Article Submitted to Bulletin of Mathematical Biology
Asymptotic analysis of noise sensitivity in a neuronal
burster
R. Kuske
1 School
1 and S.M. Baer2
of Mathematics, University of Minnesota, Minneapolis, MN 55455,
rachel@math.umn.edu
2 Department
of Mathematics, Arizona State University, Tempe, Arizona, 85287,
baer@math.la.asu.edu
Abstract
A combination of asymptotic approaches provides a new analysis of the eect of small
noise on the bursting cycle of a neuronal burster of elliptic type (Type III). The analysis
is applied to a stochastic model of an excitable spine, with an activity dependent stem
conductance, that exhibits conditional burst dynamics. First, we give an asymptotic approximation to the probability density for the state of the system. This density is used
to compute several quantities which describe the inuence of the noise on the transition
from the silent to the active phase. Second, we also use a multiscale method to provide a
reduced system for analyzing the eect of noise on the transition out of the active phase.
The combination of these two approaches results in a new framework for a quantitative
description of how noise shortens the burst cycle, which measures the signicant inuence
of small noise. For the stochastic spine model, this study suggests that small amplitude
noise can signicantly inuence the activity-dependent morphological plasticity of dendritic spines. The techniques used in this paper combine probabilistic and asymptotic
methods, and have been generalized for other noisy nonlinear systems.
1. Introduction
Bursting has been observed in individual nerve cells, neuronal networks, and in secretory
and muscle tissue (15),(18). In excitable cells, it is a periodic pattern of electrical activity
that alternates between an active phase of rapid oscillations and a silent phase of slowly
changing membrane potential (See Figure 1.1.) This paper applies and further develops
a stochastic method (20) to study the inuence of white noise on the nonlinear dynamics
of elliptic bursting, which is particularly sensitive to random uctuations. Although the
method has been used for a variety of applications (20)-(21), it is particularly useful
for problems involving slow passage through a Hopf bifurcation which is the underlying
dynamical structure of the elliptic burster. We apply the method to a stochastic model of
a dendritic spine with an activity-dependent stem conductance that exhibits conditional
burst dynamics. We investigate the degree to which small amplitude noise inuences the
activity-dependent morphological plasticity of dendritic spines.
Bursting requires at least two time scales, one on a scale of fast oscillations and the other
on the scale of slow modulations. Some bursters are generated by external forcing (11), and
1
R. Kuske: Asymptotics of a noisy burster
others, called autonomous bursters, are generated from bi-directional coupling between
the slow and fast dynamics; the slow subsystem drives and is driven by the fast subsystem.
There are several mathematical classication schemes for bursters (17),(26),(4),(10),(13).
that are based on the bifurcation structure of the fast subsystem.
In this paper we consider an elliptic burster (also called type III (4) or a Hopf/Circle
burster (17)). It is characterized by epochs of small amplitude subthreshold oscillations
before and after the active phase; the envelope of oscillations form an ellipse (40). The
dynamics of an elliptic burster is revealed by superimposing a burst solution on the bifurcation diagram of the fast subsystem, computed from static values of the slow variable.
Figure 1.1b illustrates that the solution trajectory moves slowly to the left as it spirals
into the stable steady-state within the envelope of the subcritical periodic branch. After
passing through the static Hopf point the trajectory begins to slowly unwind about the
now unstable steady-state until it destabilizes into large amplitude spikes bounded above
and below by the stable periodic branches of the bifurcation diagram. The initiation of
the large amplitude spikes mark the beginning of the active phase. The solution trajectory
now moves slowly to the right just beyond a saddle node of periodics. Note that the fast
system is bistable, and only one slow variable is needed to generate an elliptic burst. The
silent phase resumes as the trajectory winds, once again, into the steady-state.
fast
1
fast 1.0
variable
variable
0.5
0.5
0
0.0
−0.5
1
2
1.5
2
2.5
3
3.5
−0.5
0.023
0.027
0.031
4
x 10
time
(a)
(b)
slow variable
The dynamics of elliptic bursting. (a) An elliptic burst trajectory computed from (2.6)-(2.8). Small
oscillations grow in amplitude on approach to the active phase while decaying in amplitude after exiting into
the silent phase. (b) Projection of the burst trajectory onto the slow variable-fast variable (Gss -u) phase plane.
Superimposed is a bifurcation diagram of the fast subsystem, (2.6)-(2.7), for static values of the slow variable
Gss . In the silent phase the trajectory moves slowly to the left as it spirals into the stable steady-state (thin
solid line) within the envelope of the subcritical periodic branch (dashed curve). After passing through the Hopf
point the trajectory unwinds about the unstable steady-state (thin dashed line). The active phase begins when
the small amplitude oscillations destabilize into large amplitude spikes bounded above and below by the stable
periodic branches of the bifurcation diagram (solid curves). The bifurcation diagram is computed using AUTO.
Figure 1.1:
The length of the silent phase depends on (i) memory eects: deviation below the
Hopf point for the onset of the active phase is roughly equidistant to the deviation above
the Hopf point at the beginning of the silent phase, (ii) ramp speed: the speed of the
slow variable, and (iii) small amplitude uctuations. The bifurcation dynamics of the
silent phase for an elliptic burster was originally studied in (28), and then dissected in
R. Kuske: Asymptotics of a noisy burster
the context of a problem of slow passage through a Hopf bifurcation (1) (27), for an
excitable system without bursting. In the slow passage problem, the dynamics of the
slow variable is prescribed as linearly increasing in time rather than, as in the bursting
problem, as a slowly changing dependent variable of an autonomous system. Baer et
al (1) developed asymptotic methods to explore memory eects and how ramp speed
inuences delay bifurcations. They also demonstrated that the onset to large amplitude
oscillations is sensitive to small amplitude periodic and stochastic uctuations. Kuske
(20) developed an asymptotic approximation for the time-dependent probability density
function for a class of noisy bifurcation problems. The probability density was then used
to quantitatively describe the eect of the noise, that is, to calculate the reduction of
the delay in the transition to large amplitude oscillations . The question which naturally
follows is how does noise aect the solution trajectory of an elliptic burster, and can the
asymptotic methods be extended to burst problems?
Investigations into the inuence of noise on burst dynamics is not new. Theoretical
studies clearly demonstrate the inuence of channel noise on burst activity. Several models
(7),(31) have been used to demonstrate that suÆciently large channel noise can disrupt
the burst cycle of pancreatic cells, inuencing both the silent and active phases.
DeVries and Sherman (9) further showed that noise could promote bursting in weakly
coupled populations of cells (5), if the noise is not too large. By analyzing a two-cell
model, they showed that noise eectively shifts the bifurcation structure, thus achieving
the bistability necessary for bursting.
Although the dynamical structure of an elliptic burster is easily understood mathematically, only recently has it been identied in a biological system. Del Negro et al (8)
found type III bursting in neonatal rat trigeminal interneurons (TI neurons). Bursting
activity in TIs emerged as the cells progressively depolarized to potentials near and above
spike threshold (see Fig. 3c and 3d in (8)). Their data shows clear examples of growing
subthreshold oscillations prior to each active phase. They also showed that cells re intermittently when the potential was biased near threshold, suggesting sensitivity to intrinsic
and synaptic noise.
This paper quantitatively explores the eect of additive noise on an elliptic burster,
obtained by Wu and Baer (41) as a quasi-steady reduction of a model for an excitable
dendritic spine with an activity-dependent stem conductance. The reduced system is a
variant of the FitzHugh-Rinzel (FHR) model formulated in 1976 by FitzHugh and Rinzel
(unpublished) and later analyzed by Rinzel (26). The noise in the model represents small
uctuations due to channel or synaptic noise that inuence the spine head potential. The
spine head potential exhibits an elliptic burst pattern that is driven by slow oscillations
in the spine stem conductance. Changes in stem conductance result, most likely, from
changes in the structure of individual spines, in particular the geometry of the spine neck.
Noise inuences the fast membrane potential dynamics of an individual spine head which
in turn aects the slow dynamics of the stem conductance. Consequently, noise inuences
the physical structure of that spine. This paper is a rst step towards measuring this
inuence by quantitatively exploring the eect of the noise on an elliptic burster.
Several previous analyses of noise-sensitive systems such as in lasers (12), waves in
plasmas (16), and bursters (9) use a dynamical systems approach to describe potential
noise-sensitivity. To account for the eect of uctuations, they analyze the change in
the bifurcation structure for a range of deterministic perturbations. Here we provide a
probabilistic approach which is new in the context of bursting; that is, the system is
treated as a stochastic process. Standard results of stochastic analysis are combined with
3
R. Kuske: Asymptotics of a noisy burster
formal asymptotics to derive analytical expressions for quantities which measure the eect
of the noise. These results include the probability density, probability of transition, mean
transition times, and averaged equations for the deviations. In addition, the underlying
deterministic dynamics are incorporated into the asymptotic method, which allows for a
dynamical interpretation of the inuence of the uctuations.
