Noise-induced coherence and network
oscillations in a reduced bursting model
Running title: Noise-induced coherence in a reduced bursting model
Stefan Reinker
Department of Mathematics and Physics, FDM, University of Freiburg
Hermann-Herder-Str. 3
79104 Freiburg, Germany
Tel.: +49-761-2035828
Fax: +49-761-2035967
Email: reinker@physik.uni-freiburg.de
Yue-Xian Li, Rachel Kuske
Department of Mathematics,
The University of British Columbia, Vancouver, BC, Canada V6T1Z2
Abstract
The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is
reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after
a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF)
model.
Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the
occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of
the network oscillations shows a stochastic resonance-like dependence on the noise
level.
Stochastic analysis of the equations shows that the slow recovery from the spikeinduced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the
oscillation frequency strongly depends on the intrinsic time scales but changes little
with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations.
We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in
generating synchronized network oscillations with a frequency that is determined
by intrinsic properties of the neurons.
Preprint submitted to Elsevier Preprint
25 August 2005
Key words: nonlinear dynamics; random processes; neural nets; white noise;
neuroscience; synchronization; resonance; stochastic resonance
PACS: 05.45.-a, 05.40.Ca, 87.18.Sn, 87.19.
1
Introduction
In addition to the sodium and potassium currents involved in action potential
generation, many types of neurons exhibit other ion currents that generate
slower dynamics. These currents can be involved in the processing of subthreshold synaptic inputs and the generation of subthreshold frequency preferences (Destexhe et al., 1996). Many types of neurons exhibit a subthreshold
membrane resonance which is manifest as a maximal deterministic voltage response to small periodic current inputs at the resonant frequency (Hutcheon
and Yarom, 2000). This resonance can selectively amplify subthreshold periodic inputs and participate in rhythmogenesis, for example in the thalamus
(Destexhe et al., 1996). In thalamocortical neurons a low threshold calcium
current IT gives rise to membrane resonance for inputs with 3 Hz frequencies which is believed to be involved in the generation of spindle oscillations
(Contreras et al., 1996), as well as in absence epilepsy.
In neurophysiological experiments, subthreshold resonance is demonstrated
by a periodic subthreshold current stimulus which causes a maximal voltage
response at the resonant frequency. Similar to classic resonance in electrical
systems, the resonance appears as a maximum in the impedance magnitude.
Bursting is characterized by fast activity interspersed with long quiescent interburst intervals. It has been observed in many types of neurons, including
thalamic and hippocampal cells. A number of mathematical models of bursting
neurons have been developed over the past few decades. The Hindmarsh-Rose
(HR) model was derived from a physiological model of thalamic neurons which
exhibit burst firing of action potentials and subthreshold resonance. Here, we
derive an integrate-and-fire model with a subthreshold resonance that matches
the subthreshold features of the HR model of bursting neurons (Hindmarsh
and Rose, 1984). In this resonant integrate-and-fire model (RIF), bursting is
generated by a combination of a spike-induced reset mechanism and the slow
recovery of the reset variable to its equilibrium value before the burst can occur again. The simple model has the advantage that it has linear subthreshold
dynamics which make it numerically and analytically more tractable than the
HR and other current-based models.
Modified integrate-and-fire models that incorporate a subthreshold resonance
or bursting similar to our RIF model were introduced by Izhikevich (2001).
Brunel et al. (2003) and Richardson et al. (2003) created and analyzed an
2
integrate-and-fire model with Ih resonance and demonstrated that noise in conjunction with this resonance can affect the firing response rate. A more complicated nonlinear integrate-and-fire system was used by Smith et al. (2000)
to model thalamocortical neurons and burst firing. Our model also bears similarity to the IF model proposed by Liu and Wang (2001) which incorporates
spike frequency adaptation.
Noise is present in all physical systems. The important roles played by a combination of synaptic and thermic noises on information processing in neurons
has been proposed (McCormick, 1999). Through stochastic resonance (SR),
noise at an optimal level can maximize the coherence between an input signal and the output spike train. Noise helps in the detection of subthreshold
stimuli, but degrades the timing precision in response to suprathreshold stimuli (Pei et al., 1996). When amplified by noise, subthreshold oscillations and
dynamics also can have a significant influence on the coding of information in
neurons (Braun et al., 1998), and signals can be transmitted faithfully even
by highly irregular trains of action potentials (or spikes) (Wang and Wang,
2000). Thus, the noise that impacts neurons in the form of random synaptic inputs (Stacey and Durand, 2000), as well as membrane noise (Steinmetz
et al., 2000; White et al., 2000), can boost subthreshold signals. Lindner et al.
(2004) provide an extensive review of the effects of noise on neurons and the
appearance of coherence and stochastic resonance.
Neurons are usually part of a network and receive synaptic inputs from other
neurons. One prominent feature of many neural networks is the emergence of
large scale synchronization, such as appears in an EEG. Synchronized activity
is especially important in the thalamus which is involved in the generation of
the sleep/wake cycle. Some of these oscillations occur in the frequency range
of the subthreshold resonance of thalamocortical neurons. SR has also been
shown to occur in human brain waves (Mori and Kai, 2002) which suggests a
connection between noise and synchronized brain activity.
Synchronization of simple oscillators by noise has been observed and analyzed
by Neiman et al. (1999). In networks of coupled stochastic neurons, coherent
firing (Rappel and Karma, 1996) and improved detection of input signals
compared to single neurons was demonstrated (Wang and Wang, 1997). This
is called array-enhanced stochastic resonance (Lindner et al., 1995). It has
been shown in model networks that noise can induce stochastic resonance
(Hauptmann et al., 1999; Qian and Zhang, 2002) and synchronized network
activity (Acebron et al., 2004; Rappel and Karma, 1996; Tiesinga and Jose,
2000). SR and stochastic synchronization have also been shown to occur in
the crayfish mechano- and photoreceptor network (Bahar et al., 2002; Bahar
and Moss, 2003).
