THE EFFECT OF SYNAPTIC DEPRESSION ON MODEL INHIBITORY … · phenomenological models [2], [36] have...

CANADIAN APPLIEDMATHEMATICS QUARTERLYVolume 10, Number 1, Spring 2003

THE EFFECT OF SYNAPTIC DEPRESSION ON

MODEL INHIBITORY NETWORKS

Based on an invited presentation at the annual meeting of the CanadianApplied and Industrial Mathematics Society/Societe canadienne demathematiques appliquees et industrielles, Victoria, BC, June 2001.

J. GRIGULL AND F. K. SKINNER

ABSTRACT. Networks of inhibitory interneurons in thecortex generate synchronous rhythmic oscillations which are be-lieved to critically control brain output. The temporal courseof the inhibitory postsynaptic potentials is known to affect thissynchronous activity in a variety of ways, in particular its stabil-ity and frequency. Here we investigate theoretically the effectsof frequency-dependent synaptic inhibition on network dynam-ics. We show how this short-term synaptic plasticity in the formof synaptic depression confers stimulus-sensitivity on the net-work, and creates transition regimes between synchronous andasynchronous oscillatory patterns in which bursting patternsand bistability occurs. In this manner, inhibitory networks canadd further dimensions to the control of the dynamical patternsproduced by the brain.

1 Introduction. It has been several decades since rhythmic, elec-trical brain activities have been recorded at the scalp in the form of anelectroencephalogram (EEG). These activities result from interactionsoccurring at multiple brain networks that consist of organized assem-blies of neurons, and their observation indicates a significant degree ofcoherence between large populations of neurons. Furthermore, thesemacroscopic brain rhythms can be quantified and related to behaviouralstates via their frequency band [33]. However, it remains to be deter-mined how these oscillations arise.

Accepted for publication on September 20, 2001.AMS subject classification: 92C20.

Copyright c©Applied Mathematics Institute, University of Alberta.

87

88 J. GRIGULL AND F. K. SKINNER

There are two major types of neurons in the brain: principal, exci-tatory cells and GABAergic, inhibitory cells or interneurons. Togetherwith input fibers and a modulatory system, they form networks thatproduce the electrical signals recorded in EEGs [32]. It is known thatinhibitory cells play key roles in shaping the output of the excitatoryneuronal populations and thus in generating brain rhythms [5], [9], [23],[45]. Although the inhibitory cells represent a small fraction of braincells (about 10-20%), they have extensive axon arbors and are thus ableto exert strong and distributed inhibition onto excitatory cells. In thisway, they can provide precise temporal structure by phase locking thefiring of multiple pyramidal cells via synchronously depressing their ac-tivities [5].

The presence of short-term synaptic plasticity (or the fact that synap-tic transmission is not mediated by static processes of fixed strength)between neurons in several brain regions has been known for some timeand is well-established (e.g., [12], [14], [28], [19], [39]). In recent times,short-term synaptic plasticity has been shown to play essential rolesin the normal functioning of the synapse during sensory and motorprograms [10], [11], [15], [26], [38], [46]. There are many details toconsider in trying to understand the specific ways in which synapticplasticity might be manifest. For example, synaptic depression, a formof short-term plasticity, can be due to a depletion of synaptic vesiclesand/or a desensitization of postsynaptic receptors [17]. However, recentphenomenological models [2], [36] have been able to capture the essenceof synaptic transmission between pairs of neurons. In particular, thefrequency dependence of the synapse, or the fact that the amount ofdepression and facilitation seen in the postsynaptic response dependson the particular frequency or pattern of the presynaptic stimulus, hasbeen modelled. These phenomenological models have provided insightinto how such dynamic synapses might contribute functionally to neuralcoding and sensory-motor programs (e.g., [2], [6], [24], [34], [36]).

In this paper we investigate the effects of synaptic depression on thedynamics of inhibitory networks using single self-inhibited and two-cellphysiologically-based inhibitory network models. We use both simula-tions and a heuristic-analytical approach to examine the network dy-namics.

2 Model descriptions. We consider a single-compartment modelof interneurons that has been formulated recently to replicate salientproperties of hippocampal interneurons [43]. We use self-inhibitory

SYNAPTIC DEPRESSION IN INHIBITORY NETWORKS 89

single-cell networks (as a representative of a synchronously firing popu-lation) and two-cell mildly heterogeneous networks.

2.1 Interneuron model. The interneuron model below [8], [43] wasdeveloped by these authors to investigate the stability of synchronousoscillations under mild heterogeneity in the external drive, Iapp . Fromhere on we refer to the equations given below (which use a Hodgkin-Huxley formalism [16]) as the White model.