Other advantages of this approach include computational eÆciency and applicability to
similar noise-sensitive problems. To analyze noise-sensitivity in biological systems which
depend on a combination of system parameters, it is imperative to have a method that
gives mathematical expressions which can be evaluated quickly for a range of parameters.
The framework presented in this paper accomplishes this whereas most current methods,
which rely on many realizations, are costly and inaccurate in their determination of important quantities such as probability density and mean transition times. The technique
developed here can be useful in the quantifying the eect of noise on the bursting dynamics of other elliptic bursters, whether they are biological in nature or not, as well as
related types of bursters.
In Section 2 we describe the deterministic model, which was developed in (41). In Section 3 we give the model for the noisy elliptic burster, which is essentially the deterministic
model with noise introduced through the synaptic current. We provide a numerical simulation which demonstrates the role the noise can play in changing the dynamics of the
system. In Section 4 we give an asymptotic method which gives an approximation to
the probability density for the state of the system. From this probability density several
quantities are computed which describe the inuence of the noise on the transition from
the silent to the active phase. In Section 5 we use an asymptotic method based on multiscale methods which leads to reduced equations for describing the eect of noise on the
transition out of the active phase. In the Discussion we compare the eect of the noise
for these two cases.
2. The deterministic model
We outline the quasi-steady reduction of Wu and Baer (41) which leads to the model of
the elliptic burster. It is important to note that in the problem of slow passage through
a Hopf bifurcation for an excitable system, the duration of the time to onset (for the
deterministic problem) is controlled by the choice of initial condition and ramp speed.
For an autonomous burster the length of the silent phase and the speed of the slow variable
is built into the system's dynamics and is therefore controlled by a combination of system
parameters. In order to understand how noise can inuence burst dynamics it is useful to
have a model that has a straightforward biophysical interpretation that gives meaning to
the parameters. The simple variant of the FitzHugh-Rinzel model proposed by Wu and
Baer provides a clear interpretation of model parameters.
We preface our discussion of the Wu and Baer model with a brief introduction to
dendritic spines. A dendritic spine has a general knoblike appearance of a bulbous head
and a tenuous stem (see Fig. 2.1). Dendritic spines are the sites of the vast majority of
excitatory synaptic inputs in the vertebrate brain. They appear as small evaginations of
the dendritic surface of several functionally important classes of nerve cells. They are
particularly abundant in pyramidal, Purkinje, and stellate cells in the cortex. A single
Purkinje cell may have as many as 100,000 to 300,000 spines (14)(36).
Dendritic spines most likely have several functional roles. On the chemical side, dendritic spines compartmentalize calcium; calcium decay kinetics are controlled by slower
4
R. Kuske: Asymptotics of a noisy burster
diusion through the spine neck and by spine calcium pumps (42). Calcium plays a role
in spine plasticity by facilitating spine growth at moderate concentrations, but depressing
growth and eliminating spines at higher toxic concentration levels. On the electrical side,
spines may facilitate local action potential generation and propagation in the dendrite.
Experimental evidence for active channels exists: two-photon laser-scanning microscopy,
which allows uorescence imaging with high spatial resolution, has been employed successfully for the measurement of activity-induced calcium and sodium transients in individual
spines (34). This motivates us to consider the implications of repetitive spiking or oscillations originating in the spine head, including bursting oscillations. There have been
several theoretical studies devoted to the electrical interaction between many excitable
spines, for example (2) (37). These studies were motivated by models of a single excitatory
spine attached as a nonlinear boundary condition to a passive cable equation (22)(30). It
is important to understand the dynamics of a single spine before considering interactions
between many spines.
Experimental evidence is now clear that spines are highly dynamic structures. Several studies show that spines can be formed very rapidly, and that synaptic activity can
modulate changes in spine shape and the formation of new spines (23). Thus, electrical
dynamics on the msec time scale can aect and be aected by relatively slow changes
in spine morphology. Such activity-dependent morphological plasticity raises important
and obvious theoretical questions. For example, to what extent is dendritic morphology
controlled by the excitability properties of the cell, and what role does the temporal
pattern of synaptic input and synaptic noise contribute to the morphological structure?
Conversely, to what extent does dendritic spine morphology control the patterns of electrical signaling; for example can slow reversible structural changes in an individual spine
generate bursting?
To begin addressing these questions from a theoretical viewpoint, Wu and Baer examined a simple model of a single dendritic spine, with an activity-dependent stem conductance, attached to a passive cable.Wu and Baer's model is a simplied study of how
slow stem conductance changes, correlated to slow changes in spine structure, can aect
and be aected by electrical dynamics on the msec time scale. Since specic biophysical
models are not nailed down yet, they took the simplistic view that for passive spines
(with a passive shaft) activity generated from synaptic sources on the spine head tend to
shorten the necks over time (stem conductance slowly increases). If the dominant source
of current (on average) is from the dendritic shaft, necks tend to increase over time (stem
conductance decreases). For excitable spines (with a passive shaft), Wu and Baer show
that one consequence could be bursting oscillations (bursting oscillations are common in
the nervous system). Wu and Baer found that the length of a cable and the degree of
loading creates a "conditional" burster.
Specically they formulated a cable equation with a single spine boundary condition
located at X = 0 and a sealed end boundary condition at X = L, commonly used to
model no axial current ow at the dendritic terminal. They explored the threshold and
dynamical properties of a spine when a synaptic input (ISH ) and a dendritic input at
the spine base are modeled as steady current sources (See Figure 2.1 ). They examined
the possibility that the spine stem resistance (Rss) is activity-dependent. Their idealized
hypothesis was that a spine with passive membrane bombarded by synaptic activity tends
to slowly strengthen its electrical connection to the dendrite over time. However, if this
activity decreases to the extent that the dendritic current source dominates, then a passive
spine tends to become more isolated (electrically) over time. To explore this hypothesis
5
R. Kuske: Asymptotics of a noisy burster
they assumed that the time rate of change of the spine stem conductance (Gss = 1=Rss) is
proportional to the current through the stem (Iss); i.e., dGss=dt = Iss, where 1
to model the slow variation in stem conductance. Activity that drives Iss from the spine
head to the dendrite (Iss > 0) slowly increases Gss, and dendritic input that drives the
current from the dendrite to the head (Iss < 0) slowly decreases Gss. When the spine
head was made excitable, using FitzHugh-Nagumo kinetics, the cable model generated
dierent patterns of electrical activity. This activity ranged from continuous spiking to
irregular and regular bursting oscillations to constant steady-states. Transitions between
these modes of electrical activity are controlled by the electrical length of the dendrite
(3),(41). Here we interpret the time scale of a single spike to be on the msec time scale. The
important consideration in this model is that the time scale of individual spikes is much
faster than the time scale of morphological changes, which could be seconds, minutes, or
hours.
I
HEAD
ss > 0
I
6
SH
u(t)
ss < 0
I
STEM
0 |||
v (x; t)
||{
x
L
I
D
DENDRITE
A dendritic spine. In this schematic, ISH and ID denote input current into the spine head and
dendritic shaft at x=0. The stem current is Iss = Gss [u(t) v (0; t)], where Gss is the spine stem conductance.
If the potential in the head u(t) is greater (less) than the potential in the shaft v (0; t), then the stem current
Iss is positive (negative). In the activity-dependent model, if Iss is positive (negative) the stem conductance
slowly increases (decreases).
Figure 2.1:
Next they made the observation that allows the quasi-steady reduction of the full partial
dierential equation model to a system of ordinary dierential equations, where the length
L > 0 of the cable becomes a parameter of the reduced system. They observed that when
the spine head potential u(t) is a relaxation oscillator, the oscillations penetrate far into
the cable with little phase lag and for most of the cycle the dendritic potential tracks the
steady-state manifold
cosh(L X ) ;
v(X; t) = vs(u(t))
(2.1)
cosh L
where vs is the membrane potential at the spine base, which depends on u(t). The form of
Eq. (2.1) is easily derived by solving the steady-state passive cable equation, treating vs
as a constant command potential applied at X = 0 and imposing a sealed end boundary
condition at X = L.
