# Publications

## Resonant synchronization in heterogeneous networks of inhibitory neurons

R. Maex, Erik De Schutter

*Laboratory for Theoretical Neurobiology, Born-Bunge Foundation,University of Antwerp,
B-2610 Antwerp, Belgium*

### Abstract

Brain rhythms arise through the synchronization of neurons and their entrainment in a regular firing pattern. In this process, networks of reciprocally connected inhibitory neurons are often involved, but what mechanism determines the oscillation frequency is not completely understood. Analytical studies predict that the emerging frequency band is primarily constrained by the decay rate of the unitary, inhibitory postsynaptic current. We observed a new phenomenon of resonant synchronization in computer-simulated networks of inhibitory neurons in which the synaptic current has a delayed onset, reflecting finite spike propagation and synaptic transmission times. At the resonant level of network excitation, all neurons fire synchronously and rhythmically with a period about four times the mean delay of the onset of the inhibitory synaptic current. The amplitude and decay time-constant of the synaptic current have relatively minor effects on the emerging frequency band. By varying the axonal delay of the inhibitory connections, networks with a realistic synaptic kinetics can be tuned to frequencies from 40 to >200 Hz. This resonance phenomenon arises in heterogeneous networks with on average as few as five connections per neuron. We conclude that the delay of the synaptic current is the primary parameter controlling the oscillation frequency of inhibitory networks, and propose that delay-induced synchronization is a mechanism for fast brain rhythms that depend on intact inhibitory synaptic transmission.

*keywords: Computer, Delayed Response, Inhibition, Network, Oscillator, Rhythm.*

### 1. Introduction

Networks of inhibitory neurons with reciprocal synaptic connections are found from invertebrate
retina (Hartline and Ratliff, 1972) up to mammalian hippocampus (Cobb et al., 1997), striatum,
thalamus, cerebellum and neocortex (Kisvarday et al., 1993; Tamás et al., 1998; Galarreta
and Hestrin, 1999; Gibson et al., 1999; see McBain and Fisahn, 2001). In many of these brain
regions, fast oscillating local field potentials (from 40 to > 200 Hz) were recorded with
subclasses of inhibitory neurons firing phase-locked to, and often thought to generate, the rhythm
(Bragin et al., 1995; Whittington et al., 1995; Kandel and Buzsáki, 1997; Csicsvari et al.,
1999, 2003; Grenier et al., 2001). The rhythms are typically confined within a frequency band
characteristic of the brain area, the experimental procedure or the associated behavior (for
reviews see Gray, 1994; Buzsáki and Chrobak, 1995; Baçar, 1998; Farmer, 1998; Traub
et al., 1999).

If networks of inhibitory neurons are responsible for maintaining the rhythms, then the constancy
in frequency in each particular case requires either that the individual neurons have an intrinsic
resonance around the oscillation frequency or that populations of inhibitory neurons
preferentially synchronize within the recorded frequency band.

Theoretical studies demonstrated that a homogeneous pair of reciprocally connected inhibitory
neurons lacking intrinsic resonance can synchronize their spikes with zero phase lag, provided a
finite delay separates the generation of an action potential by one of the neurons and the peak of
the synaptic response, or inhibitory postsynaptic current (ipsc), in the paired neuron (van
Vreeswijk et al, 1994; Ernst et al., 1995). Most studies of inhibitory networks, analyzing the
parameter dependency of synchrony, implemented the required delay as a slow rise time of the ipsc,
thereby avoiding the analytical difficulties imposed by the discrete delays of axonal conduction
and synaptic transmission. The frequency selectivity of the rhythms, however, is not completely
understood and was attributed to the decay time-constant of the synaptic response (Wang and Rinzel,
1993; Traub et al., 1996; Wang and Buzsáki, 1996).

Here we demonstrate that the discrete delay in the onset of the ipsc, which is due to the finite
duration of axonal conduction and synaptic transmission, induces a resonance phenomenon in
inhibitory networks, and that the oscillation period at resonance is close to four times the
combined axonal and synaptic delay. A delayed ipsc onset also enables synchrony to develop in
networks with a fast rising ipsc waveform, in accordance with recent experimental findings (Bartos
et al., 2001, 2002; Carter and Regehr, 2002).

The present study elaborates on an inhibitory network model developed for reproducing 160 Hz
oscillations recorded in the cerebellar cortex of transgenic mice deficient for calretinin and
calbindin (Cheron et al., 2001; Maex et al., 2002). The observed frequency constancy made us
search for a mechanism capable of inducing resonance in spatially organized, heterogeneous
inhibitory networks.

### 2. Materials and methods

*Model neurons.* Fast-spiking model neurons had a spherical soma and an unbranched
dendrite of five cylindrical compartments. All compartments had the same surface area and a
passive membrane time-constant of 30 ms. The spike-generating inactivating Na^{+} channel
(NaF) and the delayed rectifier (Kdr) were restricted to the soma. All compartments had a
high-voltage-activated Ca^{2+} channel (CaL), A-type (KA) and combined
voltage/Ca^{2+} activated K^{+} (KC) channels, and a weak anomalous inward
rectifier (H channel). Rate constants were converted to 37 ºC, using Q_{10} = 3 for
voltage-gated and Q_{10} = 2 for ligand-gated channels (Maex and De Schutter, 1998a). The
reversal potential of the leak current was drawn from a uniform distribution between -70 and -60
mV. The neurons did not exhibit subthreshold oscillations or intrinsic membrane resonance, nor did
they fire postinhibitory rebound spikes. Suprathreshold current injection evoked narrow spikes
(0.3 ms width at -40 mV) without firing-rate adaptation (see Fig. 2A in Maex and De Schutter,
1998a). We selected this neuron model, which combines the active properties of a model cerebellar
granule cell and the passive properties of a Golgi cell, because its firing rate increased
linearly with the intensity of applied current over almost the entire dynamic range (from 15 Hz at
the brisk 65pA threshold to 1000 Hz, the maximum firing rate imposed by a 1 ms absolute refractory
period; mean slope 0.65 Hz pA^{-1} for an input resistance of 162 M½).

*Model GABA _{A} receptor synapses.* A GABA

_{A}receptor channel with a reversal potential of -70 mV was inserted on the somatic compartment. Each afferent action potential triggered a unitary conductance increase with a fixed, dual-exponential time course:

*g(t) = (w G A / n) (exp(-(t-d)/τ) - exp(-(t-d) /τ*for

_{r}))*t>d*(and zero otherwise).

