Are there bumps in cortex?
Re-evaluating the Mexican hat
Bard Ermentrout
University of Pittsburgh
Oct 2004
Are there bumps in cortex? – p.1/17
Persistent activity
Delayed response tasks
Occulomotor integrator
Head direction system
Are there bumps in cortex? – p.2/17
The standard theory
Persistent activity is an attractor
Determined by recurrent connections
Specific to the location - eg head angle, visual
space
Spatially compact and stationary
Are there bumps in cortex? – p.3/17
The standard model
Strong recurrent excitation
Stable rest state
Lateral inhibition
Amari model
ut (x, t) = −u(x, t) +
Z
∞
W (x − y)F (u(y, t)) dy
−∞
Are there bumps in cortex? – p.4/17
Amari’s bumps
u(x) =
W(x)
Z
∞
W (x − y)H(u(y) − θ) dy
−∞
u(x)
θ
x
−M
x+M
u(x) =
W(y) dy
x−M
M
Q(x)
θ
2M
x
2M
W(y) dy = Q(2M) =θ
u(M) =
0
Are there bumps in cortex? – p.5/17
Physiology and anatomy
Local excitatory connections
extend farther than local inhibitory
connections
Lateral inhibition ?
Why don’t we see bumps in slices?
Plenty of persistent activity
Mostly waves
Are there bumps in cortex? – p.6/17
Possible counter-arguments
Need some sort of modulators that aren’t in
slice
Fast linear inhibition:
u(x) = Je (x) ∗ F (u(x)) − aie Ji (x) ∗ v(x)
v(x) = aei Je (x) ∗ F (u(x))
implies
u(x) = [Je (x) − aie aei Je (x) ∗ Ji (x)] ∗ F (u(x))
which is a Mexican hat
... but inhibition is slower than excitation
Are there bumps in cortex? – p.7/17
Interactions between regions
Cortex is organized in layers over many different areas
Inputs from one area to the other are tightly focused
Projections can be reciprocal with long feedback loops
aee
a ei
Strong
focused
excitation
b ii
b ee
i
e
LAYER 1
aee
a ei
e
b ie
i
b ei
LAYER 2
Are there bumps in cortex? – p.8/17
Recurrent bistability
u0j = −uj + Fe (aee uj−1 + bee uj − bie vj )
τ vj0 = −vj + Fi (aei uj−1 + bei uj − bii vj )
Layer j receives strong excitatory projections
from layer j − 1 as well as its intrinsic local
circuit connections.
For example TC/RE <–> CTX
If aee large enough then bistable
Are there bumps in cortex? – p.9/17
Symmetric local dynamics
Assume uj = uk and vj = vk to reduce to one
pair of equations
ut = −u+Fe ((aee +bee )u−bei v)
No layer−layer
Normal
v
1.6
1.6
1.4
1.4
v
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
0
0.2
0.4
u
τ vt = −v+Fi ((aei +
0.6
0.8
1
1.2
-0.2
-0.2
0
0.2
0.4
u
0.6
0.8
1
1.2
Are there bumps in cortex? – p.10/17
Stability of symmetry



M =

Bee
Aee
Bei
Aei

Aee −Bie
0

Bee
0
−Bie 

Aei −Bii
0 
Bei
0
−Bii
decomposes into symmetric and asymmetric
#
"
"
Bee + Aee −Bie
Bee − Aee −Bie
MA =
MS =
Bei + Aei −Bii
Bei − Aei −Bii
Need detMA > 0 is detMS > 0 which is true.
Are there bumps in cortex? – p.11/17
The full network
Here are the constraints:
Local inhibition spreads less than local
excitation (anatomy)
Bumps with strong layer-layer excitation (in
vivo recordings)
No bumps without layer-layer excitation (slice)
Local waves with blocked inhibition (slice)
Are there bumps in cortex? – p.12/17
Bumps
LAYER 1
excitatory
inhibitory
LAYER 2
excitatory
inhibitory
t=0
t=20
0 < x < 100
0 < x < 100
0 < x < 100
0 < x < 100
Are there bumps in cortex? – p.13/17
Manipulation of long-range connect
LAYER 1
excitatory
inhibitory
LAYER 2
excitatory
inhibitory
t=0
weak layer−layer e to e
t=20
t=0
weak layer−layer e to i
t=20
0 < x < 100
0 < x < 100
0 < x < 100
0 < x < 100
Are there bumps in cortex? – p.14/17
No long-range, local disinhibition
excitatory
inhibitory
t=0
reduced inhibition
t=20
t=0
zero inhibition
0
x
100 0
x
t=20
100
Are there bumps in cortex? – p.15/17
Analysis
Assume weak local connections
Steady state symmetric
Are there bumps in cortex? – p.16/17
Analysis
Assume weak local connections
Steady state symmetric
u(x) = Fe (aee u(x)) + Fe0 (aee u(x))
× [bee Je (x) ∗ u(x) − bie Ji (x) ∗ v(x)]
v(x)
= Fi (aei u(x)) + Fi0 (aei v(x))
× [bei Je (x) ∗ u(x) − bii Ji (x) ∗ v(x)]
Are there bumps in cortex? – p.16/17
Analysis
Assume weak local connections
Steady state symmetric
u(x) = Fe (aee u(x)) + Fe0 (aee u(x))
× [bee Je (x) ∗ u(x) − bie Ji (x) ∗ v(x)]
v(x)
= Fi (aei u(x)) + Fi0 (aei v(x))
× [bei Je (x) ∗ u(x) − bii Ji (x) ∗ v(x)]
which simplifies to
Are there bumps in cortex? – p.16/17
Analysis cont’d
u(x) − Fe (aee u(x)) = Fe0 (aee u(x))
× [bee Je (x) ∗ u(x)
− Ji (x) ∗ bie Fi (aei u(x))]
Left-hand side is bistable between high and low values of u(x)
Take u(x) = uhi in 0 < x < a and u(x) = ulo elsewhere
For small enough this persists
Note integrals smooth out the discontinuity
Maybe use variant of Bates’ theorem?
Are there bumps in cortex? – p.17/17