CdVi

dt= Iapp − gNam3

∞h(Vi − VNa) − gKn4(Vi − VK)

− gL(Vi − VL) − Is

(1)

dh

dt=

h∞(V ) − h

τh(V )

dn

dt=

n∞(V ) − n

τn(V )

The kinetic functions of activation and inactivation are given by

m∞(V ) =1

1 + exp(

−0.08(v + 26))

h∞(V ) =1

1 + exp(

0.13(v + 38))

τh(V ) =0.6

1 + exp(

−0.12(v + 67))

n∞(V ) =1

1 + exp(

−0.045(v + 10))

τn(V ) = 0.5 +2

1 + exp(

0.045(v − 50))

where C is the capacitance, Vi is the membrane voltage of cell i, gNa ,gK , gL are the maximal conductances of sodium (Na), potassium (K)and leak (L) currents respectively, n is the activation of the K current,h is the inactivation of the Na current (m, the activation of the Nacurrent is assumed to be significantly faster than the K activation sothat its steady state value m∞ is used), m∞, n∞, h∞ are the steady-state values, τh, τn are the time constants for the Na inactivation and Kactivation respectively, and Is is the synaptic current (described below).


100

200

0 2.5 5

A

f

Iapp

B

FIGURE 1: (A) Firing frequency (f in Hz) versus external drive (Iapp

in µA / cm2) for the White model. (B) Schematic of the synaptic de-pression model.

The fixed parameters are

gNa = 30 mS / cm2 gK = 20 mS / cm2 gL = 0.1 mS / cm2

VNa = 45 mV VK = −80 mV VL = −60 mV C = 1µF/ cm2,

where Iapp is the applied current or external drive and is varied in oursimulations. Figure 1(A) shows how the frequency of the individualneuron changes with Iapp in this model. We also used the interneuronmodel developed by [40], and obtained similar results (not shown here).

2.2 Synaptic depression model. The inhibitory synaptic current Is

has the form Is = gS(V − Vs), where Vs is the synaptic reversal poten-tial, taken as −75 mV, g is the maximal synaptic conductance and S isthe synaptic gating variable. To incorporate synaptic depression in thenetwork we use the model developed in [20], [22], [36]. From here on,this synaptic current model (as given below) will be referred to as theT-M model.

dS

dt= USEF (Vj)R −

S

τs(2)

dR

dt=

1 − S − R

τD− USEF (Vj)R(3)


where Vj is the voltage of the presynaptic neuron. The synaptic kineticfunction F is given by

F (Vj) =1

1 + exp(−Vj)

as used in [43]. Without more explicit notation such as S2→1, the ex-pressions S1, R1, USE,1 and τD,1 refer to the kinetic variables and pa-rameters of the synapse that terminates on neuron 1, while S2, R2, USE,2

and τD,2 refer to the synapse terminating on neuron 2.—Implicit in theabove scheme is an activation rate constant α that is chosen to be equalto 1, as in [43]. As illustrated in Figure 1(B) (for a single self-inhibitednetwork), the variable R represents the synaptic resource in its recov-ered state, and S is the synaptic gating variable, termed the active oreffective synaptic resource by Markram and Tsodyks. USE stands forutilization of synaptic efficacy and is taken as a parameter here (for itstreatment as a continuous variable in the case of synaptic facilitationsee e.g. [37]). USE can be interpreted as the neurotransmitter releaseprobability, if synaptic depression is a presynaptic process.

1/τs is the rate of postsynaptic decay, while 1/τD is the rate of re-covery from synaptic depression. For τD ≈ 0, R instantaneously equates1 − S, once the second term on the righthand side of equation (3) van-ishes after the passing of the presynaptic spike-peak. In this case equa-tions (2), (3) effectively reduce to the standard equation for synaptickinetics used in [43]: dS

dt = USEF (Vj)(1− S)− Sτs

. Accordingly, τD ≈ 0represents the non-depressing synapse.

Table 1 displays kinetic parameters for inhibitory and depressingGABAA synapses previously used in inhibitory models and found ex-perimentally at GABAergic synapses. (Note: For the [14] experimentaldata, we have only included F2 type (i.e., depressing) synapse numberswhich these authors found to be the main type and are the ones rele-vant here. Also, two subtypes of GABAergic synapses—GABAA,fast andGABAA,slow—have been identified in electrophysiological studies on thehippocampus [27]). We explore similar parameter values for Iapp , g andτs to those used by [43] to allow comparison.

