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Mechanism for the Universal Pattern of Activity in DevelopingNeuronal Networks

Joël Tabak,1 Michael Mascagni,2 and Richard Bertram31Departments of Biological Science, 2Computer Science, and 3Mathematics, Florida State University, Tallahassee, Florida

Submitted 21 September 2009; accepted in final form 6 February 2010

Tabak J, Mascagni M, Bertram R. Mechanism for the universalpattern of activity in developing neuronal networks. J Neuro-physiol 103: 2208 –2221, 2010. First published February 17, 2010;doi:10.1152/jn.00857.2009. Spontaneous episodic activity is a fun-damental mode of operation of developing networks. Surprisingly,the duration of an episode of activity correlates with the length ofthe silent interval that precedes it, but not with the interval thatfollows. Here we use a modeling approach to explain this character-istic, but thus far unexplained, feature of developing networks. Be-cause the correlation pattern is observed in networks with differentstructures and components, a satisfactory model needs to generate theright pattern of activity regardless of the details of network architec-ture or individual cell properties. We thus developed simple modelsincorporating excitatory coupling between heterogeneous neurons andactivity-dependent synaptic depression. These models robustly gen-erated episodic activity with the correct correlation pattern. Thecorrelation pattern resulted from episodes being triggered at randomlevels of recovery from depression while they terminated around thesame level of depression. To explain this fundamental differencebetween episode onset and termination, we used a mean field model,where only average activity and average level of recovery fromsynaptic depression are considered. In this model, episode onset ishighly sensitive to inputs. Thus noise resulting from random coinci-dences in the spike times of individual neurons led to the highvariability at episode onset and to the observed correlation pattern.This work further shows that networks with widely different archi-tectures, different cell types, and different functions all operate ac-cording to the same general mechanism early in their development.

I N T R O D U C T I O N

Spontaneous activity is a fundamental property of develop-ing networks (Ben Ari 2001; Feller 1999; O’Donovan 1999),characterized by episodes of intense activity separated byperiods of quiescence. This episodic activity occurs at an earlydevelopmental stage when the networks are primarily excita-tory and plays essential roles in the development of neuronalcircuits (Hanson et al. 2008; Huberman et al. 2008; Katz andShatz 1996; Spitzer 2006). Episodic activity is also observed inother hyperexcitable circuits such as disinhibited networks(Menendez de la Prida et al. 2006; Tscherter et al. 2001). Onestriking feature of this activity is the correlation betweenepisode duration and the length of the preceding—but notfollowing—interepisode interval (Fig. 1). The spontaneousactivity of many networks exhibits this correlation pattern,including the developing spinal cord (Tabak et al. 2001),developing retina (Grzywacz and Sernagor 2000), developingcortical networks (Opitz et al. 2002), hyperexcitable hippocam-

pal slices (Staley et al. 1998), disinhibited spinal cord (Rozzoet al. 2002), and spinal cord cultures (Streit 1993; Streit et al.2001). The general occurrence of this correlation pattern sug-gests that a common mechanism of operation exists in net-works that are structurally very different (O’Donovan 1999).Here, we identify such a general mechanism and explain howthe correlation pattern is created.

Computational studies have shown that primarily excitatorynetworks with slow, activity-dependent, network depressioncan generate episodic activity (Butts et al. 1999; Kosmidiset al. 2004; Tabak et al. 2000; van Vreeswijk and Hansel 2001;Vladimirski et al. 2008). The slow depression process (Darbonet al. 2002; Fedirchuk et al. 1999) could target either synapticconnections (synaptic depression) or the ability of the neuronsto sustain spiking (cellular adaptation). Recurrent excitatoryconnectivity recruits neurons and sustains high network activ-ity, whereas the slow depression eventually shuts down activ-ity. The next episode may occur once the network has suffi-ciently recovered from depression. This general mechanismcan account for activity in a wide range of networks. Can itproduce the observed correlation pattern between the durationof episodes and interepisode intervals?

To answer this question, we use a network model of heter-ogeneous spiking neurons that is capable of generating spon-taneous episodic activity. Given its ubiquity, the correlationpattern should be generated regardless of the details of neuro-nal properties or network architecture. We therefore begin withintegrate-and-fire neurons and all-to-all excitatory connectiv-ity. Such a model network with depressing synaptic connec-tions produces the observed correlation pattern. We then showthat the correlation pattern is robust to changes in networkconnectivity and cellular properties.

To explain why the observed correlation pattern is such arobust feature of excitatory networks with slow synaptic de-pression, we use a mean field model with added noise. Theadded noise represents the random fluctuations of networkactivity that may trigger episodes. In this model, episode onsetis very sensitive to noise, whereas episode termination is not.This asymmetry in turn explains the correlation pattern. Thus amechanism based solely on recurrent excitatory connectivitywith slow activity-dependent synaptic depression can producethe correlation pattern that is the trademark of developingnetworks.

M E T H O D S

Network models

We used a network of heterogeneous integrate-and-fire neuronswith all-to-all excitatory connections as described by Vladimirski et

Address for reprint requests and other correspondence: J. Tabak, Dept. ofBiological Science, BRF 206, Florida State Univ., Tallahassee, FL 32306(E-mail: [email protected]).

J Neurophysiol 103: 2208–2221, 2010.First published February 17, 2010; doi:10.1152/jn.00857.2009.

2208 0022-3077/10 $8.00 Copyright © 2010 The American Physiological Society www.jn.org

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al. (2008). This choice is motivated by the fact that neither a particularstructure nor specific neuronal properties are necessary to generate thespontaneous activity. Heterogeneity of neuronal excitability preventsspike-to-spike synchrony and allows robust episodic behavior (Vlad-imirski et al. 2008). The voltage variation of neuron i is given by

�dVidt

� �Vi � Ii � gsyn�Vi � Vsyn� (1)

with Vi varying between 0 (resting potential) and 1 (spike threshold).The membrane time constant, �, is set to 1 so time units are relativeto the membrane time constant. Each neuron has a constant input, Ii,which sets its own excitability and is the source of heterogeneity. Thedistribution of Ii was chosen to be uniform over the interval [0.151.15] (unless mentioned otherwise), so a fraction of the neurons haveinputs Ii � 1 and they spike spontaneously. This allows the stochastictriggering of episodes. The other input to each neuron is the synapticterm �gsyn(Vi � Vsyn) with

gsyn �g� synN

�j�1

N

aj � g� syn�a� (2)

where N is the number of neurons in the network, g�syn is the maximalsynaptic activation, and aj is the synaptic drive from neuron j to all

other neurons. The term �a� �1

N�j�1N aj represents the average

activity in the network and is used as the output of the simulations.The synaptic drive from neuron j varies according to

dajdt

� �a�t��a�1 � aj� � �aaj. (3)

When neuron j fires (i.e., Vj reaches 1), its voltage is reset to 0 and keptat 0 for a refractory period Tref. At the same time, this neuron’s �a(t) isset to 1 for a brief period of time Ta; at any other time, �a(t) � 0, so thesynaptic activation coming from neuron j decays with first-order kineticswith rate constant �a. Note that all the synapses originating from a givenpresynaptic neuron are identical; this limits the number of synapticactivation variables to N. Finally, we subtract the contribution of eachneuron from its own voltage equation (no self-synapses).