1
R. Kuske: Asymptotics of a noisy burster
The input resistance (see (25)) at the spine base (for the above boundary conditions)
is
RSB = R1 coth L
(2.2)
where the resistance R1 = (2=)(Rm Ri) = (d) = depends on the cable's membrane
resistivity Rm, cytoplasmic resistivity Ri, and diameter d. Next, using conservation of
current, the current at the spine base vs(u)=RSB has two components
vs (u)
=I +I
(2.3)
1 2
RSB
3 2
ss
D
where ID is the dendritic input and Iss is the stem current,
Iss = Gss [u vs (u)]:
(2.4)
If Iss > 0, activity drives Iss from the spine head to the dendrite, and if Iss < 0, dendritic
input dominates and drives the current from the dendrite to the head. Multiplying (2.3)
by RSB , expressing RSB in terms of R1 and L (2.2), and using (2.4), they recast vs in
terms of u
Gss u + ID
:
(2.5)
vs (u) = R1
R1 Gss + tanh L
The potential in the head u(t) is determined from conservation of current. The capacitive current in the spine head must balance with the ionic current, the spine stem
current Iss, and the synaptic current ISH . Assuming that the ionic current is governed
by FitzHugh-Nagumo (FHN) kinetics, the simplied model for an excitable activitydependent spine is
du
dt
dw
dt
dGss
dt
= f (u)
= b (u
= Iss
w Iss(u)
w)
1)(u
ISH
b1
b;
where f (u) is a cubic-shaped function given by
f (u) = u(u
+
a)
0 < a < 12 ;
(2.6)
(2.7)
(2.8)
(2.9)
and Iss(u), which is dened by (2.4) with (2.5), includes all the dendritic cable parameters.
Equation (2.6) reects the current balance relation in the head. From (2.4), if Iss > 0
current passes from the head to the dendrite and if Iss < 0 current passes from dendrite
to head. The parameters b and are positive constants and w(t) represents a recovery
current which, according to (2.7), responds slowly, when b is small, to changes in u. The
activity-dependent stem conductance is modeled by (2.8). This slow equation provides
bi-direction coupling to the fast subsystem through Iss. Note from (2.8) that Iss > 0
slowly increases Gss and Iss < 0 slowly decreases Gss.
7
R. Kuske: Asymptotics of a noisy burster
8
3. Numerical exploration of a noisy elliptic burster
In this section we examine the eect of additive noise on elliptic burst dynamics by
considering the noisy Wu-Baer model
p
du = ( f (u) w Gssp(u vs (u)) + ISH )dt + 2Æ d
(3.1)
d (u; w; Gss )dt + 2Æ d
dw = b(u w))dt d (u; w; Gss )dt; b 1
dGss = Gss(u vs (u))dt d (u; w; Gss )dt; b
G u + ID
vs (u) = RSB ss
(3.2)
1 + GssRSB ; RSB = R1 coth L:
1
2
3
Here d is white noise, corresponding to a noise in the input current ISH . The model uses
1.5
1
0.5
0
−0.5
0
2
4
6
8
10
12
14
4
x 10
1.5
1
0.5
0
−0.5
0
2
4
6
8
10
12
14
4
x 10
A deterministic elliptic burster (Æ = 0 in (3.1)) (upper plot) and a realization of the burster with
noise (Æ = 2 10 4 ) (lower plot). Note that the overall eect of the noise is to shorten the length of the silent
phase, with a less dramatic eect on the active phase. Thus the percent active phase increases due to the noise.
In both we have plotted u (bursting dynamics) and 50 Gss (sawtooth shaped graph) vs. time. Here a = 0:14,
b = 0:008, = 2:54, = 5 10 4, R1 = 1=, L = 0:6, ISH = 0:04, ID = 0:2.
Figure 3.1:
white noise as a model for the combination of dierent noise sources. The noise sources
we consider could include thermal noise, channel noise arising from the stochastic nature
of voltage-dependent ionic channels, and synaptic noise due to spontaneous background
activity. For neurons, the primary source of noise comes from background synaptic input
acivity often modeled using white noise. The summation of inputs from presynaptic cells
combined with the unreliability of synaptic transmission produces incessant variations
of the membrane potential; neuroscientists call this "synaptic noise". In our model, we
appropriately add the noise to the synaptic current term that impacts the spine head
(recall that most synapses, in dendrites with spines, are on the spine head).
R. Kuske: Asymptotics of a noisy burster
When the magnitude of this noise is small, even smaller than , it can still have a
signicant eect on the length of the silent phase. Figure 3.1 compares the deterministic
dynamics with the dynamics in the presence of noise, over several burst cycles. In the
realization of the random process the period of a burst cycle decreases due to decreases in
the lengths of both the silent and active phases. The decrease in the length of the active
phase appears to be smaller on average.
These realizations suggest the following: during some initial interval of time in the silent
phase and active phase, there is little dierence between the stochastic dynamics and the
deterministic dynamics, apart from a phase shift. Eventually the noise causes a departure
from the deterministic behavior, resulting in early onsets of both the active phase and
the silent phase. Although the times for these transitions vary considerably from cycle to
cycle, we can calculate a mean length of the burst cycle by averaging over many cycles
or realizations.
9
Æ=0
L = :3
L = :6
1
(a)
0.5
Æ = :01
0
−0.5
1
5
5.5
6
6.5
(e) 0.5
4
L = :4
(b)
7
0
x 10
1
−0.5
0.5
5
5
5.5
6
6.5
8
7
(f)
4
0.5
x 10
0
0.5
−0.5
5
5.5
6
6.5
7
4
x 10
Æ = :0001
5
5.5
6
6.5
L = :6
7
4
x 10
1
(g)
0.5
1
0
0.5
−0.5
0
−0.5
7.5
Æ = :001
0
(d)
7
1
1
−0.5
6.5
4
L = :5
(c)
6
x 10
0
−0.5
5.5
5
5.5
6
6.5
7
4
x 10
5
5.5
6
6.5
7
4
x 10
Comparison of the deterministic dynamics of u as a function of time for dierent values of L
and noise level Æ . In the left column there is no noise, and the behavior changes from continuous spiking to
intermittent spiking to bursting as L increases. In the right column, the parameter L is xed, but as the
magnitude Æ of the noise increases, the burst pattern undergoes a similar transition from bursting to frequent
intermittent spiking. All plots express u as a function of time. Here a = 0:14, b = 0:008, = 2:54, = 5 10 4 ,
R1 = 1=, ISH = 0:04, ID = 0:2.
Figure 3.2:
The numerical example above suggests that in order to quantify the inuence that noise
has on elliptic burst dynamics, the analysis must be divided into two parts. The rst part
1
R. Kuske: Asymptotics of a noisy burster
is the transition from small subthreshold oscillations (near steady state) to the onset
of spiking in the active phase. In Section 4 we use an approximation of the probability
density function to determine the length of the silent phase. The second is the transition
from large amplitude spiking to the onset of the silent phase. In Section 5 we use a multiscale analysis to isolate the eects of noise, which conrms that noise has a much greater
eect on the silent phase than on the active phase of a burst.
Figure 3.2 compares the eects due to changes in the bifurcation parameter L (cable electrotonic length) with those due to changes in the noise level Æ. The simulations
demonstrate that the same qualitative behavior can be obtained from the model either by
varying the bifurcation parameter in the deterministic system or by xing the bifurcation
parameter and varying the noise level. This is particularly clear from comparisons of transitions in b)-d) with those in e)-g) in Figure 3.2. This gure motivates the development
of tools to quantitatively measure the eects of the noise on the overall burst dynamics.
Since noise can play a signicant role in changing the bursting cycle, it is necessary to have
a method for quantitatively predicting its eect. Analytical results provide an eÆcient
way to describe the eects of all parameters on the dynamics. They are often preferable
to simulations, which can be expensive, especially when the dynamical variations depend
on several signicant parameters. Furthermore, previous studies show that for similar
systems, the simulations themselves can also be very sensitive to noise (19)-(21), (27).
In the following we obtain an analytical approximation for quantifying the dynamics
of (3.1), thus providing a description in terms of all parameters. We focus on the eects
due to noise by considering a range of the parameter Æ. In particular we show that by
varying Æ, the burst dynamics can change considerably even when Æ remains small. The
asymptotic approximation provides a way to compute the average length of the burst
cycle quantitatively, and thus measure the inuence the noise on the bursting activity.
Our results show that the length of the silent phase of the bursting cycle is signicantly
reduced as the noise is increased.
4. Transition from silent to active phase
We now nd an asymptotic approximation for the probability density of the state of the
system during the silent phase and in the transition to the large amplitude spiking of
the active phase. The time-dependent probability density p(u; w; Gss ; t) is the probability
that the process (u; w; Gss ) described by (3.1) is in a certain state at time t,
p(u; w; Gss ; t)du dw dGss :=
(4.1)
Pr(u < u(t) < u + du; w < w(t) < w + dw; Gss < Gss(t) < Gss + dGss at time t)
The probability density contains all of the information about the dynamics and it can be
used to calculate many useful quantities, such as moments and transition times, as shown
below. The probability density function satises the Fokker-Planck equation which can
be derived from the system of stochastic dierential equations (SDE's) (3.1)
@p
@t
= Æ puu r ( p):
2
(4.2)
a
(See e.g. (29) for a derivation of the Fokker-Planck equation.) Here (d ; d ; d ) is the
drift in the equations (3.1), which also describe the deterministic dynamics (2.6)-(2.8).
