Gap junctions and emergent brain rhythms:
A mathematical study
Stephen Coombes
Some motivation
Gap junctions are ubiquitous
• between inhibitory cortical interneurons
• between hippocampal mossy fiber axons
• neuron to glia
• thalamo-cortical cells in LGN
connexin hemi-channels
B. W. Connors and M. A. Long, Electrical synapses in the mammalian brain, Annual Review of Neuroscience, 27 (2004), pp. 393–418.
Some motivation
Gap junctions are ubiquitous
• between inhibitory cortical interneurons
• between hippocampal mossy fiber axons
• neuron to glia
• thalamo-cortical cells in LGN
connexin hemi-channels
B. W. Connors and M. A. Long, Electrical synapses in the mammalian brain, Annual Review of Neuroscience, 27 (2004), pp. 393–418.
How do they contribute
• to phase-locking ?
Coupled Oscillator Theory
Some motivation
Gap junctions are ubiquitous
• between inhibitory cortical interneurons
• between hippocampal mossy fiber axons
• neuron to glia
• thalamo-cortical cells in LGN
connexin hemi-channels
B. W. Connors and M. A. Long, Electrical synapses in the mammalian brain, Annual Review of Neuroscience, 27 (2004), pp. 393–418.
How do they contribute
• to phase-locking ?
• to brain rhythms ??
Coupled Oscillator Theory
Beyond weak coupling ?
Some motivation
Gap junctions are ubiquitous
• between inhibitory cortical interneurons
• between hippocampal mossy fiber axons
• neuron to glia
• thalamo-cortical cells in LGN
connexin hemi-channels
B. W. Connors and M. A. Long, Electrical synapses in the mammalian brain, Annual Review of Neuroscience, 27 (2004), pp. 393–418.
How do they contribute
• to phase-locking ?
• to brain rhythms ??
Coupled Oscillator Theory
Beyond weak coupling ?
Theoretical work by
• Kopell
• Ermentrout
• Rinzel
• Chow
• Lewis
• Holmes
• Sherman
• Hansel
• Pfeuty
• Ostojic, Brunel & Hakim
• .................
Fast spiking interneuron model ?
Fast spiking interneuron model ?
Wang-Buzsaki
X. J. Wang and G. Buzsaki. Gamma oscillation by synaptic inhibition in a
hippocampal interneuronal network. Journal of Neuroscience,16:6402–
6413, 1996.
20
v
-20
-60
0
20
40
60
80
t 100
Fast spiking interneuron model ?
Wang-Buzsaki
X. J. Wang and G. Buzsaki. Gamma oscillation by synaptic inhibition in a
hippocampal interneuronal network. Journal of Neuroscience,16:6402–
6413, 1996.
20
v
-20
-60
0
20
40
60
Absolute integrate-and-fire model
80
t 100
J. Karbowski and N. Kopell. Neural Computation, 12:1573–1606, 2000.
20
v
-20
-60
0
20
40
60
80
t 100
Spike adaptation - tonic and burst firing
20
v
-20
-60
0
20
40
60
0
100
200
300
80
t 100
1
v
0
400 t 500
both tonic and bursting behavior
Mathematical
structure
gure
2:
Tonic
firing
in
the
aif
model
with
spike
adap
orporating awith
formthe
of spike
adaptation
[49]
explicit choice f (v) =the
|v
= 0.2,
v
=
1,
I
=
0.1
and
g
=
0.75.
th
a
0.4 and bursting behavior. For simplicity we
onic
a
v̇
=
|v|
+
I
−
a,
0.3
he explicit choice f (v) = |v|. In more detail w
by incorporating a form of spike adaptation [49] the aif m
subject
to a,the usual
IF reset
m
v̇
=
|v|
+
I
−
ȧ
=
−a/τ
,
a
both
tonic
and
bursting
behavior.
For
simplicity
we shal
0.1
m
m
a(T choice
) threshold
→fa(T
)+
, forweso
a /τa
with the reset
explicit
(v) = |v|.
In gmore
detail
w
0.2
t to the 0 usual
IF0.8fires
reset
mechanism
as well
v
0.4
model
tonically
as
shown
in
v̇ = |v| + I −0.4a,
ȧ = −a/τa ,
τa >
m
→ a(T ) +
g
/τ
,
for
a some ga > 0. For su
a
a
continuous)0.3dynamical system it
subject
to the as
usual
IF reset
mechanism
as well
as m
t
fires
tonically
shown
in
Fig.
2.
Since
the
plane,
where
one
can
also
plot
t
m
m
a(T ) → a(T ) + ga /τa , for
0.2 some ga > 0. For sufficie
uous)
dynamical
system
it
is
also
useful
to
vi
model fires For
tonically
as shown
in Fig.
2.gSince
themode
model
larger
values
of
the
a
0.1
where
one in
can
also
plot
the
system
nullcline
continuous)
dynamical
system
it
is
also
useful
to
view
o
Fig. 4. The
mechanism
for
th
0
plane,
where
one
can
also
plot
the
system
nullclines,
as
0
0.4
0.8 vin a
rger values reference
of ga the model
can
also
fire
to the geometry of the
0
Periodic orbits - closed form solution
−t/τa
).
period
∆
(7)
ing that v(∆) = vth .
on of ga is shown in
firing drops off with
may also construct
ch visit v < 0, period
he ideas above. The
oss v = 0.
ch
region
the separated
dynamics by
(upthetoline
threshold
reset) is gover
o two
regions
v = 0, and
so that
ynamics
(upPeriodic
threshold
and
by
ar
system.
Iftowe
denote
by -vreset)
andis vgoverned
the solution
solution
for v >
orbits
closed
form
+
−
we
denote by vthen
v− the solution
for v > gives
0 and us the close
respectively,
of parameters
+ andvariation
−t/τ
a
hen
of
gives us
the
closed
form
on variation
).
(7)
period
∆ parameters
Z t
±(t−t0 )
∓(s−t)
Z
t
v±(t−t
)e
+
e
[I − a(s)]ds,
± (t) )= v± (t0∓(s−t)
0
v
(t
)e
± that
0
ing
v(∆)+ = veth .
[I − a(s)]ds,
t0
t0
(5)
initial
data
vshown
t > t0 . For example, considering
± (t0 ) and
on
of
g
is
in
a
(t0 ) and t > t0 . For example, considering the ∆dic
tonic
solution
shown
in
Fig.
3
,
where
v
>
0
always,
t
firing
drops
off
with
on
shown
in Fig.
3
−t/τa , where v > 0 always, then we
that
, with a determined self-consistently from
t/τa a(t) = ae
may
also
constructself-consistently from a(∆) +
, with
a determined
= a, giving
ch visit v < 0, period ga
1
ga
1 a=
a = above. −∆/τ
(6)
he ideas
Thea . τa 1 − e−∆/τa .
τa 1 − e
oss
v
=
0.
e, from (5), the voltage varies according to
voltage varies according to
aτa
t
−t/τa
=−
vr e + I(e
(e − e(7) ).
vr e + I(ev(t)
− 1)
(e − 1)
e − ).
1 + τa
1 + τa
t
t
taτa
tt
−t/τa
ch
region
the separated
dynamics by
(upthetoline
threshold
reset) is gover
o two
regions
v = 0, and
so that
ynamics
(upPeriodic
threshold
and
by
ar
system.
Iftowe
denote
by -vreset)
andis vgoverned
the solution
solution
for v >
orbits
closed
form
+
−
we
denote by vthen
v− the solution
for v > gives
0 and us the close
respectively,
of parameters
+ andvariation
−t/τ
a
1gives us the closed form
hen
of
parameters
on variation
).
(7)
period ∆
Z
t
f
±(t−t0 )
∓(s−t)
Z
t
v±(t−t
)e 0.8 +
e
[I − a(s)]ds,
± (t) )= v± (t0∓(s−t)
0
v
(t
)e
± that
0
ing
v(∆)+ = veth .
t0
[I − a(s)]ds,
t0
(5)
0.6
t >
initial
data
vshown
t0 . For example, considering
± (t0 ) and
on
of
g
is
in
a
(t0 ) and t > t0 . For example, considering the ∆dic
tonic
solution
shown
in
Fig.
3
,
where
v
>
0
always,
t
0.4
frequency
firing
drops
off
with
f
on
shown
in Fig.
3
−t/τa , where v > 0 always, then we
that
, with a determined self-consistently from
t/τa a(t) = ae
may
also
constructself-consistently
, with
a determined
from a(∆) +
0.2
= a, giving
ch visit v < 0, period ga
1
ga
1 a= 0
.
a = above. −∆/τ
(6)
he ideas
Thea . τa01 − e−∆/τ
a
0.2
0.4
0.6 g a 0.8
τa 1 − e
oss
v
=
0.
e, from (5), the voltage varies according to
voltage varies according to
aτa
t
−t/τa
=−
vr e + I(e
(e − e(7) ).
vr e + I(ev(t)
− 1)
(e − 1)
e − ).
1 + τa
1 + τa
t
t
taτa
tt
−t/τa
Phase Response Curve (PRC)
A phase response curve (PRC) tabulates the
transient change in the cycle period of an
oscillator induced by a perturbation as a
function of the phase at which it is received.
R(φ)
HH
V
20
0.2
0
-20
-40
0.1
-60
-80
0
φ
2π
Easy to compute numerically
with XPPaut
0
-0.1
0
φ
1
Nice discussion at Scholarpedia :
http://www.scholarpedia.org
system ż =perturbation
F (z) with a∆z
T -periodic
solution Z(t) =
and
nfinitesimal
to the trajectory
at Z(t
time+tT=) 0.
The resu
∆z to
the trajectory
Z(t)defined
at-time
= 0.
trajectory
∆z(t)
induces
a phase shift
ast ∆θ
= The
θ(z) resulting
− θ(Z) (using
isochronal c
PRC
exact
solution
ft defined
der
in ∆z as ∆θ = θ(z) − θ(Z) (using isochronal coordinates) such
Call the orbit z = Z(t) where ż = F (z)
∆θ = "Q, ∆z#,
Introduce
a phase
∆θ = "Q,
∆z#, (isochronal coordinates) θ
(3.10)
nes the standard inner product, and Q = ∇Z θ is the gradient of θ evaluat
(Ermentrout
and Kopell 1991)
Adjoint
e shown
to
the
product,
andsatisfy
Q=∇
is theequation
gradient
of θ evaluated
at Z(t). This
Z θlinear
near equation
dQ
= D(t)Q,
D(t) = −DF T (Z(t))
dt
D(t)Q,
D(t) = −DF T (Z(t))
(3.11)
conditions ∇Z(0) θ · F (Z(0)) = 1/T and Q(t) = Q(t + T ). Here DF
evaluated
alongand
Z. Q(t)
The (vector)
is related
Q according
Z(0))
= 1/T
= Q(t +PRC,
T ). R,
Here
DF (Z)todenotes
the to th
neral
(3.10)
and R,
(3.11)
must betosolved
numerically
to obtain
PRC, say u
(vector)
PRC,
is related
Q according
to the
simple the
scaling
P [23].