3 Results. Previous modelling work by Kopell and colleagues [8],[7], [43] has shown that the frequency at which inhibitory networks syn-chronize depends in distinct ways on intrinsic and synaptic time con-stants. To summarize this and other works [42], [40], [44]:

1) The network can synchronize if the time constant of postsynaptic


α USE g τD τs = 1/β

[8] 1 ms−1 1 0–2 mS / cm2 ' 0 ms 5–50 ms[43][40] 12 ms−1 1 0.1 mS / cm2 ' 0 ms 0–30 msM

OD

EL

[20] − 0.1–0.95 − 200–800 ms ≈ 10 ms[14] 1 ms−1 0.25 0.49 706 8.3

±0.13 ±0.41nS ±405 ms ±2.2 ms[3] − − − − ≈ 10–50 ms[4] − − − 0.63 sec ≈ 2 ms[18] − − − 2 sec 2, 9 msE

XPER

IMEN

T

TABLE 1: Model and Experimental Synaptic Parameters.

inhibitory decay, τs, exceeds the intrinsic oscillatory period of theindividual neuron;

2) The synchrony is stable with respect to (mildly) heterogeneous in-trinsic oscillatory periods up to a maximal network period T , if Tscales linearly with τs;

3) The synchrony can hold in a γ frequency (≈ 40 Hz, associated withhigh cognitive processing) realm with heterogeneity; at low frequen-cies the temporal dispersion of individual spikes due to input hetero-geneity tends to break up any existing synchrony;

4) Regimes can be identified that are predictive of the amount of coher-ent activity in a heterogeneous network: the “tonic” regime, in whichinhibition is weak relative to the excitatory drive to the cells, and the“phasic” regime where T scales linearly with τs and where inhibitioncan overcome excitation and lead to suppression of activity.

3.1 Single-cell networks: A heuristic-analytical approach.

Chow and colleagues [8] showed that inhibitory network behaviour couldbe analytically understood in single self-inhibited cells via a frequencycontrol equation (FCE). They derived this equation from an integrate-&-fire model, adapted to the White model, thus preserving the physio-logical character of the period T as a function of intrinsic and synapticparameters. We modify the FCE by making the synaptic conductance,g, a function of T and τD and use the modified FCE as a tool to analyze


the frequency control of the self-inhibitory network with a depressingsynapse.

Assume that the self-inhibitory unit fires periodically. In this equilib-rium state the accumulated postsynaptic current at a depressing synapseis constant over successive interspike intervals. We use the FCE, equa-tion (29) of [8],

(4) Iapp

(

1 − exp(−T ))

− 1 = g(T )τs

τs − 1

(

exp(−T /τs) − exp(−T ))

and modify it with an expression for the rate-dependent amplitude ofthe asymptotic postsynaptic current developed by Abbott and colleagues[2], [25]:

(5) g(T ) = g01 − exp(−T /τD)

1 − d exp(−T /τD).

In the above equations Iapp , g, τs, τD and T—the firing period of theneuron—are rescaled parameters to fit an integrate-&-fire model of thephysiological neuron. We use symbols with tilde’s to denote the scaledparameters and symbols without tilde’s to represent the White modelphysiological parameters (i.e., no scaling), to avoid confusion when re-ferring to the analytical or numerical computations in this paper. (Note:In [8] their symbols with tilde’s refer to the unscaled parameters (i.e.,physiological model parameters).) g0 stands for the maximal synapticconductance. - d ∈ [0, 1] modulates the extent to which the recovery rateof a depressing synapse τD influences the FCE (no influence for d = 1,maximal influence for d = 0).

If we assume that τD � T and d = 0, then we can approximate theabove two equations to obtain a modified FCE (mFCE):

(6) Iapp

(

1 − exp(−T ))

− 1 = g0τs

τs − 1

T

τD

(


.

Let Iapp , g, τs, τD and T denote the parameters and period using theWhite model, equation (1), i.e., the unscaled model physiological param-eters. Then the rescaling scheme (as done in [8]) for the approximationby the FCE reads

Iapp =Iapp + Ir

ITg0 =

g0

gT(7)

τs =τs

τmτD =

τD

τmT =

T

τm.(8)


To study the impact of synaptic depression on frequency control in themFCE, equation (6), we choose parameter values Ir = 2, IT = 1.5,τm = 12, gT = 0.1 in the rescaling scheme above. These values aresimilar to those selected in Figures 2A–C of [8]. Here, the mFCE servesan illustrative and heuristic purpose—to get a first idea of the impactof synaptic depression on frequency control in the inhibitory network.Therefore, only a qualitative correspondence to the periods obtainedusing the White model should be expected. Choosing Iapp = 1 µA / cm2,g0 = 1 mS / cm2, and neglecting the term τs

τs−1 for simplicity, we obtain:

r(T ) = 10 ·T

τD·(


(9)

l(T ) = 1 − 2 exp(−T )(10)

as the right- and lefthand side of equation (6), respectively. The inter-section point of l(·) and r(·) is the period given by the mFCE.

