Kinetic model of exocytosis

Appendix A

Kinetic model of exocytosis

Given the kinetic model:

ξC + S −* )−

^kb

ku

S

^∗

(A.1a)

V + S

^∗

−−* )−−

^k1

k−1

VS

^{∗ k}

−→ nG

²

(A.1b)

we recall that each chemical equation represents a differential equation, hence the kinetic scheme can be rewritten in the suitable mathematical form:

d

dt [S

^∗

] = k

_b

[C]

^ξ

[S] − k

_u

[S

^∗

] (A.2a) d

dt [VS

^∗

] = (k

₋₁

+ k

₂

)[VS

^∗

] − k

₁

[V][S

^∗

] (A.2b) Then, under the hypothesis of quasi-steady state (i.e. the intermediate reaction products S

^∗

V equilibrate much faster than product nG and substrates V, S

^∗

), it follows that:

0 = k

_b

[C]

^ξ

[S] − k

_u

[S

^∗

] (A.3a)

0 = (k

₋₁

+ k

₂

)[VS

^∗

] − k

₁

[V][S

^∗

] (A.3b) Let [S]

₀

= [S] + [S

^∗

] be the total concentration of Ca

²⁺

-binding protein and ϑ = [S

^∗

]/[S]

₀

the relative fraction of its activated form. Thus, by dividing equation (A.3a) by [S]

₀

and solving for ϑ we obtain the well-known Hill equation (Stryer, 1999; Weiss, 1997):

ϑ = [C]

^ξ

ku

kb

+ [C]

^ξ

(A.4)

135

We note that for the binding reaction (A.1a) the dissociation constant is K

_d

= k

_u

/k

_b

. By substituting this latter into the previous equation (A.4) and solving for [S

^∗

], we get:

[S

^∗

]

0

= [S]

0

[C]

^ξ

K

_d

+ [C]

^ξ

(A.5)

where the “0” subscript is used hereafter to denote total concentration, in this case of activated Ca

²⁺

-binding protein S

^∗

.

We consider then reaction (A.1b) which is in the form of a standard Michaelis-Menten first-order kinetics. We first solve equation (A.2b) for [V S

^∗

] obtaining:

[VS

^∗

] = k

1

[V][S

^∗

]

k

₋₁

+ k

₂

= [V][S

^∗

]

K

_M

(A.6)

where K

_M

= (k

₋₁

+ k

₂

)/k

₁

is the Michaelis-Menten constant of reactions (A.1b). Let [V]

₀

= [V] + [S

^∗

] be the total concentration of vesicles, then [V] = [V]

₀

− [VS

^∗

]. Thus, replacing V in (A.6) by this latter expression, we get:

[VS

^∗

] = ([V]

₀

− [VS

^∗

])[S

^∗

]

K

_M

= [V]

₀

[S

^∗

] − [VS

^∗

][S

^∗

] K

_M

therefore:

[VS

^∗

] = [V]

₀

[S

^∗

]

K

_M

+ [S

^∗

] (A.7)

Assuming (implicit in the adoption of a Michaelis-Menten kinetics) that [V]

₀

<< [S

^∗

]

₀

, then [VS

^∗

] << [S

^∗

]

₀

and thus [S

^∗

]

₀

= [S

^∗

] + [VS

^∗

] ' [S

^∗

]. Accordingly, equation (A.7) can be rewritten in the form:

[VS

^∗

] ' [V]

₀

[S

^∗

]

₀

K

_M

+ [S

^∗

]

₀

(A.8)

From the law of mass action:

1 n

d

dt [G] = − d

dt [VS

^∗

] = k

₂

[VS

^∗

] hence:

d

dt [G] = nk

₂

[VS

^∗

] (A.9)

Then, we can replace [VS

^∗

] in this equation with its expression in (A.8). Following we

136

expand K

_M

and after some algebraic simplifications we obtain:

d

dt [G] = nk

2

[VS

^∗

] = nk

2

[V]

0

[S

^∗

]

0

K

_M

+ [S

^∗

]

₀

= nk

2

[V]

0

[S

^∗

]

0 k−1+k2

k1

+ [S

^∗

]

0

=

= nk

₁

k

₂

[V]

₀

[S

^∗

]

₀

(k

₋₁

+ k

₂

) + k

₁

[S

^∗

]

