Engineering, 2012, 5, 65-67
doi:10.4236/eng.2012.410B017 Published Online October 2012 (http://www.SciRP.org/journal/eng)
Copyright © 2012 SciRes. ENG
Simulation of Ultra-slow Oscillations Using the Integrate and
Fire Neuron Model
Danny Ng1, MokSiew-Ying1, C han Siow-Cheng1, Goh Sing-Yau2
1Department of Mechatronics and BioMedical Engineering, University Tunku Abdul Rahman, UTAR, Kuala Lumpur, Malaysia
2Department of Mechanical and Material Engineering, University Tunku Abdul Rahman, UTAR, Kuala Lumpur, Malaysia
Email: ngwk@utar.edu.my, moksy@utar.edu.my, chansc@utar.edu.my, gohsy@utar.edu.my
Received 2012
ABSTRACT
The Integrate and Fire (IF) neuron model wasusedto simulate ultra-slow oscillations that were observed in cortical cultures. Simula-
tion of a network with 2 sub-networks is conducted in this study. We introduced an additional equation that governs the generation
and dissipation of an inhibitory property to each of the sub-network.Sub-networks that fire at different rate are generated from the
simulation. The network activity from the simulation oscillates at frequencies that are comparable to ultra-slow oscillations observed
in cortical cultures.
Keywords: Integrate and Fire Neurons; Ultra-slow Oscillation; Simulation
1. Introduction
Oscillations play a crucial role in numerous processes of the
nervous system. Oscillations in the form of electroencephalogram
(EEG) are present in different brain structu res, wit h freq uen cies
ranging from 0.5 Hz (δ rhythm) to 40-80 Hz (γ rhythm), and
even up to 200 Hz [1]. Slow oscillations with frequency less
than 1Hz were detected in various in vivo experiments [2-4].
Ultra-slow oscillations with frequencies less than 0.01Hz were
reported in other experiments [5-8]. A recent study by Mok et
al. [5] reported on ultra-slow oscillations in MEA cultures of rat
cortical neurons. The ultra-slow oscillations were charact erized
by large synchronized bursts at the peaks and smaller bursts at
the troughs. These activity patterns emerged in cultures after
the fourth week in vitro.
In computational studies, it is possible to generate global os-
cillations in a network of inhibitory neurons [9]. Inhibitory
coupling in the network can act to synchronize the oscillatory
activity in the network [10]. Heterogeneous networks consisting
of inhibitory and excitatory neurons can exhibit a wide range of
behavior depending on the parameters and inputs given to the
network [11,12]. Activity patterns such as steady firing and
bursting can be simulated by varying the network connectivity
and fractions of endogenously active neurons [13,14]. Bursting
activit y can be ob served by havin g a balance b etween th e exci-
tation and inhibition in the network [15]. Synaptic characteris-
tics such as connection strength and synaptic depression be-
tween neurons in the model can influence the burst activity
pattern of a network [16]. For some large networks, the syn-
chronized bursting events might be classified into several dis-
tinct types based on their spatiotemporal substructures [17].
2. Simulation
In this study, numerical simulations of a network of IF neurons
were conducted in an attempt to simulateultra-slow oscillations
observed by Mok et al. [5]. All equations in this simulation
were solved using the Runge-Kutta 4th order method [18].
2.1. Neuron Model
The single neuron model introduced by Lathamet al. [13] was
used in this simulation. The time evolution equation for the
membrane potential of neuron
( )
i
vt
is,
( )
( )
( )
( )
( )
( )
,
α
i
mirita iAHPSYN
dv t
Tvt vvt vItII
dt =−−+−− (1)
whe re
m
T
is the membrane time constant, α determines the
rate of change for the membrane,
r
v
is the resting potential,
t
v
is the threshold potential,
( )
,ai
It
controls the fraction of
endogenously active cells,
AHP
I
the afterhyperpolarization
(AHP) current and
SYN
I
the synaptic current.
The AHP current is a combination of the fast AHP current
which is responsible for the refractory period and slow AHP
current which is responsible for spike frequency adaptation.
AHP
I
is given as,
(2)
whe re
k
ε
is the potassium reversal potential. ,ki
gand ,kca i
g,
the potassium conductance is governed by the time evolution
equation,
( )
,,
,
,
ki kiu
ki i
Ki
dg gtt
dt
µ
δδ
τ
=−+ −
(3)
( )
,,
,
,
kCa ikCa iu
kCa ii
kCa i
dg gtt
dt
µ
δδ
τ
−−
=−+ −
(4)
where ,Ki
τ
and
,kCa i
τ
are the time constant,
,ki
δ
and
,kCa i
δ
are the in crease in co nductance and u
i
t is the time spike occur
on neuro n i. The synaptic current
SYN
I
is given as
*
This workwas supported by a grant from the Fundamental Research Grant
Schemeunder the Ministry of Higher Education Malaysia.
D. NG ET AL.
Copyright © 2012 SciRes. ENG
66
( )
( )
,SYNiii
IvtI I
ε
= −

(5)
whe re
( )
i
vt
is the membrane potential. i
I
and
,i
I
ε
are given
as
,
()
u
ii
s ijj
ju
s
dII rw tt
dt
δ
τ
=−+ −

(6)
( )
,,
,
ii u
sijjj
ju
s
dII r wtt
dt
εε
εδ
τ
=−+ −
 (7)
where
s
τ
is the decay constant for number of open channel,
s
r
is the number of closed channel open during spike,
ij
w
the
strength of connection and
j
ε
the reversal potential.