However,
for the McKean
model
DF (Z)
piece-wise
linear and
must
be solved
numerically
to obtain
the PRC,
sayisusing
the adjoint
sed
form. Introducing
a labelling
as forlinear
the periodic
we re-write
(3
McKean
model DF (Z)
is piece-wise
and we orbit
can obtain
a
T
T
where
D
=
−A
.
The
solution
of
each
subsystem
is
given
by
Q
(t)
=
G
µ as forµ the periodic orbit we re-write (3.10) in the form
µ
labelling
µ
Qµ+1 (0),
= 1, 2, 3. Denoting
==
(q1G, qTµ2(T
) we
have the relation
olution
of for
eachµ subsystem
is given Q
by4 (T
Q4µ)(t)
µ − t)Qµ (Tµ )
. Denoting Q4 (T4q)1 =
! (q11, q2 ) we∗ have
" the relation
1
∗
1
f (vth ) − w + I + q2 g(vth , w ) = .
"
µ 1
T
1
1
∗
∗
system ż =perturbation
F (z) with a∆z
T -periodic
solution Z(t) =
and
nfinitesimal
to the trajectory
at Z(t
time+tT=) 0.
The resu
∆z to
the trajectory
Z(t)defined
at-time
= 0.
trajectory
∆z(t)
induces
a phase shift
ast ∆θ
= The
θ(z) resulting
− θ(Z) (using
isochronal c
PRC
exact
solution
ft defined
der
in ∆z as ∆θ = θ(z) − θ(Z) (using isochronal coordinates) such
Call the orbit z = Z(t) where ż = F (z)
∆θ = "Q, ∆z#,
Introduce
a phase
∆θ = "Q,
∆z#, (isochronal coordinates) θ
(3.10)
nes the standard inner product,
and Q initial
= ∇Z data
θ is the
gradient
of θ evaluat
we must choose
for Q
that guarantees
QT
(Ermentrout
and
Kopell
1991)
Adjoint
e shown
to
the
product,
andsatisfy
Q=∇
iscontinuous
theequation
gradient
of θ evaluated
at Z(t). condition
This
Z θlinear
trajectory
this
normalization
ne
near equation
at a single point in time. However, for the aif mod
dQ
is a single
in the orbit (at reset
= there
D(t)Q,
D(t)discontinuity
= −DF T (Z(t))
dt continuous.
We therefore need to establish
the con
D(t)Q,
D(t) = −DF T (Z(t))
(3.11)
+
∇
·
F
(Z(0))
=
1/T
Q(∆
Q(0).and
Introducing
components
of Q DF
as Q
Z(0)
conditions
∇Z(0) θ · F (Z(0)) ) == 1/T
Q(t) = Q(t
+ T ). Here
toTdemanding
continuity
dq1 /dq
evaluated
alongand
Z. Q(t)
Theequivalent
(vector)
is related
Qofaccording
th
Z(0))
= 1/T
= Q(t +PRC,
). R,
Here
DF
(Z)todenotes
the2 attorese
For
the orbit
given by (7)
with vthe
> PRC,
0 the say
Jacou
neral
(3.10)
and
(3.11)
must
be
solved
numerically
to
obtain
(vector) PRC, R, is related to Q according to the simple scaling
constant
matrix
»
–
P
[23].
However,
for
the
McKean
model
DF
(Z)
is
piece-wise
linear
and
must be solved numerically to obtain the PRC, say using 1the adjoint
−1
DF
=
,
sed
form.
Introducing
a
labelling
as
for
the
periodic
orbit
we
re-write
(3
McKean model DF (Z) is piece-wise linear and we can
0 obtain
−1/τa a
T
T
where
D
=
−A
.
The
solution
of
each
subsystem
is
given
by
Q
(t)
=
G
µ as forµ the periodic
µin closedµ
labelling
we equation
re-write(9)
(3.10)
form
and theorbit
adjoint
may in
be the
solved
Qµ+1 (0),
= 1, 2, 3. Denoting
(T
==
(q1G, qTµ2(T
) we
haveµthe
relation
olution
of for
eachµ subsystem
is given Q
by4−t
Q4µ)(t)
(T
)
µ − t)Q
µ
τa
t/τa
+
q
q
,
q
1 (t) = q1 (0)e
2 (t) = q2 (0)e
1 (0)
. Denoting Q4 (T4q)1 =
1
+
τ
! (q11, q2 ) we∗ have
" the relation
1
∗
1
f (vThe
)
−
w
+
I
+
q
g(v
,
w
)
=
.
2
th condition for continuity
th
of dq1T/dq2 at reset yields
"
µ
1
1
∗
∗
1
a
−t ż = F (z) with a T -periodic
t/τa
t/τ
−t
a
system
solution
Z(t)
=
Z(t
+
T
)
and
and
(9)
be solved [e
intime
closed
asresu
nfinitesimal
perturbation
∆z to
themay
at
0.
+trajectory
q1 (0)
0)e the
, adjoint
q2 (t)equation
= q2 (0)e
−t =
e form
].The
(11)
1
+
τ
a
∆z to
the trajectory
Z(t)
at
time
t
=
0.
The
resulting
trajectory
∆z(t)
induces
a phase
shift
defined
as
∆θ
=
θ(z)
−
θ(Z)
(using
c
PRC
exact
solution
τa isochronal
−t
t/τa
t/τa
+
q
(t)
= qas1 (0)e
,of dq
q
=
q
[e
−
2 (t)(using
2 (0)e
1 (0)relationship
ftq1defined
∆θ = θ(z)
− θ(Z)
isochronal
coordinates)
such
der
in ∆z
on
for
continuity
/dq
at
reset
yields
the
1
2
1
+
τ
a
Call the orbit
where
= "Q,
q2 (0)
q2 (∆)∆θ of
τ1∆z#,
a/dq2 at reset yields the rela
The condition
for =
continuity
dq
=−
, θ
(12)
Introduce
a phase
(isochronal
coordinates)
∆θ = "Q,
∆z#,
(3.10)
q1 (0)
q1 (∆)
1
+
τ
aZ θ is the gradient of θ evaluat
T
nes the standard
inner product,
and
Q
=
∇
(0)
q
(∆)
τ
we q
must
choose
initial
data
for
Q
that
guarantees
Q
2
2
a
=
=
−
,
(Ermentrout
and
Kopell
1991)
Adjoint
eormalization
shown
to
the
linear
equation
product,
andsatisfy
Q=
∇
θ
is
the
gradient
of
θ
evaluated
at
Z(t).
This
Z
continuous
trajectory
this
normalization
condition
ne
condition qgives
(0)
q
(∆)
1
+
τ
1
1
a
near equation
at a single point in time. However, for the aif mod
dQ
a discontinuity
1 T (Z(t))
whilst the
normalization
condition
gives
there
is
a
single
in the orbit (at(13)
reset
=
D(t)Q,
D(t)
=
−DF
q1 (0)[vr dt
+ I − Ta] − q2 (0)
= .
continuous.
Weτatherefore
the con
∆ needa to establish
D(t)Q,
D(t) = −DF
(Z(t))
(3.11)
1
+
∇Z(0) · F (Z(0))
= 1/T
Q(∆
)=
=+
Q(0).
Introducing
components
of Q DF
as Q
conditions
∇Z(0)
θ · F (Z(0))
1/T
and
Q(t)
= Q(t
+ T ).. Here
q
(0)[v
I
−
a]
−
q
(0)
=
2
1
r
neous solutions of (12)
and (13)
then
gives
the
adjoint
in
the
10
1.1
τ
∆
a
equivalent
to
demanding
continuity
of
dq
/dq
at
1
2
evaluated
alongand
Z. Q(t)
The (vector)
is related
Q according
th
Z(0))
= 1/T
= Q(t +PRC,
T ). QR,
Here
DF (Z)todenotes
the torese
v
For
the
orbit
given
by
(7)
with
v
>
0
the
Jaco
neral
(3.10)
and R,
(3.11)
must betosolved
numerically
to then
obtain
the
PRC,
say
u
The
simultaneous
of
(12)
and to(13)
gives
the
adj
(vector)
PRC,
related
Q according
the
simple
scaling
»is solutions
–
q
constant
matrix
5
»
–
1 piece-wise
P
[23].
However,
for
the
McKean
model
DF
(Z)
is
linear
and
κ
1
must
be solved
numerically
to obtain the PRC, say using 1the adjoint
−t
closed
form
−1
0.7
Q(t)
=
e
,
t
∈
[0,
∆),
(14)
DF
=
,
»aas
sed
form. Introducing
aa /(1
labelling
forlinear
the periodic
we re-write
(3
McKean
model
(Z)
is piece-wise
and– we orbit
can
a
+
τ
)
0 obtain
−1/τ
∆ T DF−τ
a
κ
T
1
0
−t
where
D
=
−A
.
The
solution
of
each
subsystem
is
given
by
Q
(t)
=
G
µ as forµQ(t)
µin
labelling
the periodic
orbit
we equation
re-write(9)
(3.10)
in
the
form
µ
=
e
,
t
∈
[0,
∆),
and
the
adjoint
may
be
solved
closed
−1
−τ
/(1
+
a4the
Q
(0),
for
µ
=
1,
2,
Denoting
(T
= τ=
(qa1)G, qTµq2(T
) we
have
the
relation
+
Iµ+1
− aτofa /(1
τa )] 3.. ∆
Ais plot
ofQ
adjoint
for
the
tonic
orbit
4µ)(t)
olution
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)
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0.3
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nand
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7.
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. in
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a 1 ) − w ∗a+ I + q g(v 1 , w ∗ ) =
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rossing
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ct gap junction coupling between two cells (one post- and
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uctance of the gap junction.
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pursue two approaches for studying networks of identica
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ct gap junction coupling between two cells (one post- and
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eft).
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0.4
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eft).
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eft). Right: Result for the
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cked
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riving
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he
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om linear
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asynchronous state
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re we
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omes
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ich is independent of both i and t. Hence, for an asynchronous s
ry neuron in the network is described by the same dynamical sys
mely
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ere
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ce again we may use variation of parameters to obtain a closed f
3.1
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re we will focus on a globally coupled network in the large N limit
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asynchronous
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1
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this case
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this
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re
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N
by time averages. In this case the coupling term for an asynchronous state
omes
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Z
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1
1
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X
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network
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ch is is
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lim ofof both
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0same
ry neuron
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every
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network
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ich
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v̇==|v|
|v|−−gv
gv +
+ II −
aa+
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ry neuron in the network is described by the same dynamical sys
where
ZZ ∆
mely
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1
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0
ere
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ution for the trajectory:
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±(t−t0 )/τ±
∓(s−t)/τ±
Z
v
(t)
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v
(t
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+
e
[I
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(19)f
ce again ±we may± use
tog −
obtain
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t
0 variation of parameters
namely
3.1
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asynchronous
v̇ = of
|v|the
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0,
ȧ = −
re we will focus on a globally coupled network in the large N limit
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of
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asynchronous
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v̇
=
|v|
−
gv
+
I
−
a
+
gv
ȧ
=
0 , N limit.
