The role of delays in shaping spatio-temporal
dynamics of neuronal activity in large
networks — Supplementary Information
Alex Roxin, Nicolas Brunel and David Hansel
May 23, 2005
In the rate model with threshold-linear transfer function and cosine coupling (the coupling from x to y is given by J0 + J1 cos(x − y)), the dynamics
can be written in terms of three order parameters m0 , m1 and ψ defined by
dx
m(x, t)dx
2π
Z
dx
m1 (t) =
m(x, t) cos(x − ψ(t))dx
2π
Z
dx
0 =
m(x, t) sin(x − ψ(t))dx
2π
Z
m0 (t) =
(1)
(2)
(3)
It then reads:
dx
I(x)
2π
Z
dx
ṁ1 (t) = −m1 (t) +
cos(x − ψ(t))I(x)
2π
Z
dx
ψ̇(t)m1 (t) =
sin(x − ψ(t))I(x)
h 2π
i
I(x) = I ext + J0 m0 (t − D) + J1 cos(x − ψ(t − D))m1 (t − D)
ṁ0 (t) = −m0 (t) +
Z
(4)
(5)
(6)
(7)
+
Stability of the stationary uniform state
The stability analysis of the stationary uniform state, m0 (t) = constant,
m1 = ψ = 0 yields four types of instabilities:
1
• Rate instability: J0 = 1
• Turing instability: J1 = 2
• Hopf instability with frequency ω given by ω = − tan(ωD): J0 =
1/ cos(ωD)
• Turing-Hopf instability with frequency ω given by ω = − tan(ωD):
J1 = 2/ cos(ωD)
For D ¿ 1 the oscillatory instabilities occur with frequency ω ∼ π/(2D) at
coupling strengths J0 ∼ −π/(2D), J1 ∼ −π/D.
The stationary bump and its stability
The stationary bump is characterized by m0 (t) ≡ m0 = constant, m1 (t) ≡
m1 = constant, ψ = 0. The values of m0 and m1 are determined selfconsistently [1]. The activity is non-zero in an interval [x0 − θ, x0 + θ] where
x0 is arbitrary and θ is related to J1 by the equation J1 = 4π/(2θ − sin(2θ)).
The stability analysis yields a dispersion relation:
!
Ã
1 + λ −λD
J1 θ J1 sin(2θ)
(1 + λ) −
J0 θ +
e
+
π
2
4
Ã
!
2
J0 J1 −2Dλ θ
θ sin(2θ)
2
+ 2 e
+
− sin (θ) = 0
π
2
4
2
(8)
A saddle-node bifurcation with λ = 0 (rate instability) occurs for
π
J0 =
θ − tan θ
This state can also undergo an Hopf bifurcation on a line which is obtained
by plugging λ = iω in the dispersion relation.
The oscillatory uniform (OU) state and its stability
The oscillatory uniform state is a solution of of the dynamics which has
the form m0 (t) = mh (t), m1 = ψ = 0 in which the input current I ≡
I ext −J0 mh (t−D) is negative in an interval [0, T1 ] where T1 > D and positive
in an interval [T1 , T ].
2
The OU state with R ≡ T − T1 ∈ [D, 2D]
The limit cycle
To determine mh (t) under the assumption that R ≡ T − T1 ∈ [D, 2D], we
solve the dynamical equation in the three intervals [0, T1 ], [T1 , T1 + D] and
[T1 + D, T ] successively:
• In [0, T1 ], we have I < 0, and hence
mh (t) = M e−t
(9)
• In [T1 , T1 + D], we have I > 0, I = I ext + J0 M eD−t . The condition
I(T1 ) = 0 leads to
I ext = −J0 M eD−T1
(10)
The firing rate is
³
´
mh (t) = M e−t + I ext 1 − eT1 −t + J0 M (t − T1 )eD−t
• Likewise, in [T1 + D, T ], we have I > 0,
³
I = I ext + J0 M eD−t + J0 I ext 1 − eD+T1 −t
+J02 M (t − D − T1 )e2D−t
´
(11)
(12)
The condition I(T ) = 0 leads to
³
I ext = −J0 M eD−T − J0 I ext 1 − eD−R
−J02 M (R − D)e2D−T
´
(13)
The firing rate is
³
´
mh (t) = M e−t + I ext 1 − eT1 −t + J0 M (t − T1 )eD−t
³
´
+J0 I ext 1 − eT1 +D−t − J0 I ext (t − T1 − D)eD+T1 −t
(t − T1 − D)2 2D−t
e
2
The condition mh (T ) = M leads to
+J02 M
³
(14)
´
M = M e−T + I ext 1 − e−R + J0 M ReD−T
³
´
+J0 I ext 1 − eD−R − J0 I ext (R − D)eD−R
+J02 M
(R − D)2 2D−T
e
2
3
(15)
Equations (10,13,15) lead to two equations which determine R and T as a
function of J0 :
³
1 − e−R = −J0 1 − eD−R (1 + R − D)
1 − e−T = J0 ReD−T + J02
´
(16)
(R − D)2 2D−T
e
2
(17)
Stability of the limit cycle
Assuming that:
m0 = mh + δm0
m1 = δm1
(18)
(19)
where δm0 and δm1 are small one finds:
˙ 0 = −δm0 + J0 I ext + J0 mh (t − D)
δm
h
i
+
δm0 (t − D)
(20)
h
i
˙ 1 = −δm1 + J1 I ext + J0 mh (t − D) δm1 (t − D)
δm
+
2
(21)
Integrating these equations yields:
• In [0, T1 ]
δm0 (t) = δM0 e−t
δm1 (t) = δM1 e−t
(22)
(23)
• In [T1 , T1 + D]
³
δm0 (t) = δM0 e−t 1 + J0 (t − T1 )eD
µ
δm1 (t) = δM1 e−t 1 +
´
J1
(t − T1 )eD
2
(24)
¶
(25)
• In [T1 + D, T ]
Ã
!