In a stochastic network, it is difficult to quantify synchronization and most
3
studies were performed in spontaneously oscillating systems where a phase
can be defined. When an input signal is subthreshold and periodic, stochastic phase locking to the signal can occur (Longtin, 2000; Tateno and Jimbo,
2000). Shuai and Durand (1999) were able to define a phase in the deterministic HR model, and showed that two coupled neurons even can synchronize
their chaotic oscillations. Wang et al. (2000) found CR and synchronization
of high frequency firing by noise in an HH network. In these studies network
oscillations appeared at high frequencies. The oscillation frequency increases
as the noise level increases since the threshold crossing is essential for the excitation of an excitable neuron. In contrast, synchronized network oscillations
in the delta, theta, and gamma ranges were observed in a model by Tiesinga
et al. (2001). The slow frequencies arise from additional ion currents with slow
time scales. Börgers et al. (2005) explained the genesis of gamma rhythms in
a noisy network including excitatory and inhibitory neurons.
Our main objective in this work is to examine the effects of noise on bursting
and frequency preference behaviour of single neurons and networks. In particular we want to investigate if network oscillations can occur at frequencies
related to the intrinsic frequency of the resonant neuron.
The model we derive from the reduced HR model possesses the simplest possible mechanisms required in generating noise-induced coherence resonance
with a slow oscillation frequency that is determined by the intrinsic dynamic
properties of the neuron and is almost constant over a significant range of
noise intensities.
This is an extension of our earlier work where we revealed a connection between
the subthreshold membrane resonance, noise, and information processing in
experiments and model studies on single thalamocortical neurons (Reinker
et al., 2003, 2004). With the RIF model, we are able to investigate analytically the subthreshold properties and their influence on firing pattern. We
can quantify the CR mechanism to predict the range of noise and coupling
strength for which the resonance is maximized. In a globally coupled network
of such model neurons, we study the relationship between the subthreshold
resonant frequency and that of the noise-induced coherent oscillations in a
network of such neurons.
2
The Model
The simplest possible model of a bursting neurons with a predetermined subthreshold resonance frequency is an integrate-and-fire model involving two
linear ODEs.
4
dx
= Ax + By + Isignal (t),
dt
dy
= Cx + Dy, for x < xthres ,
dt
x → xreset , y → yreset , when x reaches xthres .
(1)
(2)
(3)
Here x is a voltage-like variable and y describes the slow dynamics of an
adapting current, similar to the slow inactivation of a T-type current in thalamocortical neuron models (Huguenard et al., 1992) or the slow relaxation
variable of the HR model (Hindmarsh and Rose, 1984). In the absence of any
input (i.e. Isignal (t) = 0), the system has a single stationary equilibrium at
(0, 0). In order to test the response of the system to (current-like) inputs, a
term Isignal (t) is added in the equation for the voltage-like variable x. However,
in the study of noise-induce coherence Isignal (t) = 0 is always assumed.
As in the one-dimensional integrate-and-fire model, when x reaches a threshold
xthres , x is reset to xreset and an action potential is generated. In the HR model,
an action potential also causes an increase of the y variable. To mimic this
effect in the RIF model, we reset y to y + yreset each time when a spike occurs.
We attempt to choose the coefficients of (1)-(2) to mimic the subthreshold
behaviour of the HR model. The parameters A, B, C, and D are necessary
for matching the subthreshold properties of the HR system. The HR system
exhibits a subthreshold resonance to signals with input periods near 336 time
units (Reinker et al., 2003). In the RIF model, we attempt to include a similar
subthreshold resonance. The resonance structure of (1)-(2) can be computed
using a periodic input signal Isignal (t) = δeiwt , and linearizing the system
around its equilibrium state (0, 0),
dx
= Ax + By + δeiwt ,
dt
dy
= Cx + Dy.
dt
Letting x = x̄eiωt , y = ȳeiωt , and δ = 1 in these equations, and then dividing
by eiωt , the system becomes
x̄iω = Ax̄ + B ȳ + 1,
ȳiω = C x̄ + Dȳ.
Solving for ȳ in the second equation and eliminating ȳ in the first gives
impedRIF
BC
= x̄ = iω − A −
iω − D
·
5
¸−1
.
2.5
|imped|
2
1.5
1
0.5
0
0
500
λ
1000
1500
Fig. 1. Result of the least squares fit of the impedance magnitude of the RIF (solid
line) and HR (broken line) models with dependence on ω. The figure shows the
dependence of |impedHR | and |impedRIF | on the period λ = 2π/ω. Here, it is
important to note that the resonance curve of the HR system is dependent on
the rest level, and we fit the impedance of RIF to that of HR at the steady state
x0 = −1.44027.
This expression gives the complex impedance that describes the response of
the magnitude and phase of voltage (x) to a periodic input signal (Isignal (t) =
δeiwt ).
Now we try to obtain values for A, B, C, D such that the resting values,
resistance, and the resonance of the RIF model are similar to those of the
HR model. In order to do this matching, we minimized computationally the
difference between |impedRIF | and |impedHR |. Fig. 1 shows a match with the
parameter values A = −0.032, B = −1.3258, C = 0.00025, D = −0.001. Note
that the values of C and D are much smaller than A and B, and hence the
second equation describes slow dynamics, similar to the slow equation in the
HR model. Note also that |C| << |D| which implies that the term Cx in
the second equation has negligible significance. This was confirmed by making
C = 0 in a simplified version of this model, qualitatively the same results were
obtained (results not shown).
The resonance in Fig. 1 is strictly in the subthreshold regime analogous to
membrane resonance measured experimentally in neurons. Subthreshold membrane has been shown to influence the firing behaviour of thalamic and hippocampal neurons (Hutcheon and Yarom, 2000). Our goal is to investigate
how such subthreshold dynamic properties influence the firing of single RIF
cells and network behaviour.