1
Existence
of
the
state
Here
we
will
focus
on
a
globally
coupled
network
in
the
large
In
s
case
we
have
the
useful
result
that
network
averages
may
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repla
re
Z
this
case
we
have
the
useful
result
that
network
averages
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∆
time
averages.
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the
coupling
term
for
an
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where
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and
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re
we
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a
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1
Z
by time averages. In this case the coupling term for an ∆
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omes
v
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1
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N
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Z
v
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0
1
1
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In this
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network
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1 case
1 averages
v(t
+
j∆/N
=
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→∞ N j=1
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e again weNmay
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o
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Z
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1
ch is for
independent
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tion
thelimtrajectory:
which
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t.t. Hence,
anan
asynchronous
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v(tii +
j∆/N
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N
∆
solution
for
the
trajectory:
0same
ry neuron
describedbybythe
thesame
dynamical
syst
every
neuronininthe
thenetwork
network
dynamical
system,
j=1 is described
Z
tZ
namely
mely
ich is independent of both ±(t−t
i and t.0 )/τ
Hence,
for
an
asynchronous
s
t
∓(s−t)/τ
± ȧ = −a/τa ,
±
v̇
=
|v|
−
gv
+
I
−
a
+
gv
,
(17)
0
v̇
=
|v|
−
gv
+
I
−
a
+
gv
,
ȧ
=
−a/τ
,
±(t−t
)/τ
∓(s−t)/τ
v
(t)
=
v
(t
)e
+
e
[I
0
a
0
± inv the
±
0
± same dynamical sys
±
ry neuron
network
is
described
by
the
+
e
± (t) = v± (t0 )e
where
t0
ZZ ∆
mely
ere
t0
1
∆
v(t)dt.
(18)
v̇ = |v| − gv v+0 I=−1a + gv
ȧ = −a/τa ,
0,
v
v(t)dt.
0 = ∆ 0
re
τ
=
1/(1
∓
g)
and
I
=
I
+
gv
.
A
self-consis
±
g
0
∆
where
τ
=
1/(1
∓
g)
and
0 Ig = I + gv0 . A self-cons
±
ere
Once again we may use variation ofZ parameters
to obtain a closed form
∆
advanced-retarded
ode
self-consistent
periodic
solution
1
ce
again
use
variation
of
parameters
to
obtain
a
closed
f
(∆,
v(∆,
)wethe
is
now
obtained
from
the
simultaneous
0for
pair
vmay
)
is
now
obtained
from
the
simultaneou
solution
trajectory:
0
v0 =
v(t)dt.R ∆R
ution for the trajectory:
∆ 0Z t −1 −1 ∆
tions
v(∆)v(∆)
= vth
and
v± 0 =
v(t)dt.
For
ex
equations
=±(t−t
vth0 )/τand
vZ 0∆
=
∆
v(t)dt.
For
∓(s−t)/τ0
±
v
(t)
=
v
(t
)e
+
e
[I
a(s)]ds,
(19)f
ce again ±we may± use
tog 0−
obtain
a closed
t
0 variation of parameters
splay state as a function of coupling
0.6
Δ
3.4
v0
Δ
0.5
v0
3
0.4
2.6
2.2
0.3
0
0.2
0.4
0.6
0.8
g 1
eλT − 1
v#(λ)
0
Stability of the asynchronous state
u(t)e−λt dt is the Laplace transform of u(t). By taking the Laplace tra
we Stability
see that -this
may beapproach
written as
generalise
for synapses.
u
#(λ) = (1 − e
−λT
)#
v (λ),
Re λ > 0.
e-values as zeros of
write λ (in the right hand complex plane) as the solution to E(λ) = 0, w
e
E(λ) =
+ gλT
v#(λ)
λT
!
1
R(θ)eλθT dθ.
0
0 we see that E(0) = 0 as expected. Writing λ = ν + iω then the pa
LTsolution
of orbit of ER (ν, ω) = 0 and EI (ν, ω) =PRC
of splay
ultaneous
0, where
ER (ν, ω) = R
ν + iω). In terms of the Fourier coefficients for R(θ) and v(t) we ma
r (4.42) using
! 1 of the asynchronous state in networks
C. van Vreeswijk, Analysis
coupled oscillators,
'of strongly
Rn
λθT
Physical Review Letters, 84 (2000),
pp. 5110–5113.λT
R(θ)e
0
dθ = (e
− 1)
n
2πin + λT
spectrum for fixed g
2
ω
2
ω
0
−2
−0.5
0
ν
ga = 1.5
0.5
0
−2
−0.5
0
ν
ga = 2.5
0.5
Bifurcation
0.3
ν
0.2
ω
1.6
ω
0.1
1.4
0
1.2
-0.1
ν
-0.2
1.5
λ = ν + iω
Re (λ) = ν = 0
1
2
2.5
ga
3
3
8
?
ga
ga
6
?
synchronized bursting
2
synchronized bursting
4
asynchrony
1
2
asynchrony
0
0
0.2
0.4
0.6
0.8
g
1
0
0
50
100
150
τa
200
Synchronised bursting
N
!
1
E(t) =
vi (t)
N i=1
Mean field rhythms
S. K. Han, C. Kurrer, and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators,
Physical Review Letters, 75 (1995), pp. 3190–3193.
0.2
w
0.4
Morris-Lecar
Type I
f
0.2
0.1
0
-0.4
-0.2
0
v
0.2
0
0.07
0.075
global coupling - mean field signal as average membrane potential
I
0.08
Mean field rhythms
S. K. Han, C. Kurrer, and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators,
Physical Review Letters, 75 (1995), pp. 3190–3193.
0.2
w
0.4
Morris-Lecar
Type I
f
0.2
0.1
0
-0.4
-0.2
0
v
0.2
0
0.07
0.075
I
0.08
global coupling - mean field signal as average membrane potential
E
0.1
0
-0.1
-0.2
1000
2000
3000
t
4000
A tractable model
w
3
0.6
2
4
0.4
0.2
1
0
0
0.2
0.4
0.6
v
0.8
4
A tractable model
w
3
1
0.6
2
-0.2
0.4
0.2
4
0.6
v
1
e McKean model0.2has a nullcline with an piece-wise linear cubic shape (d
a linear one associated with ẇ = 0 (dotted blue line). Parameters are µ =
1
ine corresponds to a stable periodic orbit.
0
0
0.2
0.4
0.6
v
e two-dimensional McKean neuron take the form
µv̇ = f (v) − w + I,
ẇ = g(v, w),
w) are given by
0.8
4
[53, 65]. The equations for0.6
a single two-dimensional McKean neuron take the form
A tractable model
4 µv̇ = f (v) − w + I,
ẇ = g(v, w),
0.4
w
3 by
where the functions f (v) and g(v, w) are given
1
0.6


−v,
v 2< a/2

-0.2
0.2
0.6 v 1
f (v) = v − a, a/2 ≤ v ≤ (1 + a)/2 ,

0.4 Fig. 2.1. The phase plane 4for the McKean
1 − v,hasv a>nullcline
(1 + a)/2
model
with an piece-wise
1
-0.2
0.2
0.6
v
1
linea
green line) corresponding to v̇ = 0 and a linear one associated with ẇ = 0 (dotted blue line). P
g(v, w) = v − γw.
I = 0.5,γ = 0.5, and a = 0.25. The red line corresponds to a stable periodic orbit.
e McKean model
with an
piece-wise
cubic
shape (d
0.2has
Here,
µ >a0, nullcline
γ > 0, I is a constant
drive
and f (v) is alinear
piece-wise
linear caricature
o
Nagumo
(v)
v(1 −two-dimensional
v)(v blue
− a), whilst
g(v,Parameters
w) neuron
describes
the
linear
[53, 65]. nonlinearity
Thewith
equations
for 0=
a single
McKean
take
the
form
a linear one associated
ẇ f=
(dotted
line).
are
µ d=
1 is the function f (v) = −v + H(v −
Another
popular orbit.
choice for f (v)
ine corresponds variable.
to a stable
periodic
µv̇ =
f (v) nonlinearity
− w + I,
Heaviside
step function. The analysis of this
latter
has been pursued
0
[64, 66]. A phase-plane plot of McKean model
is w),
shown in Fig. 2.1. To genera
ẇ = g(v,
often associated with either a homoclinic bifurcation or a saddle-node on a limit
where0 the
w) are dynamics
given
v the0.8
0.2f (v) and
0.4
0.6by for
requires
thefunctions
introduction
of ag(v,
nonlinear
gating variable, as in th
 A piece-wise linear idealization of th
or the cortical neuron model of Wilson [69].
−v,
v < a/2
has already been introduced by Tonnelierand Gerstner [66], and since the nullcline
f (v) shape,
= v −ita,tooa/2
≤v≤
(1
+ a)/2 , Indeed m
in the Wilson model has a quadratic
is
easy
to
caricature.


1 −tov,be vdescribed
> (1 + a)/2
g(v, w) underlying a Type I response appear
with the simple conti
g(v, w) =%v − γw.
(v − γ1 w + b∗ γ1 − b)/γ1 , v < b
g(v, w) =
,
∗
Here, µ > 0, γ > 0, I is a constant drive and
f
(v)
is
a
piece-wise
linear
caricature
of t
(v − γ2 w + b γ2 − b)/γ2 , v ≥ b
Nagumo nonlinearity f (v) = v(1 − v)(v − a), whilst g(v, w) describes the linear dyn
e two-dimensional McKean neuron take the form
µv̇ = f (v) − w + I,
ẇ = g(v, w),
w) are given by
ts of the full nonlinear flow. To see how we do
v this it is first convenie
ż = Az + b,
z=
,
w
al linear system of the form
Periodic orbits - PWL! systems
"
× 2 matrix A has components Aij , i, j = 1, 2, and
v b is constant 2 × 1 inp
ż
=
Az
+
b,
z
=
,
3.1) may be written in the form
w
# t
2 matrix A has components Aij , i, j = 1,At
2, and b is constant 2 × 1 in
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
1) may be written in the form
0
#
t
2
real eigenvalues λ± , such that Aq± = λ± q± with
q± ∈ R , given by
At
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
$
0
2
TrA ± (TrA) − 4 det A
λ± =
, 2
eal eigenvalues λ± , such that Aq± = λ± q2± with q± ∈ R , given by
‘diagonlise’ and write G(t) in the$
computationally
useful form G(t) = P
2
TrA ± (TrA) − 4T det A
λ− ), P = [q+ , q− ], andλ±
q±== [(λ± − A22 )/A21 , 1] . If A ,has complex eigenva
d complex eigenvector is q such that Aq = 2(ρ+iω)q, q ∈ C2 . In this case G(t) =
‘diagonlise’ and write G(t) in the computationally useful form G(t) = P
" − A )/A , 1]T . If A has!complex
" eigenv
λ− ), P = [q+ , q−!],cos
and
q
=
[(λ
22
21
θ ±
− sin θ ±
0 1
2=
Rθ =
, AqP==(ρ+iω)q,
[Im(q), Re(q)]
,
complex eigenvector
is
q
such
that
q
∈
C
.
