PHYSICAL REVIEW E 84, 031935 (2011)

Renewal theory of coupled neuronal pools: Stable states and slow trajectories

Christian LeiboldDepartment Biologie II, LMU Munich, Großhadernerstrasse 2, D-82152 Planegg, Germany and

Bernstein Center for Computational Neuroscience Munich, Großhadernerstrasse 2, D-82152 Planegg, Germany(Received 15 April 2011; revised manuscript received 24 July 2011; published 30 September 2011)

A theory is provided to analyze the dynamics of delay-coupled pools of spiking neurons based on stabilityanalysis of stationary firing. Transitions between stable and unstable regimes can be predicted by bifurcationanalysis of the underlying integral dynamics. Close to the bifurcation point the network exhibits slowly changingactivities and allows for slow collective phenomena like continuous attractors.

DOI: 10.1103/PhysRevE.84.031935 PACS number(s): 87.19.lj, 87.18.Sn, 02.30.Ks

I. INTRODUCTION

Although information processing in biological neuronalnetworks is often linked to exquisitely timed individual spikes[1–3], cognitive phenomena, such as short-term memoryand decision making, are also reflected by slowly changingneuronal firing rates [4,5]. To act as such rate encoders, neuronshave to be able to gradually and persistently elevate theirfiring rate over some period of time. The mechanisms thatwere identified to underlie such ongoing firing are manifold.They reach from specific ion channels in the cell membrane(e.g., [6]) to the wiring up of respective neuronal circuits(e.g., [7]). Since the physiological impact of the neuronalcircuitry is more difficult to track down experimentally thanthat of cellular mechanisms, theoretical models have to beset up to validate network-based hypothesis of persistentfiring and to study the interplay between cellular and circuitmechanisms.

A theoretical approach to study the rate encoding offunctionally equivalent subsets of neurons is to group them intopools, thereby translating the description from the single cell tothe population level. Classically, pool dynamics are describedby a set of coupled ordinary differential equations [8–10].These approaches have been, and still are, very successfullyapplied to model neuronal processing in both cognitive andsensory cortical areas (e.g., [11,12]). Such a temporally localdescription, however, cannot account for intrinsic delays andthe dependence of spike probability on the spiking history.Therefore relating simulations of spiking neurons to suchclassical pool models is often very difficult.

These drawbacks are partially amended by pool dynamicsconsidering neuronal spiking as a renewal process [13–16].These theories provide exact descriptions of the pool firingrate if the number N of neurons in a pool tends to infinity. Adisadvantage of these models is that they are rather difficultto analyze. In particular, a theory of how to analytically treatthe dynamics, if several of such renewal pools are coupled,is not available so far. Introducing couplings is an importantnext step since it opens the way to investigate more complexnetwork topologies in the future. In this paper, I proposean extension of the stability analysis of stationary firing tomultiple coupled pools also allowing coupling delays. Thetheory is semianalytically treatable and asymptotically correctfor large pool sizes N → ∞, which distinguishes it from mostother approaches that are based on simulations of spikingneurons. Finally, to demonstrate the inner working of the

theory, it is applied to three well-studied example cases inwhich collective dynamical phenomena are known to arise.

II. MODEL

Let us consider d pools of neurons. Each pool i = 1, . . . ,d

is fully and homogeneously connected and consists of ahomogeneous population of neurons receiving identical input.The synaptic connections between the pools are determinedby the synaptic weights Wij . The neuronal properties acrosspools may be different, although this degree of freedom willnot be made use of in this paper.

Following [13,16], the pool dynamics is formulated on thelevel of a renewal process in which a hazard functionpi(t |t ′)determines the conditional probability pi(t |t ′) �t that a neuronin pool i fires in the infinitesimally short interval [t,t + �t), ifits last spike occurred at time t ′. This conditional probabilityallows one to construct the survivor function

Si(t |t ′) = �(t − t ′) exp

[−

∫ t

t ′ds pi(s|t ′)

](1)

that provides the probability that a neuron in pool i does notfire in the interval [t ′,t) if its last spike occurred at time t ′.Here, and elsewhere, � denotes the Heaviside step function�(t) = 1 for t � 0, and �(t) = 0 otherwise.