Equations 1–3 thus define the basic network model without any typeof depression. To this basic model, we can add synaptic depression orcellular adaptation. Synaptic depression is modeled by a variable, sj, forthe synapses originating from neuron j. It varies according to

dsjdt

� �s �1 � sj� � �s�t��ssj, (4)

where �s(t) � 0 at all times except for a period of time, Tdep,immediately following a presynaptic spike, during which it is set to 1.Thus spiking causes sj to decrease. This depression factor decreasessynaptic efficacy such that the effective synaptic input becomes

gsyn �g� synN

�j�1

N

ajsj (2)

The network model with synaptic depression is thus defined by Eqs.1, 2, 3, and 4. For the model with cellular adaptation, we add a slowoutward current to the voltage equation

�dVidt

� �Vi � Ii � gsyn �Vi � Vsyn� � g��i�Vi � V�� (1)

with V� 0 and the activation of the outward current, �i, varyingaccording to

d�idt

� ��t��1 � �i� � ��i (5)

with neuron i’s ��(t) switched from 0 to 1 during a period of time, T�,immediately after a spike. The network model with cellular adaptation isdefined by Eqs. 1, 2, 3, and 5. Parameters for the network models aregiven in Table 1. Simulations were run with XPPAUT (Ermentrout

epis

od

e d

ura

tio

n

preceding interval following interval

B

precedinginterval

followinginterval

A

episodeduration

FIG. 1. Correlation pattern typically observed in developing and hyperex-citable networks. A: cartoon representation of spontaneous activity, showingepisodic increases of the average firing rate in the network. B: scatter plots ofepisode duration and interepisode interval, showing a positive correlationbetween episode duration and preceding (not following) interepisode interval.Time scale ranges from 100 ms to 1 min for the episodes of activity.

TABLE 1. Parameters of the network models using integrate-and-fire neurons (normalized units)

Parameter Description Value (Model With Synaptic Depression) Value (Model With Cellular Adaptation)

Ii Input to neuron i 0.15–1.15 0.5–1.5Tref Refractory period 0.25 0.25g�syn Max. synaptic conductance 2.8 1.4Vsyn Synaptic reversal potential 5 5�a Synaptic activation rate 10 10�a Synaptic decay rate 1 1Ta Synaptic activation duration 0.05 0.05�s Synaptic recovery rate 0.004 —�s Synaptic depression rate 0.4 —Tdep Synaptic depression duration 0.05 —g�� Max. adaptation conductance — 0.5–1.5V� Reversal potential — �1�� Adaptation activation rate — 0.2�� Adaptation deactivation rate — 0.004T� Adapt. activation duration — 0.05

Times are relative to the membrane time constant, conductances are relative to the leak conductance and potentials relative to threshold voltage.

2209UNIVERSAL FEATURE OF DEVELOPING NETWORKS

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2002). The fourth-order Runge-Kutta method was used, with a time stepof 0.001. Results are presented for n � 100 neurons; simulations with30–300 neurons gave similar correlation patterns. The time courses forthe model with synaptic depression were also similar to simulations runwith 1,000 neurons (Vladimirski et al. 2008).

Mean field model

Conceptually, a mean field neuronal model represents the activityof a neuronal population averaged over the population and over ashort period of time (Gutkin et al. 2003; Wilson and Cowan 1972). Itis a useful approximation when, as in the current system, there is nostructure in the connectivity pattern. Also, in the current system, theentire neuronal population is activated almost synchronously on theepisodic time scale (but not on the spike time scale), so the average isa good indicator of network behavior. It leads to a differentialequation describing the variation of the averaged neuronal activity

�ada

dt� � a � a� �wa � �0� (6)

In this equation, a represents the averaged neuronal activity. This is

analogous to the average synaptic activation �a� �1

N�j�1N aj used

above (Eq. 2), also called synaptic drive (Pinto et al. 1996). Thisequation implies that activity tends to reach a value given by thenetwork steady-state input/output function, a�, with time constant �a.This time constant represents the network recruitment time constantand is set to 1, so all time units are relative to it. The parameter w isa measure of the connectivity in the network and is analogous to g�synin the network model. The parameter �0 sets the average cell excit-ability in the network, analogous to –I in the network model (where Iis the average of the individual neuronal inputs Ii). The term wa � �0represents the input to the network; thus the connectivity parameter wsets the amount of positive feedback within the network (Tabak et al.2000, 2006).

Equation 6 can have bistable solutions, where the activity a can beeither high (�1) or low (�0) depending on initial conditions. Togenerate episodic behavior, we add a slow synaptic depression processthat can switch the network between the high and low states. Thismodel is defined by

da

dt� �a � a� �wsa � �0� (6)

�sds

dt� �s � s��a� (7)

where s is the synaptic depression variable and s� is a decreasingfunction of a, such that s decreases during high activity episodes andrecovers during interepisode intervals. The steady-state network out-put function is a��i� � 1/�1 � e

�i/ka� and the steady-state synapticavailability is s��a� � 1/�1 � e

�a��s�/ks� (see Table 2 for parameterdescriptions).

This model can generate episodic behavior (Tabak et al. 2006), butit is deterministic and therefore produces constant durations for theepisodes and interepisode intervals. To study the variability of thesedurations, we introduce stochastic noise. This noise represents thecombination of factors that leads to variability in the durations ofepisodes and interepisode intervals. Episodes are triggered when asufficient number of cells fire in a short interval of time, leading to asynaptic event that can bring the rest of the network above threshold(Menendez de la Prida and Sanchez-Andres 1999; Wenner andO’Donovan 2001). Therefore noise represents random correlatedactivity in the network; in our network models, noisy fluctuations inaverage activity result from heterogeneity and finite size effects andnot from additional noise to the input of each individual neuron(added noise to individual neurons would mostly result in a differentshape for the network output function a�; Giugliano et al. 2004; Nesseet al. 2008). We model random correlated activity by adding Gaussianwhite noise to the activity equation, which becomes

da � ��a � a��wsa � �0��dt � n��dt (8)where n is the noise amplitude and � is a random number drawn ateach time step from a normal distribution.

Parameter values for the mean field model with noise are given inTable 2. The simulations were run with XPPAUT using the forwardEuler method with a time step of 0.05. All XPPAUT files used in thiswork are available for download from www.math.fsu.edu/�bertram/software/neuron.