The diusion coeÆcient Æ is one half the square of the coeÆcient of the noise in (3.1).
a
2
1
2
3
10
R. Kuske: Asymptotics of a noisy burster
11
In general it is diÆcult to nd an analytical solution for p(u; w; Gss ; t) from (4.2). Since
we are interested in the case of small noise Æ 1, the diusion coeÆcient in (4.2) is
small, which allows for an asymptotic approximation for the probability density function.
The underlying deterministic dynamics and the stochastic simulations of (3.1) suggest an
asymptotic form for the probability density which captures both the idea that initially the
random and deterministic dynamics are close, and that there can be signicant variation
between realizations in the transition to the active phase. A similar form was used to
study transitions in noisy delay bifurcations (20).
We assume the following form for the density
p(u; w; Gss ; t) = CeQ u;w;G ;t where
(4.3)
(u ud) g (t) (w wd ) g (t) (Gss Gssd )
Q(u; w; Gss ; t) = g (t)
2Æ
2Æ
2Æ
(u ud)(w wd)
(u ud )(Gss Gssd ) h (t) (w wd)(Gss Gssd )
h (t)
h (t)
Æ
Æ
Æ
u ud
w wd
Gss Gssd
+ q (t) Æ + q (t) Æ + q (t) Æ
+ s(t) (4.4)
Here ud, wd and Gssd give the deterministic behavior at time t, as given by (3.1) with
Æ = 0 ((2.6)-(2.8)). The factor C is a normalization constant. To motivate (4.4), consider
the case when hjm = qj = 0 for all j and m, and t is xed. Then the probability density
function is just a product of Gaussians, and u, w and Gss are independent normal random
variables with means ud (t), wd(t), and Gssd (t) and variances Æ =g (t), Æ =g (t), Æ =g (t),
respectively. If Æ gj , the variances are small and the density is sharply peaked about
u = ud , w = wd , and Gss = Gssd . That is, for Æ gj , there is a high probability
that the stochastic dynamics are close to the deterministic dynamics. Larger values of
Æ =gj indicate a larger variance about the deterministic dynamics. Then the probability
density spreads out around the deterministic dynamics, with decreased probability that
the stochastic and deterministic dynamics are close.
Of course, u, w and Gss are not independent random variables, since they are related
through (3.1). Then in general hjm 6= 0 and qj 6= 0 in (4.4). This form is similar to the
Gaussian, but includes cross terms which are necessary to completely solve for the density,
given the interactions. Since gj , hjm and qj are functions of t, the shape of the density will
vary in time. This variation captures the change in the dynamics; the stochastic process
is initially concentrated about the deterministic behavior, while at later times it deviates
from the deterministic dynamics, and the density spreads out.
In this section we outline the result for the leading order term in the asymptotic expansion of p(u; w; Gss ; t). In Appendix A we give the details for obtaining such a solution
for a more general system.
As mentioned above, we are interested in the case when Æ 1. Therefore we nd it
convenient to introduce the variables
u ud
w wd
G
Gssd
=
; =
; = ss
; = fj ; j = 1; 2; 3g (4.5)
Æ
Æ
Æ
After substituting (4.5) in (4.4), the density has the form
~
g (t)
g (t)
p(u; w; Gss ; t) P (; t) = C exp g (t)
2
2
2
h (t) h (t) h (t) + q (t) + q (t) + q (t) + s(t) (4.6)
(
ss )
2
2
1
12
2
2
13
2
2
2
23
2
1
2
3
2
2
3
2
2
2
1
2
2
2
1
2
3
1
12
1 2
13
1 3
23
2 3
2
1
1
2
1
2
2
2
2
3
3
2
1
3
2
3
R. Kuske: Asymptotics of a noisy burster
with C~ the normalization constant, scaled appropriately. As shown in Appendix A, to
solve for P we substitute (4.5) and (4.6) into (4.2), and expanding about Æ 1, we get
a system of ordinary dierential equations for the coeÆcients gj , hjm , qj , and s in the
exponent of p(u; w; Gss ; t). Once we know these coeÆcients we know the approximation to
the density. The system of equations for these coeÆcients is given in Appendix A (A.12).
For convenience, we start with an initial density p (u; w; Gss ) p(u; w; Gss ; 0),
(
w wd ) (Gss Gssd )
1
(
u
u
)
d
p (u; w; Gss ) = p
exp
(4.7)
2Æ
2Æ
2Æ
8Æ which is centered around (ud ; wd ; Gssd ) in the silent phase. The method does not depend
on this choice; another form of the initial density could easily be used in place of (4.7).
Then we use the method described above to approximate the density for later times.
We are interested in studying the eect of noise on the silent phase, and therefore we
start at the middle of the silent phase of the deterministic dynamics, corresponding to
(ud ; wd ; Gssd ) (0:0924; 0:0368; 0:0264). Together with (4.7) this provides the initial conditions for the system (A.12) for the coeÆcients gj , hjm, and qj in the approximation to
the probability density (4.3)-(4.4). The initial condition is chosen so that Gssd is near the
Hopf point, and in this case slightly below the Hopf point. One could also start with a
slightly larger value of Gssd , so that it is above the Hopf point. When Gss is above the
Hopf point, the noise would not cause growth of the oscillations, but only variations on
the order of magnitude of the standard deviation of the noise. Once Gss decreases below
the Hopf point, the dynamics would continue as shown in the simulations. By choosing
an initial condition near the Hopf point and the initial density (4.7), we model the state
of the process as Gss decreases to a value just below the Hopf point.
To determine the eect of noise on the silent phase, we rst note that the transition to
large amplitude spiking is determined by the behavior of u. This is clear from equations
(3.1), since w is essentially slaved to u, and Gss will decrease if u is not in the active
phase. It is the small random variations in the input current which can alter the time at
which u enters the active phase, and then the dynamics of w and G will follow accordingly.
Therefore we dene the transition in terms of u only. Once we have solved (A.12) and
thus approximated the density, we can calculate the time until u makes the transition to
the active phase by using the marginal probability density q(u; t) for u,
0
2
2
0
6
q(u; t)
=
2
3
Z
1Z1
1
1
2
p(u; w; Gss ; t)dwdGss
Pr(u < u(t) < u + du at time t)
2
2
(4.8)
The function q(u; t) is the probability that the process u(t) is at state u at the time
t. Thus, the size of the oscillations in u(t) are reected in the shape of q(u; t). If the
subthreshold oscillations in u(t) are small, that is near the steady state, then the probability density is concentrated about the deterministic dynamics in the silent phase. As
the subthreshold oscillations increase in amplitude during the transition to the active
phase, there is an increasing probability that u(t) takes on a large range of values, and
the probability density spreads out. Thus, the shape of the probability density q(u; t),
which indicates the variance of u(t) around the steady state, can indicate how rapidly
u(t) is making the transition to the active phase.
In Figure 4.1a we compare the asymptotic approximation of the marginal probability
density q(u; t) with an approximate density obtained from simulations at two dierent
12
R. Kuske: Asymptotics of a noisy burster
60
13
100
Æ=3
50
10
4
t = 2500
90
80
q (u; t)
q (u; t)
40
70
60
t = 500
30
Æ=5
50
10
5
40
20
30
t = 1500
10
20
Æ=3
10
0
0.02
0.04
0.06
0.08
(a)
0.1
0.12
0.14
u
0.16
0
0
0.02
0.04
10
0.06
4
Æ=5
0.08
0.1
0.12
0.14
10
0.16
4
0.18
u
(b)
(a)Marginal probability density q (u; t) for two times t = 500 and t = 1500, for Æ = 3 10 4 .
The solid line is the asymptotic approximation, and the 's represent the histogram from 5000 simulations. As
t increases, q(u; t) spreads out in the variable u, indicating u is approaching the active phase. (b) Marginal
probability density q (u; t) for t = 2500 and three dierent values of Æ , Æ = 5 10 5 (dashed line), Æ = 3 10 4
(dash-dotted line) and Æ = 5 10 4 (solid line). As Æ increases, q (u; t) spreads out in the variable u, indicating
that the noise causes u to approach the active phase. Here a = 0:14, b = 0:008, = 2:54, = 5 10 4 ,
R1 = 1=, L = 0:6, ISH = 0:04, ID = 0:2. The initial distribution is given by (4.7).
Figure 4.1:
times, normalized appropriately. Here we see good agreement between the numerical simulation and the asymptotic approximation. As time increases, the density is less concentrated about the steady state. This is consistent with Figure 1.1b, in which the amplitude
of the oscillations of u(t) increase as the system approaches the active phase, so that the
density spreads out and the variance about the steady state increases.