₀

The following substitution of equation (A.5), leads to:

nk

1

k

2

[V]

0

[S

^∗

]

0

(k

₋₁

+ k

₂

) + k

₁

[S

^∗

]

₀

= nk

₁

k

₂

[V]

₀

[S]

_0K^[C]ξ

d+[C]^ξ

(k

₋₁

+ k

₂

) + k

₁

[S]

_0K^[C]ξ

d+[C]^ξ

=

= nk

₁

k

₂

[S]

₀

[V]

₀

[C]

^ξ

(k

₋₁

+ k

₂

) (K

_d

+ [C]

^ξ

) + k

₁

[S]

₀

[C]

^ξ

=

= nk

1

k

2

[S]

0

[V]

0

[C]

^ξ

(k

₋₁

+ k

₂

) K

_d

+ (k

₋₁

+ k

₂

+ k

₁

[S]

₀

) [C]

^ξ

Accordingly, the production rate of glutamate by exocytotic release is given by:

d

dt [G] = nk

₁

k

₂

[S]

₀

[V]

₀

[C]

^ξ

(k

−1

+ k

2

) K

d

+ (k

1

[S]

0

+ k

−1

+ k

2

) [C]

^ξ

(A.10) In this form, the exocytosis production rate is in terms of n glutamate molecules per vesicle per [V]

0

available vesicles. The effective rate of exocytosis though is dimensionally expressed in 1/seconds. We need therefore to normalize equation (A.10) by n[V]

₀

. It follows that:

r

_e

= 1 n[V]

0

· d

dt [G] = k

₁

k

₂

[S]

₀

[C]

^ξ

(k

−1

+ k

2

) K

d

+ (k

−1

+ k

2

+ k

1

[S]

0

) [C]

^ξ

(A.11a) Alternatively, we can express r

e

in terms of the half maximal effective Ca

²⁺

-concentration K

_0.5

(otherwise denoted by EC

₅₀

), provided that K

_0.5

= √

^ξ

K

_d

: r

_e

= k

₁

k

₂

[S]

₀

[C]

^ξ

(k

₋₁

+ k

₂

) K

_0.5^ξ

+ (k

₋₁

+ k

₂

+ k

₁

[S]

₀

) [C]

^ξ

(A.11b) We note that thanks to the normalization we do not have to know n nor [V

₀

] whose values actually, have not been experimentally determined yet.

137

Appendix B Data fitting

Table B.1: Fitting laws

^†

Fitting law Fitting equation

^‡

Figures

Linear y(x) = Ax + B 3.20b

Power y(x) = Ax

^b

3.20a,3.21

Monoexponential y(x) = A exp(b x) 2.6b,c,d

Biexponential y(x) = A exp(b x) + C exp(d x) 2.6b,c,d Special y(x) = A + r

e

x

µ x k

_e

¶

_ξ

P

2 n=0

(−1)

ⁿ

1 + (n + 1) ξ

µ x k

_e

¶

_{n ξ}

2.6a

†

All fittings were performed by CFTOOL under MATLAB R2006a.

‡

A, B, C, d ∈ R

⁺

, b ∈ R

⁻

, n ∈ N

0

.

Special fitting (figure 2.6a only):

In this case it must be:

y

⁰

(x) = r

_e

x

^ξ

k

e^ξ

+ x

^ξ

where y ← γ (y ← g) and x ← C. Because data refer to C → 0 we can expand the right hand side of the precedent equation into Taylor series up to the second order, thus

138

obtaining

x→0

lim r

_e

x

^ξ

k

e^ξ

+ x

^ξ

= lim

x→0

r

_e

µ x

k

_e

¶

_ξ

1 + µ x

k

e

¶

_ξ

' r

_e

µ x

k

_e

¶

_ξ

à 1 −

µ x k

_e

¶

_ξ

+

µ x k

_e

¶

_2ξ

!

Then, by integration we solve for y and we get the following fitting equation:

y(x) = A + r

_e

x µ x

k

_e

¶

_ξ

X

₂

n=0

(−1)

ⁿ

1 + (n + 1) ξ

µ x k

_e

¶

_{n ξ}

139

Appendix C Software

The following XPPAUT code implements the Ca

²⁺

/IP

₃

coupled oscillator studied in Section 3.2.

