Physics 178/278 - David Kleinfeld - Fall 2004/2011; revised Winter 2014
5
Networks of Phase Coupled Neuronal Oscillators
We consider small networks or simple networks in which cells are coupled only
weakly, in the sense that then can effect each others timing but do not turn each
other on or off or, more formally, do not effect the shape of each others limit cycle.
5.1
Basic formalism
Equation of motion for a general dynamical system
~
dX
~ µ)
= F (X;
dt
(5.5)
~ is a vector that contains all the dynamical variables and the µ are
where the X
parameters. At steady state
~0
dX
~ 0 ; µ)
= F (X
dt
(5.6)
~ 0 (t + T ) = X
~ 0 (t)
X
(5.7)
where a closed orbit satisfies
See attached figure from Kuromoto’s book
~
We associate a value of ψ with each point along X(t).
Thus the multidimensional
trajectory is reduced to a single variable.
It is useful to extend the definition of ψ off of the limit cycle, or contour, C, to
~ in the tube. This
all points within a tube around C so that ψ is defined for all X
will allow us to study perturbations to the original limit cycle.
Look on a surface, denoted G, normal to and in the neighborhood of C. Let P
be a point on G and Q be the point on C, the limit cycle, that passes through the
same surface. We posit that as the trajectories evolve, the point P will approach the
closed orbit defined by C. There will be a phase difference between P and Q. This
is equivalent to an initial phase difference among the points. The main idea is that
the physical perturbation can be transformed into a phase shift along the original
limit cycle, C, if the perturbed point collapses to or forever parallels the original
limit cycle.
There are a set of points in the tube that will lead to the same phase shift. These
~ on
define a surface of constant phase shifts, that is denoted I(ψ). For all points X
I(ψ) we have
1
~
dψ(X)
=ω
dt
for the unperturbed system. But, by the chain rule,
X ∂ψ ∂Xi
dψ
=
dt
i ∂Xi ∂t
~
~ ~ ψ · dX
= ∇
X
dt
~ ~ ψ · F~ (X)
~
= ∇
(5.8)
(5.9)
X
Let’s perturb the motion by
~ → F~ (X)
~ + P~ (X,
~ X
~ 0)
F~ (X)
(5.10)
where is small in the sense that the shape of the original trajectory in unchanged
~ 0 contains all the variables that define the perturbation, e.g, the
as → 0 and X
trajectory of a neighboring oscillator and the interaction between the two oscillating
systems. Then
h
i
dψ
~ ~ ψ · F (X)
~ + P~ (X,
~ X
~ 0)
= ∇
X
dt
~ ~ ψ · P~ (X,
~ X
~ 0)
= ω + ∇
X
(5.11)
So far everything is exact, that is, all calculations are done with respect to the
perturbed orbit. The difficulty is that the orbits are not necessarily closed. But if
~ −X
~ 0 (t)| → 0 as t → ∞, the perturbation
we can make small enough so that |X(t)
will lead to a closed path. This results in periodic orbits, so that the independent
variable can now be taken as the phase, ψ, rather than time, t, where the two are
related by
ψ = 2π
t
modulo(2π)
T
(5.12)
Using
~
~ 0 (ψ)
X(t)
→X
(5.13)
we have
h
i
dψ
~~
~ X
~ 0 (ψ), X
~ 0 0 (ψ 0 )
= ω + ∇
ψ
·
P
X0 (ψ)
dt
~
≡ ω + Z(ψ)
· P~ (ψ, ψ 0 )
(5.14)
~
The term Z(ψ)
depends only on the limit cycle of the oscillator and defines the
sensitivity of the phase to perturbation. It clearly varies along the limit cycle and is
2
sometimes called a ”phase-dependent sensitivity”. It may be calculated directly by
evaluating the trajectory of points inside a tube around the original limit cycle, or
more expeditiously using a trick due to Bowtell, in which the perturbed system is
~
~ with A(t) = A(t + T ), which can be shown to
rewritten in the form ddtX = A(t)X,
have only one periodic solution. A cute way to find the periodic solution is to solve
~
the adjoint problem, ddtY = AT (t)Y~ , for which all of the solutions decay except for
~
the periodic one. From this one backs out Z(ψ).
The cool thing in that the oscillator is seen to rotate freely (ω term) with phaseshifts and frequency shifts that are determined solely by the perturbations. The term
P~ (ψ, ψ 0 ), which can be calculated from the perturbation, allows these perturbations
to be interactions with neighbors.