In Figure 4.1b we show the change in the shape of q(u; t) as a function of noise. As we
increase the noise, the probability density is less concentrated, indicating that the oscillations in u(t) are increasing as the system approaches the active phase. Then we conclude
that on any given realization, larger noise is more likely to cause a rapid transition to the
active phase, hence shortening the silent phase.
Using the probability density q(u; t), we also calculate a probability (t) that u(t) is
approaching the active phase at time t and an eective time of transition T from the silent
phase to the active phase, both dened below (see (4.9) and (4.10)). In this application
it is more convenient to use T to quantify the transition, in contrast to the mean rst
transition time, the commonly used measure of exit time, as in chemical kinetics (29) (32)
and other applications. Below we show that T has the same dependence on the noise
level as the mean rst transition time , dened below (4.11). This choice follows from
the oscillatory behavior of the elliptic burster. Since u is oscillatory it takes some values
near the steady state, as do the subthreshold oscillations of the deterministic dynamics,
ud (t). Therefore we dene the transition in terms of the dierence between the stochastic
dynamics and the deterministic dynamics which is simply ju(t) ud (t)j. This measure
follows intuitively from considering Figures 4.1 and 1.1b. Figure 4.1 shows that the size
of the subthreshold oscillations increases for increasing noise, as indicated by the shape of
0.2
1000
Æ=5
2000
10
4
t
3000
Æ=1
10
4000
4
Æ=2
10
5000
5
6000
R. Kuske: Asymptotics of a noisy burster
the probability density. Figure 1.1b shows that the increase of the threshold oscillations
corresponds to the transition to the active phase. Therefore, as the oscillations in the
stochastic process u(t) increase, there is a signicant dierence between u(t) and the
deterministic process ud (t), and u(t) exits the silent phase before ud (t) does.
1
0.9
0.8
0.7
0.6
0
d( )
c
u(t) is near the active phase. For example, in Figure (4.2) at t = 4500, the probability
(t) that ju(t) ud (t)j exceeds the cuto c is nearly 0 for Æ = 2 10 5 , while (t) 0:45
for Æ = 1 10 4 and (t) 0:9 for Æ = 5 10 4 .
( )
In order to quantify the transition to the active phase, we compare a critical value c
to the dierence ju(t) ud(t)j over a trajectory in which ud(t) remains in the silent phase.
When ju(t) ud (t)j < c, there is no signicant dierence between the stochastic and
deterministic dynamics, and u(t) is eectively still in the silent phase. If u(t) has larger
oscillations than the deterministic process, the dierence ju(t) ud (t)j will increase, and
the stochastic process approaches the active phase more rapidly than the deterministic
system. Therefore, the probability that ju(t) ud (t)j exceeds the cut-o c increases as
u(t) approaches the active phase. Therefore we dene the probability (t) that u(t) is
approaching the active phase, as the probability that ju(t) ud(t)j exceeds c. This is
easily calculated, via integration, using the probability density q(u; t) determined above:
Z
(t) =
q(u; t)du = Pr(ju(t) ud (t)j > uc ):
(4.9)
ju t u t j>u
In Figure 4.2 we plot (t) as a function of t for dierent noise levels. Note that for
small values of t, (t) 0, which indicates that the dierence between the stochastic
and deterministic behaviors is very small statistically, so that the stochastic process is
close to the deterministic dynamics of the silent phase. However, for larger values of t
the function (t) increases, so there is a larger probability that the stochastic process has
larger oscillations and is approaching the active phase. For xed time t and increased noise
level Æ, there is an increased probability that the process has moved signicantly away
from the deterministic solution, so that the increased noise increases the probability that
14
5 (dash-dotted with
Figure 4.2: In order to quantify the probability of transition, we plot (t) for Æ = 2 10
's), Æ = 1 10 4 (dash-dotted), and Æ = 5 10 4 (solid). Here a = 0:14 b = 0:008, = 2:54, = 5 10 4 ,
R1 = 1=, L = 0:6, ISH = 0:04, ID = 0:2. The initial distribution is given by (4.7). The critical cuto was
c = 0:02.
0
0.1
0.2
0.3
0.4
0.5
ρ(t)
R. Kuske: Asymptotics of a noisy burster
Here we have chosen c as a critical value which is approximately two orders of magnitude larger than Æ, i.e. O(100 Æ). The choice of c is based on the observation that once
the oscillations are suÆciently large, the noise no longer plays a signicant role in the
transition, and the process u(t) proceeds rapidly into the active phase. For larger values
of c, the shape of the curves is essentially the same, but shifted to larger values of t.
6000
T
6000
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
15
0
0
0.5
1
1.5
2
2.5
3
3.5
−4
Æ
(a)
4
4.5
5
−4
x 10
x 10
Æ
(a) The eective transition time, given by the plot T vs. Æ for (T ) = . The solid line corresponds
to = 0:2 and c = 0:02. The dashed line corresponds to = 0:7 and c = 0:01, the dash-dotted line corresponds
to = 0:5 and c = 0:01. (b) Average exit time values, , from the simulation with 5000 realizations ('s).
These values align with the dash-dotted line in (a). Here a = 0:14, b = 0:008, = 2:54, = 5 10 4, R1 = 1= ,
L = 0:6, ISH = 0:04, ID = 0:2. The initial distribution is given by (4.7).
Figure 4.3:
Above we have dened (t) as the probability that u(t) diers signicantly from ud (t).
Then, as the oscillations in u(t) increase, (t) also increases, and we can determine a
time t = T at which the dierence ju(t) ud (t)j exceeds the cuto c with a probability
. Figure (4.2) shows that it is possible to identify a time t for which (t) = , for a
given value 0 < < 1. This is the time at which the dierence ju(t) ud (t)j exceeds the
cuto c with probability , which is interpreted as the approach to the active phase with
probability . Since this time depends on the choice of we denote this time as t = T,
dening it as the eective time of transition from the silent to the active phase. Note that
this time T is obtained from the calculation of (t), and therefore it is simply dened in
terms of (t), as
Pr (ju(T ) ud(T )j > c) = (T ) = :
(4.10)
In Figure 4.3a we show T for reasonable values of and c. It is clear that T has
essentially the same behavior for dierent choices of c and , with the curve shifted up
for larger values of and/or c. Since increasing or decreasing and c will shift the
curve, the same values of T can be obtained from dierent combinations of and c.
After xing c to be two orders of magnitude larger than the noise, as described above,
there is still some exibility in the choice of which can be used to dene T, the eective
transition time. The parameter can be tuned appropriately, as follows.
(b)
R. Kuske: Asymptotics of a noisy burster
In Figure 4.3b) we compare the value of this critical time T to the mean rst transition
time , that is, the average time at which the dierence ju(t) ud(t)j rst exceeds the
cuto c. The quantity is dened as
N
1X
min juj (t) ud(t)j > c;
(4.11)
=
N
j =1
t
using N simulations, with uj (t) obtained from the j simulation of (3.1). It is clear that
the critical time T which is calculated directly from the asymptotic approximation for the
density q(u; t) has the same dependence on the noise level as the mean exit time obtained
from simulations. In practice, one can determine the appropriate by comparison of T
with for one value of Æ. Then T can be determined from the asymptotic approximation
for other values of Æ, using this value of . Since both of these quantities have the same
dependence on Æ, this gives a good approximation to over a range of noise levels, as
shown from Figure 4.3b).
th
5. Transition out of the active phase
Now we consider the eect of noise as the system leaves the active phase. Here the behavior
is dominated by large amplitude spiking, which makes the approach of the previous section
impractical. In order to isolate the eect of the noise from the large oscillations, we
consider a multiscale approach to obtain averaged equations describing the deviation
from the periodic behavior, the phase shift, and the slow increase in the conductance Gss.
This approach is motivated by the fact that the amplitude of the oscillations in the active
phase varies slowly in time, as Gss increases, as can be seen in Figure 1.1b.