The dynamic behavior of this RIF burster is characterized by three features: (i)
an integrate-and-fire mechanism in the voltage variable x that generates one
6
spike of action potential each time when xthres is reached; (ii) an instantaneous
reset of x to xreset < xthres and y to yreset > y when a spike occurs; (iii) a strong
inhibition of the voltage by the slow variable y through the term −1.3258y
that is slowly lifted as y re-approaches its equilibrium value at y = 0. We
take xthres = 1 and choose xreset = 0.9, close to the threshold, so that the
probability of a second crossing after reset is high. This allows bursts of action
potentials to occur. We reset y to y + 0.1 as observed in the HR model.
The inhibition of x by increased y terminates a burst of action potentials.
Therefore, the role of y can be described as a spike-induced inhibition of the
voltage variable x. The slow recovery of y to its equilibrium value at y = 0
plays the dominant role in determining the subthreshold frequency of the RIF
as well as the frequency of the CR.
Varying xreset between 0 and 1 influences only the number of crossings during
a burst and does not change our results. Also, as long as yreset > y after
a threshold crossing, a burst will be terminated eventually and followed by
a refractory interburst period. Thus, the results presented in this paper are
independent of the exact choice of the reset values. CR and synchronized
network oscillations can be observed for a wide range of parameter values.
3
Results
3.1 Deterministic firing in the RIF model
Without any input, the deterministic system stays at its stable fixed point
(0, 0). Continuous firing occurs for a constant, positive input Isignal as shown
in Fig. 2. The firing frequency is slow with single threshold crossings.
The number of threshold crossings in a burst depends on the relation of the
reset values xreset and yreset . After the reset, xreset and yreset determine whether
dx/dt in equation (1) is positive or negative. If dx/dt remains positive, a second
spike will occur when the threshold is reached again. For y values significantly
larger than zero, dx/dt becomes negative and the burst is terminated. This
regular burst firing behaviour reflects the bursting patterns of the HR model
and the resonant integrate-and-fire model of Izhikevich (2001).
3.2 Stochastic firing in the RIF model
When Gaussian white noise is added to the x equation (1), the RIF model becomes a two-dimensional Ornstein-Uhlenbeck process in the intervals between
7
1
0.5
0
x −0.5
−1
−1.5
0.38
0.37
0.36
0.35
0.34
y
0.33
0.32
0.31
0.3
0.29
0.28
0.27
0
200
400
600
800
1000
t
1200
1400
1600
1800
2000
Fig. 2. Trace of a realization of the deterministic RIF model with
Isignal = 0.4 = const. When x reaches xthres = 1.0, the system is reset and a new
trace starts. The concurrent increase of y makes the LHS of equation (1) negative
which causes a decrease of x away from threshold. Thus, after a burst, which here
consists of only one threshold crossing, x makes an excursion away from threshold
during which y can recover towards its equilibrium value 0, and another burst can
occur.
successive bursts of spikes. The model now becomes
dx = (−0.032x − 1.3258y + Isignal )dt + σdW,
dy = 0.001(0.25x − y)dt,
x → 0.9, y → y + 0.1, when x reaches 1.
(4)
(5)
(6)
where W is standard Brownian motion and σ is constant denoting noise intensity. The presence of noise makes it possible for x to reach the firing threshold
and evoke an action potential, even with zero or subthreshold levels of Isignal
that can not generate spikes without noise. Fig. 3 shows a trace of the stochastic model with Isignal = 0. The x variable fluctuates in response to the noise,
and threshold crossings can occur. After such a crossing the probability of
a second crossing is high for sufficiently large noise because reset is close to
threshold. Consequently, single or burst firing arises that causes steep increase
in the value of y. Higher values of y exert a strong inhibitory influence on x.
This inhibition transiently suppresses the value of x. This gives rise to a strong
refractory period during which y slowly recovers to its equilibrium value at
8
2
0
−2
x
−4
−6
−8
0.4
0.3
0.2
y
0.1
0
−0.1
−0.2
0
200
400
600
800
1000
t
1200
1400
1600
1800
2000
Fig. 3. Trace of a realization of the stochastic RIF model with σ = 0.2. The noise
acts mainly in the x direction because it is only present in the first equation, while
y dynamics are smoother.
zero with a time constant τy . During this time period, the deterministic decay
of y dominates the dynamics of the system, so that x is far below the threshold
and the stochastic fluctuations are unlikely to induce spiking. The deterministic decay of y dominates the dynamics of the system. On the subsequent
approach to steady state, sufficiently large noise can induce another burst of
threshold crossings. These dynamic features generate the burst firing rhythm
with interburst periods determined by the time scale of the slow recovery of
the y variable, τy .
These dynamics with multiple threshold crossings in a burst can be demonstrated in a interspike interval histogram (ISIH), a tool that is frequently used
in the analysis of neuronal firing. The ISIH in Fig. 4 shows a peak near 0 interspike time which corresponds to rapid consecutive threshold crossings in burst
firing. At a higher noise level, a second peak appears near 300 interspike time,
corresponding to a preferred time between bursts. This ISIH peak, however,
does not correspond to oscillation period because the bursting phase is not
counted and only the interburst time appears in the histogram. This preferred
interval is closely related to the intrinsic time scale τy that determines the
recovery of the y variable to its equilibrium value at zero. The location of the
peak moves to an interspike time near 210 with increasing noise level. Further
increase of the noise level does not cause further decrease in this interval unlike
9
0.05
rel. count
0.04
0.03
0.02
0.01
00
50 100
150 200
250 300
350 400
IS time
450 500
0
0.1
0.2
0.3
0.4
0.5
0.6
σ
Fig. 4. Interspike interval histogram of the RIF model showing the dependence of the
firing dynamics on σ. The first peak near 0 interspike interval time arises from burst
firing with small interspike times. For higher noise levels, a second peak appears in
the histogram that reflects firing at a preferred frequency, here for interspike times
between 200 and 300.
some other bursting models in which increasing noise levels result in decreasing
periods of bursting (Kuske and Baer, 2002). The shape of this low frequency
peak can be used to calculate measures of coherence and demonstrate CR (Lee
et al., 1998).
