Mathematics 562
Homework Assignment 5
Due Thursday April 7, 2016
Phase-response curve, coupled oscillators, phase-locking and synchronization
Question 1:
Consider two oscillators described by the following form of FitzHugh-Nagumo equations
ẋ = f (x) − y,
ẏ = (x + a),
(1)
where f (x) = x − x3 /3, > 0 and a are parameters. Choose = 0.01 and a = 0.8. Consider
the coupling between the two oscillators is given as c(x2 − x1 ) in the equation for x˙1 .
(a) Use XPP to calculate the phase response curve for the x-variable of the FitzHugh-Nagumo
oscillator for the parameter values above.
(b) For c = 0.5, calculate the odd part of the phase resetting function Ho (φ). Use the information to determine (i) how many phase locked solutions exist in the coupled system; (ii)
which ones are stable and which ones are unstable. (Due to a bug in the XPP, one must add
a 3rd variable in the .ode file, say ds/dt = x + y − s, so that there is enough space to save
H(φ), Ho (φ), He (φ).)
(c) Repeat (b) for c = −0.5. Explain the difference in the result.
(d) Now write a separate .ode file for the coupled system
x˙1
y˙1
x˙2
y˙2
=
=
=
=
f (x1 ) − y1 + c(x2 − x1 ),
(x1 + a),
f (x2 ) − y2 + c(x1 − x2 ),
(x2 + a),
(2)
with the same choice of parameter values. Verify the results obtained in (b) and (c) numerically.
Question 2:
Consider the network of coupled neurons described by Wang-Buzsaki model
Cm
dVi
i
= Iapp − gN a m3∞ h(Vi − VN a ) − gK n4 (Vi − VK ) − gL (Vi − VL ) − Isyn
,
dt
(i = 1, . . . , N )
with the gating variable q (q = h, n), described by
dq
= φ[αq (1 − q) − βq q]
dt
m∞ = αm /(αm +βm ) with αm = −0.1(Vi +35)/[exp(−0.1(Vi +35))−1] and βm = 4exp(−(Vi +
60)/18). Also, αh = 0.07exp(−(Vi +58)/20), βh = 1/[exp(−0.1(Vi +28))+1], αn = −0.01(Vi +
34)/[exp(−0.1(Vi + 34)) − 1], and βn = 0.125exp(−(V
i + 44)/80). Synaptic currents received
P
i
gji sj (Vi − Vsyn ), where gji are the synaptic
by the ith neuron is modeled by, Isyn =
j6=i
conductances. For excitatory coupling, Vsyn = 100 mV . For inhibitory coupling, Vsyn =
−70 mV . Synaptic gating variable is described by
dsj /dt = αF (Vj )(1 − sj ) − βsj ,
1
where F (Vj ) = 1+exp(−V
, α = 12 msec−1 and β = 0.1 msec−1 . Other parameter values
j /2)
are Cm = 1 µF/cm2 , gN a = 35 mS/cm2 , VN a = 55 mV , gK = 35 mS/cm2 , VK = −90 mV ,
gL = 0.1 mS/cm2 , VL = −65 mV , φ = 5. Iapp = 1.0µA/cm2 .
(a) Find the PRC of each individual Wang-Buzsaki oscillator for the parameter values given
above. Print a copy of the result.
(b) Consider N = 2 and that g12 = g21 = g = 0.1. Find the H(φ), He (φ), and Ho (φ) of this
model when the two oscillators are coupled by excitatory synaptic coupling. Use the result to
predict the outcome when the two are coupled by weak excitatory synaptic coupling. Print
a copy of the result.
(c) Consider N = 2 and that g12 = g21 = g = 0.1. Find the H(φ), He (φ), and Ho (φ) of this
model when the two oscillators are coupled by inhibitory synaptic coupling. Use the result to
predict the outcome when the two are coupled by weak inhibitory synaptic coupling. Print
a copy of the result.
(d) Now, consider N = 3 and that gij = g = 0.1 for all i, j = 1, 2, 3. Find the equations
governing the evolution of phase-differences between these oscillators.
(e) Based on the results in (d) and the fact that Ho (φ) is 2π-periodic and is odd, how many
phase locked solutions can occur in such a system irrespective of the nature of the oscillators
and the coupling between them.
(f) Discuss the stability of the solutions found in (c) for the system of N=3 coupled WangBuzsaki oscillators with either excitatory or inhibitory synaptic coupling based on the specific
features of Ho (φ) obtained in either (b) or (c).
(g) Write a .ode file to verify the results obtained in (b), (c), and (f).
(h) For the phase locked solutions found in (b) and (c), show numerically how the frequency
of the coupled oscillators changes relative to the frequency of a decoupled oscillator. Discuss
the relationship between this and He (φ) obtained for each case.
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