The main MSA simulator (Chapters 4–5) is available on request at the homepage of the author: http://mauriziodepitta.googlepages.com/home.

#CAIP3system.ode

#Li-Rinzel system with dynamical IP3 description (feedback formalism)

#NOTE: concentrations are in [uM], whereas the time scale is in [sec]

#Parameters p y

p tp=9.2,psi=0 p kplc=0

p rplc=0.411 p k3=.1

p c0=2.0,c1=.185,v1=6.0,v2=.11,v3=.9 p d1=.13,d2=1.049,d3=.9434,d5=.08234 p a2=.2

p Iinf=0.015 p k3K=0.4,k1=1.3 p n=0.7

p Sast=91.5,Va=1.98e-15,sigma=25 p NA=6.022e23

p zeta=1e-4,rho=4.91e-4,nves=101,Gves=60

#RHS

dca/dt=-jchan(ca,q,ip3)-jpump(ca)-jleak(ca) dq/dt=(a2*d2*(ip3+d1)/(ip3+d3))*(1-q)-a2*ca*q

140

dip3/dt=1/tp*(tp*rplc*Sast*sigma*1e6/(NA*Va)*((zeta*rho*1e3*nves*Gves*y)^n/

(k1^n+(zeta*rho*1e3*nves*Gves*y)^n))*ca^2/(kplc^2+ca^2)+(psi*ca^2/

(k3K^2+ca^2)+1-psi)*(Iinf-ip3))

#Internal functions ninf(ca)=ca/(ca+d5)

jchan(ca,q,ip3)=c1*v1*(ip3/(ip3+d1))^3*ninf(ca)^3*q^3*(ca-caER(ca)) jpump(ca)=(v3*ca^2)/(k3^2+ca^2)

jleak(ca)=c1*v2*(ca-caER(ca)) caER(ca)=(c0-ca)/c1

#Initial conditions ca(0)=.1

q(0)=.01 ip3(0)=0.015

#Settings

@ xp=ca, yp=q, xlo=0.0, ylo=0.4, xhi=1.5, yhi=1.2

@ total=1000, dt=.05, meth=cvode, bound=1000, maxstor=10000000

@ AUTOVAR=ca, AUTOXMIN=0, AUTOXMAX=1, AUTOYMIN=0, AUTOYMAX=1.5

@ DS=.001, DSMIN=.0001, DSMAX=.01

@ PARMIN=0, PARMAX=1, NORMMAX=1000, NMAX=2000, NPR=500 done

141

Appendix D Publications

De Pitt`a, M., Volman, V., Levine, H., Pioggia, G., De Rossi, D. and Ben-Jacob, E.

(2007). Coexistence of amplitude and frequency modulated intracellular calcium dy- namics. Phys. Rev. Lett. submitted.

142

Coexistence of amplitude and frequency modulated intracellular calcium dynamics

Maurizio De Pitt`a,

^{1, 2}

Vladislav Volman,

^{3, 4}

Herbert Levine,

³

Giovanni Pioggia,

²

Danilo De Rossi,

²

and Eshel Ben-Jacob

^{1, 3, ∗}

1

School of Physics & Astronomy, Tel-Aviv University, 69978 Tel-Aviv, Israel

2

Interdepartmental Research Center “E. Piaggio”, University of Pisa, 56125 Pisa, Italy

3

Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, CA 92093-0319, USA

4

Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, La Jolla, CA 92037, USA (Dated: October 6, 2007)

The complex dynamics of intracellular calcium regulates cellular responses to information encoded in extracellular signals. Here, we study the encoding of these external signals in the context of the Li-Rinzel model. We show that by control of biophysical parameters the information can be encoded in amplitude modulation, frequency modulation or mixed (AM and FM) modulation. We briefly discuss the possible implications of this new role of information encoding in the astrocytes.

PACS numbers: Valid PACS appear here

Calcium dynamics play a central role in the regulation of great variety of intra- and inter- cellular processes. Ex- amples range from egg fertilization, the secretion of reg- ulatory molecules, to the control of cardiac contraction.

Consequently, much effort has been devoted to under- stand Ca

²⁺

-related processes, both experimentally and via theoretical modelling [1]. From the perspective of physics, studies of Ca

²⁺

signaling represent a challenging example of an elaborate nonlinear system that exhibits a rich spectrum of events on many temporal and spatial scales. In particular, new findings (made possible due to advanced fluorescence techniques) call for re-examination and refinement of existing models.