Let’s look at the nature of the perturbation term. The idea is that this is small,
so that the shift in frequency on one cycle is small. We consider
ψ = δψ + ωt
(5.15)
Then the relative motion is given by
dδψ
~
= Z(ψ)
· P~ (ψ, ψ 0 )
dt
~
= Z(δψ
+ ωt) · P~ (δψ + ωt, δψ 0 + ωt)
(5.16)
This can be further simplified. To the extent that the change in ψ is small over
one cycle, i.e., dδψ
<< ω, we can average the perturbation over a full cycle. We
dt
write
dδψ
= Γ(δψ, δψ 0 )
dt
(5.17)
where
Zπ
~
dθ Z(δψ
+ θ) · P~ (δψ + θ, δψ 0 + θ)
(5.18)
Γ(δψ, δψ ) =
2π −π
The above result can be generalized to the case where the internal parameters,
~ are a bit different between oscillators, so that the underlying oscillations
i.e., the X’s
are slightly different frequency. We then have
0
dδψ
= Γ(δψ, δψ 0 ) + δω
dt
5.2
(5.19)
Simplified interaction among 2 oscillators.
We take the perturbation to be solely a function of the phase of the other oscillator.
Thus
Zπ
~
Γ(δψ, δψ ) =
dθ Z(δψ
+ θ) · P~ (δψ 0 + θ)
2π −π
0
3
(5.20)
But this is just a correlation integral that is proportion to the differences in
phase, i.e.,
Zπ
~ (θ − (δψ 0 − δψ)) · P~ (θ)
dθ Z
2π −π
So that a system of two oscillators obeys
Γ(δψ 0 − δψ) =
dδψ
= Γ(δψ 0 − δψ)
dt
dψ 0
= Γ(δψ − δψ 0 )
dt
(5.21)
(5.22)
We subtract the two equations of motion for the phase to get
d(δψ − δψ 0 )
= [Γ(δψ 0 − δψ) − Γ(δψ − δψ 0 )]
dt
≡ Γ̃(δψ 0 − δψ)
≡ −Γ̃(δψ − δψ 0 )
(5.23)
The term Γ̃(δψ − δψ 0 ) is an odd function with period T , with zeros at
x0 ≡ δψ − δψ 0 = nπ
n = 1, 2, 3, ...
(5.24)
and possibly other places. By way of analysis,
• The zeros correspond to phase locking.
• The stability depends on the sign of
< 0 implies stability with even n; attractive - phases converge.
> 0 implies stability with odd n; repulsive - phases diverge.
•
dΓ̃ dx x0
•
dΓ̃ dx x0
5.3
5.3.1
dΓ̃(x) dx x0
Examples
Two oscillators with delayed coupling.
An interesting example due to Ermentrout is to consider two oscillators that interact
by a synapse with a noninstantaneous rise time. Before we choose a realistic cell
~
model, let’s try some analytical methods and choose a form of Z(δψ)
that has
variable sensitivity along the limit cycle. The simplest choice is Z(t) = sin ωt, or
Z(δψ) = sin(δψ)
The interaction is given by an ”α” function, i.e., P (t ≥ 0) =
the substitution ψ = ωt, we have
4
(5.25)
gsynapse t −t/τ
e
.
cm
τ
With
P (δψ 0 ) = 0
δψ 0 < 0
gsynapse δψ 0 −δψ0 /ωτ
=
e
δψ 0 ≥ 0
cm ωτ
(5.26)
The convolution for Γ̃(δψ 0 −δψ) can be done by extending the range of integration
over all time, so that
Γ(δψ 0 − δψ) =
=
=
=
=
Z∞
~ (θ − (δψ 0 − δψ)) · P~ (θ)
dθ Z
(5.27)
2π 0
!
!
Z ∞
gsynapse θ
θ
0
d
ωτ
sin [θ − (δψ − δψ)]
e−θ/ωτ
cm 2π
ωτ
ωτ
0
Z ∞
Z ∞
gsynapse 1
−i(δψ 0 −δψ)
iωτ x −x
i(δψ 0 −δψ)
−iωτ x −x
ωτ
e
x dx e
e − e
x dx e
e
cm 2π
2i
0
0
!
0
0
1 e−i(δψ −δψ)
ei(δψ −δψ) Z ∞
gsynapse ωτ
−
x dx e−x
cm 2π
2i (1 − iωτ )2
(1 + iωτ )2 0
h
i
ωτ
gsynapse 2
0
0
1
−
(ωτ
)
sin(δψ
−
δψ)
+
2ωτ
cos(δψ
−
δψ)
cm 2π [1 + (ωτ )2 ]2
and thus
gsynapse ωτ [1 − (ωτ )2 ]
sin(δψ 0 − δψ)
cm π [1 + (ωτ )2 ]2
(5.28)
d(δψ − δψ 0 )
gsynapse ωτ [(ωτ )2 − 1]
=
sin(δψ − δψ 0 )
2
2
dt
cm π [1 + (ωτ ) ]
(5.29)
Γ̃(δψ 0 − δψ) =
so that
This says that, for excitatory connections (g > 0), the synchronized state, i.e.,
δψ 0 = δψ, is stable only for τ < ω1 . In contrast, for τ > ω1 the antiphastic state with
δψ 0 − δψ = ±π is stable.