(1) The parameters

*w*,

*τ*and

*d*are the relative peak conductance, decay time-constant and delay of the synaptic response (see Figure 2A), and

*t*denotes time elapsed after the rising phase of the afferent action potential crossed the -20 mV level. The parameter

*d*encompasses both axonal conduction and synaptic transmission delays. The normalization constants

*A*and

*n*(the number of afferent synapses) ensured that the summed unitary conductances had a peak amplitude equal to

*wG*when

*τ*or

*n*were varied. The dimensionless parameter

*w*is used only for reference. For

*w*= 1, the summed peak conductance

*wG*was 3 mS cm

^{-2}. The rise time-constant

*τ*determines the shape of the conductance response and was taken

_{r}*τ*/ 27.4 so as to reproduce the ipsc waveform of hippocampal pyramidal neurons (Ropert et al. 1990). For

*τ*= 3 ms, τ

_{r}equals 0.11 ms, which is comparable after temperature correction to the mean rise time-constant of 0.16 ms, recorded in basket cells of dentate gyrus at 34 ºC (Bartos et al., 2002). In selected simulations (see Figure 7

*B*), neurons reciprocally connected through GABA

_{A}receptor synapses also made electrical synapses on the same compartments (Tamás et al., 2000). These gap junctions were pure resistors with a conductance expressed as a fraction of the peak conductance of the associated chemical synapse.

*Conversion to physiological data.* Because conductances are expressed as densities over
the compartments to which the channels are attached, the output of this neuron model is largely
invariant over spatial scaling. For a neuron with total membrane area *S*, the unitary peak
conductance gunitary can be derived as:

(2)

The factor 6 in the denominator converts density over the somatic compartment to density over the
entire model neuron. Hence, taking *wG* = 3 mS cm^{-2}, the afferent synapses of a
neuron with total area *S* = 12000 µm^{2} (Bartos et al., 2001) and *n* = 60
(Wang and Buzsáki, 1996) have a unitary peak conductance of 1 nS.

*Afferent fibers.* The dendritic compartments received AMPA receptor synapses (rise and
decay time-constant: 0.03 and 0.5 ms; summed peak conductance *G* = 16.1 mS cm^{-2};
Maex and De Schutter, 1998a) from a population of afferent fibers. These fibers were Poisson
processes, all firing at the same constant rate throughout a simulation. The time histogram of
spike counts of the fiber population had a flat power spectrum. Because synapses and fibers are
computationally expensive, a diluted population of fibers was simulated. Each fiber represented
the compound spike trains from several tens of excitatory fibers and fired at a rate varied
between simulations from 50 to 12800 spikes s^{-1}.

*Model network configurations.* We first examined how the parameters of the inhibitory
synapse affect the oscillation frequency and power in a network characterized by single values for
*d*, *τ*, and *w*. We then proceeded to validate
the derived relationships on networks with distributed delays and weights, and with stochastic
connections.

In the *one-dimensional network with only nearest-neighbor coupling,* 100 model neurons were
positioned on a one-dimensional array. The neurons made autapses, with zero delay, and synapses on
the closest neighbor on each side with delay *d*. This overtly simplified network has the
advantage of being completely characterized by the single delay value *d*, the single decay
time-constant *τ* and the single synaptic weight value
*w*. We varied systematically the values of *d*,
*τ* and *w* (Figures 1, 2, 3A and 7).

In the *one-dimensional network with distributed delays,* each neuron was connected to all
neighbors within an axon or connection radius *r* (measured in units of inter-neuron distance)
with the delays *d* set proportional to the inter-neuron distances. Assuming a spacing of 300
µm between the neurons and an axonal conduction speed of 0.3 m s^{-1}, the delay to the
nearest neighbor measured 1 ms, and delays to more distant neurons were an integer multiple of
this. The mean delay of the network connections measured (*r*+1)/2. We varied systematically
the values of *r* and *τ* (Figures 3B and 4).

The *two-dimensional network,* representing circuits with a predominantly radial connection
pattern, was an array of 30 by 30 neurons. Their somata were positioned on a triagonal grid so
that each neuron had six equidistant neighbors. Connecting a neuron to all neighbors alike within
a radius *r* makes the delay distribution very skewed and biased towards the largest delay
values. Either the connection probability or the synaptic weight therefore needed to fall off with
distance from the source neuron. In the network documented in Figure 5, synaptic weight decreased
exponentially with a space constant twice the inter-neuron distance. Using *r* = 8 and a
constant connection probability of 0.6, each neuron connected on average to 115 neighbors. The
mean delay of the network was the mean *d* over all connections, each *d* weighted by
the corresponding synaptic weight *w* and with the exclusion of autapses, which did not
contribute to synchrony. We used *τ* = 3 ms.

For the effects of network size and connectivity (Figure 6), *nontopographic networks* were
simulated in which all neurons were stochastically connected with a variable probability. All
synaptic connections had the same fixed delay *d* = 1 ms and
*τ* = 3 ms.

A *population of afferent fibers* was evenly distributed over the entire network space. The
fibers had a conduction speed of 0.3 m s^{-1} and radiated 2.5 mm in a plane perpendicular
to the dendritic shafts, making en passant excitatory synapses on the dendritic compartments with
probability 0.5. There were 8100 fibers in the two-dimensional network and 4000 fibers in the
one-dimensional and nontopographic networks (except when stated otherwise), providing each neuron
on average with 424 and 320 synapses, respectively.

*Boundary conditions.* As in our previous studies, normalizing the unitary synaptic
conductance over the number of afferent synapses prevented that the boundary neurons would fire at
rates different from the network average. This normalization is justified as it is shown that our
main findings are robust to great variability among the neurons in the number of synapses (Figure
6*A*). In addition to having fewer synapses, with consequently stronger weights, neurons
positioned close to the boundaries had a selective deficit of long connections. The resulting
decrease in the mean synaptic delay, which is largest for a neuron positioned at a distance half
the connection radius from the boundary, was relatively small. For example, in the one-dimensional
network with *r* = 16, the mean delay of the inhibitory synapses onto neurons 9 and 92 was
7.2 ms instead of 8.5 ms. Although these inhomogeneities at the boundaries did have a
desynchronizing effect on the entire network, the value of the resonance frequency was hardly
affected (42.65 Hz in a 100-neuron array; 42.18 Hz for the central 100-neuron segment of a
200-neuron array; 42.57 Hz in a 100-neuron ring).

*Noise and heterogeneity.* The networks were noisy and heterogeneous, owing to the
randomness of excitation and the randomization of the resting membrane potentials (see above) and
synaptic weights (see Maex and De Schutter, 1998a). This heterogeneity made the neurons in a
disconnected network fire irregularly and with slightly different rates (coefficient of variation
(CV) of their firing rate, averaged over all excitation levels: 0.046; CV of their interspike
intervals (ISIs), averaged over all neurons: 0.155). These values typically increased in connected
networks because lateral inhibition tended to amplify differences in firing rate (CV of firing
rate in the one-dimensional network with only nearest-neighbor coupling: 0.072; CV of ISIs: 0.210).
Finally, a simulation run started from random membrane potentials distributed uniformly between
-90 and -20 mV.

*Implementation and Simulation.* Neuronal activity was calculated numerically with a
modified version of the Genesis simulator, using
Crank-Nicolson integration in 20 µs steps. We used a random number generator with two seed
variables in order to avoid correlations in firing among the afferent fibers.