In this work, we adopt a dynamical systems approach and utilize the tools of bifurcation theory in order to re- examine the Li-Rinzel (LR) model of calcium dynamics [2]. This model has been used for several cell types, most recently for astrocytes which are a predominant non- neuronal (glial) cell type known to play a crucial role in the regulation of neuronal activity [3, 4]. We found that with a specific choices of biophysical parameters, Ca

²⁺

oscillations can be amplitude modulated, frequency modulated, or both. For the specific case of astrocytes, these results can be crucial for a better understanding of synaptic regulation.

Calcium dynamics is controlled by the interplay of calcium-induced calcium release, a nonlinear amplifica- tion method dependent on the calcium-dependent open- ing of channels to Ca

²⁺

stores such as the endoplasmic reticulum (ER), and the action of active transporters

v

1

6sec

⁻¹

d

1

0.13µM v

2

0.11sec

⁻¹

d

2

1.049µM v

3

0.9µM sec

⁻¹

d

3

0.9434µM C

0

2µM d

5

0.08234µM

c

1

0.185 a

2

0.2µM

⁻¹

sec

⁻¹

K

3

0.1µM

TABLE I:

Parameters used in the original Li-Rinzel model.

(SERCA pumps) which enable a reverse flux. The dy- namical variables of LR model, that is studied here, are the free cytosolic Ca

²⁺

concentration (C), and the frac- tion of open inositol triphosphate (IP

3

) receptor sub- units, h:

C = J ˙

chan

(C, I) + J

leak

(C) − J

pump

(C) (1)

˙h = h

∞

− h τ

h

(2) The dynamics of C is controlled by 3 fluxes, correspond- ing to: 1. a passive leak of Ca

²⁺

from the ER to the cytosol, (J

leak

); 2. an active uptake of Ca

²⁺

into ER, J

pump

, due to the action of the pumps; 3. Ca

²⁺

release (J

chan

) that is mutually gated by Ca

²⁺

and the inositol triphosphate (IP3) concentration, (I):

J

_leak

(C) = v

₂

(C

₀

− (1 + c

₁

) C) J

pump

(C) = v

3

C

²

K

₃²

+ C

²

J

chan

(C, I) = v

1

m

³_∞

n

³_∞

h

³

(C

0

− (1 + c

1

) C) where the gating/inactivation variables and their time- scales are given by:

m

∞

= I I + d

1

n

∞

= C C + d

5

h

∞

= Q

2

Q

2

+ C τ

h

= 1

a

2

(Q

2

+ C) Q

2

= d

2

I + d

1

I + d

3

The level of IP

3

(I) is directly controlled by signals im- pinging on the cell from its external environment. In turn, the level of IP

₃

determines the dynamical behavior of the above model. One can therefore think of the Ca

²⁺

signal as being an encoding of information regarding the

level of IP

3

. We will show that by varying the two key

parameters of the model - K

3

(the affinity of the pump),

and C

0

(the cell-averaged resting Ca

²⁺

concentration),

the information can be encoded in amplitude modula-

tions (AM) of the calcium level, in frequency modulations

(FM), or in both (AFM).

2 The set of biophysical parameters given in Table I, as

typically used for the astrocyte case, corresponds to AM encoding. For these parameters, the phase plane and bi- furcation analysis (fig. 1) reveals that, at I ≈ 0.355µM , limit cycle Ca

²⁺

oscillations emerge through supercriti- cal Hopf bifurcation. From figure 1a it is evident that the amplitude of Ca

²⁺

oscillations increases between the two bifurcation points, while figure 1b shows that the frequency of the oscillations is almost constant - hence the term ”Amplitude Modulation”. Amplitude modu- lation by IP

3

has been observed in many experiments.

However, new findings indicate that, under some con- ditions, the variations (by external stimulation) in the level of IP

3

can also lead to frequency modulation [5, 6].

These observations motivated us to re-examine the LR model, to investigate if changes in the biophysical pa- rameters could lead to frequency-modulated dynamics.