By splitting up the joint probability ρi(t,t ′) that a neuronfired its last spike at time t ′ in a conditional and a marginaldistribution, one defines the firing probability density (or firingrate) Ai(t ′) as

ρi(t,t′) = Si(t |t ′) Ai(t

′). (2)

In what follows, the firing rate vector �A = (A1, . . . ,Ad )T

will be considered as the macroscopic dynamical variable ofinterest (although strictly speaking it is no dynamical variable).

Since ρi as a probability is normalized, the integral equation

Ni(t) =∫ t

−∞dt ′ Ai(t

′) Si(t |t ′) ≡ 1 (3)

is valid for all times t , and so defines the dynamics of thefiring rates Ai(t) via a self-consistency equation [13,16]. Forthe purpose of numerical simulation, the dynamics can bemade explicit by taking the derivative of Eq. (3) with respectto time t .

To facilitate the analysis of the integral dynamics fromEq. (3) for d coupled pools, I consider a specific model for theconditional firing probability density pi(t |t ′) that realizes the

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CHRISTIAN LEIBOLD PHYSICAL REVIEW E 84, 031935 (2011)

renewal property in the simplest nontrivial way. This modelconsists of three constituents: a synaptic coupling kernel, again function, and refractoriness.

First the activities Aj of the pools are assumed to beconvolved (∗) by a synaptic coupling kernel

κ ∗ Aj (t) =∫ t

−∞ds Aj (s) κ(t − s). (4)

If multiplied with the synaptic weight Wij the convolution inEq. (4) models the contribution of the activity of pool j to thepostsynaptic potential of neuron i in the spirit of spike responsemodels [14]. Without loss of generality, synaptic kernels areassumed to be normalized,

∫ ∞0 ds κ(s) = 1. All examples in

this paper are derived with an exponential kernel

κ(s) = �(s − �) β e−β (s−�), β > 0. (5)

The two parameters β and � are interpreted as the inverse timeconstant of the postsynaptic potential and the synaptic delay,respectively.

Second, the filter output is assumed to instantaneouslyinfluence the firing rate of the postsynaptic neuron. Thisinstantaneous action is modeled by an exponential gainfunction

fi[κ ∗ �A(t)] = ν0 exp[ �wi · (κ ∗ �A(t)) + Ii(t)], (6)

with Ii quantifying the external inputs and ν0 taking the roleof the spontaneous firing rate. The vectors �wi describe thesynaptic strength with which the other pools are connected topool i. The corresponding synaptic weight matrix Wij can thusbe derived from the elements of the individual weight vectorsas Wij = ( �wi)j .

Third, refractoriness is modeled by a multiplicative proba-bility term

g(t) = �(t − τ ) (7)

that incorporates an absolute refractory time τ , implying thattwo consecutive spikes of a neuron have to be separated atleast by this interval.

In summary, these ideas lead to the following model for thefiring probability density:

pi(t |t ′) = fi(κ ∗ A1(t), . . . ,κ ∗ Ad (t)) g(t − t ′)= :fi(κ ∗ �A(t)) g(t − t ′). (8)

III. THEORY OF LINEAR STABILITY

The pool dynamics defined by the integral equation (3) cangenerally only be treated numerically. To also allow analyticaltreatment, Eq. (3) is linearized around stationary states.