*Analysis of network activity.* We monitored network activity as the time series of spike
counts in bins of width 0.5 ms (see Figures 1 *A* and *D*). The power spectrum or
periodogram was estimated, using the Fast-Fourier-Transform algorithm, as the average of 256-point
overlapping Hann (cosine bell) windows (Press et al., 1988). In each window the mean number of
spikes was calculated first and subtracted from the individual counts (or, equivalently, power at
*f* = 0 was set zero), so that the total power almost equaled the variance of the time signal.
Coherent oscillations were quantified by the peak power and the associated center or network
frequency. Network frequency was assessed at higher resolution (512 or 1024 point windows) for
networks with a low resonance frequency.

In order to derive a network’s resonance frequency, the network was excited to various levels
by incrementing the mean firing rate of the afferent fibers by a factor of
at each new run. The obtained average
firing rates of the neurons encompassed their entire dynamic range (10 to > 900 spikes
s^{-1}). A simulation run produced at least 1000 spikes for each neuron (500 in
two-dimensional networks), and had a minimum duration of 3s. Over such long runs, power spectra
were well reproducible and independent of initial values. From the power spectra obtained at
different levels of excitation, a tuning curve was constructed after application of the following
normalization procedure. We first noticed that a disconnected network produced a power spectrum
with a single peak that was located at the mean neuronal firing rate. This peak had a height
almost proportional to firing rate. In accordance with this, the variance of the time series of
spike counts scaled almost linearly with the mean spike count. This relationship between power and
firing rate is compatible with a Poisson distribution of spike counts in the bins of the time
signal. We therefore divided the peak power obtained at each level of excitation by the associated
mean neuronal firing rate, correcting in this way for the power already present in the
disconnected network. Finally, peak power scales in a synchronous network quadratically with the
number of neurons *N*, and was therefore normalized to a network of *N*=100. All
frequency-tuning curves plot peak power, divided by the mean neuronal firing rate and by
(*N*/100)^{2}, and multiplied by a scaling factor 1000.

Resonance frequency (*f _{R}*) is the mean frequency of the tuning curve, which was
constructed from at least eight levels of excitation. Taking this weighted average reduces errors
due to sampling of the tuning curve. If the curve's central peak was positioned very
asymmetrically, the low-power tail was cut in order to avoid a systematic over- and
under-estimation of

*f*in networks with low and high

_{R}*f*, respectively. Some tuning curves had a satellite peak at lower frequencies, and

_{R}*f*may have been slightly underestimated in these networks (see e.g. Figure 6

_{R}*B*). Overall, errors on

*f*are judged to be small because of the systematic attraction of network frequency towards

_{R}*f*(see Figure 5

_{R}*B*).

Peak power is a combined metric of synchrony and rhythmicity, and quantifies the network oscillations in a manner comparable to the analysis of experimentally recorded local-field potentials. A metric frequently used in the computational literature is the coherence index (see Wang and Buzsáki, 1996; Bartos et al., 2001, 2002; or variations of it in White et al., 1998; Tiesinga et al., 2000). The coherence index (

*CI*), taking values between 0 and 1, measures exclusively synchrony and assesses, averaged over all neuron pairs of the network, the fraction of coincident spikes. The probability of coincidence increases with the bin width of the discretized spike trains, which by convention is taken one tenth of the oscillation period. Measuring only the

*CI*is justified in the above studies, because the neurons being excited by constant-current injection were almost perfect oscillators and the inhibitory synapses constituted the only synchronizing mechanism. In the present study, in contrast, excitation is provided by afferent fibers, which carry random spike trains that can contribute to the synchronization of common efferent neurons (Maex et al., 2000). We mention for each spike raster plot the

*CI*in the figure legend.

### 3.Results

We investigated the critical parameters for synchronization and frequency control in a
heterogeneous network of model inhibitory neurons lacking themselves intrinsic resonance. Randomly
firing afferent fibers excited the neurons by continually activating AMPA receptor synapses on the
dendrites. This noisy, but realistic way of driving the neurons accentuated network-induced
rhythmicity. Each neuron inhibited its neighbors, within a varying axon radius, through
GABA_{A} receptor synapses located on the somatic compartment. The delay *d*,
strength *w*. and decay time-constant τ of the inhibitory
synaptic response (the ipsc, see Figure 2*A*) were varied, and their effect on network
dynamics was compared at various levels of network excitation.

**High-frequency oscillations emerge in networks with axonal and synaptic delays**

For a better understanding of the resonance mechanism, we first illustrate the results of a
one-dimensional array in which a neuron is connected on each side only to the closest neighbor.

Figure 1*B* demonstrates waxing and waning high-frequency oscillations in a network in which
the combined synaptic and conduction delay *d* was set to 1 ms, which is a typical latency
value for inhibitory synaptic currents evoked from nearby neurons in paired recordings (Bartos et
al., 2001). As a control, no oscillations were observed in the same network if the onset of the
ipsc was instantaneous (delay *d* = 0) (Figure 1 *A,C*). This latter finding is in
accordance with analytical studies proving that synchronous firing is unstable if the rise time of
the synaptic response is shorter than the duration of the afferent spike (van Vreeswijk et al.,
1994). However, using a synaptic response function with slower rise time compared to this control,
e.g. an α -function with instantaneous onset and peak conductance
at 1 ms, did not improve synchrony in this sparsely connected network without axonal and synaptic
delays.

The oscillation frequency and power were measured at various levels of network excitation spanning
the neurons’ entire dynamic range. Figure 1*F* shows that the tuning curve of the
delayed network was centered at a resonance frequency *f _{R}* = 210 Hz, with power
falling off steeply at lower and higher levels of excitation. In the network without delays, no
robust oscillations could be evoked at any frequency (Figure 1

*E*).

At the resonant level of network excitation, the mean firing rate of the neurons was close to the oscillation frequency of the network because each neuron fired a single spike at almost every cycle (Figure 1

*D*). At nonoptimal levels of excitation, the neurons changed their firing pattern so as to maintain a network frequency deviating from their firing rate but closer to

*f*. In particular, at low excitation levels, neurons skipped cycles of the oscillation. As a consequence, adjacent neurons could fire in alternate order (antiphase synchrony) at a rate half the network frequency (open data points in Figure 1

_{R}*F*). At high excitation levels, neurons fired multiple spikes at each single cycle, without affecting much the cycle duration. This doublet or burst firing was most prominent in networks with delays larger than the present 1 ms value (not shown).