R E S U L T S

Excitatory network models with synaptic depression generatethe observed correlation pattern

Can a simple neural network model reproduce the episodicactivity and associated correlation pattern observed in many de-veloping neuronal networks? To answer this question, we use anetwork with no special architecture (all-to-all coupling) andcomposed of excitable cells with no endogenous bursting proper-ties (modeled as integrate-and-fire neurons; see METHODS). Allsynapses are excitatory and slowly depress because of presyn-aptic activity. This creates a circuit with minimal complexitywith the capacity to generate spontaneous episodic activity(Vladimirski et al. 2008). All cells are identical, except fortheir input current Ii, which is randomly chosen such that asmall fraction of the neurons are spontaneously spiking. At thebeginning of an episode, activity quickly spreads from the mostto the least excitable cells. Typically, all cells spike asynchro-nously during an episode (Vladimirski et al. 2008).

Figure 2A shows the time courses for the model average

activity �a� �1

N�j�1N aj (where aj is the synaptic drive from

neuron j, i.e., the firing pattern of that neuron filtered by thesynaptic machinery; see METHODS) and average depression vari-

able �s� �1

N�j�1N sj (where sj is the level of recovery from

depression of all synapses from neuron j; sj � 0 when thesynapses are fully depressed and sj � 1 when the synapses arefully recovered). The activity is episodic, with s� decreasingduring an episode, eventually terminating the episode whensynaptic efficacy becomes too low to sustain activity. Then

s� recovers during the interepisode interval, until a newepisode is initiated. Note that the maximal value reached by

s�, i.e., its value at episode onset, varies from episode to

TABLE 2. Parameters of the mean field model

Parameter Description Value

w Connectivity 0.8�0 Input for half activation 0.17ka Spread of a� 0.05�s Activity for half depression 0.2ks Spread of s� 0.05

Steady-state network output function is a�(i) � 1/(1 � e�i/ka) and steady-

state synaptic availability is s�(a) � 1/(1 � e(a��s)/ks).

2210 J. TABAK, M. MASCAGNI, AND R. BERTRAM


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episode. This is also visible in Fig. 2B where the a� versus

s� trajectory is plotted. There is more variability of s� atepisode onset than at episode termination. This difference invariability results in a significant positive correlation betweenepisode duration and the preceding (Fig. 2C) but not thefollowing interepisode interval (Fig. 2D). This is because alonger preceding interval (“long” on Fig. 2B) means a higher

s� at episode onset and therefore a longer episode durationbefore s� decreases enough to stop the episode. On the otherhand, because episodes terminate close to the same s� value,regardless of episode duration, there is no significant correla-tion between episode duration and the following interval.

The correlation pattern results from the large variability of

s� at episode onset compared with the low variability atepisode termination. To illustrate this difference, we plot thedistribution of the average recovery variable s� at episodeonset and episode termination. The width of the onset distri-bution (Fig. 2E) is much greater than that of the terminationdistribution (Fig. 2F). Note that this stochasticity of the episodeonset is an intrinsic property of the network, because there isno added source of noise in the model (Thivierge and Cisek2008). A fraction of the neurons spike slowly during therecovery period, so an episode can be triggered at any time if

enough neurons fire together to trigger a wave of recruitment ofother neurons. This seems to be a stochastic event, and theprobability that such an event occurs increases with recoverytime.

Thus the simple excitatory network model generates spon-taneous episodic activity with the correlation pattern observedexperimentally. This correlation pattern is the result of thegreater variability of episode onset compared with episodetermination. To test whether the correlation pattern is robust,we ran 10 simulations with different, randomly chosen distri-butions of input current Ii. The correlation coefficients betweenepisode duration and interepisode intervals are plotted on Fig. 3A.In all cases, episode onset was more variable than episode termi-nation, resulting in a significant correlation between episodeduration and preceding interval, but not with following interval.

Robustness of the correlation pattern to changes in cellularand network properties

Is the correlation pattern still observed if some features ofthe model are modified? Our hypothesis is that as long as thetwo key features of the model—recurrent excitatory connec-tivity and slow activity-dependent depression of network ex-

0 500 1000 1500 20000

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< a

>, <

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>

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epis

ode

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epis

ode

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freq

uenc

y

sd = 0.020

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freq

uenc

y

sd = 0.0006sh

ort

long

A B

C D

E F

FIG. 2. Episodic activity and correlationpattern of the excitatory network of inte-grate-and-fire (I&F) neurons with synapticdepression and all-to-all coupling. A: timecourses of activity, a� (thin, black curve),and average recovery from depression, s�(thick, light green curve). Note how thevalue of s� at episode onset varies fromepisode to episode, while the value of s�at episode termination remains almost con-stant. B: “phase plane” representation, where

a� is plotted as a function of s� foreach time point shown in A. This defines atrajectory with the bottom part of the trajectorycorresponding to the interepisode intervals, theupper part corresponding to the episodes andthe right and left parts corresponding to thetransitions between episodes and silent inter-vals (arrows indicate direction of movement).This representation clearly shows the greatervariability of episode onset than episode ter-mination. Short, episode starting after a shortinterval; long, episode starting after a longinterval. C: the correlation between episodeduration and preceding interepisode interval ishigh. D: absence of correlation between epi-sode duration and following interepisode inter-val. E: wide distribution of s� values atepisode onset. F: narrow distribution of s�values at episode termination. sd, SD of thecorresponding distribution.



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citability—are present, the correlation pattern will be observed.To test this, we alter some features of the network—one at atime—and check if the same correlation pattern is present.Note that we limit the scope of this study to all-to-all andrandom connectivity patterns.

First, we change the connectivity of the network. We do notconsider networks with ordered topologies because there is nocommon structure in the many developing and other hyperexcit-able networks that exhibit the correlation pattern (O’Donovan1999). Instead, we replace the all-to-all coupling connectivity ofour network model by random connectivity whereby each neuronprojects to 10% of the population. This randomly connectedsparse network exhibits episodic activity with a high degree ofvariability at episode onset, resulting in the same robust correla-tion pattern as the fully connected networks (Fig. 3B). The samerobust correlation pattern is also obtained with connection prob-ability decreased to 5%.

Next, we replace the integrate-and-fire neurons (with all-to-all coupling) with Hodgkin-Huxley–type neurons (see appen-dix for the formulation of this model). That is, we change theneuronal model from an integrator to a resonator (Izhikevich2001). This network requires a high degree of heterogeneity inthe distribution of input current to generate episodic activity,and the activity is less regular (both onset and termination aremore variable than with I&F neurons). The baseline level canalso vary during the interepisode interval (Fig. 4, A and B) insome simulations, which was never observed with the networkof I&F neurons with all-to-all coupling. Nevertheless, asshown in Figs. 3C and 4, C and D, this model network stillproduces activity with the variability of s� much higher atepisode onset (Fig. 4E) than episode termination (Fig. 4F),leading to a robust correlation between episode duration andthe preceding, but not the following, interepisode interval.

