomena
Visual images from within
Common in hallucinogenic drugs
Premigrainous auras
Flicker/pressure phosphenes
The Vision of Shamans – p.12/35
Entoptic phenomena
Visual images from within
Common in hallucinogenic drugs
Premigrainous auras
Flicker/pressure phosphenes
The Vision of Shamans – p.12/35
Form constants
Spiral/vortex
Funnel/tunnel
Cobwebs/filigrees
Exploding lightrays
Mosaics
The Vision of Shamans – p.16/35
Retino-cortical transform
a
e
log(e)
cortexa π/2
−π/2
−π/2
π/2
retina
(e, a) −→
(
λ log(1 + e/e0),−λea
e0 + e
)
The Vision of Shamans – p.17/35
Recapitulating
There are common patterns to shamanisticart
Transform to geometric patterns in cortex
How do these patterns arise?
The Vision of Shamans – p.20/35
Recapitulating
There are common patterns to shamanisticart
Transform to geometric patterns in cortex
How do these patterns arise?
The Vision of Shamans – p.20/35
Recapitulating
There are common patterns to shamanisticart
Transform to geometric patterns in cortex
How do these patterns arise?
The Vision of Shamans – p.20/35
Inside the box
computer
display
fast camera
Tsodyks et al Science 1999
2 mm
Spontaneous activityshows spatialperiodicity
Similar to evokedactivity
Visual system is poisednear “instability”
The Vision of Shamans – p.21/35
Why doesn’t it always happen?
Cortex is poised near instability
Manipulation must push it past the point
Drugs, flicker, pressure should be enough
The Vision of Shamans – p.23/35
The local equations . . .
E I
eecc ie
cei cii
decayrate
changeof activity
excitatorycoupling
inhibitorycoupling
sensoryinputoutput
dIdt
= _ + Fi ( E _ I + )τi cei cii TiI
dEdt
= _ E + Fe( E _ I + )τe cee ie Tec
The Vision of Shamans – p.24/35
. . . in a spatial array
τe
dEjk
dt= −Ejk + Fe[
∑
j′,k′
Wee(j − j′, k − k′)Ej′,k′
− Wie(j − j′, k − k′)Ij′,k′ + Te(t)]
τi
dIjk
dt= −Ijk + Fi[
∑
j′,k′
Wei(j − j′, k − k′)Ej′,k′
− Wii(j − j′, k − k′)Ij′,k′ + Ti(t)]
The Vision of Shamans – p.25/35
Dynamic instability
There is a constant equilibrium state
This can be made unstable
Translation and rotational symmetry forcesthe patterns
The Vision of Shamans – p.26/35
The Underlying Mechanism
positive
negative
inte
ract
ion
stre
ngth
+ + +0 0 __
__
+
space
Lateral Inhibition
The Vision of Shamans – p.27/35
How this works
while surrounding regionis depressed
Slight inhomogeneity
is amplified bylocal excitation
in turn, amplifyingfarther regions
and so on .....leading to a final patterned state
and depressing their neighbors
+++
+ +___ _ _
_
The Vision of Shamans – p.28/35
Then what?
Near the transition all dynamics is the same!
Nonlinear analysis is needed
Amplitude equations: E(x, y) ≈ r cos nx + s sinny
r′ = r(p − ar2− bs2) s′ = s(p − as2
− br2)
s = 0, r > 0 r = 0, s > 0
a < b a < b
r > 0, s > 0
a > b
The Vision of Shamans – p.29/35
Drugs
Mescaline, LSD, etc have common molecularmechanism
Bind to special serotonin receptors in brain
Increase in glutamate production =⇒greater excitation
Blocked by 5HT2A antagonists
High occurrence of 5HT2A in schizophrenia
The Vision of Shamans – p.30/35
Pressure phosphenes
E
I
Reduce
activitybackground
dI/dt=0
dE/dt=0
Pressure on optic nerve
suppression of inputs - like sensory deprivation
Paradoxical excitation
The Vision of Shamans – p.31/35
Flicker instability
time
P
P/2
Lateral Inhibitory Network
Each cell has periodically dampedimpulse response
Spatially uniform periodic input withdouble the natural frequency
Simulation
The Vision of Shamans – p.33/35
Other examples
Transition from rest to oscillation
Transition from asynchrony to synchrony(temporal patterns)
Spots vs Stripes
The Vision of Shamans – p.34/35
Other examples
Transition from rest to oscillation
Transition from asynchrony to synchrony(temporal patterns)
Spots vs Stripes
The Vision of Shamans – p.34/35
Other examples
Transition from rest to oscillation
Transition from asynchrony to synchrony(temporal patterns)
Spots vs Stripes
The Vision of Shamans – p.34/35
Conclusions
Dynamic instabilities underly many naturalpatterns
Idea of “lateral inhibition” is very generic
Local dynamics looks the same under themicroscope
The Vision of Shamans – p.35/35
Conclusions
Dynamic instabilities underly many naturalpatterns
Idea of “lateral inhibition” is very generic
Local dynamics looks the same under themicroscope
The Vision of Shamans – p.35/35