**Figure 1.**

Resonance in a delayed inhibitory network. The activity of a one-dimensional,
nearest-neighbor-coupled network of 100 neurons is illustrated for two delay values of the
synaptic response: d = 0 ms (left panels, A, C, E) and d = 1 ms (right panels, B, D, F). The
synaptic decay time-constant t was 3 ms. A and B, Time histogram of the number of spikes generated
by the entire population of neurons (bin width 0.5 ms). Horizontal bar indicates zero level. C and
D, Raster plot of spikes fired during the time interval indicated by the horizontal bar in A and
B. Each dot represents a spike (horizontal axis, time; vertical axis, position of the neuron along
the array). In the network with d = 1 ms (D, CI = 0.110), most neurons fired at each oscillation
cycle and hence the network frequency (187 Hz) was close to the mean neuronal firing rate (181±10
spikes/s). In the network with zero delay (C, CI = 0.087), spikes were produced at a similar rate
(172±16 spikes/s) but their spatiotemporal patterning was lost. E and F, Insets show periodograms
(units (0.5ms)-2) of the spike count histograms shown in A and B, averaged over 7s. Tuning curves
plot peak power and network frequency obtained at various levels of network excitation (filled
circle in each tuning curve: data from A and B). In the simulations represented by open points,
adjacent neurons fired in anti-synchrony (van Vreeswijk et al., 1994), and hence network frequency
was twice the mean neuronal firing frequency.

**The delay parameter determines the resonance frequency**

Several studies of inhibitory networks demonstrated that the oscillation frequency decreases
when the strength, decay time-constant or delay of the inhibitory synaptic response are increased
(Bush and Sejnowski, 1995; Traub et al., 1996; Wang and Buzsáki, 1996; Pauluis et al.,
1999; Bartos et al., 2002; Liley et al., 2002). Figure 2 illustrates the complementary effects of
these parameters, i.e. the effects on a network's frequency-tuning curve.

The frequency tuning was almost invariant over the value *w* of the synaptic strength (Figure
2*B*). In the network with a resonance frequency *f _{R}* = 210 Hz, illustrated
above, a variation in peak conductance by a factor of 8 suppressed the mean firing rate on average
by 51%, but

*f*decreased only 14% (from 225 to 183 Hz). We disregard this weight parameter further and present the tuning curve with largest power for networks simulated at various weight values. Increasing or decreasing the exponential decay time-constant τ of the postsynaptic current displaced the tuning curve to lower and higher frequencies, respectively (Figure 2

_{R}*C*). The resonance frequency

*f*decreased from 338 Hz at τ = 0.75 ms to 149 Hz at τ = 12 ms. This effect was small, however, compared to the inverse relationship obtained between

_{R}*f*and the delay

_{R}*d*of the synaptic response (Figure 2

*D*). Varying

*d*from 0.25 to 4 ms decreased

*f*from 536 to 65 Hz.

_{R}Over a broad range of values of the delay

*d*and the decay time-constant τ,

*f*approximated 1/(4

_{R}*d*) (Figure 3

*A*). At the lowest delay values (0.25 and 0.5 ms), the measured

*f*was less than 1/(4

_{R}*d*), as the network frequency approached 1000 Hz, the maximal neuronal firing rate. Networks with a (single) large delay value, on the other hand, could have multimodal tuning curves because at high excitation levels the neurons started firing bursts, in particular when τ was much less than 4

*d*(the oscillation period at resonance). Such unrealistic networks did not always have a clear optimal frequency, and their data were not included in Figure 3

*A*(but see next section).

A different value of τ was optimal for different values of

*d*(Figure 3

*A*), a non-optimal τ driving

*f*off the predicted frequency of

_{R}*1/(4d)*with a considerable loss of power (Figure 2

*C*). For example, oscillations about 250 Hz (obtained with

*d*= 1 ms) were optimal with τ = 0.75-3 ms. Oscillations about 62.5 Hz (

*d*= 4 ms) required a τ of 6-12 ms, a value comparable to the recorded 10 ms decay of (probably compound) ipscs during hippocampal gamma oscillations (Traub et al., 1996). With τ

*≤*3 ms, networks with only-nearest-neighbor coupling were unable to generate oscillations slower than 100 Hz.

**Figure 2.**

The dependence of resonance frequency on the parameters of the
inhibitory synaptic response. A, The unitary inhibitory synaptic response, which is a transient
conductance increase of the GABAA receptor channel (bold curve), follows an action potential of
the presynaptic neuron after a delay d, reaches (relative) peak conductance w, and decays
exponentially with time-constant t. The middle trace is the (vertically offset) somatic
postsynaptic potential. B-D, Tuning curves of a network with only nearest-neighbor coupling for
various values of w, t and d. For each parameter set, the resonance frequency is indicated by the
corresponding symbol at the top of the panel. B, Tuning curves for four values of w (d = 1 ms; t =
3 ms; w = 1 corresponds to a synaptic peak conductance density G = 3 mS cm-2). C, Tuning curves
for five values of t, indicated in ms (d = 1 ms; w = 1 except for t = 12 ms where w = 0.5). D,
Tuning curves for five values of d, indicated in ms (w = 1; t = 3 ms except for d = 4 ms where t =
6 ms).

**Distributed delays increase the robustness of frequency control**

Increasing the connection radius *r* of the one-dimensional network, with each neuron now
connected on each side to its *r* nearest neighbors, improved the synchronization process as
assessed from a rise in the *CI* from 0.11 to > 0.3 (Figure 4*A*). Networks with larger
connection radii produced oscillations of greater amplitude and lower frequency (Figure 4*B*).
The period at resonance was equal to about four times the *mean* delay *d* of the
synaptic response (Figure 3*B*).

Increasing the connection radius made the oscillations more robust at nonoptimal values of
τ. As each neuron was connected to neighbors positioned at various
distances, within a fixed axon radius *r* and with delays proportional to distance, a
rectangular distribution of delays was formed. Inhibition through synapses with distributed delays
made resonance less dependent on the appropriate value of τ
because synchronously firing neurons at increasing distances activated their synapses with
staggered delays. More particularly, in a linear array, the most distant afferent neuron would
activate its synapse with a delay almost twice the mean delay, i.e. halfway in the oscillation
cycle. This temporal summation of ipscs induced a more lasting inhibition in the postsynaptic
neuron, comparable to that induced by the activation of a single slowly decaying synapse. As a
consequence, with a realistic decay time-constant as small as 1.5-3 ms (Bartos et al., 2001;
Carter and Regehr, 2002), resonance could be induced in the entire gamma-frequency range and
beyond, by varying only the size of the axonal connection radius *r* (Figures 3*B* and
4*B*).

The same relationship between mean delay and resonance was observed in two-dimensional networks
provided the connection probability between two neurons, or the connection weight, tapered off
with distance so as to obtain a uniform distribution of synaptic delays, or weighted delays. For
example, the tuning curve in Figure 5*A* peaked at 78 Hz, i.e. at a period four times the
mean weighted delay of network connections (3.2 ms). The tuning was sharp, oscillations being only
produced within a narrow frequency band around *f _{R}* for levels of excitation
ranging from 12 to 375 spikes s

^{-1}, i.e. for all but the two rightmost excitation levels plotted in Figure 5

*B*. Hence, although resonance was achieved when the neuronal firing rate equaled the oscillation frequency, oscillations at frequencies close to

*f*were generated over a broad domain of firing rates, encompassing almost the entire dynamic range of the neurons. Indeed, levels of excitation that produced average firing rates from 22 to 913 spikes s

_{R}^{-1}in a disconnected network evoked oscillations in the connected network restricted to a 47-149 Hz band (Figure 5

*B*).