2.2. Network Model
Network connections are formed randomly between neurons.
Connectivity bias [13] are introduced in the network through
j
j
j
T
ET
E IT
K
PN NB
=+
(8)
j
j
j
j
TT
IT
E IT
K
PNN
B
B
=+
(9)
where
j
ET
P
and
j
IT
P
are the connection probability for exci-
tatory and inhibitory neurons, j
T
K is the number of postsy-
naptic neuron,
I
N
is the number of inhibitory neurons,
E
N
is the number of excitatory neurons,
j
T
B
is the connection
bias and j
T is the type of postsynaptic neuron. The connec-
tion probability with inhibitory neuron will be higher if
j
T
B
>
1 whereas connection probability with excitatory neuron will be
higher if
j
T
B
< 1. Neuron s in the network are allowed to con-
nect to all other neurons.
2.3. Sub-networks
Single-unit activity from the experiment showed that some
neurons fire continuously while others fire only at the peaks.
Similar burst motifs were observed in experiment conducted by
Mok et al. [5] and Volman et al. [17]. Based on the activity
pattern of the single-unit activity and the burst motifs [5], we
post ulate t hat th ere are t wo or more di sti nct sub-networks in the
cortical culture.
For the purpose of the present simulations, the neurons are
divided into 2 sub-networks.Neurons are assigned randomly to
the sub-networks through the Bernoulli processwith
1s
N
pN
=
(10)
whe re
p
is the probability of success,
1s
N
is the number of
neurons in sub-network 1, N is the total number of neurons. A
success in the trial will place the neuron in sub-network 1 whe-
reas a fail i n the trial will place the neuron in sub-network 2.
An equation describing the generation and dissipation of an
inhibition property,
( )
,0
in u
i
i
dtt
dt
ϕ
µ
ϕ
ϕϕ
δδ
τ
=
=
=−+ −
(11)
is assigned to each of the sub-network.
ϕ
increaseswhen
neuron spikes in the sub-network and decays with a time con-
stant,
ϕ
τ
. The increase in inhibition property is given by
ϕ
δ
.
When
ϕ
reaches anupper threshold (
)
u
T
, the neurons within
the sub-network stop firing. The neurons start to fire again
when the inhibitory property has dissipated to alower threshold
(
)
l
T
. In this manner we will have neurons in each sub-network
firing at a different r ate.
3. Results and Discussion
The parameters in Ta ble 1 are used in the simulation to repro-
duce the bursting activity seen in the experiments. Figure 1
show simulated activities with frequencies similar to those ob-
served in the experiments. For the simulation, a slow
ϕ
τ
is
assigned to sub-network 1 to simulate a neuron group that ac-
tive only at the peaks whereas a fast
ϕ
τ
is assigned to sub-
network 2 to simulate a neuron group that is active all the time.
Figure 2 s hows simulat ed activit ies compared to anot her set of
experiment. With a different number of neuron in the
sub-networks, peak activity width of around 100s can be ob-
tained from the simulation. Upper threshold,
3.8
u
T=
an-
dlower threshold,
0.1
l
T=
are used for the simulations.
If the properties of the 2 sub-networks are different, it is
possible to simulate one sub-network that fires continuously
while another sub-network fires periodically.The time period of
the ultra-slow oscillations in experiments by Mok et al. [5] is
much larger than previously reported and cannot be generated
by the standard IF model. The current simulations of the ul-
tra-slow oscillations are the results of 2 sub-networks firing at
different rates.
Tabl e 1. Par amet e rs use d f or s imu lation of neu ron activity.
Parameter Value
m
T
10ms
r
v
-65mV
t
v
-50mV
α
1/15
k
ε
-80mV
s
τ
3ms
s
r
0.1mS
K
τ
30ms
k
δ
1mS
k Ca
δ
0.2mS
Number of Neuron 10000
Con ne ctio n pe r Neur o n 2000
Excitatory Neuron 2000
Inhibitory Neuron 8000
VEPSP 1mV
VIPSP -1.5mV
Bi 0.8
Be 1.2
Imax 3.8-5
D. NG ET AL.
Copyright © 2012 SciRes. E NG
67
Fig ure 1. Simulation of activity generated with parameters:
ϕ
δ
=
0.07, upper threshold,
u
T
= 3.8, lower threshold,
l
T
= 0.1, sub-
network 1: number of neuron = 2000,
φ
τ
= 30000, s ub-network 2:
number of neurons = 8000,
φ
τ
= 1000 compared with experimen-
tal results. The time bin for spikes is 10ms.
Fig ure 2. Simulation of activity generated with parameters:
ϕ
δ
=
0.07, upper threshold,
u
T
= 3.8, lower threshold,
l
T
= 0.1, sub-
network 1: number of neuron = 3000,
φ
τ
= 30000, sub-network 2:
number of neurons = 7000,
φ
τ
= 1000 compared with experimen-
tal results. The time bin for spikes is 10ms.
It is likely that these ultra-slow oscillations are controlled by
biochemical processes and/or network structures in the cortical
culture. Fu rther experi ments are n eed ed t o bett er unders tan d the
underlying biochemical processes that cause theseultra-slow
oscillations.
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