A. Stationary states

In this paper only constant stationary states are considered,i.e., the firing rate solution �A(t) = �A∗ = const is constant intime. This of course also restricts the study to constant inputsIi(t) = Ii = const (which can be relaxed a bit by consideringpiecewise constant inputs). In such a regime, according toEq. (8), the firing probability density pi and hence the survivorfunction only depend on the time difference t − t ′ to the last

spike. Particularly, one finds

S∗i (t − t ′) = �(t − t ′) exp[−fi( �A∗) G(t − t ′)] (9)

with

G(s) = �(s − τ ) (s − τ ). (10)

From the dynamical equation (3), one readily obtains thefixed-point condition

(A∗i )−1 =

∫ ∞

0ds exp[−fi( �A∗) G(s)] = τ + (fi( �A∗))−1,

(11)which is solved by

A∗i = fi( �A∗)

1 + τ fi( �A∗). (12)

In Fig. 1, the fixed-point states from Eq. (12) are comparedto simulations of neurons firing according to the renewalprocess from Eq. (8). For the specified model, excitatory-coupled pools exhibit bistability, in which the two stable fixedpoints are separated by an unstable one [Figs. 1(A)–1(C)].However, if several pools are coupled, the stability propertiesof the fixed points are not readily available from inspectingthe isoclines [Figs. 1(D)–1(F)]. Thus it is necessary toderive a formal stability criterion based on linear perturbationtheory.

B. Perturbations

Stability analysis of the stationary states �A∗ is done bylinearizing the integral equation (3). For small perturbations

�α(t) = �A(t) − �A∗ (13)

one obtains the linear relation

0 =d∑

j=1

∫ t

−∞dt

δNi(t)

δAj (t)αj (t) , i = 1, . . . ,d. (14)

Equation (14) thus imposes a condition for permitted per-turbations �α(t). Since �A∗ is a stationary solution, �α(t) = 0 isalways permitted, so the search has to be restricted to nontrivialperturbations �α(t) �= 0.

To allow further analysis of Eq. (14), one requires an explicitexpression of the functional derivative, which, for the presentmodel, is

δNi(t)

δAj (t−x)= S∗

i (x) δij − A∗i

∫ ∞

0du S∗

i (u)∫ t

t−u

dsδpi(s|t − u)

δAj (t − x)

= S∗i (x) δij − A∗

i Wij fi( �A∗)∫ ∞

0du S∗

i (u)

×∫ u

0ds g(s) κ(x−u+s) . (15)

This expression reveals that the functional derivative has noexplicit dependence on time t , which justifies to introduce theabbreviation �ij (x) := δNi (t)

δAj (t−x) . The integrals in Eq. (14) arethus convolutions and the problem can be analyzed more easilyin the Laplace domain.

To this end, one assumes that the perturbations haveexponential time dependence and, without loss of generality,

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FIG. 1. (Color online) Multistability of excitatory-coupled pools. (A) Spike raster plot of one simulated pool with 100 neurons (W = 30,I = 2). The thick blue bar on the time axis indicates the time interval at which the neurons received an additional stimulus. (B) Rate A(t)derived from simulation in (A). Red dashed lines indicate the predicted stationary states, which are (C) the points of intersection between theidentity (thin line) and the right hand side of Eq. (12) (thick line). Red discs correspond to the stable states from B. (D) Same as (A) for two

pools with W = ( 30 −1010 30 ), and I = (2,1)T. (E) Same as (B) for �A(t) from (D). (F) Isoclines for A1 (black) and A2 [green (gray)]. Red discs

indicate stationary solutions from (E).

start at time t = 0;

�α(t) = �(t) exp(−z t) �α(z). (16)

Note that according to this definition, and though in contrastto most other approaches on linear stability analysis, a positivereal part of z indicates a stable perturbation, whereas anegative real part of z indicates instability. Using Eq. (16),the convolution integrals in Eq. (14) read∫ t

−∞dt

δNi(t)

δAj (t)αj (t) =

∫ t

0dt �ij (t − t) αj (t)

=∫ t

0dt ′ �ij (t ′) αj (t − t ′)

= αj (z) e−z t

∫ t

0dt ′ �ij (t ′) ez t ′ . (17)

As a consequence, if �ij is integrable, the convolution integralon the left-hand side exists independently of z and t . Theintegrability conditions for the �ij s imply that for t → ∞, the

integral over such � is bounded by∣∣∣∣ ∫ t

0dt ′ �(t ′) − L1

∣∣∣∣ < L2 e−tλ , (18)

for some maximal constant λ > 0. Stability of the stationarysolution is then determined by the integral on the right-handside of Eq. (17) for t → ∞. Two cases have to be discerned:

(1) If Re(z) − λ � 0, the integral diverges like L2 et(Re(z)−λ).This divergence is balanced by the exponential prefactore−z t , such that the overall convolution integral convergeslike e−λ t independently of z. As a result, Eq. (14) doesnot impose any constraint on z and all perturbations withRe(z) − λ � 0 are permitted. Since in particular Re(z) > 0, allthose perturbations are stable. As an intuitive interpretation,Re(z) � λ means that the perturbations die out faster [at rateRe(z) > 0] than the integral (at rate λ > 0) and thus do notcontribute in the limit t → ∞.

(2) If Re(z) − λ < 0, the integral on the right-hand sideexists for all t . One thus can properly define the Laplace trans-form �(z) = ∫ ∞

0 dx�(x) exz, and, requiring independence oftime (i.e., disregarding the prefactor ezt ), Eq. (14) turns into a

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−1 −0.5 00

0.5

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R e (z)

Nullclines det[ Λ(z) ]

FIG. 2. (Color online) Complex roots. (Top) Real and imaginary part [color (gray) coded] of det �(z) for uncoupled pools W = 0. Thickblack lines indicate respective nullclines. Right: roots of det �(z) (black squares) are determined by the intersection of the nullclines (red forreal part, blue for imaginary part). (Bottom) Introducing synaptic self-coupling W = 30 limits the range of slow real parts to Re(z) < β = 0.05.Left: The singularity in κ(z) guarantees a zero crossing of the real part at Im(z) = 0 (the two right-most panels are equivalent to top).

linear matrix equation,

0 = �(z) �α(z). (19)

Being a homogeneous linear equation for α(z), the existenceof nonzero perturbations imposes the condition

det[�(z)] = 0. (20)

The roots z of Eq. (20) determine the possible perturbationmodes α(z) �= 0 at the respective fixed point. The real partRe(z) determines the stability of the mode: Re(z) > 0 accountsfor a stable mode, whereas Re(z) < 0 identifies an unstablemode. A nonvanishing imaginary part Im(z) �= 0 indicatesan oscillatory perturbation, with period 2π/Im(z). SinceRe(z) − λ < 0, stable perturbations decay more slowly thanthe convolution integrals in Eq. (14). I thus call them slowmodes.

Using the specific model assumptions for the functions f

and g one finds explicit forms for the survivor function,

S∗i (z) = 1

fi( �A∗) − z

(fi( �A∗)

ezτ − 1

z+ 1

)(21)

and the matrix

�ij (z) = δij S∗i (z) − A∗

i Wij κ(z)

fi( �A∗) − z. (22)

The feasibility condition from Eq. (20) is hence equivalent to

det

[(fi( �A∗)

ezτ − 1

z+ 1

)δij − κ(z) Mij

]= 0 (23)

with (M)ij = Mij = A∗i Wij . Let us next consider two special

cases in which Eq. (23) becomes particularly simple.

No refractoriness. If the refractory time is chosen τ = 0,the neurons are allowed to fire with infinite rate. In that case,the first term in Eq. (23) simplifies to the unit matrix 1 and thefeasibility condition can be translated to

det(κ(z)−1 1 − M) = 0. (24)

This equation can be readily solved for those values of z atwhich κ−1(z) equals an eigenvalue μ of M . For the specificchoice of an exponential coupling kernel, one finds

κ(z) = e� z

1 − z/β(25)

and thus the feasibility condition reads

e−� z (1 − z/β) = λ ∈ σ (M). (26)

It is shown in the Appendix that (1) Eq. (26) has exactlyone root for real positive eigenvalues μ of M . Moreover, itis shown that (2) if the modulus |μ| < 1, a root z has alwayspositive real part and the perturbation is stable, and (3) z has anonzero imaginary part only if also μ has a nonzero imaginarypart. Imaginary parts of the eigenvalues thus directly indicateoscillatory behavior of the perturbations.