Figures 5

*C*and

*D*illustrate in greater detail some observations already mentioned for the one-dimensional network. At resonance (Figure 5

*C*), the mean neuronal firing rate (73.4 spikes s

^{-1}) was close to the network frequency on the periodogram (78 Hz,

*Ca*). The neurons fired a single spike at each cycle, producing a single peak at 13 ms on the averaged interspike-interval histogram (

*Cb*). The raster plot (

*Cc*) and action potential trajectories (

*Cd*) further demonstrate that synchrony was very robust though not very precise (CI = 0.287). At the next lower level of excitation (Figure 5

*D*), network frequency (72 Hz,

*Da*) was higher than the average firing rate (45 spikes s

^{-1}) and close to

*f*(83 Hz). The interspike interval histogram (

_{R}*Db*) showed a main peak at 25 ms, i.e. about twice the period at resonance. The synchrony induced at the onset of excitation rapidly faded away (

*Dc*), but 72 Hz oscillations kept waxing and waning throughout the simulation (

*Dd*).

**Figure 3.**

Predicted versus measured resonance frequency. The resonance frequency fR derived from the tuning
curves (horizontal axis) is compared to 1/(4d) (vertical axis). The diagonal lines are the
identity curves, indicating perfect prediction. A, Network with only nearest-neighbor coupling,
for various delays d of the synaptic response (from 0.25 to 16 ms). Symbols represent different
values of the decay time-constant t, indicated in ms. For each (d, t) pair, the network with
weight w producing maximal power was selected. The effect of varying w, which is not illustrated,
was always smaller than the effect of varying t. B, Network with d = 1 ms from A, for increasing
degrees of connectivity (axon connection radii r from 1 to 16) and hence for increasing values of
the mean delay d (from 1 to 8.5 ms). At the highest connectivity (r = 16), resonance was clustered
between 34 Hz (t = 24 ms) and 44 Hz (t = 1.5 ms).

**Figure 4.**

Inverse relationship between the connection radius r of the one-dimensional network and the
resonance frequency fR. A, Spike raster plots compared for various axonal connection radii r (in
units of interneuron distance; t = 3 ms; w = 1). The raster plots (width 100 ms) were obtained at
firing rates close to resonance for r = 16 (Aa, 47 Hz oscillations, CI = 0.303), r = 8 (Ab, 70 Hz,
CI = 0.302), r = 4 (Ac, 125 Hz, CI=0.234) and r = 2 (Ad, 172 Hz, CI = 0.153). Same format as in
Figure 1D, which shows the raster plot for r = 1. B, The frequency-tuning curves of networks with
the indicated connection radii r. Tuning curves for r %/1-15£ 4 were obtained with a 100-neuron
array. The tuning curves for r = 8 and r = 16 were calculated from the central 100-neuron segment
of a 200-neuron array.

**Resonance arises in networks with sparse, asymmetrical connections **

In order to determine the critical synaptic number for resonance, the mean number of
connections per neuron was varied, while total synaptic weight was kept constant, in a network
without topographic ordering, i.e. with each neuron able to connect to each other neuron with a
single delay *d* = 1 ms. A network with a mean number of five synapses per neuron produced at
resonance a power close to that achieved in the fully connected network (Figure 6*A, open
circles*). This threshold did not depend on the connection rule, because power hardly increased
when the network was randomly connected using the restriction that all neurons received the same
number of synapses (Figure 6*A, black diamonds*). This absolute threshold for synchronization
was further independent of the size of the network and the summed synaptic weight, and it was
always less than the number of synapses needed to obtain synchrony in a network without delays
(Figure 6*A, dashed curve*).

The tuning curves (Figure 6*B*) of the sparsely connected network (5 synapses per neuron,
*open squares*, *f _{R}* = 225 Hz) and the fully connected network (100 synapses,

*black squares*,

*f*= 214 Hz) were centered at about the same resonance frequency as the tuning curve of the network with only nearest-neighbor coupling, shown in Figure 1

_{R}*F*(

*f*= 210 Hz). The sparsely connected network was the more narrowly tuned (Figure 6

_{R}*B*) because antiphase synchronization produced a broader fall-off at low frequencies in the fully connected network.

**figure 5**

Resonance in a two-dimensional network. A, Tuning curve of a triagonal network of 900 neurons,
with a mean weighted delay d = 3.2 ms (decay time-constant t = 3 ms). B, Frequency range of the
emerging network oscillations. Over the various levels of excitation, the mean neuronal firing
rate varied from 12 to 598 spikes s-1 (horizontal axis). The resulting network oscillations were
clustered on the frequency axis (vertical axis) around the resonance frequency (fR = 83 Hz; range
47-149 Hz), except at the two strongest levels of excitation, where power was almost zero. The
arrows labeled C and D indicate the levels of network excitation illustrated further in panels C
and D, respectively. C, Description of the firing pattern at about the resonant level of
excitation (mean firing rate 73.4 spikes s-1). Ca, Periodogram of network activity over a
simulation run of 8 s duration. Cb, Mean interspike interval (ISI) histogram averaged over all 900
neurons. Cc, Time histogram of population spike counts (bin width 0.5 ms; first peak truncated at
200 spikes) and raster plot of the individual spikes fired during the first 200 ms of the
simulation (CI = 0.287). Cd, Membrane potential traces of 9 randomly selected neurons (the neurons
with array indices 100, 200, ... 900). Horizontal lines indicate the zero and -60mV levels.
Unequal spike heights are a graphical sampling artifact. D, Description of the firing pattern at
the next lower level of excitation (mean firing rate 45.0 spikes s-1). Da, Periodogram, multiplied
by a factor of ten, over a simulation run of 12 s duration. Db, Mean ISI histogram. Dc, Time
histogram of population spike counts and raster plot of the individual spikes fired during the
first 200 ms of the simulation (CI = 0.142). Dd, Population spike count histogram and raster plot
of the individual spikes fired between 8.2 and 8.4 s after the start of the simulation (CI =
0.092, or 0.147 after correction for the mismatch between firing rate and network frequency
(Tiesinga and Jos?, 2000)).

**The effects of noise and heterogeneity**

The randomization of synaptic weights and resting membrane potentials and the randomness in the
activation of excitatory synapses made the present network heterogeneous and noisy.

We excited the nontopographic network in a noiseless manner by injecting a constant current into
each neuron. The raster plots in Figures 6 *C* and *D* compare the firing patterns of
the sparsely (6*C*) and fully connected network (6*D*) during excitation by randomly
firing fibers (*Ca* and *Da*) or an equivalent constant current (*Cb* and *Db*).
Removing the noise improved synchrony, as assessed from the increase in CI from 0.178 (*Ca*)
to 0.381 (*Cb*) and from 0.237 (*Da*) to 0.465 (*Db*).