According to bifurcation theory, a nonlinear system can exhibit frequency modulation in the presence of saddle- node bifurcation [7]. The latter describes a transition of a system from an excitable state (in which there are 3 fixed points - stable, unstable and a saddle) to a limit cy- cle. At a certain value of the control parameter I = I

sn

, the stable and saddle fixed points coalesce and the only remaining attractor is a limit cycle. The frequency of the oscillations of the limit cycle can be very sensitive to the distance from the bifurcation point (I-I

sn

in the LR model), whereas the amplitude remains almost constant [8].

We have explored the range of parameters for which the LR system can exhibit a saddle-node bifurcation with the level of IP

3

being the control parameter.We found that K

3

(the affinity of the active SERCA pump) can regulate the switching between the AM and the FM en- coding dynamics. We further discovered the existence of a new dynamical regime in which the variations in the IP

3

are co-encoded both in amplitude and frequency modulations (AF modulation). This AFM dynamics ex- ists for higher levels of the cell-averaged resting Ca

²⁺

concentration C

₀

(table II). We now present a physical picture of these different regimes and then comment on the implications of these results for the specific case of astrocytes. We note that there have been some earlier indications that the LR model can encompass excitable behavior [9, 10], but these works did not present either a complete analysis nor a biophysical picture.

We begin with the well studied and simpler AM dy- namics that corresponds to higher values of K

₃

. As we mentioned before, in this case a limit cycle emerges through a supercritical Hopf bifurcation in which a sin- gle stable fixed point becomes unstable. In fig. 1c we show the nullclines of h and C when the fixed point is unstable. In this case the properties of the limit cycle can simply be understood from linear stability analysis of the unstable fixed point near the bifurcation [7]. In fig. 1d we plot the calcium fluxes (for two different val-

ues of I) as determined by setting h to h

∞

(C); these fluxes capture the fast time scale response of the system close to the fixed point, since the rate of receptor change is slower than that of the Ca

²⁺

concentration. The fixed point occurs at high calcium, where the slope of the efflux J

rel

= J

chan

+ J

leak

is negative. At values of Ca

²⁺

above the unstable fixed point, J

rel

< J

pump

, and there is a re- turn flow in the (h, C) phase plane towards small values of C and vice versa below the fixed point. The instability, then, gives rise to a dynamical flow that does not exhibit any strong positive amplification and instead is due to the slow dynamics. This fixes the frequency to be close to that of the h time delay which does not vary much across the oscillatory regime. Conversely, the amplitude is relatively free to vary and hence the system exhibits amplitude modulation in response to IP

3

variation.

From the above analysis, it is clear that the key to get frequency modulation is to increase the pumping rate so as to make the stable fixed point occur at small calcium, on the rising part of the efflux curve. In fig. 2 we show the dramatic changes in the nature of the bifurcations for lower values of K

3

. First, as shown in figs. 2a,b, the limit cycle has almost constant amplitude yet strong frequency modulation as function of IP

3

. To understand the results we consider the h and C nullclines (shown in fig. 2c) and note that there is a stable fixed point at low levels of Ca

²⁺

and that it is close to a saddle-point.

Inspection of the characteristics of the Ca

²⁺

fluxes shown in fig. 2d reveals that at the saddle-point the slope of the J

rel

characteristic is steeper than that of the J

pump

characteristic. In this situation, a finite yet relatively small deviation away from the stable fixed point crosses the saddle-point separatrix and leads to a large excursion in the phase plane. For this reason, the Hopf bifurcation of the stable fixed point (which still occurs before I

sn

for this set of parameters) must now be subcritical and different deviations (i.e. different values of I) will lead to trajectories with similar amplitudes, as this is determined by the global flow. At the same time, the flow field in the vicinity of the stable fixed point and the saddle-point is very weak, and hence the time of the excursions is very sensitive and can be effectively modulated by the IP

3

. This dynamics shows ”Frequency Modulation”. Thus, we have shown that two classes of Ca

²⁺

dynamics (AM and FM) can be exhibited by astrocytes in responses to IP

3

variations. These two classes are conceptually analogous to classes of neural excitability [7].