In
this
case
G(t)
sin θ
cos θ
ω
% ρ%
ts of the full nonlinear flow. To see how we do
v this it is first convenie
ż = Az + b,
z=
,
w
al linear system of the form
Periodic orbits - PWL! systems
"
× 2 matrix A has components Aij , i, j = 1, 2, and
v b is constant 2 × 1 inp
ż
=
Az
+
b,
z
=
,
3.1) may be written in the form
w
# t
2 matrix A has components Aij , i, j = 1,At
2, and b is constant 2 × 1 in
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
1) may be written in the form
0
# t
G(t) and K(t) can be calculated in closed 2form
real eigenvalues λ± , such that Aq± = λ± q± with
q± ∈ R , given by
At
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
$
0
2
TrA ± (TrA) − 4 det A
λ± =
, 2
eal eigenvalues λ± , such that Aq± = λ± q2± with q± ∈ R , given by
‘diagonlise’ and write G(t) in the$
computationally
useful form G(t) = P
2
TrA ± (TrA) − 4T det A
λ− ), P = [q+ , q− ], andλ±
q±== [(λ± − A22 )/A21 , 1] . If A ,has complex eigenva
d complex eigenvector is q such that Aq = 2(ρ+iω)q, q ∈ C2 . In this case G(t) =
‘diagonlise’ and write G(t) in the computationally useful form G(t) = P
" − A )/A , 1]T . If A has!complex
" eigenv
λ− ), P = [q+ , q−!],cos
and
q
=
[(λ
22
21
θ ±
− sin θ ±
0 1
2=
Rθ =
, AqP==(ρ+iω)q,
[Im(q), Re(q)]
,
complex eigenvector
is
q
such
that
q
∈
C
.
In
this
case
G(t)
sin θ
cos θ
ω
% ρ%
ts of the full nonlinear flow. To see how we do
v this it is first convenie
ż = Az + b,
z=
,
w
al linear system of the form
Periodic
orbits - PWL2 ! systems
A11 + A22
2
,
ω = A11 A22 − A12 A21 − ρ > "0.
(3
× 2 matrix
A has components Aij , i, j = 1, 2, and
2
v b is constant 2 × 1 inp
ż
=
Az
+
b,
z
=
,
3.1) may be written in the form
w
cessary for carrying out computations, is given in Appendix A
# t
A has components
Aij , i,ofj choice
= 1,At
2, it
and
bconvenient
is constantto2 break
× 1 in
it2 matrix
of the
piece-wise
linear
model
is
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
1)
written
the formis governed by a linear dynamical
onmay
eachbepiece
the in
dynamics
system.
0
cus on the type
shown ininFigures
and
#
G(t) of
andperiodic
K(t) canorbits
be calculated
closed 22.1
form
t 2.2. In b
real
eigenvalues
λ± , such
that Aq
q± µ
with
q±. .∈. ,R
, We
given
by the ti
At
± = λ±by
consider
four
distinct
pieces,
labelled
=
1,
4.
denote
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
tes as
Tµ . For 4each
piece
write$zµ (t) =
Gµ (t)zµ (0) + K0µ (t)bµ with
Introduce
labels
andwe
write
2
TrA ± (TrA) − 4 det A
λ± =
, 2
(3.2) under the replacement
of A by Aµ . For the McKean
model we h
eal eigenvalues λ± , such that Aq± = λ± q2± with q± ∈ R , given by
‘diagonlise’
and #write G(t) in "the$
computationally
useful form G(t) = P
#
"
2
TrA −1/µ
± (TrA)
− 4T det A
1/µ
−1/µ
−1/µ
λ=
q±A
[(λ± − A22 )/A21 , 1] , . If A ,has complex eigenva
λ±
== =
− ), P = [q+ , q− ], and
,
(3
1
2
2
2
−γ
1 eigenvector
−γ
d complex
is q such that 1Aq = (ρ+iω)q,
q ∈ C . In this case G(t) =
‘diagonlise’ and write G(t) in the computationally useful form G(t) = P
" − A )/A , 1]T . If A has!complex
" eigenv
λ− ), P = [q+ , q−!],cos
and
q
=
[(λ
22
21
θ" ±
− sin θ ±#
0 1
"
#
"
#
2=
R
=
,
P
=
[Im(q),
Re(q)]
,
θ
complex
eigenvector
is
q
such
that
Aq
=
(ρ+iω)q,
q
∈
C
.
In
this
case
G(t)
(I − a)/µ
(1
+
I)/µ
I/µ
sin θ
cos θ
ω
% ρ%
, b2 =
, b4 =
,
(3
0
0
0
ts of the full nonlinear flow. To see how we do
v this it is first convenie
ż = Az + b,
z=
,
w
al linear system of the form
Periodic
orbits - PWL2 ! systems
A11 + A22
2
,
ω = A11 A22 − A12 A21 − ρ > "0.
(3
× 2 matrix
A has components Aij , i, j = 1, 2, and
2
v b is constant 2 × 1 inp
ż
=
Az
+
b,
z
=
,
3.1) may be written in the form
w
cessary for carrying out computations, is given in Appendix A
# t
A has components
Aij , i,ofj choice
= 1,At
2, it
and
bconvenient
is constantto2 break
× 1 in
it2 matrix
of the
piece-wise
linear
model
is
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
1)
written
the formis governed by a linear dynamical
onmay
eachbepiece
the in
dynamics
system.
0
cus on the type
shown ininFigures
and
#
G(t) of
andperiodic
K(t) canorbits
be calculated
closed 22.1
form
t 2.2. In b
real
eigenvalues
λ± , such
that Aq
q± µ
with
q±. .∈. ,R
, We
given
by the ti
At
± = λ±by
consider
four
distinct
pieces,
labelled
=
1,
4.
denote
z(t) = G(t)z(0) + K(t)b,
G(t) = e ,
K(t) =
G(s)ds.
tes as
Tµ . For 4each
piece
write$zµ (t) =
Gµ (t)zµ (0) + K0µ (t)bµ with
Introduce
labels
andwe
write
2
TrA ± (TrA) − 4 det A
λ± =
, 2
(3.2) under the replacement
of A by Aµ . For the McKean
model we h
eal eigenvalues λ± , such
w that Aq± = λ± q2± with q± ∈ R , given by
3
0.6
‘diagonlise’
and #write G(t) in "the$
computationally
useful form G(t) = P
#
"
2
2
TrA −1/µ
± (TrA)
− 4T det A
1/µ
−1/µ
−1/µ
λ=
q±A
[(λ± − A422 )/A21 , 1] , . If A ,has complex eigenva
λ0.4±
== =
− ), P = [q+ , q− ], and
,
(3
1
2
2
2
−γ
1 eigenvector
−γ
d complex
is q such that 1Aq = (ρ+iω)q,
q ∈ C . In this case G(t) =
‘diagonlise’ and write0.2G(t) in the computationally useful form G(t) = P
" − A )/A 1, 1]T . If A has!complex
" eigenv
λ− ), P = [q+ , q−!],cos
and
q
=
[(λ
22
21
θ" 0±
− sin θ ±#
0 1
"
#
"
#
2=
R
=
,
P
=
[Im(q),
Re(q)]
,
θ
complex
eigenvector
is
q
such
that
Aq
=
(ρ+iω)q,
q
∈
C
.
In
this
case
G(t)
(I − a)/µ
(1
+
I)/µ
I/µ
sin θ
cos θ
ω
% ρ%
, b2 =
, b0.4
, 0.8
(3
4 =
v
0
0.2
0.6
0
0
0
b"2 = ∗ #
, b"4 = ∗ #
,
(3.9)
b
−
b/γ
b
−
b/γ
/γ2
2
(1 + I)/µ
(I − a)/µ 1
, b2 = ∗
, b4 = ∗
, periodic
(3.9)
Continuous
and
b #− b/γ2 "
− b/γ21 "
1b
1
2
"
#
ucing two voltage thresholds vth and vth , where (v#th
, vth
) = (a/2, (1 +
(1 + I)/µ
(I − a)/µ
(I
− 2a)/µ
1
,(b,initial
b(1
, 2 Type
b4 = I∗ model,
,
(3.9)
=(vth∗•, vChoose
)
=
+
a)/2)
that
we
can
2 =
1 for the
1
2means
∗
data
th
b −
b/γand
b
−
b/γ
b −voltage
b/γ2 thresholds
g two
vth
vth , where
(v
,
v
)
=
(a/2,
(1
+
2
1
th
th
1
∗
∗
hoosing
initial
data
such
that
z
(0)
=
(v
,
w
)
(with
w
as
yet
undeter2
1
th
I model, means that we can
h , vth ) = (b, (1 + a)/2) for the Type 1
2
1
2
0)
+
K
(T
)b
,
for
µ
=
1,
2,
3.
The
‘times-of-flight’
T
are
determined
1
∗
∗
.
Introducing
two
voltage
thresholds
v
and
v
,
where
(v
,
v
) = (a/2, (1 +
µ
µ
µ
µ
1ng initial data such that z1 (0) = (v ,th
th
th
th
w
)
(with
w
as
yet
undeter“Times
of
flight”
th2by threshold 1crossings
•
2 determined
1
1
2
conditions:
v
(T
=
v
,
v
(T
)
=
v
,
v
(T
)
=
v
,
and
v
(T
)
=
v
. can
odel
and
(v
,
v
)
(b,
(1
+
a)/2)
for
the
Type
I
model,
means
that
we
1 µ
1 = 1, 2,
2 ‘times-of-flight’
3 3
4 4
th 3. 2The
th
th
th
th
Kµ (Tµ )bµ , th
for
T
are
determined
µ∗
1
∗
∗
rbit
by
choosing
initial
data
such
that
z
(0)
=
(v
,
w
)
(with
w
found
by
solving
w
(T
)
=
w
(0),
thus
yielding
w
and
1
1period
4 ) = v12 , v 1(T ) = vth
itions: v (T ) = v 2 ,4v (T
, and v (T ) the
=asvyet
. undeter,
1
1
th
2
2
th
3
3
th
4
4
th
(T
)z
(0)
+
K
(T
)b
,
for
µ
=
1,
2,
3.
The
‘times-of-flight’
T
are
determined
µ µ µ
µ µ µ
µ
∗
nd by solving w4 (T4 ) = w1 (0),
thus yielding
w and 1the period
2
2
1
crossing
conditions:
v1 (T1 ) = vth , v2 (T2 ) = vth , v3 (T3 ) = vth , and v4 (T4 ) = vth
.