The residual asynchrony of the fully connected network (*Db*) must be attributed to the
heterogeneity of neurons and synapses. The homogeneous network produced a CI = 0.98 (not shown).
We quantified in an indirect way the present degree of heterogeneity by increasing the variability
of current intensity among the neurons of the homogeneous network. Distributed currents drawn from
a Gaussian distribution with a CV equal to 0.02 reduced the CI from 0.98 to 0.453, a value
comparable to the CI of 0.465 in the raster plot of Figure *Db*.

The noise level can also be changed by varying the number of afferent fibers. Because the membrane
potential has a lower variance when large numbers of afferents excite a neuron through weak
synapses, disconnected neurons fire more regularly when the number of fibers increases, whereas
smaller numbers of fibers, activating stronger synapses, synchronize the disconnected neurons more
precisely (see e.g. Maex and De Schutter, 1998a,b; Maex et al., 2000). We examined which effect
predominates in the generation of network oscillations and always observed that power increased
with the size of the fiber population (Figure 7*A*). In the limit of an infinite number of
fibers, excitation by fibers is identical to the injection of a constant current. The tuning curve
constructed by current injection did not differ in shape from that obtained with afferent fibers
(Figure 7*B, black squares*, *f _{R}* = 209 Hz).

Varying the degree of heterogeneity altered the peak power over almost two orders of magnitude without affecting much

*f*, although there was a tendency for low-frequency oscillations to be more robust than high-frequency oscillations.

_{R}**Adding gap junctions stabilizes the oscillations but does not change
f_{R}**

Closely spaced inhibitory neurons form dendritic gap junctions (> 60% of neuron pairs separated
< 50 µm: Galarreta and Hestrin, 1999; Gibson et al., 1999; all reciprocally connected pairs in
Galarreta and Hestrin, 2002). In accordance with previous modeling studies (Traub et al., 2001;
Bartos et al., 2002), the power of the network oscillations increased in a non-frequency-selective
manner when adjacent neurons were connected by purely resistive electrical synapses (Figure 7*B,
open diamonds*). Hence, although the electrotonically transmitted afferent spike conceals the
initial phase of the ipsp (see Tam‡s et al., 2000), this apparent increase in delay of the ipsp
onset does not decrease the resonance frequency of the network (*f _{R}* = 210 Hz).

**Figure 6**

The threshold number of synapses for resonance is small. A, The power at about the resonant level
of network excitation (see arrow in B) is compared for two stochastic connection rules in a
network of 100 neurons with varying degree of connectivity (open circles and black diamonds; delay
d = 1 ms and decay time-constant t = 3 ms). Either each neuron was connected with a fixed
probability to each other neuron, producing a binomial distribution of synaptic numbers (black
diamonds; horizontal axis gives mean number of synapses per neuron) or, alternatively, for each
neuron a fixed number of afferents was randomly selected (open circles). In the control
simulations (dashed curve), the synaptic response was modeled as an a-function with zero delay and
a time-constant of 1 ms, producing a peak conductance at 1 ms and a resonance frequency at 300 Hz.
B, Frequency-tuning curve of the sparsely connected network with on average 5 synapses per neuron
(open squares; see left arrow in A) and of the fully connected network (black squares; 100
synapses, see right arrow in A). C, Raster plots of spikes in the sparsely connected network (mean
number of 5 synapses per neuron). Note that neurons 11 and 53 were not connected to by any other
neuron using this stochastic connection rule. The neurons were excited either by afferent fibers
(Ca; CI = 0.178) or by the intrasomatic injection of the equivalent constant current (Cb; CI =
0.381). D, Raster plots for the fully connected network excited by afferent fibers (Da; CI =
0.237) and current injection (Db; CI = 0.465).

**Figure 7**

Noise, heterogeneity and gap junctions modulate the power over more than an order of magnitude
without affecting the resonance frequency. Tuning curves were calculated by simulating the
one-dimensional network with only nearest-neighbor coupling (d = 1 ms, t = 3 ms); plain curves are
the tuning curve from Figure 1 F. All tuning curves yield a resonance frequency between 205 and
210 Hz. A, The effect of varying the size of the population of afferent fibers (from 1000 to 8000).
B, Tuning curve constructed by incrementing the intensity of a fixed constant current injected
into each neuron's soma (black squares), and tuning curve obtained during fiber excitation after
the addition of electrical synapses between mutually inhibitory neurons (open diamonds).
Electrical synapses were pure conductors, with a conductance 0.4 times the unitary peak
conductance of the chemical synapses between the same compartments.

**The mechanism of synchronization and frequency tuning**

Pairs of reciprocally connected inhibitory neurons tend to fire synchronously if each neuron,
upon firing, resets in an appropriate way the firing cycle of the paired neuron. In the present
model neuron, activation of an inhibitory synapse delayed the generation of a spike, and the
resulting increase of the interspike interval was greater the later in the firing cycle the
synapse was activated (Figure 8*A*, and the phase response curve in 8*B*). As a
consequence, of a pair of neurons reciprocally connected through inhibitory synapses, the neuron
firing earlier received inhibition later and postponed its next spike more (Figure 8*C*).
Repetition of this mechanism progressively leads to a more precise synchronization of successive
pairs of spikes (Ernst et al., 1995).

More precisely, let two mutually inhibitory neurons, denoted by superscripts *1* and *2*,
fire regularly with the same period *T* in a disconnected network. If the neurons generated
their most recent spikes, indicated by subscript *i*, at *t _{i}^{1}* and

*t*=

_{i}^{2}*t*+ Δ

_{i}^{1}_{i}, with Δ

*, then they receive inhibition in a connected network at time*

_{i}< d*t*and

_{i}^{2}+ d = t_{i}^{1}+ Δ_{i}+ d*t*, (3)

_{i}^{1}+ di.e. with a phase equal to

*φ*and

^{1}= (t_{i}^{1}+ Δ_{i}+ d - t_{i}^{1}) / T = (d + Δ_{i}) / T*φ*. (4)

^{2}= (t_{i}^{1}+ d - t_{i}^{2}) / T = (d - Δ_{i}) / TIf the phase response curve has a linear form

*(t*,

_{i+1}- t_{i}- T) / T = b + a φthen the neurons will generate their next spikes at time

*t*and

_{i+1}^{1}= t_{i}^{1}+ T + T (b + a (d + Δ_{i}) / T)*t*, (5)

_{i+1}^{2}= t_{i}^{2}+ T + T (b + a (d - Δ_{i}) / T)giving

*Δ*. (6)

_{i+1}= t_{i+1}^{2}- t_{i+1}^{1}= Δ_{i}(1 - 2a)The neurons synchronize asymptotically if ½ Δ

*, which requires*

_{i+1½ <}½ Δ_{i½}*a < 1*(assuming

*a > 0*). Figure 8

*C*depicts the sequence of events for phase response curves with slope

*a < 0.5*. For

*0.5 < a < 1.0*, the paired neurons still converge to synchrony but the order of their spikes reverses at each cycle.