3.2 Single self-inhibitory network exhibits two regimes. We nowmake use of the framework developed above. The solutions of T derivedfrom the mFCE, equation (6), imply that synaptic depression imposes atransition in the dependency of the firing period on the synaptic decaytime constant. Figure 2 shows the period, T , versus the synaptic decaytime constant, τs. Upon introducing and increasing the synaptic de-pression time constant, τD, the linear (“phasic”) relation splits up intoan almost constant domain and a linear domain; for large values of τD

the relationship is almost constant (“tonic”) over the whole range of τs.“Phasic” and “tonic” are used here to keep consistent with the termi-nology introduced by [43]. Note that the mFCE predicts the existenceof a bistable regime (for τD = 15 in Figure 2), in which two solutionscoexist, one of which is sensitive with respect to changes in τs, whereasthe other one is not. This is illustrated later using the White model.

This division of regimes can be explained by considering that theeffect of an increase in τs on the period T is twofold: Let R be thesynaptic reservoir variable averaged over the period T and choose τ ′

s >τs. If, in a first step, T1 denotes the period associated with τ ′

s andR, we clearly have T1 > T by prolonged inhibition. The time T1 −

T permits the synapse to recover more from depression, leading to areservoir variable R1 > R. This, in turn, enhances the inhibition beyondthe effect implied by τ ′

s > τs alone: If T2 denotes the period resultingfrom both effects of an increase in τs, we obtain: T2 > T1 > T . In thisrelation, the right-hand inequality represents the prolonged inhibition,


4

8

12

2 4 6

4

8

12

2 4 6

4

8

12

2 4 6

T~

τ~s

τD=40~

=15Dτ~

D=10τ~

FIGURE 2: Period (T ) versus synaptic decay time constant (τs) re-lationship with changing synaptic depression (in rescaled parameters)according to equations (9) and (10). The solutions T , as dependent onτs, are shown for three different values of τD, as indicated: For τD = 10(τD ≈ 100ms) the τs− T relation changes from being almost constant toa linear dependency upon increasing τs. For τD = 15 (τD ≈ 200ms), theintersection points of r(·) and l(·) lie on an S-shaped curve. For τD = 40(τD ≈ 500ms), the period is independent of τs over the physiologicalrange (τs < 70ms).


while the left-hand inequality represents the recruitment of inhibition. Inthe almost constant (“tonic”) domain both effects T → T1 and T1 → T2

are negligible. At moderate values of τs, however, the synapse recoverssufficiently so as to make the effect of any further increases in τs beingfelt by the postsynaptic neuron: T becomes sensitive to τs and the τs/T -relation enters the linear (“phasic”) domain.

3.3 Sensitivity profiles change with synaptic depression. Thesensitivity of a model neuron with respect to the applied current is usu-ally represented by the stimulus-frequency diagram. Model neurons areclassified as either Type I or Type II, depending on whether the periodicactivity starts with zero (saddle-node bifurcation) or positive (Hopf bi-furcation) frequency upon increasing the external stimulus, Iapp , beyondthe threshold of action potential initiation [30]. The White model usedhere is of Type I. To capture adaptation and accommodation, the func-tional corollaries of synaptic depression, we use the stimulus-elasticityof the frequency (f), which we simply term the elasticity, defined as:

(11) Elas(f, Iapp) =df

dIapp

Iapp

f.

In other words, maximal elasticity values imply that the largest fre-quency response is obtained for a minimal stimulus change. The elastic-ity is closely related to variables used in psychophysical descriptions ofperceptual adaptation, such as the Weber-Fechner law and this signifi-cance has been noted [2].

We now use our mFCE described above to investigate the stimulus-sensitivity of the frequency. We choose Ir = 0.7 (in the rescaling scheme,other parameters the same) and rewrite equation (6) as:

Iapp + 0.7

1.5·(

1 − exp(−T ))

− 10 ·T

τD·(


− 1 ≡ 0.