Ensure
periodicity
•
solution are also possible. Two of these involve only a single
∗ threshold
then be found by solving w
1 4 (T4 ) = w1 (0), 2thus yielding w and the period
stion
through
thepossible.
section v Two
= vthof
, but
notinvolve
v = vthonly
andaanother
which crosses
are also
these
single threshold
w
Calling
these
thev subperiodic
orbit respectively
1 and
2
3 supra-threshold
ough
the
section
=
v
,
but
not
v
=
v
and
another
which
crosses
th possible. Two th
periodic
solution
are
also
of
these
involve
only
a
single threshold
0.6
using
the the
approach
(and
notation)
The 2orbit
sub-threshold
orbit is
1 above.
ing
these
suband
supra-threshold
periodic
respectively
ch crosses through the section
1
∗v =2vth , but not v = vth and1another which crosses
{1,
4} 1approach
with z1 (0)(and
=
(vnotation)
, w ), subject
to
v1sub-threshold
(T1 ) = vth = orbit
v4 (T4is) and
th
gv the
above.
The
4
=0.4vth . Calling these the sub- and supra-threshold periodic orbit respectively
1
∗is specified by the restriction
1
+
T
.
The
right
orbit
µv4=
{2,
3} with
4
}
with
z
(0)
=
(v
,
w
),
subject
to
v
(T
)
=
v
=
(T
)
and
olved for
using
the
approach
(and
notation)
above.
The
sub-threshold
orbit is
1
1
1
4
th
th
2
=
v
(T
)
and
w
(T
)
=
w
(0),
so
that
T
=
T
+
T
final
)
=
v
1 by
∗the
1. The
3
3
3
3
2
2
3
th
.
The
right
orbit
is
specified
restriction
µ
=
{2,
3}
with
0.2
on
µ
=
{1,
4}
with
z
(0)
=
(v
,
w
),
subject
to
v
(T
)
=
v
=
v
4
1
1 0 1
4 (T4 ) and
th
th
2 = T + Tand
thresholds,
is wdefined
by
simply
by
v(T
)
=
v
and
v(T
)
=
v(0)
th
The
right
orbit
is
specified
by
the
restriction
µ
=
{2,
3} with
= v13 (T3 )4 .and
(T
)
=
w
(0),
so
that
T
=
T
+
T
.
The
final
vTth
1
3 3
2
2
3
2
0 that
orbit.
We
shall
anw3orbit
harmonic,
because
shape
willfinal
=call
v3 (Tsuch
(Tby
w2)(0),
= its
T
T3 . The
tsholds,
to v02 (Tand
)
=
v
is thdefined
by
simply
v(T
= vsoth
andTv(T
) 2=+
v(0)
2
3 ) and
3) =
0isolated point
of
ODEs.
Note
however
that
it
will
only
exist
at
an
ross
any
thresholds,
and
is
defined
by
simply
by
v(T
)
=
v
v(T ) = v(0)
t. We0 shall0.2call such
orbitv harmonic,
because its shape
th and will
0.4 an 0.6
0.8
eough
the the
coefficient
matrix
has
purely
complex
eigenvalues
= 0, will
orbit. We
shall A
call
such
an orbit
harmonic,
because(TrA
its shape
, b"
,
(3.9) 1/
"/γ2 , b"2 = #
#
4 =
∗
∗
#
"
#
b
−
b/γ
b
−
b/γ
2
1
(1 + I)/µ
(I −−
a)/µ
(1
+
I)/µ
(I
a)/µ
, b2 = ∗
,b b=
, periodic
(3.9)
Continuous
and
4 =
∗
,
,
b #− b/γ2 " 4
− b/γ21 "
∗
∗
1b
1
2
"
#
#
ucing
two
voltage
thresholds
v
and
v
,
where
(v
,
v
) = (a/2, (1 +
b (I−
b/γ
b
−
b/γ
with
th
th
th
th
2
1
(1 + I)/µ
(I − a)/µ
− 2a)/µ
1
,(b,initial
b(1
, 2 Type
b4 = I∗ model,
,
(3.9)
=(vth∗•, vChoose
)
=
+
a)/2)
that
we
can
2 =
1 for the
1
2means
∗
data
th
b −
b/γand
b
−
b/γ
b −voltage
b/γ2 thresholds
g two
vth
vth , where
(v
,
v
)
=
(a/2,
(1
+
2
1
th
th
1
∗
∗
hoosing
initial
data
such
that
z
(0)
=
(v
,
w
)
(with
w
as
yet
undeter2
1
th
1 the Type2I model, means 1that we
2 can
, vth ) = (b, (1 + a)/2) for
hage
thresholds
v
and
v
(v
,
v
1 ,∗ where
2
1
2 ) = (a/2
0)
+
K
(T
)b
,
for
µ
=
1,
2,
3.
The
‘times-of-flight’
T
are
determined
1
∗
.
Introducing
two
voltage
thresholds
v
and
v
,
where
(v
,
v
) = (a/2, (1b+
th
th
th
th
µ
µ
µ
µ
1ng initial data such that z1 (0) = (v ,th
th
th
th
w
)
(with
w
as
yet
undeter1
“Times
of
flight”
determined
by
threshold
crossings
th
•
2
2
1
1
1
2
conditions:
v
(T
=
v
,
v
(T
)
=
v
,
v
(T
)
=
v
,
and
v
(T
)
=
v
. can
odel
and
(v
,
v
)
(b,
(1
+
a)/2)
for
the
Type
I
model,
means
that
we
1 µ
1 = 1,
2The
2 Type
3
4 4 that
(b,
+µ , a)/2)
for
I3 model,
means
w
th 3.the
th
th
th
th
th
Kµ (1
(Tµ )b
for
2,
‘times-of-flight’
T
are
determined
µ
1
∗
∗
∗
rbit
by
choosing
initial
data
such
that
z
(0)
=
(v
,
w
)
(with
w
asvyet
undeterfound
by
solving
w
(T
)
=
w
(0),
thus
yielding
w
and
the
2 4 4
1
1period
1(T ) =∗vth
12 , v 1
∗
itions:
v
(T
)
=
v
,
v
(T
)
=
v
,
and
v
(T
)
=
.
1 1 that
2 (0)
2
th
th (v3 3, w )
th(with4 w4 asthyet und
data
such
z
=
1
The ‘times-of-flight’
Tµ are determined
µ (Tµ )zµ (0) + Kµ (Tµ )bµ , for µ = 1, 2, 3.th
∗
nd by solving w4 (T4 ) = w1 (0),
thus yielding
wA3and
period
and
=
A
and
b
=
2
2
1the
1
1
3
crossing
conditions:
v
(T
)
=
v
,
v
(T
)
=
v
,
v
(T
)
=
v
,
and
v
(T
)
=
v
.
Ensure
periodicity
•
2 2
3 3
4 determ
4
th ‘times-of-flight’
th
th
th
,solution
for µare=also
1,possible.
2, 13. 1 The
T
are
µ∗ threshold
Two of these involve only a single
then be found
by solving w
w and
the period
for
the
McKean
1 4 (T4 )2 = w1 (0),a)/2)
2thus yielding
2
1
stion
through
the
section
v Two
=)vth
, but
not
v =(T
vthonly
and
another
whichvcrosses
(T
)
=
v
,
v
(T
=
v
,
v
)
=
v
,
and
are
also
possible.
of
these
involve
a
single
threshold
1
2 2
3 3
4 (T4 ) =
th
th
th
w
Calling
these
thev subperiodic
orbit
respectively
1 and
2 parameterize
a
periodic
3 supra-threshold
∗
ough
the
section
=
v
,
but
not
v
=
v
and
another
which
crosses
th
periodic
solution
are
also
possible.
Two th
of these
involve only
a and
single threshold
Yielding
ving
w
(T
)
=
w
(0),
thus
yielding
w
the
p
0.6
4
4
1
using
the
approach
(and
notation)
above.
The
sub-threshold
orbit
is
1
2orbit
ingcrosses
these through
the sub-the
and
supra-threshold
periodic
respectively
mined)
and
z
(0)
=
ch
section
v
=
v
,
but
not
v
=
v
and
which
crosses
2
µ+1
1
∗
1another
th
th
{1,
4} 1approach
with z1 (0)(and
=
(vnotation)
, w ), subject
to
v1sub-threshold
(T1 ) = vth = orbit
v4 (T4is) and
th
gv the
above.
The
4
=0.4vth . Calling these the sub- and supra-threshold periodic orbit respectively
by
solving
the
threshold
1
∗is specified by the
1 period
+
T
.
The
right
orbit
restriction
µ
=
{2,
3} with
and
the
4
} withfor
z
(0)
=
(v
,
w
),
subject
to
v
(T
)
=
v
=
v
(T
)
and
olved
using
the
approach
(and
notation)
above.
The
sub-threshold
orbit is
1
1
1
4
4
th
th
2
=
v
(T
)
and
w
(T
)
=
w
(0),
so
that
T
=
T
+
T
.
The
final
)
=
v
1 by
∗the
1single
lso
possible.
Two
of
these
involve
only
a
thre
3
3
3
3
2
2
3
th
.
The
right
orbit
is
specified
restriction
µ
=
{2,
3}
with
0.2
A
periodic
solution
ca
on
µ
=
{1,
4}
with
z
(0)
=
(v
,
w
),
subject
to
v
(T
)
=
v
=
v
(T
)
and
4
1
1
1
4
4
th
th
0
$
2
thresholds,
and
is
defined
by
simply
by
v(T
)
=
v
and
v(T
)
=
v(0)
1
2
th
4
T
=
T
+
T
.
The
right
orbit
is
specified
by
the
restriction
µ
=
{2,
3} with
=
v
(T
)
and
w
(T
)
=
w
(0),
so
that
T
=
T
+
T
.
The
final
v
1
4
1
3 v
3 = v 3 , but
3
2
2 another
3
th
section
not
v
=
v
and
which
cr
T
=
T
.
th
th
2
0 that
We
shall
anw3orbit
harmonic,
because
shape
willfinal
=call
v3 (Tsuch
(Tby
w2)(0),
Tv(T
= its
T
T3 . The
torbit.
to v02 (Tand
vthdefined
sholds,
by
simply
v(T
= vsoth
andµ=1
) 2µ
=+
v(0)
2 ) =is
3 ) and
3) =
0isolated
the
suband
supra-threshold
periodic
orbit
respect
of
ODEs.
Note
however
that
it
will
only
exist
at
an
point
ross
any
thresholds,
and
is
defined
by
simply
by
v(T
)
=
v
and
v(T
) = v(0)
t. We0 shall0.2call such
an
orbit
harmonic,
because
its
shape
will
th
v 0.8
0.4
0.6
Three
other
types
o
eough
the the
coefficient
matrix
has
purely
complex
eigenvalues
= 0, will
orbit. We
shall A
call
such
an orbit
harmonic,
because(TrA
its shape
Orbit and period
0.6
w
0.4
0.2
I
0
0.2
0.4
v
0.6
30
T
20
10
0
0.08
0.1
0.12
0.14
I 0.16
ż =a Fdynamical
(z) with asystem
T -periodic
solution
Z(t)
Z(t + T solution
) and Z(t
wesystem
consider
ż = F
(z) with
a T=
-periodic
nfinitesimal perturbation ∆z to the trajectory Z(t) at time t = 0. The resu
∆z to the perturbation
trajectory PRC
Z(t)
time
t = 0.