First we identify a slow time T = t, and write the dependent variables in terms of
both the original fast time t and the slow time T .
u(t; T ) = U (t + (T )) + A(T )W 0 (t + (T ))
(5.1)
0
w(t; T ) = W (t + (T )) A(T )U (t + (T ))
(5.2)
D
E
z (T ) = Gss (t)
(5.3)
The functions U and W give the deterministic limit cycle solution for xed Gss = Gss,
that is, for = 0 in (2.6)-(2.7),
U 0 (t) = ( f (U ) W Gss (U vs (U )) + ISH
W 0 (t) = b(U W ))
(5.4)
The equations (5.1)-(5.2) express u and w in terms of the deterministic limit cycle solution
U and W , which is a function of the original time scale, modied by a slowly varying
phase shift (T ) and a small correction with slowly varying amplitude A(T ). The form
of (5.1) and (5.2) is similar to that used for a study of coupled limit cycle oscillators
near a critical point in (24). The transformation (5.1)-(5.2) is typically used to extract
the oscillatory behavior from the solution, leaving relatively simple behavior of the slowly
varying dynamics, described by the equations for A(T ) and (T ). As shown in Appendix
B, substitution of (5.1)-(5.2) into the equations for u and w results in cancellations of
the leading order terms, leaving equations for A and which are in terms of the slow
time scale. The function z(T ) is obtained by averaging Gss over the period of U and W ,
0
0
16
R. Kuske: Asymptotics of a noisy burster
indicated by , so that z(T ) is a slowly varying function. It could be anticipated that
the average of Gss varies on the slow time scale T , since the right hand side of (2.8) has
a factor of . We discuss the averaging below and in detail in Appendix B. Note that vs
depends on Gss as well (3.2), so that Gss = Gss in vs(U ) in (5.4).
In our analysis, we focus on the behavior of A, which measures the deviation from the
large oscillations of the active phase, described by the periodic functions U and W . For
A small, the dynamics are essentially the oscillations of the active phase, and for A large,
the dynamics dier signicantly from the active phase oscillations, indicating a return to
the silent phase.
Our goal is to look for approximate stochastic equations for A(T ), (T ), coupled to the
averaged equation for Gss:
dA A (A; ; z (T ))dT + A d (T )
(5.5)
d (A; ; z (T ))dT + d (T )
(5.6)
h D
E
D
E
i
dz = z U vs (U ) + W 0 vs0 (U )W 0 z (T )A(T ) dT
(5.7)
Here and are independent Brownian motions on the slow time scale T . The functions
A , A , and are derived in Appendix B. The method is based on using Ito's formula
and the original system (3.1) to obtain equations for these coeÆcients. We also use the
fact that U and W are periodic functions in the fast time scale t, over which we average.
Then we obtain the coeÆcients in (5.5)-(5.6)
1 hD(W 0) f 0(U ) W 0U 0 + bU 0 (W 0 + U 0)EA(T )+
(5.8)
A =
D E
0
1
2
1
2
2
M
D
z (T )A(T ) W 02
v0 (U )(W 0 )2
E
+
D
U 00 W 0
E 2
W 00 U 0 2
D
E
1
W 0U
(5.9)
A = p
MN
1 hDW 00U 0 U 00W 0EA(T ) + z(T )A(T )DU 0 W 0(1 v0 (U ))E
=
s
M
D
E
i
+ (U 0W 0)f 0(U ) (U 0 ) bW 0(W 0 + U 0) A(T )+
(5.10)
Æ 1 D 0 E
UW
(5.11)
= p
MN
D
E
M = U0 + W 0
(5.12)
D
E
N = U +W
(5.13)
At rst this may not appear to be an improvement over the original system. However,
there are several advantages to looking at the system (5.5)-(5.7). First, since this system
is in terms of the slow time T , simulation over the length of the burst is much faster
than for a simulation of (3.1). One must simulate over the length of the active phase of
(3.1) with a suÆciently small time step. Due to signicant variability in the system, this
must be repeated thousands of times to get reasonably good statistics. In contrast to the
original systems which is in terms of t, the simulation of (5.5)-(5.7) over the same length
of time is much faster. Since the time scale is based on t = T=, the number of time
steps needed is reduced by factor of O( ). This system does not resolve the detail of the
oscillations, but describes variation on the T scale. The drift A and have relatively
s
Æ
2
2
2
2
2
1
17
R. Kuske: Asymptotics of a noisy burster
simple behavior in time, i.e. increasing or decreasing rather than oscillatory. This system
is proposed for describing the average behavior of the system, or the behavior over many
simulations over the length of the burst interval. It is not appropriate for resolving the
detail of a single realization.
18
3
2
H (T )A
1
0
−1
−2
−3
100
150
200
250
300
350
time steps
The dynamics of the term H (T )A= from the drift in the equation (5.14) for the slow amplitude
A(T ). The solid line is the deterministic behavior, and the dotted lines are three realizations for Æ = 1 10 4 .
Figure 5.1:
The deterministic function remains close to zero for the interval shown, except at the very end of the interval
when it begins to grow. That is, at the end of interval the deterministic dynamics move away from A(T ) = 0,
which corresponds to the transition out of the active phase. In contrast, the random dynamics show considerable
variability from zero, which indicates that the dynamics are not dominated by the function H (T )A=, but rather
the other terms from the drift and diusion coeÆcients play a signicant role. Then these additional terms,
which are proportional to Æ , can cause an earlier transition out of the active phase than in the case of no noise
(Æ = 0). Thus the slowly varying amplitude A(T ) in the presence of noise can grow signicantly in magnitude,
thus indicating an early exit from the active phase in the presence of noise.
A closer look at the equations for A and reveals the eect of the noise. We focus
on the equation for A, since the behavior of A indicates the deviation from the periodic
behavior of U and W . If A remains small, then the large oscillations of the active phase
persist. If A increases in magnitude, then these large oscillations are disrupted, and the
system returns to the silent phase. The equation for A has the form
H (T )A
Æ
dA =
+ K dT + pÆ K d(T )
(5.14)
2
1
2
Here K and K are constants which depend on the average of U and W and their
derivatives (5.8)-(5.9). First we consider the drift, H (T )A +(Æ =)K . It would appear
that this drift is large, due to the factor of . However, H (T ) is small, and jAj decays
exponentially when H (T ) is negative. Furthermore, since we have assumed that Æ =
O() or smaller, the second term in the drift is not large. Without the noise, jAj will
grow (decay) if H (T ) is positive (negative). Therefore, when H (T ) is suÆciently large in
1
2
1
1
2
1
R. Kuske: Asymptotics of a noisy burster
magnitude and negative, A will remain small on the average. If H (T ) is positive, or if it
suÆciently small so that H (T )A= is negligible with respect to the second term Æ K , the
second term in the drift has a greater inuence, and jAj will increase in magnitude.
In Figure 5.1 we show a graph of H (T )A=, which describes the dynamics of the slow
variable A(T ) in the absence of noise (solid line). Note that in the multiscale approach
we treat H (T ) as a constant with respect to t, but it is in fact a slowly varying function
of T . From Figure 5.1 we see that the behavior of H (T )A= changes considerably in the
presence of noise (dotted lines), which indicates signicant inuence from the additional
term K Æ in the drift, and from the diusion term. That is, even with small noise, there is
variability from the deterministic behavior. Then there is an opportunity for a reduction
of the length of the active phase.
19
2
1
2
1
1
0.9
0.8
Average length
0.7
of the stochastic
0.6
active phase
0.5
(scaled)
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
−4
Æ
x 10
Graph of the average length of the stochastic active phase scaled as a fraction of the length of
deterministic active phase vs. the size of the noise. The 's are obtained from the simulation of the fast system
(3.1) with 5000 realizations. The exit time was dened as the time when the magnitude of the oscillations was
smaller than .3. The initial condition is taken for u, w, Gss in the beginning of the active phase. The solid line
gives the results from the simulation of the slow system (5.5)-(5.7), obtained by the averaging method discussed
in this section and in Appendix B. The exit time was dened in terms of jAj > :25. The initial condition was
A = 0:001 and = 0. Here a = 0:14, b = 0:008, = 2:54, = 5 10 4, R1 = 1=, L = 0:6, ISH = 0:04,
ID = 0:2.
Figure 5.2:
Now consider the diusion term.
By considering the system on the slow time T , the
p
noise now has a coeÆcient of Æ= , which describes the eective inuence of white noise,
originally on the t scale, over a long time t = T=. This term shows the accumulated
eect of the noise over the time scale t = T=, so that eectively the variance of the noise
isp larger than in the original system (3.1). It is not surprising that we have a factor of
, since (t) and = (t) have the same statistics, for a Brownian motion (29).
Therefore both the drift and diusion in the equation for A are inuenced by the size of
the noise in the original system.
1 2
R. Kuske: Asymptotics of a noisy burster
Now we compare a simulation of the averaged equations (5.5)-(5.7) with that of the
original system (3.1). We wish to identify the inuence of the noise on the transition
out of the active phase. For (3.1) we identify the exit from the burst as the time at
which the oscillations change from large to small, that is, the time at which oscillations
with amplitude below a critical level c persist. For the averaged system (5.5)-(5.13), we
identify the transition as the time when A is O(1), that is, A exceeds a critical value Ac.