3.3 Network Behaviour
Neurons are always part of an interconnected ensemble and the properties of
individual neurons can give rise to emergent behaviour of neuronal networks.
The idea that noise can increase coherence in network dynamics was analyzed
by Zhou et al. (2001). Here, we have a neuron model with a slow time scale
and we set out to investigate if such a system can give rise to slow oscillations
in the network.
In order to study a network of RIF neurons, we model synaptic coupling in a
network of N cells by simply resetting xi to xi + ∆ when a threshold crossing
occurs in neuron j (j 6= i). This corresponds to the case of all-to-all coupling
with an identical coupling strength ∆ (Izhikevich, 2001). For positive values
of ∆, the coupling is analogous to depolarizing synaptic inputs, which brings
the neuron closer to threshold. Note that the coupling is instantaneous, i.e.
the firing of one neuron causes an immediate jump in xi by a value of ∆ for
all other connected neurons.
Typically, one excitatory synaptic input is not enough to kick a resting neuron
over threshold from rest, and consequently ∆ should be small. Then
10
A
50
45
40
35
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25
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B
0
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7000
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9000
0
1000
2000
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7000
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9000
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neuron number
40
35
30
25
20
15
10
5
0
C
50
45
40
35
30
25
20
15
10
5
0
t
Fig. 5. Raster plots of simulations of 50 RIF neurons. With the dependence on the
noise level (A: σ = 0.08, B: 0.12 C: 0.9), synchronized bands appear at intermediate
noise levels and are stable over a wide range of σ values. Here ∆ = 0.06.
xi (t + dt) = xi (t) +
yi (t + dt) = yi (t) +
Z
Z
t+dt
t
X
j6=i
t+dt
t
(A(xi (τ ) + Byi (τ ))dτ +
C(xi (τ ) + Dyi (τ )dτ,
∆δ(t − tj ) + σWi (t),
(7)
(8)
i = 1, . . . , N , with global coupling strength ∆, where tj denotes the times
neuron j crosses its threshold and fires an action potential.
Figure 5 demonstrates that such a network of 50 pulse coupled RIF neurons is
able to produce synchronized activity. When the noise level is low (Fig. 5A),
there is not much overall activity and only occasional threshold crossings oc11
0.1
rel. count
0.08
0.06
0.04
0.02
00
200
400
600
IS time
800
1000
0
0.05
0.1
0.2
0.25
0.3
σ
0.15
Fig. 6. Cumulative interspike interval histogram from a network of 50 RIF neurons,
dependent on the noise level σ. When the noise is strong enough, spiking activity
splits into bursts with short interspike times and interburst firing which stems from
the synchronized activity of the network. This second interspike peak is located
near 300 time units, and does not change significantly over intermediate levels.
Here, ∆ = 0.06.
cur, denoted by a black diamond symbol. For higher noise in Fig. 5B, there is
sustained activity and the firing events tend to be arranged in vertical bands.
These bands are made up of multiple action potentials or bursts of spikes,
which are synchronized through the whole network. The intervals between the
oscillatory bands are approximately 300 time units long which, is close to the
subthreshold resonance (λ = 336). The stochastic oscillations are destroyed
at higher noise levels (Fig. 5C). This demonstrates that synchronized stochastic network oscillations occur preferably at intermediate noise strength. This
regular synchronized activity occurs only in the presence of noise, because the
deterministic network is quiescent and cannot generate a spike in the absence
of noise.
The cumulative ISIH in Figure 6 shows that the period of the network oscillation does not change strongly with increasing noise for low or intermediate
levels. At low noise, the firing is irregular with few interspike events, while
for intermediate noise levels a peak near 300 time units appears. This interval
corresponds to the interburst time which is the length of the quiescent period
between two bursts. For increasing noise, the location of this peak only shifts
slightly towards shorter interspike times. This is analogous to the coherence
resonance of a single RIF neuron, see Figure 4.
The appearance of these stochastic oscillations is also dependent on the coupling strength ∆, which corresponds to synaptic coupling strength, as demonstrated in Figure 7. For fixed noise level σ and small ∆ (Fig. 7A), the stochastic
threshold crossings of individual neurons cannot evoke whole network oscillations. However when ∆ is larger (Fig. 7B), synchronized oscillations occur
similar to Figure 5. With increasing ∆, the synchronized activity persisted
and the frequency of the network oscillations does not change (Fig. 7C).
12
A
50
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0
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neuron number
45
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C
50
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5
0
t
Fig. 7. Raster plots of simulations of 50 RIF neurons, for different coupling strengths
∆ (A: ∆ = 0.06, B: 0.1, C: 0.2; σ = 0.08 = const.). Synchronized network oscillations occur when ∆ is large enough, and persists for higher coupling strengths.
In order to quantify the stochastic network synchronization we introduce a
measure that uses the spectrum of the mean network dynamics. Figure 8A
shows the x-trace averaged from all neurons of a network that exhibits synchronized stochastic oscillations. This trace shows smooth periodic oscillations.
The strong periodicity also is visible in the power spectrum of the average
trace as shown in Figure 8B. The main power spectrum peak appears at a
wavelength λ of 366 time units which is the average period of the network oscillations. Consequently we can use the power at this wavelength, that is the
area under the main peak, to define a measure S of network synchronization.
Because S is defined based on the averaged trace of all the neurons, it measures synchronization and would be 0 in a network of independently oscillating
13
A
10
0
−10
x
−20
−30
−40
−50
−60
0
500
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1500
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2500
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0
500
1000
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3000
2
1.5
y
1
0.5
0
B
t
4
10
2
Power
10
0
10
−2
10
−4
10
−6
10
0
100
200
300
400
500
λ
600
700
800
900
1000
Fig. 8. A: Average x and y traces of 50 neurons with σ = 0.1, ∆ = 0.06, cf. Figure 5.