Interestingly, synchronous, all inhibitory (g < 0). networks are observed experimentally at high frequencies. This is consistent with
inhibitory
|gsynapse
| ωτ [(1 − ωτ )2 ]
d(δψ − δψ 0 )
sin(δψ − δψ 0 )
=
2
2
dt
cm
π [1 + (ωτ ) ]
5.3.2
(5.30)
Two identical Hodgkin Huxley oscillators.
How well does the above analysis hold with more realistic cells.
~ = (V, h, m, n)T ,
Recall the Hodgkin Huxley equations for a point neuron, where X
i.e.,
5
−rm ∂V (t)
=
g N a m3 h(V − VN a ) + g K n4 (V − VK ) + g leak (V − Vl ) + Isyn
∂t
2πaτ
dh(V, t)
h∞ (V ) − h(V, t)
=
dt
τh (V )
(5.31)
dm(V, t)
m∞ (V ) − m(V, t)
=
dt
τm (V )
dn(V, t)
n∞ (V ) − n(V, t)
=
dt
τn (V )
Hansel and later van Vreeswijk considered two Hodgkin Huxley cells with a
synaptic current given by
Isyn = −Gsyn [V (t) − Vsyn ]
spikes
X
f (t − ti )
(5.32)
i
~ + t) is found
where he used an ”alpha” function for f (t). The form of Z(ψ
from directly evaluating perturbations to the limit cycle, which is found from the
Hodgkin-Huxley equations with Isyn = 0. The perturbation is given by
0
P (ψ + t, ψ + t) = −Gsyn [V (ψ + ωt) − Vsyn ]
spikes
X
f (ψ 0 + ωt − ωti )
(5.33)
i
The systematics as a function of α for fixed ω were explored by van Vreeswijk
The phase as a function of Ie xt, really ω, for fixed α were explored by Hansel.
He also examined where in the cycle the neuron is most sensitive to perturbations.
5.3.3
Two oscillators with different intrinsic frequency.
We take
Γ(δψ − δψ 0 ) ≡ −Γ0 sin(δψ − δψ 0 )
(5.34)
Then
dδψ
= Γ0 sin(δψ 0 − δψ) + δω
dt
dδψ 0
= Γ0 sin(δψ − δψ 0 ) + δω 0
dt
The system will phase lock, for which
strength can satisfy
6
dδψ
dt
=
dδψ 0
,
dt
(5.35)
so long as the interaction
Γ̃ = Γ0 sin(δψ 0 − δψ) − Γ0 sin(δψ − δψ 0 )
= −2Γ0 sin(δψ − δψ 0 ) = δω − δω 0
(5.36)
(5.37)
2Γ0
>1
|δω 0 − δω|
(5.38)
or
The phase shift is just
0
δω 0 − δω
2Γ0
−1
δψ − δψ = sin
!
(5.39)
and the frequency under phase lock is
δω + δω 0
(5.40)
2
The above are the two quantities are the ones measured in the lab!
Outside of the phase locked region, the system undergoes quasiperiodic motion
with a time varying phase shift given by
ωobserved = ω +
q
δψ − δψ 0 = 2 tan
 (δω −
−1 
δω 0 )2
−
4Γ20
tan
(δω−δω 0 )2 −4Γ20 t
2
δω − δω 0

5.3.4
√

+ 2Γ0 


(5.41)
Chain of oscillators with δω ∝ ∆x: The example of Limax.
X
dδψx
Γ(δψx − δψx0 )
= δωx +
dt
x6=x0
(5.42)
δωx ∝ x + constant
(5.43)
with
When the system locks, there is a single frequency, but a gradient of phase shifts
x
x
with ∆ψ
given by a monotonic function of x, like ∆ψ
∝ constant, i.e., the phase
dx
dx
shift appears as a traveling wave. The data from Limax shows traveling waves
and a gradient of intrinsic frequencies. The article by Ermentrout and Kleinfeld
summarizes this and other data.
5.3.5
Two oscillators with propagation delays.
We again take
Γ(δψ − δψ 0 ) ≡ −Γ0 sin(δψ − δψ 0 )
Then
7
(5.44)
dδψ
= Γ0 sin(δψ 0 (t − τD ) − δψ(t)) + δω0
dt
dδψ 0
= Γ0 sin(δψ(t − τD ) − δψ 0 (t)) + δω0
dt
(5.45)
where the frequencies δω0 are assumed to be equal. We assume a solution of the
form
δω = δω0 − Γ0 cos α sin δωτD
(5.46)
This is satisfied for
(
α=
0 if cos ωτD ≥ 0
π if cos ωτD < 0
Thus we observe both frequency shifts and potential phase shifts. The synπ
. The details of this relation will
chronous stare is stable only for 0 < τD < 2δω
change if the symmetry of the waveform changes, but the gist is correct.
8