(12)

Here all parameters except Iapp are rescaled parameters of the mFCE

(equation (6)). It is easy to see that dfdIapp

Iapp

f = −dT

dIapp

Iapp

T holds for

f = 1000/T . Using the implicit function theorem we obtain:

Elas(f, Iapp) =2Iapp ·

(

1−exp(−T ))

3T (exp(−T )·(

2(Iapp+0.7)

3 + 10τD

(1−T ))

+exp(−T /τs)·(

10/τD ·(T /τs−1)) .


Elasticity tells us about the profile of sensitivity of a particular system(in our case, inhibitory networks) over a range of incoming stimuli. Us-ing our mFCE we can investigate the dependence of the elasticity profileon τD. The time constant τD can be thought of as a phenomenologicalcontrol parameter for synaptic depression: The higher the time constant,the lower the steady-state postsynaptic current. Using equation (3.3),we illustrate in Figure 3 what the mFCE predicts for the dependence ofthe sensitivity on τD. It can be seen that τD “allocates” the maximallysensitive response of the neuron to different regions of the external cur-rent. For low τD (≈ modest synaptic depression) the sensitivity peaksat high values of Iapp—in this range equation (12) determines a solution

of about T = 2. In the employed rescaling scheme this corresponds to afrequency of approximately 40 Hz. For higher τD, e.g., for a doubling ofτD (meaning more synaptic depression), the sensitivity peaks at lowervalue of Iapp . At this peak solutions of about T ≈ 2.7, which correspondto a period of ≈ 30 Hz after rescaling are obtained.

3.4 Synaptic depression confers network stimulus-sensitivity.

Let us now return to the White model to examine the effect of synapticdepression. For any given exterior current Iapp above firing thresholdthe frequency of the self-inhibitory neuron is reduced as compared withthe free neuron (Figure 4(A)). Note that self-inhibition changes the con-cave stimulus-frequency (Iapp − f) relation into a nearly linear one. Ifsynaptic depression is now included in the synapse, a range of convexityis introduced into the stimulus-frequency relation (Figure 4(A) ii)–iv).This is because at low input currents and low frequencies the inhibitorysynapse recovers completely between two subsequent spikes, and a largetime constant τs prevents incremental increases in Iapp from being feltby the neuron. At higher values of the applied current Iapp furtherincreases enhance the frequency as a result of two effects:

1) the higher excitatory drive due to Iapp ;2) the reduction of the self-inhibition induced by any individual spike in

a train due to synaptic depression—that is, the inhibitory synapticimpact flattens out at an asymptotic value.

The neuron enters the convex realm in the Iapp−f curve, as decreases inIs sufficient to release the neuron from inhibition are transferred from thetail of the inhibitory current’s decay curve to its steeper parts by synapticdepression. Thus, synaptic depression endows the self-inhibitory neuronwith a high stimulus elasticity (as defined by equation (11)).


0

2

4

6

8

1 1.5 2 2.5app

=10D

τ

=20D

τ~

~

)appElas(f,I

I

FIGURE 3: Stimulus-sensitivity changes with synaptic depression. Us-ing a heuristic-analytic approach, the elasticity, Elas(f, Iapp), is deter-mined for two different synaptic depression time constants. τs = 2(τs ≈ 25ms) and τD = 10 or 20 (τD ≈ 120 or 250ms, respectively).


50

1000 100 200 300 400 500

30

60

90

120100

200

0 2.5 5

free

f

τ =50s

ττD=500

τs

τD

Iapp

τD=200τs =10

T

=50=200τD

τs =50s

τ =10s

B

i) ii)

iii) iv)

A

FIGURE 4: Transition domains in the self-inhibitory unit using theWhite model. (A) Firing frequency (f in Hz) versus injected current(Iapp in µA/ cm2). i) without synaptic depression; free: without inhibi-tion (g = 0mS / cm2); with self-inhibition: g = 1mS / cm2, USE = 0.5;τs = 10ms and τs = 50ms. ii–iv) with self-inhibition and synapticdepression for g = 1mS / cm2 and USE = 0.5 and parameter values forτs and τD in ms as indicated. (B) Transition domains for variationsin synaptic decay (τs) and depression (τD) time constants in the self-inhibitory unit. Changes in the period, T , are plotted versus the synap-tic time constants (both in ms). Notice the presence of almost constantand linear domains as predicted using a heuristic-analytical approach(see Figure 2). Other parameters are Iapp = 2µA/ cm2, g = 1mS / cm2,USE = 0.5.