The synchronized network oscillation appears as a periodicity. B: Power spectrum
from A plotted against the wavelength λ. The peak at λ = 366 reflects the period
of network oscillations. The synchronization measure S is defined as the area under
the peak.
cells.
The network synchronizes at intermediate noise levels (cf. Figure 5), and consequently the synchronization measure goes through a maximum as shown
in Figure 9, demonstrating a stochastic resonance-like phenomenon. Synchronization increases abruptly when the noise strength is sufficiently large and S
quickly reaches its maximum. Then, increased noise degrades synchronization,
leading to a slow decay of S. Even at high noise, when no synchronized oscillations are apparent in a raster plot, some degree of the preferred oscillation
frequency persists. Here, it is interesting to note that only low noise is required
to synchronize a network of 50 neurons when compared to the noise required
for CR in a single neuron, cf. Figure 4.
14
A
50
synchronization
45
40
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30
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5
0
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0
0.1
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1
460
440
420
400
λ
380
360
340
320
300
280
260
240
σ
Fig. 9. A: Synchronization measure S in dependence on the noise level. S shows
a stochastic-resonance-like maximum at intermediate noise. B: Peak wavelength
in power spectrum of the network average as a function of the noise level. When
synchronization is apparent in the raster plots (for σ between approximately 0.1
and 0.85), the power spectrum peak wavelength decreases.
A plot of the peak wavelength in Figure 9B shows that the synchronized
oscillations are faster with higher noise, although the range is limited between
250 and 440 time units. This is in contrast to the constant interspike time from
the ISIH (Figure 6). However, the power spectrum measures the wavelength
of the entire oscillation while the ISIH does not take into account the length
of the bursts. The sharp increase in λ at lower values of σ is caused by a
significant increase in the length of the plateau bursting phase between two
successive recovery phases (see the significantly longer durations of the last
three bursts in Fig. 3). The duration of the bursts is reduced by increasing the
noise intensity, because threshold crossing occur more often leading to a faster
increase in y. For fixed noise intensity, it can also be reduced by increasing
the value of yreset or by reducing the value of xreset .
15
4
Stochastic Analysis
Because of the simple linear form, a mathematical analysis in the single neuron case can give a closed form solution for the probability distribution of the
system described by the RIF equations without a threshold. During the interburst recovery period the system is far away from threshold (Fig. 3) and it is
appropriate to analyze the unconstrained equations as an approximation for
the isolated neuron. Furthermore, we show how this probability density can
be used to analyze the influence of the magnitude of the noise and the coupling, described by σ and ∆ respectively, on the transition of the system from
quiescent to regular bursting behaviour. In particular, we describe the mechanism which explains why the synchronized burst pattern is observed only for
intermediate values of σ when the coupling is small, as shown in Figure 5, or
for increased coupling when the noise level is small, as in Figure 7. We also
explain why long transient periods can be observed for intermediate values
of the coupling, and why these transients do not appear for larger values of
the coupling, as in Figure 7. A similar analysis of stochastic and coherence
resonance in the single FHN neuron model was performed by Pikovsky and
Kurths (1997).
4.1 Probability density
The probability density function for the state (xj , yj ) of cell j at time t is
defined as
P (x, y, t)∆x∆y = Prob(x ≤ xj < x + ∆x, y ≤ yj < y + ∆y), at time t.
For the solution of the unlimited stochastic differential equation (4)- (5) this
probability density is a Gaussian, which, following Risken (Risken, 1989) see
also (Brunel and Sergi, 1998), can be determined by solving the associated
Fokker-Planck equation (FPE):
∂P
∂
∂
σ2 ∂ 2P
= LF P P = − (Ax + By)P −
(Cx + Dy)P +
.
∂t
∂x
∂y
2 ∂x2
There is only one diffusion term because noise is present only in the x equation.
Initial conditions for P at an initial time t0 are given by a delta distribution
at the deterministic starting point (x(t0 ), y(t0 )),
P (x, y, t0 ) = δ(x − x(t0 ))δ(y − y(t0 )).
We consider initial conditions at t0 = 0 as well as reset times t0 following an
action potential, where we take x(t0 ) = xreset and y(t0 ) = yreset .
16
Then, the equation can be solved by expressing P through its Fourier transform P̃ with respect to (x, y), which fulfills a similar FPE. Using the Ansatz
that P̃ is also a Gaussian, the moments Mi (t) and variances Nij (t), i, j = 1, 2,
of P̃ are given by a system of ordinary differential equations
dM1
dt
dM2
dt
dN11
dt
dN12
dt
dN22
dt
= AM1 + BM2 ,
= CM1 + DM2 ,
= 2AN11 + 2BN12 + σ 2 ,
= CN11 + (A + D)N12 + BN22 ,
= 2CN12 + 2DN22 .
which can be solved analytically using the Green’s function
 

A B  
G(t) = exp 
 t .

C D
Then, inverting the Fourier transform, P is given by
P (x, y, t) =
√
1
2π N11 N22 − N12 N21

exp −

(x − M1 )2
(9)
N2
N11
)
2(1 − N1112
N22
1


2N12 (x − M1 )(y − M2 ) (y − M2 )2 
,
+
−
N11 N22
N22
a bivariate Gaussian distribution centred around (M1 , M2 ) with variances N11
2
and N12 and correlation coefficient N12
/N11 N22 . Note that the differential
equations for the first moments have the same coefficients as the deterministic
equation (1, 2).
A plot of M1 (t), M2 (t) in Figure 10 shows the time course of the two moments
starting from x(t0 ) = 0.9, just below threshold, and y(t0 ) = 0.1 and 0.2. For
both initial values of y, the mean of x, given by M1 , decreases steeply at the
beginning, falling well below 0 before approaching the equilibrium state after
about 300 time units. There is even a small overshoot of the steady state for
300 < t < 350 which is not visible in the plot.