Finally, we change the slow negative feedback process thatterminates the episodes. For this last model, synapses do notdepress, but each individual neuron has a slow outward currentwith activation �j (see METHODS). This model generates episodicactivity similar to the model using synaptic depression (Fig. 5A),

but with the average cellular depression �� 1

N�j�1N �j in-

creasing during an episode and recovering (decreasing) duringthe interepisode interval. This network model tends to producevery regular activity because the adaptation process slowsdown the faster firing cells (the cells with higher input current)more than the slower cells. This tends to homogenize thenetwork. To limit this effect, we chose the conductance pa-rameter for the adaptation process randomly, over a wideinterval, for each cell. Simulation results for the model withcellular adaptation are shown in Figs. 3D and 5. Although thesame pattern of correlation is generally observed, the variabil-ity at onset relative to termination is less (Fig. 5, E and F) thanfor the other model networks. In a minority of cases, thisresulted in the absence of a significant correlation betweenepisode duration and preceding interval or the presence of asignificant correlation between episode duration and followinginterval (Fig. 3D). Thus the correlation pattern seems lessrobust in the model with cellular adaptation.

Emergence and disappearance of the correlation pattern ascellular excitability is increased

The results shown in Figs. 2–4 show that simple excitatorynetwork models with slow synaptic depression can generateepisodic activity with the observed correlation between episodeduration and interepisode interval. We note that episodic ac-tivity is observed for a wide range of parameter values (Vlad-imirski et al. 2008) and that we did not need to tune parametervalues to obtain the correlation pattern shown in the precedingsections, suggesting that it is also a robust feature of the episodicactivity generated by these networks. In this section, we focus onthe all-to-all coupled network of Hodgkin-Huxley–type neuronswith synaptic depression (Figs. 3C and 4) and test the robustnessof the correlation pattern on parameter values. A complete param-eter study is outside the scope of this paper; here, we describe theeffects of cell excitability on the correlation pattern, because this

previous following

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A Synaptic depressionco

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FIG. 3. Robustness of the correlation pattern to model implementation. Each plot indicates the correlation coefficient between episode duration and precedingor following interepisode interval obtained from 10 simulations. Filled circles indicate significant correlation (P 0.01). A: results from a network of I&Fneurons with synaptic depression and all-to-all coupling. Input currents were uniformly distributed on the interval [0.15, 1.15] (the minimal input current thatresults in spiking, or rheobase, is 1). Each simulation corresponds to a different, randomly chosen, distribution of bias currents. B: results from a network of I&Fneurons with synaptic depression as in A, but with sparse (10%) connectivity. Here, all simulations used the same, uniformly spaced distribution of bias currenton [0.15, 1.15], but differed in the connectivity matrix. For each simulation, each neuron was assigned to project to 10 randomly chosen postsynaptic neurons.C: results from a network of Hodgkin-Huxley neurons with synaptic depression and all-to-all coupling. Bias currents were uniformly distributed on the interval[�10, 5] (rheobase �3 A/cm2). Each simulation corresponds to a different, randomly chosen, distribution of bias currents. D: results from a network of I&Fneurons with cellular adaptation and all-to-all coupling. Both bias currents and strength of the adaptation process g� were uniformly distributed on the interval[0.5, 1.5]. Each simulation corresponds to 2 different, randomly chosen, distributions for bias currents and g�.



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is motivated by experimental findings (Menendez de la Prida et al.2006).

We noticed that simulations resulting in low correlationsbetween episode duration and preceding interepisode intervalare often the ones for which cell excitability (i.e., input current)in the network is the lowest. This is in agreement with exper-imental findings from disinhibited hippocampus slices (Me-nendez de la Prida et al. 2006), which showed no correlationbetween episode duration and interepisode interval, unlesscellular excitability was increased using a high extracellularconcentration of potassium ions. Figure 6A shows the timecourse of network activity and slow recovery from depres-sion for a network with all cells receiving an additionalhyperpolarizing current of �1.2 A/cm2. In this case, thereare practically no episodes starting before the recovery fromsynaptic depression reaches its asymptotic level. Thus allepisodes start around the same value of s�, so they allhave a similar duration regardless of the preceding interepi-sode interval. Therefore there is no correlation betweenepisode duration and preceding interepisode interval (Fig.6B) or episode duration and the following interepisodeinterval (Fig. 6C).

As the additional bias current is made less hyperpolarizing,episodes can more often start before maximal recovery isreached, increasing the variability of s� at episode onset(Fig. 6D, green dotted curve representing the ratio between theSD of s� at onset and the SD of s� at termination). As

s� at onset becomes more variable relative to s� attermination, the episodes that start at higher s� have a longerduration. That is, a correlation between episode duration andinterepisode interval appears (Fig. 6D, ●). As the bias currentis increased to depolarizing values, episode onset is facilitatedand becomes less variable relative to episode termination. Thecorrelation between episode duration and preceding intervaltherefore becomes lower. Further increasing the bias currentled to activity with less clearly defined episodes, so the detec-tion of episode onset and termination became ambiguous andthe correlation pattern could not be determined. Although notshown here, we obtained similar results with I&F neurons, thatis, low correlation (or no correlation) for very low cellularexcitability, strong correlation at intermediate excitability andagain lower correlation for high cell excitability.

Thus the mechanism of synaptic depression reproduces theexperimental finding that a correlation pattern emerges when the

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0.1

0.15

0.2


freq

uenc

y

sd = 0.096

0 0.1 0.2 0.3 0.4

0.05

0. 1

0.15

0.2


freq

uenc

y

sd = 0.004

A B

C D

E F

< a

>, <

s >

< s >

< a

>

FIG. 4. Episodic activity and correlationpattern of the excitatory network of Hodgkin-Huxley (HH) neurons with synaptic depres-sion and all-to-all coupling. A: time courses ofactivity, a� (thin, black curve), and averagerecovery from depression, s� (thick, lightgreen curve). B: “phase plane” trajectory.C: the correlation between episode dura-tion and preceding interepisode interval ishigh. D: absence of correlation betweenepisode duration and following interepi-sode interval. E: wide distribution of s�values at episode onset. F: narrow distri-bution of s� values at episode termina-tion.



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cellular activity level of a hyperexcitable network is increased.Also, the correlation pattern is robust to changes in cell excitabil-ity, because a significant correlation is observed for most of therange of bias current that supports episodic activity. We obtainedsimilar results when we varied synaptic efficacy instead of cellularexcitability (data not shown). This robustness to parameter values,together with the robustness to model choices (Fig. 3), imply thatthe correlation pattern is a general feature of excitatory networkswith activity-dependent depression.

Mean field model with noise reproduces thecorrelation pattern

Figures 2, 4, and 5 suggest that the mean activity and meandepression variables may provide sufficient information to under-stand the correlation patterns generated by the network models.Thus we now use a mean field model that describes the globalactivity of the whole population in the network in terms of themean firing rate (averaged over the population and over thesynaptic time scale) a and a mean depression variable s (seeMETHODS). Although such models may not capture all the effects ofcellular heterogeneity in a network (Vladimirski et al. 2008), they

qualitatively describe many features of episodic activity (Lathamet al. 2000; Tabak et al. 2000) and can be easily analyzed.

