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A brief introduction to A brief introduction to Computational Neuroscience Computational Neuroscience Aaditya Rangan Courant, NYU
A brief introduction toA brief introduction to Computational NeuroscienceComputational Neuroscience
Aaditya RanganCourant, NYU
Visual PathwayVisual Pathway
Eye: Retinal CellsEye: Retinal Cells
Black and White Color Sensitive(found in most animals!)
Brain = 1.5mTONS of neurons! (okay, 1.5 pounds)Brain = 1.5mTONS of neurons! (okay, 1.5 pounds)
nucleus
soma
dendrite
axon
sheath
Schwann cell
node
axon terminal
Cerebral Cortex
Retinal Receptive Field
Cortical Receptive Field
Thalamic Receptive Field
Cones Rods
Off-center
On-center
45 degree slant135 degree slant
Rods
=
=
Off-center=
At low contrast, and over intermediate time-scales,
Both the retina and the thalamus function (approximately) as linear transducers
However, there is substantial connectivity and nonlinear processing within the cortex
LineLine--Motion IllusionMotion Illusion
Experiment — Line Motion Illusion
Grinvald et al.
Stimulus
ExperimentVoltage
ExperimentVoltage
Stimulus
Time
Brain = 1.5mTONS of neurons! (okay, 1.5 pounds)Brain = 1.5mTONS of neurons! (okay, 1.5 pounds)
nucleus
soma
dendrite
axon
sheath
Schwann cell
node
axon terminal
soma
axon
inputs
outputs
input
/
( ) ( ) ( ) ! Pulse Coupled spik
Nonlineare at
1( ) ( ) ( )
( ) EX
t LEAK LEAK EX
k
EX
EX EX
THRESHOLD
kt
k
t
EX
EX
V t G V V G V V
V V T
G t G t t T
t e τ
δτ
α μ −
∂ = − − − −
> →
∂ = − + −
⎡ ⎤= ⎣ ⎦
∑
Simple Neuronal Model Simple Neuronal Model –– Integrate and FireIntegrate and Fire
spike
THRESHOLDV
time
EXV
LEAKV
INV
V1
G1
V2
G2
Simple Neuronal Model Simple Neuronal Model –– Integrate and FireIntegrate and Fire
Model Cortex
2mm
Pref
erre
d O
rient
atio
n
Actual Cortex ~25mm2
,Inhib
( , ) ( ) ( ) ( ) ( ) ! Pulse Coupled spike at
Nonlinear
( , ) ( )i j
t i i LEAK i LEAK i i IN i i EX i i EX
ki TH
GABA
GABA
NM
GABA GABA
RESHOLD i
LOCAL
AMPA DA
kt i i x x j
j k
V x t G V V G V V G V V G V V
V V T
G x t S K t Tα∈
∂ = − − − − − − − −
> →
∂ = −∑{ }
{ }
{ }
Excit
itory
atory
Excita
,
tory,
( , ) (Stimulus) ( )
( , ) ( )
LGN
i i j
i j
NMDA NMDA
LOCAL kt i i x x x j
j k
LONG RANGE kt i
AMP
i
A AMPA
x x jj
AMPA
A
k
NMD
G x t F S K t T
G x t S K t T
α
α∈
−
∈
∂ = + −
∂ = −
∑
∑ ∑
∑ ∑
spike
THRESHOLDV
GABAα
AMPAα
NMDAα
timeLONG RANGEK −LOCALK
time
corti
cal s
pace
Reasonable Neuronal Model Reasonable Neuronal Model –– Integrate and Fire (Integrate and Fire (33 conductance typesconductance types))
7ms
3ms
80ms
EXV
LEAKV
INV
Stimulus
Experiment
Model
Time
Voltage
Voltage
ModelNMDA
Line Motion Illusion Stimulus
model experiment
Stimulus
Experiment
Model
Voltage
Voltage
ModelNMDA
Drifting SquareStimulus
Input? What does the Thalamus do?
77 968768584839 116106 1250ms 135291910
30 110spikes/sec
Firing
Rate:
LGN Time Kernel
Firin
g R
ate
16 36 100Time (ms)
1ox1o
LGN Spatial Kernel
Mechanisms — Intermittently De-Suppressed system
. . .
Time
VisualInput
Input LG
N fromC
orticalR
esponse(S
pikes)
Cortical
NM
DAN
Cortical
VoltageP
rofile
96877768584839 116106 1250
g
Predictions:
• Not a ‘wave’ of synaptic delays
• LGN time course + Cortical recruitment + Priming effect
• NMDA time-scale crucial
• Not sensitive to Sharp Edges
• Not sensitive to Linearity of Bar
• Proximity matters
• Lower contrast delays recruitment
What is going on?
Predicted line-motion-like variants
Hallucinations induced by Lysergic acid diethylamide ?
Orientation Selectivity
Grinvald et al.
Sur et al.
Orientation Domains approximately 0.5mm across
Retinotopy: a map from retina to V1
Large region dedicated to the fovea (F)
Visual field
Visual field
Visual cortex
Visual cortex
( )logire r iθ θ→ +
Could cortical interconnectivity be responsible for spiral hallucinations?(see Bressloff, Cowan et al.)
Coarse-grainedpatches
Embedded I&F neurons
1Retinotopy−
modelmodel
Artists depictionLong rangeconnections
Linking a Neuronal response to visual input does NOT imply function!