Larger y starting values cause greater excursions of x from threshold. The
time scale before returning to rest, however, does not vary with the starting
value. M2 also recovers towards its rest value 0 with the same time constant
17
5
M
1
0
−5
0
100
200
300
400
500
600
0
100
200
300
400
500
600
0
100
200
300
400
500
600
0
100
200
300
400
500
600
0
100
200
300
t
400
500
600
M
2
−10
0
N
11
0.2
0.1
0
0.0006
N
12
0.0004
0.0002
0
N
22
0.00002
0.00001
0
Fig. 10. Plot of the moments M1 (t) and M2 (t) (above) and the variances Nij (t)
(below) of the probability distribution of the RIF model, starting from x = 0.9 and
y = 0.2 (black line), and x = 0.9 and y = 0.1 (broken line).
as M1 . The behaviour of these first moments is independent of the noise level
σ, so the single cell does not show qualitatively different behaviours for varying noise levels. The variances N11 and N22 shown in Figure 10 simply grow
monotonously and approach a non-negative steady state value. Because of the
identical coefficients in the deterministic and the moment equations, the time
scale of the stochastic dynamics is of the same order as that of the subthreshold
resonance.
The unconstrained solution P (x, y, t), defined by its moments shown in Figure
10 does not give the distribution of the system throughout threshold firing,
particularly during burst firing. However, it can be used to understand the
critical values of noise σ and coupling strength ∆ which lead to the regular
behaviour in Figures 5 and 7.
18
4.2 Transition to synchronized bursting
As described above, the initiation of a synchronized network oscillation is
abrupt. We want to derive a quantitative description for this transition to the
bursting phase. This can be achieved by using the probability distribution of
an isolated single neuron to estimate the threshold crossing probability and
the effect of coupling on the firing probability of other neurons in the network.
After a burst, that is following the last threshold crossing at time t 0 , y(t0 ) is
large which forces x to drop sharply (as shown in Figure 8A) so that threshold
crossings are unlikely for a significant time interval. During the interburst
period, the dynamics of the individual elements are isolated as long as no
threshold crossings occur.
Then P gives a good approximation for the probability density of a single
cell during the interburst time. With initial condition (xreset , y(t0 )) we can
compute the probability density from (9). Since the threshold is defined in
terms of x it is convenient to consider the marginal density for xj , that is
pj (x, t) =
Z
∞
−∞
P (x, y, t)dy = √
1
2
e−(x−M1 ) /2N11 ,
2πN11
(10)
and in this regime pj (x, t) provides a good basis for a description of the system.
In the interburst period M1 decreases away from threshold and the corresponding variance of x, N11 , is small. Because the variance N11 (t) grows with the
same time constant as that with which M1 (t) returns to rest, see Figure 10, a
threshold crossing can become more likely again for t > 300 time units after
initiation of the interburst period. This probability depends on the magnitude
of the noise σ.
The probability of a threshold crossing for the j th cell is written simply in
terms of the probability that xj > xthres , denoted qj (t), which can be found
by integrating pj (x, t)
qj (t) = 1 −
Z
xthres
−∞
pj (x, t)dx.
The quantity qj (t) can be viewed as the fraction of the traces in the unconstrained and uncoupled system which are above threshold at time t. For the
single neuron case, we show in Figure 11A and B the behaviour of pj (x, t)
as a function of x and σ. Figures 11C and D demonstrate how qj (t) varies
over time and with σ. Note that qj (t) is directly related to the portion of the
tail of the density pj (x, t) which exceeds xthres . For increased values of σ, the
variance of xj increases, so that a larger portion of the tail of pj (x, t) falls in
the range x > xthres .
19
A
B
1.4
2
1.2
1.5
0.8
j
p (x)
pj(x)
1
0.6
0.4
1
0.5
0.2
0
−2
0
1
x
0
−2
2
D
0.05
0.04
0.03
0.03
0
1
x
2
j
q (t)
−1
0.05
0.04
j
q (t)
C
−1
0.02
0.01
0
0.02
0.01
0
200
t
400
600
0
0
200
t
400
600
Fig. 11. A: Graph of pj (x, t) for σ = 0.12 at t = 200 and t = 330. The marginal
density is concentrated about the mean M1 with variance N11 as shown in Figure 10.
B: Graph of pj (x, t) for t = 330 and for σ = 0.08 (solid line) σ = 0.12 (dash-dotted
line), and σ = 0.2 (dashed line). C: Graph of qj (t) corresponding to pj (x, t) in B.
The solid line, for σ = 0.08 is indistinguishable from zero for all t. The dash-dotted
line corresponds to σ = 0.12 and the dashed line corresponds to σ = 0.2. D: Graph
of qj (t) corresponding to shifting the mean for pj (x, t) in B by ∆ = 0.06. Initial
conditions were x=0.9, y = 0.2 for all graphs in this figure.
For small σ, qj (t) is close to 0 for all time as shown in Figure 11C. That is,
an isolated cell is very unlikely to cross threshold, as follows from the fact
that the mass of pj (x, t) is concentrated on values of x < xthres . However, for
larger σ and large t > 300, qj (t) increases. This corresponds to the time when
x again approaches its first moment equilibrium M1 = 0, shown in Figure 10.
Even though qj (t) for a single neuron is still small for σ = 0.12 (qj (t) =
O(10−3 ) for t > 300), this increase is important for the mechanism of burst
initiation in the entire network. The rise in qj for t >200 with a maximum
for t > 300 demonstrates the relation between the subthreshold frequency
with the time scale of the refractory period. Only for t > 300 will there be a
significant chance for firing and initiating a burst. The similarity between the
curves in Figures C and D, where the moment curve is shifted by ∆ = 0.15,
also demonstrates why the reset value xreset is not important for the qualitative behaviour and synchronized burst firing. Comparison of these theoretical
curves pj and qj with simulations of an isolated cell system without coupling
showed qualitatively similar curves which are not given in Figure 11 for clarity.