We modified a mean field model of an excitatory network withsynaptic depression (Tabak et al. 2006) by adding noise to themean activity to generate variability in episode duration andinterepisode interval (METHODS). This noise is meant to reflectcoincident firing of a subset of neurons in the finite networkpopulation. If several cells spike within a short period of time,they create a small bump in the mean activity level in the network.The noise we add to the global activity therefore represents thefluctuations caused by correlated activity of subsets of the network(Doiron et al. 2006; Holcman and Tsodyks 2006).

First, we show that the mean field model with noise gener-ates episodic activity with correlation patterns similar to thoseof the network models used above. Figure 7 shows the activityand correlation patterns produced by the mean field model. Theactivity is episodic (Fig. 7A), with slow oscillations of thesynaptic recovery variable s. Furthermore, the variability of s atepisode onset is much greater than its variability at episodetermination (Fig. 7, A, B, E, and F), leading to a correlationbetween episode duration and preceding—not following—

0 500 1000 1500 20000

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< θ >

155 160 165 17043

44

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epis

ode

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tion R = 0.82

155 160 165 17043

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epis

ode

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tion R = 0.01

0.26 0.28 0.3 0.32 0.34 0.360

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freq

uenc

y

sd = 0.0048

0.52 0.54 0.56 0.58 0.60

0.2

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θ value at episode termination

freq

uenc

y

sd = 0.0007

A B

C D

E F

< a

>, <

θ >

< a

>

FIG. 5. Episodic activity and correlationpattern of the excitatory network of I&F neu-rons with cellular adaptation and all-to-all cou-pling. A: time courses of activity, a� (thin,black curve), and average adaptation ��(thick, light green curve). Note that the activityappears more regular than for the previousexamples (Figs. 2 and 4). B: “phase plane”trajectory. C: the correlation between episodeduration and preceding interepisode interval ishigh. D: absence of correlation between epi-sode duration and following interepisode inter-val. E: distribution of �� values at episodeonset. F: narrow distribution of �� values atepisode termination.



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interepisode interval (Fig. 7, C and D). These results arequalitatively similar to the ones obtained for the network modelwith spiking neurons (Fig. 2). Therefore the mean field modelcaptures not only the mechanism for generating spontaneousepisodic activity but also for producing the observed correla-tion pattern.

Qualitative explanation for the correlation pattern

We now take advantage of the simplicity of the mean fieldmodel to perform a geometric analysis of the effects of thenoise that lead to the correlation pattern shown in Fig. 7, C andD. More sophisticated tools have been used previously (Limand Rinzel 2010; Pedersen and Sorensen 2006). Our analysisuses the fact that one model variable (a) is fast, whereas theother (s) is slow.

The mechanism underlying episodic activity in the deter-ministic version of our mean field model can be visualized

using a phase portrait (Fig. 8A). We start by treating s as aconstant. The S-shaped curve represents the possible steadystates of the activity for each value of s; this is the a-nullclineand is defined by a � a�(wsa � �0), where a� is the steady-state output function of the network (see METHODS). For a rangeof values of s, there are three steady states, one on the low branchwith a � 0 (the network is silent), one on the high branch witha � 1 (network active), and one on the middle branch (dashed).Unlike the high and low steady states, the steady state on themiddle branch is unstable and serves as a threshold for thenetwork activity: at a given s, if a is above the threshold value,the activity will quickly equilibrate to the high state; if a isbelow threshold, it will fall to the low state. The S-shapedsteady-state curve allows hysteresis to occur when s is allowedto vary. Thus when s slowly varies (according to Eq. 7), theactivity becomes episodic, switching between the high and lowbranch of the a-nullcline. For example, when activity is high,

0 2000 4000 6000 8000 10000 12000 14000

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2000 4000 6000 8000 10000130

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epis

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tion R = 0.25

2000 4000 6000 8000 10000130

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epis

ode

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tion R = -0.39

-1 -0.5 0 0.5 1 1.5 20

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bias current (µA/cm2)

corr

elat

ion

0

1

2

3

sd r

atio

(Lo

g)

A

B C

D

< a

>, <

s >

FIG. 6. Dependence of the correlation pat-tern on cell excitability. A: time course ofactivity (a�, black) and recovery from de-pression (s�, thick light green) for a networkof 100 HH neurons with all-to-all coupling andsynaptic depression. Input currents to the neu-rons are uniformly distributed on the interval[�10, 5], and an additional bias current of�1.2 A/cm2 is applied to all cells of thenetwork. Note that episodes occur after randomamounts of time once s� has reached itsmaximal value. B: absence of correlation be-tween episode duration and preceding interepi-sode interval. C: absence of correlation be-tween episode duration and following interepi-sode interval. D: correlation coefficientbetween episode duration and preceding in-terepisode interval (circles, black line) andvariability of s� at episode onset relative tovariability at episode termination (dotted greenline), plotted as a function of the bias current.The relative variability is measured as the logof the ratio of the SD of s� at onset and attermination. For the most negative bias cur-rents ( �1 A/cm2), the correlation is notsignificant because the variability of s� atepisode onset is low. As the variability at onsetincreases, the correlation between episode du-ration and interepisode interval becomes sig-nificant and remains high until higher values ofbias current lead to lower relative variability of

s� at episode onset.



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synapses depress so s decreases. The state of the network,represented by a point in the (a, s) plane, tracks the high branchmoving to the left, until it reaches the point where the high andmiddle branch meet [high knee (HK)]. Beyond this point, thereis only one steady state, on the lower branch. Thus activityfalls down and the system then tracks the low branchmoving to the right, as s recovers. This is the silent phase.Beyond the point where the low and middle branches meet[low knee (LK)], the system jumps back to the high branch,starting the next episode, and the cycle repeats.

How does this picture change when noise is added to thesystem? We first consider perturbations to the deterministicsystem and ask whether a sudden input that provokes a jump inactivity level can make the system switch between the active/silent phases. In Fig. 8A, we consider two cases, where thesystem is in either the active or the silent phase and approach-ing the transition point (HK or LK). For the perturbation toinduce a transition, it must bring the system across the middlebranch. If the system is in the silent phase, a small jump in theactivity will successfully bring it above threshold (upwardarrow) and thus switch it to the active state. On the other hand,if the system is in the active phase, a comparatively largedecrease in activity will be necessary to bring it below thresh-old (downward arrow), because the a-nullcline (the S-curve)

has a rounder shape at the high knee. Thus it is more difficultto perturb the system from the active to silent phase than fromthe silent to the active phase. This implies that noise of a givenamplitude will be able to initiate an episode of activity for arelatively wide range of s values compared with the range of svalues where episodes terminate.