In Figure 5.2 we compare the length of the active phase for the stochastic dynamics as a
fraction of the active phase for the deterministic dynamics as we increase the noise level
Æ. We see that there is good agreement between the original model (3.1) and the reduced
system (5.5)-(5.13), even though the reduced system depends only on the average of the
deterministic oscillations and the exit times for the fast and slow variables are dened in
terms of two dierent quantities.
6. Discussion
The new method developed in this paper is used to quantify the eect of noise on the
dynamics of a type III (elliptic) burster. This is an important problem, since this kind
of bursting, and its sensitivity to noise, has been observed experimentally (8). For convenience, we analyze an idealized model of an activity-dependent spine, attached to a
passive cable. Our simulations in Section 3 illustrate how small amplitude synaptic noise
can change the burst dynamics of the spine head membrane potential, similar to changes
in burst dynamics as a response to varying the electrotonic length L of the cable. Thus,
we quantify, for this idealized model, how small amplitude noise inuences the activitydependent spike dynamics of a single dendritic spine.
Using a combination of asymptotic approaches we analyzed how small amplitude noise
can inuence transitions between the active and silent phases of the burst. It remains, in
this study, to compare the eect of the noise in the transition from the silent to active
phase, studied in Section 4, with the eect of the noise on the the transition out of the
active phase, studied in Section 5. We see that the noise plays a more signicant role
in the transition into the active phase. This is not surprising, since in the silent phase
the conductance Gss drops below a critical value, and the subthreshold oscillations are
unstable. Since Gss varies slowly, there is a long period in which the small oscillations
are in the unstable regime. Without the noise the memory eect causes a delay of the
onset of the active phase. With the noise this delay is reduced. This was discussed in
detail for other delay bifurcations in (1), (41), (20). Using Figures 1.1 and 4.3 and 5.2
we can estimate that mean reduction of the length of the silent and active phases due to
the noise. In the rst half of the silent phase, Gss is above the critical Hopf point Gcss,
so that the oscillations decrease, approaching the attracting steady-state. In this region
small noise will have little inuence, except for small oscillations. However, for Gss < Gcss,
these small oscillations are unstable, and the noise causes a more rapid approach to the
active phase. Figure 4.3 shows that the second half of the silent phase can be reduced by
as much as 85-90% for Æ = :0005. For the parameters considered in Figure 4.3, the average
length of the silent phase is approximately 7000+T . For other ranges of parameter values
corresponding to bursting, the method can be applied easily to obtain the same type of
asymptotic approximations for quantifying the eects of the noise.
In contrast to the results of Section 4, it can be seen from Figure 5.2 that the length of
the active phase is reduced by less that 10-25% by the same noise levels as in Figure 4.3.
This result could be anticipated from the dynamics described by the averaged equations
20
R. Kuske: Asymptotics of a noisy burster
for the slowly varying amplitude A and phase in the active phase. Since A is the amplitude of the correction to the deterministic periodic spiking of the the active phase, when
A becomes large, the active phase is disrupted. In Section 5 we see that the deterministic
behavior of A corresponds to the decay of this amplitude for most of the deterministic
active phase (see Figure 5.1). Because of this decay, the addition of small noise does not
cause a disruption of the large oscillations of the active phase, but only a small deviation from the deterministic behavior. For a large part of the active phase, this results in
variability of the phase and some variation of A about A = 0, so that the behavior of u
is very close to the deterministic behavior with a phase shift. Later in the active phase,
the deterministic dynamics are no longer attracting; that is, perturbations about A = 0
do not decay. Then the noisy perturbations result in the growth ofjAj, thus causing an
earlier departure from the active phase. Since this eect occurs only later in the active
phase, the spiking behavior persists for a signicant portion of the deterministic active
phase. As the noise increases, both the drift and diusion in A are increased, so that the
noise is more likely to cause a transition from the active phase to the silent phase. Similar
numerical demonstrations of the dynamics are shown in (6).
Summarizing, noise has a greater eect on reducing the silent phase than on reducing
the active phase. This eect can be quantied in terms of parameter values through
the methods described in Sections 4 and 5. In particular, the increased noise can result
in the active phase occupying a larger percentage of the burst cycle. In this sense the
analysis also gives a qualitative picture of the transition from bursting to intermittent
spiking due to increased noise, even though the analysis is limited to analysis of the
bursting phenomenon. In our application to dendritic spines, we have taken a rst step in
quantifying the eects of noise on synaptic plasticity. We have found that small amplitude
noise can signicantly change the fast membrane dynamics of an individual spine which in
turn aects the slow dynamics of the stem conductance. If changes in stem conductance
are dependent on changes in stem structure, our results suggest that for a population of
spines, small amplitude noise could modify the population's pattern of electrical activity,
resulting in an overall change in spine morphology.
The method of this paper has advantages over other studies of noise-sensitive dynamics.
The asymptotic approximation gives the dependence of the dynamics on both the size of
the noise and the other parameters in the model through mathematical expressions which
can be evaluated quickly for a range of parameters. In contrast, a simulation requires many
realizations to accurately approximate quantities such as the probability density and mean
transition times, since the noise-sensitivity of the system depends on a combination of
system parameters. The eÆciency of the approach of this paper is crucial in studying
biological systems which depend on a combination of system parameters,
The techniques applied here are not conned to this problem; they can be used to
quantitatively describe the bursting dynamics of other elliptic bursters, biological or not,
in the presence of noise. We expect that they can be extended to related types of bursters.
Furthermore, the method of this paper can be applied to a general nonlinear system, in
contrast to approximate linearized systems or models of a special form, (see, for example,
(32)-(33), (38)- (39) and references therein). The approach potentially could be used in
larger systems, such as compartmental models. Finally, since the method includes the
behavior of the underlying dynamics as a basis for the asymptotic approximation, it
provides a qualitative picture of the noisy dynamics in addition to quantitative measures
of the process.
21
R. Kuske: Asymptotics of a noisy burster
22
7. Acknowledgements
The rst author (R. K.) was supported in part by an NSF grant, DMS-0072311. Both
authors wish to thank the referees for their comments, which were helpful in clarifying
the paper's message.
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9
36
7
A. General derivation of asymptotic approximation to the
density
Consider which satises the systems of SDE's
p
d = dt + 2d ;
= fyi ; i = 1; : : : ; ng
(A.1)
where d = fiWi; i = 1; : : : ; ng. The Wi's are n independent Brownian motions
and i are constants. We seek an asymptotic approximation for the probability density
function p( ; t). Note that in (3.1), = (d ; d ; d ), = (u; w; Gss ), = (Æ; 0; 0), which we
y
y
a
W
y
W
y
a
1
2
3
y
24
R. Kuske: Asymptotics of a noisy burster
use in the derivation below. This probability density function satises the Fokker-Planck
equation which can be derived from the system of SDE's (A.1),
@p
@t
=Æ
2
@2p
@y12
r (ap)
(A.2)
(see e.g. (29) for a derivation of the Fokker-Planck equation) Here is the drift in the
equation (A.1). Also note that the diusion coeÆcient is small for Æ 1 in (A.2). We
assume a Gaussian type form for the density p( ; t):
p( ; t) = CeQ y;t ; where,
(A.3)
2
3
n
X
(y yi;d) + X h (t) (yi yi;d)(yj yj;d) q (t) yi yi;d 5 + s(t):
4gi (t) i
Q( ; t) =
ji
i
2Æ
Æ
Æ
i
j<i
(A.4)
Here yi;d is the deterministic behavior of yi, with dynamics described by _d = ( d )
((A.1) with ).
As mentioned above, we are interested in the case when Æ 1. Therefore we nd it
convenient to introduce the change of variables
y y
i = i i;d ; i = 1; : : : n; p(y; t) = P (; t);
(A.5)
Æ
Under the change of variables, the Gaussian-type form (A.3) for p( ; ) becomes
a
y
y
(
)
2
y
2
2
=1
y
a y
0
y t
8
n
<X
P (; t) = C exp : [ gi (t)i =2 +
2
i=1
X
j<i
hij (t)i j
9
=
qi (t)i ] + s(t) :
;
(A.6)
In terms of P (; t), the Fokker-Planck equation (A.2) becomes
@ P @
1 r (AP );
+
r
(A.7)
P = r P
@t @t
Æ
where r is the gradient with respect to the variables, and A is the vector obtained by
making the change of variables (A.5) in the drift from (A.1). Substituting P (; t) into
(A.7) we obtain
n
X
X
1
[( Aj + yj;d0(t))(j gj qj + j hjm)] +
(A.8)
0 =
2
a
Æ j =1
n "
X
j =1
gj0 (t)
(j gj (t)
j 6=m
j
2
0
0
2 + qj (t)j + s (t)
X
m6=j
j
m<j
h0jm (t)m
1
@ Aj
m hjm + qj ) gj +
Æ @
2
j
Note that A is a function of Æ, so for Æ 1 we expand
Aj = Aj + ÆAj : : : ; where
Aj Aj = Aj jÆ ; Aj = @@Æ
0
0
X
1
=0
1
Æ=0
(A.9)
(A.10)
(A.11)
25
R. Kuske: Asymptotics of a noisy burster
and substitute (A.10) into the equation (A.8). Here Aj = j ( d (t)), and from the deterministic system, d0(t) = ( d (t)). Also, Aj = Æ( j ( d ))y , so terms like Æ Aj are
actually O(1). The leading order term with coeÆcient Æ is then Aj yj;d0 (t), which
vanishes by the equation for the deterministic dynamics. Then the terms from the O(1)
part of the equation give a sum of powers of i and crossterms ij . We set the coeÆcients
of like powers of i equal to zero in order to obtain equations for gj , hjm and qj . We also
neglect terms with coeÆcients Æk for k 1.