Most interestingly, we predict that there can exist a

third class of Ca

²⁺

dynamics which occurs when the level

of C

0

(the cell-averaged resting Ca

²⁺

concentration) is

increased, as is illustrated in the modulation map in fig

3. This new dynamics is the AFM that we have already

mentioned, in which the IP

3

variations modulate both

the amplitude and the frequency of the oscillations. This

dynamical response exists when the Hopf bifurcation is

not yet strongly subcritical but we can nonetheless feel

3 Fixed d

5

= 0.2 µM v

2

= 2 · 10

⁻³

s

⁻¹

K

3

= 0.051 µM

Variable Range Range Range

d

5

[µM] – – 0.107 0.161 0.085 0.127

v

2

[s

⁻¹

] 0.175 0.262 – – 0.178 0.266

K

3

[µM] 0.134 0.201 – – – –

v

3

[µMs

⁻¹

] 0.149 0.595 0.139 0.555 0.144 0.577 C

0

[µM] 2.949 4.424 2.115 3.172 3.037 4.555 TABLE II:

Parameter ranges for the coexistence of amplitude and frequency modulation of Ca²⁺response in Li-Rinzel model.

the influence of saddle-node coalescence. In such a case, the emerging oscillations would show significant variabil- ity both in amplitude and frequency. Technically, the transition from the AM to FM regimes occurs via a char- acteristic codimension-2 bifurcation sequence which com- prises the following steps: first, the lower supercritical Hopf point changes into a subcritical one via Bautin bi- furcation; elsewhere in the parameter space, a cusp bifur- cation generates the SNIC and saddle-node bifurcations which are responsible for variable period oscillations [11].

It follows that the extent of parametric region between the Bautin and cusp bifurcations determines the range of coexistence.

Our prediction regarding the existence of the AFM dy- namics can in principle be directly tested experimentally by using un-caging techniques to directly control the level of IP

3

. For in vivo physiology, IP

3

dynamics itself might be linked to Ca

²⁺

by a chain of biophysical pathways [12], but nonetheless one should expect to see both types of modulation as a function of external drivers of IP

3

; for the astrocyte, this external driver can be for example the rate of synaptic transmission as measured by gluta- mate receptors on the astrocytic process which envelopes the synaptic cleft. In particular, our study implies the possibility that the experimentally observed variability of Ca

²⁺

responses in astrocytes might be due to the in- herent dynamical properties of the cell in addition to the existence of complex feedback loops [13–15] not taken into account in the simple LR scheme.

So far we discussed the ideal deterministic calcium dy- namics. In real cells stochastic fluctuations can play im- portant role and strongly affect the dynamical behavior.

In particular, it has been suggested that many cell types can exhibit spontaneous oscillations even below of the Hopf bifurcation, due to the random openings of clusters of small numbers of IP

3

Rs. Indeed stochastic calcium events occurring in the absence of external stimulation were observed in astrocytes [1, 16, 17]. Since the rate of such openings will depend on the IP

3

concentration, the cell could exhibit a noisy from of FM even without saddle-node structure. This point can also be checked experimentally to test the new picture presented here.

Extending previous studies, we showed the existence of three distinct classes of information-bearing modulations of cellular Ca

²⁺

in response to variations in the IP

3

that

0.01 0.34 0.67 1

0.04 0.23 0.43 0.62

[IP

₃

] [µM]

[Ca

2+

] [ µ M]

0.43 0.57 0.71 10

11.33 12.67 14

[IP

₃

] [µM]

Period [sec]

[Ca

²⁺

] [µM]

inactivation, h

0 0.23 0.47 0.7

0.4 0.67 0.93 1.2

0.010 0.1 1 10

0.4 0.8 1.2

[Ca

²⁺

] [µM]

flux [ µ M/s]

(a) (b)

(c) (d)

FIG. 1: The Li-Rinzel model. (a) Bifurcation diagram for the original set of parameters of the Li-Rinzel model: (−) stable fixed points, (· · · ) unstable ones, (•) stable limit cycles, (◦) unstable ones. Oscillations are born via supercritical Hopf bifurcation at [IP

3

]' 0.355 µM and die via subcritical Hopf bifurcation at [IP

3

]' 0.637 µM. While the amplitude changes, the frequency is nearly constant (b). (c) The nullclines (green for h and orange for C for the case of unstable fixed point. (d) At basal IP

3

levels (' 0.015 µM) J

pump

(red curve) intersects J

rel

(−) at a calcium-level such that J

_rel⁰

< 0. This situation also occurs at higher IP

3

level when J

rel

becomes bell-shaped.