However, this is not the only factor explaining the highervariability of s values at episode onset compared with episodetermination. The noise term is in the differential equationdescribing activity. To understand its contribution in the frame-work of deterministic dynamical systems, we consider noise asan extra input to the network, an input that varies quickly withtime but whose effects are integrated over time. If that extrainput was constant, �i, it would affect the shape of thea-nullcline, now defined by a � a�(wsa � �0) � �i. Thisaffects the low knee of the S-curve more than the high knee, asshown on Fig. 8B. That is, an input to the network has a greatereffect on the transition point at episode onset than at episodetermination. If �i is small, the change in the horizontal positionof the knee, �sk, is given approximately by �sk/�i � �sk/ak,where sk is the value of s at the knee, and ak is the activity levelat the knee (as shown in appendix). Thus the ratio of thechanges in the low and high knee postitions caused by a smallperturbing input is �slk/�shk � (slk/shk)(ahk/alk). For the case

0 1000 2000 30000

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0.3 0.4 0.5 0.6 0.7 0.80

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sa

200 250 300 350

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epis

ode

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tion R = 0.97

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epis

ode

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tion R = 0.03

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freq

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sd = 0.013

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freq

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A B

C D

E F

FIG. 7. Episodic activity and correlation pattern of the ex-citatory mean field model with synaptic depression. A: timecourses of activity, a (thin, black curve), and average recoveryfrom depression, s (thick, light green curve). The value of s atepisode onset varies from episode to episode, whereas the valueof s at episode termination remains almost constant, as for thenetwork of spiking neurons (Fig. 2). Noise amplitude was 0.01.B: phase plane representation, where a is plotted as a functionof s for each time point shown in A. C: the correlation betweenepisode duration and preceding interepisode interval is high.D: absence of correlation between episode duration and follow-ing interepisode interval. E: wide distribution of s values atepisode onset. F: narrow distribution of s values at episodetermination. These distributions compare with the ones shownin Figs. 2 and 4.



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shown in Fig. 8B, this ratio is 17.4, so the effect of an input onthe low knee is �17 times greater than the effect on the highknee.

If the input is not constant but noisy, the a-nullcline isconstantly deformed. As shown in Fig. 8B, this will have amuch greater effect at episode onset than episode termination.As an illustration, in Fig. 8C, we plot the time course of thepositions of the knees slk and shk on the s-axis during the simu-lation shown in Fig. 7A (a portion of the time course of s from Fig.7A is also shown as a thick green curve). Because the knees’positions can vary with a large amplitude, for illustration pur-

poses, we filtered them according to �kdLK

dt� slk � LK, where

LK is the filtered low knee position (and with a similar equationfor the high knee HK), with a time constant �k � 10. It is clear thatthe low knee varies more widely than the high knee, with aratio of SD sd(LK)/sd(HK) � 17.4, as predicted from ouranalysis with a small constant input (Fig. 8B). Consequently,

during the silent phase, if the low knee’s position slk fallsbelow the current value of s and does not immediately comeback up, an episode can be triggered even if s is far fromhaving fully recovered. Similarly, if the knee’s position movesup, the occurrence of an episode can be delayed. These early ordelayed transitions occur to a much lesser extent for episodetermination. Thus the difference in the spread of the distribu-tion of the low and high knee locations (Fig. 8, D and E) canexplain the difference in the spread of the values of s at episodeonset and termination (Fig. 7, E and F).

Finally, we note that the higher sensitivity of the lower knee toperturbations (Fig. 8B) does not occur for a model where the slownegative feedback process is cellular adaptation instead of synap-tic depression (Tabak et al. 2006). In the case of cellular adapta-tion, both knees are equally affected by noise, so the correlationpattern mainly depends on the shape of the a-nullcline, which inturn depends on the shape of the network steady state input/outputfunction a�. For a network of I&F neurons, a� has a sharp

0.7 0.74 0.78 0.820

0.1

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Low Knee

freq

uenc

y

sd = 0.019

0.3 0.35 0.4 0.450

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High Knee

freq

uenc

y

sd = 0.0011

1200 1400 1600 1800 2000 2200 2400 2600

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time

s

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1

s

a

0.2 0.4 0.6 0.8 10

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1

s

a

LK

HK

+∆i -∆i

A B

C

D E

LK

HKFIG. 8. Phase plane argument for higher

variability at episode onset. A: the trajectoryof the deterministic model tracks low andhigh branches of the a-nullcline (theS-shaped curve). Episode onset and termina-tion occur at the knees of the S-curve: LK,low knee, the transition point from silent toactive phase; HK, high knee, the transitionpoint from active to silent phase. Note thatthe trajectory slightly overshoots the knees.When the system is close to the low kneeduring the silent phase, a small perturbation(short upward arrow) brings it above thresh-old (dashed portion of the S-curve). Whenthe system is close to the high knee duringthe episode, a stronger perturbation (longdownward arrow) is needed to bring it belowthreshold. B: effect of an input on the a-nullcline. Thick curve, control position as inA; thin curves, effect of small input � �iwith �i � 0.01. The low knee is moreaffected than the high knee. The ratio oftheir horizontal displacements is given by�slk/�shk � (slk/shk)(ahk/alk) � 17.4 for theparameters given in Table 2. C: portion ofthe time course of s from Fig. 7A (thickgreen curve), together with the time coursesof the knees positions slk (LK) and shk (HK).The knees positions are filtered according to�kdLK/dt � slk � LK (and similarly for thehigh knee). Episode onset occurs when spasses above slk, whereas episode termina-tion occurs when s falls below shk. D: dis-tribution of all slks obtained during the entiresimulation. E: distribution of all shks ob-tained during the entire simulation. The ratioof their SD was 17.4. Compare these distri-butions with the distributions shown in Fig.7, E and F.



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increase at low activity and a smooth saturation at high activity,which results in a nullcline with a flatter shape at the bottom(sharp low knee, round high knee, as in Fig. 8A). This leads to asignificant correlation between episode duration and preceding,but not following, interval (data not shown, but observed for thenetwork case, Fig. 5, C and D). However, for a symmetrical a�such as the one used here, the shape of the a-nullcline is symmet-rical too, unlike the nullcline obtained with synaptic depression(Fig. 8A). In that case, noise can induce transitions between activeto silent phase and silent to active phase with equal probability.This results in a weak but significant correlation between episodeduration and both the preceding and following intervals (Lim andRinzel 2010; Tabak et al. 2007). Thus when the slow negativefeedback process is cellular adaptation, the correlation patternmay depend on network parameters. In contrast, for synapticdepression, even if we choose a� such that the S-shaped nullclineis symmetrical, a strong correlation between episode duration andpreceding (not following) interval is still observed because of thehigh sensitivity of the low knee to perturbations.

D I S C U S S I O N

The correlation between episode duration and previous—notfollowing—interepisode interval may represent a signature ofdeveloping networks (O’Donovan 1999), but its cause has notbeen studied until now. Here, we show that two characteristicfeatures of developing networks are sufficient to produceactivity with this correlation pattern: recurrent excitatory con-nectivity and activity-dependent synaptic depression. We showthat this pattern can emerge naturally in diverse developingnetworks because the correlation pattern is robust to changes inconnectivity or cell properties. The correlation pattern is adirect consequence of the high sensitivity to noise of episodeonset compared with episode termination.