While the quantities pj (x, t) and qj (t) describe an isolated cell only, they
can also be used to investigate the initial firing and feedback which provides
the mechanism for transition to the burst period. In a system of N isolated
20
σ
0.08
0.12
0.9
qj
0.000005
0.0016
0.35
µ
0.00026
0.08
17.4
Table 1
Values of qj and µ for an interval of length T = 1 corresponding to the different
values of the noise as in Figure 5. The parameter µ(T ) approximates the average
number of crossings in the interval for 50 isolated neurons.
neurons, it is convenient to define µ(T ), the average number of neurons that
cross the threshold x = xthres spontaneously in a given time interval of length
T . µ can be calculated approximately simply by
µ ≈ N q j T.
(11)
The definition of µ(T ) is just T multiplied by the mean of a binomial random
variable with parameters N = 50 and qj which is the mean of qj (t) over the
time interval. Note that for times t > 300 qj (t) has reached its maximum
and stays roughly constant at values near the maximum, as shown in Figure
11. Thus µ gives a measure of how likely the isolated neurons are to cross
threshold. In Table 1 we give values of µ over a brief interval of length T = 1
at t = 325 for different levels of σ (cf. Figure 11C).
When the noise is sufficiently strong, we observe that µ exceeds 1 so that
the approximation based on an isolated neuron no longer holds. However, this
approximation shows that for large noise the neurons are very likely to cross
the threshold even without coupling.
Now we can use qj (t) and µ(T ) to demonstrate how threshold crossings of a
small number of neurons can push the system into a bursting state. The mechanism can be summarized as a shift in pj (x, t), and consequently an increase
in qj (t) through the coupling parameter ∆. For every threshold crossing of
neuron i, xi (t) > xthres , xj (t) is incremented by ∆. This gives an effective shift
in the density pj (x, y, t), similar to an increase of the mean M1 by ∆, so that a
larger portion of the tail of the density pj (x, t) exceeds x = xthres . Then qj (t)
increases as shown in Figure 11D.
For ∆ = 0.06 the shift in pj (x, t) gives an increased maximum value of qj (t)
for t > 300 and σ = 0.12, as can be seen from comparing Figure 11C and
Figure 11D. Hence, if one of the neurons fires, the probability that one of
the remaining neurons fires increases. In Table 2 we show the impact of one
neuron firing on the quantitative measures, the average probability of firing,
qj and the average number of firings µ(T ) in a period of length T = 1 for the
remaining 49 cells.
This argument can be repeated to demonstrate how the feedback through the
21
σ
0.08
0.12
0.9
qj
0.000016
0.0028
0.34
µ
0.00078
0.138
16.6
Table 2
Values of qj and µ for an interval of length T = 1 following the threshold crossing
of one neuron, which results in a shift of ∆ of the mean for pj (t). The value of
µ(T ) approximates the average number of crossings in the interval for 49 remaining
neurons. The results are shown for the different values of the noise, compare to
Figure 5.
coupling shifts pj (x, t) and thus increases qj (t) and µ, which can initiate the
burst phase. Note that for larger noise levels, there is little change in q j (t) due
to the shift since the number of independent firings is already large.
4.3 Influence of noise
The initiation of a synchronized burst depends on the noise strength σ as seen
in Figure 5. For small σ in A, qj (t) is small (O(10−5 ) for σ = 0.08) so that
even if pj (t) is shifted by ∆ due to a random threshold crossing, the increase
in qj (t) is negligible and it is unlikely that any other neurons will fire because
µ(T ) remains small. Thus for low noise levels, the network cannot sustain
synchronized burst firing and only isolated firing occurs as shown in Figure
5A. In contrast, for large σ, such as σ = 0.9 (Figure 5C), the density pj (t) has
a large variance and qj (t) gives an increased probability for threshold crossing.
For example for 300 < t < 350, the mean number of cells firing in a unit time
interval exceeds 15. Because of this high probability of a number of neurons
firing nearly simultaneously, the coupling does not have to be strong to induce
firing in all cells of the network. Individual cells are likely to fire even without
a large shift in pj (x, t) and qj (t) induced by the coupling.
This independent firing however does not synchronize the cells for larger σ
(0.9 in Figure 5C) but only leads to uncorrelated firing at high frequencies.
In contrast, for intermediate values of σ, the cells are less likely to fire by
themselves, but following the firing of a single cell, the probability that others
follow is close to 1, which provides the route to the synchronization.
Naturally, µ(T ) depends on the size of the network N , and in larger networks
there will be more random threshold crossings which may initiate a burst.
Thus, the critical values of σ which is necessary to generate synchronized
network oscillations varies with network size. However, the basic mechanism
of oscillogenesis remains the same.
22
4.4 Influence of the coupling
The appearance of synchronized network oscillations not only depends on the
noise strength σ but also on the coupling strength ∆ as shown in Figure 7.
At low noise σ = 0.08, regular oscillations appear when the coupling is strong
enough to stabilize bursting. Here the mechanism of oscillogenesis is slightly
different but can again be explained through the quantities pj (x, y, t), qj (t),
and µ as defined above.
In Figure 7A threshold crossings are rare, since pj (x, t) is concentrated on
values well below xthres which implies that qj (t) is very small for all t, even
when shifted by ∆. However, in Figure 7B for increased ∆ = 0.1, a regular
bursting pattern appears after a long transient period of isolated stochastic
firing. This results only from a slightly larger ∆ as in Figure 7B, where the
increased coupling shifts pj (x, t) sufficiently to result in a higher probability
qj (t) of other cells firing following the spontaneous firing of a single cell. Then
eventually a sufficient number of neurons will fire and the coupling provides
the feedback to synchronize the whole system. Here σ and ∆ can be relatively
small and the critical feedback occurs with a small probability, so that when
the initial condition is y(0) = 0 it can take a long time until the bursting
phase is observed.