0

1

210 20 30 40 50

2

4

6

8

10

0

1

20 100 200 300 400 500

2

4

6

8

Dτ

)appElas(f,I )appElas(f,I

τs

)appk*dln(I )appk*dln(I

A B

FIGURE 5: Elasticity in the self-inhibitory unit using the White model.(A) Dependence of the elasticity profile on τD(ms). The y-axis shows

Iapp on a logarithmic scale, where the increment dln(Iapp) :=d ln(Iapp )

dIappis

fixed at 0.25, i.e. each point represents an increase in Iapp by 25% of thepreceding value. Iapp begins at the bifurcation value of spike initiation,

which is set to 1 in determining the increments I(k+1)app −Ik

app logarithmi-cally. Elas(f, Iapp) is approximated by 4 ·

(

f(Ik+1app ) − f(Ik

app))

/f(Ikapp),

where Ikapp adopts the values −0.62, −0.37, −0.06, 0.33, 0.82, 1.43, 2.19,

3.14, 4.33µA/ cm2 successively for k = 0 . . . 8 and I9app = 5.82µA/ cm2.

Other parameters are g = 1mS / cm2, τs = 50ms, USE = 0.5. (B) De-pendence of the elasticity profile on τs(ms). Iapp is increased as in (A).Other parameters are g = 1 mS / cm2, τD = 200ms and USE = 0.5.


Figure 4(B) displays the dependence of the firing period T on bothsynaptic time constants, τs and τD, in our model simulations. Withoutsynaptic depression, i.e. for τD ≈ 0, the τs − T relationship is linear.Upon introducing synaptic depression into the model, i.e. choosing τD

equal to several hundred msec, the relation splits up into two separatedomains of almost constant and linear relationships as predicted usinga heuristic-analytic approach (see above). In essence, the reason forthese different domains is due to the twofold effect described earlier.For example, a smaller τD allows a larger R to develop in the given timewhich in turn allows a larger inhibitory current to be produced and tobe able to affect network period (i.e., a linear τs − T relationship forlower τD values).

The elasticity of the neuron with a depressing self-inhibitory synapseis shown in the sensitivity profiles over a range of τs and τD values (inFigures 5(A) and (B)). The peak in the back of the figure is due to athreshold phenomenon for the type-I neuron. The simulations indicatethat with synaptic depression, i.e. for τD > 0, a second peak emerges inthe Elas(f, Iapp) profiles. In Figure 5(A), we see that for low values of τD

this peak is located at high values of Iapp and vice versa, as indicatedusing the heuristic-analytical approach (see Figure 3). For fixed τD,Figure 5(B) shows that an increase in τs allocates the elasticity peak toregions of higher Iapp . The elasticity peaks reflect the changing shapes ofthe Iapp−f curve in Figure 4(A). Consider a fixed intrinsic frequency, asset by Iapp in our model. Then we see that a transition from an almostconstant to a linear τs − T relationship occurs, as synaptic depressionis introduced into the model. Specifically, Figure 4(B) shows that thethreshold value in τs, beyond which the period T depends linearly on τs,varies with the time constant τD. The latter determines the durationand thus the severity of synaptic depression. If τs is held fixed and Iapp

allowed to vary, the Iapp − f curve assumes an S-like shape producingan elasticity peak for intermediate Iapp values. As Figures 4(A) (iii),(iv) and 5(A) show, this elasticity peak shifts to lower values of Iapp forhigher synaptic depression, because the domain of frequency over whichsynaptic depression weakens inhibition is reached earlier with highervalues of τD. A similar reasoning explains the rightward shift of theelasticity for higher values of τs which is displayed in Figures 4(A) (ii),(iii) and 5(B). These results clearly demonstrate that synaptic depressionendows the neuron with a higher overall sensitivity that can be fine-regulated or allocated to different stimuli realms by a proper choice oftemporal kinetic parameters.


3.5 Linking to two-cell heterogeneous networks. [43] previouslyshowed that consideration of the single self-inhibited unit is predictiveof the stability of synchronous oscillations in mildly heterogeneous net-works (defined as <≈ 5% intrinsic frequency differences). Simulations oftwo-cell mutually inhibitory networks with mild heterogeneity indicatesthat predictive information from the single cell network is still possible,but in a different way.