What causes higher variability at episode onset?

In developing spinal and hippocampal networks, episodes(spinal cord) or network bursts (hippocampus) of activity arestochastic events, triggered by random fluctuations in basalactivity (Menendez de la Prida and Sanchez-Andres 1999;Wenner and O’Donovan 2001). In the heterogeneous networksof excitatory neurons with synaptic depression used here,episode onset only occurs once a small subpopulation ofneurons with intermediate excitability becomes active (Tso-dyks et al. 2000; Vladimirski et al. 2008; Wiedemann andLuthi 2003). The recruitment of this subpopulation seems to bea stochastic event, requiring the synchronous firing of some ofthe spontaneously spiking cells. Thus the variability observedat episode onset results from heterogeneity (Thivierge andCisek 2008) of cell excitability. In comparison, episode termi-nation is highly predictable, the variability of the averagedepression variable being about one order of magnitude less atepisode termination than episode onset.

To explain this asymmetry between episode onset and epi-sode termination, we used a mean field model with noise. Thestochastic recruitment of a subpopulation of neurons caused byvariations in the degree of synchrony between spontaneouslyfiring cells cannot be incorporated directly in a simple meanfield model. Instead, we modeled the varying degree of corre-lated activity by adding noise to the mean activity a (Holcman

and Tsodyks 2006; Venzl 1976). The mean field model withnoise exhibits a similar correlation pattern and asymmetrybetween episode onset and termination as the network models.Therefore details of individual neuron interactions are notnecessary to produce the correlation pattern, which simplyresults from population dynamics and activity fluctuations.

Here, we considered noise as either a perturbing stimulus ora rapidly varying input to gain intuition into the production ofthe correlation pattern using phase plane analysis. This geo-metric analysis suggests that the greater variability of the levelof recovery from depression at episode onset is caused by tworeasons: 1) a smaller stimulus is required to reset the systemfrom the silent to active than from active to silent phase (Fig.8A); and 2) the sensitivity of the transition point to perturba-tions is one order of magnitude greater at episode onset thantermination (Fig. 8B). Note that 1) is dependent on the shape ofthe network steady state input/output relationship, which it-self is determined by the individual cell properties, degreeof heterogeneity, and synaptic properties. On the other hand,2) requires that the slow process that terminates the episodebe a divisive factor, like synaptic depression.

If the slow negative feedback process is instead subtractive,like cellular adaptation, the correlation pattern can only bepresent because of 1). Thus for a network with cellular adap-tation, the presence of the correlation pattern is not a robustfeature and depends on cell and synaptic properties. If thenetwork input/output function is a symmetrical sigmoid such asthe one used in our mean field model, the correlation pattern isnot present (Lim and Rinzel 2010; Tabak et al. 2007). Thenetwork simulations with cellular adaptation exhibit the corre-lation pattern (Fig. 5) because the input/output function of thenetwork is asymmetrical, being much sharper at low activitythan high activity levels. Thus if the episodic activity generatedby an excitatory network does not exhibit the correlationpattern, our results suggest that the slow feedback processresponsible for episode termination might be of the subtractivetype, i.e., not synaptic depression.

Although these results were obtained using the framework ofdeterministic dynamics to provide an intuitive geometric ex-planation for the correlation pattern, we note that a stochasticanalysis, in the context of single cell oscillations, providessimilar results (Lim and Rinzel 2010). This analysis shows thatthe spread of the distribution of the slow variable at thetransitions between silent and active phases (such as Fig. 7, Eand F) depends on two factors. These two factors are equiva-lent to points 1) and 2) above (shown in appendix).

Does the mechanism described here occur inbiological networks?

Could a different mechanism than the one presented hereproduce the correlation pattern? We are aware of one alterna-tive mechanism that might generate a similar correlation pat-tern. Suppose that there is a slow, unobserved rhythm thatunderlies the appearance of episodes of activity. Episodes canoccur at any phase of this rhythm, but their likelihood andduration are greater before or at the peak of the slow underly-ing rhythm. An episode preceded by a short silent phase wouldmore likely occur before the peak of the slow oscillation andwould be shorter than an episode occurring after a long silentphase (more likely to occur around the peak), hence a corre-



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lation between episode duration and preceding silent phase. Onthe other hand, episode duration would not affect the followingsilent phase, so little correlation would be observed betweenthem. Thus this mechanism might reproduce the observedcorrelation pattern, if certain conditions are met. However, thepresence of the underlying oscillation was not detected in datafrom the chick embryonic spinal cord. This mechanism alsorequires more assumptions than the mechanism presented here.

The mechanism presented in this paper is simple and pro-duces a correlation pattern that is robust, mostly independent ofanatomical or biophysical details or parameter values. We usedall-to-all coupling and random coupling (as well as moreordered architecture, data not shown), integrate-and-fire, andHodgkin-Huxley neurons and found that for all of these net-works the correlation pattern was produced in a similar way.Parameter tuning was not required, and we found that a widerange of parameter values supporting episodic activity pro-duced the correlation pattern. In short, the correlation patternemerges naturally from excitatory model networks with slowactivity-dependent synaptic depression. This mechanism istherefore likely to occur in biological networks regardless ofthe actual biophysical processes found in these networks.Synaptic depression may be caused by transmitter depletion(Jones et al. 2007; Staley et al. 1998), chloride extrusion (Chuband O’Donovan 2001; Marchetti et al. 2005), or activation ofGABAB receptors (Menendez de la Prida et al. 2006), but thesedifferent biophysical processes all underlie the same basicmechanism and therefore should all contribute to produce thesame correlation pattern.

Furthermore, our model makes predictions, some of whichhave been verified experimentally. The high sensitivity to biasinput at episode onset exhibited by the mean field modeltranslates into a high sensitivity of the interepisode interval(but not episode duration) to cell excitability (Tabak et al.2006). This is also observed in the network models (data notshown). Thus if developing and other hyperexcitable networksgenerate the correlation pattern according to the mechanismdescribed here, they should respond to acute changes in exper-imental manipulations affecting cell excitability with a largechange in interepisode interval but a smaller change in episodeduration. This has been shown in bursting hippocampal net-works (Staley et al. 1998). Our model also predicts (Fig. 6) thatthe correlation pattern may be low or absent when cell excit-ability is so low that most episodes do not start before fullrecovery. This is observed experimentally in hyperexcitablehippocampus networks that do not exhibit the correlationpattern unless cell excitability is raised by increasing theextracellular potassium concentration (Menendez de la Prida etal. 2006). More generally, the correlation pattern is observedfor most of the range of parameter values that support episodicactivity, except toward the edges of that range (i.e., at high orlow network excitability). This predicts that the correlationpattern may be weak or absent early in development when theactivity just emerges or later in development as spontaneousactivity disappears.