− T1 − D)2 2D
1 + J0 (t − T1 )e +
δm0 (t) = δM0 e
(26)
e
2
Ã
!
2
J1
−t
D
2 (t − T1 − D) 2D
1 + (t − T1 )e + J1
δm1 (t) = δM1 e
(27)
e
2
8
−t
D
4
(t
J02
Hence, we have
δm0 (T ) = δm0 (0)β0
δm1 (T ) = δm1 (0)β1
(28)
(29)
where the Floquet multipliers β0 and β1
β0 = 1
(30)
Ã
2
β1 = e−T 1 +
J1 D
(R − D) 2D
Re + J12
e
2
8
!
(31)
The homogeneous oscillatory solution is stable for |β1 | < 1.
We get three instability lines: two for β1 = 1:
J1 = 2J0
(32)
Re
(R − D)2
−D
J1 = −2J0 − 4
(33)
and a period doubling instability (β1 = −1) given by
J1 = 2
Re−D
e−D q 2
±
2
R − 2(R − D)2 (1 + eT )
(R − D)2
(R − D)2
Small D limit
When D ¿ 1, we rescale T1 = τ1 D, T = τ2 D, J0 = γ0 /D, J1 = γ1 /D. In the
limit D → 0, we find that τ1,2 are given by:
√
1 − γ0 + 2γ0 − 1
ρ ≡ τ2 − τ1 =
(34)
γ0
µ
¶
ρ
ρ
1+
(35)
τ2 = −γ0
3
2
τ1 = τ 2 − ρ
(36)
The consistency condition ρ < 2 gives γ0 < −4. In the whole range γ0 < −4
ρ (duration of the interval when I > 0) is between 2 and 1. Hence the
calculation is valid in this range.
The Floquet multipliers are:
β0 = 1
β1 = 1 +
(37)
γ1 ρ
+
2
γ12 (ρ
5
2
− 1)
+ O(D)
8
(38)
β1 = −1 (period doubling bifurcation) when
γ1 = −
2ρ −
q
4ρ2 − 16(ρ − 1)2
(ρ − 1)2
The OU state with R ≡ T − T1 ∈ [2D, 3D]
In the case R ≡ T − T1 ∈ [2D, 3D] one has to consider the dynamics in four
distinct intervals. Solving the dynamics in these intervals yields the following
equations which determine T and T1 a function of J0 :
³
1 − e−R = −J0 1 − eD−R (1 + R − D)
Ã
´
(R − 2D)2
)
1−e
(1 + R − 2D +
2
(R − D)2 2D−T
= J0 ReD−T + J02
e
2
(R − 2D)3 3D−T
+J03
e
6
+J02
1 − e−T
2D−R
!
(39)
(40)
The stability analysis can be performed in a similar way as above: One
finds that β0 = 1, and that
β1 = e
−T
Ã
(R − D)2 2D
(R − 2D)3 3D
J1
e + J13
e
1 + ReD + J12
2
8
48
!