No coupling. If pools are considered as uncoupled, i.e.,W = 0, then the feasibility conditions from Eq. (23) maps to

d∏i=1

(fi( �A∗)

ezτ − 1

z+ 1

)= 0. (27)

Hence the roots z of this condition are given by

ezτ − 1

z= −f −1

i ( �A∗), i = 1, . . . ,d. (28)

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FIG. 3. (Color online) Population oscillations. (A1−5) Population rate from network simulations for five different inhibitory synapticcoupling strengths |W |. (B1−5) Roots (squares) of the feasibility conditions for the weights from A. Color (gray) code corresponds to Fig. 2,top right. (C) Empirical firing rate (black dots) and stationary state (red solid line) from the fixed point equation (12). (D) Real part of the rootof Eq. (23). (E) Imaginary part (divided by 2π , red) of the root of Eq. (23) and empirical oscillation frequency (black dots).

For each value of fi( �A∗) Eq. (28) can have multiple complexroots, as shown in the upper panel of Fig. 2.

General case. In general, neurons are refractory τ > 0 andcoupled W �= 0 and hence the feasibility condition can only beevaluated numerically. Most notably also for arbitrarily smallpositive weights Wij > 0 the pole of κ at z = β [Eq. (25)]introduces a slow root with Re(z) < β (Fig. 2, bottom).This shows that positive feedback, no matter how small,always introduces a slow time scale. Since the integrabilityof the Laplace transform of � requires Re(z) < β, the rootsof the uncoupled network are no longer effective in that caseand the network dynamics permits all (stable) perturbations attime scales faster than 1/β.

Here and unless otherwise mentioned the following pa-rameters are used: the decay rate of the coupling kernelβ = 1/(20 ms), indicating a neuronal integration time constantof 20 ms; the refractory time τ = 3 ms, imposing a maximalfiring rate of 333 Hz; the synaptic delay � = 2 ms.

IV. APPLICATIONS

Our understanding of dynamical systems is dominated byconcepts derived from studying ordinary differential equa-tions, such as, e.g., phase space trajectories and fixed points.Integral dynamics such as the one presented here do not allowusing these intuitive ideas, at least in the strict sense. In thissection, I will introduce three examples of collective neuronal

phenomena that have been (partly extensively) studied in thepast to illustrate how the theory presented here applies to them.Interestingly, the slow stable modes turn out to be quite wellreflected by the classical concept of a phase space.

A. Gamma rhythm d = 1

As an example for a homogeneously coupled populationof neurons (a single pool, d = 1), I reconsider inhibitorynetworks that generate oscillations in the gamma range(30–100 Hz). All neurons receive the same positive driveI = 50, and the synaptic weight W is used as a parameterto control the population oscillation. In line with a multitudeof previous studies (e.g., [16–19]), Fig. 3(A) shows thatoscillation frequency of a simulated pool of N = 4,500neurons decreases with increasing inhibitory weight |W |.

In the language of the present theory, the dependenceof oscillation frequency on synaptic coupling is reflectedby the corresponding roots of the feasibility condition fromEq. (23). All roots have a negative real part and a nonvanishingimaginary part [Fig. 3(B)] indicating that the stationarystates are unstable and that the unstable perturbations exhibitoscillatory nature. A comparison between the stationary rateA∗ from Eq. (12) and the mean spike rate of the simulationsshows excellent agreement [Fig. 3(C)] despite the fast speedsof divergence from the asynchronous state, as indicated by thestrongly negative real parts of the roots in Fig. 3(D). Only for

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FIG. 4. (Color online) Line attractor. (A1−3) State spaces for three different I/E ratios (as indicated). Squares indicate stable stationarysolutions, discs indicate unstable stationary solutions. Black thick and green thin lines mark the isoclines (below 33 Hz); cf. Fig. 2. Note thataxes are logarithmically scaled. (B1−3) Simulations corresponding to the respective situations in (A). Black (upper) and green (lower) tracesrepresent the rates derived from the two individual pools. The blue (middle) trace is the average activity of the system. Gray vertical bars markthe times of the stimuli. The colors (gray) of levels of the horizontal lines correspond to the stationary states from (A). (C),(D) Bifurcationdiagrams. (C) Stationary states as a function of the I/E ratio [colors (gray) of levels from (A) and (B). (D) Roots of the feasibility conditionfrom Eq. (23).

very low values of |W | does the stationary rate overestimatethe simulation results, as the strict upper bound of the firingrate induces a bias.