As developing networks mature, a refinement of the synaptictopology takes place, creating specific network structures. Thischange in structure might affect the correlation pattern. Re-placing long-range, random connections with short-range con-nections can switch the activity pattern of a network fromsynchronized episodes to propagating waves (Netoff et al.

2004). Although spontaneous waves in two-dimensional net-works may exhibit the correlation pattern typical of developingnetworks (Grzywacz and Sernagor 2000), the effect of networkstructure on the correlation pattern may weaken some of ourpredictions and should be studied.

Significance of the findings

Neuronal networks are complex and plastic, and they displaya variety of structures and functions. Can we find unifyingconcepts of neuronal network activity? This study strengthensthe view that developing networks may have a common modeof operation. Not only does the combination of fast excitationand slow activity-dependent depression generate spontaneousepisodic activity in model networks, but it also naturallyproduces the correlation pattern observed in many developingnetworks, regardless of model details. This suggests that net-works with different architectures, different components, anddifferent functions all share a common mechanism of operationearly in development. This could form the basis of a generalframework for the study of neuronal network operations.

As networks mature, GABAergic and glycinergic transmissionswitch from functionally excitatory to inhibitory, silencing epi-sodic activity. This switch, together with the refinement of syn-aptic connectivity, leads to different structures and functions in thevarious networks. Nevertheless, mechanisms for rhythm genera-tion in many mature networks rely on recurrent excitation withslow synaptic depression or cellular adaptation. This is the case ofthe slow wave cortical oscillations during sleep (Compte et al.2003; Sanchez-Vives and McCormick 2000), the inspiratoryrhythm (Rubin et al. 2009), and the pulsatile secretion of oxytocinduring lactation (Rossoni et al. 2008). The same mechanism alsoreadily emerges when the balance between excitation and inhibi-tion is broken (Darbon et al. 2002; Golomb et al. 2006; Staley etal. 1998). Interestingly, nonrhythmic cortical circuits may alsorely on a recurrent excitatory core (Yuste et al. 2005). Wespeculate that much progress in our understanding of neuronalnetworks may be obtained by asking how network structure andinhibitory connectivity alters the basic rhythm generation mech-anism found in immature networks.

A P P E N D I X

Network model with Hodgkin-Huxley–type neurons

Here, we use a reduced Hodgkin-Huxley model for each neuronwith two variables: voltage (V) and activation of delayed rectifierpotassium current (n) (Rinzel 1985). Each neuron is described by thefollowing two equations

CdVidt

� �gleak�Vi � Vleak�

�g�Nam�3 �Vi��0.8 � ni��Vi � Vleak� � g�Kni

4�Vi � Vleak�

� Ii � gsyn�Vi � Vsyn�

dnidt

� �n�Vi��1 � ni� � �n�Vi�ni

where the synaptic conductance is given by gsyn �g�synN

�j�1N ajsj as forthe integrate-and-fire network model, with aj (synaptic drive from



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neuron j) and sj (depression variable for neuron j) calculated asdescribed in METHODS (Eq. 3 and 4) with the �(t) functions nowreplaced by �(V) � 1/(1 � e(Vthresh �V)/kv). Voltage unit is milli-volts, time unit is milliseconds, and the capacitance, conductances,and currents are normalized with respect to surface (with C � 1F/cm2). Integration time step was 0.01 ms. A computer code,along with all parameter values, is available for download fromwww.math.fsu.edu/�bertram/software/neuron.

Sensitivity of the knees’ position to perturbation

We add a perturbing input i to the network activity equation as follows

da

dt� �a � a��wsa � �0� � i (A1)

and ask how this input affects the horizontal location of the knees (i.e.,the value of s at episode onset and termination), as shown on Fig. 8B.The a-nullcline is the curve s(a) for which a is at steady state (da/dt �0). Thus it is defined by

a � i � a��wsa � �0� (A2)

To find how this curve is affected by i, we differentiate Eq. A2 withrespect to i. This results in

�a

�i� 1 � �ws �a

�i� wa

�s

�i�a��wsa � �0� (A3)

The knees are the special solutions of Eq. A2 for which the two sides of theequation are not only equal but have the same slope, that is, same derivativewith respect to a. Thus at the knees, differentiating Eq. A2 gives

1 � wsa��wsa � �0� (A4)

Substituting in Eq. A3, we obtain

�s

�i� �

s

a�at the knees� (A5)

We see that a knee’s position can be highly sensitive to i if activityat the knee is very low; this is the case for the low knee. Activity ishigh at the high knee, so the high knee is not very sensitive to i (cf.Fig. 8B). This is true regardless of the shape of the network outputfunction a�. Here we only assume that a� is differentiable.

Distribution of the slow variable at the transitions

Lim and Rinzel reformulated the equation describing the dynamics ofthe fast variable (a in our case) using a potential function U(a,s) � �[a �a�(wsa � �0)]da. For any fixed s, this function has a double-well shape.The stable steady states of the activity now correspond to the minima ofthe potential and the unstable (middle) steady state corresponds to themaximum separating the two wells. Noise-induced transitions betweenthe low and high states of activity pictured on Fig. 8A are now transitionsbetween the two potential wells. The distribution of the slow variable atthe transitions can be obtained using the probability of transitions abovethe energy barrier �U(s) (Kramers rate). If the noise amplitude is smallsuch that all transitions occur around s � s* and the time scale of s is closeto the time scale of the transitions, the SD of s at the transitions can beapproximately expressed as

sd � Kn2��d�U�s*�ds

� (A6)where K is a constant and n is the noise amplitude. The denominatorrepresents the sensitivity of the depth of the potential well to the slowprocess, which is

d�U�s*�

ds� � waa��wsa � �0��da (A7)

where the bounds of integration are the values of a at the bottom of thewell and at the top of the energy barrier. Because the transitions occurnear the knees of the a-nullcline, we can use Eq. A4 and substitute it

in Eq. A7 to getd�U�s*�

ds� ��a/s�da. Making the approximation

that a and s are close to their values at the knees before the transition,we finally find that the SD of s at the transitions is proportional to1

�a

skak

.

The first term is the inverse of the size of the jump required to crossthe middle steady state shown in Fig. 8A, and the second term is theabsolute sensitivity of the knee position to a perturbation (cf. Eq. A5)represented graphically in Fig. 8B. Both terms are larger at lowactivity level, so the transition from silent to active phase is moresensitive to noise than the transition from active to silent phase.

The second term is the largest and therefore the asymmetricsensitivity of the knees to perturbations is the main factor in theasymmetric sensitivity of the transitions to noise. We emphasize thatthe term s/a appears because the slow negative feedback process is ofthe divisive type. If the model uses cellular adaptation (subtractivefeedback) instead of synaptic depression as the slow process, the terms/a is replaced by the parameter w (the connectivity), so there is noasymmetry between high and low activity. In that case, the only factorthat can cause asymmetry in the sensitivity to noise is 1/�a.

G R A N T S

This work was supported by National Institute on Drug Abuse GrantDA-19356.

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