For (3.1) the resulting equations are
g_
(A.12)
2 = g d +g +d h +d h
g_
2 = g d +h +d h +d h
g_
2 = g d +h +d h +d h
h_ = 2g h + h d + g d + h d + g d + h d + h d
h_ = 2g h + h d + g d + h d + g d + h d + h d
h_ = 2h h + h d + g d + h d + g d + h d + h d
q_ = 2g q q d
qd
qd
q_ = 2h q q d
qd
qd
q_ = 2h q q d
qd
qd
s_ = q d y d y d y g
@dj
; m = 1; 2; 3:
(A.13)
djm =
@ym
0
y
1
2
3
a y
1 1
2 2
3 3
a
i
(1)
2
(1)
3
y
y
1
i
1
(1)
2
1
(2)
2
12
(2)
1
12
(2)
3
23
(3)
2
13
(3)
1
13
(3)
2
23
12
a
13
1
12
(1)
12 1
(2)
1 1
(2)
12 2
(1)
2 2
(2)
13 3
(1)
23 3
13
1
13
(1)
13 1
(3)
1 1
(3)
13 3
(1)
3 3
(3)
12 2
(1)
23 2
13
12
13
(3)
23 3
(1)
1 1
(2)
3 3
(1)
2 2
(2)
23 2
1 1
2
12 1
(2)
1 1
(2)
2 2
(2)
3 3
3
13 1
(3)
1 1
(3)
2 2
(3)
3 3
2 2
3 3
(3)
2 2
(3)
12 1
(2)
13 1
(1)
3 3
1
1 1
0
i
0
12
2
1
(
0
1
)
where dj for j = 1; 2; 3 are given in (3.1).
B.
Derivation of slow time, averaged equations
We obtain equations for the drift and diusion coeÆcients in the approximate SDE's for
A and in the active phase (5.5)-(5.6). We write u and w in terms of the deterministic
periodic functions U (t) and W (t) (5.4) and the slow variables A(T ) and (T ) for T = t.
u(t; T ) = U (t + (T )) + A(T )W 0 (t + (T ))
w(t; T ) = W (t + (T )) A(T )U 0 (t + (T ))
(B.1)
To emphasize that we consider the case when A(T ) is small, we write A(T ) = a(T ).
When A(T ) is no longer small, that is, when a(t) becomes large, then the system is no
longer well described by the oscillations U (t) and W (t) and it leaves the active phase.
We assume the form of equation (5.5)-(5.6) for A(T ) and (T ), and derive equations for
the coeÆcients A, A, and in (5.5)-(5.6) using two equations for du. This method
is similar to a reduction of a stochastic partial dierential equation via a projection to
a system stochastic ordinary dierential equation in the context of metastable interface
dynamics (35). We also write Gss = z(T ), treating it as a function of the slow time T
only, as follows from equation (2.8) which has on the right hand side.
26
R. Kuske: Asymptotics of a noisy burster
First we use Ito's formula (29), which gives
@u
@u 1 @ u
0
00
du = U (t + (T )) + a(T )W (t + (T )) + A + + dt
@A
@ 2 @
2
2
2
dw
=
@u
@u
+A @A
d (T ) + d (T )
@
1
2
W 0 (t + (T )) a(T )U 00 (t + (T )) + @w
+
A
@A
@w 1 2 @ 2 w
+ @ 2 @2
(B.2)
dt
@w
d (T ) + d (T )
(B.3)
+A @w
@A
@
Ito's formula is essentially the chain rule for stochastic dierential equations. In the spirit
of multiscale analysis, we have treated functions of the slow time T as independent of the
fast time t. Here we have left out the second derivatives of w and u with respect to A
since they are zero.
We have two other equations for du and dw from (3.1), which are obtained by substituting (B.1) into the right-hand sides of (3.1). They are
du =
f (U ) W f 0 (U )a(T )W 0 + a(T )U 0
z (T ) U vs (U ) + a(T )W 0 (1 vs0 (U )) + ISH dt + Æd + O() (B.4)
dw = b U W + a(T )W 0 + a(T )U 0 dt
(B.5)
The O() are the terms which are nonlinear in A = a, and we neglect them below under
the assumption that A is small.
Now we equate W 0du U 0 dw and U 0du + W 0dw obtained from (B.2)-(B.3) and (B.4)(B.5). In order to compare these expressions, we have replaced dT = dt in (B.2)-(B.3)
and we use d (t) = = d (T ) in (B.4)-(B.5). From the two expressions for W 0du U 0 dw
we obtain
0
0
((U ) + (W ) )( A dT + Ad (T )) = a(T ) W 0 f 0(U )W 0 + U 0
1
2
1 2
2
2
1
z (T )(W 0 )2 (1 vs0 (U ))
bU 0 (W 0 + U 0 )
0
1 (U 00 W 0 W 00U 0)dT + ÆW
p
d (T ) + O()
(B.6)
2 From the two expressions for U 0 du + W 0dw we obtain
0
0
((U ) + (W ) )( dT + d (T )) = a(T ) W 00U 0 + U 00W 0 + U 0 ( f 0(U )W 0 + U 0)
2
2
2
z (T )(W 0 U 0 )(1
2
v0 (U ))
s
+
bW 0 (W 0 + U 0 )
1 (U 00 U 0 + W 00W 0) dT + pÆ U 0d (T ) + O()
(B.7)
2 Here (T ) is a white noise process on the slow time scale, and we have written dt = dT .
Now we introduce
W (t)
U (t)
dW (T ) +
dW (T ) where
(B.8)
d^(T ) =
NZ
N
1 P U + V dt
N =
(B.9)
2
1
P
2
0
2
2
27
R. Kuske: Asymptotics of a noisy burster
where W (T ) and W (T ) are two independent Brownian motions. Then E [^] = 0 and
E [^ ] = (U + V )T . Here we anticipate that we will average over the period P of U (t)
and W (t) below, treating t and T as independent time scales. Averaging (B.8) in this way
gives
1 Z P E [^] dt = 0
2
1
2
2
2
P
0
1 Z P E [^ ] dt = 1 NT = T
P
N
(B.10)
2
0
Therefore we replace (T ) in (B.6) and (B.7) with ^(T ) as an approximation to the noise,
and then average the equations over the fast time scale below.
In (B.6)-(B.7) we have functions which depend on the fast time scale, U (t) and W (t)
and their derivatives, and all other terms depend on the slow time scale. Equating the
respective drift and diusion terms and averaging the functions of t over the period P of
U and W , we obtain
M
A
= a(T )
Z P
0
W0
z (T )a(T )
f 0(U )W 0 + U 0 dt
Z P
0
(W 0 )2(1
v0 (U ))dt
s
+ 21 W 00U 0 U 00W 0dt
Z P
1
Æ
0
p
W Udt dW (T )
MA d (T )) =
N 2
Z P
a(T )
Z P
0
(B.11)
0
1
M
=
a(T )
Z P
0
[W 00U 0
z (T )a(T )
Z P
0
Z P
U 00 W 0]dt + a(T )
(W 0 U 0)(1
0
M
=
Z P
0
2
0
(U 0 ) + (W 0) dt
2
Z P
0
U0
f 0(U )W 0 + U 0 dt
vs0 (U ))dt
+a(T ) bW 0(W 0 + U 0)dt
Z
1
Æ P 0
p
M d (T ) =
U W dt dW (T )
N 2
(B.12)
1
0
bU 0 (W 0 + U 0 )dt
(B.13)
(B.14)
(B.15)
2
Here we have left out terms like R P UU 0dt and R P U 0U 00 dt which vanish for periodic U .
Also, we average the equation for Gss over the period of U and W , setting Gss = z(T ),
Z P
1
dz = z (T )
(U v (U ))dT + O():
(B.16)
0
P
0
0
s
with the O() terms are given in (5.7). Then the system (B.11)-(B.16) is identical to the
equations (5.8)-(5.13) given in Section 5.
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