In the aftermath of a bursting period at t = t0 , the average value of y(t0 ) is
large (see Figure 8A) which accounts for the drop in the average x(t). In the
ensuing interburst period we again can restrict our analysis to an isolated cell,
but pj (x, t) has to be evaluated with a different initial condition, x(t0 ) = xreset ,
and a range of y values. Figure 12 shows qj (t) for different values of y(t0 ). For
all y(t0 ) values, qj stays close to 0 until t exceeds 300. Then qj and hence the
probability of a threshold crossing increases sharply for t > 300 time units after
a burst. The maximal probability is strongly dependent on y(t0 ) which relates
to the number of threshold crossings of the cell during the previous burst.
Then qj (t) is sufficiently large so that a few cells will cross the threshold, shift
the densities pj (x, t) towards x = xthres for the remaining cells, and drive the
system into the bursting state. Thus, following the initial bursting period, the
system enters the next bursting phase after the recovery period.
For large enough values of the coupling, ∆ = 0.2 in Figure 7C, the transient
period of isolated firing is reduced since the firing of a few cells gives a large
enough shift to pj (x, t) and qj (t) to start a burst. Note also that in Figure 7
the noise is relatively small (σ = 0.08). Thus the noise level is not large enough
to disrupt the synchronized entry into the bursting phase because the noise is
not large enough to allow independent threshold crossings of many cells, as in
Figure 5C.
23
−5
3.5
x 10
3
2.5
j
q (t)
2
1.5
1
0.5
0
0
100
200
300
400
500
600
t
Fig. 12. Graph of qj (t) for different initial values of y(t0 ), y(t0 ) = 0.2 (dash-dotted
line) and y(t0 ) = 1.0 (solid line) and σ = 0.08, x(t0 ) = xreset . qj (t) increases sharply
for t > 300, cf. Figure 8. The peak height of qj (t) varies strongly with the initial y
value.
5
Discussion and Conclusion
We derived a resonant integrate-and-fire model that resembles subthreshold
and firing properties of the HR model. Simulations of this stochastic model
show stochastic firing properties and coherence resonance. The preferred frequency in the stochastic firing stems from an interplay between the stochastic
threshold crossing and reset during a burst and the deterministic dynamics
which generate a long refractory period.
In our studies of a network of pulse-coupled RIF neurons, which is quiescent without noise, we found synchronized oscillations that are evoked and
sustained by noise. Noise evokes firing, and under excitatory coupling, this
stochastic activity can synchronize to an oscillatory state that involves the
whole network. These stochastic oscillations take the form of bursts of action potentials, simultaneously in all cells, with an interburst interval close to
the subthreshold resonance period of the single neuron model. The network
oscillation frequency decreases with increasing noise.
We defined a measure of network synchronization based on the frequency
content of the network average dynamics. This measure reveals a stochastic
resonance-like behaviour of the noise-induced network synchronization with a
maximum at intermediate noise levels. A main observation is that in neurons
with a slow time scale, noise can induce network synchronization at approximately the resonant frequency.
In the two-dimensional linear RIF model without reset it is possible to ob24
tain an analytical description of the solution as a time dependent stochastic
process. This solution reveals that the equations for the first moments are
independent of the noise level and have the same form and coefficients as the
deterministic RIF equations which give rise to the subthreshold resonance.
Consequently, the frequency of CR and synchronized network oscillations are
closely related since they both depend on the time constants of the underlying
deterministic system.
The results show the value of a simple model which allows a mathematical
analysis of the features of the more complicated models. We are able to derive
simple analytical quantities which describe the mechanism for the transition
to synchronized bursts and provide measures for the influence of the noise and
the coupling levels, complementary to the SR and synchronization measures
from computations. The RIF model also is an obvious choice for simulations
of large networks of resonant neurons because of its computational simplicity.
The reduced RIF model of bursting neuron we derived in this paper can be
further simplified into the minimal model given below.
dx = (−0.01x − y)dt + σdW,
dy = −0.001ydt,
x → 0.9, y → y + 0.1, when x reaches 1.
This minimal model is characterized by the “spike-reset-recover” mechanism.
It contains three different time scales 1, 0.01, and 0.001. The slowest time
scale characterizes the recovery time τy required for y to return to its equilibrium value. The intermediate time scale 0.01 is small as compared to a noise
intensity of σ = 0.1 or larger so that noise dominates when the system is near
its equilibrium at (0, 0). The largest time scale 1 characterizes the inhibitory
feedback from variable y to variable x so that any significant increase of y
causes a strong inhibition of x, pulling it away from its equilibrium value at
zero. Since the recovery is slow, x will remain suppressed before y slowly recovers its equilibrium value at zero. During this refractory period, a noise level
around 0.1 is too small to have any detectable influence on the dynamics of the
system. This is the simplest possible system in which noise-induced coherence
can occur with a frequency that is closely related to the intrinsic time scale
governing the recovery from a refractory state.
In a number of networks showing coherence resonance, the oscillation frequency increases almost linearly as the noise level increase (Wang and Wang,
1997, 2000; Wang et al., 2000). This suggests that the network frequency is
not determined by intrinsic properties of the neurons but by the noise level.
In our ‘spike-reset-recover” model, the frequency of the emergent oscillation
is mostly determined by the intrinsic subthreshold properties of the bursting
neurons. A qualitatively similar but more complicated mechanism has been
25
proposed recently to explain the background gamma rhythmicity in a cortical
network involving excitatory and inhibitory neurons (Börgers et al., 2005).
Our work demonstrates a new aspect of information processing in neurons
by a combination of stochastic resonance and slow ion currents. The resulting
stochastic preferred frequency could be important for the integration of inputs
under in vivo noisy conditions and rhythmogenesis in neuronal networks.
The observed stochastic synchronization in the RIF network suggests a role for
the mechanism that produces subthreshold resonance in the generation and
control of thalamic oscillations. For example, synchronization in the thalamus
is critical for brain function, including conscious behaviour and sleep states.
A combination of noise and resonance properties could control the frequency
and stability of these oscillations. Increased synchronization based on combined membrane resonance and SR may facilitate generalized absence seizures,
a condition when the cortico-thalamocortical system displays increased synchronization and disrupted brain function (Bal et al., 2000).
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