β1 is always larger than −1. The instabilities lines for which β1 = 1 are given
by J1 = 2J0 , and the other one by solving the quadratic equation
e
2D (R
#
"
(R − 2D)3
(R − D)2
− 2D)3 2
J1 + e−D
J0 e D −
J1 +
24
12
4
(R − D)2 J0 e−D (R − 2D)3 e2D J02
R−
+
= 0 (41)
2
6
The traveling waves and their stability
The traveling waves are characterized by m0 = constant, m1 = constant,
ψ(t) = vt. The profile and the velocity of the waves can be determined by
solving self-consistently the fixed point equations of the dynamics [1]. In
6
particular one finds that the speed of the wave is given by v = − tan(vD)
and that at time t the total input to the neurons is supra-threshold only for
those in the range [x0 + ψ(t) − θ, x0 + ψ(t) + θ] where θ is given as a function
of J1 by
2π
(42)
J1 =
cos(vD)(θ − sin(θ) cos(θ))
The dispersion equation for the instabilities of the waves can be obtained
analytically. From this relation one finds that a rate instability occurs if
π
J0 =
θ − tan θ
One also finds oscillatory instabilities with an unstable mode with frequency
ω. The value of J0 and ω at the instability onset are given by
[A(ω) − d(θ)B(ω)]
π iωD
e (1 + iω)
θ
[A(ω) − c(θ)B(ω)]
2
2
A(ω) = (1 + iω) + v − (1 + iω + v 2 )e−iωD
J0 =
B(ω) = e
−iωD
³
2
1 + iω + v − (1 + v )e
θ + sin θ cos θ
θ − sin θ cos θ
θ2 + θ sin θ cos θ − 2 sin2 θ
d(θ) =
θ(θ − sin θ cos θ)
c(θ) =
2
−iωD
´
(43)
(44)
(45)
(46)
(47)
Note that J0 and ω depend on J1 since θ and J1 are related by Eq. (42).
Phase diagram in J0 − J1 plane as a function of
delay
Phase diagrams for various values of the delays D = 0, 0.05, 0.1, 0.2, 0.5, 1,
2, 5 are shown in Figures 1, 2.
References
[1] D. Hansel and H. Sompolinsky. Modeling feature selectivity in local cortical circuits. In C. Koch and I. Segev, editors, Methods in Neuronal
Modeling. MIT press, Cambridge, MA, 2nd edition, 1998.
7
D=0
D=0.05
20
20
SB
SB
OB
0
OU+SB
0
-20
-20
SU
J1
J1
SU
-40
-40
-60
-60
-80
-80
OU
OU+TW
TW
SW
-100
-100
-80
-60
-40
J0
-20
0
-100
-100
20
-80
-60
-40
J0
D=0.1
-20
0
20
0
20
D=0.2
20
20
OB
OB
SB
OU-OB
SB
0
OU+SB
0
OU-SB
SU
SU
OU
-20
OU
-20
OU+TW
J1
J1
TW
-40
-40
SW
OU-TW
TW
OU-SW
TW-SW
-60
SW-LW
SW
OU
-60
LW
OU-A
SW
SW
-80
-80
OU-SW
A
-100
-100
-80
A
OU-SW
-60
-40
J0
-20
0
20
-100
-100
-80
-60
-40
J0
-20
Figure 1: Phase diagram for short delays D = 0, 0.05, 0.1, 0.2. States are
named as in Figure 1 of the paper. In all diagrams except the one for D = 0.1,
only analytically determined lines are shown.
8
D=0.5
D=1
40
40
30
30
20
OB
20
SB
OU +
SB
10
SB
10
J1
J1
OU+SB
0
0
SU
SU
OU
OU
-10
-10
TW
OU+TW
SW
-20
-20
SW
-30
-40
-40
OU
+ TW
TW
-30
-20
-30
-10
0
-40
-40
10
-30
-20
J0
-10
0
10
J0
D=5
D=2
40
40
30
30
OU+SB
OU+SB
20
20
10
10
SB
J1
J1
SB
0
SU
0
SU
TW
OU
TW
OU
-10
OU+TW
-10
SW
-20
OU+TW
SW
-20
-30
-40
-40
-30
-30
-20
-10
0
10
J0
-40
-40
-30
-20
-10
0
J0
Figure 2: Phase diagram for longer delays D = 0.5, 1, 2, 5. States are named
as in Figure 1 of the paper. In all diagrams, only analytically determined
lines are shown. Note the change of scale compared with figure 1.
9
10
50
100
150
200
time
time (ms)
Figure 3: Left: ‘Lurching’ waves in the rate model for J0 = −10, J1 = −62
with D = 0.1. Right: Transient waves in the NSN. The waves are unstable
I
E
to a pair of oscillatory bumps. Parameters: N=2000, pE
0 = p0 = 0.2, p1 = 0,
I
p1 = 0.1, gmax,E = 0.01, gmax,I = 0.028, νext = 500, gmax,ext = 0.01 and
δ = 2.0 ms. The raster plot of the inhibitory population is shown.
10