The immediate effect of synaptic depression consists in diminishingthe inhibition and is equivalent to mapping g-values from the phasicrealm of the unit without synaptic depression towards low, i.e., “tonic”,values for the unit with synaptic depression. Figures 6(A) and (B) can beseen to constitute a phasic relationship between τs and T for a mildly het-erogeneous two-cell network with synaptic depression. In Figure 6(C),an alternating bursting dynamical behaviour emerges, and Figure 6(D)shows one cell suppressing the other. Figure 6(E) displays the realm ofsynchronous activity in the mildly heterogeneous two-cell network overthe same range of parameter values for g, τs, τD and Iapp as in Fig-ure 4(B) (where self-inhibitory network simulations were done with theWhite model). The linear domain (“phasic”) of the self-inhibitory unitcorresponds roughly to the domain of suppression in Figure 6(E) whereasthe almost constant domain (“tonic”) corresponds to synchronous activ-ity in the two-cell network. This is surprising only on the first view: Ag-value, whose transform under equation (5) lies in the phasic domain ofthe self-inhibitory unit using the White model, can be associated withsynchronous activity in the two-cell network with synaptic depression (asin Figures 6(A) and (B)). The almost constant domain in Figure 4(B) isindeed indicative of a regime in which the only heterogeneity of both neu-rons is given by Iapp,1, Iapp,2. The linear domain in the τs/T -relationshipof the self-inhibitory unit with depression, by comparison, is a domainof added heterogeneity in that the dynamics of the inhibitory currentbecomes significant in the frequency control. In other words, the linearpart in the τs − T -relationship represents a domain of higher sensitiv-ity with respect to τs, compared to the almost-constant domain. Thus,the heterogeneity in Iapp,1, Iapp,2 is mild only with respect to a givenfrequency. The twofold effect of an increase in τs, as discussed above,transfers the absolute difference Iapp,1 − Iapp,2 towards lower activitylevels of both neurons, where the same difference entails largely differ-ent relative effects on both neurons and destabilizes synchrony (whensynaptic depression is present). This explains why the linear domainseen with the self-inhibitory unit corresponds approximately to a do-main of suppression and the almost-constant domain to synchronous


activity.

Therefore, the informative value of the self-inhibitory unit with synap-tic depression lies in the prediction and localization of a transition do-main with enhanced sensitivity, that separates domains of synchronyand suppression in the heterogeneous two-cell network with synapticdepression.

3.6 Alternating bursting and bistability with synaptic depres-

sion. The heuristic-analytic approach employed above indicates that aregion of bistability can occur in our inhibitory network models withsynaptic depression. Furthermore, in linking an understanding of thesingle self-inhibited unit with two-cell networks above (Figure 6), wefind that alternating bursting patterns can occur. We now illustratethis more fully using the White model.

Heterogeneity, as modeled by Iapp,1 ≈ Iapp,2, in which we have fixedand low absolute differences Iapp,1−Iapp,2 is assumed. With this hetero-geneity, we find that network dynamics in which one cell tonically firesand suppresses the other (see Figure 6(D)) gives way to synchronous fir-ing (see Figures 6(A) and (B)) at higher values of Iapp,1 and Iapp,2 sincethere is now a smaller difference in intrinsic frequencies. This is shownin Figure 7. We further find that the transition between synchronousfiring and suppression is bistable, as illustrated in Figure 7, where ‘I’refers to dynamical behaviour in which one cell fires and suppresses thatother, ‘II’ refers to synchronous firing, and ‘III’ refers to an alternatingbursting pattern (also shown in Figure 6(C)).

This bistability gives rise to a frequency transition zone (FTZ) whichis characterized as follows: Upon increasing (Iapp,1, Iapp,2), the networkpasses through a zone of alternating bursting dynamics before switchingto synchrony (Figure 7 (i)), whereas the reversed paradigm of decreasing(Iapp,1, Iapp,2) values induces the network to switch directly from syn-chrony to suppression (Figure 7 (ii)). The FTZ is given by those pairs(Iapp,1, Iapp,2) for which the entries in the diagrams of Figure 7 (i) and7 (ii) differ. The bistability is shown in the voltage versus time plots ofFigure 7 (iii) where ‘b’ and ‘d’ patterns derive from the same parameterset. It is interesting to note that with synaptic depression the alternat-ing bursting pattern occurs in the direction of increasing rather thandecreasing values of the excitation parameter Iapp . By this, synapticdepression endows the inhibitory network with a desynchronizing mech-anism in its output, that is effective for increasing excitation levels, butit tends to conserve an already existent network synchrony with respectto decreases in the excitation level.


FIGURE 6: Two-cell network with mild heterogeneity and synapticdepression using the White model: Dependence of the synchrony anddynamics on synaptic time constants. (A)–(D): Voltage traces V (inmV) vs. time (in ms) for τs = 10, 30, 50, 70ms respectively; (E) dis-plays the period T (in ms) of the two-cell network and the type ofdynamics along the two time constants τD(ms) and τs(ms). Darkcolumns represent phase-locked synchronous firing of both neurons asin (A) and (B), shaded columns represent alternating bursting as in(C), and white columns stand for lack of synchrony or suppression as in(D). Other parameters are: Iapp,1 = 2.1µA / cm2, Iapp,2 = 2 µA / cm2,g = 1 mS / cm2, USE = 0.5 and τD = 300ms in (A)–(E).