The Z(t)
resulting
trajectory
finitesimal
∆zatto
the
trajectory
at time
t = 0. The re
exact
solution
∆z(t) induces a phase shift defined as ∆θ = θ(z) − θ(Z) (using isochronal c
ft defined
as a∆θ
= θ(z)
− θ(Z)
(using
isochronal
z(t)
induces
phase
shift
defined
as ∆θ
= θ(z) −coordinates)
θ(Z) (using such
isochronal
der Call
in ∆z
der in ∆zthe orbit z = Z(t) where ż = F (z)
∆θ =coordinates)
"Q, ∆z#,
Introduce
a
phase
(isochronal
θ
∆θ = "Q, ∆z#,
(3.10)
∆θ = "Q, ∆z#,
nes the standard inner product, and
Q
=
∇
θ
is
the
gradient
of
θ
evaluat
Z
(Ermentrout
andθKopell
1991)
Adjoint
product,
and Q =inner
∇Z θ product,
is the gradient
of=θ∇evaluated
at
Z(t). This
nes
the
standard
and
Q
is
the
gradient
of θ evalua
Z
e shown to satisfy the linear equation
near
equation
shown
to satisfy the linear equation
dQ
T
= D(t)Q,
D(t) = −DF T(Z(t))
dQ
T
dt−DF
D(t)Q,
D(t) =
(Z(t)) D(t) = −DF (Z(t)) (3.11)
= D(t)Q,
dt
conditions ∇Z(0) θ · F (Z(0)) = 1/T and Q(t) = Q(t + T ). Here DF
Z(0))
=
1/T
and
Q(t)
=
Q(t
+
T
).
Here
DF
(Z)
denotes
the
conditions
∇
θ
·
F
(Z(0))
=
1/T
and
Q(t)
=
Q(t
+
T
).
HeretoDF
Z(0)
evaluated along Z. The (vector) PRC, R, is related to Q according
th
(vector)
PRC,
related
QPRC,
according
the
simple
scaling
evaluated
along
Z. isThe
(vector)
R, is to
related
to Q according
to tu
neral
(3.10)
and R,
(3.11)
must
betosolved
numerically
to obtain
the
PRC, say
must
be
solved
numerically
tobe
obtain
the
PRC,
sayisusing
the adjoint
neral
(3.10)
and
(3.11)
must
solved
numerically
topiece-wise
obtain
the
PRC,and
say
P
[23].
However,
for the
McKean
model
DF (Z)
linear
McKean
model for
DFthe
(Z)
is piece-wise
and iswepiece-wise
can obtain
a and
[23].
However,
McKean
model
DFperiodic
(Z)
linear
sed
form.
Introducing
a labelling
as
forlinear
the
orbit
we re-write
(3
labelling
as−A
forT .the
periodic
orbit
we
re-write
(3.10)
in the
form
T
ed
form.
Introducing
a
labelling
as
for
the
periodic
orbit
we
re-write
(
where
Dµ =
The
solution
of
each
subsystem
is
given
by
Q
(t)
=
G
µ
µ
µ
T
Tsubsystem is given by Qµ (t) = G (Tµ − t)Qµ (Tµ )
olution
of
each
µ2is
here
D
=
−A
. The
solution
of each
given
bythe
Qµrelation
(t) = G
Qµ+1 (0),
1, 2, 3.
Denoting
Q4 (Tsubsystem
) we
have
µ for µµ=
4 ) = (q1 , q
.QDenoting
Q (T ) = (q , q2 ) we have
Q4the
(T4 )relation
= (q1 , q2 ) we have the relatio
µ+1 (0), for 4µ =4 1, 2, 3.1 Denoting
"
q1 !
1
1
∗
∗
1
f
(v
)
−
w
+
I
+
q
g(v
,
w
)
=
.
"
2
1
th
th
!
"
1
∗
qµ1 1 1 ∗
T1
∗
∗
1
consider
a(z)
dynamical
=solution
(z) with
a
ż =aFollowing
Fdynamical
(z) with[12]
asystem
Twe
-periodic
solution
Z(t)
=
Z(t +ż T
)Fand
wesystem
consider
ż = F
with
a Tsystem
-periodic
Z(t
nfinitesimal
perturbation
∆z
to
the
trajectory
Z(t)
at
time
t
=
0.
The
resu
ned
by
the
isochrons
of
an
attracting
limit
cycle.
introduce PRC
an
infinitesimal
∆z
thet trajectory
∆z to the perturbation
trajectory
Z(t)
atto
time
t =perturbation
0.
The Z(t)
resulting
trajectory
finitesimal
∆z
the
trajectory
at to
time
= 0. The Z(
re
exact
solution
∆z(t)
induces
a phase
shift
defined
as
∆θ+a=
θ(z)
− θ(Z)
(using
isochronal
c−
with
a
T
-periodic
solution
Z(t)
=
Z(t
T
)
and
z(t)
=
Z(t)
+
∆z(t)
induces
phase
shift
defined
as
∆θ
=
θ(z)
ft defined
as a∆θ
− θ(Z)
(using
isochronal
z(t)
induces
phaseθ(z)
shift
defined
as ∆θ
= θ(z) −coordinates)
θ(Z) (using such
isochronal
der Call
inZ(t)
∆z
=0.Z(t)
tory
at orbit
time
The
resulting
that ttoz=first
order
in ∆z ż trajectory
= F (z)
where
der in ∆zthe
= θ(z) − θ(Z) (using isochronal ∆θ
coordinates)
such
=
"Q,
∆z#,
∆θ
=
"Q,
∆z#,
Introduce
a phase
coordinates)
θ
∆θ = "Q,
∆z#, (isochronal
(3.10)
∆θ = "Q, ∆z#,
nes the standard
inner
product,
and
Q
=
∇
θ
is
the
gradient
of
θ
evaluat
Z
where
"·,
·#
defines
the
standard
inner
product,
and
Q
=
∇
θ
is
Z
(Ermentrout
and
Kopell
1991)
Adjoint
product,
and Q =inner
∇Z θ product,
is the gradient
of=θ(3.10)
evaluated
at
Z(t). This
∆z#,
nes
the
standard
and
Q
∇
θ
is
the
gradient
of θ evalua
Z
e shown to satisfy
the
linear
equation
gradient can be shown to satisfy the linear equation
near
equation
shown
to satisfy the linear equation
∇Z θ is the gradient
dQof θ evaluated at Z(t). This
dQ T
= D(t)Q,
D(t) = −DF
(Z(t))
= TD(t)Q,
D(t) = −DF
dQ
T
dt−DF
D(t)Q,
D(t) =
(Z(t)) D(t) = −DF
dt
= D(t)Q,
(Z(t)) (3.11)
dt
T
subject
the conditions
∇Z(0)
θ · F=
(Z(0))
= T1/T
and Q(t)
conditions
∇
F (Z(0))
= 1/T and
Q(t)
Q(t +
). Here
DF
Z(0) θ · to
=
−DF
(Z(t))
(3.11)
Z(0))
=
1/T
and
Q(t)
=
Q(t
+
T
).
Here
DF
(Z)
denotes
the
conditions
∇
θ
·
F
(Z(0))
=
1/T
and
Q(t)
=
Q(t
+
T
).
Here
DF
Z(0)
Jacobian
of
F
evaluated
along
Z.
The
(vector)
PRC,
R, istorela
evaluated along Z. The (vector) PRC, R, is related to Q according
th
(vector)
PRC,
R,
isThe
related
tosolved
QPRC,
according
to
the
simple
scaling
evaluated
along
Z.
(vector)
R,
is
related
to
Q
according
to tu
R
=
QT
.
In
general
(3.10)
and
(3.11)
must
be
solved
numerically
neral
(3.10)
and
(3.11)
must
be
numerically
to
obtain
the
PRC, say
d Q(t) = Q(t + T ). Here DF (Z) denotes the
must
be
solved
numerically
tobe
obtain
the
PRC,
say
using
the model
adjoint
neral
(3.10)
and
(3.11)
must
solved
numerically
to
obtain
the
PRC,
say
routine
in
XPP
[23].
However,
for
the
McKean
DFand
(Z)
Another
pwl
system
with
4
labels!
P
[23].
However,
for
the
McKean
model
DF
(Z)
is
piece-wise
Qlinear
, is related to Q according to the simple scaling
McKean
model
DFthe
(Z)
is piece-wise
linear
andaiswe
can
obtain
athe peri
T
solution
in
closed
form.
Introducing
labelling
as
for
[23].
However,
for
McKean
model
DF
(Z)
piece-wise
linear
and
3
sed
form.
Introducing
a
labelling
as
for
the
periodic
orbit
we
re-write
(3
merically to obtain the PRC, say using the adjoint
T
labelling
as
for
the
periodic
orbit
we
re-write
(3.10)
in of
the
form
T
T
Q̇
=
D
Q
,
where
D
=
−A
.
The
solution
each
subsystem
ed
form.
Introducing
a
labelling
as
for
the
periodic
orbit
we
re-write
(
µ
µ
µ
µ
where
D
=
−A
.
The
solution
of
each
subsystem
is
given
by
Q
(t)
=
G
µ
µ
DF (Z) µis piece-wise
linear and we can obtain a T
µ
µ
T
2(T
Tsubsystem
olution
of
each
is
given
by
Q
(t)
=
G
(T
−
t)Q
)
T
with
Q
(T
)
=
Q
(0),
for
µ
=
1,
2,
3.
Denoting
Q
(T
) ==(qG
µ
µ
µ
µ
µ
µ
µ+1
4
4(t)
1
µ
here
D
=
−A
.
The
solution
of
each
subsystem
is
given
by
Q
4
Q
(0),
for
µ
=
1,
2,
3.
Denoting
Q
(T
)
=
(q
,
q
)
we
have
the
relation
µ
µ
µ+1
4
4
1
2
µ
he periodic orbit we re-write (3.10) in the form
.QDenoting
Q4µ(T=4 )1,=2,(q
q2 ) we
the
relation
1 ,Denoting
T have
(0),
for
3.
Q
(T
)
=
(q
,
q
)
we
have
the
relatio
T
µ+1
4
4
1
2
!
"
ubsystem
is
given
by
Q
(t)
=
G
(T
−
t)Q
(T
)
q
1
Solve using q1 ! µ 1
µ"
µ 1µ
µ
1
1 ∗ + I + q2 g(v 1 ,
∗
1 f (v∗
)
−
w
th) =
th
f
(v
)
−
w
+
I
+
q
g(v
,
w
.