The |W | dependence of the mean rate suggests thatthe dynamics is still governed by the asynchronous(unstable) stationary state. It is thus justified to alsocompare the oscillation frequency of the simulated networkwith the imaginary part of the root of Eq. (23): For smallmodulus of the weight |W | (fast oscillations) the imaginarypart turns out to be a good predictor [Fig. 3(E)]. However, asthe modulus |W | increases, the empirical oscillation frequencybecomes only about half of the predicted one, indicating thatthe stability analysis only qualitatively captures the networkdynamics in this regime.

B. Line attractors d = 2

Irregular-asynchronously firing pools of neurons can onlyencode inputs by their mean firing rate. It is thereforeinteresting to study if the linear perturbation theory canpredict whether two pools can fire with a continuum of firingrates. A dynamical system that exhibits such a continuumof stable states is called a line attractor. In neurobiology, line

attractors are considered to be necessary for integrating stimuliand keeping the integral value in short-term memory for asufficiently long period of time. Line attractors have beensuggested to stabilize eye position [7] and to perform pathintegration [20].

Here a line attractor is considered to consist of two poolswith positive self-coupling and negative cross coupling. The

specific choice of the coupling matrix is W = 50 ( 1 −ρ−ρ 1 ), in

which ρ measures the ratio between inhibitory and excitatorycoupling strength (I/E ratio). In addition, the pools receivehomogeneous excitatory drives �I = (2,2)T.

The stationary states are then analyzed in dependence onthe I/E ratio ρ (Fig. 4). For weak inhibition (small ρ) thesystem exhibits five stationary solutions: two stable oscillatorysolutions for which one of the pools is silent and the other poolfires at maximal rate, a symmetric stable solution in whichboth pools fire at the same low rate, and two unstable solutionsthat separate the basins of attraction of the stable solutions[Fig. 4(A1)]. At some critical I/E ratio ρc the symmetricstationary solution loses stability, i.e., the real part of the rootof Eq. (23) becomes zero and the isoclines become parallelat the stationary point [Fig. 4(A2)]. For ρ > ρc the symmetric

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(D)(C)

(E) (F)

FIG. 5. (Color online) Slow transients. (A),(C),(E) Firing rates of three coupled pools from stochastic simulations. Pool 1 received astrong stimulus of amplitude I1 = 6.5 between 450 and 500 ms. The coupling strength k between the first and the second pool is k = 30 [in(A)], k = 20 [in (C)], and k = 15 [in (E)]. The rates are averaged over 10 ms. (B),(D),(F) Two different projections (blue lines with dots)of the activity variables from the simulations in (A),(C),(E). The arrow in the left graphs indicate the time course. The temporal separation ofthe dots is 10 ms. Theoretically obtained fixed points are plotted as orange squares (stable) and yellow discs (unstable). The arrows indicatethe direction and magnitude of the slow stable and unstable modes. Arrows are drawn black if the imaginary part of the corresponding root ofEq. (23) is zero. Red (gray) arrows indicate oscillatory modes with nonzero imaginary part of the root. Here, the refractory time is τ = 10 ms,and the coupling kernel has a time constant of 1/β = 30 ms.

stationary state is unstable and spawns two new neighboringstable solutions [Fig. 4(A3)]. The resulting bifurcation diagram[Figs. 4(C) and 4(D)] thereby reminds one of a pitch forkbifurcation.