The frequency tuning of the network oscillations, and more particularly the 1/(4

*d*) frequency observed for resonance, can be understood from the effect of inhibition on a neuron's instantaneous firing probability. When the 1

^{st}neuron of a pair has fired, the 2

^{nd}neuron lowers its firing rate following the onset of inhibition, i.e. after an interval equal to the delay

*d*. Alternatively, the 2

^{nd}neuron may have fired first, preceding the 1

^{st}neuron within an interval of the same duration, so that the 2

^{nd}neuron will preferentially fire in an interval [-

*d*,

*d*] around each spike of the 1

^{st}neuron (Figure 8

*D*). The sine wave covering best this

*2d*window of increased firing has a

*4d*period, and resonance occurs when both neurons, driven by the appropriate amount of excitatory input, fire on average with

*4d*intervals. In addition, for

*4d*to be the optimal period, inhibition must ensure that the

*2d*windows of increased firing alternate with

*2d*intervals of suppression of firing. This effect of the duration of inhibition explains the existence of an optimal decay time-constant for each delay in Figure 3

*A*.

**Figure 8**

The mechanism of synchronization and frequency tuning. A, Activation of an inhibitory synapse
resets the firing cycle of the model neuron. The synapse is activated after 15 (upper membrane
potential trace), 27 (middle trace) and 35 ms (lower trace) in a 36 ms firing cycle. Dots indicate
time of synaptic activation and vertical dashed line gives end of cycle in the absence of
inhibition. B, Phase response curve, plotting the delay in spike generation following the
activation of an inhibitory synapse at varying time instants (Rinzel and Ermentrout, 1999).
Abscissa and ordinate variables are expressed as a fraction of the duration of the firing cycle.
The strength of the activated synapse was 1 or 3 mS cm-2 (filled and open symbols, respectively).
A phase resetting curve parallel to the diagonal line corresponds to a fixed delay elapsing
between the activation of the inhibitory synapse and the generation of the next spike. C,
Heuristic diagram of the synchronization mechanism. In one cycle, the interval between a pair of
spikes from mutually inhibitory neurons (vertical bars) decreases from Dti < d to Dti+1 (see
Results). The broken vertical bars represent the expected spike times in the absence of inhibition.
D, Relative timing of spikes from two adjacent neurons in the simulated network of Figure 1 (d = 1
ms). Each spike of the reference neuron is represented as a dot, positioned according to the
absolute time interval made with the previous (Dt+, horizontal axis) and next spike (Dt-, vertical
axis) of a neighbor neuron. Most spikes led or lagged the neighbor neuron by an interval less
than the delay d of the synaptic response (shaded areas).

### 4. Discussion

Delayed reciprocal inhibition is able to synchronize, with zero phase lag, homogeneous networks
of pulse-coupled oscillators and was proposed to be a mechanism of neuronal synchronization (van
Vreeswijk et al., 1994; Ernst et al., 1995). Simulating heterogeneous hippocampal interneuron
networks, Bartos et al. (2002) concluded that ‘a rapid inhibitory signal generated with a
certain delay is a very effective synchronization signal‘. The present study demonstrates
that the resulting oscillations in network activity are limited to a narrow frequency band
constrained by the ipsc latency but rather insensitive to the ipsc strength and decay
time-constant. Small, realistic axonal and synaptic delays, considered negligible compared to the
low-pass time-constant of the neuronal membrane (Manor et al., 1991), have a dramatic effect on
the frequency spectrum (Figure 1). The induced synchrony may be less precise than that which was
achieved without explicit delays in more homogeneous networks (Wang and Buszáki, 1996), but
the robustness of the oscillations, their sharp tuning and frequency constancy, and their
emergence in sparsely connected networks favor a model of delay-induced synchrony for fast brain
rhythms. The extreme frequency control in these networks can be appreciated from the fact that
variations in network heterogeneity did not affect the resonance frequency even when power changed
by almost two orders of magnitude.

Oscillations obeying a 1/(4*d*) rule are observed in many biological systems (May, 1976;
Glass and Mackey, 1988; MacDonald, 1989). In previous, analytical models of oscillatory behavior
in populations of neurons, delays due to signal propagation along axons and dendrites were lumped
into the time-constants of linear or nonlinear differential equations (Wilson and Cowan, 1972;
Freeman, 1975). However, the incorporation of discrete delay variables can profoundly alter the
dynamics of a system (MacDonald, 1989). Linear first-order delay systems can oscillate with a
4*d* period, and nonlinear delay differential equations can exhibit, at a critical delay
value, a bifurcation from a steady state to a limit cycle solution with period roughly equal to
four times the delay period. In the Appendix, we derive a formal explanation for our present
findings.

**Sources of delays in neural networks**

The present results hold irrespective of the source of the delay between the timing of an
inhibitory neuron's action potential and the onset of the current response in the paired
postsynaptic neuron. Because the range of the delays that neural circuits are able to produce
constrains the frequency range of the oscillations to which the present model may apply, we
briefly discuss some prevalent mechanisms of delayed inhibition.

Although inhibitory neurons are fast processing units, with an axon tree often confined within 600
µm from the soma (Buhl et al., 1994; Sik et al., 1995), some (partly myelinated) axons can extend
several millimeters in hippocampus (Sik et al., 1994) and neocortex (Kisvarday et al., 1993; Gupta
et al., 2000). Salin and Prince (1996) electrically evoked in neocortical pyramidal cells *in
vitro* monosynaptic ipscs with latencies of 1 to > 6 ms, and derived slow speeds for spike
propagation (0.06-0.2 m s^{-1}, mean 0.1). From our simulation data, a mean latency of 4
ms induces resonance at 62.5 Hz (Figure 3), which is within the frequency range of the
oscillations recorded in visual (Gray and Viana di Prisco, 1997) and auditory neocortex (Brosch et
al., 2002).

The reported latencies of unitary ipscs, evoked by firing an impaled presynaptic neuron, are
usually much shorter (mean 0.8 ms for pairs < 200 µm apart in dentate gyrus, Bartos et al., 2001),
but paired recordings may be biased to small interelectrode distances at which the probability of
finding connected neurons is highest. Ipsp latencies of 3.2-8.6 ms (mean 5.4) were measured for
pairs 153 to 445 µm apart in striatum (Tunstall et al., 2002), and in hippocampus unitary ipscs
between lacunosum-moleculare interneurons and pyramidal cells showed latencies, albeit at room
temperature, of 2.4-7.2 ms (mean 4.2; Bertrand and Lacaille, 2001).