FIGURE 7: Bistability in the two-cell network with synaptic depression.Using the White model with synaptic depression; g = 1.5mS / cm2,USE = 0.5, τD = 300ms. Throughout we set Iapp,2 = Iapp,1 −

0.1µA / cm2, where Iapp,1 is shown on the x-axis and τs varies from10 to 50ms on the y-axis in (i)–(ii). (i) Iapp,1 and Iapp,2 are enhancedby 0.5µA / cm2 every 2000ms. (ii) Iapp,1 and Iapp,2 are diminished by0.5µA / cm2 every 2000ms. In both cases the dynamical behaviour isrecorded after transients. (iii) Voltage (mV) versus time (ms) plots ofboth neurons for the Iapp -path shown below and indicated by a, b, c

and its reverse c, d, e in (i) and (ii), respectively, τs = 20 ms. Boxedregions illustrate the various patterns after transients have died away.


4 Discussion. We have investigated the impact of synaptic depres-sion on the synchronicity and firing frequency in self-inhibited single celland mutually inhibitory two-cell network models with mild heterogene-ity. With the self-inhibited single cell network model (also represent-ing a homogeneous network of synchronized cells), we showed that thesensitivity of the network (i.e., its elasticity as determined from thef − Iapp curve) can be understood by the two temporal determinantsof the synaptic kinetics, τs and τD. Specifically, as the time to recov-ery from synaptic depression increases, the required input stimulus toinvoke maximal sensitivity decreases. We also showed that the singlecell network behaviour predicts for two-cell networks the approximatelocation of where dynamical patterns would switch from synchronous tosuppressive behaviour and where bistability with more complex dynam-ics, alternating bursting, could occur. In other work [13], we describethis alternating bursting pattern more fully as well as the situation ofnon-mild heterogeneity.

Work by [2], [36] shows that synaptic depression provides a mech-anism by which a neuron can become sensitive to a wider frequencyrange of input afferents. In other words, depressing synapses can de-tect changes in presynaptic frequency over several orders of magnitudeeffectively due to its frequency-dependent responses. This aspect is cap-tured in our study as the elasticity of frequency with respect to stimulus.Indeed, we would like to suggest that the observed dependence of theelasticity profile on the time constants of synaptic kinetics, τs and τD,functionally “allocates” inhibitory networks with different synaptic ki-netics as detectors to different excitation levels. One such inhibitory net-work detects a change in the excitation level of its maximal sensitivity bylosing synchrony at particular frequencies, another one—with a differ-ent combination of τD, τs—does the same at a different excitation level.This elasticity or sensitivity allocation may complement the organizingprinciples of GABAergic interneurons in terms of frequency-dependentsynaptic kinetics recently found by [14] and termed “GABA-groups”.

Temporal coherence (or synchrony) is a well-recognized brain activitythat possibly contributes to neuronal coding [29]. Given the myriad ofintrinsic and synaptic heterogeneities and details that brain networkshave, the mechanisms of generating this coherence or synchronous firingare likely to be complex and diverse. Theoretical mechanisms deter-mined in smaller networks can be helpful in understanding dynamicalbehaviours in larger networks. For example, the seminal work by [41]that used a two-cell mutually inhibitory network to show that synchronywas possible in purely inhibitory networks, given appropriate balances


between intrinsic and synaptic properties has been invoked in largernetwork studies such as [35].

4.1 Closing remarks and future work. Synaptic plasticity is thebasis of several modelling studies addressing learning and memory inneuronal networks [1], and there is a recognized need for stable activity.Interactions between short and long-term plasticity have been shownto be present [21], [31], [46] so that studies such as these become evenmore significant for an understanding of information coding in the brain.Furthermore, the observed desynchronizing effect of the burst-generatingdynamics (Figures 6 and 7) deserves closer attention in a future study,focusing on the possible protective effects of inhibitory networks withsynaptic depression against epileptiform activity elicited by rising exci-tation levels.

Acknowledgements Supported by CFI/ORDCF, CIHR, NSERC, andDCIEM of Canada. F.K.S. is an MRC Scholar and a CFI Researcher.

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Toronto Western Research Institute, University Health Network,

Depts. of Medicine (Neurology), Physiology, and Institute of

Biomaterials and Biomedical Engineering, University of Toronto,

Toronto, Ontario, Canada

E-mail address: [email protected]

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