Linear perturbation analysis thus predicts that at the criticalI/E ratio ρc the system should behave as a line attractor. Thisprediction was tested by three simulations for ρ < ρc, ρ ≈ ρc,and ρ > ρc. The setting was such that after an initializationphase of 500 ms, the pools received four additional currentstimuli �I (k) = a (1, − 1)T,k = 1, . . . ,4, with amplitude a =0.25 for small time intervals of 20 ms at times t stim(k) = k ·500 ms. For ρ < ρc (where the symmetric stationary state isstable), the firing rates of the pools always decays back tothe symmetric state [yellow in Fig. 4(B1)], i.e., the systemdid not show considerable short-term memory. For ρ > ρc,where the symmetric stationary state is unstable, the firing ratesdrift away from the symmetric stationary state and approachone of the stable asymmetric states [orange in Fig. 4(B3)].Only at the critical I/E ratio ρ = ρc, the firing rates remainat different levels for a duration that considerably exceeds

the coupling time constant of 20 ms: In the vicinity of thesymmetric stationary state, the system acts as a continuousattractor [Fig. 4(B2)].

To conclude, linear perturbation theory is able to alsoquantitatively identify network states in which the systemintegrates a stimulus. These continuous attractor states areparticularly sensitive to noise. The pools thus have to be eitherlarge (as in the present example), or the intrinsic time constants(e.g., 1/β) have to be long.

C. Trajectories in activity space d = 3

The example of the line attractor shows that pool dynamicscan act on time scales that are much slower than the intrinsiccellular processes such that the temporal evolution of poolfiring rates is a true network phenomenon. One thus may askwhether such coupled pools may also be able to representlong time intervals themselves, making use of the time ittakes to follow a specific trajectory, such as a limit cycleor a slow transient sequence of activity states. Such activity

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sequences have been investigated in various flavors in the past,e.g., [21–23]. These models consider sequential activationof neuron populations either as a complex spatiotemporalmemory trace that is stored in the network, or as a short-term memory buffer for sensory stimuli. To the best of myknowledge, no model so far was able to provide a macroscopicdescription of activation sequences of recurrently coupledpools that is rigorously derived from microscopic networksof spiking neurons in continuous time.

In this section, an implementation of such a slow trajectoryis presented for the example of three coupled pools. The firstpool thereby represents a bistable switch (cf. Fig. 1) that actsas a short-term memory buffer for a brief (50 ms) input pulse.Once activated, the first pool excites a second pool, whichexhibits a slow buildup in rate that acts as a time delay. Finally,a third inhibitory pool is driven by the second pool. The thirdpool inhibits the first pool and stops the short-term memorybuffer. As a result, the dynamical system is set back into alow-activity stable state. The dynamics is realized using theweight matrix

W =⎛⎝70 0 −20

k 35 00 50 0

⎞⎠ , (29)

in which we use the weight k between the first and thesecond pool as parameter to analyze the system behavior.Additionally, all pools receive a constant drive �I = 3

2 (1,1,1)T .Typical trajectories from cellular simulation are shown inFig. 5.

The duration of the transient burst of activity can beadjusted by the coupling strength k. If k is large the delaybetween activation of the first and the second pool is relativelyshort and so is the burst duration. The “trajectory” in the�A space is dominated by two stationary states with high

rates. For large k > k(1)c these stationary states do not actually

exist, although their “ghosts” still mediate slow trajectories[Figs. 5(A) and 5(B)]. At some critical k = k(1)

c the high-ratestationary states appear [Figs. 5(C) and 5(D)] and the dynamicsslows down further for k < k(1)

c . Below a second critical valuek < k(2)

c < k(1)c , one of the stationary states becomes stable and

thus the transient behavior disappears [Figs. 5(E) and 5(F). Inthis regime, the dynamics is dominated by the bistability ofthe first pool.

To conclude, linear stability analysis of the normalizationcondition from Eq. (3) allows qualitative statements about thedynamics of the high-dimensional dynamical system, similarto classical systems of ordinary differential equations [24],although, strictly speaking, the notion of phase spaces andtrajectories is incorrect for an integral dynamics. For example,trajectories may intersect as the dynamics is not local intime and, similarly, the arrows in Fig. 5 do not representa flow field but the direction of slow stable and unstablemodes.