In addition, it is the mean distance to the entire set of postsynaptic neurons that determines the
resonance frequency. This mean distance can be estimated from the experimentally derived
connection probability function, provided the probability at each distance is corrected for the
varying numbers of neurons available. For axons with a disk- or sphere-like arborization, the
number of candidate target neurons increases linearly or quadratically with distance from the
source neuron. Hence, if the connection probability is observed to fall off according to an
exponential function with space constant δ, the actual mean
distance increases from δ to 2δ
and 3 δ for axons branching in two and three dimensions,
respectively.

The present mechanism is not restricted to purely inhibitory networks. The 1/(4*d*) rule
generalizes to circuits with intercalated excitatory neurons (for example
I_{1}->E_{1}->I_{2}->E_{2->}I_{1} instead of
I_{1}->I_{2}->I_{1}). Here resonance is predicted to arise at an
oscillation period equal to four times the delay of the disynaptic response
I_{1}->E_{1}->I_{2}. If the response delay is much greater for excitatory
than for inhibitory connections, then the circuit will resonate with a period approximately four
times the delay of excitation and, consequently, the excitatory neurons will lead by one quarter
cycle the inhibitory neurons. Such a phase relationship has been observed in olfactory systems
(Freeman, 1975; Bazhenov et al., 2001) and hippocampus (Csicsvari et al. 2003).

Finally, circuits with excitatory feedback can generate trains of spikes with delays of tens to
hundreds of milliseconds. In visual cortex, feedback excitation was proposed to provide the long
delays of inhibition needed to generate directionally selective responses to slowly moving stimuli
(down to <1 Hz) (Maex and Orban, 1996). It is noteworthy in this respect that the property of
directional selectivity, which can be implemented in a circuit with lateral inhibition, exhibits
the same *f=1/(4d)* relationship, *f* being the temporal frequency of the moving
stimulus evoking maximally selective responses (van Santen and Sperling, 1985).

**Predictions**

The present results and the above arguments lead to the following predictions, some of which are substantiated by recent findings. The delay of mono- or polysynaptic inhibition is the critical parameter determining the resonance frequency of an inhibitory network. Hence oscillations with different characteristic frequencies are expected to be generated by microcircuits involving different types of interneurons (see Klausberger et al., 2003). In these interneurons, and in their target projection neurons, ipscs are predicted to be evoked with variable latencies, centered about a quarter of the oscillation period (Traub et al., 1996). As the multitude of microcircuits in the brain could generate a continuum of delays, with probably slight differences emerging between different areas or developmental stages, oscillations in the nervous system could form a continuum rather than being divided into a few discrete frequency bands (see Csicsvari et al., 1999; Grenier et al., 2001). Interneurons involved in circuits with low characteristic frequencies are predicted to fire multispike bursts during each cycle. Finally, although the delay of reciprocal inhibition might be poorly amenable to experimental manipulation, changing the size of the stimulus would vary the effective mean delay. Stimuli exciting a focus smaller than the connection radius of an inhibitory network do not recruit long-distance neuron pairs, leading to faster oscillations than predicted from the mean delay of the circuit (see Fig. 4E in Bartos et al., 2002). Fast synchronous oscillations are therefore predicted to be more narrowly localized than low-frequency oscillations, although long-distance excitatory connections may contribute to this difference (Ermentrout and Kopell, 1998; Pauluis et al., 1999; Kopell et al., 2000).

**Conclusion**

We propose that resonant synchronization induced by the delay of (mono- or polysynaptic) inhibitory connections contributes to the emergence and the frequency tuning of all types of oscillations in which inhibitory synapses are involved, and that delay-induced synchronization should be considered for fast oscillations in particular (> 40 Hz). Some types of fast oscillations appear to persist however in the absence of synaptic transmission (200 Hz 'ripples'), and axo-axonal gap junctions may be essential for their generation (Schmitz et al, 2001). For other types of oscillations, additional tuning mechanisms may be important such as intrinsic neuronal resonance and synaptic dynamics (Gupta et al., 2000; Beierlein et al., 2003), which were not included in the present model.

**Appendix**

Let *A(t)* be the population activity at time *t*, *F* the neuronal transfer
function, *I* the constant level of external input, *K(t)* the synaptic impulse response
and *d* the combined axonal and synaptic delay, then the dynamics of a fully connected,
homogeneous network can be described by:

*λ dA/dt + A (t) = F (I — K • A
(t - d)),* (A1)

where • denotes convolution in time (Gerstner, 2000). During asynchronous firing the
population activity is considered constant over time: *A (t) = C = F (I Ð C)*. Linearization
about the asynchronous state yields:

*λ dA/dt + A(t) = F (I — C + C
— K • A(t - d))**
Å F(I - C) + F’(I - C) (C — K • A(t - d)). * (A2)

The linearized equation, after substituting

*y(t)*for

*A (t) - F(I - C)*and

*a*for the derivative of

*F*is:

*λ dy/dt + y(t) = - a K • y(t — d).*(A3)

We take

*K(t) = (1/τ) exp (-t/τ)*and follow the reasoning developed in MacDonald (1989, p. 85), looking for a periodic solution

*y = exp (iωt)*, with ω = 2π

*/ T*. We consider two cases. The first case explains the observation in Figure 3

*B*that for networks with low resonance frequency,

*f*is slightly greater than 1/(4

_{R}*d*) but almost completely independent of the decay time-constant τ. The second case matches the observation that networks with a high resonance frequency can have an

*f*less than or equal to 1/(4

_{R}*d*), which in addition is more dependent on the value of τ.

At low resonance frequencies (*case 1*), the synaptic kinetics is much faster than both
the membrane time-constant λ and the oscillation period *T*,
so that the delayed feedback can be considered to be approximately instantaneous. This reduces Eq.
A3 to l dy/dt + y(t) = - a y(t - d). (A4) The resulting transcendental characteristic
equation

(A5)

has real roots for *cos (ω d) = -1/a *and *sin
(ω d) = ω
λ / a*, or

*tan (ω d) = -
(ω d) (λ /d)*.
(A6)

Provided *a>1*, solving for ω *d* yields
π / 2 < ω <
π, giving *2 d < T < 4 d*. Hence, *f _{R}* >
1/(4

*d*) but independent of t.

At high resonance frequencies and hence high levels of excitation (*case 2*), a
neuron’s membrane potential is strongly depolarized and voltage deflections are small
(although sufficient to pass the firing threshold) due to the low-pass characteristics of the
membrane. Hence the contribution of the leak current to membrane dynamics can, approximately, be
ignored. Eq. A3 now reduces to

*λ dy/dt = - a K • y(t —
d). * (A7)

The resulting transcendental characteristic equation is

(A8)

For very fast synapses in the limit τ* -> 0*, the solution is
ω *d* = π*/ 2*, or *T
= 4d*. For τ > 0, the present case is an instance of a
distributed delay (the synaptic kernel) with a gap (the synaptic delay *d*), and *T >
4d* with *T* increasing for increasing τ (MacDonald,
1989).

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