" have
)1 = (q1 ,∗q2 ) we
the
relation
2
1
th
th
µ
"
qµ1 ! 1 1 ∗
T1
∗
∗
1
PRC
0.7
R
200
v
0.6
100
0.5
0
0.4
-100
0.3
0
1.5
3
4.5
t
2.3
Networks of interacting phase oscillators
Weak coupling
Theorem: Phase equations for oscillatory networks
Consider a family of weakly connected systems
Ẋi = Fi(Xi) + !G(X),
i = 1, . . . , n
such that each equation in the uncoupled system (! = 0) has an exponentially orbitally stable
limit cycle γi ⊂ Rm having natural frequency Ωi "= 0. Then the oscillatory weakly connected
system can be reduced to a phase model of the form
θ̇i = Ωi + !gi(θ1, . . . , θn),
θi ∈ S1,
i = 1, . . . , n
defined on the n-torus T n = S1 × . . . × S1. ie there is an open neighbourhood W of M =
γ1 × . . . × γn ⊂ Rmn and a continuous function h : W → T n that maps solutions of the full
model to those of the phase model. !
The proof of this theorem is based around the fact that the direct product of hyperbolic limit
cycles M = γ1 × . . . × γn is a normally hyperbolic invariant manifold (Limit cycles are hyperbolic
if Floquet multipliers are not on the unit circle). The invariant manifold theorem guarantees the
persistence of a manifold M!, ! close to M. The restriction of the dynamics to M (non-zero !)
then has something to say about the dynamics on M! for small enough !.
Since γi is homemorphic to S1, we can parameterise it using the phase variable θi ∈ S1, θi = Ωi:
Then we have
Γi : S1 → γi,
dΓi(θi(t))
Γi(θi(t)) = xi(t) ∈ γi,
t ∈ [0, 2π/Ωi]
2.3
Networks of interacting phase oscillators
Weak coupling
Theorem: Phase equations for oscillatory networks
Consider a family of weakly connected systems
Ẋi = Fi(Xi) + !G(X),
i = 1, . . . , n
such that each equation in the uncoupled system (! = 0) has an exponentially orbitally stable
limit cycle γi ⊂ Rm having natural frequency Ωi "= 0. Then the oscillatory weakly connected
system can be reduced to a phase model of the form
θ̇i = Ωi + !gi(θ1, . . . , θn),
es defined
by on the n-torus T
θi ∈ S1,
i = 1, . . . , n
n
= S1 × . . . × S1. ie there is an open neighbourhood W of M =
n !
γ1 × . . . × γn ⊂ Rmn and a continuous function
h
:
W
→
T
i i i i that maps solutions of the full
model to those of the phaseimodel.
i !
2 product of hyperbolic limit
The proof of this theorem is based around the fact
that
the
direct
i i i
cycles M = γ1 × . . . × γn is a normally hyperbolic invariant manifold (Limit cycles are hyperbolic
if Floquet multipliers are not on the unit circle). The invariant manifold theorem guarantees the
persistence of a manifold M!, ! close to M. The restriction of the dynamics to M (non-zero !)
then has something to say about the dynamics on M! for small enough !.
i parameterise
i itiusing ithe phase variable θi ∈ S1, θi = Ωi:
Since γi is homemorphicito S1, we can
Ω F (Γ (θ ))
R (θ ) =
|F (Γ (θ )|
Drive
θ̇ = Ω + $R (θ )G (Γ (θ))
Γi : S1 → γi,
Γi(θi(t)) = xi(t) ∈ γi,
t ∈ [0, 2π/Ωi]
alysis of such systems we consider the following restricte
Then we have
dΓi(θi(t))
PRC
gular limit µ → 0 [18, 21]. Introducing the conductance strengt
Averaging
ay write such a network dynamics
in the form
N
"
dθi
1
1
= +
gij H(θj − θi ),
dt
T
N j=1
so-called phase interaction function. For gap junction coupling th
H(θ) =
#
0
T
Q (t)(v(t + θT ) − v(t), 0)dt,
T
riodic solution of (2.1) and Q(t) is the associated adjoint. It is con
the 2 × 1 vectors z and Q and write
"
"
zn e2πint/T ,
Q(t) =
Qn e2πint/T .
z(t) =
n
n
gular
limit
Introducingthethe
conductance
strengt
singular
limitµ µ→→00[18,
[18, 21].
21]. Introducing
conductance
strength
bet
Averaging
eaymay
write
sucha anetwork
network dynamics
dynamics
ininthe
write
such
theform
form
NN
"
dθ
1
1
i
dθi = 1 + 1 "gij H(θj − θi ),
dt = T + N j=1 gij H(θj − θi ),
dt
T
N
j=1
thePhase
so-called
phase
interaction
function.
For
gap
junction
coupling
this
is
interaction function
so-called phase interaction function. For gap junction coupling th
#
H(θ) =#
H(θ) =
T
T
0
QT (t)(v(t + θT ) − v(t), 0)dt,
Q (t)(v(t + θT ) − v(t), 0)dt,
T
a periodic solution of (2.1) 0and Q(t) is the associated adjoint. It is convenie
or
the
2
×
1
vectors
z
and
Q
and
write
riodic solution of (2.1) and Q(t) is the associated adjoint. It is con
"
"
the 2 × 1 vectors
z
and
Q
and
write
2πint/T
2πint/T
zn e
,
Q(t) =
Qn e
.
z(t) =
z(t) =
"n
n
zn e2πint/T ,
"
n
Q(t) =
n
Qn e2πint/T .
gular
limit
Introducingthethe
conductance
strengt
singular
limitµ µ→→00[18,
[18, 21].
21]. Introducing
conductance
strength
bet
Averaging
eaymay
write
sucha anetwork
network dynamics
dynamics
ininthe
write
such
theform
form
NN
"
dθ
1
1
i
dθi = 1 + 1 "gij H(θj − θi ),
dt = T + N j=1 gij H(θj − θi ),
dt
T
N
j=1
thePhase
so-called
phase
interaction
function.
For
gap
junction
coupling
this
is
interaction function
so-called phase interaction function. For gap junction coupling th
#
H(θ) =#
H(θ) =
T
T
0
QT (t)(v(t + θT ) − v(t), 0)dt,
Q (t)(v(t + θT ) − v(t), 0)dt,
T
0and Q(t) is the associated adjoint. It is convenie
a periodic
solution
of
(2.1)
For convenience introduce Fourier series representation
or
the
2
×
1
vectors
z
and
Q
and
write
riodic solution of (2.1) and Q(t) is the associated adjoint. It is con
!
"
"
the 2 × 1 vectors
z
and
Q
and
write
2πinθ
2πint/T
2πint/T
zn e=
, n e Q(t) =
Qn e
.
z(t) = H(θ)
H
z(t) =
"n
n
zn e
n
2πint/T
,
"
n
Q(t) =
n
Qn e2πint/T .
gular
limit
Introducingthethe
conductance
strengt
singular
limitµ µ→→00[18,
[18, 21].
21]. Introducing
conductance
strength
bet
Averaging
eaymay
write
sucha anetwork
network dynamics
dynamics
ininthe
write
such
theform
form
NN
"
dθ
1
1
i
dθi = 1 + 1 "gij H(θj − θi ),
dt = T + N j=1 gij H(θj − θi ),
dt
T
N
j=1
thePhase
so-called
phase
interaction
function.
For
gap
junction
coupling
this
is
interaction function
so-called phase interaction function. For gap junction coupling th
#
H(θ) =#
H(θ) =
T
T
0
QT (t)(v(t + θT ) − v(t), 0)dt,
Q (t)(v(t + θT ) − v(t), 0)dt,
T
0and Q(t) is the associated adjoint. It is convenie
a periodic
solution
of
(2.1)
For convenience introduce Fourier series representation
or
the
2
×
1
vectors
z
and
Q
and
write
riodic solution of (2.1) and Q(t) is the associated adjoint. It is con
!
"
"
the 2 × 1 vectors
z
and
Q
and
write
2πinθ
2πint/T
2πint/T
zn e=
, n e Q(t) =
Qn e
.
z(t) = H(θ)
H
z(t) =
"n
zn e
n
2πint/T
,
"
n
Q(t) =
Qn e2πint/T .
For the pwl
model we can obtainnthe Fourier
n
coefficients in closed form (spare the details!)
Phase interaction function H
14
H
10
6
2
-2
0
0.2
0.4
0.6
0.8
θ
1
interaction
functions
Figure
4.1.McKean
In this model
example
we see
t
4.1. In this example
weofsee
that the
admits
a sta
Global
coupling
large
N a stable an
eonous
I model
admits
a stable
anti-synchronous
solution.
solution,
whilst
the
Type and
I model
admits
1
hr gglobally
=
g
the
system
(4.2)
is
S
×T
equivariant.
By the(4.2)
equivari
coupled networks Nwith gij
= g the system
is S
ij
ric solutions
describing synchronous,
splay, anddescribing
cluster states
hing
lemma maximally
symmetric solutions
sync
nchronous
state
with
= 0,synchronous
Ω = 1/T andstate
therewith
is a single
z
Synchrony
(relative):
i (t) the
ed
to be generic
[5]. φFor
φi (t) =
!
H (0)and
of multiplicity
N −λ 1.= Hence
for the
examples
in
Figure
alue
aneigenvalue
eigenvalue
−gH ! (0)
of multiplicity
N
− 1.
(multiplicity N-1)
stable
synchronous
solution
whilst
the
Type
I
model
does
not.
"
that the McKean model has a stable
synchronous
solution
e eigenvalues are given by λn = g j H ! (j/N )(e2πinj/N − 1)/N
y state of the form φi = i/N the eigenvalues are given by λn
w of the stability of cluster states we refer the reader to [11]. In
. . . , N − 1. For a recent review of the stability of cluster st
ult that (for global coupling) network averages may be replaced
N → ∞ we have the useful result that (for global coupling)
verages:
1
N
N
#
1
F (jT /N ) =
T
j=1
$
0
T
F (t)dt =1
N
F#
1
F (jT /N ) =
lim
N →∞ N
T
j=1
0
$
T
(4.
F (t)
0
F (t + T ). Hence in the large N limit the collective frequency o
by T-periodic function F (t) = F (t + T ). Hence in the larg
me
ned by G(φ) = 0 where G(φ) = g[H(−φ) − H(φ)] and φ is the
a recent
review
of thewe
stability
of the
cluster
states
we
referadmits
the
reader
interaction
functions
of
Figure
In
this
example
we
see
t
4.1.
In
this
example
see
that
model
a
sta
!4.1.McKean
ondition for stability is simply G (φ) < 0. By symmetry the pha
Global
coupling
and
large
N
the
useful
result
that
(for
global
coupling)
network
averages
may
b
eenchronous
I model
admits
a
stable
anti-synchronous
solution.
onous
solution,
whilst
the
Type
I
model
admits
a
stable
an
state (φ = 1/2) are guaranteed to exist. In Figure 4
1
hrn gglobally
=
g
the
system
(4.2)
is
S
×T
equivariant.