V. DISCUSSION

This paper presents a theoretical approach to analyzethe firing of coupled pools of neurons. A stability criterionis derived for stationary states, which requires finding thecomplex roots of a matrix determinant. These roots indicate

slow stable and unstable modes, i.e., activity features thatgo beyond the single-cell time scales and as such representcollective dynamical phenomena.

The theory is applied to well-studied examples, whichparticularly reveal slow dynamical features. It is importantto understand which collective mechanisms can underly suchslow time scales, because they are not readily accessibleto experimental measurements. Slow cellular time scales,however, are much easier to track down experimentally, andthus possible additional network mechanisms are often notrecognized.

So far, the presented theory only applies to simple (one-dimensional) neuron models, that cannot extend to relevantintrinsic behaviors such as adaptation or slow positive self-feedback. Particularly, such interplay between cellular andnetwork mechanisms may be important to understand slowneuronal processing [7]. An extension to high-dimensionalneuron models has been achieved for a pool model withoutexplicit renewal property [25]. For the present theory suchextensions are not yet available as they require extending therenewal assumption, one of the corner stones of the approach.Yet it seems possible but tedious to generalize the dynamicalequation (3) to many previous spike times.

Slow time scales of neuronal network activity have beendescribed in many contexts. Aside from the ones mentioned(integration of retinal slip and path integration), such activitypatterns are generally assumed to occur for most short-termmemory tasks. A well-studied example is decision making,where a slow time evolution of firing rates has been observedin the prefrontal cortex preceding the decision [4,26]. Morerecently, short-term memory has also been connected to short-term synaptic plasticity [27,28], which means that there thememory is not represented by long-lasting firing, but by cell-(or synapse)-intrinsic dynamical processes. In order to studythe interaction of the two mechanisms (synaptic and network),future work requires also incorporating synaptic short-termplasticity into the renewal pool description presented in thispaper.

ACKNOWLEDGMENTS

The author is grateful to Chun-Wei Yuan for comments onthe manuscript. This work was funded by the German Ministryof Education and Research (BMBF) Bernstein Fokus NeuronalBasis of Learning (Grant No. 01GQ0981) and BernsteinCenter for Computational Neuroscience Munich (Grant No.01GQ0440).

APPENDIX

In the case of nonrefractory neurons τ = 0 the permittedslow stable and unstable perturbations are determined as theroots z = x + i y of(

1 − z

β

)e−� z = μ = |μ| eiarg(μ) (A1)

with μ being an eigenvalue of the matrix Mij = Ai Wij .Moreover, integrability of the Laplace transform requires

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Re(z) < β. Equation (A1) can be split up into modulus andphase. For the modulus one obtains

y±(x) = ±β

√[|μ|2 e2x � −

(1 − x

β

)2 ], (A2)

which defines two functions y+(x) > 0 and y−(x) =−y+(x). Note that y± are only defined for real parts atwhich (

1 − x

β

)< |μ| ex �. (A3)

For |μ| < 1, the possible x values are thus restricted to positivevalues and the perturbations are stable. If the eigenvalue μ isreal and positive, μ = 1 yields the solution z = 0 and thus

marks the verge of stability, separating unstable (μ > 1) andstable (μ < 1) perturbations.

The phase part of Eq. (A1) yields the implicit equations

�y±(x) + arctan

(y±(x)

β − x

)= −arg(μ). (A4)

More specifically, as complex eigenvalues of real matrices M

always come in pairs of complex conjugates, for any solutionwith imaginary part y there is also one with imaginary part−y (corresponding to the complex conjugate eigenvalue).The phase equation (A4) also reveals that, since � > 0,the imaginary part y± of the solution vanishes only forarg(μ) = 0 and then the real part is uniquely determined by1 − x/β = μe�x . Nonzero imaginary parts of μ always alsoresult in imaginary parts of z.

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