0.010 0.51 1 1.5 0.43

0.87 1.3

[IP

₃

] [µM]

[Ca

2+

] [ µ M]

0.410 0.71 1 1.3 20

40 60

[IP

₃

] [µM]

Period [sec]

[Ca

²⁺

] [µM]

inactivation, h

0 0.17 0.33 0.5

0.6 0.73 0.87 1

0.010 0.1 1 10

0.4 0.8 1.2

[Ca

²⁺

] [µM]

flux [ µ M/s]

(a) (b)

(c) (d)

FIG. 2: An excitable version of the Li-Rinzel model.

(a) Bifurcation diagram and period (b) diagram of an ex- citable version of LR model with K

3

= 0.051 µM. In this case, four bifurcations exist: a saddle-node and a saddle-node on invariant circle (SNIC) at [IP

3

]' 0.479 µM and [IP

3

]' 0.526 µM, and two subcritical Hopf bifurcations at [IP

3

]' 0.51 µM and [IP

3

]' 0.857 µM. (c) Between the two saddle-node bifur- cations nullclines intersect in three points which are a stable focus (•) and an unstable node (¤) separated by a saddle (M).

(d) J

pump

(magenta curve) now intersects J

rel

at lower Ca

²⁺

.

4

0 0.06 0.12

1 3 5

AM FM

AFM

K3 [µM]

C0 [µM]

FIG. 3: Schematic modulation map. The map illustrates the regime of existence of the classes (AM, FM and AFM) of dynamical response. The 3 points correspond to the 3 cases shown in figure 4.

0 120 240 360 480

time [sec]

0 0.5 1 1.5

[Ca2+] [µM]

0 0.5 1 1.5

[Ca2+] [µM]

0 0.25 0.5 0.75

[Ca2+ ] [µM]

(a)

(b)

(c)

(d)

FIG. 4: Different types of excitability. Proper tuning of parameters allows for different Ca

²⁺

-responses for a generic IP

3

-stimulus of the type depicted in (d). The original param- eter set of values (K

3

= 0.1 µM, C

0

= 2 µM) shows (a) am- plitude variability in oscillations that occurs at roughly fixed frequency. (b) An excitable LR model version (K

3

= 0.051 µM, C

0

= 2 µM) displays oscillations with variable frequency but nearly constant amplitude. In spite of the different na- ture of the bifurcations through which oscillations emerge in these two cases, there is a chance of having both AM and FM features by simultaneously adjusting at least two parameters as shown in (c) for K

3

= 0.051 µM and C

0

= 4 µM. Stimulus:

(a) ∆IP

3

= 0.4, 0.1, 0.1, 0.5 µM; (b) ∆IP

3

= 0.4, 0.2, 0.2, 0.5 µM; (c) ∆IP

3

= 0.17, 0.06, 0.06, 0.5 µM.

can be accessed by varying the affinity of a SERCA pump and the cell-averaged resting Ca

²⁺

concentration. AM and FM encoding of information has been extensively studied in the context of communication theory. More recently, mixed A/F modulation (MM) has received re- newed interest as it has advantages in various tasks such as the coordination of informational input from multiple channels. Of course, these systems operate by modulat-

ing a carrier signal which is always active, even when nothing is being transmitted. In contrast, the intracel- lular Ca

²⁺

dynamics uses a principle which we can refer to as bifurcation-based encoding. Here, we mean that the baseline level I

O

is set to be sufficiently close to a bifurcation point so that the variations in IP

3

caused by external signals regularly cross that point. In the AM class, the peak value of calcium encodes the information;

in the FM class, variations in the IP

3

will trigger bursts of Ca

²⁺

spikes (see fig. 4) with information encoded in inter-spike intervals. In the mixed AFM mode, both fea- tures contain information and these can be separately decoded by different downstream effectors with different calcium responses. For the astrocyte, we might expect that the short-time scale effectors (mostly sensitive to the number of pulses) are involved in feedback to the local synapse whereas the long-time scale ones (which integrate the total signal) coordinate information with other astrocytes via intercellular signaling.

The authors would like to thank V. Parpura, G.

Carmignoto, M. Zonta, B. Ermentrout, B. Sautois and N. Raichman for insightful conversations. VV was sup- ported by ICAM Travel Award. This research has been supported in part by the NSF-sponsored Center for The- oretical Biological Physics (grant nos. PHY-0216576 and PHY-0225630) and by the Tauber Fund at Tel-Aviv Uni- versity.

∗

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