By the
equivari
functions
of
Figure
4.1.
In
this
example
seesystem
that
the
McKea
coupled networks Nwith gij
= gwethe
(4.2)
is S
ij
$
N
ric
solutions
describing
synchronous,
splay,
cluster states
ution,
whilst
the
Type
I
model
admits
a stableand
anti-synchronous
hing lemma maximally
symmetric
describing
sync
1 T solutions
1 #
F (jT
/N
)
=
F
(t)dt
=
F
limstate
0
1 single z
nchronous
with
φ
(t)
=
0,
Ω
=
1/T
and
there
is
a
Synchrony
(relative):
i
coupled
with
g
=
g
the
system
(4.2)
is
S
×T
equivar
ed
to beNnetworks
generic
[5].
For
the
synchronous
state
with
φi (t) =
→∞ N
T
ij
N
0
!
j=1
! for the examples
H maximally
(0)and
of multiplicity
N −λ
1.= Hence
in
Figure
ma
symmetric
solutions
synchronous,
splay
alue
aneigenvalue
eigenvalue
−gHdescribing
(0) of multiplicity
N
− 1.
(multiplicity N-1)
stable
synchronous
solution
whilst
the
Type
I
model
does
not.
generic
[5].
For
the
synchronous
state
with
φ
(t)
=
0,
Ω
=
1/T
a
unction
F
(t)
=
F
(t
+
T
).
Hence
in
the
large
N
limit
the
collective
i
"
that the McKean model has a stable
synchronous
solution
!
2πinj/N
!by λn = g
e
eigenvalues
are
given
H
(j/N
)(e
−for
1)/N
upling)
is
given
by
an
eigenvalue
λ
=
−gH
(0)
of
multiplicity
N
−
1.
Hence
j
y state of the form φi = i/N the eigenvalues are given bythe
λn
wMcKean
of the stability
of cluster
states
we refersolution
the reader
to [11].
In
model
has
a
stable
synchronous
whilst
the
Typ
"
. . . , N − 1. For a recent 1review of the stability of cluster
st
!
ult
that
global
theSplay:
form(for
φi =
i/N coupling)
theΩeigenvalues
given by may
λn =be
g replaced
H (j/
= +network
gH0 ,are averages
j
N → ∞ we have the useful
T result that (for global coupling)
1. For a recent review of the stability of cluster states we refer th
verages:
λ
=
−2πingH
we have
the
useful
result
that
(for
global
coupling)
network
avera
n
−n
$
N
T
#
1
$ T1
$
N
T
#
F (jT /Ng ) = !
F (t)dt
=
F
(4.
0
1
2πint/T1
H (t/T
dt = −2πingH
.
N j=1 λn =
T
−n
0 N)elim
F
(jT
/N
)
=
F
(t)
$
T
T 0
1
1 #N →∞ N
T
0
j=1
F (jT /N ) =
F (t)dt = F0
lim
N →∞
T 0collective frequency o
Fstable
(t + Tif). Hence
in N
thej=1
large N limit the
by T-periodic function F (t) = F (t + T ). Hence in the larg
me
ned by G(φ) = 0 where G(φ) = g[H(−φ) − H(φ)] and φ is the
a recent
review
of thewe
stability
of the
cluster
states
we
referadmits
the
reader
interaction
functions
of
Figure
In
this
example
we
see
t
4.1.
In
this
example
see
that
model
a
sta
!4.1.McKean
ondition for stability is simply G (φ) < 0. By symmetry the pha
Global
coupling
and
large
N
the
useful
result
that
(for
global
coupling)
network
averages
may
b
eenchronous
I model
admits
a
stable
anti-synchronous
solution.
onous
solution,
whilst
the
Type
I
model
admits
a
stable
an
state (φ = 1/2) are guaranteed to exist. In Figure 4
1
hrn gglobally
=
g
the
system
(4.2)
is
S
×T
equivariant.
By the
equivari
functions
of
Figure
4.1.
In
this
example
seesystem
that
the
McKea
coupled networks Nwith gij
= gwethe
(4.2)
is S
ij
$
N
ric
solutions
describing
synchronous,
splay,
cluster states
ution,
whilst
the
Type
I
model
admits
a stableand
anti-synchronous
hing lemma maximally
symmetric
describing
sync
1 T solutions
1 #
F (jT
/N
)
=
F
(t)dt
=
F
limstate
0
1 single z
nchronous
with
φ
(t)
=
0,
Ω
=
1/T
and
there
is
a
Synchrony
(relative):
i
coupled
with
g
=
g
the
system
(4.2)
is
S
×T
equivar
ed
to beNnetworks
generic
[5].
For
the
synchronous
state
with
φi (t) =
→∞ N
T
ij
N
0
!
j=1
! for the examples
H maximally
(0)and
of multiplicity
N −λ
1.= Hence
in
Figure
ma
symmetric
solutions
synchronous,
splay
alue
aneigenvalue
eigenvalue
−gHdescribing
(0) of multiplicity
N
− 1.
(multiplicity N-1)
stable
synchronous
solution
whilst
the
Type
I
model
does
not.
generic
[5].
For
the
synchronous
state
with
φ
(t)
=
0,
Ω
=
1/T
a
unction
F
(t)
=
F
(t
+
T
).
Hence
in
the
large
N
limit
the
collective
i
"
that the McKean model has a stable
synchronous
solution
!
2πinj/N
!by λn = g
e
eigenvalues
are
given
H
(j/N
)(e
−for
1)/N
upling)
is
given
by
an
eigenvalue
λ
=
−gH
(0)
of
multiplicity
N
−
1.
Hence
j
y state of the form φi = i/N the eigenvalues are given bythe
λn
wMcKean
of the stability
of cluster
states
we refersolution
the reader
to [11].
In
model
has
a
stable
synchronous
whilst
the
Typ
"
. . . , N − 1. For a recent 1review of the stability of cluster
st
!
ult
that
global
theSplay:
form(for
φi =
i/N coupling)
theΩeigenvalues
given by may
λn =be
g replaced
H (j/
= +network
gH0 ,are averages
j
N → ∞ we have the useful
T result that (for global coupling)
1. For a recent review of the stability of cluster states we refer th
verages:
λ
=
−2πingH
we have
the
useful
result
that
(for
global
coupling)
network
avera
n
−n
$
N
T
#
1
$ T1
$
N
T
#
F (jT /Ng ) = !
F (t)dt
=
F
(4.
0
1
2πint/T1
H (t/T
dt = −2πingH
.
N j=1 λn =
T
−n
0 N)elim
F
(jT
/N
)
=
F
(t)
$
T
T 0
#
→∞splay
1 unstable T 0
1
N j=1
SynchronousNand
state
F (jT /N ) =
F (t)dt = F0
lim
N →∞
T 0collective frequency o
Fstable
(t + Tif). Hence
in N
thej=1
large N limit the
by T-periodic function F (t) = F (t + T ). Hence in the larg
me
Beyond weak coupling
Synchrony - existence as for uncoupled model
Stability - Floquet theory shows restabilisation with increasing g
Beyond weak coupling
Synchrony - existence as for uncoupled model
Stability - Floquet theory shows restabilisation with increasing g
Splay - analysis as for absolute integrate-and-fire model
4
ω
8
ω
0
−4
−0.6
0
ν
weak g
0.4
0
−8
−1
0
1.5
ν
strong g
Understanding rhythms
17
FROM PHASE-LOCKING TO BURSTING
0.6
w
0.6
w
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.4
0.6
v
w
0.4
0.6
0.4
v
0.6
v
1
!0.5
0
0.6
0.4
0.2
0.6
w
0.4
v
0.6
0.65
0.6
E
0.5
E
0.45
200
E
0.5
600
1000
1400
1800 t
200
600
1000
1400
1800 t
200
600
1000
1400
1800 t
Fig. 5.1. Top: A family of coexisting unstable orbits in the Type I model; synchronous (green), splay (blue), subthreshold splay (light blue) and harmonic splay (red). Here g = 0.1, I = 0.085 and other parameters as in Figure 2.2. left:
µ < 1 − g (µ = 0.89). middle: µ = 1 − g (µ = 0.9). right: µ < 1 − g (µ = 0.91). Middle: Numerical simulation (after
dropping transients) with N = 100 neurons showing a pseudo color plot of the triple (θ, vi , wi ), where θ = t/∆ mod 1 for
Understanding rhythms
17
FROM PHASE-LOCKING TO BURSTING
0.6
w
0.6
w
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.4
0.6
v
w
0.4
0.6
0.4
v
0.6
v
1
!0.5
0
0.6
0.4
0.2
w
0.6
0.4
v
0.65
E
0.6
0.6
E
0.5
E
0.6
0.45
0.5
0.5200 600 1000 1400 1800 t 200 600 1000 1400 1800 t 200 600 1000 1400 1800 t
E Fig. 5.1. Top: A family of coexisting unstable orbits in the Type I model; synchronous (green), splay (blue), subthreshold splay (light blue) and harmonic splay (red). Here g = 0.1, I = 0.085 and other parameters as in Figure 2.2. left:
200
600
1400Middle: Numerical1800
t
µ<1
− g (µ = 0.89). middle:
µ = 1 − g (µ = 0.9).1000
right: µ < 1 − g (µ = 0.91).
simulation (after
dropping transients) with N = 100 neurons showing a pseudo color plot of the triple (θ, vi , wi ), where θ = t/∆ mod 1 for
Future Challenges
Future Challenges
Wilson-Cowan style model with gaps?
“Equation Free Modelling” - role of architectecture,
and allowing spatio-temporal pattern analysis
Future Challenges
Wilson-Cowan style model with gaps?
“Equation Free Modelling” - role of architectecture,
and allowing spatio-temporal pattern analysis
Gaps are not static conductances
Voltage gated channel models
Future Challenges
Wilson-Cowan style model with gaps?
“Equation Free Modelling” - role of architectecture,
and allowing spatio-temporal pattern analysis
Gaps are not static conductances
Voltage gated channel models
Influenced by neuromodulators
eg cannabinoids
Future Challenges
Wilson-Cowan style model with gaps?
“Equation Free Modelling” - role of architectecture,
and allowing spatio-temporal pattern analysis
Gaps are not static conductances
Voltage gated channel models
Influenced by neuromodulators
eg cannabinoids
Gaps on distal dendrites
already known to “tune” network dynamics
2nd Annual meeting of the UK Mathematical
Neuroscience Network: 22-25 Mar 2009, Edinburgh.
http://icms.org.uk/workshops/mathneuro2009
Ad Aertsen
Michael Breakspear
Carson Chow
Geoff Goodhill
Vincent Hakim
Viktor Jirsa
Carlo Laing
Peter Latham
Andre Longtin
Stefano Panzeri
David Pinto
Horacio Rotstein
Andrey Shilnikov
Dan Tranchina
Krasimira Tsaneva-Atanasova
Carl Van Vreeswijk